SAS Model - Scale-Adaptive Simulation

Hybrid RANS-LES approach for scale-adaptive turbulent flow simulation

1. Model Overview

  • Scale-Adaptive Simulation (SAS) bridging RANS and LES through automatic scale switching.
  • Hybrid approach that adapts between RANS mode for attached flows and LES mode for separated regions.
  • von Kármán length scale modification enabling transition to turbulence-resolving mode when appropriate.
  • Industrial efficiency providing LES-like accuracy in critical regions while maintaining RANS economy elsewhere.

2. Key Advantages

  • Automatic switching: No user intervention required for RANS-LES transition in flow regions.
  • Scale resolution: Resolves unsteady large-scale structures in separated and swirling flows.
  • Computational efficiency: More economical than pure LES while capturing critical unsteady physics.
  • Industrial applicability: Practical for complex geometries without interface definition requirements.

3. Performance Characteristics

  • Accuracy: Superior to RANS for unsteady flows while maintaining computational feasibility.
  • Computational cost: 2-5x more expensive than RANS, significantly cheaper than LES.
  • Time dependency: Inherently unsteady requiring transient simulation approach.
  • Mesh requirements: Intermediate between RANS and LES with focus on separation regions.

4. Industry Applications

Automotive: External aerodynamics, wake analysis, drag and noise prediction
Aerospace: Aircraft wake vortices, buffeting analysis, control surface effectiveness
Turbomachinery: Unsteady blade interactions, secondary flow analysis, loss mechanisms
Industrial Mixing: Complex mixing processes, heat and mass transfer enhancement
Wind Engineering: Bluff body flows, wind loading on structures, urban aerodynamics
Energy Systems: Heat exchanger performance, thermal mixing, combustion applications

5. When to Choose SAS Over Other Models

  • Unsteady separation: When large-scale unsteady structures are critical to the physics.
  • RANS limitations: Where steady RANS models fail to capture essential flow features.
  • LES constraints: When LES is too expensive but unsteady physics is important.
  • Mixed flow regimes: Flows with both attached and separated regions requiring different approaches.
  • Industrial complexity: Complex geometries where automatic adaptation is preferred over manual interface definition.

6. When NOT to Use SAS

Steady attached flows: Simple attached flows where RANS provides adequate accuracy
Tight computational budgets: When the 2-5x cost increase over RANS is prohibitive
High-frequency phenomena: When very small-scale turbulence resolution is critical
Steady-state requirements: Applications requiring only time-averaged flow properties
Wall-bounded flows: Simple channel or pipe flows without significant separation

7. Setup and Mesh Guidelines

Mesh Requirements:

  • Near-wall resolution: y⁺ < 1 preferred for accurate boundary layer resolution
  • Separation region refinement: Higher density in expected unsteady separation zones
  • Streamwise resolution: Adequate to capture developing unsteady structures
  • Time step: CFL < 1 for accurate unsteady resolution

Solver Settings:

  • Time integration: Second-order implicit schemes for temporal accuracy
  • Spatial discretization: Low-dissipation schemes to preserve unsteady content
  • Initialization: Start from converged RANS solution when available
  • Convergence: Monitor unsteady statistics for flow development

8. Performance Optimization

Computational Strategy:

  • RANS initialization: Use converged RANS as starting point for faster development
  • Adaptive time stepping: Optimize time step for CFL and accuracy requirements
  • Parallel efficiency: Good scaling characteristics for domain decomposition

Result Analysis:

  • Statistical convergence: Ensure adequate sampling time for mean and RMS quantities
  • Scale switching: Verify appropriate RANS-LES transitions in flow regions
  • Unsteady validation: Compare unsteady signatures with experimental data

9. Common Issues and Solutions

Scale Switching Problems:

  • Problem: Model not switching to LES mode in separated regions
  • Solution: Check mesh resolution and von Kármán length scale implementation

Convergence Difficulties:

  • Problem: Excessive computational time for flow development
  • Solution: Use RANS initialization and optimize time step selection

Numerical Stability:

  • Problem: Solution divergence in high Reynolds number flows
  • Solution: Reduce time step, improve mesh quality, check boundary conditions

Historical Context and Development

The Scale-Adaptive Simulation (SAS) model was developed by Menter and Egorov (2005, 2010) as an advanced hybrid RANS-LES approach that automatically adapts the level of turbulence resolution based on local flow characteristics. Unlike traditional hybrid methods requiring interface definitions, SAS provides seamless transition between RANS and LES modes.

