The smallest scale occurring in a turbulent flow (also called the Kolmogorov scale) remains far larger than the free molecular path. Turbulence is therefore a continuum phenomenon, which can be described by the Navier-Stokes equations. For the sake of clarity, only the incompressible governing equations will be derived in this review. In order to achieve notational brevity, the Einstein notation will be used. This convention implies that if an index variable appears several times in a single term, it actually represents a summation over all values of the index. For example, given two vectors (a₁, a₂, a₃) and (b₁, b₂, b₃), Some additional exercises are provided in the example section. Using the exact same process, the Navier-Stokes equations can be reduced to Looking at Eq.(2↑) and Eq.(3↑), the system consists of four unknowns (p, u₁, u₂ and u₃) and four equations. Given appropriate boundary conditions, the flow could theoretically be resolved. This is actually done in research laboratories, and is called Direct Numerical Simulation (DNS). In DNS, the full Navier-Stokes equations are solved numerically, without making any assumption or implementing any turbulence model. This method is very accurate (sometime called “numerical experiments"), but limited due to its extreme computational cost. More details about DNS are given in the final section of this review.

A very important question arises from this previous observation. What are the outputs expected from the simulation? If one’s only concern is the mean drag coefficient of a wing, why bother calculating all the fluctuations? Instead of solving the full unsteady Navier-Stokes equations, it would be much easier to average the flow field. This concept is called Reynolds averaging, where each variable (eg. p) is decomposed into a mean (eg. P) and a fluctuating component (eg. p’). The mean value can be computed using time averaging (suitable for stationary turbulence), spatial averaging (suitable for homogeneous turbulence) or ensemble averaging (combination of both). This approach is similar to that used in small perturbation analysis. In this case however, the perturbation and mean component have the same order of magnitude. Introducing the Reynolds decomposition in Eq.(2↑) and (3↑), and averaging the whole equations once more, the Reynolds Averaged Navier-Stokes (RANS) equations are obtained. This process is fully detailed in the example section. where the over-line in the very last term denotes an averaged quantity. The RANS equations are very similar to the Navier-Stokes equations, except for the very last term in Eq.(6↑). This new term has the dimension of a stress, and is therefore called the Reynolds stress tensor. It is symmetric, hence it introduces six additional unknowns to the problem. Therefore, it appears that the RANS equations are unclosed (more unknowns than equations). In order to solve the system, the Reynolds stress tensor will need to be estimated.