KepsilonAlgorithm

turbulence modeling using the k-epsilon model

The K-epsilon turbulence model is one of the common models for including turbulence into a CFD simulation. In MESHFREE, it can be incorporated for chambers of LIQUID and GASDYN type in the KindOfProblem definition. For all boundary elements also boundary conditions BC_eps and BC_k must be defined. Also positive initial values for k and epsilon must be chosen, e.g.

KOP(iChamber) = ... TURBULENCE:k-epsilon ... # necessary to invoke the turbulence model, works only for LIQUID and GASDYN INITDATA($MAT$,%ind_k%) = 1.0e-6 INITDATA($MAT$,%ind_eps%) = 1.0e-6 BC_k($whatEver$) = (...) BC_eps($whatEver$) = (...)

Optinally we have the models

# k-epsilon variants KOP(iChamber) = ... TURBULENCE:k-epsilon ... # trigger k-epsilon turbulence model, implicit Euler time integration, works only for LIQUID and GASDYN KOP(iChamber) = ... TURBULENCE:k-epsilon:explicit ... # trigger k-epsilon turbulence model, explicit Euler time integration, works only for LIQUID and GASDYN KOP(iChamber) = ... TURBULENCE:k-epsilon:BDF2 ... # trigger k-epsilon turbulence model, BDF2 time integration, works only for LIQUID and GASDYN KOP(iChamber) = ... TURBULENCE:k-epsilon:RNG ... # trigger k-epsilon turbulence model KOP(iChamber) = ... TURBULENCE:k-epsilon:RNG:explicit ... # trigger k-epsilon turbulence model KOP(iChamber) = ... TURBULENCE:k-epsilon:RNG:BDF2 ... # trigger k-epsilon turbulence model KOP(iChamber) = ... TURBULENCE:k-epsilon:realizable ... # trigger k-epsilon turbulence model KOP(iChamber) = ... TURBULENCE:k-epsilon:realizable:explicit ... # ttrigger k-epsilon turbulence model KOP(iChamber) = ... TURBULENCE:k-epsilon:realizable:BDF2 ... # trigger k-epsilon turbulence model # k-omega variants KOP(iChamber) = ... TURBULENCE:k-omega:wilcox ... # trigger k-omega-model, implicit Euler time integration KOP(iChamber) = ... TURBULENCE:k-omega:wilcox:explicit ... # trigger k-omega-model, explicit Euler time integration KOP(iChamber) = ... TURBULENCE:k-omega:wilcox:BDF2 ... # trigger k-omega-model, BDF2 time integration KOP(iChamber) = ... TURBULENCE:k-omega:mentersst ... # trigger k-omega-model KOP(iChamber) = ... TURBULENCE:k-omega:mentersst:explicit ... # trigger k-omega-model KOP(iChamber) = ... TURBULENCE:k-omega:mentersst:BDF2 ... # trigger k-omega-model

k-omega-modelling

The k-omega model is, numerically, mapped into the k-epsilon solution procedure of MESHFREE. There is a unique way of mathematical mapping, decribed in DOCUMATH_NumericalIntegrationOfTurbulence.pdf Thus, if "TURBULENCE:k-omega" is chosen, MESHFREE produces results for %ind_k% and %ind_eps% , indeed. In order to reconstruct the values of omega, it is simply done by the mathematical correspondance \begin{align} \omega = \frac{1}{c_\mu} \frac{\epsilon}{k}\end{align} hence one could simply self-compute omega by

CODI_eq($someMaterial$,%indU_omega%) = [ Y%ind_eps% / ( Y%ind_KEPS_cMue% * Y%ind_k% ) ]

More Information

See DOCUMATH_NumericalIntegrationOfTurbulence.pdf for a detailed discussion of how MESHFREE incorporates the k-epsilon turbulence model. For some specific derivation of the heat source triggered by turbulence, see DOCUMATH_DerivationOfEnergyEquationWithTurbulence.pdf .

Relevant Indices

%ind_k% :: turbulent kinetic energy
%ind_eps% :: dissipation rate
%ind_NUE_turb% :: turbulent viscosity, $ \mu_{turb}= C_\mu \frac{k^2}{\epsilon}$, section 1.1 in DOCUMATH_NumericalIntegrationOfTurbulence.pdf
%ind_ETA_eff% :: effective viscosity $ \eta_{eff} = \eta + \rho \mu_{turb} = \eta_{laminar} + \eta_{turb}$
%ind_tauW% :: effective turbulent wall stress $ \tau_W$ , section 1.6 and 1.7 in DOCUMATH_NumericalIntegrationOfTurbulence.pdf
%ind_tauWv(1)%, %ind_tauWv(2)%, %ind_tauWv(3)% :: vector of the effective turbulent wall stress $ {\mathbf \tau}_W$, with the correlation $ \| \mathbf{\tau}_W \| = \tau_W$, see section 1.6 and 1.7 in DOCUMATH_NumericalIntegrationOfTurbulence.pdf
%ind_KEPS_cMue% :: value of $ C_\mu$ computed by the particular turbulence model in use. classical k-epsilon model: $ C_\mu = 0.09$
%ind_KEPS_C1% :: effective value of $ C_{1 \epsilon}$ computed by the particular turbulence model in use. classical k-epsilon model: $ C_{1 \epsilon} = 1.44$ . Please see the PDE for $ \epsilon$ in DOCUMATH_NumericalIntegrationOfTurbulence.pdf .
%ind_KEPS_C2% :: effective value of $ C_{2 \epsilon}$ computed by the particular turbulence model in use. classical k-epsilon model: $ C_{2 \epsilon} = 1.92$ . Please see the PDE for $ \epsilon$ in DOCUMATH_NumericalIntegrationOfTurbulence.pdf .
%ind_KEPS_G% :: effective value of the turbulent production basis $ G = S^2$ . Especially, use this variable to poistprocess the averaged/net production values along the turbulent wall boundaries.

Additional constraints and restrictions

Compressibility constraint by Sarkar and Lakshmanan :: see CompressibilitySarkarAndLakshmanan
Realizibility constraint by Durbin :: see MESHFREE.Solvers.Numerics.LIQUID.Algorithms.KepsilonAlgorithm.RealizabilityDurbin

List of members:
CompressibilitySarkarAndLakshmanan
RealizabilityByDurbin	Realizability constraint by Durbin