Fluid model

IncompressibleNavierStokesSPH

Incompressible Navier-Stokes equations discretized with smoothed particle hydrodynamics (SPH).

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Overview

IncompressibleNavierStokesSPH is the particle-based SPH fluid model...

Use cases

This model is appropriate for:

SPH verification with pressure and viscosity
meshfree compressible-fluid experiments
testing particle-based viscosity and pressure pushers

Governing equations

PDEs solved by model:

Incompressible Navier-Stokes:

ρ (\partial_{t} u + u \cdot \nabla u) = - \nabla p - \nabla \cdot σ \nabla \cdot u = 0 .

where $σ$ is the viscous stress tensor.

The viscous stress tensor for a Newtonian fluid is given by:

σ = - μ (\nabla u + (\nabla u)^{T} - \frac{2}{3} (\nabla \cdot u) I)

where $μ$ is the dynamic (shear) viscosity and $I$ is the identity tensor.

Normalization

The characteristic speed is thermal:

u^= v^_{th} .

Discretization

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

PushEta
PushVxB (if with_B0 is True)
PushVinViscousPotential (if with_viscosity is True)
PoissonSolve
PushVinForceField

Diagnostics

The following scalars are tracked during simulation:

SPH kinetic energy: en_kin

Examples

Incompressible shear relaxation

A viscous fluid between no-slip walls starts with a shear flow and a compressive wave. Incompressible SPH with a pressure projection removes the wave at once and lets the shear decay viscously, at the exact rate. Before the pressure projection, $u_x(x,y,0)=0.5\sin(\pi y)+0.2\sin(2\pi x),\qquad u_y(x,y,0)=0,$ with $n=1$ , viscosity $\nu=0.1$ , and no-slip walls at $y=0,1$ .