Computational Fluid Dynamics (CFD) relies heavily on turbulence models to accurately simulate fluid flow in various engineering applications. Among these models, the Shear Stress Transport (SST) turbulence model stands out for its versatility and accuracy in predicting complex flow phenomena, especially in aerodynamic and hydrodynamic scenarios. This article delves into the intricacies of the Menter SST turbulence model, exploring its equations, variations, and practical considerations for implementation.
What is the Shear Stress Transport (SST) Turbulence Model?
The SST model is a two-equation eddy-viscosity model, a refined descendant of the k-omega (k-ω) family of turbulence models. Developed by Florian Menter in the early 1990s, it combines the strengths of both the k-omega and the k-epsilon (k-ε) models. The SST model cleverly utilizes a blending function to switch between the k-omega formulation near walls, where it performs well in resolving viscous sublayer effects, and the k-epsilon formulation in the freestream, avoiding the k-omega model’s sensitivity to inlet freestream turbulence properties. This hybrid approach makes the SST model more robust and widely applicable compared to its predecessors.
The foundation of the SST model, like other linear eddy viscosity models, is the Boussinesq assumption. This constitutive relation simplifies the Reynolds stress tensor, ![tau{ij}], by relating it to the mean strain rate tensor, ![(S{ij})], and the turbulent eddy viscosity, ![(mu_t)], as shown in the equation above. This simplification is crucial for making turbulence modeling computationally tractable in engineering simulations.
Standard Menter SST Two-Equation Model
The “standard” SST model, also referred to as SST (or SSTm for a common approximation described later), is the original and most widely used version. It builds upon the Menter Baseline (BSL) model but introduces key differences in one constant and the turbulent eddy viscosity calculation. The model is governed by two transport equations, one for the turbulent kinetic energy (k) and another for the specific dissipation rate (ω). These equations, in conservation form, are:
In these equations:
- ![(rho)] is the density.
- ![(u_j)] represents the velocity components.
- ![(mu)] is the dynamic viscosity.
- ![(mu_t)] is the turbulent eddy viscosity.
- ![(cal P)] is the production of turbulent kinetic energy.
- ![(beta^*), (beta), (sigmak), (sigma{omega}), (gamma), (sigma_{omega 2})] are model constants.
- ![(F_1)] is the blending function.
The production term, ![(cal P)], is defined as:
Where the Reynolds stress tensor, ![(tau{ij})], and the mean strain rate tensor, ![(S{ij})], are given by:
The turbulent eddy viscosity, ![(mu_t)], a crucial parameter in the SST model, is computed as:
Here, ![(a_1)] is a constant, and ![(Omega)] is the vorticity magnitude. The blending function, ![(F_1)], and another function, ![(F_2)], are defined to ensure the model behaves like k-omega near walls and k-epsilon away from walls:
In these functions, ![(nu)] is the kinematic viscosity, and d is the distance to the nearest wall.
To prevent excessive turbulence production in stagnation regions, a production limiter is often applied, replacing ![(cal P)] in the k-equation with ![{rm min}(cal P, 20 beta^* rho omega k)].
The model constants are blended using ![(phi = F_1 phi_1 + (1-F_1) phi_2)], where ![(phi_1)] and ![(phi_2)] represent constants associated with the k-omega and k-epsilon models, respectively. The specific values for these constants are provided in the original article for detailed implementation.
Variations of the SST Model
Over time, several variations of the standard SST model have been developed to address specific limitations or improve performance in certain flow regimes. Some notable variations include:
SST with Vorticity Source Term (SST-V)
In SST-V, the production term ![(cal P)] is approximated using vorticity magnitude ![(Omega)], which is often readily available in CFD codes. This approximation can simplify implementation and sometimes improve numerical stability, especially in hypersonic flows with strong shock waves. The production term in SST-V is:
SST with Kato-Launder Source Term (SST-KL)
The SST-KL variation employs the Kato-Launder correction, modifying the production term to ![(mu_t S Omega)], where ![(S)] is the strain rate magnitude. This modification aims to reduce excessive turbulence production in stagnation regions, similar to the production limiter but through a different approach.
SST-2003
The SST-2003 version introduces minor refinements to the standard SST model based on a decade of industrial experience. The key changes include:
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Eddy Viscosity Definition: Uses strain rate magnitude ![(S)] instead of vorticity magnitude ![(Omega)] in the eddy viscosity calculation.
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Production Limiter: Applied to both k and omega equations with a modified constant.
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Constant Modifications: Slight adjustments to some model constants.
SST with Controlled Decay (SST-sust)
SST-sust addresses the issue of non-physical decay of turbulence in the freestream for external aerodynamic flows. It adds “sustaining” terms to both the k and omega equations, effectively maintaining freestream turbulence levels. The modified equations are:
These additional terms, involving ambient values ![(k{rm amb})] and ![(omega{rm amb})], counteract the dissipation of turbulence in regions far from walls.
SST with Rotation/Curvature Correction (SST-RC) and SST-RC-Hellsten
To improve accuracy in flows with rotation or curvature, corrections have been incorporated into the SST model. SST-RC multiplies the production term ![(cal P)] by a rotation-sensitive function ![(f_{r1})], which depends on strain rate, vorticity, and rotation rate. SST-RC-Hellsten simplifies this correction by modifying the destruction term in the omega equation with a function ![(F_4)] based on the ratio of vorticity to strain rate.
Implementation Notes and Considerations
Implementing the SST model and its variants in CFD codes requires careful attention to several details:
- Boundary Conditions: Appropriate boundary conditions for k and omega are crucial. The original article provides recommended values for wall and farfield conditions. SST-sust, in particular, requires different farfield conditions to be effective.
- Compressibility Effects: For compressible flows, the SST model is typically implemented with density variations and thermodynamic properties accounted for, as described in resources like “Implementing Turbulence Models into the Compressible RANS Equations.”
- Approximations (SSTm, SSTs, SSTe): The article mentions different approximations related to the ![(2/3) overline rho k delta{ij}] term in the Reynolds stress and the production term. SSTm (most common) ignores the ![(2/3) overline rho k delta{ij}] term and approximates production as ![(P = mut S^2)]. SSTs includes the ![(2/3) overline rho k delta{ij}] term but still approximates production. SSTe uses the exact production expression. The choice of approximation can affect accuracy and computational cost.
- OpenFOAM: Special notes are available for users implementing SST models in OpenFOAM, addressing specific implementation aspects in this popular open-source CFD software.
Conclusion
The Shear Stress Transport (SST) turbulence model, in its standard form and various modifications, represents a powerful and versatile tool for CFD simulations. Its ability to blend the advantages of k-omega and k-epsilon models, along with the refinements introduced in its variations, makes it a popular choice for a wide range of engineering applications, especially those involving complex flows with separation, rotation, and curvature. Understanding the nuances of the SST model, its equations, and implementation considerations is essential for engineers and researchers seeking accurate and reliable CFD predictions.
For further in-depth information and the specific equations and constants, refer back to the original NASA Langley Research Center’s Turbulence Modeling Resource page.