Theoretical Foundation

The SAS concept emerged from recognizing limitations in both pure RANS and LES approaches for industrial applications. While RANS models excel in attached boundary layers but struggle with separated flows, LES provides excellent unsteady resolution but at prohibitive computational cost for entire industrial configurations.

The breakthrough innovation involves modifying the turbulence model's dissipation based on the von Kármán length scale, allowing the model to automatically detect when the mesh is fine enough to resolve unsteady turbulent structures and switch to a scale-resolving mode.

Recent developments have enhanced the SAS formulation for specific applications including combustion, multiphase flows, and rotating machinery, establishing it as a practical hybrid approach for complex engineering problems.

Mathematical Foundation and Model Formulation

The SAS model is based on the k-ω SST framework with an additional source term that enables scale-adaptive behavior through dynamic modification of the turbulent dissipation rate.

Base k-ω SST Equations with SAS Source:

$$\frac{\partial k}{\partial t} + U_j \frac{\partial k}{\partial x_j} = P_k - \beta^* k \omega + \frac{\partial}{\partial x_j}\left[\left(\nu + \sigma_k \nu_T\right) \frac{\partial k}{\partial x_j}\right]$$
$$\frac{\partial \omega}{\partial t} + U_j \frac{\partial \omega}{\partial x_j} = \alpha \frac{\omega}{k} P_k - \beta \omega^2 + Q_{SAS} + \frac{\partial}{\partial x_j}\left[\left(\nu + \sigma_\omega \nu_T\right) \frac{\partial \omega}{\partial x_j}\right]$$

SAS Source Term:

$$Q_{SAS} = \max\left[0, \zeta_2 \kappa S^2 \left(\frac{L}{L_{vK}}\right)^2 - C_{SAS} \frac{2k}{\sigma_\Phi} \max\left(\frac{1}{\omega^2} \frac{\partial \omega}{\partial x_j} \frac{\partial \omega}{\partial x_j}, \frac{1}{k^2} \frac{\partial k}{\partial x_j} \frac{\partial k}{\partial x_j}\right)\right]$$

von Kármán Length Scale:

$$L_{vK} = \kappa \frac{S}{\sqrt{0.5}\left|\frac{\partial^2 U_i}{\partial x_j \partial x_k}\right|}, \quad S = \sqrt{2S_{ij}S_{ij}}$$

Turbulent Length Scale:

$$L = \frac{\sqrt{k}}{\beta^{*1/4} \omega}$$

Scale-Adaptive Parameter:

$$\zeta_2 = 3.51, \quad C_{SAS} = 2.0, \quad \sigma_\Phi = 2/3$$

Variable Definitions:

Q_SAS: SAS source term (1/s³)
L: Turbulent length scale (m)
L_vK: von Kármán length scale (m)
S: Strain rate magnitude (1/s)
κ: Kármán constant (≈ 0.41)
ζ₂: SAS model constant
C_SAS: SAS limiter constant
Constant Value Physical Significance
ζ₂ 3.51 SAS production coefficient
C_SAS 2.0 SAS limiter coefficient
σ_Φ 2/3 SAS diffusion coefficient
κ 0.41 von Kármán constant
β* 0.09 k destruction coefficient

Scale-Adaptive Mechanism and Physics

The SAS model operates by comparing the turbulent length scale L with the von Kármán length scale L_vK. When the mesh is fine enough to resolve unsteady structures (L > L_vK), the SAS source term activates, reducing the turbulent viscosity and allowing the flow to develop LES-like behavior.

RANS Mode Operation

In attached boundary layers and regions where the mesh is too coarse for scale resolution, the ratio L/L_vK remains small, the SAS source term is inactive, and the model operates in standard k-ω SST mode, providing efficient RANS-like behavior with high turbulent viscosity damping unsteady fluctuations.

LES Mode Transition

When separation occurs or in regions with adequate mesh resolution, velocity gradients increase significantly, reducing L_vK. As L/L_vK increases beyond the threshold, Q_SAS becomes positive, increasing ω and reducing turbulent viscosity, allowing natural unsteady turbulent structures to develop.

von Kármán Length Scale Physics

The von Kármán length scale represents the size of the largest turbulent eddies that can be sustained by the local velocity field. It is computed from the strain rate and second derivatives of velocity, making it sensitive to flow curvature and separation, exactly where scale-resolving capabilities are most beneficial.

