def Pvec(Q, MP):
""" Vector of primitive variables
NOTE: Uses atypical ordering
"""
P = State(Q, MP)
ret = Q.copy()
ret[1] = P.p()
ret[2:5] /= P.ρ
if THERMAL:
ret[14:17] /= P.ρ
return ret
def F_cons(Q, d, MP):
ret = zeros(NV)
P = State(Q, MP)
ρ = P.ρ
p = P.p()
E = P.E
v = P.v
vd = v[d]
ρvd = ρ * vd
ret[0] = ρvd
ret[1] = ρvd * E + p * vd
ret[2:5] = ρvd * v
ret[2 + d] += p
if VISCOUS:
A = P.A
σ = P.σ()
σd = σ[d]
ret[1] -= dot(σd, v)
ret[2:5] -= σd
Av = dot(A, v)
ret[5 + d] = Av[0]
ret[8 + d] = Av[1]
ret[11 + d] = Av[2]
if THERMAL:
J = P.J
T = P.T()
q = P.q()
ret[1] += q[d]
ret[14:17] = ρvd * J
ret[14 + d] += T
if MULTI:
λ = P.λ
ret[17] = ρvd * λ
return ret
def M_prim(Q, d, MP):
""" The system jacobian of the primitive system
NOTE: Uses typical ordering
"""
P = State(Q, MP)
ρ = P.ρ
p = P.p()
A = P.A
v = P.v
c0 = c_0(ρ, p, A, MP)
ret = v[d] * eye(NV)
ret[0, 2 + d] = ρ
ret[1, 2 + d] = ρ * c0**2
ret[2 + d, 1] = 1 / ρ
if VISCOUS:
σ = P.σ()
dσdρ = P.dσdρ()
dσdA = P.dσdA()
ret[1, 2:5] += σ[d] - ρ * dσdρ[d]
ret[2:5, 0] = -1 / ρ * dσdρ[d]
ret[2:5, 5:14] = -1 / ρ * dσdA[d].reshape([3, 9])
ret[5 + d, 2:5] = A[0]
ret[8 + d, 2:5] = A[1]
ret[11 + d, 2:5] = A[2]
if THERMAL:
dTdρ = P.dTdρ()
dTdp = P.dTdp()
T = P.T()
ch = c_h(ρ, T, MP)
ret[1, 14 + d] = ρ * ch**2 / dTdp
ret[14 + d, 0] = dTdρ / ρ
ret[14 + d, 1] = dTdp / ρ
return ret
def gauge_plot(uList, MPs):
dt = 5e-6 / 100
x = [0.0018125 + 0.006 + 0.003625 * i for i in range(5)]
n = len(uList)
nx, ny = uList[0].shape[:2]
x_ = linspace(0, 5e-6, n)
vy = zeros([5, n])
py = zeros([5, n])
ρy = zeros([5, n])
Σy = zeros([5, n])
# j0 = int(ny / 2)
j0 = 0
for j in range(n):
for i in range(5):
i0 = int(x[i] / 0.03 * nx)
Q = uList[j][i0, j0]
P = State(Q, MPs[1])
vy[i, j] = P.v[0]
py[i, j] = P.p()
ρy[i, j] = P.ρ
Σy[i, j] = P.Σ()[0, 0]
x[i] += P.v[0] * dt
figure(1)
for i in range(5):
plot(x_, vy[i])
figure(2)
for i in range(5):
plot(x_, py[i])
figure(3)
for i in range(5):
plot(x_, ρy[i])
figure(4)
for i in range(5):
plot(x_, -Σy[i])
def exact_euler(n, t, x0, QL, QR, MPL, MPR):
""" Returns the exact solution to the Euler equations at (x,t),
given initial states PL for x<x0 and PR for x>x0
"""
ret = zeros([n, len(QL)])
PL = State(QL, MPL)
PR = State(QR, MPR)
ρL = PL.ρ
ρR = PR.ρ
uL = PL.v[0]
uR = PR.v[0]
pL = PL.p()
pR = PR.p()
c0L = c_0(ρL, pL, eye(3), MPL)
c0R = c_0(ρR, pR, eye(3), MPR)
p_ = p_star(PL, PR, MPL, MPR)
u_ = u_star(p_, PL, PR, MPL, MPR)
print('Interface:', u_ * t + x0)
for i in range(n):
x = (i + 0.5) / n
S = (x - x0) / t
if (S < u_):
if (p_ < pL): # Left fan
if (S < uL - c0L):
ret[i] = QL
else:
STL = u_ - c0_star(p_, PL, MPL)
if (S < STL):
ret[i] = Wfan(S, PL, MPL, c0L)
else:
r_ = r_star_fan(p_, PL, MPL)
v_ = array([u_, 0, 0])
ret[i] = Cvec(r_, p_, v_, MPL)
else: # Left shock
SL = uL - Q(p_, PL, MPL) / ρL
if (S < SL):
ret[i] = QL
else:
r_ = r_star_shock(p_, PL, MPL)
v_ = array([u_, 0, 0])
ret[i] = Cvec(r_, p_, v_, MPL)
ret[i, -1] = -1
else:
if (p_ < pR): # Right fan
if (uR + c0R < S):
ret[i] = QR
else:
STR = u_ + c0_star(p_, PR, MPR)
if (STR < S):
ret[i] = Wfan(S, PR, MPR, -c0R)
else:
r_ = r_star_fan(p_, PR, MPR)
v_ = array([u_, 0, 0])
ret[i] = Cvec(r_, p_, v_, MPR)
else: # Right shock
SR = uR + Q(p_, PR, MPR) / ρR
if (SR < S):
ret[i] = QR
else:
r_ = r_star_shock(p_, PR, MPR)
v_ = array([u_, 0, 0])
ret[i] = Cvec(r_, p_, v_, MPR)
ret[i, -1] = 1
return ret