Reynolds-Averaged Navier-Stokes (RANS) Equations

The core idea behind RANS is Reynolds decomposition, where an instantaneous quantity is split into its time-averaged and fluctuating parts. For velocity, this is \(u_i(x,t) = \bar{u}_i(x) + u’_i(x,t)\). When this is substituted into the Navier-Stokes equations and the equations are time-averaged, the non-linear convective term generates a new term, \( -\rho \overline{u’_i u’_j} \), known as the Reynolds stress tensor. This tensor represents the net transfer of momentum due to turbulent fluctuations.

The appearance of this unknown tensor leads to the ‘closure problem’ of turbulence: there are more unknowns than equations. To solve the system, the Reynolds stresses must be related to the mean flow quantities through a turbulence model. The most common approach is the Boussinesq hypothesis, which assumes the Reynolds stresses are proportional to the mean strain rate, introducing an ‘eddy viscosity’ or ‘turbulent viscosity’. This is analogous to how molecular viscosity relates stress to strain rate in laminar flow. Turbulence models, such as the popular k-ε (k-epsilon) and k-ω (k-omega) models, are sets of additional transport equations used to compute this eddy viscosity throughout the flow field. For example, the k-ε model solves for the turbulent kinetic energy (k) and its rate of dissipation (ε). RANS provides a good balance of accuracy and computational cost for many engineering applications, as it avoids the prohibitive expense of resolving all turbulent eddies directly.