The Attractors of the Rayleigh-Bénard Flowof ararefied Gas

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1 The Attractors of the Raleigh-Bénard Flowof ararefied Gas S.Stefanov, V. Roussinov and C. Cercignani Inst. of Mech., Bulg. Acad. of Sciences, Sofia, Bulgaria Dipart. di Matematica, Polit. di Milano, Ital Abstract. We have studied the Raleigh-Bénard flow of a rarefied gas for Kn [. 3,4 2 ], Fr [.,.5 3 ] and a fixed temperature ratio T c /T h =.. The calculations are performed b both the DSMC and the numerical solution of Navier-Stokes equations, with a remarkable agreement between the methods. We exhibit chaotic behavior and also a hsteresis ccle. INTRODUCTION The formation and development of convection flows in a fluid confined between two horiontal parallel plates with the bottom plate heated from below is a classical problem known in hdrodnamics as the Raleigh-Bénard problem. The first numerical calculations of the RB convection of a rarefied gas performed either b the DSMC method ([, 2]) or b solving the BGK equation [3] show that the transition from pure conduction to convection occurs for temperature gradients larger than a certain critical value and for sufficientl low Knudsen numbers. A variet of studies [2, 3, 4, 5, 6, 7] report about a stable vortex formation for Knudsen numbers Kn (= l /L) =..5 and various magnitudes of temperature ratio and Froude number Fr (= Vth 2/gL). Here g is the acceleration of gravit, V th the most probable molecular speed, L the distance between the hot (with temperature T h ) and cold (with temperature T c ) plates. These investigations can be separated into two groups according on the aim pursued b the authors. In the first group of articles [, 4, 5] the authors tr to eliminate the densit gradient b choosing the acceleration of gravit g consistent with constant densit in the convection-free (pure conduction) solution. Thus, the flow is assumed to be ver close to the Boussinesq approximation conditions and, consequentl, the Raleigh number Ra (in this case, the unique non-dimensional parameter determining the transition from pure conduction to a convection flow) can be used and compared with the critical value Ra c (for no-slip boundar conditions Ra c = 78) obtained b linear stabilit analsis of the Oberbek-Boussinesq equation. In this approach, however, the choice of the governing parameters is restricted. The authors of the second group of articles[2, 3, 6, 8] consider the problem for a set of freel varing independent parameters and investigate the effect of gas stratification (a natural formulation for a rarefied gas). Thus, the densit of the purel conduction state might increase when moving toward the cold plate in the case of weak gravit, increase when moving toward the hot plate for strong gravit or, as shown further in the paper, be non-monotonic for some intermediate values. All these papers show that for such conditions the onset of instabilit cannot be determined b a single non-dimensional parameter Ra. In the present research we investigated numericall the RB flow for a set of the non-dimensional parameters in the intervals Kn [. 3,4 2 ], Fr [.,.5 3 ]. For most of the computations the third nondimensional parameter, the temperature ratio was fixed to T c /T h =., corresponding to a large temperature difference ( T h serves as reference temperature), for which the RB sstem is believed to reach most of the possible final states (attractors).

