containing a stably stratified brine solution, diverse flow regimes have been observed ... the buoyancy ratio have been revealed to possess multiple steady flows.
The multiplicity of steady flows in confined with lateral heating N. Tsitverblit
double-diffusive
convection
and E. Kit
Faculty of Mechanical Engin’eering, Tel-Aviv University, Ramat-Aviv, Israel
(Received 17 June 1992; accepted 21 December 1992) In a number of studies concerning the lateral heating of a vertical rectangular enclosure containing a stably stratified brine solution, diverse flow regimes have been observed for different values of the buoyancy ratio. In this Brief Communication, certain intervals of the buoyancy ratio have been revealed to possess multiple steady flows. The presence of the multiple solutions was demonstrated to be underlain by complex bifurcation phenomena including numerous limit points and the symmetry-breaking bifurcation. The results appear to be insightful for the understanding of various findings of previous works.
Thorpe et al. ’ were the first to inquire about the instability as a reason for layer formation in a stably stratified solution with horizontal temperature gradient applied. A more comprehensive linear stability analysis was carried out by Hart,’ who obtained the full marginal stability diagram illustrating the exchange in the mechanisms of shear and double-diffusive instabilities. In his later work, Hart3 extended his preceding study to consider the nonlinear behavior of disturbances and revealed the possibility of a subcritical finite-amplitude instability. In subsequent experimental and numerical studies (see Chen et al.,4 Wirtz et a1.,5 Wirtz and Reddy,6 Lee et al,,’ Lee and Hyun,8’9 and references therein), both the temporal evolution of the systems of layers and the developed states have been established to depend essentially on the buoyancy ratio. In particular, “successive” and “simultaneous” modes of layer formation were distinguished, flows possessing different numbers of cells with and without a region of almost motionless fluid in the middle of an enclosure were observed. It is worth being mentioned that the horizontal boundaries employed in some of the works (i.e., Refs. 1, 3-6, and 8) were actually impermeable to solutes rather than those having a constant difference in their solute concentrations (Refs. 7 and 9). Consequently, the developed systems of layers observed in such works were not, rigorously speaking, steady states. However, the large time scales of diffusion of the solutes used there justify quasisteady interpretations usually given to the results. In the present work, we have examined numerically the steady solutions for the range of relatively small Rayleigh numbers. The purpose of this Brief Communication is to report about the existence of multiple steady solutions (i.e., distinct solutions being present for the same numbers of the buoyancy ratio) engendered by the steady bifurcation phenomenon. We also present evidence for the existence of the symmetry-breaking bifurcation. Thus the bifurcation phenomenon is demonstrated to play a fundamental role in the formation of the flows being qualitatively akin to the ones previously observed for various buoyancy ratios. The aspect ratio y=H/d (His the height of an enclosure, d its width) was set to y= 3, the same as that used by Lee et al. ,’ and the salt concentration Rayleigh number 1062
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was specified to be Ras=gB&(&e/az>/kfl=30 000; g being the gravitational acceleration, fi the coefficient of solutal expansion, (&c/dz> the imposed solute concentration gradient, k, the diffusivity of salt, and Y the kinematic viscosity. The Prandtl number is Pr=v/kr= 6.7; the Here k, is the diffiSchmidt number is Sc=v/ks=677. sivity of heat. The Boussinesq approximation of the steady two-dimensional Navier-Stokes equation in the conventional vorticity-streamfunction formulation, together with the energy and salinity diffusion equations have been discretized by central finite differences to obtain a finitedimensional system of equations. The boundary conditions incorporated into these equations were as follows. The horizontal walls were assumed to have constant concentrations of salt and to be adiabatic. At the vertical walls, a temperature difference was specified at each step, and these boundaries were taken to be insulating