Shockwaves

Normal Shock Waves

pygasflow.shockwave.pressure_ratio(M1, gamma=1.4)

Compute the static pressure ratio \(\frac{P_{2}}{P_{1}}\).

Parameters

M1array_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M \ge 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Pressure Ratio \(\frac{P_{2}}{P_{1}}\)

pygasflow.shockwave.temperature_ratio(M1, gamma=1.4)

Compute the static temperature ratio \(\frac{T_{2}}{T_{1}}\).

Parameters

M1array_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M \ge 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Temperature Ratio \(\frac{T_{2}}{T_{1}}\)

pygasflow.shockwave.density_ratio(M1, gamma=1.4)

Compute the density ratio \(\frac{\rho_{2}}{\rho_{1}}\).

Parameters

M1array_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M \ge 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Density Ratio \(\frac{\rho_{2}}{\rho_{1}}\)

pygasflow.shockwave.total_pressure_ratio(M1, gamma=1.4)

Compute the total pressure ratio \(\frac{P_{2}^{0}}{P_{1}^{0}}\).

Parameters

M1array_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M \ge 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Total Pressure Ratio \(\frac{P_{2}^{0}}{P_{1}^{0}}\)

pygasflow.shockwave.total_temperature_ratio(M1, gamma=1.4)

Compute the total temperature ratio \(\frac{T_{2}^{0}}{T_{1}^{0}}\).

Parameters

M1array_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M \ge 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Total Temperature Ratio \(\frac{T_{2}^{0}}{T_{1}^{0}}\) (spoiler: always equal to 1 :P )

pygasflow.shockwave.rayleigh_pitot_formula(M1, gamma=1.4)

Compute the ratio \(\frac{P_{t2}}{P_{1}}\), between the stagnation pressure behind a normal shock wave and the static pressure ahead of the shock wave.

Parameters

M1array_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M \ge 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Ratio \(\frac{P_{t2}}{P_{1}}\)

References

“Equations, Tables and Charts for compressible flow”, NACA R-1135, 1953

pygasflow.shockwave.entropy_difference(M1, gamma=1.4)

Compute the dimensionless entropy difference, \(\frac{s_{2} - s_{1}}{C_{p}}\). Eq (3.60) Anderson’s.

Parameters

M1array_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M \ge 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Dimensionless Entropy Difference, \(\frac{s_{2} - s_{1}}{C_{p}}\)

pygasflow.shockwave.mach_downstream(M1, gamma=1.4)

Compute the downstream Mach number M2. Note that this function can also be used to compute M1 given M2.

Parameters

M1array_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Because this function can be used to compute M1 given M2, it will not perform a check wheter \(M_{1} \ge 1\). Be careful on your use!

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Downstream Mach Number M2

pygasflow.shockwave.m1_from_pressure_ratio(ratio, gamma=1.4)

Compute M1 from the pressure ratio.

Parameters

ratioarray_like

Pressure Ratio \(\frac{P_{2}}{P_{1}}\). If float, list, tuple is given as input, a conversion will be attempted. Must be \(\frac{P_{2}}{P_{1}}\) >= 1.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Upstream Mach number M1.

pygasflow.shockwave.m1_from_temperature_ratio(ratio, gamma=1.4)

Compute M1 from the temperature ratio.

Parameters

ratioarray_like

Temperature Ratio \(\frac{T_{2}}{T_{1}}\). If float, list, tuple is given as input, a conversion will be attempted. Must be \(\frac{T_{2}}{T_{1}}\) >= 1.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Upstream Mach number M1.

pygasflow.shockwave.m1_from_density_ratio(ratio, gamma=1.4)

Compute M1 from the density ratio.

Parameters

ratioarray_like

Density Ratio \(\frac{\rho_{2}}{\rho_{1}}\). If float, list, tuple is given as input, a conversion will be attempted. Must be \(1 \le \frac{\rho_{2}}{\rho_{1}} < \frac{\gamma + 1}{\gamma - 1}\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Upstream Mach number M1.

pygasflow.shockwave.m1_from_total_pressure_ratio(ratio, gamma=1.4)

Compute M1 from the total pressure ratio.

Parameters

ratioarray_like

Total Pressure Ratio. If float, list, tuple is given as input, a conversion will be attempted. Must be \(0 \le \frac{P_{2}^{0}}{P_{1}^{0}} \le 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Upstream Mach number M1.

pygasflow.shockwave.m1_from_m2(M2, gamma=1.4)

Compute M1 from the downstream Mach number M2.

Parameters

M2array_like

Downstream Mach Number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(\frac{\gamma - 1}{2 \gamma} < M_{2} < 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Upstream Mach number M1.

Oblique Shock Waves

pygasflow.shockwave.theta_from_mach_beta(M1, beta, gamma=1.4)

Compute the flow turning angle Theta accordingly to the input parameters.

Parameters

M1array_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M_{1} \ge 1\).

betafloat

Shock wave angle in degrees.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Flow angle Theta [degrees]

pygasflow.shockwave.beta_from_mach_theta(M1, theta, gamma=1.4)

Compute the shock angle Beta accordingly to the input parameters.

