Category An Introduction to THE THEORY OF. AEROELASTICITY

A question arises naturally in connection with the study of tail buffeting: What happens to the wing that produces the turbulent wake? A separ­ated flow over the wing creates aerodynamic forces that must be regarded as stochastic processes. The wing’s elastic response is necessarily also stochastic and therefore may be properly called buffeting. However, as the unsteady aerodynamic forces are produced by the wing itself, in a flow that otherwise is steady, the motion of the wing may, according to the definition of § 9.4, be called flutter. The nature of the wing motion also borders between buffeting and stall flutter described in the preceding sections. Sometimes the motion is quite random, sometimes it is quite regular, and sometimes it is a mixture of the two: e. g., random in bending, while more or less a regular sinusoidal oscillation in torsion. In order to distinguish the last feature from those described in the preceding sections, we shall use a new term: buffeting flutter.

Buffeting flutter of a wing is a problem of grave concern in transonic flight. It frequently causes wing and aileron damages.

Regarding buffeting flutter as a mixture of buffeting and stall flutter, we anticipate a situation as follows. Consider a cantilever wing. The buffeting boundary of the wing will be similar to that of the tail, because of their common origin in flow separation. For a hypothetical wing this boundary is sketched in Fig. 9.12 (cf. Fig. 9.4). On the other hand, if the stall-flutter characteristics of the wing is similar to that shown in Fig. 9.6, and if we plot the stall-flutter boundary with angle of attack as

ordinate and Mach number as abscissa, the result of Fig. 9.12 may be obtained. The relative position of these two boundaries depends on the wing planform, airfoil section, wing-fuselage junction geometry, etc. The boundaries as shown in Fig. 9.12 intersect each other. The wing motion beyond the buffet boundary is random in amplitude and frequency, and in general is predominantly bending, with very small torsional motion. As the stall-flutter boundary is reached, torsional motion of more or less regular amplitude and frequency takes place, while random motion in bending may continue.

One may question the definitiveness of the terms buffeting, stall flutter, and buffeting flutter. Indeed, since fluctuations in the flow, whether

caused by the wing itself or by other means, are the basic reason of all these phenomena, it is impossible to distinguish them with absolute clarity. Therefore they should be interpreted only as descriptive terms useful for engineering purpose: Buffeting characterizes the irregularity of the motion, stall flutter the more or less regular oscillations, and buffeting flutter a mixture of the two in different degrees of freedom.

As mentioned before, buffeting flutter is a serious problem in high-speed flight. Flight at high subsonic Mach numbers is often troubled by strong shock-wave formation and associated boundary-layer separation. A tendency of strong pressure pulsation over the wing exists, particularly in the region of the shock wave. This pressure pulsation becomes irregular when the angle of attack is sufficiently large. The intensity of this pressure pulsation can be mitigated by reducing the thickness of the airfoil and limiting the angle of attack.

As the basic cause of buffeting at high Mach number is the shock-wave formation and boundary-layer separation, the buffeting boundary should be related closely to the characteristics of the curve of lift coefficient versus Mach number. The “Mach number of divergence” of the wing (cf. § 4.3) may serve as a good estimate of the buffeting boundary at high speed. For the same reason, by reducing the wing thickness, by a proper control of the leading-edge camber which reduces the possibility of flow separation at the leading edge in moderate angles of attack, by sufficient sweepback, by using smaller aspect ratio, and by other auxiliary means, one can hope to design a wing that is practically free from buffeting throughout the transonic speed range. In supersonic flight, when the shock wave is attached to the leading edge of the wing, buffeting does not seem to be a serious problem.

An allied problem of buffeting of aircraft structures due to fluctuating loads induced by a jet is discussed by Miles.9 42

The phenomena of buffeting, stall flutter and buffeting flutter are profoundly interesting physical phenomena which still defy satisfactory mathematical analysis.

According to Eq. 1 of § 9.4, the critical stall-flutter speed can be raised by increasing the frequency of free torsional oscillation of the blade, i. e., by increasing the torsional stiffness and reducing the mass moment of inertia.

Stall flutter can be delayed if the airfoil can be prevented from stalling. Thus, in designing propellers, care must be exercised to choose the proper airfoil section, and, if possible, the working angle of attack should be limited to below the stalling angle.

It is shown by Theodorsen and Regier4,36 that the divergence speed of a propeller is an important parameter to consider in connection with the stall flutter (see § 4.8). If the rotational speed of a propeller is so high that the relative wind speed is close to the critical-divergence speed, the
blade will be twisted excessively, possibly beyond the stalling angle, and thus causing stall flutter. Theodorsen and Regier show that, for several models of wind-tunnel propellers tested in the high-angle-of-attack range (with initial settings below the stalling angle), flutter invariably occurs at a speed substantially below the classical flutter speed. The angle of attack of the blade at which flutter occurs appears to be nearly constant and independent of the initial blade setting. Apparently the blade simply

twists to the stalling angle, and flutter starts. Furthermore, Theodorsen and Regier show that the classical flutter speed and the divergence speed of a propeller are approximately the same because of the centrifugal force effect. The problem of predicting propeller flutter is thus resolved pri­marily into the calculation of the speed at which the propeller will be twisted to stall. This can be done by methods of Chapter 3.

As to the practical design measures to raise the critical stall-flutter speed, we may quote an interesting case reported by Sterne and Brown.7-141

They tested on a spinning tower a variable-pitch propeller fitted with compressed wood blades. The airfoil sections of the blade inboard of the 75 per cent radius were of conventional Clark shape, but outboard of the 75 per cent radius they had an undercamber. Serious flutter was encountered when the blades were set at angles larger than 20° at the 70 per cent radius. But, when the blades were set at 19° and less, there was no evidence of flutter. The difference in the flutter characteristics at the blade-angle settings of 20° and 19° was very pronounced. It appeared that the flutter was stall flutter. A pitch setting of 20° corresponded to a lift coefficient CL of about 1.0 at the tip sections. The propeller was then modified by the removal of part of the leading edge of the sections out­board of 75 per cent radius, and reshaping the cross sections near the blade tips into the conventional Clark form. This modification had the effect of bringing the center of pressure further back, relative to the centroids of the inboard sections, so that the torsional deflection of the blades became smaller. The modified propeller was then retested. No flutter was encountered for blade settings up to 22° at the 70 per cent radius.

So far our attention is directed toward stall flutter at low speed of flow (incompressible fluid). The stall-flutter problem of high-speed aircraft is complicated not only by the effect of the compressibility of the air, but also by the geometrical factors often associated with high-speed wing designs: low aspect ratio, thin wing sections, sweep angle, large masses attached to the wings, etc. To determine the effects of these items on the stall-flutter characteristics of a wing is a challenging problem for future research.

