def viscous_shock_exact(x, Ms, MP, μ, center=0):
""" Returns the density, pressure, and velocity of the viscous shock
(Mach number Ms) at x
"""
x -= center
γ = MP.γ
ρ0 = 1
p0 = 1 / γ
if Ms == 2:
L = 0.3
elif Ms == 3:
L = 0.13
x = min(x, L)
x = max(x, -L)
c0 = c_0(ρ0, p0, eye(3), MP)
a = 2 / (Ms**2 * (γ + 1)) + (γ - 1) / (γ + 1)
Re = ρ0 * c0 * Ms / μ
c1 = ((1 - a) / 2)**(1 - a)
c2 = 3 / 4 * Re * (Ms**2 - 1) / (γ * Ms**2)
def f(z):
return (1 - z) / (z - a)**a - c1 * exp(c2 * -x)
vbar = brentq(f, a + 1e-16, 1)
p = p0 / vbar * (1 + (γ - 1) / 2 * Ms**2 * (1 - vbar**2))
ρ = ρ0 / vbar
v = Ms * c0 * vbar
v = Ms * c0 - v # Shock travelling into fluid at rest
return ρ, p, v
def Xi2(P, d, MP):
ρ = P.ρ
p = P.p()
A = P.A
c0 = c_0(ρ, p, A, MP)
if THERMAL:
ret = zeros([5, 4])
else:
ret = zeros([5, 3])
ret[0, 0] = ρ
ret[1, d] = ρ * c0**2
if VISCOUS:
σ = P.σ()
dσdρ = P.dσdρ()
ret[1, :3] += σ[d] - ρ * dσdρ[d]
ret[2:, :3] = A
if THERMAL:
T = P.T()
dTdp = P.dTdp()
ch = c_h(ρ, T, MP)
ret[1, 3] = ρ * ch**2 / dTdp
return ret
def c0_star(p_, P, MP):
r = P.ρ
p = P.p()
y = MP.γ
pINF = MP.pINF
c0 = c_0(r, p, eye(3), MP)
return c0 * pow((p_ + pINF) / (p + pINF), (y - 1) / (2 * y))
def Y_matrix(P, MP, sgn):
ρ = P.ρ
A = P.A
σρ = P.dσdρ()
σA = P.dσdA()
e0 = array([1, 0, 0])
Π1 = σA[0, :, :, 0]
tmp = zeros([5, 5])
tmp[:3, 0] = -σρ[0]
tmp[0, 1] = 1
tmp[:3, 2:5] = -Π1
if THERMAL:
tmp[3, 0] = P.dTdρ()
tmp[3, 1] = P.dTdp()
tmp[4, 0] = -1 / ρ
tmp[4, 2:5] = inv(A)[0]
else:
p = P.p()
σ = P.σ()
c0 = c_0(ρ, p, A, MP)
B = zeros([2, 3])
B[0, 0] = ρ
B[1] = σ[0] - ρ * σρ[0]
B[1, 0] += ρ * c0**2
rhs = column_stack((-σρ[0], e0))
C = solve(Π1, rhs)
BA_1 = dot(B, inv(A))
Z = eye(2) - dot(BA_1, C)
W = concatenate([eye(2), -BA_1], axis=1)
tmp[3:5] = solve(Z, W)
b = zeros([5, n1])
b[:n1, :n1] = eye(n1)
X = solve(tmp, b)
Ξ1 = Xi1(P, 0, MP)
Ξ2 = Xi2(P, 0, MP)
Ξ = dot(Ξ1, Ξ2)
w, vr = eig(Ξ)
D_1 = diag(1 / sqrt(w.real))
Q_1 = vr
Q = inv(Q_1)
Y0 = dot(Q_1, dot(D_1, Q))
return sgn * dot(Y0, dot(Ξ1, X))
def f1(z, P, MP):
r = P.ρ
p = P.p()
y = MP.γ
pINF = MP.pINF
c0 = c_0(r, p, eye(3), MP)
if (z > p):
temp = sqrt(A0(r, y) / (z + B(p, y, pINF)))
return (1 - (z - p) / (2 * (z + B(p, y, pINF)))) * temp
else:
temp = pow((z + pINF) / (p + pINF), -(y + 1) / (2 * y))
return temp * c0 / (y * (p + pINF))
def f0(z, P, MP):
r = P.ρ
p = P.p()
y = MP.γ
pINF = MP.pINF
c0 = c_0(r, p, eye(3), MP)
if (z > p):
temp = sqrt(A0(r, y) / (z + B(p, y, pINF)))
return (z - p) * temp
else:
temp = pow((z + pINF) / (p + pINF), (y - 1) / (2 * y))
return 2 * c0 / (y - 1) * (temp - 1)
