... ... @@ -24,8 +24,57 @@ * * \copydoc RANSModel * * These models calculate the eddy viscosity with two additional PDEs, * one for the turbulent kinetic energy (k) and for the dissipation (epsilon). * The low-Reynolds k-epsilon models calculate the eddy viscosity with two additional PDEs, * one for the turbulent kinetic energy (k) and for the dissipation (\f$\varepsilon \f$). * The model uses the one proposed by Chien \cite Chien1982a. * A good overview and additional models are given in Patel et al. \cite Patel1985a. * * The turbulent kinetic energy balance is identical with the one from the k-epsilon model, * but the dissipation includes a dampening function (\f$D_\varepsilon \f$): * \f$\varepsilon = \tilde{\varepsilon} + D_\varepsilon \f$: * * \f[ * \frac{\partial \left( k \right)}{\partial t} * + \nabla \cdot \left( \textbf{v} k \right) * - \nabla \cdot \left( \left( \nu + \frac{\nu_\text{t}}{\sigma_\text{k}} \right) \nabla k \right) * - 2 \nu_\text{t} \textbf{S} \cdot \textbf{S} * + \tilde{\varepsilon} * + D_\varepsilon * = 0 * \f]. * * The dissipation balance is changed by introducing additional functions * (\f$E_\text{k}\f$, \f$f_1 \f$, and \f$f_2 \f$) to account for a dampening towards the wall: * \f[ * \frac{\partial \left( \tilde{\varepsilon} \right)}{\partial t} * + \nabla \cdot \left( \textbf{v} \tilde{\varepsilon} \right) * - \nabla \cdot \left( \left( \nu + \frac{\nu_\text{t}}{\sigma_{\varepsilon}} \right) \nabla \tilde{\varepsilon} \right) * - C_{1\tilde{\varepsilon}} f_1 \frac{\tilde{\varepsilon}}{k} 2 \nu_\text{t} \textbf{S} \cdot \textbf{S} * + C_{2\tilde{\varepsilon}} f_2 \frac{\tilde{\varepsilon}^2}{k} * - E_\text{k} * = 0 * \f]. * * The kinematic eddy viscosity \f$\nu_\text{t} \f$ is dampened by \f$f_\mu \f$: * \f[ * \nu_\text{t} = C_\mu f_\mu \frac{k^2}{\tilde{\varepsilon}} * \f]. * * The auxiliary and dampening functions are defined as: * \f[ D_\varepsilon = 2 \nu \nicefrac{k}{y^2} \f] * \f[ E_\text{k} = -2 \nu \frac{\tilde{\varepsilon}}{y^2} \exp \left( -0.5 y^+ \right) \f] * \f[ f_1 = 1 \f] * \f[ f_2 = 1 - 0.22 \exp \left( - \left( \frac{\mathit{Re}_\text{t}}{6} \right)^2 \right) \f] * \f[ f_\mu = 1 - \exp \left( -0.0115 y^+ \right) \f] * \f[ \mathit{Re}_\text{t} = \frac{k^2}{\nu \tilde{\varepsilon}} \f] * . * * Finally, the model is closed with the following constants: * \f[ \sigma_\text{k} = 1.00 \f] * \f[ \sigma_\varepsilon =1.30 \f] * \f[ C_{1\tilde{\varepsilon}} = 1.35 \f] * \f[ C_{2\tilde{\varepsilon}} = 1.80 \f] * \f[ C_\mu = 0.09 \f] */ #ifndef DUMUX_LOWREKEPSILON_MODEL_HH ... ...