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12.10.4 Enhanced Wall Treatment

Enhanced wall treatment is a near-wall modeling method that combines a two-layer model with enhanced wall functions. If the near-wall mesh is fine enough to be able to resolve the laminar sublayer (typically $y^+ \approx 1$), then the enhanced wall treatment will be identical to the traditional two-layer zonal model (see below for details). However, the restriction that the near-wall mesh must be sufficiently fine everywhere might impose too large a computational requirement. Ideally, then, one would like to have a near-wall formulation that can be used with coarse meshes (usually referred to as wall-function meshes) as well as fine meshes (low-Reynolds-number meshes). In addition, excessive error should not be incurred for intermediate meshes that are too fine for the near-wall cell centroid to lie in the fully turbulent region, but also too coarse to properly resolve the sublayer.

To achieve the goal of having a near-wall modeling approach that will possess the accuracy of the standard two-layer approach for fine near-wall meshes and that, at the same time, will not significantly reduce accuracy for wall-function meshes, FLUENT can combine the two-layer model with enhanced wall functions, as described in the following sections.



Two-Layer Model for Enhanced Wall Treatment


In FLUENT's near-wall model, the viscosity-affected near-wall region is completely resolved all the way to the viscous sublayer. The two-layer approach is an integral part of the enhanced wall treatment and is used to specify both $\epsilon$ and the turbulent viscosity in the near-wall cells. In this approach, the whole domain is subdivided into a viscosity-affected region and a fully-turbulent region. The demarcation of the two regions is determined by a wall-distance-based, turbulent Reynolds number, Re $_y$, defined as

 {\rm Re}_y \equiv \frac{\rho y \sqrt{k}}{\mu} (12.10-18)

where $y$ is the normal distance from the wall at the cell centers. In FLUENT, $y$ is interpreted as the distance to the nearest wall:
 y \equiv \min_{{\vec r}_w \in {\Gamma}_w} \Vert{\vec r} - {\vec r}_w\Vert (12.10-19)

where ${\vec r}$ is the position vector at the field point, and ${\vec r}_w$ is the position vector on the wall boundary. ${\Gamma}_w$ is the union of all the wall boundaries involved. This interpretation allows $y$ to be uniquely defined in flow domains of complex shape involving multiple walls. Furthermore, $y$ defined in this way is independent of the mesh topology used, and is definable even on unstructured meshes.

In the fully turbulent region ( ${\rm Re}_y > {\rm Re}_y^*$; ${\rm Re}_y^*= 200 $), the $k$- $\epsilon$ models or the RSM (described in Sections  12.4 and 12.7) are employed.

In the viscosity-affected near-wall region ( ${\rm Re}_y < {\rm Re}_y^*$), the one-equation model of Wolfstein [ 406] is employed. In the one-equation model, the momentum equations and the $k$ equation are retained as described in Sections  12.4 and 12.7. However, the turbulent viscosity, $\mu_t$, is computed from
 \mu_{t,{\rm 2layer}} = \rho \; C_{\mu} \ell_{\mu} \sqrt{k} (12.10-20)

where the length scale that appears in Equation  12.10-20 is computed from [ 54]
 \ell_{\mu} = y {C_{\ell}}^* \left(1 - e^{-{\rm Re}_y/A_{\mu}}\right) (12.10-21)

The two-layer formulation for turbulent viscosity described above is used as a part of the enhanced wall treatment, in which the two-layer definition is smoothly blended with the high-Reynolds-number $\mu_t$ definition from the outer region, as proposed by Jongen [ 168]:
 \mu_{t,{\rm enh}} = \lambda_\epsilon \mu_t + (1 - \lambda_\epsilon) \mu_{t,{\rm 2layer}} (12.10-22)

where $\mu_t$ is the high-Reynolds-number definition as described in Section  12.4 or 12.7 for the $k$- $\epsilon$ models or the RSM. A blending function, $\lambda_\epsilon$, is defined in such a way that it is equal to unity far from walls and is zero very near to walls. The blending function chosen is
 \lambda_\epsilon = \frac{1}{2} \left[1 + \tanh \left(\frac{{\rm Re}_y - {\rm Re}_y^*}{A}\right)\right] (12.10-23)

