k-ω SST Model - CFD Handbook

1. Model Overview

Advanced hybrid model switching between k-ω (near walls) and k-ε (freestream) formulations.
Resolves boundary layer from wall to edge without wall functions.
Industry gold standard for complex flows with separation and adverse pressure gradients.
Built-in shear stress limiter prevents overprediction in strained flows.

2. Key Advantages

Exceptional near-wall accuracy: Direct integration to wall surface (y⁺ < 1).
Separation prediction: Superior performance in adverse pressure gradients.
Freestream independence: Eliminates k-ω model's sensitivity to inlet conditions.
Robust convergence: Stable across wide range of flow conditions.
Heat transfer accuracy: Reliable prediction of wall heat transfer coefficients.

3. Computational Requirements

Mesh requirements: Fine near-wall mesh with y⁺ < 1 for optimal accuracy.
Computational cost: 20-30% higher than k-ε due to additional equation complexity.
Memory usage: Additional storage for blending functions and wall distance.
Convergence time: May require more iterations due to near-wall coupling.

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 SST Over k-ε

Wall-bounded flows: When accurate wall shear stress and heat transfer are critical.
Separation-prone flows: Diffusers, airfoils at high angles, curved channels.
Pressure-driven flows: Where adverse pressure gradients dominate.
Mixed flow types: Combination of attached and separated flow regions.
High accuracy requirements: When ±5% accuracy is needed vs ±15% for k-ε.

6. When NOT to Use SST

Simple internal flows: Straight ducts, pipes - k-ε is adequate and faster

Preliminary design: Early concept studies where speed matters more than precision

Large domains: External flows around buildings - computational cost prohibitive

Highly swirling flows: Cyclones, swirl combustors - consider RSM instead

Transitional flows: Use SST Transition model variant instead

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 k-ω SST (Shear Stress Transport) model was developed by Florian Menter in 1994 to combine the best features of k-ω and k-ε formulations while addressing their individual limitations.

Timeline and Evolution

The development of the SST model represents a systematic evolution in turbulence modeling. In 1988, Wilcox developed the k-ω model, which demonstrated superior near-wall behavior compared to k-ε formulations but suffered from excessive sensitivity to freestream ω values. This limitation was identified by Menter in 1993, who recognized that the k-ω model's performance degraded significantly when freestream ω values were not properly specified.

The breakthrough came in 1994 when Menter introduced the BSL (Baseline) k-ω model, which incorporated a blending function to seamlessly transition between k-ω and k-ε formulations. The same year saw the development of the complete SST model with its characteristic shear stress limiter. Subsequent refinements included the SST-2003 version with improved corner flow prediction capabilities, and the SST-V variant in 2009 featuring vorticity source terms for enhanced rotation effects modeling.

Mathematical Foundation and Hybrid Formulation

The SST model employs a blending function F₁ to seamlessly transition between k-ω formulation near walls and k-ε behavior in the outer region. This hybrid approach combines the superior near-wall treatment of the k-ω model with the freestream independence of the k-ε model.

Turbulent Kinetic Energy Transport Equation:

$$\frac{\partial(\rho k)}{\partial t} + \frac{\partial(\rho k u_i)}{\partial x_i} = \frac{\partial}{\partial x_j}\left[\left(\mu + \frac{\mu_t}{\sigma_k}\right) \frac{\partial k}{\partial x_j}\right] + \tilde{P}_k - \beta^* \rho k \omega$$

Specific Dissipation Rate Transport Equation:

$$\frac{\partial(\rho \omega)}{\partial t} + \frac{\partial(\rho \omega u_i)}{\partial x_i} = \frac{\partial}{\partial x_j}\left[\left(\mu + \frac{\mu_t}{\sigma_\omega}\right) \frac{\partial \omega}{\partial x_j}\right] + \alpha \frac{\omega}{k}\tilde{P}_k - \beta \rho \omega^2 + 2(1-F_1)\frac{\rho}{\sigma_{\omega 2}}\frac{1}{\omega}\frac{\partial k}{\partial x_j}\frac{\partial \omega}{\partial x_j}$$

Modified Turbulent Viscosity (SST Limiter):

$$\mu_t = \frac{\rho a_1 k}{\max(a_1 \omega, \Omega F_2)}, \quad \text{where } \Omega = \sqrt{2\Omega_{ij}\Omega_{ij}}$$

