Solvers

Isentropic Solver

pygasflow.solvers.isentropic.isentropic_solver(param_name, param_value, gamma=1.4, to_dict=False)

Compute all isentropic ratios and Mach number given an input parameter.

Parameters

param_namestring

Name of the parameter given in input. Can be either one of:

• 'm': Mach number

• 'pressure': Pressure Ratio \(\frac{P}{P_{0}}\)

• 'density': Density Ratio \(\frac{\rho}{\rho_{0}}\)

• 'temperature': Temperature Ratio \(\frac{T}{T_{0}}\)

• 'crit_area_sub': Critical Area Ratio \(\frac{A}{A^{*}}\) for subsonic case.

• 'crit_area_super': Critical Area Ratio \(\frac{A}{A^{*}}\) for supersonic case.

• 'mach_angle': Mach Angle in degrees.

• 'prandtl_meyer': Prandtl-Meyer Angle in degrees.

param_valuefloat/list/array_like

Actual value of the parameter. If float, list, tuple is given as input, a conversion will be attempted.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be > 1.

to_dictbool, 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

marray_like

Mach number

prarray_like

Pressure Ratio \(\frac{P}{P_{0}}\)

drarray_like

Density Ratio \(\frac{\rho}{\rho_{0}}\)

trarray_like

Temperature Ratio \(\frac{T}{T_{0}}\)

prsarray_like

Critical Pressure Ratio \(\frac{P}{P^{*}}\)

drsarray_like

Critical Density Ratio \(\frac{\rho}{\rho^{*}}\)

trsarray_like

Critical Temperature Ratio \(\frac{T}{T^{*}}\)

ursarray_like

Critical Velocity Ratio \(\frac{U}{U^{*}}\)

arsarray_like

Critical Area Ratio \(\frac{A}{A^{*}}\)

maarray_like

Mach Angle

pmarray_like

Prandtl-Meyer Angle

Examples

Compute all ratios starting from a single Mach number:

>>> from pygasflow import isentropic_solver
>>> isentropic_solver("m", 2)
[2.0, 0.12780452546295096, 0.2300481458333117, 0.5555555555555556, 0.24192491286747442, 0.36288736930121157, 0.6666666666666667, 2.3515101530718505, 1.6875000000000002, 30.000000000000004, 26.379760813416457]

Compute all parameters starting from the pressure ratio:

>>> isentropic_solver("pressure", 0.12780452546295096)
[1.9999999999999996, 0.12780452546295107, 0.23004814583331185, 0.5555555555555558, 0.24192491286747458, 0.3628873693012118, 0.6666666666666669, 2.3515101530718505, 1.6874999999999993, 30.00000000000001, 26.379760813416446]

Compute the Mach number starting from the Mach Angle:

>>> results = isentropic_solver("mach_angle", 25)
>>> print(results[0])
2.3662015831524985

Compute the pressure ratios starting from two Mach numbers:

>>> results = isentropic_solver("m", [2, 3])
>>> print(results[1])
[0.12780453 0.02722368]

Compute the pressure ratios starting from two Mach numbers, returning a dictionary:

>>> results = isentropic_solver("m", [2, 3], to_dict=True)
>>> print(results["pr"])
[0.12780453 0.02722368]

Fanno Solver

pygasflow.solvers.fanno.fanno_solver(param_name, param_value, gamma=1.4, to_dict=False)

Compute all Fanno ratios and Mach number given an input parameter.

Parameters

param_namestring

Name of the parameter given in input. Can be either one of:

• 'm': Mach number

• 'pressure': Critical Pressure Ratio \(\frac{P}{P^{*}}\)

• 'density': Critical Density Ratio \(\frac{\rho}{\rho^{*}}\)

• 'temperature': Critical Temperature Ratio \(\frac{T}{T^{*}}\)

• 'total_pressure_sub': Critical Total Pressure Ratio \(\frac{P_{0}}{P_{0}^{*}}\) for subsonic case.

• 'total_pressure_super': Critical Total Pressure Ratio \(\frac{P_{0}}{P_{0}^{*}}\) for supersonic case.

• 'velocity': Critical Velocity Ratio \(\frac{U}{U^{*}}\).

