Numerical Simulation of Three-Dimensional Viscous Flows with ...

1336
ISMAIL-ZADE et al.
∂ψ
Γ( x 3 = 0,
x 3 = l 3 ) : ψ 1 ψ 2 0, 3 ∂ 2 ψ
-------- 0, 1 ∂ 2 ψ
= = = ---------- = ---------- 2
= 0 .
∂x 3
In the case of no-slip conditions, we set the following conditions:
The initial conditions for temperature, density, and viscosity are set as follows:
0 0
0
Here, the prescribed functions T *
, ρ *
, and µ *
define the temperature and thermally unperturbed density
and viscosity at the initial moment.
Equations (7)–(12), combined with the boundary and initial conditions, uniquely determine the unknown
functions u 1 , u 2 , u 3 , T, ρ, and µ within Ω at any t ≥ 0. Equations (13)–(15) (with prescribed functions T, ρ *
,
and µ *
) do not uniquely determine ψ 1 , ψ 2 , and ψ 3 under boundary conditions of any form. This is explained
by the fact that u can be expressed in terms of the corresponding vector potential y only up to the gradient
of an arbitrary differentiable scalar function y, because u = curly = curl(y + ∇ϕ), which implies that
u = curly 1 and u = curly 2 , where y 1 = y 2 + ∇ϕ. When a potential y satisfies these relations, the potential
y + ∇ϕ, where ϕ is an arbitrary sufficiently smooth function of x ∈ Ω with a compact support in Ω, satisfies
these relations as well. Since ϕ is a compactly supported function on Ω, the gradient ∇ϕ does not contribute
to the boundary conditions for y. In particular, this implies that Eqs. (9)–(15), combined with the boundary
and initial conditions, do not uniquely determine the unknown functions ψ 1 , ψ 2 , and ψ 3 , whereas the
unknown functions T, ρ *
, and µ *
(and, therefore, T, ρ, and µ) are uniquely determined within Ω at any t ≥ 0.
For our purposes, any potential found by solving the equations above is suitable, because the same velocity
field u is obtained.
4. VARIATIONAL EQUATION OF THE PROBLEM
To apply a finite element method, we replace Eqs. (13)–(15) with an equivalent variational equation. We
multiply Eqs. (13)–(15) by the components ω i of a test vector function w satisfying the conditions set for
the vector function y. Performing these operations and using the boundary conditions for the desired and
test vector functions, we obtain the variational equation
which can be compactly rewritten as
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 41 No. 9 2001
∂x 3
2
∂x 3
2
∂ψ
Γ( x 1 = 0,
x 1 = l 1 ) : ψ 1 ψ 2 ψ 3 0, 2 ∂ψ
= = = -------- = -------- 3
= 0 ,
∂x 1 ∂x 1
∂ψ
Γ( x 2 = 0,
x 2 = l 2 ) : ψ 1 ψ 2 ψ 3 0, 1 ∂ψ
= = = -------- = -------- 3
= 0 ,
∂x 2 ∂x 2
Γ( x 3 = 0,
x 3 = l 3 ) : ψ 1 = ψ 2 = ψ 3 = 0,
∂ψ 1 ∂ψ
-------- = -------- 2
∂x 3 ∂x 3
= 0 .
For the temperature on the side faces on Ω, we set zero heat-flux conditions (as in a homogeneous Neumann
problem). On the top and bottom faces of Ω, the following conditions for temperature are prescribed
(as in a nonhomogeneous Dirichlet problem):
Γ( x 1 = 0,
x 1 = l 1 ) : ∂T/∂x 1 = 0, t ≥ 0,
Γ( x 2 = 0,
x 2 = l 2 ) : ∂T/∂x 2 = 0, t ≥ 0,
Γ( x 3 = 0) : T( t, x 1 , x 2 , 0) = T 1 ( t, x 1 , x 2 ), t ≥ 0,
Γ( x 3 = l 3 ) : T( t, x 1 , x 2 , l 3 ) = T 2 ( t, x 1 , x 2 ), t ≥ 0.
T( 0,
x) = T 0 *
(), x ρ *
( 0,
x) = ρ 0 *
(), x µ *
( 0,
x) = µ 0 *
(), x x ∈ Ω.
∫µ ( 2e 11 ẽ + 11 2e 22 ẽ + 22 2e 33 ẽ + 33 e 12 ẽ + 12 e 13 ẽ + 13 e 23 ẽ ) dx
23 = – La∫ρω 3 dx,
Ω
E( yw , ) = L( w) for any admissible ω = ( ω 1 , ω 2 , ω 3 ).
Ω
(17)