Fluid model

ViscousEulerSPH

Euler equations with viscosity discretized with smoothed particle hydrodynamics (SPH).

Overview

ViscousEulerSPH is the particle-based SPH fluid model with optional

pressure and viscosity contributions. It is intended for meshfree fluid experiments rather than FEEC-based field simulations.

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:

Continuity:

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

Momentum:

ρ (\partial_{t} u + u \cdot \nabla u) = - \nabla (ρ^{2} \frac{\partial U ( ρ , S )}{\partial ρ}) - \nabla \cdot π

Entropy:

\partial_{t} S + u \cdot \nabla S = 0

where $S$ denotes the entropy per unit mass and $π$ 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.

The internal energy per unit mass can be defined in two ways:

isothermal : U (ρ, S) = κ (S) lo g ρ

polytropic : U (ρ, S) = κ (S) \frac{ρ ^{γ - 1}}{γ - 1}

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)
PushVinSPHpressure (if with_p is True)
PushVinViscousPotential (if with_viscosity is True)

Diagnostics

The following scalars are tracked during simulation:

SPH kinetic energy: en_kin

Examples

Dam break

A dense fluid column collapses under gravity in a closed box. Smoothed particle hydrodynamics follows the collapse, the wave across the box and the settling of the fluid, using markers alone. The column starts at rest with $n(x,y,0)=\begin{cases}0.1,&0\le x<0.25,\ 0\le y<1,\\0,&\text{elsewhere},\end{cases}$ in the unit square, under gravity $\mathbf{g}=(0,-10,0)$ .

Gas expansion into vacuum

An isothermal gas initially occupies the left part of a one-dimensional box and expands into the vacuum. Smoothed particle hydrodynamics follows the rarefaction wave and the gas streaming into the vacuum, and the result is compared with the exact self-similar solution. The initial state is $n(x,0)=\begin{cases}1,&0\le x<1.5,\\0,&1.5<x\le8,\end{cases}\qquad u_x(x,0)=0,$ with isothermal sound speed $c_s=\sqrt{\kappa}=1$ .

SPH velocity diffusion

A sinusoidal velocity field diffuses on a periodic interval. The binned SPH current is compared with the exact viscous decay of the mode. The initial state is $u_x(x,0)=0.5\sin(2\pi x),\qquad n(x,0)=1,$ on $0\le x<1$ , with viscosity $\mu=0.05$ . The reference decay rate is $\Gamma=\tfrac43\mu(2\pi)^2$ .