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A More Rigorous Test Problem For Viscous Hydrodynamics Codes

Alexander J. Dittmann, Geoffrey Ryan

Abstract

We advocate for a more stringent test problem for codes that aim to solve the equations of viscous hydrodynamics. Specifically, we discuss a nonuniform-density version of the common (uniform-density) Gaussian velocity shear test, where density gradients transverse to the direction of velocity shear cause the velocity profile to drift over time. By employing a nonunifom density, this test provides a test that the full viscous stress (and velocity shear) tensors are calculated correctly from the conserved variables, and checks the correctness of the fluxes and source terms calculated therefrom. In Appendix A, we present a detailed exposition of the Navier Stokes equations, particularly their fluxes and source terms in a variety of common coordinate systems.

A More Rigorous Test Problem For Viscous Hydrodynamics Codes

Abstract

We advocate for a more stringent test problem for codes that aim to solve the equations of viscous hydrodynamics. Specifically, we discuss a nonuniform-density version of the common (uniform-density) Gaussian velocity shear test, where density gradients transverse to the direction of velocity shear cause the velocity profile to drift over time. By employing a nonunifom density, this test provides a test that the full viscous stress (and velocity shear) tensors are calculated correctly from the conserved variables, and checks the correctness of the fluxes and source terms calculated therefrom. In Appendix A, we present a detailed exposition of the Navier Stokes equations, particularly their fluxes and source terms in a variety of common coordinate systems.
Paper Structure (8 sections, 35 equations, 2 figures)

This paper contains 8 sections, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A Cartesian Velocity Shear Test with Nonuniform Background Density
  3. Examples
  4. Summary
  5. Compressible Navier-Stokes In Curvilinear Coordinates
  6. Cartesian Coordinates
  7. Cylindrical Polar Coordinates
  8. Spherical Polar Coordinates

Figures (2)

  • Figure 1: Snapshots of $v_y$ at $t=0.02$ for simulations of a $\kappa=15$ test problem. The color scale is the same in each panel, such that red colors indicate velocities in excess of the maximum of the analytical solution.
  • Figure 2: The $L_1$ error of the $y-$velocity in each simulation, as a function of the cell size at $x=0,5,\,y=0$ (note that while the Disco and Cartesian simulations used uniformly sized cells, the cylindrical Athena++ simulations had $\Delta r \propto r\Delta \phi \propto r$). At $\kappa=0$, each code converges at second order. However, for $\kappa=15$, Disco v2016 fails to converge to the analytical solution.