• 'friction_sub': Critical Friction parameter \(\frac{4 f L^{*}}{D}\) for subsonic case.

• 'friction_super': Critical Friction parameter \(\frac{4 f L^{*}}{D}\) for supersonic case.

• 'entropy_sub': Entropy parameter \(\frac{s^{*} - s}{R}\) for subsonic case.

• 'entropy_super': Entropy parameter \(\frac{s^{*} - s}{R}\) for supersonic case.

param_valuefloat/list/array_like

Actual value of the parameter. If float, list, tuple is given as input, a conversion will be attempted.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be > 1.

to_dictbool, 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

marray_like

Mach number

prsarray_like

Critical Pressure Ratio \(\frac{P}{P^{*}}\)

drsarray_like

Critical Density Ratio \(\frac{\rho}{\rho^{*}}\)

trsarray_like

Critical Temperature Ratio \(\frac{T}{T^{*}}\)

tprsarray_like

Critical Total Pressure Ratio \(\frac{P_{0}}{P_{0}^{*}}\)

ursarray_like

Critical Velocity Ratio \(\frac{U}{U^{*}}\)

fpsarray_like

Critical Friction Parameter \(\frac{4 f L^{*}}{D}\)

epsarray_like

Critical Entropy Ratio \(\frac{s^{*} - s}{R}\)

Examples

Compute all ratios starting from a Mach number:

>>> from pygasflow import fanno_solver
>>> fanno_solver("m", 2)
[2.0, 0.408248290463863, 0.6123724356957945, 0.6666666666666667, 1.6875000000000002, 1.632993161855452, 0.3049965025814798, 0.523248143764548]

Compute the subsonic Mach number starting from the critical friction parameter:

>>> results = fanno_solver("friction_sub", 0.3049965025814798)
>>> print(results[0])
0.6572579935727846

Compute the critical temperature ratio starting from multiple Mach numbers for a gas having specific heat ratio gamma=1.2:

>>> results = fanno_solver("m", [0.5, 1.5], 1.2)
>>> print(results[3])
[1.07317073 0.89795918]

Compute the critical temperature ratio starting from multiple Mach numbers for a gas having specific heat ratio gamma=1.2, returning a dictionary:

>>> results = fanno_solver("m", [0.5, 1.5], 1.2, to_dict=True)
>>> print(results["trs"])
[1.07317073 0.89795918]

Rayleigh Solver

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_namestring

Name of the parameter given in input. Can be either one of:

• 'm': Mach number

• 'pressure': Critical Pressure Ratio \(\frac{P}{P^{*}}\)

• 'density': Critical Density Ratio \(\frac{\rho}{\rho^{*}}\)

• 'velocity': Critical Velocity Ratio \(\frac{U}{U^{*}}\).

• 'temperature_sub': Critical Temperature Ratio T/T for subsonic case.

• 'temperature_super': Critical Temperature Ratio \(\frac{T}{T^{*}}\) for supersonic case.

• 'total_pressure_sub': Critical Total Pressure Ratio \(\frac{P_{0}}{P_{0}^{*}}\) for subsonic case.

• 'total_pressure_super': Critical Total Pressure Ratio \(\frac{P_{0}}{P_{0}^{*}}\) for supersonic case.

• 'total_temperature_sub': Critical Total Temperature Ratio \(\frac{T_{0}}{T_{0}^{*}}\) for subsonic case.

• 'total_temperature_super': Critical Total Temperature Ratio \(\frac{T_{0}}{T_{0}^{*}}\) for supersonic case.

• 'entropy_sub': Entropy parameter \(\frac{s^{*} - s}{R}\) for subsonic case.