to salt. All the walls were assumed no-slip. The Euler-Newton continuation method (see, for example, Cliffe,*’ Winters,” and references therein) was applied to trace out the solution of the finite-dimensional system of equations as the thermal Rayleigh number Rar=gaATd3/k,Y is varied. Here a is the coefficient of thermal expansion, AT is the temperature difference between the vertical walls. During this procedure, the presence of limit points was detected by noticing the failure of convergence of the Newton method. Limit points were rounded using the Kelleri arclength continuation algorithm. Part of the bifurcation diagram schematically illustrating the results obtained at this stage is presented in Fig. 1. Here 4(x) stands for a functional representing the variation of a measure of the symmetric component of the solution vector, and so two asymmetric branches, being the result of symmetry-breaking bifurcation point Bl, are shown as one Al. For the clarity of the figure, 4(x> was intentionally depicted to increase any time after a limit point is passed irrespective of the real behavior of the functional chosen. When limit points Ll,... ,LlO are successively passed by, two end cells with a stagnant region between them, two cells occupying the entire slot, four cells, three cells, and ultimately one well-mixed cell are observed. These flows are presented in Fig. 2. A remarkable feature of all the flows illustrated in this figure is that convective motion
0699-8213/93/041062-03$06.00
@I 1993 American Institute of Physics
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200
--.--L----L300 RaT
400
500
*
FIG. 1. A schematic diagram of the variation of a measure of the symmetric component 4(x) of solution vector x with thermal Rayleigh number Rar. Sl,...,Sll (the solid lines) are symmetric branches; Al (the dotted line) is the asymmetric branch; Ll,...,LIO-limit points; O-approximate places of the Jacobian sign alterations.
almost does not affect the temperature field which remains very close to conductive due to the low intensity of convection at the small Rayleigh numbers. The changes in the salinity field, however, are a lot more tangible owing to the much smaller diffusivity of salt. It is interesting that such a distinction in the sensitivities to convective motion can be considered as an illustration of the utterly different diffusivities of temperature and salt underlying the doublediffusive phenomenon. As the thermal Rayleigh number Rar is increased slightly from zero, small cells at the top and bottom, the existence of which is due to the presence of the horizontal walls, become evident. These cells are surrounded by the streamlines contiguous to the boundaries. The motion delineated by these streamlines is very weak, which is indicated by the absence of almost any distortions both in the isotherms and in the lines of constant salinity; the latter
(4
(b)
(‘4
(4
(e)
(9
(h)
0)
FIG. 2. Steady solutions. Y: streamlines; S: the lines of constant salinity; T: isotherms. Left wall is at higher temperature. All the contours are equally spaced between their maximum and minimum values. (a) Rar=20, branch Sl; (b) Ra,=195, branch S2; (c) Rar=360, branch S3; (d) Rar=368, branch S5; (e) Rar=480, branch S6; (f) Rar=287, branch S8; (g) Rar=4025, branch Sll; (h) Rar= 183, branch Al-; (i) Rar= 150, branch Al + . 1063
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clearly reflect a vertical gradient imposed [Fig. 2(a)]. The cells start growing after limit point Ll is rounded. Over branch S2 [Fig. 2(b)] and up to the vicinity of limit point L3 at branch S3 [Fig. 2(c)], however, a region of stagnant fluid between them is still present. Branch S4 is characterized by the two cells, of relatively high intensity, cuddling together near the center of the enclosure. Thus the space between each of the end walls and the central region, where the cells are located, is cleared out. This space is gradually taken up by two additional cells progressively arising at the end walls after limit point L4 is passed. Four welldeveloped cells form for larger Ra, along branch S5 [Fig. 2(d)]. They are slowly weakened as limit point L5 is approached. Immediately after this point has been passed, the four-cell solution collapses into a one-cell flow [Fig. 2(e)]. This flow persists only over a very short interval of the thermal Rayleigh number, below which the formation of