The turbulent kinetic energy transport equation, \(k\), [[41], Eq. 3]:
\[ \partial_t k + u_i \partial_{x_i} k = P - \epsilon + \partial_{x_j} \left\{ \left( \nu + \frac{\nu_t}{\sigma_k} \right) \partial_{x_j} k \right\} \]
where
| \(k \) | = | Turbulent kinetic energy [ \(\text{m}^2 \text{s}^{-3}\)] |
| \(P \) | = | Turbulent kinetic energy production rate [ \(\text{m}^2 \text{s}^{-3}\)] |
| \(\epsilon \) | = | Turbulent kinetic energy dissipation rate [ \(\text{m}^2 \text{s}^{-3}\)] |
| \(\nu \) | = | Kinematic viscosity of fluid [ \(\text{m}^2 \text{s}^{-1}\)] |
| \(\nu_t \) | = | Turbulent viscosity [ \(\text{m}^2 \text{s}^{-1}\)] |
| \(\sigma_k \) | = | Turbulent Prandtl number for \(k\) [-] |
The turbulent kinetic energy dissipation rate transport equation, \(\epsilon\), [[41], Eq. 4]:
\[ \partial_t \epsilon + u_i \partial_{x_i} \epsilon = \frac{C_{\epsilon_1} P}{T} - \frac{C_{\epsilon_2} \epsilon}{T} + \partial_{x_j} \left\{ \left( \nu + \frac{\nu_t}{\sigma_\epsilon} \right) \partial_{x_j} \epsilon \right\} \]
where
| \(T \) | = | Modelled turbulent time scale [ \(s\)] |
| \(C_{\epsilon_*} \) | = | Empirical model constants [-] |
| \(\sigma_\epsilon \) | = | Turbulent Prandtl number for \(\epsilon\) [-] |
The normalised wall-normal fluctuating velocity scale transport equation, \(\phi\), [[41], Eq. 17]:
\[ \partial_t \phi + u_i \partial_{x_i} \phi = f - P \frac{\phi}{k} + \frac{2 \nu_t}{k \sigma_k} \partial_{x_j} \phi \partial_{x_j} k + \partial_{x_j} \left\{ \left( \frac{\nu_t}{\sigma_k} \right) \partial_{x_j} \phi \right\} \]
where
| \(f \) | = | Elliptic relaxation factor [ \(s^{-1}\)] |
| \(\phi \) | = | Normalised wall-normal fluctuating velocity scale [-] |
| \(\sigma_\phi \) | = | Turbulent Prandtl number for \(\phi\) [-] |
The elliptic relaxation factor equation, \(f\), [[41], Eq. 18]:
\[ L^2 \partial^2_{x_j} f - f = \frac{1}{T} (C_1 - 1) \left[ \phi - \frac{2}{3} \right] - C_2 \frac{P}{k} - 2 \frac{\nu }{k} \partial_{x_j} \phi \partial_{x_j} k - \nu \partial^2_{x_j} \phi \]
where
| \(L \) | = | Modelled turbulent length scale [ \(m\)] |
| \(C_{f_*} \) | = | Empirical model constants [-] |
The turbulent time scale equation, \(T\), [[41], Eq. 7]:
\[ T = \max \left[ \frac{k}{\epsilon}, 6.0 \sqrt{\frac{\nu}{\epsilon}} \right] \]
The turbulent length scale equation, \(L\), [[41], Eq. 7]:
\[ L = C_L \max \left[ \frac{k^{1.5}}{\epsilon}, C_\eta \left( \frac{\nu^3}{\epsilon} \right)^{0.25} \right] \]
where
| \(C_L \) | = | Empirical model constant [-] |
| \(C_\eta \) | = | Empirical model constant [-] |
The turbulent viscosity equation, \(\nu_t\), [[41], p. 173]:
\[ \nu_t = C_\mu \phi k T \]
where
| \(C_{\mu} \) | = | Model coefficient for the turbulent viscosity [-] |
| \(\nu_t \) | = | Turbulent viscosity [ \(\text{m}^2 \text{s}^{-1}\)] |
The main differences between the implemented and original governing equations are as follows:
nu into DphitEff even though it is not present in [[41], Eq. 17] (This change has provided higher level of resemblance to benchmarks for the tests considered, particularly for the peak skin friction, yet pressure-related predictions were unaffected. Users can switch off nu in DphitEff by using includeNu entry in kEpsilonPhitFCoeffs as shown below in order to follow the reference paper thereat. includeNu is left true by default.)The implementation of the \(\epsilon\) transport equation:
\[ \ddt{\alpha \rho \epsilon} + \div \left(\alpha \rho \u \epsilon \right) - \laplacian \left(\alpha \rho D_\epsilon \epsilon \right) = \alpha \rho C_{\epsilon_1} \frac{G}{T} - \left( \left\{ \frac{2}{3} C_{\epsilon_1} \right\} \left\{ \alpha \rho \div \u \right\} \epsilon \right) - \left( \alpha \rho \frac{C_{\epsilon_2}}{T} \epsilon \right) + S_{\text{fvOptions}} \]
where
| \(\alpha \) | = | Phase fraction of the given phase [-] |
