RANS equations and the closure problem

Reynolds decomposition and averaging

Each flow variable is split into a mean and a fluctuation:

$$u=\bar u+u',\qquad p=\bar p+p',\qquad \bar u=\frac1T\int_0^Tu\,dt,\qquad \overline{u'}=0$$

Rules: $\overline{\bar u}=\bar u$, $\overline{u'}=0$, but $\overline{u'v'}\neq0$ in general. For unsteady mean flow, ensemble averaging is used.

RANS equations

Substitute into the incompressible continuity and Navier-Stokes equations and average:

Continuity:

$$\frac{\partial\bar u_i}{\partial x_i}=0$$

Momentum:

$$\rho\left(\frac{\partial\bar u_i}{\partial t}+\bar u_j\frac{\partial\bar u_i}{\partial x_j}\right)=-\frac{\partial\bar p}{\partial x_i}+\frac{\partial}{\partial x_j}\left(\mu\frac{\partial\bar u_i}{\partial x_j}-\rho\overline{u'_iu'_j}\right)$$

The new term $-\rho\overline{u'_iu'_j}$ is the Reynolds stress tensor: 6 independent components (symmetric) which represent the extra momentum transport by turbulence. The term comes from the averaging of the non-linear convective term.

Closure problem

The averaged equations have 4 equations (continuity + 3 momentum) but 10 unknowns (4 mean variables + 6 Reynolds stresses). Writing equations for the stresses brings in triple correlations $\overline{u'u'u'}$, and so on: every level introduces more unknowns than equations. The set cannot be closed without turbulence models that relate the Reynolds stresses to mean quantities.

Boussinesq hypothesis

Reynolds stresses are assumed to act like viscous stresses, proportional to the mean strain rate, through an eddy viscosity $\mu_t$:

$$-\rho\overline{u'_iu'_j}=\mu_t\left(\frac{\partial\bar u_i}{\partial x_j}+\frac{\partial\bar u_j}{\partial x_i}\right)-\frac23\rho k\delta_{ij},\qquad k=\tfrac12\overline{u'_iu'_i}$$

The problem then becomes one of finding $\mu_t$: zero-equation (mixing length), one-equation (Spalart-Allmaras) and two-equation (k-$\varepsilon$, k-$\omega$) models. Alternatively, Reynolds stress models (RSM) solve transport equations for each stress, without the isotropy assumption.