34.4.3. Quasi 1-dimensional Euler equations for Perfect Ideal Gas#
Unsteady Euler equations for quasi-1 dimensional flows read
\[\begin{split}
\partial_t \begin{bmatrix} \rho A \\ \rho u A \\ \rho e^t A \end{bmatrix} +
\partial_z \begin{bmatrix} \rho u A \\ \frac{( \rho u A )^2}{\rho A} + p A \\ \frac{ (\rho e^t A + p A) (\rho u A) }{ \rho A } \end{bmatrix}
= \begin{bmatrix} 0 \\ p \partial_z A \\ 0 \end{bmatrix}
\end{split}\]
Method 1. The same conservative variables as the 1-dimensional flow solver are used \((\rho, m, E^t)\), and non-uniform
appears as a source in momentum equation
appears in discretization for cells \(A_i\) and as a factor multiplying fluxes \(A_{i+1/2} F_{i+1/2}\)
\[\mathbf{u}_i^{n+1} = \mathbf{u}_i^{n} - \frac{\Delta t}{\Delta A_i} \left( \mathbf{F}_{i+1/2} A_{i+1/2} - \mathbf{F}_{i-1/2} A_{i-1/2} \right) + \Delta t \, \mathbf{s}_i\]
Method 2. Another approach would be using \((\rho A, m A, E^t A)\) as conservative variables.
todo
34.4.3.1. Useful classes for numerical methods#
34.4.3.1.1. Basic libraries#
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import numpy as np
import matplotlib.pyplot as plt
34.4.3.1.2. Parent class#
\(\texttt{HyperbolicSystem1d()}\) class with common functions (or just their templates) required to implement a finite volume method solver.
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class HyperbolicSystem1d():
"""
Analytical quantities of a Hyperbolic linear system in a 1-dimensional domain
"""
def __init(self, **params):
pass
def flux(self,):
""" Analytical flux, f """
pass
def A(self,):
""" Advection matrix, A """
pass
def s(self,):
""" Eigenvalues of matrix, s """
pass
def R(self,):
""" Matrix of right eigenvectors, R """
pass
def L(self,):
""" Matrix of left eigenvectors, L=inv(R) """
pass
def spectrum(self,):
""" Spectrum of matrix A() """
return self.s(), self.R(), self.L()
def absA(self, u):
""" A = R * S * L, |A| = R * |S| * L """
return self.R(u) @ np.diag(np.abs(self.s(u))) @ self.L(u)
def roe_intermediate_state(self,):
""" Roe intermediate state for Roe linearization """
pass
34.4.3.1.3. Euler equations for perfect ideal gas class#
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class Euler1dPIG(HyperbolicSystem1d):
"""
Analytical quantities of a Hyperbolic linear system in a 1-dimensional domain
Physical quantities:
- conservative: (rho, m, Et)
- physical/convective (rho, u, e) or (rho, u, s)
Parameters:
- gamma: cp/cv ratio
PIG (Perfect Ideal Gas)
gamma = cp / cv --> cv = 1 / ( gamma - 1 ) * R
cp - cv = R --> cp = gamma / ( gamma - 1 ) * R
p = rho * R * T = rho * R/cv e = (gamma-1) * rho * e = (gamma-1) * E
e = cv * T
"""
def __init__(self, gamma=5./3., name=""):
""" """
