Is it called a two-equation model because it solves two equations for $k$ and $\varepsilon$?

Exactly. Based on the eddy viscosity hypothesis (Boussinesq hypothesis), it approximates the Reynolds stress tensor with an eddy viscosity coefficient $\mu_t$. The form is as follows: $$ -\rho\overline{u_i'u_j'} = \mu_t\left(\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i}\right) - \frac{2}{3}\rho k \delta_{ij} $$

Standard k-epsilon model

Q: Professor, the standard k-ε model is the most famous model in the world of turbulence, right? Could you explain its fundamentals from the beginning?

It is a two-equation turbulence model proposed by Launder and Spalding (1974). It solves transport equations for turbulent kinetic energy $k$ and its dissipation rate $\varepsilon$. It has a history of being the most widely used model in industrial CFD.

Q: What is the form of the transport equation for each, $k$ and $\varepsilon$?

First, the equation for turbulent kinetic energy $k$: $$ \frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho U_j k)}{\partial x_j}=\frac{\partial}{\partial x_j}\left[\left(\mu+\frac{\mu_t}{\sigma_k}\right)\frac{\partial k}{\partial x_j}\right]+P_k-\rho\varepsilon $$

Q: Why does it over-predict swirling flows?

It's because $C_\mu$ is fixed as a constant 0.09. In regions with strong swirl or curvature, the effective $C_\mu$ should be smaller, but the standard model cannot account for this. This is the point improved upon in the Realizable k-ε model.

Category: Fluid Analysis (CFD) | Integrated 2026-04-06

CAE visualization for k epsilon standard theory - technical simulation diagram

Standard k-ε Model

Standard k-epsilon: Theoretical Foundations

Overview

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Professor, the standard k-ε model is the most famous model in the world of turbulence, right? Could you explain the basics to me again from the beginning?

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It's a two-equation turbulence model proposed by Launder and Spalding (1974). It solves the transport equations for turbulent kinetic energy $k$ and its dissipation rate $\varepsilon$. It has a history of being the most widely used model in industrial CFD.

🧑‍🎓

Is it called a two-equation model because it solves two equations, $k$ and $\varepsilon$?

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Exactly. Based on the eddy viscosity hypothesis (Boussinesq hypothesis), it approximates the Reynolds stress tensor with the eddy viscosity coefficient $\mu_t$. In other words, it takes this form.

$$ -\rho\overline{u_i'u_j'} = \mu_t\left(\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i}\right) - \frac{2}{3}\rho k \delta_{ij} $$

Transport Equations

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What do the transport equations for $k$ and $\varepsilon$ look like?

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First, the equation for turbulent kinetic energy $k$.

$$ \frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho U_j k)}{\partial x_j}=\frac{\partial}{\partial x_j}\left[\left(\mu+\frac{\mu_t}{\sigma_k}\right)\frac{\partial k}{\partial x_j}\right]+P_k-\rho\varepsilon $$

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Next, the equation for dissipation rate $\varepsilon$.

$$ \frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial(\rho U_j\varepsilon)}{\partial x_j}=\frac{\partial}{\partial x_j}\left[\left(\mu+\frac{\mu_t}{\sigma_\varepsilon}\right)\frac{\partial\varepsilon}{\partial x_j}\right]+C_{\varepsilon 1}\frac{\varepsilon}{k}P_k - C_{\varepsilon 2}\rho\frac{\varepsilon^2}{k} $$

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Here, the eddy viscosity is defined as follows.

$$ \mu_t = \rho C_\mu \frac{k^2}{\varepsilon} $$

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The production term $P_k$ is written using the mean strain rate tensor $S_{ij}$ as follows.

$$ P_k = \mu_t S^2, \quad S = \sqrt{2S_{ij}S_{ij}} $$

Model Constants

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Please tell me the standard values for each constant.

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The constants for the standard k-ε model are as follows. These are values determined by Launder-Spalding from data such as decaying homogeneous isotropic turbulence and the log law.

Constant	Value	Basis for Determination
$C_\mu$	0.09	Consistency in the log-law region
$C_{\varepsilon 1}$	1.44	Experiments on uniform shear flow
$C_{\varepsilon 2}$	1.92	Decaying grid turbulence experiments
$\sigma_k$	1.0	Empirical value for turbulent diffusion
$\sigma_\varepsilon$	1.3	Empirical value for turbulent diffusion

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Are we not allowed to change these constants?

