kOmegaSST 在 foam-extend-4.0 中的模型为
\rho\frac{\partial{ k}}{\partial t} + \rho\boldsymbol{u}\cdot \nabla k
=
\nabla\cdot \left( \rho D_k \nabla k \right)
- \frac{2}{3} \rho k \left( \nabla \cdot\boldsymbol{u} \right)
+ \rho\min\left( G,\, c_1\beta^{*} \omega k\right)
+ S_k
式中， D_k=\alpha_k \times \nu_t + \nu ， G=\nu_t \mathbf{S}^2
\rho \frac{\partial{\omega}}{\partial t} + \rho\boldsymbol{u}\cdot \nabla (\omega )
=
\nabla\cdot \left( \rho D_\omega \nabla \omega \right)
+ \frac{\rho \gamma \min\left( G,\, c_1\beta^{*} \omega k\right)}{\nu_t}
- \frac{2}{3} \rho \gamma \omega \left( \nabla\cdot \boldsymbol{u} \right)
- \rho \beta \omega^2
- \rho \left(F_1 - 1\right) C\! D_{k\omega}
+ S_\omega
式中，
D_\omega=\alpha_\omega \times \nu_t + \nu\\
F_1 = \tanh(arg_1^4)\\
arg_1 = \min\left\{ \min\left[\max\left(\frac{\sqrt{k}}{\beta^* \omega d},\,\frac{500\nu}{d^2\omega} \right), \,\frac{4\rho\sigma_{\omega2} k}{C\! D_{k\omega}^+ d^2} \right], 10\right\}\\
C\! D_{k\omega}^+ = \max\left(C\! D_{k\omega} ,\, 10^{-10}\right)\\
C\! D_{k\omega} = 2\rho\sigma_{\omega2}\dfrac{1}{\omega}\frac{\partial k}{\partial x_j} \frac{\partial \omega}{\partial x_j}\\
\nu_t = a_1 \frac{k}{\max (a_1 \omega_, b_1 F_{23} \mathbf{S})}\\
F_{2} = \tanh(arg_2^2)\\
arg_2 = \min\left[ \max\left(\frac{2\sqrt{k}}{\beta^* \omega d} ,\, \dfrac{500\nu}{d^2\omega}\right),\, 100\right]\\
F_3 = 1-\tanh(arg_3^4)\\
arg_3 = \min\left(\frac{150\nu}{\omega d^2}, \, 10 \right)\\
F_{23} = F_2 F_3,\, \text{if}\, F_3\neq 0
式中， d 是网格中心点到最近的 wall 的距离， \mathbf{S}=\sqrt{2\left|\dfrac{1}{2}\left(\nabla\boldsymbol{u} + (\nabla\boldsymbol{u})^T \right)\right|^2} ，
对于系数 \alpha_k 和 \alpha_\omega ， \gamma
\alpha_k =\alpha_{k1} F_1 + \alpha_{k2}(1-F_1) ,\\
\alpha_\omega =\alpha_{\omega1} F_1 + \alpha_{\omega2}(1-F_1), \\
\gamma=\gamma_1 F_1 + \gamma_2 (1-F_1).
| α_{k1} | α_{k2} | α_{ω1} | α_{ω2} | β_1 | β_2 | γ_1 | γ_2 | \beta^∗ | a_1 | b_1 | c_1 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.85 | 1.0 | 0.5 | 0.856 | 0.075 | 0.0828 | 5/9 | 0.44 | 0.09 | 0.31 | 1.0 | 10.0 |
k 的估计
k=\frac{3}{2}\left(I u_{ref} \right)^2
其中 I 为湍流强度， u_{ref} 为参考速度。
\omega 的估计
\omega = \frac{k^{0.5}}{C_{\mu} L}
式中， C_μ=0.09 ， L 为参考长度.
在 foam-extend-4.0 版本的 kOmegaSST 中关键的代码如下：
void kOmegaSST::correct()
{
// Bound in case of topological change
// HJ, 22/Aug/2007
if (mesh_.changing())
{
bound(k_, k0_);
bound(omega_, omega0_);
}
RASModel::correct();
if (!turbulence_)
{
return;
}
if (mesh_.changing())
{
y_.correct();
}
const volScalarField S2(2*magSqr(symm(fvc::grad(U_))));
volScalarField G("RASModel::G", nut_*S2);
// Update omega and G at the wall
omega_.boundaryField().updateCoeffs();
const volScalarField CDkOmega
(
(2*alphaOmega2_)*(fvc::grad(k_) & fvc::grad(omega_))/omega_
);
const volScalarField F1(this->F1(CDkOmega));
// Turbulent frequency equation
fvScalarMatrix omegaEqn
(
fvm::ddt(omega_)
+ fvm::div(phi_, omega_)
+ fvm::SuSp(-fvc::div(phi_), omega_)
- fvm::laplacian(DomegaEff(F1), omega_)
==
gamma(F1)
*min
(
S2,
(c1_/a1_)*betaStar_*omega_*max(a1_*omega_, b1_*F23()*sqrt(S2))
)
- fvm::Sp(beta(F1)*omega_, omega_)
- fvm::SuSp
(
(F1 - scalar(1))*CDkOmega/omega_,
omega_
)
);
omegaEqn.relax();
// No longer needed: matrix completes at the point of solution
// HJ, 17/Apr/2012
// omegaEqn.completeAssembly();
solve(omegaEqn);
bound(omega_, omega0_);
// Turbulent kinetic energy equation
fvScalarMatrix kEqn
(
fvm::ddt(k_)
+ fvm::div(phi_, k_)
+ fvm::SuSp(-fvc::div(phi_), k_)
- fvm::laplacian(DkEff(F1), k_)
==
min(G, c1_*betaStar_*k_*omega_)
- fvm::Sp(betaStar_*omega_, k_)
);
kEqn.relax();
solve(kEqn);
bound(k_, k0_); // 限制器
// Re-calculate viscosity
// Fixed sqrt(2) error. HJ, 10/Jun/2015
nut_ = a1_*k_/max(a1_*omega_, b1_*F23()*sqrt(S2));
nut_ = min(nut_, nuRatio()*nu()); // 限制湍流粘度，缺省最大值 nuRatio = 1E+06
nut_.correctBoundaryConditions();
}
bound() 是 OpenFOAM 里面的限制器，文件是 src/finiteVolume/cfdTools/general/bound/bound.C，其部分代码如下：
void Foam::bound(volScalarField& vsf, const dimensionedScalar& vsf0)
{
scalar minVsf = min(vsf).value();
if (minVsf < vsf0.value())
{
Info<< "bounding " << vsf.name()
<< ", min: " << minVsf
<< " max: " << max(vsf).value()
<< " average: " << gAverage(vsf.internalField())
<< endl;
vsf.internalField() = max
(
max
(
vsf.internalField(),
fvc::average(max(vsf, vsf0))().internalField()
// Bug fix: was assuming bound on zero. HJ, 25/Nov/2008
*pos(vsf0.value() - vsf.internalField())
),
vsf0.value()
);
vsf.correctBoundaryConditions();
vsf.boundaryField() = max(vsf.boundaryField(), vsf0.value());
}
}
参考文献
- F. R. Menter, M. Kuntz, R. Langtry. Ten Years of Industrial Experience with the SST Turbulence Model, in in Turbulence, Heat and Mass Transfer 4, 2003, pp. 625-632
- https://turbmodels.larc.nasa.gov/sst.html