# Speed-related 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.

### Drag of Well Known Bodies

#### Flat Plate Drag

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 Kármán-Schoenerr, 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 real-life 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. Semi-empirical correlations of the type shown above have been proposed (Green, Hoerner, Winter-Gaudet, etc.) to reduce the compressible skin friction coefficient to an incompressible one by using the free stream Mach number.

#### Circular Cylinder

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 semi-empirical formulas for the CD. The experimental drag of Fig. 3 can be fitted with a simple equation.

#### Sphere Drag

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 Cox-Crabree, 1965).

Figure 5: CD of a sphere at supersonic speeds

### Drag at Very Low 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, micro-propellers, and micro-air vehicles (MAV).

#### 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 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

#### Drag at Supersonic Speeds

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.

#### 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.

#### 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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