# Turbulence Modeling

Three-dimensional industrial scale problems are concerned with the time averaged (mean) flow, not the instantaneous motion. The preferred approach is to model turbulence using simplifying approximations, and not resolve it.

Turbulence modeling is a procedure to solve a modified set of the Navier-Stokes equations by means of developing a mathematical model of the turbulent flow that represents the time-averaged characteristics of the flow. Turbulence modeling is used to compute the impact of eddies on the mean flow field. This approach is based on the assumption that the turbulent eddy motion is “universal” and can be related to the large-scale average motion.

Over the course of the past few decades, turbulence models of various complexities have been developed. Depending on the simplifications made to the Navier-Stokes equations, the turbulence models can be classified as shown below.
Large Eddy Simulation (LES)
LES solves the filtered Navier-Stokes equations to resolve eddies down to the inertial range and it uses subgrid models to account for the influence of eddies in the dissipative range. The computing requirement is substantially less than that of DNS but is still not practical for many industrial applications containing wall bounded flows.
Common types include:
• Smagorinsky model
• Germano dynamic model
Hybrid
Hybrid simulations are the bridge between LES and RANS by utilizing RANS for attached boundary layers and LES for separated flow regions. In general, hybrid simulations need spatial filtering processes to determine the local sub grid turbulent viscosity. Compared to LES and DNS, hybrid simulations are much more tractable since the numerical requirement is less severe than the other two approaches.
Common types include:
• Detached Eddy Simulation (DES)
• Delayed DES (DDES)
• Improved DDES (IDDES)
• Wall Modeled LES (WMES)
• Zonal LES
Reynolds-averaged Navier-Stokes (RANS)
RANS simulations solve directly for the time averaged flow and model the effects of turbulent eddies on the mean flow. This method is the most computationally efficient CFD approach. Since most engineering problems are concerned with the time-averaged properties of the flow, this approach is used most frequently in the industry. The Reynolds averaging procedure introduces additional unknowns into the Navier Stokes equations, and it is thus necessary to develop additional turbulence model equations to close the set. These additional model equations can be categorized into turbulence models that use the Boussinesq assumption and turbulence models that do not use the Boussinesq assumption. The turbulence models with the Boussinesq assumption have two steps to compute Reynolds stresses for the RANS equations. First, turbulence models are needed for the computations of the eddy viscosity, second, the eddy viscosity is used for the estimation of the Reynolds stress with the Boussinesq assumption. The turbulence models without the Boussinesq assumption, for example, Reynolds stress models or nonlinear eddy viscosity models, determine the Reynolds (turbulent) stresses explicitly by solving an equation for each stress component.
Common types include:
• Seven equation models
• Reynolds stress model (RSM)
• One, two and three equation models + Boussinesq
• v2f, zeta-f
• k-ε, k-ω, SST
• Spalart-Allmaras (SA)
Figure 1 shows turbulence models and their corresponding energy spectrum ranges for modeling. For example, RANS models rely on their transport equations to model the entire wave number range, while LES needs a subgrid model to model behaviors of eddies in the dissipative range, but explicitly resolves the large eddies. The hybrid RANS/LES approach needs a model to cover the inertial range and the dissipative range.
As expected, computing requirements for these models differ since their models cover a different turbulent spectrum for the range of wave numbers. Spalart (2000) summarized resource requirements for each model, as shown below. Strictly speaking, DNS is not a turbulence model since it is resolving all scales of motion, however, it is included for comparison purposes. Grid numbers and time steps were estimated for a clean wing (Spalart, 2000). Grid numbers and time steps increase from RANS to DNS.
Table 1. CFD Methods (Spalart, 2000)
CFD methods Reynolds Number Dependence Empiricism Grid Time steps
DNS strong none 1016 107.7
LES weak weak 1011.5 106.7
DES (hybrid) weak strong 108 104
RANS weak strong 107 103