Parameters

M1array_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M_{1} \ge 1\).

thetafloat

Flow turning angle in degrees.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

Dictionary of Shock angle beta [degrees] if it exists, else NaN: {"weak": beta_weak, "strong": beta_strong}.

pygasflow.shockwave.beta_from_upstream_mach(M1, MN1)

Compute the shock wave angle beta from the upstream Mach number and its normal component.

Parameters

M1array_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M_{1} \ge 1\).

MN1array_like

Normal Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be MN1.shape == M1.shape.

Returns

outndarray

Shock angle Beta [degrees]

pygasflow.shockwave.normal_mach_upstream(M1, beta=None, theta=None, gamma=1.4, flag='weak')

Compute the upstream normal Mach Number, which can then be used to evaluate all other ratios.

Parameters

M1array_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M_{1} \ge 1\).

betafloat, optional

The shock wave angle in degrees. If beta=None you must give in theta.

thetafloat, optional

The flow deflection angle in degrees. If theta=None you must give in beta.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

flagstring, optional

Can be either 'weak' or 'strong'. Default to 'weak'. Chose what value to compute if theta is provided.

Returns

outndarray

Normal Mach number upstream of the shock wave. If theta is given, and flag="both" it returns a dictionary of Normal Mach numbers: {"weak":weak_MN1, "strong":strong_MN1}.

pygasflow.shockwave.get_upstream_normal_mach_from_ratio(ratioName, ratio, gamma=1.4)

Compute the upstream Mach number given a ratio as an argument.

Parameters

ratioNamestring

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

• 'pressure': \(\frac{P_{2}}{P_{1}}\)

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

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

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

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

ratioarray_like

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

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

outndarray

The upstream Mach number.

pygasflow.shockwave.get_ratios_from_normal_mach_upstream(Mn, gamma=1.4)

Compute the ratios of the quantities across a Shock Wave given the Normal Mach number.

Parameters

Mnarray_like

Normal Mach number upstream of the shock wave. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M_{1} \ge 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

prarray_like

Pressure ratio across the shock wave.

drarray_like

Density ratio across the shock wave.

trarray_like

Temperature ratio across the shock wave.

tprarray_like

Total Pressure ratio across the shock wave.

mn2array_like

Normal Mach number dowstream of the shock wave.

pygasflow.shockwave.maximum_mach_from_deflection_angle(theta, gamma=1.4)

Compute the maximum Mach number from a given Deflection angle theta.

Parameters

thetafloat

Deflection angle in degrees. Must be \(0 \le \theta \le 90\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

Mfloat

The maximum Mach number for the specified theta.

pygasflow.shockwave.mimimum_beta_from_mach(M1)

Compute the minimum shock wave angle for a given upstream Mach number.

Parameters

Marray_like

Upstream Mach number. Must be >= 1.

Returns

beta: float

Shock wave angle in degrees

pygasflow.shockwave.max_theta_from_mach(M1, gamma=1.4)

Compute the maximum deflection angle for a given upstream Mach number.

Parameters

M1array_like

Upstream Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M_{1} \ge 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

Theta_maxndarray

Maximum deflection angle theta in degrees

pygasflow.shockwave.beta_from_mach_max_theta(M1, gamma=1.4)

Compute the shock wave angle beta corresponding to the maximum deflection angle theta given an upstream Mach number.

Parameters

M1array_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M_{1} \ge 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

Betaarray_like

The shock angle in degrees.

pygasflow.shockwave.beta_theta_max_for_unit_mach_downstream(M1, gamma=1.4)

Compute the shock maximum deflection angle, theta_max, as well as the wave angle beta corresponding to the unitary downstream Mach number, \(M_{2} = 1\).

Parameters

M1array_like

Upstream Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M_{1} \ge 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

Betaarray_like

The shock angle in degrees corresponding to \(M_{2} = 1\).

Theta maxarray_like

The maximum deflection angle in degrees corresponding to \(M_{2} = 1\).

pygasflow.shockwave.mach_from_theta_beta(theta, beta, gamma=1.4)

Compute the upstream Mach number given the flow deflection angle and the shock wave angle.

Parameters

thetaarray_like

Flow deflection angle in degrees. If float, list, tuple is given as input, a conversion will be attempted. Must be \(0 \le \theta \le 90\).

betaarray_like

Shock wave angle in degrees. If float, list, tuple is given as input, a conversion will be attempted. Must be \(0 \le \beta \le 90\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

Machndarray

The upstream Mach number.

pygasflow.shockwave.shock_polar(M1, gamma=1.4, N=100)

Compute the ratios (Vx/a*), (Vy/a*) for plotting a Shock Polar.

Parameters

M1float

Upstream Mach number of the shock wave. Must be > 1.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Nint, optional

Number of discretization steps in the range [2, 2*pi]. Must be > 1.

