The Attractors of the Rayleigh-Bénard Flow of a Rarefied
Gas
S.Stefanov∗ , V. Roussinov∗ and C. Cercignani†
∗
Inst. of Mech., Bulg. Acad. of Sciences, Sofia, Bulgaria
†
Dipart. di Matematica, Polit. di Milano, Italy
Abstract. We have studied the Rayleigh-Bénard flow of a rarefied gas for Kn ∈ [1.0×10−3 , 4×10−2 ] , Fr ∈ [1.0×10−1 , 1.5×
103 ] and a fixed temperature ratio Tc /Th = 0.1. The calculations are performed by 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
hysteresis cycle.
INTRODUCTION
The formation and development of convection flows in a fluid confined between two horizontal parallel plates with the
bottom plate heated from below is a classical problem known in hydrodynamics as the Rayleigh-Bénard problem.
The first numerical calculations of the RB convection of a rarefied gas performed either by the DSMC method
([1, 2]) or by 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 sufficiently low Knudsen numbers . A variety
of studies [2, 3, 4, 5, 6, 7] report about a stable vortex formation for Knudsen numbers Kn (= `0 /L) = 0.01 − 0.05 and
various magnitudes of temperature ratio and Froude number Fr (= Vth2 /gL). Here g is the acceleration of gravity, Vth
the most probable molecular speed, L the distance between the hot (with temperature Th ) and cold (with temperature
Tc ) plates. These investigations can be separated into two groups according on the aim pursued by the authors. In the
first group of articles [1, 4, 5] the authors try to eliminate the density gradient by choosing the acceleration of gravity
g consistent with constant density in the convection-free (pure conduction) solution. Thus, the flow is assumed to be
very close to the Boussinesq approximation conditions and, consequently, the Rayleigh 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 Rac (for no-slip boundary conditions Rac = 1708) obtained by linear stability
analysis 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 freely varying
independent parameters and investigate the effect of gas stratification (a natural formulation for a rarefied gas). Thus,
the density of the purely conduction state might increase when moving toward the cold plate in the case of weak gravity,
increase when moving toward the hot plate for strong gravity 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 instability cannot be determined
by a single non-dimensional parameter Ra.
In the present research we investigated numerically the RB flow for a set of the non-dimensional parameters in
the intervals Kn ∈ [1.0 × 10−3 , 4 × 10−2 ] , Fr ∈ [1.0 × 10−1 , 1.5 × 103 ]. For most of the computations the third nondimensional parameter, the temperature ratio was fixed to Tc /Th = 0.1, corresponding to a large temperature difference
( Th serves as reference temperature), for which the RB system is believed to reach most of the possible final states
(attractors).
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
194
0.035
MC(rolls)
FD(rolls)
chaotic
wavy
Ra =1708
m
z =1.0
min
z =0.0
0.03
min
0.015
0.01
convective states
Kn
0.02
pure−conductive state
0.025
pure−conductive state
0.005
0
−1
10
0
10
1
2
10
10
3
10
4
10
Fr
FIGURE 1. The zone of convection computed by DSMC (circles) and FD (other markers) methods within a range of Knudsen
and Froude numbers and a fixed temperature ratio r = 0.1. The bold solid and dashed lines are analytical estimates bounding the
zone.
NUMERICAL RESULTS
The domain in the parameter space defined above covers the Bénard instability zone down to Kn = 10−3 . The fact that
the onset of instability occurs for small Knudsen numbers suggests that the RB instability in a rarefied gas can be used
to investigate the agreement between Navier-Stokes and Boltzmann equations for small Knudsen number flows. This
was first checked on simple unsteady one-dimensional gas flows in our previous paper [9] and the results were very
encouraging. In the present research we extend this approach to a more complicated case of study, namely, the onset
of transition from a pure conduction state to a convection state in the two-dimensional Rayleigh-Bénard system of a
“hard sphere” gas. We also treat the possible long term states of developed convection. For that purpose we use two
principally different numerical approaches: particle simulation by the DSMC method [10] and Navier-Stokes finite
difference (FD) calculations [9]. A comparative analysis of the results obtained by both methods is carried out for the
domain of governing parameters indicated above.