Blending Functions (F₁ and F₂):

$$F_1 = \tanh\left[\min\left(\max\left(\frac{\sqrt{k}}{\beta^* \omega y}, \frac{500\mu}{\rho y^2 \omega}\right), \frac{4\rho k}{\sigma_{\omega 2} D_{\omega}^+ y^2}\right)^4\right]$$

$$F_2 = \tanh\left[\left[\max\left(\frac{2\sqrt{k}}{\beta^* \omega y}, \frac{500\mu}{\rho y^2 \omega}\right)\right]^2\right]$$

Cross-Diffusion Term (Key SST Innovation):

$$D_{\omega}^+ = \max\left(2\frac{\rho}{\sigma_{\omega 2}}\frac{1}{\omega}\frac{\partial k}{\partial x_j}\frac{\partial \omega}{\partial x_j}, 10^{-10}\right)$$

This term eliminates k-ω model's sensitivity to freestream ω values by transforming k-ω into k-ε formulation away from walls.

Comprehensive Term Definitions and Physical Meaning

ω: Specific dissipation rate [1/s] $$\omega = \frac{\epsilon}{C_\mu k}, \quad \text{relates to turbulent frequency}$$

Physical meaning: Inverse of turbulent time scale, represents frequency of energy-containing eddies. Higher ω indicates faster turbulent mixing.

F₁: Blending function (wall-to-freestream transition) $$F_1 = 1 \text{ near walls (k-ω)}, \quad F_1 = 0 \text{ in freestream (k-ε)}$$

Physical meaning: Switches between k-ω formulation near walls (superior viscous sublayer treatment) and k-ε in outer region (freestream independence).

F₂: Shear stress limiter activation function $$F_2 = 1 \text{ in boundary layers}, \quad F_2 = 0 \text{ in free shear flows}$$

Physical meaning: Activates shear stress limiter only where needed (attached boundary layers), preventing overprediction in adverse pressure gradients.

P̃ₖ: Limited production term $$\tilde{P}_k = \min(P_k, 10\beta^* \rho k \omega)$$

Physical meaning: Prevents excessive production in stagnation regions, maintaining realizability constraints on turbulent kinetic energy.

a₁: Shear stress limiter constant (0.31) $$\tau_{12} = \mu_t \frac{\partial u}{\partial y} \leq a_1 \rho k$$

Physical meaning: Enforces Bradshaw's assumption that shear stress is proportional to turbulent kinetic energy in boundary layers.

Cross-diffusion: Model transformation mechanism $$2(1-F_1)\frac{\rho}{\sigma_{\omega 2}}\frac{1}{\omega}\frac{\partial k}{\partial x_j}\frac{\partial \omega}{\partial x_j}$$

Physical meaning: Mathematically transforms k-ω equations to k-ε formulation in freestream, eliminating ω sensitivity issues.

Turbulent Scales and SST Scale-Resolving Behavior

The SST model's hybrid formulation provides scale-appropriate behavior from viscous sublayer to outer boundary layer region.

Near-Wall Scale Resolution (k-ω region):

$$y^+ = \frac{y u_\tau}{\nu} < 1, \quad \text{viscous sublayer integration}$$

Direct integration to wall without wall functions, resolving viscous length scale.

Outer Layer Scale Behavior (k-ε transformation):

$$\ell_t = \frac{\sqrt{k}}{\omega} \rightarrow \frac{k^{3/2}}{\epsilon}, \quad \text{turbulent length scale transition}$$

Automatic transition to k-ε length scale formulation in freestream regions.

Shear Layer Scale Limitation:

$$\tau_{wall} \propto \rho k, \quad \text{Bradshaw's constraint on Reynolds stress}$$

Ensures proper scaling of turbulent stresses in adverse pressure gradient boundary layers.

Model Constants: Blending and Physical Significance

Blended Constants (φ = F₁φ₁ + (1-F₁)φ₂):

$$\text{Set 1 (k-ω): } \sigma_{k1} = 0.85, \sigma_{\omega 1} = 0.5, \beta_1 = 0.075, \alpha_1 = 0.5532$$ $$\text{Set 2 (k-ε): } \sigma_{k2} = 1.0, \sigma_{\omega 2} = 0.856, \beta_2 = 0.0828, \alpha_2 = 0.4403$$ $$\text{Universal: } \beta^* = 0.09, \kappa = 0.41, a_1 = 0.31$$

Constant Derivations and Relationships:

α coefficients from k-ε constants: $$\alpha = \frac{\beta}{\beta^*} - \frac{\sigma_\omega \kappa^2}{\sqrt{\beta^*}}$$

Ensures equivalent behavior to k-ε model in freestream when F₁ = 0.

