EulerFlow

Description

Solve the Euler (Compressible, inviscid Navier Stokes) equations in cartesian, cylindrical, and spherical coordinates in python. Validation is done through the self-similar to the solution, described as the Taylor-Von Neumann-Sedov solution.

Numerical Methods for the Euler Solution

Example

Import Libraries

For this simple example, we're going to use SedovBlast, which handles setting up the dimensionless problem. Using SI units to solve the Euler equations slows down the problem significanly and leads to numerical instability. It utilizes the Jameson-Shmidt-Turkel finite volume scheme for spatial discretization and scipy.integrate.solve_ivp for the time integration. The boundary conditions at the origin are reflective and transmissive at the exit. Unit tests have shown that the wave transmits easily through the end and that the wave reflects at the origin.

from EulerFlow import SedovBlast

Set Up Blast Scenario

LenScale__m = 1    # length scale of the problem
DomainLen__m = 10   # size of the domain
PAmb__Pa = 101325   # ambient air pressure
PExpl__Pa = 20*PAmb__Pa # Explosive pressure
RExpl__m = 3        # radius of explosion
tFin__s  = 0.010    # final simulation time
rhoAmb__kgpm3=1.225 # ambient air density
orders = 2          # order of solution

We look at a blast solition over the domain 0 to 10 meters. We scale the parameters by 1 meter, but values 1-10 are acceptable. We set the ambient pressure and density to STP: PAmb__Pa=101325 and rhoAmb__kgpm3=1.225. The Explosion is centered at the origin at t=0, and has a radius of 3 meters. We allow the solution to solve for 10 miliseconds, and set order=2 (spherical coordinates). order=0 indicates cartesian coordinates, and order=1 indicates cylindrical coordinates.

Run Simulation

SedovBlast converts the SI coordinates, time, and thermodynamic parameters into dimensionless units. SedovBlast.solve() takes the argument of method, which is the time integration scheme for scipy.integrate.solve_ivp. The solution should converge in a few seconds.

Blast = SedovBlast(LenScale__m, DomainLen__m, RExpl__m, PExpl__Pa, tFin__s,
                P0__Pa=PAmb__Pa, rho0__kgpm3=rhoAmb__kgpm3, order=orders)
Blast.solve(method='RK45')

Analyze Results

We can then view the fields as density plots or at discrete times.

Blast.dispFields()      ## density plots of the field
Blast.plotDiscTimes()   ## plott at discrete times

Taylor-Von Neumann-Sedov Analytical Solution

To-Do

Other goals I have with this project:

• solve the euler equation with other flux integration schemes (Roe, etc)
• simple PDEs as unit tests such as the wave and diffusion equation
• Basset-Oseen equation (transient, compressible, particle-laden flow)
• Coupled Euler equations with combustion
• Full Navier-Stokes equations
• quasi-1D Euler, with cross-sectional area variation
• etc