Turbulence modeling in RANS (Reynolds-Averaged Navier-Stokes) solves the closure problem by introducing turbulent viscosity \(\mu_t\). We analyze the two most common models:

1. Standard \(k-\epsilon\) Model

This model solves two transport equations: one for turbulent kinetic energy \(k\), and one for its dissipation rate \(\epsilon\). The turbulent viscosity is defined as:

\[\mu_t = \rho C_m \frac{k^2}{\epsilon}\]

It is highly robust for free-shear flows but performs poorly in near-wall regions with adverse pressure gradients, failing to predict separation accurately.

2. Menter\'s \(k-\omega\) Shear Stress Transport (SST)

The SST model uses standard \(k-\omega\) in the near-wall region and transitions to standard \(k-\epsilon\) in the free-stream. A blending function \(F_1\) coordinates this transition:

\[\mu_t = \frac{\rho a_1 k}{\max(a_1 \omega, S F_2)}\]

SST predicts boundary layer separation and shock-induced stall with high fidelity, making it the industry standard for aerodynamics.

References

  • Menter, F. R. (1994). Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 32(8), 1598-1605.
  • Wilcox, D. C. (1998). Turbulence Modeling for CFD. DCW Industries.