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Cone Flow Relations

 import minuteman.cpg.cone_flow as cone_flow

Solve for the flow solution of a right cone at zero degrees angle of attack for a calorically perfect gas.

High-Level API

minuteman.cpg.cone_flow.lookup_solution_by_cone_angle 

lookup_solution_by_cone_angle(
    cone_angle: Floatlike,
    mach_upstream: Floatlike,
    specific_heat_ratio: Floatlike = 1.4,
    shock_type: ObliqueShockType = ObliqueShockType.weak,
) -> ConeFlowSolution

Solve a cone flow problem with a known cone angle, \(\theta_c\)

Parameters:

  • cone_angle (Floatlike) –

    cone angle, \(\theta_c\) [radians]. Bounds: \((0, \theta_{c,max}]\)

  • mach_upstream (Floatlike) –

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

  • specific_heat_ratio (Floatlike, default: 1.4 ) –

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

  • shock_type (ObliqueShockType, default: weak ) –

    shock type - you almost always want weak

Returns:

Raises:

minuteman.cpg.cone_flow.lookup_solution_by_shock_angle 

lookup_solution_by_shock_angle(
    shock_angle: Floatlike, mach_upstream: Floatlike, specific_heat_ratio: Floatlike = 1.4
) -> ConeFlowSolution

Solve a cone flow problem with a known shock angle, \(\theta_s\)

Parameters:

  • shock_angle (Floatlike) –

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

  • mach_upstream (Floatlike) –

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

  • specific_heat_ratio (Floatlike, default: 1.4 ) –

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

Returns:

Raises:

minuteman.cpg.cone_flow.lookup_solution_by_surface_mach 

lookup_solution_by_surface_mach(
    surface_mach: Floatlike, mach_upstream: Floatlike, specific_heat_ratio: Floatlike = 1.4
) -> ConeFlowSolution

Solve a cone flow problem with a known surface Mach number, \(M_c\)

Parameters:

  • surface_mach (Floatlike) –

    Mach number at the surface of the cone, \(M_c\). Bounds: \([M_2, M_1]\), where \(M_2\) is the Mach number downstream of a normal shock.

  • mach_upstream (Floatlike) –

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

  • specific_heat_ratio (Floatlike, default: 1.4 ) –

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

Returns:

Raises:

Low-Level API

minuteman.cpg.cone_flow.solve_taylor_maccoll_by_cone_angle 

solve_taylor_maccoll_by_cone_angle(
    cone_angle: Floatlike,
    mach_upstream: Floatlike,
    specific_heat_ratio: Floatlike,
    shock_type: ObliqueShockType,
) -> tuple[NDArrayFloat, NDArrayFloat, NDArrayFloat]

Solve the Taylor-Maccoll equations for a given cone angle, \(\theta_c\)

Parameters:

  • cone_angle (Floatlike) –

    cone angle, \(\theta_c\) [radians]

  • mach_upstream (Floatlike) –

    Upstream Mach number, \(M_1\)

  • specific_heat_ratio (Floatlike) –

    ratio of specific heats, \(\gamma\)

  • shock_type (ObliqueShockType) –

    shock type, strong or weak

Returns:

minuteman.cpg.cone_flow.solve_taylor_maccoll_by_shock_angle 

solve_taylor_maccoll_by_shock_angle(
    shock_angle: Floatlike, mach_upstream: Floatlike, specific_heat_ratio: Floatlike
) -> tuple[NDArrayFloat, NDArrayFloat, NDArrayFloat]

Compute the solution to the Taylor Maccoll equations for a given shock angle, \(\theta_s\).

Parameters:

  • shock_angle (Floatlike) –

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

  • mach_upstream (Floatlike) –

    upstream Mach number, \(M_1\)

  • specific_heat_ratio (Floatlike) –

    ratio of specific heats, \(\gamma\)

Returns:

Raises:

  • ValueError –

    polar velocity is positive (should be negative by convention)

  • SolveIVPError –

    IVP solver failed, check inputs

minuteman.cpg.cone_flow.solve_taylor_maccoll_by_surface_mach 

solve_taylor_maccoll_by_surface_mach(
    surface_mach: Floatlike, mach_upstream: Floatlike, specific_heat_ratio: Floatlike
) -> tuple[NDArrayFloat, NDArrayFloat, NDArrayFloat]

Compute the solution to the Taylor-Maccoll equations for a given Mach number at the surface of the cone, \(M_c\).

Parameters:

  • surface_mach (Floatlike) –

    Mach number at the surface of the cone, \(M_c\)

  • mach_upstream (Floatlike) –

    upstream Mach number, \(M_1\)

  • specific_heat_ratio (Floatlike) –

    ratio of specific heats, \(\gamma\)

Returns:

Raises:

minuteman.cpg.cone_flow.cone_shock_angle_maxes 

cone_shock_angle_maxes(
    mach_upstream: Floatlike, specific_heat_ratio: Floatlike
) -> tuple[float, float]

Compute the max cone angle \(\theta_{c,max}\) for a given upstream condition before the shock detaches, as well as the shock angle at that max cone angle condition, \(\theta_{s,max}\)

Parameters:

  • mach_upstream (Floatlike) –

    Upstream Mach number, \(M_1\)

  • specific_heat_ratio (Floatlike) –

    Ratio of specific heats, \(\gamma\)

Returns:

  • tuple[float, float] –

    (max cone angle \(\theta_{c,max}\), shock angle for the max cone angle, \(\theta_{s,max}\))

