Mathematical modeling and simulation in hydrodynamic stability

= N(v>,
(1)

where ip is a vector valued variable in general. L, M and V are linear homogeneous differential operators and R is the control parameter. N(ip,

) is a nonlinear operator which typically is bilinear in the variable (p. Although the surfaces of homogeneity could be cylindrical or spherical surfaces, we focus on the planar case and adopt a Cartesian system of coordinates x,y,z with the ^-coordinate in the direction of inhomogeneity. Accordingly the operators of equations (1) may depend on z, but not on x,y or t. All considerations discussed in the following can be easily carried over to problems that are spherically or cylindrically symmetric. The stability of the basic state with respect to infinitesimal disturbances
(2)

can be assumed where r is the position vector and where the wave vector / has a vanishing z-component. The minimum value Rc for which the real part ar of the growthrate of the strongest growing solution of the linear problem vanishes is called the critical value and corresponds to a minimizing wave vector lc. We may choose the

19 ^/-coordinate in the direction of the vector lc which allows us to write the solution

P = ^2amn exp{imay}gn(z)

(3)

m,n

where the functions gn{z) denote a complete system of functions which satisfy all boundary conditions of

Table 1: Symmetry properties of two-dimensional rolls translation in time: d

O

Fig. 2: Region of stable convection rolls in the R-a-P-parameter space. The stability region is bounded by several surfaces corresponding to in­ stabilities listed in table 2.

21 For the determination of the coefficients amn in the representation (3) it is necessary to project the basic equation (1) onto the space of the expansion functions introduced in (3). The infinite system of nonlinear algebraic equations that is obtained in this way can be solved through a Newton-Raphson iteration method after a suitable trun­ cation has been introduced. Following early work on this subject one may neglect all coefficients and corresponding equations whose subscripts satisfy the condition

n + m> NT

(4)

After the same finite system of equations for the coefficients amn has been solved for given values of R and a, the solution can be compared with the solution obtained in the case when NT is replaced by NT — 2. If the change in typical physical properties described by the solution is sufficiently small, the solution can be regarded as a good approximation. Otherwise NT must be increased.

3. T e r t i a r y Solutions In contrast to the rather universal form (3) of the secondary solutions, the tertiary solutions bifurcating from them reflect the specific physical conditions of the problem and thus exhibit wide variety of shapes and styles. In order to analyze the instabilities of the solutions (3) we superimpose infinitesimal disturbances of the form

(5)

m,n

After introducing this ansatz into the equation L

(6)

and projecting it onto the space of expansion functions we obtain a linear algebraic eigenvalue problem for the coefficients amn with the growthrate a as eigenvalue. Of primary interest are the growthrates a with largest real part ar as a function of the wavenumbers b and d. Whenever there exists a positive oy, the stationary solution

22 evident, two or more symmetries are usually broken, but a number of symmetries are still preserved which tend to characterize the bifurcating three-dimensional tertiary solution. The two-dimensional Eckhaus-instability usually does not lead to a new solution, but instead tends to replace a roll solution in the unstable region with one in the stable region. The actual evolution of this instability can be rather complex 3 . An important role is played by the neutral disturbance mode .

d

(7)

which satisfies (6) with a = 0. This disturbance corresponds to an infinitesimal dis­ placement of the stationary pattern in the y-direction. Some instabilities such as the zig-zag- or the oscillatory instability (in the case of stress-free boundaries) correspond to a small modification of the mode (7).

Table 2: Symmetries Broken by Bifurcations from Convection Rolls Broken inversion translation transverse transverse longitud. Symmetries about axis in time reflection translation periodicity Properties of cnjLO 6^0 d^O a>mn ^ 0 for 0>mn i1 m + n = odd disturbances Q>—mn Eckhaus Instab. X X Crossroll Inst. X CR X Knot-Instab. KN X X Even Blob-Inst. EB X X Odd Blob-Inst. OB X X X Oscillatory Inst. OS X X X Zig-Zag-Instab. ZZ X X Skewed Var. Inst. SV X X X Osc.Skewed Var.I X X X X

As the three-dimensional solutions bifurcate from the roll solution, they can be de­ scribed by expressions of the form SP = ^2 aimnexp{ilaxx

+

imayy}gn(z)

(8)

l,m,n

where we have assumed that the instability of interest has a vanishing imaginary part &i of the growthrate, i.e. the bifurcation is monotonic. When an instability with a finite value of <7t occurs, it typically leads to travelling waves propagating along the #-axis which can also be described by the representation (7) if x is replaced by x = x — ct. Numerous tertiary solutions can be described in this way; a partial list is given in table 3. In addition to the usual conditions Olmn =

Q--1-

(*indicates complex conjugate)

23 Table 3: Examples for Tertiary Solutions and Their Symmetries in Terms of the Complex Coefficients Tertiary Solution Reflection Symmetries Inversion Symmetry knot-/bimodal convection a-lmn = aimn = of the tertiary solution, say the velocity component in the z-direction, instead of introducing a cum­ bersome notation of the various symmetries. The wavenumber ay in expression (8) is not necessarily identical with the wavenumber a of the original roll representation (3). If a subharmonic instability occurs with b = a / 2 , the representation (8) can be used as for example in the case of the staggered vortex solution that replaces transverse rolls in an inclined fluid layer 4 .

4. Quarternary Solutions and Higher Order Bifurcations After the onset of three-dimensional tertiary solutions the continuous spatial symme­ tries such as the invariance with respect to the translation along the roll axis have been broken and have been replaced by reflection symmetries and inversion symme­ tries such as those shown in table 3. The stability of the steady or travelling tertiary solutions can be investigated through the superposition of infinitesimal disturbances just as in the case (5) of the two-dimensional solution. Using the general Floquet ansatz

we arrive at a linear homogeneous system of equations for the unknown coefficients o-imn with the growthrate a as eigenvalue. Since the same truncation parameter NT

24 must be used as for the representation (8) of the tertiary solution, but since no sym­ metry properties can be assumed for the complex coefficients a/ mn in the general case, the stability analysis is much more demanding in terms of computer resources than the determination of the tertiary solution. Fortunately, many of the most interesting instabilities do not tend to change the periodicity interval in the x,y-plane and thus correspond to eigenvalues a for which the real parts
(10)

The system of differential equations is obtained, just as in the case of the algebraic equations of tertiary solutions (8) through projections of the equations of motion onto the space of the expansion functions. Alternatively, time periodic solutions of form (10) can be generated through a Fourier expansion M

a>imn(t)= ^2 a>imnPexp{iput}

(11)

p=-M

where the complex coefficients a/ mnp obey nonlinear algebraic equations which can be solved through a Newton-Raphson iteration method. The truncation parameter M must be chosen sufficiently high such that the solution does not change significantly when M is replaced by M—1. The frequency LJ can be determined most easily through the choice of a particular phase, say ai0U = ai 0 i_i. The equation corresponding to the latter coefficient thus becomes available for the unknown CJ. Once the coefficients (11) including u have been computed, the stability of solutions of the form (10) can be investigated, just as in the case of the tertiary solutions. The Fourier method offers advantages in this respect, since a stability analysis is not as easily possible when the coefficients aimn(t) are computed through forward integration in time. The Fourier method has recently been applied by De la Torre and Busse 5 in the case of thermal convection in a Hele-Shaw-channel. In the following some typical results in the case of Rayleigh-Benard convection in a fluid layer heated from below will be described.

25

Fig. 3: Lines of constant vertical velocity in the planes z — 0 (upper left) and z — —0.4 (lower left) and isotherms in the planes z = 0 (upper right) and z = -0.4 (lower right) in the case of steady knot convection with ax = 1.5, oty = 2.2 for R — 60000, P = 2.5. Dashed lines indicate negative values and the dotted line corresponds to zero. 5. Steady and Oscillatory Knot Convection Since the density variation in the Rayleigh-Benard layer are small, the velocity field u satisfies the equation V • u = 0 in good approximation, and the general representation u = V x (V x k
(12)

can be used where k is the vertical unit vector parallel to the 2-axis and (p,t/> are bounded functions with vanishing average over the ar,?/-plane. The function U de­ scribes the profile of a mean flow along the x-axis of the convection rolls. In the case of steady knot convection, U always vanishes identically. All velocities and frequencies mentioned in the following will be measured in thermal units based on the thermal diffusivity K of the fluid and the depth h of the layer. In addition to the variables the dimensionless deviation 0 of the temperature from the static profile of pure

26 conduction is needed for the description of convection. Since rigid boundaries with fixed temperatures are assumed at the top and bottom of the layer, the boundary conditions

= 0 = U = O a t z = ±(13) must be assumed and the expansion functions in representations (8) and (10) must be selected accordingly. Referring to the original work6 for details we describe here only some typical results. Knot convection represents the transition from convection in the form of thermal sheets, as in the case of roll like motions, to convection in the form of thermal plumes. Hot and cold fluid from the respective thermal boundary layer is collected into cylin­ drical columns of fluid that moves to the opposite boundaries. This process is evident in the left plots of figure 3. The collection of fluid through the finger like ejections from the thermal boundary layers as indicated in the upper right plot give rise to the typical knot like appearance of this type of convection in shadowgraph pictures 7 . The concentration of the buoyancy force into narrow columns leads to an acceleration of the fluid particles as they approach the other boundary and to a thinning of the opposite thermal boundary layer. This process permits a very efficient heat transfer as indicated in the lower right plot leaving a maximum area of the thermal boundary layer for the process of collection of fluid into the plumes. The stability analysis for steady knot convection is particularly simple in the limit of vanishing b and c?, since there are three symmetry properties available according to table 3. The stability equations thus separate into those for disturbances which share the respective symmetry with the steady solution and those which exhibit the opposite symmetry. Denoting by OSC, for instance, the class of disturbances with vanishing coefficients a/ mn for / + m -f n = even and with the properties a/mn = —a_/ mn , a/ mn = a>i-mn, we arrive at a total of 2 3 disturbance classes denoted by ECC, ESC, ECS, ESS, OCC, OSC, OCS, OSS

(14)

which can be analyzed separately. The letter E, 0,5, and C obviously stand for even, odd, sine and cosine like symmetry. Depending on the Prandtl number and the wavenumbers ax,ay of the steady knot solution, different classes of disturbances correspond to the highest real part crr of the growthrate cr. In the case P = 4, disturbances of the type ESC are the strongest grow­ ing ones. Their evolution to finite amplitude can be followed through time integration of the coefficients aimn(t) in the representation (10). Examples of such computations are shown in figures 4 through 6. Although there is already an increase in the heat

27

Fig. 4: Oscillatory knot convection with ax = 1.5, ay = 2.0 arising from the ESCtype instability in the case R = 36000, P = 4. Equidistant plots in time are shown in each column from top to bottom corresponding to a half period 0.020 of oscillation. Lines of constant vertical velocity in the planes z = 0 and z = —0.4 and isotherms in the same planes are shown (left to right). The x-coordinates run downward from 0 to 27r/ax.

28

Fig. 5: Isotherms (left) and lines of constant d

F i g . 6: The Nusselt number as a function of time for the case of figures 4 and 5.

29

Fig. 7: Lines of constant vertical velocity in the planes z — 0 (upper left) and z = -0.4 (lower left) and isotherms, 6 = const., in the planes z = 0 (upper right) and z = -0.4 (lower right) in the case of travelling wave convection with ax = 2.3, ay = 2.2 for R = 8000, P = 0.71. transport by knot convection in comparison to that of rolls, a slight further increase occurs through the onset of oscillation. The separation of hot and cold fluid from the respective boundary layers is now modulated in time such that the blobs of fluid that are hotter or colder than the average temperature at their position travel around the convection cell as can be seen in figures 4 and 5. These time sequences of plots visualize only half a period of the knot oscillation, but the other half cycle is evident from the symmetry of the problem. The Nusselt number which is defined as the ratio

30

Fig. 8: Asymmetric travelling wave convection with ax = 2.6, ay = 2.2 arising from the OO-type instability in the case R = 10000, P = 0.71. Equidistant plots in time are shown in each row from left to right corresponding to a period of the Nusselt number. Lines of constant vertical velocity in the planes z = 0 (top row) and z = -0.4 (2. row), isotherms 0 = const, in the planes z = 0 (3. row) and z = -0.4 (4. row), and lines of constant ip in the plane z = 0 (bottom row) are shown.

31

Fig. 0: The Nusselt number as a function of time for the case of figure 8. of the heat transports with and without convection is shown in figure 6 for the same half period as the plots of igures 4 and 5. Like all spatially averaged properties the Nusselt number exhibits half the period of the velocity held.

©. Travelling W a v e C o n v e c t i o n Travelling wave convection arises from the oscillatory instability of convection rolls and can be observed in iuid layers with Prandtl numbers of the order unity or less. A typical solution is shown in figure ? which "corresponds closely to experimental pictures 8 . The representation of the travelling wave solution can be written in the form ^

=

] C ^lmn

cm

'««(*-<*)+«'"»* s i n f a * ( * - c t ) ] { 5 / cos maya-\r(l-si)

sin

mayy}gn(z)

l,m,n

(15) with corresponding expressions for # , # . In the real notation of expression (15) the subscripts 1, m ran through all non-negative integers while n runs through ail pos­ itive integers. The symmetries of the travelling wave solution are expressed by the

32 conditions aimn = ciimn = 0 for m + n = odd ai

= 1(1+ (-!)')

(£) (E)

(16a) (16b)

The disturbances <£ corresponding to the special limit d = b = 0 in representation (9) can be separated into four classes depending whether they share the property (16a) or the property (16b) with the travelling wave solution (E) or exhibit the opposite symmetry (O). In the latter case the property a/ mn = a/ mn = 0 for m + n = even holds or the definition si = | ( 1 — (—l)1) applies. The analysis of the stability with respect to disturbances of the classes EE,EO,OE,00

(17)

yields the result that for Prandtl numbers of the order unity the OO-disturbances are growing most strongly. For example for the parameters of the solution in figure 7 the critical Rayleigh number Rm for the onset of the OO-inst ability is 8412 corresponding to LJ = 12.37. As OLX is increased to 2.6 the value of Rm drops to 7946 corresponding to u) = 10.52. The stability calculations have been done for unrestricted mean flow in the terminology of an earlier paper 9 . Slight differences in the values of Rm are due to higher truncations used for the present computations. As the disturbances grow, they lead to asymmetric distortions of the travelling wave as shown in figure 8 through a time sequence of plots. The wave is no longer shape preserving and integral properties such as the Nusselt number become time depen­ dent. The Nusselt number corresponding to the time series of figure 8 is shown in figure 9. As in the case of oscillatory knot convection, a period of the Nusselt number corresponds to half a period in the pattern oscillation, where a shift in the x-direction must be taken into account. The period of the oscillation, Tp = 0.5656, is somewhat decreased relative to the period 2ir/u> at the onset of the OO-instability. Because the periods are related to the circulation times in the rolls, they tend to decrease with increasing Rayleigh number. All travelling wave phenomena in dissipative systems are associated with a mean flow. In the case of the stationary travelling wave shown in figure 7, the mean flow is steady and directed in the direction of propagation of the wave. After the transition to asymmetric travelling waves the mean flow becomes time dependent just as the Nusselt number. Moreover, an oscillatory mean flow, V, in the ^-direction perpendic­ ular to the axis of the original rolls develops. It is caused by the asymmetry of the wave which shifts periodically from one side to the other. As can be seen from figure 10, the oscillatory ^-component of the mean flow is antisymmetric with respect to the middle plane of the layer, while the component along the axis of the rolls is symmetric.

33

Fig. 10: Mean flows U (in the x-direction, symmetric with respect to z = 0) and V (in the y-direction, antisymmetric with respect to z = 0) in the case of asymmetric travelling wave convection. The profiles correspond to the times t = 0 (solid lines), t = T/S (dashed lines), t = T/4 (dash-dotted lines), t = 3T/8 (dash-double dotted lines) and t = T/2 (dotted line) with the period T given by 0.3714. The parameter R = 12000, ax = 2.0, ay = 2.2, P = 0.71 differ somewhat from the case of figures 8 and 9. 7. Discussion Only a few examples of the vast possibilities for tertiary, quarternary and higher order solutions that are spatially periodic in the planar, cylindrical, or spherical surfaces of homogeneity have been discussed in the preceding sections. There are many reasons why those solutions are not often observed in experiments even if they are stable with respect to infinitesimal disturbances. First, the periodic solutions do not satisfy the boundary conditions at the unavoidable side walls of the experimental apparatus. These effects can be minimized through the use of large aspect ratios. But some more subtle effects induced by the side walls can usually still be recognized. Secondly, disturbances which induce bifurcating flow states occur inhomogeneously and the re­ sulting pattern often do not heal. Only in the case of rolls in strongly anisotropic

34 systems is the tendency towards a regular arrangement sufficiently strong such that a constant wavelength is realized. In the case of more complex patterns the mani­ fold of competing solutions is much larger and the basin of attraction of a particular twice periodic solution is much smaller. Nevertheless, the Taylor-Couette system and the Rayleigh-Benard layer show numerous regular tertiary and even quarternary flow states 9 * 10 ' 11 ' 12 .

References 1. A. Schluter, D. Lortz and F. Busse, J. Fluid Mech. 23 (1965) 129. 2. E. Weisshaar, F.H. Busse and M. Nagata, J. Fluid Mech. 226 (1991) 549. 3. L. Kramer, H.R. Schober and W. Zimmermann, Physica D 31 (1988) 212. 4. M. Nagata and F.H. Busse, J. Fluid Mech. 135 (1983) 1. 5. M. de la Torre Juarez and F.H. Busse, J. Fluid Mech. 292 (1995) 305. 6. R.M. Clever and F.H. Busse, J. Fluid Mech. 198 (1989) 345. 7. F.H. Busse and R.M. Clever, J. Fluid Mech. 91 (1979) 319. 8. V. Croquette and H. Williams, Phys. Rev. A 39 (1989) 2765. 9. R.M. Clever and F.H. Busse, J. Fluid Mech. 201 (1989) 507. 10. D. Coles, J. Fluid Mech. 21 (1965) 385. 11. C D . Andereck, S.S. Liu and H.L. Swinney, J. Fluid Mech. 164 (1986) 155. 12. F.H. Busse and J.A. Whitehead, J. Fluid Mech. 47 (1971) 305. 13. F.H. Busse and J.A. Whitehead, J. Fluid Mech. 66 (1974) 67.

35 DERIVATIONS AND SIMULATIONS OF EVOLUTION EQUATIONS OF WAVY FILM FLOWS ALEXANDER L. FRENKEL AND K. INDIRESHKUMAR Department of Mathematics, University of Alabama Tuscaloosa, Alabama 35487

Abstract Wavy flows of viscous films on solid surfaces are considered. The focus is on approximate decriptions which are hinged on a single evolution equation. Perturbative approaches to constructing such theories are discussed. For several film flows, evolution equations obtained—along with the validity conditions of those theories—with the multiparametric perturbation approach are reviewed. The results of their three-dimensional numerical simulations on extended spa­ tial intervals are discussed. Some unresolved fundamental questions concerning such film-flow studies are posed and discussed.

1

Introduction: tions

Film flows and evolution equa­

Our subject matter is the flow of liquid films on solid surfaces. The force driving the flow is usually gravity and/or an externally applied pressure. One boundary surface of the liquid layer is its interface with the supporting solid, the other a fluid interface. If the ambient fluid is a dynamically passive gas, the film has a free surface—as in flows down inclined planes or vertical cylinders. Otherwise, the film has a fluidfluid interface with the surrounding active fluid. Such is the case in the so-called core-annular flow (see e.g. [51]), which is a two-fluid flow in a pipe. One encounters such film flows in nature and in everyday life. Controllable flows of films are used in industry. Naturally, they have been studied for quite some time (see, for example, [69, 52, 71, 70, 72] for a considerable history of the film flow investigation). Currently, they continue to be a subject of growing research activity (see e.g. [15] for a recent review of work on a film flow down a vertical wall). Of course, in principle the film dynamics is known: it is given by the familiar NavierStokes (NS) equations [supplemented with appropriate boundary conditions (BC)]. Usually, however, one is interested less in the equations per se than in the dynamical fields—velocities, pressures, the position of the interface, etc.—which are unknown solutions of the NS equations. These are far from being easy to solve: indeed, one has to deal with a system of several coupled partial differential equations (PDE) which is additionally complicated with the moving boundary—whose dynamical PDE itself involves unknown fields, the fluid velocities. Even with the most powerful

36 modern computers, such a full NS problem is too difficult to solve, especially—as is frequently needed—in extended space-time domains. Therefore, one would like to have a simpler, solvable description of evolution, provided that it yields a sufficiently good approximation to the exact solution of the NS problem. Ideally, each such simplified dynamical system should be accompanied by validity conditions, such that if the flow parameters satisfy these conditions, the solutions are guaranteed to be good approximations to the corresponding exact NS evolutions. The most favorable case of such a simplification is the one in which the problem reduces to a single PDE—which as a rule is the one for the thickness of the film. If the solution is found, the theory yields the velocities and pressures as explicit expressions in terms of the film thickness. However, even if one succeeds in obtaining such a simplified evolution equation (EE), it still cannot, in practice, be solved analytically; so one turns to computers. Over a number of years, we engaged in such mathematical modeling and numerical simulations of film flows. Below, we discuss some particular problems we considered, methods which have been developed, the results obtained, and also some difficult questions which have not yet been answered. (Some of the results presented here have not been published before.) The work of other researchers is touched upon only inasmuch as it was judged relevant to those issues. Accordingly, our reference list is far from being comprehensive. We sincerely apologize to those colleagues whose relevant work may have been inadvertently overlooked here. The rest of this paper is organized as follows. In the next section, we introduce our multiparametric perturbation (MP) approach and discuss its advantages over previous perturbative methods. In section 3, we discuss the resulting evolution equations and conditions of their validity obtained with this approach, as well as the results of numerical simulations of the evolution equations for various film flows. In section 4, some unanswered fundamental questions pertaining to these film-flow studies are discussed. The last section summarizes the paper.

2 2.1

Perturbation approaches An example of the less-formal 2-D vertical film flow

MP approach

at

work:

We consider a layer, of an average thickness h0 (the overbar here and below indicates a dimensional quantity), of an incompressible Newtonian liquid—of a density p, viscosity /Z, and surface tension a—which flows under the action of gravity (whose acceleration is denoted g ) down a vertical solid plane (also called "the wall". Here we confine ourselves to this particular case of an inclined film for the sake of simplicity, and only consider two-dimensional (2-D) wavy flows. Some more general results are given in section 3.1; we refer the reader to [48] for more results and the theory for the general case, a 3-D flow down an inclined plane.). There is a well-known

37 time-independent, "Nusselt's", solution of the NS problem for the vertical film. The thickness of the Nusselt film is constant (hence, Nusselt's flow is also referred to as a "flat-film" solution). The only nonzero component of velocity is the downward one. It only changes across the film, starting from the zero value at the wall. The free-surface value Uv of the Nusselt velocity is Uv = gh\/(2v) (where v — ~pj p is the kinematic viscosity). We nondimensionalize all quantities with units based on p, /i 0 , and Uv. (As we will see below, exactly two independent basic parameters appear in the dimensionless equations and boundary conditions; one can choose e.g.. the Reynolds number R = h0Uv/i>, and the Weber number W — crR/2 as such basic parameters.) The x axis of our system of coordinates is normal to the solid plane and directed away from it; the y axis is in the spanwise (i.e. horizontal and parallel to the wall) direction; and the z axis is directed downward. [A subscript x, y, z, or (time) t, will always—whether preceded by a comma or not—indicate the differentiation with respect to that variable.] For simplicity (as was mentioned above), we first consider the 2-D flow, so that v, the y component of velocity, is zero. The x and z components of velocity are denoted, respectively, u and w. In a coordinate system moving (with respect to the laboratory reference frame) with a velocity V in the z-direction, the NS equations (for velocities measured in the laboratory frame) written in the dimensionless form are (see e.g. [2]) ut + uux + wuz - Vuz = -px + (uxx + uzz)/R,

(1)

wt + uwx + wwz - Vwz = -pz + 2/R + (wxx + wzz)/R.

(2)

The continuity equation is ux + wz = 0.

(3)

The boundary conditions are as follows. The no-slip conditions at the solid surface are u — w = 0 at x = 0. (4) The tangential-stress balance condition at the free surface is Puhz + pi 3 [l - hz2] - p33hz = 0

(5)

and the normal-stress one is b n + Pas/**2 - 2p13hx][l + hz2}^2

= ahzz.

(6)

Here, the stress components pij are pn = -p + 2ux/R,

p33 = ~P + 2wz/R,

P13 = (uz + wx)/R.

(7)

38 Finally, the kinematic condition at the free surface is ht + whz - Vhz = u.

(8)

Let Wo denote the Nusselt velocity for an imaginary film of constant thickness h which is equal to the local thickness h{z, t) of the real film (which is in agreement with experiments [1]): WQ = 2hx — x2. [Note that WQ depends not only on x but also on z and t—through h(z, t)] The reference normal component of velocity, wo, is chosen to satisfy incompressibility (3); i.e., it is defined as the solution of u0x = —w0z, with the no-slip boundary condition u0 = 0 at x = 0. This solution clearly is u0 = —x2hz. The above NS problem becomes an exact one for a new set of unknowns—/z, ui, w\, and p—by the substitution u = uQ + u\ and w = Wo + Wi (the remaining unknowns p and h are not changed). For example, the kinematic boundary condition at the free surface x = h [Eq. (8)] becomes ht + {w0 + Wi- V)hz - u0 - ui = 0.

(9)

If one looks into derivations of known evolution equations (EEs), one realizes that each of them is invariably obtained from exactly a kinematic condition of the type (9), which can be done in the following way. First, the NS equations are simplified by discarding some terms, to arrive at essentially ordinary differential equations (ODEs) for the velocities and pressure as functions of the streamwise coordinate x [with the other independent variables entering as parameters only, through h(z,i)]. It is straightforward to solve these simple (albeit nonhomogeneous) ODEs, since their coefficients are constant. Then, one proceeds to substitute the resulting solutions for velocities (in terms of h) into the kinematic condition. This yields a closed PDE for h, i.e. the evolution equation. In our derivation, we follow this recipe as well, but we take care to discard only those terms in the NS equations which must be dropped if one is to obtain solvable ODEs. Since ui, Wi, and p are to be found by reducing the NS equations to an ODE in x for each of these dependent variables, it is clear that the equation for p is the (reduced) x-NS equation: indeed, this is the only equation (of the full NS problem) containing a derivative of p with respect to x. Consequently, u\ has to be determined from the incompressibility condition (since it contains Uix), which can be done after finding w\ from the z-NS equation. The order in which these equations are solved can, in principle, play a role, since in solving for a particular variable, one has to neglect the terms containing those of the other variables which have not been determined prior to that. After a little investigation, one can see that the most natural order is as follows. First, one finds p from the x-NS equation, with the BC coming from the normalstress balance condition at the free surface [for simplicity, the pressure of the ambient gas has been neglected in the formulation (1-8)]. In this problem for p, one clearly has to discard all terms containing one or more of the unknowns u\ and w\. Next,

39 the z-NS equation is recast into an ODE for w\. Here, one has to discard only those terms containing u\\ the p-term can be retained, as it is now a known expression in terms of ft. But all terms containing wi, except for the viscous one with w\xx, must be discarded as well— otherwise one does not get a constant-coefficient ODE in x. Let us denote by A the characteristic amplitude of the surface deviation 77 = ft — 1, and let T and L be, respectively, the characteristic time- and length-scales. Then, for example, from the condition that the neglected viscous term should be smaller than the one retained, W\zz 1 (indeed, ft ~ A for A ^> 1; and the velocities change from being zero at x = 0 to their full magnitudes at x — ft; hence, the characteristic lengthscale of change is ft, which is ~ A). Also, d/dt ~ 1/T, d/dz ~ 1/L, etc. One can see that, similar to the small-slope condition above, the condition (R/T)(l + A) 2
{l + A)W L3

(1 + A ) 4 # ( 1 + A ) 2 L ' L2


(10)

[One notes that all three of these conditions can be obtained e.g. already from the requirement of negligibility of the term containing U\z in the BC for W\, Eq. (5).] Due to these conditions, even some terms which can be retained in the equations—and handled, in principle, without any difficulty—can be shown to be actually negligible. Also, some other terms are estimated to lead to a negligible contribution to the final evolution equation and therefore can be discarded as well. As a result, one can see that the expression for p is in fact the solution of the problem [see Eq. (1) and (6)] Px = [uoxx/R =] ~

2hz/R,

p = [2u0x/R - ahzz =] - 4hhz/R

- ahzz

(x = ft),

that is p = -2(x + h)hz/R

- ahzz.

(11)

Note that this pressure contains a viscous contribution (the first term on the righthand-side) in addition to the usual surface-tension part. The equation for Wi is the (simplified) z-NS equation:

40 Wlxx = R{uoW0x + w0w0z - 2w0z +pz+

Wot) -

w0zz.

(Note that we use the system of coordinates of a moving reference frame, whose z—directed velocity with respect to the solid plane is V = 2—that is equal to the well-known phase velocity of the infinitesimal waves; all the velocities, however, are measured with respect to the laboratory frame.) We also have to satisfy the following boundary conditions: first, W\ = 0 at x = 0; and second, from the tangential-stress balance equation at x = h (5), we have Wix = —u0z — 2u0xhz -f 2w0zhz. The solution of this problem is Wl

= —Rhhz

- — [hzz + Rhz] - x2 [hhzz + Whzzz

+x \bh2hzz + 2Whhzzz

+ 2Rh2hz - \Rh*hz

+

(hz)2\

+ 10h(hz)2] .

(12)

Next, the equation for ui derives from the continuity equation: u\x = —W\z, where w\ is already known (12); and the no-slip BC requires that u\ = 0 at x = 0. The solution of this problem is

ui = -^R(hhz)z - y {5 (h2hzz)z

+ y [hzzz + Rhzz] + y [~(/*2W + Whzzzz]

+ 2W [{hhzzz)z\ + 2R (h2hz)z

- | f l (h%)z

+ 10 (hhz2)x}

. (13)

One substitutes the expressions (12) and (13) for velocities in terms of h (taken at x = h) into the kinematic condition (9), which yields the EE

ht-^(h5)tz

+ 2(h?-l)hz

+R

l*H.-±#H.

+1 [h3Whzzz]z + 2h*hzzz = 0.

(14)

We have omitted all of the several dispersive terms except for a single one—the last term on the left-hand-side (l.h.s.) of Eq. (14)—since in the limit of small deviations 77 they turn out to be negligible [see Eq. (15) below]; and if the amplitude of 77 is not small, the equation, in any case, can only be good as a qualitative model, as we will also discuss below. The solutions of this equation yield a good approximation to the true evolutions of the thickness h—for some time, at least—as long as the conditions (10) are satisfied. Using those conditions and estimating all of the members of the EE (14) in terms of the current amplitude A, the (current) lengthscale L, and the basic parameters, one can see the following facts: If A is not small, the equation can be written as simply

41 ht + 2(h2 — l)hz = 0 (or just ht-\-2h2hz = 0 in the laboratory reference frame). Indeed, all other terms of the EE (14) turn out to be much smaller than the two selected into the shortened equation above. The solutions of this short equation (called "the simplest hyperbolic equation" in [96]) are well known to exhibit steepening of the wavefronts that leads to breaking of the waves in a finite time. It follows that the EE (14) cannot be valid globally, i.e. for all time. But one can easily see that this EE is equivalent to the well-known Benney equation: The former transforms into the latter (in the laboratory reference frame and with the dispersion term omitted) by the substitution of the term —2h2hz in place of ht (this trick was called the "trade of timefor space-derivative" in [2]) into the second term of (14). Hence, the large-amplitude waves can be approximated by the Benney equation at best for a finite time—a fact that was established before in [35]. Thus, here from a different direction, we arrive at the same conclusion: a (film thickness) evolution equation which would provide a good time-uniform approximation to large-amplitude waves on films flowing down vertical (as well as inclined, as is discussed below) planes, does not exist. (This is in contrast to films flowing down vertical cylinders, where such an equation does exist: it has been obtained in [34].) For a quantitatively good approximation of such largewave regimes of planar films, one has to be content with a less drastic simplification of the original NS problem, which would contain a system of at least two coupled PDEs—such as those studied e.g. by Chang, Demekhin, and their co-workers (see e.g. [17, 16]). The domain of validity of such evolution systems may clearly be larger than that of theories hinged on a single evolution equation; however, the difficulty faced in attempting to solve them is correspondingly more severe, and indeed can approach the formidable difficulty of the original NS system. [Note that the two equations, (14) and the Benney one, are not necessarily equiv­ alent when the validity conditions (10) break down. In fact, there are indications that our equation might make a better model; we will discuss this point in section 3.1.1.] The case of small amplitude waves, (rj ~ ) A
+ 2r)zzz = 0,

(15)

o

the evolution equation for the small-amplitude waves. This equation has appeared before in [93]. [It is, however, a particular case of the more general equation (31) below (which was derived in [48, 47]) that allows for an arbitrary inclination of the film plane to the horizontal and also for the dependence of waves on the spanwise coordinate y] We use the inequality A
42 the local-validity conditions for the small-amplitude EE (15): max(W7L 3 , R/L,

1/L2, A) < 1.

(16)

We call these conditions local (in time) since they involve the parameters L and A which, in contrast to the basic parameters R and W, can change with time. Indeed, due to the dissipativeness of the EE (15), the system evolves towards an attractor and thus essentially forgets the initial conditions. On the attractor, there can be fluctuations, but no systematic change in time is possible. Then, following the ideas of [6, 37], the second-derivative term of the Eq. (15)—which term is well known to lead to the growth of disturbances—should be of the same order of magnitude as the stabilizing, fourth-derivative one (while the third-derivative term is purely dispersive: it makes only a purely imaginary contribution to the linear-theory growth rate). Hence, the (dimensionless) characteristic lengthscale at large times, La , is estimated to be La = [W7(4fl/5)] 1/2 . Similarly, the asymptotic magnitude of the characteristic amplitude Aa is deter­ mined by the balance between the nonlinear "convective" term and either the disper­ sive term or the capillary one (whichever is larger): Aa = max(V7/L 3 , 1/1%). Using these estimates, the condition (16) can be written as max( W/L^ R/La, 1/L2a)
< 1 and

Rx = R/La < 1.

(17)

These conditions involve only the basic (time-independent) parameters of the flow. If the basic parameters satisfy these conditions, the EE (15) is valid (yields a good approximation) for all time. Hence, these conditions may be termed the globalvalidity condition. We can transform Eq. (15) to a "canonical " form-which would contain a mini­ mum of "tunable" constants—by rescaling rj = Nrj, z = Laz, and t = Tat. We take N = W/{6L3a) and Ta = {3W/2)/(AR/5)2. Dropping the tildes in the notations of variables, the resulting canonical form of the small-amplitude evolution equation is rjt + rjrjz + r\zz + rjzzzz + Xqzzz = 0,

(18)

where A 3/y/W{4R/b). The way we have derived the above theory is in fact a refinement of the lessformal version of the multiparametric perturbation (MP) approach used before in [6, 31, 34, 37].

43

2.2

The leading two orders of the fully proach: the strongly dispersive case

formal

MP

ap­

In the more formal version of the method (see [31, 32, 34]), one represents the vari­ ables of the problem as multiple series in powers of two (or more in other physical problems) independent perturbation parameters (this is why we call the method multiparametric/ The choice of the appropriate perturbation parameters depends on the magnitude of the dispersion constant A in Eq. (18): (i) If A ~ 1, the appropriate small parameters are A2 and Ri of Eq. (17). Then the evolution equation and the approximate expressions for velocities and pressure follow from the leading-order equations [which couple together the leading-order coefficients of the (A2, JR1)-power series of the unknowns]. There is no need for the more formal procedure unless one is concerned with higher-order corrections to the leading-order results. (ii) When A 1, we introduce e = 1/A

(e < 1)

A = x/A2"

(A < 1).

and It is easy to see that in terms of the independent small parameters £ and A, other basic parameters are given by the following expressions: R = RoAe, W = WoA _1 £, and a{= 2W/R) = a0A~2, where Ro = 15/4, W0 = 3, and a0 = 8/5. The EE (18) can be rescaled to the form rjt + rjrjz + e(j]zz + rjzzzz) + rjzzz = 0.

(19)

The leading order equation—which follows by neglecting the terms multiplied by e—is the Korteweg-de Vries (KdV) one,

vt + wnz + Vzzz = o.

(20)

However, as was shown some time ago [55, 31], the dissipative terms of Eq. (19), although they are of a higher order, nevertheless play an important role. Namely, it is well known that there is a one-parameter family of soliton solutions to the KdV equation, and the smaller-amplitude solitons have a greater width. If one starts with a wide soliton as an initial condition to Eq. (19), the stabilizing fourth-derivative term is much smaller than the destabilizing second-derivative one, and as a result the

44 amplitude of the soliton will slowly grow. The soliton width quickly adjusts to the changing amplitude. The width continues to decrease in this manner until the two dissipative terms of (19) are finally in balance. Thus, these small terms determine the final width, and hence the amplitude and the velocity, of the soliton solution. Hence, one needs to employ the formal version of the MP approach in order to determine if the higher-order dissipative terms of Eq. (19) have been found correctly [the answer will be seen to be affirmative in the case under consideration; however, in some cases of the inclined-film, problem the formal analysis can reveal [61] cer­ tain corrections to the coefficients obtained with the less-formal version of the MP approach]. This formal MP procedure is as follows. 2.2.1

P o w e r series

One starts by substituting into the NS problem (l)-(9) the power series p = A2£-1^PijAV, 4

ft-l

(21)

tei = A X ; W y A V ,

(22)

«i = A 8 £ t f y A V ,

(23)

2

= 7y = A ^ i f i , A V ,

(24)

V = £V^AV,

(25)

n

where Yl = E H ' W O O) ■ ^ these series, V^ are constants, and the coefficient functions (CF)—P{j, Wij, etc.—depend on (i) x (except for Hij), (ii) the rescaled downstream coordinate £, such that

d__Ad_

dz

af

and (iii) on a sequence of time-variables of different scales—Too, TQI, etc.—such that

The exponents of A and e in front of sums in these series are determined from the results of the preceding less-formal MP analysis (similar to e.g. [32, 34]). The series for the reference velocities are easily determined from the expressions u0 = —Ax2r)^ and w0 = (2x — x2) + 2x7), by substituting into them the series (24) for rj. Each NS equation acquires the form in which the l.h.s is a (double) series in the powers of A and e, and the r.h.s. is 0. Requiring that each coefficient of the l.h.s. is equal to zero separately, we obtain a sequence of problems for the coefficient functions. They can be solved one after another (the higher-order problems contain as their coefficients the CFs found as a result of solving the lower-order problems). Thus, the MP approach yields a theory which is "consistent to all orders in perturbations", in the terminology of Ref. [10].

