Turbulence Models Database

A comprehensive reference for selecting and configuring turbulence models in computational fluid dynamics (CFD) simulations.

Overview of Turbulence Modeling

Turbulence is a chaotic, three-dimensional, time-dependent flow phenomenon characterized by random fluctuations in velocity, pressure, and other flow quantities. Direct numerical simulation (DNS) of turbulent flows is computationally prohibitive for most engineering applications, necessitating turbulence modeling approaches.

Modeling Approaches Hierarchy

1. DNS (Direct Numerical Simulation): Resolves all turbulent scales, no modeling
2. LES (Large Eddy Simulation): Resolves large scales, models small scales
3. RANS (Reynolds-Averaged Navier-Stokes): Models all turbulent scales
4. Laminar: No turbulence modeling

RANS Turbulence Models

RANS models solve time-averaged equations and model turbulent fluctuations using the Reynolds stress concept. They are the most widely used in industrial CFD due to computational efficiency.

k-ε Models (k-epsilon)

The k-ε family models turbulent kinetic energy (k) and its dissipation rate (ε).

Standard k-ε

Characteristics:

• Two-equation model
• Robust and widely validated
• Good for free shear flows and fully turbulent flows
• Poor for flows with strong adverse pressure gradients
• Not suitable for low-Reynolds number flows without modifications

Best Applications:

• Fully turbulent flows
• Free shear layers, mixing layers, jets
• Flow in ducts and channels (far from walls)
• Industrial flows with high Reynolds numbers

Limitations:

• Overpredicts separation
• Poor near-wall performance without wall functions
• Inaccurate for swirling flows
• Stagnation point anomaly

Wall Treatment:

• Requires y+ > 30 (typically 30-300) with wall functions
• Not suitable for wall-resolved simulations

RNG k-ε

Characteristics:

• Derived using Renormalization Group theory
• Improved performance for swirling flows and streamline curvature
• Better handles low-Reynolds number effects
• Modified ε equation improves accuracy for rapidly strained flows

Best Applications:

• Flows with strong streamline curvature
• Swirling and rotating flows
• Transitional flows (with enhanced wall treatment)
• Separated flows (better than standard k-ε)

Improvements over Standard k-ε:

• Additional term in ε equation for rapid strain
• Modified turbulent viscosity formula
• Better prediction of near-wall flows

Wall Treatment:

• Can use wall functions (y+ > 30)
• Enhanced wall treatment allows y+ ≈ 1

Realizable k-ε

Characteristics:

• Ensures mathematical realizability constraints
• Variable Cμ coefficient
• Improved prediction of spreading rate for planar and round jets
• Better performance for rotating flows and boundary layers under strong adverse pressure gradients

Best Applications:

• Flows with rotation and recirculation
• Boundary layers with strong pressure gradients
• Separated flows
• Jets and mixing layers

Advantages:

• More accurate for complex flows than standard k-ε
• Superior prediction of jet spreading rates
• Better captures effects of streamline curvature

Wall Treatment:

• Standard wall functions (y+ > 30)
• Enhanced wall treatment available (y+ ≈ 1)

k-ω Models (k-omega)

The k-ω family models turbulent kinetic energy (k) and specific dissipation rate (ω).

Standard k-ω (Wilcox)

Characteristics:

• Two-equation model
• Superior near-wall treatment without wall functions
• Accurate for adverse pressure gradients
• Sensitive to freestream values of ω
• Good for transitional flows

Best Applications:

• Low-Reynolds number flows
• Transitional flows
• Flows with adverse pressure gradients
• Aerodynamic flows (airfoils, wings)
• Wall-bounded flows

Limitations:

• Highly sensitive to freestream ω values
• Less accurate in free shear flows compared to k-ε
• Can be numerically stiff

Wall Treatment:

• Integrates to the wall (y+ ≈ 1 required)
• No wall functions needed for near-wall region

k-ω SST (Shear Stress Transport)

Characteristics:

• Blends k-ω near walls with k-ε in freestream
• Insensitive to freestream values
• Accounts for transport of turbulent shear stress
• Modified turbulent viscosity formulation
• Industry standard for aerodynamics

Best Applications:

• Aerodynamic flows (external aerodynamics)
• Flows with adverse pressure gradients and separation
• Transonic flows
• Heat transfer problems
• Turbomachinery

