RCAIDE.Library.Methods.Aerodynamics.Common.Drag.compressibility_drag

compressibility_drag#

compressibility_drag(state, settings, geometry)[source]#

Computes compressibility drag coefficient for full aircraft including volume drag effects.

Parameters:

  • state (Data) –

    Flight conditions and aerodynamic state containing:
    • conditions.freestream.mach_numberfloat

      Freestream Mach number [unitless]

    • conditions.aerodynamics.coefficients.lift.totalfloat

      Total lift coefficient [unitless]

  • settings (dict) –

    Aerodynamic analysis settings containing:
    • supersonic.begin_drag_rise_mach_numberfloat

      Mach number at which drag rise begins [unitless]

    • supersonic.end_drag_rise_mach_numberfloat

      Mach number at which drag rise ends [unitless]

  • geometry (Data) –

    Aircraft geometry containing:
    • reference_areafloat

      Reference area for drag coefficient calculation [m²]

    • wingslist
      List of wing objects containing:
      • sweeps.leading_edgefloat

        Leading edge sweep angle [radians]

      • thickness_to_chordfloat

        Thickness-to-chord ratio [unitless]

Returns:

Results are stored in state.conditions.aerodynamics.coefficients.drag.compressible.total

Return type:

None

Notes

This function calculates the compressibility drag coefficient using empirical correlations based on wing geometry and flight conditions. The calculation accounts for the critical Mach number and the drag rise characteristics of swept wings.

Major Assumptions

  • Compressibility effects are primarily due to wing geometry

  • Critical Mach number correlation is valid for typical transport aircraft

  • Cubic spline blending smooths transition between subsonic and supersonic regimes

Theory

The critical Mach number is calculated using a regression fit from AA241:

\(M_{cc} \cos(\Lambda) = 0.922 - 1.154(t/c) - 0.305 C_L + 0.333(t/c)^2 + 0.467(t/c)C_L + 0.087 C_L^2\)

where:

  • \(M_{cc}\) is the critical Mach number

  • \(\Lambda\) is the leading edge sweep angle [radians]

  • \(t/c\) is the thickness-to-chord ratio corrected for sweep

  • \(C_L\) is the lift coefficient corrected for sweep

The corrected thickness-to-chord ratio is:

\((t/c)_{eff} = \frac{t/c}{\cos(\Lambda)}\)

The corrected lift coefficient is:

\(C_{L,eff} = \frac{C_L}{\cos^2(\Lambda)}\)

The divergence ratio is:

\(M/M_{cc} = \frac{M}{M_{cc}}\)

The compressibility drag coefficient follows Shevell’s correlation:

\(\Delta C_{D,comp} = 0.0019 \left(\frac{M}{M_{cc}}\right)^{14.641} \cos^3(\Lambda)\)

Definitions

‘Compressibility Drag’

Additional drag caused by compressibility effects as the aircraft approaches and exceeds the critical Mach number.

‘Critical Mach Number’

The freestream Mach number at which the local flow over some part of the aircraft first reaches sonic velocity.

‘Drag Rise’

Rapid increase in drag coefficient as the aircraft approaches and exceeds the critical Mach number.

References

[1] Stanford AA241 Lecture Notes [2] Shevell, R. S. (1989). “Fundamentals of Flight.” Prentice Hall.

See also

RCAIDE.Library.Components.Wings.Main_Wing, RCAIDE.Library.Components.Wings.Blended_Wing_Body, RCAIDE.Library.Methods.Utilities.Cubic_Spline_Blender

subsonic_compressibility_drag(state, settings, geometry)[source]#

Computes compressibility drag coefficient for full aircraft including volume drag effects.

Parameters:

  • state (Data) –

    Flight conditions and aerodynamic state containing:
    • conditions.freestream.mach_numberfloat

      Freestream Mach number [unitless]

    • conditions.aerodynamics.coefficients.lift.totalfloat

      Total lift coefficient [unitless]

  • settings (dict) –

    Aerodynamic analysis settings containing:
    • supersonic.begin_drag_rise_mach_numberfloat

      Mach number at which drag rise begins [unitless]

    • supersonic.end_drag_rise_mach_numberfloat

      Mach number at which drag rise ends [unitless]

  • geometry (Data) –

    Aircraft geometry containing:
    • reference_areafloat

      Reference area for drag coefficient calculation [m²]

    • wingslist
      List of wing objects containing:
      • sweeps.leading_edgefloat

        Leading edge sweep angle [radians]

      • thickness_to_chordfloat

        Thickness-to-chord ratio [unitless]

Returns:

Results are stored in state.conditions.aerodynamics.coefficients.drag.compressible.total

Return type:

None

Notes

This function calculates the compressibility drag coefficient using empirical correlations based on wing geometry and flight conditions. The calculation accounts for the critical Mach number and the drag rise characteristics of swept wings.

