Turbulence Chapter 2: Turbulence Anisotropy in RANS
This chapter introduces Reynolds-Stress Models (RSM), a second-order turbulence closure that abandons the Boussinesq hypothesis and directly solves transport equations for the individual Reynolds stresses. The chapter explains why anisotropy matters, how the Reynolds-stress transport equations are constructed and closed, and why the pressure–strain correlation is the central modeling challenge. Practical variants of RSM and their numerical behavior in industrial CFD are also discussed.
Motivation: Beyond Eddy Viscosity
2.1 Why Two-Equation Models Fail
Two-equation RANS models assume:
Turbulence is isotropic at small scales
Reynolds stresses align with the mean strain rate
This breaks down in flows dominated by:
Strong curvature or rotation
Swirl and secondary motion
Rapid changes in strain rate
Stress-induced secondary flows in ducts
In such cases, turbulence remembers its history and responds in directions not aligned with the local strain field .
2.2 Core Idea of RSM
Instead of modeling Reynolds stresses algebraically:
RSM solves a transport equation for each stress component
No assumption of isotropy is made
Turbulence anisotropy is resolved explicitly
This shifts the modeling effort from:
“How should stresses depend on strain?”
to
“How do stresses evolve in space and time?”
Reynolds-Stress Transport Equation: Structure and Meaning
3.1 What Is Being Transported
Each component of the Reynolds-stress tensor follows a transport balance that includes:
Convection by the mean flow
Production by mean velocity gradients
Redistribution via pressure–strain interaction
Dissipation by small-scale turbulence
Molecular and turbulent diffusion
Physically, this means:
Each stress component has its own dynamics
Stress orientation and magnitude evolve independently
3.2 Why This Is More Physical
Compared to eddy-viscosity models:
Stress evolution responds immediately to curvature and rotation
Flow history effects are retained
Large eddies are no longer tied solely to local strain
This is the key reason RSM succeeds in complex 3D flows.
Closure Problem Revisited
Although RSM resolves stresses directly:
Several higher-order terms remain unclosed
These terms require modeling assumptions
The most important modeled terms are:
Dissipation tensor
Turbulent transport
Pressure–strain correlation
Among these, the pressure–strain term dominates model behavior.
Dissipation Modeling
5.1 Physical Role
Dissipation represents:
Energy transfer from large turbulent structures to small scales
Conversion of kinetic energy into heat by viscosity
At small scales, turbulence tends toward isotropy.
5.2 Modeling Assumption
Most RSM formulations assume:
Dissipation is isotropic away from walls
Dissipation rate comes from a separate scale equation (ε or ω)
Near walls:
Turbulence becomes strongly anisotropic
Corrections or damping functions are required
Turbulent Transport Modeling
6.1 Physical Interpretation
Turbulent transport represents:
Redistribution of stresses by large eddies
Non-local transport of turbulence properties
Because triple correlations are involved:
Direct modeling is not feasible
Gradient-diffusion assumptions are used
6.2 Limitations
This modeling step:
Reintroduces a form of eddy-viscosity logic
Assumes separation between transport scales and stress variation scales
Despite this, it performs reasonably well in practice for many flows.
Pressure–Strain Correlation: The Heart of RSM
7.1 What Pressure–Strain Does
Pressure–strain interaction:
Redistributes turbulent kinetic energy among components
Drives turbulence toward or away from isotropy
Acts on the same order of magnitude as production
It does not create or destroy energy — it redistributes it.
7.2 Slow vs Rapid Response
The pressure–strain term is split conceptually into:
Slow (return-to-isotropy) contribution
Dominant in weakly strained flows
Drives turbulence toward isotropy over time
Rapid contribution
Dominant in rapidly strained or curved flows
Responds directly to mean velocity gradients
Captures curvature, rotation, and swirl effects
7.3 Why This Is Hard to Model
Pressure depends on the entire velocity field
Involves non-local, two-point correlations
Limited experimental and DNS guidance
As a result:
Most differences between RSM variants come from pressure–strain modeling.
Reynolds-Stress Anisotropy
8.1 Anisotropy Tensor
Turbulence anisotropy is quantified by:
Deviations of stresses from isotropic form
A traceless, symmetric anisotropy tensor
This tensor:
Has five independent components
Encodes directionality of turbulence
8.2 Lumley Triangle
Using invariants of the anisotropy tensor:
Turbulent states can be plotted in a realizability map
Physical turbulence must lie inside this domain
Important limiting states:
Isotropic turbulence
Axisymmetric turbulence
Two-component and one-component turbulence
This framework is central for:
Model validation
Realizability enforcement
Practical RSM Variants (ANSYS Fluent)
9.1 Linear Pressure–Strain Model (LRR)
Most robust and widely used
Includes wall-reflection effects
Can be used with wall functions or enhanced wall treatment
Best for:
General industrial applications
Moderate curvature and swirl
9.2 Quadratic Pressure–Strain Model
Higher fidelity in complex strain environments
Better prediction of rotating and axisymmetric flows
Requires wall-function meshes
Limitation:
No low-Re formulation available in Fluent
9.3 Stress–Omega Model
Uses ω as scale equation
Natural near-wall behavior
Excellent for curvature and swirling flows
Often the most reliable RSM option when wall resolution is available.
9.4 Stress-BSL Model
Reduces free-stream sensitivity
Blends robustness and near-wall performance
Good compromise for industrial meshes
Boundary Conditions and Near-Wall Treatment
RSM requires:
Boundary conditions for each Reynolds-stress component
A scale equation boundary condition
Near walls:
Wall-function or enhanced wall treatments are used
Low-Re RSM requires careful mesh design (y⁺ ≈ 1)
Incorrect near-wall treatment is a common cause of failure.
Numerical Behavior and Stability
11.1 Computational Cost
Compared to two-equation models:
~50–60% higher CPU cost
More memory usage
Stronger non-linear coupling
11.2 Convergence Challenges
RSM is:
Less dissipative numerically
More sensitive to mesh quality
More prone to instability
Best practices:
Start from a converged simpler RANS solution
Use conservative under-relaxation
Avoid excessive mesh skewness
When RSM Is Worth It
RSM is justified when:
Secondary flows drive performance
Swirl or rotation dominates
Stress anisotropy matters physically
Typical applications:
Cyclones
S-ducts and intakes
Curved ducts with sharp corners
Tip vortices and rotating machinery
For many flows:
Accuracy gain over SST or k-ω may be modest
Cost and convergence risk must be weighed
Engineering Intuition
RSM resolves how turbulence is oriented, not just how strong it is
Pressure–strain redistribution is the key physical mechanism
Better physics ≠ easier convergence
RSM should be used selectively, not by default
A good rule:
If the flow physics depends on turbulence directionality, RSM is often the right tool.
Study Priorities
If time is limited, focus on:
Why Boussinesq fails
Structure of the Reynolds-stress transport equation
Role of pressure–strain correlation
Turbulence anisotropy and Lumley triangle
Differences between RSM variants
Key Takeaways
RSM abandons isotropic eddy viscosity assumptions.
Reynolds stresses are transported, not prescribed.
Pressure–strain modeling controls accuracy.
Anisotropy is central to secondary flows and swirl.
RSM is powerful but computationally demanding.
Correct near-wall treatment is essential.