Transport Velocity Formulation for EDAC

[LasNikas]

The Transport Velocity Formulation (TVF) was first introduced by Adami for the standard WCSPH. Ramachandran applied the TVF for the EDAC method. He suggested to use the standard EDAC for external flow problems and the EDAC TVF for internal flow problems.
Basically the transport velocity is obtained from the momentum velocity at each time step using

The challenge here was to swap the integrator $\tilde{u}$ to $u$. Modifying the u vector is only allowed with a callback or a step_limiter!. So, I basically switch v at each time step like

v[ndims(system) + i, particle] = v[i, particle]

To verify this method, I added a few examples.

Channel flow through periodic array of cylinder (Adami 2013)

I'm not really sure if I'm using the right color map. But what's interesting is that the particle arrangement of TP implementation is way better than the one from Adami or Ramachandran.

TP:

Adami:

Ramachandran:

Taylor Green vortex (Ramachandran 2019)

The solid lines (red and lila) are TP's errors. Note, that the error of TP's standard edac scheme is (conspicuously) smaller than the error of the Ramachandran paper.

Lid-driven cavity (Adami 2013)

The following plots show the steady velocity field on the horizontal and vertical center line.
Note, I extracted the Ghia data from this figure. So, it is quite inaccurate and not really comparable.

Reynolds number 100

Reynolds number 1000

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src/schemes/fluid/entropically_damped_sph/rhs.jl	0.00%	32 Missing ⚠️
src/schemes/fluid/transport_velocity.jl	0.00%	20 Missing ⚠️

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[svchb]

I think we should add these types of validation simulation setups including the plots to maybe a separate folder so that these can be reused. Using trixi_include to use the example code in these setups. Might also be a good start for tutorials.