45 2.2.2

Leading-order system

One can readily see that the kinematic condition (9) in the leading order A3 only determines that Voo = 2. But from the A5-order of that equation, the leading-order EE is to come: Ht + AHH^ - U00{x = 1) = 0. (26) [(i) For brevity, we denote HQ0 as simply H and r 00 as r. (ii) Note that the term Vio#io£ appears in this equation; this term is eliminated by choosing V10 = 0.] To express the last member of this equation in terms of H, one proceeds through the equations of the NS problem, one after another. Equation (1) in the order A 2 e _ 1 yields a first-order ODE in x for P0o> viz. P0Qx = — {2/R0)H^. The appropriate BC comes from the same order of the normal-stress Eq. (6): P0o{x = 1) = 2u0x[2,-i\/Ro = -AH^/RQ (where a new type of notation has been used: UQX[2, —1] is the notation for the CF of A2e~l in the power-series expression of u0x, etc.) The solution is P0o = — {^/Ro)H^{x + 1). Next, Woo is found from the z-NS equation (2) in order [3,-1], W00xx = — {Ax + 2)H#. The BCs are (i) the no-slip Eq. (4) in order [4,0] and (ii) the tangentialstress Eq. (5) in order [3,-1]: W00x{x = 1) = H#. This yields the solution Woo = — ( | x 3 + x2 — 5x)Htf. From the incompressibility, Eq. (3) in order [5,0], one finds Uw = | ( x 4 + 2x3 — 15x2)Htf£. Substituting this expression (evaluated at x = 1) into Eq. (26), we obtain the leading-order EE: Ht + AHHz + 2HK£. = 0.

2.2.3

The ^-correction t o the leading order

The ^-correction to the EE involves H0i and comes from the kinematic condition, Eq. (9), when taken one order in £ higher than before, viz. in the order [5,1]. Preliminary, one finds P 0 i, Woi, and UQI by going through the NS equations in the same sequence as above but taking each at the next order in e\ e.g. the z—NS equation (2) is used in the order [3,0]. In the end, one arrives at the equation HlT + 4 # i # e + AHHU + 2HUK +Hn - V2lHz + J^ROHK

+ |wbfT«« = 0

(27)

(where we have simplified the notations H0i and r 0 i to Hi and r l 5 respectively). This can be considered to be a nonhomogeneous equation for Hu and then the constant V21 can be tuned in order that the solvability condition on the forcing part (the sum of terms that do not contain Hi) be satisfied. (The exact form of that condition is determined only after the BC in f are stipulated; then, it can be obtained e.g. by multiplying the equation by H, integrating over the coordinate f and the fast time r by parts, and using the BC.) We note that, if necessary (which does not seem to be

46 the case in the present problem), one could use a more complicated arrangement—of the type which was indicated already in [34] and is used for the time t here—for the coordinate z as well, such that one would have

dz

^

aiij

If the power series (24) for rj is substituted directly into Eq. (9), the leading order [5,0] yields the KdV equation (20), and the higher order [5,1] leads to (19). This means that all the coefficients appearing in the equation (15) (including those of the— possibly, small—dissipative terms) are correct even in the case of large dispersion.

2.3

Earlier lidity

perturbation conditions

approaches

and the problem

of va­

To discuss the advantages of the MP approach, we will consider other perturbation schemes used in derivations of evolution equations (see also section 4.1). 2.3.1

Long-wave approach

In early derivations (e.g. [2, 84, 75, 62, 41, 63]), starting from the pioneering work of Benney [11], the formal series were in powers of a single local "long-wave" parameter a = 1/L, where L is a local streamwise lengthscale of the type introduced above. The principal unknown, the film thickness /i, remains, at least initially, whole (i.e. without being expanded in the power series); and each basic parameter of the NS problem is assigned a certain order (e.g. the surface tension can be a ~ a~2)—in most cases, just implied to be of order a 0 . The EE for h is obtained in the form of an infinite power series NQh + aNxh + a2N2h + • • • = 0

(28)

where Ni (i = 0,1,2, • • •) are differential operators (containing the derivatives with respect to the temporal and spatial variables). Usually, the series is truncated to retain only the first two—or, less commonly, three—terms. If one thinks about the possible conditions of validity, it is clear that the global-validity conditions cannot be formulated in this approach in principle, since the perturbation parameter is not a global, basic one. The question of local validity depends on the nature of the series which can be convergent or, alternatively, merely asymptotic (see also section 4.1). In the latter case it is only possible to assert that the approximation is good for a smaller than some threshold value a0 < 1, but the exact value of aQ remains unknown since the asymptotic series are merely formal ones. [There are examples (see e.g. [74]) where the threshold value of the asymptotic parameter is really very small in comparison to 1.] In contrast, in the case of convergent series (and provided

47 the coefficients are of the order of magnitude 1), one has a geometric-series estimate for the remainder of the series in question, and the condition that the leading-order truncation yields a good approximation (i.e. one with the relative error being much smaller than unity) is just a
Single-parameter approach

If there are basic parameters of a nonzero order of magnitude, such as a = OM{&~2) above, then one can redefine the perturbation parameter to be a basic rather than a local one; e.g. A = a - 1 / 2 —as was the case in the work [89] on the vertical planar films and in the paper [85] on a flow down a cylinder. They, in contrast to the long-wave approach described above, do represent the thickness h as a perturbation series similar to those of other unknown fields. Independent variables are as well rescaled with some powers of the perturbation parameter. We call this technique the (global) single-parameter (SP) approach. Here (assuming convergent series as discussed above and in section 4.1), one can argue that A
48 tion approaches are briefly discussed below in section 4.1.

3 3.1

Some results Three-dimensional

inclined-film

flow

The general 3-D flow can be considered quite similarly to the 2-D flow treatment in section 2.1. In addition, one can allow the plane to be inclined to the horizontal through an arbitrary angle 9 (between 0 and 7r/2); the vertical film of section 2.1 is just a particular case, with 9 = n/2. This most general case was studied recently in [48] where the reader can find more details. Here, we briefly discuss some of the results. 3.1.1

General evolution equation; impossibility of a single-equation de­ scription of large-amplitude regimes for large times

For 9 < 7r/2, there is a component of gravity perpendicular to the plane. This results in a nonzero pressure in the flat-film solution. Accordingly, the locally Nusselt pressure po is introduced in addition to the locally Nusselt components of velocity, the streamwise one w0 and the cross-film one UQ. These are given by the same expressions as above (see section 2.1), w0 = 2hx—x2 and u0 = —x2hz [now, however, the thickness may depend on the spanwise coordinate y as well: h = h(y,z, £)]; similarly, one defines po = 2(h — x) cot9/R. Here, the Reynolds number R is defined with the generalized interfacial Nusselt velocity U = ghl sin 9/(2u) [above, we denoted Uv the value of U for the vertical case, sin# = 1]; also, U (along with h0 and p) is used here to nondimensionalize the formulation of the problem equations, the same way as Uv was used for the vertical case above. The new dependent variables are thus wi, v, w\, and pi[= p — p0]. One first solves the x-NS equation, with the normal-stress BC at the free surface, to find pi: Pi = -2(x + h)hz/R - aV2h (here V 2 = d2/dy2 + d2/dz2). This pressure clearly contains a viscous contribution in addition to the usual surface-tension part (and the reference pressure p0 is of a purely hydrostatic origin). Next, the equations for v, W\, and Ui are solved, in that order. We do not write these results here; they can be found in [48]. Finally, by substituting the velocities (taken at x = h) into the kinematic equation, one arrives at the EE which is the generalization of (14) above: ht - ^(h%

+ 2(h2 - l)hz + R ^-hAhz -

- | v • [h3(cot0 - WV2)Vh]

+ 2hiV2hz

^h6hz^ = 0.

(29)

49 Also, the theory yields the local-validity conditions [a generalization of Eq. (10)]: \(1 + A)W L3

(1 + A)cot0 (1 + A ) 4 # (1 + A) 2 1 < 1, L L ' L2

(30)

where we have assumed, for simplicity, that the lengthscale in the y direction is not much smaller than L, the lengthscale of the solution change in the z direction; such so far has been the case in all experiments. [A one-dimensional version of the EE (29)—which also lacked the last, odd-derivative term—appeared before, e.g. in [62]. Also, similar to Eq. (14), we have omitted all the dispersive terms other than the last term on the l.h.s. of Eq. (29).] The solutions of this equation yield a good approximation to the true evolutions of the thickness h as long as the conditions (30) are satisfied. Similar to our discussion of Eq. (14), the two equations, (29) and the Benney one, are not necessarily equivalent when the conditions (30) break down, and there are indications that our equation has a chance to avoid the explosive solutions which are known (see e.g. [50, 81, 83]) to mar the Benney equation. Indeed, it is the term with the highest power of h (in fact, h6) in its coefficient which causes the explosion in the Benney equation; but our /i 6 -term enters with the opposite sign to the one in the Benney equation. Thus, although neither of the two equations is capable of a quantitatively good description of the large-time behavior, the EE (29) might be a better choice to be used as a qualitative model for large-amplitude waves. This possibility might make this equation worthwhile of a further investigation (which we have not attempted as yet). 3.1.2

Evolution equation for small-amplitude regimes

For small amplitude-regimes, we have rj = h — 1 0. We will always assume this condition to hold. It follows that R > (5/4) cot 6 > cot 6. One can use this inequality together with the condition A < 1 to transform the general conditions (30), which leads to the local-validity conditions for the small-amplitude EE (31): max(W7L 3 , R/L,

1/L 2 , A) < 1.

(32)

The analysis of the derivation of the EE (31) shows that the third, destabilizing term has its origins in the inertia terms of the NS equations. The two stabiliz­ ing terms, the fourth and the fifth, are due to the hydrostatic and capillary (i.e.

50 surface-tension) parts of the pressure, respectively. Finally, the last, odd-derivative, dispersive term is generated by the viscous part of the pressure. Such a term also appeared in the EE which was obtained by Topper and Kawahara [93] who assumed the plane to be nearly vertical (as a result, the hydrostatic term was absent in that equation): They used the small angle of the plane with the vertical as their (single) perturbation parameter. Our derivation shows that this assumption is unnecessary. One can see that for an arbitrary inclination 0 (and any value of W), provided R is close to Rc = 5 cot 0/4 (so that 5 is sufficiently small), the equation (31) can be good, with its dispersive term being not small. At the same time the hydrostatic term can be large (but one needs a sufficiently large spanwise lengthscale Y for this, viz. Y ^> Z). If the dispersive term is omitted in the equation of Topper and Kawahara, it becomes the one obtained by Nepomnyashchy [77]. The one-dimensional version of the latter is just the Kuramoto-Sivashinsky equation [64, 87]. Thus, all of these equations—as well as the Zakharov-Kuznetsov equation [98] (see also [65])—can be obtained as certain limiting cases of the EE (31). Because the EE (31) is dissipative, the system evolves towards an attractor. Thus, it essentially forgets the initial conditions. On the attractor, there cannot be any systematic change in time (although fluctuations around the constant averages may have arbitrarily large amplitudes). Consequently (as was first argued in [6, 37]), the destabilizing inertia term and the stabilizing, capillary one should be of the same order of magnitude. Hence, an estimate of the (dimensionless) characteristic lengthscale at large times, L a , follows: La = (W/S)1/2. (33) Similarly, the balance between the nonlinear "convective" term and either the disper­ sive term or the capillary one (whichever is larger) determines the asymptotic magni­ tude of the characteristic amplitude: Aa = max(W r /L^, 1/1%). With these estimates, one transforms the condition (16) to the form max( W/L\, R/La, \/L2a) 5. Thus the global-validity conditions can be written as Eq. (17) again, but with the generalized definition of La given by Eq. (33). The transformation of Eq. (31) to a "canonical" form is accomplished by rescaling 3 3W/252. v = Nrj, z = Laz, y = Lay, and t = T a t,with N = W/(6L a) and Ta = The resulting small-amplitude evolution equation (with the tildes in the notations of variables omitted) is Vt + Viz + Vzz ~ Wyy + V477 + AV277Z = 0.

(34)

(with definitions K = cot 9/5 and A = 3/y/WS). This is the generalization of Eq. (18). Looking at the linear stability properties of Eq. (34), one substitutes the normal mode rj oc exp(s£ - iujt) expi(jy + kz) into the linearized version of Eq. (34). This yields the growth rate: s = -K (jf - (j)4 + [l - 2 (j)2] k2 - kA. From ds/dk2 = 0, we

51 find the streamwise wavenumber kmax corresponding to the maximum growth rate (at fixed j): k2max = 1/2 - j 2 . Hence, the maximum growth rate smax(K,]j) is Smax = 1/4 - j 2 ( l + *).

(35)

One can see that, for every fixed j , this smax is a (linearly) decreasing function of K. The results of the linear stability theory are useful for understanding certain results of numerical simulations (see [47, 48]) of EE (34). The same problem has been treated earlier by Krishna and Lin [63] with the LW approach (see also [84]). They derived a Benney-type equation in the series form (28). They had cited explicitly about one hundred terms of that equation, among which all the terms of Eq. (34) can be found. However, as the MP derivation establishes, only a small number of those terms are essential and should be retained in the equation. All the other terms can be neglected under conditions (32) or (17), i.e. when the equation can yield a good approximation. 3.1.3

Numerical studies of evolution equation

We [48] have numerically simulated the dissipative-dispersive equation (34) with pe­ riodic boundary condition. Below, we discuss some preliminary results. It seems plausible that the results will be insensitive to the exact form of the boundary condi­ tions and the size of the integration domain provided the dimensions of the domain are sufficiently large; namely it should include many "elementary structures" (we note that similar simulations give "surprisingly good results" [44] for certain problems of the boundary-layer wave transitions). Accordingly, we solved Eq. (34) on extended spatial intervals, 0 < y < 2Kp and 0 < z < 2irq, with p >> 1 and q ^> 1. We used spatial grids of up to 256 x 256 nodes and employed the Fourier pseudospectral method for spatial derivatives, with appropriate dealiasing. Time march­ ing was done (in the Fourier space) with a third-order Adams-Bashforth and/or Runge-Kutta methods. The results were verified by refining the space grids and time steps; by observing the volume conservation, / rjdydz = 0; etc. Some of our initial conditions were motivated by the experiments [72]: these initial conditions modeled the inlet conditions of their experiments [see Eq. (36) below]. Below, we discuss some results of our numerical simulations, such as (i) the effect of dispersion on the large-time behavior of the film surface near the attractor and (ii) the dependence of the transient states on the wavenumber of the initial "forcing", as compared to the experiments of Liu et al. [72]. 3.1.4

Unusual patterns on strange attractors for strongly dispersive falling films

In this section, we consider the most unexpected result of our numerical simulations. Namely, during our studies of the strongly dispersive cases of Eq. (34), A ^> 1, we

52 found highly nontrivial patterns (of the film surface) persisting for all large times. The system evolves toward these self-organized states starting from the small-amplitude (3-D) white-noise surface (whereas with the same, random initial conditions, look­ ing at the large-time states of the surface in the case of small dispersion, we see no ordered patterns). This contrasts the spatial patterns studied up to now in fluiddynamical experiments—as well as in solid state physics, nonlinear optics, chemistry, and biology—which invariably were almost periodic, at least locally (see e.g. [22]). Our results indicate that patterns of quite a different nature can exist on attractors of driven dissipative-dispersive systems. These patterns [48, 47] consist of two subpatterns of localized soliton-like deviations of the surface. One of these subpatterns is a V-shaped array of larger-amplitude bulges on the film surface. Those move in a "sea" of smaller-amplitude bumps which constitute the second subpattern. Each of the two subpatterns spans the entire domain of the flow and moves as a whole along the z coordinate axis. However, the velocity of the bulge formation is different from that of the bump subpattern, so one subpattern "percolates" through the other. Figures 1 through Fig. 3 illustrate these points. A snapshot of the film surface at t — 3200, for a vertical film (i.e., K = 0) and with A = 50 (and p = q — 16) appears in Fig. 1. (Note that in Fig. 1 the amplitudes of the structures—which in reality are all small-slope—are exaggerated, due to a different scale of the rr-axis.) The evolution of "energy" E oc / r}2dydz shown in Fig. 2 demonstrates the time t = 3200 the film waves have approached their asymptotic state in which there is no further systematic change (although chaotic undulations of the energy are evident). One can see the above-mentioned two-stream nature of the pattern in Figures 3(a) and 3(b). The V-shaped formation consisting of 13 large bulges (see also Fig. 1) moves down between the two snapshots shown [the earlier-time one in Fig. 3(a) and a later-time snapshot in Fig. 3(b)]. The small-amplitude background subpattern moves uniformly as well, but in the opposite direction (in the moving reference frame of the observer). One can see also that the bump subpattern slowly changes with time. (We have also made a computer animation in which one can clearly see these motions of the two subpatterns.) It had transpired that one component of this dynamical, spatiotemporal pattern, viz. the bulges, has already been observed by the authors of Ref. [92], who performed the simulations of the vertical-film equation with A = 25 for p = q = 64/(27r). (In fact, they simulated a dissipative-dispersive equation [93] in which the K term of Eq. (34) was missing, which happened because of a certain—unnecessary, as we discussed above—assumption on which their derivation was based.) However, as far as we know, the authors of [92] used only contour plots as their graphics tools, and overlooked the bump subpattern. Thus, the entire complex, dynamical character of the two-phase pattern remained undiscovered. Figs. 2 and 3 confirm the estimates of characteristic quantities obtained with the term-balancing considerations (see [48, 47] for more on this point). For the 1-D version of Eq. (34), a perturbation theory of weak interactions (e.g. [9, 8, 42, 27, 56]) of the "pulses" (see [55]) was developed. It is based on soliton

53

Figure 1: Snapshot of the pattern on an attractor of Eq. (34) with K = 0 (i.e. the scaled surface of a film flowing down a vertical plate, here—down the page, with illumination from the top left), A = 50, and periodic boundary conditions on 0 < y,z < 32n: a view in an oblique direction, for i — 3200.

F i g u r e 2: Evolution of the surface deviation "energy" Ji]2dydz from an, initial smallamplitude "white-noise" surface to an. attractor of Eq. (34). The snapshots of Figs. 1 and 3 were taken near the end of this run.

54

Figure 3: (a) Cross-stream view of the pattern shown in Fig. 1. i;;: Ji.-.m? as in (a) but after a time interval At = 0.25. Note that the vertical motion of t :>-■ ■'• ■■ h.-.ped formation of bulges from (a) to (b) is in the opposite direction to that of the n\n\u) lottem. solutions of the ODE describing the pulse in a certain reference frame, The soliton, or at least its small outer "tails"—which are responsible for the weak interactions— are found analytically. In contrast, no analytic solution is available for a solitary 3-D bulge. Therefore, the prospects for an interaction theory of the 3-D bulges appear to be uncertain at best. One observes that the bumps incessantly collide with the bulges. These interac­ tions are seen to be (almost) reversible, like interactions of KdV pulses (e.g. [970. This contrasts with the irreversible coalescences of 2-D pulses discovered in [38] and [591 for highly nonlinear dissipative equations. It has been implicitly assumed (see section 4.1) in the derivation of the globalvalidity conditions for the EE (34) that the solutions of that equation remain bounded. Our simulations justify that assumption. Also, the amplitude of the solutions turns out to be of the same order of magnitude as its estimate obtained by the painvise balance of terms in the evolution equation. These simulations also confirm that the large-time behavior of the solutions of (34) is essentially insensitive to the initial conditions: Every solution evolves toward an attraetor (whose nature is solely determined by the basic parameters of the XS problem). Similar to Eq. (34), we [29, 36] have derived an evolution equation for a film which lows down a vertical cylinder (see Eq. (46) in section 3.2.2). The only essential difference between the two EE is the opposite sign of the K term. However, this term disappears in the case of a vertical planar film, as well as in the limit of an ininitely large radius of the cylinder. According to our simulations, the corresponding #~~2~ term of the annular-film equation (in which the variable <j> is changed to y = 60, so that 0 < y < 27r6 = 2wp) is sufficiently small even for p as small as p = 55 so that the results essentially coincide with those of the K = 0 version of Eq. (34). So, physical

55 experiments with films flowing down vertical cylinders can verify the theory which leads to the planar-film EE (34). However, the cylinder should be sufficiently long so that the waves propagating from the inlet could have enough time to approach the attractor. It is not difficult to find that with h0 ~ 1 mm, the radius ~ 1 cm, and with the restricting conditions (17), in order for the cylinder to be not too long, the liquid should be sufficiently viscous, in fact several hundred times as viscous as water. This can be easily achieved with e.g. a glycerin-water solution. [It is interesting that one can see a straight row of bulges in the photograph of a film flowing down a cylinder in Fig. 2 of Ref. [12]; however, the jRi-condition of validity (17) was not strictly satisfied there.] 3.1.5

Transient patterns: Qualitative agreement of simulations w i t h ex­ periments

As was mentioned above, the large-time behavior of the film is insensitive to the ini­ tial conditions. In contrast, the transient states do "remember" the initial condition: the film surface exhibits a variety of patterns as one varies the initial conditions. We studied the transient states primarily in connection with the recent experiments of Liu et al. [72], in which the flow (down an inclined plane) was perturbed at its inlet with sinusoidal pressure variations having a fixed frequency / . In some of their experiments, in addition to this single-frequency "forcing", they used a secondary forcing at the subharmonic frequency / / 2 , with an amplitude considerably smaller than that of the primary forcing. (The objective for the secondary forcing was to enhance possible broad-band subharmonic resonances). We modeled this inlet (pos­ sibly, two-frequency) forcing by the initial condition rj(y,z;t

=

0) = Af cos(nfz) + Ah[cos{2nfz

- fa) + cos(3nfz - fa)]

+Ah[cos(rifZ — fa) + cos(2nfZ — fa) + cos(3n/^ — fa)] cosy +Asub[cos(0.brifZ — fa) + cos(0.5n/£ — fa) cosy] H-small white noise,

(36)

(where y = y/p, z = z/q, and fa are independently generated random phases), with Ah
56 observed in the simulations turn out to be quite similar to their counterparts in the physical experiments [72], When the wavenumber k of the primary forcing is close to the neutral wavenumber (i.e. one for which the growth rate s = 0), which is k = 1 (for the 2-D modes, j = l ) , and in the presence of the secondary forcing at the subharmonic frequency [i.e. Asub ^ 0 in Eq. (36)], we observe in our simulations that the oblique subharmonic interacts very strongly with the fundamental. Fig. 4 shows this for an idealized initial condition, 77(2/, z\ t = 0) = Af cos(rifz) + Agub cos(0.5n/£) cos y (37) with Af = 1.2, Asub = 0.05, and n / = 15 (while q = 16). Figure 4(a) shows the strong interaction with the exchange of energy between the fundamental and the (detuned) oblique subharmonics. Plotted there are the "energies" Emtn of the Fourier modes, defined as ■t^m,n = | tiffin \

where amn is the Fourier coefficient: 71 =

Y,arnnei{my/P+nZ/q).

The surface exhibits checkerboard patterns such as the one in Fig. 4(b). One can see that it resembles the checkerboard pattern of "cat eyes" in Fig. 12 of Ref. [72]. However, if one employs the full initial forcing (36), then other modes become signif­ icant, as is evident in Fig. 5(a), and while the period doubling is obvious there, only an imperfect checkerboard pattern is observed. (This was further explained in [48] with a simple analysis of the spatial structure of the principal modes.) However, the property of period-doubling remains very robust when the forcing is sufficiently close to the neutral wavenumber. But as the forcing wavenumber decreases away from the neutral, the intensity of the interaction between the fundamental and the subhar­ monic dies out. Qualitatively similar observations were reported in the experiments by Liu et al [72]. With one-frequency initial forcing, there is a certain stability window: When the forcing wavenumber k is between 0.77 and 0.84 and K > 1.55, the amplitude growth saturates and the final state is a stationary propagating 2-D wave. These stationary waves consist mainly of the linearly unstable fundamental and a few of its stable overtones with smaller amplitudes. Such stationary waves were constructed by a number of researchers (e.g. [24, 49, 75, 94, 95]) for different film evolution equations; however, it appears that most of the huge number of such stationary solutions are significantly unstable; only a small number of them are observed in experiments— and only when artificial forcing is present. In our case, the stationary waves are well approximated as two-mode equilibria consisting of the unstable fundamental and one stable overtone. Assuming for simplicity that dispersion is negligible, we approximate

57

F i g u r e 4: Subharmonic interaction of Fourier modes in a run with the initial condition (37) whose frequency of primary forcing is close to the neutral one (q = 16, nf = 15). (a) Evolution of energies of the principal Fourier modes (the numbers shown in the legend next to each line are the corresponding values of m and n, in this order). Note that since n / is an odd number, there are two "almost-subharmonic" modes, one with n — 7 and the other with n = 8. (b) Checkerboard pattern of the film surface for t = 21 (cf Fig. 12 of [72]). The interval between two neighboring isothickness contours is 0.6.

Figure 5: Similar to Fig. 4 but for a run starting from the general initial condition (36), with gr = 47 and nf = 46. Note that the checkerboard-Uke pattern in (b) is not perfect, but the streamwise period-doubling is clear (t = 14.6; only one quarter of the streamwise extent of the periodicity domain is shown; the interval between two neighboring isothickness contours is 0.6).

58 rj as the Fourier series truncated at the Nth member, r](z,t) = Y/Mt)einkz

+ c.c.,

(38)

n=l

and obtain from Eq. (34), for N = 2, the Galerkin system dAi/dt = siAi — ikA\A2 and dA2/dt = s2A2 — ikAiAi, where sn = n2k2(l — n2k2). Hence, for a stationary state (dAn/dt — 0), and assuming A\ to be real, we find Ai = yjsi\s2\/k-

A2 =

-isi/k.

Similarly, a three mode equilibrium including the third harmonic A 3 ^ 0, can be found analytically (see [48]). To analyze the linear stability of such equilibria, one considers a normal mode of the Bloch type (introduced a long time ago for the stationary wavefunctions of an electron in the periodic field of the crystal lattice; see e-g. [66]), I

H ~ ePt+iHJy+KZ) Y^ Bmeirnkz.

(39)

m=—I

After substituting rj = rj0+ H into Eq. (34) and linearizing it, one obtains an eigenvalue problem for P. [Such "Floquet" analysis was applied before to film waves (e.g. in [49, 17]), and was used earlier in other hydrodynamic problems, see e.g. [44, 57]. We (see e.g. [33, 39, 88, 99]) believe that in this context it is more appropriate to mention Bloch's rather than Floquet's name.] Such an analysis [48] confirms the stability to all 3-D disturbances for the values of the parameters indicated above. For the forcing frequencies slightly outside of the stability window, it yields growth rates which lead to valid estimates of the lifetimes of the quasi-stationary states. However, often times such a linear stability analysis fails to predict what patterns will emerge, because the nonlinearity quickly makes dominant other modes than those which govern the linear stage of instability. Thus, the usefulness of the linear theory is quite limited here. In Ref. [48], the reader can find more results (some of them presented in graphics form) of the analysis and numerical simulations of the stationary waves and their stability. In addition, there are results of simulations of "natural" waves, i.e. those which develop from random initial conditions, corresponding to experiments without any forcing. When the forcing wavenumbers fall below the stability window, we observe evolu­ tion which resembles the "synchronous instability" of experiments [72]. Figure 6(a) comes from a run with the K, = 0.18 and A = 0.50, the values calculated with the parameters of the corresponding physical experiment. Evidently, the oblique modes with ra = 2 become important; as a result, a quasi-stationary 3-D state develops. The resulting film surface pattern is shown in Fig. 6(b). It should be compared to the experimental pattern found in Fig. 18 of [72]. Some further discussion of the

59

F i g u r e 6: 3-D "synchronous instability" of initially (almost) 2-D waves, Eq. (36) Asub = 0, at lower forcing frequencies (q = 47, n / = 26). (a) Evolution of energies of principal Fourier modes. Note the quasistaionary-wave state beginning at some time t = 120. (b) The surface pattern at t = 164 (K = 0.18, A = 0.05: cf Fig. 1(a) of [72]). interval between two neighboring isothickness contours is 0.4.

with four after The

F i g u r e 7: "Solitary wave" pattern in a run with a low-frequency forcing (the initial con­ dition (36), with q = 47 and nj = 4; K = 0.56, A = 0.057). (a) Evolution of energies of some principal Fourier modes (the numbers shown in the legend next to each line are the corresponding values of m and n, in that order), (b) The wave profile at t = 19.3. Only a part of the train of four solitary waves is shown (cf Fig. 2(b) of [72]).

60 synchronous patterns is found in Ref. [47] (They have been also observed [49] in simulations of the Benney equation on nearly-minimal domains of periodicity.) When lowering the wavenumbers of the one-frequency forcing even further, we find that the film surface develops solitary wave-like transient patterns (see Fig. 7). To a certain degree, these patterns resemble those seen in Fig. 2 of Ref. [72] obtained in low-frequency forcing experiments [however, those experimental solitary waves are seen to have large amplitudes which do not quite satisfy the condition A < 1 , which further lowers the expectations of agreement with simulations of Eq. (34)]. One concludes that these results of simulations of Eq. (34) qualitatively agree with the experimental findings of [72] regarding the observed types of transient patterns on inclined films as well as the correspondence between the pattern types and the ranges of forcing frequency.

3.2 3.2.1

Flow down a vertical

fiber

Large-amplitude regimes

A theory (of such film flows down vertical cylinders) hinged on a single evolution equations was constructed in [34]. It was motivated by the experiments [82] on flows down vertical fibers. The less-formal version of the MP approach was used to derive the evolution equation for the 2-D waves, ht + 2h2hz + S [h3{hz + hzzz)]z

= 0.

(40)

Here the modified Weber number S = 2ah0/{3pgb3)

(41)

where b is the radius of the cylinder and the other notations have the same meaning as in section 2.1. The nondimensionalization is based on b, p, and the "Nusselt" velocity = ghl/(2u) (where v — ~p/ p is the kinematic viscosity); however, h is in units h0. For S = O M ( 1 ) , the following parameters were identified which are required to be small for the (global) validity of the theory: the aspect ratio /3 = ho/b and Ri = ghQ/(2u2b) = (Uvh0/v)l3, the modified Reynolds number (the notations here are different from [34]). The appropriate formal MP series (in powers of (3 and Ri) were also pointed out, and the expressions for velocities and pressure in terms of the film thickness were given. For S
+ S [(1 + Xf{Xz

+ Xzzz)]z

= 0.

(42)

Also, one can analyze restrictions on the experimental parameters following from the validity conditions for practically available working liquids [34].

61

Figure 8: Evolution of the surface deviation governed by Eq. (42) for a large value of the control parameter (S — 3.0), with a "white-noise" small-amplitude initial condition on an extended interval (q = 8.0). After (by t = 12) the initial pulse growth has saturated and a pulse train has formed, a pulse grows by a cascade of coalescences. (From [59].) In [59], we reported extensive numerical simulations of the EEs (40) and (42) on extended spatial domains 0 < z < 2irq (where 7 < q < 41) with periodic boundary conditions. We used (pseudospectral) numerical methods similar to those of section 3.1.3 including the appropriate ("2/5") dealiasing; details can be found in [59]. When a simulation started from small-amplitude random initial conditions, the large-time behavior depended mainly on the value of the control parameter S (pro­ vided the periodicity interval was sufficiently large, i.e. q > 1). For very small S, the surface evolution has (chaotic) KS character (not shown; see e.g. [19, 20, 46, 80]), which is in accordance with theoretical considerations (see [59]). For larger S = O M ( 1 ) , first a train of saturated pulses with OM0) amplitudes grows, and the further slow evolution proceeds by various interactions of the pulses. The most remarkable discovery in this respect was the irreversible coalescence of pulses; a pulse can grow to a large amplitude by a cascade of such coalescences, as in Fig. 8. Earlier, such coalescences were observed in physical experiments [70], but to the best of our knowl­ edge the work [59] was the first observation of pulse coalescences in any numerical simulations of PDEs. It turned out that the pulse coalescences could happen only for sufficiently large 5, greater than certain critical value Sc. (At intermediate values of 5, the pulse interactions have the character of particle-like, quasielastic collisions,

62

Figure 9: Pulse train evolution for two intermediate (subcritical) values of S in (42). (a) S = 0.4 (q = 8.0). (b) S = 0.8 (q = 20.0). (Prom [59].)

Figure 10: Growth of the amplitude hm of a pulse in a simulation of Eq. (40) (5 = 2.0; q = 16.0). (From [59].)

63 as in Fig. 9) The simulation results led to Sc = 1 ± 0.1. From Eq. (41), one obtains hoc = l.5Scpgb3/a.

(43)

This is in the form of the Quere law, who found that the critical average film thickness, defined as a thickness value below which no discernible change (in time) of the local film thickness is registered, is proportional to the cube of the fiber radius and does not depend on the film viscosity. In fact, he gave precisely the formula (43) with a single difference: it had a constant he called a in place of (1.5 5 C ), and Quere's experimental value was a = 1.4 ± 0 . 1 . Thus there is an agreement between our simulations and the experiments [82]. Figure 10 indicates that an isolated mature pulse does not grow between coalescences, contrary to some assertions of Ref. [53] who also simulated Eq. (40) (see [59] for further discussions regarding that work). At S > 1, the EE (40) was noted to have the Hammond [43] limit, ht + S[h3(hx

+ hmj\x

= 0.

(44)

This EE describes the film evolution in the absence of both gravity and the average axial flow. However, our results with S even as large as 20 (the largest value we were able to employ) were different from those for the Hammond equation (a notable difference was that for Eq. (40) the pulse coalescences persisted for large values of S). This suggests that the Hammond equation is inapplicable for cases with even very weak (but nonzero) average flow. The EE (40) can be formally obtained from a Benney-type evolution equation derived in [35] with a LW approach. Before that work, such an equation for the annular films, the analog of the well-known Benney equation for planar films (see also section 3.1.1), was missing in the literature. The key to its derivation in [35] was the assumption of large radius of the cylinder. Previously [68, 2], this problem was considered without that assumption, which led, after very involved calculations, to extremely complicated equations (and the reader should be warned that there are typos in the results of Ref. [68]). One can try to obtain the large-radius EE (40) as a limit of their results, but such a derivation appears to be very complicated and less illuminating than the one in [35] (the authors attempted the former derivation once, but had to give up). We note that in [35], only the first two orders of the Benney-type equation were given explicitly (we remind the reader that any Benney-type equation has the form of an infinite power series in the small longwave parameter a = h0/l
+ ±E(h% + | W 2 V[/i 3 V( A

+ V2/*)]) + ■ ■ • = 0. (45)

[Here (h, t, z) are in units of (h0, h0/UvJ), where 7 is the characteristic lengthscale in the axial direction, E = 7/5, and W2 = Wa2.} By now, the next-order, a 2 -correction

64 to this equation [such as that found by Nakaya [75] for the planar Benney equation] is also available [73]. One of its members can be shown to correspond to the dispersive term of the small-amplitude EE considered immediately below. 3.2.2

Small-amplitude evolution equation

The first EE for small-amplitude waves on falling annular films was obtained in early 1980s by Shlang and Sivashinsky [85] who used a SP approach. However, by applying the MP machinery, we [36] found that there is a more general equation of that kind; its canonical form is (cf section 3.1.2) r]t + Wz + Vzz + $~2V + V4rj + W2rjz 2

= 0.

(46) 2

2

Here is the azimuthal angle, the constant $ = l + 86 /(5a), $<_= 3b/(W^ ), V = $~ld2/d(j)2 + d2/dz2 and —in terms of the original, based on h0 and Uv, units—the new units for [z,rj,t] are [Lc = b/$V2,Nc = W/(6L 3 ),T C = 3L*/(2W)]. The global-validity conditions for this equation are A2EE1/L2<1

and

Rle = W/Lzc4Zl.

(47)

The equation of Shlang and Sivashinsky [85] is obtained if the last term in Eq. (46) is omitted. Their derivation corresponds to the following choice of powers of the single perturbation parameter (say, A,
0M(A~1), Tc = 0M(A~2).

(48)

It follows that $ = 0 M (A°), i.e. the $-term should be kept in the EE (46), but the last term can be neglected since W (= aR/2) = O M ( A " 2 ) and therefore ^ is small: ^ = ^ M ( A 1 ) , < ; 1. However (as was discussed in general in section 2.3) these SP conditions are unnecessarily restrictive. There are many other choices of SP powers which satisfy the less restrictive MP conditions (47). For example, changing just two of the SP stipulations (48), to R ~ A1 and Tc ~ A - 3 , yields the general dissipativedispersive equation (46). Another choice, R ~ A - 1 / 2 ("large" R\) and Tc ~ A - 3 gives the EE of [85] again (but with the amplitude A ~ y/X instead of their A ~ A). The presence of two terms in the expression $ = 1 4- 8b2/(5a) reflects the fact that there are two destabilizing influences, which are (i) inertia [the same as for the planar film, Eq. (34)] and (ii) the transverse-curvature part of the capillary pressure (which does not exist for the planar film). Their balance with the stabilizing longitudinal-curvature part of the capillary pressure determines the longitudinal (i.e. axial) lengthscale. If a<62, the capillary destabilization is negligible in comparison with the curvature one, and the EE reduces to that of the planar film (34). One can readily find that in this case

65 $ = 0M{b2/cr) and Lc = 0M{\fcr); also, the condition of negligibility of dispersion, \I> < 1, can be written (recall the relation W = Ra/2) as l/v / W r R < 1, the same as for the vertical planar film (in this case, the validity conditions (47) reduce to R/y/a « 1 ) . In the opposite case, i.e. when \/WR is sufficiently small, so that the dispersive effects cannot be neglected, the validity condition is just a >• 1. From the above considerations, the "cylindricity" of the film can play a role only when b is sufficiently small—0M(y/cf) or less. Otherwise, we have essentially the problem of a planar vertical film (with the restriction that the boundary condition in the spanwise direction must be periodic.) Such appears to be the case in all experiments documented in the literature (e.g. [62, 12, 54]) with the exception of [82]. The analysis of the above condition for the practically available liquids shows that for the cylinder curvature to be essential, the cylinder should be a microscopic fiber or the film a very viscous liquid (or both). In this case, Lc = 0M(b). The condition of dispersion negligibility becomes b/W < 1. When curvature and dissipation are essential factors, the (single) validity condition is W/b3 1. (This has in fact been used to simplify expressions for coefficients of Eq. (46): in some places where (b 4-1) enters, we changed it to just b.) For the equation of [85], that is Eq. (46) without the dispersion term, 3-D sim­ ulations on extended domains were done by Deissler et al. [23]. It is not surprising that in those simulations the azimuthal dependences developed: for their values of parameters, there are linearly unstable non-axisymmetric modes, as one can easily see from the linear stability analysis. However, only axisymmetric modes are unsta­ ble for $ < 2. Nevertheless, the azimuthal structure appeared in our simulations for $ as small as <$ = 1.1 (see Fig. 11). The explanation is that some modes, of the azimuthal wavenumber equal to unity, are only "weakly stable": their "decay" rate is exactly zero (see Fig. 12), i.e. they are neutrally stable (or, one can say as well, neutrally unstable). Therefore, the nonlinear interaction with unstable axisymmetric modes can easily grow these nonaxisymmetric ones. However, the modes with azimuthal wavenumber higher than unity are strongly stable and hence do not grow. We believe this is the reason that we did not observe any coherent structures—such as the bulges of Fig. 1— in our strong-dispersion

86

Figure 11: (a) Snapshot of the surface of a film (flowing down a cylinder) which develops from a small-amplitude white-noise initial surface in a simulation of evolution governed by Eq. (40). Due to nonlinearity, aziniuthal variations have grown, despite the stability of nonaxisymmetric normal modes of the linear theory, (b) Evolution of modal energies; the snapshot of (a) corresponds to the end of this run.

