2-dimensional Euler equations

Compressible Euler flow in cylindrical symmetry

Solve the Euler equations of compressible fluid dynamics in 2D r-z coordinates:

\[\begin{split}\rho_t + (\rho u)_x + (\rho v)_y & = - \rho v / r \\ (\rho u)_t + (\rho u^2 + p)_x + (\rho uv)_y & = -\rho u v / r \\ (\rho v)_t + (\rho uv)_x + (\rho v^2 + p)_y & = - \rho v^2 / r \\ E_t + (u (E + p) )_x + (v (E + p))_y & = - (E + p) v / r.\end{split}\]

Here \(\rho\) is the density, (u,v) is the velocity, and E is the total energy. The radial coordinate is denoted by r.

The problem involves a planar shock wave impacting a spherical low-density bubble. The problem is 3-dimensional but has been reduced to two dimensions using cylindrical symmetry.

This problem demonstrates:

  • how to incorporate source (non-hyperbolic) terms using both Classic and SharpClaw solvers
  • how to impose a custom boundary condition
  • how to use the auxiliary array for spatially-varying coefficients

Output:

../../_images/pyclaw_examples_euler_2d__plots_shock_bubble_interaction_frame0000fig0.png ../../_images/pyclaw_examples_euler_2d__plots_shock_bubble_interaction_frame0004fig0.png ../../_images/pyclaw_examples_euler_2d__plots_shock_bubble_interaction_frame0010fig0.png

Source:

#!/usr/bin/env python
# encoding: utf-8
r"""
Compressible Euler flow in cylindrical symmetry
===============================================

Solve the Euler equations of compressible fluid dynamics in 2D r-z coordinates:

.. math::
    \rho_t + (\rho u)_x + (\rho v)_y & = - \rho v / r \\
    (\rho u)_t + (\rho u^2 + p)_x + (\rho uv)_y & = -\rho u v / r \\
    (\rho v)_t + (\rho uv)_x + (\rho v^2 + p)_y & = - \rho v^2 / r \\
    E_t + (u (E + p) )_x + (v (E + p))_y & = - (E + p) v / r.

Here :math:`\rho` is the density, (u,v) is the velocity, and E is the total energy.
The radial coordinate is denoted by r.

The problem involves a planar shock wave impacting a spherical low-density bubble.
The problem is 3-dimensional but has been reduced to two dimensions using 
cylindrical symmetry.

This problem demonstrates:

    - how to incorporate source (non-hyperbolic) terms using both Classic and SharpClaw solvers
    - how to impose a custom boundary condition
    - how to use the auxiliary array for spatially-varying coefficients
"""

from __future__ import absolute_import
import numpy as np
from clawpack import riemann
from clawpack.riemann.euler_5wave_2D_constants import density, x_momentum, y_momentum, \
        energy, num_eqn
from six.moves import range

gamma = 1.4 # Ratio of specific heats

x0=0.5; y0=0.; r0=0.2


def ycirc(x,ymin,ymax):
    if ((x-x0)**2)<(r0**2):
        return max(min(y0 + np.sqrt(r0**2-(x-x0)**2),ymax) - ymin,0.)
    else:
        return 0

def qinit(state,rhoin=0.1,pinf=5.):
    from scipy import integrate

    gamma1 = gamma - 1.

    grid = state.grid

    rhoout = 1.
    pout   = 1.
    pin    = 1.
    xshock = 0.2

    rinf = (gamma1 + pinf*(gamma+1.))/ ((gamma+1.) + gamma1*pinf)
    vinf = 1./np.sqrt(gamma) * (pinf - 1.) / np.sqrt(0.5*((gamma+1.)/gamma) * pinf+0.5*gamma1/gamma)
    einf = 0.5*rinf*vinf**2 + pinf/gamma1
    
    X, Y = grid.p_centers

    r = np.sqrt((X-x0)**2 + (Y-y0)**2)

    #First set the values for the cells that don't intersect the bubble boundary
    state.q[0,:,:] = rinf*(X<xshock) + rhoin*(r<=r0) + rhoout*(r>r0)*(X>=xshock)
    state.q[1,:,:] = rinf*vinf*(X<xshock)
    state.q[2,:,:] = 0.
    state.q[3,:,:] = einf*(X<xshock) + (pin*(r<=r0) + pout*(r>r0)*(X>=xshock))/gamma1
    state.q[4,:,:] = 1.*(r<=r0)