β* = 0.09: Universal constant linking k and ω $$\beta^* = C_\mu^{k-\epsilon} = 0.09$$

Maintains consistency with k-ε model's C_μ constant.

a₁ = 0.31: Bradshaw's constant $$\frac{-\overline{u'v'}}{k} \approx 0.31 \text{ in equilibrium boundary layers}$$

Based on experimental observation that shear stress correlates with turbulent kinetic energy.

Model Validation and Calibration Database

The SST model has been extensively validated against fundamental flow configurations that test specific aspects of the model's hybrid formulation. Flat plate boundary layers provide zero pressure gradient validation, confirming the model's ability to predict canonical attached flows. Adverse pressure gradient flows from the ERCOFTAC database demonstrate the model's superior performance in separating boundary layers compared to standard k-ε models.

Airfoil flows using NACA series at various angles of attack validate the model's transition prediction and separated flow behavior. Backward-facing step configurations test separation and reattachment prediction capabilities, while free jets and mixing layers verify the model's freestream independence - a critical improvement over the original k-ω formulation.

Advanced Theoretical Framework and Fundamental Assumptions

Bradshaw's Assumption and Shear Stress Limitation:

$$\frac{-\overline{u'v'}}{k} = a_1 = 0.31 \text{ (in equilibrium boundary layers)}$$

The SST model enforces this through the modified viscosity formulation:

$$\mu_t = \frac{\rho a_1 k}{\max(a_1 \omega, \Omega F_2)} \Rightarrow \tau_{12} = \mu_t S \leq a_1 \rho k$$

Realizability and Positivity Constraints:

The SST model maintains physical realizability through several mechanisms:

Production limiting: P̃_k prevents excessive k production
Shear stress limiting: Prevents unrealistic Reynolds stress levels
Positive definiteness: All transport coefficients remain positive

Wall Distance Dependency and y⁺ Insensitivity:

$$y^+ = \frac{y u_\tau}{\nu}, \quad \text{wall units}$$

Unlike k-ε, SST integrates to the wall (y = 0) with boundary conditions:

$$k|_{wall} = 0, \quad \omega|_{wall} = \frac{60\nu}{\beta_1 y_1^2}$$

Where y₁ is distance to first grid point. This eliminates wall function dependence.

Transformation to k-ε Formulation:

In the freestream (F₁ = 0), the model becomes equivalent to k-ε:

$$\epsilon = \beta^* k \omega, \quad \mu_t = \rho C_\mu \frac{k^2}{\epsilon}$$

This transformation is achieved through the cross-diffusion term, ensuring freestream independence.

Advanced Model Variants and Modern Improvements

Viscous Sublayer Integration:

The SST model resolves the complete boundary layer structure:

$$\text{Viscous sublayer: } y^+ < 5, \quad u^+ = y^+$$ $$\text{Buffer layer: } 5 < y^+ < 30, \quad \text{transition region}$$ $$\text{Log layer: } y^+ > 30, \quad u^+ = \frac{1}{\kappa}\ln(y^+) + B$$

Near-Wall Asymptotic Behavior:

$$k \sim y^2, \quad \omega \sim y^{-2} \text{ as } y \to 0$$

These asymptotic behaviors are automatically satisfied by the SST formulation.

Mesh Requirements for Accurate Wall Treatment:

y⁺ < 1: Recommended for optimal accuracy
y⁺ < 2: Acceptable for engineering accuracy
Minimum 10-15 points: In boundary layer for resolution
Growth ratio < 1.2: In near-wall region

Wall Boundary Conditions Implementation:

$$k_{wall} = 0, \quad \frac{\partial k}{\partial n}\bigg|_{wall} = 0$$ $$\omega_{wall} = \frac{60\nu}{\beta_1 (\Delta y_1)^2}, \quad \text{for } y_1^+ < 2$$

Advanced Model Variants and Modern Extensions

SST-2003 (Improved Corner Flow Prediction):

Enhanced version addressing corner flow stagnation issues:

$$\tilde{P}_k = \mu_t S^2 - \frac{2}{3}\mu_t S_{kk}\frac{\partial u_k}{\partial x_k} - \frac{2}{3}\rho k \frac{\partial u_k}{\partial x_k}$$

Includes proper treatment of dilatational effects and compressibility.

SST-V (Vorticity Source Term):

$$S_\omega^{vorticity} = 2\rho\chi\Omega^2, \quad \chi = \frac{1}{3}\left(\frac{\Omega}{\sqrt{S^2 + \Omega^2}}\right)^3$$

Improves prediction in flows with significant rotation or curvature effects.