Figure 12: Maximum (over the axial wavenumbers) growth rate of normal modes (with the indicated values of the aziniuthal wavenumber m) of linearized Eq. (46), as a function of the "curvature parameter' #.

67 simulations, ^ = 10, for $ < 4 : as was mentioned before, Fourier modes with many azimuthal wavenumbers (and many axial ones as well) are needed in order to make such a coherent structure.

3.3

Small-amplitude

waves in core-annular

flows

A perfect core-annular flow (CAF) is axially symmetric and unidirectional: Two immiscible fluids fill a circular pipe and flow parallel to its axis. The interface between the annular fluid and the core one is a perfect cylindrical surface. However, it is well known that the surface tension at a cylindrical interface is a destabilizing factor (e.g. [25]). The jump in viscosity at the interface can be destabilizing as well. The linear-stability studies of CAFs have a long history (see e.g. [52]). In [51], the numerical studies of D. D. Joseph and his collaborators are presented in detail. They have greatly extended our knowledge of linear stability properties in different domains of the space of basic parameters (see also [90]). They also studied CAFs experimentally, and compared the results with the linear the­ ory. However, their nonlinear Ginzburg-Landau-type theory yields nearly monochro­ matic waves modulated on large lengthscales. Such waveforms were never observed in experiments—presumably because the parametric domains of validity were extremely narrow. We [37] have suggested a nonlinear MP theory for core-annular film flows (CAFFs; in which the annular fluid is a film, i.e. its thickness is much smaller than the core radius) which yielded interfacial waveforms in closer resemblance to those observed in experiments (this work is discussed in [51] as well). The paper [37] was, to the best of our knowledge, the first nonlinear study of CAFFs from the first principles of hydrodynamics. (The nonlinear evolution of core-annular configurations with no basic flow was considered earlier by Hammond [43].) In [37], we studied the horizontal CAFF, driven by an axial pressure gradient, and such that the effects of gravity are negligible [which is the case when a certain— called Bond's—number (whose definition can be found e.g. in [37]) is small, due to either the closeness of the densities or the small thicknesses of the liquid layers]. We derived (with the less-formal version of the MP approach) an EE for small-amplitude regimes in which the influence of the core disturbances on the film can be neglected. This EE turned out to have the structure of the KS equation. The global-validity conditions were obtained as three inequalities which can be written in the form h

i

6<<X and

ah3

Tr

W<
v

(where b is the pipe radius, UQ the velocity of the basic interface, and the rest of the notations are as before; all quantities here are dimensional, but, for brevity, we have

68 omitted the overbars). This work has demonstrated that the presence of basic flow can lead to saturation of the capillary instability, and thus keep from breaking down the core-annular topological type of the arrangement of the fluids. Similar KS evolution equations were obtained for small-amplitude cases of twofluid horizontal flows [6, 86, 45]. In [6], it was shown that the basic plane Couette flow can defy the Rayleigh-Taylor instability, thus keeping as topologically invariable the arrangement in which a film of a lighter fluid is at the bottom. (Similar results were obtained for a sheared film of a single fluid destabilized by molecular forces [5]).The other two papers considered a plane Poiseuille flow of two fluids in a channel (see [52], section IV.8, for a review). Recently, Benney-type equations were obtained for two-fluid flow in an inclined channel by Tilley et al. [91]. Such equations for largeamplitude cases can only be good as qualitative models, as was discussed above. A more realistic core-annular problem than in [37], viz. a problem of CAFF with nonequal viscosities, was solved in [31] with the MP approach (later this solution was reformulated with an SP approach by Papageorgiou et al. [79] whose numer­ ical simulations of the EE of [31] had confirmed the conclusions of [31].) In that work, we had shown that the core disturbances can make a nonlocal contribution expressed by an integral-operator term in the evolution equation. (It was not written explicitly there—because of space limitations required by the publisher, but it was clearly indicated in [31] how to obtain that integral term "...in terms of the (confluent hypergeometric) Rummer's function..." [31].) By using certain properties of the (dispersive-dissipative) integral term, we es­ tablished several domains in the space of basic parameters for which no break-up occurs, so that the flow keeps its core-annular character—including some cases in which the Reynolds number of the core was much greater than unity. (In fact, each such domain is given by the corresponding conditions of global validity of the theory.) When this core Reynolds number is much less than unity, the integral term of the EE, as we pointed out in [31], "...simplifies and is found in terms of modified Bessel functions...". The idea of the formal MP approach was first introduced in that work. We also pointed out the importance of the dissipative terms even when they are much smaller than the dispersive terms coming from the nonlocal, integral member of the EE (similar to [55]). The possibility of a linear-mechanism stabilization (by the viscosity stratification) in the case of a more viscous film was demonstrated. A generalization of this work to the case of rotating CAFF was undertaken in [21].

69

3.4

Vertical and horizontal amplitude waves

core-annular

flows

with

large-

A nonlinear evolution equation for 2-D waves in a vertical CAFF driven by gravity and/or vertical pressure gradient were cited in [38]: ht + (U- Q)hhz + 2Qh2hz + S [h3(hz + hzzz)]z

= 0.

(49)

Here the units of the film thickness h, basic velocity of the interface U, time t, and axial coordinate z are the Nusselt film thickness /i 0 , the Nusselt velocity Uv = ghl/(2v), the time h0/Uv, and the tube radius, respectively; Q = 1 — p c /p, where the last term is the core-to-film ratio of densities; and S is a modified Weber number S = af33 /(3pvUv) where a is the surface tension (we have omitted for brevity the overbars in notations of the dimensional quantities) and /3 is the ratio of film thickness to the tube radius. One [28] can obtain this equation, valid in certain domains of parameters, either directly from the NS problem with a MP approach, or formally from the 3-D Benneytype equation [28] ht + (U- Q)hhz + 2Qh2hz

+a (J^R(h6hz)z + ^W2V[h3V(P2h + V2h)]\ + • • • = 0

(50)

[here (h, t, z) are in units of (h0, h0/Uv,l), where I is the characteristic lengthscale in the axial direction, E = 7/6, and W2 = Wa2) which can be derived [28] analogous to the similar equation [35] for the vertical annular film [see Eq. (45) in section 3.2.1]. In the regimes Eq. (49) describes, the effect of the core disturbances is negligible; additionally, inertia effects are negligible. (This equation is good for CAFFs in ver­ tical capillaries, but cannot be expected to describe the "bamboo waves" discovered in "macroscopic" experiments [7]: indeed, as is easy to estimate, the inertia effects are important in those experiments.) When the pressure gradient is absent, and in the limit of a vanishing core, pc = 0, one finds that U ~ 1 and Q = 1, and Eq. (49) reduces to the Eq. (40) of a falling single film. In another limit, g = 0, Eq. (49) reduces to the EE of the horizontal CAFF which first appeared in [3], ht + hhz + SG [h3(hz + hzzz)]z

= 0.

(51)

Here the Weber-type number So is defined similar to S in Eq. (49) but with the basic interface velocity UQ instead of Uv. The same change of basic velocities is used in definitions of the units for the basic velocity of the interface U, time t, and axial coordinate z. Note that the structure of this EE differs from that of the falling-film EE (40) by the power of h in the coefficient of hz (the advective term) only. In [38], we simulated EEs (49) and (51) on extended spatial intervals with peri­ odic boundary conditions, similar to the work [59] for the falling-film equation (40).

70

Figure 13: Development of a pulse train and the cascade of pulse coalescences in the horizontal core-annular flow governed by Eq. (51) with So = 0.5 (q = 10.0). (From [38].)

Figure 14: Coalescences of pulses in a vertical core-annular flow governed by Eq. (49) with U = 2.5 and Q = -0.5, for S = 2.0 (q = 8.0). (From [38].)

71 [Also, later the same results appeared in [58] (with some elaborations and along with numerical simulations of some model equations of the horizontal CAFF)]. For the horizontal-CAFF equation (51), we detected some errors in earlier finite-difference short-interval simulations of [3]. In general, as the control parameter increased, we observed the same windows of qualitatively different behaviors as for the EE (40) (see [59]), but with lower boundaries between the neighboring windows. In particular, the coalescences of two pulses one of which is sufficiently larger than the other (see Fig. 13) occurred for So as small as the critical value So = 0.15. Cascades of such coalescences can result in formation of "giant" collar-like bulges of the film, which eventually may become "lenses" spanning the entire cross-section of the capillary tube. The lens formation can be regarded as a result of a pinch-off of the core, such as those observed in physical experiments [3, 4] for even very small average thicknesses, such as /3 = 0.04. The mechanism in Ref. [40] (proposed there for a core-annular configuration without any mean flow), if applied to the CAFF, would predict no pinch-off for (3 < 0.12. Our results suggest that coalescences can raise a local mean thickness over the threshold value, allowing the mechanism [40] to pinch-off the core. Thus, the coalescences of pulses explain the formation of lenses in experiments [3, 4], which formerly remained unexplained. Such coalescences have been observed [38] for general vertical CAFFs as well (see Fig. 14).

4 4.1

Some unresolved questions concerning foundations of the film flow research On foundations of perturbation in film flow studies.

methods

used

Although the history of the use of perturbation methods in nonlinear studies of film flows is at least several decades long, in all cases we know of they have been used in only formal and heuristic ways, quite far from being mathematically rigorous. Every time that a formal power series is obtained to represent a solution, one can ask whether that series is convergent and what is its radius (or radii, in the case of multiple series) of convergence. If it is divergent, the series can be asymptotic to the exact solution and thus still useful for approximating it. The worst possibility is that the series is not even asymptotic to the true solution; in that case, using it to represent the true solution can be plainly misleading. Even if the series is asymptotic, this asymptoticness can be uniform or not. To the best of our knowledge, such questions have never even been mentioned in the literature on film flows. We believe it is important to discuss these questions even if few, if any, of them can be answered at the present time. (For terminology we use and a good introduction to perturbation methods, the reader can consult the book by Murdock [74]; see also [60, 76, 26].)

72 It is clear that in all papers which employ the perturbative power series, it is im­ plicitly assumed that those are at least asymptotic to the true solutions. Considering the long-wave theory, the perturbation parameter a is based on the initial lengthscale (rather than on basic parameters). The global, i.e. uniform in time, validity of (i.e. a good approximation by) the series in a must include their validity at large times, at the attractor—on which the characteristic lengthscale of the solutions may be quite different from their initial lengthscale. The attractor lengthscale is deter­ mined by the basic parameters, and the initial parameters are essentially irrelevant to it. Therefore it is clear that the global-validity conditions (see section 2.1) cannot even be formulated in terms of the longwave parameter. Proceeding to theories based on basic perturbation parameters, let us speak for simplicity of a single-parameter perturbation approach, where the series are in powers of a parameter (say) e. If a theory is merely asymptotic, all one can assert is that the approximation is good for e less than some £0, and there are examples when the threshold value e0 is extremely small (e.g. [74, 67]) so that just e
73 proach. The (global) validity conditions assume convergent (multiple) series of the geometric type. Since such a convergence has not been proved, the method remains heuristic, just as the longwave and the single-parameter approaches. [However, as we argued in the Introduction, the advantage of the MP approach is that it yields the validity conditions (albeit heuristic as well), which are much less restrictive than those following from the SP derivation.] In the less-formal MP approach demonstrated in the Introduction and used in most of our theories, the validity conditions follow in fact from what is termed "the apparent consistency of the basic simplification procedure" in the excellent book [67]. However, in Chapter 6 of that book, it is demonstrated (albeit for algebraic examples only) that sometimes the presence of the apparent consistency does not imply that the solution of the simplified equations is a good approximation to the exact solution of the original, full problem. Therefore, in principle, a rigorous proof is required that our validity conditions really guarantee the good approximation. Until such a proof is available, the less-formal MP remains only heuristic as well. Even when the perturbation series are just asymptotic rather than convergent, the MP approach can be preferable since a single MP theory is equivalent to infinitely many SP ones (with different scalings of the basic parameters as powers of the single perturbation parameter). However, the MP theory uses the assumption that if a series is equal to zero, then each coefficient separately must be equal to zero. The proof is well known for the usual asymptotic series containing a single perturbation parameter, but one can see that it cannot be immediately generalized to the case of asymptotic series with many independent asymptotic parameters. Nevertheless, the uniqueness of asymptotic series can be proved at least for the case of two perturbation parameters [30]. Finally, the questions touched upon in this section are not specific to the theories hinged on a single evolution equation, but are as well relevant to theories based on less drastic simplifications of the NS problem, such as the boundary-layer system of Refs. [17, 16]. Indeed, such theories can only be justified by a perturbation theory using one or more small parameters.

4.2

On numerical

simulations

of evolution

equations

Most simulations in the film flow literature use periodic boundary conditions, which make the problem easier. As was mentioned before, it is desirable that the interval of periodicity be sufficiently large, containing many elementary structures of a char­ acteristic lengthscale; then one can hope that the result do not depend on the size of the interval, except for the edge effects. However, in any case, the problem with periodic BC is an initial value problem, whereas in the physical experiments with the convectively unstable [72] films an important phenomenon is the downstream amplification of the inlet noise. This can be modeled with different BC [23, 14], corresponding to the spatial (as opposite to

74 temporal) development of instability. Though we do not expect a significant difference between the two kinds of numerical simulations, the spatial one and the temporal one that we have been doing (indeed, one can think of a wave packet which is emitted at the inlet; after that, in its co-moving reference frame, it just develops from that initial condition, as it is advected downstream by the film flow), still it would be interesting to make the spatial 3-D simulations of our evolution equations and compare them with the periodic-BC-simulation results. (One source of a possible difference is the fact that with the periodic conditions, a structure exiting from the downstream end of the interval exits the upstream end, and thus can catch up with another, slower moving structure, which originally was upstream of the first structure, so that the two would never interact on an unlimited interval; whereas they come to interact on the limited interval of periodicity.) Such 3-D spatial-evolution simulations have never been done before with the more complicated boundary-layer systems, because of computational difficulties (see [14] for their 2-D simulations). However, for a single evolution equation they are quite feasible, as is evidenced by such simulations [23] of the equation of Ref. [85]. Another question one should be aware of concerns long-time simulations of a system evolution to a strange attractor. Since there is a sensitive dependence of a trajectory on the attractor to the initial conditions, the computed trajectory close to the attractor can relatively fast wander away from the true trajectory starting from the same initial conditions. This can happen no matter how good the numerical method is, just because of the unavoidable round-off errors. (This is a kind of the illconditioning problem.) Nevertheless, we believe it can happen that at the same time the computed chaotic trajectory reflects all the important (statistical?) properties of the true trajectory, so that the computed results are "good" for all practical purposes. We have not seen this problem discussed in the literature. In simulations of small-amplitude EEs of flows down cylinders it was always as­ sumed that the waves are axisymmetric because there are no unstable nonaxisymmetric modes in the linear theory. However as we mentioned above, some first (i.e. with m = 1) nonaxisymmetric modes can be only weakly, neutrally, stable, so that nonlinearity can easily excite them. This circumstance should be kept in mind for the future studies of the falling annular films.

5

Summary

For different film flows, we have discussed theories hinged on a single evolution PDE. Such theories can be obtained by the multiparametric perturbation approach. Its ad­ vantage over earlier perturbation methods is that along with an evolution equation— and explicit expressions for velocities and pressures in terms of film thickness—it also yields conditions on parameters under which the theory provides a good approxima­ tion of the wavy film evolution.

75 Evolution equations are now available, with known parametric domains of their validity (local and global), for small-amplitude regimes of planar—inclined or vertical—film flows. Such theories, each hinged on a single evolution equation of the film thickness, have also been obtained for waves in the general vertical core-annular film flow, and in its limiting cases—the flow of a free-surface film down a cylinder and the horizontal core-annular flow; in all these cases the evolution equations are available for both the small-amplitude and large-amplitude (but still small-slope) regimes. The large-amplitude regimes of planar-film flows cannot be quantitatively approximated by a single evolution equation of the film thickness. The simulations of the Benney-type equations (which equations are now available for the cylindrical cases as well—for the large-radius falling annular film and for the general vertical core-annular flow) should be used as qualitative models only. The small-amplitude evolution equations are relatively amenable to fullydimensional numerical simulations on extended spatiotemporal domains. These can lead to insights useful for the studies of even large-amplitude waves. Simulations of the equation for small-amplitude regimes of a strongly dispersive falling planar film exhibit formation of unusual two-stream 3-D patterns at large times, near a strange attractor. Some large-amplitude, highly-nonlinear equations have been simulated on ex­ tended spatiotemporal domains for two-dimensional waves. Here, the most signifi­ cant finding appears to be the phenomenon of irreversible coalescences of interacting solitary pulses. There are several agreements of these theories with corresponding experiments: quantitatively with the experiments on flows on vertical fibers [82]—whose parameters did satisfy the conditions of validity of the theory; and qualitatively with the inclinedfilm experiments [72] whose parameters were somewhat outside of the domain of theory validity. Also, the collision of pulses may qualitatively explain the phenomenon of core pinch-off in the capillary core-annular flow [4] (where the pulses grow beyond the domain of the theory validity). Many fundamental questions in the study of film flows have not yet been answered. Nevertheless, significant progress is under way, and many interesting insights and findings can be expected in the near future.

6

Acknowledgments

This work has been supported in part by DOE Grant DE-FG05-90ER14100. In our research, we have used the computing facilities of the Alabama Supercomputer Au­ thority and the National Energy Research Supercomputing Center of the Department of Energy.

76

7

References

[1] S. V. Alekseenko, V. Y. Nakoryakov, and B. G. Pokusaev. Wave formation on a vertical falling liquid film. AIChE J., 31:pp. 1446-1460, 1985. [2] R. W. Atherton and G. M. Homsy. On the derivation of evolution equations for interfacial waves. Chem. Engng Commun., 2:57-77, 1976. [3] R. W. Aul. The motion of drops and long bubbles through small capillaries: Coalescence of drops and annular film stability. PhD thesis, Cornell University, 1989. [4] R. W. Aul and W. Y. Olbricht. Stability of a thin annular film in a pressuredriven, low Reynolds number flow through a capillary. J. Fluid Mech., 215:585599, 1990. [5] A. J. Babchin, A. L. Frenkel, B. G. Levich, and G. I. Sivashinsky. Flow-induced nonlinear effects in thin liquid film stability. Ann. NY Acad. ScL, 404:426-427, 1983. [6] A. J. Babchin, A. L. Frenkel, B. G. Levich, and G. I. Sivashinsky. Nonlinear saturation of Rayleigh-Taylor instability. Phys. Fluids, 26:3159-3161, 1983. [7] R. Bai, K. Chen, and D. D. Joseph. Lubricated pipelining, stability of coreannular flow. Part V: experiments and comparison with theory. J. Fluid Mech., 240:97-132, 1992. [8] N. J. Balmforth. Solitary waves and homoclinic orbits. Annu. Rev. Fluid Mech., 27:335, 1995. [9] N. J. Balmforth, G. R. Ierely, and R. Worthing. Pulse dynamics in an unstable medium. Institute for Fusion Studies (University of Texas, Austin) Preprint IFSR#708, May 1995. [10] S. E. Bechtel, J. Z. Cao, and M. G. Forrest. Practical application of a higher order perturbation theory for slender viscoelastic jets and fibers. J. Non-Newtonian Fluid Mech., 41:201-273, 1992. [11] D. J. Benney. Long waves in liquid films. J. Math. Phys., 45:150-155, 1966. [12] A M. Binnie. Experiments on the onset of wave formation on a film flowing down a vertical plane. J. Fluid Mech., 2:551-553, 1957. [13] J. P. Burelbach, S. G. Bankoff, and S. H. Davis. Nonlinear stability of evaporat­ ing/condensing liquid films. J. Fluid Mech., 195:463-494, 1988. [14] H. -C. Chang, E. A. Demekhin, and E. Kalaidin. Simulation of noise-driven wave dynamics on a falling film, 1995. Submitted to AIChE Journal. [15] H.-C. Chang. Wave evolution on a falling film. Annu. Rev. Fluid Mech., 26:103136, 1994. [16] H.-C. Chang, M. Cheng, E. A. Demekhin, and D. I. Kopelevich. Secondary and tertiary excitation of three-dimensional patterns on a falling film. J. Fluid Mech., 270:251-275, 1994. [17] H.-C. Chang, E. A. Demekhin, and D. I. Kopelevich. Nonlinear evolution of waves on a vertically falling film. J. Fluid Mech., 250:433-480, 1993.

77 [18] K. Chen and D. D. Joseph. Lubrication theory and long waves. Preprint, 1990. [19] P. Collet, J.-P. Eckmann, H. Epsten, and J. Stubbe. Global attracting set for Kuramoto-Sivashinsky equation. Commun. Math. Phys., 152:203-214, 1993. [20] P. Const antin, C. Foias, B. Nicolaenko, and R. Temam. Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. Springer, New York, 1989. [21] A. V. Coward and P. Hall. On the nonlinear interfacial instability of rotating core-annular flow. Theoret. Comput. Fluid Dynamics, 5:269-289, 1993. [22] M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium. Rev. Mod. Phys, 65:851-1112, 1993. [23] R. J. Deissler, A. Oron, and Y. C. Lee. Evolution of two-dimensional waves in externally perturbed flow on a vertical cylinder. Phys. Rev. A, 43:4558-4561, 1991. [24] E. A. Demekhin, G. Yu. Tokarev, and V. Ya. Shkadov. Hierarchy of bifurcations of space-periodic structures in a nonlinear model of active dissipative media. Physica D, 52:338-361, 1991. [25] P. G. Drazin and W. H. Reid. Hydrodynamic Stability. Cambridge University Press, 1981. [26] M. Van Dyke. Perturbation Methods in Fluid Mechanics. Parabolic Press, Palo Alto, 1975. [27] C. Elphick, G. R. Ierley, O. Regev, and E. A. Spiegel. Interacting localized structures with Galilean invariance. Phys. Rev. A, 44:1110-1112, 1991. [28] A. L. Frenkel. On evolution equations for core-annular film flows. To be pub­ lished. [29] A. L. Frenkel. On wavy flow of films down a vertical cylinder. To be published. [30] A. L. Frenkel. Uniqueness of multiparametric asymptotic series. To be published. [31] A. L. Frenkel. Nonlinear saturation of core-annular flow instabilities. In Proc. Sixth Symp. on Energy Engineering Sciences, CONF-8805106 (Argonne Natl. Lab., Argonne, Illinois), pages 100-107, 1988. [32] A. L. Frenkel. On asymptotic multiparameter method: Nonlinear film rupture. In Proc. Ninth Symp. on Energy Engineering Sciences, CONF-9105116 (Argonne Natl. Lab., Argonne, Illinois), pages 185-192, 1991. [33] A. L. Frenkel. Stability of an oscillating Kolmogorov flow. Phys. Fluids A, 3:1718-1729, 1991. [34] A. L. Frenkel. Nonlinear theory of strongly undulating thin films flowing down a vertical cylinder. Europhys. Lett, 18:583-588, 1992. [35] A. L. Frenkel. On evolution equations for thin films flowing down solid surfaces. Phys. Fluids A, 5:2342-2347, 1993. [36] A. L. Frenkel. Dissipative-dispersive evolution equation for falling annular films. Bull. Am. Phys. Soc, 39:1856, 1994. [37] A. L. Frenkel, A. J. Babchin, B. G. Levich, T. Shlang, and G. I. Sivashinsky. An­ nular flow can keep unstable flow from breakup: nonlinear saturation of capillary

78 instability. J. Colloid Interfac. Sci., 115:225-233, 1987. [38] A. L. Frenkel and V. I. Kerchman. On large amplitude waves in core-annular flows. In Proc. 14th IMACS Congress on Computations and Applied Mathematics (ed. W. F. Ames), volume 2, pages 397-400, Atlanta, 1994. [39] A. L. Frenkel and G. Rudenko. Instabilities of space-time-periodic "triangulareddy" flow. To be published. [40] P. A. Gauglitz and C. J. Radke. An extended evolution equation for liquid film breakup in cylindrical capillaries. Chem. Engng Sci., 43:1457-1465, 1988. [41] B. Gjevik. Occurence of finite amplitude surface waves on falling liquid films. Phys. Fluids, 13:1918-1925, 1970. [42] K. A. Gorshkov and L. A. Ostrovskii. Interactions of solitons in non-integrable systems: direct perturbation method and applications. Physica D, 3:424-438, 1981. [43] P. S. Hammond. Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe. J. Fluid Mech., 137:363-384, 1983. [44] T. Herbert. Secondary instabilities of boundary layers. Annu. Rev. Fluid Mech., 20:487-526, 1988. [45] A. P. Hooper and R. Grimshaw. Nonlinear instability at the interface between two viscous fluids. Phys. Fluids, 28:3, 1985. [46] J. M. Hyman, B. Nicolaenko, and S. Zaleski. Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces. Physica D, 23:265292, 1986. [47] K. Indireshkumar and A. L. Frenkel. On wavy flow of films down an inclined plane. To be published. [48] K. Indireshkumar and A. L. Frenkel. Spatiotemporal patterns in a 3-D film flow. Submitted to "Advances in Multi-Fluid Flows" (ed. Y. Y. Renardy), SIAM, 1996 (Proceedings of Joint AMS-IMS-SIAM Summer Research Conference, Seattle, July 1995). [49] S. W. Joo and S. H. Davis. Instabilities of three-dimensional viscous falling films. J. Fluid Mech., 242:529-547, 1992. [50] S. W. Joo, S. H. Davis, and S. G. Bankoff. On falling film instabilities and wave breaking. Phys. Fluids, 3:231-232, 1991. [51] D. D. Joseph and Y. Renardy. Fundamentals of Two-Fluid Dynamics, volume II: Lubricated Transport, Drops, and Miscible Liquids. Springer, New York, 1993. [52] D. D. Joseph and Y. Renardy. Fundamentals of Two-Fluid Dynamics, volume I: Mathematical Theory and Applications. Springer, New York, 1993. [53] S. Kalliadasis and H.-C. Chang. Drop formation during coating of vertical fibres. J. Fluid Mech., 261:135-168, 1994. [54] P. L. Kapitza and S. P. Kapitza. Wave flow of thin layers of a liquid fluid:III. Experimental study of undulatory flow conditions. Zh. Exp. Teor. Fiz. 19:105, 1949. Also in Collected Papers of P. L. Kapitza (ed. D. Ter Haar), vol. 2 pp.

79 690-709. Pergamon, 1965. [55] T. Kawahara. Formation of saturated solitons in a nonlinear dispersive system with instability and dissipation. Phys. Rev. Lett, 51(5):381-383, 1983. [56] T. Kawahara and S. Toh. Pulse interactions in an unstable dissipative-dispersive nonlinear system. Phys. Fluids, 31:2103-2111, 1988. [57] R. E. Kelly. On the stability of an inviscid shear layer which is periodic in space and time. J. Fluid Mech., 27:657-689, 1967. [58] V. I. Kerchman. Strongly nonlinear interfacial dynamics in core-annular flows. J. Fluid Mech., 290:131-166, 1995. [59] V. I. Kerchman and A. L. Frenkel. Interactions of coherent structures in a film flow: Simulations of a highly nonlinear evolution equation. Theoret. Comput. Fluid Dynamics, 6:235-254, 1994. [60] J. Kevorkian and J. D. Cole. Perturbation Methods in Applied Mathematics. Springer-Verlag, New York, 1981. [61] I. L. Kliakhandler and G. I. Sivashinsky. Viscous damping of short-wavelength disturbances in down flowing liquid films. Submitted to Phys. Fluids. [62] W. B. Krantz and S. L. Goren. Finite-amplitude, long waves on liquid films flowing down a plane. Ind. Engng Chem. Fundam., 9:107-113, 1970. [63] M. V. G. Krishna and S. P. Lin. Nonlinear stability of a viscous film with respect to three-dimensional side-band disturbances. Phys. Fluids, 20:1039-1044, 1977. [64] Y. Kuramoto and T. Tsuzuki. Persistent propagation of concentration waves in disspative media far from thermal equilibrium. Progr. Theor. Phys., 55:356-369, 1976. [65] E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov. Soliton stability in plasmas and hydrodynamics. Phys. Rep., 142:103-165, 1986. [66] E. M. Lifshitz and L. P. Pitaevsky. Statistical Physics, Part 2: Theory of Con­ densed Matter (in Russian; vol. 9 of Landau and Lifshitz's Theoretical Physics). Nauka, Moscow, 1978. [67] C. C. Lin and L. A. Segel. Mathematics Applied to Deterministic Problems in the Natural Sciences. SIAM, Philadelphia, 1988. [68] S. P. Lin and W. C. Liu. Instability of film coating of wires and tubes. AIChE J., 21:775-782, 1975. [69] S. P. Lin and C. Y. Wang. Modeling wavy film flows. In Encyclopedia of Fluid Mechanics (ed. N. P. Cheremisinoff), vol. 1, pp. 931-951. Gulf, Houston, 1985. [70] J. Liu and J. P. Gollub. Solitary wave dynamics of film flows. Phys. Fluids, 6:1702-1712, 1994. [71] J. Liu, J. D. Paul, and J. P. Gollub. Measurments of the primary instabilities of film flows. J. Fluid Mech., 220:69-101, 1993. [72] J. Liu, J. B. Schneider, and J. P. Golub. Three dimensional instabilities of flowing films. Phys. Fluids., 7:55-67, 1995. [73] M. Lukin and A. L. Frenkel. On an evolution equation for a film flowing down a vertical cylinder. To be published.

80 [74] J. A. Murdock. Perturbation:: Theory and Methods. John Wiley, New York, 1991. [75] C. Nakaya. Long waves on a thin fluid layer flowing down an inclined plane. Phys. Fluids, 18:1407-1420, 1975. [76] A. H. Nayfeh. Perturbation Methods. Wiley-Interscience, New York, 1973. [77] A. A. Nepomnyashchy. Three-dimensional spatially-periodic motions in liquid films flowing down a vertical plane. Hydrodynamics (Russian) Perm, 7:43-52, 1974. [78] A. Oron and P. Rosenau. On a nonlinear thermocapillary effect in thin liquid layers. J. Fluid Mech., 273:361-374, 1994. [79] D. T. Papageorgiou, C. Maldarelli, and D. S. Rumschitzki. Nonlinear interfacial stability of core-annular film flows. Phys. Fluids A, 2:340, 1990. [80] D. T. Papageorgiou and Y. S. Smyrlis. The route to chaos for the KuramotoSivashinsky equation. Theoret. Comput. Fluid Dynamics, 3:15-42, 1991. [81] A. Pumir, P. Manneville, and Y. Pomeau. On solitary waves running down an inclined plane. J. Fluid Mech., 135:27-50, 1983. [82] D. Quere. Thin films flowing on vertical fibers. Europhys. Lett, 13:721-725, 1990. [83] P. Rosenau, A. Oron, and J. M Hyman. Bounded and unbounded patterns of the Benney equation. Phys. Fluids A, 4:1102-1104, 1992. [84] G. J. Roskes. Three dimensional long waves on a liquid film. Phys. Fluids, 13:1440-1445, 1970. [85] T. Shlang and G. I. Sivashinsky. Irregular flow of a liquid film down a vertical column. J. Physique, 43:459-466, 1982. [86] T. Shlang, G. I. Sivashinsky, A. J. Babchin, and A. L. Frenkel. Irregular wavy flow due to viscous stratification. J. Physique, 46:863-866, 1985. [87] G. I. Sivashinsky. Nonlinear analysis of hydrodynamic instability in laminar flames. Ada Astronau,, 4:1175-1206, 1977. [88] G.I. Sivashinsky and A. L. Frenkel. On negative eddy viscosity under conditions of isotropy. Phys. Fluids A, 4:1608-1610, 1992. [89] G. I. Sivashinsky and D. M. Michelson. On irregular wavy flow of a liquid film down a vertical plane. Prog. Theor. Phys., 63:2112, 1980. [90] M. K. Smith. The axisymmetric long-wave instability of concentric two-phase pipe flow. Phys. Fluids A, 1:494-506, 1989. [91] B. S. Tilley, S. H. Davis, and S. G. Bankoff. Nonlinear long-wave stability of superposed fluids in an inclined channel. J. Fluid Mech., 277:55-83, 1994. [92] S. Toh, H. Iwasaki, and T. Kawahara. Two-dimensionally localized pulses of a nonlinear equation with dissipation and dispersion. Phys. Rev. A, 40:5472-5475, 1989. [93] J. Topper and T. Kawahara. Approximate equations for long nonlinear waves on a viscous film. J. Phys. Soc. Japan, 44(2):663-666, 1978. [94] Yu. Ya. Trifonov and O. Yu. Tsvelodub. Nonlinear waves on the surface of a

81 falling liquid film. Part I. J. Fluid Mech., 229:531-554, 1991. [95] O. Yu. Tsvelodub and L. N. Kotychenko. Spatial wave regimes on a surface of thin viscous liquid film. Physica D, 63:361-377, 1993. [96] G. B. Whitham. Linear and Nonlinear Waves. John Wiley and Sons, New York, 1974. [97] N. J. Zabusky and M. D. Kruskal. Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett., 15:240-243, 1965. [98] V. E. Zakharov and E. A. Kuznetsov. Three-dimensional solitons. Sov. Phys. JETP, 39:285-286, 1974. [99] X. Zhang. On linear and nonlinear stability theory of periodic flows of incom­ pressible fluids. PhD thesis, University of Alabama, 1995.

83 INSTABILITIES D U E TO SORET DIFFUSION COUPLED TO T H E M O R P H O L O G Y OF A S O L I D - L I Q U I D I N T E R F A C E

LAYACHI HADJI Department of Mathematics The University of Alabama Tuscaloosa, Alabama 35487

ABSTRACT This paper aims to discuss the influence of thermal Soret diffusion on the in­ stabilities which can occur when a dilute binary mixture is solidified. The discussion is treated from two points of view. First the modification of the condition for the onset of constitutional supercooling in the presence of the Soret effect during the unidirectional solidification of a dilute binary alloy at constrained velocity is considered. Next, the importance of the coupling be­ tween Soret-driven convection and solidification in a Benard-like geometry is analyzed.

1.

Introduction

The classical experiments of Soret * have demonstrated that an imposed tem­ perature gradient can induce the formation of a concentration gradient in a binary mixture that is held in a closed container. A typical experiment consists of placing an initially uniform salt solution in a vertical cylindrical tube whose top is main­ tained at a higher temperature than its bottom. A condition of steady state is approximated by allowing the experiment to run for a long time (up to 25 days). It is observed that an accumulation of salt takes place at the lower boundary; the amount of salt that has piled up at the cold boundary is found to increase with

84 increasing initial salt concentration. These experimental findings have been com­ plemented by a derivation of a mathematical model to describe this newly dicovered effect; namely the concentration flux which appears in the classical 'Fickian' diffu­ sion equation has an extra term. This added term is proportional to the temperature gradient and on some mixture dependent constant. A careful theoretical analysis of this phenomenon (DeGroot and Mazur2) reveals that the extra flux term depends in a nonlinear fashion upon the concentration and takes the form -Dyc(l — c)VT> where Dj is the thermal diffusion coefficient due to the Soret effect, which could be either positive or negative, c is the concentration and V T represents the tempera­ ture gradient. The significance of the Soret effect in a large variety of phenomena has been recognized only in fairly recent times. Some examples include separation of mix­ tures in thermal diffusion columns 3 , thermal reactions 4, planetary dynamics 5 , electrochemical processes 6 , solid state physics 7 , magma chamber dynamics 8 and fusion-reactor technology 9 . Hurle and Jakeman10 stimulated a surge in research activity in the area of Soret induced convection. Much of the recent considerable interest in this area stems from the rich variety of spatial and temporal structures that can be observed in convection experiments near threshold11. The purpose of this paper is to present a general account of the influence of the Soret effect in solidification processes. This account will show that the importance of the Soret effect in solidification processes remains poorly understood. To date only few studies exist on this topic, and due to my own interests, only those dealing specifically with morphological or fluid instabilities will be discussed in detail. The motivation for considering the Soret effect has to do with the fact that solidifica­ tion systems are often closed to mass flow across their boundaries. Given that the temperature gradient across the binary alloy is the only external control parameter, the concentration gradients that subsequently form in the liquid phase are mainly due to two sources: the first one being the interfacial solute segregation that is associated with the moving solid front and the other mechanism depends on the Soret effect. The plan for the remainder of this paper is as follows: The influence of the thermal Soret diffusion on the interfacial instabilities is discussed in section 2. We first review existing results which incorporate the Soret effect in a linear fashion, then we derive an approximate condition for the onset of constitutional supercooling which takes into account nonlinear Soret effect. The modifications of the critical solidification velocity and of the critical wavelength are discussed. The coupling between Soret induced convection and solidification is discussed in section 3 and some concluding remarks are presented in section 4.

85 2.

Morphological Instabilities

2.1 Linear Soret effect The few relevant studies that exist in the literature differ only on the form for the flux relation. Hurle 12 , in his analysis of the morphological instability associated with the directional solidification of a binary alloy, took into account the effects of thermal Soret diffusion by assuming a relation of the form T = -DVc

- DTc0{l

- c 0 )VT,

(1)

where c0 is a constant reference concentration. His detailed stability calculations show that the influence of the Soret effect on the onset of morphological instability appears in a modified relation for the interfacial concentration gradient. His relation for the threshold velocity has the following asymptotic value y

UlC^+U, c

( )

K)

Utc + U*

where the U-s are functions of the various physical parameters of the problem, and the only contribution of the Soret effect appears in the expression for Ui. Realistic numerical values of these coefficients are such that Vc is a monotonically decreas­ ing function of c^. Therefore, the inclusion of the Soret effect did not affect the qualitative from of the relationship for the critical velocity. However, the inclusion of the Soret effect had a dramatic influence on the critical wavelength for systems exhibiting very small partition coefficients (k
- DT{fa +

fac)VT.

(3)

They consider two models: Model I, with fa = c^ and fa = 0, corresponds to the model of Hurle 12 and Model II with fa = 0 and fa = 1. They have performed a lin­ ear stability analysis in order to derive the conditions for the onset of morphological instability and predictions from the two models are compared. No significant differ­ ences between the two models are found for Dj < 0 or for Dj > 0 and large growth velocities. However, for Dj > 0, their calculations show an increase of the critical solidification velocity with the concentration CQQ for small growth velocities. This re­ sult is in contrast to the predictions from model I which show a monotonic decrease of the solidification velocity with concentration. They also report the possibility of oscillatory instability. For some parameter ranges, these oscillatory modes of insta­ bility precede the stationary modes and hence become the physically relevant ones.

