RANS and turbulence closure

Reynolds-averaged Navier–Stokes (RANS) describes the mean flow. Turbulence models close the additional terms that appear when the equations are averaged.

Average the nonlinear term

At a fixed position, decompose an instantaneous quantity into its mean and its fluctuation.

\phi(t)=\overline{\phi}+\phi'(t), \qquad \overline{\phi'}=0.

The overbar denotes the Reynolds average. For statistically stationary flow this can be estimated with a sufficiently long time average. For an unsteady mean, an ensemble average can retain time dependence.

Apply the same split to velocity, $u_i=\overline{u_i}+u_i'$ . The nonlinear momentum flux averages to

\overline{u_i u_k}=\overline{u_i}\,\overline{u_k}+\overline{u_i'u_k'}.

The covariance term survives even though the individual fluctuations average to zero. For an incompressible fluid with constant density and viscosity, the averaged momentum equation in the same conservation notation is

\begin{aligned} \frac{\partial(\rho\overline{u_k})}{\partial t} +\frac{\partial}{\partial x_i}\left( \rho\overline{u_i}\,\overline{u_k} -\mu\frac{\partial\overline{u_k}}{\partial x_i} \right) &=-\frac{\partial\overline p}{\partial x_k} +\overline{F_k} -\frac{\partial}{\partial x_i} \left(\rho\overline{u_i'u_k'}\right). \end{aligned}

Here $i$ is summed, $k$ selects the momentum component, and the averaged velocity still satisfies $\nabla\cdot\overline{\mathbf u}=0$ . The Reynolds stress $-\rho\overline{u_i'u_k'}$ has six independent components in three dimensions. An eddy-viscosity closure approximates it in terms of the mean strain $S_{ij}=(\partial_j\overline{u_i}+\partial_i\overline{u_j})/2$ .

-\overline{u_i'u_j'}=2\nu_t S_{ij}-\frac{2}{3}k\delta_{ij}, \qquad k=\tfrac12\overline{u_i'u_i'}.

In this closure $k$ denotes turbulent kinetic energy, and $\nu_t$ is kinematic eddy viscosity.

Compressible RANS commonly uses density-weighted (Favre) averages, $\widetilde\phi=\overline{\rho\phi}/\overline\rho$ . The full compressible energy and stress closures are more involved.

What the common models solve

Closure	Additional transport	Why you might use it
Spalart–Allmaras (SA)	One equation for a modified eddy-viscosity variable	A common external-aero baseline for attached or mildly separated flows. Standard SA assumes fully turbulent flow.
Menter SST	Two equations, for turbulent kinetic energy $k$ and specific dissipation rate $\omega$	Blends near-wall $k$ – $\omega$ behavior with a transformed outer $k$ – $\varepsilon$ formulation and limits eddy viscosity. A common baseline for adverse pressure gradients.
Reynolds-stress model	Transport for individual stress components and a scale variable	Can represent turbulence anisotropy beyond a scalar eddy viscosity. More equations and additional closures are needed.
Transition models	Model-specific transition correlations or transport equations, often coupled to SST	Useful to investigate low-Reynolds-number airfoils where laminar separation and transition affect drag. Freestream turbulence, roughness and trip assumptions matter.

The form of a closure equation

For example, in a fully turbulent SA form with trip terms disabled, the working variable $\tilde\nu$ evolves by advection, production, diffusion and wall-related destruction. A useful schematic grouping of its transport equation is

\partial_t\tilde\nu+\overline{u_j}\,\partial_j\tilde\nu = P_{\tilde\nu}+D_{\tilde\nu}-W_{\tilde\nu}, \qquad \nu_t=\tilde\nu f_{v1}.

The wall-related destruction term contains a factor proportional to $(\tilde\nu/d)^2$ , with $d$ the wall distance. This dependence is what the original SA-based DES modifies. SST instead solves transport equations for both $k$ and $\omega$ , blending coefficients through a wall-sensitive function and adding cross diffusion in the $\omega$ equation. The linked TMR pages give the full coefficients and distinguish the variants.

What the output means

A steady RANS solution targets a steady mean. For cruise loads it can be a useful first baseline. Validate near stall or when flow is strongly separated. Report the specific closure variant, wall treatment, mesh sensitivity and what the benchmark actually measures.

Average the nonlinear term

What the common models solve

The form of a closure equation

What the output means

Sources