Reynolds-averaged Navier-Stokes equations

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The Reynolds-averaged Navier-Stokes (RANS) equations are time-averaged ^[1] 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 an incompressible flow of Newtonian fluid, these equations can be written as,

$\rho \frac{\partial \bar{u_i}}{\partial t} + \rho \frac{\partial \bar{u_j} \bar{u_i} }{\partial x_j} = \rho \bar{f_i} + \frac{\partial}{\partial x_j} \left[ - \bar{p}\delta_{ij} + \mu \left( \frac{\partial \bar{u_i}}{\partial x_j} + \frac{\partial \bar{u_j}}{\partial x_i} \right) - \rho \overline{u_i^\prime u_j^\prime} \right ]$

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.

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 ( $u^\prime$ ).^[2] Thus,

$u(\mathbf{x},t) = \bar{u}(\mathbf{x}) + u^\prime(\mathbf{x},t) \,$ ^[3]

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 ( $ρ$ ), velocity ( $u$ ), pressure ( $p$ ), etc.) and $s$ is one of the independent variables ( $x, y, z, or t$ ) 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 ^[4] for an incompressible Newtonian fluid are:

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

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

Subsituting, $u_i = \bar{u_i} + u_i^\prime, p = \bar{p} + p^\prime$ , etc. ^[5] 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, ^[6]

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

On further manipulations this yields,

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

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 of strain tensor.

^ The true time average ( $\bar{X}$ ) of a variable ( $x$ ) is defined by,

$\bar{X} = \lim_{T \to \infty}\frac{1}{T}\int_{t_0}^{t_0+T} x\, dt$

In general for a time-average to be useful quantity, it is required that the average ( $\bar{X}$ ) be independent of the starting time ( $t 0$ ). This constraint is important for otherwise using time-averaging would be meaning less. This implies that the average value ( $\bar{X}$ ) is independent of time ( $t$ ). Since it is not possible to integrate over an infinte time period, it is necessary to restict the integration to some finite, yet large time interval. This interval is so selected that the term $\bar{X}$ is independent of the length of the interval $(T)$ . However, the independence from $t 0$ can no longer be ensured. Only in case of steady flows will $\bar{X}$ be independent of both $t 0$ and $T$ . Thus,

$\bar{X}(t_0) = \frac{1}{T}\int_{t_0}^{t_0+T} x\, dt$
^ By definition, the mean of the fluctuating quantity is zero( $\bar{u^\prime} = 0$ ).
^ Some authors prefer using $U$ instead of $\bar{u}$ for the mean term (since an overbar is used to represent a vector). Also it is common practice to represent the fluctuating term $u^\prime$ by $u$ , even though $u$ refers to the instantaneous value. This is possible because the two terms do not appear simultaneously in the same equation. to avoid confusion we will use $u, \bar{u}, \mbox{ and }u^\prime$ to represent the instantaneous, mean and fluctuating term.
^ The equations are expressed in tensor notation, which greatly simplifies the maths.
^

$\frac{\partial \left( \bar{u_i} + u_i^\prime \right)}{\partial x_i} = 0$

$\frac{\partial \left( \bar{u_i} + u_i^\prime\right)}{\partial t} + \left( \bar{u_j} + u_j^\prime\right) \frac{\partial \left( \bar{u_i} + u_i^\prime\right)}{\partial x_j} = \left( \bar{f_i} + f_i^\prime\right) - \frac{1}{\rho} \frac{\partial \left(\bar{p} + p^\prime\right)}{\partial x_i} + \nu \frac{\partial^2 \left( \bar{u_i} + u_i^\prime\right)}{\partial x_j \partial x_j}$

Time averaging these equations yields,

$\overline{\frac{\partial \left( \bar{u_i} + u_i^\prime \right)}{\partial x_i}} = 0$

$\overline{\frac{\partial \left( \bar{u_i} + u_i^\prime\right)}{\partial t}} + \overline{\left( \bar{u_j} + u_j^\prime\right) \frac{\partial \left( \bar{u_i} + u_i^\prime\right)}{\partial x_j}} = \overline{\left( \bar{f_i} + f_i^\prime\right)} - \frac{1}{\rho} \overline{\frac{\partial \left(\bar{p} + p^\prime\right)}{\partial x_i}} + \nu \overline{\frac{\partial^2 \left( \bar{u_i} + u_i^\prime\right)}{\partial x_j \partial x_j}}$

Note that the nonlinear terms (like $\overline{u_i u_i}$ ) can be simplified to, $\overline{u_i u_i} = \overline{\left( \bar{u_i} + u_i^\prime \right)\left( \bar{u_i} + u_i^\prime \right) } = \overline{\bar{u_i}\bar{u_i} + \bar{u_i}u_i^\prime + u_i^\prime\bar{u_i} + u_i^\prime u_i^\prime} = \bar{u_i}\bar{u_i} + \overline{u_i^\prime u_i^\prime}$
^ This follows from the mass conservation equation which gives,

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