Flow Similarity

Consider two different flow fields over two different bodies. By definition, different flows are dynamically similar if:

1. The streamline patterns are geometrically similar.

2. The distributions of V/ Voo, p/p^, T/Tetc., throughout the flow field are the same when plotted against common nondimensional coordinates.

3. The force coefficients are the same.

Actually, item 3 is a consequence of item 2; if the nondimensional pressure and shear stress distributions over different bodies are the same, then the nondimensional force coefficients will be the same.

The definition of dynamic similarity was given above. Question: What are the criteria to ensure that two flows are dynamically similar? The answer comes from the results of the dimensional analysis in Section 1.7. Two flows will be dynamically similar if:

1. The bodies and any other solid boundaries are geometrically similar for both flows.

2. The similarity parameters are the same for both flows.

So far, we have emphasized two parameters, Re and Mx. For many aerodynamic applications, these are by far the dominant similarity parameters. Therefore, in a lim­ited sense, but applicable to many problems, we can say that flows over geometrically similar bodies at the same Mach and Reynolds numbers are dynamically similar, and hence the lift, drag, and moment coefficients will be identical for the bodies. This is a key point in the validity of wind-tunnel testing. If a scale model of a flight vehicle is tested in a wind tunnel, the measured lift, drag, and moment coefficients will be the same as for free flight as long as the Mach and Reynolds numbers of the wind-tunnel test-section flow are the same as for the free-flight case. As we will see in subse­quent chapters, this statement is not quite precise because there are other similarity parameters that influence the flow. In addition, differences in freestream turbulence between the wind tunnel and free flight can have an important effect on Co and the maximum value of CL. However, direct simulation of the free-flight Re and Мж is the primary goal of many wind-tunnel tests.

Example 1.4

Consider the flow over two circular cylinders, one having four times the diameter of the other, as shown in Figure 1.20. The flow over the smaller cylinder has a freestream density, velocity and temperature given by p, V, and Гь respectively. The flow over the larger cylinder has a freestream density, velocity, and temperature given by p2, V2, and T2, respectively, where p2 = pi/4, V2 = 2Vi, and T2 = AT. Assume that both ц and a are proportional to T1’2. Show that the two flows are dynamically similar.

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.



Flow Similarity


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.}




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.}


Note that for most conventional flight situations, the magnitude of L and W is much larger than the magnitude of T and D, as indicated by the sketch in Figure 1.23. Typically, for conventional cruising flight, L/D ~ 15 to 20.

For an airplane of given shape, such as that sketched in Figure 1.24, at given Mach and Reynolds number, Cl and CD are simply functions of the angle of attack, a of the airplane. This is the message conveyed by Equations (1.42) and (1.43). It is a simple and basic message—part of the beauty of nature—that the actual values of CL and Cd for a given body shape just depend on the orientation of the body in the flow, i. e., angle of attack. Generic variations for CL and Cd versus a are sketched in Figure 1.25. Note that CL increases linearly with a until an angle of attack is reached when the wing stalls, the lift coefficient reaches a peak value, and then drops off as a is further increased. The maximum value of the lift coefficient is denoted by Ci milx, as noted in Figure 1.25.

The lowest possible velocity at which the airplane can maintain steady, level flight is the stalling velocity, Vstaii; it is dictated by the value of Cl, max, as follows.6 From the definition of lift coefficient given in Section 1.5, applied for the case of level flight where L = W, we have

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




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.


Flow Similarity


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*.,







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)










For a given airplane, without the aid of any artificial devices, Cz,,max is determined purely by nature, through the physical laws for the aerodynamic flowfield over the airplane. However, the airplane designer has some devices available that artificially increase CL, mm beyond that for the basic airplane shape. These mechanical devices are called high-lift devices-, examples are flaps, slats, and slots on the wing which, when deployed by the pilot, serve to increase CLлшх, and hence decrease the stalling speed. High-lift devices are usually deployed for landing and take-off; they are discussed in more detail in Section 4.11.

