Elliptic Blending Model

The Elliptic Blending turbulence model solves transport equations for the turbulent kinetic energy $k$ , the turbulent dissipation rate $ε$ , the normalized (reduced) wall-normal stress component $φ = \bar{ϑ^{2}} / k$ , and the elliptic blending factor $α$ in order to determine the turbulent eddy viscosity.

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.

Model Variants

Two variants of the Elliptic Blending model are implemented in Simcenter STAR-CCM+:


Model Variant	Abbreviation
Standard Elliptic Blending	EBS
Lag Elliptic Blending	EBL

Standard Elliptic Blending

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 Elliptic Blending

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.

Relation for Turbulent Viscosity

The turbulent eddy viscosity $μ_{t}$ is calculated as:

Figure 1. EQUATION_DISPLAY

μ_{t} = ρ C_{μ} φ k \min (T, \frac{C_{T}}{\sqrt{3} C_{μ} φ S})

(1265)

where:

$ρ$ is the density.
$C_{μ}$ and $C_{T}$ are Model Coefficients.
$T$ is the turbulent time scale.
$S$ is given by Eqn. (1129).

The turbulent time scale is calculated as:

Figure 2. EQUATION_DISPLAY

T = \sqrt{T_{e}^{2} + C_{t}^{2} \frac{ν}{ε}}

(1266)

where:

$T_{e} = \frac{k}{ε}$ is the large-eddy time scale.
$C_{t}$ is a Model Coefficient.
$ν$ is the kinematic viscosity.

Transport Equations

The transport equations for the four variables $k$ , $ε$ , $φ$ , and $α$ are:

Figure 3. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ k) + \nabla\cdot (ρ k \bar{v}) = \nabla\cdot [(\frac{μ}{2} + \frac{μ_{t}}{σ_{k}}) \nabla k] \begin{matrix} + P_{k} - ρ (ε - ε_{0}) + S_{k} \end{matrix}

(1267)

Figure 4. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ ε) + \nabla\cdot (ρ ε \bar{v}) = \nabla\cdot [(\frac{μ}{2} + \frac{μ_{t}}{σ_{ε}}) \nabla ε] + \frac{1}{T_{e}} C_{ε 1} P_{ε} - C_{ε 2}^{*} ρ (\frac{ε}{T_{e}} - \frac{ε_{0}}{T_{0}}) + S_{ε}

(1268)

Figure 5. EQUATION_DISPLAY

\frac{\partial}{\partial t} \begin{matrix} (ρ φ) + \nabla\cdot (ρ φ \bar{v}) = \nabla\cdot [(\frac{μ}{2} + \frac{μ_{t}}{σ_{φ}}) \nabla φ] \end{matrix} \begin{matrix} + ρ \frac{ε_{0} φ_{0}}{k_{0}} + P_{φ} + S_{φ} \end{matrix}

(1269)

Figure 6. EQUATION_DISPLAY

\nabla\cdot (L^{2} \nabla α) \begin{matrix} = α - 1 \end{matrix}

(1270)

where:

$\bar{v}$ is the mean velocity.
$μ$ is the dynamic viscosity.
$P_{k}$ , $P_{ε}$ , and $P_{φ}$ are Production Terms.
$C_{ε 1}$ , $C_{ε 2}^{*}$ , $σ_{k}$ , $σ_{ε}$ , and $σ_{φ}$ are Model Coefficients.
$S_{k}$ , $S_{ε}$ , and $S_{φ}$ are the user-specified source terms.

$L$ is the turbulent length scale calculated as:

Figure 7. EQUATION_DISPLAY

L = C_{L} \sqrt{\frac{k^{3}}{ε^{2}} + {C_{η}}^{2} \sqrt{\frac{ν^{3}}{ε}}}

(1271)

where $C_{L}$ and $C_{η}$ are Model Coefficients.

