Fluid model

VariationalPressurelessFluid

Pressure-less fluid equations discretized with a variational method.

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Overview

This is the pressureless limit of the variational fluid hierarchy. It

is intended as a reduced benchmark and as a simple transport model with conservative density and momentum updates.

Use cases

This model is appropriate for:

pressureless compressible benchmarks
testing the minimal variational fluid update chain
reduced transport problems without thermodynamics

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) = 0

Normalization

The flow speed is normalized with the Alfvén speed:

u^= v^_{A} .

Discretization

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

Diagnostics

The following scalars are tracked during simulation:

Kinetic energy: kinetic_energy

Examples

Pressureless density transport

A density ripple travels once around a periodic box at constant speed. VariationalPressurelessFluid has no pressure force, so the exact nonlinear solution keeps the profile unchanged. Compare its shape, mass and kinetic energy. The initial data are $\rho(x,0)=1+0.2\cos x,\qquad \mathbf{u}(x,0)=(0.5,0,0),$ giving $\rho(x,t)=1+0.2\cos(x-0.5t)$ on $0\leq x<2\pi$ .