Flow Similarity

Geometrically similar bodies

Figure 1.20 Example of dynamic flow similarity. Note that as part of the definition of dynamic similarity, the streamlines (lines along which the flow velocity is tangent at each point) are geometrically similar between the two flows.

Figure 1.22 The NACA variable density tunnel (VDT). Authorized in March of 1921,

the VDT was operational in October 1922 at the NACA Langley Memorial Laboratory at Hampton, Virginia. It is essentially a large, subsonic wind tunnel entirely contained within an 85-ton pressure shell, capable of 20 atm. This tunnel was instrumental in the development of the various families of NACA airfoil shapes in the 1920s and 1930s. In the early 1940s, it was decommissioned as a wind tunnel and used as a high-pressure air storage tank. In 1983, due to its age and outdated riveted construction, its use was discontinued altogether. Today, the VDT remains at the NASA Langley Research Center; it has been officially designated as a National Historic Landmark. (Courtesy of NASA.}

Screen

Figure 1.23 Schematic of the variable density tunnel. (From Baals, D. D. and Carliss, W. R„ Wind Tunnels of NASA, NASA SP-440, 1981.}

extreme measures sometimes taken in order to simulate simultaneously the free-flight values of the important similarity parameters in a wind tunnel. Today, for the most part, we do not attempt to simulate all the parameters simultaneously; rather, Mach number simulation is achieved in one wind tunnel, and Reynolds number simulation in another tunnel. The results from both tunnels are then analyzed and correlated to obtain reasonable values for CL and CD appropriate for free flight. In any event, this example serves to illustrate the difficulty of full free-flight simulation in a given wind tunnel and underscores the importance given to dynamically similar flows in experimental aerodynamics.

Design Box I

Lift and drag coefficients play a strong role in the preliminary design and performance analysis of airplanes. The purpose of this design box is to enforce the importance of CL and Ct) in aeronautical engineering; they are much more than just the conveniently defined terms discussed so far—they are fundamental quantities, which make the difference between intelligent engineering and simply groping in the dark.

Consider an airplane in steady, level (horizontal) flight, as illustrated in Figure 1.24. For this case, the weight W acts vertically downward. The lift L acts vertically upward, perpendicular to the relative wind Vx (by definition). In order to sustain the airplane in level flight,

L = W

The thrust T from the propulsive mechanism and the drag D are both parallel to Vk,. For steady (unaccelerated) flight,

T = D

L _ W _ 2 W

qxS qxS PocV^S

[1.45]

6 The lowest velocity may instead by dictated by the power required to maintain level flight exceeding the power available from the powerplant. This occurs on the "back side of the power curve." The velocity at which this occurs is usually less than the stalling velocity, so is of academic interest only. See Anderson, Aircraft Performance and Design, McGraw-Hill, 1999, for more details.

Figure 1.24 The four forces acting on an airplane in flight.

Figure 1.25 Schematic of lift and drag coefficients versus angle of attack; illustration of maximum lift coefficient and minimum drag coefficient.

Solving Equation (1.45) for V4*.,

2W

PooSCL

[1.46]

For a given airplane flying at a given altitude, W, p, and S are fixed values; hence from Equation (1.46) each value of velocity corresponds to a specific value of CL. In particular, will be the smallest when CL is a maximum. Hence, the stalling velocity for a given airplane is determined by C;. max from Equation (1.46)

2W

[1.47]

К..П

PooSCl.1

a Decreasing

Schematic of the variation of lift coefficient with flight velocity for level flight.

Figure 1.26

i. e., the CD curve in Figure 1.27 will ride high on the graph, as denoted by the dashed curve in Figure 1.27. An aerodynamically efficient shape, however, will produce the requisite values of CL prescribed by Figure 1.26 with much lower drag, as denoted by the solid curve in Figure 1.27. An undesirable by-product of the high-drag shape is a lower value of the maximum velocity for the same maximum thrust, as also indicated in Figure 1.27.

Finally, we emphasize that a true measure of the aerodynamic efficiency of a body shape is its lift-to-drag ratio, given by

L _ qocSCL _ Cl D qooSCp Co

[1.51]

Since the value of CL necessary for flight at a given velocity and altitude is determined by the airplane’s weight and wing area (actually, by the ratio of W/S, called the wing loading) through the relationship given by Equation (1.45), the value of L/D at this velocity is controlled by CD, the denominator in Equation (1.51). At any given velocity, we want L/D to be as high as possible; the higher is L/D, the more aerodynamically efficient is the body. For a given airplane at a given altitude, the variation of L/D as a function of velocity is sketched generically in Figure 1.28. Note that, as Voo increases from a low value, L/D first increases, reaches a maximum at some intermediate velocity, and then decreases. Note that, as increases, the angle of attack of the airplane decreases, as explained earlier. From a strictly aerodynamic consideration, L/D for a given body shape depends on angle of

a Decreasing

Figure 1.27 Schematic of the variation of drag coefficient with flight velocity for level flight. Comparison between high and low drag aerodynamic bodies, with the consequent effect on maximum velocity.

attack. This can be seen from Figure 1.25, where Cl and Co are given as a function of a. If these two curves are ratioed, the result is L/D as a function of angle of attack, as sketched generically in Figure 1.29. The relationship of Figure 1.28 to Figure 1.29 is that, when the airplane is flying at the velocity that corresponds to (L/£>)raax as shown in Figure 1.28, it is at the angle of attack for (L/£>)max as shown in Figure 1.29.

In summary, the purpose of this design box is to emphasize the important role played by the aerodynamic coefficients in the performance analysis and design of airplanes. In this discussion, what has been important is not the lift and drag per se, but rather CL and CD. These coefficients are a wonderful intellectual construct that helps us to better understand the aerodynamic characteristics of a body, and to make reasoned, intelligent calculations. Hence they are more than just conveniently defined quantities as might first appear when introduced in Section 1.5.

For more insight to the engineering value of these coefficients, see Anderson, Aircraft Performance and Design, McGraw-Hill, 1999, and Anderson, Introduction to Flight, 4th edition, McGraw-Hill, 2000. Also, home­work problem 1.15 at the end of this chapter gives you the opportunity to construct specific curves for CL, CD, and L/D versus velocity for an actual airplane so that you can obtain a feel for some real numbers that have been only

Figure 1.28

Schematic of the variation of lift-to-drag ratio with flight velocity for level flight.

Figure 1.29 Schematic of the variation of lift-to-drag ratio with angle of attack.

generically indicated in the figures here. (In our present discussion, the use of generic figures has been intentional for pedagogic reasons.) Finally, an historical note on the origins of the use of aerodynamic coefficients is given in Section 1.13.

Figure 1.30 Cessna 560 Citation V.