$ε_{0}$ , $φ_{0}$ , and $k_{0}$ are the ambient turbulence values 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:

Figure 8. EQUATION_DISPLAY

T_{0} = \max (\frac{k_{0}}{ε_{0}}, C_{t} \sqrt{\frac{ν}{ε_{0}}})

(1272)

Production Terms

The formulation of the production terms $P_{k}$ , $P_{ε}$ , and $P_{φ}$ depends on the Elliptic Blending model variant:


Model Variant	$P_{k}$	$P_{ε}$
EBS	Figure 9. EQUATION_DISPLAY $G_{k} + G_{b} - ϒ_{M}$ (1273)	Figure 10. EQUATION_DISPLAY $G_{k} + {C_{ε 3} G}_{b} + \frac{1}{C_{ε 1}} E$ (1274)
EBL	Figure 9. EQUATION_DISPLAY $G_{k} + G_{b} - ϒ_{M}$ (1273)


Model Variant	$P_{φ}$
EBS	Figure 11. EQUATION_DISPLAY $- \frac{φ}{k} (G_{k} + G_{b}) + ρ (1 - α^{3}) f_{w} + {ρ α}^{3} f_{h}$ (1275) where: $f_{w} = - \frac{1}{2} \frac{φ}{T_{e}}$ $f_{h} = - \frac{1}{T} (C_{1} - 1 + C_{2} \frac{G_{k} + G_{b}}{ρ ε}) (φ - \frac{2}{3})$
EBL	Figure 12. EQUATION_DISPLAY $- (2 - C_{ε 1}) \frac{φ}{k} (G_{k} + G_{b}) + ρ (1 - α^{3}) f_{w} + ρ α^{3} f_{h}$ (1276) where: $f_{w} = - (C_{ε 2} - 1 + 5 - \frac{1}{C_{μ}}) \frac{φ}{T_{e}}$ $f_{h} = - \frac{1}{T_{e}} (C_{1} + C_{ε 2} - 2 + C_{1}^{} \frac{G_{k} + G_{b}}{ρ ε}) φ + \frac{C_{P 3}}{T_{e}} + \frac{C_{3}^{}}{\sqrt{2}} φ S + \frac{1}{S^{2} T_{e}} [\frac{2}{C_{μ}} (1 - C_{4}) A S - \frac{2}{C_{μ}} (1 - C_{5}) A \hat{W}] : S$ $A$ is the Reynolds-stress anisotropy tensor that is defined as: $A = - 2 \frac{μ_{t}}{ρ k} [S + \frac{2 - 2 C_{5}}{C_{1} + C_{1}^{*} + 1} \frac{2}{\sqrt{(S + \hat{W}) : (S + \hat{W})}} (S \hat{W} - \hat{W} S)]$ where $S$ is given by Eqn. (1130). $\hat{W}$ is the modified absolute vorticity tensor given by: $\hat{W} = \tilde{W} - W^{s}$ where $\tilde{W}$ is given by Eqn. (1133) and $W^{s}$ is the Spalart-Shur tensor given by: $W^{s} = \frac{1}{S^{2}} (S \frac{D S}{D t} - \frac{D S}{D t} S)$

where:

$C_{1}$ , $C_{1}^{*}$ , $C_{2}$ , $C_{3}^{*}$ , $C_{4}$ , $C_{5}$ , $C_{ε 2}$ , $C_{ε 3}$ , $C_{μ}$ , and $C_{P 3}$ are Model Coefficients.

the contributions to the production terms are:


	Description	Formulation	Where:
$G_{k}$	Turbulent production	Figure 13. EQUATION_DISPLAY $μ_{t} S^{2} - \frac{2}{3} ρ k \nabla\cdot \bar{v} {- \frac{2}{3} μ}_{t} {(\nabla\cdot \bar{v})}^{2}$ (1277)	-
$G_{b}$	Buoyancy production	Figure 14. EQUATION_DISPLAY $β \frac{μ_{t}}{\Pr_{t}} (\nabla \bar{T} \cdot g)$ (1278)	$β$ is the coefficient of thermal expansion. For constant density flows using the Boussinesq approximation, $β$ is user-specified. For ideal gases, $β$ is given by $β = - \frac{1}{ρ} \frac{\partial ρ}{\partial \bar{T}}$ . $\Pr_{t}$ is the turbulent Prandtl number. $\bar{T}$ is the mean temperature. $g$ is the gravitational vector.
$ϒ_{M}$	Compressibility modification (Sarkar et al. [314])	Figure 15. EQUATION_DISPLAY $\frac{ρ C_{M} k ε}{c^{2}}$ (1279)	$C_{M}$ is a Model Coefficient. $c$ is the speed of sound.
$E$	Additional production	Figure 16. EQUATION_DISPLAY $C_{k} {(1 - α)}^{3} {ν μ_{t} \frac{k}{ε} [\nabla\cdot (‖ 2 S n ‖ n)]}^{2}$ (1280)	$C_{k}$ is a Model Coefficient. $n$ represents the wall normal direction.

