A mathematical and numerical framework for the analysis of compressible thermal convection in gases at very high temperatures

Abstract

The relevance of non-equilibrium phenomena, nonlinear behavior, gravitational effects and fluid compressibility in a wide range of problems related to high-temperature gas-dynamics, especially in thermal, mechanical and nuclear engineering, calls for a concerted approach using the tools of the kinetic theory of gases, statistical physics, quantum mechanics, thermodynamics and mathematical modeling in synergy with advanced numerical strategies for the solution of the Navier-Stokes equations. The reason behind such a need is that in many instances of relevance in this field one witnesses a departure from canonical models and the resulting inadequacy of standard CFD approaches, especially those traditionally used to deal with thermal (buoyancy) convection problems. Starting from microscopic considerations and typical concepts of molecular dynamics, passing through the Boltzmann equation and its known solutions, we show how it is possible to remove past assumptions and elaborate an algorithm capable of targeting the broadest range of applications. Moving beyond the Boussinesq approximation, the Sutherland law and the principle of energy equipartition, the resulting method allows most of the fluid properties (density, viscosity, thermal conductivity, heat capacity and diffusivity, etc.) to be derived in a rational and natural way while keeping empirical contamination to the minimum. Special attention is deserved as well to the well-known pressure issue. With the application of the socalled multiple pressure variables concept and a projection-like numerical approach, difficulties with such a term in the momentum equation are circumvented by allowing the hydrodynamic pressure to decouple from its thermodynamic counterpart. The final result is a flexible and modular framework that on the one hand is able to account for all the molecule (translational, rotational and vibrational) degrees of freedom and their effective excitation, and on the other hand can guarantee adequate interplay between molecular and macroscopic-level entities and processes. Performances are demonstrated by computing some incompressible and compressible benchmark test cases for thermal (gravitational) convection, which are then extended to the high-temperature regime taking advantage of the newly developed features

A Numerical Study of Antisymmetric-Symmetric Mode of RB Convection through on Time Evolution

“…This mode exhibits an odd number of rolls along x and an even number of cells in the perpendicular direction; Some steady and oscillatory 2D regimes for specific case is also considered in the study of Marcello Lappa [24] (cavity filled with a silicone oil with Pr= 15) They found that flow becomes oscillatory for Ra= 5 × 105. Lappa [15] shows that when the behavior can be considered incompressible (very small value of the temperature gradient, 1K only) and the average temperature is much smaller than the vibrational characteristic temperature, the emerging solution is very similar to that obtained by Goldhirsch et al [25] . In this work even if the temperature gradient is equal to 10 K, then the density does not undergo any significant change whereas specific heat remains constant in this interval.…”

“…temperature pulsations. Lappa [14] provided a comparative and critical review of secondary and oscillatory instabilities in thermal gravitational convection through examination of some existing studies, however in his recent works he [15] moved beyond the Boussinesq approximation; the resulting method allows the fluid flow properties to derive in a rational way by keeping a minimum empirical contamination.…”

On the highly unsteady dynamics of multiple thermal buoyant jets in cross flows

“…These techniques have enjoyed a widespread use in the literature especially for categories of flows intrinsically unsteady for which they enable cost-effective solutions by virtue of recent progresses in the capacity of computers and the increasing success of direct numerical simulation in capturing bifurcations and instability phenomena 136,137 . Additional information on this class of methods can be found in Refs [138][139][140][141][142], while for a review of recent advances with regard to their application to situations in which the density cannot be considered constant (Boussinesq approximation no longer valid), the reader may consider Refs [137,120] and references therein. In the present work, this method has been implemented using schemes explicit in time for the integration of both the momentum and energy equations.…”

On the oscillatory modes of compressible thermal convection in inclined differentially heated cavities

“…Despite its perceived practical importance, the problem has not been investigated sufficiently. Among other things, from a more academic standpoint (beyond practical applications), the work set forth in this paper might also be regarded as a specific attempt to bridge theoretically two lines of inquiry previously running in parallel in the literature, namely that devoted to the assessment of NOB effects in horizontal or vertical rectangular cavities (Lappa [38] and references therein) and that focusing on the influence of arbitrary angles of inclination on the emerging dynamics.…”