86

Figure 1: Sketch of a binary fluid undergoing directional solidification. Figure 1: Sketch of a binary fluid undergoing directional solidification. A comparison of the critical wavelengths between the two models is mot carried out. 2,2 Nonlinear Soret effect 2,2 Nonlinear Soret effect In the following section, we shall derive an approximate relation for the onset of constitutional supercooling that is valid for nonlinear Soret effect by uti&ing thermodynamic arguments 14 . We consider the directional solidification of a dilute binary alloy in such a way that the solid-liquid interface is moving vertically upward with velocity V. See Fig. 1 for a schematic diagram, The system is completely described by the heat conduction equations in the solid and liquid layers and the solute diffusion equation in the liquid which incorporates the Soret effect,

dTLL _ 3T djiL _ _22 ^ _di _ V. _dz =~KL li,L) K 'L V dTs 0TS _

m_ m di

dz

Ks

T'

(4a) (4a) (46)

( 4(4c) C) C)VTL] ¥i -" Jz % Tz '" '' ~ ~ C)VTL] ' ' where TL and Ts are the temperature in the liquid and solid phases, respectively, c is the concentration of solute, D is the solutal diffusion coefficient, S is the Soret coefficient, KS and KL are the thermal diffusion coefficients in the solid and liquid phases, respectively and V denotes the gradient vector. Eq. (4c) is obtained Vv

==

Vv

[[ D SDc(l DV V Cc ++ SDc{1

87 from a mass balance over a differential element of volume in the liquid phase and the term between brackets that appears on the right-hand side of Eq. (4c) repre­ sents the sum of the flux due to the compositional gradient and the flux that arise due to the temperature gradient (Soret effects). Note that 5 = — DT/D in Eq. (4c). The corresponding boundary conditions, appropriate to a system that is closed to mass flow, are TL = TOQ)

S c ( l - c ) ^ — = 0, OZ

z = oo,

(5a)

Oz

at the solid-liquid interface located at z = 0, the continuity of temperature implies TL = Ts.

(56)

The interface temperature is obtained from the phase diagram for a dilute binary alloy. This boundary condition is a coupling source between the temperature and concentration fields at the interface, and takes into account the dependence of the freezing temperature of the mixture on the interface curvature (the GibbsThompson effect),

TL = TM + mc- TM1(±- + - U JL Hi

(5c)

til

The assumption of local thermodynamic equilibrium at the solid-liquid interface leads to the following connection condition for the concentration in the solid phase, cs = kc.

(bd)

The conservation of heat at the interface leads to a balance between the jump in heat flux and latent heat release, {KsVTs

- KLVTL)

• n = £ ( v • n).

(5e)

Finally, the conservation of solute at the interface, with the added assumption of no solute diffusion in the solid phase, yields, c(l - *)(v • n) = - [ P V c + DSc{l - c)VTL) • n.

(5/)

Here TM is the freezing temperature of the pure solvent, K§ and KL are the thermal conductivities in the solid and liquid phases, respectively; k is the partition coef­ ficient, n is the unit normal vector pointing into the liquid, v is the local growth velocity, m is the liquidus slope obtained from the idealized phase diagram, C is the latent heat of fusion per unit volume and 7 is the solid-liquid surface free energy per unit area; Ri and R2 are the principal radii of curvature of the interface (defined so that a positive undercooling is associated with a convex crystal).

88 In this work, we investigate the onset of constitutional supercooling in the liquid adjacent to the interface during the directional solidification of a dilute binary alloy at constrained velocity V with the added assumption that the liquid exhibits Soret effects. Since we are only interested in determing the conditions for the onset of constitutional supercooling, it is sufficient to consider the following simplified system obtained by evoquing the quasi-steady state assumption and by considering the fact that K/D « K,/D > 1, L

= 0, n n'

(6a) (6a)

d Ts 0 , 2 = 0,

MM (66) (66)

dz2 dz*


Vdc „drr„

dF++Dd-z ^ ^

+5 s +

K }

,dT n JTLL A

(6c) (6c)

[c(1 Tz -c)-dT SI[c{1 - c) -d7 = 0
subject to the boundary conditions TL = Tco ao,

^ =S c ( l - c ) ^ , dz dz dz dz

z = oo,

T = TTss =T =TMM ++ mcmc-l(±+ -J-), h, TLL = !(-£- + C^Ri R2' CyRi R2' KsSGs GS - KLGL = CV,

, == 0o 0o x

z z = 0, 0,

(7a) (7a) (76) (76) (7c) (7c)

c(l - Jb)V k)V = D^- - DSc{l DSc(l - c)^, 2 = 0. dz dz where Gs and GL are the temperature gradients in the solid and liquid phases, respectively. Equations (6a) and (6b) are integrated to give rL(*) = T, + GLL*, rL{z) *,

(8a) (8a)

Ts{z) = Tj T, + GGsz, sz,

(86) (86)

where T, is the temperature at the interface; Eq. (6c) then becomes cfc

V dc

„„ d , ,

+SG 1 d?++Dd-z DTz+SG^< ^<1-^ -^

= °'

(9)

«

Integration of Eq. (9) yields ^ + ^ ^c-SG c - S G LLcc(l-c) ( l - c ) = B,

(10) (10)

where B is an integration constant. Upon evaluation of Eq. (10) at infinity, we obtain C . B = 7J ^c x B = °°-

(11) (11)

89 Finally, the solution of Eq. (10) yields C{Z) c{z) =

l - e x« xP p( -( /- / a« )

((12) 12)

''

where C c++ ==

2 a + P)c<»], /J)cO0], c_ c_ == ^^ [[ -- aa -- Ny/d* A0(a ++ /?)c 0)c„], v/a +4/?(a + 2yj[-<* / 3 [ ~ Q ++ \f<**+W /a 2 ++ 4/?(a 0O],

(13a) V /J a = 13 a= D~ 1)~ ' '

(13c) ( 13fc )

SGi '" ^P == SGL

c M - fcc+

a=

(13c) fi

y/cP + Afta+

(13d) (13d)

fic

Note that the derivation is valid for parameter ranges for which both c+ and c_ are positive quantities. The thermodynamic freezing temperature ahead of the interface is obtained from the boundary condition, Eq. (7b), with the concentraion c given by Eq. (12), „_, _, , c + - Q n c -_eexxpp (( - ^ 2 / qa ) 7 (14) T = *TM +m m l - f i e exxpp((-_V2 /aa )) - r «TMI £ K>^ (14) *>f = " + ~ where /C is the mean interface curvature. The freezing temperature of the pure solvent, 7M, can be written in terms of the temperature at the interface as, Tj - mc(0) +TM1/C llC TM = T,

TI-- "w^'J^] = T, ^ " ^ 1 + l Ti M, ^K. ^ .

(15) (15)

Upon substituting Eq. (15) into Eq. (14), we obtain the following expression for the freezing temperature in the liquid ahead of the interface 3) = 7} mt-y^^ T/ T'-m-[-T3n

_ _ fOi e x p ((--**//a«))

]

"'

((16))

The condition for the existence of constitutional supercooling is then determined by imposing

S><*-

(IT)

and the condition for the onset of morphological instability is obtained by replacing the inequality symbol by an an equalilty sign. This translates into the following nonlinear relation between the various parameters 2 2 ===Q[(Q =fi[(fi +)c___- -i(c_ ++)+- - - -G -)22 ===,=, 4> +++)c_ ( c _ __++)] G LL{\ ( l - - -t)

(18) (18)

90 where <j> represents the degree of constitutional supercooling. The threshold values are obtained as the roots of Eq. (18). The numerical task of determining the zeros is accomplished by a bisection method that is based on Brent's method. This method is very efficient in detecting all zero crossings even for functions that are discontinu­ ous or whose curvature changes rapidly near a root. The root of interest is the solid­ ification velocity and its dependence on the bulk solute concentration with the rest of the physical parameters fixed. The following thermophysical properties, which are appropriate for a tin-bismuth alloy13, are considered: D = 1.8 x 10~ 5 cm 2 /s, k = 0.29, and m = —2.2. Two cases are considered. First the solidification ve­ locity is solved as a function of the bulk solute concentration with the rest of the parameters fixed. For S > 0, we have fixed two pairs of values for S and GL- The first pair is S — 0.01 and GL = 180 K/s, and the second pair is S = 0.0001 and GL = 180 K/s. The results, shown in Fig. 2, depict neutral curves that are qualita­ tively similar to those obtained by Hurle12 and Van Vaerenbergh et a/.13. However, a quantitative difference exists between their results and ours, namely, our neutral curve lies above theirs. This can be seen by comparing their asymptotic value for large c^ of 1.6^/m/s to our asymptotic value of about 10//m/s. The critical velocity is also found to decrease with decreasing values of the Soret coefficient. Fig. 3 de­ picts the neutral curve for negative Soret coefficient for S = —0.01 and GL = 180 K/cm. We note that, in this case, the nonlinear equation has two roots and only in a narrow range of values for c^. Our neutral curve differs significantly from the curve obtained by Hurle12 but shares some similarities with the curve corresponding to a linear thermotransport term. 13 Second, in order to clearly depict the modification of the threshold values for the onset of the constitutional supercooling by Soret diffusion, we have plotted the neutral curves in the (c^ — GL) parameter plane. Fig. 4 shows, on the same graph, a plot of the neutral curve for S = 0.01 and V = 200//m/s as obtained from the solution of Eq.(15) and a plot of the neutral curve without Soret diffusion which is described by 14 , GLDk C (19a) ~ = -(l-k)VmThe case for negative Soret coefficient is shown in Fig. 6 for S = —0.01 and V = 200/im/s. We note that, for positive Soret coefficient, higher temperature gradients are required for the onset of instability when Soret diffusion is included. Thus, the Soret effect has a stabilizing effect in this case. The opposite scenario holds, however, for negative Soret coefficients when realistic temperature gradients (GL > 20K/cm) are considered. The changes that occur in the concentration profile due to the inclusion of the Soret effect will also induce a change in the degree of supercooling (j>) Eq. (18),

91

woo

Figure 2: Neutral stability curve in the (c^ — V/D) plane for parameter values S = 10" 2 and S = 1G~4 (insert). which in turn affects the critical wavelength of the instability. This is given by \TM1

A = 2TTJ

(196)

where A is the wavelength of a sinusoidal deformation to the solid-liquid interface. We observe from Fig. 4 that positive Soret coefficient is associated with an increase in the degree of supercooling, and hence leads to a decrease in the wavelength at the onset of the instability. The opposite scenario holds for S < 0 and realistic tem­ perature gradients. Note that the regions of constitutional supercooling lie above the curves shown in the stability graphs. 3.

Soret-driven Convection Effects

The coupling between Soret-driven convection and solidification has been exam­ ined by Hadji and Schell15 and Zimmermann et a/.16. We refer the reader to other references in the latter paper. They investigated the influence of Soret induced con­ vection on the morphology of a solid-liquid interface. They utilized a Benard-like set up in which the top plate was maintained at a temperature that is low enough to allow for the formation of a thin layer of ice. The solid-liquid interface that appears in the process is assumed to be rigid and planar in the absence of convection, but is allowed to deform under the influence of the convective currents in the liquid.

92

Figure 3:

2

Neutral stability curve in the (c^ — V/D) plane for parameter values

s = -io- .

Figure 4: Neutral stability curve in the (Gi — CQQ) plane for parameter values S = 0.01 and V = 200^m/«.

93

Figure 5: Neutral stability curve in the (G, - Coo) plane for parameter values S = -0.01 and V = 200f/m/$. 3.1

Formulation

Consider a thin layer of a dilute binary mixture contained between two rigid and isothermal plates that are separated by the distance d. Fig. 6 shows a schematic diagram. The temperature gradient across the plates is selected so as to allow for the solidification of an upper portion of the fluid mixture. The solid-liqid interface that separates the lower liquid phase from the solid layer is assumed to be planar in the conduction state but deformable under the influence of the convective currents. The mathematical model is based on the conservation equations for heat, solute, mass and momentum, which in the Boussinesq approximation, take the form16, ^

01

^

+

v

• VTL = /c L V 2 T L ,

+ v • VC = DV • [VC - SC(l - C)Vr L ], V • v = 0,

?£ + v • v = — V p + i/V2v + [aT(TL - TR) + as{C - CR)]gk,

(20a)

(206) (20c) (20rf)

where TL, C and v are the temperature, concentration and velocity in the liquid phase, respectively; aT and as are the thermal and solutal expansivity coefficients, respectively; g is the gravitational constant, k is a unit vector in the upward vertical

94

Figure 6: Sketch of a solid-liquid interface that forms in a binary mixtures between two plates. direction and TR and CR are reference values. Only heat conduction is assumed to take place in the solid, and this is accounted for by dTs di

KSV2TS.

(21)

Eqs, (20-21) are supplemented by the following boundary conditions. The kinematic condition and the no slip conditions at the interface yield v « n = v x n = 0,

(22a)

where n is the unit normal vector to the interface. The interface temperature is obtained from an idealized phase diagram and is described by TL=TM

+ mC.

(226)

The conservation of mass which takes into account the thermotransport due to the Soret effect takes the form C^l

= -D[VC

+ SC(l - C ) V I i ] ■ n/|n|

(22c)

where |n| represents the magnitude of the vector n. The balance between the production of latent heat £ and the interfacial heat flux yields £jt

=

lKsVTs^KLVTL].m\m\

(22d)

95 The continuity of temperature at the interface implies TL = TS}

(22c)

and the assumption of a lower plate that is impermeable, rigid and isothermal leads to T, = T0,

v = 0,

and

— + 5(7(1 - C)-^-

= 0,

(22/)

At the upper plate, located at z = d + h) the temperature is fixed TS = TU

(22g)

where Xi is assumed to be lower than the freezing temperature of the mixture. The governing equations are then made non-dimensional by using h^ for length scale where hi is the height of the fluid layer bounded above by the planar solidliquid interface and below by the lower plate; time is scaled by the thermal dif­ fusion time. Several parameters appear in the nondimensionalization, namely, the Rayleigh number R, the separation ratio S, the Lewis number r, the dimensionless solid layer height A and the dimensionless liquidus slope 6. The results of the linear and nonlinear studies are discussed in the next sections. 3.2 Linear Stability Results Zimmermann et a/.16 carried a detailed linear stability analysis of the conduc­ tion state. They consider a non-linear relationship for the thermotransport term in the concentration equation which led to an expression for the concentration profile that is non linear in the vertical variable. As expected, their results show that the bifurcation to a state of convection depends on the separation ratio, namely that a critical value for the separation ratio 5* exists below which the bifurcation is to oscillatory convection and above which it is to steady convection. The initial concentration C 0 appears in the results for the effects of diffusion and conduction. It is found that the critical Rayleigh number for the onset of convection /? c , the critical wavenumber kc and the critical frequency UJC depend on Co- For the case of steady convection, S < 5*, both Rc and kc decrease with Co. However, for the case of oscillatory convection, RC) kc and u>c all increase with Co- The dependence of the critical parameters on the initial concentration can be explained by the fact that an increase in initial concentration leads to an increase in the effective Soret coefficient. The effect of the solidified layer is represented by the parameter A. For S > S«, both Rc and kc are found to decrease with A in agreement with the results of Hadji and Schell 15 . An explanation of this result is stated as follows: As the parameter A increases, the solid phase begins to act as the upper boundary

96 for the fluid layer. Since the thermal conductivity of the solidified layer is lower than that of the plate, a temperature fluctuation carried in the upward direction is less easily released through the effective upper boundary. Zimmermann et a/16 have also noticed that the critical Rayleigh number and associated wavenumber are only slightly affected by increases in latent heat. The impact of an increased latent heat is, however, noticeable on the value of the angular frequency at critical. Hadji17 considers the case of plates that are both nearly impermeable and poor conductors of heat compared to the fluid. The reformulated problem is characterized by two coefficients: a heat transfer Biot number 7 and a mass transfer /?. For S > 0, which corresponds to steady convection, it is found that the variation of the critical wavenumber kc with the separation ratio S depends on 7 and /9. Namely, it is observed that kc decreases with S for 7 > P and increases with S for 7 < /?. The magnitude of kc becomes vanishingly small as both 7 and /3 approach zero. This case corresponds to a system that is completely insulated and impermeable. The critical Rayleigh number is found to decrease monotonically with decreasing values of 7 and /3. 3.3 Nonlinear results Hadji and Schell15 have carried a weakly nonlinear investigation of the interac­ tion between steady Soret driven convection and solidification. In order to make their mathematical analysis tractable, they assume that thermal diffusion due to the Soret effect is independent of the concentration. The basic state is, in this case, linear in the vertical variable. Their linear stability analysis shows that, for a relatively large positive separation ratio, the conduction state is unstable to long wavelength disturbances and that the bifurcation to convection is stationary. In this limit, an expansion in the small wavenumber is possible. Hadji and Schell15 consider a double expansion in the small amplitude and the small wavenumber in order to analyze the stability of the interfacial patterns. The analysis leads to the prediction of a bifurcation diagram that is characterized by two bistable regions. First a stable structure consisting of down-hexagons appears through a subcritical bifurcation. This convective structure coexists with the static state in a very small range of Rayleigh numbers. Second, squares become stable at higher Rayleigh numbers and coexist with the down-hexagons in a small parameter range. It is also shown that these convectives structures, down-hexagons and squares, are imprinted as patterns on the solid-liquid interface through the melting action of the convective currents. The nonlinear stability results demonstrate that the appearance of the down-hexagons is due to the coupling between the convective flow in the melt with the deformations in the solid-liquid interface. Hadji17 relaxes the small amplitude assumption and derives the following evolu-

97 tion equation that is valid for larger supercritical region,

^

= -TX(S,

A, T) W

- (/ir/*o)V 2 „/ + VHr, • Vuf -0f + y r[VHf ■ VH\VH\2f

+V2„/|V„/|2] - ^0^\yHf

■ V„(VV) + (V2„/)2] -

15V2H(»?/),

(23a)

where Xlb>A>T>-lm+

2315

4 5 ( 1 + .1)

and

'-why***"-

J

'

(236)

<23c)

The symbols that appear in Eqs. (19) are the dimensionless solid layer thickness A, the separation ratio 5, the Lewis number r, the dimensionless liquidus slope £, the leading order interface deviation 77, the leading order perturbation to the concentra­ tion profile / and // is a parameter measuring the deviation of the Rayleigh number from its critical value. The derivation of the nonlinear evolution equation is based on a long wavelength expansion and is valid for parameters A, S and r for which \ is positive. The evolution equation depicts the coupled effects of solidification and thermosolutal convection due to the Soret effect. Hadji17 considers the numerical solution of the following scaled one dimensional version of the evolution equation, Eq. (23), using an explicit finite difference scheme, Q /

■QT = -fxxxx

-rp(fxfXx)x

- %fxx - Pf - Pfxfxxx + [ ( / I ) 3 ] I

+ 1 5 p ( / / „ ) „ + qUxf - 1 5 9 ( / 2 ) « ,

(24)

where the coefficients p and q represent some complicated functions of the param­ eters. Eq. (24) is solved numerically on the interval of length Sn (4 characteristic dimensionless length scales) with periodic boundary conditions. Their parametric study shows that increases in A lead to wider up-flow regions of warm fluid sepa­ rated by narrow down-flow regions of cold fluid. In turn, this flow pattern induces variations in curvature in the solid-liquid interface. An explanation of these results is stated as follows: The deformations in the interface introduce variations in the depth on the fluid whose stability must be characterized by an effective Rayleigh number which is dependent on the horizontal variable. In regions where the depth has increased, 7/ > 0, both the liquid depth and the temperature difference are larger than those required for convection onset. The opposite scenario holds in re­ gions where 77 < 0. Thus, a periodic array of unstable regions separated by stable regions forms. See Fig. 7. The convection currents that initiate in the unstable regions penetrate the neighboring stable regions resulting in the formation of wide

38

Figure 7: Plots of the cell pattern for parameter values S = 0.035, r = 0.001, A = 0.1 and 0 = 1.4 for ( = 0 (top) and ( = 500 (bottom), regions of cold fluid separated by narrow regions of warm luid. Jin and Hadji 18 re-examine the solution of Eq. (24) using a fully implicit finite difference scheme combined with Newton linearization with coupling. They have. obtained results which were unobtainable when a fully explicit scheme was used. Mainly, it is found that, for large values of the parameter (", the solid-liquid interface exhibits a cellular morphology consisting of thin fingers that extend deep into the liquid. See Fig. 8. This morphology is similar to the thin interfacial fingers which have been observed experimentally by Jackson 19 during the directional solidification of tetrabromomethane, CBr^, Jin and Hadji18 have shown that this morphology bifurcates subcritically from the planar state. 3.4 Experimental

validation

Zimmermann ti el.16 have undertaken an experimental verification of their analyt­ ical results. They have selected to use alcohol-water mixtures in their experiments for several reasons. Besides the fact that they exhibit strong Soret effects and are suitable for experimental visualizations due to their transparency, they also offer the added advantages that their physical properties remain nearly constant within the range of temperature used and that all alcohol is rejected upon freezing. The latter requirements are needed in order to eliminate any non-Boussinesq effects and for the experimental conditions to be in agreement with the assumptions made in the derivation of the analytical results. Their linear stability results agree with their experimental findings when the dependence of the separation ratio on concentration is taken into account. Indeed, Marcher and Miiller20 have shown that the variation

99

Figure 8: Plot of the steady state solution of Eq. (24) for 5 = 0.035, r = 0.001, A — 0.1, /? = 1.4 and £ = 500. Dotted line represents the initial condition 0.1 cos a;. of concentration plays a dominant role in the stability analysis and must be included in the calculations. Zimmermann et al.16 have also noticed that in the presence of solidification (A ^ 0) and for sufficiently low mean temperature in the mixture, the state of travelling waves that is observed when ice is absent, becomes unstable and gives way to a state of stationary convection. The authors attribute this result to the presence of the solid-liquid interface and other possible non-Boussinesq effects. The visualizations of the solid-liquid interface using interferometry and photogra­ phy reveal that the morphology consists of hexagonal-like patterns. For a thin solid layer, they report observing an interface that consists of ice-regions surrounded by ice free regions. This scenario is consistent with a convection flow in the form of hexagons with down flow at their centers. These experimental observations, even though valid for negative separation ratio, seem to corraborate the results of Hadji and Schell 15 , namely, that the coupling between Soret driven convection and the deformations of the solid-liquid interface yields a flow pattern consisting of downhexagons. 4.

Concluding R e m a r k s

In this paper, an attempt was made at summarizing existing results dealing with the role of thermotransport in solidification processes. The main focus was on the

100 influence of the Soret effect on the instabilities usually encountered in solidification systems, namely, interfacial instabilities during the unidirectional solidification of a binary alloy at constant speed and fluid instabilities induced by Soret driven con­ vection in the presence of freezing. The summary of results has shown that many aspects arising from the Soret effect are nt>t yet fully resolved. These results also show that in order to bring to light the significance of Soret diffusion in solidification processes, the nonlinear form of the thermotransport term must be considered. We are currently attempting the complete solution of the stability problem taking into account the nonlinear form of the thermotransport term and the coupled morpho­ logical and convective modes of instability. 5.

Acknowledgements

I am grateful to Dr R.F. Sekerka for pointing out to me references 12 and 13. I also thank Dr L. Nastac, Mr A. Catalina and Mr J. Leon for many stimulating discussions.

6.

References

1. 2.

M.C. Soret, Arch. Sci. Phys. Nat. Geneve 2 (1879) 48. S.R. DeGroot and P. Mazur, Non-equilibrium Thermodynamics (Dover, New York, 1984), Chap. XI, Sec. 7. D. R. Calwell and S. A. Eide, Deep Sea Res. 32 (1985) 965. D. Gutkowicz-Kruskin and J. Ross, J. Chem. Phys 72 (1980) 3577. A. Rice, Geophysical Surveys 7 (1985) 303. J. F. Smalley, R. A. Mac Farquhar and S. W. Feldberg, J. Eleciroanal. Chem. 256 (1988) 21. S. A. Akbar, M. Kaburagi and H. Sato, J. Phys. Chem. Solids 4 8 (1987) 579. C. R. Carrigan and R. T. Cygan, J. Geophys. Res. 91 (1986) 1451. G. R. Longhurst, J. Nuclear Materials 131 (1985) 61. D. T. J. Hurle and E. Jakeman, J. Fluid Mech. 4 7 (1971) 667; D. T. J. Hurle and E. Jakeman, Phys. Fluids 12 (1969) 2704. P. Bigazzi, S. Ciliberto and V. Croquette, J. Phys. (Paris) 51 (1990) 611; E. Knobloch, Phys. Rev. A 40 (1989) 1549. D.T.J. Hurle, J. Crystal Growth 61 (1983) 463. S. Van Vaerenbergh, S.R. Coriell, G.B. McFadden, B.T. Murray and J.C.Legros, J. Crystal Growth 147 (1995) 207. R.F. Sekerka, in Crystal growth: an introduction, ed. P. Hartman (NorthHolland Publ. Co., 1973) p. 403. L. Hadji and M. Schell, Phys. Fluids A 2 (1990) 1597.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

101 16. 17. 18. 19.

G. Zimmermann, U. Miiller and S.H. Davis, J. Fluid Mech. 238 (1992) 657. L. Hadji, Phys. Rev. E 4 7 (1993) 1078. X. Jin and L. Hadji, Phys. Rev. E 50 (1994) 2361. K.A. Jackson, in Solidification, eds. T.J. Hughel and G.F. Boiling (American Society of Metals, Metals Park, OH, 1971), p. 133. 20. C. Karcher and U. Miiller, Int. J. Heat and Mass Transfer 3 7 (1994) 2517.

103

THE INSTABILITY OF FINITE AMPLITUDE WAVES IN STRONG VISCID AND INVISCID SHEAR

W. R. C. PHILLIPS Department of Mechanical and Aeronautical Engineering Clarkson University, Potsdam NY 13699-5725, USA

ABSTRACT Finite-amplitude initially spanwise-independent two-dimensional rotational waves and their nonlinear interaction with unidirectional 0(1) shear flows are reviewed. Of particular in­ terest is the instability of the flow to longitudinal vortex form. The wave-mean interaction is described by a generalized Lagrangian-mean formulation in combination with a separate theory to account for the back effect of the developing mean flow on the wave field. Both the inviscid and viscid eigenvalue problems relevant to the instability are outlined, along with detailed results for uniform shear in the case of the former and periodic Poiseuille flow in the case of the latter.

1. Introduction Shear flows that comprise both mean and fluctuating parts are commonplace in Na­ ture and engineering, and often interact to form longitudinal vortices. Examples include Langmuir circulations12, which occur beneath wind driven surface waves, stream wise rolls in the atmosphere over surface waves 23 or mountainous terrain13,21, and quasi-streamwise vortices in laboratory boundary layers (both laminar10 and turbulent3), free shear layers2 and Stokes layers8. Important in each case is an understanding of the nonlinear processes that couple the mean and fluctuating motions, and moreover the secondary and possible tertiary phenomena attributable to the nonlinear rectification of those oscillatory motions: e.g. modifications to the mean flow as a result of the waves; and the back effect, if any, of those mean flow modifications on the wave field. Quantities that follow individual fluid particles are crucial to such studies and such quantities are poorly described by the Eulerian equations of mean motion. Indeed, Eulerianmean vorticity as defined by Reynolds averaging has no simple conservative properties, even when viscosity is ignored and thus acts to conceal the role played by nonlinear recti­ fication (in its guise as Stokes drift) in vortex line deformation. A more rational way to separate 'wave' from 'mean flow' and to define wavemean interactions was developed by Andrews & Mclntyre 1 in the form of the generalized Lagrangian-mean (GLM) equations. These equations describe the back effect of oscillatory

104

disturbances upon the mean state and are exact provided the mapping between the true Lagrangian and the reference GLM remains invertible. Of course GLM still describes mean motions and is therefore conceptually equivalent to Reynolds averaging, but it describes Lagrangian aspects of the motion from an Eulerian framework and is consequently able to capture structural aspects of the flow. The object of the present work is to review the application of GLM to describe mean structures in unidirectional 0(1) shear layers (be they viscid or inviscid) owing to the pres­ ence of neutral finite-amplitude rotational waves that are initially spanwise independent. Wave-mean interactions of this ilk were first investigated by Craik4 and Leibovich 11 , who each sought to model Langmuir circulations. They considered 0(e) neutral irrotational waves interacting with an 0(e 2 ) unidirectional Eulerian mean shear flow and found the interaction unstable to longitudinal vortex form via an instability now known as CL2, or Craik-Leibovich type-2. CL2 continues to operate in 0(e) mean shear flows with only minor modifications to the theory, but that is not the case for strong 0(1) shear flows, where the back effect of the mean flow modification upon the wave field must be explicitly calculated (Craik6). In essence, waves do not drive CL2 (Mclntyre & Norton14) but act through the pseudomomentum as a catalyst. This means that the magnitude of the mean flow modification is bound not by the strength of the waves but by the magnitude of the pre-existing vorticity in the initial state. With sufficiently strong pre-existing vorticity, therefore, the mean flow modification acts to distort the waves. Of course the detailed kinematics of the instability mechanism are less clear with 0(1) shear than with 0(e 2 ), though the seminal idea of the CL2 instability remains within the theory. In order to construct an inviscid theory for 0(1) shear flows in the presence of 0(e) rotational neutral waves, Craik6 employed the GLM equations and found that the resulting eigenvalue problem for longitudinal vortices is far more complex than its counterpart for weaker shear; requiring inter alia, a further differential equation to account for wave distor­ tion. That notwithstanding, he was able to obtain definite results analytically to demonstrate the existence of longitudinal vortex instability when the spanwise spacing of the vortices is small, and this same technique was extended to a different, wider class of flows, by Phillips & Shen18, who found that CL2 continues to operate. Numerical results by Phillips & Wu 20 and Phillips et aP1 concur, and further indicate that wave distortion acts (i), to diminish catalytic action for all but the shortest waves; and (ii), to suppress the instability markedly if the waves are sufficiently long. We begin with a brief review of GLM (§2) and then specialize the GLM-equations to the problem of 0(e) neutral waves interacting with an 0(1) unidirectional shear flow (§3). The waves are initially two-dimensional. An appropriate numerical scheme is sketched in §4 and detailed results for the case of uniform shear in §5. The relevant viscous eigenvalue problem is outlined in §6. 2. The generalized Lagrangian-mean equations Andrews & Mclntyre's 1 generalized Lagrangian-mean equations are an exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances

105 upon the mean state. The formulation is based upon an exact Lagrangian-mean operator ( ) , corresponding to any given Eulerian-mean operator ( ), through an exact disturbanceassociated particle displacement field £(x, t) and is valid provided the mapping X H X + { is invertible. In consequence dependent variables are given an Eulerian description with position x and time t as independent variables. The Lagrangian-mean velocity u L is then the velocity field describing trajectories about which the fluctuating particle motions have zero mean, when any averaging process is applied. For homentropic flows of constant density p in a non-rotating reference frame the GLM momentum equation is DL(uf - p{) + u^(uLk - pk) + 7T, = -Xu

(1)

where repeated indices imply summation, commas denote partial differentiation and (xi, x2, x3) = (x,y,z). The operator DL is defined as DL = d/dt + vJ-d/dxj, and the vector wave property pi, the pseudomomentum per unit mass, is Pi = -tj,iu£3,

(2)

with DL(j = uly Finally Xi are dissipative terms while

p

z

where u^ is the actual fluid velocity and $ L is the force potential per unit mass. We shall restrict attention to fluids in which all mean quantities except possibly the mean pressure V are independent of the stream wise direction; then, with $ L = 0, IT reduces tCPL. 3. In viscid 0(1) shear and 0(e) waves Consider the interaction between an inviscid 0(1) primary shear flow and two di­ mensional straightcrested periodic waves that propagate in (or opposite to) the direction of the flow. Then in a reference frame that moves in the x-direction with the phase speed of the waves and with space coordinates (x, y, z), the primary shear flow is [u(z), 0,0], where u is the Eulerian-mean velocity profile in [z\ ,z2]. The waves are initially independent of the span wise coordinate and are of constant amplitude with slope characterized by the small pa­ rameter e; moreover they induce, in most circumstances, an 0(e 2 ) pseudomomentum field [pi,0,0]. Envisage now small spanwise-periodic perturbations with streamwise averaged Eu­ lerian velocity components of the form (u, v, w) = £Re{e^eWl/[u(;z), -eiv(z),

ew(z)]}

(3)

which, provided the amplitude field of the waves is steady, satisfy continuity correct to 0(e 2 ) as Iv + u>,3 = 0. Here a is the growth rate of the spanwise perturbation and 6 is a

106 second small parameter that measures the strength of this motion relative to the primary shear flow; 6 is assumed sufficiently small that linearization with respect to it yields a good approximation to the equations governing the spanwise periodic disturbance. Observe that velocity perturbations in the y and z directions are weaker, by a factor of e, than the x velocity perturbation; this is necessary in order to ensure the GLM equations yield nontrivial solutions for boundary conditions of the form6 w = 0 at z — z\, z2. Accordingly a = eov In such cases the GLM equations reduce to cj\U — —wu, and

10,33+* [—i

l\w =

(4) pj.

(5)

Here prime denotes d/dz and pi has been expanded as Pl

= e2P? + c 2 «te{e' t e i , y p 1 (^)} + 0(e\ e36, e2S2).

(6)

So P® is the 0(e2) component of pseudomomentum5

A° = 4 { I A T + " 2 6 2 } ,

(7)

l u u in which <j>(z) and a denote the eigenfunction and wavenumber of the primary wave field which together satisfy the Rayleigh equation, u(" - a2) - u" = 0.

(8)

Note that P® is unbounded in the vicinity of critical layers (u = 0) unless 4> oc um (m > 1). Such behavior is an indication that the mapping upon which GLM is based has broken down and thus that, except perhaps at a rigid boundary, critical layers should be avoided. The 0(e2S) spanwise-periodic perturbation of pseudomomentum, R e { e 1 ^ } , arises because the emerging secondary Eulerian velocity field distorts the primary wave field. But because the GLM formulation provides no direct means of evaluating p1, a separate examination of the wave field is necessary. In carrying out that examination, Craik6 notes that the 0(6) spanwise periodic z-velocity, u, causes the significant part of distortion of the wave field. Then n = A(z)u(z)

+ B(z)u'(z)

+ Rt{C(z)j>(z) + V(z)4'(z)},

(9)

where A, By C and V are functions which are independent of a while (z) relates to the 0(e6) spanwise periodic wave field modification and satisfies the Rayleigh-Craik equation, u[^

- (a2 + l2)]$ - u"$ = - u [ ^ - (a2 + l2)] + u»*.

(10)

Thus given the primary Eulerian-mean shear flow u(z), the primary wave-field eigenfunction <j>(z) and appropriate boundary conditions, the eigenvalue problem for <jx is completely specified by the coupled system (4), (5) and (10) together with (7) and (9).

107

Craik6 has further shown that when P > a2 and a = 0(1) non-trivial solutions to (4), (5) and (10) with homogeneous (i.e. = w = 0 on each wall) boundary conditions exist provided Im{/ (1 + \k)1/2dz} = —

(fe=lor2),

(11)

for some non-negative integer N. Here + H) ± [<7-4(G + Hf -

Ai,2 = \{-a-\G

H(,)S!*g#

and

to?H)i},

G W s -„»*'(!£)'.

(12)

(13)

4. Numerical Solution to the eigenvalue problem Galerkin techniques may be used to solve the eigenvalue problem (4), (5) and (10) for the eigenvalue ai which, because (4) and (5) are real, may be real, imaginary or a com­ plex conjugate pair. Complete details are given in Phillips & Wu20. Briefly, we suppose u(z) and <j>(z) can be expanded in linearly independent complete sets of basis functions U{(z) and <j>i(z) as M

A

Hz) = J2 biui(z)>

M

kz) = Y, bM+ii(z),

( 14 )

i'=l

i=\

where 6, and &M+* are the expansion coefficients of u(z) and <j>(z) respectively, and M is a positive integer. The basis functions were chosen to satisfy the boundary conditions of u and , viz u(^i), u(z2) = 0; <j>(zi), ^>(z2) = 0. Substitution of (14) into (4), (5) and (10) then leads to the residual functions Ru(z) and R<(>(z) which must satisfy the inner products (Ru(z),uj(z))=0]{R^(z)^j(z))=0. A 2xM order linear eigenvalue problem for A2, where A = l/a\, results as C = \2M,

(15)

where C and M, are 2Mx2M order matrices with components that are MxM order submatrices; specifically, the components are functions of a and /, but not a\. Of interest is the largest real value of cr\ for each pair (a, /) and the eigenfunctions u and <£. Chebyshev polynomials were used as basis functions and the accuracy provided by the M = 20 expansion was considered adequate for our purposes. All computations were performed on a DECstation 5000/200 using double precision arithmetic with IMSL routines to solve the eigenvalue problem (15).

108

Figure 1: Instability map of a.: against ~/."? for uniform shear between rigid wavy walls: shaded, region denotes instability ( Phillips & ShenlN). 5. Uniform shear between rigid wavy walls Much can be learned from the simple case of uniform shear interacting with two dimensional wavelike disturbances uo (which satisfy (8)) of the form oo -- v ~ '' -\- ■">' '"' (~. .i constant). Note that two very different classes of Rayieigh waves are admissible: those for which ao diverges and those for which ao converges with jo.:| as :n: ! — ?-,:. Phillips & Shen18 denote the former 'type-1' and the later, 'type-2* waves. SJ The limit I'2 > n'2, n = (){ 1). In order to proceed analytically. Craik1' employed (12) and (13) to consider the case ■i = 0, and showed that type-1 waves destabilize the flow for n.;- t ( — 1.0) and accordingly type-2 waves for ail n;; > 0. Phillips & Shen have applied the same technique to oilier combinations of -. and .1 There findings are plotted in figure 1. in which the shaded portion denotes instability. Note that points of symmetry at - / J = n 1 are clearly evident, as is the obverse {i.e. ^ / .i = 0) of Craik's result. Note also that ~ / .j = —1 is stable for all ^ : | greater than zero. Interestingly. - j3 — — I is the only situation that admits, at least in the limit an —^ 0, the amplitude ao —r 0. giving rise to what is tantamount to plane Couette flow subject to infinitesimal waves, an inviscid configuration long known to be stable (see Drazln & Reid'). in short, instability to longitudinal vortex form occurs whenever the local wave amplitude \ao\ everywhere (Ucn-ast s in the direction of increasing \u\. but occurs only for

109

Figure 2: Curves of G\ against a in the limit I2 —► oo, both with and without wave distortion (Phillips & Wu20). sufficiently long waves when \a<j>\ everywhere increases in the direction of increasing \u\. Indeed, waves are stable to longitudinal vortex form wherever in the direction of increasing mean flow, the relative increase in wave amplitude exceeds the relative increase in mean flow18 and this is so for all wavelengths when 7//? = — 1. But although bounds deduced from (12) and (13) are useful to determine whether the flow is stable or unstable to longitudinal vortex form, these equations alone do not yield the growth rate or resolve whether wave distortion acts to enhance or inhibit the instability. Phillips & Wu 20 determined to answer both questions and, because (3 — 0 depicts instability over the widest range of az, chose it as the test case. They then considered the problem both with and without wave distortion. When wave distortion is ignored, only (4) and (5) with A = 0 need be solved; moreover from (11) the largest upperbound for G\ is then seen to occur when 01

A2 e - c *(l + 2a + 2a 2 + -c*3)2 o

(a ^ 0,

I2 -> 00).

(16)

Unfortunately (11) proved quite a challenge to solve for the wave-distortion case and Phillips & Wu chose instead to solve (4), (5) and (10) in the appropriate limit. It transpires that the distortion and non-distortion results are markedly different, as we see in figure 2. Indeed the two cases are asymptotically equivalent only as az —> +00 and it is evident that wave distortion plays an increasingly greater role as az decreases from +00.