• 'entropy_super': Entropy parameter \(\frac{s^{*} - s}{R}\) for supersonic case.

param_valuefloat/list/array_like

Actual value of the parameter. If float, list, tuple is given as input, a conversion will be attempted.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be > 1.

to_dictbool, 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

marray_like

Mach number

prsarray_like

Critical Pressure Ratio \(\frac{P}{P^{*}}\)

drsarray_like

Critical Density Ratio \(\frac{\rho}{\rho^{*}}\)

trsarray_like

Critical Temperature Ratio \(\frac{T}{T^{*}}\)

tprsarray_like

Critical Total Pressure Ratio \(\frac{P_{0}}{P_{0}^{*}}\)

ttrsarray_like

Critical Total Temperature Ratio \(\frac{T_{0}}{T_{0}^{*}}\)

ursarray_like

Critical Velocity Ratio \(\frac{U}{U^{*}}\)

epsarray_like

Critical Entropy Ratio \(\frac{s^{*} - s}{R}\)

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]

Shockwave Solvers

pygasflow.solvers.shockwave.shockwave_solver(p1_name, p1_value, p2_name='beta', p2_value=90, gamma=1.4, flag='weak', to_dict=False)

Try to compute all the ratios, angles and mach numbers across the shock wave.

Remember: a normal shock wave has a wave angle beta=90 deg.

Parameters

p1_namestring

Name of the first parameter given in input. Can be either one of:

• 'pressure': Pressure Ratio \(\frac{P_{2}}{P_{1}}\)

• 'temperature': Temperature Ratio \(\frac{T_{2}}{T_{1}}\)

• 'density': Density Ratio \(\frac{\rho_{2}}{\rho_{1}}\)

• 'total_pressure': Total Pressure Ratio \(\frac{P_{2}^{0}}{P_{1}^{0}}\)

• 'm1': Mach upstream of the shock wave

• 'mn1': Normal Mach upstream of the shock wave

• 'mn2': Normal Mach downstream of the shock wave

• 'beta': The shock wave angle [in degrees]. It can only be used if p2_name='theta'.

• 'theta': The deflection angle [in degrees]. It can only be used if p2_name='beta'.

If the parameter is a ratio, it is in the form downstream/upstream:

p1_valuefloat

Actual value of the parameter.

p2_namestring, optional

Name of the second parameter. It could either be:

• 'beta': Shock wave angle.

• 'theta': Flow deflection angle.

• 'mn1': Input Normal Mach number.

Default to 'beta'.

p2_valuefloat, optional

Value of the angle in degrees. Default to 90 degrees (normal shock wave).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be > 1.

flagstring, optional

Chose what solution to compute if the angle ‘theta’ is provided. Can be either 'weak' or 'strong'. Default to 'weak'.

to_dictbool, 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

m1float

Mach number upstream of the shock wave.

mn1float

Normal Mach number upstream of the shock wave.

m2float

Mach number downstream of the shock wave.

mn2float

Normal Mach number downstream of the shock wave.

betafloat

Shock wave angle in degrees.

thetafloat

Flow deflection angle in degrees.

prfloat

Pressure ratio across the shock wave.

drfloat

Density ratio across the shock wave.

trfloat

Temperature ratio across the shock wave.

tprfloat

Total Pressure ratio across the shock wave.

Examples

Compute all ratios across a normal shockwave starting with the upstream Mach number:

>>> from pygasflow import shockwave_solver
>>> shockwave_solver("m1", 2)
[2.0, 2.0, 0.5773502691896257, 0.5773502691896257, 90.0, 5.847257748779064e-15, 4.5, 2.666666666666667, 1.6874999999999998, 0.7208738614847455]

Compute all ratios and parameters across an oblique shockwave starting from the shockwave angle and the deflection angle:

>>> shockwave_solver("theta", 8, "beta", 80)
[1.511670289641015, 1.4887046212366817, 0.7414131402857721, 0.7051257983356364, 80.0, 7.999999999999998, 2.418948357506694, 1.84271116608139, 1.312711618636739, 0.9333272472012358]

Compute the Mach number downstream of an oblique shockwave starting with multiple upstream Mach numbers:

>>> results = shockwave_solver("m1", [1.5, 3], "beta", [60, 60])
>>> print(results[2])
[1.04454822 1.12256381]

Compute the Mach number downstream of an oblique shockwave starting with multiple upstream Mach numbers, returning a dictionary:

>>> results = shockwave_solver("m1", [1.5, 3], "beta", [60, 60], to_dict=True)
>>> print(results["m2"])
[1.04454822 1.12256381]

pygasflow.solvers.shockwave.conical_shockwave_solver(M1, param_name, param_value, gamma=1.4, flag='weak', to_dict=False)

Try to compute all the ratios, angles and mach numbers across the conical shock wave.