Advantages:

• Combines strengths of k-ω (near-wall) and k-ε (far-field)
• Accurate separation prediction
• Not sensitive to freestream turbulence values
• Robust and reliable

Limitations:

• Requires fine near-wall mesh (y+ ≈ 1)
• More computationally expensive than standard models
• Can underpredict separation in some cases

Wall Treatment:

• Designed for low-Reynolds number (y+ ≈ 1)
• Automatic wall functions available for coarse meshes
• Best results with wall-resolved mesh

Spalart-Allmaras

Characteristics:

• One-equation model (solves for modified turbulent viscosity)
• Designed for aerodynamic flows
• Low computational cost
• Good for wall-bounded flows
• Limited for free shear flows and decaying turbulence

Best Applications:

• Aerospace applications
• External aerodynamics (airfoils, wings, fuselages)
• Mild separation and attached flows
• Transonic flows

Advantages:

• Computationally efficient (one equation)
• Robust and stable
• Good near-wall behavior
• Well-suited for structured meshes

Limitations:

• Not suitable for complex flows with multiple physics
• Limited accuracy for free shear flows
• Not ideal for internal flows
• Poor for flows with large separation regions

Wall Treatment:

• Designed for low-Reynolds number (y+ ≈ 1)
• Can use wall functions for coarser meshes

Reynolds Stress Models (RSM)

Characteristics:

• Seven-equation model (6 Reynolds stresses + ε or ω)
• Solves transport equations for each Reynolds stress component
• Accounts for anisotropy of turbulence
• Most complex RANS approach

Best Applications:

• Highly swirling flows
• Flows with strong streamline curvature
• Complex 3D flows
• Rotating flows and cyclone separators
• Flows where turbulence anisotropy is critical

Advantages:

• Most accurate RANS model for complex flows
• Captures turbulence anisotropy
• No isotropic eddy viscosity assumption

Limitations:

• Most computationally expensive RANS model
• Convergence can be challenging
• Requires very good mesh quality
• More sensitive to numerical settings

Large Eddy Simulation (LES)

Overview

LES resolves large turbulent eddies directly while modeling small-scale (subgrid-scale) turbulence. Provides time-accurate flow structures.

Characteristics:

• Spatially filtered Navier-Stokes equations
• Resolves energy-containing eddies
• Models universal small-scale turbulence
• Requires 3D time-dependent simulation

Mesh Requirements:

• Very fine mesh (Δx, Δy, Δz ≈ local turbulent length scale)
• Isotropic or near-isotropic cells in turbulent regions
• y+ < 1 for wall-resolved LES
• Wall-modeled LES: y+ can be 30-100

Computational Cost:

• 10-100x more expensive than RANS
• Scales as Re^(9/4) for channel flows
• Requires long simulation times for statistical convergence

Subgrid-Scale Models

Smagorinsky-Lilly

• Classic algebraic model
• Cs ≈ 0.1-0.2 (model constant)
• Overly dissipative near walls

Dynamic Smagorinsky

• Computes Cs dynamically
• More accurate than standard Smagorinsky
• Self-adapting to flow conditions

WALE (Wall-Adapting Local Eddy-viscosity)

• Better near-wall behavior
• Returns correct y³ scaling near walls
• No dynamic procedure needed

Kinetic Energy Subgrid-Scale

• One-equation model for subgrid kinetic energy
• More accurate but more expensive

Applications of LES

Best suited for:

• Acoustics (noise prediction)
• Combustion and reacting flows
• Complex unsteady flows
• Flows with large-scale instabilities
• Vortex shedding and wake flows
• Mixing problems

Not recommended for:

• Steady-state problems
• High-Reynolds number wall-bounded flows (prohibitive cost)
• Industrial simulations with limited resources

Hybrid RANS-LES Methods

DES (Detached Eddy Simulation)

• RANS near walls, LES in separated regions
• Good for massively separated flows
• More affordable than pure LES

DDES (Delayed DES)

• Improved shielding of boundary layer
• Prevents premature switch to LES mode

SDES (Shielded DES)

• Further improvements to RANS-LES interface
• Better suited for attached flows

SAS (Scale-Adaptive Simulation)