a three-cell flow is observed. These three cells are consequently modified [an example of the three-cell flow is given in Fig. 2(f)] when continued through branches S6, S7, S8, S9, and SlO and become quite weak in the vicinity of limit point LlO. They finally collide into one cell when branch Sl 1 is reached. Along this branch, the cell becomes more and more intensive as RaT is increased. For the high Rayleigh numbers, where the steady solution is already unique, the convective motion associated with this cell affects not only the lines of constant salinity, which are already seen to be nearly homogeneously distributed throughout the whole cavity [Fig. 2(g)], but also the isotherms, demonstrating rather strong undulations compared to other flows described above. Thus when the buoyancy ratio is either large enough or sufficiently small, the solution is unique. In the region of intermediate values of the buoyancy ratio, however, two distinct possibilities are realizable-the solutions having cells with and without a region of stagnant flow between them. The latter flows can be of two-, four-, or three-cell types. It is interesting that such a combination of events is qualitatively similar to that observed by Lee et ali’ though the Rayleigh numbers used by these authors are much larger. However, the application of continuation methods enabled us to uncover several intervals of Rar within which multiple solutions exist. For example, when 252 < Ra, < 259 and 350 < Rar < 364, at least seven steady solutions are present. Moreover, it can be noticed that below the value of Ra, associated with limit point L3, the solutions containing a part of a cavity at rest can still be found while above this value, only the flows with the convective cells occupying the entire enclosure are present. Allowing for the distinct ranges of the Rayleigh numbers, this situation is reminiscent of the “successive” and “simultaneous” mechanisms of layer formation observed in Refs. 4,5,7, and 8 and some other works, below and above the critical value of an appropriately defined Rayleigh number, respectively. Another interesting result worth emphasizing is that the onset of the first instability is essentially due to end effects. It is caused by the breaking of the central symmetry between the end cells at symmetry-breaking bifurcation Brief Communications
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point Bl. Examples of one of the asymmetric flows arising from B 1 are illustrated in Figs. 2 (h) and 2 (i) . It is worthwhile to note that large time solutions, which can be regarded only as quasi-steady states when the horizontal walls are assumed impermeable, persistently preserved asymmetric traits also in the time-dependent computations of Heinrich,13 when the temperature difference was applied asymmetrically with respect to the initial state. Asymmetric features can also be noticed in one of the flows, being exactly steady, presented by Lee et al.’ The sign of the Jacobian determinant was monitored at any step of the continuation procedure and, in addition to Bl, determinant sign alterations were noticed several times inside certain intervals between successive limit points. We anticipate these alterations in the Jacobian sign to be also associated with symmetry-breaking bifurcation points. The negative Jacobian sign is indicative of the presence of a negative eigenvalue in the set of eigenvalues of the system of the linearized governing equations. Therefore a solution is necessarily unstable inside such an interval. The full study of the structure of steady solutions, including the calculation of the asymmetric solutions, the examination of stability of all the solutions described above, as well as the circumstantial analysis of the nature of the instabilities, is currently in process. Tanny and Tsinober’4*‘5 studied the dynamics of double-diffusive +yers in a wide tank heated from the vertical wall. They found the dependence of the thickness of layers on the Rayleigh number, defined similarly to that of Chen et a1.,4 to be irreproducible in spite of almost identical conditions in a particular set of their experiments. The authors called such a behavior “chaotic.” On the other hand, the critical values of their temperature and salinity Rayleigh numbers were compared, also using a kind of the quasi-steady approach, with those of Hart’ and turned out to be in surprisingly