The term flutter is applied categorically to oscillations of an elastic body in a flow that is steady in the absence of the body. If, during part or all of the time of oscillation the flow is separated, then the flutter phenomenon exhibits some characteristics different from those discussed in Chapters 5-7 and is called a stall flutter.

Stall flutter is a serious aeroelastic instability for rotating machineries such as propellers, turbine blades, and compressors, which sometimes have to operate at angles of attack close to the static stalling angle of the blades. Airplane wings and tails rarely suffer from stall flutter. How­ever, the trend toward thin wing sections and large wing span has increas­ingly made stall flutter of wings a serious concern for the design of high­speed aircraft.

The phenomenon of stall flutter can be best illustrated by the spinning test of a propeller. Generally, when the speed of a propeller is gradually increased, the appearance of a peculiar noise normally associated with propeller-blade flutter can be quite definitely determined. As the speed is increased beyond a value at which flutter can first be detected aurally, considerable weaving of the propeller tips can be detected visually by an observer in the plane of the propeller disk. The blade motions can be recorded by attaching, to the blades, strain gages which measure the amplitude of strain as the propeller speed changes.

Figure 9.6 shows a typical result of the spinning test of a particular propeller obtained by Sterne.7140 The abscissa is the blade angle at 70 per cent radius of the propeller; the ordinate is the propeller speed. Above the solid curve is a region of flutter, below it, there is no flutter: Over the wide range of blade settings of this particular propeller, there are two very abrupt changes of flutter speed. These two abrupt changes divide the range of blade settings at which the propeller is tested into three regions—conveniently described as fine pitch, medium pitch, and coarse pitch. The characteristics in each region are as follows:

At very fine pitch settings, corresponding approximately to zero lift angle at the blade tips, there is no flutter in the speed range of the tests. In the fine-pitch range, the critical speed first decreases as the pitch angle is increased, and then tends to a constant value at larger angles of attack.

In the medium-pitch region, the critical flutter speed is constant, but is much lower than that in the fine – and coarse-pitch regions.

In the coarse-pitch region, the flutter speed is constant and is approx­imately equal to that at the coarser end of the fine-pitch region.*

Clearly, there are two distinct types of flutter. The blade settings in the medium-pitch range correspond to the stalling angles of the tip sections. In the medium-pitch range the blade stalls over part of the cycles of oscillation. In the coarse-pitch region the blade remains stalled through­out the cycles. f The flutter in the medium – and coarse-pitch regions is

Blade angle at 0.7 radius

Fig. 9.6. Propeller flutter. Critical flutter speed against blade angle at 0.7
radius. Steady-state stalling angle of airfoil equal to 12 degrees, correspond-
ing to a blade angle of 20 degrees at 0.7 radius of propeller. Measurements
by Sterne, Ref. 7.140. (Courtesy of the Aeronautical Research Council.)

stall flutter. That in the fine-pitch region is classical flutter. The flutter motions of the blade in both regions are nearly sinusoidal.

In general, as the angle of attack is raised through the stall region, the following characteristics may be observed:

* This equality of the flutter speed in the fine-pitch and coarse-pitch regions must be regarded as a mere coincidence in this particular example.

t Stall flutter in the coarse-pitch range, i. e. at high angles of attack, resembles a forced vibration. In fact, if the speed of flow is gradually raised beyond the stall- flutter boundary at such a high angle of attack, the intensity of flutter increases rapidly at first, reaches a maximum, then decreases to practically flutter-free condition. The reason is that, at such high angles of attack, the predominant frequency in the wake (vortex street, cf. § 9.3) becomes clearly defined. The stall flutter of the wing is simply the reaction to the creation of the vortices in the wake. The phenomenon resembles the oscillation of smokestacks as analyzed in § 2.4.

1. The flutter speed drops severely.

2. The flutter frequency rises slowly toward the natural torsional oscillation frequency of the blade in still air.

3. The torsional motion predominates. Whereas in classical flutter the torsional strain and bending strain are of the same order of magnitude, in stall flutter the amplitude of the bending oscillation becomes negligibly small in comparison with that of the torsion, although the axis of rotation is in general not the elastic axis.7124

4. Usually the flutter speed reaches a minimum and rises again as the blade becomes completely stalled. The flutter amplitude becomes very small in the coarse-pitch region. The range of the angle of attack in which the violent low-speed flutter persists increases as the elastic axis is moved backward along the chord.

5. There is a large phase shift at the transition from classical flutter to stall flutter. The phase difference between the bending and torsion drops by about 45°, sometimes vanishing completely.

6. Variation of the structural properties of the airfoils has very different effects on the stall flutter as compared with the classical flutter. Change of inertia-axis location has little effect on stall flutter, which could occur even when the inertia axis lies ahead of the elastic axis when it is impossible to obtain classical flutter. The ratio of the uncoupled bending and torsion frequencies in still air has little effect on stall flutter, for which the critical speed is often higher when the ratio is equal to 1 than at other values (whereas at low angles of attack there is usually a minimum flutter speed when the bending and torsion frequencies are equal). In one case, Stiider9-26 found the critical speed to be the lowest when the wing is restrained from translatory motion and is allowed only freedom in pitch.

Stiider9-26 was the first one who studied stall flutter experimentally in great detail. In a series of tests he examined the field of flow around an airfoil oscillating about the stalling angle. His result shows that, in a stroke of increasing amplitude, the separation is delayed to an angle of attack appreciably greater than that for a stationary airfoil. On the return movement, re-establishment of a smooth flow is also delayed. Stiider calls this an “aerodynamic hysteresis” and concludes that it is the basic cause for stall flutter. This observation is supported by the earlier works of Farren and others (see § 15.4).

Thus the influence of flow separation on stall flutter is revealed through the hysteresis effect. Since the aerodynamic characteristics is nonlinear, the principle of superposition does not hold. The aerodynamic forces corresponding to different modes of motion cannot be added. Hence, an extensive experimental evaluation of the aerodynamic forces is neces­sary (see surveys by Victory,9,27 Halfman,9-22 and Bratt15,121)-

A sketch showing typical forms of the hysteresis loop corresponding to simple-harmonic oscillations in pitch is given in Fig. 9.7. At smaller angles of attack, the loop is elliptical, as predicted by the linearized theory. At angles of attack in the neighborhood of the static stalling angle, the loop is of the shape of a figure 8 (the left-hand side loop may become very small or disappear entirely, leaving a loop which indicates all positive work). At large angles of attack, the loop becomes oval again. The exact shape of the loops depend on the location of the axis of rotation,

the reduced frequency, the amplitude of oscillation, the Reynolds number, and the airfoil shape.

The area within each loop represents the work done by the air on the airfoil in each cycle ‘of oscillation. A positive sign in front of W in Fig.