# Speed of sound, a, and pressure, p, are functions of the TD state
# ...add functions fun_a(...), fun_p(...)
# - which physical variables?
# - equations of state for different fluids (PIG, VdW,...)
self.gamma = gamma
self.gamma_1 = self.gamma - 1
self.name = name
def fun_p(self, u):
"""
Pressure as a function of conservative variables
p = rho R T = rho R/cv e = (gamma-1) rho e = (gamma-1) ( Et - .5* m^2/rho)
"""
return self.gamma_1 * ( u[2] - .5 * u[1]**2 / u[0] )
def phys_from_cons(self, u):
"""
rho = rho
u = m / rho
p = (gamma-1) * E
= (gamma-1) * ( Et - .5 * rho * u^2 )
= (gamma-1) * ( Et - .5 * m^2 / rho )
"""
return np.array([ u[0], u[1]/u[0], self.fun_p(u) ])
def cons_from_phys(self, p):
"""
rho = rho
m = rho * u
Et = rho * ( e + .5 * u^2 )
= E + .5 * rho * u^2
= p / ( gamma - 1 ) + .5 * rho * u^2
"""
Et = p[2]/self.gamma_1 + .5 * p[0] * p[1]**2
return np.array([ p[0], p[0]*p[1], Et])
def flux(self, u):
""" Numerical flux
u: conservative variables (rho, m, Et)
F = ( m, rho*u^2+p(...), u*(Et+p(...)) )
"""
vel = u[1]/u[0]
p = self.fun_p(u)
return np.array([ u[1], u[0]*vel**2+p, vel*(u[2]+p) ])
def A(self, u):
"""
Convection matrix, A; input: conservative variables
A = [[ 0, 1, 0 ],
[ u^2+p_rho, 2u+p_m, p_Et ],
[-u*ht+u*p_rho, ht+u*p_m, u*(1+p_Et)]]
"""
vel = u[1]/u[0]
p = self.fun_p(u)
ht = ( u[2] + p ) / u[0]
p_rho = .5 * self.gamma_1 * vel**2
p_m = - self.gamma_1 * vel
p_Et = self.gamma_1
return np.array(
[[ .0, 1.,.0 ],
[ -vel**2+p_rho, 2*vel+p_m, p_Et ],
[ vel*(-ht+p_rho), ht+vel*p_m, vel*(1+p_Et) ]])
def s(self, u):
"""
Eigenvalues of A; input: conservative variables
s = [u-a, u, u+a]
"""
vel = u[1]/u[0]
p = self.fun_p(u)
# Speed of sound, a^2 = gamma R T = gamma p / rho
a = np.sqrt(self.gamma * self.fun_p(u)/u[0])
return np.array([vel-a, vel, vel+a])
def R(self, u):
"""
Right eigenvectors of A; input: conservative variables
R = ...
"""
vel = u[1]/u[0]
p = self.fun_p(u)
# Speed of sound, a^2 = gamma R T = gamma p / rho
a = np.sqrt(self.gamma * self.fun_p(u)/u[0])
ht = ( u[2] + p ) / u[0]
p_rho = .5 * self.gamma_1 * vel**2
p_m = - self.gamma_1 * vel
p_Et = self.gamma_1
return np.array([
[1, vel-a, ht-vel*a],
[1, vel , vel**2-p_rho/p_Et],
[1, vel+a, ht+vel*a]]).T
def L(self, u):
"""
Left eigenvectors of A; input: conservative variables
L = ...
