Turbulence Models Db

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.