5 Numerical Simulation of Taylor Vortex Flows …
49
E v =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
0
σ xx
τ xy
τ xz
d x
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
, F v =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
0
τ xy
σ yy
τ yz
d y
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
, G v =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
0
τ xz
τ yz
σ zz
d z
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
d x = uσ xx + vτ xy + wτ xz + q x ,
d y = uτ xy + vσ yy + wσ yz + q y ,
d z = uτ xz + vτ yz + wσ zz + q z ,
div V =
∂u
∂ x
+
∂v
∂ y
+
∂w
∂z
,
σ xx =
μ
Re
2
∂u
∂ x
−
2
3
div V
, σ yy =
μ
Re
2
∂v
∂ y
−
2
3
div V
, σ zz =
μ
Re
2
∂w
∂z
−
2
3
div V
,
τ xy =
μ
Re
∂u
∂ x
+
∂v
∂ y
, τ xz =
μ
Re
∂u
∂ x
+
∂w
∂z
, τ yz =
μ
Re
∂v
∂ y
+
∂w
∂z
,
q x =
γ
γ −1
μ
Re Pr
∂ T
∂ x
, q y =
γ
γ −1
μ
Re Pr
∂ T
∂ y
, q z =
γ
γ −1
μ
Re Pr
∂ T
∂z
.
The dimensionless variables are determined via the «
»-marked dimensional
variables:
t =
p
o
ρ
o
t
L , X =
X
L , V =
ρ
o
p
o
V
,
ρ =
ρ
ρ
o
, p =
p
p
o
, T =
T
T
o
, μ =
μ
μ
o
.
The low index “ o ” is the value of the parameter before the beginning of rotation.
Here, L
is the characteristic dimension.
It is assumed that the Prandtl number Pr =
μc p
λ
is constant. Here, μ, c p , λ
are the coefficients of heat capacity, viscidity, and heat conductivity (Pr = 0.72
in calculations), respectively. Re =
√
p
0 ρ
0 L
μ
0
is the Reynolds number.
The system of differential equations is supplemented by the state equation: p =
ρRT, where T is the temperature, and R is the gas constant. The state equation in a
dimensionless form is as follows: p = ρT.
Thus, the complete system of the Navier–Stokes equations for perfect gas flows
in the absence of external mass forces is given. It is assumed that the heat may enter
the medium only as a result of thermal conductivity. The flows considered below are
subsonic, and the compressibility effects are not of greater importance, but the use of
the compressible gas model allows to apply the numerical method and the programs
[11] developed for the simulation of viscous gas flows.
The use of the Cartesian coordinate system for the adequate description of the
flows with complex topology in solving the problems by the finite-difference methods
seems difficult for two reasons. One reason is that the interpolation procedures to get
boundary conditions are necessary. Also there are some difficulties in the description
of the computational grid because the computational grid is not rectangular. We will
49
E v =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
0
σ xx
τ xy
τ xz
d x
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
, F v =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
0
τ xy
σ yy
τ yz
d y
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
, G v =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
0
τ xz
τ yz
σ zz
d z
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎭
,
d x = uσ xx + vτ xy + wτ xz + q x ,
d y = uτ xy + vσ yy + wσ yz + q y ,
d z = uτ xz + vτ yz + wσ zz + q z ,
div V =
∂u
∂ x
+
∂v
∂ y
+
∂w
∂z
,
σ xx =
μ
Re
2
∂u
∂ x
−
2
3
div V
, σ yy =
μ
Re
2
∂v
∂ y
−
2
3
div V
, σ zz =
μ
Re
2
∂w
∂z
−
2
3
div V
,
τ xy =
μ
Re
∂u
∂ x
+
∂v
∂ y
, τ xz =
μ
Re
∂u
∂ x
+
∂w
∂z
, τ yz =
μ
Re
∂v
∂ y
+
∂w
∂z
,
q x =
γ
γ −1
μ
Re Pr
∂ T
∂ x
, q y =
γ
γ −1
μ
Re Pr
∂ T
∂ y
, q z =
γ
γ −1
μ
Re Pr
∂ T
∂z
.
The dimensionless variables are determined via the «
»-marked dimensional
variables:
t =
p
o
ρ
o
t
L , X =
X
L , V =
ρ
o
p
o
V
,
ρ =
ρ
ρ
o
, p =
p
p
o
, T =
T
T
o
, μ =
μ
μ
o
.
The low index “ o ” is the value of the parameter before the beginning of rotation.
Here, L
is the characteristic dimension.
It is assumed that the Prandtl number Pr =
μc p
λ
is constant. Here, μ, c p , λ
are the coefficients of heat capacity, viscidity, and heat conductivity (Pr = 0.72
in calculations), respectively. Re =
√
p
0 ρ
0 L
μ
0
is the Reynolds number.
The system of differential equations is supplemented by the state equation: p =
ρRT, where T is the temperature, and R is the gas constant. The state equation in a
dimensionless form is as follows: p = ρT.
Thus, the complete system of the Navier–Stokes equations for perfect gas flows
in the absence of external mass forces is given. It is assumed that the heat may enter
the medium only as a result of thermal conductivity. The flows considered below are
subsonic, and the compressibility effects are not of greater importance, but the use of
the compressible gas model allows to apply the numerical method and the programs
[11] developed for the simulation of viscous gas flows.
The use of the Cartesian coordinate system for the adequate description of the
flows with complex topology in solving the problems by the finite-difference methods
seems difficult for two reasons. One reason is that the interpolation procedures to get
boundary conditions are necessary. Also there are some difficulties in the description
of the computational grid because the computational grid is not rectangular. We will