A Crash Introduction to Near-wall Treatment in RANS Turbulence Modelling

In the near-wall region, the flow property changes dramatically. The flow region can be divided into three layers,

• the viscous sublayer($y^{+}<10$), where the flow is laminar, and the viscosity is dominant
• the buffer layer ($10<y^{+}<30$), which is a transitional region and no one knows what is happening there
• the log-law layer($30<y^{+}<300$), where the flow is fully turbulent

There are two kinds of ways to do near-wall treatment; one is called Resolved BL, solving the PDE up to the viscous sub-layer, the other called Wall Functions, having everything below $y^{+}=30$ modelled by algebraic equations instead of solved by PDEs.

Boundary Layer Resolving

If what happens in the viscous sublayer is what you care (as in aerodynamic drag, turbo-machinery blade performance, heat transfer), you would prefer a resolved boundary layer, which is also computationally expensive. When dealing with resolved BL, take notice that

• You should mesh all the way to the wall, both covering the whole boundary layer sufficiently and achieving the certain $y^{+}$ criterion. This kind of fine mesh is called Low-Re mesh, requiring

  • 10 ~ 20 layers to resolve the boundary layer and cover a little bit thicker to ensure the boundary layer of growing fully,

    • For wall-bounded flows, the turbulent viscosity $\nu_{\mathrm{t}}$ (or nut) has a maximum value in the middle of the boundary layer, which can be used to estimate the thickness of the BL
    • For free shear flows, this is hard to achieve, because the position of the shear layer is not known
  • The first grid from the wall placed at $y^{+} \approx 1$ for high resolution
  • 10 layers clustered for $y^{+} < 6$ to get better results,
  • Expansion ratio $\leqslant 1.1$ for transition models, and additional mesh refinement at regions where laminar separation occurs
• The turbulence model should be valid throughout the near-wall region. Bear in mind that some models (High-Re) like the k-epsilon family, RSM, LES are sound only for core flows while some (Low-Re) are applicable throughout the whole boundary layer like k-omega and S-A.

Wall Functions

If the wall turbulence is not of a big concern, use wall functions to cut the computational cost. When using wall functions,

• We do not resolve the viscous sublayer and the buffer layer. Instead, we bridge the log-law region directly to the wall with the first layer of cells at the wall.
• The High-Re turbulence models can be used.

• Place the first point outside $y^{+}=30$ or use $y^{+}$-independent/insensitive wall functions

  • It is interesting to know that excessive near-wall refinement should result in deteriorated results. However, taking a second thought, it is understandable when you imagine the first grid locates at the viscous sublayer or the buffer layer, where the wall function is not valid, and the rest Low-Re region would be resolved by High-Re models. Actually, $y^{+}<15$ would result in unbounded errors in shear stress and heat transfer and most wall functions are $y^{+}$-sensitive.
  • There are exceptions. For example, when using scalable wall functions, which is $y^{+}$-independent and valid in the whole boundary layer, $y^{+}$ value from 1 to 300 can be used, but not all turbulence models support it. In Ansys FLUENT, some innovative $y^{+}$-insensitive wall functions are also available.
  • In any case, avoid using $5<y^{+}<30$ because it is a transitional region and no one knows what happens there.

The workflow of Near-wall Treatment in RANS Turbulence Modelling

The simulation requirement determines the near-wall treatment method(Resolved BL or Wall Functions), calling for the responding mesh types(High-Re or Low-Re). Each mesh has its corresponding turbulence model that produces the most desirable results.

• Decide whether to use Low-Re approach

  • For thermal boundary layers, a low-Re approach is recommended
  • For steady RANS simulations, it is not computationally expensive to use Low-Re approach
  • For unsteady RANS simulations, a low-Re mesh would bring about strict time-step restrictions. Although large CFL numbers in URANS simulation can also be used, it is still tempting to use a high-Re mesh in order to save the computational resource.
• Rule-of-the-thumb estimation of the first layer thickness using the desired $y^{+}$ value: $$y =\frac{\mu{}y^{+}}{\rho{}U_{\tau}}$$ $$U_{\tau} = \sqrt{\frac{\tau_w}{\rho}}, \tau_{w} = 0.5C_f\rho{}U_{\infty}^2, C_f = 0.058Re^{-0.2}, Re = \frac{\rho{}UL}{\mu}$$

• Reconnaissance by fire: Use a coarse mesh, run the simulation for a few steps or iterations and get an estimate of

  • the $y^{+}$ value, which we cannot know a priori because of the unknown friction velocity
  • the BL thickness by observing the nut contour
• Modify the mesh if necessary. Cover the BL sufficiently then achieve the certain $y^{+}$ criterion
• Run the simulation, and check again

It is an iterative process and very time consuming, but gives desirable results.