The skin friction coefficients are sometimes based on experimental data for flat plates with various amounts of roughness. In the present method, experimental results for turbulent flat plates are fit and combined with basic laminar flow boundary layer theory to produce the data in the figure below. The data apply to insulated flat plates with transition from laminar to turbulent flow specified as a fraction of the chord length (xt / c = 0 represents fully turbulent flow.) The data are total coefficients; that is, they are average values for the total wetted area of a component based on the characteristic length of the component.
When the skin friction is plotted on a log-log scale the curves are nearly straight lines, but the actual variation of cf is more pronounced at lower Reynolds numbers.
For fully turbulent plates, the skin friction coefficient may be approximated by one of several formula that represent simple fits to the experimentally-derived curves shown in the above figure. For incompressible, flow:
The logarithmic fit by von Karman seems to be a better match over a larger range of Reynolds number, but the power law fit is often more convenient. (Note that the log in the above expression is log base 10, not the natural log, denoted ln here.)
In the computation of Reynolds number, Re = r V l / m, the characteristic length, l, for a body (fuselage, nacelle) is the overall length, and for the aerodynamic surfaces (wing, tail, pylon) it is usually the exposed mean aerodynamic chord. The values of density (r), velocity (V), and viscosity m are obtained from standard atmospheric conditions at the point of interest. For our purposes we often use the initial cruise conditions. Atmospheric data may be computed in the atmospheric calculator included here.
Experimental measurements of skin friction coefficient compared with curve fits. Note scatter and transition between laminar and turbulent flow.
It is, for all practical purposes, impossible to explicitly define the incremental drags for all of the protruding rivets, the steps, the gaps, and bulges in the skin; the leakage due to pressurization; etc. Instead, in the method of these notes, an overall markup is applied to the skin friction drag to account for drag increments associated with roughness resulting from typical construction procedures. Values of the roughness markup factor have been determined for several subsonic jet transports by matching the flight-test parasite drag with that calculated by the method described in these notes. The values so determined tend to be larger for smaller airplanes, but a 6%-9% increase above the smooth flat values shown in the figure is reasonable for initial design studies. Carefully-built laminar flow, composite aircraft may achieve a lower drag associated with roughness, perhaps as low as 2-3%.
The drag assigned to roughness also implicitly accounts for all other sources of drag at zero lift that are not explicitly included. This category includes interference drag, some trim drag, drag due to unaligned control surfaces, drag due to landing gear door gaps, and any excess drag of the individual surfaces. Consequently the use of the present method implies the same degree of proficiency in design as that of the airplanes from which the roughness drag correlation was obtained.
The friction coefficient is affected by Mach number as well. The figure below shows that this effect is small at subsonic speeds, but becomes appreciable for supersonic aircraft. For this course, the effect may be approximated from the plot below, but a computational approach is described by Sommer and Short in NACA TN 3391 in 1955. The idea is that aerodynamic heating modifies the fluid properties. If one assumes a wall recovery factor of 0.89 (a reasonable estimate), and fully-turbulent flow, the wall temperature may be estimated from:
An effective incompressible temperature ratio is defined:
When the viscosity ratio is given by the Sutherland formula (with T in units of °R):
The effective Reynolds number becomes:
So, the compressible skin friction coefficient is then given by:
where c'finc is the incompressible skin friction coefficient, computed at the Reynolds number R'.
Finally, the ratio of compressible Cf to incompressible Cf at the same Reynolds number is:
The net result is shown in the plot below.
Note that the difference in Cf between Mach 0 and Mach 0.5 is about 3%.
A program for computing Cf is available here.