Reynolds Stress Model - CFD Handbook

1. Model Overview

Seven-equation RANS model solving directly for all six Reynolds stress components plus dissipation rate.
Most advanced turbulence closure capturing full anisotropic turbulence behavior without eddy viscosity assumptions.
Superior accuracy for complex flows with strong streamline curvature, swirl, and rotation effects.
Research-grade model providing detailed insight into turbulence structure and physics.

2. Key Strengths

Anisotropic turbulence: Directly computes all six Reynolds stress components without isotropic assumptions.
Complex flow accuracy: Superior performance in flows with strong streamline curvature, swirl, and rotation.
No eddy viscosity limitation: Eliminates Boussinesq approximation constraints for highly anisotropic flows.
Secondary flow prediction: Captures corner flows, duct flows with secondary circulation patterns.
Research capabilities: Provides detailed insight into turbulence structure and physics.

3. Limitations

Computational cost: 3-4x higher than two-equation models due to seven transport equations.
Memory requirements: Significant storage for six Reynolds stress components plus dissipation rate.
Convergence challenges: More difficult convergence due to equation coupling and stiffness.
Numerical stability: Requires careful under-relaxation and initialization for stable solutions.

4. Home Appliance Applications

Washing Machines: Drum flow analysis, water distribution patterns, detergent mixing efficiency

Dishwashers: Spray arm performance, water jet impingement, cleaning effectiveness zones

Cooktops: Precise heat transfer analysis, airflow patterns around burners, ventilation efficiency

Vacuum Cleaners: Complex internal flows, separation in curved ducts, filtration system optimization

Range Hoods: Extraction efficiency, flow separation over surfaces, noise prediction

Dryers: Airflow uniformity, heat distribution, lint transport analysis

5. When to Choose RSM Over Eddy Viscosity Models

Strongly anisotropic flows: When Reynolds stress anisotropy is critical to flow physics.
Swirling and rotating flows: Cyclones, turbomachinery, and system rotation effects.
Complex 3D separation: Where crossflow effects and corner flows dominate.
Buoyant flows: Natural convection with strong stratification and secondary flows.
Curved boundaries: Flow over curved surfaces with streamline curvature effects.

6. When NOT to Use RSM

Simple attached flows: Straight ducts, pipes - simpler models are adequate and much faster

Preliminary design: Early concept studies where computational cost outweighs accuracy benefits

Large domain studies: Massive computational cost - use eddy viscosity models instead

Limited computational resources: RSM requires 3-5x more CPU time and memory

Convergence issues: Complex coupling can lead to numerical instabilities

7. Setup and Mesh Guidelines

Mesh Requirements:

y⁺ < 1: Essential for accurate wall treatment (target y⁺ ≈ 0.5).
Boundary layer resolution: Minimum 15-20 cells across boundary layer.
Growth ratio: < 1.2 in near-wall region, < 1.3 elsewhere.
Aspect ratio: < 1000 in boundary layer regions acceptable.

Solver Settings:

Discretization: Second-order upwind minimum for all equations.
Convergence: Residuals < 10⁻⁵, monitor integrated quantities.
Under-relaxation: k: 0.6, ω: 0.6, reduce if convergence issues.

Advanced Options:

Curvature correction: Enable for vacuum cleaners and highly curved flows.
Production limiter: Usually enabled by default, prevents unrealistic production.
Low-Re corrections: Consider for transitional flows or very fine meshes.

8. Performance Expectations

Accuracy Levels:

Wall shear stress: ±3-5% for attached flows, ±5-10% for separated flows.
Heat transfer: ±5-8% for most engineering applications.
Pressure drop: ±5-10% depending on geometry complexity.
Separation point: ±10-15% for airfoil-type flows.

Computational Overhead:

vs k-ε: 20-30% longer solution time, 2-3x finer mesh required.
Memory: 15-20% higher due to additional variables and functions.
ROI: Justifiable when wall effects or separation are critical to design.

9. Common Pitfalls and Solutions

Mesh-Related Issues:

Problem: y⁺ > 5 causing poor wall treatment. Solution: Refine near-wall mesh.
Problem: Excessive aspect ratios causing convergence issues. Solution: Improve mesh quality.

Convergence Issues:

Problem: ω equation diverging. Solution: Reduce under-relaxation, check inlet ω values.
Problem: Oscillating residuals. Solution: Enable production limiter, check mesh quality.

Physical Issues:

Problem: Overpredicted separation. Solution: Check curvature correction, consider transition model.
Problem: Poor swirl prediction. Solution: Consider Reynolds stress model instead.