| \(\rho \) | = | Density of the fluid [ \( \text{kg} m^{-3} \)] |
| \(S_{\text{fvOptions}} \) | = | Source terms introduced by fvOptions dictionary |
The implementation of the \(k\) transport equation:
\[ \ddt{ \alpha \rho k } + \div \left( \alpha \rho \u k \right) - \laplacian \left( \alpha \rho D_k k \right) = \alpha \rho G - \left( \left[ \frac{2}{3} \alpha \rho \div \u k \right] \right) - \left( \frac{\alpha \rho}{T} k \right) + S_{\text{fvOptions}} \]
where
| \(G \) | = | Anisotropic contribution part of the turbulent kinetic energy production rate, i.e. \( P = G - 2/3 k \div \u \) |
The implementation of the \(f\) elliptic relaxation equation:
\[ - \laplacian f = - \left( \frac{1}{L^2} f \right) - \left( (C_{f_1} - 1) \frac{\phi - \frac{2}{3}}{T} - \frac{C_{f_2} G}{k} + C_{f_2} \frac{2}{3} \div \u - \frac{2 \nu (\nabla \phi \cdot \nabla k )}{k} - \nu \laplacian \phi \right)\frac{1}{L^2} \]
The implementation of the \(\phi\) transport equation:
\[ \ddt{\alpha \rho \phi} + \div \left(\alpha \rho \u \phi \right) - \laplacian \left(\alpha \rho D_\phi \, \phi \right) = \alpha \rho f - \left( \alpha \rho \left\{ \frac{G}{k} - \frac{2}{3} \div \u - \frac{2 \nu (\nabla \phi \cdot \nabla k )}{k \sigma_k \phi } \right\} \phi \right) + S_{\text{fvOptions}} \]
The implementation of the \( T \) equation:
\[ T = \max \left( \frac{k}{\epsilon}, C_T \frac{\sqrt{\max\{\nu, 0\}}}{\epsilon} \right) \]
The implementation of the \(L\) equation:
\[ L = C_L \max \left( \frac{k^{1.5}}{\epsilon}, C_\eta \left[ \frac{(\max\{\nu, 0\})^3}{\epsilon} \right]^{0.25} \right) \]
The model coefficients are [[41], Eqs. 19-20]:
\[ C_\mu = 0.22; \quad C_{\epsilon_1} = 1.4(1.0 + 0.05\sqrt{1.0/\phi} ); \quad C_{\epsilon_2} = 1.9; \quad C_T = 6.0; \quad C_L = 0.25;\\ C_{f_1} = 1.4; \quad C_{f_2} = 0.3; \quad C_\eta = 110.0; \quad \sigma_k = 1.0; \quad \sigma_\epsilon = 1.3; \quad \sigma_\phi = 1.0. \]
Please note that [[41], p. 176] stated that the model constants above have been calibrated to match the flow statistics of the smooth-wall plane channel flow at \(\text{Re}_\tau = 395\).
The model can be enabled by using constant/turbulenceProperties dictionary:
simulationType RAS;
RAS
{
// Mandatory entries
RASModel kEpsilonPhitF;
// Optional entries
turbulence on;
printCoeffs on;
kEpsilonPhitFCoeffs
{
includeNu true; // include nu in DphitEff()
Cmu 0.22; // Turbulent viscosity constant
Ceps1a 1.4; // Model constant for epsilon
Ceps1b 1.0; // Model constant for epsilon
Ceps1c 0.05; // Model constant for epsilon
Ceps2 1.9; // Model constant for epsilon
Cf1 1.4; // Model constant for f
Cf2 0.3; // Model constant for f
CL 0.25; // Model constant for L
Ceta 110.0; // Model constant for L
CT 6.0; // Model constant for T
sigmaK 1.0; // Turbulent Prandtl number for k
sigmaEps 1.3; // Turbulent Prandtl number for epsilon
sigmaPhit 1.0; // Turbulent Prandtl number for phit = sigmaK
}
Please note that \( \phi \) is a symmetric, and \( f \) is an asymmetric matrix.
For \( k \) and \( \epsilon \) fields, the initial conditions can be estimated by using the recommendations made for the kEpsilon model.
For \( \phi \):
object phit; dimensions [0 0 0 0 0 0 0]; internalField uniform 0.66;
For \( f \):
object f; dimensions [0 0 -1 0 0 0 0]; internalField uniform 1.0;
For \( k \) and \( \epsilon \) fields, please use the boundary conditions that are available for the kEpsilon model.
For \( \phi \) and \( f \):
Inlet/Outlet:
Walls:
For example, \( \phi \):
object phit;
dimensions [0 0 0 0 0 0 0];
wall
{
type fixedValue;
value uniform 1e-10;
}
"(inlet|outlet)"
{
type zeroGradient;
}
For example, \( f \):
object f;
dimensions [0 0 -1 0 0 0 0];
wall
{
type fixedValue;
value uniform 0;
}
"(inlet|outlet)"
{
type zeroGradient;
}
Source code:
References:
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