Source code for aerosandbox.library.aerodynamics.normal_shock_relations

# From https://www.grc.nasa.gov/WWW/K-12/airplane/normal.html


[docs]def mach_number_after_normal_shock(
    mach_upstream,
    gamma=1.4,
):
    """
    Computes the mach number immediately after a normal shock wave.

    Args:
        mach_upstream: The mach number immediately before the normal shock wave.
        gamma: The ratio of specific heats of the fluid. 1.4 for air.

    Returns: The mach number immediately after the normal shock wave.

    """
    gm1 = gamma - 1
    m2 = mach_upstream**2

    return ((gm1 * m2 + 2) / (2 * gamma * m2 - gm1)) ** 0.5


[docs]def density_ratio_across_normal_shock(mach_upstream, gamma=1.4):
    """
    Computes the ratio of fluid density across a normal shock.

    Specifically, returns: rho_after_shock / rho_before_shock

    Args:
        mach_upstream: The mach number immediately before the normal shock wave.
        gamma: The ratio of specific heats of the fluid. 1.4 for air.

    Returns: rho_after_shock / rho_before_shock

    """
    return ((gamma + 1) * mach_upstream**2) / ((gamma - 1) * mach_upstream**2 + 2)


[docs]def temperature_ratio_across_normal_shock(mach_upstream, gamma=1.4):
    """
    Computes the ratio of fluid temperature across a normal shock.

    Specifically, returns: T_after_shock / T_before_shock

    Args:
        mach_upstream: The mach number immediately before the normal shock wave.
        gamma: The ratio of specific heats of the fluid. 1.4 for air.

    Returns: T_after_shock / T_before_shock

    """
    gm1 = gamma - 1
    m2 = mach_upstream**2
    return ((2 * gamma * m2 - gm1) * (gm1 * m2 + 2)) / ((gamma + 1) ** 2 * m2)


[docs]def pressure_ratio_across_normal_shock(mach_upstream, gamma=1.4):
    """
    Computes the ratio of fluid static pressure across a normal shock.

    Specifically, returns: P_after_shock / P_before_shock

    Args:
        mach_upstream: The mach number immediately before the normal shock wave.
        gamma: The ratio of specific heats of the fluid. 1.4 for air.

    Returns: P_after_shock / P_before_shock

    """
    m2 = mach_upstream**2
    return (2 * gamma * m2 - (gamma - 1)) / ((gamma + 1))


[docs]def total_pressure_ratio_across_normal_shock(mach_upstream, gamma=1.4):
    """
    Computes the ratio of fluid total pressure across a normal shock.

    Specifically, returns: Pt_after_shock / Pt_before_shock

    Args:
        mach_upstream: The mach number immediately before the normal shock wave.
        gamma: The ratio of specific heats of the fluid. 1.4 for air.

    Returns: Pt_after_shock / Pt_before_shock

    """
    return density_ratio_across_normal_shock(
        mach_upstream=mach_upstream, gamma=gamma
    ) ** (gamma / (gamma - 1)) * (
        (gamma + 1) / (2 * gamma * mach_upstream**2 - (gamma - 1))
    ) ** (
        1 / (gamma - 1)
    )


if __name__ == "__main__":

[docs]    def q_ratio(mach):
        return (
            density_ratio_across_normal_shock(mach)
            * (
                mach_number_after_normal_shock(mach)
                * temperature_ratio_across_normal_shock(mach) ** 0.5
            )
            ** 2
        )

    q_ratio(2)