Linear Euler 2D - Plane Wave Propagation Tutorial

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 prescribed 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 . Model boundaries are all tagged with the SELF_BC_PRESCRIBED flag to implement prescribed 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 and Boundary Conditions

The initial and boundary conditions are set using an exact solution,

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

where

• \(\bar{p} = 10^{-4}\) is the amplitude of the sound wave
• \(x_0 = y_0 = 0.2\) defines the center of the initial sound wave
• \(L = \frac{0.2}{2\sqrt{\ln{2}}}\) is the half-width of the sound wave
• \(k_x = k_y = \frac{\sqrt{2}}{2}\) are the \(x\) and \(y\) components of the wave number

The model domain

Pressure field for the initial condition, showing a plane wave with a front oriented at 45 degrees

Pressure field at t=0.5 computed with SELF

How we implement this

You can find the example file for this demo in the examples/linear_euler2d_planewave_propagation.f90 file. This file defines the lineareuler2d_planewave_model module in addition to a program that runs the propagating plane wave simulation.

The lineareuler2d_planewave_model module defines the lineareuler2d_planewave class, which is a type extension of the lineareuler2d class that is provided by SELF. We make this type extension so that we can

• add attributes ( kx and ky ) for the x and y components of the plane-wave wave number
• add attributes ( x0 and y0 ) for the initial center position of the plane-wave
• add an attribute ( p ) for the pressure amplitude of the wave
• add an attribute ( L ) for the half-width of the plane wave
• override the hbc1d_Prescribed type-bound procedure to set the boundary condition to the exact solution

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_planewave_propagation

This will run the simulation from \(t=0\) to \(t=1.0\) and write model output at intervals of \(Δ t_{io} = 0.05\).

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