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 riemann_constraints(P, sgn, MP, left=False):
""" K=R: sgn = -1
K=L: sgn = 1
NOTE: Uses atypical ordering
Extra constraints are:
dΣ = dΣ/dρ * dρ + dΣ/dp * dp + dΣ/dA * dA
dT = dT/dρ * dρ + dT/dp * dp
v*L = v*R
J*L = J*R
"""
_, Lhat, Rhat = eigen(P, 0, False, MP, values=False, right=True, left=left,
typical_order=False)
if left:
Lhat = left_riemann_constraints(P, Lhat, sgn)
ρ = P.ρ
A = P.A
σρ = P.dσdρ()
σA = P.dσdA()
e0 = array([1, 0, 0])
Π1 = σA[0, :, :, 0]
tmp = zeros([5, 5])
tmp[:3, 0] = -σρ[0]
tmp[0, 1] = 1
tmp[:3, 2:5] = -Π1
if THERMAL:
tmp[3, 0] = P.dTdρ()
tmp[3, 1] = P.dTdp()
tmp[4, 0] = -1 / ρ
tmp[4, 2:5] = inv(A)[0]
else:
p = P.p()
σ = P.σ()
c0 = c_0(ρ, p, A, MP)
B = zeros([2, 3])
B[0, 0] = ρ
B[1] = σ[0] - ρ * σρ[0]
B[1, 0] += ρ * c0**2
rhs = column_stack((-σρ[0], e0))
C = solve(Π1, rhs)
BA_1 = dot(B, inv(A))
Z = eye(2) - dot(BA_1, C)
W = concatenate([eye(2), -BA_1], axis=1)
tmp[3:5] = solve(Z, W)
b = zeros([5, n1])
b[:n1, :n1] = eye(n1)
X = solve(tmp, b)
Rhat[:5, :n1] = X
Ξ1 = Xi1(P, 0, MP)
Ξ2 = Xi2(P, 0, MP)
Q, Q_1, _, D_1 = decompose_Ξ(Ξ1, Ξ2)
Y0 = dot(Q_1, dot(D_1, Q))
Rhat[11:n5, :n1] = -sgn * dot(Y0, dot(Ξ1, X))
Rhat[:11, n1:n2] = 0
Rhat[11:n5, n1:n2] = sgn * dot(Q_1, D_1)
return Lhat, Rhat
def left_eigenvectors(P, d, MP, typical_order):
ρ = P.ρ
A = P.A
σA = P.dσdA()
σρ = P.dσdρ()
Π1 = σA[d, :, :, 0]
Π2 = σA[d, :, :, 1]
Π3 = σA[d, :, :, 2]
Ξ1 = Xi1(P, d, MP)
Ξ2 = Xi2(P, d, MP)
Q, Q_1, D, D_1 = decompose_Ξ(Ξ1, Ξ2)
n1, n2, n3, n4, n5, n6 = get_indexes()
e0 = array([1, 0, 0])
L = zeros([17, 17])
tmp1 = dot(Q, Ξ1)
tmp2 = -dot(Q[:, :3], Π2) / ρ
tmp3 = -dot(Q[:, :3], Π3) / ρ
tmp4 = dot(D, Q)
L[:n1, :5] = tmp1
L[n1:n2, :5] = tmp1
L[:n1, 5:8] = tmp2
L[n1:n2, 5:8] = tmp2
L[:n1, 8:11] = tmp3
L[n1:n2, 8:11] = tmp3
L[:n1, 11:n5] = tmp4
L[n1:n2, 11:n5] = -tmp4
if THERMAL:
tmp = inv(A)[0]
L[8, 0] = -1 / ρ
L[8, 2:5] = tmp
L[8, 5:8] = dot(tmp, solve(Π1, Π2))
L[8, 8:11] = dot(tmp, solve(Π1, Π3))
L[15, 15] = 1
L[16, 16] = 1
else:
σ = P.σ()
p = P.p()
c0 = c_0(ρ, p, A, MP)
B = zeros([2, 3])
B[0, 0] = ρ
B[1] = σ[0] - ρ * σρ[0]
B[1, 0] += ρ * c0**2
C = zeros([3, 2])
C[:, 0] = -solve(Π1, σρ[0])
C[:, 1] = solve(Π1, e0)
BA_1 = dot(B, inv(A))
Z = eye(2) - dot(BA_1, C)
X = zeros([2, 14])
X[:, :2] = eye(2)
X[:, 2:5] = -BA_1
X[:, 5:8] = -dot(BA_1, solve(Π1, Π2))
X[:, 8:11] = -dot(BA_1, solve(Π1, Π3))
L[6:8, :14] = solve(Z, X)
for i in range(6):
L[n3 + i, 5 + i] = 1
if typical_order:
L = reorder(L.T).T
if not THERMAL:
L = L[:14, :14]
return L
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
