Computation of convective flows using the ... - Springer Link

Computational Mathematics and Modeling, Vol. I0, No. 2, 1999 COMPUTATION OF CONVECTIVE FLOWS USING THE QUASIHYDRODYNAMIC EQUATIONS

T. G. Elizarova, I. S. Kalachinskaya, A. V. Klyuchnikova, and Yu. V. Sheretov

UDC 518.2:533.6

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 ~, (1) ~, + div (~ | t~) + l ~ p = l d i v tr + div ((~ | ~) + (H | ~ ) ) - ~flT, (2) Po P0 Tt + 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 = z l ( P o C p ) 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

+=+((,

(4,

Here j = p 0 ( ~ - ~ ) is the mass flux density vector of the

liquid, and b = rl[(V|

|

viscous Navier-Stokes stresses [1], in which rl is the coefficient of dynamic viscosity.

is the tensor of The parameter

z = const. > 0 is interpreted as a certain characteristic time, and is chosen as z = 0 ! (Pock), where c s is the speed of sound in the liquid at temperature T0. The density of the liquid P0 is assumed constant. The dissipative function in Eq. (3) is omitted. In the limit as z ~ 0 the system (1)--(3) becomes the system of Boussinesq equations [1]. 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

(~-

($)

m). ~ = 0,

where ~ is the outward unit normal vector to the surface. These boundary conditions make the system (1)-(3) closed with respect to the unknown functions: thb~e|O~ity ~=ff(~;t); pressure p = p(~,t), and temperature T = T(~,t). The system of equations in the two-dimensional ease. We now write the system (1)-(4) in dimensionless form for the case of two-dimensional nonstationary flows: t

au av

a t" au

au

a ( au

~ x + ~yy= ~:0~x~U-.~-fx-',-V~y-.t--~x)+7o ~ytU~x

~+~(u )+

ap

/

+ V~y + ~ - - Gr T,)

Re~t,&) ~ee~l%-) Ree~t.&)+ a

au

z0~[,(

@ Re t )

au

~ +-~-GrT)]

ov

,a

av

t

&

dx

o~

(7)

v

+'o-~[u(u"~+v--~+-~-Gr' +'~177162176 )]

~

(6)

"

a~T') a IT( ~ _~ ] ~p]] o ~ IT( M3v av ap ~-~-+-~')+%-~L~,U--~+Vdy+ax).J+%-~Ll~U'~x+V-~y+~y-GrT)j "(9)

=vr--l/a~T

Here Re, Pr, and Gr are the Reynolds, Prandtl, and Grashof numbers respectively, ~:0 is the characteristic time, also written in dimensionless form. The method of dedimensionalizing the system of equations (6)-(9) and the conditions on the boundary are determined by the specific problem and will be described below. 3. The computational algorithm, For numerical solution of the system of equations written above we use the finite difference method. The spatial derivatives in the system (6)-(9) are approximated on a uniform spatial grid by central differences with order O(h 2) as h --4 0. The quantities here are computed at the nodes of the difference grid. The values at the half-integer nodes are defined as half the sum of their values at the adjacent integer indices. The mixed derivatives are approximated using the values of the quantities at the centers of the cells, which are computed as one-fourth of the sum of the values of these quantities at the comers. The time derivatives are approximated to first order by forward differences. The boundary of the computational region is situated at the half-integer nodes of the grid. The approximation of the boundary conditions for the speed and temperature is carried out by computing the corresponding derivatives with second-order precision and is guaranteed by the introduction of additional layers of fictitious nodes over the outer boundaries of the computational region. At each time step the pressure field is found from the velocity field and the temperature by solving the Poisson equation a2p O2p_ 1 (au Ov'~ 0 ( Ou 0u'~ a ( ~v Ov "~ ax2 + ~ - - - / - - + - - / - - - / u - - + v - - / - m / u ~ + - - - Gr T), (10)

9ot.Ox ay) Oxt. Ox

Oy) ayt, Ox v y

which follows from (6) and is approximated like the equations of motion. The boundary conditions for the pressure are approximated to second-order precision by extrapolating the Poisson equation to the boundary. To solve (1) we apply the method of the preconditioned generalized method of coupled gradients [9]. The preconditioner is constructed using the pointwise incomplete decomposition of the matrix of the linear system of equations A2 =/7. At the upper right computational point the pressure is held constant and equal to one at all times. The rate of convergence of the iterations in solving the Poisson equation (10) determines the effectiveness of the algorithm as a whole. The conditions for halting the recursion has the form

161

ev=I~ij(Prx,O+pyy,o+fij)2h2