Physical Processes

\[\frac{\partial}{\partial t} (state)_i = \sum_j (process)_{ij}\]

Resolved Momentum Advection

Contribution to the instantaneous momentum equation:

\[\begin{aligned} \frac{\partial}{\partial t} u_i &= - u_j \frac{\partial u_i}{\partial x_j} + \dots \quad &\text{Convection Form} \\ &= - \frac{\partial u_i u_j}{\partial x_j} + \dots \quad &\text{Divergence Form} \\ &= - \frac{1}{2} \left( \frac{\partial u_i u_j}{\partial x_j} + u_j \frac{\partial u_i}{\partial x_j} \right) + \dots \quad &\text{Skew-Symmetric Form} \\ &= - u_j \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) - \frac{\partial}{\partial x_i} \frac{u_j u_j}{2} + \dots \quad &\text{Rotation Form} \end{aligned}\]

Contribution to the mean momentum equation:

\[\begin{aligned} \frac{\partial}{\partial t} \overline{ u_i } = - \overline{ u_j \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) } - \frac{\partial}{\partial x_i} \frac{\overline{u_j u_j}}{2} + \dots \end{aligned}\]

\[\begin{aligned} \frac{\partial}{\partial t} \overline{ u_i } = - \frac{\partial \overline{u_i}\,\overline{u_j}}{\partial x_j} - \frac{\partial \overline{u_i^\prime u_j^\prime}}{\partial x_j} + \dots \end{aligned}\]

Contribution to turbulent momentum equation:

\[\begin{aligned} \frac{\partial}{\partial t} u_i^\prime = - \frac{\partial}{\partial x_j} \left( u_i^\prime \overline{u_j} + \overline{u_i} u_j^\prime + u_i^\prime u_j^\prime - \overline{u_i^\prime u_j^\prime} \right) + \dots \end{aligned}\]

Contribution to the mean kinetic energy equation:

\[\begin{aligned} \frac{\partial}{\partial t} \frac{\overline{u_i}^2}{2} &= - \overline{u_i} \frac{\partial \overline{u_i}\,\overline{u_j}}{\partial x_j} - \overline{u_i} \frac{\partial \overline{u_i^\prime u_j^\prime}}{\partial x_j} + \dots \\ &= \frac{\partial}{\partial x_j} \left( - \frac{\overline{u_i}^2}{2} \overline{u_j} - \overline{u_i} \overline{u_i^\prime u_j^\prime} \right) + \overline{u_i^\prime u_j^\prime} \frac{\partial \overline{u_i}}{\partial x_j} + \dots \end{aligned}\]

Contribution to the turbulent kinetic energy equation:

\[\begin{aligned} \frac{\partial}{\partial t} \frac{\overline{u_i^\prime u_i^\prime}}{2} = \frac{\partial}{\partial x_j} \left( - \frac{\overline{u_i^\prime u_i^\prime}}{2} \overline{u_j} - \frac{1}{2} \overline{u_i^\prime u_i^\prime u_j^\prime} \right) - \overline{u_i^\prime u_j^\prime} \frac{\partial \overline{u_i}}{\partial x_j} + \dots \end{aligned}\]

Subgrid-Scale Momentum Advection

Contribution to the instantaneous momentum equation:

\[\frac{\partial}{\partial t} u_i = - \frac{\partial \tau_{ij}^{sgs}}{\partial x_j} + \dots\]

Contribution to the mean momentum equation:

\[\frac{\partial}{\partial t} \overline{u_i} = - \frac{\partial \overline{\tau_{ij}^{sgs}}}{\partial x_j} + \dots\]

Contribution to turbulent momentum equation:

\[\frac{\partial}{\partial t} u_i^\prime = - \frac{\partial {\tau_{ij}^{sgs}}^\prime}{\partial x_j} + \dots\]

Contribution to the mean kinetic energy equation:

\[\frac{\partial}{\partial t} \frac{\overline{u_i}^2}{2} = \frac{\partial}{\partial x_j} \left( - \overline{u_i} \overline{\tau_{ij}^{sgs}} \right) + \overline{\tau_{ij}^{sgs}} \frac{\partial \overline{u_i}}{\partial x_j} + \dots\]

