Linear Euler 2D - Plane Wave Reflection 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 north, west, and south boundaries and a no-normal-flow boundary condition on the east boundary.

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 on the south, north, and west edges of the domain are all tagged with the SELF_BC_PRESCRIBED flag to implement prescribed boundary conditions; the east boundary is tagged with SELF_BC_NONORMALFLOW 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 and Boundary Conditions

The initial and prescribed boundary conditions are set using an exact solution. The exact solution is found using the method of images where a no-normal-flow wall is placed at \(x=1\) . We define the solution as the sum of an incident wave and a reflecting wave

\[ \vec{s} = \vec{s}_i + \vec{s}_r \]

where

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

is the incident wave, and

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

is the reflecting wave.

The parameters used in the exact solution are as follows :

• \(\bar{p} = 10^{-4}\) is the amplitude of the sound wave
• \(x_0 = y_0 = 0.2\) 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
• \(c = 1\) is the speed of sound.

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

Pressure field at t=0.75 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
• ad 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.