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Numerical Heat Transfer, Part B: Fundamentals
An International Journal of Computation and Methodology
Volume 72, 2017 - Issue 6
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Original Articles

A simple gas kinetic scheme for simulation of 3D incompressible thermal flows

, , &
Pages 450-468 | Received 04 Sep 2017, Accepted 14 Nov 2017, Published online: 14 Dec 2017
 

ABSTRACT

In this work, a simple gas kinetic scheme is presented for solving the 3D incompressible thermal flow problems. In the scheme, the macroscopic-governing equations are discretized by finite volume method, and the numerical fluxes at cell interface are reconstructed by the local solution of the Boltzmann equation. To compute the numerical fluxes, two equilibrium distribution functions are introduced. One is the sphere function for calculating the fluxes of mass and momentum equations, and the other is the D3Q6 discrete velocity model for evaluating the flux of energy equation. Using the difference of equilibrium distribution functions at the cell interface and its surrounding points to approximate the nonequilibrium distribution function, and at the same time considering the incompressible limit, the numerical fluxes of macroscopic governing equations at the cell interface can be given explicitly and concisely. Numerical results showed that the present scheme can predict accurately the thermal flow properties at a wide range of the Rayleigh numbers.

Nomenclature

c=

peculiar velocity of particles

cs=

sound speed

D=

space dimension

=

discrete particle velocity

f=

density distribution function

feq=

equilibrium state for f

fE=

buoyancy force

Fn=

numerical flux vector

g=

gravity acceleration

gS=

sphere function

=

temperature distribution function

=

equilibrium state for

I=

unit tensor

L=

characteristic length

Ma=

Mach number

n=

unit normal vector

Nf=

number of the faces

Nu=

Nusselt number

p=

pressure

Pr=

Prandtl number

Q=

source term

Ra=

Rayleigh number

Si=

area of the ith interface

t=

time

T=

temperature

Tm=

average temperature

u=

velocity vector

u0=

reference velocity

Vc=

characteristic thermal velocity

W=

conservative variable vector

β=

thermal expansion coefficient

Δ t=

streaming time step

κ=

thermal diffusivity

μ=

dynamic viscosity

ν=

kinematic viscosity

=

particle velocity

ρ=

density

=

collision time scale for

=

collision time scale for f

=

moment vector

=

volume of cell I

Subscripts=
I=

cell index

1, 2, 3=

coordinate component

α=

particle index

Superscripts=
face=

cell interface

L=

left side of cell interface

R=

right side of cell interface

sph=

spherical surface

Nomenclature

c=

peculiar velocity of particles

cs=

sound speed

D=

space dimension

=

discrete particle velocity

f=

density distribution function

feq=

equilibrium state for f

fE=

buoyancy force

Fn=

numerical flux vector

g=

gravity acceleration

gS=

sphere function

=

temperature distribution function

=

equilibrium state for

I=

unit tensor

L=

characteristic length

Ma=

Mach number

n=

unit normal vector

Nf=

number of the faces

Nu=

Nusselt number

p=

pressure

Pr=

Prandtl number

Q=

source term

Ra=

Rayleigh number

Si=

area of the ith interface

t=

time

T=

temperature

Tm=

average temperature

u=

velocity vector

u0=

reference velocity

Vc=

characteristic thermal velocity

W=

conservative variable vector

β=

thermal expansion coefficient

Δ t=

streaming time step

κ=

thermal diffusivity

μ=

dynamic viscosity

ν=

kinematic viscosity

=

particle velocity

ρ=

density

=

collision time scale for

=

collision time scale for f

=

moment vector

=

volume of cell I

Subscripts=
I=

cell index

1, 2, 3=

coordinate component

α=

particle index

Superscripts=
face=

cell interface

L=

left side of cell interface

R=

right side of cell interface

sph=

spherical surface

Additional information

Funding

The research is supported by the National Research Foundation Singapore, Sembcorp Industries Ltd and National University of Singapore under the Sembcorp-NUS Corporate Laboratory.

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