Computation of convective flows using the quasihydrodynamic equations

Abstract

We discuss the results of numerical modeling of problems of gravitational and thermocapillary convection in rectangular cavities with small Prandtl numbers. The results are obtained using the quasihydrodynamic system of equations and compared with the results of computations using the Navier-Stokes system. Three tables, 12figures. Bibliography: 19 titles.Introduction. The Navier-Stokes equations are widely used to describe the convective flow of a viscous incompressible liquid [1], [2]. In the present paper we consider a different approach based on the use of quasihydrodynamic equations. The quasihydrodynamic system was introduced to describe the flow of a viscous incompressible liquid and studied in [3]-[5]. This system is dissipative and has a number of exact physically meaningful solutions. Mathematical analysis of the quasihydrodynamic system that models the flow of a viscous compressible heat-conducting medium was given in [4]. In particular, the equation for balance of thermodynamic entropy with a nonnegative dissipative function was obtained, the theorem on increase of total thermodynamic entropy in adiabatically isolated regions was proved, and a laminar boundary layer approximation was constructed. A physical derivation of quasihydrodynamic systems with an explanation of the meaning of all quantities and parameters occurring in them was published in [5].The possibility of using quasihydrodynamic equations for numerical modeling of flows of a viscous incompressible liquid in a cavern [6] and to study the Boussinesq convection in a square cavity [7] had been demonstrated previously.A system of quasihydrodynamic equations related to the quasihydrodynamic system has been successfully applied in numerical modeling of flows of a rarefied gas [8].The quasihydrodynamic system of equations. The quasihydrodynamic system in Boussinesq approximation was derived in [5] and has the form div ~ = div ~,Po P0 T t + div (fiT) = div (fiT) + r div (VT).(3) Here P0 is the mean value of the density, ~ = ~(.~,t) is the hydrodynamic velocity, p = p(Y,t) is the excess of pressure above the hydrostatic value, T = T(.Tc, t) is the deviation of the temperature from its mean value To, fl > 0 is the coefficient of thermal expansion of the liquid, ~, is the acceleration of gravity, r=zl(PoCp) is the coefficient of thermal diffusivity, and Z is the coefficient of thermal conductivity. The quantity po E is interpreted as the mean momentum of a unit volume of liquid. The vector ~ is computed from the formula As boundary conditions for the system (1)-(4) in a closed region we use the conditions for velocity and temperature adopted in the Navier-Stokes theory, supplemented by the condition of mass impermeability in the form Translated from Problemy Matematicheskoi Fiziki, 1998, pp. 193-208. 160 1046-283X/99/t002-0160522.00 9 Kluwer Academic/Plenum Publishers

Fluid mechanics in fluids at rest

“…Rather, the coefficients must be obtained independently of LIT from other sources. These sources include (i) Klimontovich [34] and [35]; (ii) experiment [36]; (iii) molecular theories, such as Burnett's [10] or Grad's [10] solutions of the Boltzmann equation, including extensions of the Boltzmann equation by Dadzie-Reese et al [37,38]; (iii) statistical-mechanical theories [39][40][41][42][43]; (iv) macroscopic theories, either rational, such as the equidiffuse model [44], or ad hoc [45][46][47][48]; (v) simulation, etc.…”