On the Rayleigh–Bénard problem in the continuum limit

Manela, A.; Frankel, I.

doi:10.1063/1.1861876

Cited by 27 publications

(28 citation statements)

References 18 publications

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“…A similar effect was found in other analyses of rarefied gas flows, including the Rayleigh-Bénard (Stefanov et al 2002;Manela & Frankel 2005) and Taylor-Couette (Yoshida & Aoki 2006;Manela & Frankel 2007) problems. Interestingly, it was shown that suppression of instability in the Taylor-Couette problem is brought about by increased rates of dissipation associated with aerodynamic heating of the fluid, similarly to what has been observed in the present study.…”

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confidence: 60%

The effect of compressibility on the stability of wall-bounded Kolmogorov flow

Manela

Zhang

2012

J. Fluid Mech.

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We extend the stability analysis of incompressible Kolmogorov flow, induced by a spatially periodic external force in an unbounded domain, to a compressible hardsphere gas confined between two parallel isothermal walls. The two-dimensional problem is studied by means of temporal stability analysis of a 'slip flow' continuumlimit model and the direct simulation Monte Carlo (DSMC) method. The neutral curve is obtained in terms of the Reynolds (Re) and Knudsen (Kn) numbers, for a given non-dimensional wavenumber (2πn) of the external force. In the incompressible limit (Kn, Kn Re → 0), the problem is governed only by the Reynolds number, and our neutral curve coincides with the critical Reynolds number (Re cr ) calculated in previous incompressible analyses. Fluid compressibility (Kn, Kn Re = 0) affects the flow field through the generation of viscous dissipation, coupling flow shear rates with irreversible heat production, and resulting in elevated bulk-fluid temperatures. This mechanism has a stabilizing effect on the system, thus increasing Re cr (compared to its incompressible value) with increasing Kn. When compressibility effects become strong enough, transition to instability changes type from 'exchange of stabilities' to 'overstability', and perturbations are dominated by fluctuations in the thermodynamic fields. Most remarkably, compressibility confines the instability to small (O(10 −3 )) Knudsen numbers, above which the Kolmogorov flow is stable for all Re. Good agreement is found between 'slip flow' and DSMC analyses, suggesting the former as a useful alternative in studying the effects of various parameters on the onset of instability, particularly in the context of small Knudsen numbers considered.

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confidence: 60%

The effect of compressibility on the stability of wall-bounded Kolmogorov flow

Manela

Zhang

2012

J. Fluid Mech.

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“…A direct and interesting idea is to study the transport properties for different scales of vortex in RB convection. However, DSMC results 14,25 and linear stability analysis 26 based on compressible Navier-Stokes equations have showed that there is a critical scale of the vortex about 30 times of molecular mean free paths in RB convection, that is, the thermal convection will not occur if the characteristic length of the RB system is less than the critical scale. Therefore, the phenomenon of diffusion reduction does not exist in RB convection.…”

Section: Diffusive Transport Properties In Rayleigh-bénard Convecmentioning

confidence: 99%

Effects of convection and solid wall on the diffusion in microscale convection flows

2010

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The diffusive transport properties in microscale convection flows are studied by using the direct simulation Monte Carlo method. The effective diffusion coefficient D ‫ء‬ is computed from the mean square displacements of simulated molecules based on the Einstein diffusion equation D ‫ء‬ = ͗⌬x 2 ͑t͒͘ / 2t. Two typical convection flows, namely, thermal creep convection and RayleighBénard convection, are investigated. The thermal creep convection in our simulation is in the noncontinuum regime, with the characteristic scale of the vortex varying from 1 to 100 molecular mean free paths. The diffusion is shown to be enhanced only when the vortex scale exceeds a certain critical value, while the diffusion is reduced when the vortex scale is less than the critical value. The reason for phenomenon of diffusion reduction in the noncontinuum regime is that the reduction effect due to solid wall is dominant while the enhancement effect due to convection is negligible. A molecule will lose its memory of macroscopic velocity when it collides with the walls, and thus molecules are hard to diffuse away if they are confined between very close walls. The RayleighBénard convection in our simulation is in the continuum regime, with the characteristic length of 1000 molecular mean free paths. Under such condition, the effect of solid wall on diffusion is negligible. The diffusion enhancement due to convection is shown to scale as the square root of the Péclet number in the steady convection regime, which is in agreement with previous theoretical and experimental results. In the oscillation convection regime, the diffusion is more strongly enhanced because the molecules can easily advect from one roll to its neighbor due to an oscillation mechanism.

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“…(The full system of equations can be found, for example, in (11)- (14) of Manela and Frankel, 2005a). These are supplemented by the perturbation equations of motion (see Equations (14) and (15)) and the boundary conditions prescribing the vanishing of temperature and velocity perturbations at y ¼ 0, 1.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…The linear temporal stability of this reference state is analyzed assuming that it is perturbed by small spatially harmonic perturbations. By the transverse symmetry of the problem (Manela and Frankel, 2005a) we use a two-dimensional description in the Cartesian coordinates (x, y) whose origin lies on the lower wall and where x is a horizontal coordinate in the wave-vector direction. Accordingly, each of the above-mentioned fields is generically represented by the sum…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…Based on these analyses the prevailing view is that under these conditions the Boussinesq approximation applies to a compressible fluid provided that a generalized Rayleigh number based on the potential (i.e., the excess of the actual over the adiabatic) temperature gradient is employed (Tritton, 1988). The RB problem for a compressible fluid subject to an arbitrary temperature difference has been addressed in the continuum limit of slightly rarefied gases (Sone et al, 1997;Stefanov et al, 2002;Manela and Frankel, 2005a). The stability problem is here characterized by a pair of parameters, e.g., Fr and the Knudsen number, rather than just by Ra.…”

Section: Introductionmentioning

confidence: 99%

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From the Generalized Boussinesq Approximation to the Marginally Super-Adiabatic Limit

Manela¹,

Frankel²

2009

Chemical Engineering Communications

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The prevailing view of the Rayleigh-Bénard problem in compressible fluids is that for small temperature differences the Boussinesq approximation holds, provided that it is based on a modified Rayleigh number incorporating the potential-temperature gradient. However, for small values of the latter, the onset of convection is characterized by distinct non-Boussinesq features. We consider the linear temporal stability problem and identify the origin of the nonuniformity in the convection and compression-work terms of the perturbation energy balance. We thereby regularize the transition with diminishing potential-temperature gradient from the generalized Boussinesq approximation to the limit when the temperature gradient is only marginally super-adiabatic. It is demonstrated that this transition is accomplished in two phases. Initially, the critical Rayleigh number rapidly increases, which is accompanied by only slight variations of the corresponding wave number. Subsequently, with further diminishing potential-temperature gradients, the critical wave number rapidly increases as well, and the resulting convection becomes effectively confined to a narrowing fluid layer adjacent to the upper wall.

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On the Rayleigh–Bénard problem in the continuum limit

Cited by 27 publications

References 18 publications

The effect of compressibility on the stability of wall-bounded Kolmogorov flow

The effect of compressibility on the stability of wall-bounded Kolmogorov flow

Effects of convection and solid wall on the diffusion in microscale convection flows

From the Generalized Boussinesq Approximation to the Marginally Super-Adiabatic Limit

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