# Reynolds-averaged Navier-Stokes (RANS) equations #turbulence #derivation #in-progress We start with the incompressible Navier-Stokes (NS) equations:![[Navier-Stokes equations#^ec02dc]] Incompressibility: ![[Incompressibility#^768316]] Then, we perform a [[Reynolds decomposition]] of the velocity $u$ and pressure $p$, and assume that the flow is [[Stationary random functions|stationary]]. $u_i = \overline{u}_i + u_i', \quad p = \overline{p} + p', \quad \frac{\partial \overline{u}_i}{\partial t} = 0$ Substituting these expressions in our original NS equations and averaging, we obtain $ \begin{align} &\overline{\frac{\partial}{\partial t}\left(\overline{u_i} + u_i'\right)} + \overline{\left(\overline{u_j} + u_j'\right)\frac{\partial}{\partial x_j}\left(\overline{u_i} + u_i'\right)} = -\overline{\frac{1}{\rho}\frac{\partial}{\partial x_i}\left(\overline{p} + p'\right)} + \overline{\nu\frac{\partial^2}{\partial x_j \partial x_j}\left(\overline{u_i} + u_i'\right)}\\ \implies&\cancel{\frac{\partial\overline{u_i}}{\partial t}} + \frac{\partial\cancel{\overline{u_i'}}}{\partial t} + \overline{u_j}\frac{\partial\overline{u_i}}{\partial x_j} + \overline{u_j}\frac{\partial\cancel{\overline{u_i'}}}{\partial x_j} + \cancel{\overline{u_j'}}\frac{\partial\overline{u_i}}{\partial x_j} + \overline{u_j'\frac{\partial u_i'}{\partial x_j}} = \\&-\frac{1}{\rho}\frac{\partial \overline{p}}{\partial x_i} + \frac{1}{\rho}\frac{\partial \cancel{\overline{p'}}}{\partial x_i} + \nu\frac{\partial^2 \overline{u_i}}{\partial x_j\partial x_j} + \nu\frac{\partial^2 \cancel{\overline{u_i'}}}{\partial x_j\partial x_j} \end{align} $ In the above equations, we have used the facts that the flow is stationary, so the mean flow is time-independent, and that the average of any fluctuating quantity is zero. We are left with $ \overline{u_j}\frac{\partial \overline{u_i}}{\partial x_j} + \overline{u_j'\frac{\partial u_i'}{\partial x_j}} = -\frac{1}{\rho}\frac{\partial \overline{p}}{\partial x_i} + \nu\frac{\partial^2 \overline{u_i}}{\partial x_j\partial x_j} $ [[MHD RANS equations|Generalization of the RANS equations to MHD]] ## Explainer video (under construction) ![[slide1.mp4]]