We first compare the numerical results obtained by DSMC method and finite difference (FD) computations in a twodimensional domain with aspect ratio 2 and delineate the zone of the Rayleigh-Bénard convection within the domain
of Knudsen and Froude numbers as pointed out above and for a fixed temperature ratio r = Tc /Th = 0.1. The computed
zone of instability is presented in Fig. 1 by a gray shaded area. The neutral curve separating the outer area of final
pure conduction from the inner one of final convection is determined approximately by 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 by
circles and squares respectively. As can be seen the onset of the Bénard instability with regard to the Knudsen numbers
is about Kn = 0.029 for the Monte Carlo simulations and Kn = 0.028 for the finite difference computations; this can
be accepted as a good enough coincidence between the two approaches.
The agreement between the solutions, not unexpectedly, improves when the Knudsen number decreases. Down to
Kn = 0.01 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 gravity is neglected. For Kn = 0.005 and
small Fr a four-roll configuration was also observed (Fig. 2). Our calculations confirmed the results reported by Sone
et. al. [6], and Golshtein, Elperin [8] about the presence of co-existing attractors of the RB flow of a rarefied gas.
Moreover, trying to locate the line separating convection from pure conduction more accurately we have found that
195
0.9
0.8
0.7
z
0.6
0.5
0.4
0.3
0.2
0.1
0.2
FIGURE 2.
0.4
0.6
0.8
1
y
1.2
1.4
1.6
1.8
The final four-roll state computed by DSMC (a) and FD (b) methods for Kn = 0.005, Fr = 0.95 and r = 0.1.
0.16
FD
MC
0.14
A
0.12
max|w(y,z)|
0.1
0.08
0.06
D
0.04
B
0.02
C
0
1.2
FIGURE 3.
Kn = 0.02.
1.25
1.3
1.35
1.4
Fr
1.45
1.5
1.55
1.6
The hysteresis loop computed by DSMC (circles) and FD (triangles) methods for small Fr numbers and a fixed
there exist hysteresis loops for fixed Kn when by increasing and reversely 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 zone of convection. Analytically, an upper bound
can be determined by investigating the zone of stability of pure conduction on the basis of the continuum model. A
196
6
cold wall
5
1
Fr=10
4
Fr=5x101
3
Q
Fr=102
2
Fr=103
1
0
Fr=1.5x103
hot wall
−1
perturbations
−2
1
10
2
10
3
10
t
FIGURE 4. The heat flux behavior for a set of large Froude numbers. The Knudsen number and the temperature ratio are fixed
(Kn = 0.001 and r = 0.1).
further analysis shows that the condition delineates a line which follows very close the left (small Fr) boundary of the
computed zone of instability. So, both left and right conditions can serve to determine of a rough exterior bound which
envelopes the zone of instability from the side of small and large Froude numbers respectively.
We then analyse the results obtained for the lowest Knudsen number considered (Kn = 0.001) where a set of
qualitatively new regimes and final states have been found. Due to the extremely 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 only (Fr = 1.0). The finite difference computations for the continuum model are performed for a
variety of Froude numbers starting from a large value Fr = 1.5 × 103 where the attractor is a pure-conduction state.
By decreasing the Froude number the final state of the system 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 = 0.9 − 1.0, then stabilizes as a stable six-roll configuration for Fr = 0.8 and, finally, becomes again
pure-conduction for Fr < 0.8. The comparison between the two models performed for Fr = 1.0 showed that both
methods exhibit a secondary instability with a final flow regime of a permanently chaotic vortex formation (a similar
result has been obtained by Bird [11] for the Taylor-Couette cylindrical flow by using the DSMC method).
The time evolution of the non-dimensional heat fluxes at the walls computed for the continuum model by 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 (10 ≤ Fr ≤ 1500) for which the slope of the density profile of a pure conduction regime would be positive.
Figure 5 shows the heat flux profiles for the cases (1.0 ≤ Fr ≤ 2.0) with a non-monotonic or negative (Fr = 1.0) slope
of the density profile.
