k-epsilon and k-omega SST models

Both are two-equation eddy viscosity models: two transport equations give a velocity scale and a length (or time) scale, from which the eddy viscosity $\mu_t$ is found (Boussinesq hypothesis).

Standard k-$\varepsilon$ (Launder-Spalding)

Solves transport equations for turbulent kinetic energy $k$ and its dissipation rate $\varepsilon$:

$$\mu_t=\rho C_\mu\frac{k^2}{\varepsilon},\quad C_\mu=0.09$$ $$\frac{D(\rho k)}{Dt}=\nabla\cdot\left[\left(\mu+\frac{\mu_t}{\sigma_k}\right)\nabla k\right]+P_k-\rho\varepsilon$$ $$\frac{D(\rho\varepsilon)}{Dt}=\nabla\cdot\left[\left(\mu+\frac{\mu_t}{\sigma_\varepsilon}\right)\nabla\varepsilon\right]+C_{1\varepsilon}\frac\varepsilon kP_k-C_{2\varepsilon}\rho\frac{\varepsilon^2}{k}$$

with $C_{1\varepsilon}=1.44,\ C_{2\varepsilon}=1.92,\ \sigma_k=1.0,\ \sigma_\varepsilon=1.3$. Valid for fully turbulent flow, so it uses wall functions near walls.

k-$\omega$ SST (Menter)

A hybrid: the Wilcox k-$\omega$ model near walls, which is switched by blending functions $F_1,F_2$ to a transformed k-$\varepsilon$ model in the free stream. Uses $\omega=\varepsilon/(C_\mu k)$ (specific dissipation rate) and limits the shear stress in adverse pressure gradients through

$$\mu_t=\frac{\rho a_1k}{\max(a_1\omega,\ SF_2)}$$

(the "shear stress transport" limiter, $a_1=0.31$).

Comparison

Neither model captures anisotropy or strong swirl well; for those, RSM or LES is preferred.