"""
vel = u[1]/u[0]
p = self.fun_p(u)
# Speed of sound, a^2 = gamma R T = gamma p / rho
a = np.sqrt(self.gamma * self.fun_p(u)/u[0])
ht = ( u[2] + p ) / u[0]
p_rho = .5 * self.gamma_1 * vel**2
p_m = - self.gamma_1 * vel
p_Et = self.gamma_1
return np.array([
[ p_rho + vel*a , p_m - a , p_Et ],
[-2*(p_rho-a**2),-2*p_m ,-2*p_Et ],
[ p_rho - vel*a , p_m + a , p_Et ]
]) / ( 2. * a**2 )
def spectrum(self, u):
""" Spectrum of matrix A; input: conservative variables """
return self.s(u), self.R(u), self.L(u)
def roe_intermediate_state(self, u_0, u_1):
"""
For p-sys:
rho_roe = ANY! = ... CHOOSE ONE: average = 0.5*(rho_0+rho_1)
vel_roe = (sqrt(rho_0)*vel_0+sqrt(rho_1)*vel_1)/(sqrt(rho_0)+sqrt(rho_1))
- convective/physical: (rho, vel) = ( rho_roe, vel_roe )
- conservatvie : (rho, mom) = ( rho_roe, rho_roe*vel_roe)
F_roe(u) ~ dFdu * du ~ A(u_roe(u0,u1)) * (u_1 - u_0)
"""
#> Density (for a PIG, it's arbitrary once consistency is provided)
rho_roe = .5*(u_0[0]+u_1[0])
vel_0, vel_1 = u_0[1]/u_0[0], u_1[1]/u_1[0]
p_0, p_1 = self.fun_p(u_0), self.fun_p(u_1)
ht_0, ht_1 = ( u_0[2] + p_0 ) / u_0[0], ( u_1[2] + p_1 ) / u_1[0]
#> Velocity and total enthalpy (Roe linearization)
vel_roe = (vel_0*np.sqrt(u_0[0])+vel_1*np.sqrt(u_1[0])) / \
(np.sqrt(u_0[0])+np.sqrt(u_1[0]))
ht_roe = ( ht_0*np.sqrt(u_0[0])+ ht_1*np.sqrt(u_1[0])) / \
(np.sqrt(u_0[0])+np.sqrt(u_1[0]))
#> Enthalpy and pressure (PIG)
h_roe = ht_roe - .5 * vel_roe**2
# p = rho R T = rho R/cP h = (gamma-1)/gamma * rho * h
p_roe = self.gamma_1/self.gamma * rho_roe * h_roe
#> Momentum and total energy
m_roe = rho_roe * vel_roe
Et_roe = rho_roe * ht_roe - p_roe
return np.array([rho_roe, m_roe, Et_roe])
def absA_entropy_fix(self, u, delta=.1):
""" """
#> Eigenvalues s1 = u-a, s2 = u, s3 = u+a
eigvals = self.s(u)
#> Speed of sound a = .5 * ( s3 - s1 )
a = .5 * ( np.max(eigvals) - np.min(eigvals) )
#> Entropy fix
entropy_evals = np.abs(eigvals)
entropy_evals[entropy_evals < delta*a] = \
.5 * entropy_evals[entropy_evals < delta*a]**2 / ( delta * a ) + .5 * ( delta * a )
return self.R(u) @ np.diag(entropy_evals) @ self.L(u)
def roe_flux(self, u_0, u_1):
""" """
u_roe = self.roe_intermediate_state(u_0, u_1)
return .5 * (self.flux(u_1) + self.flux(u_0) + \
self.absA_entropy_fix(u_roe) @ ( u_0 - u_1 ) )
def flux_at_boundary(self, boundary, u):
"""
boundary: boundary conditino dict
u: conservative variables (rho, m, Et) at boundary cell
"""
if ( boundary["type"] == "wall" ):
u_boundary = u.copy(); u_boundary[1] = -u[1]
if ( boundary["nor"] < 0 ):
return self.roe_flux(u_boundary, u)
else:
return self.roe_flux(u, u_boundary)
elif ( boundary["type"] == "inflow" or boundary["type"] == "outflow" ):
delta_v = self.L(u) @ ( self.cons_from_phys(boundary["phys_ext"]) - u )
eigval = self.s(u)
delta_v[ eigval*boundary["nor"] > 0 ] = .0
u_flux = u + self.R(u) @ delta_v
return self.flux(u_flux)
34.4.3.2. Numerical simulation#
34.4.3.2.1. System of equations#
Initialized the system of equations to be solved
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# Number of unknown fields, nu
#> P-system, Shallow Water
# - conservative : u = (rho, mom)
# - physical/convective: p = (rho, vel), with mom = rho*vel
#> Euler
# - conservative : u = (rho, mom, Et)
# - physical/convective: p = (rho, vel, e) or = (rho, vel, s), with mom = rho*vel
gamma = 5. / 3.
