Fluid model

VariationalCompressibleFluid

Fully compressible fluid equations discretized with a variational method.

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Overview

VariationalCompressibleFluid is the entropy-based compressible fluid

model in the variational family. It is the natural non-magnetic counterpart to the full variational MHD models.

Use cases

This model is appropriate for:

compressible fluid benchmarks with entropy transport
testing the variational fluid propagator stack
non-magnetic hydrodynamics with conservative 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) + ρ \nabla \frac{( ρ U ( ρ , s ))}{\partial ρ} + s \nabla \frac{( ρ U ( ρ , s ))}{\partial s} = 0

Entropy:

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

where the internal energy per unit mass is $U (ρ) = ρ^{γ - 1} exp (s / ρ)$ .

Normalization

The model uses Alfvén-speed scaling for the flow variables together

with separate units for internal energy and entropy:

u^= v^_{A}, U^= K, s^= ρ^C_{v} .

Discretization

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

Diagnostics

The following scalars are tracked during simulation:

Kinetic energy: en_U
Thermodynamic energy: en_thermo
Total energy: en_tot

Examples

Acoustic pulse

A Gaussian bump in the density of a compressible gas splits into two half-height pulses that travel in opposite directions at the speed of sound. Struphy's variational discretization follows them around the periodic box, conserving energy, and they agree with d'Alembert's solution. The periodic initial density uses ten Gaussian-weighted modes: $n(x,0)=1+\sum_{j=1}^{10}a_j\cos(jx),\qquad a_j=\frac{0.02\sqrt{2\pi}}{2\pi}e^{-j^2/8},$ with $\mathbf{u}(x,0)=0$ on $0\le x<2\pi$ . The pulse has width $\sigma=0.5$ and sound speed $c_s=1$ .