Returns

(Vx/a*)ndarray [1 x N]

x-coordinate for the shock polar plot

(Vy/a*)ndarray [1 x N]

y-coordinate for the shock polar plot

pygasflow.shockwave.pressure_deflection(M1, gamma=1.4, N=100)

Helper function to build Pressure-Deflection plots.

Parameters

M1float

Upstream Mach number. Must be > 1.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Nint, optional

Half number of points discretizing the range [0, theta_max]. This function compute N points in the range [0, theta_max] for the ‘weak’ solution, then compute N points in the range [theta_max, 0] for the ‘strong’ solution.

Returns

thetaarray_like

Deflection angles

prarray_like

Pressure ratios computed for the given Mach and the above deflection angles.

Conical Flow

pygasflow.shockwave.taylor_maccoll(theta, V, gamma=1.4)

Taylor-Maccoll differential equation for conical shock wave.

Parameters

thetafloat

Polar coordinate, angle in radians.

Vlist

Velocity Vector with components:

• \(V_{r}\): velocity along the radial direction

• \(V_{\theta}\): velocity along the polar direction

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

dV_dthetalist

Taylor-Maccoll differential equation.

pygasflow.shockwave.nondimensional_velocity(M, gamma=1.4)

Compute the Nondimensional Velocity given the Mach number.

Parameters

Marray_like

Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be M >= 0.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

Varray_like

Nondimensional Velocity

pygasflow.shockwave.mach_from_nondimensional_velocity(V, gamma=1.4)

Compute the Mach number given the Nondimensional Velocity.

Parameters

Varray_like

Nondimensional Velocity. If float, list, tuple is given as input, a conversion will be attempted. Must be V > 0.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

Marray_like

Mach number

pygasflow.shockwave.mach_cone_angle_from_shock_angle(M, beta, gamma=1.4)

Compute the half-cone angle and the Mach number at the surface of the cone. NOTE: this function is undecorated, hence no check is performed to assure the validity of the input parameters. It’s up to the user to assure that.

Parameters

Mfloat

Upstream Mach number. Must be > 1.

betafloat

Shock Angle in degrees. Must be \(\mu \, \text(Mach Angle) \le beta \le 90\). NOTE: no check is done over beta. If an error is raised during the computation, make sure beta is at least ever so slightly bigger than the mach angle.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

Mcfloat

Mach number at the surface of the cone

theta_cfloat

Half-cone angle in degrees.

pygasflow.shockwave.shock_angle_from_mach_cone_angle(M1, theta_c, gamma=1.4, flag='weak')

Compute the shock wave angle given the upstream mach number and the half-cone angle.

Parameters

M1array_like

Upstream Mach number. Must be > 1.

theta_cfloat

Half cone angle in degrees. Must be \(0 < \theta_{c} < 90\)

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

flagstring, optional

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

Returns

Mcfloat

Mach number at the surface of the cone, computed.

theta_cfloat

Half cone angle of the cone in degrees (which was provided in input).

betafloat

Shock wave angle in degrees.

pygasflow.shockwave.shock_angle_from_machs(M1, Mc, gamma=1.4, flag='weak')

Compute the shock wave angle given the upstream mach number and the mach number at the surface of the cone.

Parameters

M1array_like

Upstream Mach number. Must be > 1.

Mcfloat

Mach number at the surface of the cone.

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

flagstring, optional

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

Returns

Mcfloat

Mach number at the surface of the cone, computed.

theta_cfloat

Half cone angle of the cone in degrees.

betafloat

Shock wave angle in degrees.

pygasflow.shockwave.max_theta_c_from_mach(M1, gamma=1.4)

Compute the maximum cone angle and the corresponding shockwave angle for a given upstream Mach number.

Parameters

M1array_like

Upstream Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M_{1} \ge 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

Mcndarray

Mach number at the cone surface

Theta_c_maxndarray

Maximum cone angle theta_c in degrees

betandarray

Shockwave angle corresponding to the maximum cone angle and provided Mach number.

pygasflow.shockwave.beta_theta_c_for_unit_mach_downstream(M1, gamma=1.4)

Given an upstream Mach number, compute the point (beta, theta_c) where the downstream Mach number is sonic. WARNING: this procedure is really slow!

Parameters

M1array_like

Upstream Mach number. If float, list, tuple is given as input, a conversion will be attempted. Must be \(M_{1} \ge 1\).

gammafloat, optional

Specific heats ratio. Default to 1.4. Must be \(\gamma > 1\).

Returns

theta_cndarray

Cone angle theta_c in degrees.

betandarray

Shockwave angle corresponding in degrees.

pygasflow.shockwave.load_data(gamma=1.4)

The beta_theta_c_for_unit_mach_downstream function is really slow in computing the data. Often, that data is needed in a plot. Here, a few precomputed tables has been provided.

Parameters

gammafloat, optional

The specific heat ratio.

Returns

M1ndarray

The precomputed upstream Mach numbers

betandarray

The shockwave angle associate to \(M_{2} = 1\) for the precomputed M1

theta_cndarray

The half-cone angle associate to \(M_{2} = 1\) for the precomputed M1