Historical Context and Development

The Reynolds Stress Model (RSM) represents the most advanced level of RANS turbulence closure, directly solving transport equations for all six independent Reynolds stress components plus the dissipation rate, eliminating the eddy viscosity hypothesis entirely.

Timeline and Evolution

The theoretical foundation for RSM was established in the 1970s with the work of Hanjalic and Launder (1972), who derived the exact transport equations for Reynolds stresses. The Launder, Reece, and Rodi (1975) model provided the first practical implementation with closure approximations for pressure-strain correlations.

Subsequent developments included the Speziale, Sarkar, and Gatski (1991) model with improved pressure-strain modeling, the omega-based RSM variants by Wilcox (1993), and modern implementations like the Stress-Omega model. Despite computational advances, RSM remains the most resource-intensive RANS approach, reserved for cases where Reynolds stress anisotropy is critical.

Mathematical Foundation and Transport Equations

The Reynolds Stress Model solves seven transport equations: six for the independent Reynolds stress components and one for either dissipation rate or specific dissipation rate, providing complete closure without eddy viscosity assumptions.

Reynolds Stress Transport Equation:

$$\frac{\partial(\rho \overline{u_i'u_j'})}{\partial t} + \frac{\partial(\rho U_k \overline{u_i'u_j'})}{\partial x_k} = P_{ij} + D_{ij} + \Pi_{ij} - \varepsilon_{ij}$$

Production Term (Exact):

$$P_{ij} = -\rho \overline{u_i'u_k'} \frac{\partial U_j}{\partial x_k} - \rho \overline{u_j'u_k'} \frac{\partial U_i}{\partial x_k}$$

Pressure-Strain Correlation (Modeled):

$$\Pi_{ij} = \overline{p' \left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)} = \Pi_{ij,1} + \Pi_{ij,2} + \Pi_{ij,w}$$

Linear Pressure-Strain Model (Rotta):

$$\Pi_{ij,1} = -C_1 \rho \varepsilon \left(\frac{\overline{u_i'u_j'}}{k} - \frac{2}{3}\delta_{ij}\right)$$

Rapid Distortion (Isotropization of Production):

$$\Pi_{ij,2} = -C_2 \left(P_{ij} - \frac{2}{3}P \delta_{ij}\right)$$

Dissipation Rate Transport Equation:

$$\frac{\partial(\rho \varepsilon)}{\partial t} + \frac{\partial(\rho U_k \varepsilon)}{\partial x_k} = \frac{\partial}{\partial x_k}\left[\left(\mu + \frac{\mu_t}{\sigma_\varepsilon}\right) \frac{\partial \varepsilon}{\partial x_k}\right] + C_{1\varepsilon}\frac{\varepsilon}{k}P - C_{2\varepsilon}\rho\frac{\varepsilon^2}{k}$$

Variable Definitions:

u'ᵢu'ⱼ: Reynolds stress tensor components (m²/s²)

Pᵢⱼ: Production tensor (exact, no modeling) (kg/(m·s³))

Πᵢⱼ: Pressure-strain correlation (modeled) (kg/(m·s³))

εᵢⱼ: Dissipation tensor (kg/(m·s³))

k: Turbulent kinetic energy = ½u'ᵢu'ᵢ (m²/s²)

ε: Dissipation rate (m²/s³)

Constant	LRR Model	SSG Model	Physical Significance
C₁	1.8	1.7	Return to isotropy coefficient
C₂	0.6	1.05	Rapid distortion coefficient
C₁ε	1.44	1.44	ε production coefficient
C₂ε	1.92	1.92	ε destruction coefficient
σε	1.3	1.3	Turbulent Prandtl number for ε

Physical Interpretation and Anisotropy Modeling

The Reynolds Stress Model represents the pinnacle of RANS modeling by directly computing the anisotropic Reynolds stress tensor without the fundamental limitation of the eddy viscosity hypothesis.

Reynolds Stress Anisotropy

Unlike eddy viscosity models that assume isotropic turbulent viscosity, RSM captures the full anisotropic nature of Reynolds stresses. The stress tensor is not constrained to be proportional to the strain rate tensor, enabling prediction of complex flow phenomena like secondary flows and stress-driven separation.

Pressure-Strain Correlation Physics

The pressure-strain correlation Πᵢⱼ represents the redistribution of energy among different stress components and the return to isotropy. The Rotta term (Π₁) drives stresses toward isotropy, while the rapid distortion term (Π₂) accounts for the immediate response to mean flow deformation.