#> Area
# A(x) = a + b * cos(2*pi*x)
# Amax = a + b
# Amin = a - b
# a = ( Amax + Amin ) / 2
# b = ( Amax - Amin ) / 2
A_min_max_ratio = 1. / 2
a_coeff = .5 * ( 1 + A_min_max_ratio )
b_coeff = .5 * ( 1 - A_min_max_ratio )
A_fun = lambda x: a_coeff + b_coeff * np.cos(2 * np.pi * x)
dA_fun = lambda x: -2. * np.pi * b_coeff * np.sin(2 * np.pi * x)
#> Parameters and initialization of the system
# S = Psys1d(a=1); nu = 2
# S = ShallowWater1d(g=1); nu = 2
# S = Psys1dLinearized(a=1); nu = 2
S = Euler1dPIG(gamma=gamma,); nu = 3
34.4.3.2.2. Domain#
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#> Parameters
x0, x1 = 0., 1. # coords of left and right boundaries
ne = 50 # n. of elements
#> Domain
xs = np.linspace(x0, x1, ne+1) # coord of the cell boundaries
xc = .5 * ( xs[:-1] + xs[1:] ) # coord of the cell centers
dx = xs[1:] - xs[:-1] # volume of the cells
#> Quasi-1 dimensional flows
A_cell = A_fun(xc)
dA_cell = dA_fun(xc)
A_face = A_fun(xs)
34.4.3.2.3. Boundary conditions#
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#> Boundary conditions
# boundary_0 = { # left boundary
# "type": "wall",
# "nor" : -1.
# }
# boundary_1 = { # right boundary
# "type": "wall",
# "nor" : 1.
# }
boundary_0 = { # left boundary
"type": "inflow",
"phys_ext": np.array([1., 1.2, .5]), # rho_ext, u_ext, p_ext
"nor": -1. # nx
}
boundary_1 = { # right boundary
"type": "outflow",
# "phys_ext": [ rho_ext, u_ext, p_ext ],
# "phys_ext": np.array([.1, .0, .1]), # 0. No shock in the divergent
# "phys_ext": np.array([.5, .0, .5]), # 1. Shock in the divergent at x~0.9
"phys_ext": np.array([.75, .0, .75]), # 2. Shock in the divergent at x~0.7
# "phys_ext": np.array([1., .0, 1.]), # 3. No supersonic flow
"nor": 1. # nx
}
34.4.3.2.4. Initial conditions#
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#> Initial conditions, in physical variables
# #> Initial condition 0: uniform
# rhoL, rhoR = 1.0, 1.0
# velL, velR = 0.0, 0.0
# pL , pR = 1., 1.
#> Initial condition 1: u=0, T uniform, pL != pR
rhoL, rhoR = 1., 1.
velL, velR = 0., 0.
pL , pR = .5, .5
rhoL, momL, EtL = S.cons_from_phys(np.array([rhoL, velL, pL]))
rhoR, momR, EtR = S.cons_from_phys(np.array([rhoR, velR, pR]))
#> Write initial conditions into uc0 array
uc0 = np.zeros((nu,ne))
uc0[0,:] = np.where( xc<.5, rhoL, rhoR )
uc0[1,:] = np.where( xc<.5, momL, momR )
uc0[2,:] = np.where( xc<.5, EtL , EtR )
34.4.3.2.5. Simulation parameters#
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#> Time
t0, t1, dt = 0., 5., .0025 # 25
nt = int((t1-t0)/dt)+1
tv = np.arange(nt)*dt
#> Numerical schemes
#> Time integration
time_integration = 'ee' # ...
numerical_flux = 'roe' # ...