Parameters

M1float

Upstream Mach number. Must be M1 > 1.

param_namestring

Name of the parameter given in input. Can be either one of:

• 'mc': Mach number at the cone’s surface.

• 'theta_c': Half cone angle.

• 'beta': shock wave angle.

param_valuefloat

Actual value of the parameter. Requirements:

• Mc >= 0

• 0 < beta <= 90

• \(0 < \theta_{c} < 90\)

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be > 1.

flagstring, optional

Can be either 'weak' or 'strong'. Default to 'weak' (in conical shockwaves, the strong solution is rarely encountered).

to_dictbool, 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

mfloat

Upstream Mach number.

mcfloat

Mach number at the surface of the cone.

theta_cfloat

Half cone angle.

betafloat

Shock wave angle.

deltafloat

Flow deflection angle.

prfloat

Pressure ratio across the shock wave.

drfloat

Density ratio across the shock wave.

trfloat

Temperature ratio across the shock wave.

tprfloat

Total Pressure ratio across the shock wave.

pc_p1float

Pressure ratio between the cone’s surface and the upstream condition.

rhoc_rho1float

Density ratio between the cone’s surface and the upstream condition.

Tc_T1float

Temperature ratio between the cone’s surface and the upstream condition.

Examples

Compute all quantities across a conical shockwave starting from the upstream Mach number and the half cone angle:

>>> from pygasflow import conical_shockwave_solver
>>> conical_shockwave_solver(2.5, "theta_c", 15)
[2.5, 2.1179295900668067, 15.0, 28.45459370447941, 6.229019180107892, 1.488659699248579, 1.3262664608044694, 1.122443900410184, 0.9936173734022627, 1.8051864085591218, 1.5220731269187135, 1.186005045771711]

Compute the pressure ratio across a conical shockwave starting with multiple upstream Mach numbers and Mach numbers at the cone surface:

>>> results = conical_shockwave_solver([2.5, 5], "mc", 1.5)
>>> print(results[5])
[ 3.42459174 18.60172442]

Compute the pressure ratio across a conical shockwave starting with multiple upstream Mach numbers and Mach numbers at the cone surface, but returning a dictionary:

>>> results = conical_shockwave_solver([2.5, 5], "mc", 1.5, to_dict=True)
>>> print(results["pr"])
[ 3.42459174 18.60172442]

De Laval Solver

pygasflow.solvers.de_laval.find_shockwave_area_ratio(Ae_At_ratio, Pe_P0_ratio, R, gamma)

Iterative procedure to find the critical area ratio where the shock wave happens.

Parameters

Ae_At_ratiofloat

Area ratio between the exit section of the divergent and the throat section.

Pe_P0_ratiofloat

Pressure ratio between the back (= exit) pressure to the reservoir pressure.

Rfloat

Specific Gas Constant.

gammafloat

Specific Heats ratio.

Returns

Asw/Atfloat

The critical area ratio Asw/At

Examples

>>> from pygasflow.solvers import find_shockwave_area_ratio
>>> find_shockwave_area_ratio(20, 0.1, 287, 1.4)
14.247517873372033

class pygasflow.solvers.de_laval.De_Laval_Solver(gas, geometry, input_state, Pb_P0_ratio=None)

Solve a De Laval Nozzle (Convergent-Divergent nozzle), starting from stagnation conditions and the geometry type.

Examples

Visualize all the quantities along the length of a TOP nozzle using air and a back-to-stagnation pressure ratio of 0.12.

