K-Epsilon Turbulence

The K-Epsilon turbulence model is a two-equation model that solves transport equations for the turbulent kinetic energy $k$ and the turbulent dissipation rate $ε$ in order to determine the turbulent eddy viscosity.

Various forms of the K-Epsilon model have been in use for a number of decades, and it has become the most widely used model for industrial applications. Since the inception of the K-Epsilon model, there have been countless attempts to improve it. The most significant improvements have been incorporated into Simcenter STAR-CCM+.

High Reynolds Number Approach

The original K-Epsilon turbulence model by Jones and Launder [306] was applied with wall functions. This high Reynolds number approach was later modified to take into account the blocking effects of the wall (viscous and buffer layer) using a low Reynolds number approach and a two-layer approach.

Low Reynolds Number Approach

The most common approach is to apply damping functions to some or all of the coefficients in the model ( $C_{μ}$ , $C_{ε 1}$ , and $C_{ε 2}$ ). These damping functions modulate the coefficients as functions of a turbulence Reynolds number, often also incorporating the wall distance. Dozens of models incorporating damping functions have been proposed in the literature, and the Standard K-Epsilon Low-Re model has been incorporated into Simcenter STAR-CCM+.

For details, see K-Epsilon Model Variants.

Two-Layer Approach

The two-layer approach, first suggested by Rodi [313], is an alternative to the low Reynolds number approach that allows the K-Epsilon model to be applied in the viscous-affected layer (including the viscous sub-layer and the buffer layer).

In this approach, the computation is divided into two layers. In the layer next to the wall, the turbulent dissipation rate $ε$ and the turbulent viscosity $μ_{t}$ are specified as functions of wall distance. The values of $ε$ specified in the near-wall layer are blended smoothly with the values computed from solving the transport equation far from the wall. The equation for the turbulent kinetic energy is solved across the entire flow domain. This explicit specification of $ε$ and $μ_{t}$ is arguably no less empirical than the damping function approach, and the results are often as good or better.

Several types of two-layer formulations have been proposed, and three have been implemented in Simcenter STAR-CCM+, two for shear-driven flows and one for buoyancy-driven flows:

Shear Driven (Wolfstein)
Shear Driven (Norris-Reynolds)
Buoyancy Driven (Xu)

In Simcenter STAR-CCM+, the two-layer formulations work with either low Reynolds number type meshes $y^{+} \sim 1$ or wall-function type meshes $y^{+} > 30$ .

Of the two models for shear-driven flows, neither is clearly better. The Wolfstein model is activated by default, but the Norris-Reynolds model is offered as an option since some users find it preferable. Only use the Xu natural-convection model for buoyant flows.

K-Epsilon Model Variants

The following variants of the K-Epsilon model are implemented in Simcenter STAR-CCM+:


Model Variants
Standard K-Epsilon
Standard K-Epsilon Two-Layer
Standard K-Epsilon Low-Re
Realizable K-Epsilon
Realizable K-Epsilon Two-layer
EB K-Epsilon
Lag EB K-Epsilon
EB K-Epsilon Detached Eddy

Standard K-Epsilon

The Standard K-Epsilon model is a de facto standard version of the two-equation model that involves transport equations for the turbulent kinetic energy $k$ and its dissipation rate $ε$ . The transport equations are of the form suggested by Jones and Launder [306], with coefficients suggested by Launder and Sharma [309]. Some additional terms have been added to the model in Simcenter STAR-CCM+ to account for effects such as buoyancy and compressibility. An optional non-linear constitutive relation is also provided.

Standard K-Epsilon Two-Layer

The Standard K-Epsilon Two-Layer model combines the Standard K-Epsilon model with the two-layer approach.

The coefficients in the models are identical, but the model gains the added flexibility of an all- $y^{+}$ wall treatment.

Standard K-Epsilon Low-Re

The Standard K-Epsilon Low-Reynolds Number model combines the Standard K-Epsilon model with the low Reynolds number approach.

The low Reynolds number model by Lien and others is dubbed the “Standard Low-Reynolds Number K-Epsilon Model” because it has identical coefficients to the Standard K-Epsilon model, but it provides more damping functions. These functions enable it to be applied in the viscous-affected regions near walls. This model is recommended for natural convection problems, particularly if the Yap correction is invoked. (See [311].)

Realizable K-Epsilon

The Realizable K-Epsilon model contains a new transport equation for the turbulent dissipation rate $ε$ [315]. Also, a variable damping function $f_{μ}$ —expressed as a function of mean flow and turbulence properties—is applied to a critical coefficient of the model $C_{μ}$ . This procedure lets the model satisfy certain mathematical constraints on the normal stresses consistent with the physics of turbulence (realizability). This concept of a damped $C_{μ}$ is also consistent with experimental observations in boundary layers.

This model is substantially better than the Standard K-Epsilon model for many applications, and can generally be relied upon to give answers that are at least as accurate. Both the standard and realizable models are available in Simcenter STAR-CCM+ with the option of using a two-layer approach, which enables them to be used with fine meshes that resolve the viscous sublayer.

Realizable Two-Layer K-Epsilon

The Realizable Two-Layer K-Epsilon model combines the Realizable K-Epsilon model with the two-layer approach.

The coefficients in the models are identical, but the model gains the added flexibility of an all- $y^{+}$ wall treatment.

EB K-Epsilon

The concept of elliptic relaxation was proposed by Durbin [331] for Reynolds-stress models. The initial model required the solution of six additional transport equations, but this number was later reduced to a single additional equation. The model was later simplified by Manceau and Hanjalić [334] to make it more industry-friendly.

The elliptic relaxation model led to the development of some two-equation eddy viscosity models, starting with the $\overline{ϑ^{2}} – f$ model. One of the most robust models, according to its authors, is the $B L - \overline{ϑ^{2}} / k$ model of Billard and Laurence [330]. Important modifications to this model make it truly robust for complex flow geometries.

The benefits of using the Billard - Laurence model are:

An improvement on the existing realizable $k - ε$ model in terms of accuracy, especially in the near-wall region.
An improvement in terms of stability over the SST $k - ω$ model.

Lag EB K-Epsilon

The Lag Elliptic Blending model combines the Standard Elliptic Blending model with the stress-strain lag concept originally proposed by Revell et al. [336].

In flow regions where non-equilibrium effects result in a misalignment of the principal components of the stress and strain-rate tensors, linear eddy viscosity models tend to over-predict the production of $k$ . To overcome this effect, the Lag Elliptic Blending model incorporates the angle between these components. Additional terms model the effects of anisotropy—similar to non-linear constitutive relations—and curvatures and rotational effects—similar to curvature correction. These terms are directly embedded in the transport equation for the reduced stress function $φ$ [335].

The Lag Elliptic Blending model provides a good predictive capability for separated or unsteady flow (such as vortex shedding), or when the flow is subject to rotation or strong streamline curvature.

EB K-Epsilon Detached Eddy

See EB K-Epsilon Detached Eddy Model.