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.

This model is appropriate for:

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}

The characteristic speed is thermal:

u^= v^_{th} .

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

struphy.propagators.push_eta.PushEta
struphy.propagators.push_vxb.PushVxB (if with_B0 is True)
struphy.propagators.push_vin_sph_pressure.PushVinSPHpressure (if with_p is True)
struphy.propagators.push_vin_viscous_potential.PushVinViscousPotential (if with_viscosity is True)

Create and initialize the viscous Euler SPH model:

from struphy.models import ViscousEulerSPH

model = ViscousEulerSPH()

model.euler_fluid.var