9.7 indicates an energy gain by the airfoil; a negative sign, an energy loss. The oscillation is aerodynamically unstable with respect to pitch if the net gain of energy during each cycle is positive.

Since for a stalled airfoil the aerodynamic force produced by a simple – harmonic motion, though periodic, is not exactly simple harmonic, the work per cycle must be obtained by graphical integration of experimental results. In Fig. 9.8 is shown the work per cycle in pure pitching motion computed922 from the data obtained by Bratt and his associates9-27 (airfoil’s steady-state stalling angle 12°, elastic-axis location mid-chord, Reynolds number 1.42 X IQ5). In this figure, oq denotes the mean angle of attack, Да denotes the difference between the mean angle of attack and the steady-state stalling angle. A positive value of Да means that а,- is above the steady-state stalling angle. The amplitude of pitching oscilla­tion is approximately 6°. Points above the W — 0 line correspond to aero – dynamically unstable oscillations. A curve corresponding to the linearized

two-dimensional theory is added for comparison. When ос,- is small, the experimental work per cycle agrees closely with the theoretical prediction (cf. Figs. 20-26 of Ref. 9.22).

Figure 9.9 shows the work per cycle in pure translation obtained by Halfman9-22 for a wing designated as “blunt,” whose steady-state stalling

angle is 15°, at a Reynolds number near 10®. It is seen that the work per cycle remains negative for the range of reduced frequencies tested. The oscillation in pure translation is therefore stable, a conclusion in agree­ment with the tests by von Karman and Dunn2 22 (§2.5).

Because of the nonlinear nature of the aerodynamic response, the British and American experimental results cannot be compared with each

other quantitatively, since the Reynolds number, the airfoil shape, and the reduced frequency range are different. But, using Да as a basis,

Да = а, — a6tall

where аг is the mean angle of attack and astall is the steady-state stalling angle, Halfman9-22 obtains an interesting qualitative comparison. From Halfman’s results9-22 in the lower к range and the British data9-27 in the higher к range, Fig. 9.10 is constructed. This figure shows qualitatively the way in which work per cycle varies with к and a. First, for a low angle of attack, the curves of work per cycle against к remain negative, quite close to the theoretical curves. At given values of к the work per

cycle gradually approaches zero as Да increases. When Да is zero, a positive work area appears at lower values of k. As Да increases further, the maximum value of positive work increases, whereas the range of positive work is narrowed but continues to move to higher values of k.

Since stall flutter is, predominantly torsional, it may be assumed, as a crude approximation, that the flutter frequency is the same as the natural vibration frequency of torsion in still air, and that stall flutter will occur when the aerodynamic work per cycle for pure pitch becomes positive and greater than the energy dissipated by the structure. From Fig. 9.10 it is seen that there is a region on the (к, Да) plane in which the work per cycle is positive. For each Да, this region is bounded by two critical values of k. Let the upper critical value of к be denoted by ker. For a given airfoil, at a given angle of attack, the value of к gradually decreases as the speed of flow gradually increases (k — oo when U = 0, the frequency со being that of the natural torsional vibration). When U reaches Ucr, where

U

сг Ъ л’сг

a pitching oscillation of the airfoil causes no exchange of energy with the airstream, and gives a crude estimation of the critical stall flutter speed. More accurate methods of analysis are discussed in Refs. 9.22, 9.27.

Figure 9.11 shows a summary of the kcr values obtained in various laboratories. The effects of location of the axis of rotation, airfoil shape, Reynolds number R, and amplitude of oscillation 0 are revealed by the wide variation among the results.

The similarity of the wake behind a stalled airfoil and that behind a circular cylinder suggests at once a “vortex-shedding” theory of tail buffeting. In Abdrashitov’s analysis91 the wing wake is regarded as a well developed vortex street in which the vortices are regularly spaced, and the buffeting motion of the tail is computed as a forced vibration. The random characteristics of tail buffeting are neglected.

A different idealization is made by Shih-chun Lo9’12 who regards the wing wake as a vortex sheet—an “interface” without thickness, across which the flow undergoes a sudden change in velocity and density. Velocity potential is assumed to exist on both sides of the interface. It is shown that the vortex sheet reacts strongly with an oscillating airfoil located in the neighborhood of the interface, particularly for higher values of the reduced frequency. Large reduction of flutter speed, due to the presence of an interface, is indicated. It is impossible to verify experi­mentally Lo’s mathematical theory, because such an idealized interface cannot be produced. However, wind-tunnel tests made by Dankworth and Walker,9-6 using a “barrier” which blocks partly the flow in the working section, do not indicate any significant change in flutter speed of an airfoil when the distance between the airfoil and the “interface” (which is taken as the tangent to the barrier) exceeds 20 per cent of the chord length of the airfoil. When this distance is less than 20 per cent of the chord length, irregular motion of the airfoil is observed which is clearly due to turbulences in the jet mixing region.

Liepmann911 points out that buffeting is the response of an elastic body to a turbulent flow, and hence is a stochastic process. A correct theory of buffeting, therefore, must account for the turbulent characteristics of the oncoming flow. If the power spectrum of the turbulences in the flow is known, the intensity of the buffeting motion can be calculated in a

manner similar to the gust-response calculation of the previous chapter, provided that the intensity of turbulence is so low and the amplitude of the wing motion is so small that the wing is never stalled during buffeting. If stalling did occur, even only for a fraction of the time, the relation among the lift, the turbulent fluctuations, and the wing motion becomes nonlinear, and the calculation of the buffeting intensity is more difficult. We must point out that, turbulence being a stochastic process, large – velocity fluctuation may exist, though infrequently, even if the root mean square of the velocity fluctuation is small. Hence the chance of the wing’s being stalled always exists.

One feature of the flow in the wake of a stalled wing seems undisputed. As with flow in the wake of a circular cylinder (see § 2.3), there may exist one or more frequencies at which the power spectrum has sharp peaks. These “predominant” frequencies are those at which the kinetic energy of the turbulent motion is more or less concentrated. Experimental values of the predominant frequency in the wake behind flat plates and airfoils in incompressible fluids are given by Page and Johansen;9-39 Tyler;9,40 Blenk, Fuchs, and Liebers;9,36 and Dunn and Finston.9,38 All agree that, for angles of attack above about 30°, the reduced frequency, at the predominant peak of the power spectrum, is nearly a constant, independent of the angle of attack, provided that the projected length of the chord normal to the direction of flow is taken as the characteristic length. If the chord length is c and the angle of attack is oc, the projected length is c sin a. The reduced frequency that remains constant is nc sin a/ U, where n is the wave number (cycles per second) of the vortices, and U is the mean speed of flow. There are numerical discrepancies between various authors as shown in Table 9.1.