The constant $A$ determines the width of the blending function. By defining a width such that the value of $\lambda_\epsilon$ will be within 1% of its far-field value given a variation of $\Delta {\rm Re}_y$, the result is
 A = \frac{\vert\Delta {\rm Re}_y\vert}{\tanh (0.98)} (12.10-24)

Typically, $\Delta {\rm Re}_y$ would be assigned a value that is between 5% and 20% of ${\rm Re}_y^*$. The main purpose of the blending function $\lambda_\epsilon$ is to prevent solution convergence from being impeded when the $k$- $\epsilon$ solution in the outer layer does not match with the two-layer formulation.

The $\epsilon$ field is computed from
 \epsilon = \frac{k^{3/2}}{\ell_{\epsilon}} (12.10-25)

The length scales that appear in Equation  12.10-25 are again computed from Chen and Patel [ 54]:
 \ell_{\epsilon} = y {C_{\ell}}^* \left(1 - e^{-{\rm Re}_y/A_{\epsilon}}\right) (12.10-26)

If the whole flow domain is inside the viscosity-affected region ( ${\rm Re}_y < 200$), $\epsilon$ is not obtained by solving the transport equation; it is instead obtained algebraically from Equation  12.10-25. FLUENT uses a procedure for the $\epsilon$ specification that is similar to the $\mu_t$ blending in order to ensure a smooth transition between the algebraically-specified $\epsilon$ in the inner region and the $\epsilon$ obtained from solution of the transport equation in the outer region.

The constants in the length scale formulas, Equations  12.10-21 and 12.10-26, are taken from [ 54]:
 {C_{\ell}}^* = \kappa C_{\mu}^{-3/4}, \; \; \; A_{\mu} = 70, \; \; \; A_{\epsilon} = 2 {C_{\ell}}^* (12.10-27)



Enhanced Wall Functions


To have a method that can extend its applicability throughout the near-wall region (i.e., laminar sublayer, buffer region, and fully-turbulent outer region) it is necessary to formulate the law-of-the wall as a single wall law for the entire wall region. FLUENT achieves this by blending linear (laminar) and logarithmic (turbulent) laws-of-the-wall using a function suggested by Kader [ 170]:

 u^+ = e^\Gamma u_{\rm lam}^+ + e^{\frac{1}{\Gamma}} u_{\rm turb}^+ (12.10-28)

where the blending function is given by:
 \Gamma = - \frac{a (y^+)^4}{1 + b y^+} (12.10-29)

where $a = 0.01$ and $b = 5$. Similarly, the general equation for the derivative $\frac{d u^+}{d y^+}$ is
 \frac{d u^+}{d y^+} = e^\Gamma \, \frac{d u_{\rm lam}^+}{d y^+} + e^\frac{1}{\Gamma} \, \frac{d u_{\rm turb}^+}{d y^+} (12.10-30)

This approach allows the fully turbulent law to be easily modified and extended to take into account other effects such as pressure gradients or variable properties. This formula also guarantees the correct asymptotic behavior for large and small values of $y^+$ and reasonable representation of velocity profiles in the cases where $y^+$ falls inside the wall buffer region ( $3 < y^+ < 10$).

The enhanced wall functions were developed by smoothly blending an enhanced turbulent wall law with the laminar wall law. The enhanced turbulent law-of-the-wall for compressible flow with heat transfer and pressure gradients has been derived by combining the approaches of White and Cristoph [ 402] and Huang et al. [ 149]:
 \frac{d u_{\rm turb}^+}{dy^+} = \frac{1}{\kappa y^+} \left[S' (1 - \beta u^+ - \gamma(u^+)^2) \right]^{1/2} (12.10-31)

where
 S' = \left\{ \begin{array}{ll} 1 + \alpha y^+ & \mbox{for... ...ha y^+_s & \mbox{for $y^+ \ge y^+_s$} \end{array} \right. (12.10-32)

