Reynolds-averaged Navier–Stokes equations

The Reynolds-averaged Navier–Stokes (RANS) equations are time-averaged equations of motion for fluid flow. They are primarily used while dealing with turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate averaged solutions to the Navier–Stokes equations. For a stationary, incompressible flow of Newtonian fluid, these equations can be written as :

$rho\; frac\{partial\; bar\{u\}\_j\; bar\{u\}\_i\; \}\{partial\; x\_j\}\; =\; rho\; bar\{f\}\_i\; +\; frac\{partial\}\{partial\; x\_j\}\; left[-\; bar\{p\}delta\_\{ij\}\; +\; mu\; left(frac\{partial\; bar\{u\}\_i\}\{partial\; x\_j\}\; +\; frac\{partial\; bar\{u\}\_j\}\{partial\; x\_i\}\; right)\; -\; rho\; overline\{u\_i^prime\; u\_j^prime\}\; right\; ].$

The left hand side of this equation represents the change in mean momentum of fluid element due to the unsteadiness in the mean flow and the convection by the mean flow. This change is balanced by the mean body force, the isotropic stress due to the mean pressure field, the viscous stresses, and apparent stress $left(-\; rho\; overline\{u\_i^prime\; u\_j^prime\}\; right)$ due to the fluctuating velocity field, generally referred to as Reynolds stresses.

Derivation of RANS equations

The basic tool required for the derivation of the RANS equations from the instantaneous Navier–Stokes equations is the Reynolds decomposition. Reynolds decomposition refers to separation of the flow variable (like velocity $u$) into the mean (time-averaged) component ($bar\{u\}$) and the fluctuating component ($u^prime$). Thus,

$u(mathbf\{x\},t)\; =\; bar\{u\}(mathbf\{x\})\; +\; u^prime(mathbf\{x\},t)\; ,$

where, $mathbf\{x\}\; =\; (x,y,z)$ is the position vector.

The following rules will be useful while deriving the RANS. If $f$ and $g$ are two flow variables (like density ($rho$), velocity ($u$), pressure ($p$), etc.) and $s$ is one of the independent variables ($x,y,z,\; mbox\{\; or\; \}\; t$) then,

$overline\{overline\{f\}\}\; =\; bar\{f\}$

$overline\{f+g\}\; =\; bar\{f\}\; +\; bar\{g\}$

$overline\{overline\{f\}g\}\; =\; bar\{f\}bar\{g\}$

$overline\{fg\}\; ne\; bar\{f\}bar\{g\}$

$overline\{frac\{partial\; f\}\{partial\; s\}\}\; =\; frac\{partial\; bar\{f\}\}\{partial\; s\}$

Now the Navier–Stokes equations of motion for an incompressible Newtonian fluid are:

$frac\{partial\; u\_i\}\{partial\; x\_i\}\; =\; 0$

$frac\{partial\; u\_i\}\{partial\; t\}\; +\; u\_j\; frac\{partial\; u\_i\}\{partial\; x\_j\}$

= f_i - frac{1}{rho} frac{partial p}{partial x_i} + nu frac{partial^2 u_i}{partial x_j partial x_j}

Substituting, $u\_i\; =\; bar\{u\_i\}\; +\; u\_i^prime,\; p\; =\; bar\{p\}\; +\; p^prime$, etc. and taking a time-average of these equations yields,

$frac\{partial\; bar\{u\_i\}\}\{partial\; x\_i\}\; =\; 0$

$frac\{partial\; bar\{u\_i\}\}\{partial\; t\}$

+ bar{u_j}frac{partial bar{u_i} }{partial x_j} + overline{u_j^prime frac{partial u_i^prime }{partial x_j}} = bar{f_i} - frac{1}{rho}frac{partial bar{p}}{partial x_i} + nu frac{partial^2 bar{u_i}}{partial x_j partial x_j}

The momentum equation can also be written as,

$frac\{partial\; bar\{u\_i\}\}\{partial\; t\}$

+ frac{partial bar{u_j} bar{u_i} }{partial x_j} = bar{f_i} - frac{1}{rho}frac{partial bar{p}}{partial x_i} + nu frac{partial^2 bar{u_i}}{partial x_j partial x_j} - frac{partial overline{u_i^prime u_j^prime }}{partial x_j} On further manipulations this yields,

$rho\; frac\{partial\; bar\{u\_i\}\}\{partial\; t\}$

+ rho frac{partial bar{u_j} bar{u_i} }{partial x_j} = rho bar{f_i} + frac{partial}{partial x_j} left[- bar{p}delta_{ij} + 2mu bar{S_{ij}} - rho overline{u_i^prime u_j^prime} right ]

where, $bar\{S\_\{ij\}\}\; =\; frac\{1\}\{2\}left(frac\{partial\; bar\{u\_i\}\}\{partial\; x\_j\}\; +\; frac\{partial\; bar\{u\_j\}\}\{partial\; x\_i\}\; right)$ is the mean rate of strain tensor.

Finally, since integration in time removes the time dependence of the resultant terms, the time derivative must be eliminated, leaving:

$rho\; frac\{partial\; bar\{u\_j\}\; bar\{u\_i\}\; \}\{partial\; x\_j\}$

= rho bar{f_i} + frac{partial}{partial x_j} left[- bar{p}delta_{ij} + 2mu bar{S_{ij}} - rho overline{u_i^prime u_j^prime} right ]

Notes