For smaller angles of attack, the variations of the values of the reduced frequency nc sin ct/U given by various authors become large, and seem strongly affected by the Reynolds number.* For angles of attack near and below the stalling angle, the reduced frequency nc sin a/17 decreases. The results of Dunn and Finston9,38 are shown in Fig. 9.5. Tyler’s result is similar, but the absolute values are smaller, whereas Blenk’s result shows very little decrease in the reduced frequency nc sin a/U for a down to 10°.

It may be noticed that the results as shown in Fig. 9.5 suggest that, for angles of attack a less than about 17°, the predominant vortex frequency is nearly proportional to a. Hence, a reduced frequency based on the chord length itself, ncjU, will remain nearly a constant. The average value of ncjU, for a between 8 to 17°, is approximately 0.54.

* See Chuan and Magnus.9,21 The reduced frequency nc sin a/U increases from 0.112 at R = 9.3 x 101 to 0.178 at 7? = 21 x 104, in the range of a between 18 and 29°.

(a) For Flat Plate

It may also be noted that the value of nc sin ocjU in Fig. 9.5, when multiplied by 277, give values of reduced frequency of comparable magni­tude to those shown in Fig. 2.4 for the wake of a circular cylinder.

Since tail buffeting is associated with flow separation over parts of the airplane ahead of the tail, it can be avoided by keeping the operation attitude of an airplane below the separation limits.

Figure 9.2 shows a plot of the lift coefficient versus the Mach number at various angles of attack of an NACA 2409-34 airfoil.9-15 It is seen that, at each angle of attack, the lift coefficient increases with increasing Mach number until a maximum is reached; then it drops sharply with further increase in speed. This drop is associated either with a strong shock-wave formation or with flow separation. The airfoil is said to be shock-stalled when severe drop of CL occurs.

At lower Mach number, Fig. 9.2 shows that a maximum CL is reached at certain angle of attack. This angle (of the order of 12° for the NACA 2409-34 airfoil) is the stalling angle in the usual sense.

When the high-angle-of-attack stall and the shock-stall limits are plotted together, a figure showing the boundaries of the lift coefficients beyond which stall occurs is obtained (Fig. 9.3).

It is known that stall at high angle of attack is associated with flow separ­ation. But a shock wave does not necessarily initiate complete breakoff of the flow from the airfoil. Moreover, high angle of attack and high speed of flow are not the only causes of separation. Improper fillets at the

wing-fuselage junction, etc., cause separation readily. Thus for a given airplane, the boundaries indicating the onset of separation cannot be determined from the wing alone.

Nevertheless, for a given airplane, a boundary on a CL vs. M chart can. be determined which separates the region of possible tail buffeting

from that of a smooth potential flow. Such a tail-buffeting boundary (see, for example, Fig. 9.4) resembles in appearance the stall boundary of Fig. 9.3. It is determined by flight testing.916

It should be noticed that two curves are shown in Fig. 9.4, which gives the buffeting boundaries for the horizontal tail of a fighter-type airplane that has a low-drag wing section. One curve, labeled “abrupt pull-ups” was obtained by pulling up the airplane abruptly at various altitudes and Mach numbers, the degree of abruptness being limited by the inertia,

control power, and the stability of the airplane. The other curve was obtained by gradual stalls made in turns. The difference between these two curves indicates the effect of the rate of change of angle of attack on the flow separation, which will be discussed further in Chapter 15. For the test airplane from which Fig. 9.4 was obtained, buffeting of the horizontal tail in abrupt pull-ups occurred simultaneously with the attain­ment of maximum normal force at Mach numbers below about 0.64 (solid curve in figure). Above this value of Mach number, tail buffeting in abrupt pull-ups occurred before the attainment of the maximum normal force (dotted curve). In the range tested, altitude, and hence the Reynolds number, had no effect on the tail-buffeting boundary determined in abrupt, pull-ups.

Fig. 9.4. Buffeting boundaries obtained in abrupt and gradual stalls for a test airplane. The airplane normal force coefficient CNA is defined as the ratio n IVKqS), where n is the load factor, i. e. the airplane normal acceleration measured at the center of gravity expressed as multiples of the gravitational acceleration. W is the airplane gross weight, q is the dynamic pressure, S is the wing area. The normal force (perpendicular to thrust line), rather than the lift (perpendicular to relative wind direction), is used because it is more convenient to be defined and measured in transient conditions. In steady flight Слл = CL. Flight measurements by Stokke and Aiken, Ref. 9.16.

(Courtesy of the NACA.)

Since in horizontal steady flight the lift L balances the weight W of the airplane, we have

W — L — pU*SCL (1)

Hence, in terms of the Mach number,

W_ 2_ __ W 2 8 pa2 S

where a denotes the speed of sound, p denotes the static pressure, and у = CJCV denotes the ratio of specific heats at constant pressure and constant volume.

Thus the relation between the lift coefficient and the Mach number depends on the wing loading W/S and the static pressure p, which in turn depends on the altitude z. A curve showing the relation 2 appears as a dotted line in Fig. 9.3. When such curves are plotted on Fig. 9.4, the permissible range of airplane flight speed as limited by tail buffeting will be given by the intersections of the tail-buffeting boundaries with these curves. Since the atmospheric pressure p decreases with altitude, it is seen that the permissible range of flight speeds becomes narrower as the altitude increases. The segments of the M2CL = const lines between the tail buffeting boundary indicate the maneuverability of the airplane as far as tail buffeting is concerned. Clearly, it decreases with increasing altitude.

9.1 BUFFETING PHENOMENON

Buffeting is an irregular motion of a structure or parts of a structure in a flow, excited by turbulences in the flow.

Historically, the term buffeting was originated in connection with an accident of a commercial airplane (type Junkers F13) at Meopham, England, on July 21, 1930. Four passengers and two pilots were killed. Eye witnesses to the accident could report only seeing the airplane enter a cloud, hearing a loud noise almost immediately, and seeing the fragments fall to the ground.9,2 The unusual circumstances of the accident led scientific organizations in England and Germany to undertake detailed investigations of the possible causes. The British Aeronautical Research Committee conducted extensive laboratory investigations and concluded that the most probable cause of the accident was “buffeting” of the tail. In these wind-tunnel-model investigations it was established that at large angles of attack the tail vibrated intensely but irregularly, and the “mean” amplitude of the vibration increased with the increase of speed of flow. An investigation of the meteorological circumstances at Meopham indi­cated the presence of strong rising air currents. Hence, an explanation of the cause of the Meopham accident was offered as follows: The air­plane, flying horizontally at high speed, suddenly entered a region of strong rising gusts; as a result there was a sharp increase in the angle of attack, with the formation of flow separation over the wing. The tail, situated in the wing wake, was subjected to intense forced vibrations caused by the turbulences in the separated flow, which brought about the accident. The term buffeting was used by the British investigators and was explained as irregular oscillations of the tail unit, in which the stabilizer bent rapidly up and down and the elevator moved in an erratic manner.

