# Reynolds Stress - Reynolds Averaging of The Navier–Stokes Equations

Reynolds Averaging of The Navier–Stokes Equations

For instance, for an incompressible, viscous, Newtonian fluid, the continuity and momentum equations—the incompressible Navier–Stokes equations—can be written as

and

$rho frac{Du_i}{Dt} = -frac{partial p}{partial x_i} + mu left( frac{partial^2 u_i}{partial x_j partial x_j} right),$

where is the Lagrangian derivative or the substantial derivative,

Defining the flow variables above with a time-averaged component and a fluctuating component, the continuity and momentum equations become

and

Examining one of the terms on the left hand side of the momentum equation, it is seen that

$left( overline{u_j} + u_j' right) frac{partial left( overline{u_i} + u_i' right)}{partial x_j} = frac{partial left( overline{u_i} + u_i' right) left( overline{u_j} + u_j' right)}{partial x_j} - left( overline{u_i} + u_i' right) frac{partial left( overline{u_j} + u_j' right)}{partial x_j},$

where the last term on the right hand side vanishes as a result of the continuity equation. Accordingly, the momentum equation becomes

$rho left = -frac{partial left( bar{p} + p' right) }{partial x_i} + mu left.$

Now the continuity and momentum equations will be averaged. The ensemble rules of averaging need to be employed, keeping in mind that the average of products of fluctuating quantities will not in general vanish. After averaging, the continuity and momentum equations become

and

$rho left = -frac{partial bar{p}}{partial x_i} + mu frac{partial^2 overline{u_i}}{partial x_j partial x_j}.$

Using the chain rule on one of the terms of the left hand side, it is revealed that

where the last term on the right hand side vanishes as a result of the averaged continuity equation. The averaged momentum equation now becomes, after a rearrangement:

where the Reynolds stresses, are collected with the viscous normal and shear stress terms, .