In a CFD simulation (see FMK in the 3DEXPERIENCE and PowerFLOW) that involves wall-bounded flows, it is crucial to accurately predict the behavior of the turbulent boundary layer. Accurate prediction of the turbulent boundary layer dictates the ability of the CFD simulation to accurately predict forces exerted by the fluid on the solid surfaces such as lift and drag, flow separation, and how turbulence is generated, transported, and dissipated.
In a typical boundary layer, the flow quantities of interest such as velocity and temperature are equal to the freestream value away from the wall, and 0 close to the wall. As you get closer to the wall, the gradients of these quantities are quite large, and hence a fine mesh resolution is needed very close to the wall to accurately capture these gradients.
Wall-Resolved VS Wall-Modeled Approaches
A wall-resolved approach (shown above) ensures that the mesh is fine enough near the walls such that y+ < 1.
The definitions of y+ and u+ are given below.
y+: This is the nondimensional distance away from the wall (normal to the wall), defined as
where,
y is the dimensional distance away from the wall
ν is the kinematic viscosity of the fluid
u+: This is the nondimensional velocity parallel to the wall, defined as
where,
uτ = √τw/ρ, the friction velocity
u is the fluid velocity
τw is the wall shear stress
ρ is the fluid density
This guarantees that the viscous sublayer is directly resolved, providing highly accurate results. However, this approach requires a very high mesh count, which can be computationally expensive, especially for complex geometries. It may also lead to elements with high aspect ratios, which can impact the numerical stability of the CFD solver.
Although this approach provides a highly accurate solution, a fine mesh resolution near the walls results in a very high mesh count. This is feasible for simple flow scenarios but can be computationally expensive and prohibitive for complex cases and geometries. A fine mesh resolution near the walls also leads to cells or elements with high aspect ratios, which can affect the numerical stability of your CFD solver.
A wall-modeled approach (shown above) offers a more computationally efficient alternative. Instead of resolving the boundary layer down to y+ < 1. This approach assumes that the first cell center falls within the logarithmic region of the boundary layer, typically in the range 30 < y+ < 300 . This is achieved using wall functions, which approximate the near-wall flow behavior using empirical and semi-empirical relations.
Law of the wall
Wall functions are derived from the law of the wall, whic describes the velocity profile in the near-wall region of turbulent flows. As long as the boundary layer is fully turbulent and the flow is attached, all turbulent boundary layers follow a universal behavior, described by the u+ (nondimensional velocity) vs y+ (nondimensional distance from the wall) profile. This profile is shown below (red curve denotes the turbulent boundary layer profile, whereas blue curves denote the viscous sublayer and the log-law layer curves).
Fig: u+ vs y+ profile for a turbulent boundary layer.
(source: aokomoriuta（青子守歌）, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons)
The law of the wall is given by:
U+ = y+ y+ < 5 (Viscous sublayer) 
30 < y+ < 300 (Low-law region)
Where κ and C are constants, with widely reported values of κ ~ 0.4 and C ~ 5.0 for a smooth wall, determined from experimental data of turbulent boundary layers.
Within the buffer region (5 < y+ < 30), most CFD codes use a proprietary blending function for a smooth transition between the viscous sublayer and log-law layer.
Types of Wall Functions
Standard Wall Functions
These are based on the log-law and assume that the near-wall region follows a universal profile. They work well for high-Re flows but may become inaccurate for flows with strong pressure gradients, separation, or adverse effects.
Continuous Wall Function
Instead of using separate equations for different regions, a continuous wall function provides a single equation that smoothly transitions across the near-wall region. An example is Spalding’s Law, given by:
In addition, most CFD solvers also use enhanced wall treatments, which are hybrid near-wall modeling approaches designed to work across a wide range of y+ values, allowing CFD solvers to transition seamlessly between low-Re (fully resolved) and high-Re (wall-function-based) approaches. These methods improve accuracy in complex turbulent flows, particularly those involving flow separation, pressure gradients, and heat transfer.
Limitations of using the wall function approach
Although wall functions provide significant savings in computational costs, they also have some limitations. They rely on assumptions that may not hold in flows with strong pressure gradients, separation, or adverse conditions such as flow reversal. For example, in highly separated flows or cases with significant three-dimensional effects, wall functions may introduce inaccuracies compared to wall-resolved approaches. Moreover, applying wall functions incorrectly, such as using them when y+ is too low, can lead to incorrect predictions of wall shear stress and heat transfer.
Final Thoughts
Choosing between a wall-resolved and a wall-modeled approach depends on the trade-off between accuracy and computational resources. High-fidelity simulations with sufficient computational power can benefit from resolving the boundary layer fully, while industrial and engineering applications often rely on wall functions to achieve reasonable accuracy with practical computational costs. Understanding the strengths and weaknesses of both approaches is key to making informed decisions in CFD simulations involving turbulent boundary layers.
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