110

F i g u r e 3: Contours of constant growth rate cr\ in a-l space with stability boundaries along wnicn G\ is not constant. The boundary connecting lR to aR separates instability to longitudinal vortex form (I) with same of alternating sign (II); the dashed curve separates II from a region which is stable to longitudinal vortex form (III) (Phillips &Wu 2 0 ).

5.2 The long-wave long-wave limit Phillips & Wu further show that a\ increases monotonically to an upperbound with increasing /, except in the long-wave long-wave limit, figure 3. Here they find that of the 2N roots for cri, only 23 are real and 2K are imaginary (J, K integer); the remainder are complex conjugate pairs. Specifically, J = 0 for a < aR and K = 0 for a > a/, where aR « 0.76 and cti « —0.61. Moreover 3 = 1 (K = 1) at a = aR-\- (a = a/—) and increases with increasing (decreasing) a until eventually J — N (K = N), at which point all the eigenvalues are real (imaginary). Finally, while the real parts of the complex conjugate pairs are positive when N > J > 0, they change from positive to negative in «/ < oc < aR and remain negative for N > K > 0. So on writing <jx — OLCV{ q= iacj!, the right-hand-side of (3) should be interpreted as 6Re{eac> V ^ a c ^ [ u ( ; z ) , -eiv(s), ew{z)]}. The curve connecting lR to aR is shown, with impinging contours of constant (T\, in I-a space in figure 3. Note that the lR-aR-cxxrvG is not a line of constant cra but rather a demarcation between eigenvalues for GX for which J = 0 and J = 1. The /^-a^-curve also depicts marked suppression in catalytic action beyond that for J > 1; so although the flow remains unstable to longitudinal vortex form on the long wave side of the curve, the vortices are relatively weak rolls. On the J > 1 side of the curve, the instability is dominated by eigenmodes which are real; the real parts of the complex conjugate pairs, on

111 the other hand, are tertiary at best. But only complex conjugate pairs occur on the J = 0 side of the curve where the instability not only grows, so long as OLCV{ > 0, but is subject to a standing oscillation owing to the ±acvT contribution: that is, the longitudinal vortices stand in space and alternate in sign; there is also the possibility that a +acvr or —acvr alone each give rise to spanwise propagating rolls with equal and opposite phase speed. Of course on fixing / on the lR-aR-curve and further decreasing a, we ultimately reach a point at which OLCV{ < 0 for all eigenmodes; here the flow is neutrally stable to longitudinal vortex form. The neutral curve is depicted by dashes and indicates that az ~ —0.5 as P —> oo and that az = a\z when / = 0. 5.3 Non-uniform shear Phillips & Shen18 extend the analytical techniques introduced by Craik6 in the limit I > a, a = 0(1) to deduce that type-1 and/or type-2 waves are unstable to longitudinal vortex form in the presence of a wide variety of mean velocity profiles, from boundary layers to mixing layers. Indeed the implication is that CL2 is ubiquitous to all wave-mean interactions that satisfy the criteria stated in §5.1. Further, in order to deduce more detail, several cases of particular physical interest were treated numerically: to wit, an exponentially decaying velocity profile beneath surface gravity waves 20 and power-law and logarithmic-law velocity profiles as would occur in atmospheric boundary layers over random terrain21. The gross features in each case are much as those in the case of uniform shear, although of course details vary. One point of particular interest is that the fastest growth rate occurs (in both cases) when a = 0(1) for all/. 2

6. The viscous eigenvalue problem Although the CL2 instability is inviscid, it can at times be modified by viscosity and the appropriate eigenvalue problem that accounts for viscosity is given by Phillips 17 . Here it is best to work in terms of the associated velocity field Qi = u + Ji —pi, which must be determined as part of the problem; here d is the generalized Stokes drift. Indeed the problem comprises two parts: (i), a two-dimensional primary instability composed of equilibrated waves interacting with a mean flow often vastly different from its unperturbed laminar forbear. And (ii), the three dimensional secondary instability which manifest as longitudinal vortices. Here we shall outline only the secondary linear instability. 6.1 Secondary flow Here the perturbation equations take the form [D2-l2-a?]u

=

faw,

[D2 _ /2 _ < ] [ D 2 _ /2]t a = - ^ [ / f u - Qifc]

(17)

(18)

112 where av = <XJ\, D = d/dz and to avoid annihilation of the 0(e) growth rate predicted by inviscid theory, i/a/C = 0(e), where C is a typical phase speed; appropriate boundary conditions at rigid boundaries are: u = w = Dw = 0

on

(z =

zi,z2).

Accordingly px is described through the 0(e6) spanwise periodic wave field modi­ fication, <£ as 17 (Qr - i c ^ ) M ^ - Q»4 + uM - u"<j> = T ^ M 2 fa (19) the operators are L = ~ M - (Qi - icj")

and M = D 2 - (a2 + I2).

Finally, the 0(e2S) correction to pseudomomentum is p, = A1(z)u(z) + A2{z)u'{z)+Rt{M*)k*) + M*)Uih

( 2 °)

where Ai, A2, A3 and A4 are functions which are independent of a\. Also required in (20) \&U\y viz17 2

LWi = -

3

/2

^jL^ +-T—JJ(U^-UV-Q^).

(21)

Note that in contrast to its inviscid counterpart which is algebraic, (21) is an ordinary dif­ ferential equation. For that reason the Ai's (i = 1,4) in (20) should not be confused with Craik's A, B, C and V, although of course (20) recovers Craik's 6 form in the appropriate limit. Thus given the eigenfields Qi and and appropriate boundary conditions, the eigen­ value problem for G\ is completely specified by the coupled system (17), (18), (19) and (21). Finally, although the viscous eigenvalue problem is significantly more complex than its in­ viscid counterpart, it is nevertheless suitable for numerical treatment by methods similar to those outlined in §4. 6.2 Periodic Poiseuille flow Phillips 15 ' 16 and Phillips & Tu19 consider the development of secondary instabilities via the CL2 mechanism on a nonlinear equilibrium solution in periodic Poiseuille flow. They choose a nonlinear equilibria whose streamwise and spanwise average (of velocity and kinetic energy) mimics that of fully developed turbulent channel flow. To proceed, however, they exclude wave distortion, arguing that it should not greatly affect the secondary neutral linear stability curve. The secondary instability, which of course manifests as longitudinal vortices, yields the neutral curve of figure 4, which depicts the spanwise periodicity of the vortices (£+, in wall units) as a function of Reynolds number (ReT, based upon channel half width and friction velocity). Further, to compare with Direct numerical simulations, they include the k = (0,1,1), (0,2,1) and (0,3,1) modes (implying streamwise vortices) calculated by Sirovich et aP2, along with Jiminez & Moin's 9 minimum width; that is the

113

Figure 4: Neutral secondary curves for periodic Poiseuilleflow:o, Sirovich et aP1; +, Jiminez &Moin9. (Phillips &Tu19, Phillips16) minimum periodic channel width necessary for turbulence to sustain itself at ReT — 180. All lie close to the first neutral mode whose wavelength is ci that of streaks (viz 50-200 wall units). Moreover, in addition to the first mode, a higher-order mode is evident that is linearly most unstable at a Reynolds number higher than critical but with a much lower spanwise wavelength, about 40 wall units, a tad more than twice the empirical optimal spacing of riblets, a drag reducing device. Phillips & Tu19 conjecture that riblets excite this mode of instability. The vortices so formed are smaller in diameter than their flat plate counterparts and do not extend much beyond the buffer region. Detailed nonlinear calculations following their lifecycle suggest that their growth rate, while still exponential, is significantly less than their larger counterparts, possibly because of viscous damping. The net result is a reduced production of turbulent kinetic energy. 7. Summary The generalized Lagrangian-mean equations are a powerful and useful tool for de­ scribing the interaction of rotational finite amplitude waves with shear flows of all orders. Here we restricted our review to 0(1) shear flows, as these are common in a variety of geophysical and engineering situations. In such situations two-dimensional waves interact with the mean flow to catalyze an instability - known as Craik-Leibovich type-2 - to longitu­ dinal vortex form. The resulting spanwise periodic secondary flow, however, is sufficiently strong to distort the original wave field and this in turn affects the growth rate of the insta-

114 bility. Results to date indicate that wave distortion acts to inhibit the instability for all but the shortest waves. Results for both inviscid and viscid interactions are presented, although more em­ phasis is given to the former case which has been more thoroughly explored. 8. Acknowledgement This work was supported by National Science Foundation grant CTS 9008477.

9. References 1. Andrews, D. G. & Mclntyre, M. E., An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech., 89 (1978) 609-646. 2. Bernal, L. P. & Roshko, A., Stream wise vortex structure in plane mixing layers. J. Fluid Mech., 170 (1986) 499-526 3. Corrsin, S., Some current problems in turbulent shear flows, In Symposium on Naval hydrodynamics (Publn. 515, NAS-NRC, Washington, DC, 1957), pp373. 4. Craik, A. D. D., The generation of Langmuir circulations by an instability mechanism, J. Fluid Mech., 81 (1977) 209-223. 5. Craik, A. D. D., The generalized Lagrangian-mean equations and hydrodynamic stability. J. Fluid Mech., 125 (1982) 27-35. 6. Craik, A. D. D., Wave-induced longitudinal-vortex instability in shear layers. J. Fluid Mech., 125 (1982) 37-52. 7. Drazin, P. G. & Reid, W.H., Hydrodynamic (Cambridge University Press, 1981).

Stability,

8. Hino, M. Sawamoto, M. & Taksu, S., Experiments on transition in an oscillatory pipe flow. J. Fluid Mech., 75 (1976) 193-207. 9. Jiminez, J & Moin, P. , The minimal flow unit in near wall turbulence. J. Fluid Mech., 225 (1991) 213-240. 10. Klebanoff, P. S. Tidstrom, K. D. & Sargent, L. M., The three dimensional nature of boundary layer instability. J. Fluid Mech., 12 (1962) 1-34. 11. Leibovich, S, Convective instability of stably stratified water in the ocean. J. Fluid Mech. 82 (1977)561-585. 12. Leibovich, S., The form and dynamics of Langmuir circulations. Ann. Rev. Fluid Mech., 15 (1983) 391-427. 13. LeMone, M., The structure and dynamics of horizontal roll vortices in the planetary boundary layer. J. Atmos. Sci. 30 (1973) 1077-1091.

115 14. Mclntyre, M. E. & Norton, W. A., Dissipative wave-mean interactions and the transport of vorticity or potential vorticity. J. Fluid Mech., 212 (1990) 4 0 3 ^ 3 5 . 15. Phillips, W.R.C., On the etiology of shear layer vortices Theoret. Comput. Fluid Dynamics, 2 (1991) 329-338. 16. Phillips, W. R. C., The genesis of longitudinal vortices in free and bounded shear layers. In Eddy structure identification in free turbulent shear flows ed J. Bonnet & M. Glauser, (Kluwer, 1993), pp35-41. 17. Phillips, W. R. C , On finite amplitude rotational waves in viscous shear flows. J. Fluid Mech., (1995) (submitted). 18. Phillips, W.R.C. & Shen, Q., On a family of wave-mean shear interactions and their instability to longitudinal vortex form. Stud. Appl. Math, (1995) (to appear) 19. Phillips, W.R.C. & Tu, H.Y., Why Riblets work. Bull. Am. Phys. Soc, 37 (1992) 1794. 20. Phillips, W.R.C. & Wu. Z., On the instability of wave-catalysed longitudinal vortices in strong shear J. Fluid Mech. 272 (1994) 235-254. 21. Phillips, W.R.C. Wu. Z. & Lumley, J. L., On the formation of longitudinal vortices in atmospheric boundary layers over random terrain. J. Atmos. Sci. (1995) (submitted). 22. Sirovich, L, Ball, K.S., & Keefe, L.R., Plane waves and structures in turbulent channel flow. Phys. Fluids A, 2, (1990) 2217-2226. 23. Thompson, T. W, Liu, W T. & Weissman, D. E., Synthetic aperture radar observation of ocean roughness from rolls in an unstable marine boundary layer. Geophys. Res. Lett., 10 (1983) 1172-1175.

11.7

NONLINEAR STABILITY ANALYSIS AND MODELING FOR CONVECTIVE FLOWS

D.N. RIAHI Department of Theoretical and Applied Mechanics University of Illinois 216 Talbot Laboratory 104 S. Wright Street Urbana, Illinois 61801 USA ABSTRACT This chapter reviews nonlinear hydrodynamic stability analysis and modeling for several convective flow problems which deal with the preferred patterns in horizontal layers and in spherical shells, stability regions of realizable solutions, steady and oscillatory solutions, simple and mixed type solutions, effects of external constraints of rotation and magnetohydrohynamics and directional solidification, and for cases described by discrete modal and wave packet representation of the flow solutions. Modeling is accomplished by making use of certain mathematical methods such as perturbation technique, multiple scales and scaling analysis. In addition, the latest research results for each particular convective flow problem, accomplished by the author, are also presented.

1. Introduction Nonlinear stability analysis is an important and essential part of nonlinear hydrodynamic stability theory1 and is relatively well established for weakly nonlinear convective flows2-4. The problems of preferred solutions of nonlinear convective flows are fundamental problems of importance in many hydrodynamical research problems with useful applications in engineering, geophysics, astrophysics and environmental sciences. Such problems are heavily relied on nonlinear stability analysis. Malkus and Veronis 2 developed a method of analysis to determine the steady solutions of weakly nonlinear stability system due to three-dimensional disturbances superimposed on motionless steady basic state solution of convective flows in horizontal layers with isothermal boundaries. Schluter, Lortz and Busse3 developed a method which determines such general solutions and then investigates the stability of these solutions with respect to arbitrary threedimensional infinitesimal disturbances. Busse and Riahi4 extended such method to cases with nearly insulating boundaries using double series expansions in powers of amplitude e of convection and Biot number Bi of the boundaries. Until recently the stable solutions of weakly nonlinear stability system for convective flow disturbances in horizontal layers were known to be due to two-dimensional rolls, squares and hexagonal cells. Some examples are given below. For convection in a symmetric layer with high conducting boundaries, solutions due to two-dimensional rolls are preferred3. For the case of asymmetric layer with high conducting boundaries, solutions due to hexagonal cells are preferred for sufficiently small e, solutions due to rolls

118

are preferred for larger values of e and there is a hysteresis region where both solutions due to these two types of cells can be stable5. For convection in a symmetric layer with low conducting boundaries, nonlinear stability analysis leads to the results4*6 that square pattern convection is preferred, provided Prandtl number Pr is not too small. Otherwise, rolls are preferred7 ♦For the symmetric layer, for either high or low conducting boundaries, the results of the nonlinear stability analysis for the stability regions of the preferred convection modes in the Rayleigh number R, horizontal wave number a -plane were determined8"10. For the case of asymmetric layers with low Z?,11"14, hexagons are preferred for sufficiently small £, square are preferred for larger values of e and there is a hysteresis region where both squares and hexagons can be stable. For the case where asymmetric layer is due to internal heating15*16, the preferred hexagons have upward motion at the cells' centers and downward motion along the cells' boundaries (referred to as up-hexagons structure) for large Pr, while the preferred hexagons have downward motion at the cells' centers and up-ward motion along the cells' boundaries (referred to as down-hexagons structure) for small Pr. For internal cooling, the opposite is true. For convection in a layer with nearly insulating boundaries and with general variable fluid properties and internal heating, the nonlinear stability analysis predicts that the layer can be composed of multiple hexagonal cells in the vertical direction, where the sign of vertical motion may be opposite at the centers of any two adjacent cells in the vertical direction 17. For dominant thermocapillay effects up-hexagons are preferred for high Pr and small e, while down-hexagons are preferred for low Pr and small £18«19- For Maragoni convection in the presence of gravity20-21 stable patterns can be either only hexagons or both hexagon and squares (for low B-). When both hexagons and squares are stable, there is a region of overlap where hysteresis effects exist. In all cases where hexagonal structure can be preferred, finite amplitude disturbances dominated over the infinitesimal disturbances in the nonlinear stability analysis. The flow patterns for the horizontal layer cases discussed above are all called regular patterns. Here by a regular pattern we mean pattern of the horizontal structure of the flow solution whose wave number vectors Kn (n = -N,...,-1,1,...,N, JV is a positive integer) have all the same magnitude and where the angles co between any two consecutive wave vectors have all the same value5. Rolls, squares and hexagons all fall in such category and correspond, respectively, to N = 1, 2 and 3 of regular patterns. However, recent nonlinear stability analysis of convective flows in horizontal layers with boundary corrugations22"25 led to interesting results for the first time that non-regular flow patterns can be preferred in some ranges of surface corrugated amplitude even if such roughness elements represent a regular pattern, and that subcritical instability of the governing nonlinear stability system is possible for such non-regular solutions. Examples of nonregular patterns are those due to rectangular (non-square) cells (N = 2, co = co* and 180° -fi>J\where 0(G)? (l80° and co°Y * 90°) and six-sided polygonal (non-hexagon) cells (N = 3, co = col co°
119

instability of the nonlinear stability system is possible26. Nonlinear stability analysis for convective flows in spherical shells can lead to important solutions with useful applications, as well. Such a convective flow is one of the fundamental problems of geophysical fluid dynamics and has attracted extensive attention among geophysicists interested in convective processes and their associated flow structures. Chandrasekhar in his well known book27 discussed the main geophysical motivation for the interest in such problem, namely the hypothesis of convection in the earth's mantle. When a homogeneous fluid contained in a spherical shell is heated from below for spherically symmetric distribution of gravity and heat sources, a basic static state exists. When the basic temperature gradient exceeds a critical value, an infinite number of convection patterns becomes possible, most of which are physically identical and representing the same solution because they can be transformed into one another by a rotation of sphere. This orientational degeneracy of the solutions reflects the isotropy and homogeneity of the problem and will not be removed by the consideration of nonlinear stability system of the problem. After all the solutions that represent transformations of other solutions have been eliminated, there remains a finite number of solutions corresponding to physically different patterns. Since these solutions differ in their physical properties, the solution degeneracy can be removed by the consideration and analysis of the nonlinear stability system of the problem. In general, one of the solution is preferred and describes the realized pattern of convection. The problems of nonlinear stability of convection in spherical shells for cases where a single degree / of spherical harmonics yields the lowest Re of the Rayleigh number R were investigated by Busse28, Busse and Riahi29'30, Riahi31 and Riahi et al.32. The problems of nonlinear stability of convection in spherical shells for cases where two degrees / and /* = / +1 yield the lowest Re of R were investigated by Busse and Riahi33, Riahi and Busse34 and Riahi35. This subject is far from being well understood. In the weakly nonlinear regime, however, the beginning of a general theory based on nonlinear stability analysis of the convective flow systems in spherical shells appears to be emerged28'35. Another interesting area where nonlinear stability analysis has been quite successful is nonlinear convective flows in the melt during the directional solidification of a binary alloy36. The nonlinear stability analysis here is led to some nonlinear evolution equations which are based on long wavelength theory4'36-38. Hurle et al.39 investigated the linear stability of the melt during solidification of a binary alloy. They calculated the critical conditions for the onset of stationary solutal convection and found that long wavelength modes are preferred when the segregation coefficient K is small. This result motivated Riahi40 to investigate small amplitude nonlinear stability of the problem using the theory developed by Busse and Riahi4. He found, in particular, that nonlinear stability analysis predicted the preference of solutal convection solution in the melt in the form of hexagons for sufficiently small e and that the effective depth of the melt (over which the basic solute gradient is significant) and K can affect the flow features rather strongly. The shape of the solid-melt interface can closely be related to the realizable flow pattern caused by the nonlinear convective flow instability of the governing system41. Systems with small K(K < 0.01) have great practical interest since they become nearly pure substances upon solidification. Small-amplitude theory40 was based on a single horizontal wave number a which was found to be of order K^ for small K. Riahi42 extended the small-amplitude

120

theory to the moderate-(or even large) amplitude regime, where the horizontal wave numbers ccn(n = 1,2,...) that contributed to the horizontal planform functions satisfied the relationship an = riHK^, where the parameters T]n were assumed to be of the order unity and independent of K. This moderate-amplitude theory for the nonlinear stability system of the convective flows was developed first by Riahi43 in the context of a Marangoni convective flow problem. In contrast to the small-amplitude case, it was found that there was no subcritical instability and that nonlinear stability analysis predicted that squares could be the only stable flow pattern for a significant range of amplitudes that widened as K decreased or as peclet number increased. Nonlinear evolution equations development based on the proper scaling and the nonlinear stability analysis of the governing solidification system was carried out by several authors recently4450. Riley and Davis45 investigated the problem of nonlinear hydrodynamic stability of the melt during the directional solidification of a binary alloy with small K. They applied some proper scaling of the variables and parameter and used them in the nonlinear stability system of the threedimensional disturbances. They derived a nonlinear evolution equation governing the spatial structure of the convective flow of the melt and found that a stable solution of the equation did not exist. Similar results were found earlier by Young and Davis44 for the solution of essentially the same type of nonlinear evolution equation. Nonlinear stability system of the solutal convective flow in the melt under a solidifying surface and rotating about the vertical axis was investigated by Riahi46'49 in the limit of small K. Applying scaling analysis in his nonlinear stability system, he derived, in particular, a nonlinear evolution equation governing the cellular structure of the flow of the melt. This equation incorporated convective instabilities and rotational effects. It was found that sufficiently large rotation rate stabilized the solution of the evolution equation. Nonlinear solutal convective flow in the melt under a horizontal solidifying surface and in the presence of a magnetic field was studied by Riahi47-48'50 in the limit of small K. Proper scaling analysis was carried out for the governing nonlinear magnetohydrodynamic stability system of equations for the three-dimensional disturbances and, in particular, a nonlinear evolution equation governing the flow structure of the melt was derived. This equation incorporated convective instabilities and magnetic field effects. The presence of a strong vertical magnetic field was found to inhibit the onset of the cellular structure, while presence of a strong horizontal magnetic field was found to eliminate the three-dimensional solutions. Consequently, sufficiently strong inclined magnetic field stabilized the two-dimensional solution of the convective flow. 2.

Steady Flow

We consider a horizontally infinite layer of fluid of average depth d and heated from below. The Oberbeck-Boussinesq approximation for the momentum, continuity and heat equations can be written in the following non-dimensional form: M—

+ W-VJW = - V P + BA:+V 2 W,

(la)

121

Vw = 0,

(lb)

I Ik + -' r = R" ^+ V 2 R

(1C)

These equations are indeed the ones for the disturbances superimposed on the basic motionless state, and the analysis for such nonlinear system is given below. Here u is the velocity vector, P is the modified pressure, B is the deviation of temperature from its basic value, K is a unit vector in the vertical direction, Pr = ji/k is the Prandtl number, /z is the kinematic viscosity, k is the thermal diffusivity, R = pATgd3/(kfj) is the Rayleigh number, AT is the temperature difference across the layer, /J is the coefficient of thermal expansion and g is acceleration due to gravity. The boundary conditions, for the case where the lower boundary is corrugated, are in the following forms22'23'51*52. u = B- SRh = 0 at the lower wall, z = 8h,

(2a)

u = B = 0 at the upper wall, z = 1.

(2b)

Here h{x,y) is a given spatially non-uniform function of the horizontal variables x and y describing the shape of the corrugated surface and 8 is a constant modulation parameter which is assumed to be small. Following these latter references, we assume that the modulation parameter 8 equals £3, where e is the amplitude of convection. This is a particular case, appropriate for resonant wavelength excitation case52 and consistent with a Landau type weakly nonlinear formulation, and it turns out that it leads to preference of a number of unusual flow patterns23. The function h(xyy) is actually assumed to have the following arbitrary representation

h(x,y)= X * W -

X LC^exp/^.A

(3)

where L is a constant, r is the horizontal position vector, i is the imaginary number -J-lT Nw is a positive integer which may tend to infinity, and the horizontal wave number vectors K^ satisfy the properties £
(4)

The coefficients C^ are constants and satisfy the condition

%cib)c?b) = i,c?b) = c!»,

(5)

where the asterisk indicates the complex conjugate. The general solution of the linear problem, derived from Eq. (l)-(2) and designated by superscript zero quantity, for marginal convection instability can be written in the form U[')=F(z,a)ftCmW„

(6a)

122 where WH = expliKH-r j with \ = af KHK = 0, K. n =-AT n

(6b)

represent solution of the problem A2Wm+a2WH=0.

(6c)

Here A2 is the two-dimensional horizontal Laplacian, Kn are the horizontal wave vectors of the convective flow and a is the wave number. The constants CH satisfy the condition

I|C„|2=1

(7)

n=-N

and the constant N is a positive integer. Only the vertical component of velocity is given by Eq. (6b) because the other dependent variables can be easily derived from Eq. (6a) 3 . The conditions given in Eq. (6b) and (7) are imposed to ensure that Eq. (6a) is real. All the solutions of the linear problem correspond to the same eigenvalue R(oc), which is a continuous function of a and reaches its minimum value Re at the critical ac of the wave number. The linear problem as presented partly by Eq. (6a), is infinitely degenerate. The infinite degeneracy of the linear eigenvalue problem is two fold, corresponding to the free choice of the directions of the wave vectors Kn as well as to the arbitrariness of the coefficients Cn and the integer N. Part of such degeneracy can be removed by the nonlinear processes. It can be shown that only a small subset of solutions of type Eq. (6a) corresponds to the solutions of the steady case in the limit where amplitude of convection approaches zero. Another part of the degeneracy corresponds to the translations and rotations of given solutions. This part of the degeneracy arises from the horizontal homogeneity and isotropy of the problem and cannot be removed by the nonlinearities. But, even if this part of the degeneracy is neglected, still there exists an infinite degeneracy. The physical principle that selects one or more preferred solutions of the nonlinear system is the concept of stability. Only the solution that is stable with respect to arbitrary three-dimensional disturbances is physically realizable. Stability analysis, in most cases, predicts a single stable form of convection, while in other cases it may depend on the history of the experiment. Nonlinear analysis is based on the expansion of the type w z =a/ z (o) + e 2 ^ 1 ) +... R = Rc+eRil)+e2R{2)+...

(8a)

(8b)

Using Eq. (8) in Eq. (1) and (2) leads to a series of systems of inhomogeneous equations to orders £n(n>2) m Solvability condition requires that the inhomogeneity is orthogonal to all solutions of the adjoint homogeneous problem. Such condition yields the following system

123 m 2 eR eb^C,CpS^K,+ £*<» + et?Rp\C R«XCmn = eb^Cfi K-Kp-K £ .n) ps(Kl+

-e%LCnJdCt)8(K^-Kn)

+

^[Ao\Cnf +

X
V'

(9) (9)

•/

where &, it,, A), andfc are constants, A is a function of the angles between the wave vectors and JO JOifa^O, if a ^ 0, 55(a) (a) =\\iia = 0. Q. =llifa = The constant b is the asymmetry parameter and is often very small. It typically represents the temperature dependence of material properties such as viscosity. If fc = 0,then* (11 =0. The general manifold of the solutions of Eq. (9) contains both regular and nonregular solutions. For zero modulation parameter, L = 0, the cases N = 1,2,3 of regular K -vector distribution correspond to realizable patterns and the following results generally hold: Rolls and square pattern are symmetric, while hexagons are not. R{n) = 0 for N = 1,2, provided n is odd. Since R{2) > 0 in general, supercritical convection is expected for N = 1,2. R{n) * 0 for N = 3 with odd n unless b = 0. Thus £R{l) may have positive or negative sign. For eR{1) < 0, subcritical convection is possible. For non-zero L and for the case where K%> * Kn for all m and n, the results are the same as the L = 0 case. For non-zero L andfor the case where Kn = K™ for at least some m and n, regular or nonregular solutions of different patterns can be realizable depending on the magnitude of L and on the number of m and n for which the above vector equality holds. Such realizable patterns have at least one vector in the direction of K^ for some m. For sufficiently large L, R{2) < 0 and, thus, the corresponding solution is" subcritical. To isolate the preferred stable patterns, the stability analysis is needed to be under taken. Assuming time dependent coefficients Cn(t) are in place of the steady coefficients eCn, the system of equations ^

=

-k^CnC^s(K^"M-X C„C^S(K'$-

b^C,CpS^Kl+Kp-K^ K K„) [-(R - Rc)k5 + Ao Ao n) + [-(*

|c/+ZA.*.|W]c.,H<~, m=l

"

(10) ao,

''

is obtained, wherefc3,kA, and ks are constants. The stability problem can be treated as a special case of Eq. (10) by setting CB=£Cn+Cnexp(or) CB=£Cn + Cnexp(or)

(11) (11)

and linearizing the equation with respect to the coefficients C„. Here a = ar + id, is the

124

complex growth rate of the disturbances and is the eigenvalue of the resulting linear homogeneous system of the equations. When or <0 for all possible o\ the steady solution is stable. Otherwise, it is unstable. In the case of high conducting boundaries and zero L, only rolls or hexagonal cells are possibly stable5. In the case of poorly conducting boundaries and L = 0, only squares or hexagonal cells are possibly stable4*37' provided Prandtl number Pr is not too small. Otherwise, rolls or hexagonal cells are possibly stable7. In the case of finite conducting boundaries, for a symmetric layer, and L = 0, only rolls or square cells are possibly stable depending on whether the conductivity ratio y of fluid to boundary is beyond or below some critical order one value6*53'54 and provided that Pr is not too small. Otherwise, rolls are the only stable solutions6*7. For L * 0 case 23 , there exists positive constants /,. f/ = l, 2, 2r, ...,m; m = Nw -1] such that /._! y). Otherwise, l2r=h' For l2r /m, multi-modal (N = m +1) pattern convection is preferred. Extension to non-zero asymmetric layer case26 is found to lead to dependence of the preferred flow patterns on b as well. The case of non-resonant wavelength excitation, where the wavelength of the wth mode of the modulation is not equal to the critical wavelength for the onset of classical convection for any m, was investigated by Pal and Kelly55 and by Riahi22-25 Pal and Kelly55 considered the onset of two-dimensional convection when the one-dimensional variations of the temperatures of the walls are in the special form of a sine wave. They found that the modulation can be stabilizing. Riahi22"25*56*57 extended the work by Pal and Kelly55 to arbitrary three-dimensional flows and an arbitrary one or two-dimensional nonuniform temperature boundary modulations. A double expansion of the dependent variables and R in powers of e and S was used. It was found that 0(e2)<S<0(e)

(12)

and \K^\<2K

(13)

for at least some m. For the case where K^-K^ =0 holds M times multi-modal convection patterns (N = 2, ..., 2M) are the preferred patterns for a^ = -Jin, provided flow is subcritical. Otherwise, multi-modal - M l patterns are all possible. For other cases, where K^- K^ = 0 or a^ } = -Jin may not hold, then multi-

125 modal (2 < N < 2iV(fc)) patterns are all possible, and thus the preferred pattern is one of these due to the initial condition. 3. Oscillatory Flow A typical oscillatory flow case is low Pr(0 < Pr < 0.677) Rayleigh-Benard convection rotating about the vertical axis 27 . Riahi 58 ' 60 investigated the problem of weakly nonlinear oscillatory convection in a horizontal low Pr fluid layer heated from below and rotating about a vertical axis. A multi modal discrete analysis was performed and two and three-dimensional solutions in the form of standing waves were studied for isothermal boundaries. The stability of all such solutions was investigated with respect to both standing wave and traveling wave disturbances. The results of the stability and the nonlinear analyses for various values of rotation parameter r and Pr(0 < Pr < 0.677) indicated that all the solutions, which were determined based on a discrete modal approach, were unstable. Later Riahi 61 ' 62 applied two procedures which led to stable oscillatory patterns. These two procedures are briefly described here. The first procedure 61 is based on the formal perturbation theory which makes use of the so-called multiple-scales technique usually applied directly to differential equations. This technique recognizes explicitly in the dependent variables that changes are occurring on two scales: the fast oscillations in space and time of the waves and the slow variations of the wave number, frequency of the oscillations and the amplitude of the waves. A perturbation parameter e is introduced which measures the ratio of the fast to slow scales. Next, the dependent variables, R and the frequency are expanded in powers of e. To lowest order in e, the dependent variables are assumed to be product of an exponential periodic function in x, y and r, a z-dependence function and an amplitude function A which is a function of slow variables xs, ys, and ts in the horizontal and in time. Following Newell and Whitehead 63 , the critical wave vector, at which R attains its minimum value Re, is taken to be (ac, 0) so that an 0(e) band of allowed modes in the x -direction implies .an 0(Ve) band of allowed modes in the v-direction since neutral modes lie on circle \K\ = a c 6 3 . The main concern was in regard to finite amplitude two-dimensional oscillatory solution as discrete modal analysis of the problem 59 ' 60 indicated that disturbances acting on two-dimensional oscillatory roll solutions have the highest growth rate among all the disturbances acting on two and three-dimensional solutions. The wave number and the frequency of the linear system are related by a linear dispersion relation, and they are taken critical values ac and coc at which R = Rc. The waves encountered in the present problem are of the socalled dispersive type where a local disturbance disperses into a wave train 64 - If the disturbance is initially localized in space, then it propagates like a wave packet of the most unstable modes traveling with the group velocity G. A general property of the dispersive wave is that the energy propagation of the wave (group velocity) differs from the velocity of the wave train (phase velocity C). It is found from the dispersion relation that G*C for the present problem. It should be noted that the linear system leaves A undetermined since terms of space or time rate of change of A are of order e2 or higher. The solvability conditions for the second order system of equations in e imply that A propagates with speed G, but its value vary at a rate which is due to variations with respect to ys. Hence A = A(xs -Gt5, ys, ts). The results suggest that one should use the following slowly

126

varying quantities: § = e(x - Gt), Jey, *!=*', xx = et,*T=eX TT = e\ e(x-Gt) rj ==Vey. 9 7]

((4) ((4)

and proceeds as before. In the order e, the linear system is obtained which is the same as before. In the order e2, the solvability conditions lead to a parabolic equation describing the dependence of A on xx and 776«65. In the order ez the solvability conditions lead to the following equation6-65:

£-£-4-w>

(15)

B = SRh(x,t) at the lower boundary, z = 0,

(16a)

B = 0 at the upper boundary, z = 1,

(16b) (16b)

The equation (15) subjected to a forcing term and periodic initial condition lead to stable solutions61*63'65. Such stable solutions do not exist in the discrete modal approach 58»60. Hence, wave packet approach described above can provide physically realizable oscillatory flow patterns in contrast to the discrete modal approach which fails to do so for the present problem. The second approach, which leads to stable oscillatory patterns62, is aimed at examining the effects of a spatially and/or temporally periodic boundary condition upon convection. It turns out that the following two general types of variation in the boundary condition lead to the same qualitative results23'52 for the case where the convective flow is considered in a range close to the onset of convection based on the classical theory52'60: (I) a bounding surface which is a flat plane but has temperature which varies non-uniformly in space and/or time. (II) Surface which has constant temperature but is corrugated such that the gap size between the two walls varies non-uniformly in space and/or time. In the actual analysis of the problem62 the second type of boundary condition is used. Hence, it is assumed that

where h(xj) = V2cos(ax)[Ls5 + L0 sin(oA)],

(17) (17)

Ls andL0 are two constants of order one, a and w are the wave number and the frequency of the boundary modulation, Lis=5LsiL*0=SL0 and it is assumed that the amplitude of the modulation is small (S « l). The particular form Eq. (17) is assumed for h(x,t) since it is the simplest expression which includes both temporal and spatial modes appropriate for oscillatory standing wave and steady condition. It turns out that the parameter Ls is effective for steady case, while Lo is effective for oscillatory case. The steady component in (17) is included in the analysis to see its role in the oscillatory convection regime and also to suppress instabilities which occur in steady convection at high r 5 4 ' 6 6 " 6 9. The analysis for the critical regime is considered23'52'70 where R « Re and 5 = e\ This case corresponds to the resonant wavelength and period

127

excitation where a* = ac and co* = coc. The same nonlinear theory as in the case with only spatial boundary modulation23 is applied which leads to the following results: (I) For steady convection case, there can be more than one stable solution for sufficiently large and positive values of Ls. In particular, rolls pattern can be preferred and Kuppers-Lortz instability of steady two-dimensional rolls66 can be suppressed for given r and Pr and for sufficiently large and positive Ls. (II) For standing wave convection case, there may be more than one stable solution for sufficiently large \L0\ and negative values of Lo. In particular, instabilities of the standing waves due to standing wave disturbances can be suppressed for given rand Pr and for sufficiently large and negative L0. (Ill) For standing wave convection subjected to traveling wave disturbances, there may be more than one stable solution for sufficiently large values of LQ with appropriate sign62. In particular, instabilities of the standing waves due to traveling wave disturbances can be suppressed for given r and Pr and for sufficiently large L0 with appropriate sign. These results together with more detailed analysis23-62 indicate that the preferred pattern is due to the stable rolls solution whose wave vector is along the one of the boundary modulation and for which R is minimum for steady convection case, while initial condition can determine the preferred oscillatory pattern for wave convection case. Also type of modulation mode is significant only if it is acceptable by the linear theory. For example, steady mode of modulation is ineffective for the case where the linear theory predicts that oscillatory solutions are preferred. 4.

Stability Region

In this section, we are concerned with the range of wave numbers a of the realized convective flow. Let us designate ac to be the wave number at the onset of convective flow. In the case a*ac, the most strongly growing disturbances, which make the growth rate in the stability system maximum, have wave number a which do not equal a. Here, we consider such case and include a nonlinear stability analysis for the case of convective flow in a plane layer with poorly conducting boundaries and in the limit of sufficiendy large Pr4'9. In an earlier analysis of a related problem with isothermal boundaries, Busse8 determined the stability region for the steady two-dimensional cells via a stability criterion. In the present case, we find a more complicated stability criterion for the stability region of square cells which are the preferred flow structure here. Although Busse's stability region depends on the Prandt number Pr and decreases slightly with increasing Pr, Riahi's stability region9 is independent of Pr, provided Pr does not tend to zero as was pointed out in Busse and Riahi (1980). Furthermore, in contrast to the case with isothermal boundaries, the stability region in the case with poorly conducting boundaries9'10 is located to the left of the line a = ac in the (R>a) space. We consider an infinite horizontal layer of fluid of depth d bounded by two infinite half spaces with the thermal conductivity small compared to that of the fluid. In the steady state, a constant heat flux transverses the system such that the temperatures 7J and T2 are attained at the upper and lower boundaries of the fluid layer. Under the usual Boussinesq approximation, the non-dimensional form of the nonlinear stability system for the notionless base state, derived from the three-dimensional disturbance equations for momentum, heat and conservation of mass, can be simplified by using the general

128 representation for the divergence-free velocity vector as the sum of the poloidal and toroidal components 5 * 27 . The detailed analysis indicates that the contribution of the toroidal component vanishes in the limit of sufficiently large Pr, which we assume here. Vertical component of the double curl of the momentum equation together with the heat equation then yield the following governing equations: W

= £,

(18a)

V2B-RA2V-=-

= 8VVB9 (18b) dt " where 8 = V x V x K, A2 is horizontal Laplacian and V is the poloidal function satisfying the relation pVxVx^y.