    #Now compute average density for the cells on the edge of the bubble
    d2 = np.linalg.norm(state.grid.delta)/2.
    dx = state.grid.delta[0]
    dy = state.grid.delta[1]
    dx2 = state.grid.delta[0]/2.
    dy2 = state.grid.delta[1]/2.
    for i in range(state.q.shape[1]):
        for j in range(state.q.shape[2]):
            ydown = Y[i,j]-dy2
            yup   = Y[i,j]+dy2
            if abs(r[i,j]-r0)<d2:
                infrac,abserr = integrate.quad(ycirc,X[i,j]-dx2,X[i,j]+dx2,args=(ydown,yup),epsabs=1.e-8,epsrel=1.e-5)
                infrac=infrac/(dx*dy)
                state.q[0,i,j] = rhoin*infrac + rhoout*(1.-infrac)
                state.q[3,i,j] = (pin*infrac + pout*(1.-infrac))/gamma1
                state.q[4,i,j] = 1.*infrac


def auxinit(state):
    """
    aux[1,i,j] = radial coordinate of cell centers for cylindrical source terms
    """
    y = state.grid.y.centers
    for j,r in enumerate(y):
        state.aux[0,:,j] = r


def incoming_shock(state,dim,t,qbc,auxbc,num_ghost):
    """
    Incoming shock at left boundary.
    """
    gamma1 = gamma - 1.

    pinf=5.
    rinf = (gamma1 + pinf*(gamma+1.))/ ((gamma+1.) + gamma1*pinf)
    vinf = 1./np.sqrt(gamma) * (pinf - 1.) / np.sqrt(0.5*((gamma+1.)/gamma) * pinf+0.5*gamma1/gamma)
    einf = 0.5*rinf*vinf**2 + pinf/gamma1

    for i in range(num_ghost):
        qbc[0,i,...] = rinf
        qbc[1,i,...] = rinf*vinf
        qbc[2,i,...] = 0.
        qbc[3,i,...] = einf
        qbc[4,i,...] = 0.


def step_Euler_radial(solver,state,dt):
    """
    Geometric source terms for Euler equations with cylindrical symmetry.
    Integrated using a 2-stage, 2nd-order Runge-Kutta method.
    This is a Clawpack-style source term routine, which approximates
    the integral of the source terms over a step.
    """
    dt2 = dt/2.

    q = state.q
    rad = state.aux[0,:,:]

    rho = q[0,:,:]
    u   = q[1,:,:]/rho
    v   = q[2,:,:]/rho
    press  = (gamma - 1.) * (q[3,:,:] - 0.5*rho*(u**2 + v**2))

    qstar = np.empty(q.shape)

    qstar[0,:,:] = q[0,:,:] - dt2/rad * q[2,:,:]
    qstar[1,:,:] = q[1,:,:] - dt2/rad * rho*u*v
    qstar[2,:,:] = q[2,:,:] - dt2/rad * rho*v*v
    qstar[3,:,:] = q[3,:,:] - dt2/rad * v * (q[3,:,:] + press)

    rho = qstar[0,:,:]
    u   = qstar[1,:,:]/rho
    v   = qstar[2,:,:]/rho
    press  = (gamma - 1.) * (qstar[3,:,:] - 0.5*rho*(u**2 + v**2))

    q[0,:,:] = q[0,:,:] - dt/rad * qstar[2,:,:]
    q[1,:,:] = q[1,:,:] - dt/rad * rho*u*v
    q[2,:,:] = q[2,:,:] - dt/rad * rho*v*v
    q[3,:,:] = q[3,:,:] - dt/rad * v * (qstar[3,:,:] + press)


def dq_Euler_radial(solver,state,dt):
    """
    Geometric source terms for Euler equations with radial symmetry.
    This is a SharpClaw-style source term routine, which returns
    the value of the source terms.
    """
    q   = state.q
    rad = state.aux[0,:,:]

    rho = q[0,:,:]
    u   = q[1,:,:]/rho
    v   = q[2,:,:]/rho
    press  = (gamma - 1.) * (q[3,:,:] - 0.5*rho*(u**2 + v**2))

    dq = np.empty(q.shape)

    dq[0,:,:] = -dt/rad * q[2,:,:]
    dq[1,:,:] = -dt/rad * rho*u*v
    dq[2,:,:] = -dt/rad * rho*v*v
    dq[3,:,:] = -dt/rad * v * (q[3,:,:] + press)
    dq[4,:,:] = 0