Exact Production Term

The production term Pᵢⱼ requires no modeling assumptions, representing the exact extraction of turbulent energy from the mean flow. This provides superior accuracy in complex flows where production mechanisms vary significantly among stress components.

Wall Effects and Near-Wall Treatment

Near solid boundaries, additional wall reflection terms (Πᵢⱼ,w) account for the redistribution effects of viscous pressure fluctuations. These terms ensure proper stress behavior at walls, particularly important for heat transfer and skin friction prediction.

Model Variants and Closure Approaches

Several RSM variants exist, differing primarily in their pressure-strain correlation models and near-wall treatment approaches.

Linear Pressure-Strain Models

The Launder, Reece, and Rodi (LRR) model provides a linear relationship between pressure-strain correlation and stress anisotropy. The Speziale, Sarkar, and Gatski (SSG) model improves upon LRR with better performance in homogeneous flows and realizability constraints.

Quadratic Pressure-Strain Models

More advanced models include quadratic terms in the anisotropy tensor, providing better representation of strong anisotropic flows but at increased computational cost and complexity.

Wall Treatment Approaches

Low-Reynolds number: Direct integration to wall with damping functions
Wall function approach: Enhanced wall functions accounting for anisotropy
Two-layer approach: RSM in outer region, one-equation model near wall

Omega-Based RSM

Alternative formulations solve for specific dissipation rate ω instead of ε, providing better near-wall behavior similar to k-ω models while maintaining the anisotropic stress predictions of RSM.

Model Applications and Validation Database

RSM excels in flows where Reynolds stress anisotropy significantly affects the mean flow development, particularly in cases where eddy viscosity models fundamentally fail.

Successful Applications

Strongly swirling flows with cyclone-type geometries
Flows with system rotation and Coriolis effects
Secondary flow development in non-circular ducts
Buoyancy-driven flows with strong stratification
Complex 3D separation with crossflow effects

Benchmark Test Cases

Standard validation includes the square duct secondary flow, rotating channel flow, impinging jets, backward-facing step with ribs, and various swirling flow configurations. RSM generally shows superior performance compared to eddy viscosity models in these anisotropic flow scenarios.

Known Limitations

Computational cost: 3-5x more expensive than two-equation models
Convergence challenges due to strong coupling between equations
Pressure-strain modeling remains largely empirical
Can be overly sensitive to boundary conditions and numerical schemes

Advanced Topics and Modern Developments

Contemporary RSM research focuses on improving pressure-strain correlation models, developing more efficient numerical algorithms, and extending RSM to complex physical phenomena.

Realizability and Consistency

Modern RSM implementations ensure realizability constraints are satisfied, preventing non-physical states where normal stresses become negative or the stress tensor loses positive semi-definiteness. This involves careful design of pressure-strain correlations and dissipation modeling.

Elliptic Relaxation Approach

The v²-f model and elliptic relaxation RSM variants solve additional elliptic equations to better represent near-wall turbulence anisotropy without explicit wall distance dependencies, improving robustness in complex geometries.

Machine Learning Enhancement

Recent developments incorporate machine learning to improve pressure-strain correlation modeling, using high-fidelity DNS/LES data to develop data-driven closure models that outperform traditional algebraic formulations.

Hybrid RANS-LES Applications

RSM serves as the RANS component in advanced hybrid approaches, particularly for flows where anisotropic effects in attached regions significantly influence the separated regions that transition to LES treatment.

Numerical Implementation and Best Practices

Successful RSM implementation requires careful attention to numerical methods, convergence strategies, and computational resource management due to the model's complexity and strong coupling.

Solution Strategies

Coupled solution of stress equations to reduce iterations
Conservative under-relaxation factors (0.3-0.5) for stress equations
Careful initialization with simpler model results
Monitoring of stress realizability during solution process

Mesh Requirements

RSM typically requires finer meshes than eddy viscosity models due to the complexity of stress gradient resolution. Near-wall treatment depends on the chosen variant (y⁺ < 1 for low-Re models, y⁺ ≈ 30 for wall functions).

Boundary Conditions

Proper specification of Reynolds stress boundary conditions requires understanding of flow physics. Inlet profiles should reflect realistic turbulence anisotropy when available from experiments or higher-fidelity simulations.

Computational Efficiency

Modern implementations optimize memory usage through efficient stress tensor storage and leverage parallel computing architectures. Despite advances, RSM remains computationally demanding and should be reserved for cases where anisotropic effects are crucial to flow physics.