(18c)

Following Busse and Riahi4, the conditions for V and B at rigid boundaries are * = 0 , B = B-,% = P%-atz = ±\, (19) oz dz dz 2 where P « 1 is assumed and Be describes the deviation from the static temperature distribution in the space |z|>l/2. The linear planform function w(x,y) has the representation given by the (x, v) dependence of u[°^ given by (6a) 71 . Since the finite amplitude steady system here is identical to that discussed in Busse and Riahi 4 , it is found again that ccc is proportional to p^. Thus it is assumed that a2 = 7]2y, where y = P% and rj is of the order unity for the convection modes of physical interest. We then seek solutions to the nonlinear stability system in terms of a double series in powers of / a n d e. In particular, R is given in the following form V

R=

^emYnRW.

(20)

m,n=0

For the linear system, the solutions for V and B are determined 9 , and it is found that the linear eigenvalue problem leads to the following results: 7]c = (462/17)**, Ri0) = 720, R® = 720(2/ta" 1 + ^ " V ) / " 1 .

(21)

Minimizing tf„=/?<°>+ )*<'>

(22)

with respect to a yields Rc = 720(1 + 3m;1), K = minfl 0 . v

'

(23)

a

Using (21)-(22), the approximate expression of R0 for small values of | a - c t c | can be written as

129

*.=**+/*.(«-«J2.

(24a)

where ld2Ro = 3Rl0)7i;\ (24b) ^.=r2 da2 . For the nonlinear stability system, it was found that the expression e2yR^) is the largest non-zero term in the nonlinear domain (m > 1), in the expression (20) for R. The expression for R^ is found after forming the solvability condition for the heat equation in the order £3y2. After some considerable algebra9, this condition was found to yield the following system ^CH = a'Rf^-X^h^CSiC^w:

WmW,Wp)

I*P

+3C„}/10I, n = -N,...,-1,1,...,N

(25)

where ^

= (^^«)<*" 2 and

angular bracket indicates horizontal average. It was shown in Busse and Riahi4 that the preferred stable solution for the present flow is that due to squares for sufficiently small e. In order to determine the region of stability of squares in (a,R) space, we consider the following system of equations for the time dependent infinitesimal disturbances V and B which are superimposed on the solution of the nonlinear stability system in the form of squares W4V = B,

(26a)

(V2 - G)B - RA2V = 8VVB + 8VVB,

(26b)

V = — = B-Be =— ( £ - i r ) = 0atz = ±l/2. (26c) dz dzK ' Here Be describes the temperature perturbation for the disturbances in the space \Z\ > 1/2, and we have introduced a growth rate a by % = oV. The Eqs. (26) are solved by expansions in powers of sandy of the dependent variables V,£,Vand£, and of parameters R and a. In particular, a is given in the following form

c= ££"y"ai").

(27)

m,n=0

In the lowest order of e and y solutions for V and B are given in terms of a horizontal planform function vv(x,y), and it is found that

130

cr
(28) (28)

*(*.?) = I.C.W.= £C„ exp(^.. A

(29) (29)

The function w{x,y) is defined by n n

H H

where the horizontal wave number vectors KH for the infinitesimal disturbances are assumed to be in the direction of KH, have'the same magnitude a and satisfy the properties KnHKK = 0, K_n = -K -K„. H.

(30) (30)

The coefficients Cn satisfy the conditions

£c„c; = i. c;=c.„.

(31)

It is assumed that the horizontal wave numbers of the steady squares and the disturbances differ from ac only by a small amount of the order ey^ and that t « « + +(ey/2). . I the order y solutions to Eqs. (26) are found9, and the existence condition yields oi=0. oi=0.

(32)

In the order y2 the existence condition yield 2 cxi2)2) 7 V<

4 3 -2fiyx - aa*n? = -2ayx rj; + «a22 ^J^>/^ V * !0") .•

(33)

This result implies that in the limit a - » a , cci2f> >>00ffoorraa>>aa....

(34)

In the order ey solutions to Eqs. (26) are found9, and the existence condition yields <x. cr # ww=0.

(35)

In the order e2/2 the solvability condition for Eqs. (26) yields a lengthy expression for a{2)9 which will not be given here. Next we define 2

2

<7< = a( >2>++c7(V. <7<>2>=<X< <7
(36) (36)

The aim is then to find region in (/?, a) space above neutral linear stability curve where the solution to the nonlinear stability system (squares) is stable (cr(2) < o) . Forming the expression for
131

£ . = £ . + ?,+ £,

(37)

are satisfied. An extensive search for the possible disturbances that satisfy the expression (36) for CT(2) leads to two cases which lead qualitatively different results. Case A) Disturbances that satisfy the relation K^Km±K^Kx

or Kn=Km±K2+K2.

(38)

Using (38) in (36), in the limit a -» a, we find from the characteristic equation for the growth rate
occf (37 - 22a~lac)/l5 < 0.

(39)

Case B) Disturbances that satisfy the relation Kn=Km±Kx±Kx

or Kn = Km±K2±K2.

(40)

Using (40) in (36) and following the same procedure as in the Case A, we find that the disturbances of type B are stronger than those of type A, and that, in the limit a —> a, a (2) is non-positive for a
(41)

From the above analysis, it is seen that the stability criteria (39) and (41) determine the region of the stable steady solution to the original nonlinear stability system. Such stable steady solution in the present problem under discussion describes cellular convection in the form of squares with wave number a. 5. Spherical Layer In this section we consider nonlinear stability analysis for convective flow in a fluid layer between two rigid concentric spherical surfaces. This problem is of importance in many areas of geophysics and astrophysics. The early linear analysis of the problem27 has been motivated by the hypothesis of earth's mantle convection, and increasing evidence for solid-state convection in the earth and in other terrestrial planets has stimulated nonlinear stability analysis of the problem28'29*31. An example of the application of the problem in astrophysics is that of a convective flow zone in a star whose physical conditions are spherically symmetric if the rotation rate is sufficiently small. The nonlinear stability system to be introduced and analyzed below is based on the assumption that the basic motionless state is spherically symmetric. Such assumption leads to general analysis for the stability system which can, in principle, be applied to other mechanical systems with spherical symmetry, and that many results about the preferred structure of the solutions can be obtained without reference to particular physical problem.

132

We consider a spherical fluid layer of thickness h which is bounded by two concentric spherical surfaces with radii rjn and (r0 + l)h.. The fluid is subject to a spherically symmetric gravity force and contains a spherically symmetric distribution of heat sources. The properties of the core inside the fluid layer are also assumed to exhibit spherical symmetry. Due to its symmetry the problem permits a steady static solution of pure conduction with a temperature AT across the fluid layer. The basic equations in the Boussinesq approximation can be written in the following non-dimensional form29:

(V - Pr"1 I J V 2 ^ - Ry^L, B/(r0 + ±) = Pr"1 • V x fv x (ux (v x u) YL [ V2 - ~ ) B + r-lT{r)L^

= u-WB,

(42a) (42b)

where u, as before, is the divergence free velocity vector, r is a unit vector in the radial direction, B is the deviation of the temperature from the dimensionless static temperature distribution T(r), r is the radial variable, O is the poloidal component of u related to u by u = V xfa xr®\ + V xry/y

(43)

yr is the toroidal component of u whose consideration is not needed in the present analysis27*29, P and R are the Prandtl number and the Rayleigh number defined by R = ag0M3/(kti),P

= ti/k,

a is the coefficient of thermal expansion, jj. is the kinematic viscosity, g0 is the gravitational acceleration at r = r0 + \ such that the function a(r) becomes r = ra+j, -g0 y(r) r(r0 - j) is the gravity force, a prime denotes derivative with respect to r, k is thermal diffusivity, t is the time variable and the operator L^ represents the negative two-dimensional Laplacian on the unit sphere, i.e. in spherical co-ordinates L*i = —

smfl-

sin0<90 dd sin2 6 d and B29. Solutions of Eqs. (42) are obtained by expanding the dependent variables and R in powers of the amplitude e of convection. In order to investigate the stability of any solution of the problem, infinitesimal disturbances are superimposed on the steady solution of Eqs. (42). All the linear, nonlinear and stability analyses are based on the approach due to Schluter et al.3, Busse5 and Busse and Riahi4. Hence, detailed analysis will not be presented here. The general solution of the linear version of Eqs. (42) is given by

133 ),

(44a)

m=-l

with C.„=C;,X|C„| =1

(44b)

m=-l

where Ytm denotes spherical harmonics of degree / and with similar form of solution for £ ( 0 ) . Here superscript (0) denotes linear solution and star denotes complex conjugate. The degree / assumes the role of the wave number a of the plane layer, but instead of the continuous dependence, R(a)21, the Rayleigh number determined by the solution of the linear version of Eq. (42) assumes discrete values corresponding to all integers / > 1. It should be noted that the dependent variables and R are already expanded in powers of e

(*H^M*>K R = Rc + £K(1) + e2R{2) +...

(45a) (45b)

The symmetry properties of the solutions (44) depend on the degree /. Because of the property for any spherical harmonic that its values at opposite points on the sphere are equal, with the same or opposite sign depending whether / is even or odd, Ytm{n-0,

0 + 7r) = (-l)'y,"(0 f 0)

(46)

solutions (44) with odd / describes convection that is symmetric in up-and downgoing motions. Solutions (44) with even / can describe convection that is not symmetric in upand downgoing motions. Because of the geometry of spherical layers, asymmetries are likely to play a much stronger role than in plane layers. However, since the coefficient 7?(1) in (45b) is determined by expressions of the form of averaged over the sphere of triple products of spherical harmonics 28 , R^ must vanish for odd Z29. By the same reasoning it can be shown that all coefficients R^ with odd n vanish in the case of odd /. This result can also be shown from the fact that in the case of odd / the coefficients /? (n) must vanish for odd n, since the reversal of the sign of e must not change expression (45b). Since R{2) > o, in general, subcritical finite amplitude convection does not occur for odd /. Since /? (n) with odd n does not vanish in general for even /, there always exist two physically different solutions for a given pattern. The two solutions differ by the sign of the coefficient CH and have the same relationship to one another as the up- and downhexagon solutions for plane layers. The solution with eR^ < o are possible for subcritical values of R. The solution that maximizes /?(1) is unique at sufficient low values of R and is likely to be preferred over an extended range of R. Since # ( 2 ) is typically positive in bifurcation problems characterized by a minimum of the control parameter R for the physical onset of bifurcation, subcritical bifurcation can only take place for /?(1) < o, i.e. in the case of even /. This gives a slight edge to even / modes over odd / modes. In addition, a negative £ft(1) causes an onset of hysteresis of the kind discussed in the case of

134

hexagon pattern convection in plane layers5. Because of the minimum Rayleigh number Rm at which steady convection exist differs from Re for even / in general, the degree / of the realized pattern may differ from that predicted by linear theory. In order to describe several distinct preferred patterns that have been discovered so far28-31'33'34, it is appropriate to write down the expression for the spherical planform function W^O,^) introduced in (44a) in the following form W5(0,0)s j^aJV™

= £a M cosm0|(2/ + l)

m=o

m—o

(2-Smo)(l-m)\/(l

+

m)fpr(cosd)=Jlam m-o

cosm0Pzm(cos0)

(47a)

where the functions />m(cos0) are the associated Legendre polynomials and the constant coefficients am satisfy the normalization condition

iX=l-

(47b)

It turns out that the axisymmetric solution cc20=l, a,.=o(j = l,...,/)

(48)

represent a solution of the solvability conditions at orders e2 and e3 for all values of /28,29. But there does not seem to exist any other solution that can be defined independently of /. Each / must thus be discussed separately. Before we discuss the realizable patterns for several values of /, it should be noted that the linear system is self-adjoint in the special case when = 028«72.

(49)

For self-adjoint case /?(1) = O 28 . Hence, flow patterns for odd / have been obtained, based on the solvability conditions at order e3 which determine /?(2), for self-adjoint case, while flow patterns for even / have been obtained, based on the solvability conditions at order e2 which determine R?\ for non-self-adjoint case. For / = 129,73? m e solvability conditions at order e3 yield a2=l-a20.

(50)

The possible choices of coefficients admitted by (50) correspond to all possible inclination of the axisymmetric solution (48) with respect to the axis of the coordinate system within the plane (j> = o. The solvability conditions do not constrain the manifold of solutions (47a) because each one of the latter represents a transformation of the axisymmetric solution in the case / = !. Thus the stability problem becomes trivial in this case because of the

135

absence of competing states of convection. For / = 2 28 ' 29 and non-self adjoint case, the solvability condition at order e2 28 yields a2 = 2{l + 2a0){l-a0)/3,

a2 = ( l - a j / 3 ±

(51)

for arbitrary values of ao with \oco\< 1- comparison with (48) shows that solution (51) represent the axisymmetric solution rotated about the axis 0 = f, 0 = ±f. Hence the axisymmetric solution is the only solution in this case which satisfy the solvability condition. This is a non-trivial result, since (48) contains other patterns, which are excluded by the solvability condition. For / = 2 and self-adjoint case, there are two distinct solutions but pattern degeneracy cannot be removed since solvability conditions vanish identically29. Since the self-adjoint case represents a relatively special problem, Busse and Riahi29 did not consider this case further. However, detailed mathematical discussion of / = 2 case has been given by other authors74*75. For / = 3 29 , there exists three distinct solutions. One is the axisymmetric solution which is found to be unstable. The second one exhibits a tetrahedral symmetry and the solvability conditions at order e3 yield a\ = 1, a0 = ax = a3 = 0.

(52)

The last one exhibits a purely sectorial structure. Both of these later solutions my be stable depending on details of the problem. However, for case of a thin spherical fluid layer29, it is found that only the solution (52) exhibiting the tetrahedral symmetry is physically possible for / = 3. For / = 4 28 , the solvability condition at order e2 yield 3 distinct solutions, which are not preferred28, and the following preferred solution which minimizes R «o = *(*)*• «4 = Hi)*. «. = «2 = «3 = 0-

(53)

Solution Eq. (53) exhibits the symmetries of a cube with opposite sides facing the poles or of an octahedron with opposite vortices pointing towards the poles28. For / = 5 31 , the solvability conditions at order £3 yield six distinct solutions. However, the following solution, which transports the maximum amount of heat, is preferred <*o = }. OL\ = f, a.t = 0 (i = 1,2,3,5).

(54)

Solution (54) resembles symmetries of a polyhedron which consists of a triangular prism plus two tetrahedrons. For / = 6 28 , the solvability condition at order e2 yield eight distinct solutions. However, the following solution, which minimizes R, is preferred a0=i(ll)i,a3 = | ( ^ ) i ,

136 « 6 = - f ( T ) i , « J = ^ = 1*2,4,5).

(55)

Solution (55) exhibits the symmetries of a dodecahedron as well as symmetries of an icosahedron which has faces in the form of equilateral triangles. In the cases of / > 7, the determination of all the solutions due to solvability conditions at orders e2 or e 3 becomes more difficult and solutions for this range of / values have not yet been obtained. It is expected that the preferred solution in the limit / -» oo approaches the hexagon solution28. It seems that a hexagonal pattern can be closely approximated on a spherical surface as some natural phenomenon suggests 76 , despite the fact that an exact hexagonal pattern cannot be realized on a spherical surface according to Euler's theorem 28 . Nearly hexagonal pattern corresponding to large values of / are also observed when a thin elastic spherical shell buckles under the influence of a uniform pressure 77 . 6.

Mixed Solutions

With the same analysis to that for the simple patterns cases discussed in the previous section, the finite amplitude and stability analyses are done for the cases when the difference between two linear eigenvalues R for different values of / tends to zero. Previous analysis28*29-31 of patterns corresponding to a single / is extended to the case of two participating classes of spherical harmonics with degrees / and /* = / +1, and have focused the investigations on the cases for which / = 1,2,3,4 so far 30,33-35,78,79. The case of thin shell is used both as a simple and appropriate model for more complicated cases of shells of finite size and to make the mathematical formulation of the problem as general as possible. The governing equations are the same as Eq. (42a) and Eq. (42b). The limit of a thin fluid shell with poorly conducting boundaries is assumed which permits simple analytical expressions for the radial dependence of all dependent variables. Both selfadjoint 34 ' 35 and non-self-adjoint29'33 cases are investigated. The boundary conditions for rigid boundaries of uniform thermal conductivity's on both sides of the fluid layer are <& = — = 0 a t r = r „ r o + l . | U

C

^ , Z ? = £ , a t r = r 0 ,r 0 + l.

(56a) (56b)

Here Be is the deviation from the static temperature distribution outside the fluid layer and C, which is the ratio of the thermal conductivity outside to that inside the fluid layer, is assumed to be small (C « 1). It was shown by Busse and Riahi 29 that Be affects the problem implicitly through B and only in the solvability condition for the order C 4/3 . The governing system (42) considered in the limit of small C and a perturbation parameter X = C*

(57)

is introduced. At the same time limit of a thin shell is assumed such that rQ tends to infinity

137

with C"K:

fc+±r=*?2A,

(58)

where rj is a parameter of the order unity which can be varied arbitrarily. The expansions (45) are then used in Eqs. (42). The linear system has the general solution, with two participating classes of harmonic's with degrees / and f = / +1, of the form O(0) = 4F(r,/)W,(0,0) + A2F(rX)\Vr ( M

(59)

and with similar form for £ (0) . Here Ax and A2 are two amplitude parameters, and the expression for Wr (0,) given by (47a), provided / and a are replaced, respectively, by /* and new constant coefficients a*m. These coefficients satisfy (47b), provided / and am are replaced, respectively, by /* and a*m. The existence condition for the solution of the linear problem4 leads to the following result fl(0)(f ,77) = * (0) (Z,TJ) + &e + /z2£2 = 720

r

/

< 2/ + 1 ^ 1+ 462 /(/ + l)77j 2 +0(A ) + jU1e + Ai2e2>

(60)

where /?(0) is the Ravleigh number for the linear system, and it is assumed that /?(0)(/,7j) can differ from /T0)(l*,il) only by two terms of orders eande 2 designated by faemd/J^e2, respectively. It turns out that for non-self-adjoint case only the term fJ^ed^ * 0) need to be considered, while for self-adjoint case only the term /Mle2(fil = 0) is effective ( ^ * 0). The minimum Re as a function of / is achieved by 1 = 1 (77 > (77/17)*), 1 = 2 [(77/17)* > 77 > (77/68)*], = 3 [(77/68)* > 7] > (77/170)*],

(61)

and so on, by increasing values of / as r0 increases for a fixed C. It is seen from (61) that if 77 satisfies an inequality in (61), then a single value of / niinimizes Ri0\ However, if 77 is sufficiently close to an equality in (61), then the difference between the values of #(0) for two neighboring degrees /and/* can become sufficiently small (of order fixe + fx2e2 or less) and we obtain the singular case which is presented in this section. Let us first consider the non-self-adjoint case33. The solvability conditions at order 2 e yield

4*(,)a„ = AX Xa.^wZ-WW) n,p=0

138

+AlM2 j^aX^W^W^W^),

(62a)

n,p=o

A.R^al = 2\AJA, ^ajx^WJT^W^

(62b)

n,p=0

where angular bracket denotes average over the spherical surface and Aff.(l,2,3) denote some constants which depend only on / and /*. The following variational functional are then determined from Eq. (62)33: (R - rt^ = Nrf + A ^ 2 ,

(63a)

(R - R;)^ = IN^A,,

(63b)

Where Rt andRr designate /?(0)(/,7]), and /?(0)(f,7j), respectively, and the coefficients N^i = 1,2,3) contain information from Eqs. (62) which deals with the patterns of the bifurcating solutions. The fact that in the case of the functional (63) two wavelengths participate instead of one in the case of simple patterns permits larger values for the extremum. The effect of the combination of two wavelengths is most pronounced in the case / = 1, /* = 2 where the wavelengths differ by a factor of 2. The extremalising solution is axisymmetric in this case and is given by \Wt = (A/7/4)P2°, A2Wr = l ^ 0 , R* = V7/5,

(64)

where R* denotes the extremum value of the functional (63). This solution describes a strong rising motion at one pole and a broad descending motion spread over the major part of the spherical surface including the other pole and the equatorial region. For the case / = 2, /* = 3, it is found that a three-cell pattern extremalizes (63): Wi =i(7/3)*A°. AiWr =!(!)*A 3 cos30, R* =f > /f.

(65)

The four-cell pattern is obtained for / = 3, 1* = 4: A ^ . = ( f ) * % cos20, \Wt = ^ [ P 4 ° - (f fP44 cos40], /?*=V91/11.

(66)

This pattern exhibits the symmetry of a tetrahedron. A seven-cell pattern becomes realized when we consider the case / = 4, /* = 5: AlWl = V77/150P4°, A2Wr = V73/150 F55 cos50, R* = 2-V462/65.

(67)

In contrast to the realized patterns discussed above, the seven-cell pattern is no longer

139 symmetric with respect to each of the cells. This property will become the rule as the preferred patterns are sought for higher values of /. In the asymptotic case of large /, /* the preferred pattern will consist of a network of hexagonal cells with some pentagonal cells interdispersed33. Let us now consider the self-adjoint case 34 ' 35 ' 78 . The solvability conditions at order e3 yield the expressions for /? (2) which are lengthy34 and will not be given here. In general, several solutions of the form (59) satisfy the solvability conditions. To distinguish the physically realizable solution(s) among all the steady solutions which satisfy these conditions, the stability of these steady solutions with respect to arbitrary three-dimensional disturbances are investigated, and the stable solutions are determined for / = l(f = 2) 3 4 . It is of interest to note that the solvability conditions for the steady solution and for the stability equations have also been derived from an extremum principle34. A functional was discovered 34 whose stationary value(s) correspond to steady solutions of the nonlinear system and those solutions for which the stationary value is a maximum are stable. For the self-adjoint case only the cases / = i(f = 2) and / = 2(/* = 3) have been investigated so far. For 1 = 1, /* = 2, there are three distinct mixed solutions. Solution I is an unstable axisymmetric solution given by a\ = a*l = 1, ax = a* = a2 = 0.

(68)

Solution II (so called tennis ball solution33) is given by <x\ = a\ = 1, ax = a*0 = a* = 0.

(69a)

It is stable if ^ > 0.0017 faf A}.

(69b)

Solution El is given by a\ = a*\ =1, ccx = a*0 = a2 = 0

(70a)

and is stable if -O.OlOmTj 2 ^ 2 < ju2 < -0.00407XrfAt.

(70b)

For / = 2, /* = 3, there are five distinct mixed solutions35. No stability analysis for these solutions have yet been done, but it is found that the following non-axisymmetric solution carries the maximum amount of heat among all the other solutions and thus appears to be preferred31. <x\ = a\ = 1, a0 = ax = a\ = a\ = a\ = 0.

(71)

140

7.

Flow During Solidification

In this section we consider nonlinear stability analysis for convective flow of the melt during directional solidification for small segregation K «1. Systems with K < 0.01 have great practical interest since they become nearly pure substances upon solidification. We analyze the problem in the presence of either rotation or magnetohydrodynamic effects which were shown by Riahi47"49 to be able to stabilize the solution to the nonlinear stability system. Such solution is known to be unstable in the absence of such external We consider the problem of solutal convection in a horizontal layer of a binary alloy melt, rotating about a vertical axis with constant rotate rate £2 and permeated by magnetic field B, from which a semi-infinite slab of crystal is being grown. The magnetic field is assumed to be composed of a uniform external field \B2 y+ B3 z in plane of yoz and a perturbation field b*>. The governing hydromagnetic system orequations and boundary conditions for the flow velocity w, magnetic field and the solute concentration, under the usual Boussinesq approximation, are considered in a coordinate system moving at velocity V0 (crystal growth rate) with the solidification front at the vertical variable z = 0. These equations and the appropriate boundary conditions are given in Riahi^-**. They are lengthy and will not be repeated here. We shall make our analysis here specific and simpler by considering zero rotation case. In the above description of the problem, y and z are unit vectors along horizontal axis y and vertical axis z, respectively. Non-dimensional variables are used in which length, velocity, pressure, time, magnetic field and solute concentration are scaled, respectively, by d/V0, V0, pV02, d/V?, B3 and (1 - K)CJK. Here d is the diffusivity coefficient of solute, p is a constant reference density, K = CslCm> is the segregation coefficient, Cs is the solute concentration in the solid at interface z = 0, Cm is the solute concentration in the melt at z = 0, and C is the solute concentration in the bulk of the melt. It turns out that C = Cs in the present problem. Another assumption to be made here is that the adjoining medium at the upper boundary (z = oo) is a non-conductor medium, while the crystal is a high conductor. The non-dimensional parameters of the nonlinear stability system, which is the above described hydromagnetic equations and the boundary conditions, are the ratio B23=B2/B3,K, the solutal Rayleigh number R = pg{\-K)C00d2l(idKV*\ the Chandrasekhar number Q = ji!%d>faJmV*)9 theeiffusi^^ field to solute concentration and the Schmidt number Sc = jn/d. Here p is the fractional change in density due to a change in the solute concentration, gis acceleration due to gravity, ju is the kinematic viscosity, and ft is the magnetic permeability. The governing nonlinear stability system, in the limit of Sc = oo, can be converted into the following form: A2(V40-?C) = -QT

L

+ BB232:^-\E IE+MSE-VSEYA ;A A 22V V*2[ D D+ + S(SE'VSEYZ[ L V dy) ~\~ ~ ) ~J

(72a) (72a)

141

A2 fTV4 + V 2 D - | - V 2 l £ + V2fD + ^ | - l o j = 8ld<&V5E-8EV8®YzJ (V2 + D - %)C - exp(-z) A2
(72b) (72c)

O = D
(72d)

|4>|(oo, |£|(oo, c = 0 as z -» oo.

(72e)

Here D = d/dz, C is the deviation of solute concentration from the basic concentration C0, and <X> and E are the Poloidal components of u and b given by (w,&) = 5(0,£), S = V x V x z .

(73)

As in the non-magnetic field case45, the toroidal components of u and b are insignificant48 and are not included in (72)-(73). The simplification of the limit of Sc = <*> is assumed here since the results given in Hurle et al.39 for the regime of practical interest Sc »1 indicate that the limit of Sc = ©° is expected to be achieved so long as Sc > 10. First the linear stability analysis of (72) is considered by using the infinitesimal perturbation method: as was used by Riley and Davis45 for the case without magnetic field. We analyze the linear version of the stability system (72) by using normal modes in the horizontal variables x and y and in time t and by assuming that the critical wave number approaches zero as K —»039*40. We, thus, assume that JC, v and t dependence of the dependent variables is like expf at + iJC- r I, where r = (JC, V),
(74a) (74b) (74c)

and R0 = R(a = 0). The results (74) imply that o~ is real and independent of B23 and that it is impossible to have positive a for r < 1. Hence, all the solutions decay to zero for r < 1 as t increases. The non-trivial solutions, thus, correspond to cases where r > 1. It is also of interest to note that the condition of thermal diffusivity dt less than 7] is amply

142

satisfied for terrestrial materials80. Also dt » dso. Hence we shall assume the condition that x »1. For marginal stability (74a) yields RO = M{Q,T)

(75)

and the linear version of the evolution equation, to be introduced later in this section, leads us readily to the following expression for R: R/Ro = Ky*[Ky* + //£*) + 0(a 3 ),

(76)

where the expression for / is quite lengthy47 and will not be given here. Minimizing R given by (76) with respect to K% then yield Re = Ro{\ + 2K*) + 0(Ky<), ccc = (K/tf

(77)

We find that both Re and ac increase with Q, but they are independent of B^. Following Riley and Davis45 for the nonlinear stability system for small K9 we let K = e2KltR = R0(l + ey),

(78)

where 0 < e « l . The linear stability analysis discussed above suggests to introduce rescaled variables (jc , ,/) = eK(jc,v),z' = z,r / = e2r, (<*>',£') = £(,£), C' = C.

(79)

These rescaled variables and parameters are then used in (72) and the primes are dropped for simplifying the notation. Next, we seek a solution in terms of a series in powers of eYl:

(O.E,C) = X ^ ^ . ^ . C ^ ) .

(80)

Using (80) in the rescaled form of the nonlinear system, we find the following solutions in the orders e, em and e2: *i = % = %2 = £3/2 = 0, ( C ^ ) = [A(x,v,r), B(x,y,r)]exp(-z), (
(81)

where the expressions for the functions F2, H2, and G2 are given in Riahi48, and A andB are two amplitude functions. Integrating Eq. (72c), in the order e2, in the vertical direction from z = 0 t o z = «> and using (81) yield (75). In the limit of T » l , (75) yield

143

Q*2f. Ra0=2[l + Q^.

(82)

It is seen from (82) that R0 increases with Q but is independent of B23. For Q -> 0, the expression for R0 approaches the value 2 in agreement with the critical solute Rayleigh number computed in the limit K -* 0 and Sc -> ~ by Hurle et al.3 for the problem with no magnetic field. In the order e5/2(72) yield solution48 which depend on B and f. Integrating Eq. (72c) in the order e5'2 in the vertical direction from z = 0 to z = « and using (81) and solution in the order e5/2 together with the boundary conditions for C5/2, yield

^-o= °-

((83) 83)

dA

Hence, the amplitude function A does not vary with respect to y. In the order e3 Eqs. (72a,b) yield solution^ which depend on A and f . Integrating (72c) in the order e3 with respect to z from z = 0 to z = ~ aad ussng ((81 )nd solution in the order e3 together with the boundary conditions for C3, yield the following nonlinear evolution equation for the amplitude A: „„K A

dA dA

dd22AA

,d ,d44AA

11 dd

[|(^)] + , A f = 0, W*'h*4-°-

(84) (84)

where the lengthy expression for the coefficient / is given in Riahi48 and lx is a constant. Using (83), (84) together with appropriate boundary conditions for f atjc = ±~ and }> = ±oo° yield ^ = 0. dy dy

(85)

The results (83) and (85) imply that disturbance are two-dimensional in (x,z) plane. Thus, presence of a horizontal magnetic field (along y-axis) eliminates three-dimensional disturbances. Next, we reduce the number of coefficients in (84) first by choosing y = 1 which defines e for a given R. Then set 1/2

^ l==S/l,t 5//,r = = f/,x = Jc(/) . K tl,x=x(lf-

(86) (86)

A + A^ Aiiii++ 0.5(A 0.5(A2)H 0, A;!+SA + SA + + AH + )H == 0,

(87)

Using (85-(86), (84) becomes

where a subscript iorx means
144

S = Kl/e\

(88)

larger than the physicochemical value K by an amount which is determined by R and Q. Young and Davis44 considered Eq. (87) in the non-magnetic field case subject to the periodic boundary conditions in the internal 0 < x < Jc0. They solved equation (87) using an explicit finite difference scheme for various values of x0 chosen as multiples of the wavelength 2->/2;r which gives maximum amplification in the linear theory of Eq. (87). This linear theory give S < 0.25 for instability. Figure 7 in Young and Davis44 presents the calculated A for S = 0.2, 0.24 and 0.4. When S > 0.25, initial disturbances decay to zero and the zero basic state value (A = 0) is regained. For S < 0.25, the system is unstable and a cellular structure forms. For S « 0.25, the results reported in Young and Davis 44 indicate an apparent secondary instability where the tips of the cells show a tendency to split. It is clear from these results the presence of a strong vertical magnetic field can stabilize the system, as is the case in the presence of a strong vertical rotation49, and can inhibit the onset of the cellular structure by effectively magnifying the segregation coefficient. The case of presence of combined rotation and magnetic field is presently under investigation by Riahi81. It is of interest to note that the following evolution equation for the function B is obtained from the order e1/2 system by applying the same procedure as that applied to the order e3 system which determined the evolution equation for A: Bt+SB + Bn + B^+lAB)-^.

(89)

This equation is linear in B. For S < 0.25, A * 0 and the solution for B is not needed since A is the leading order amplitude. For S > 0.25, A = 0 and Eq. (89) yields solution which decays to zero. 8. Some Concluding Remarks We reviewed nonlinear stability analysis for a number of convective flow problems dealing with the preferred patterns in horizontal layers and in spherical shells, stability region of realizable solutions with different wave-lengths, steady and oscillatory solutions, simple and mixed type solutions, effects of external constraints and solidification, and for cases described by discrete modal and wave packet representation of the flow solutions. We found that certain mathematical methods, such as perturbation, multiple scales and scaling procedure are quite effective to carry out the analysis for the nonlinear system. While the results based on the analyses presented in this review article are general in the sense that they require only the horizontal symmetry of the basic state in the horizontal layer cases and the spherical symmetry of the basic state in the spherical shell cases, they are limited in most cases by the condition that only small deviations from the basic state are permitted. But since most of the results are based on the analysis that is mainly concerned with symmetry properties, those results are hardly affected by the condition of small e. The symmetry of the patterns based on the full nonlinear stability system cannot be changed by the contributions of higher order in e unless further bifurcation takes place. Because of

145

the exceptional symmetry of such preferred patterns it is expected that those patterns retain their form and distinction for an extensive range of R. Although there have been very few studies so far on the problems of wave packet convection patterns as well as problems of convective flows influenced by the boundary modulation effects, the results of the nonlinear stability analyses presented here indicate that the importance of such problems should not be under estimated. Wave packet convective flows, which are generally dispersive65, provide a larger manifold of realizable patterns due to solutions of initial value problems, while a discrete modal approach often fails to provide such manifold. Convective flow models with boundary corrugation or modulation effects can provide a large manifold of realizable solutions and can provide an operating procedure to control instabilities and the flow structure. The results of nonlinear stability system for convective flows in spherical shells reviewed here has been based on the analyses of much more general subject of solutions generated by bifurcations from spherical symmetric basic states which has received much attention in the recent literature. The problem of degeneracy and its removal by nonlinear effects dealt in the analyses is common to all eigenvalue problems with spherical symmetry. Related problems which can be investigated using the analysis of the type used here include buckling of spherical shells, cracking patterns of symmetrically cooling solid spheres and oscillations of non-rotating stars.

146 References 1. P.G. Drazin and W.H. Reid, Hydrodynamic Stability (Cambridge Univ. Press, Cambridge, U.K., 1980), p. 527. 2. W.V.R. Malkus and G. Veronis, / . Fluid Mech., 4, (1958) 225. 3. A. Schluter, D. Lortz and F.H. Busse, / . Fluid Mech., 23, (1965) 129. 4. F.H. Busse and N. Riahi, J. Fluid Mech., 96, (1980) 243. 5. F.H. Busse, / . Fluid Mech., 30, (1967) 625. 6. N. Riahi, / . Fluid Mech., 152, (1985) 113. 7. N. Riahi, ZAMP, 31, (1980) 261. 8. F.H. Busse, In Instability of Continuous Systems, ed. H. Leipholz (1971) 41. 9. N. Riahi, Development in TAM ,11, (1982a) 357. 10. N. Riahi, / . Phys. Soc. Japan, 51, (1982b) 4104. 11. N. Riahi, Proc. Symp. Fluid Dyn., (Urbana, EL) (1984a) 315. 12. D.N. Riahi, Proc. Conf. Math Appl. Fluid Mech. and Stability (Troy, NY) (1986a) 257. 13. D.N. Riahi, Proc. AIAA/ASME Heat Trans, and Thermophys. Conf. (Boston, MA), 56 (1986b) 41. 14. D.N. Riahi, Acta Mech, 60, (1986c) 143. 15. N. Riahi, / . Phys. Soc. Japan, 53, (1984b) 4169. 16. D.N. Riahi, Int. J. Non-linear Mech., 21, (1986d) 97. 17. D.N. Riahi, / . Math. Phys. Sci., 22, (1988a) 161. 18. D.N. Riahi, Acta Mech., 64, (1986e) 155. 19. D.N. Riahi, Acta Mech.,71, (1988b) 249. 20. D.N. Riahi, Proc. SECT AM XIII (USQSC), (1986f) 91. 21. D.N. Riahi, J. Phys. Soc. Japan, 56, (1986g) 3515. 22. D.N. Riahi, CAM Conf on Wave Phenomena (Edmonton, Canada) (1992a) 145. 23. D.N. Riahi, / . Fluid Mech., 246 (1993a) 529. 24. D.N. Riahi, TAM-Report #757 (TAM Dept., UIUC, Urbana, IL) (1994a). 25. D.N. Riahi, Proc. Royal Soc. London series A, (1995a) in press. 26. D.N. Riahi, Bifurcation Phenomena and Chaos in Thermal Convection, ASME, HTD-Vol. 214/A MD-Vol. 138 (1992b) 69. 27. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, (Clarendon Press, Oxford, U.K., 1961), p. 654. 28. F.H. Busse, J. Fluid Mech., 72, (1975) 67. 29. F.H. Busse and N. Riahi, / . Fluid Mech., 123, (1982) 283. 30. F.H. Busse and N. Riahi, / . Tera Cognita, 4, (1984) 240. 31. N. Riahi, / . Phys. Soc. Japan, 53, (1984c) 2506. 32. N. Riahi, G. Geiger and F.H. Busse, Geophys. Astrophys. Fluid Dyn., 20, (1982) 307. 33. F.H. Busse and D.N. Riahi, / . Nonlinearity, 1, (1988) 379. 34. D.N. Riahi and F.H. Busse, ZAMP, 39, (1988) 699. 35. D.N. Riahi, Proc. SECT AM XVII, (Hot Springs, AK., USA) (1994c) 469. 36. S.H. Davis, / . Fluid Mech., 212, (1990) 241. 37. M.R.E. Proctor, / . Fluid Mech., 113, (1981) 469.

147 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.

66. 67. 68. 69. 70. 71. 72. 73. 74. 75.

C J . Chapman and M.R.E. Proctor, / . Fluid Mech., 101, (1980) 759. D.TJ. Hurle, EJakeman and A.A. Wheeler, Phys. Fluids, 26, (1983) 624. D.N. Riahi, Phys. Fluids, 31, (1988c) 27. S.H. Davis, U. Muller and C. Dietsche, / . Fluid Mech., 144, (1984) 133. D.N. Riahi, Phys. Fluids A, 3, (1991) 2816. D.N. Riahi, Int. J. Engr. Sci., 27, (1989a) 689. G.W. Young and S.H. Davis, Physical Rev. B, 34, (1986) 3388. D.S. Riley and S.H. Davis, Physica D., 39, (1989) 231. D.N. Riahi, Phys. Rev. B, 44, (1992c) 4170. D.N. Riahi, / . Math. Phys. Sci., 26, (1992d) 429. D.N. Riahi, Proc. ASME Symp. on Bifurcation and Chaos (Anaheim, CA), Vol. 214/A MD-Vol. 138, (1992e) 93. D.N. Riahi, Acta Mech., 99, (1993b) 95. D.N. Riahi, Int. J. Engr. Sci., 30, (1993c) 551. R.E. Kelly and D. Pal, Proc. Heat Transf. and Fluid Mech. Inst. (Stanford University Press, Stanford, CA., 1976), 1. R.E. Kelly and D. Pal, / . Fluid Mech., 86, (1978) 433. D.N. Riahi, / . Fluid Mech., 129, (1983) 153. D.N. Riahi, Phys. Fluids A, 2, (1990) 353. D. Pal and R.E. Kelly, Proc. 6th Intl. Heat Transf. Conf. (Toronto, Canada, 1978) 235. D.N. Riahi, / . Abstracts (AMS) (1995b) 398. D.N. Riahi, submitted for publication (1995c). N. Riahi, / . Aust. Math. Soc. (Series B), 25, (1984d) 406. D.N. Riahi, Bull. Am. Phys. Soc, 34, (1989b) 2284. D.N. Riahi, Proc. Royal Soc. London Series A, 436, (1992f) 33. D.N. Riahi, Proc. of Pan Am. Cong. ofAppl. Mech. (Brazil), (1993d) 327. D.N. Riahi, Proc. of First World Cong. Nonlinear Analysis (Tampa, Fl) (1995d) in press. A.C. Newell and J.A. Whitehead, / . Fluid Mech., 39, (1969) 279. G.B. Whitham, Linear and Nonlinear Waves (Wiley, NY, 1974), p. 636. D.N. Riahi, Nonlinear Dispersive Waves: The Math. Theory with Applications to Fluids and Plasmas ed. L. Debnath , (World Sci. Publishing Co., 1992) (1992g) 417. G. Kuppers and D. Lortz, / . Fluid Mech., 35, (1969) 609. G. Kuppers, Phys. Lett. A, 32, (1970) 7. R.M. Clever and F.H. Busse, / . Fluid Mech., 94, (1979) 609. N. Riahi, ZAMP, 33, (1982c) 81. J. Tavantzis, E.L. Reiss and B J . Matkowsky, SIAM J Appl. Math., 34, (1978) 322. F.H. Busse, Rep. on Prog, in Phys., 41, (1978) 1929. D.D. Joseph and S. Carmi, / . Fluid Mech., 26, (1966) 769. G. Geiger, Ph.D. Dissertation (University of Munich, 1977). P. Chossat, SIAM J Appl. Math., 37, (1979) 624. M. Golubitsky and D.G. Schaeffer, Commun. Pure Appl. Math., 4, (1982) 500.

