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Oblique Shock Relations

 import minuteman.cpg.oblique_shock as oblique_shock

This module computes flow parameters of a 2D, stationary, calorically perfect oblique shocks.

High-Level API

minuteman.cpg.oblique_shock.lookup_table_by_deflection_angle 

lookup_table_by_deflection_angle(
    deflection_angle: ArraylikeFloat,
    mach_upstream: ArraylikeFloat,
    specific_heat_ratio: ArraylikeFloat = 1.4,
    shock_type: ArraylikeObliqueShockType = ObliqueShockType.weak,
) -> ObliqueShockTable

Look up the oblique shock properties from a known flow deflection angle \(\theta\)

Parameters:

  • deflection_angle (ArraylikeFloat) –

    deflection angle, \(\theta\) [radians]. Bounds: \((0, \theta_{max}]\)

  • mach_upstream (ArraylikeFloat) –

    upstream Mach number \(M_1\). Bounds: \((1, \infty)\)

  • specific_heat_ratio (ArraylikeFloat, default: 1.4 ) –

    ratio of specific heats, \(\gamma\). Bounds: \((1, 1.67]\)

  • shock_type (ArraylikeObliqueShockType, default: weak ) –

    shock type

Returns:

Raises:

minuteman.cpg.oblique_shock.lookup_table_by_mach_upstream_normal 

lookup_table_by_mach_upstream_normal(
    mach_upstream_normal: ArraylikeFloat,
    mach_upstream: ArraylikeFloat,
    specific_heat_ratio: ArraylikeFloat = 1.4,
) -> ObliqueShockTable

Look up oblique shock table by the normal component of the upstream Mach number, \(M_{n1}\)

Parameters:

  • mach_upstream_normal (ArraylikeFloat) –

    normal component of the upstream Mach number, \(M_{n1}\). Bounds: \((1, M_1)\)

  • mach_upstream (ArraylikeFloat) –

    upstream Mach number, \(M_1\). Bounds: \((1, \infty)\)

  • specific_heat_ratio (ArraylikeFloat, default: 1.4 ) –

    ratio of specific heats, \(\gamma\). Bounds: \((1, 1.67]\)

Returns:

Raises:

minuteman.cpg.oblique_shock.lookup_table_by_shock_angle 

lookup_table_by_shock_angle(
    shock_angle: ArraylikeFloat,
    mach_upstream: ArraylikeFloat,
    specific_heat_ratio: ArraylikeFloat = 1.4,
) -> ObliqueShockTable

Look up the oblique shock properties from a known shock angle, \(\beta\)

Parameters:

  • shock_angle (ArraylikeFloat) –

    shock angle, \(\beta\) [radians]. Bounds: \([\arcsin\left(\frac{1}{M1}\right), 90^\circ]\)

  • mach_upstream (ArraylikeFloat) –

    upstream Mach number, \(M_1\). Bounds: \((1, \infty)\)

  • specific_heat_ratio (ArraylikeFloat, default: 1.4 ) –

    ratio of specific heats, \(\gamma\). Bounds: \((1, 1.67]\)

Returns:

Raises:

Low-Level API

minuteman.cpg.oblique_shock.check_deflection_angle 

check_deflection_angle(
    deflection_angle: ArraylikeFloat,
    mach_upstream: ArraylikeFloat,
    specific_heat_ratio: ArraylikeFloat,
) -> None

Ensure deflection angle \(\theta\) is within bounds or throw an error.

Parameters:

  • deflection_angle (ArraylikeFloat) –

    deflection angle \(\theta\) [radians]

  • mach_upstream (ArraylikeFloat) –

    upstream Mach number, \(M_1\)

  • specific_heat_ratio (ArraylikeFloat) –

    ratio of specific heats, \(\gamma\)

Raises:

minuteman.cpg.oblique_shock.check_shock_angle 

check_shock_angle(shock_angle: ArraylikeFloat, mach: ArraylikeFloat) -> None

Ensure shock angle \(\beta\) is within bounds or throw an error.

