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.
Discretization
To compute the non-linear advection term, the velocity components $u_i$ as well as the vorticity components $ω_i$ are transformed to the physical domain.
The contributions of the advection term become
\[\begin{aligned} \frac{∂}{∂t} u₁(ζ_C) &= … + u₂(ζ_C) \, ω₃(ζ_C) - \frac{u₃(ζ_C⁻) \, ω₂(ζ_C⁻) + u₃(ζ_C⁺) \, ω₂(ζ_C⁺)}{2} \\ \frac{∂}{∂t} u₂(ζ_C) &= … + \frac{u₃(ζ_C⁻) \, ω₁(ζ_C⁻) + u₃(ζ_C⁺) \, ω₁(ζ_C⁺)}{2} - u₁(ζ_C) \, ω₃(ζ_C) \\ \frac{∂}{∂t} u₃(ζ_I) &= … + \frac{u₁(ζ_I⁻) + u₁(ζ_I⁺)}{2} \, ω₂(ζ_I) - \frac{u₂(ζ_I⁻) + u₂(ζ_I⁺)}{2} \, ω₁(ζ_I) \end{aligned}\]
where the values that are defined at the “wrong” set of nodes have been interpolated.
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:
\[\begin{aligned} \frac{\partial}{\partial t} \frac{\overline{u_i^\prime u_i^\prime}}{2} = … + \frac{\partial}{\partial x_j} \left( - \frac{\overline{u_i^\prime u_i^\prime}}{2} \overline{u_j} - \frac{1}{2} \overline{u_i^\prime u_i^\prime u_j^\prime} \right) \underbrace{- \overline{u_i^\prime u_j^\prime} \frac{\partial \overline{u_i}}{\partial x_j}}_{\text{TKE production}} \end{aligned}\]
Discrete Conservation Properties
The momentum advection term preserves the total kinetic energy to machine precision in the absence of grid stretching and time-integration errors.
With the definition
\[2K\!E = \sum_{ζ_C} \left( u₁(ζ_C)² + u₂(ζ_C)² + \frac{u₃(ζ_C⁻)² + u₃(ζ_C⁺)²}{2} \right) = \sum_{ζ_C} u₁(ζ_C)² + \sum_{ζ_C} u₂(ζ_C)² + \sum_{ζ_I} u₃(ζ_I)²\]
we need
\[\frac{∂}{∂t} K\!E = \sum_{ζ_C} u₁ \frac{∂u₁}{∂t} + \sum_{ζ_C} u₂ \frac{∂u₂}{∂t} + \sum_{ζ_I} u₃ \frac{∂u₃}{∂t} = 0\]
for discrete energy conservation (to machine precision) in the absence of time-differencing errors. If grid stretching is used, we need to include the vertical grid spacing $Δζ \, ∂x₃/∂ζ$ in the sum. We get slightly different definitions depending on whether we evaluate the grid spacing at $ζ_C$ or $ζ_I$ for the $u₃²$-contribution.
Refer to Mansour, Moin, Reynolds & Ferziger 1979 for a general discussion of discrete conservation properties and how to derive them.
Analytically, we can easily show that the velocity–vorticity cross-product in the rotational form of the advection term conserves energy since
\[\frac{∂}{∂t} \frac{u_i^2}{2} = … − u_i u_j \left(\frac{∂u_i}{∂x_j} − \frac{∂u_j}{∂x_i}\right)\]
results in a net zero contribution as each term in the sum over $i$ and $j$ is either zero or appears with both positive and negative sign. The contribution is therefore zero at every point (i.e. locally) and therefore globally too.
For the discretized term, this is less obvious as the $ij$ and $ji$ contributions can have different discretization errors. With the numerical methods employed here, the global kinetic energy is conserved to machine precision as long as no grid stretching is used.
To demonstrate the net zero contribution on the staggered grid, we can rely on
\[\begin{aligned} &\sum_{ζ_I} u₃(ζ_I) \left( \frac{u₁(ζ_I⁻) + u₁(ζ_I⁺)}{2} \, ω₂(ζ_I) - \frac{u₂(ζ_I⁻) + u₂(ζ_I⁺)}{2} \, ω₁(ζ_I) \right) = \\ &\sum_{ζ_C} u₁(ζ_C) \frac{u₃(ζ_C⁻) \, ω₂(ζ_C⁻) + u₃(ζ_C⁺) \, ω₂(ζ_C⁺)}{2} - \sum_{ζ_C} u₂(ζ_C) \frac{u₃(ζ_C⁻) \, ω₁(ζ_C⁻) + u₃(ζ_C⁺) \, ω₁(ζ_C⁺)}{2} \end{aligned}\]
to show that the contributions still cancel in the global sum (assuming $u_3$ is zero at the boundaries).
Note that we also rely on results being unaffected by the order in which triple-products are computed, e.g. $u₁(ω₂u₃) = (u₁ω₂)u₃$. Due to truncated wavenumbers this is not true in general, but it does hold for the horizontal average, i.e. the $(0, 0)$ wavenumber pair, since the truncated wavenumbers of the first product are too large to contribute to the mean of the second product. The discrete energy conservation therefore holds independent of whether the advection term is dealiased or not.
In the presence of grid stretching, the $∂x₃/∂ζ$-factors do not cancel and the energy conservation is not exact.