Acoustics in one dimension

The linearized acoustic equations can be derived from Navier-Stokes equations (which we solve in the example The compressible Navier-Stokes equations).

Considering an ideal gas with negligible viscosity Navier-Stokes equations are reduced to the compressible Euler equations:

\[\begin{eqnarray} &&\frac{D p}{Dt} +\gamma p \nabla \cdot \vec{v} = 0,\\ &&\frac{D \vec{v}}{Dt} + \frac{1}{\rho} \nabla p = 0, \\ &&\frac{D \rho}{Dt} + \rho \nabla \cdot \vec{v} = 0. \end{eqnarray}\]

Here we consider the one dimensional case and small perturbations around a known flow state \((1/\gamma, a(x), \rho_0(x))\). That is we replace

\[\begin{equation} p \rightarrow 1/\gamma + p(x,t), \ \ v \rightarrow a(x) + v(x,t), \ \ \rho \rightarrow \rho_0(x) + \rho(x,t), \end{equation}\]

and neglect small quadratic terms. The resulting linearized equations are

\[\begin{eqnarray} p_t + a(x) p_x + v_x &=& - a'(x), \\ v_t + a(x) v_x + \frac{p_x}{\rho_0(x)} &=& - a(x) a'(x), \\ \rho_t + a(x) \rho_x + \rho_0(x) v_x &=& - \rho_0'(x) a(x) - a'(x) \rho_0(x). \end{eqnarray}\]

This module considers three different cases

  1. Zero mean flow, \(a(x) = 0\) and constant mean density, \(\rho_0(x) = 1\).
  2. Constant mean flow, \(a(x) = a\) and constant density, \(\rho_0(x) = 1\).
  3. Variable mean flow with variable density.

Case 1, zero mean flow and constant mean density

For this case the equations for the pressure and velocity are reduced to

\[\begin{eqnarray} p_t &=& - v_x, \\ v_t &=& - p_x. \end{eqnarray}\]

The evolution of the above system using wall boundary conditions \(v = 0\) on both sides (which by using the PDE also means \(p_x = 0\)) and with the exact solution

\[\begin{eqnarray} &&p(x,t) = \cos(k \pi t) \cos(k \pi x), \\ &&v(x,t) = \sin(k \pi t) \sin(k \pi x), \end{eqnarray}\]

is implemented in the file /chides/acoustic/1D/acoustics1d_wall.f90

  1. Add description of PDE.
  2. Add description of enforcing BC.

Case 2, Constant mean flow, \(a(x) = a\) and constant density \(\rho_0(x) = 1\)

For this case the equations for the pressure and velocity are reduced to

\[\begin{eqnarray} p_t + a p_x + v_x &=& 0, \\ v_t + a v_x + p_x &=& 0, \\ \end{eqnarray}\]

For this problem we consider periodic boundary conditions, as implemented in /chides/acoustic/1D/acoustics1d_per.f90 and characteristics based non-reflecting boundary conditions as implemented in /chides/acoustic/1D/acoustics1d_cbc.f90.

  1. Add description of CBC, equations.
  2. Add description of CBC, how the extrapolation is done.

Case 3, Varying mean flow and density with source terms

For this case the equations for the pressure and velocity are reduced to

\[\begin{eqnarray} p_t + a(x) p_x + v_x &=& - a'(x) + \tilde{f}_p(x,t), \\ v_t + a(x) v_x + \frac{p_x}{\rho_0(x)} &=& - a(x) a'(x) + \tilde{f}_v(x,t), \\ \end{eqnarray}\]

which we reformulate as

\[\begin{eqnarray} p_t + a(x) p_x + v_x &=& f_p(x,t), \\ v_t + a(x) v_x + s(x) p_x &=& f_v(x,t), \end{eqnarray}\]

For this problem we consider periodic boundary conditions, as implemented in /chides/acoustic/1D/acoustics1d_var.f90.

  1. Add description of how cofs(a,s,f_p,f_v,mp,dx,x,t) is used.
  2. Explain polynomial multiplication.
  3. Explain point_to_Taylor.
  4. Explain the RK4 procedure.