Commit e391680c authored by Matthieu Schaller's avatar Matthieu Schaller
Browse files

Added a python script to generate the analytical solution of the Sedov blast.




Former-commit-id: 9c158c61d428ca48595d0fdc16505cb793cd93d2
parent 9494cf3b
"""
Peter Creasey
p.e.creasey.00@googlemail.com
solution to the Sedov problem
based on the C code by Aamer Haque
"""
from scipy.special import gamma as Gamma
from numpy import power, arange, empty, float64, log, exp, pi, diff, inner, outer, array
def calc_a(g,nu=3):
"""
exponents of the polynomials of the sedov solution
g - the polytropic gamma
nu - the dimension
"""
a = [0]*8
a[0] = 2.0 / (nu + 2)
a[2] = (1-g) / (2*(g-1) + nu)
a[3] = nu / (2*(g-1) + nu)
a[5] = 2 / (g-2)
a[6] = g / (2*(g-1) + nu)
a[1] = (((nu+2)*g)/(2.0+nu*(g-1.0)) ) * ( (2.0*nu*(2.0-g))/(g*(nu+2.0)**2) - a[2])
a[4] = a[1]*(nu+2) / (2-g)
a[7] = (2 + nu*(g-1))*a[1]/(nu*(2-g))
return a
def calc_beta(v, g, nu=3):
"""
beta values for the sedov solution (coefficients of the polynomials of the similarity variables)
v - the similarity variable
g - the polytropic gamma
nu- the dimension
"""
beta = (nu+2) * (g+1) * array((0.25, (g/(g-1))*0.5,
-(2 + nu*(g-1))/2.0 / ((nu+2)*(g+1) -2*(2 + nu*(g-1))),
-0.5/(g-1)), dtype=float64)
beta = outer(beta, v)
beta += (g+1) * array((0.0, -1.0/(g-1),
(nu+2) / ((nu+2)*(g+1) -2.0*(2 + nu*(g-1))),
1.0/(g-1)), dtype=float64).reshape((4,1))
return beta
def sedov(t, E0, rho0, g, n=1000, nu=3):
"""
solve the sedov problem
t - the time
E0 - the initial energy
rho0 - the initial density
n - number of points (10000)
nu - the dimension
g - the polytropic gas gamma
"""
# the similarity variable
v_min = 2.0 / ((nu + 2) * g)
v_max = 4.0 / ((nu + 2) * (g + 1))
v = v_min + arange(n) * (v_max - v_min) / (n - 1.0)
a = calc_a(g, nu)
beta = calc_beta(v, g=g, nu=nu)
lbeta = log(beta)
r = exp(-a[0] * lbeta[0] - a[2] * lbeta[1] - a[1] * lbeta[2])
rho = ((g + 1.0) / (g - 1.0)) * exp(a[3] * lbeta[1] + a[5] * lbeta[3] + a[4] * lbeta[2])
p = exp(nu * a[0] * lbeta[0] + (a[5] + 1) * lbeta[3] + (a[4] - 2 * a[1]) * lbeta[2])
u = beta[0] * r * 4.0 / ((g + 1) * (nu + 2))
p *= 8.0 / ((g + 1) * (nu + 2) * (nu + 2))
# we have to take extra care at v=v_min, since this can be a special point.
# It is not a singularity, however, the gradients of our variables (wrt v) are.
# r -> 0, u -> 0, rho -> 0, p-> constant
u[0] = 0.0; rho[0] = 0.0; r[0] = 0.0; p[0] = p[1]
# volume of an n-sphere
vol = (pi ** (nu / 2.0) / Gamma(nu / 2.0 + 1)) * power(r, nu)
