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12.7.4 Modeling the Pressure-Strain Term



Linear Pressure-Strain Model


By default in FLUENT, the pressure-strain term, $\phi_{ij}$, in Equation  12.7-1 is modeled according to the proposals by Gibson and Launder [ 119], Fu et al. [ 114], and Launder [ 192, 193].

The classical approach to modeling $\phi_{ij}$ uses the following decomposition:

 \phi_{ij} = \phi_{ij,1} + \phi_{ij,2} + \phi_{ij,w} (12.7-4)

where $\phi_{ij,1}$ is the slow pressure-strain term, also known as the return-to-isotropy term, $\phi_{ij,2}$ is called the rapid pressure-strain term, and $\phi_{ij,w}$ is the wall-reflection term.

The slow pressure-strain term, $\phi_{ij,1}$, is modeled as
 \phi_{ij,1} \equiv - C_1 \rho \; \frac{\epsilon}{k} \left ... ...erline{u'_{i} u'_{j}} - \frac{2}{3} \delta_{ij} k \right ] (12.7-5)

with $C_1$ = 1.8.

The rapid pressure-strain term, $\phi_{ij,2}$, is modeled as
 \phi_{ij,2} \equiv - C_2 \left [ (P_{ij} + F_{ij} + G_{ij} - C_{ij}) - \frac{2}{3} \delta_{ij} (P + G - C) \right ] (12.7-6)

where $C_2$ = 0.60, $P_{ij}$, $F_{ij}$, $G_{ij}$, and $C_{ij}$ are defined as in Equation  12.7-1, $P = \frac{1}{2} P_{kk}$, $G = \frac{1}{2} G_{kk}$, and $C = \frac{1}{2} C_{kk}$.

The wall-reflection term, $\phi_{ij,w}$, is responsible for the redistribution of normal stresses near the wall. It tends to damp the normal stress perpendicular to the wall, while enhancing the stresses parallel to the wall. This term is modeled as
$\displaystyle \phi_{ij,w}$ $\textstyle \equiv$ $\displaystyle C'_1 \frac{\epsilon}{k} \left( \overline{u'_{k} u'_{m}} n_k n_m \... ...2} \overline{u'_{j} u'_{k}} n_i n_k \right) \frac{C_{\ell} k^{3/2}}{\epsilon d}$  
    $\displaystyle + \; C'_2 \left( \phi_{km,2} n_k n_m \delta_{ij} - \frac{3}{2} \p... ...k - \frac{3}{2} \phi_{jk,2} n_i n_k \right) \frac{C_{\ell} k^{3/2}}{\epsilon d}$  
$\displaystyle \phantom{xx}$     (12.7-7)

where $C'_1 = 0.5$, $C'_2 = 0.3$, $n_k$ is the $x_k$ component of the unit normal to the wall, $d$ is the normal distance to the wall, and $C_{\ell} = C_{\mu}^{3/4}/\kappa$, where $C_{\mu} = 0.09$ and $\kappa$ is the von Kármán constant (= 0.4187).

$\phi_{ij,w}$ is included by default in the Reynolds stress model.



Low-Re Modifications to the Linear Pressure-Strain Model


When the RSM is applied to near-wall flows using the enhanced wall treatment described in Section  12.10.4, the pressure-strain model needs to be modified. The modification used in FLUENT specifies the values of $C_1$, $C_2$, $C'_1$, and $C'_2$ as functions of the Reynolds stress invariants and the turbulent Reynolds number, according to the suggestion of Launder and Shima [ 195]:

$\displaystyle C_1$ $\textstyle =$ $\displaystyle 1 + 2.58 A \sqrt{A_2} \left\{1 - \exp \left[-(0.0067 {\rm Re}_t)^2 \right] \right\}$ (12.7-8)
$\displaystyle C_2$ $\textstyle =$ $\displaystyle 0.75 \sqrt{A}$ (12.7-9)
$\displaystyle C'_1$ $\textstyle =$ $\displaystyle - \frac{2}{3} C_1 + 1.67$ (12.7-10)
$\displaystyle C'_2$ $\textstyle =$ $\displaystyle \max \left[ \frac{ \frac{2}{3} C_2 - \frac{1}{6} }{C_2}, 0 \right]$ (12.7-11)

with the turbulent Reynolds number defined as ${\rm Re}_t = (\rho k^2/\mu \epsilon)$. The parameter $A$ and tensor invariants, $A_2$ and $A_3$, are defined as
$\displaystyle A$ $\textstyle \equiv$ $\displaystyle \left[1 - \frac{9}{8}\left(A_2 - A_3\right)\right]$ (12.7-12)
$\displaystyle A_2$ $\textstyle \equiv$ $\displaystyle a_{ik} a_{ki}$ (12.7-13)
$\displaystyle A_3$ $\textstyle \equiv$ $\displaystyle a_{ik} a_{kj} a_{ji}$ (12.7-14)

$a_{ij}$ is the Reynolds-stress anisotropy tensor, defined as
 a_{ij} = -\left( \frac{- \rho \overline{u'_i u'_j} + \frac{2}{3} \rho k \delta_{ij}}{\rho k}\right) (12.7-15)

The modifications detailed above are employed only when the enhanced wall treatment is selected in the Viscous Model panel.