Contribution to the turbulent kinetic energy equation (could use $S_{ij}^\prime$ for the last term, since the SGS-tensor is symmetric):

\[\frac{\partial}{\partial t} \frac{\overline{u_i^\prime u_i^\prime}}{2} = \frac{\partial}{\partial x_j} \left( - \overline{u_i^\prime {\tau_{ij}^{sgs}}^\prime} \right) + \overline{ {\tau_{ij}^{sgs}}^\prime \frac{\partial u_i^\prime}{\partial x_j} } + \dots\]

Molecular Diffusion

\[\frac{\partial}{\partial t} \varphi = D_\varphi \frac{\partial^2 \varphi}{\partial x_i^2}\]

Contribution to the instantaneous momentum equation:

\[\frac{\partial}{\partial t} u_i = \nu \frac{\partial^2 u_i}{\partial x_j^2}\]

Contribution to the mean momentum equation:

\[\frac{\partial}{\partial t} \overline{u_i} = \nu \frac{\partial^2 \overline{u_i}}{\partial x_j^2}\]

Contribution to the turbulent momentum equation:

\[\frac{\partial}{\partial t} u_i^\prime = \nu \frac{\partial^2 u_i^\prime}{\partial x_j^2}\]

Contribution to the mean kinetic energy equation:

\[\begin{aligned} \frac{\partial}{\partial t} \frac{\overline{u_i}^2}{2} &= \nu \frac{\partial^2}{\partial x_j^2} \frac{\overline{u_i}^2}{2} - \frac{\partial \overline{u_i}}{\partial x_j} \frac{\partial \overline{u_i}}{\partial x_j} \\ &= \frac{\partial}{\partial x_j} \left( 2 \nu \overline{u_i} \overline{S_{ij}} \right) - 2 \nu \overline{S_{ij}} \, \overline{S_{ij}} \end{aligned}\]

Contribution to the turbulent kinetic energy equation:

\[\begin{aligned} \frac{\partial}{\partial t} \frac{\overline{u_i^\prime u_i^\prime}}{2} &= \nu \frac{\partial^2}{\partial x_j^2} \frac{\overline{u_i^\prime u_i^\prime}}{2} \underbrace{ - \nu \overline{ \frac{\partial u_i^\prime}{\partial x_j} \frac{\partial u_i^\prime}{\partial x_j} } }_{\text{pseudo-dissipation}} \\ &= \nu \frac{\partial^2}{\partial x_j^2} \frac{\overline{u_i^\prime u_i^\prime}}{2} + \nu \frac{\partial^2 \overline{u_i^\prime u_j^\prime}}{\partial x_i \partial x_j} - 2 \nu \overline{S_{ij}^\prime S_{ij}^\prime} \\ &= \frac{\partial}{\partial x_j} \left( 2 \nu \overline{u_i^\prime S_{ij}^\prime} \right) \underbrace{ - 2 \nu \overline{S_{ij}^\prime S_{ij}^\prime} }_{\text{dissipation}} \end{aligned}\]

Mass Conservation & Pressure

\[\begin{aligned} \frac{\partial u_i}{\partial x_i} &= 0 \\ \frac{\partial}{\partial t} u_i &= - \frac{\partial \phi}{\partial x_i} + \dots \\ \frac{\partial^2 \phi}{\partial x_i^2} &= \frac{\partial}{\partial x_i} \dots \end{aligned}\]

Contribution to the instantaneous momentum equation:

\[\frac{\partial}{\partial t} u_i = - \frac{\partial \phi}{\partial x_i} + \dots\]

Contribution to the mean momentum equation:

\[\frac{\partial}{\partial t} \overline{u_i} = - \frac{\partial \overline{\phi}}{\partial x_i} + \dots\]

Contribution to the turbulent momentum equation:

\[\frac{\partial}{\partial t} u_i^\prime = - \frac{\partial \phi^\prime}{\partial x_i} + \dots\]

Contribution to the mean kinetic energy equation:

\[\frac{\partial}{\partial t} \frac{\overline{u_i}^2}{2} = - \frac{\partial \overline{u_i} \overline{\phi}}{\partial x_i} + \dots\]

Contribution to the turbulent kinetic energy equation:

\[\frac{\partial}{\partial t} \frac{\overline{u_i^\prime u_i^\prime}}{2} = - \frac{\partial \overline{u_i^\prime \phi^\prime}}{\partial x_i} + \dots\]

General Notes

• in products of the form $\overline{a^\prime b^\prime}$, one of the terms could be replaced with the instantaneous value instead of the turbulent part only