# Near-Wall Modeling

For internal wall bounded flows, proper mesh resolution is required in order to calculate the steep gradients of the velocity components, turbulent kinetic energy, dissipation, as well as the temperature.

Furthermore, the first grid locations from the wall and the stretching ratio between subsequent points are also determinant factors that affect solution accuracy. Since boundary layer thickness is reduced as the Reynolds number increases, the computational cost increases for higher Reynolds number flows due to the dense grid requirements near the walls. Additionally, it is hard to determine the first grid locations for complex three dimensional industrial problems without adequate testing and simulation.

For high Reynolds number flows, it is numerically efficient if flows within boundary layers are modeled rather than resolved down to the wall. This suggests that a coarse mesh is used at the expense of numerical accuracy when compared to fully wall resolved approaches.

## Wall Function

Figure 1 shows velocity profiles over a flat plate under the zero pressure gradient.

Velocity profile and wall distances are scaled by the friction velocity (${u}_{\tau }=\sqrt{\frac{{\tau }_{w}}{\rho }}$ where ${\tau }_{w}$ is the wall shear stress and $\rho$ is the fluid density). This is referred to as the log-law velocity profile. As shown in Figure 1, the velocity profiles are divided into three distinguished regions, the viscous sublayer, the buffer layer and the logarithmic layer.

The wall function utilizes the universality of the log-law velocity profile to obtain the wall shear stress. This wall function approach allows the reduction of mesh requirement near the wall as the first mesh location is placed outside the viscous sublayer.

Overall, the log-law based wall model is economical and reasonably accurate in most flow conditions, especially for high Reynolds number flows. However, it tends to display poor performance in situations with low Reynolds number flows, strong body forces (rotational effect, buoyancy effect), and massively separated flows with adverse pressure gradients.

### Velocity Profile in the Viscous Sublayer (y+ < 5)

In the viscous sublayer, the normalized velocity profile (${U}^{+}$) has a linear relationship with the normalized wall distance. (1) ${U}^{+}={y}^{+}$

where ${U}^{+}=\frac{\overline{u}}{{u}_{\tau }}$ is the velocity ($\overline{u}$) parallel to the wall, normalized by the friction velocity. The friction velocity is defined as ${u}_{\tau }=\sqrt{\frac{{\tau }_{w}}{\rho }}$, ${\tau }_{w}$ is the wall shear stress and $\rho$ is the fluid density. ${y}^{+}=\frac{\rho {u}_{\tau }y}{\mu }$ is the normalized wall distance (or wall unit). The distance from the wall is $y$ and $\mu$ is the fluid dynamic viscosity.

### Velocity Profile in the Logarithmic Layer (30 < y+ < 500)

In the logarithmic layer the velocity profile can be given by a logarithmic function (2) ${U}^{+}=\frac{1}{\kappa }\mathrm{log}\left({y}^{+}\right)+B$

where $\kappa$= 0.4 is the Von Kármán constant and B = 5.5 is a constant.

The wall shear stress can be estimated from the above equation via the iterative solution procedure.

### Velocity Profile in the Buffer Layer (5 < y+ < 30)

Since two equations mentioned before are not valid in the buffer layer a special function is needed to bridge the viscous sublayer and the logarithmic layer. Details are not covered here.

### Kinematic Eddy Viscosity

The kinematic eddy viscosity can be obtained as ${\nu }_{t}=\kappa y{u}_{\tau }$.

## Two Layer Wall Model

The two equation turbulence models based on eddy frequency (ω) and the Spalart-Allmaras (SA) model do not require special wall treatments to solve the boundary layer as these models are valid through the viscous sub layer.

However, the two equation turbulence models based on turbulence dissipation rate (ε) need additional functions to simulate the near wall effects. The two layer wall model is one of them. In the two layer model, turbulent kinetic energy is determined from the turbulent kinetic energy transport equation, while dissipation rate is resolved with a one equation turbulence model (Wolfstein, 1969) in the near wall, viscous-affected regions where $R{e}_{y}=\frac{\rho y\sqrt{k}}{\mu }<200$. (3) $\epsilon =\frac{{k}^{3/2}}{{l}_{\epsilon }}$
where
• ${l}_{\epsilon }={C}_{l}y\left(1-\mathrm{exp}\left[-\frac{R{e}_{y}}{{A}_{\epsilon }}\right]\right)$,
• ${C}_{l}=\kappa {C}_{\mu }{}^{-3/4}$,
• $\kappa$ is the Von Kármán constant,
• ${A}_{\epsilon }$ = 5.08 (Chen and Patel, 1988).
The eddy viscosity can be written as (4) ${\nu }_{t}={C}_{\mu }{l}_{\mu }\sqrt{k}$
where
• ${l}_{\mu }={C}_{l}y\left(1-\mathrm{exp}\left[-\frac{R{e}_{y}}{{A}_{\mu }}\right]\right)$,
• ${A}_{\mu }$ = 70.