4.9.4 Modeling the Pressure-Strain Term

Linear Pressure-Strain Model

By default in ANSYS FLUENT, the pressure-strain term, $\phi_{ij}$, in Equation  4.9-1 is modeled according to the proposals by Gibson and Launder [ 108], Fu et al. [ 104], and Launder [ 176, 177].

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

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

with $C_1$ = 1.8.

The rapid pressure-strain term, $\phi_{ij,2}$, is modeled as

where $C_2$ = 0.60, $P_{ij}$, $F_{ij}$, $G_{ij}$, and $C_{ij}$ are defined as in Equation  4.9-1, $P = \frac{1}{2} P_{kk}$, $G = \frac{1}{2} G_{kk}$, 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

$$\phi_{ij,w} \equiv C'_1 \frac{\epsilon}{k} \left(\overline{u'_{k} u'_{m}} n_k n_m \... ...2} \overline{u'_{j} u'_{k}} n_i n_k \right) \frac{C_{\ell} k^{3/2}}{\epsilon d}$$

$$+ \; C'_2 \left(\phi_{km,2} n_k n_m \delta_{ij} - \frac{3}{2} \p... ...k - \frac{3}{2} \phi_{jk,2} n_i n_k \right) \frac{C_{\ell} k^{3/2}}{\epsilon d}$$

$$\phantom{xx}$$ (4.9-7)

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  4.12.4, the pressure-strain model needs to be modified. The modification used in ANSYS 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 [ 179]:

$$C_1 = 1 + 2.58 A {A_2}^{0.25} \left\{1 - \exp \left[-(0.0067 {\rm Re}_t)^2 \right] \right\}$$ (4.9-8)

$$C_2 = 0.75 \sqrt{A}$$ (4.9-9)

$$C'_1 = - \frac{2}{3} C_1 + 1.67$$ (4.9-10)

$$C'_2 = \max \left[ \frac{ \frac{2}{3} C_2 - \frac{1}{6} }{C_2}, 0 \right]$$ (4.9-11)

with the turbulent Reynolds number defined as ${\rm Re}_t = (\rho k^2/\mu \epsilon)$. The flatness parameter $A$ and tensor invariants, $A_2$ and $A_3$, are defined as

$$A \equiv \left[1 - \frac{9}{8}\left(A_2 - A_3\right)\right]$$ (4.9-12)

$$A_2 \equiv a_{ik} a_{ki}$$ (4.9-13)

$$A_3 \equiv a_{ik} a_{kj} a_{ji}$$ (4.9-14)

$a_{ij}$ is the Reynolds-stress anisotropy tensor, defined as

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

Quadratic Pressure-Strain Model

An optional pressure-strain model proposed by Speziale, Sarkar, and Gatski [ 334] is provided in ANSYS 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 dialog box.

This model is written as follows:

where $b_{ij}$ is the Reynolds-stress anisotropy tensor defined as

The mean strain rate, $S_{ij}$, is defined as

The mean rate-of-rotation tensor, $\Omega_{ij}$, is defined by

The constants are

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 dialog box.

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 [ 379]. 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 dialog box and requires no treatments of wall reflections. The closure coefficients are identical to the $k$- $\omega$ model (Section  4.5.1), however, there are additional closure coefficients, $C_1$ and $C_2$, noted below.

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  4.9-4 can be re-written for the low-Re stress-omega model such that wall reflections are excluded:

Therefore,

where $D_{ij}$ is defined as

The mean strain rate $S_{ij}$ is defined in Equation  4.9-18 and $\beta^*_{RSM}$ is defined by

where $\beta^*$ and $f_{\beta}^*$ are defined in the same way as for the standard $k-\omega$, using Equations  4.5-16 and 4.5-22, respectively. The only difference here is that the equation for $f_{\beta}^*$ uses a value of 640 instead of 680, as in Equation  4.5-16.

The constants are

The above formulation does not require viscous damping functions to resolve the near-wall sublayer. However, inclusion of the viscous damping function [ 379] could improve model predictions for certain flows. This results in the following changes:

where $\hat{\alpha}$, $\hat{\beta}$, and $\hat{\gamma}$ would replace $\hat{\alpha_0}$, $\hat{\beta_0}$, and $\hat{\gamma_0}$ in Equation  4.9-21. The constants are

Inclusion of the low-Re viscous damping is controlled by enabling Low-Re Corrections under k-omega Options in the Viscous Model dialog box.