def conical_deflection_Mach_sigma(Mach, sigma, gamma=1.4, tol=1.0e-6):
def rkf45(F, x, y, h):
# Runge-Kutta-Fehlberg formulas
C = [37./378, 0., 250./621, 125./594, 0., 512./1771]
D = [2825./27648, 0., 18575./48384, 13525./55296, 277./14336, 1./4]
n = len(y)
K = numpy.zeros((6,n))
K[0] = h*F(x,y)
K[1] = h*F(x + 1./5*h, y + 1./5*K[0])
K[2] = h*F(x + 3./10*h, y + 3./40*K[0] + 9./40*K[1])
K[3] = h*F(x + 3./5*h, y + 3./10*K[0]- 9./10*K[1] + 6./5*K[2])
K[4] = h*F(x + h, y - 11./54*K[0] + 5./2*K[1] - 70./27*K[2] + 35./27*K[3])
K[5] = h*F(x + 7./8*h, y + 1631./55296*K[0] + 175./512*K[1] + 575./13824*K[2] + 44275./110592*K[3] + 253./4096*K[4])
# Initialize arrays {dy} and {E}
E = numpy.zeros((n))
dy = numpy.zeros((n))
# Compute solution increment {dy} and per-step error {E}
for i in range(6):
dy = dy + C[i]*K[i]
E = E + (C[i] - D[i])*K[i]
# Compute RMS error e
e = math.sqrt(sum(E**2)/n)
return dy, e
def rhs(phi, data):
th = data[0]
ma = data[1]
k = 1.-(ma*degree.sin(phi-th))**2
rhs=numpy.zeros(2)
rhs[0] = -degree.sin(th)*degree.cos(phi-th)/degree.sin(phi)/k
rhs[1] = math.pi/180.*degree.sin(th)*degree.sin(phi-th)/degree.sin(phi)/k*ma*(1.+.5*(gamma-1)*ma*ma)
return rhs
th = deflection_Mach_sigma(Mach, sigma, gamma)
ma = downstream_Mn(Mach*degree.sin(sigma), gamma)/degree.sin(sigma-th)
phi = sigma
thma = numpy.array([th, ma])
h = -phi/10.
conv = phi
while (abs(conv) >= tol):
dthma, err = rkf45(rhs, phi, thma, h)
# Accept integration step if error e is within tolerance
if err <= tol:
conv = thma[0] - phi
if conv/(h-dthma[0]) < 1:
h = h*conv/(h-dthma[0])
else:
thma = thma + dthma
phi = phi + h
print phi, thma
else:
h = 0.9*h*(tol/err)**0.2
print "new h ",h
deflection = .5*(thma[0] + phi)
return deflection
def rhs(phi, data):
th = data[0]
ma = data[1]
k = 1.-(ma*degree.sin(phi-th))**2
rhs=numpy.zeros(2)
rhs[0] = -degree.sin(th)*degree.cos(phi-th)/degree.sin(phi)/k
rhs[1] = math.pi/180.*degree.sin(th)*degree.sin(phi-th)/degree.sin(phi)/k*ma*(1.+.5*(gamma-1)*ma*ma)
return rhs