About the same time, German scientists Blenk, Hertel, and Thalau0,5 conducted a series of laboratory and flight tests, using the same type of airplane as the one involved in the Meopham accident. The laboratory investigations showed that it was possible for the airplane to buffet, but in actual flight, except during a steep dive, buffeting of sufficient intensity to endanger the structures of the tail was not observed. For this reason,

Blenk concluded that the Meopham accident was probably caused by overstressing of the wing due to high gust or maneuvering load. Buffeting was investigated by means of motion pictures in flight. The following conclusions were reached:

1. The tail surfaces, at large angles of attack of the wing, entered a region of vortices springing from the intersection of the wing and fuselage,

and vibrations were observed. The vortices arose on both sides of the fuselage, usually in an unsymmetrical manner. The tail vibrations were irregular in amplitude and frequency; and, as a rule, large amplitudes were rare and continued only for a very short time.

2. The recorded amplitudes, plotted against the corresponding fre­quencies, showed clearly three resonances. These resonance frequencies corresponded with the first three natural frequencies in the dynamic tests in the hangar.

3. The amplitudes increased slowly with the flight speed.

In 1933, Duncan and his associates published two papers on buffet­ing.9-7’98 In an attempt to separate numerous factors, Duncan investi­gated the case where the airfoil creating the disturbances was of infinite aspect ratio and where a “detector” (which played the part of a buffeting tail surface) was so arranged that its only possible movement was bending about a pair of flexural hinges, and, hence, it had only a single natural frequency. The oscillations of the detector were recorded optically. The relative position of the detector and the wing in front could be changed both vertically and horizontally. Hence, it was possible to explore a whole region behind the-wing. The most interesting feature of these tests is illustrated in Fig. 9.1, in which contours of equal “buffeting intensity” are plotted. Here the “buffeting intensity” means the greatest total range of movement of the detector during a standard exposure of the camera. In the same figure is shown the width of the wake as obtained by a total – head tube.[26] Note that the buffeting-intensity contours do not coincide with the total-head wake.

Since these early researches show that tail buffeting is a result of flow separation, it is clear that tail buffeting can be prevented by preventing separation. This can be effected by a proper filletf at the wing-fuselage junction, by boundary-layer control, and by limiting the operating con­ditions of the airplane. Tail buffeting can also be avoided by locating the airfoil outside the region where disturbances exist.

For modern high-speed airplanes, flow separation over the wing or the wing-fuselage junction at transonic speeds of flight causes a very serious

Schlieren photograph of the eddying wake following a shock-induced
flow separation. The dark lines are shock waves. (Courtesy of
National Physical Laboratory, England; photo by D. W. Holder)

Schlieren photograph of a 10 per c:ent scale model of the Nimbus spacecraft with Snap-19 generator in the 50-inch Hypersonic Tunnel В at Mach number 8 and Reynolds number of 0.42 X 10c per fool. Photo by Optical Systems group. Von Karman Gas Dynamics Facility, ARO, Inc. Arnold Engineering Development Center, Arnold Air Force Station, Tennessee.

problem of tail buffeting. Tail buffeting of very high intensity has been observed. The general feature as demonstrated in Fig. 9.1 remains in a transonic flow, although the contour lines are different. The intensity of turbulence and the location of the highest intensity change with the wing thickness, camber, and angle of attack. Separation of the flow at the leading edge of a thin wing at moderate angles of attack causes the most severe tail buffeting. Control over such leading-edge separation, for instance, by proper camber near the leading edge, will be most helpful in reducing the intensity of tail buffeting at transonic speeds.

The general picture is therefore as follows: Whenever separation occurs in a flow, the turbulence level increases. If an airfoil is situated in a turbulent flow, it buffets. The term buffeting, however, will not be limited to airfoils. The oscillation of a smokestack in the wake of another smokestack is also an example of buffeting.

In § 9.2 the limits of the flight speed and the angle of attack beyond which tail buffeting may occur are discussed. In § 9.3 some remarks on the theories of tail buffeting are given. Buffeting of an airfoil caused by flow separation over the airfoil itself, in a flow which is otherwise free from turbulence, will be discussed in § 9.6.

As an illustration of the statistical approach, let us consider the motion of an airplane in response to atmospheric turbulences. For simplicity, let us use the assumptions listed in § 8.2 (except 6), that the airplane may be regarded as a rigid body, that the forward velocity U can be regarded as a constant, that the disturbed motion is symmetrical with respect to the longitudinal plane of symmetry, and that the pitching motion can be neglected and only the translational motion normal to the flight path is of significance. Moreover, we shall assume that only the velocity com­ponent of turbulence normal to the flight path need be considered. The effect of other velocity components will be neglected.

The characteristics of atmospheric turbulences may depend on the geographic and weather conditions. However, Clementson8-47 has shown, that the correlation functions (and the power spectra) of atmospheric turbulences, in several different conditions (unstable air mass, water-land discontinuities, thunderstorms, and mountainous terrain), are, aside from a constant multiplier, remarkably similar to each other. They differ essentially only in intensity.

The following analysis will not refer to any particular weather condition but to isotropic turbulences as measured in wind tunnels.

We assume, therefore, that the mean value of the gust fluctuation is zero and that the gust has a power spectrum given by p2{co) of Eq, 26 of § 8.5, which can be written as

where w2 is the mean square intensity of the gust, l is the scale of turbu­lence, and U is the speed of the general flow, i. e., the speed of flight of the airplane. The scale of turbulence is defined by the area under the normalized correlation curve

Let us consider first an airfoil strip of unit span in a two-dimensional flow. Let the mean flow velocity be U which is parallel to the mean posi­tion of the airfoil chord. The airfoil is assumed thin, and the amplitude of its motion is assumed small in comparison with the chord. On the mean velocity of flow U is superposed a small turbulent fluctuation w, uniform across the span, and normal to the chord. The turbulent motion in the fluid may be regarded as transported with the mean flow, so that w is given by a stationary random function

The lift that acts on the airfoil is partly due to the disturbance w and partly due to the motion of the airfoil itself. Within the framework of the linearized theory the lift induced by these two parts are superposable. The equation of motion of the airfoil (having the translational degree of freedom) can be written as

mz – f L(z) = — L(w) (4)

where L(w) indicates the lift induced by w(t), and L(z) that due to the motion of the airfoil. Regarding — L(w) as a known external force, Eq. 4 is the same as Eq. 10 of § 6.8, describing the disturbed motion of an air­foil. Thus it is seen that the problem of the response of an airfoil to a turbulent flow can be separated into two parts:[25] (1) the determination of the lift produced by the turbulent flow on an airfoil in a steady flow, and (2) the determination of the disturbed motion of the airfoil due to an exciting force, with the flow regarded as uniform and without turbulence.

