Spectral Particle Methods

2018 SPHeric meeting

Galway, IE

Daniel Duque & Javier Calderon-Sanchez
CEHINAV, UPM, Spain
daniel.duque@upm.es

Why Fourier

$$\frac{d \mathbf{u}}{d t} =- \nabla p + \nu \nabla^2 \mathbf{u} + \mathbf{g}$$ Pressure: $$\nabla\cdot\mathbf{u}=0$$

$$\frac{d \mathbf{u}_\mathbf{q}}{d t} =- i \mathbf{q} p_\mathbf{q} -\nu q^2 \mathbf{u}_\mathbf{q} +\mathbf{g}_\mathbf{q}$$ $$\mathbf{q} \cdot \mathbf{u}_\mathbf{q} = 0$$

... but, of course
$d \mathbf{u}/d t$ is a total derivative (hence, use particles)

$\mathbf{u}_\mathbf{q}$ should be computed on a mesh (hence, use a mesh)

A compromise: use both, Particle-in-Cell style (also, pFEM v2.0)

Procedure: onto mesh and back

TG errors

doi: 10.1142/S021987621850130X

Procedure: onto mesh and back

Zalesak disk test Zalesak disk test
Zalesak disk test Zalesak disk test

Quick check: TG vortices

TG errors

TG vortices go turbulent

Kolmogorov force

Kolmogorov force

Kolmogorov force

Kolmogorov force

Onset of 2D turbulence


Original idea: P. Branson et al. CC BY-NC 4.0

Onset of 2D turbulence


Original idea: P. Branson et al. CC BY-NC 4.0. Ending: J.L. Cercos-Pita

2D turbulence: velocity

v pdf

2D turbulence: acceleration

a pdf

2D turbulence: power spectrum

power spectrum

Membranes: 2D fluids

Biological membrane

Landau double-well

Double well potential and force

Equilibrium profile: $\mu=0$

profile

Membrane segregation

If interested

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