    return dq

def setup(use_petsc=False,solver_type='classic', outdir='_output', kernel_language='Fortran',
        disable_output=False, mx=320, my=80, tfinal=0.6, num_output_times = 10):
    if use_petsc:
        import clawpack.petclaw as pyclaw
    else:
        from clawpack import pyclaw

    if solver_type=='sharpclaw':
        solver = pyclaw.SharpClawSolver2D(riemann.euler_5wave_2D)
        solver.dq_src = dq_Euler_radial
        solver.weno_order = 5
        solver.lim_type   = 2
    else:
        solver = pyclaw.ClawSolver2D(riemann.euler_5wave_2D)
        solver.step_source = step_Euler_radial
        solver.source_split = 1
        solver.limiters = [4,4,4,4,2]
        solver.cfl_max = 0.5
        solver.cfl_desired = 0.45

    x = pyclaw.Dimension(0.0,2.0,mx,name='x')
    y = pyclaw.Dimension(0.0,0.5,my,name='y')
    domain = pyclaw.Domain([x,y])

    num_aux=1
    state = pyclaw.State(domain,num_eqn,num_aux)
    state.problem_data['gamma']= gamma

    qinit(state)
    auxinit(state)

    solver.user_bc_lower = incoming_shock

    solver.bc_lower[0]=pyclaw.BC.custom
    solver.bc_upper[0]=pyclaw.BC.extrap
    solver.bc_lower[1]=pyclaw.BC.wall
    solver.bc_upper[1]=pyclaw.BC.extrap
    #Aux variable in ghost cells doesn't matter
    solver.aux_bc_lower[0]=pyclaw.BC.extrap
    solver.aux_bc_upper[0]=pyclaw.BC.extrap
    solver.aux_bc_lower[1]=pyclaw.BC.extrap
    solver.aux_bc_upper[1]=pyclaw.BC.extrap

    claw = pyclaw.Controller()
    claw.solution = pyclaw.Solution(state,domain)
    claw.solver = solver

    claw.keep_copy = True
    if disable_output:
        claw.output_format = None
    claw.tfinal = tfinal
    claw.num_output_times = num_output_times
    claw.outdir = outdir
    claw.setplot = setplot

    return claw

    
def setplot(plotdata):
    """ 
    Plot solution using VisClaw.
    """ 
    from clawpack.visclaw import colormaps

    plotdata.clearfigures()  # clear any old figures,axes,items data
    
    # Pressure plot
    plotfigure = plotdata.new_plotfigure(name='Density', figno=0)

    plotaxes = plotfigure.new_plotaxes()
    plotaxes.title = 'Density'
    plotaxes.scaled = True      # so aspect ratio is 1
    plotaxes.afteraxes = label_axes

    plotitem = plotaxes.new_plotitem(plot_type='2d_schlieren')
    plotitem.plot_var = 0
    plotitem.add_colorbar = False
    

    # Tracer plot
    plotfigure = plotdata.new_plotfigure(name='Tracer', figno=1)

    plotaxes = plotfigure.new_plotaxes()
    plotaxes.title = 'Tracer'
    plotaxes.scaled = True      # so aspect ratio is 1
    plotaxes.afteraxes = label_axes

    plotitem = plotaxes.new_plotitem(plot_type='2d_pcolor')
    plotitem.pcolor_cmin = 0.
    plotitem.pcolor_cmax=1.0
    plotitem.plot_var = 4
    plotitem.pcolor_cmap = colormaps.yellow_red_blue
    plotitem.add_colorbar = False
    

    # Energy plot
    plotfigure = plotdata.new_plotfigure(name='Energy', figno=2)

    plotaxes = plotfigure.new_plotaxes()
    plotaxes.title = 'Energy'
    plotaxes.scaled = True      # so aspect ratio is 1
    plotaxes.afteraxes = label_axes

    plotitem = plotaxes.new_plotitem(plot_type='2d_pcolor')
    plotitem.pcolor_cmin = 2.
    plotitem.pcolor_cmax=18.0
    plotitem.plot_var = 3
    plotitem.pcolor_cmap = colormaps.yellow_red_blue
    plotitem.add_colorbar = False
    
    return plotdata

def label_axes(current_data):
    import matplotlib.pyplot as plt
    plt.xlabel('z')
    plt.ylabel('r')
    

if __name__=="__main__":
    from clawpack.pyclaw.util import run_app_from_main
    output = run_app_from_main(setup,setplot)