According to the results in § 8.6, if the power spectrum of the turbulence w{t) is jPgUSt(w), that of the lift would be

Pm(°>) = (5)

where xa{w) denotes the square of the absolute value of the frequency response (admittance) of the lift to a sinusoidal gust. The power spectrum of the airfoil acceleration would be

Accfa) = У.1т) УЛт) ^gust(w) (6)

where xs{p) denotes the square of the absolute value of the frequency response of the acceleration to a sinusoidal lift force. Hence, the problem is finally resolved to the determination of the quantities Хз(ы) and Xa(w)- If we introduce the strip assumption that each section of an airfoil can be regarded as a two-dimensional airfoil, neglecting the effect of finite span, then the above argument can be applied to the whole airplane. We shall consider this problem in greater detail below.

Lift Due to a Sinusoidal Gust. Let the coordinate axes be fixed on the airfoil, with the origin x = 0 located at the mid-chord point. Let the vertical gust velocity be given by the expression

w(x, t) = w0 eim{t-xlV) (7)

which shows a sinusoidal gust pattern moving past the airfoil with a speed U. In § 13.4, it will be shoVn that the lift induced by w(x, t) on a two-dimensional airfoil of unit span is

L = 2rrpbUw0 ешф(к) (8)

where p is the air density, b is the semichord length, к is the reduced

frequency соЬ/и, and

* ПГЇ5 <9>

The resultant lift acts through the Vrcliord point from the leading edge. The factor 2прЫ1ф(к) represents the frequency response (admittance) of the lift to the gust.

Admittance of the Airplane. The equation of motion of a two-dimen­sional airfoil subject to a sinusoidal force has been derived in § 6.8. In a single degree of freedom of vertical translation, Eq. 10 of § 6.8 gives

-L(t) + P(t) (10)

where m = the mass of the airfoil

r = a dimensionless time = Utjb z! — dzjdx, г" = dh/di2

The lift per unit span on the two-dimensional wing is given by an integral in Eq. 8 of § 6.8. Neglecting the lift on the tail surfaces, and using strip assumption, we can write the lift од the airplane as

where S = wing area, and Ф(т — т0) is the Wagner’s function.

In a simple-harmonic motion

2 = z0eimt = z0eikT, P = Р0еш = P0eikT (11)

the lift can be written as (compare Eqs. 7 and 9 of § 6.9)

L{r) = ^ S j^2C(Ar) j z, – k2 eikT (12)

Hence, the equation of motion is

{- m ~ k* + ^ I [2 am – ^]) z0 = P0 (13)

The quantity in { } is the complex impedance from P(r) to z(r). Its inverse, multiplied by — со2 == — (Uk/b)2, gives the admittance of the acceleration z. Introducing the airplane density ratio к as a parameter,

The exact integration of the above expression is difficult because of the complicated manner in which the Bessel functions are involved. To obtain an approximate solution, let us note that in the full range (0, oo) G2 is much less than F2. Hence we may neglect terms involving G2 in Eqs. 17. Moreover, since

while in practice, к varies from a number of order 40 for trainers and transports to 150 for high-speed fighters, a fair approximation of Eq. 17 is simply the quasi-steady result

The variation of /(a, s’) is shown in Fig. 8.11. Note that z2 -> 0 both when s -> 0 and when s-*■ oo. Hence, as the scale of turbulence becomes either negligibly small or infinitely large in comparison with the wing chord, the intensity of the acceleration experienced by the airplane will tend to zero. This can be expected, because, when the wing chord is very large in comparison with the turbulence scale, the “gusts” are smoothed out by canceling each other over the wing. On the other hand, when the chord length is very small in comparison with the scale of turbulence, the airfoil behaves quasi-stationarily. A rigid airplane having only the trans­lational degree of freedom will experience no acceleration in a steady flight; hence, the limiting case z2->0ass->0.

The importance of the critical speeds of aeroelastic stability on the dynamic-stress problem is evident from the airplane admittance to the sinusoidal lift. For example, when flutter speed is approached, %s(w) will become very great, thus causing very large dynamic stresses.

The mean square value of the acceleration z is thus known as a function of the airplane speed, mass, and size, and the intensity and scale of the turbulence. From the assumption of stationariness of the gusts, the time averages are equal to the ensemble averages. Hence, z2 is also the standard deviation of an ensemble of gust responses. According to the interpretation given in § 8.4, it gives some idea of the dispersion of the induced acceleration caused by the gusts. The mean value being zero,

and the standard deviation being known, two of the most important para­meters of the probability distribution of the gust response are determined.

In order to compare Eq. 23 with Eq. 10 of § 8.2, we may assume к 1 and reduce Eq. 23 into the form

V£ __ brp^w2US 1I(a, s) g 2 mg ^ n

whence, upon defining (z2)I,2/g as the mean acceleration Дn, and (w2)l,‘ as a mean gust speed w0, and replacing 2n by the lift curve slope dCLjda, we obtain8-44

Comparing this with Eq. 10 of§ 8.2, we see that they are identical but for the factor V/(a, s)/tt, which may be identified with the usual “gust allevia­tion factor.” Williams8-44 points out that, if we use the peak value of the
7(oc, s) curve in (24), a proper account of the effect of airplane mass ratio on gust response is obtained. Moreover, the existence of a peak in the /(oc, s) curve means that an airplane responds more readily to a scale of turbulence which is a constant multiple of wing chord, the constant being dependent upon the mass ratio. This explains a rather interesting experi­mental result that the so-called “gust gradient distance” is more closely related to the wing chord, rather than to the meteorological conditions.

If the probability-distribution function of the gust response is deter­mined, the probability of encountering an acceleration of a specified mag­nitude can be found according to statistical methods. The probable number of times when the dynamic stresses in the structure exceed a specified stress level can be computed. Such information will be useful in designing an aircraft with respect to fatigue and service life. Engin­eering applications of this nature are discussed in Refs. 8.39-8.44, 8.63 and 8.86.

To obtain complete information about the probability distribution of dynamic stresses, higher-order correlation functions would have to be computed (see § 8.5). Such calculations are usually quite involved (see Mazelsky8 83). In practice, however, some idea about the probability distribution can often be obtained by experimental means, and a know­ledge about the mean value and the root-mean-square deviation of the random variable is sufficient for engineering purpose.