• RANS-based but resolves large unsteady structures
• Automatic adjustment to resolved scales

Model Selection Criteria

Flow Type Classification

Internal Flows

Examples: Pipes, ducts, channels, valves, pumps Recommended models:

• k-ε Realizable (general purpose)
• k-ω SST (with heat transfer or separation)
• Standard k-ε (simple fully turbulent)

External Flows

Examples: Airfoils, vehicles, buildings, external aerodynamics Recommended models:

• k-ω SST (industry standard)
• Spalart-Allmaras (aerospace)
• Realizable k-ε (blunt bodies)

Free Shear Flows

Examples: Jets, wakes, mixing layers Recommended models:

• Realizable k-ε
• Standard k-ε
• RNG k-ε

Separated Flows

Examples: Flow over backward-facing step, airfoil stall Recommended models:

• k-ω SST (best for mild-moderate separation)
• DES/DDES (massive separation)
• LES (if resources available)

Rotating/Swirling Flows

Examples: Turbomachinery, cyclones, swirl burners Recommended models:

• RNG k-ε
• RSM
• k-ω SST

Reynolds Number Considerations

Low Re (Re < 10⁴):

• Low-Re k-ω or k-ω SST
• Spalart-Allmaras
• May need transitional models

Moderate Re (10⁴ < Re < 10⁶):

• Most RANS models applicable
• k-ω SST for aerodynamics
• Realizable k-ε for internal flows

High Re (Re > 10⁶):

• Standard k-ε (with wall functions)
• k-ω SST
• Realizable k-ε

Very High Re (Re > 10⁷):

• Wall function approaches necessary
• Standard k-ε
• Realizable k-ε

Mesh Requirements and y+ Values

Wall Functions Approach

y+ range: 30 < y+ < 300 (ideally 30-100) Models:

• Standard k-ε
• Realizable k-ε
• RNG k-ε (with standard wall functions)

Advantages:

• Coarser mesh acceptable
• Lower computational cost
• Suitable for high-Re flows

Limitations:

• Less accurate near-wall gradients
• Not suitable for low-Re or transitional flows
• Poor for flows with separation or reattachment

Wall-Resolved Approach

y+ range: y+ ≈ 1 (first cell) Models:

• k-ω SST
• Standard k-ω
• Spalart-Allmaras
• Low-Re k-ε variants

Requirements:

• Very fine near-wall mesh
• At least 10-15 cells in boundary layer
• y+ < 1 for first cell
• Growth ratio ≤ 1.2 near wall

Advantages:

• Accurate near-wall resolution
• Captures boundary layer accurately
• Better for heat transfer
• Handles low-Re and transitional flows

Limitations:

• High cell count
• Increased computational cost
• Mesh generation more complex

Enhanced Wall Treatment

y+ range: y+ < 5 or 30 < y+ < 300 (adaptive) Models:

• Realizable k-ε with EWT
• RNG k-ε with EWT

Advantages:

• Flexibility in mesh resolution
• Blends wall functions and low-Re formulation
• Handles variable y+ in domain

y+ Guidelines by Application

Computational Cost Comparison

Relative cost (normalized to standard k-ε = 1):

Model Constants and Parameters

Standard k-ε Constants

• Cμ = 0.09
• C1ε = 1.44
• C2ε = 1.92
• σk = 1.0
• σε = 1.3

RNG k-ε Constants

• Cμ = 0.0845
• C1ε = 1.42
• C2ε = 1.68
• σk = 0.7179
• σε = 0.7179
• η0 = 4.38
• β = 0.012

Realizable k-ε Constants

• C1ε = 1.44
• C2 = 1.9
• σk = 1.0
• σε = 1.2
• Cμ = variable (function of strain rate and rotation)

Standard k-ω Constants

• α = 5/9
• β = 0.075
• β* = 0.09
• σk = 2.0
• σω = 2.0

k-ω SST Constants

k-ω inner:

• α1 = 5/9
• β1 = 0.075
• σk1 = 2.0
• σω1 = 2.0

k-ε outer (transformed):

• α2 = 0.44
• β2 = 0.0828
• σk2 = 1.0
• σω2 = 1.168

Other:

• β* = 0.09
• a1 = 0.31 (SST limiter constant)

