Momentum Advection
This page describes a “process”. Refer to Physical Processes for general information about processes and their implementation.
The momentum advection process represents the advective transport of $u_i$ (i.e. momentum normalized by density) in rotation form, i.e.
\[\frac{∂u_i}{∂t} = … − u_j \left(\frac{∂u_i}{∂x_j} − \frac{∂u_j}{∂x_i}\right) − \frac{∂}{∂x_i} \frac{u_j u_j}{2} \,.\]
Only the first term is computed here while the second term is assumed to become part of a modified pressure handled separately.
The process is discretized by computing the vorticity in the frequency domain, then computing the velocity–vorticity product in the physical domain, and finally transforming the result back to a frequency-domain representation. If other processes (e.g. a StaticSmagorinskyModel) require physical-domain velocity gradients at the same resolution, the vorticity is computed from those to minimize the number of transforms.
If the level of dealiasing is set to at least :quadratic, the physical-domain products are equivalent to frequency-domain convolution and no additional aliasing errors are introduced.
The process relies on boundary conditions for $u₃$.
BoundaryLayerDynamics.Processes.MomentumAdvection — TypeMomentumAdvection(; dealiasing = :quadratic)Advective transport of momentum ($u_i$, i.e. normalized by density) in rotation form. Only the velocity–vorticity products are computed, while the gradient of kinetic energy is assumed to be handled as part of a modified pressure term.
Keywords
dealiasing: The level of dealiasing, determining the physical-domain resolution at which multiplications are performed. The default:quadraticsetting relies on the “3/2-rule” to avoid aliasing errors for the products that are computed as part of the momentum advection term. Alternatively,dealiasingcan be set tonothingfor a physical-domain resolution that matches the frequency-domain resolution (rounded up to an even value), or to a tuple of two integers to set the resolution manually.
Contributions to Budget Equations
Contribution to the instantaneous momentum equation:
\[\begin{aligned} \frac{\partial}{\partial t} u_i &= … - u_j \frac{\partial u_i}{\partial x_j} \quad &\text{Convection Form} \\ &= … - \frac{\partial u_i u_j}{\partial x_j} \quad &\text{Divergence Form} \\ &= … - \frac{1}{2} \left( \frac{\partial u_i u_j}{\partial x_j} + u_j \frac{\partial u_i}{\partial x_j} \right) \quad &\text{Skew-Symmetric Form} \\ &= … - u_j \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) - \frac{\partial}{\partial x_i} \frac{u_j u_j}{2} \quad &\text{Rotation Form} \end{aligned}\]
Contribution to the mean momentum equation:
\[\begin{aligned} \frac{\partial}{\partial t} \overline{ u_i } = … - \overline{ u_j \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) } - \frac{\partial}{\partial x_i} \frac{\overline{u_j u_j}}{2} \end{aligned}\]
\[\begin{aligned} \frac{\partial}{\partial t} \overline{ u_i } = … + \frac{\partial}{\partial x_j} \left( − \overline{u_i}\,\overline{u_j} − \overline{u_i^\prime u_j^\prime} \right) \end{aligned}\]
\[\begin{aligned} \frac{\partial}{\partial t} \overline{ u_i } = … - \frac{\partial \overline{u_i}\,\overline{u_j}}{\partial x_j} - \frac{\partial \overline{u_i^\prime u_j^\prime}}{\partial x_j} \end{aligned}\]
Contribution to turbulent momentum equation:
\[\begin{aligned} \frac{\partial}{\partial t} u_i^\prime = … - \frac{\partial}{\partial x_j} \left( u_i^\prime \overline{u_j} + \overline{u_i} u_j^\prime + u_i^\prime u_j^\prime - \overline{u_i^\prime u_j^\prime} \right) \end{aligned}\]
Contribution to the mean kinetic energy equation:
\[\begin{aligned} \frac{\partial}{\partial t} \frac{\overline{u_i}^2}{2} &= … - \overline{u_i} \frac{\partial \overline{u_i}\,\overline{u_j}}{\partial x_j} - \overline{u_i} \frac{\partial \overline{u_i^\prime u_j^\prime}}{\partial x_j} \\ &= … \frac{\partial}{\partial x_j} \left( - \frac{\overline{u_i}^2}{2} \overline{u_j} - \overline{u_i} \overline{u_i^\prime u_j^\prime} \right) \underbrace{+ \overline{u_i^\prime u_j^\prime} \frac{\partial \overline{u_i}}{\partial x_j}}_{\text{TKE production}} \end{aligned}\]
Contribution to the turbulent kinetic energy equation:
\[\begin{aligned} \frac{\partial}{\partial t} \frac{\overline{u_i^\prime u_i^\prime}}{2} = … + \frac{\partial}{\partial x_j} \left( - \frac{\overline{u_i^\prime u_i^\prime}}{2} \overline{u_j} - \frac{1}{2} \overline{u_i^\prime u_i^\prime u_j^\prime} \right) \underbrace{- \overline{u_i^\prime u_j^\prime} \frac{\partial \overline{u_i}}{\partial x_j}}_{\text{TKE production}} \end{aligned}\]