and
$\displaystyle \alpha$ $\textstyle \equiv$ $\displaystyle \frac{\nu_w}{\tau_w u^*}\frac{dp}{dx} = \frac{\mu}{\rho^2(u^*)^3} \frac{dp}{dx}$ (12.10-33)
$\displaystyle \beta$ $\textstyle \equiv$ $\displaystyle \frac{\sigma_t q_w u^*}{c_p \tau_w T_w} = \frac{\sigma_t q_w}{\rho c_p u^* T_w}$ (12.10-34)
$\displaystyle \gamma$ $\textstyle \equiv$ $\displaystyle \frac{\sigma_t (u^*)^2}{2 c_p T_w}$ (12.10-35)

where $y^+_s$ is the location at which the log-law slope will remain fixed. By default, $y^+_s = 60$. The coefficient $\alpha$ in Equation  12.10-31 represents the influences of pressure gradients while the coefficients $\beta$ and $\gamma$ represent thermal effects. Equation  12.10-31 is an ordinary differential equation and FLUENT will provide an appropriate analytical solution. If $\alpha$, $\beta$, and $\gamma$ all equal 0, an analytical solution would lead to the classical turbulent logarithmic law-of-the-wall.

The laminar law-of-the-wall is determined from the following expression:
 \frac{du^+_{\rm lam}}{dy^+} = 1 + \alpha y^+ (12.10-36)

Note that the above expression only includes effects of pressure gradients through $\alpha$, while the effects of variable properties due to heat transfer and compressibility on the laminar wall law are neglected. These effects are neglected because they are thought to be of minor importance when they occur close to the wall. Integration of Equation  12.10-36 results in
 u^+_{\rm lam} = y^+ \left(1 + \frac{\alpha}{2} y^+ \right) (12.10-37)

Enhanced thermal wall functions follow the same approach developed for the profile of $u^+$. The unified wall thermal formulation blends the laminar and logarithmic profiles according to the method of Kader [ 170]:
$\displaystyle T^+ \equiv \frac{\left(T_w - T_P\right) \rho c_p u^*}{\dot q}$ $\textstyle =$ $\displaystyle e^{\Gamma} T_{\rm lam}^+ + e^\frac{1}{\Gamma} T_{\rm turb}^+$ (12.10-38)

where the notation for $T_P$ and $\dot q$ is the same as for standard thermal wall functions (see Equation  12.10-5). Furthermore, the blending factor $\Gamma$ is defined as
$\displaystyle \Gamma$ $\textstyle =$ $\displaystyle - \frac{a ({\rm Pr} \, y^+)^4}{1 + b {\rm Pr}^3 \, y^+}$ (12.10-39)

where ${\rm Pr}$ is the molecular Prandtl number, and the coefficients $a$ and $b$ are defined as in Equation  12.10-29.

Apart from the formulation for $T^+$ in Equation  12.10-38, enhanced thermal wall functions follow the same logic as for standard thermal wall functions (see Section  12.10.2), resulting in the following definition for turbulent and laminar thermal wall functions:
 T^+_{\rm lam} = {\rm Pr} \left( u^+_{\rm lam} + \frac{\rho u^*}{2 \dot q} u^2 \right) (12.10-40)


 T^+_{\rm turb} = {\rm Pr}_{\rm t} \left\{ u^+_{\rm turb} + ... ... Pr}_{\rm t}} - 1 \right) (u_c^+)^2 (u*)^2 \right] \right\} (12.10-41)

where the quantity $u_c^+$ is the value of $u^+$ at the fictitious "crossover" between the laminar and turbulent region. The function $P$ is defined in the same way as for standard wall functions.

A similar procedure is also used for species wall functions when the enhanced wall treatment is used. In this case, the Prandtl numbers in Equations  12.10-40 and 12.10-41 are replaced by adequate Schmidt numbers. See Section  12.10.2 for details about species wall functions.

The boundary condition for turbulence kinetic energy is the same as for standard wall functions (Equation  12.10-9). However, the production of turbulence kinetic energy $G_k$ is computed using the velocity gradients that are consistent with the enhanced law-of-the-wall (Equations  12.10-28 and 12.10-30), ensuring a formulation that is valid throughout the near-wall region.

figure   

The enhanced wall treatment is available with the following viscous models:
  • K-epsilon

  • Reynolds Stress
Enhanced wall functions are available with the following viscous models:
  • Spalart-Allmaras

  • K-omega

  • Large Eddy Simulation


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