Parameters:

Raises:

minuteman.cpg.oblique_shock.deflection_angle_by_shock_mach 

deflection_angle_by_shock_mach(
    shock_angle: ArraylikeFloat, mach_upstream: ArraylikeFloat, specific_heat_ratio: ArraylikeFloat
) -> NDArrayFloat

Compute the deflection angle \(\theta\) for a given shock angle \(\beta\) and upstream Mach number \(M_1\). This is the \(\theta\)-\(\beta\)-\(M\) relation (Eq. 4.17 in 1).

Parameters:

Returns:

  • NDArrayFloat –

    Flow deflection angle, \(\theta\) [radians]

minuteman.cpg.oblique_shock.deflection_angle_max 

deflection_angle_max(
    mach_upstream: ArraylikeFloat, specific_heat_ratio: ArraylikeFloat
) -> NDArrayFloat

Compute the maximum flow deflection angle \(\theta_{max}\) [radians] for a given upstream Mach number \(M_1\).

Parameters:

  • mach_upstream (ArraylikeFloat) –

    upstream Mach number, \(M_1\)

  • specific_heat_ratio (ArraylikeFloat) –

    ratio of specific heats, \(\gamma\)

Returns:

  • NDArrayFloat –

    Max flow deflection angle, \(\theta_{max}\) [radians]

minuteman.cpg.oblique_shock.deflection_angle_sonic 

deflection_angle_sonic(
    mach_upstream: ArraylikeFloat, specific_heat_ratio: ArraylikeFloat
) -> NDArrayFloat

Compute the flow deflection angle \(\theta\) [radians] such that the downstream Mach number is sonic (\(M_2=1\)).

Parameters:

  • mach_upstream (ArraylikeFloat) –

    upstream Mach number, \(M_1\)

  • specific_heat_ratio (ArraylikeFloat) –

    ratio of specific heats, \(\gamma\)

Returns:

  • NDArrayFloat –

    Deflection angle \(\theta\) [radians] yielding sonic flow

minuteman.cpg.oblique_shock.mach_downstream_by_postshock 

mach_downstream_by_postshock(
    mach_downstream_normal: ArraylikeFloat,
    shock_angle: ArraylikeFloat,
    deflection_angle: ArraylikeFloat,
) -> NDArrayFloat

Compute the downstream Mach number, \(M_2\)

Parameters:

  • mach_downstream_normal (ArraylikeFloat) –

    component of downstream Mach number normal to the shock, \(M_{n2}\)

  • shock_angle (ArraylikeFloat) –

    shock angle, \(\beta\) [radians]

  • deflection_angle (ArraylikeFloat) –

    flow deflection angle, \(\theta\) [radians]

Returns:

Raises:

minuteman.cpg.oblique_shock.mach_downstream_normal_component 

mach_downstream_normal_component(
    mach_upstream_normal: ArraylikeFloat, specific_heat_ratio: ArraylikeFloat
) -> NDArrayFloat

Compute the component of downstream Mach number normal to the shock, \(M_{n2}\)

Parameters:

  • mach_upstream_normal (ArraylikeFloat) –

    component of upstream Mach number normal to the shock, \(M_{n1}\)

  • specific_heat_ratio (ArraylikeFloat) –

    ratio of specific heats, \(\gamma\)

Returns:

  • NDArrayFloat –

    Component of downstream Mach number normal to the shock, \(M_{n2}\)

minuteman.cpg.oblique_shock.mach_upstream_normal_component 

mach_upstream_normal_component(
    mach_upstream: ArraylikeFloat, shock_angle: ArraylikeFloat
) -> NDArrayFloat

Compute the normal component of the upstream Mach number, \(M_{n1}\)

Parameters:

Returns:

  • NDArrayFloat –

    Normal component of the upstream Mach number, \(M_{n1}\)

minuteman.cpg.oblique_shock.shock_angle_by_deflection_mach 

shock_angle_by_deflection_mach(
    deflection_angle: ArraylikeFloat,
    mach_upstream: ArraylikeFloat,
    specific_heat_ratio: ArraylikeFloat,
    shock_type: ArraylikeObliqueShockType = ObliqueShockType.weak,
) -> NDArrayFloat

Compute the oblique shock angle \(\beta\) [radians] for a given deflection angle \(\theta\) and upstream Mach number \(M_1\).

