Navier Stokes Vorticity Model

This is a discretization of the incompressible Navier Stokes equations using the Discrete Exterior Calculus.

The formulations are based on those given by Mohamed, Hirani, Samtaney (in turn from Marsden, Ratiu, Abraham).

However, different choices in discretization are chosen for purposes of brevity, to demonstrate novel discretizations of certain operators, and to demonstrate the automated Decapodes workflow.

The full code that generated these results is available in a julia script.

We give the vorticity formulation of the inviscid incompressible Navier-Stokes momentum equation as follows:

eq11_inviscid_poisson = @decapode begin
  d𝐮::DualForm2
  𝐮::DualForm1
  ψ::Form0

  ψ == Δ⁻¹(⋆(d𝐮))
  𝐮 == ⋆(d(ψ))

  ∂ₜ(d𝐮) ==  (-1) * ∘(♭♯, ⋆₁, d̃₁)(∧ᵈᵖ₁₀(𝐮, ⋆(d𝐮)))
end

Our initial conditions here are Point vortices:

function point_vortex(pnt::Point3D, cntr::Point3D, p::PointVortexParams)
  gcd = great_circle_dist(pnt,cntr)
  p.τ / (cosh(3gcd/p.a)^2)
end

Based on the configuration, you can see different results that match the expected solutions from the literature.

Here is one set of results from using the inviscid Poisson formulation:

We can visualize the distribution of vorticity at the $\theta = 0.4$ latitude. The difference between the distributions at $t=0$ and $t=12$ is accumulated error.

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