Spalart-Allmaras Constants

• cb1 = 0.1355
• cb2 = 0.622
• σ = 2/3
• κ = 0.41 (von Karman constant)
• cw1 = cb1/κ² + (1 + cb2)/σ
• cw2 = 0.3
• cw3 = 2.0
• cv1 = 7.1

Wall Functions vs Wall-Resolved

Standard Wall Functions

Theory:

• Based on law of the wall
• Assumes equilibrium boundary layer
• Logarithmic law: u+ = (1/κ)ln(y+) + B

Requirements:

• 30 < y+ < 300
• Equilibrium turbulent boundary layer
• No significant pressure gradients

When to use:

• High-Re fully turbulent flows
• Simple geometries
• Limited computational resources
• Steady-state simulations

Limitations:

• Inaccurate for adverse pressure gradients
• Poor for separation and reattachment
• Not suitable for heat transfer predictions
• Fails in transitional flows

Scalable Wall Functions

Improvements:

• Avoid deterioration for fine meshes
• y+ insensitive formulation
• Better for y+ < 30

Non-Equilibrium Wall Functions

Improvements:

• Account for pressure gradient effects
• Improved separation prediction
• Better suited for complex flows

When to use:

• Flows with pressure gradients
• Separation and reattachment
• Complex geometries

Enhanced Wall Treatment (EWT)

Characteristics:

• Two-layer approach
• Blends wall functions (high y+) with low-Re formulation (low y+)
• Adaptive based on local y+

When to use:

• Variable mesh resolution
• Uncertainty in y+ values
• Complex geometries with varying resolution

Wall-Resolved (Low-Re)

Requirements:

• y+ ≈ 1 (ideally y+ < 1)
• 10-15+ cells in boundary layer
• Growth ratio ≤ 1.2 near wall
• Integration to wall (no wall functions)

When to use:

• Accurate heat transfer required
• Separation prediction critical
• Low-Re or transitional flows
• Aerodynamic design optimization

Models requiring wall-resolved:

• k-ω SST (optimal)
• Standard k-ω
• Spalart-Allmaras (optimal)
• LES

Turbulence Model Selection Decision Tree

START: What is your flow problem?
│
├─ Need time-accurate unsteady structures?
│  │
│  ├─ YES: Go to LES/Hybrid
│  │  │
│  │  ├─ Can afford very fine mesh and long run time?
│  │  │  ├─ YES: LES (wall-resolved or wall-modeled)
│  │  │  └─ NO: DES/DDES (massively separated flows) or SAS
│  │  │
│  │  └─ Is flow primarily attached?
│  │     ├─ YES: URANS (k-ω SST or Realizable k-ε)
│  │     └─ NO: DES/DDES
│  │
│  └─ NO: Continue to RANS selection
│
├─ Flow type?
│  │
│  ├─ External aerodynamics (airfoils, vehicles, aircraft)
│  │  └─ k-ω SST (first choice) or Spalart-Allmaras
│  │     Required: y+ ≈ 1, wall-resolved mesh
│  │
│  ├─ Internal flows (pipes, ducts, channels)
│  │  │
│  │  ├─ Heat transfer important?
│  │  │  ├─ YES: k-ω SST (y+ ≈ 1)
│  │  │  └─ NO: Continue
│  │  │
│  │  ├─ Separation or adverse pressure gradients?
│  │  │  ├─ YES: k-ω SST (y+ ≈ 1) or Realizable k-ε (EWT)
│  │  │  └─ NO: Realizable k-ε (wall functions, y+ = 30-100)
│  │  │
│  │  └─ Simple fully turbulent?
│  │     └─ Standard k-ε (wall functions, y+ = 30-100)
│  │
│  ├─ Free shear flows (jets, wakes, mixing)
│  │  └─ Realizable k-ε or RNG k-ε
│  │
│  ├─ Rotating/swirling flows (turbomachinery, cyclones)
│  │  │
│  │  ├─ Simple rotation?
│  │  │  └─ RNG k-ε or k-ω SST
│  │  │
│  │  └─ Complex 3D rotation with strong curvature?
│  │     └─ RSM (if resources available) or RNG k-ε
│  │
│  └─ Separated flows (backward step, airfoil stall)
│     │
│     ├─ Mild-moderate separation?
│     │  └─ k-ω SST (y+ ≈ 1)
│     │
│     └─ Massive separation?
│        └─ DES/DDES or LES (if affordable)
│
├─ Can you achieve y+ ≈ 1 near walls?
│  │
│  ├─ YES: k-ω SST, Spalart-Allmaras, or Low-Re k-ε
│  │
│  └─ NO: Must use wall functions
│     └─ Realizable k-ε or RNG k-ε (with EWT if possible)
│
└─ Special considerations:
   │
   ├─ Transitional flows (low Re)? → k-ω SST + transition model
   ├─ Compressible/high-speed? → k-ω SST or Spalart-Allmaras
   ├─ Buoyancy-driven? → Realizable k-ε or k-ω SST with buoyancy terms
   ├─ Multiphase flows? → Realizable k-ε or k-ω SST
   └─ Limited resources? → Standard k-ε or Spalart-Allmaras

