---
title: "k-epsilon and k-omega SST models"
subject: "Applied Computational Fluid Dynamics"
chapter: 5
key: "semester_7/applied_computational_fluid_dynamics:k-epsilon-k-omega-sst-models"
source: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/k-epsilon-k-omega-sst-models
site: BCT Engineering
---

# k-epsilon and k-omega SST models

Applied Computational Fluid Dynamics · Chapter 5: Turbulence Modeling

- Asked in: year not given, 6 marks
- Video: [k-ε vs k-ω SST | When to Use Which Turbulence Model? | CFD Simplified](https://www.youtube.com/watch?v=L_1Gh7VGI8k) — CFD Simplified
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/k-epsilon-k-omega-sst-models
- Whole chapter: https://www.bctengineer.com/md/semester_7/applied_computational_fluid_dynamics/chapter-5

## Question

Explain the standard k-$\varepsilon$ and the k-$\omega$ SST turbulence models. Differentiate between them with respect to equations, near-wall treatment, strengths and limitations.

## Short answer

Standard k-ε solves transport for k and ε, uses wall functions, and is robust for fully turbulent flows but poor in separation. SST k-ω blends k-ω near-wall integration (y+≈1) with k-ε in the free stream, includes a shear-stress limiter, and captures adverse pressure gradients and separation much better.

## Answer

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

| Point | Standard k-$\varepsilon$ | k-$\omega$ SST |
|---|---|---|
| Variables | $k,\ \varepsilon$ | $k,\ \omega$ |
| Near wall | Needs wall functions ($y^+$ 30-300) | Integrates to the wall ($y^+\approx1$); also works with wall functions |
| Adverse pressure gradient, separation | Poor; delays separation | Good; shear stress limiter |
| Free-stream sensitivity | Low | Low (due to blending) |
| Cost and robustness | Robust, cheap, widely used | Slightly costlier, more mesh needed at the wall |
| Best for | Fully turbulent internal flow, mixing, jets, industrial flows | External aerodynamics, boundary layers, turbomachinery, heat transfer |
| Limitations | Over-predicts k in stagnation regions; poor for swirl, strong curvature, low-$Re$ | Needs fine near-wall mesh; still isotropic eddy viscosity |

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

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How does the near-wall treatment differ between standard k-epsilon and k-omega SST?
   A. k-epsilon integrates to the wall; SST needs wall functions
   B. k-epsilon uses wall functions (y+ 30-300); SST can integrate to the wall (y+ about 1)
   C. Both require y+ above 300
   D. Neither uses y+
   Answer: B. Standard k-epsilon is for fully turbulent flow and uses wall functions; SST uses Wilcox k-omega near the wall and can resolve it. (Key point: Near-wall treatment)

2. What does k-omega SST do?
   A. Uses only k-epsilon everywhere
   B. Blends Wilcox k-omega near walls to transformed k-epsilon in the free stream, and limits shear stress in adverse pressure gradients
   C. Solves all six Reynolds stress transport equations
   D. Uses a zero-equation mixing length
   Answer: B. Blending functions F1 and F2 switch between models, and the shear stress limiter improves separation prediction. (Key point: SST hybrid nature)

3. Which is correct about adverse pressure gradients and separation?
   A. Standard k-epsilon predicts separation well; SST delays it
   B. SST predicts it well; k-epsilon is poor and delays separation
   C. Both are excellent
   D. Neither is affected
   Answer: B. k-epsilon delays separation, while SST's shear-stress limiter handles adverse pressure gradients better, at higher near-wall mesh cost. (Key point: Strengths and limits)

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