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12.10.1 Overview

Turbulent flows are significantly affected by the presence of walls. Obviously, the mean velocity field is affected through the no-slip condition that has to be satisfied at the wall. However, the turbulence is also changed by the presence of the wall in non-trivial ways. Very close to the wall, viscous damping reduces the tangential velocity fluctuations, while kinematic blocking reduces the normal fluctuations. Toward the outer part of the near-wall region, however, the turbulence is rapidly augmented by the production of turbulence kinetic energy due to the large gradients in mean velocity.

The near-wall modeling significantly impacts the fidelity of numerical solutions, inasmuch as walls are the main source of mean vorticity and turbulence. After all, it is in the near-wall region that the solution variables have large gradients, and the momentum and other scalar transports occur most vigorously. Therefore, accurate representation of the flow in the near-wall region determines successful predictions of wall-bounded turbulent flows.

The $k$- $\epsilon$ models, the RSM, and the LES model are primarily valid for turbulent core flows (i.e., the flow in the regions somewhat far from walls). Consideration therefore needs to be given as to how to make these models suitable for wall-bounded flows. The Spalart-Allmaras and $k$- $\omega$ models were designed to be applied throughout the boundary layer, provided that the near-wall mesh resolution is sufficient.

Numerous experiments have shown that the near-wall region can be largely subdivided into three layers. In the innermost layer, called the "viscous sublayer'', the flow is almost laminar, and the (molecular) viscosity plays a dominant role in momentum and heat or mass transfer. In the outer layer, called the fully-turbulent layer, turbulence plays a major role. Finally, there is an interim region between the viscous sublayer and the fully turbulent layer where the effects of molecular viscosity and turbulence are equally important. Figure  12.10.1 illustrates these subdivisions of the near-wall region, plotted in semi-log coordinates.

Figure 12.10.1: Subdivisions of the Near-Wall Region
figure

In Figure  12.10.1, $y^+ \equiv \rho u_{\tau} y/\mu$, where $u_{\tau}$ is the friction velocity, defined as $\sqrt{\frac{\tau_w}{\rho}}$.



Wall Functions vs. Near-Wall Model


Traditionally, there are two approaches to modeling the near-wall region. In one approach, the viscosity-affected inner region (viscous sublayer and buffer layer) is not resolved. Instead, semi-empirical formulas called "wall functions'' are used to bridge the viscosity-affected region between the wall and the fully-turbulent region. The use of wall functions obviates the need to modify the turbulence models to account for the presence of the wall.

In another approach, the turbulence models are modified to enable the viscosity-affected region to be resolved with a mesh all the way to the wall, including the viscous sublayer. For purposes of discussion, this will be termed the "near-wall modeling'' approach. These two approaches are depicted schematically in Figure  12.10.2.

Figure 12.10.2: Near-Wall Treatments in FLUENT
figure

In most high-Reynolds-number flows, the wall function approach substantially saves computational resources, because the viscosity-affected near-wall region, in which the solution variables change most rapidly, does not need to be resolved. The wall function approach is popular because it is economical, robust, and reasonably accurate. It is a practical option for the near-wall treatments for industrial flow simulations.

The wall function approach, however, is inadequate in situations where the low-Reynolds-number effects are pervasive in the flow domain in question, and the hypotheses underlying the wall functions cease to be valid. Such situations require near-wall models that are valid in the viscosity-affected region and accordingly integrable all the way to the wall.

FLUENT provides both the wall function approach and the near-wall modeling approach.



Wall Functions


Wall functions are a collection of semi-empirical formulas and functions that in effect "bridge'' or "link'' the solution variables at the near-wall cells and the corresponding quantities on the wall. The wall functions comprise

Depending on the turbulent model you choose, FLUENT offers three to four choices of wall function approaches:


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