2 MC(rolls) FD(rolls) chaotic wav Ra m =78 min =. min =. Kn.2.5. pure conductive state convective states pure conductive state Fr FIGURE. The one of convection computed b DSMC (circles) and FD (other markers) methods within a range of Knudsen and Froude numbers and a fixed temperature ratio r =.. The bold solid and dashed lines are analtical estimates bounding the one. NUMERICAL RESULTS The domain in the parameter space defined above covers the Bénard instabilit one down to Kn = 3. The fact that the onset of instabilit occurs for small Knudsen numbers suggests that the RB instabilit in a rarefied gas can be used to investigate the agreement between Navier-Stokes and Boltmann equations for small Knudsen number flows. This was first checked on simple unstead one-dimensional gas flows in our previous paper [9] and the results were ver encouraging. In the present research we extend this approach to a more complicated case of stud, namel, the onset of transition from a pure conduction state to a convection state in the two-dimensional Raleigh-Bénard sstem of a hard sphere gas. We also treat the possible long term states of developed convection. For that purpose we use two principall different numerical approaches: particle simulation b the DSMC method [] and Navier-Stokes finite difference (FD) calculations [9]. A comparative analsis of the results obtained b both methods is carried out for the domain of governing parameters indicated above. We first compare the numerical results obtained b DSMC method and finite difference (FD) computations in a twodimensional domain with aspect ratio 2 and delineate the one of the Raleigh-Bénard convection within the domain of Knudsen and Froude numbers as pointed out above and for a fixed temperature ratio r = T c /T h =.. The computed one of instabilit is presented in Fig. b a gra shaded area. The neutral curve separating the outer area of final pure conduction from the inner one of final convection is determined approximatel b points bounding the domain of convection (i.e. these are the last points where a convection is detected): the Monte Carlo and FD data are marked b circles and squares respectivel. As can be seen the onset of the Bénard instabilit with regard to the Knudsen numbers is about Kn =.29 for the Monte Carlo simulations and Kn =.28 for the finite difference computations; this can be accepted as a good enough coincidence between the two approaches. The agreement between the solutions, not unexpectedl, improves when the Knudsen number decreases. Down to Kn =. we found that in all computed cases the final states of convection were stable two-roll configurations if the initial state of the gas is in thermal equilibrium with the hot wall and gravit is neglected. For Kn =.5 and small Fr a four-roll configuration was also observed (Fig. 2). Our calculations confirmed the results reported b Sone et. al. [6], and Golshtein, Elperin [8] about the presence of co-existing attractors of the RB flow of a rarefied gas. Moreover, tring to locate the line separating convection from pure conduction more accuratel we have found that

3 FIGURE 2. The final four-roll state computed b DSMC (a) and FD (b) methods for Kn =.5, Fr =.95 and r = FD MC.2 A. max w(,) D.2 B C Fr FIGURE 3. Kn =.2. The hsteresis loop computed b DSMC (circles) and FD (triangles) methods for small Fr numbers and a fixed there exist hsteresis loops for fixed Kn when b increasing and reversel decreasing the Froude number Fr the state is varied within a certain interval (Fig. 3). Thus the separation (neutral) line that we have determined could be considered as a rough lower bound enveloping the right (large Fr) side of the one of convection. Analticall, an upper bound can be determined b investigating the one of stabilit of pure conduction on the basis of the continuum model. A

4 6 cold wall 5 Fr= 4 Fr=5x Q 3 2 Fr= 2 Fr= 3 hot wall Fr=.5x 3 perturbations FIGURE 4. The heat flux behavior for a set of large Froude numbers. The Knudsen number and the temperature ratio are fixed (Kn =. and r =.). t further analsis shows that the condition delineates a line which follows ver close the left (small Fr) boundar of the computed one of instabilit. So, both left and right conditions can serve to determine of a rough exterior bound which envelopes the one of instabilit from the side of small and large Froude numbers respectivel. We then analse the results obtained for the lowest Knudsen number considered (Kn =.) where a set of qualitativel new regimes and final states have been found. Due to the extremel intensive DSMC computations needed to reach the long time behavior, the results for the molecular model have been obtained for one of the most interesting cases onl (Fr =.). The finite difference computations for the continuum model are performed for a variet of Froude numbers starting from a large value Fr =.5 3 where the attractor is a pure-conduction state. B decreasing the Froude number the final state of the sstem goes from a stable two-roll configuration to a tworoll configuration with periodic and quasi-periodic regimes of oscillations, then degenerates to a permanent chaotic behaviour for Fr =.9., then stabilies as a stable six-roll configuration for Fr =.8 and, finall, becomes again pure-conduction for Fr <.8. The comparison between the two models performed for Fr =. showed that both methods exhibit a secondar instabilit with a final flow regime of a permanentl chaotic vortex formation (a similar result has been obtained b Bird [] for the Talor-Couette clindrical flow b using the DSMC method). The time evolution of the non-dimensional heat fluxes at the walls computed for the continuum model b the FD method is presented in Figs. 4 and 5 for a set of Froude numbers. Figure 4 shows the heat flux behavior for large Froude numbers ( Fr 5) for which the slope of the densit profile of a pure conduction regime would be positive. Figure 5 shows the heat flux profiles for the cases (. Fr 2.) with a non-monotonic or negative (Fr =.) slope of the densit profile. The orbit of the established state in the reduced phase plane (Q =,Q = ) (Fig.8), when compared with the orbits in the cases Fr =.5 (Fig. 6) and Fr =.2 (Fig. 7), gives an impression of the chaotic character of the flow attractor. In the theor of nonlinear dnamic sstems, the long time solutions of this kind are known under the name of

