Rayleigh Solver
Rayleigh ratios plot
Before explaining the solver, just a quick sanity check with the ratios plot.
[1]:
import numpy as np
import matplotlib.pyplot as plt
from pygasflow import rayleigh_solver
gamma = 1.4
M = np.linspace(1e-05, 5, 100)
results = rayleigh_solver("m", M, gamma)
plt.plot(M, results[1], label=r"$p / p^{*}$")
plt.plot(M, results[2], label=r"$\rho / \rho^{*}$")
plt.plot(M, results[3], label=r"$T / T^{*}$")
plt.plot(M, results[4], label=r"$P0 / P_{0}^{*}$")
plt.plot(M, results[5], label=r"$T0 / T_{0}^{*}$")
plt.plot(M, results[6], label=r"$U / U^{*}$")
plt.plot(M, results[7], label=r"$(s^{*}-s)/R$")
plt.xlim(0, max(M))
plt.ylim(0, 3)
plt.xlabel("M")
plt.ylabel("ratios")
plt.grid(which='major', linestyle='-', alpha=0.7)
plt.grid(which='minor', linestyle=':', alpha=0.5)
plt.legend(loc='upper right')
plt.suptitle("Rayleigh Flow")
plt.title("$\gamma$ = " + str(gamma))
plt.show()
Examples
This solver allows to quickly compute the Mach number, all the ratios and parameters (entropy paramater) by specifying what parameter we do know.
Let’s read the solver’s documentation:
[2]:
help(rayleigh_solver)
Help on function rayleigh_solver in module pygasflow.solvers.rayleigh:
rayleigh_solver(param_name, param_value, gamma=1.4, to_dict=False)
Compute all Rayleigh ratios and Mach number given an input parameter.
Parameters
----------
param_name : string
Name of the parameter given in input. Can be either one of:
* ``'m'``: Mach number
* ``'pressure'``: Critical Pressure Ratio P/P*
* ``'density'``: Critical Density Ratio rho/rho*
* ``'velocity'``: Critical Velocity Ratio U/U*.
* ``'temperature_sub'``: Critical Temperature Ratio T/T for
subsonic case.
* ``'temperature_super'``: Critical Temperature Ratio T/T* for
supersonic case.
* ``'total_pressure_sub'``: Critical Total Pressure Ratio P0/P0*
for subsonic case.
* ``'total_pressure_super'``: Critical Total Pressure Ratio P0/P0*
for supersonic case.
* ``'total_temperature_sub'``: Critical Total Temperature Ratio T0/T0*
for subsonic case.
* ``'total_temperature_super'``: Critical Total Temperature Ratio
T0/T0* for supersonic case.
* ``'entropy_sub'``: Entropy parameter (s*-s)/R for subsonic case.
* ``'entropy_super'``: Entropy parameter (s*-s)/R for supersonic case.
param_value : float/list/array_like
Actual value of the parameter. If float, list, tuple is given as
input, a conversion will be attempted.
gamma : float, optional
Specific heats ratio. Default to 1.4. Must be > 1.
to_dict : bool, optional
If False, the function returns a list of results. If True, it returns
a dictionary in which the keys are listed in the Returns section.
Default to False (return a list of results).
Returns
-------
m : array_like
Mach number
prs : array_like
Critical Pressure Ratio P/P*
drs : array_like
Critical Density Ratio rho/rho*
trs : array_like
Critical Temperature Ratio T/T*
tprs : array_like
Critical Total Pressure Ratio P0/P0*
ttrs : array_like
Critical Total Temperature Ratio T0/T0*
urs : array_like
Critical Velocity Ratio U/U*
eps : array_like
Critical Entropy Ratio (s*-s)/R
Examples
--------
Compute all ratios starting from a single Mach number:
>>> from pygasflow import rayleigh_solver
>>> rayleigh_solver("m", 2)
[2.0, 0.36363636363636365, 0.6875, 0.5289256198347108, 1.5030959785260414, 0.793388429752066, 1.4545454545454546, 1.2175752061512626]
Compute the subsonic Mach number starting from the critical entropy ratio:
>>> results = rayleigh_solver("entropy_sub", 0.5)
>>> print(results[0])
0.6634188478510624
Compute the critical temperature ratio starting from multiple Mach numbers
for a gas having specific heat ratio gamma=1.2:
>>> results = rayleigh_solver("m", [0.5, 1.5], 1.2)
>>> print(results[3])
[0.71597633 0.79547115]
Compute the critical temperature ratio starting from multiple Mach numbers
for a gas having specific heat ratio gamma=1.2, returning a dictionary:
>>> results = rayleigh_solver("m", [0.5, 1.5], 1.2, to_dict=True)
>>> print(results["trs"])
[0.71597633 0.79547115]
This is just a pretty print function…
[3]:
def print_rayleigh(M, prs, drs, trs, tprs, ttrs, urs, eps):
print("M \t\t {}".format(M))
print("P/P* \t\t {}".format(prs))
