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 a stationary, incompressible flow of Newtonian fluid, these equations can be written in Einstein notation 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 owing 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 owing to the mean pressure field, the viscous stresses, and apparent stress  \left( - \rho \overline{u_i^\prime u_j^\prime} \right) owing to the fluctuating velocity field, generally referred to as the Reynolds stress. This nonlinear Reynolds stress term requires additional modeling to close the RANS equation for solving, and has led to the creation of many different turbulence models.

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). (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}.


 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 ]

\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} 
+ 2\mu \bar{S_{ij}}
- \rho \overline{u_i^\prime u_j^\prime} \right ].

Additional Notes

(1) 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.

For this to be a well-defined term, the limit ( \bar{X} ) must be independent of the initial condition at t0. In the case of a chaotic dynamical system, which the equations under turbulent conditions are thought to be, this means that the system can have only one strange attractor, a result that has yet to be proved for the Navier-Stokes equations. However, assuming the limit exists (which it does for any bounded system, which fluid velocities certainly are), there exists some T such that integration from t0 to T is arbitrarily close to the average. This means that given transient data over a sufficiently large time, the average can be numerically computed within some small error. However, there is no analytical way to obtain an upper bound on T.

(2) By definition, the mean of the fluctuating quantity is zero( \bar{u^\prime}  = 0).

(3) 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.

(4) The equations are expressed in tensor notation, which greatly simplifies the maths.

(5) :  \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}

(6) 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