Turbulence modeling is an attempt to devise a number of partial differential equations for turbulent-flow calculation, based on appropriate approximations of the exact Navier–Stokes equations. In the approach of the Reynolds-averaged Navier–Stokes equations (RANS) the starting point is the Reynolds decomposition of the flow variables into mean and fluctuating parts, where the insertion of the Reynolds-decomposed variables into the Navier–Stokes equations—followed by an averaging of the equations themselves—gives rise to the Reynolds-stress tensor, an unknown term that has to be modeled in order for the RANS equations to be solved. The problem of the closure of the system of the Navier–Stokes equations essentially consists in this operation. Different classes of RANS turbulence models exist:
i. Zero-equation models. In this class, only a system of partial differential equations (PDEs) for the mean field is solved, and no other PDEs are used.
ii. One-equation models. With respect to (i), this class involves an additional transport equation for the calculation of the turbulence velocity scale, usually cast in terms of the average turbulent kinetic energy (k).
iii. Two-equation models. With respect to (ii), this class involves an additional transport equation for the calculation of the turbulence length scale, usually cast in terms of the scalar dissipation rate of turbulent kinetic-energy (ε).
iv. Stress-equation models. With respect to (i), this class involves a number of additional transport equations for the components of the Reynolds-stress tensor 𝜏𝑖𝑗 and one for
22 the scalar dissipation rate (ε). For this reason, models of class (iv) are called ( 𝜏𝑖𝑗 – ε) models.
The following is a brief overview of commonly employed models in modern engineering applications.
Spalart–Allmaras (S–A). The Spalart–Allmaras model is a one-equation model that solves a modelled transport equation for the kinematic eddy turbulent viscosity. The Spalart–Allmaras model was designed specifically for aerospace applications involving wall-bounded flows and has been shown to give good results for boundary layers subjected to adverse pressure gradients. It is also gaining popularity in turbo-machinery applications.
K–ε (k–epsilon). k-epsilon (k-ε) turbulence model is the most common model used in computational fluid dynamics (CFD) to simulate mean flow characteristics for turbulent flow conditions. It is a two-equation model which gives a general description of turbulence by means of two transport equations (PDEs).
Advantages of the k-epsilon model are-
• Robust model
• Widely used despite the known limitations of the model
• Easy to implement
• Computationally cheap
• Valid for fully turbulent flows only
• Suitable for initial iterations, initial screening of alternative designs, and parametric studies
Disadvantages of the k-epsilon model are-
• Performs poorly for complex flows involving severe pressure gradient, separation, and strong streamline curvature.
• Lack of sensitivity to adverse pressure gradients
• Numerical stiffness when equations are integrated through the viscous sub layer which are treated with damping functions that have stability issues
K–ω (k–omega). In computational fluid dynamics, the k–omega (k–ω) turbulence model is a common two-equation turbulence model that is used as a closure for the Reynolds-averaged Navier–Stokes equations (RANS equations). The model attempts to predict turbulence by two partial differential equations for two variables, k and ω, with the first variable being the
23 turbulence kinetic energy (k) while the second (ω) is the specific rate of dissipation.
Advantages of the k- ω model are-
• Superior performance for wall-bounded boundary layer, free shear, and low Reynolds number flows
• Suitable for complex boundary layer flows under adverse pressure gradient and separation (external aerodynamics and turbo-machinery)
• Can be used for transitional flows (though tends to predict early transition) Disadvantages of the k- ω model are-
• Separation is typically predicted to be excessive and early
• Requires mesh resolution near the wall
SST (Menter’s Shear Stress Transport).Shear Stress Transport (SST) is a variant of the standard k–ω model. It is a two equations model. This model combines the original Wilcox k–ω model for use near walls and the standard k–ε model away from walls using a blending function, and the eddy viscosity formulation is modified to account for the transport effects of the principle turbulent shear stress.
Advantages of the SST model are-
• Offers similar benefits as standard k–ω
• Accounts for the transport of turbulent shear stress and gives highly accurate predictions of the onset and the amount of flow separation under adverse pressure gradients
• Recommended for high accuracy boundary layer simulations.
Disadvantages of the SST model are-
• Dependency on wall distance makes this model less suitable for free shear flows compared to standard k-w
• Requires mesh resolution near the wall.
• A Reynolds Stress model may be more appropriate for flows with sudden changes in strain rate or rotating flows while the SST model may be more appropriate for separated flows.
Reynolds stress equation model. The Reynolds stress equation model (RSM), also referred to as second moment closure model, is the most complete classical turbulence modelling approach. Popular eddy-viscosity based models like the k–ε (k–epsilon) model and the k–ω (k–omega) models have significant shortcomings in complex engineering flows. This arises
24 due to the use of the eddy-viscosity hypothesis in their formulation. For instance, in flows with high degrees of anisotropy, significant streamline curvature, flow separation, zones of recirculating flow or flows influenced by rotational effects, the performance of such models is unsatisfactory. In such flows, Reynolds stress equation models offer much better accuracy.
Despite having some limitations, because of the cheapness of the computation and worldwide uses, k-ε turbulence model is used in the present work.