Turbulence Chapter 1: Review of RANS-Boussinesq Models & Statistical Turbulence Description
Turbulence modeling is critical in Computational Fluid Dynamics (CFD), enabling engineers to approximate complex flow behaviors without resorting to computationally infeasible Direct Numerical Simulation (DNS)
1. Fundamentals of Turbulence Modeling
Key Characteristics of Turbulence
Turbulence is a highly complex and chaotic phenomenon, but it follows three fundamental principles:
Three-dimensionality: Turbulence occurs in all three spatial dimensions, driven by vortex stretching.
Chaos and unpredictability: Even with fully known initial conditions, small perturbations grow exponentially, making exact predictions impossible.
Energy cascade: Large turbulent eddies transfer energy to progressively smaller ones, eventually dissipating as heat.
Why Do We Model Turbulence?
Solving the full Navier-Stokes equations for turbulent flows (DNS) is computationally infeasible due to the vast range of length and time scales.
Instead, Reynolds-Averaged Navier-Stokes (RANS) equations are used, averaging turbulence effects while requiring closure models to approximate unknown Reynolds stresses.
2. Review of Basic RANS-Boussinesq Models
The Closure Problem in RANS Modeling
RANS equations introduce Reynolds stresses, additional unknowns that must be modeled.
This leads to the need for turbulence models, which approximate turbulence effects without resolving all turbulent structures.
One-Equation vs. Two-Equation Models
One-equation models (e.g., Spalart-Allmaras) solve a single equation for turbulent viscosity, making them simple but limited in complex flows.
Two-equation models (e.g., k-ε, k-ω) solve for turbulent kinetic energy (k) and an additional variable (ε or ω) to provide better accuracy in more complex scenarios.
3. Statistical Description of Turbulence & Limitations of RANS Models
Limitations of the Boussinesq Approximation
The Boussinesq hypothesis assumes Reynolds stresses are proportional to the mean strain rate. While effective in simple flows, it fails in:
Flows with rapid strain rate changes (e.g., boundary layer separation and reattachment)
Highly three-dimensional or swirling flows (e.g., curved ducts, rotating flows)
Flows with strong streamline curvature
Reynolds Stress Models (RSM)
Unlike two-equation models, RSM directly solves transport equations for Reynolds stresses, making it more accurate for anisotropic turbulence.
RSM is computationally expensive but necessary for flows with swirling motion, secondary flows, and strong curvature effects.
Turbulence Production Limiters
Standard RANS models may artificially generate turbulence in regions where it shouldn’t exist (e.g., stagnation points, diffusers).
Production limiters prevent excessive turbulence generation, improving accuracy in engineering applications.
4. Improved RANS-Boussinesq Models & Refinements
Realizable k-Epsilon Model
Improves standard k-ε by ensuring physical realizability constraints (e.g., preventing negative turbulent kinetic energy).
Modifies turbulent viscosity and dissipation terms, enhancing performance in highly strained flows.
RNG k-Epsilon Model
Uses renormalization group (RNG) theory to improve accuracy in rapidly strained flows.
Features a modified dissipation equation and adjustable turbulent viscosity for swirling flows.
Best suited for transitional flows, impinging jets, and complex shear flows.
Curvature Correction in Turbulence Models
Standard RANS models don’t handle streamline curvature effects well.
Curvature correction adjusts turbulence production to enhance or reduce turbulence based on flow curvature:
Concave curvature → Increases turbulence (e.g., inside bends)
Convex curvature → Reduces turbulence (e.g., outside bends)
Crucial for modeling vortex flows and swirling motions accurately.
5. Practical Application of RANS Models in CFD
Comparing Common Turbulence Models
Model: Spalart-Allmaras
Best For: External aerodynamics (airfoils, wings)
Strengths: Simple, robust, accurate for attached flow
Weaknesses: Poor for separated flows
Model: k-ε
Best For: General-purpose, industrial flows
Strengths: Well-tested, stable
Weaknesses: Weak near walls, bad at predicting separation
Model: k-ω (SST)
Best For: Wall-bounded flows, aerodynamics
Strengths: Great near-wall accuracy, less diffusive
Weaknesses: Sensitive to freestream conditions
Importance of y+ in Turbulence Modeling
y+ < 5 → Near-wall resolved models (SST k-ω, LES, DNS)
y+ 30-100 → Wall functions (k-ε)
Too high y+ (100+) → Likely inaccurate results
Bridging RANS and DNS with LES & DES
LES resolves large eddies, models small ones → high accuracy, expensive
DES blends RANS (near-wall) and LES (far-field) → cheaper than full LES, ideal for high-Reynolds number external flows
6. Best Practices in Fluent for RANS Simulations
Key Steps to Set Up a Turbulence Model in Fluent
Ensure a high-quality mesh (refined near walls, avoiding skewed elements).
Pick the right turbulence model (based on flow type & y+ values).
Check near-wall treatment (wall functions vs. resolving boundary layers).
Set boundary conditions properly (e.g., turbulence intensity, length scale).
Choose an appropriate solver scheme (pressure-velocity coupling, discretization).
Monitor residuals and key flow variables to check for convergence.
Conclusions
This chapter has demonstrated the evolution of turbulence modeling, from basic RANS closures to improved models that address their limitations.
The complexity of turbulent flows requires careful selection of models based on flow characteristics, computational resources, and accuracy needs. Advanced approaches, such as curvature correction and Reynolds stress models, offer more precision but come at a higher computational cost. Meanwhile, hybrid models like LES and DES serve as a bridge between RANS and DNS, balancing accuracy and feasibility.
The practical aspects of CFD—such as mesh independence testing, proper near-wall treatment, and careful post-processing—are just as critical as the choice of turbulence model itself.
Mastering these techniques ensures that CFD simulations are both accurate and computationally efficient, leading to better engineering decisions and designs.