14 GE \(k-\omega\) Turbulence Model
Source-aligned revision. This chapter has been rebuilt from the current GE-RANS implementation. The earlier placeholder compressibility expression has been removed. The equations below distinguish the active SST-like core, optional low-Reynolds modifications, Kato-Launder production, production limiting, curvature corrections, free-stream modification and axisymmetric terms.
14.1 Coupled transport equations
The implemented turbulence block transports turbulent kinetic energy \(k\) and specific dissipation rate \(\omega\):
\[\frac{\partial(\rho k)}{\partial t}+\nabla\cdot(\rho\mathbf u k)=\nabla\cdot(\Gamma_k\nabla k)+G_k-Y_k.\](14.1)\[\frac{\partial(\rho\omega)}{\partial t}+\nabla\cdot(\rho\mathbf u\omega)=\nabla\cdot(\Gamma_\omega\nabla\omega)+G_\omega-Y_\omega+D_\omega+Q_{SAS}.\](14.2)\(Q_{SAS}=0\) unless the optional scale-adaptive extension is enabled; its exact implementation is documented in Chapter 15.
14.2 SST blending and local coefficients
\[Re_t=\frac{\rho k}{\mu\omega}.\](14.3)The cross-diffusion sensor used by the first blending function is
\[CD_{k\omega}=\max\!\left(\frac{\nabla k\cdot\nabla\omega}{\omega},10^{-10}\right).\](14.4)\[F_1=\tanh\!\left\{\left[\min\!\left(\max\!\left(\frac{\sqrt{k}}{0.09\omega d},\frac{500\nu}{d^2\omega}\right),\frac{2k}{d^2CD_{k\omega}}\right)\right]^4\right\}.\](14.5)\[F_2=\tanh\!\left\{\left[\max\!\left(\frac{2\sqrt{k}}{\beta_{s2}\omega d},\frac{500\nu}{d^2\omega}\right)\right]^2\right\},\qquad \beta_{s2}=0.09.\](14.6)The standard blended coefficients are \(\sigma_k=F_1\sigma_{k1}+(1-F_1)\sigma_{k2}\) and \(\sigma_\omega=F_1\sigma_{\omega1}+(1-F_1)\sigma_{\omega2}\), with current constants \(\sigma_{k1}=0.85,\sigma_{k2}=1,\sigma_{\omega1}=0.5,\sigma_{\omega2}=0.856\).
\[\Gamma_k=\mu+\sigma_k\mu_t,\qquad \Gamma_\omega=\mu+\sigma_\omega\mu_t.\](14.7)14.2.1 Optional low-Reynolds coefficient modification
With the low-Reynolds option off, \(\alpha_s=1\) and \(\beta_s=0.09\) because both blended endpoint values are equal. When it is on, the near-wall endpoint coefficients are modified using \(Re_t\):
\[\alpha_{s1}^{LR}=\frac{\beta/3+Re_t/Re_k}{1+Re_t/Re_k},\qquad Re_k=6,\qquad \alpha_s=F_1\alpha_{s1}^{LR}+(1-F_1)\alpha_{s2}.\](14.8)\[\beta_{s1}^{LR}=0.09\frac{4/15+(Re_t/Re_\beta)^4}{1+(Re_t/Re_\beta)^4},\qquad Re_\beta=8,\qquad \beta_s=F_1\beta_{s1}^{LR}+(1-F_1)\beta_{s2}.\](14.9)14.3 Turbulent time scale and eddy viscosity
CMPS first evaluates the strain magnitude. In planar flow, \(S^2=2\mathbf S:\mathbf S\). In axisymmetric flow, the radial contribution is added:
\[S_{axi}^2=S_{2D}^2+2\left(\frac{u_r}{r}\right)^2.\](14.10)\[T_{SST}=\min\!\left(\frac{\alpha_s}{\omega},\frac{a_1}{SF_2}\right),\qquad a_1=0.31.\](14.11)When the realizable time-scale option is enabled, an additional upper limit is used:
\[T=\min\!\left(T_{SST},\frac{C_T}{\sqrt3\,S}\right),\qquad C_T=0.6.\](14.12)Otherwise \(T=T_{SST}\). Eddy viscosity follows directly:
\[\mu_t=\rho kT.\](14.13)If the resulting viscosity ratio exceeds the configured maximum, the implementation raises the effective \(\omega\) and reevaluates \(F_2,T,\mu_t\):
