Category Dynamics of. Atmospheric Flight

542 Dynamics of atmospheric flight

where W = mean wind, h = height above ground, WG and hQ are the speed and height outside the friction layer, and a and ha depend on surface roughness (see Fig. 9.36).

COMPONENT INTENSITIES:

In the layer below about 300 ft:

о^.Оъ’.Оъ = 1:0.8:0.5

where a12 = (щ2), Xj is in the wind direction, and x3 is vertical. Above this height the a, i tend toward equality at about 1000 ft.

SPECTRUM SHAPES:

The one-dimensional spectra are given by the von Karman equations (13.2, 16 and 17e), with a in Фй replaced by Thus because of (13.2,20) there are three different one-dimensional spectra.

SHEAR AND CROSS SPECTRA:

The turbulent shearing stress in the boundary layer results in nonvanishing %wз, but symmetry requires = м2мз = 0. The data suggests

uxuz — О-Зп^з

in which a representative value of y0 is 0.5.

INTEGRAL SCALES:

For the boundary layer as a whole.

іц = 20 лД

L^i — LZ1 — 0.4Й-

Above about 200 ft, slightly better values are given by

£n = 4.2Г73 L21 = L31 = 2.1A0’73

Turbulence near the ground is of the boundary-layer variety (see Fig. 9.36), being variable with height and anisotropic. A model for this case should ideally give the following:

(i) Variation of mean wind with height as function of ground roughness.

(ii) Variation with height of %2, ада2, w32.

(iii) Variation with height of all significant scales.

(iv) The form of the spectrum function 6ip or the correlation function Ri}.

Since the scales are much smaller than at high altitude, it becomes more important to have the two-dimensional spectrum functions 02), which

enable both streamwise and spanwise variations of turbulent velocity to be
taken into account. The anisotropy also leads to the nonvanishing of the correlation R13 = (UjU^ (where xx is in the wind direction and x3 vertical), which is simply related to the turbulent shear stress (Reynolds stress) in the boundary layer.

An interesting fact about low-altitude turbulence is the existence of a gap in the spectrum at a rather useful location. There is considerable evidence to show that the spectrum of wind speed measured by van der Hoven is representative, Fig. 13.7. This is a spectral density of horizontal wind speed taken as a function of time at a fixed point. The gap occurs for periods

greater than 6 min, or frequencies less than 10 cycles per hr. The lobe on the right corresponds to the turbulent energy of interest for flight (cf. Fig. 13.6).

The extensive information available on the wind-induced turbulence near the ground—much of it inconclusive and even contradictory—has recently been reviewed in refs. 13.7, 13.8. From these we adopt the following model as a reasonable representation of presently-available information: the turbulence is Gaussian, stationary, and homogeneous w. r.t. horizontal translations; it is anisotropic, but the one-dimensional spectra display isotropic behavior at the highest wave numbers; the turbulence is symmetric w. r.t. vertical planes.

The experimental data on turbulence in clear air and in thunderstorms, and from altitudes below 5000 to 40,000 ft have been reviewed by Houbolt
et al. (ref. 13.5). They have examined it from the standpoint of scale, in­tensity, shape of one-dimensional spectra, homogeneity, isotropy, and normality. Their general conclusion is that an adequate model for analysis purposes is the simplest one described above—isotropic, homogeneous, Gaussian, and frozen. The intensity a varies from very small to as much as 16 fps, and the scale is large, typically of order L = 5000 ft. The one­dimensional spectrum function that best fits the data for the vertical com­ponent of turbulence is the von Karman spectrum

(13.2,16)

a = 1.339

This spectrum function yields Ф.—• as Qj —> oo, a condition required

to satisfy the Kolmogorov law in the so called inertial subrange (ref. 13.6). The energy spectrum function and some useful two – and one-dimensional spectraf of the von Karman model are

(a)

(‘ь)

(13.2,17)

(с)

{d)

(е)

(/)

The inverse Fourier integrals of Фи and Ф33 provide the associated corre­lation functions (ref. 13.5).