Major Assumptions

  • Compressibility effects are primarily due to wing geometry

  • Critical Mach number correlation is valid for typical transport aircraft

  • Cubic spline blending smooths transition between subsonic and supersonic regimes

Theory

The critical Mach number is calculated using a regression fit from AA241:

\(M_{cc} \cos(\Lambda) = 0.922 - 1.154(t/c) - 0.305 C_L + 0.333(t/c)^2 + 0.467(t/c)C_L + 0.087 C_L^2\)

where:

  • \(M_{cc}\) is the critical Mach number

  • \(\Lambda\) is the leading edge sweep angle [radians]

  • \(t/c\) is the thickness-to-chord ratio corrected for sweep

  • \(C_L\) is the lift coefficient corrected for sweep

The corrected thickness-to-chord ratio is:

\((t/c)_{eff} = \frac{t/c}{\cos(\Lambda)}\)

The corrected lift coefficient is:

\(C_{L,eff} = \frac{C_L}{\cos^2(\Lambda)}\)

The divergence ratio is:

\(M/M_{cc} = \frac{M}{M_{cc}}\)

The compressibility drag coefficient follows Shevell’s correlation:

\(\Delta C_{D,comp} = 0.0019 \left(\frac{M}{M_{cc}}\right)^{14.641} \cos^3(\Lambda)\)

Definitions

‘Compressibility Drag’

Additional drag caused by compressibility effects as the aircraft approaches and exceeds the critical Mach number.

‘Critical Mach Number’

The freestream Mach number at which the local flow over some part of the aircraft first reaches sonic velocity.

‘Drag Rise’

Rapid increase in drag coefficient as the aircraft approaches and exceeds the critical Mach number.

References

[1] Stanford AA241 Lecture Notes [2] Shevell, R. S. (1989). “Fundamentals of Flight.” Prentice Hall.

See also

RCAIDE.Library.Components.Wings.Main_Wing, RCAIDE.Library.Components.Wings.Blended_Wing_Body, RCAIDE.Library.Methods.Utilities.Cubic_Spline_Blender

transonic_lift_wave_drag(conditions, settings, geometry)[source]#

Computes transonic lift wave drag coefficient using empirical correlations.

Parameters:

  • conditions (Data) –

    Flight conditions containing:
    • freestream.mach_numberfloat

      Freestream Mach number [unitless]

    • aerodynamics.coefficients.lift.totalfloat

      Total lift coefficient [unitless]

    • aerodynamics.coefficients.lift.spanwisefloat, optional

      Spanwise lift distribution [unitless]

  • settings (dict) –

    Analysis settings containing:
    • use_surrogatebool

      Flag to use surrogate model

    • vortex_distributionData, optional

      Vortex distribution data containing: - chord_lengths : array

      Chord lengths at spanwise stations [m]

      • leading_edge_sweepsarray

        Leading edge sweep angles [radians]

      • chord_widthsarray

        Chord widths at spanwise stations [m]

  • geometry (Data) –

    Aircraft geometry containing:
    • reference_areafloat

      Reference area [m²]

Returns:

CD_wave_transonic – Transonic lift wave drag coefficient [unitless]

Return type:

float

Notes

This function calculates the transonic lift wave drag using either a surrogate model or detailed spanwise analysis based on shock wave formation from Lock (1986). A generic airfoil (RAE 5225) is assumed for the Cp data.

Major Assumptions

  • Surrogate model is valid for Mach 0.7-0.95 range

  • Shock wave formation depends on local pressure coefficient

  • Normalized curvature factor is constant (0.23)

  • Spanwise analysis accounts for sweep effects

Theory

For surrogate model: \(C_{D,wave} = f(C_L) \cdot (12.5M - 8.75)\)

For detailed analysis: \(C_{D,wave} = \sum_{i=1}^{n} C_{D,wave,i} \cdot \frac{S_i}{S_{ref}}\)

where each segment’s wave drag is: \(C_{D,wave,i} = \frac{\cos^4(\Lambda)}{\kappa} \cdot 0.243 \cdot \left(\frac{1+0.2M\cos(\Lambda)}{M\cos(\Lambda)}\right)^3 \cdot (M_1^* - 1)^4 \cdot \frac{2-M_1^*}{M_1^*(1+0.2M_1^{*2})} \cdot 0.5\)

The local Mach number before shock is: \(M_1^* = \sqrt{\frac{5+M^2\cos^2(\Lambda)}{(1+0.7M^2C_p)^{2/7}} - 5}\)

Definitions

‘Transonic Wave Drag’

Wave drag occurring in the transonic regime (M ≈ 0.7-0.95).

‘Shock Wave’

Discontinuity in flow properties caused by compressibility effects.