Raises:

minuteman.cpg.cone_flow.deflection_angle_by_velocity_components 

deflection_angle_by_velocity_components(
    polar_angle: NDArrayFloat, velocity_radial: NDArrayFloat, velocity_polar: NDArrayFloat
) -> NDArrayFloat

Compute the flow deflection angle \(\psi\) at all polar angles \(\theta\)

Parameters:

  • polar_angle (NDArrayFloat) –

    polar angle \(\theta\) [radians]

  • velocity_radial (NDArrayFloat) –

    nondimensional radial velocity, \(V'_r\)

  • velocity_polar (NDArrayFloat) –

    nondimensional polar velocity, \(V'_{\theta}\)

Returns:

  • NDArrayFloat –

    Flow deflection angle \(\psi\) at all polar angles post-shock

minuteman.cpg.cone_flow.mach_from_nondimensional_velocity 

mach_from_nondimensional_velocity(velocity: Any, specific_heat_ratio: Any) -> Any

Compute Mach number \(M\) from nondimensional velocity \(V'\)

Parameters:

  • velocity (Any) –

    nondimensional velocity, \(V'\)

  • specific_heat_ratio (Any) –

    ratio of specific heats, \(\gamma\)

Returns:

  • Any –

    Mach number \(M\)

minuteman.cpg.cone_flow.nondimensional_velocity_from_components 

nondimensional_velocity_from_components(velocity_radial: Any, velocity_polar: Any) -> Any

Compute the nondimensional velocity \(V'\) useful to nondimensionalizing the Taylor Maccoll equations from its radial and polar components.

Parameters:

  • velocity_radial (Any) –

    nondimensional radial velocity \(V'_r\)

  • velocity_polar (Any) –

    nondimensional polar velocity \(V'_{\theta}\)

Returns:

  • Any –

    Nondimensional velocity \(V'\)

minuteman.cpg.cone_flow.nondimensional_velocity_from_mach 

nondimensional_velocity_from_mach(mach: Any, specific_heat_ratio: Any) -> Any

Compute the nondimensional velocity \(V'\) useful to nondimensionalizing the Taylor Maccoll equations.

\(V' = V / V_{max}\) in chapter 10 of [1]. \(V_{max}\) is a max theoretical velocity if the flow were expanded to 0 K.

Parameters:

  • mach (Any) –

    Mach number \(M\)

  • specific_heat_ratio (Any) –

    ratio of specific heats, \(\gamma\)

Returns:

  • Any –

    Nondimensional velocity \(V'\)

minuteman.cpg.cone_flow.nondimensional_velocity_polar 

nondimensional_velocity_polar(velocity: Any, shock_angle: Any, deflection_angle: Any) -> Any

Compute the nondimensional polar velocity \(V'_{\theta}\) for conic flow

The quantity is negative, as the \(+V'_{\theta}\) axis in the coordinate system is positive pointing away from the body.

Parameters:

  • velocity (Any) –

    nondimensional velocity, \(V'\)

  • shock_angle (Any) –

    shock angle \(\theta_s\) [radians]

  • deflection_angle (Any) –

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

Returns:

  • Any –

    Polar component of nondimensional velocity, \(V'_{\theta}\)

minuteman.cpg.cone_flow.nondimensional_velocity_radial 

nondimensional_velocity_radial(velocity: Any, shock_angle: Any, deflection_angle: Any) -> Any

Compute the nondimensional radial velocity \(V'_r\) for conic flow

The quantity is positive in the downstream direction

Parameters:

  • velocity (Any) –

    nondimensional velocity, \(V'\)

  • shock_angle (Any) –

    shock angle \(\theta_s\) [radians]

  • deflection_angle (Any) –

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

Returns:

  • Any –

    Radial component of nondimensional velocity, \(V'_r\)

Data Structures

minuteman.cpg.cone_flow.ConeFlowSolution dataclass

Flowfield solution for cone flow for a calorically perfect gas.

Array-like quantities vary as a function of the polar angle, \(\theta\).

mach_upstream: float instance-attribute

Upstream mach number, \(M_1\)

polar_angle: NDArrayFloat instance-attribute

Polar or cross-flow angle decreasing from the shock to the cone surface, \(\theta\) [radians]

shock_angle: float property

Shock angle, \(\theta_s\) [radians]

cone_angle: float property

Cone half-angle, \(\theta_c\) [radians]

specific_heat_ratio: float instance-attribute

Ratio of specific heats, \(\gamma\)

flow_angle: NDArrayFloat instance-attribute

Flow angle w.r.t. the cone axis, \(\psi\) [radians]

velocity_radial: NDArrayFloat instance-attribute

Nondimensional radial velocity, \(V'_r\)

velocity_polar: NDArrayFloat instance-attribute

Nondimensional polar velocity, \(V'_{\theta}\)

velocity: NDArrayFloat instance-attribute

Nondimensional velocity magnitude, \(V'\)

mach: NDArrayFloat instance-attribute

Downstream mach number, \(M\)

pressure_ratio: NDArrayFloat instance-attribute

Pressure ratio, \(p / p_1\)

temperature_ratio: NDArrayFloat instance-attribute

Temperature ratio, \(T / T_1\)

density_ratio: NDArrayFloat instance-attribute

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

total_pressure_ratio: float instance-attribute

Total pressure ratio, \(p_0 / p_{01}\). This value is a constant

Error Types

minuteman.cpg.cone_flow.SolveIVPError

Bases: Exception

solve_ivp call failed

Theory

References

  1. Anderson, J. D., Jr. (2003). Modern compressible flow: With historical perspective (3rd ed.). McGraw-Hill.
  2. Sims, J. L. (1964). Tables for supersonic flow around right circular cones at zero angle of attack.