Algebraic stress models describe the Reynolds stress tensor with a series expansion. As can be seen in Eq.(1↓), the Boussinesq assumption (Eq.(↓)) is only reflected in the first order term: A and B are functions of the mean strain rate and vorticity tensors, and include six closure coefficients. The model is based on the homogeneous part of the dissipation rate ϵ̃ = ϵ − 2ν(∂k^1 ⁄ 2 ⁄ ∂y)². There is a wide range of methods available in the literature to determine the numerous coefficients (Wallin and Johansson (2000) [13]). This non-linear formulation delivers the model from its local isotropy assumption. However, the local equilibrium hypothesis is still considered, so that history effects remain mostly ignored.

Reynolds stress models (also called second order closure models) directly solve additional equations for the Reynolds stresses. The latter are derived by considering moments of the Navier-Stokes equation. with and This approach is capable of handling anisotropic turbulence and considers the effects of flow history. However, the price to pay is a dramatic increase in computational cost. In three dimensions, six partial differential equations are introduced, in addition to the standard Navier-Stokes equations. Moreover, the equations for k and ϵ must also be incorporated. The review of Speziale (1995) is recommended for a more advanced discussion on Reynolds stress models [11].

Direct numerical simulation is a very accurate method that consists in solving numerically the full unsteady Navier-Stokes equation. No assumption is made, and the flow is solved from the large eddies down to the Kolmogorov scale. Almost all quantities can be determined by averaging the results. DNS is computationally very expensive, especially at high Reynolds number. Indeed, as the Reynolds number increases, the smallest scales become smaller and smaller. The computational domain must be large enough to contain the big eddies, while being composed of cells smaller than the smallest eddies. Several decades will probably pass before DNS becomes manageable for full aircraft configurations. Nevertheless, it remains a great research tool. Using DNS, it becomes possible to study the very nature of turbulence, performing "numerical experiments" that would be extremely challenging in an actual wind tunnel environment.

As discussed in the previous sections, turbulence can be resolved or modeled. Resolving the turbulence yields excellent results, but increases dramatically the computational cost. However, Kolmogorov hypothesised that at sufficiently large Reynolds number, the small scales would become locally isotropic. To date, this assumption still has not been proven right, but has not been discarded. Modern DNS and experimental results seem to suggest the validity of the approximation [12], but additional research is required to confirm this statement. The idea behind large eddy simulation is as follow: since the smallest scales are "universal", why not resolve the large scales (which contain the energy) and model the smaller eddies? Typically, the transition from resolution to modeling occurs when the length scale reaches the order of the grid size. If the latter is sufficiently small, the local isotropy assumption should therefore provide reliable results. A filter is applied to the Navier-Stokes equation in order to discard the scales that will be judged “small enough" to be modeled. A generalized spatial filter can be defined as follow