3D Euler Sedov (with symmetry)#

This problem solves the 3D Euler equations in conservative form. The gas dynamics is governed by a system of PDE

\[\begin{split}\frac{\partial }{\partial t} \begin{bmatrix}\rho \\ \rho u_x \\ \rho u_y \\ \rho u_z\\ \rho E \end{bmatrix} + \frac{\partial }{\partial x} \begin{bmatrix}\rho u_x \\ \rho u_x^2 +p \\ \rho u_x u_y \\ \rho u_x u_z\\ (\rho E+p)u_x \end{bmatrix} + \frac{\partial }{\partial y} \begin{bmatrix}\rho u_y \\ \rho u_x u_y \\ \rho u_y^2 +p \\ \rho u_y u_z\\ (\rho E+p)u_y \end{bmatrix} + \frac{\partial }{\partial z} \begin{bmatrix}\rho u_z \\ \rho u_x u_z \\ \rho u_y u_z \\ \rho u_z^2 +p\\ (\rho E+p)u_z \end{bmatrix}= 0\end{split}\]

where the pressure \(p\) is related to the conserved quantities through the equation of the state

\[p=(\gamma -1)(\rho E-\frac{1}{2}\rho (u_x^2 + u_y^2 + u_z^2)).\]

  • Initial conditions (primitive variables):

    • a high pressure concentrated small spherical region of radius \(R = 3 \min(dx, dy)\)

    • \(\left\{\begin{matrix}\rho =1, u = 0, v = 0, p = ((\gamma-1)0.851072)/(4 \pi R^3); & r\leq R \\ \rho =1, u = 0, v = 0, p = 2.5\cdot 10^{-5}; & r>R \end{matrix}\right.\)

    • These are used to create the initial conditions in conservative variables.

  • By default, \(\gamma = 1.4\)

  • Domain is \([0.0, 1.2]^3\) with reflective BC on \(x=y=z=0\) and homogeneous Neumann on others.

  • Typically, integration is performed over \(t \in (0, 1.)\).

Caution

Currently, this problem only works for first order and Weno3 inviscid flux reconstruction.

Mesh#

python3 pressio-demoapps/meshing_scripts/create_full_mesh_for.py \
        --problem sedov3dsym_s<stencilSize> -n Nx Ny Nz --outDir <destination-path>

where

  • Nx, Ny, Nz is the number of cells you want along \(x\), \(y\), \(z\) respectively

  • <stencilSize> = 3 or 5

  • <destination-path> is where you want the mesh files to be generated

Important

When you set the <stencilSize>, keep in mind the following constraints (more on this below):

  • InviscidFluxReconstruction::FirstOrder requires <stencilSize> >= 3

  • InviscidFluxReconstruction::Weno3 requires <stencilSize> >= 5

C++ synopsis#

#include "pressiodemoapps/euler3d.hpp"

int main(){
  namespace pda = pressiodemoapps;

  const auto meshObj = pda::load_cellcentered_uniform_mesh_eigen("path-to-mesh");

  const auto probId = pda::Euler3d::SedovSymmetry;
  const auto scheme = pda::InviscidFluxReconstruction::FirstOrder; //or Weno3
  auto problem      = pda::create_problem_eigen(meshObj, probId, scheme);
  auto state      = problem.initialCondition();
}

Python synopsis#

import pressiodemoapps as pda

meshObj = pda.load_cellcentered_uniform_mesh("path-to-mesh")

probId  = pda.Euler3d.SedovSymmetry
scheme  = pda.InviscidFluxReconstruction.FirstOrder # or Weno3
problem = pda.create_problem(meshObj, probId, scheme)
state   = problem.initialCondition()