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As a rule, it's basic practice not to change them. However, the "round jet correction" is known, where $C_{\varepsilon 1}$ is changed to 1.60 to match the spreading rate of a round jet. This is knowledge sometimes used in practical work.

Strengths and Weaknesses

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Could you summarize the strengths and weaknesses of the standard k-ε model?

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Strengths:

Robust and easy to converge
Extensive track record in industrial internal flows (pipes, ducts)
Low computational cost
Easy to set initial conditions (calculate $k$, $\varepsilon$ from turbulence intensity and scale)

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Weaknesses:

Overpredicts turbulent kinetic energy in swirling flows and flows with strong curvature
Cannot accurately capture separation under adverse pressure gradients
Requires wall functions near walls (standard model is high-Reynolds number type)
Unsuitable for strongly anisotropic turbulence due to the limitations of the isotropic eddy viscosity hypothesis

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Why does it overpredict in swirling flows?

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It's because $C_\mu$ is fixed at the constant 0.09. In regions with strong swirl or curvature, the effective $C_\mu$ should be smaller, but the standard model cannot reflect this. This is the point improved upon in the Realizable k-ε model.

Coffee Break Yomoyama Talk

The Weight of the Number C_μ=0.09 — The Perseverance of Launder and Spalding

The constant C_μ=0.09 for the standard k-ε model is a value proposed in a paper published by Brian Launder and D. B. Spalding in 1972. This number was not derived theoretically; it was determined through painstaking fitting to multiple experimental datasets such as pipe turbulence, boundary layers, and backward-facing steps. At the time, supercomputers didn't exist, and numerical calculations were done via batch processing with punch cards. Stories remain of them waiting days for results to come back and then fine-tuning the constants. It was this persevering calibration work that created the trust that leads to the phrase "first, try k-ε" even half a century later.

Computational Methods for Standard k-epsilon

Discretization by Finite Volume Method

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Please explain how the transport equations for the standard k-ε model are solved in a CFD solver.

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In CFD, the Finite Volume Method (FVM) is standard. The transport equations for $k$ and $\varepsilon$ are integrated over cell volumes and converted to fluxes on faces using Gauss's divergence theorem. The general choice of discretization schemes is as follows.

Term	Recommended Scheme	Remarks
Convection term	Second Order Upwind	Suppresses numerical diffusion
Diffusion term	Central Difference	Second-order accuracy is standard
Temporal term (unsteady)	Second Order Implicit	Balances stability and accuracy

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Is first-order upwind no good?

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First-order upwind has large numerical diffusion, significantly degrading prediction accuracy in turbulent fields. It's particularly problematic in separation regions and shear layers. However, a technique sometimes used in practice is to run only the initial few iterations with first-order upwind when convergence is difficult, then switch to second-order.

Coupling with SIMPLE-type Algorithms

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What is the relationship between pressure-velocity coupling and the k-ε equations?

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In pressure-based solvers like SIMPLE, SIMPLEC, PISO, etc., they are solved in the following order within one iteration.

1. Solve momentum equations (tentative update of velocity field)

2. Solve pressure correction equation

3. Correct velocity and pressure

4. Solve $k$ equation

5. Solve $\varepsilon$ equation

6. Update $\mu_t$

7. Convergence check → If not converged, return to 1

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$k$ and $\varepsilon$ are typically solved sequentially using a segregated approach. In density-based solvers (coupled), all variables may be solved simultaneously.

Relaxation Factor Settings

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Are relaxation factors important?

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Very important. The recommended Under-Relaxation Factor (URF) for the standard k-ε model is something like this.

Variable	Recommended URF (Steady)	Remarks
Pressure	0.3	Lower to 0.2 if convergence is slow
Momentum	0.7
$k$	0.8
$\varepsilon$	0.8
Turbulent Viscosity Ratio	1.0	Usually no need to change

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What happens if you lower the relaxation for $k$ or $\varepsilon$ too much?

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Convergence becomes extremely slow. Also, the update of $\mu_t$ is delayed, which can cause oscillations due to misalignment with the velocity field. If divergence is unavoidable, one method is to lower only the URF for $\varepsilon$ to around 0.5.

Boundary Conditions

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How do you set the turbulence boundary conditions at the inlet?

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It's common to calculate them from the Turbulence Intensity $I$ and the turbulent length scale $l$ (or hydraulic diameter $D_H$).

$$ k = \frac{3}{2}(U_{avg} \cdot I)^2 $$

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