Model Coefficients


Coefficient	EBS	EBL
$C_{1}$	1.7	1.7
$C_{1}^{*}$	-	0.9
$C_{2}$	0.9	-
$C_{3}$	-	0.8
$C_{3}^{*}$	-	0.65
$C_{4}$	-	0.625
$C_{5}$	-	0.2
$C_{ε 1}$	1.44	1.44
$C_{ε 2}$	1.83	1.9
$C_{ε 2}^{*}$	See Free-Stream Option.	$C_{ε 2}$
$C_{ε 3}$	See K-Epsilon Model—Model Coefficients.
$C_{k}$	2.3	2.3
$C_{L}$	0.164	0.164
$C_{M}$ (Sarkar)	2	2
$C_{μ}$	0.22	0.22
$C_{η}$	75	75
$C_{P 3}$	-	See Damping Function.
$C_{T}$	1	1
$C_{t}$	4	4
$σ_{ε}$	1.5	1.2
$σ_{φ}$	1	1
$σ_{k}$	1	1

Free-Stream Option

One important difference between the Standard Elliptic Blending model and other two-equation models is the definition of the coefficient of the destruction term

C_{ε 2}^{*}

. This term is defined as a function of the turbulent kinetic energy gradient, to reduce the

C_{ε 2}

value in the defect layer, where the free-stream is starting to dominate. However, for some external aerodynamic cases, this formulation can cause instabilities in the free-stream. For such cases, a constant

C_{ε 2}

value is recommended:


Free-stream Option	$C_{ε 2}^{*}$
Variable C2e Option	Figure 17. EQUATION_DISPLAY $C_{ε 2} + α^{3} (1 - C_{ε 2}) \tanh [{\| \frac{\partial}{\partial x_{j}} (\frac{μ_{t}}{ρ σ_{k}} \frac{\partial k}{\partial x_{j}}) ε^{- 1} \|}^{3 / 2}]$ (1281)
Off	Figure 18. EQUATION_DISPLAY $C_{ε 2}$ (1282)

Damping Function

For turbulence models that resolve the viscous- and buffer-layer, damping functions mimic the decrease of turbulent mixing near the walls. For the Lag Elliptic Blending model, the kinematic blocking of the wall is implemented within the Model Coefficient

C_{P 3}

as:

Figure 19. EQUATION_DISPLAY

C_{P 3} = \frac{f_{μ}}{C_{μ}} (\frac{2}{3} - \frac{C_{3}}{2})

(1283)

where $C_{3}$ is a Model Coefficient and the damping function $f_{μ}$ is defined as:

Figure 20. EQUATION_DISPLAY

f_{μ} = \frac{η + α^{3}}{\max (η, 1.87)}

(1284)

and time-scale ratio $η$ is calculated as:

Figure 21. EQUATION_DISPLAY

η = S T_{e}

(1285)

Changes From Base Model

The main changes from the original model proposed by Billard and Laurence [330] are:

Removal of an additional cross-diffusion term in the $φ$ transport equation, for stability reasons.
The constant in the definition of $C_{ε 2}^{*}$ , which acts on its decay in the defect layer, has been increased from 0.4 to 1.
The extra-dissipation term $E$ , which accounts for viscous wall-effects, was moved from the $k$ -equation back to the $ε$ -equation. This change is for stability reasons and to avoid unwanted relaminarization effects in some scenarios.