The idea of statistical treatment is not new. The first application of statistical theory to dynamics goes back to Lord Rayleigh. It forms the basis of Taylor8-54 and von Karman’s8-51 theory of turbulence. It is also the foundation of the theory of Brownian motion, of noise in electric and acoustical systems, and of certain aspects of astrophysics. The mathe­matical theory of the stationary time series was developed by Wiener and Kintchine. The type of analysis used in this section was first made by Lin.8-82 its application to aeroelastic problems was first pointed out by Liepmann.9,11

The functions we are concerned with are functions of time and depend on chance. Such a function is known as a stochastic process. A sto­chastic process bears the same relation to a definite function that a random variable does to a definite number. Whereas an ensemble for a random variable may be regarded as the results of observation of a random experiment, an ensemble for a stochastic process should be regarded as a large number of experiments carried out under similar conditions, in which each experiment provides a function of time.

The application of the theory of stochastic process to dynamic problems raises many fundamental questions regarding the philosophy of structural design. Its successful application in the fields of engineering seismology,

gust measurement,8-41~8-43 fatigue-life studies, landing-impact loads,870 and rough-water operations of seaplanes,8-74~8-78 etc., may be noted. In many such applications it is not permissible to consider the random process as stationary. Examples of dynamic response analysis involving nonstationary stochastic process can be found in Ref. 8.70 in which the dynamic loads problem in aircraft structures during landing is treated.

Suppose that the response у and the excitation / are connected by the linear differential equation

(4)

Iff(t) represents a stationary fluctuation, it does not tend to zero as t oo and its Fourier transform does not exist. To overcome this difficulty the truncated function used in the last section can be used here again. Let f{t) be truncated in such a manner that it is zero outside an interval (— T, T). The Fourier transform of the truncated Д/) can be defined provided that /(/) is absolutely integrable in (— T, T) and has bounded variations. Let

the same interval. Comparing Eq. 3 with Eq. 5 of § 8.5, and following the reasoning of that section, we obtain

(6)

where p(co) is the power spectrum of fit). Equation 6 shows that the power spectrum of the response is equal to that of the forcing function divided by the square of the absolute value of the impedance. The phase relationship between the excitation and the response, represented by the argument of the complex number Z(/co), is completely obliterated in the power-spectra relations.

Equation 6 may be applied to general dynamic systems for which the impedance Z(ico) can be defined. It holds generally for all dynamic systems described by differential equations, linear integral equations and linear integral-differential equations that occur in aeroelasticity.

Let u(t) represent a random function of time, such as a velocity com­ponent at a point in a turbulent flow, the force acting in the landing-gear strut of an airplane during landing impact, the lift acting on an airfoil in passing through a gust, etc. By saying that u(t) is a random function, we mean that at a given time t the value of и is not predictable from the data of the problem but takes random values which are distributed according to certain definite probability laws. We shall assume that the probability laws describing the randomness are determined by the data of the problem. Such a determination can sometimes be made through a suitable theoretical model, but in general it has to be obtained by experiments.

A set of observations forms an “ensemble” of events. In the example of gusts, each gust record, such as the one given in Fig. 8.6, is a member in an ensemble of such records. Imagine that a large number of observa­tions be made simultaneously under similar conditions. Let the records be numbered and denoted by uft), u2(t), • • •, uN(t). If the total number of records is N, the following averages may be formed

/wwvjV

u{t) = [uft) + «2(0 + ‘ ‘ • + uN(t)]/N

www’ r

ut) = Wit) + «до + • ■ • + uNt)W (l)

Assume that «(/) and ut) , etc., tend to definite limits, respectively, as N -> 00; then these limiting values are ensemble averages of the random process «(0-

In a similar manner, ensemble averages of the following nature can be formed:

N

^WVWV ллл/wv J ^

u{t)u{t + t) = lim — > «ДК(г + т) (2)

2V—>cc -/V

г = 1 N

u(t)u(t + rx) ■ ■ ■ u(t + rm) = lim — / «г(0«;0 + Tj) • • • U,{t + Tm)

ДГ-*0О A Z—t

г = 1 (О

These are ensemble averages of the correlation functions of the random function u(t).

For a complete statistical description of a random process, ensemble averages of all orders are required. However, for practical dynamic-load problems in aeroelasticity, often the most important information is

TRANSIENT LOADS, GUSTS

ЛАЛЛ VWVW J

afforded by the mean value u(t) and the mean “intensity” (ut)) For

these simplest kinds of average quantity, analysis can proceed in a simple

manner. By definition, it is clear that u(t) and ([«(r) — м(Г)]2)1/г are the mean and the standard deviation of the random function u{t) at any instant t.

If the ensemble averages of a function u(t) are independent of the variable t, then u(t) is said to be stationary in the ensemble sense.

A different kind of average is the well-known concept of time average. Thus, the mean and the mean square of a function of time u(t) over a time interval 2T, are

Similarly, higher-order averages u%t) (p = 3, 4, • • •) can be formed,

provided that the definition integrals converge. If the time averages tend

to be independent of t and T when T is sufficiently large, then u(t) is also

said to be stationary, but in the sense of time-average. We shall write in

____ 2′ ………………………….

this case the limiting value of uv{t) as T > oo as up(t).

The property of stationariness says, in effect, that all time instants are similar as far as the statistical properties of и are concerned. This suggests that the results of averaging over a large number of observations could be obtained equally well by averaging over a large time interval for one observation. In other words, for a stationary random process, one expects the time averages to be equal to the ensemble averages, and we do not have to distinguish the concepts of stationariness in time average or in ensemble average. The study of the exact conditions under which this equivalence of the time average over one observation and the ensemble average over many observations will indeed be true is called the ergodic theory. In aeroelastic problems which we are going to consider, this equivalence may be assumed.

It is natural to extend the above concepts to random functions of space. In the example of gusts, we may have to consider the velocity fluctuation u, itself a vector, as a function of space and time x, y, z, t. If records of the velocity fluctuation were taken over different regions of space and if the ensemble averages over space are independent of the spatial coordinates of the regions, then the velocity fluctuation is said to be homogeneous.

There are many flows, of interest to aeroelasticity, that are approx­imately homogeneous and stationary when the time and distance scales

are properly chosen. Such is the case of atmospheric turbulence within a suitable expanse of time and space.

The Power Spectrum. Let us consider a stationary random process u(t) which has a mean value m equal to zero[23] and define the mean square of u(t) over an interval 2T as in Eq. 4. Let u(t) be so “truncated” that it becomes zero outside the interval (— T, T) (Fig. 8.8). Let the truncated function be written as uT(t).

