Turbulence Chapter 3: Near-Wall Modeling
This chapter focuses on turbulence behavior in the near-wall region and explains why special modeling strategies are required close to solid boundaries. The physical structure of the turbulent wall layer is derived using scaling arguments, leading to the identification of the viscous sublayer, buffer layer, logarithmic region, and outer layer. Based on this structure, different near-wall modeling approaches are introduced, including wall functions, low-Reynolds-number models, and blended or zonal treatments. Practical implications for mesh design, accuracy, and robustness are emphasized.
Why Near-Wall Modeling Is Critical
Walls are the primary source of:
Vorticity generation
Turbulence production
Momentum and heat transfer
Engineering quantities of interest such as:
Skin friction
Pressure drop
Heat transfer coefficient
Separation and reattachment
are all controlled by near-wall physics .
However:
Velocity and turbulence gradients near walls are extremely steep
Fully resolving these gradients is often too expensive
Near-wall modeling exists to balance accuracy vs computational cost.
Structure of Turbulent Flow Near a Smooth Wall
3.1 Quasi One-Directional Flow Assumption
Away from stagnation and separation:
Mean velocity is primarily tangential to the wall
Wall-normal velocity is small
Pressure gradient normal to the wall is negligible
This allows a simplified, locally one-dimensional description of the wall layer .
3.2 Competing Shear Mechanisms
Two shear stresses act near the wall:
Viscous shear stress (molecular)
Turbulent shear stress (Reynolds stress)
Their relative importance varies with distance from the wall:
Very close to wall → viscous stress dominates
Farther away → turbulent stress dominates
This competition defines the layered structure of the wall region.
Wall-Normal Scaling and Friction Velocity
The near-wall region is governed by:
Wall shear stress
Kinematic viscosity
These define the friction velocity, which sets the velocity scale for near-wall turbulence .
Using this scale, wall-normal distance and velocity are expressed in non-dimensional wall units, leading to universal behavior independent of the outer flow — as long as equilibrium assumptions hold.
The Four Near-Wall Regions
5.1 Viscous Sublayer
Very close to the wall
Turbulent fluctuations are suppressed
Flow behaves essentially laminar
Velocity varies linearly with distance
Here:
Viscous shear carries nearly all the wall stress
Turbulence models are not valid
5.2 Buffer Layer
Transitional region between viscous and turbulent dominance
Turbulence production and dissipation are not in equilibrium
Strong intermittency and bursting events occur
This is the most difficult region to model accurately.
5.3 Logarithmic (Log) Layer
Turbulent shear stress dominates
Viscous effects are negligible
Velocity follows a logarithmic variation with wall distance
Key properties:
Turbulent production ≈ dissipation
Turbulent kinetic energy is approximately uniform
Reynolds stresses are nearly constant
5.4 Outer (Defect) Layer
Flow begins to feel the outer geometry and pressure gradients
Universal scaling breaks down
Flow becomes problem-dependent
Universality of the Wall Layer
Within a wide range of wall distances:
Mean velocity and turbulent stresses depend only on wall distance and viscosity
The structure is independent of Reynolds number and geometry
This universality is the theoretical basis for wall functions .
However, it holds only for:
Equilibrium boundary layers
Mild pressure gradients
Attached flow
Why Standard Turbulence Models Fail Near Walls
Most RANS models assume:
Isotropic turbulence
Equilibrium between production and dissipation
Fully turbulent flow
Near walls:
Turbulence is highly anisotropic
Dissipation dominates close to the wall
Length scales collapse
As a result:
Standard k-ε and RSM equations cannot be integrated directly to the wall without modification .
Near-Wall Modeling Strategies
Two broad approaches are used.
8.1 Wall-Function Approach
Wall functions:
Avoid resolving the viscous sublayer
Use empirical laws to bridge from the wall to the log layer
Place the first computational point inside the logarithmic region
Advantages:
Low mesh requirement
Very robust for high-Re flows
Limitations:
Valid only for equilibrium flows
Sensitive to mesh placement
Poor for separation, curvature, and low-Re flows
8.2 Low-Reynolds-Number Modeling
Low-Re approaches:
Resolve the entire boundary layer
Integrate turbulence equations down to the wall
Apply damping functions to turbulence quantities
Advantages:
Physically consistent
Accurate for complex near-wall flows
Limitations:
Requires very fine near-wall mesh
Higher computational cost
Standard Wall Functions
Standard wall functions:
Assume first cell center lies in the log layer
Impose logarithmic velocity profile
Compute turbulence quantities using local equilibrium assumptions
Major drawback:
Solution quality degrades under mesh refinement if the first cell enters the viscous sublayer .
This violates a core numerical principle: solutions should improve with mesh refinement.
Thickness of the Log Layer and Reynolds Number Effects
The log layer thickness depends strongly on Reynolds number:
Very thick at high Re (external aerodynamics)
Very thin at moderate and low Re (turbomachinery, internal flows)
Consequences:
Correct wall-function placement becomes difficult
Standard wall functions become unreliable for many engineering flows .
Scalable Wall Functions
Scalable wall functions:
Prevent the first grid point from entering the viscous sublayer
Artificially shift the wall location when mesh is too fine
Benefits:
Reduce sensitivity to y⁺
Improve robustness
Limitation:
Still rely on equilibrium assumptions
Do not replace proper boundary-layer resolution
Non-Equilibrium Wall Functions
Non-equilibrium wall functions:
Account for pressure gradients
Relax the assumption of local equilibrium
Improve predictions for separation and reattachment
They extend applicability but:
Still fail in strongly non-equilibrium flows
Remain approximate models .
y⁺-Insensitive (Blended) Wall Treatments
Blended approaches aim to:
Be insensitive to first-cell y⁺
Work across a wide range of mesh resolutions
Key idea:
Blend viscous-sublayer and log-layer formulations
Switch dynamically based on local flow conditions
Examples:
Enhanced Wall Treatment (k-ε)
Automatic Wall Treatment (k-ω, SST, RSM)
Two-Layer Zonal Concept
The domain is split into:
Viscosity-affected near-wall zone
Fully turbulent outer zone
Different models are used in each region:
Simple turbulence model near the wall
High-Re model in the core
Zoning is:
Automatic
Based on local turbulent Reynolds number
This approach combines robustness with physical realism.
Engineering Intuition
Near-wall modeling controls drag, heat transfer, and separation
Wall functions trade physics for robustness
Low-Re models trade cost for accuracy
Blended methods are the industrial sweet spot
Rule of thumb:
If wall behavior matters, mesh and near-wall treatment matter more than the turbulence model itself.
Study Priorities
If short on time, focus on:
Structure of the turbulent wall layer
Physical meaning of viscous, buffer, and log layers
Assumptions behind wall functions
Why standard wall functions fail under refinement
Difference between wall functions and low-Re approaches
Key Takeaways
Turbulent near-wall flow has a layered structure.
Universal wall behavior enables wall functions.
Standard RANS models are invalid very close to walls.
Wall functions reduce cost but rely on strong assumptions.
Low-Re models resolve physics but require fine meshes.
Blended approaches offer the best compromise for industry.