Creeping Flows
At lower speeds we find many insects. Flows at Re < 10 are also called creeping
flows, which are not considered properly aerodynamic.
The drag characteristics at low speeds are strongly affected by the laminar
separation and by viscous skin friction, according to a physics explained in the low
speed chapter.
The drag coefficient can take very unusually high values, that are approximated
with the Oseen formula at Re < 1 and by the Klaycho formula at Re < 400.
For extensive low Reynolds data consult Clift et. al, 1978.
Drag reduction at low speeds is a very open problem in aerodynamics, that only
recently has become object of analysis, mainly spurred by technological advances in
solar powered flight, high altitude flight, unmanned vehicles, model airplains, and
more.
Drag at Transonic Speeds
At transonic speeds there are local buckets of supersonic flow
delimited by
shock waves. Shock waves and shock-induced boundary layer separation are a consistent
source of drag at these speeds. A typical example of how the drag increases is given
by the divergence Mach number for a airfoil section (below)
Figure 6: Transonic drag rise
At a certain Mach number that depends on the airfoil and the angle of attack, a wave
drag starts to build up because of the increasing effect of the shock wave. Once the
flow is fully supersonic, the drag coefficient falls. The climb shown in Fig. 6 can
be pushed toward higher Mach numbers with supercritical airfoils.
Airfoils at Transonic Speeds
A case of particular interest is that of the airfoil section, whose transonic drag
rise is dependent on the angle of attack. An example is shown in Fig. 7 below.
Figure 7: Transonic drag rise, with alfa as parameter
Military Aircraft
Military aircraft feature external stores and weapons systems that can change
dramatically the performance of the aircraft. Here only a comparative effect
will be shown for some selected configurations, Fig. 8.
Figure 8: Transonic drag rise, with alfa as parameter
Methods for reducing the drag at transonic speeds include the use of
As in the case of lower speeds, drag is produced by viscosity and vorticity
release. There is one more component, called wave drag, peculiar to supersonic
flows. In general the total drag will consists of the skin friction (viscous) drag,
the induced drag (as in subsonic flows), the (supersonic) drag due to volume, and the
(supersonic) wave drag due to lift.
Supersonic flows are considered well behaved and more stable, as compared with
transonic flows, because the problem of the shock at the wall is eliminated.
Effect of Nose Bluntness
Bodies of minimum drag at supersonic and hypersonic speeds have a blunted nose.
The radius of a blunt body is an essential parameter in determining the heat flux.
Figure 9: Hypersonic CD for sphere and cone
Supersonic Area Rule
The problem of computing and minimizing the wave drag is fairly complicated, because of
several different sources (listed above), and because of conflicting constraints.
A general practice is the supersonic area ruling: The wave drag is
minimized if the distribution of cross-sectional area along the longitudinal axis is
a smooth function. The combination of wing-body interference, in fact, can be reduced
to a slender body optimum drag problem, for which the solution is known (Sears-Haack,
1947; von Kármán, 1948).
Elliptic Wings
The wave drag due to lift is minimized when the loading on each oblique plane is
elliptical. The wave drag due to volume is at a minimum when each equivalent body of
revolution (opportunely defined) is a Sears- Haack body.
Overall minimum induced drag can be obtained with an oblique wing of elliptical
planform having elliptical loading (R.T. Jones, von Kármán). Elliptical loading
distribution can be obtained by twisting the wing.
Another approach to drag minimization is the use of flow-reversal
theorems ( von Kármán, Hayes, Jones, Graham et. al.). See
Ashley-Landhal (1965) and Heaslet-Spreiter (1953) for details.
Related Material
Selected References
- White FM, Viscous Fluid Flow , McGraw-Hill, New York, 1974.
- Hoerner SF, Fluid Dynamic Drag, Hoerner Fluid Dynamics, 1965.
- AGARD, Aircraft Drag Prediction and Reduction, AGARD Report R-723, 1985.
- Ashley H, Landhal M, Aerodynamics of Wings and Bodies, Addison-Wesley,
Reading, MA, 1963.
- Clift R, Grace JR, Weber ME, Bubbles, Drops, and Particles, Academic Press,
New York, 1978.
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