\[\omega\leftarrow\frac{\rho k\alpha_s}{\mu(\mu_t/\mu)_{max}}.\](14.14)14.4 Turbulence production
Away from the wall-function cell treatment, the baseline production is
\[G_k^{(0)}=\mu_tS^2.\](14.15)When Kato-Launder production is enabled, the strain-vorticity form is used instead:
\[G_k^{(0)}=\mu_tS\Omega,\](14.16)where \(\Omega\) is the stored vorticity magnitude. If the SSTm production option is off, the dilatation correction is then applied:
\[G_k^{(1)}=G_k^{(0)}-\frac23\left(\rho k\,\nabla\cdot\mathbf u+\mu_t(\nabla\cdot\mathbf u)^2\right).\](14.17)For axisymmetric flow, the divergence in this expression is \(\nabla\cdot\mathbf u=(\nabla\cdot\mathbf u)_{2D}+u_r/r\). The optional production limiter and nonnegativity enforcement give
\[G_k=\max\!\left(0,\min\!\left(G_k^{(1)},10Y_k\right)\right).\](14.18)If the production limiter is disabled, only the final nonnegative bound remains. Near walls, CMPS uses the wall-function production based on wall shear rather than this cell-center strain formula.
14.5 Dissipation, omega production and free-stream modification
Without free-stream modification,
\[Y_k=\rho\beta_s k\omega.\](14.19)With free-stream modification, the stored ambient values \(k_0,\omega_0\) shift the decay:
\[Y_k=\rho\beta_s(k\omega-k_0\omega_0).\](14.20)The specific-dissipation production is built from the already limited \(G_k\):
\[G_\omega=\gamma\rho\frac{G_k}{\mu_t}.\](14.21)With low-Reynolds modification off, \(\gamma=F_1(5/9)+(1-F_1)0.44\). With it on, the near-wall endpoint becomes
\[\gamma_1^{LR}=\frac{5}{9}\frac{1/9+Re_t/Re_\omega}{(1+Re_t/Re_\omega)\alpha_s},\qquad Re_\omega=2.95,\qquad \gamma=F_1\gamma_1^{LR}+(1-F_1)0.44.\](14.22)The omega dissipation coefficient is \(\beta=F_1(0.075)+(1-F_1)(0.0828)\). Therefore
\[Y_\omega=\rho\beta\omega^2\](14.23)without free-stream modification, and
\[Y_\omega=\rho\beta(\omega^2-\omega_0^2)\](14.24)when it is enabled.
14.6 Cross diffusion
\[D_\omega=2(1-F_1)\rho\sigma_{\omega2}\frac{\nabla k\cdot\nabla\omega}{\omega}.\](14.25)14.7 Curvature corrections
14.7.1 Smirnov-Menter production correction
The source forms strain and rotation tensors
\[\mathbf S=\frac12(\nabla\mathbf u+\nabla\mathbf u^T),\qquad \mathbf\Omega=\frac12(\nabla\mathbf u-\nabla\mathbf u^T).\](14.26)\[r^*=\frac{|S|}{|\Omega|+\epsilon},\qquad D=\frac12(|S|^2+|\Omega|^2)+\epsilon.\](14.27)Using the steady material strain derivative \(D\mathbf S/Dt\), the rotation sensor is
\[\widetilde r=\frac{2(\mathbf\Omega\mathbf S):(D\mathbf S/Dt)}{D^3(|\Omega|+\epsilon)}.\](14.28)\[f_{r1}=(1+C_{r1})\frac{2r^*}{1+r^*}\left[1-C_{r3}\arctan(C_{r2}\widetilde r)\right]-C_{r1}.\](14.29)\[F_{rot}=\max\!\left(0,\min(C_{max},f_{r1})\right),\qquad G_k\leftarrow F_{rot}G_k.\](14.30)The current defaults are \(C_{r1}=1,C_{r2}=12,C_{r3}=1,C_{max}=1.25\). Implementation note: the audited helper currently constructs a 2-by-2 internal velocity-gradient tensor for this correction; the manual therefore does not claim a separately verified full-3D Smirnov-Menter implementation.
