K-Epsilon Model

Model Variants

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

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 $\epsilon$ . 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 $\epsilon$ [315]. Also, a variable damping function $f_{\mu }$ —expressed as a function of mean flow and turbulence properties—is applied to a critical coefficient of the model $C_{\mu }$ . 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 $C_{\mu }$ 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 K-Epsilon Two-Layer

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.

Relation for Turbulent Viscosity

The turbulent eddy viscosity ${\mu }_t$ is calculated as:

where:

• $\rho$ is the density.
• $C_{\mu }$ is a Model Coefficient.
• $f_{\mu }$ is a Damping Function.
• $T$ is the turbulent time scale.

The turbulent time scale is calculated using Durbin's realizability constraint [321] or the Vorticity Limiter realizability option [310] which is only applicable to Volume of Fluid (VOF) multiphase wave flow.

where:

• $T_e=\frac{k}{\epsilon }$ is the large-eddy time scale.
• $C_t$ , and $C_T$ are Model Coefficients.
• ${\lambda }_2\frac{C_{ϵ2}}{C_{ϵ1}}$ is the vorticity coefficient. See Model Coefficients
• $\nu$ is the kinematic viscosity.
• $S$ is given by Eqn. (1129).

Transport Equations

The transport equations for the kinetic energy $k$ and the turbulent dissipation rate $\epsilon$ are:

where:

• $\overline{v}$ is the mean velocity.
• $\mu$ is the dynamic viscosity.
• ${\sigma }_k$ , ${\sigma }_{\epsilon }$ , $C_{\epsilon 1}$ , and $C_{\epsilon 2}$ are Model Coefficients.
• $P_k$ and $P_{\epsilon }$ are Production Terms.
• $f_2$ is a Damping Function.
• $S_k$ and $S_{\epsilon }$ are the user-specified source terms.

${\epsilon }_0$ is the ambient turbulence value in the source terms that counteracts turbulence decay [316]. The possibility to impose an ambient source term also leads to the definition of a specific time-scale $T_0$ that is defined as:

where:

• $C_t$ is a Model Coefficient.

Production Terms

The formulation of the production terms $P_k$ and $P_{\epsilon }$ depends on the K-Epsilon model variant:

where:

• $C_{\epsilon 3}$ is a Model Coefficient.
• $f_c$ is the curvature correction factor given by Eqn. (1287).
• the contributions to the production terms are:

  	Description	Formulation	where:
  $G_k$	Turbulent production	$${\mu }_t{S}^2-\frac{2}{3}\rho k\nabla \cdot \overline{\mathbf{\text{v}}}{-\frac{2}{3}\mu }_t{(\nabla \cdot \overline{\mathbf{\text{v}}})}^2$$ (1181)	-
  $G_b$	Buoyancy production	$$\beta \frac{{\mu }_t}{{\mathrm{Pr}}_t}(\nabla \overline{T}\cdot \mathbf{\text{g}})$$ (1182)	$\beta$ is the coefficient of thermal expansion. For constant density flows using the Boussinesq approximation, $\beta$ is user-specified. For ideal gases, $\beta$ is given by $\beta =-\frac{1}{\rho }\frac{\partial \rho }{\partial \overline{T}}$. ${\mathrm{Pr}}_t$ is the turbulent Prandtl number. $\overline{T}$ is the mean temperature. $\mathbf{\text{g}}$ is the gravitational vector.
  $G_{nl}$	“Non-linear” production	$$\left({\mathbf{\text{T}}}_{\mathrm{RANS},NL}\right):\nabla \overline{\mathbf{\text{v}}}$$ (1183)	${\mathbf{\text{T}}}_{\mathrm{RANS},NL}$ is the non-linear contribution to the Constitutive Relationship.
  $G\prime$	Additional production	$$D{f}_2(G_k+2\mu \frac{k}{d^2})\exp (-ER{e}_d^2)$$ (1184)	$D$ and $E$ are Model Coefficients. $f_2$ is a Damping Function. $d$ is the distance to the wall. ${\mathrm{Re}}_d$ is given by Eqn. (1134).
  ${\Upsilon }_M$	Compressibility modification (Sarkar et al. [314])	$$\frac{\rho C_{M}k\epsilon }{c^2}$$ (1185)	$C_M$ is a Model Coefficient. $c$ is the speed of sound.
  ${\Upsilon }_y$	Yap correction [319]	$$C_w\frac{{\epsilon }^2}{k}\text{max}[\left(\frac{l}{l_{\epsilon }}-1\right){\left(\frac{l}{l_{\epsilon }}\right)}^2,0]$$ (1186)	$C_w$ is a Model Coefficient. $l$ and $l_{\epsilon }$ are length scales defined as $l=\frac{k^{\frac{3}{2}}}{\epsilon }$ and $l_{\epsilon }=C_{l}d$, where $C_l$ is a Model Coefficient.

Damping Functions

For turbulence models that resolve the viscous- and buffer-layer (SKE LRe), damping functions mimic the decrease of turbulent mixing near the walls. For the RKE and RKE 2L models, damping functions enforce realizability.

where:

• $C$ , $C_{d0}$ , $C_{d1}$ , and $C_{d2}$ are Model Coefficients.
• ${\mathrm{Re}}_t$ and ${\mathrm{Re}}_d$ are given by Eqn. (1135) and Eqn. (1134), respectively.

• $S$ and $\mathbf{\text{W}}$ are given by Eqn. (1130) and Eqn. (1132), respectively.

• $S^{*}=S-\frac{1}{3}tr\left(S\right)I$

Model Coefficients

$C_{\epsilon 3}$ (all model variants)

The available literature is not clear as to the specification of this coefficient. By default, it is computed according to [305] as:

$$C_{\epsilon 3}=\tanh \frac{\left|v_b\right|}{\left|u_b\right|}$$

(1191)

where $v_b$ and $u_b$ are the velocity components parallel and perpendicular to the gravitational vector $\mathbf{\text{g}}$.

This formulation tends to set the coefficient to zero outside natural convection boundary layers.

Alternatively, $C_{\epsilon 3}$ can be taken as constant everywhere, or specified depending on the buoyancy production term $G_b$ as follows:

$$C_{\epsilon 3}=\{\begin{array}{c}1\text{ for }{G}_b\ge 0\\ 0\text{ for }{G}_b<0\end{array}$$

(1192)