Standard k-epsilon model
Theory and Physics
Overview
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?
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 $\varepsilon$?
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.
Transport Equations
What do the transport equations for $k$ and $\varepsilon$ look like?
First, the equation for turbulent kinetic energy $k$.
Next, the equation for dissipation rate $\varepsilon$.
Here, the eddy viscosity is defined as follows.
The production term $P_k$ is written using the mean strain rate tensor $S_{ij}$ as follows.
Model Constants
Please tell me the standard values for each constant.
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 |
Are we not allowed to change these constants?
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
Could you summarize the strengths and weaknesses of the standard k-ε model?
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)
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
Why does it overpredict in swirling flows?
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.
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.
Physical Meaning of Each Term
- Temporal term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, the water comes out spluttering and unstable, but after a while, it becomes a steady flow, right? This term describes that "period of change." The pulsation of blood flow from a heartbeat, the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost drops significantly, the basic CFD strategy is to first try solving it as steady-state.
- Convection term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misunderstanding: "Convection and conduction are similar things" → They're completely different! Convection is carried by flow, conduction is transmitted by molecules. There's an order of magnitude difference in efficiency.
- Diffusion term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while it naturally mixes, right? That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is large, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, convection overwhelms and diffusion plays a supporting role.
- Pressure term $-\nabla p$: When you push the plunger of a syringe, the liquid shoots out forcefully from the needle tip, right? Why? Because the plunger side is high pressure, the needle tip is low pressure—this pressure difference provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow arises where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: The "pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, mixing up absolute/gauge pressure might be the cause.
- Source term $S_\phi$: Heated air rises—why? Because it becomes lighter (less dense) than its surroundings, so it's pushed up by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by source terms. What happens if you forget a source term? In natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian fluid assumption: Linear relationship between shear stress and strain rate (non-Newtonian fluids require viscosity models)
- Incompressibility assumption (for Ma < 0.3): Treat density as constant. For Mach numbers above 0.3, consider compressibility effects
- Boussinesq approximation (Natural Convection): Consider density changes only in the buoyancy term, using constant density in other terms
- Non-applicable cases: Rarefied gases (Kn > 0.1), supersonic/hypersonic flows (shock capturing required), free surface flows (VOF/Level Set, etc. required)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C |
| Viscosity coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Criterion for laminar/turbulent transition |
| CFL number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
Discretization by Finite Volume Method
Please explain how the transport equations for the standard k-ε model are solved in a CFD solver.
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 |
Is first-order upwind no good?
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
What is the relationship between pressure-velocity coupling and the k-ε equations?
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
$k$ and $\varepsilon$ are typically solved sequentially using a segregated approach. In density-based solvers (coupled), all variables may be solved simultaneously.
Relaxation Factor Settings
Are relaxation factors important?
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 |
What happens if you lower the relaxation for $k$ or $\varepsilon$ too much?
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
How do you set the turbulence boundary conditions at the inlet?
It's common to calculate them from the Turbulence Intensity $I$ and the turbulent length scale $l$ (or hydraulic diameter $D_H$).
$$ k = \frac{3}{2}(U_{avg} \cdot I)^2 $$Related Topics
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How do you set the turbulence boundary conditions at the inlet?
It's common to calculate them from the Turbulence Intensity $I$ and the turbulent length scale $l$ (or hydraulic diameter $D_H$).
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