148

76. D. Thompson, On Growth and Form (Cambridge University Press, Cambridge, U.K., 1942), p. 250. 77. W.T. Koiter, Proc. Koninklijke Nederland Akad. Van Wetenschapen, B72, (1969) 40. 78. D.N. Riahi and F.H. Busse, Proc. ASME Winter Meeting (Boston, MA) (1987) 109. 79. L. Hadji and D.N. Riahi, Proc. SECT AM XVII (Hot Springs, AK) (1994) 444. 80. S.R. Coriell, M.R. Cordes, W.J. Boettinger and R.F. Sekerka, / . Crystal Growth, 49, (1981) 13. 81. D.N. Riahi, In preparation (1995e).

149 MODELING AND SIMULATION FOR PRIMARY INSTABILITIES IN SHEAR FLOWS

D.N. RIAHI Department of Theoretical and Applied Mechanics 216 Talbot Laboratory, University of Illinois 104 S. Wright St., Urbana, IL 61801

ABSTRACT

This chapter reviews primary hydrodynamic instabilities and their mathematical modeling and numerical simulation for several important shear flow problems of practical importance. The specific problems considered here deal with instabilities which may take place in a number of shear flow problems such as bounded and unbounded mixing layers, rotating disk flows, flows over wings and boundary layers. Modeling is accomplished by making use of certain mathematical methods such as perturbation techniques, multiple scales and method of stationary phase. Computation is done using finite differences, spectral, collocation and Galerkin methods. In addition, the latest research results, for each particular shear flow problem, accomplished by the author, are also presented.

1. Introduction This chapter deals with a review of the fundamental understanding of the primary instabilities of a number of shear flows which are of importance in relation to practical applications of such flows. A number of primary instabilities of the shear flows lead to completely different form of such flows with different behavior and in certain cases can facilitate transition to turbulence. We shall present in this chapter a review of some of the mathematical modelings and numerical simulations that have been done for the primary instabilities which may take place in bounded and unbounded mixing layers, plane wake flows, rotating disk flows, boundary layers, flows over wings and in channel flows. Lie and Riahi1 investigated linear spatial primary instabilities of viscous flows in both bounded and unbounded mixing layers for arbitrary values of the ratio A between the difference and the sum of the velocities of the two flow streams. They applied spectral and Galerkin methods with expansions in Chebyshev polynomials and some mathematical analysis and modeling to determine the critical values of the Reynolds number at which the primary instability arises first in the mixing layers. They detected new modes of viscous instability, and they studied the effects of the presence of solid boundaries on the primary instability processes. Earlier, there was some studies on the linear temporal viscous primary instability of unbounded mixing layers for X = l 2 . However, it is known 3-5 that the spatial instability is generally preferred over the temporal instability in mixing layers, particulary for the primary instability and the early development of the mixing layers which are mostly influenced by the spatial events. The parameter A also plays a significant role in

150 mixing layers. For example, it can have significant effects on the critical Reynolds number for the onset of secondary motion1, and the spreading rate of the mixing layers decrease with A 6 ' 7 . Numerical simulations in mixing layers are particularly efficient if they are based on spectral methods 1 with expansions of disturbance variables in Chebyshev polynomials which require less computer time and storage to achieve accurate results when compared with those required by finite difference methods and is known to lead to accurate results in other problems as well8. More recent modeling search for mixing layers9 led to the conclusion that a mathematically rigorous approach for a time evolving base flow in a mixing layer seems to be the singular value decomposition of the linear operator transforming some initial perturbations into the solution at a given later time. Another important type of free shear flow problems are those of plane wake flows10. In a series of papers, Kuah and Riahi 11-13 reported results for the computational investigations of primary instabilities in plane wakes behind a flat plate parallel to the upstream uniform flow. Such shear flow system is an important flow in practice and, in particular, is of interest in aerodynamics and Naval Sciences. Just as in the mixing flow cases discussed in the previous paragraph, an improved fundamental understanding of the primary instabilities and early transition processes would open significant possibilities to the design engineers to delay transition and control or promote turbulence in such shear flow systems. The results reported in the above references 1113 are based on viscous modes of instabilities which are in contrast to most of the investigations of plane wakes due to in viscid modes of instabilities14-18. Boundary layer flow over a rotating disk is an important prototype shear flow system to study the fundamental aspects of the cross flow instability mechanism which is operative in many three-dimensional boundary layer flows with aerodynamic applications. There have been notable recent contributions on the rotating-disk primary instability aspects both experimentally19 and theoretically 20-23 . All these theoretical studies were based on linear stability theory. Very few nonlinear investigations have been done to date. Malik24 used a spectral method in a Navier-Stokes simulation and found some realistic mode interactions similar to those found experimentally25 in a swept wing flow. Itoh26 predicted the supercritical stability of rotating disk flow using a weakly nonlinear model based on the Orr-Sommerfeld equation. MacKerrel 27 used an asymptotic analysis as the Reynold's number R approaches infinity to find that nonlinear effects can be destabilizing for some small wavelength modes near the lower branch of the neutral stability curve, although these modes may not be preferred at such large R. More recently Vonderwell and Riahi 28 carried out a rigorous weakly nonlinear study based on the full governing equations of the preferred disturbances for the rotating disk primary instability problem. They found, in particular, that subcritical bifurcation exists in rotating disk flow, at least for the preferred primary instability modes near the minimum of the upper branch of the neutral stability curve 2 2 . It should be noted that the above study was based on the assumption that disturbance consists of a monochromatic wave with constant amplitude for all the horizontal variables. More realistic is the consideration of localized disturbances whose behavior can be studied by using the wave packet approach 29-31 . Riahi and Vonderwell32 investigated wave packet development of primary instability modes of rotating disk flow using the method of multiple scales. The system of equations for small amplitude disturbances was solved by using the weakly nonlinear theory to determine the wave packet solution which represented the time and space variations of the wave pattern. The

151 amplitude of disturbance satisfied a nonlinear partial differential equation, a generalization of the time dependent equation and of the linear model of earlier works30"32. The solution of the amplitude equation was found to become more and more concentrated at the center of the wave packet disturbance becoming infinite there in a finite time. This bursting type behavior suggested possible fast transition to turbulence in rotating disk flow. A general weakly nonlinear and non-parallel boundary layer theory for primary instabilities, which include those in two-dimensional Blasius flows, three-dimensional flows over infinite and finite span wings, was developed recently 33-34 . This theory is an extension of an earlier spectral theory 35 and provide dominant contributions to the nonlinear terms by subharmonic components of primary instability modes at various frequencies. The results are consistent with experimental observations5*36. Recently some progress has been made on fundamental understanding of the roles and mechanism played by the primary instability modes and their interactions in threedimensional compressible boundary layer flows over swept wings 37-38 . Asymptotic and triple-deck investigation of the primary modes of instability for such flows37 indicated that there are basically two kinds of modes. Inviscid modes having wavelengths scaled on the boundary layer thickness which are relevant upper branch modes. Wall modes having a triple deck structure which are relevant lower branch modes. Stationary and long time scale non-stationary modes are preferred. Compressibility effects lead to several types of primary wall modes which depend on particular disturbance density and temperature scaling. Value of the wave number for the wall modes was found to depend critically on the wall values of the mean flow velocity gradients, density and temperature as well as on the wall value of the viscosity and the viscosity gradient, while value of the wave number for the inviscid modes depends on the mean flow quantities throughout the boundary layer. Primary instability modes that satisfy the triad resonance conditions were investigated using collocation and multiple scale methods 38 . The most critical resonant triads were determined, and the roles played by the primary modes and their interactions in evaluating the compressibility effects were detected. Primary instability of two and three dimensional rotating or non-rotating channel flow modes can be of algebraic and exponential linear or nonlinear types for non-rotating and rotating flow cases, respectively 39 . The primary instability modes found for the rotating system is stronger, but less probable, than the corresponding one which occurs for the non-rotating system. 2. Mixing Layers We shall first review and discuss the primary instabilities in the confined mixing layers 1 . The well known tangent-hyperbolic profile 2'40 u(y) is often used as the basic flow profile for a mixing layer generated by two parallel streams with different velocities ux and u2 and with an assumed arbitrary velocity ratio A = (ux - u2)/(ul +u2) for ul>u2. Here v is the cross stream (transverse) variable. Guided by previous inviscid instability results 40 , y = ±5 can be chosen to represent the upper and lower limits of v of the flow field (base flow plus the primary instability modes). As a computational modeling approach to the problem, spectral method, used earlier by Orszag for the channel flow8, can be applied along with the expansions in Chebyshev Polynomials 41 of the eigenfunctions for the Orr-Sommerfeld equation to be discussed below. This suggests that

152 u and its second derivative u" with respect to v which are needed in the governing equation for the primary instability modes to be discussed below can be expressed in terms of Chebyshev Polynomials Tm(y/5) as «(y) = l + i 5 „ r . ( y / 5 ) ,

(la)

m=0

«"(>) = i t t f r / s ) , M<|

(ib)

m=0

where am and a^ are constants for a given A. Here a prime (') represents derivative with respect to y. The stream function of the primary spatial instability mode is assumed of the form 0(y)exp[/(ccc - wtj\ in terms of a complex wave number a , phase velocity c = w/a, real frequency w, eigenfunction 0(y), streamwise variable x, pure imaginary number / = V-T and time variable t. Since Squire's theorem 42 holds only two-dimensional modes of instability are of interest here which are the most critical ones. Also only spatial instability modes are considered here due their dominant effects in the early development of free shear flows 1 ' 42 . Using the above description for the stream function of a primary instability mode in the Navier-Stokes equations and eliminating the pressure terms with the aid of the continuity equation for the incompressible fluid flow cases leads us to the so-called OrrSommerfeld equation which is given below 0(/v)(y) - 2 a 2 0 " (y) + a > ( y ) = iaR {[u(y) - c\t"(y) ~ a2(/>(y)] - u"(yMy)}

§

(2)

where R = AuO/n *s t n e Reynolds number, Au = ul-u2, 0 = A M / ( 4 M ^ ) (momentum thickness40) and JLL is the kinematic viscosity. The Eq. (2) together with (1) indicate that
(3a)

m=0

"(y/5)=fja^Tm(y/5l W

(y/5)=ia
(3b) (3c)

m=0

where certain formulas which relates the constant coefficients a{J} and a(4) to a (for a given X) are derived and given by Lie and Riahi1. Using these formulas, (2)-(3) and the identity41 TmTH=[TH+m + TH_m]/2, T_K = TKforK>0,

(4)

the truncated form of the Eq. (2) is derived1 which is used for the computational investigation. In such truncation the actual value of °o in (1) and (3) is adequately chosen as M = 37 and N = 38*, respectively. By analyzing the possible combination of Tn, Tn+m and Tn_m, the truncated form of (2) is simplified to the following form which includes a

153 new parameter n for 7\ and a constant coefficient H . (a 0 ,« p ..., a 38 ,/?,w,a,A) which represents the functional coefficient of T. l£HH.(a09al,...9ax,R,wfa,X)T.=0.

(5)

Since 7\ is the complete orthogonal basis of eigenspace, we can formulate a matrix operation through the system H. = 0 for each n which is equivalent to the product of each n-th row Hn(R,w,ayA) of a matrix #39*39 and a column vector of the unknown coefficient a /"/i N

= 0 orH ~ 39x39

xA

=0.

~ 39x1

(6)

% Due to nontrivial coefficient set a n ,« = 0,1,2,...,38, the criterion for the numerical computation of Eq. (6) is that the determinant of # 3 9 x 3 9 equals zero; * W

39X39=0.

(7)

We now take into account the effects of the boundary conditions for the eigenfunction of the primary instability modes in confined mixing layers which are 0 = 0' = O at v/5 = ± l .

(8)

Using the properties of the Chebyshev Polynomials

r„(±i)=(±i)\Tn'(±i) = (±irV,

(9 a,b)

the boundary conditions (8) can be written in the forms 38

38

2X=0. n=0 I«„H)" =0 38

J,n\=0,

I

(10 a,b)

38

5> 2 (-l) ( --\=0.

(10c,d)

n=0

n=0

Following the spectral tau method8, we replace the last four rows in H by these four boundary conditions. We then apply the criterion (7) to find the critical Reynolds number at the onset of the secondary motion due to the primary instability modes. This completes modeling of confined mixing layers based on the spectral tau method. We now consider modeling of confined mixing layers based on Galerkin's method. By extending the expression for even mode of eigenfunction8, certain formulas are constructed1 which satisfy the boundary conditions for both even and odd modes of eigenfunctions

154

{y/5)= Ianqn{y/5), y{yl5)=ia?qn{yl%

(11 a) (lib)

n=4

)

^ (3'/5)=£«i\(y/5),

(lie)

n=4

where / qAyl

/5)=

{ ^ / 5 ) - " ( f ) 2 7 i ( v / 5 ) + [(f) 2 -l]T 0 (y/5), »:4,6.8,.-. { r ^ / 5 ) - i ( n 2 - l ) r 3 ( v / 5 ) + 1 ( ^ - 9 ) ^ ( ^ / 5 ) , n: 5,7,9,...

(12)

By using (12) in (2), truncating the series expansion at n = 38 and with some manipulations, the governing equations for both even and odd modes are obtained1. In direct analogy to the spectral tau modeling, we obtain a similar criterion of zero determinant for a matrix H*(R,w,a,X) det|/f*l

= 0, n = 4,5,6,...,38.

(13)

It should be noted that through Galerkin's method we only have to deal with a 35 x 35 matrix which saves some CPU time for the computation. Due to the two extra terms in the formulas of eigenfunction qn, the disadvantage is extra coding needed to convert the equations for the modes into the matrix H*. The results based on Galerkin and spectral tau modeling for the critical R at the onset of secondary motion for confined mixing layers1 indicate very good agreement. Let us now consider unbounded layer cases. After investigations of several choices for the proper mathematical expression of unbounded boundaries, a best possible one, which, in particular, leads to the closest eigenvalue to the in viscid case40, was chosen1. Using (2) together with the asymptotic expressions for the basic flow40 and eigenfunction 0, we have (d2 Y d2 \ {dy \dy ) u = (1 + AS) + 2AS£(-l) n exp(-Srtv), y -> S<~,

(14 b)

n=l

= £/„exp[-5(n + a)y]+£g„exp[-5(n + *: 2 )4 n=0

(14c)

n=0

where S = ±1 as y -» ±00,
(15 a)

n=l

K2 = a2 + tf{a(l + SX) -w\

(15 b)

and fn andgn are constant coefficients. Using (14 b,c) and setting the coefficients of exp(-Say) and exp(-SAT2y) in the resulting equation zero, the formulas for fn and gn are

155 derived1. These formulas are lengthy and are given in Lie and Riahi1- Now make use of the boundary conditions as v -> ±°o where u" - 0 and exp(-Srtv) -» 0. Then Eq. (14 a) can be reduced approximately to

a2

=0

(16)

+K =

(17)

$- lw-^) A solution to Eq. (16) satisfies the equation

Ty+a\i

^ °'

which implies that + (a + K2)() = 0iity = ooi

(iga)

and " - a2 + (a + K2){-<j>' + a(/>) = 0 at y = -<».

(18 b)

Using (14 c) and the approximate expressions for fn and gln in (18) yield relation between f0 and gl0 for v/5 = ±1. Next, this relation is used in (14 c) and then take derivatives of (14 c) with respect to v. After some manipulation the results for the approximate forms for 0 and ' at v/5 = ±1 are obtained1 which can be used as the set of inhomogeneous boundary conditions appropriate for the free boundary conditions for the unbounded mixing layers. These lengthy results are reported in Lie and Riahi1. Following the procedures for the bounded mixing layer cases, we end up with the matrix equation for the unbounded mixing layer cases which describes the truncated expansions at N = 38 ^ 3 9 x 3 9 * ^ 9 x 1 = / o ^39x1'

(19)

Here only the last four entries in the array of B are nonzero which are due to the conditions for 0 and <j>' and y/5 = ±1. The numerical procedure for modeling cases described so far is presented in details elsewhere1. Here we briefly review only the main aspects of the numerical procedure. The coefficients am and a{2) introduced in (1) which affect the solution to (2) are computed by applying the particular Gaussian quadrature formula1 to (1). By reducing R gradually and applying to condition (7), one can locate the critical Reynolds number, Re, which is achieved for a particular ar = real (a), for given A, by checking that at = imaginary part of a equals zero. Searching process for Re can be achieved by using a Newton-Raphson scheme1. The results of the numerical computations1 indicate that the primary instabibty mode at the onset of the secondary motion holds the same character for both bounded and unbounded cases, although Re for unbounded case is much smaller than the corresponding one for the bounded case. The effect of boundaries is essentially to restrain the growth of the primary instability, so that Re becomes very small for the unbounded cases. In both bounded and unbounded cases, the wave number and frequency of the critical instability mode are very close and they held essentially the same character. It was found that the

156 primary viscous instability modes are more critical and unstable than the corresponding primary inviscid instability modes, and the most dangereous primary instability modes are intrinsically viscous in nature and are due to intermediate values of the frequency w. 3. Rotating Disk Flows We consider the problem of incompressible fluid over an infinite horizontal plane rotating about a vertical axis with angular velocity Q. It is convenient to use cylindrical coordinates r*,0,z\ with z = 0 being the plane of the disk. Let the over-bar variables P and U = («,v,vv) denote the primary steady base flow quantities for pressure and velocity in the radial (r*), azimuthal (0) and axial (z*) directions, respectively, in the rotating coordinate system. The well known Von-Karman's exact solution of the NavierStokes equations43 is obtained by setting u = r*QF(z% v = r*ClG(z), w = (MG)* #(*)» p = p/*lP(z),

(20a,d)

where z = z*(£l/jj) • Here ji is the kinematic viscosity and p is a constant reference density. The functions F , G , / / a n d P satisfy a well known system 43 whose solution is easily determined. To study the stability of such solution with respect to super imposed small disturbances, we first non-dimensionalize the governing Navier-Stokes equations by using (fi/tif* as the reference length, r*Q as the reference velocity, and P 2 .Q 2 as the reference pressure, where r* denotes the radial location near which the analysis is made28. The primary instability mode variables w,v,n\ and/? superimposed on the base flow variables (20) are used to satisfy the Navier-Stokes equations in cylindrical coordinates. The resulting equations for the primary instability mode variables, as functions of nondimensional space and time variables (r,0,z) and t can be simplified by applying the usual quasi-parallel flow approximation22. Here we are interested in the stability near a radial location such that the Reynolds number, R = r*(Q.//i) , is the minimum Reynolds number, Re, below which no stationary primary mode is amplified. The quasi-parallel approximation, then, amounts to substituting Re for r, since r* corresponds to r = R. Further simplification is obtained by neglecting terms of order R~5 and smaller. The final equations for the primary instability modes are lengthy28 and will not be given here. Next, the method of multiple scales is applied. It is assumed that R does not deviate much from Re and a slowly varying time variable ts is defined ^(fl-jg/*,,

(21a)

ts = e\

(21b)

where e is a small perturbation parameter and /^ is a constant. The sign and magnitude of #2 is to be selected so as to insure that e2 remains small and positive. It is then assumed that the solution to the governing system for the primary mode varies on two different time scales such that


dt

dts

(22)

157 Applying the weakly nonlinear theory 44-45 , we pose expansions in powers of e of the form F = e{EAFn) + e2(E2A2F22+\A\2F20) + £ 3 (£ 3 A 3 F 33 + |A| 2 EAF 3 1 ) + C.C,

(23)

where F is any primary mode variable, E = exp[/(ar + fiRcO - wf)], A = A(ts), \A\2 = AA where the over-bar indicates complex conjugate, and c.c. indicates the complex conjugate of the entire preceding expression. Next, use (23) in the governing system and collecting coefficients of powers of e leads to the systems of orders e\ e2 and £3 whose complete forms and solutions are discussed in details in Vonderwell and Riahi28. An equation for A is obtained by applying an existence condition28 for the solution of the order e3E system. The equation for A and its complex conjugate are then simplified to ^ - = M + c|A|2A,

(24 a)

-^- = bA + c\A\2A. dts

(24 b)

where b and c are constants whose complete expressions are given in Vonderwell and Riahi 28 . Multiplying Eq. (24 a) by A, Eq. (24 b) by A, and summing yields a Landautype equation42

J-|A|2=2/>,|A|2+2c,|A|\

(25)

where twice the real parts of b and c, 2b r , and 2cr, are recognized as the linear growth rate of \A\ , and the negative of the Landau constant42, respectively. A Chebyshev collocation method was used to solve the systems for the base flow (20) and for the primary mode in the orders £ and e2 as well as the adjoint system to the linear system for the primary mode whose solution were needed to determine the various results including the values for b and c. In each case the physical domain, 0 < z < z ^ , was mapped onto a computation domain, |7]| < 1, by 2 8 £ = 1.8(1 + ^/(1.18-77).

(26)

Adjoint and linear systems for the primary instability modes represent eigenvalue problems with identical eigenvalues. Both were solved by using (26) and about 181 Gauss-Lobatto points28. The minimum Reynolds number for neutral stability of stationary (w = 0) modes and the associated wave numbers were found to be 28 Rc= 275.36, a = 0.38315, 0 = 0.078232.

(27 a,b,c)

These results are in agreement with those obtained by use of the finite difference method22. The same collocation simulation and mapping (27) with 181 Gauss-Lobatto points in the

158

computational domain were used to determine the solutions in the order e2 for the primary instability mode systems. The constants b and c are determined by evaluating their expressions 28 using Simpson's rule on 180 intervals to approximate the integrations28. The constants are b = (2.8993 x 10"5 - 2.5127 x lO-5*)/^,

(28 a)

c = 0.09759 + 5.1792/.

(28 b)

Since the Landau constant, -2c r , is negative, it is known 42 that |A| increases superexponentially for /^ > 0 (R > Rc). For /^ > 0 the solution breaks down after a finite time 28 ' 42 and (29) \A\^>ooasts^>:±-ln\l+_ 2br \ crK, where Ao is some initial value of \A\. It is anticipated that in this case there is a fast transition to turbulence28'42. For ft, < 0(ft < Rc), subcritial instability occurs since / (30) \A\ -> oo as ts -» lnl ^ — I A2-4? 2b, for AQ > Ac, where Ac = (-br/cr) . It is suggested46 that in this case there should be a lower critical value /^ of ft below which the bifurcated primary instability solution does not exist so that values of \A\ begin to increase with R. It is expected that the base flow is globally asymptotically stable with respect to all primary instability modes for R< Rc. So far we have assumed a primary instability mode consisting of a monochromatic wave with constant amplitude for all rand 0. More realistic is the consideration of localized primary instability modes whose properties can be studied by using the wave packet modeling approach29-32. Riahi30 and Vonderwell and Riahi31 used the method of multiple scales29'47 to examine the linear primary rotating disk flow instability, in order to make a better evaluation of the details of evolution of wave packet disturbances and their behavior which develops in both space and time. They assumed/? > Rc but R does not deviate much from Rc and defined slowly varying quantities 6S = £0, ts = et, E2R2 = R-Rc,

(31 a,b,c)

where £ is a small perturbation quantity and ft, is a positive constant. It is assumed that the primary instability modes vary on two different length and time scales d

d +£

M-*TO M-/

d

d

d

*->*

+fi

d

,„

*

(32a b)

^

'

The primary instability mode variables are expanded in powers of e and are used in the linear instability system. To lowest order in e it is assumed that any primary mode variable fx is of the form f^A{es,ts)f,{z)E,

(33)

159 and the goal has been to determine the form of the amplitude function A. To order e2 the existence condition yield dA

_ dA

+ G

^

_

= 0

^

„.. (34)

-

where Rf-Gg is the group velocity in the azimuthal direction. This equation implies A = A{ds - Gets) and suggests that instead of (31a,b) one should use the slowly varying quantities T = £2f

£ = e(e-Get),

(35a,b)

and proceed as before. The solution varies on two different time and length scales d

d

d

+e

(36a)

Te^to %> d

d

2

d

^

d

n^****-***

(36b)

To orders e and e2 one obtains the same linear eigenvalue problem as before, and the solutions are found. To order e3 the existence condition yield the equation for A dA

d2A

JT*W

,-_

'

( }

where ax and b are constant30. If the initial disturbance is centered at 0 = 0, r = /^, the solution for small T will be concentrated near £ = 0. Using the method for separation of variables, the solution for (37) is found to be A = A0 exp(fcT)T_>* exp(-^ 2 /4a 1 r),

(38)

where \ is a constant and the condition that A —> 0 as % —> °° is used to derive (38). This condition as well as the numerical computation of a{30 indicated that the real part of Oj is indeed positive. It is found that disturbance spreads out as T increases and at the center it grows exponentially, after an initial decrease produced by r~^ factor, rate of growth being proportional to (R - Rc). Riahi and Vonderwell32 extended the above analysis to nonlinear and more general propagating motion of the primary wave packet modes. It is found appropriate to define the slowly varying quantities (35) and r) = e{r-Grt\

(39)

where £ is defined in (31 c) and G = (GrGe) is the group velocity vector in the non-axial direction. Apply the nonlinear theory29, we pose expansion of the form (23) where now A = A(£,77,T). Substituting such expressions into the primary instability mode system and collecting coefficients of powers e leads to systems to orders eJe2E1£2E2ye2E° and e3E which are needed to determine an equation for A. The first four systems together with the linear adjoint system are solved by using a Chebyshev collocation method28. The solutions to these systems are then used in the solvability condition for the system to order e3E to

160 determine the following partial differential equation for A: dA

a

d2A

d2A

„

d2A

'W+2a2^+aiW)

= bA + c\A?A,

(40)

where the coefficients a,. (i = 1,2,3), b and c are constants. Define

£,=(S + M)/2, H=(« + J,l?)/2,

(41a)

where ^ and Aj are the roots of the quadratic equation a3A2+2^ +^ = 0

(41b)

for X. Using (41), (40) is converted to the form d2A

dA a-

= M + c|A|2A, a^a}-al/a3.

(42 a,b)

Linear solution of (42) is of the form A = (A0/r)exp(bT)exp[-ri£J(aT)l

(43)

where AQ is a constant. The maximum of the real amplitude of the wave packet derived from (43) decreases initially and then amplifies. In the time that elapses before the wave packet begins to grow it will propagate in the radial direction a distance leading to a value of Rc found to be closer to the experimental value48 than that due to the linear stability theory based on the monochromatic wave disturbances28. Using (43), the conditions for (42) can be written as A = \r~l Qxp[-T]^J(ar)] as T->0

(44 a)

|A|->0 as |77,£|->~.

(44 b)

System (42) and (44) is then well posed. For the purpose of the present discussion, we consider the case where d/drj = 0. Then (40) or (42) becomes ^ - a

l

^ = bA + c\A?A,

(45)

where the coefficients take the values a{ = R^\ b = (2.899 - 2.513/)/^(l0- 5 ), c = 0.098 + 5.179/,

(46a,b,c)

and Rc = 275.36 28 . The equation (45) is of the general mathematical form studied by Hocking and Stewartson 49 for various values of the coefficients. They found that the solution of the equation of the type (45) becomes more and more concentrated at the center (^ = 0) of the wave packet, becoming infinite there in a finite T , provided R^ is not too small. Otherwise, no bursting type of behavior is exhibited by A. Studies based on a less realistic case of a monochromatic wave disturbance28 do not exhibit this later result. The

161 results of the bursting for order one values of /^ are suggestive of possible fast transition to turbulence due to primary instability modes. It should be noted that a number of features used in the present model are determined before experimentally48 such as the results that the stationary spiral vortices are the primary instability mechanism which propagate and grow as wave packets, and the wave patterns from each point source spreads out circumferentially downstream of the source. 4. Spectral Theory In this section we review a spectral theory developed first by Plaschko and Hussain 35 for two dimensional shear layers and more recently extended 33 ' 34 to threedimensional shear layers such as slowly divergent free shear layers and boundary layers. Although there have been a number of theoretical modelings for secondary instabilities in shear layers 50 ' 51 , the spectral theory 33-35 , to be reviewed here, is apparently the only credible theory which provides some significant insights into the nature of the primary instability modes in non-parallel shear layers, and the results are consistent with experimental observations5'36. Consider a slowly diverging base flow field (w0, v0, w0) = u0 (JC, v, z). Here (JC, y, z) are the space variables along the streamwise, transverse and spanwise directions, respectively. Designate the flow divergence parameter by e and define xs = ex and zs - ez as the slowly varying streamwise and spanwise coordinates, under the assumption that nonparallel effects are equally important along the x and z directions. Based on the quasi-parallel assumption22'42, u0 = (u.evQ, w0) where («o,v0,w0) *S a n OI"der one quantity in each component. Next, superimpose three-dimensional primary instability modes on the base flow. Subtracting the non-dimensional forms of the governing Navier-Stokes and continuity equations for the base flow from the corresponding ones for the total flow (base flow + primary instability modes) lead to the following stability system for the primary modes f — + W 0 - V - - V 2 ] W + W - V W 0 + V P = -M-VW,

(47 a)

V w = 0,

(47b)

w = 0 at y = yx,y2,

(47 c)

where u = (w,v,w) and P are the velocity vector and pressure for the primary instability modes, R is the appropriately defined Reynolds number of the flow, and yl and v2 are the values of v at the lower and upper boundaries of the flow, respectively. Now expand the primary mode dependent variables in powers of e with the assumption that nonlinear and nonparallel effects are equally important. L p ) = / M p ^ l + e 2 f« 2 ,P 2 j + ...

(48)

Using (48) and (47) and collecting terms of equal powers in £, yield systems to various orders in e. The order e system 3334 admits solution of the form

162

UiiXs.y.ZstW.P)]

r

Pi(xs>y>zsw>P) \ +pz - wt]}dwdp,

(49)

where w is the frequency, P is the span wise wave number and g(xs,zs>w,P) is a function to be determined later. The fast variable in (49) is defined to be g/e. The solution of the form (49) is given in general form to cover the following three special cases: (i) Twodimensional primary instability modes acting on two-dimensional base flow 35 . Here zs = 0, P = 0 and (49) reduces to a single integral over frequencies. An example for this case is two-dimensional primary instability of Blasius flow, (ii) Three-dimensional primary instability modes acting on three-dimensional z -independent base flow. Here zs = 0 in (49). An example for this case is three-dimensional boundary layer primary instabilities of flow over an infinite span wing, (MI) Three-dimensional primary instability modes acting on three-dimensional z-dependent base flow. Here P = 0 and (49) reduces to a single integral over frequencies. An example for this case is three-dimensional boundary layer primary instabilities of flow over a finite span wing. Using (49) in the e order system yield the system for \ul,Pl

. This system represents the three-dimensional

extension of the well known Orr-Sommerfeld equation governing a linear primary instability mode in a parallel flow if one defines —— and /?+— as the streamwise and dxs

dzs

spanwise wave numbers, respectively 33-35 . In order to determine the weakly varying amplitude function Al(xs,zs,w9P), given in (49), one need to consider the e2 order system and determines its solution 33>35. Also, one need to consider the adjoint eigenvalue problem of the linear system which is a system for the adjoint variables \ux, Px adjoint linear solution is of the form (49), provided

AjWpAjPi

33 35

" . The

is replaced by

M 1 (X 5 ,V,Z 5 ,W,J8)P 1 (^ 5 ,V,Z S ,W,^)J. Applying the solvability condition to the system for the y -dependence functions of the e2 order system33-35, yield the following integro partial differential equation for Al: k(xSJzSyWiP)—L + l(xs1zSyWip)-^ + m(xSizs1w,P)Al dxs dzs

=

[exp{-ig/£)]'\ZLH{xs>Zs^P>wi>Pi)A{xs^s^ ■txp[i[g(xs,zs,wiypl)

+ g(xs,zs,w-wl,p-p])]/e}dvv]dpi,

(50)

163 where the coefficients kj,m, and H depend on the solutions of the adjoint and e order systems, but they are independent of A, 33 " 35 . Equation (50) is a kind of generalized Landau equation where the nonlinear term in the right hand side represents nonlocal interactions between different wave modes over the frequency and spanwise wave numbers domain of primary instability waves. Equation (50) is too-complicated and no analytic solution appears to be possible, except for large x and z. However, this equation can be solved by numerical iterative process starting with the solution of the linear part of (50). Let us now consider the phase function g given in (50) and define § = a(*,z s ,w,j8), § = 7(*5,z,w\jB).

(51a,b)

as the local wave numbers along x and z axes, respectively. It then follows from (51) that, to the leading orders, a is a function of x only and y is a function of z only. Therefore -g{xs,zs,w,p)

= jXoa(x\zs,w,p)dx'

+

jj(xs,z\w,p)dz''

(52)

Now the oscillating term, which appear in the integral in the right hand side of (50), is of the form e x p | i j J a r ( ^ z 5 , w P A ) + a r (jc / ,z 5 ,w-w I ,i8-A)] dx^i^r^z^w^P^y^z^w-w^P-P^dz^

(53)

where ar and yr are the real parts of a and y, respectively. We use the convention from the parallel flow stability theory 2 ' 42 that ar and yr are positive for positive frequencies. Thus, for locations far downstream (JC,Z » 1), the integral in (53) takes large oscillations and we can use the method of stationary phase 52 to determine the dominant contribution of the nonlinear term in (50). The dominant contribution comes from frequencies w i = w'^ and wave numbers at which the phase in (53) is stationary 33-35 . Using this result in (50) yield k(x9z9w,P)^

+ l(x,z,w,P)^

+ m(x9z,w,p)Ai =

r(x,z,w,j3)A 2 (^z,f4),

(54)

where T is independent of y^ 33-35 . This equation can be solved easily by using the method of characteristics34-35• It is seen from (54) that, for given w and /?, the dominant nonlinear contribution comes from the subharmonic mode at the frequency w/2 and at the spanwise wave number P/2. This result is consistent with experimental data 5 ' 36 and agrees with the theory due to Plasckho and Hussain35. Previous nonlinear theories 35 ' 53 failed to predict this role of the subharmonic. Let us now pay attention to the fact that the experimental results36 also indicate dominance of other subharmonic modes at frequencies

164 w/n(n>2) for slightly larger values of R. The theory developed by Plaschko and Hussain35 is unable to predict the dominance of any subharmonic mode other than just one at w/2 and, indeed, solvability conditions at orders en(n > 2) of that theory may contradict each other. In particular, the solvability condition in the order e3 imposes further condition on Ay which may not be obeyed by (50). However, the reformulated and generalized theory due to Riahi 33-34 is capable of predicting the dominance of the subharmonics at frequencies w/n(n>2) and at spanwise wave numbers p/n(n>2) and there are no contradictions between the solvability conditions at different orders in e. As explained in Riahi 33 ~34, the main reason for the deficiency of the original theory35 has been the fact that only a particular solution for the e2 order system was considered and a large subset of frequencies contributing to the most general solution for the e2 order system was discarded in that theory Riahi33"34 considered the most general solution to the e2 order system with inclusion of all the frequencies and the spanwise wave numbers which contribute to such solution. It turns out that, due to contribution of all the frequencies and spanwise wave numbers, the solvability condition at each order en(n>2) leads to an equation for an amplitude function An_x which is related to Am(m2) order system, based on the reformulated and generalized theory 33-34 , leads to an equation for A^^. The nonlinear part of this equation contains an oscillating term of the form exp|ijJa r (jc / ,z 5 ,w 1 ,j3 1 ) + a r (jc / ,z 5 ,w 5 ,i3 2 ) + ...+ ar(^Z5.w w .pi9 - _0 + a r ( ^ z s , w - l V f j 8 - I A ) ] dxf + if0[yr(xs,z\wlypi)-hyr(xsiz\w2ip2)^... {xs,z\wn_vpn_l)^y(xs,z\w-n^wnp-n^pi

+ rr dz*}-

(55)

Using the method of stationary phase to determine the dominant contribution of the nonlinear term in the equation for ^4n_, and apply similar procedure to that used for ( 5 0 ) 3 3 " 3 4 , we find the phase is stationary at wl=w2=... = wn_l=w/n and Px = P2 = ••• = Pn-\ = P/n. Hence for given w and /?, the dominant nonlinear contribution in the equation for An_l comes from the subharmonic mode at the frequencies w/n and at the spanwise wave number P/n. 5.

