Turbulence Chapter 2: Turbulence Anisotropy in RANS
Chapter 2 of the Turbulence course focuses on Reynolds-Stress Models (RSM) and their role in capturing anisotropic turbulence
Reynolds-Stress Models: Why Go Beyond Eddy Viscosity?
Q: Why do we need Reynolds-Stress Models (RSM)?
Most common RANS turbulence models rely on the Boussinesq approximation, which assumes that turbulence acts like an isotropic eddy viscosity. This is fine for simple shear flows, but fails in cases with:
Strong curvature (e.g., S-ducts)
Rotation or swirl (e.g., cyclone separators)
Secondary flows due to anisotropy (e.g., rectangular ducts)
Simple analogy: Imagine trying to model a crowd moving through a twisty hallway using one rule: "people follow the person ahead." That works fine in a straight corridor, but breaks down when people swirl, hesitate, or change directions due to corners or obstacles. You need a smarter model that tracks what each direction is doing.
The Reynolds-Stress Transport Equation
Q: Why is the pressure-strain term so important, and why can't we calculate it directly?
The pressure-strain term redistributes energy between components of turbulence. It drives turbulence toward isotropy, smoothing out directional imbalances created by production.
It also depends on fluctuating pressure, which in turn depends on all velocity fluctuations — it's a non-local, unclosed term. Direct calculation would require knowing the full turbulence field. That's why it's modeled using empirical approaches.
Linear vs. Quadratic Pressure-Strain Models
Q: What’s the difference between a linear and quadratic pressure-strain model?
Linear models assume that the redistribution is directly proportional to existing stress imbalances. They're simpler and work near walls but can fail in rotating or swirling flows.
Quadratic models go a step further, accounting for non-linear interactions and providing better predictions in more complex flows (e.g., axisymmetric contraction, impingement).
However, quadratic models often require wall-function meshes and can be harder to converge.
EARSM: A Compromise Between Two Worlds
Q: What is the EARSM model and why is it useful?
EARSM (Explicit Algebraic RSM) aims to capture anisotropy without solving six transport equations. It approximates RSM behavior using algebraic expressions derived from assumptions about equilibrium.
Benefits:
More accurate than eddy viscosity models in anisotropic flows
Faster and more robust than full RSM
Downsides:
Slightly more expensive than standard 2-equation models
Still lacks some fidelity in strongly 3D flows (e.g., tip vortices)
When Should You Use RSM?
RSM models are computationally expensive and hard to converge. But in some cases, they are essential:
Cyclone separators
S-ducts with separation and secondary flows
Sharp corner ducts where anisotropy dominates
If anisotropy drives the physics, RSM gives better predictions. Otherwise, SST or Realizable k-epsilon might suffice.
Stability Tips for RSM Simulations
Use good mesh quality with smooth transitions and inflation layers
Start with a simpler model (e.g., SST) and switch to RSM after a few hundred iterations
Use first-order discretization for turbulence equations initially
Adjust under-relaxation factors (URFs), especially for pressure and momentum
Use Scalable Wall Functions early on to stabilize wall shear stress predictions
Monitor local residuals to find divergence sources