On the other extreme of flight velocity, the maximum velocity for a given airplane with a given maximum thrust from the engine is determined by the value of minimum drag coefficient, CD min, where Со, тіп is marked in Figure 1.25. From the definition of drag coefficient in Section 1.5, applied for the case of steady, level flight where T = D, we have

Подпись: [1.48]Подпись:D _ T _ IT

Чос s qxs PocV^S

Solving Equation (1.48) for 14c,

2 T

Рос SC о

Flow Similarity Подпись: [1.50]

For a given airplane flying at maximum thrust Гтах and a given altitude, from Equation (1.49) the maximum value of Vx corresponds to flight at CD, min

From the above discussion, it is clear that the aerodynamic coefficients are important engineering quantities that dictate the performance and design of airplanes. For example, stalling velocity is determined in part by Ci max, and maximum velocity is determined in part by C0 min.

Broadening our discussion to the whole range of flight velocity for a given airplane, note from Equation (1.45) that each value of Vx corresponds to a specific value of CL. Therefore, over the whole range of flight velocity from Tstaii to Fmax, the airplane lift coefficient varies as shown genetically in Figure 1.26. The values of CL given by the curve in Figure 1.26 are what are needed to maintain level flight over the whole range of velocity at a given altitude. The airplane designer must design the airplane to achieve these values of CL for an airplane of given weight and wing area. Note that the required values of Cl decrease as Vx increases. Examining the lift coefficient variation with angle of attack shown in Figure 1.26, note that as the airplane flies faster, the angle of attack must be smaller, as also shown in Figure 1.26. Hence, at high speeds, airplanes are at low a, and at low speeds, airplanes are at high a; the specific angle of attack which the airplane must have at a specific Vx is dictated by the specific value of CL required at that velocity.

Obtaining raw lift on a body is relatively easy—even a barn door creates lift at angle of attack. The name of the game is to obtain the necessary lift with as low a drag as possible. That is, the values of CL required over the entire flight range for an airplane, as represented by Figure 1.26, can sometimes be obtained even for the least effective lifting shape—just make the angle of attack high enough. But CD also varies with as governed by Equation (1.48); the generic variation of Co with is sketched in Figure 1.27. A poor aerodynamic shape, even though it generates the necessary values of CL shown in Figure 1.26, will have inordinately high values of CD,

Flow Similarity


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




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



Flow Similarity


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



Flow Similarity


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.


Consider an executive jet transport patterned after the Cessna 560 Citation V shown in three – view in Figure 1.30. The airplane is cruising at a velocity of 492 mph at an altitude of 33,000 ft, where the ambient air density is 7.9656 x 10~4 slug/ft3. The weight and wing planform areas of the airplane are 15,000 lb and 342.6 ft2, respectively. The drag coefficient at cruise is 0.015. Calculate the lift coefficient and the lift-to-drag ratio at cruise.

Подпись: Example 1.6Solution

The units of miles per hour for velocity are not consistent units. In the English engineering system of units, feet per second are consistent units for velocity (see Section 2.4 of Reference 2). To convert between mph and ft/s, it is useful to remember that 88 ft/s = 60 mph. For the present example,

Vx> = 492(H) = 721.6 ft/s

Flow Similarity Подпись: 0.21

From Equation (1.45),

From Equation (1.51),

Подпись:L _ CL _ 0.21 Ъ ~ ~C~D ~~ 0.015

Remarks: For a conventional airplane such as shown in Figure 1.30, almost all the lift at cruising conditions is produced by the wing; the lift of the fuselage and tail are very small by comparison. Hence, the wing can be viewed as an aerodynamic “lever.” In this example, the lift-to-drag ratio is 14, which means that for the expenditure of one pound of thrust to overcome one pound of drag, the wing is lifting 14 pounds of weight—quite a nice leverage.

Flow Similarity


Figure 1.30 Cessna 560 Citation V.


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