good agreement. In view of this, it seems reasonable to anticipate that the abundance of the steady solutions in the vertical enclosure problem be somehow reflected in the quasi-steady configurations considered by Tanny and Tsinober, and consequently, “chaotic behavior” have its virtual origin in the multiplicity of steady solutions uncovered in this Brief Communication. Several tests have been made to ascertain the credence of the numerical code used. When applied to the problem of the BCnard convection in a box, good agreement with the results of Cliffe and Winters16 was established. The curvilinear version of the code was successfully tested against the results of Cliffe,” for the tlow in the Taylor experiment, and those of Winters,” regarding the curvilinear channel flow. In addition, when the salinity Rayleigh numbers were small enough, no instability was observed up to the values of Ra, being very large. The value of Rar for the first bifurcation in the double-diffusive region was found to increase as Ras was increased. This is qualitatively similar to the results of Hart.” Quantitative discrep-
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ancies are easily explained by the presence of the end walls. The results presented here were verified on several grid sizes: 21x21, 33~33, and 53x53, for the formulation where the aspect ratio was present in the equations as an explicit parameter, and 21 X 65, 33 X 101, and 41 X 125, for that having the aspect ratio being expressed in terms of grid sizes. All these grids gave qualitatively identical and quantitatively very close results as to the steady solutions the locations of limit points, and the alterations of the Jacobian signs. All the calculations were based on the MA32 sparse matrix code elaborated by I. Duff at Computer Science and Systems Division at AERE (Atomic Energy Research Establishment ) , Harwell. ACKNOWLEDGMENTS
The authors are grateful to Professor A. Tsinober for his very useful comments. This research was supported by the Gordon Institute for Energy Studies. ‘S. A. Thorpe, P. K. Hutt, and R. Soulsby, “The effect of horizontal gradients on thertiohaline convection,” J. Fluid Mech. 38, 375 (1969). 2J. E. Hart, “On sideways diffusive instability,” J. Fluid Mech. 49, 279 (1971). ‘J. E. Hart, “Finite amplitude sideways diffusive convection,” J. Fluid Mech. 59, 47 (1973). 4C. F. Chen, D. G. Briggs, and R. A. W&z, “Stability of thermal convection in a salinity gradient due to lateral heating,” Int. J. Heat Mass Transfer 14, 57 (1971). ‘R. A. Wiitz, D. G. Briggs, and C. F. Chen, “Physical and numerical experiments in layered convection in density-stratified fluid,” Geophys. Fluid Dyn. 3, 265 (1972). 6R. A. Wirtz and C. S. Reddy, “Experiments in convective layer formation and merging in a differentially heated slot,” J. Fluid Mech. 91,451 (1979). 7J. Lee, M. T. Hyun, and Y. S. Kang, “Confined natural convection due to lateral heating in a stably stratified solution,” Int. J. Heat Mass Transfer 33, 869 ( 1990). ‘5. W. Lee and J. M. Hyun, “Tie-dependent double diffusion in a stably stratified fluid under lateral heating,” Int. J. Heat Mass Transfer 34, 2409 (1991). 9J. W. Lee and J. M. Hyun, “Double-diffusive convection in a cavity under a vertical solutal gradient and a horizontal temperature gradient,” Int. J. Heat Mass Transfer 34, 2423 (1991). ‘OK. A. Cliie, “Numerical calculations of two-cell and single-cell Taylor flows,” J. Fluid Mech. 135, 219 (1983). “K. H. Winters, “A bifurcation study of laminar flow in a curved tube of rectangular cross-section,” J. Fluid Mech. 180. 343 (1987). “H. B. Keller, “Numerical solution of bifurcation and nonlinear eigenvalue problems,” in Applications of Bifurcation Theoty, edited by P. H. Rabinowitz (Academic, New York, 1977). 13J. C Heinrich, “A finite element model for double diffusive convection,” Int. J. Num. Methods Eng. 20, 447 (1984). 14J. Taimy and A. B. Tsinober, “The dynamics and structure of doublediffusive layers in sidewall-heating experiments,” J. Fluid Mech. 196, 135 (1988). “J. Tanny and A. B. Tsinober, “On the behavior of a system of doublediffusive layers during its evolution,” Phys. Fluids A 1, 606 (1989). 16K. A. Cliffe and K. H. Winters, “The use of symmetry in bifurcation calculations and its application to the Btnard problem,” J. Comput. Phys. 67, 310 (1986).
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