Quick Selection Guide by Application

Aerospace

Model: k-ω SST or Spalart-Allmaras y+: ≈ 1 Rationale: Accurate prediction of boundary layers, separation, and pressure distribution

Automotive (External)

Model: k-ω SST y+: ≈ 1 Rationale: Separation prediction, drag/lift accuracy

HVAC / Building Ventilation

Model: Realizable k-ε with wall functions y+: 30-100 Rationale: Large domains, computational efficiency, adequate accuracy

Turbomachinery

Model: k-ω SST y+: 1-2 Rationale: Adverse pressure gradients, rotation, heat transfer

Combustion

Model: Realizable k-ε or LES y+: Depends (wall functions for RANS, y+ < 1 for LES) Rationale: Mixing, turbulence-chemistry interaction

Heat Exchangers

Model: k-ω SST or Realizable k-ε with EWT y+: < 5 or wall-resolved Rationale: Heat transfer accuracy critical

Mixing / Chemical Reactors

Model: Realizable k-ε or RSM y+: 30-100 Rationale: Capturing mixing patterns, turbulence anisotropy

Hydraulics (Dams, Spillways)

Model: Realizable k-ε or k-ω SST y+: Variable Rationale: Free surface flows, separation, aeration

Environmental (Atmospheric flows)

Model: Standard k-ε or Realizable k-ε y+: Wall functions Rationale: Large scale, computational cost, atmospheric boundary layer

Best Practices

General Guidelines

1. Start simple: Begin with simpler models (k-ε, k-ω SST) before trying complex models
2. Validate mesh: Ensure y+ values are appropriate for chosen model
3. Check mesh quality: Turbulence models are sensitive to mesh quality
4. Monitor residuals: Turbulence equations often converge slower than momentum
5. Use appropriate boundary conditions: Turbulent intensity and length scale matter
6. Verify sensitivity: Test mesh independence and model sensitivity

Boundary Conditions

Inlet:

• Turbulent intensity: I = 0.16 Re^(-1/8) for fully developed pipe flow
• Low turbulence: I = 0.1% - 1%
• Medium turbulence: I = 1% - 5%
• High turbulence: I = 5% - 20%
• Turbulent length scale: l ≈ 0.07 × characteristic length

Wall:

• No-slip condition
• Wall functions or wall-resolved (model dependent)
• Roughness can be specified (equivalent sand-grain roughness)

Outlet:

• Zero gradient (outflow)
• Fixed pressure

Convergence Tips

1. Initialize properly: Use potential flow or previous solution
2. Under-relax initially: Start with URF = 0.3-0.5 for turbulence equations
3. Gradually increase: Increase URF as solution stabilizes
4. Coupled solvers: Often help with k-ω models
5. Monitor flow features: Check separation, reattachment, vortex shedding
6. Residuals alone insufficient: Verify force/flux convergence

Common Pitfalls

1. Incorrect y+: Most common error; check and adjust mesh
2. Poor mesh quality: High skewness, aspect ratio issues
3. Inadequate refinement: In regions of high gradients
4. Wrong model for application: Using k-ε for separated flows
5. Freestream sensitivity: k-ω without SST correction
6. Ignoring validation: Always compare with experiments or benchmarks

Summary Table: RANS Model Comparison

References

See reference.md for detailed equations, validation cases, and academic references.