Fluid model

PressureLessSPH

Particle discretization of pressureless Euler flow with external forcing.

Overview

PressureLessSPH is a meshfree particle model for a pressureless fluid.

It is primarily useful as a simple SPH benchmark or as a reduced particle transport model in a prescribed force field or potential.

Use cases

This model is appropriate for:

meshfree pressureless-flow benchmarks
straight-line particle transport with external forcing
testing SPH particle loading and diagnostics

Governing equations

PDEs solved by model:

Continuity:

\partial_{t} ρ + \nabla \cdot (ρ u) = 0

Momentum:

\partial_{t} (ρ u) + \nabla \cdot (ρ u \otimes u) = F

where $F$ is an external force.

Normalization

Velocity and field normalizations:

u^= 1 m/s, F^= \frac{m n ^ u ^}{t ^} .

Discretization

Time integration is performed by the following propagators (in sequence):

Diagnostics

The following scalars are tracked during simulation:

SPH kinetic energy: en_kin

Examples

SPH in a Beltrami force field

Pressureless SPH markers circulate through a stationary cellular flow driven by a prescribed Beltrami potential. The computed marker velocity is compared with the exact Eulerian velocity field, while conservation of each marker's kinetic-plus-potential energy checks the orbit integration. Initially $n=1$ and $\mathbf{u}(x,y,0)=(-\cos(\pi x)\sin(\pi y),\sin(\pi x)\cos(\pi y),0).$ The prescribed potential is $V(x,y)=\tfrac12[\sin^2(\pi x)+\sin^2(\pi y)]$ on $[-0.5,0.5]^2$ .

Zel'dovich caustic

A sinusoidal velocity field makes a pressureless gas collapse: the density steepens until it diverges at a caustic, after which the fluid streams through itself. Smoothed particle hydrodynamics follows the collapse and the multi-stream phase space, and is compared with the exact solution of the Lagrangian map. Initially $n(x,0)=1,\qquad u_x(x,0)=0.5\sin(2\pi x),$ on $0\le x<1$ . The exact particle map is $x(q,t)=q+0.5t\sin(2\pi q)$ (modulo one), with first caustic at $t_c=1/\pi$ .