from pygasflow.nozzles import CD_TOP_Nozzle
from pygasflow.solvers import De_Laval_Solver
from pygasflow.utils import Ideal_Gas, Flow_State
# Set air as the gas to use in the nozzle
gas = Ideal_Gas(287, 1.4)
# stagnation condition
upstream_state = Flow_State(p0=8 * 101325, t0 = 303.15)
Ri = 0.4            # Inlet Radius
Rt = 0.2            # Throat Radius
Re = 1.2            # Exit (outlet) Radius
theta_c = 40        # half cone angle of the divergent
theta_N = 15        # half cone angle of the convergent
Rbt = 0.75 * Rt     # Junction radius between the convergent and divergent
Rbc = 1.5 * Rt      # Junction radius between the "combustion chamber" and convergent
K = 0.8             # Fractional Length of the TOP nozzle
geom_type = "axisymmetric"  # geometry type
geom_top = CD_TOP_Nozzle(Ri, Re, Rt, Rbc, theta_c, K, geom_type, 1000)
nozzle_top = De_Laval_Solver(gas, geom_top, upstream_state)
nozzle_top.plot(0.12)

(Source code, png, hires.png, pdf)

Compute the critical area ratio where the shockwave is located:

>>> from pygasflow.nozzles import CD_TOP_Nozzle
>>> from pygasflow.solvers import De_Laval_Solver
>>> from pygasflow.utils import Ideal_Gas, Flow_State
>>> gas = Ideal_Gas(287, 1.4) # Set air as the gas to use in the nozzle
>>> upstream_state = Flow_State(p0=8 * 101325, t0 = 303.15) # stagnation condition
>>> Ri, Rt, Re = 0.4, 0.2, 1.2
>>> theta_c, theta_N = 40, 15
>>> Rbt, Rbc = 0.75 * Rt, 1.5 * Rt
>>> geom_top = CD_TOP_Nozzle(Ri, Re, Rt, Rbc, theta_c, 0.8, "axisymmetric")
>>> nozzle_top = De_Laval_Solver(gas, geom_top, upstream_state)
>>> results = nozzle_top.compute(0.12)
>>> print(results[-1])
12.329635184167444

De_Laval_Solver.critical_area

Returns the critical area

De_Laval_Solver.critical_density

Returns the critical density

De_Laval_Solver.critical_pressure

Returns the critical pressure

De_Laval_Solver.critical_temperature

Returns the critical temperature

De_Laval_Solver.critical_velocity

Returns the critical velocity

De_Laval_Solver.inlet_area

Returns the inlet area

De_Laval_Solver.outlet_area

Returns the outlet area

De_Laval_Solver.limit_pressure_ratios

Returns

r1float

Exit pressure ratio corresponding to fully subsonic flow in the divergent (when M=1 in A*).

r2float

Exit pressure ratio corresponding to a normal shock wave at the exit section of the divergent.

r3float

Exit pressure ratio corresponding to fully supersonic flow in the divergent and Pe = Pb (pressure at back). Design condition.

pygasflow.solvers.de_laval.De_Laval_Solver.compute(self, Pe_P0_ratio)

Compute the flow quantities along the nozzle geometry.

Parameters

Pe_P0_ratiofloat

Back to Stagnation pressure ratio. The pressure at the exit plane coincide with the back pressure.

Returns

Lnp.ndarray

Lengths along the stream flow.

area_ratiosnp.ndarray

Area ratios along the stream flow.

Mnp.ndarray

Mach numbers.

P_ratiosnp.ndarray

Pressure ratios.

rho_ratiosnp.ndarray

Density ratios.

T_ratiosnp.ndarray

Temperature ratios.

flow_conditionstring

The flow condition given the input pressure ratio.

Asw_At_ratiofloat

Area ratio of the shock wave location if present, otherwise return None.

pygasflow.solvers.de_laval.De_Laval_Solver.flow_condition(self, gas, upstream_state, Pb_P0_ratio)

Return the flow condition inside the nozzle given the Back to Stagnation pressure ratio.

Parameters

Pb_P0_ratiofloat

Back to Stagnation pressure ratio.

Returns

Flow_Conditionstr

pygasflow.solvers.de_laval.De_Laval_Solver.plot(self, Pb_P0_ratio=None, show=True, size=None, title=None)

Visualize the nozzle geometry, the Mach number and the ratios along the length of the nozzle.

Parameters

Pb_P0_ratiofloat, optional

Back to Stagnation pressure ratio. Default to None.

size(width, height) or None, optional

Set the figure size, in inches.

titlestr or None

Title of the overall plot. Default to None, in which case a default title will be added, showing the type of the geometry and the flow condition.