Pressure/Continuity
This page describes a “process”. Refer to Physical Processes for general information about processes and their implementation.
The pressure solver enforces the continuity equation by subtracting the gradient of a pressure-like variable $ϕ$ from the momentum equation, i.e.
\[\frac{∂u_i}{∂t} = … - \frac{∂ϕ}{∂x_i} \quad \text{such that} \quad \frac{∂u_i}{∂x_i} = 0\]
with the additional constraint that $\int\int ϕ(x₁, x₂, x₃(0)) \, \mathrm{d}x₁ \mathrm{d}x₂ = 0$ to fix the absolute value of $ϕ$.
After discretization, the computation becomes
\[\begin{aligned} f_i(\hat{u}_1^{κ₁κ₂ζ_C}) &= - ∂₁(κ₁) \, \hat{ϕ}^{κ₁κ₂ζ_C} \\ f_i(\hat{u}_2^{κ₁κ₂ζ_C}) &= - ∂₂(κ₂) \, \hat{ϕ}^{κ₁κ₂ζ_C} \\ f_i(\hat{u}_3^{κ₁κ₂ζ_I}) &= - ∂₃(ζ_I) \left(\hat{ϕ}^{κ₁κ₂ζ_I^+} - \hat{ϕ}^{κ₁κ₂ζ_I^−}\right) \end{aligned}\]
such that
\[∂₁(κ₁) \, \hat{u}_1^{κ₁κ₂ζ_C} + ∂₁(κ₁) \, \hat{u}_2^{κ₁κ₂ζ_C} + ∂₃(ζ_C) \left( \hat{u}₃^{κ₁κ₂ζ_C^+} - \hat{u}₃^{κ₁κ₂ζ_C^−} \right) = 0\]
with the constraint that $\hat{ϕ}⁰⁰⁰=0$.
The process is implemented as a Projection. It relies on boundary conditions for $u₃$.
BoundaryLayerDynamics.Processes.Pressure — TypePressure(batch_size = 64)Transport of momentum by a pressure-like variable that enforces a divergence-free velocity field.
Arguments
batch_size::Int: The number of wavenumber pairs that are included in each batch of the tri-diagonal solver. The batching serves to stagger the computation such that different MPI ranks can work on different batches at the same time.
Contributions to Budget Equations
Contribution to the instantaneous momentum equation:
\[\frac{\partial}{\partial t} u_i = … - \frac{\partial \phi}{\partial x_i}\]
Contribution to the mean momentum equation:
\[\frac{\partial}{\partial t} \overline{u_i} = … - \frac{\partial \overline{\phi}}{\partial x_i}\]
Contribution to the turbulent momentum equation:
\[\frac{\partial}{\partial t} u_i^\prime = … - \frac{\partial \phi^\prime}{\partial x_i}\]
Contribution to the mean kinetic energy equation:
\[\frac{\partial}{\partial t} \frac{\overline{u_i}^2}{2} = … - \frac{\partial \overline{u_i} \overline{\phi}}{\partial x_i}\]
Contribution to the turbulent kinetic energy equation:
\[\frac{\partial}{\partial t} \frac{\overline{u_i^\prime u_i^\prime}}{2} = … - \frac{\partial \overline{u_i^\prime \phi^\prime}}{\partial x_i}\]