4.12.4 Enhanced Wall Treatment

Enhanced wall treatment is a near-wall modeling method that combines a two-layer model with so-called enhanced wall functions. If the near-wall mesh is fine enough to be able to resolve the viscous sublayer (typically with the first near-wall node placed at $y^+ \approx 1$), 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, 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 the intermediate meshes where the first near-wall node is placed neither in the fully turbulent region, where the wall functions are suitable, nor in the direct vicinity of the wall at $y^+ \approx 1$, where the low-Reynold-number approach is adequate.

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, ANSYS 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 ANSYS 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

where $y$ is the wall-normal distance calculated at the cell centers. In ANSYS FLUENT, $y$ is interpreted as the distance to the nearest wall:

where ${\vec r}$ is the position vector at the field point, and ${\vec r}_w$ is the position vector of 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.

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  4.4 and 4.9) are employed.

In the viscosity-affected near-wall region ( ${\rm Re}_y < {\rm Re}_y^*$), the one-equation model of Wolfstein [ 382] is employed. In the one-equation model, the momentum equations and the $k$ equation are retained as described in Sections  4.4 and 4.9. However, the turbulent viscosity, $\mu_t$, is computed from

where the length scale that appears in Equation  4.12-22 is computed from [ 51]

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 [ 153]:

where $\mu_t$ is the high-Reynolds-number definition as described in Section  4.4 or 4.9 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 away from walls and is zero in the vicinity of the walls. The blending function has the following form:

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

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 value of $\mu_t$ obtained in the outer layer does not match with the value of $\mu_t$ returned by the Wolfstein model at the edge of the viscosity-affected region.

The $\epsilon$ field in the viscosity-affected region is computed from

The length scales that appear in Equation  4.12-27 are computed from Chen and Patel [ 51]:

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  4.12-27. ANSYS FLUENT uses a procedure for the blending of $\epsilon$ 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 Equations  4.12-23 and 4.12-28, are taken from [ 51] and are as follows:

Enhanced Wall Functions

To have a method that can extend its applicability throughout the near-wall region (i.e., viscous 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. ANSYS FLUENT achieves this by blending the linear (laminar) and logarithmic (turbulent) laws-of-the-wall using a function suggested by Kader [ 155]:

where the blending function is given by:

where $a = 0.01$ and $b = 5$.

Similarly, the general equation for the derivative $\frac{d u^+}{d y^+}$ is

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 [ 378] and Huang et al. [ 134]:

where

and

$$\alpha \equiv \frac{\nu_w}{\tau_w u^*}\frac{dp}{dx} = \frac{\mu}{\rho^2(u^*)^3} \frac{dp}{dx}$$ (4.12-35)

$$\beta \equiv \frac{\sigma_t q_w u^*}{c_p \tau_w T_w} = \frac{\sigma_t q_w}{\rho c_p u^* T_w}$$ (4.12-36)

$$\gamma \equiv \frac{\sigma_t (u^*)^2}{2 c_p T_w}$$ (4.12-37)

where $y^+_s$ is the location at which the log-law slope is fixed. By default, $y^+_s = 60$. The coefficient $\alpha$ in Equation  4.12-33 represents the influences of pressure gradients while the coefficients $\beta$ and $\gamma$ represent the thermal effects. Equation  4.12-33 is an ordinary differential equation and ANSYS 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:

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  4.12-38 results in

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 [ 155]:

$$T^+ \equiv \frac{\left(T_w - T_P\right) \rho c_p u_{T}}{\dot q} = e^{\Gamma} T_{\rm lam}^+ + e^\frac{1}{\Gamma} T_{\rm turb}^+$$ (4.12-40)

where the notation for $T_P$ and $\dot q$ is the same as for standard thermal wall functions (see Equation  4.12-6). Furthermore, the blending factor $\Gamma$ is defined as

$$\Gamma = - \frac{a ({\rm Pr} \, y^+)^4}{1 + b {\rm Pr}^3 \, y^+}$$ (4.12-41)

where ${\rm Pr}$ is the molecular Prandtl number, and the coefficients $a$ and $b$ are defined as in Equation  4.12-31.

Apart from the formulation for $T^+$ in Equation  4.12-40, the enhanced thermal wall functions follow the same logic as for standard thermal wall functions (see Section  4.12.2), resulting in the following definition for turbulent and laminar thermal wall functions:

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 the 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  4.12-42 and 4.12-43 are replaced by adequate Schmidt numbers. See Section  4.12.2 for details about the species wall functions.

The boundary conditions for the turbulence kinetic energy are similar to the ones used with the standard wall functions (Equation  4.12-10). 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  4.12-30 and 4.12-32), ensuring a formulation that is valid throughout the near-wall region.

The enhanced wall treatment is available with the following turbulence closures: k- $\epsilon$ models Realizable k- $\epsilon$ based DES model Reynolds Stress Transport models The enhanced wall functions are available with the following turbulence models: Spalart-Allmaras model k- $\omega$ models k- $\omega$ based DES model Large Eddy Simulation However, the enhanced wall functions are not available with Spalart-Allmaras model.