Quadratic Pressure-Strain Model


An optional pressure-strain model proposed by Speziale, Sarkar, and Gatski [ 352] is provided in FLUENT. This model has been demonstrated to give superior performance in a range of basic shear flows, including plane strain, rotating plane shear, and axisymmetric expansion/contraction. This improved accuracy should be beneficial for a wider class of complex engineering flows, particularly those with streamline curvature. The quadratic pressure-strain model can be selected as an option in the Viscous Model panel.

This model is written as follows:

\phi_{ij} = - \left(C_1 \rho \epsilon + C_1^* P\right) b_{ij}... ... + \left(C_3 - C_3^* \sqrt{b_{ij} b_{ij}}\right) \rho k S_{ij}


 + C_4 \rho k \left(b_{ik} S_{jk} + b_{jk} S_{ik} - \frac{2}... ...\rho k \left(b_{ik} \Omega_{jk} + b_{jk} \Omega_{ik}\right) (12.7-16)

where $b_{ij}$ is the Reynolds-stress anisotropy tensor defined as
 b_{ij} = -\left( \frac{- \rho \overline{u'_i u'_j} + \frac{2}{3} \rho k \delta_{ij}}{2 \rho k}\right) (12.7-17)

The mean strain rate, $S_{ij}$, is defined as
 S_{ij} = \frac{1}{2}\left(\frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right) (12.7-18)

The mean rate-of-rotation tensor, $\Omega_{ij}$, is defined by
 \Omega_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) (12.7-19)

The constants are

C_1 = 3.4, \; C_1^* = 1.8, \; C_2 = 4.2, \; C_3 = 0.8, \; C_3^* = 1.3, \; C_4 = 1.25, \; C_5 = 0.4

The quadratic pressure-strain model does not require a correction to account for the wall-reflection effect in order to obtain a satisfactory solution in the logarithmic region of a turbulent boundary layer. It should be noted, however, that the quadratic pressure-strain model is not available when the enhanced wall treatment is selected in the Viscous Model panel.



Low-Re Stress-Omega Model


The low-Re stress-omega model is a stress-transport model that is based on the omega equations and LRR model [ 403]. This model is ideal for modeling flows over curved surfaces and swirling flows. The low-Re stress-omega model can be selected in the Viscous Model panel and requires no treatments of wall reflections. The closure coefficients are identical to the $k$- $\omega$ model (Section  12.5.1), however, there are additional closure coefficients, $C_1$ and $C_2$, defined in Section  12.7.4.

The low-Re stress-omega model resembles the $k$- $\omega$ model due to its excellent predictions for a wide range of turbulent flows. Furthermore, low Reynolds number modifications and surface boundary conditions for rough surfaces are similar to the $k$- $\omega$ model. Equation  12.7-4 can be re-written for the low-Re stress-omega model such that wall reflections are excluded:

 \phi_{ij} = \phi_{ij,1} + \phi_{ij,2} (12.7-20)

Hence,

\phi_{ij} = - \left(C_1 \rho \epsilon + C_1^* P\right) b_{ij}... ... + \left(C_3 - C_3^* \sqrt{b_{ij} b_{ij}}\right) \rho k S_{ij}


 + C_4 \rho k \left(b_{ik} S_{jk} + b_{jk} S_{ik} - \frac{2}... ...\rho k \left(b_{ik} \Omega_{jk} + b_{jk} \Omega_{ik}\right) (12.7-21)

where $b_{ij}$ is the Reynolds-stress anisotropy tensor defined as
 b_{ij} = -\left( \frac{- \rho \overline{u'_i u'_j} + \frac{2}{3} \rho k \delta_{ij}}{2 \rho k}\right) (12.7-22)

The mean strain rate, $S_{ij}$, is defined in Equation  12.7-18 and the mean rate-of-rotation tensor, $\Omega_{ij}$, is defined by Equation  12.7-19.

The constants are

C_1 = 3.4, \; C_1^* = 1.8, \; C_2 = 4.2, \; C_3 = 0.8, \; C_3^* = 1.3, \; C_4 = 1.25, \; C_5 = 0.4

Near-wall treatment options in the Viscous Model panel are not available with the low-Re stress-omega model.

Model Constants


C_1 = 1.8,\;\; C_2 = 0.52


\alpha^*_\infty = 1,\;\; \alpha_\infty = 0.52,\;\; \alpha_0... ...beta^*_\infty = 0.09,\;\; \beta_i = 0.072, \;\; R_\beta = 8


R_k = 6,\;\; R_\omega = 2.95,\;\; \zeta^* = 1.5,\;\; {\rm M}_{t0} = 0.25,\;\; \sigma_k = 2.0, \;\; \sigma_\omega = 2.0

Wall Boundary Conditions

The wall boundary conditions for the low-Re stress-omega equation in the RSM models are treated in the same way as the $k$ equation in the $k$- $\omega$ models.

FLUENT defines the value of $\omega$ at the wall as

 \omega_w = \frac{\rho \, (u^*)^2}{\mu} \omega^+ (12.7-23)

where $\omega^+$ is dimensionless and is defined as
 \omega_w^+ = \left\{ \begin{array}{ll} \left(\frac{50}{k_s... ...\\ \frac{500}{k_s^+} & k_s^+ \ge 25 \end{array} \right. (12.7-24)

where
 k_s^+ = \frac{\rho k_s u^*}{\mu} (12.7-25)

and $k_s$ is the roughness height.


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