Turbulence Chapter 3: Near-Wall Modeling
Chapter 3 was the exploration of turbulence modeling in one of the trickiest zones: the region right next to the wall. This is where sharp gradients, subtle balances, and small-scale interactions shape the drag, heat transfer, and flow separation we ultimately care about.
The Structure of the Near-Wall Region
Turbulent flow near walls is not uniform - it has distinct layers, each dominated by different physics:
Viscous sublayer (y-plus below 5):
This is the very thin region closest to the wall, where molecular viscosity dominates, and turbulence is damped out. The mean velocity increases linearly with distance from the wall.Buffer layer (y-plus between 5 and 30):
This is a transition zone where both viscous and turbulent stresses are important. Neither purely viscous nor purely turbulent - it’s a mixed region where the flow starts to shift.Log-layer (y-plus between 30 and 300):
Here, turbulent shear stress dominates, and viscous effects are negligible. The mean velocity follows a logarithmic pattern that’s been confirmed across many flows and experiments, making it a reliable anchor for models.Outer (defect) layer:
Farther from the wall, the flow transitions to match the outer conditions. This region adjusts the inner near-wall behavior to the larger free-stream flow.
Understanding these layers is crucial because each region has different dominant mechanisms, and modeling approaches must respect these boundaries.
Scaling and Non-Dimensionalization
To uncover universal behaviors, we use non-dimensional variables. The two key ones are:
The dimensionless velocity, called U-plus, which scales the velocity by the friction velocity (a value derived from the wall shear stress).
The dimensionless distance, called y-plus, which scales the physical distance from the wall using the friction velocity and the fluid’s viscosity.
Expressing velocity and distance in these scaled terms reveals patterns that hold across vastly different flows. For example, in the near-wall region, the velocity profile is linear in U-plus versus y-plus, but farther out, it follows a logarithmic trend.
This non-dimensional scaling underpins most of the near-wall models we use, allowing us to apply a general law - like the logarithmic velocity profile - even if the specific geometry or Reynolds number varies.
Modeling Strategies: Wall Functions and Low-Reynolds Approaches
In practice, engineers must choose how to handle the near-wall region when running simulations. There are two strategies:
Wall functions:
This approach avoids resolving the steep near-wall gradients by placing the first grid point relatively far from the wall, typically where y-plus is greater than 30. We then apply empirical formulas to estimate the missing near-wall physics, especially the velocity and turbulence behavior in the log-layer. This method assumes local equilibrium, meaning the turbulence production and dissipation are balanced, and it works best when there are no strong pressure gradients, flow separation, or curvature effects.Low-Reynolds-number models:
Here, the solver fully resolves down to the wall by placing the first grid point at a y-plus close to 1. This approach makes no assumptions about the near-wall behavior and directly calculates the details - but it demands a much finer mesh, increasing computational cost.
To make things more robust, hybrid methods have been developed:
Scalable wall functions introduce a minimum cutoff value for y-plus, so the log-law isn’t misapplied if the mesh is too fine.
Non-equilibrium wall functions relax some of the local equilibrium assumptions and include pressure gradient effects, making them more accurate in complex flows.
Enhanced Wall Treatment (EWT) blends low-Reynolds and wall-function models, dynamically adapting to the mesh resolution and local flow conditions.
These advanced strategies help balance accuracy and cost, especially when dealing with large or complex geometries where mesh control is difficult.
Rough Wall Effects
In reality, surfaces are rarely perfectly smooth, and roughness has a major impact on near-wall turbulence. Roughness disrupts the viscous sublayer, enhances momentum and heat transfer, and increases the wall shear stress.
We quantify roughness in CFD using two parameters:
The equivalent sandgrain roughness height, which represents the average size of roughness elements on the surface.
The dimensionless roughness height, called k-plus, which scales the roughness size using the friction velocity and viscosity.
Depending on the value of k-plus, surfaces are categorized as hydraulically smooth (if k-plus is below about 5), transitional (if between roughly 5 and 70), or fully rough (if above 70).
Rough walls cause the logarithmic velocity profile to shift downward by an amount called the roughness function. This means that, at the same wall-normal distance, the velocity is lower than over a smooth wall. Accounting for this shift is essential for accurate predictions of drag, pressure loss, and thermal performance.
Practical Considerations: Mesh and Grid Sensitivity
While refining the mesh often improves simulation accuracy, this isn’t always true when using standard wall functions. These functions assume that the first grid point is placed in the log-layer. If mesh refinement pushes the first point into the buffer layer or viscous sublayer, the assumptions break, and the results can deteriorate.
To avoid this, it’s good practice to estimate the first-layer cell height before meshing. This involves calculating the desired physical height that corresponds to a target y-plus value (typically 1 for low-Reynolds models or around 50 if using wall functions). This estimation requires knowledge of the fluid’s properties and an estimate of the friction velocity, which can be derived from empirical skin friction coefficients or simple flat-plate calculations.
For cases where the mesh quality varies or where different regions require different treatments, blended models like Enhanced Wall Treatment or two-layer zonal models can provide an adaptive approach that switches between wall functions and low-Reynolds modeling depending on local conditions.
Final Takeaway
Near-wall modeling is where turbulence theory, empirical modeling, and practical engineering converge. Whether you’re using wall functions, enhanced treatments, or fully resolving the boundary layer, success depends on understanding the layered structure of the near-wall region, knowing the assumptions and limits of each modeling approach, and carefully preparing your mesh.
In turbulent flows, the wall isn’t just a boundary - it’s where the real complexity unfolds.