Scale-Adaptive Simulation (SST-SAS):

$$Q_{SAS} = \max\left[2\rho\zeta_2\kappa S^2\left(\frac{L}{L_{vk}}\right)^2 - C_{SAS}\frac{2\rho k}{\sigma_\phi}\max\left(\frac{|\nabla\omega|^2}{\omega^2}, \frac{|\nabla k|^2}{k^2}\right), 0\right]$$

Allows model to adapt between RANS and LES-like behavior based on flow unsteadiness.

Compressibility Corrections:

$$M_t = \sqrt{\frac{2k}{a^2}}, \quad \text{turbulent Mach number}$$ $$\beta^* = \beta_0^*[1 + \chi_t M_t^2], \quad \chi_t = 1.5 \text{ for } M_t > 0.25$$

Transition Modeling Extensions:

$$\gamma-Re_{\theta t} \text{ model: } \gamma \text{ (intermittency)}, \, Re_{\theta t} \text{ (transition momentum thickness Reynolds number)}$$

Enables prediction of laminar-turbulent transition in boundary layers.

Advanced Numerical Implementation and Best Practices

Robust Source Term Treatment:

$$S_k = \tilde{P}_k - \beta^* \rho k \omega \rightarrow \tilde{P}_k - \beta^* \rho \omega \cdot k$$ $$S_\omega = \alpha \frac{\omega}{k}\tilde{P}_k - \beta \rho \omega^2 \rightarrow \alpha \frac{\tilde{P}_k}{k} - \beta \rho \omega \cdot \omega$$

Implicit treatment of destruction terms ensures numerical stability.

Blending Function Evaluation and Computational Considerations

The evaluation of blending functions F₁ and F₂ requires careful numerical treatment to ensure robust performance across diverse computational domains. Wall distance computation presents the primary computational challenge, particularly for complex geometries where efficient algorithms such as fast marching methods or Eikonal solvers are essential to maintain reasonable computational overhead while ensuring accuracy near geometric singularities.

Gradient limiters play a crucial role in preventing excessive gradients during F₁ and F₂ calculation, which can lead to numerical instabilities or convergence difficulties. Smoothing techniques are employed to ensure continuous blending behavior across the computational domain, preventing artificial discontinuities that could compromise solution quality or convergence characteristics.

Cross-Diffusion Term Treatment and ω Equation Stability

$$\text{Cross-diffusion term: } 2(1-F_1)\frac{\rho}{\sigma_{\omega 2}}\frac{1}{\omega}\frac{\partial k}{\partial x_j}\frac{\partial \omega}{\partial x_j}$$

The cross-diffusion term represents a unique numerical challenge in SST model implementation, requiring careful treatment to maintain positivity and avoid division by zero operations. Advanced discretization schemes must ensure that the term's contribution remains physically meaningful while maintaining numerical stability, particularly in regions where ω approaches zero or exhibits sharp gradients.

Advanced Solution Strategies and Convergence Enhancement

Contemporary CFD solvers implement sophisticated solution strategies to enhance SST model convergence characteristics. Coupled k-ω solving demonstrates superior convergence behavior compared to segregated approaches, particularly for flows with strong coupling between the turbulent variables. This approach treats the k and ω equations as a coupled system, reducing the iteration count required for convergence and improving overall robustness.

Comprehensive monitoring strategies encompass both scaled and absolute residuals, typically requiring convergence criteria below 10⁻⁵ for engineering accuracy. Additionally, integral monitoring of key flow quantities including mass flow rates, pressure drops, and heat transfer coefficients provides essential verification of solution convergence beyond simple residual criteria.

Mesh Quality Requirements and Computational Grid Considerations

The SST model's near-wall integration capability places stringent requirements on computational mesh quality, necessitating careful attention to multiple geometric parameters:

Near-wall spacing: First cell height must satisfy y⁺ < 1 for accurate wall integration
Aspect ratio control: Maintain ratios below 1000 in near-wall regions for gradient accuracy
Orthogonality: Overall mesh skewness should remain below 0.9 for accurate computations
Boundary layer resolution: Minimum 15-20 cells within the boundary layer
Growth rate: Geometric progression with ratios limited to 1.2 in near-wall regions

Implementation Robustness and Numerical Stability

Production-grade SST implementations incorporate multiple robustness enhancements including production limiting to prevent non-physical turbulence generation in stagnation regions, realizability constraints to maintain positive definiteness of turbulent quantities, and adaptive under-relaxation strategies that automatically adjust based on local convergence behavior. These features ensure reliable performance across the wide range of flow conditions encountered in engineering applications.