Linear Euler 2D - Spherical Sound Wave Example

This tutorial will walk you through using an example program that uses the LinearEuler2D class to run a simulation with the linear Euler equations for an ideal gas in 2-D. This example is configured using the built in structured mesh generator with no-normal-flow boundary conditions on all domain boundaries.

Problem statement

Equations Solved

In this example, we are solving the linear Euler equations in 2-D for an ideal gas, given by

\[ \vec{s}_t + \nabla \cdot \overleftrightarrow{f} = \vec{q} \]

where

\[ \vec{s} = \begin{pmatrix} \rho \\ u \\ v \\ p \end{pmatrix} \]

and

\[ \overleftrightarrow{f} = \begin{pmatrix} \rho_0(u \hat{x} + v \hat{y}) \\ p \hat{x} \\ p \hat{y} \\ \rho_0c^2(u \hat{x} + v \hat{y}) \end{pmatrix} \]

\[ \vec{q} = \vec{0} \]

The variables are defined as follows

• \(\rho\) is a density anomaly referenced to the density \(\rho_0\)
• \(u\) and \(v\) are the \(x\) and \(y\) components of the fluid velocity (respectively)
• \(p\) is the pressure
• \(c\) is the (constant) speed of sound.

Model Domain

The physical domain is defined by \(\vec{x} \in [0, 1]\times[0,1]\). We use the StructuredMesh routine to create a domain with 20 × 20 elements that are dimensioned 0.05 × 0.05 . All model boundaries are all tagged with the SELF_BC_NONORMALFLOW flag to implement no-normal-flow boundary conditions.

Within each element, all variables are approximated by a Lagrange interpolating polynomial of degree 7. The interpolation knots are the Legendre-Gauss points.

Initial Conditions

The initial condition is set using

\[ \begin{pmatrix} ρ \\ u \\ v \\ P \end{pmatrix} = \begin{pmatrix} \frac{1}{c^2} \\ 0 \\ 0 \\ 1 \end{pmatrix} \bar{p} e^{-\left( \frac{ (x-x_0)^2 + (y-y_0)^2 }{L^2} \right)} \]

The parameters used in the exact solution are as follows :

• \(\bar{p} = 10^{-4}\) is the amplitude of the sound wave
• \(x_0 = y_0 = 0.5\) defines the center of the initial sound wave
• \(L = \frac{0.06}{\sqrt{\ln{2}}}\) is the half-width of the sound wave
• \(c = 1\) is the speed of sound.

This initial condition is similar to the spherical sound wave on pg. 218 of Kopriva (2009), "Implementing Spectral Methods for Partial Differential Equations"

Pressure field for the initial condition, showing a spherical sound wave centered in the domain.

Pressure field at t=0.5 computed with SELF

Pressure field at t=1 computed with SELF

How we implement this

You can find the example file for this demo in the examples/linear_euler2d_spherical_soundwave_closeddomain.f90 file. This examples using the LinearEuler2D class as-is to write a simple program that simulates the expansion of a spherical sound-wave in a closed domain.

Running this example

Note

To run this example, you must first install SELF . We assume that SELF is installed in path referenced by the SELF_ROOT environment variable.

To run this example, simply execute

${SELF_ROOT}/examples/linear_euler2d_spherical_soundwave_closeddomain

This will run the simulation from \(t=0\) to \(t=1.0\) and write model output at intervals of \(Δ t_{io} = 0.1\).

During the simulation, tecplot (solution.*.tec) files are generated which can easily be visualized with paraview.