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maccormack.py
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import numpy as np
import matplotlib.pyplot as plt
## Setup
lx = 2
CFL = 0.9
T = 300 # temperatura
cv = 0.718
k_1 = .4 # k-1 onde k e cp/cv
n = 100 # No de elementos
steps = 40
# saida
P = np.zeros((steps,n+2))
Rho = np.zeros((steps,n+2))
U = np.zeros((steps,n+2))
## Volumes
# (n+2) adicionara 2 volumes fantasma
elem = np.array([[0]*2]*(n+2) ,dtype=np.float64)
Q = np.array([[0]*3]*(n+2) ,dtype=np.float64)
F = np.array([[0]*3]*(n+2) ,dtype=np.float64)
# indice linha = Nº do volume
# se vizinho = -1, entao e' contorno (fantasma)
elem[0,0] = -1
elem[n-1,1] = -1
for i in range(1,n-1):
elem[i,0] = i-1; elem[i,1] = i+1;
dx = lx/n;
vol = dx*1; # volumes sempre iguais
## C.I.
# p = rho = 1 para x < 1, p = rho = 2 para x >= 1, u = 0
u = 0;
for i in range(n+2):
if i*dx > 1:
rho = 1; p = 1;
else:
rho = 2; p = 2;
e = p/(k_1)
Q[i,0] = rho
Q[i,1] = rho*u
Q[i,2] = e
F[i,0] = Q[i,1]
F[i,1] = rho*u**2+p
F[i,2] = (e+p)*u
P[0,i] = p;
Rho[0,i] = rho;
#fantasma
F[0,0] = -F[1,0]
F[0,1] = -F[1,1]+2*1 #2x a pressao 1
F[0,2] = -F[1,2]
F[n+1,0] = -F[n,0]
F[n+1,1] = -F[n,1]+2*2
F[n+1,2] = -F[n,2]
Qp = np.copy(Q) #passo preditor
# MacCormack
print('{} {}'.format(p,rho))
c = np.sqrt(1.4*p/rho)
dt = CFL*dx/(c) # alterar?
a1 = dt/vol
a2 = dt/(2*vol)
S = 1 # superficie unitaria
dtmin = 10
dtmax = 0
for step in range(1,steps):
# print(dt)
# i denota o elemento
for i in range(1,n+1): #preditor
Qp[i,:] = Q[i,:] - a1*(F[i,:]*S-F[(i-1),:]*S)
# atualiza valores de fluxo preditor
for i in range(1,n+1):
Rho[step,i] = Qp[i,0]
P[step,i] = k_1*Rho[step,i]*( Qp[i,2]/Rho[step,i] - (1/2)*(Qp[i,1]/Qp[i,0])**2 )
F[i,0] = Qp[i,1]
F[i,1] = (Qp[i,1]**2/Q[i,0]) + P[step,i]
F[i,2] = (Qp[i,2]+P[step,i])*Qp[i,1]/Q[i,0]
# atualiza fantasmas
F[0,0] = -F[1,0]
F[0,1] = -F[1,1]+2*P[step,1] #2x a pressao 1
F[0,2] = -F[1,2]
F[n+1,0] = -F[n,0]
F[n+1,1] = -F[n,1]+2*P[step,n]
F[n+1,2] = -F[n,2]
for i in range(1,n+1): #corretor
Q[i,:] = 0.5*(Q[i,:]+Qp[i,:])-a2*(F[i+1,:]*S-F[i,:]*S)
# atualiza valores de fluxo coretor
for i in range(1,n+1):
# print('{} {} {} \n'.format(Q[i,0],Q[i,1],Q[i,2]))
Rho[step,i] = Q[i,0]
# print('{} {} {}'.format( k_1*Rho[step,i] , Q[i,2]/Rho[step,i] , (1/2)*(Q[i,1]/Q[i,0] )**2))
U[step,i] = Q[i,1]/Q[i,0]
P[step,i] = k_1*Rho[step,i]*( Q[i,2]/Rho[step,i] - (1/2)*(U[step,i])**2 )
F[i,0] = Q[i,1]
F[i,1] = (U[step,i]**2*Q[i,0]) + P[step,i]
F[i,2] = (Q[i,2]+P[step,i])*U[step,i]
dt = CFL*dx/(c+np.amax(np.abs(U[step,:])))
if dt < dtmin:
dtmin = dt
if dt > dtmax:
dtmax = dt
a1 = dt/vol
a2 = dt/(2*vol)
# atualiza fantasmas
F[0,0] = -F[1,0]
F[0,1] = -F[1,1]+2*P[step,1] #2x a pressao 1
F[0,2] = -F[1,2]
F[n+1,0] = -F[n,0]
F[n+1,1] = -F[n,1]+2*P[step,n]
F[n+1,2] = -F[n,2]
# print('{} {} {}'.format(np.amax(np.abs(Q)),np.amax(U),dt))
if np.amax(Q) > 10**6:
print('{}'.format(step))
break
# if not(step % 100):
# plt.plot(Q[:,0])
# plt.plot(Q[:,1])
# plt.plot(Q[:,2])
# plt.show()
# time.sleep(1)
LX = np.linspace(0,lx,n)
plt.plot(LX,U[0,1:n+1],'k')
plt.plot(LX,U[int(steps/3),1:n+1],'b')
plt.plot(LX,U[int(steps*2/3),1:n+1],'g')
plt.plot(LX,U[steps-1,1:n+1],'r')
plt.show()
plt.plot(LX,P[0,1:n+1],'k')
plt.plot(LX,P[int(steps/3),1:n+1],'b')
plt.plot(LX,P[int(2*steps/3),1:n+1],'g')
plt.plot(LX,P[steps-1,1:n+1],'r')
plt.show()