‘Pressure Coefficient’

Dimensionless pressure difference normalized by dynamic pressure.

References

[1] Lock, R. C. (1986). “The Prediction of the Drag of Aerofoils and Wings at High Subsonic Speeds.” Aeronautical Journal. [2] Harris, C. D. (1990). “NASA Supercritical Airfoils.” NASA TP 2969.

supersonic_lift_wave_drag(conditions, configuration, geometry)[source]#

Computes supersonic lift wave drag coefficient using JAXA methodology.

Parameters:

  • conditions (Data) –

    Flight conditions containing:
    • freestream.mach_numberfloat

      Freestream Mach number [unitless]

    • aerodynamics.coefficients.lift.totalfloat

      Total lift coefficient [unitless]

  • configuration (dict) – Aircraft configuration settings

  • geometry (Data) –

    Aircraft geometry containing:
    • wingslist
      List of wing objects containing:
      • chords.rootfloat

        Root chord length [m]

      • spans.projectedfloat

        Projected span [m]

      • aspect_ratiofloat

        Aspect ratio [unitless]

Returns:

cd_lift_wave – Supersonic lift wave drag coefficient [unitless]

Return type:

float

Notes

This function calculates the supersonic lift wave drag using the JAXA methodology based on wing geometry and lift coefficient.

Major Assumptions

  • JAXA methodology is valid for supersonic speeds

  • Only main wing contributes to lift wave drag

  • Wing geometry parameters are sufficient for calculation

  • Empirical correlation is valid for typical supersonic aircraft

Theory

The supersonic lift wave drag coefficient is:

\(C_{D,wave,lift} = C_L^2 \cdot \frac{\beta^2}{\pi} \cdot p \cdot \frac{s}{l} \cdot K_w\)

where:

  • \(\beta = \sqrt{M^2-1}\) is the Prandtl-Glauert factor

  • \(p = \frac{2}{AR} \cdot \frac{s}{l}\) is the wing parameter

  • \(x = \beta \cdot \frac{s}{l}\) is the normalized span parameter

  • \(K_w\) is the wave drag factor calculated from empirical correlation

Definitions

‘Supersonic Wave Drag’

Wave drag occurring at supersonic speeds (M > 1.0).

‘JAXA Methodology’

Empirical method for calculating supersonic wave drag developed by JAXA.

‘Prandtl-Glauert Factor’

Compressibility correction factor for supersonic flow.

References

[1] Yoshida, K. “Supersonic drag reduction technology in the scaled supersonic experimental airplane project by JAXA.”

supersonic_volume_wave_drag(conditions, settings, vehicle)[source]#

Computes supersonic volume wave drag coefficient based on aircraft geometry.

Parameters:

  • conditions (Data) – Flight conditions (not used in calculation)

  • settings (dict) – Analysis settings (not used in calculation)

  • vehicle (Data) –

    Aircraft geometry containing:
    • reference_areafloat

      Reference area for drag coefficient [m²]

    • lengthfloat

      Total aircraft length [m]

    • maximum_cross_sectional_areafloat

      Maximum cross-sectional area [m²]

    • fuselageslist
      List of fuselage objects containing:
      • widthfloat

        Width of the fuselage [m]

Returns:

CD_wave_vol – Supersonic volume wave drag coefficient [unitless]

Return type:

float

Notes

This function calculates the supersonic volume wave drag based on aircraft geometry using empirical correlations from Wright Laboratory procedures.

Major Assumptions

  • Volume wave drag is independent of Mach number

  • Aircraft volume can be approximated by cylindrical geometry

  • Maximum fuselage width represents the characteristic dimension

  • Empirical correlation is valid for typical supersonic aircraft

Theory

The volume wave drag coefficient is:

\(C_{D,wave,volume} = C_{D,wave,volume,frontal} \cdot \frac{A_{max}}{S_{ref}}\)

where the frontal area wave drag is: \(C_{D,wave,volume,frontal} = 24 \cdot \frac{V}{L^3}\)

The aircraft volume is approximated as: \(V = \frac{3}{16} \pi^2 R_{max}^2 L\)

where:

  • \(R_{max}\) is the maximum fuselage radius

  • \(L\) is the total aircraft length

  • \(A_{max}\) is the maximum cross-sectional area

  • \(S_{ref}\) is the reference area

Definitions

‘Volume Wave Drag’

Wave drag caused by aircraft volume in supersonic flow.

‘Frontal Area Wave Drag’

Volume wave drag normalized by frontal area.

‘Characteristic Length’

Representative length scale for volume wave drag calculation.

References

[1] Sieron, T. R., et al. (1993). “Procedures and design data for the formulation of aircraft configurations.” WRIGHT LAB WRIGHT-PATTERSON AFB OH.