#> Time loop
uuc = np.zeros((nt, nu, ne)) # Array to store solution, for small dimensional pbs
uc = uc0.copy()
uuc[0,:,:] = uc
34.4.3.2.6. Time loop#
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#> Time loop
it = 0
while it < nt-1:
t = tv[it] # time, if there's some function of time
#> Evaluate fluxes at internal boundaries
if numerical_flux == 'roe':
flux = np.array([ S.roe_flux(uc[:,ia], uc[:,ia+1]) for ia in range(ne-1) ]).T
#> Flux at boundaries - Treat boundary conditions
flux_0 = S.flux_at_boundary(boundary_0, uc[:, 0])
flux_1 = S.flux_at_boundary(boundary_1, uc[:,-1])
# flux = np.append(np.append(flux_0, flux), flux_1)
flux = np.concatenate((flux_0[:,np.newaxis], flux, flux_1[:, np.newaxis]), axis=1)
#> Time integration
if time_integration == 'ee':
dt = tv[it+1] - tv[it]
# Pressure (PIG), p = (gamma-1) * ( Et - .5 * m^2 / rho )
p_cell = S.gamma_1 * ( uc[2,:] - 0.5 * uc[1,:]**2 / uc[0,:] )
#> Source
source = np.zeros((nu,ne)); source[1,:] = p_cell * dA_cell
#> Update cell values with flux and source contributions
uc += ( flux[:,:-1] * A_face[:-1] - flux[:,1:] * A_face[1:] ) \
* dt / ( dx * A_cell ) + source * dt
it += 1 # Updating counter
uuc[it,:,:] = uc.copy() # Storing results
34.4.3.2.7. Post-processing#
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fig, ax = plt.subplots(1,5, figsize=(20,5))
iax = 0
ax[iax].imshow(uuc[:,0,:], aspect="auto")
# ax[0].set_colorbar()
ax[iax].set_xlabel('x')
ax[iax].set_ylabel('t')
ax[iax].set_title('Density')
iax = 1
ax[iax].imshow(uuc[:,1,:], aspect="auto")
# ax[0].set_colorbar()
ax[iax].set_xlabel('x')
ax[iax].set_ylabel('t')
ax[iax].set_title('Momentum')
iax = 2
ax[iax].imshow(uuc[:,2,:], aspect="auto")
# ax[0].set_colorbar()
ax[2].set_xlabel('x')
ax[iax].set_ylabel('t')
ax[iax].set_title('Total Energy')
iax = 3
ax[iax].imshow(uuc[:,1,:]/uuc[:,0,:], aspect="auto")
# ax[0].set_colorbar()
ax[iax].set_xlabel('x')
ax[iax].set_ylabel('t')
ax[iax].set_title('Velocity')
# pressure
# p = (gamma-1) * ( Et - .5 * m^2 / rho )
iax = 4
ax[iax].imshow(S.gamma_1 * (uuc[:,2,:]-.5*uuc[:,1,:]**2/uuc[:,0,:]), aspect="auto")
# ax[0].set_colorbar()
ax[iax].set_xlabel('x')
ax[iax].set_ylabel('t')
ax[iax].set_title('Pressure')
Text(0.5, 1.0, 'Pressure')
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n_plots = 10
i_dt = int(np.floor(nt/n_plots))
fig, ax = plt.subplots(n_plots, figsize=(5,20))
for i in range(n_plots):
p = S.gamma_1 * (uuc[i_dt*i,2,:]-.5*uuc[i_dt*i,1,:]**2/uuc[i_dt*i,0,:])
ax[i].plot(xc,uuc[i_dt*i,0,:], label="rho")
ax[i].plot(xc,uuc[i_dt*i,1,:], label="m")
# ax[i].plot(xc,uuc[i_dt*i,2,:], label='Et')
ax[i].plot(xc,uuc[i_dt*i,1,:]/uuc[i_dt*i,0,:], label="v")
ax[i].plot(xc, p, label="p")
ax[i].plot(xc, np.sqrt(S.gamma * p / uuc[i_dt*i,0,:]), label="a")
ax[i].grid()
ax[i].legend(bbox_to_anchor=(1.05, 1.05), loc='upper left')
# Pressure: p = (gamma-1) * ( Et - .5 * m^2 / rho )
# rho, m, Et = 1, 1, 2
# gamma = 5/3
# -> u = 1
# -> p = 2/3 * ( 2 - .5 * 1 ) = 2 / 3 * 1.5 = 1