# note we choose to evaluate the integral in this way because the
# volumes of the first few elements (i.e near v=vmin) are shrinking
# very slowly, so we dramatically improve the error convergence by
# finding the volumes exactly. This is most important for the
# pressure integral, as this is on the order of the volume.
# (dimensionless) energy of the model solution
de = rho * u * u * 0.5 + p / (g - 1)
# integrate (trapezium rule)
q = inner(de[1:] + de[:-1], diff(vol)) * 0.5
# the factor to convert to this particular problem
fac = (q * (t ** nu) * rho0 / E0) ** (-1.0 / (nu + 2))
# shock speed
shock_speed = fac * (2.0 / (nu + 2))
rho_s = ((g + 1) / (g - 1)) * rho0
r_s = shock_speed * t * (nu + 2) / 2.0
p_s = (2.0 * rho0 * shock_speed * shock_speed) / (g + 1)
u_s = (2.0 * shock_speed) / (g + 1)
r *= fac * t
u *= fac
p *= fac * fac * rho0
rho *= rho0
return r, p, rho, u, r_s, p_s, rho_s, u_s, shock_speed
def test():
""" draw a 3d sedov solution """
import pylab as pl
gamma = 5.0/3.0
r,p,rho,u,r_s,p_s,rho_s,u_s,shock_speed = \
sedov(t=0.05, E0=5.0, rho0=5.0, g=gamma)
print 'rho shock', rho_s
print 'p shock', p_s
print 'u shock', u_s
print 'r shock', r_s
print 'Dimensionless var (E/rho) t^2 r^-5', (5.0 /5.0)* 0.05**0.4 * r[-1]**-1.0
vols = (4/3.0)*pi*r*r*r
dv = vols.copy()
dv[1:] = diff(dv)
# thermal and kinetic energy
te = (p*dv/(gamma-1))
ke = (rho*u*u*0.5*dv)
energy = te.sum() + ke.sum()
mass = 0.5*inner(rho[1:]+rho[:-1],dv[1:])
print 'density', mass / (4/3.0 * pi * r_s**3)
print 'energy', energy
print 'shock speed', shock_speed
pl.plot(r/r_s,rho/rho_s, label=r'$\rho/\rho_s$')
pl.plot(r/r_s,p/p_s, label=r'$p/p_s$')
pl.plot(r/r_s,u/u_s, label=r'$u/u_s$')
pl.legend(loc='upper left')
pl.show()
def test2():
""" test momentum and mass conservation in 3d """
import pylab as pl
r,p,rho,u,r_s,p_s,rho_s,u_s,shock_speed = \
sedov(t=0.05, E0=5.0, rho0=5.0, g=5.0/3.0,n=10000)
dt = 1e-5
r2,p2,rho2,u2 = sedov(t=0.05+dt, E0=5.0, rho0=5.0, g=5.0/3.0, n=9000)[:4]
# align the results
from numpy import interp, gradient
p2 = interp(r,r2,p2)
rho2 = interp(r,r2,rho2)
u2 = interp(r,r2,u2)
# mass conservation
pl.plot(r, -gradient(rho*u*r*r)/(r*r*gradient(r)), 'b', label=r'$\frac{1}{r^2}\frac{\partial}{\partial r} \rho u r^2$')
pl.plot(r, (rho2-rho)/dt, 'k', label=r'$\frac{\partial \rho}{\partial t}$')
# momentum conservation
pl.plot(r, -gradient(p)/gradient(r), 'g',label=r'$-\frac{\partial p}{\partial r}$')
pl.plot(r, rho*((u2-u)/dt+u*gradient(u)/gradient(r)), 'r',label=r'$\rho \left( \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial r} \right)$')
pl.legend(loc='lower left')
pl.show()
def test3():
""" draw a 2d sedov solution """
import pylab as pl
r,p,rho,u,r_s,p_s,rho_s,u_s,shock_speed = sedov(t=1.2, E0=1, rho0=1, g=5.0/3.0, nu=2)
print 'rho shock', rho_s
print 'p shock', p_s
print 'u shock', u_s
print 'r shock', r_s
area = pi*r*r
dv = area.copy()
dv[1:] = diff(dv)
# thermal and kinetic energy
te = (p*dv/(5.0/3.0-1))
ke = (rho*u*u*0.5*dv)
#pl.plot(arange(te.size), ke, 'x')
#pl.show()
print 'r0', r[:2]
energy = te.sum() + ke.sum()
mass = 0.5*inner(rho[1:]+rho[:-1],dv[1:])
print 'density', mass / (pi * r_s**2)
print 'energy', energy
print 'shock speed', shock_speed
#pl.plot(r/r_s,rho/rho_s, 'b,',label=r'$\rho/\rho_s$')
#pl.plot(r/r_s,p/p_s,'r',label=r'$p/p_s$')
#pl.plot(r/r_s,u/u_s, 'g,',label=r'$u/u_s$')
# pl.plot(r,rho,'b',label='$\\rho$')
pl.plot(r,u,'b',label='$u$')
# pl.plot(r,p,'b',label='$P$')
pl.legend(loc='upper left')
# pl.xlim(0,2)
# pl.ylim(0,5)
pl.show()
if __name__=='__main__':
#test()
#test2()
test3()
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