Automatic Adaptation Criteria

The transition between RANS and LES modes is governed by local flow physics rather than geometric criteria. This enables the model to automatically activate scale resolution in separated regions, wake flows, and areas of high unsteadiness while maintaining RANS efficiency in attached boundary layers.

Model Validation and Performance Assessment

The SAS model has been extensively validated across a range of separated flow configurations, demonstrating superior performance compared to steady RANS while maintaining computational efficiency relative to LES.

Benchmark Test Cases

  • Periodic hill flow at Re_H = 10,600 (scale-resolving validation)
  • Backward-facing step with various expansion ratios
  • Flow around circular cylinder at subcritical Reynolds numbers
  • Ahmed body automotive configuration for wake prediction

Industrial Applications

  • External vehicle aerodynamics with wake analysis
  • Turbomachinery unsteady blade interactions
  • Heat exchanger thermal mixing enhancement
  • Wind engineering and bluff body flows

Performance Characteristics

SAS provides significant improvement over RANS in predicting separation location, reattachment length, and unsteady force coefficients. Computational cost typically ranges from 2-5 times steady RANS, depending on the extent of scale-resolving regions, making it more affordable than pure LES for engineering applications.

Limitations and Considerations

  • Mesh quality requirements higher than RANS but lower than LES
  • Time step restrictions for accurate unsteady resolution
  • Statistical convergence requirements for unsteady quantities
  • Potential for delayed transition to scale-resolving mode

Implementation and Numerical Considerations

Successful SAS implementation requires careful attention to mesh design, time integration, and spatial discretization to ensure proper activation of scale-resolving capabilities while maintaining numerical stability.

Mesh Design Strategy

The mesh should be designed with RANS-quality resolution in attached regions and enhanced resolution in areas where scale resolution is expected. This hybrid approach optimizes computational resources while ensuring adequate capturing of unsteady phenomena in critical flow regions.

Time Integration Requirements

Second-order implicit time integration is recommended with time steps chosen to maintain CFL numbers below unity in scale-resolving regions. The time step should resolve the characteristic frequencies of large-scale unsteady structures while maintaining stability in RANS regions.

Spatial Discretization Considerations

Low-dissipation spatial schemes are crucial for preserving unsteady content in scale-resolving regions. Second-order upwind or higher-order schemes help maintain the balance between numerical stability and unsteady structure preservation.

Initialization and Convergence

  • Initialize from converged steady RANS solution when possible
  • Allow sufficient flow development time for unsteady structures
  • Monitor SAS source term activation in expected regions
  • Verify statistical convergence of mean and fluctuating quantities

Advanced Topics and Future Developments

Current research focuses on enhancing SAS performance for specific applications and developing improved formulations for challenging flow configurations.

Combustion Applications

SAS extensions for reacting flows incorporate modifications to handle density variations and chemical source terms, enabling application to combustion systems where large-scale mixing structures significantly influence reaction rates and emissions formation.

Multiphase Flow Enhancement

Developments for multiphase applications address interface dynamics and bubble/droplet interactions, where scale-resolving capabilities provide significant advantages over traditional RANS approaches for predicting mass transfer and mixing characteristics.

Machine Learning Integration

Research explores machine learning-enhanced SAS formulations that adapt model constants based on local flow features, potentially improving automatic detection of regions requiring scale resolution and optimizing the RANS-LES transition process.

Wall Function Developments

Advanced wall function formulations for SAS applications aim to reduce near-wall mesh requirements while maintaining accuracy in scale-resolving regions, extending the method's applicability to higher Reynolds number industrial flows.

Anisotropic SAS Formulations

Next-generation SAS models incorporate anisotropic length scale detection, allowing different scale-resolving behavior in different spatial directions based on local flow physics, potentially improving performance in complex three-dimensional separated flows.

Industrial Implementation Advances

Production-level implementations focus on robustness enhancements, automatic mesh adaptation based on SAS activation patterns, and hybrid parallelization strategies that optimize computational efficiency for mixed RANS-LES domains in large-scale industrial applications.