Flows Over Swept Wings

Another important example of shear flow with practical applications is that of flow over swept wings. Such flows in aerodynamic applications are essentially within threedimensional boundary layers which are subjected to primary crossflow instability producing cross flow modes as well as primary viscous instability producing the TollmeinSchlichting (TS) modes. TS modes are the primary instability modes in two-dimensional boundary layers. For three-dimensional boundary layer flows over swept wings, crossflow modes (CF) dominate at the leading edge of the wing, while TS modes show larger growth rates in the mid-chord region. In between, there is a region where both CF

165 and TS modes are present. We shall first briefly review the asymptotic analysis of the linear primary instability modes of the three dimensional compressible boundary layer over a swept wing which has recently been done 37 and led to some understanding of the behavior and characteristics of such primary modes. The governing system for the linear primary instability modes of the present problem are given by several authors 38 ' 54-55 and will not be given here. Consider a coordinate system with x-axis along the normal chord, the z-axis along the span and the y-axis normal to the surface of the wing. The boundary layer on the wing is assumed slightly non-parallel, and the non-parallel effects do not enter the governing linear primary stability system to the leading terms. The non-dimensional form of the governing primary stability system contains mean flow variables as coefficients, four non-dimensional paramenters R, Pr, M and / , and the expressions for the viscosity ju and the conductivity K and for their first and second derivatives with respect to temperature which are given by Sutherland type laws 56 . Here R is the Reynolds number, Pr is the Prandtl number (defined based on the free stream quantities), M is the Mach number (defined based on the free stream quantities) and y is the ratio of specific heats. It is assumed that R is sufficiently large. Basically there are two sets of primary modes: inviscid modes which are useful to describe the upper branch disturbances and wall modes which are useful to describe the lower branch disturbances57. For the inviscid modes the appropriate small parameter £ is given by £ = R~y\

(56)

and any dependent variable F of the primary modes is of the form F = f{y)exV{(i/e3)[ja(x,e)dx

+ zP(e) - e'M^l

(57)

where T is time, a is the streamwise wave number, p is the spanwise wave number, w is the frequency and f(y) is the y-dependence part of F. We restrict our attention to neutral disturbances. The boundary layer for these modes is divided into an inviscid zone of thickness 0(e 3 ) and a thin viscous zone of thickness 0(e 4 ). In the inviscid zone, the governing equation are simplified. These equations have coefficients which depend on the mean flow quantitites. The equations are solved using spectral collocation method. The mean flow data used in the computation is based on the boundary layer flow on an infinite span 23° swept wing with suction, M = 0.82 and chord Reynolds number of 2 x 107. The boundary layer edge velocities are determined by integration of the surface Euler equations knowing the pressure coefficient. The mean flow solution is found using the compressible boundary layer finite difference code of Iyer 58 ' 37-38 . Since the long time scale modes are found to be preferred, the results are independent of w to the leading terms. Computational results for the eigenvalue problem of the inviscid zone led to the values for the wave number and the wave angle arctan (p/oc) at several nodes37. It turns out that it is possible to have more than one solution at a node. Each of such solution at a particular node may correspond to a particular w which can be confirmed only by some higher order analysis and calculation57. In the viscous zone, the equations are simplified and solved using matching condition with the inviscid zone solution at the upper boundary. Next, the wall modes are considered, where the appropriate small parameter £ is given by

166 £ = R~yi\

(58)

and any dependent variable F of these modes is of the form F = f{y)exp{(i/e4)[ja(x,e)dx

+ zP(e) - e4Tw^)]}-

(59)

These modes have a triple-deck23 structure. In the upper deck of thickness 0(e 4 J, the variables are properly scaled and both viscous and conduction effects are negligible. The governing equations are then simplified and solutions for the dependent variables are found 37 . In the main deck of thickness 0(£ 8 ), the variables are properly scaled and both viscous and conduction terms are negligible. The governing equations are then simplified, and solutions for the dependent variables which match those for the upper deck at the upper bound are found. An interesting result for the main deck is that alternative scalings for the density and temperature variables of the primary modes are possible which can lead, to leading order terms, to zero vertical velocity (vertical vorticity mode) or zero temperature (isothermal mode) or zero vertical velocity and temperature (vertical vorticity and isothermal mode). It turns out that the solution to all the components of the velocity vector do not match the no slip condition at the wall which requires consideration of a lower deck zone. In the lower deck of thickness 0(e 9 ), the variables are properly scaled and the governing equations are simplified and then solved. The requirement of no slip condition at the wall and the matching condition with the main deck solution at the upper boundary lead to the results for the wave number and the wave angle37. The complete incompressible and compressible Navier-Stokes, continuity, state and energy equations for the primary instability modes have also been solved numerically 38 ' 59-60 to determine the solutions for such modes in the three-dimensional boundary layers over an infinite span swept wing. Since such systems have coefficients which depend on the steady state base flow solution of the governing Navier-Stokes, continuity, state (for compressible case) and energy equations, such governing equations, subjected to the boundary condition for an adiabatic wing with weak suction 59 , were solved first for flow over a 23° swept infinite span wing 38 ' 60 . A finite difference code 58 , fourth order accurate in the v -direction (wall normal direction) and second order accurate in the x -direction, was used to determine the base flow solution for a flow with free-stream Mach number of 0.82 38 ' 60 . It should be noted that both baseflow and stability systems are derived in a systematic way by the following procedure 38 ' 60 . First, introduce the slowly varying variables xs = ex, zs = £z, ts = £?,

(60 a,b,c)

where the small parameter e characterizes the magnitude of the disturbance quantity and we assume 59 that it simultaneously characterizes the nonparallelism of the boundary layer. Next, pose as a solution to the basic governing system for total quantities (base flow + disturbance) the form F = F0{xs,y) + eFl(x,y9zJ9xSJzsts)

+ e2F2(xty9zXxS9zS9ts)9

(61)

167

where F is any dependent variable, F0 is the variable for the base flow and FY and F2 are the variables for the disturbance. For the total viscosity we assume the form

A£ = ,io v0(r") +... 0) + A ^7; +

(62) (62)

with a similar expansion for the total conductivity. Here T denotes temperature. Substituting these forms into the system for the total quantities, the base flow system is obtained to the zeroth order in e. To the first order in £ the linear stability system is obtained which is appropriately simplified by assuming the following form for F{: ^^x ==^n(^,y,z 5 ,r 5 )exp(/0i**), n ), Fi.(xs>y.Zs'h)*M

(63) (63)

where the phase function (/>n satisfies = a

x

z

x

W X [\j£Zs)< syPn{ n{s^ S>Zs)\ s\wn{xs,zs)\

(64) (64)

Here, an,pn, and wn are the streamwise wave number, spanwise wave number and frequency of the disturbance, respectively. The linear stability system for dependent variables like Fln is an eigenvalue problem. Once an and pn are selected, a non-trivial solution exists only for certain values of wn. The linear stability system is solved by the collocation method 38 ' 60 for both incompressible and compressible cases. Both the incompressible and compressible systems result in a linear stability eigenvalue problem which has a dispersion relation of the form (65) (65)

,Pn,w„,R) = 0. D(an,3„,w„,R) 0.

The real part of the frequency, wnr of the disturbance remains constant downstream, and for the special case of spatial instabilities with zero spanwise growth rate on an infinite span swept wing, pnr is fixed as well. The chordwise wave numbers, anr, will change as the boundary layer grows, however, as will the growth rate given by -ani. Once the values of wnr and pnr are specified, a search for the remaining eigenvalues must be undertaken at each node of interest. A Newton-Raphson search is used to find the correct value of In addition to the individual primary instability modes determined from the linear stability system described in the previous paragraph, instability modes that satisfy the triad resonance conditions can also be determined using the wave interaction modeling 38 ' 42 ' 60 . Using (61)-(62) in the system for the total quantities and retain only term of order e 2 lead to a system whose solvability condition lead to a system for the amplitudes an(xs,zsts), provided solution to the e order linear system is assumed to be in the form of

,=i^,,,ssW*)K(*,,«) ,=i^„, w*)K(*,,«)

(66)

in place of (63). To form a triad, it is required that (a^P3r,w3 (cc,r,P wlr) }r,wrir)) = (alr,plr lrWlr

+ (a2r,p2r,w2r)

+ e(<7 e(a a,<7 a,a^a p,a^, w),

(67) (67)

168 where the last term in the right hand side of (67) represents a small deviation from perfect resonance. The solvability condition for the e2 order system is a system of three coupled first order partial differential equations in xs,zs and ts for an(n = 1,2,3) with quadratic nonlinearities in an3S>60. In actual computation 38 ' 60 , the special case of an infinite span wing and spatial stability theory are considered, so that the solvability system reduces to a system of first order ordinary differential equations in xs for an(n = 1,2,3). To make the interaction coefficients of the nonlinear terms in the solvability system unique, the eigenfunctions were normalized by setting the eigenfunction magnitude of all three members (n = 1,2,3) equal to one, and then preserves the eigenfunctions magnitudes downstream. With initial values of an specified, the system for an is integrated using a standard fourth-order variable interval Runge-Kutta method 38 ' 60 . The results of the computations indicate that if the interaction coefficient for a mode is small, then the mode's amplitude can be boosted only if it is much smaller than the amplitude of the other two members of triad. The combinations of two TS modes and one CF mode always show a very strong crossflow interaction coefficient and very small TS interaction coefficients. Thus a small amplitude CF mode may be boosted after the TS modes have reached some finite amplitude. A comparison between the results for the incompressible and compressible stability system cases indicate that the popular and simpler incompressible modeling may not yield meaningful results and, thus, the compressibility effects must be taken into account in the stability system. 6.

Some Concluding Remarks

In this chapter we reviewed the structure and characteristics of the primary instability modes and their important effects on the evolution of the fluid flows in several basic shear flow systems. In particular, we emphasized the roles of the primary viscous and spatial instability modes in confined and unbounded mixing layers, the subcritical and bursting behavior of primary instability modes, in the form of either monochromatic waves of wave packets, in the rotating disk flows with subsequent possible fast transition to turbulence due to such modes, the dominance of primary instability subharmonic modes in non-parallel shear layers, triple-deck structure and long time scale behavior of primary instability modes in three-dimensional boundary layer flows over airfoils and large growth rates of primary instability crossflow modes due to triad interactions with primary TS modes in flows over swept wings. In the last several decades there have been significant progress in understanding the shear flow instabilities and their roles in important practical applications for flow control and for turbulence promotion. Most of the large amount of work in the shear flow instability area that have been reported by many authors in the literature so far, have been on the secondary instabilities61 in the boundary layers and free shear flows, and primary instabilities have been referred to as long as their relation to the secondary instability modeling and simulation were concerned. Despite this, the importance of primary instability modes by themselves cannot be under estimated. The present review chapter, which is devoted entirely to primary instabilities, demonstrates the significance and important roles of primary instabilities on the overall shear flow behavior and on the past, present and future evolution of shear flows. Primary instability modes are essential parts

169 of any shear flow instability mechanism and, indeed, secondary and higher instability modes cannot exist without the presence of the primary ones. Finally it should be noted that there are other important basic shear flow systems, such as those of wake flows and channel flows referred to only briefly in the section 1, or those of jets and pipe flows which were not even referred to in this chapter. However, such flow systems can be studied using the same types of mathematical and computational methods introduced in this chapter that were found to be so effective to determine the results for the primary instabilities. Also certain types of primary instabilities such as primary algebraic instabilities62 can be operative not only in channel flows but in any wall shear flows. In regard to wake flows, a separate article63 is devoted to the computational simulations of primary instabilities in planewake flows using techniques which can also be employed in other free shear flow systems.

170 References 1. K.H. Lie and D.N. Riahi, Int. J. Eng. Sci. 26 (1988) 163. 2. R. Betchov and W.O. Criminale, Stability of Parallel Flow (Academic Press, NY, 1966). 3. A. Michalke, 7. Fluid Mech. 23 (1965) 521. 4. P. Freymuth, J. Fluid Mech. 25 (1966) 683. 5. R. W. Miksad, J. Fluid Mech. 56 (1972) 695. 6. G. Brown and A. Roshko, /. Fluid Mech. 83 (1974) 641. 7. D. Oster and I. Wyganski, J. Fluid Mech. 123 (1983) 91. 8. S.A. Orszag, J. Fluid Mech. 50 (1971) 689. 9. D.N. Riahi, R.D. Moser and F. Waleffe, Proceed - Summer Research at Center for Turbulence Research (Stanford, CA) (1990) 205. 10. A.S. Burger, Laminar Wakes (American Elsevier Publishing Co., NY 1971). 11. T.H. Kuan and D.N. Riahi, TAM Report #703 (UIUC 1992) 1. 12. T.H. Kuan and D.N. Riahi, Develop, in TAM 17 (1994) 557. 13. T.H. Kuan and D.N. Riahi, Proceed. IMAC 14th World Cong, on Comput. and Appl. Math. (Atlanta, GA) I (1994) 275. 14. H. Sato and K. Kuriki, /. Fluid Mech. 11 (1961) 321. 15. G.E. Mattingly and W.O. Criminale, J. Fluid Mech. 51 (1972) 233. 16. E. Meiburg and J.C. Lasheras, /. Fluid Mech. 190 (1988) 1. 17. D.T. Papageorgiou and F.T. Smith, Proc. R. Soc. Lond. A 419 (1988) 1. 18. D.T. Papageorgiou and F.T. Smith, /. Fluid Mech. 208 (1989) 67. 19. S.P. Wilkinson and M.R. Malik, AIAA paper no. 83-1760 (1983). 20. M.R. Malik, S.P. Wilkinson and S.A. Orszag, AIAA J. 19 (1981) 1131. 21. L. Mack, AIAA paper no. 85-0490 (1985). 22. M.R. Malik, J. Fluid Mech. 164 (1986) 275. 23. P. Hall, Proc. R. Soc. Lond. A. 406 (1986) 93. 24. M.R. Malik, AIAA paper no. 86-1129 (1986). 25. W.S. Saric and L.G. Yeates, AIAA paper no. 85-0493 (1985). 26. N. Itoh, In Laminar-Turbulent Transition, ed. V.R. Kozlov (1984) 463. 27. S. MacKerrell, Proc. R. Soc. Lond. A. 413 (1987) 497. 28. M.P. Vonderwell and D.N. Riahi, Int. J. Eng. Sci. 31 (1993) 549. 29. K. Stewartson and J.T. Stuart, J. Fluid Mech. 48 (1971) 525. 30. D.N. Riahi, Instability and Transition, Vol. II, eds. M.Y. Hussaini and R.G. Voight (1990) 160. 31. M.P. Vonderwell and D.N. Riahi, Bull. APS. 35 (1990) 2322. 32. D.N. Riahi and M.P. Vonderwell, Proceed. Pan Am. Cong. Appl. Mech. (Brazil) (1993) 331. 33. D.N. Riahi, Proceed. IMAC 14th World Cong. Comput. and Appl. Math. (Atlanta, GA) I (1994) 421. 34. D.N. Riahi, Submitted for publication (1995). 35. P. Plaschko and A.K.M.F. Hussain, Phys., Fluids A. 27 (1984) 1603. 36. Z.D. Husain and A.K.M.F. Hussain, AIAA J. 21 (1982) 1512. 37. D.N. Riahi and M.P. Vonderwell, Proceed, of NASA Langley Workshop on Transition, Turbulence and Combustion (Hampton, VA) I (1994) 235.

171 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63.

M.P. Vonderwell and D.N. Riahi, submitted for publication (1995). D.N. Riahi, submitted for publication (1995). P.A. Monkewitz and P. Huerre, Phys. Fluid 25 (1982) 1137. L. Fox and LB. Parker, Chebyshev Polynomials in Numerical Analysis (Oxford Univ. Press, Oxford, U.K., 1968), 205. P.G. Drazin and W.H. Reid, Hydrodynamic Stability (Cambridge Univ. Press, Cambridge, U.K., 1980) 527. H. Schlichting, Boundary Layer Theory (McGraw-Hill Book Co., 1979) p. 817. J.T. Stuart, J. Fluid Mech. 4 (1958) 1. J.T. Stuart, J. Fluid Mech. 9 (1960) 353. L.D. Landau and E.M. Lifschitz, Fluid Mechanics, 2nd edition (Pergamon Press, NY, 1987), 539. A. Davey, L.M. Hocking and K. Stewartson, J. Fluid Mech. 63 (1974) 529. S.P. Wilkinson and M.R. Malik, AIAA J. 23 (1985) 588. L.M. Hocking and K. Stewartson, Proc. R. Soc. Lond. A. 326 (1972) 289. T. Herbert, AIAA paper no. 85-0489 (1984). T.M. Fischer and U. Dallman, AIAA paper no. 84-0009 (1984). G.F. Carrier, M. Krook and C.E. Pearson, Functions of a Complex Variable (McGraw-Hill Book Co., NY, 1966), 438. D.J. Benney and S.A. Maslowe, Stud. Appl. Math. 54 (1975) 181. N.M. El-Hady, AIAA paper no. 80-1374 (1980). L.M. Mack, AGARD report no. 709 (1984). F.M. White, Viscous Fluid Flow, 2nd Edition (McGraw-Hill Book Co., NY, 1991), 614. D.N. Riahi, In preparation (1995). V. Iyer, NASA contractor report 4269 (1990). N.M. El-Hady, Phys. Fluids A 1 (1989) 549. M.P. Vonderwell, Ph.D. Dissertation, UIUC, Illinois (1994). T. Herbert, Ann. Rev. Fluid Mech. 20 (1988) 487. M.T. Landahl, J. Fluid Mech. 98 (1980) 243. T.H. Kuan and D.N. Riahi, in preparation (1995).

173

GORTLER VORTICES WITH SYSTEM ROTATION Abdelfattah Zebib Rutgers University Department of Mechanical & Aerospace Engineering Piscataway, NJ 08855-0909, USA and Alessandro Bottaro Ecole Polytechnique Federate de Lausanne IMHEF-DME CH-1015 Lausanne, Switzerland and Barbro G.B. Klingmann Volvo Aerospace Corporation Space Propulsion Division S-461 81 Trollhattan, Sweden

ABSTRACT Spatial development of longitudinal vortices is determined by a linear, streamwise marching, stability approach as well as by a nonlinear, finite-volumes computational method. The basic flow is a Blasius boundary layer on a rotating concave surface, with the rotation vector parallel to the span. It is shown that linear marching results agree well with nonlinear spatially developing solutions up to fairly large disturbance amplitudes. It is also shown that while rotation influences the growth of the "mushroom" structures (rotation in the sense of the basic flow enhances the development of the vortices while counter-rotation dampens the vortices), the nonlinear saturation energy for a given spanwise wavelength is not dependent on rotation. Wavelength selection based on maximum linear amplification is investigated within a range of rotation numbers including zero, and is found to be rather weak, particularly in the case of counter-rotation. Under random disturbance inflow conditions, perturbation energies decrease initially in a linear filtering phase (which does not depend on rotation, but is a function of the inlet noise distribution) until coherent patches of vorticity near the wall emerge and can be amplified by the instability mechanism. The average wavenumber found is close to that predicted by the linear criterion.

174 1.

Introduction

Spatial development of the boundary layer flow over a concave surface subject to system rotation, with axis of rotation perpendicular to the plane of the base motion, is the subject of this paper. In addition to applications in rotating machinery, interest in vortices developing along curved boundary layers stems from their similarities with coherent structures in the near-wall region of turbulent shear flows1. Experiments 2 have provided a basis for the computational studies of, among others 3 ' 4 . While computations have faithfully reproduced laboratory results when experimentally established initial and boundary conditions are used, only theory 3 ' 5 has provided some criteria for the wavelength selection. It has been argued that a wavelength in the neighborhood of the curve of maximum linear amplification has a good chance of being selectively amplified. This has been confirmed in experiments where the upstream flow perturbations are randomly distributed along the span, but it is at variance with other experimental evidence that indicates that different wavelengths are selected in different experiments 2 ' 6 " 8 . In fact, the available literature for the nonrotating Gortler flow provides average values of the selected span wise wavenumbers that differ by up to a factor of 10. This fact indicates two things: the downstream vortex development is strongly influenced by the inlet flow and disturbance field (typical of a convective instability), and the wavelength selection mechanism is weak. In the present paper we report results from numerically solving the steady, nonlinear equations of motion to assess the influence of rotation on the spatial development of the flow along a concave wall, and the wavelength selection mechanism. These equations are parabolic, and are thus solved by straightforward marching 3 . In addition, the linearized version of these equations is solved by a finite difference-spectral marching procedure. Linear and nonlinear results, including those from local nonparallel linear stability 9 , for different rotation numbers complement and confirm each other convincingly; both codes are then used to address issues of wavelength selection.

2.

Mathematical Formulation

We consider the rotating boundary layer flow over the concave wall with constant radius of curvature R subjected to system rotation Q. with the axis of rotation coinciding with the axis of curvature (see Figure 1). Following9, we take / to be a typical length along the wall and define the curvature parameter e, the Reynolds number Re and the Gortler number G by: e = /fl-i,

Re = ^ - , v

G2 = £Re"2,

(1)

175 where Uoo is a free stream velocity and v is the kinematic viscosity. The theory is developed for 8 -» 0, Re -» «> such that G = 0(1). In a cylindrical coordinate system (r, 9, 0 with corresponding velocity components (vr, vG, v^) we introduce the dimensionless boundary layer coordinates x, y, and z: y=-^^Rel/2,

z = ^Rel/2.

(2)

and the dimensionless velocities u, v, w; and pressure p:

•■-£■

Figure 1.

v = - u ^Re'/2 >

w = y^Re I / 2 ,

P=

pU 2 >

Re.

(3)

Sketch of the problem.

where p is the density and p' is the dimensional pressure. When these variables are introduced into the Navier-Stokes equations, written in the rotating frame of reference of Figure 1, we find the leading system: du 5x

3v 5y

3w _ ~dz~ '

(4)

176

d

r

r [

U

^ ^

+ V

^ ^

d2u 3 2 u

d,

d

+ W

3, 3i]V

-,~ r> + G2u2 =

d d d, r [U^-+V^-+W^-JW L J dx

dy

dz

3D 3 2 V 3 2 V - _ ^ry - ^ + ^2 + ^2-2R0G2u'

3p 3 2 w 3 2 w = - V : + -r-=r + -=-7T. dz 3 y 2 9Z2

/CN

(6)

,_. >

v(7)

The boundary conditions associated with the parabolic system (4) - (7) are: u=v=w=0

aty = 0,

(8a)

uy=v =w=0

asy-»oo.

(8b)

where the infinite field was truncated at y e = 50. It also was checked that the adoption of conditions at y e such as 3 : u = 1, Vy = 0, w = 0, produced the same results as the boundary conditions (8b). Most computations are performed for one-half wavelength (7t/oc, where a is the wavenumber at x = 1), and along z the symmetry conditions u z = v z = w = 0,

for z = 0,7t/oc,

(8c)

are adopted. The numerical solutions of the nonlinear parabolic system is accomplished by a finite volume procedure 10 . Briefly, the y-z computational domain is divided into rectangular cells with the grid points located at the geometric centers of these small cells. Additional boundary points are included to incorporate the boundary conditions. The discretized equations are obtained by integrating the conservation equations over control volumes having these cells as a base and extending a distance Ax in the streamwise direction. Local linear y and z dependence in any of the primitive variables is assumed, resulting in a second order accurate scheme. Staggered location for the y-z velocity components is adopted to avoid unrealistic pressure fields and associated numerical instabilities. Streamwise marching is accomplished through a fully implicit first order forward Euler scheme. For the parameter values of this work, we have determined that a 71x25 y-z grid (stretched in y and uniform in z) and Ax = 0.04 are adequate for computations covering one-half wavelength. Numerous computations were also performed on a finer grid (121x50 and Ax = 0.01); each one of these latter calculations required of the order of 15 CPU hours on the Cray-2 supercomputer at EPFL. We also consider the linear spatial development of a perturbed Blasius profile:

177 u = (U(x,y) + ui(x,y,z), V(x,y) + vi(x,y,z), wi(x,y,z))

(9)

where (U, V) is the Blasius solution, which remain unchanged in the limit e -* 0. The linear stability equations are obtained by linearizing equations (4) - (7), and then reducing the system to a two-equation set in which the pressure and the spanwise disturbance velocity component are eliminated. The resulting equations for the perturbation velocities ui and vi are equivalent to those given by Hall 11 , with an additional rotation term. They are solved by a downstream marching procedure employing a second order accurate finite-difference scheme in x and Chebyshev collocation in y12> 13 > 14 . A local nonparallel analysis is also performed 9 ' 14 . It provides local growth rates, a, and eigenfunctions which can be used as initial conditions for both linear and nonlinear spatial developments.

3.

Linear and Nonlinear Development of the Vortices

It has been shown experimentally2 that the initial spatial development of Gortler vortices is steady. This development depends on a large number of variables, such as the shape and the streamwise position / of the initial disturbances. Other parameters at play fix the values of Re, G and Ro. Whereas the locally scaled wavenumber a increases as x 1 / 2 and the local G as x 3/4 , the dimensionless wavelength parameter A, defined as

A = ^ ' V ^ = G(-) 1 - 5 v

V"

(10)

a

remains constant at all streamwise locations (unless merging and/or splitting of vortices takes place and the physical wavelength A,' of the vortices is modified). Here we consider the influence rotation would have on previously reported experiments 2 and computations 3 . The detailed measurements 2 were made at Uoo = 500 cm/s, R = 320 cm, v = 0.15 cm 2 /s, for a vortex pair with a wavelength A/ of 1.8 cm, giving A = 450. The simulations are started at / = 40 cm, so that at x = 1, G = 6.756, a = 0.3824, and an initial disturbance (ui, vi, wi) composed of the local linear eigenfunctions for Ro = 0 with u i m a x = 0.042 is used 3 to correspond to the experimental data 2 . Figure 2 illustrates the "mushroom" growth corresponding to Ro between -0.2 and 0.2. Here contour plots of constant u are shown at 5 axial locations. The results obtained at Ro = 0 are in excellent agreement with previously published results 3 . The stabilizing (destabilizing) influence of negative (positive) rotation which is predicted by the local linear analysis in 9 is verified by the nonlinear calculations. The penetration of low momentum fluid into the free stream, indicated by the 0.9 u-level, is seen to increase (decrease) according to Ro > 0 (< 0). Further evidence of this influence can be seen from the "peak" and "valley" wall shear stress variation with x in Figure 3. At the mushroom

178

Figure 2.

Development of vortices in x corresponding to different values of Ro and for a disturbance wavenumber a = 0.3824. At x = 1, G = 6.756 and a perturbation proportional to the local linear eigenfunction at Ro = 0 normalized according to ui = 0.042 is assumed. Deeper penetration of low momentum fluid into the free stream at larger values of Ro is observed. The y-z grid used is 121x50, with Ax = 0.01; the range of y in the plot is [0,20] and the plots are scaled correctly. Spacing between neighboring isolines is 0.1.

179 stem (peak) positive rotation first decreases the wall gradient, with a subsequent increase which is more rapid the larger the rotation number. In the valley, the spatial development is also faster at large positive Ro, and the maximum value attained is higher. Far downstream, the shear stresses at the peak and valley locations reach almost constant values which depend weakly on Ro.

Figure 3.

Variation of wall shear stress with x for the same conditions of Figure 1. The destabilizing influence of rotation can be inferred from the decrease (increase) in shear stress at peak (valley) locations.

The growth of the perturbation energy with x as function of Ro (all other parameters fixed as stated above) is compared in Figure 4 to linear marching results. The nonlinear perturbation energy is defined as: a ^e E = 5 f fui 2 dydz,

(ID

180

Figure 4.

Growth of the perturbation kinetic energy E as defined in Eqns. (11) and (12) corresponding to nonlinear and linear marching predictions, respectively.

and the curves of linear growth shown in Figure 4 are computed according to X

E = E 0 e 2 fadx,

(12)

where Eo is the kinetic energy of the perturbation at x = 1. It can be seen that the linear marching results shown in Figure 4 faithfully reproduce the nonlinearly computed growth rates until quite large values of E (local linear results9 slightly overestimate the growth of the Gortler vortices14). Far downstream, nonlinear saturation leads to a departure from the linearly predicted energy growth. An interesting observation to be made from Figure 4 is that, despite the large influence of rotation on the initial development, the vortices have the same perturbation energy regardless of Ro, once the nonlinear saturation stage is reached.

181 4.

Wavelength Selection

According to the local linear theory 9 , the most amplified wavelength A lies in the neighborhood of 260 (160) for Ro = -0.3 (0.3). In the following parametric study, the linear marching code was used to obtain solutions for a large number of A's and Ro's.

Figure 5.

Amplification rates a for different A at different constant values of G.

Each run was initiated at G = 1 with the solution of the local stability problem. It was noted that varying the starting point of the downstream marching did not significantly affect the results. Figure 5 shows how the amplification rates vary with A and Ro, at fixed values of G. In all cases, the dependence of a on A (wavelength selection) is weak at low G where the linear amplification starts, giving similar amplification rates within a wide range of A's between 100 and 2000. As G increases, the region of intensively

182 amplified A's becomes slightly narrower, particularly for positive Ro. Some nonlinear calculations did indeed show that vortices with widely varying A had practically equal amplification rates, and more so for Ro < 0 than for Ro > 0. With increasing Ro the most amplified wavelengths shift towards lower values. This is seen clearly in Figure 6, where the most unstable A (i.e. the A where a is locally maximum at a given downstream position, G), is plotted as a function of Ro. When Ro < 0 the range of the most unstable A's varies quite broadly with G, whereas the variation is contained for Ro > 0; for example, when Ro = 0.5, the most amplified A is in a narrow range centered on A = 160, whereas for Ro = 0, it varies between A = 185 and 240 as G increases from 2 to 15. This fact could be interpreted as an indication that the wavelength selection mechanism is weaker at negative Ro.

Figure 6.

4.1

Variation of the most amplified wavelength with Ro, at different values of G. The crosses correspond to results of section 4.2, for natural wavelength selection, when the vortices begin to appear.

Nonlinear Development

It is by now clear that a system subject to controlled perturbations can amplify vortices of almost any desired wavelength within a large range. On the other side, a randomly distributed field of incoming disturbances can be expected to develop into vortices with the linearly most amplified wavelength. This can be verified by considering a long (in z) test section and applying at the entrance of the computational domain a white noise perturbation field. This is the strategy adopted in the following nonlinear calculations. These runs start from G = 6.756 with a computational box of dimensionless spanwise length equal to 89.7. A 72x152 mesh was found adequate for these simulations (a finer

183 mesh was also employed to verify the accuracy of the results). We performed two calculations both starting from the same global disturbance amplitude, but with different random noise distribution with Ro = 0.3. The perturbation energies in Figure 7 differ between the two cases, except clearly for x = 1, and also the saturation levels reached are slightly different. The latter is an effect of the different filtering selection actuated by the equations in the linear regime when E decreases, and of the different average wavelengths achieved in the two cases in the nonlinear stages. Note, however, that the streamwise extent of the initial energy-decreasing phase is the same (from x = 1 to about x = 1.5) for the two different random noise distributions. The streamwise velocity fields are shown in Figure 8 at five different x values; the spatial evolution produced are qualitatively similar. Most of the vortices are bent towards a side evidencing a disparity in streamwise vorticity levels on either side of the vortex pair.

Figure 7.

Perturbation energy versus x, for two different inlet noise distributions but same initial amplitude levels; Ro = 0.3.

Such asymmetries are due to interactions originating from an Eckhaus instability4. The main difference between the two cases lies in the average wavelength produced; the flow field on the right side of Figure 8 (corresponding to the dotted curve in Figure n) presents consistently one less upwash region than the case on the left. Simple hand counting of the visible upflows in the cross-sections indicates that there are 11, 10 and 9 pairs for x equal to, respectively, 2.4, 2.8 and 3.2 (flow field on the right). Even this count is approximate, since some vortices do not have "companion".

184

Figure 8.

Contours of the streamwise velocity fields for the two cases of Figure 7 (the left side corresponds to the dotted curve in Figure 7). The plots are scaled correctly.

Starting with the same white noise but with different amplitudes, the energy of the disturbance decreases (Figure 9a); this occurs over an extent which is independent of the initial amplitudes.

Figure 9.

a. b.

Perturbation energy versus x, Ro = 0.3. Different initial amplitude levels. Collapse of the curves when a "virtual origin" is introduced.

185 The perturbation amplitudes chosen on the different velocity components are 1%, 4.2%, 10% and 33.3%, corresponding to curves 1, 2, 3 and 4, respectively. Thus, the boundary layer equations act as a linear filter between x = 1 and xo = 1.52 (vertical line in the figure) for whatever initial perturbation amplitude. Quasi-exponential growth follows xo according to A_ = e2a( V x 0 ),

(13)

where A is the amplitude of the energy perturbation, a is the linear growth rate, and xj > xo, i = 1, 2, 3, 4, is the streamwise location where the level A is achieved for any initial level Aoi. Thus, we find, using the fact that Aoi = 0.014, A02 = 0.233, A03 = 1.329 and A04 = 16.130 at the common minimum point of the three curves, xo = 1.52, the relations: xi -X3= 1.61 (xi -X2) xi - X4 - 2.50 (xi - x 2 )

(14)

This scenario is confirmed by sliding curve 2 horizontally a distance L until it coincides, for x large enough, with curve 1. When curve 3 is likewise shifted a distance 1.61 L to the right and curve 4 a distance 2.5 L, we find the nice collapse indicated in Figure 9b. The overlap of the four curves for some x range till the nonlinear saturation is a powerful result which indicates conclusively that a linear receptivity operation is actuated by this flow when subject to steady, random perturbations.

4.2

Effect of Rotation

The effect of rotation on this selection process is investigated from numerical experiments starting from the same random noise distribution and level as case 1 above, three cases have been compared: Ro = - 0.3, 0, and 0.3. In Figure 10 we have plotted isolines of the streamwise velocity for these three cases at different downstream positions x. As already shown, the vortices start growing at random locations in z, and vortices with different individual wavelengths and amplitudes coexist at each x. The average wavelength at the x-position where the vortices first appear can be obtained by simply counting the number of outflow regions; this gives A = 125 (for Ro = 0.3), 181 (Ro = 0) and 250 (Ro = -0.3). Because of the irregularity of the vortices, the value of the average wavelength is approximate to = ±10%. It is also clear (Figure 7) that different inlet noises would produce slightly different results. The average "natural" wavelengths obtained in this way are in agreement, although slightly lower, with those predicted by linear stability (shown as crosses in Figure 6). This confirms that, on the average, the linearly most amplified wavelength is selected when the flow is subjected to random inlet perturbations.

186

A notable difference between the three cases computed is the fact that A changes significantly for the positive rotation case (Figure 10c), in the range of x considered, because of merging of vortices.

Figure 10.

a. Isolines of the streamwise velocity for Ro = -0.3 at, from the top, x = 1.6, 2, 2.4, 2.8, 3.2. The isolines are spaced 0.1 apart. Note that, for comparison purposes, a small horizontal arrow has been drawn in correspondence to the edge of the undisturbed Blasius boundary layer, i.e. i\ = 5, at each x station. b. (next page) Ro = 0, same x values as Figure 10 a. c. (next page) Ro = 0.3, same x values as Figure 10 a. A tendency towards further merging events, for x > 3.2, manifests itself.

For Ro = 0.3, there are 13 vortex pairs at x = 2 (average A is 125), 12 at x = 2.4 (A = 138), 11 at x = 2.8 (A = 157) and 10 at x = 3.2 (A = 181). To a lesser degree this also occurs for Ro = 0, where only one merging event occurs for 2.2 < x < 2.4, whereas the average wavelength remains constant for the negative value of Ro. This points out the fact that the case Ro = 0.3 is more susceptible to an Eckhaus instability than Ro = -0.3; hence, in the nonlinear regime for Ro > 0 an Eckhaus criterion should be used to interpret the selected wavelength.

187

188 The vortex merging at positive Ro can be clearly observed both in the contours of u in Figure 10c and in the streamwise vorticity field of Figure 11. A clear example is the merging of two vortex pair near the center of the cross-section at x = 2.4 into one pair at x = 2.8; a similar phenomenon is also produced between x = 2.8 and 3.2. Both events are marked by vertical arrows in Figure 11 for easy identification. Typically, the merging involves two vortex pairs, where one pair is much stronger than its neighbor. The stronger pair "jumps" above the weaker one with a spanwise shift of half a wavelength. In so doing, at first one cell of the weaker pair is annihilated and secondly two co-rotating cells (originally belonging to two different pairs), placed one above the other, merge into one. The end result is one strong pair (central pair at x = 3.2) with the upwash region somewhat inclined, and a tilting of the neighboring vortices.

Figure 11.

Streamwise vorticity field in the cross-section at the same x values as Figure 10, Ro = 0.3. Isolines spacing are 0.03 (for x = 1.6), 0.6 (x = 2), 1.2 (x = 2.4) and 2 (x = 2.8, 3.2); zero lines are not drawn. The horizontal arrows mark the edge of the undisturbed Blasius boundary layer, i.e. r| = 5.

189 5.

Concluding remarks

In this paper, results for nonlinear spatially developing Gortler vortices subject to system rotation have been presented. Comparisons with recent nonparallel linear theories, using both local and nonlocal solution procedures, have also been shown. It is shown how positive rotation enhances the development of the vortices while negative rotation retards their spatial development. We have performed numerical experiments to study the natural formation of steady Gortler vortices in a long (along the span) box, subject to random inlet perturbation fields. Initially, the flow fields go through a linear receptivity phase which selects the modes to be amplified independently of the initial perturbation amplitude level. However, different inflow random noise distributions yield different vortical structures downstream, but approximately same average wavelengths. This indicates that the initial filtering phase is similar for different initial conditions. The natural average wavelengths obtained are very close to those predicted by the linear analysis. As the vortices develop into the strongly nonlinear stage, the wavelength increases due to merging of the most narrowly spaced vortices, a process related to the Eckhaus instability. This is seen most clearly at positive Ro, for which the nonlinear development of the vortices is enhanced. Reference 14 gives more details and results about this problem.

6.

Acknowledgements

We thank Professor I. L. Ryhming for initiating this collaborative effort. A.Z. has been supported during his stay at EPFL by an ERCOFTAC Visitor Grant. A.B. acknowledges the Swiss National Fund, grant no. 21-36035.92, for travel support associated with this research. This work was also supported by the Swedish Board of Technical Development (NUTEK), the Swedish Technical-Scientific Council (TFR) and an ERCOFTAC Visitor Grant, through which the stay of B.G.B.K. at the EPFL was made possible. Cray-2 computing time for this research was generously provided by the Service Informatique CentraleofEPFL.

7.

References

1.

R.F. Blackwelder, Analogies between transitional and turbulent boundary layers. Phys. Fluids 26, 2807, (1983).

2.

J.D. Swearingen and R.F. Blackwelder, The growth and breakdown of streamwise vortices in the presence of a wall. /. Fluid Mech. 182, 255, (1987).

3.

K. Lee and J.T.C. Liu, On the growth of mushroomlike structures in nonlinear spatially developing Gortler vortex flow. Phys. Fluids A 4, 95, (1992).

190 4.

Y. Guo and W.H. Finlay, Wavenumber selection and irregularity of spatially developing nonlinear Dean and Gortler vortices. J. Fluid Mech. 264, 1, (1994).

5.

J.M. Floryan and W.S. Saric, Wavelength selection and the growth of Gortler vortices. AIAAJ. 22, 1529, (1984).

6.

I. Tani, Production of longitudinal vortices in the boundary layer along a curved wall. J. Geophysical Research 67, 3075, (1962).

7.

H. Bippes and H. Gortler, Dreidimensionale Storungen in der Grenzschicht an einer konkaven Wand. Acfa Mechanica 14, 251, (1972).

8.

H. Bippes, Experimental study of the laminar-turbulent transition of a concave wall in a parallel flow. NASA TM. 75243, (1978).

9.

A. Zebib and A. Bottaro, Gortler vortices with system rotation: Linear theory. Phys. Fluids A 5, 1206, (1993).

10.

S.V. Patankar and D.B. Spalding, A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer 15, 1787,(1972).

11.

P. Hall, The linear development of Gortler vortices in growing boundary layers. J. Fluid Mech. 130,41,(1983).

12.

F.P. Bertolotti, Linear and nonlinear stability of boundary layers with streamwise varying properties. Ph.D. Thesis, The Ohio State University, (1991).

13.

B.G.B. Klingmann, Linear parabolic stability equations applied to threedimensional perturbations in a boundary layer. Rep. T-93-12. IMHEF-DME, Swiss Federal Institute of Technology, Lausanne, (1993).

14.

A. Bottaro, B.G.B. Klingmann and A. Zebib, Gortler Vortices with system rotation. Submitted to Theor. & Comp. Fluid Dyn. (1995).