The Fourier integral of uT(t) exists provided that the absolute value of uT(t) is integrable and uT(t) has bounded variation,

uT(t) = –== f FT(co) еІШІ dco (5)

V2lT J – со

where

FT(co) = —U Г uT(t) e~imt dt (6)

v2tt J-t

If FT*(co) denotes the complex conjugate, then FT*{— со) == FT(co) since uT(t) is real. It is well known (Parseval theorem) that

f uT2(t) dt = f uT2(t) dt = f FT(co)2 dco (7)

J— oo J — T J— да

Hence

___ 1 Гда /*да <sr (fjy)|2

ut) = lim — FT (со) 12 dco = lim —^——– dco (8)

r—>co сіл J —со T—> CO 00 Z

For a stationary random process for which u2(t) exists, the left-hand side of the above equation tends to a constant as T oc; hence let

The function p(oj) is called the power spectrum or the spectral density of u(t). When u{t) is resolved. into harmonic components by a Fourier an­alysis, as in Eq. 5, the element p(co) dm gives the contribution to u2(t) from components having frequencies ranging from m to c/j + dm. The integral

/•to

p(m) dm represents the contribution to u2(t) from frequencies less than m. Jo

Correlation Functions. Consider again a stationary random process u(t), and define the average value

—————- 1 Г

W(t) = «(0 u(t + t) = lim —; u(t) u(t + t)dt (11)

37—► oo і J — T

ір(т) as a function of т is a correlation function defined in the sense of time average. For a stationary random process it is the same as that defined in Eq. 2.

It follows from the definition of correlation function that

№ = Ш (12)

Since the average values of a stationary random process are independent of the origin of t, it follows immediately that

УІт) = У (— r) (13)

Furthermore, if u(t) is a continuous function of t, then

lim u(t)[u(t + t + h) — u(t + t)] = 0

Й-И)

since the factor in [ ] tends to zero as h -> 0. This may be written as

lim [tp(r + h) — y<t)] = 0 (14)

A->0

showing that ір(т) is continuous at all values of t. According to the Schwarzian inequality,

y(r) < [u2(t) • u2(t + r)]1’2 = [?/>(0) • W(0)p‘

we have

In general, random processes having zero mean values satisfy the condition

lim yj(r) = 0 (16)

T—> CO

which means that the function «(/) at two instants separated by a long time interval are uncorrelated with each other.

The correlation function y(r) when plotted against r is generally a bell­shaped curve. The interval of т in which y(r) differs significantly from zero is a measure of the “scale” of the random process. In the example of atmospheric turbulence, such a scale may be thought of as representing the mean size of eddies.

Relation between the Power Spectrum and the Correlation Function. The Fourier transform of y(r) is

ф(а>) = -)= Г w(T)e-imrdr (17)

V2lT J – 00

Since y(r) is an even function of т, ф(со) can be expressed as a real-valued integral

/2 Г®

ф{со) — ^ — J y(r) cos сот (It

because е~ыЛ = cos сот — г sin сот and the imaginary part of Eq. 17 vanishes. The inverse transform is –

І2 f®

w(t) — – ф(со) cos сот dco

^ – it Jo

When t 0, the left-hand side tends to ut); hence

m[24](0 =J – Ф(ю) dco

A comparison with Eq. 10 shows that, aside from the numerical factor V2/77, ф(со) is identical with the power spectrum p(co) defined before. A formal proof for the identity of ^ — ф(со) with p(co) can be constructed.

Hence, we obtain the reciprocal relations between the power spectrum and the correlation function

Equations 5 through 19 hold for any stationary function u{t). Now consider an ensemble of functions и ft), и ft), ■ • •. Each of these func­tions defines its p{m) and ір(т), which can be averaged over the ensemble.

The resulting p(co) will be called the spectral density or the power spectrum of the random process. For a stationary random process it follows from Eqs. 18 and 19 that the correlation function and the power spectrum are each other’s Fourier cosine transform.

Example. When

y(t) = A + В sin (ay + °0

we have

У2 = A2 + B2

w(T) = + t) = A2 +B2 cos ay

and, if we define a unit-impulse function 6(t) as in Eq. 14 of § 8.1, but

Л OO

impose further the condition d(t) — <5(— t) so that 6(t)dt — then

Jo

p(co) = 2A2d(co) — 132S(oj — co0)

This example shows that, if the mean value of a function is not zero, and if the function is periodic, the power spectrum will have singular peaks of the well-known Dirac й-function type.

Example of Wind-Tunnel Turbulences. Consider the velocity fluctua­tions in the flow in a wind tunnel. Let the deviations from the respective mean values of the velocity components be written as иъ u2, u3. These are functions of space and time (x, y, z; t). Similar to the time correlation function гр(г), the spatial correlation functions such as

Щ(х, у, z; t)u3(x + r, y,z;t)

1 Г (20)

= lim — щ(х, у, z; t)u,(x + r, y,zt) dx dy dz (i, j =1,2, 3)

F->co r JV

may be formed. If the field of flow is homogeneous, such a correlation function is well defined and is a function of r and t only. It is possible in wind-tunnel work to determine experimentally the general correlation function

Щ(хъ уъ 2X; t^ufxi, уг, z2; t2) (/, j = 1, 2, 3) (21)

which, for a stationary and homogeneous turbulence field, depends only on the relative position of the points (aq, ylt Zj) and (x2, y2, z2), and the time interval t2 — tx, and not on the absolute location of the points or time.

Thus a large number of space and time correlation functions can be defined among various velocity components. It is here that the simplifying concept of an isotropic turbulence field enters. As was first shown by G. I. Taylor8 54 and Th. von Karman,8-51 an isotropic turbulence field is one that can be specified by two “principal” velocity correlations, in a way similar to that two principal stresses at a point in an isotropic elastic solid define the state of stress at that point. The two principal velocity correlations are denoted by / and g and are defined pictorially in Fig. 8.9. /(/•) is the correlation function between the same velocity component measured at two points at a distance r apart, lying along a line in the direction of the velocity component. g(r) is the correlation function between points at a distance r apart, lying along a direction normal to the direction of the velocity component. For example, let us write u, v, w in place of иъ иг, щ, then

u(x, y, г; t)u(x + r, y,z;t) = /0)

u(x, y, 2; t)u(x, у + r, z;t) = g(r)

v(x, y, 2; t)v(x + r, y,z;t) = g(r)

If the field of fluctuation u, v, w is superimposed upon a mean flow of velocity U is the x direction, and if к2/!/2, t>2/ C/2, w2/t/2 are small quantities, it is possible to interchange time and space variables and consider the turbulence as simply being transported along with the velocity U in the x direction. Hence the time т which enters into the time correlation function can be replaced by r/С/, where r is chosen along the x axis. We may write

u(t)u(t + t) = xp-tir) = f(rU)

v(t)v(t + т) = У’2(т) = g(jU)

Experiments8-1 have shown that in wind-tunnels the turbulences are nearly isotropic and the correlation functions f(r) and g(r) have the form

(24)

Fig. 8.9. Correlation functions / and g in isotropic turbulences.

and the power spectra are

AH =A(°)

1 + 3f2a аИ=М°)(ГТ1^

where

The functions fir), g(r), and p2(oi) are normalized and depicted in

Figs. 8.9 and 8.10.