A new eddy viscosity model is formulated. To remove the problem of previous eddy viscosity models that infer the turbulent length scale from the gradient of the mean velocity, the present model adopts the turbulent kinetic energy equation to find the turbulent length scale. Developed turbulence model is a two-equation model that consists of transport equations for the eddy viscosity and the turbulent kinetic energy. Unlike the dissipation rate and the turbulent kinetic energy, it is difficult to derive the exact transport equation of the eddy viscosity from the Navier-Stokes equations. In this study, the transport equation of the eddy viscosity is obtained by a transformation of Wilcox``s κ-ω model. The closure coefficients appearing in the new model equations are determined based on experiments and numerical optimizations.
The eddy viscosity and the turbulent kinetic energy are zero at the wall. And the model equations should be modified to yield the asymptotic behaviors of $k~y^2$ and $v_t~y^3$ in the near-wall region. In previous eddy viscosity models, such asymptotic behaviors were ascertained by adopting appropriate damping functions in the near-wall region. However, in this study,
$v(∂\sqrt{k}/∂y)^2$ and $ν(∂\sqrt{ν_t}/∂y)^2$ are added to model equations to guarantee such behaviors. This new $ν_t-κ$ model is applied to fully developed channel flows and flat plate boundary layers and the results show good agreements with the experimental data. In the simulation of the Samuel and Joubert``s experiment with increasing adverse pressure gradients, the present model demonstrates better predictions of friction coefficients and mean velocities than the κ-ω model and the Spalart-Allmaras model.
In addition, in order to find problems in applying the present model to free shear flows, a plane far-wake, a two-dimensional mixing layer, a plane jet, and a round jet were calculated. Spreading rates of all free shear flows were
over-predicted. It is well known th...