This is the lesser-known \(\beta\)-\(\theta\)-\(M\) relation (Eq. 4.19-4.21 of 1)

Parameters:

  • deflection_angle (ArraylikeFloat) –

    flow deflection angle, \(\theta\) [radians]

  • mach_upstream (ArraylikeFloat) –

    upstream Mach number, \(M_1\)

  • specific_heat_ratio (ArraylikeFloat) –

    ratio of specific heats, \(\gamma\)

  • shock_type (ArraylikeObliqueShockType, default: weak ) –

    Oblique shock type (weak or strong).

Returns:

minuteman.cpg.oblique_shock.shock_angle_max 

shock_angle_max(mach_upstream: ArraylikeFloat, specific_heat_ratio: ArraylikeFloat) -> NDArrayFloat

Compute the shock angle \(\beta_{max}\) [radians] that yields the max deflection angle \(\theta_{max}\) for a given Mach number \(M\) (slide 11 of 2).

Parameters:

  • mach_upstream (ArraylikeFloat) –

    upstream Mach number, \(M_1\)

  • specific_heat_ratio (ArraylikeFloat) –

    ratio of specific heats, \(\gamma\).

Returns:

  • NDArrayFloat –

    Maximum shock angle \(\beta_{max}\) [radians] that is still attached

minuteman.cpg.oblique_shock.shock_angle_sonic 

shock_angle_sonic(
    mach_upstream: ArraylikeFloat, specific_heat_ratio: ArraylikeFloat
) -> NDArrayFloat

Compute the oblique shock angle \(\beta\) [radians] which will yield a sonic downstream Mach number, \(M_2=1\) (slide 29 of 2).

Parameters:

  • mach_upstream (ArraylikeFloat) –

    upstream Mach number, \(M_1\)

  • specific_heat_ratio (ArraylikeFloat) –

    ratio of specific heats, \(\gamma\)

Returns:

  • NDArrayFloat –

    Shock angle \(\beta\) [radians] yielding sonic flow

Data Structures

minuteman.cpg.oblique_shock.ObliqueShockTable dataclass

Oblique shock table for a calorically perfect gas

mach_upstream: NDArrayFloat instance-attribute

Upstream mach number, \(M_1\)

mach_downstream: NDArrayFloat instance-attribute

Downstream mach number, \(M_2\)

mach_upstream_normal: NDArrayFloat instance-attribute

Normal component of upstream Mach number, \(M_{n1}\)

mach_downstream_normal: NDArrayFloat instance-attribute

Normal component of downstream Mach number, \(M_{n2}\)

deflection_angle: NDArrayFloat instance-attribute

Deflection angle, \(\theta\)

shock_angle: NDArrayFloat instance-attribute

Shock angle, \(\beta\)

temperature_ratio: NDArrayFloat instance-attribute

Temperature ratio, \(T_2 / T_1\)

pressure_ratio: NDArrayFloat instance-attribute

Static pressure ratio, \(p_2 / p_1\)

density_ratio: NDArrayFloat instance-attribute

Density ratio, \(\rho_2 / \rho_1\)

total_pressure_ratio: NDArrayFloat instance-attribute

Total pressure ratio, \(p_{02} / p_{01}\)

specific_heat_ratio: NDArrayFloat instance-attribute

Ratio of specific heats, \(\gamma\)

minuteman.cpg.oblique_shock.ObliqueShockType

Bases: IntEnum

Oblique shock type (weak or strong shock)

strong = 0 class-attribute instance-attribute

Strong shock

weak = auto() class-attribute instance-attribute

Weak shock (most common in nature)

References

  1. Anderson, J. D., Jr. (2003). Modern compressible flow: With historical perspective (3rd ed.). McGraw-Hill.
  2. Whitmore, Stephen. Section 8 Lecture 3: Supersonic Flow Around a Blunt Body [[PDF slides]]. MAE 5420, Utah State University. URL