14.7.2 Hellsten omega-dissipation correction
The alternative curvature option modifies omega dissipation through
\[R=\frac{\Omega}{S},\qquad Ri=R(R-1),\qquad F_4=\frac{1}{1+1.4Ri},\qquad Y_\omega\leftarrow F_4Y_\omega.\](14.31)14.8 Turbulent heat and species transport
The turbulence closure also supplies the turbulent conductivity and species diffusivity used by the carrier transport equations:
\[k_t=\frac{\mu_t C_p}{Pr_t},\qquad Pr_t=0.85\ \text{by default}.\](14.32)\[D_t=\frac{\mu_t}{\rho Sc_t},\qquad Sc_t=0.9.\](14.33)The viscous face operator uses harmonic interpolation of effective viscosity/conductivity between the two cells and retains the compact non-orthogonal correction plus the strong two-point implicit contribution.
14.9 Axisymmetric turbulence source terms
Besides the strain/divergence additions already shown, the radial momentum source includes modified-pressure and viscous-dilatation contributions. With cell volume \(V\), radius \(r\) and \(\mu_{eff}=\mu+\mu_t\),
\[S_{r,k}=-V\frac{2}{3r}\rho k.\](14.34)\[S_{r,\mu}=-V\frac{2}{3r}\mu_{eff}\left[(\nabla\cdot\mathbf u)_{2D}+\frac{u_r}{r}\right].\](14.35)The circumferential viscous stress contribution used by the energy equation is
\[\tau_{\theta\theta}=\mu_{eff}\left[2\frac{u_r}{r}-\frac23\left((\nabla\cdot\mathbf u)_{2D}+\frac{u_r}{r}\right)\right],\](14.36)\[S_E^{axi}=-V\frac{u_r\tau_{\theta\theta}}{r}.\](14.37)14.10 Turbulent viscous/thermal energy flux
The viscous face energy flux contains thermal conduction and, when viscous dissipation is enabled, viscous work. In compact continuum notation the corresponding energy contribution is
\[\mathbf F_E^{visc}=-k_{eff}\nabla T+\boldsymbol\tau_{eff}\cdot\mathbf u,\qquad k_{eff}=k_l+k_t.\](14.38)The implemented face discretization uses harmonic interpolation of the effective conductivity and the arithmetic face velocity in the viscous-work term. Viscous dissipation is enabled by default in the current solver options.
14.11 Inlet and initialization relations
At an inflow face the current implementation uses the boundary-normal transport speed \(V_n=\dot m/(\rho_B A_f)\). If turbulent kinetic energy is not prescribed directly, the intensity route gives
\[k_B=\frac32(V_n I)^2.\](14.39)The specific-dissipation variable can then be prescribed directly, inferred from a turbulence length scale, or inferred from a prescribed turbulent-viscosity ratio:
\[\omega_B=\omega_{prescribed}.\](14.40)\[\omega_B=\frac{\sqrt{k_B}}{C_\mu^{1/4}L_t},\qquad C_\mu=0.09.\](14.41)\[\omega_B=\frac{\rho_B k_B}{\mu\,(\mu_t/\mu)_{prescribed}}.\](14.42)The inlet turbulence flux also contributes the isotropic turbulent pressure \(p_t=2\rho_Bk_B/3\). For a far-field inlet, the energy flux includes the associated turbulent-pressure work in addition to turbulent-energy transport.