Speedrelated Drag
Summary
Speed is related to the flow regime: laminar, transitional, and
turbulent. This is a major problem in all aerodynamic systems. Laminar boundary
layers are characterized by minimum skin friction drag. Laminar boundary layers are
generally assumed to keep laminar at Reynolds numbers
, to be
transitional at about , and turbulent above this value.
The actual transitional Reynolds numbers may depend on the specific case and
several side constraints.
Flat plate, circular cylinder, sphere and cones have been widely studied over the years, and
the amount of data collected is staggering: in particular drag data are available
from the smallest Reynolds numbers (unity and below), to the largest Mach numbers
(hypersonic speeds). The data witness the importance of this set of bodies as a
limiting case of real life problems.
The effect of the velocity (or Reynolds number) on the behavior of the drag
coefficient of a flat plate for both laminar and turbulent incompressible flow
is shown in the figure below. The turbulent drag has been computed with various
theories (von KármánSchoenerr, Prandtl Schlichting, White).
Figure 1: Computed flat plate CD at subsonic speeds
Current laminar wings have drag coefficients closer to the laminar curve (Blasius
theory) than to the turbulent curve. The problem is, however, far more complex, since
reallife flows involve a range of Reynolds numbers with transitional boundary
layers.
The laminar curve in Fig. 1, though, can be considered a practical barrier of
the skin friction drag.
The effect of turbulent transition on the flat plate drag coefficient is shown
schematically in Fig 2. (incompressible flow)
Figure 2: Real flat plate CD at subsonic speeds
At transonic and supersonic speeds the problem is complicated by the temperature
gradient in the boundary layer. Semiempirical correlations of the type shown above
have been proposed (Green, Hoerner, WinterGaudet, etc.) to reduce the compressible
skin friction coefficient to an incompressible one by using the free stream Mach
number.
Flat plate drag calculations at supersonic Mach numbers were first performed
by van Driest, 1952. For details on high speed drag on a flat plate see White, 1974.
There is a large body of investigations on cylinders at all speeds and all aspect
ratios, with fixed or rotating bodies. The infinite cylinder (e.g. cylinder of very
large L/D) is one of the most amusing problems in fluid dynamics. Its rational study
was first performed by von Kármán (1911), who investigated the
appearance of the so called vortex trail (or vortex street), while studying
the advantages of streamlined bodies for drag reduction.
The following considerations will be restricted to the drag characteristics as
function of the Reynolds number. Fig. 3 below shows a classic summary of cylinder
drag coefficients, from the creeping flow domain (see below) to large Reynolds
numbers. Speeds are intended as subsonic at all cases. The data show a
drag crisis at about Re=500,000.
Figure 3: Cylinder CD at subsonic speeds
The technical literature reports a large number of semiempirical formulas for
the CD. The experimental drag of Fig. 3 can be fitted with a simple
equation.
The finite cylinder is not less interesting. Actually, it features a great variety
of wake flow patterns, instabilities and drag coefficients (Williamson, 1996).
Fig. 4 shows the behavior of the drag coefficient for a sphere at subsonic
speeds. The surface finish has been found of extreme importance in imparting
aerodynamic characteristics. The two curves on the graphic refer to two different
surface conditions. When the surface is rough, turbulent transition occurs earlier,
and so does the drag drop. This feature is fully exploited in golf balls (Metha,
1985).
Figure 4: CD of a sphere at subsonic speeds
Experiments on spheres have been perfomed up to M=12.15 in freon (to the
author’s knowledge.) The figure below shows the CD behavior at supersonic and
hypersonic speeds (data elaborated from CoxCrabree, 1965).
Figure 5: CD of a sphere at supersonic speeds
Very low speeds are characteristic of flows at Reynolds numbers less than a
50,000. Some airfoils still work as at Reynolds numbers as low as 30,000. Yet they
become increasingly inefficient at lower speeds. This range is also that of the
model airplanes, micropropellers, and microair vehicles (MAV).
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.
At transonic speeds there are local buckets of supersonic flow delimited by
shock waves. Shock waves and shockinduced 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.
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 crosssectional area along the longitudinal axis is
a smooth function. The combination of wingbody interference, in fact, can be reduced
to a slender body optimum drag problem, for which the solution is known (SearsHaack,
1947; von Kármán, 1948).
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 flowreversal
theorems ( von Kármán, Hayes, Jones, Graham et. al.). See
AshleyLandhal (1965) and HeasletSpreiter (1953) for details.
Related Material
Selected References
 White FM, Viscous Fluid Flow , McGrawHill, New York, 1974.
 Hoerner SF, Fluid Dynamic Drag, Hoerner Fluid Dynamics, 1965.
 AGARD, Aircraft Drag Prediction and Reduction, AGARD Report R723, 1985.
 Ashley H, Landhal M, Aerodynamics of Wings and Bodies, AddisonWesley,
Reading, MA, 1963.
 Clift R, Grace JR, Weber ME, Bubbles, Drops, and Particles, Academic Press,
New York, 1978.
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