tdynreference manual
The turbulence modelling approach develop in Tdyn, is based on the concept of a dynamic turbulent viscosity, μτ. This relates the turbulent stresses appearing in the RANS equations to the averaged velocity gradients (i.e. the rate of strain) in direct analogy to the classical interpretation of viscous stresses in laminar flow by means of the fluid viscosity, μ. For example. in a shear layer where the dominant velocity gradient is ∂u/∂y (u is the averaged velocity in the principal direction of flow and y is the cross-stream co-ordinate) the turbulent shear stress is given as μτ·∂u/∂y.
The viscous analogy can be extended to the interpretation of the turbulent energy fluxes using the so-called Reynolds extended analogy. From dimensional considerations μτ/ρ is proportional to V·L, where V is a velocity scale and L is a length scale of the larger turbulent motions (called the mixing length in so-called mixing length models). Both the velocity scale V and the length scale L are determined by the state of turbulence. Several ways to determine V and L fields have been implemented in Tdyn.
The simplest prescription of V and L is with the so-called algebraic (or zero-equation) class of models.
These assume, that V and L can be related by algebraic equations to the local properties of the flow. This is fairly straightforward for simple flows but can often be difficult in geometrically complex configurations.
Algebraic models of turbulence have the virtue of simplicity and are widely used with considerable success for simple shear flows such as attached boundary layers, jets and wakes. For more complex flows where the state of turbulence is not locally determined but related to the upstream history of the flow a more sophisticated prescription is required.
The one-equation models attempt to improve on the zero-equation models by using an eddy viscosity that no longer depends purely on the local flow conditions but takes into account where the flow has come from, i.e. upon the flow history.
In these models, V is identified with k1/2, where k is the kinetic energy per unit mass of fluid arising from the turbulent fluctuations in velocity around the averaged velocity. A transport equation for k can be derived from the Navier-Stokes equations and it is the single transport equation in the one-equation model.
In shear-layer type flows, and especially in regions close to a wall, it is often possible to algebraically prescribe L with reasonable confidence. In geometrically complex configurations it is difficult to prescribe the L, because it is dependent on non-local quantities, such as a boundary layer thickness, displacement thickness, etc., and it introduces similar uncertainties as in an algebraic turbulence model.
Spalart and Allmaras [11] have devised an alternative formulation of a one-equation model appropriate for aerodynamic flows, which determines the turbulent viscosity directly from a single transport equation for μτ. This model is quite successful in practical turbulent flows in external airfoil applications. It is not well suited for more general flows, as it leads to serious errors even for simple shear flows (round jet).
For general applications, it is usual to solve two separate transport equations to determine V and L, giving rise to the name two-equation model. In combination with the transport equation for k, an additional transport equation is solved for a quantity, which determines the length scale L. This class of models is the best known and the most widely used in industrial applications since it is the simplest, level of closure which does not require geometry or flow regime dependent input.
The most popular version of two equation models is the k-ε model, where ε is the rate at which turbulent energy is dissipated to smaller eddies. A modelled transport equation for ε is solved and then L is determined as Cμ·k3/2/ε where Cμ is usually taken as a constant.
The second most widely used type of two-equation model is the k-ω model, where ω is a frequency of the large eddies. A modelled transport equation for ω is solved and L is then determined from k1/2/ω. The k-ω model performs very well close to walls in boundary layer flows, particularly under strong adverse pressure gradients. However it is very sensitive to the free stream value of ω and unless great care is taken in setting this value, spurious results can be obtained in both boundary layer flows and free shear flows. The k-ε model is less sensitive to free stream values but is often inadequate in adverse pressure gradients. Tdyn incorporates a variant of this model (the k-ω SST model) that tries to circumvent this problem by retaining the properties of k-ω close to the wall and gradually blending into the k-ε model away from the wall. This model has been shown to eliminate the free stream sensitivity problem without sacrificing the k-ω near wall performance.
The performance of two-equation turbulence models deteriorates when the turbulence structure is no longer close to local equilibrium. This occurs when the production of turbulence energy departs significantly from the rate at which it is dissipated at the small scales (i.e. ε), or equivalently when dimensionless strain rates (i.e. absolute value of the rate of strain times k/ε) become large. The so-called SST (shear stress transport) variation of the k-ω model leads to marked improvements in performance for non-equilibrium boundary layer regions such as those found close to separation. However, such modifications should not be viewed as a universal cure. For example SST is less able to deal with flow recovery following re-attachment.