The orbit of the established state in the reduced phase plane (Qz=0 (t), Qz=1 (t)) (Fig.8), when compared with the
orbits in the cases Fr = 1.5 (Fig. 6) and Fr = 1.2 (Fig. 7), gives an impression of the chaotic character of the flow
attractor.
In the theory of nonlinear dynamic systems, the long time solutions of this kind are known under the name of
197
6
5
cold wall
4
Fr=2.0
Fr=1.5
Q
3
2
1
Fr=1.0
0
−1
hot wall
−2
1
10
2
10
t
FIGURE 5. The heat flux behavior for a set of small Froude numbers. The Knudsen number and the temperature ratio are fixed
(Kn = 0.001 and r = 0.1).
“strange attractors”. The existence of “strange attractors”, obtained by 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 = 6.0 on the chaotic convection
for the basic case (Kn = 0.001, Fr = 1.0, r = 0.1). The computations have been carried out for the continuum model
by using FD method on a 1200 × 200 grid. The run starts from a randomly disturbed pure conduction state.
A series of snapshots presented in Figs. 9 show some typical evolutions of the velocity vector field. A later
excitement of the instability occurs due to the initial condition of pure conduction; this helps avoiding the appearance
of a wave travelling between the walls due to gravity as occurs in the basic case. The larger aspect ratio A = 6.0 helps
keep longer (t > 150.0) the initial symmetry with respect to the midline y = L0 /2 ( Fig. 9, 3rd snapshot), in comparison
with the case with a smaller aspect ratio. In the end, the flow looses unavoidably the symmetry and degenerates to
chaotic spatiotemporal structures that are observed in the case with aspect ratio A = 2.0.
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 frequently raised whether the chaotic solutions
of the Navier-Stokes equations produce a turbulent behavior adequate to reality. In our opinion, the qualitative
and quantitative similarity of the results presented in this paper, obtained by 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 binary collisions is able
to reproduce macroscopic instability patterns quite accurately.
198
2.49
2.48
2.47
2.46
Qz=1(t)
2.45
2.44
t>600.0
2.43
2.42
2.41
t=195.0
2.4
2.39
2.45
FIGURE 6.
2.5
2.55
2.6
2.65
Qz=0(t)
2.7
2.75
2.8
2.85
The final orbit traced by the convection RB system in the plane (Qz=0 (t), Qz=1 (t)) for Fr = 1.5
1.48
500<t<1500
1.46
1.44
Qz=1(t)
1.42
1.4
1.38
1.36
1.34
1.1
FIGURE 7.
1.2
1.3
1.4
1.5
Qz=0(t)
1.6
1.7
1.8
1.9
The attractor’s orbit in the phase plane (Qz=0 (t), Qz=1 (t)) for Fr = 1.2.
Acknowledgments
Two of the authors, S.S. and V.R., would like to acknowledge the financial support provided by the Bulgarian
Ministry of Education and Sciences with Grant No. MM806/98. The research of C.C. was performed in the frame of
European TMR (contract n. ERBFMRXCT97O157) and was also partially supported by MURST of Italy.
199
0.78
500<t<2000
0.76
0.72
Q
z=1
(t)
0.74
0.7
0.68
0.66
0.66
FIGURE 8.
0.68
0.7
0.72
0.74
Qz=0(t)
0.76
0.78
0.8
0.82
The orbit traced by the RB system in the phase plane (Qz=0 (t), Qz=1 (t)) for the case (Kn = 0.001, Fr = 1.0, r = 0.2).
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200
1
0.8
z
0.6
0.4
0.2
0
t=76.5
0.2
0
1
2
3
y
4
5
6
1
0.8
z
0.6
0.4
0.2
t=100.0
0.2
0
0
1
2
3
y
4
5
1
2
3
y
4
6
1
0.8
z
0.6
0.4
0.2
t=200.0
0.2
0
0
5
6
1
0.8
z
0.6
0.4
0.2
0
t=300.0
0.2
0
1
2
3
y
4
5
6
1
0.8
z
0.6
0.4
0.2
0
FIGURE 9.
t=400.0
0.2
0
1
2
3
y
4
5
6
The five pictures show some typical evolutions of the velocity vector field for an aspect ratio A = 6.
201