5 6 5 cold wall 4 Fr=2. 3 Fr=.5 Q 2 Fr=. hot wall 2 2 FIGURE 5. The heat flux behavior for a set of small Froude numbers. The Knudsen number and the temperature ratio are fixed (Kn =. and r =.). t strange attractors. The existence of strange attractors, obtained b two methods having rather different bases, opens a new path for the investigation of the transition to turbulence at the edge between molecular and continuum descriptions. The last numerical result we present refers to the influence of a larger aspect ratio A = 6. on the chaotic convection for the basic case (Kn =., Fr =., r =.). The computations have been carried out for the continuum model b using FD method on a 2 2 grid. The run starts from a randoml disturbed pure conduction state. A series of snapshots presented in Figs. 9 show some tpical evolutions of the velocit vector field. A later excitement of the instabilit occurs due to the initial condition of pure conduction; this helps avoiding the appearance of a wave travelling between the walls due to gravit as occurs in the basic case. The larger aspect ratio A = 6. helps keep longer (t > 5.) the initial smmetr with respect to the midline = L /2 ( Fig. 9, 3rd snapshot), in comparison with the case with a smaller aspect ratio. In the end, the flow looses unavoidabl the smmetr and degenerates to chaotic spatiotemporal structures that are observed in the case with aspect ratio A = 2.. CONCLUDING REMARKS In spite of the numerous results of Navier-Stokes calculations for various incompressible viscous flows showing chaotic behavior for long times, it turns out that the question is frequentl raised whether the chaotic solutions of the Navier-Stokes equations produce a turbulent behavior adequate to realit. In our opinion, the qualitative and quantitative similarit of the results presented in this paper, obtained b using both molecular and continuum calculations gives an important argument for a positive answer: both approaches exhibit the basic properties of the transition to a chaotic fluid motion. In particular, we want to stress the fact that a model with binar collisions is able to reproduce macroscopic instabilit patterns quite accuratel.

6 Q = t> t= Q = FIGURE 6. The final orbit traced b the convection RB sstem in the plane (Q =,Q = ) for Fr = <t< Q = Q = FIGURE 7. The attractor s orbit in the phase plane (Q =,Q = ) for Fr =.2. Acknowledgments Two of the authors, S.S. and V.R., would like to acknowledge the financial support provided b the Bulgarian Ministr of Education and Sciences with Grant No. MM86/98. The research of C.C. was performed in the frame of European TMR (contract n. ERBFMRXCT97O57) and was also partiall supported b MURST of Ital.

7 .78 5<t< Q = Q = FIGURE 8. The orbit traced b the RB sstem in the phase plane (Q =,Q = ) for the case (Kn =.,Fr =.,r = ). REFERENCES. A. Garcia, in Microscopic simulation of Complex Flows, M. Mareschal, ed., New York: Plenum, 99, p S. Stefanov and C. Cercignani, Euro. J. of Mechanics B/Fluids,, 543 (992). 3. H. Sugimoto, Y. Sone, K. Aoki and H. Motohashi, in Rarefied Gas Dnamics 9, J. Harve and G. Lord, eds., Oxford: Oxford Universit Press, 995, Vol. I, T. Watanabe, H. Kaburaki and M. Yokokawa, Phs. Rev. E, 49, 46 (994). 5. C. Robinson and J. Harve, in Rarefied Gas Dnamics Smposium 2, Ching Shen, ed., Beijing: Peking Universit Press, 997, pp Y. Sone, K. Aoki and H. Sugimoto, Phs. Fluids, 9, 3898 (997). 7. J. Struckmeier, SIAM J. Sci. Comput. 2, 435 (2). 8. E. Golshtein and T. Elperin, J. Thermophsics and Heat Transfer,, 25 (996). 9. S. Stefanov, P. Gospodinov and C. Cercignani, Phs. Fluids,, 289 (998).. G. Bird, Molecular Gas Dnamics and the Direct Simulation of Gas Flows, Oxford: Clarendon Press, G. Bird, in: Rarefied Gas Dnamics Smposium 2, Ching Shen, ed., Beijing: Peking Universit Press, 997, pp

8 t= t= t= t= t= FIGURE 9. The five pictures show some tpical evolutions of the velocit vector field for an aspect ratio A = 6.