print("rho/rho* \t {}".format(drs))
print("T/T* \t\t {}".format(trs))
print("P0/P0* \t\t {}".format(tprs))
print("T0/T0* \t\t {}".format(ttrs))
print("U/U* \t\t {}".format(urs))
print("(s*-s)/R \t {}".format(eps))
print()
[4]:
result = rayleigh_solver('m', 2)
print_rayleigh(*result)
M 2.0
P/P* 0.36363636363636365
rho/rho* 0.6875
T/T* 0.5289256198347108
P0/P0* 1.5030959785260414
T0/T0* 0.793388429752066
U/U* 1.4545454545454546
(s*-s)/R 1.2175752061512626
[5]:
r1 = rayleigh_solver('pressure', 0.36363636363636365)
r2 = rayleigh_solver('density', 0.6875)
r3 = rayleigh_solver('temperature_super', 0.5289256198347108)
r4 = rayleigh_solver('total_pressure_super', 1.5030959785260414)
r5 = rayleigh_solver('total_temperature_super', 0.793388429752066)
r6 = rayleigh_solver('velocity', 1.4545454545454546)
r7 = rayleigh_solver('entropy_super', 1.2175752061512626)
print_rayleigh(*r1)
print_rayleigh(*r2)
print_rayleigh(*r3)
print_rayleigh(*r4)
print_rayleigh(*r5)
print_rayleigh(*r6)
print_rayleigh(*r7)
M 2.0
P/P* 0.36363636363636365
rho/rho* 0.6875
T/T* 0.5289256198347108
P0/P0* 1.5030959785260414
T0/T0* 0.793388429752066
U/U* 1.4545454545454546
(s*-s)/R 1.2175752061512626
M 2.0
P/P* 0.36363636363636365
rho/rho* 0.6875
T/T* 0.5289256198347108
P0/P0* 1.5030959785260414
T0/T0* 0.793388429752066
U/U* 1.4545454545454546
(s*-s)/R 1.2175752061512626
M 2.0000000000000004
P/P* 0.3636363636363635
rho/rho* 0.6875
T/T* 0.5289256198347106
P0/P0* 1.5030959785260414
T0/T0* 0.7933884297520659
U/U* 1.4545454545454546
(s*-s)/R 1.2175752061512637
M 1.9999999999986215
P/P* 0.3636363636367889
rho/rho* 0.6875000000001437
T/T* 0.5289256198352189
P0/P0* 1.5030959785245759
T0/T0* 0.7933884297523423
U/U* 1.4545454545451506
(s*-s)/R 1.2175752061490703
M 2.0
P/P* 0.36363636363636365
rho/rho* 0.6875
T/T* 0.5289256198347108
P0/P0* 1.5030959785260414
T0/T0* 0.793388429752066
U/U* 1.4545454545454546
(s*-s)/R 1.2175752061512626
M 1.9999999999999998
P/P* 0.3636363636363637
rho/rho* 0.6875
T/T* 0.5289256198347109
P0/P0* 1.503095978526041
T0/T0* 0.793388429752066
U/U* 1.4545454545454546
(s*-s)/R 1.2175752061512626
M 1.9999999999986215
P/P* 0.3636363636367889
rho/rho* 0.6875000000001437
T/T* 0.5289256198352189
P0/P0* 1.5030959785245759
T0/T0* 0.7933884297523423
U/U* 1.4545454545451506
(s*-s)/R 1.2175752061490703
Should you wish to solve for more than one Mach number of ratio at the same time, you can do:
[6]:
result = rayleigh_solver('m', [0.5, 1, 2, 4])
print_rayleigh(*result)
M [0.5 1. 2. 4. ]
P/P* [1.77777778 1. 0.36363636 0.1025641 ]
rho/rho* [2.25 1. 0.6875 0.609375]
T/T* [0.79012346 1. 0.52892562 0.16831032]
P0/P0* [1.1140525 1. 1.50309598 8.22684925]
T0/T0* [0.69135802 1. 0.79338843 0.58908613]
U/U* [0.44444444 1. 1.45454545 1.64102564]
(s*-s)/R [ 1.39984539 -0. 1.21757521 3.95954318]
We can do the same with different parameters, but we must be carefoul. In the following example, we ask the solver to solve for the supersonic case, given the entropy ratios computed just above:
[7]:
result = rayleigh_solver('entropy_super', [1.39984539, -0., 1.21757521, 3.95954318])
print_rayleigh(*result)
M [2.1148006 1. 2. 4. ]
P/P* [0.33051777 1. 0.36363636 0.1025641 ]
rho/rho* [0.67649772 1. 0.6875 0.609375 ]
T/T* [0.4885719 1. 0.52892562 0.16831032]
P0/P0* [1.63407725 1. 1.50309598 8.22684924]
T0/T0* [0.77132324 1. 0.79338843 0.58908613]
U/U* [1.47820159 1. 1.45454546 1.64102564]
(s*-s)/R [ 1.39984539 -0. 1.21757521 3.95954318]
Therefore, it is not a surprise that the Mach number for the \(\frac{s^{*} - s}{R} = 1.39984539\) is \(M = 2.1148006\).
This holds true for every parameter that does have distinct formulation for subsonic and supersonic case.
[8]:
result = rayleigh_solver('temperature_super', [0.79012346, 1., 0.52892562, 0.16831032])
print_rayleigh(*result)
M [1.42857142 1. 2. 4.00000003]
P/P* [0.62222223 1. 0.36363636 0.1025641 ]
rho/rho* [0.7875 1. 0.6875 0.609375]
T/T* [0.79012346 1. 0.52892562 0.16831032]
P0/P0* [1.08917256 1. 1.50309598 8.22684945]
T0/T0* [0.92718569 1. 0.79338843 0.58908613]
U/U* [1.26984127 1. 1.45454545 1.64102564]
(s*-s)/R [ 0.35002326 -0. 1.21757521 3.95954321]
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