RCAIDE.Library.Methods.Aerodynamics.Common.Drag.compressible_mixed_flat_plate

compressible_mixed_flat_plate#

compressible_mixed_flat_plate(Re, Ma, Tc, xt)[source]#

Computes the compressible skin friction coefficient for a flat plate with mixed laminar-turbulent boundary layer.

Parameters:

  • Re (float) – Reynolds number based on chord length [unitless]

  • Ma (float) – Freestream Mach number [unitless]

  • Tc (float) – Freestream static temperature [K]

  • xt (float) –

    Turbulent transition point as a proportion of chord length [unitless]

    • 0.0 indicates fully turbulent flow

    • 1.0 indicates fully laminar flow

    • Values between 0.0 and 1.0 indicate mixed flow

Returns:

  • cf_comp (float) – Compressible skin friction coefficient [unitless]

  • k_comp (float) – Compressibility correction factor [unitless]

  • k_reyn (float) – Reynolds number correction factor [unitless]

Notes

This function calculates the skin friction coefficient for a flat plate accounting for compressibility effects, mixed laminar-turbulent boundary layers, and surface roughness typical of transport aircraft. The method combines laminar and turbulent correlations with compressibility corrections based on temperature effects.

Major Assumptions

  • Reynolds number between 10^5 and 10^9

  • Turbulent transition point between 0 and 1

  • Surface roughness typical of transport aircraft

  • Adiabatic wall conditions

  • Perfect gas behavior

Theory

The effective transition length is calculated as:

\(\theta = \frac{0.671 x_t}{\sqrt{Re_{x_t}}}\)

\(x_{eff} = (27.78 \theta Re^{0.2})^{1.25}\)

where \(Re_{x_t} = Re \cdot x_t\) is the Reynolds number at transition.

The effective Reynolds number for turbulent flow is:

\(Re_{eff} = Re(1 - x_t + x_{eff})\)

The laminar skin friction coefficient is:

\(C_{f,lam} = \frac{1.328}{\sqrt{Re_{x_t}}}\)

The turbulent skin friction coefficient is interpolated from empirical data accounting for typical transport aircraft surface roughness.

The mixed flow skin friction coefficient is:

\(C_{f,mixed} = C_{f,lam} x_t + C_{f,turb}(1 - x_t + x_{eff}) - C_{f,start} x_{eff}\)

The compressibility correction accounts for temperature effects:

\(T_w = T_c(1 + 0.178 M^2)\)

\(T_d = T_c(1 + 0.035 M^2 + 0.45(\frac{T_w}{T_c} - 1))\)

\(k_{comp} = \frac{T_c}{T_d}\)

The Reynolds number correction is:

\(Re_d = Re(\frac{T_d}{T_c})^{1.5}(\frac{T_d + 216}{T_c + 216})\)

\(k_{reyn} = (\frac{Re}{Re_d})^{0.2}\)

The final compressible skin friction coefficient is:

\(C_{f,comp} = C_{f,mixed} \cdot k_{comp} \cdot k_{reyn}\)

Definitions

‘Mixed Boundary Layer’

Boundary layer that transitions from laminar to turbulent flow at some point along the surface.

‘Skin Friction Coefficient’

Dimensionless measure of the shear stress at the wall normalized by dynamic pressure.

‘Compressibility Correction’

Factor accounting for the effects of compressibility on boundary layer properties.

References

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