Elliptic Blending Model

The Elliptic Blending RST model of Manceau and Hanjalić [334] is a low-Reynolds number model that is based on a inhomogeneous near-wall formulation of the quasi-linear quadratic pressure-strain term. The blending function is used to blend the viscous sub-layer with the log-layer formulation of the pressure-strain term. This approach requires the solution of an elliptic equation for the blending parameter $α$ . The version of the model implemented in Simcenter STAR-CCM+ was revised by Lardeau and Manceau [341].

The pressure-strain model of Manceau and Hanjalić is based on a blending of near-wall and quadratic pressure-strain models for the pressure-strain and dissipation:

Figure 1. EQUATION_DISPLAY

\underset{̲}{ϕ} - \underset{̲}{ε} = (1 - α^{3}) ({\underset{̲}{ϕ}}^{w} - {\underset{̲}{ε}}^{w}) + α^{3} ({\underset{̲}{ϕ}}^{h} - {\underset{̲}{ε}}^{h})

(1332)

where the blending parameter $α$ is the solution of the elliptic equation:

Figure 2. EQUATION_DISPLAY

α - L^{2} \nabla^{2} α = 1

(1333)

In the outer region, the quasi-linear version of the Sarkar, Speziale, and Gatski [345] model is used:

Figure 3. EQUATION_DISPLAY

\begin{matrix} {\underset{̲}{ϕ}}^{h} = - [C_{1} ρ ε + C_{1 s} t r (P + G)] A + (C_{3} - C_{3 s} \sqrt{A : A}) ρ k S \\ + C_{4} ρ k (S \cdot A + A \cdot S - \frac{2}{3} A : S I) + C_{5} ρ k (W \cdot A + A \cdot W^{T}) \end{matrix}

(1334)

where:

$ρ$ is the density.
$k$ is the turbulent kinetic energy given by Eqn. (1313).
$C_{1}$ , $C_{1 s}$ , $C_{3}$ , $C_{3 s}$ , $C_{4}$ , and $C_{5}$ are Model Coefficients.
$P$ is given by Eqn. (1315).
$G$ is given by Eqn. (1316).
$A$ is given by Eqn. (1330).
$S$ is given by Eqn. (1130).
$W$ is given by Eqn. (1132).
$I$ is the identity tensor.

In the near-wall layer:

Figure 4. EQUATION_DISPLAY

\begin{matrix} {\underset{̲}{ϕ}}^{w} = - 5 ρ \frac{ε}{k} [R \cdot N + N \cdot R - \frac{1}{2} R : N (N + I)] \end{matrix}

(1335)

where $N = n \otimes n$ is computed from the wall-normal vector $n$ , defined as:

Figure 5. EQUATION_DISPLAY

n = \frac{\nabla α}{\sqrt{\nabla α \cdot \nabla α}}

(1336)

For the dissipation rate, the formulations for the outer region and the near-wall layer are:

Figure 6. EQUATION_DISPLAY

{\underset{̲}{ε}}^{h} = \frac{2}{3} ε I

(1337)

Figure 7. EQUATION_DISPLAY

{\underset{̲}{ε}}^{w} = R \frac{ε}{k}

(1338)

The turbulent length-scale $L$ is defined as:

Figure 8. EQUATION_DISPLAY

L = C_{l} \max (\frac{k^{3 / 2}}{ε}, C_{η} \frac{ν^{3 / 4}}{ε^{1 / 4}})

(1339)

where:

$C_{l}$ and $C_{η}$ are Model Coefficients.
$ν$ is the kinematic viscosity.

For the Elliptic Blending model, the eddy viscosity $μ_{t}$ (Eqn. (1312)) is redefined as:

Figure 9. EQUATION_DISPLAY

μ_{t} = ρ C_{μ} k T

(1340)

where:

$C_{μ}$ is a Model Coefficient.
$T$ is the turbulent time-scale calculated as:
Figure 10. EQUATION_DISPLAY

$T = \max (\frac{k}{ε}, C_{t} \sqrt{\frac{ν}{ε}})$
(1341)

where $C_{t}$ is a Model Coefficient.

An additional source term is also added to the transport equation for $ε$ (Eqn. (1169)), in order to reproduce the correct near-wall behavior of the dissipation rate:

Figure 11. EQUATION_DISPLAY

E = A_{1} ν (R : N) \frac{k}{ε} (1 - α^{3}) {[\nabla (‖ S . n ‖ n)]}^{2}

(1342)

where $A_{1}$ is a Model Coefficient.

Model Coefficients


$A_{1}$	$C_{1}$	$C_{1_{s}}$	$C_{3}$	$C_{3_{s}}$	$C_{4}$	$C_{5}$
0.115	1.7	0.9	0.8	0.65	0.625	0.2


$C_{l}$	$C_{t}$	$C_{η}$
0.133	6	80