№ = ^-£Ak>A{1) (a)

J (13.2,18)

g(£) = ±- СА[КуД) – иКуД)] (Ъ)

А (з)

t Note that the spectra used herein are two-sided, such that for example <r2 = //П]. In ref. 13.5 and in many others, one-sided spectra are used that are double those herein, and the integration is from zero to infinity.

where £ — |/(«L), and Г, К denote gamma and Bessel functions, respectively.

With the typical value L = 5000 ft, the longitudinal and lateral one­dimensional spectra are as shown in Fig. 13.6. With this form of plot­ting, ОФ vs. log10 Q, the area under an element of the curve is dA = const X QФ(1|Q) dQ = const x Ф(О) dQ which is proportional to the contribu­tion of the bandwidth dQ to o’2. Hence the shape of the curve truly shows the turbulent energy distribution.

The peak of £21Ф33 occurs at LQ1 = 1.33, which shows directly how scale affects the spectrum. It also yields the “dominant wavelength,” i. e.

Thus for turbulence of 5000 ft scale, the dominant wavelength is about 41- miles, and the energy level is down by a factor of 25 at a wave length of 100 ft, the order of the size of an airplane.

For comparison, the ranges of О associated with typical rigid-body and structural-mode frequencies are indicated on Fig. 13.6. These show what relative excitation levels of these modes are to be expected from turbulence of this scale. The spectrum shifts without change of shape to the right for smaller L and to the left for larger. For example at a scale L = 500 ft, the spectra move to the right by one decade inO, and by two decades for L = 50 ft. This drastically alters the relative intensity of excitation of the various rigid-body and elastic modes. We shall see later that the difficulty of com­puting the response in any mode is very much affected by the wavelength X associated with it. If very large compared to the dimensions of the airplane, the simplest analysis results. On the other hand, for structural modes of relatively short wavelength this condition is not met, and more sophisticated analysis is needed.

There is an intuitive notion of the scale of turbulence. Clearly there are significant differences of “size” between the turbulence in the wing boundary layer, in the wake of the airplane, and in the atmosphere itself. These differ­ences are quantified by a definition of integral scale derived from the corre­lation function. Thus let

lu = -2

и і Jo

be a line integral on the axis. It might be called “the j scale of the і velocity component.” There are in general nine such scales, e. g. for ux measured along the xx axis, or «3 measured along the x2 axis, etc.

A second notion of scale derives from the spectral representation of turbulence. The wavelength at which the energy density peaks (see Fig. 13.6) is also a scale parameter, and for any given spectrum shape is uniquely related to L (defined below).

In isotropic turbulence, only two different scales are found, associated with the basic correlations / and g, and these are of course simply the areas

under the / and g curves. Because the maximum ordinate is unity, Li} is equal to the width of a rectangle that contains the same area as the corre­lation curve—i. e. it is a measure of the spatial extent of significant correlation. The two scales are

L = Lijt і = j = area under /(£) = longitudinal scale L’ = Li}, і Ф j = area under g(£) = lateral scale

The continuity condition (13.2,7) yields L — 2L’.

The situation with respect to scale is unfortunately more complicated in the ground boundary layer where isotropy does not hold.

Although there is some evidence that atmospheric turbulence is not necessarily normal, or Gaussian (ref. 13.1), many researchers have concluded that it is for practical purposes in many situations. There are great gains in simplicity in calculating the probabilities of exceeding given stress or motion levels if the process is Gaussian (see Sec. 2.6), for then one needs only the information given by the spectral distribution of the variables in question. We therefore assume that the random functions we have to deal with have normal distributions. (This assumption only enters when probabilities are being calculated, not correlations and spectra.)

The most general case, covered by (13.2,2 to 4) allows the turbulence statistics to vary from point to point and time to time—i. e. R{j and ві} are functions of the base point r and base time t. One assumption made almost universally is that there is no dependence on t, i. e. that the turbulence is a stationary process. A second widely employed assumption is that the turbulence is effectively homogeneous i. e. that Ri} and di} are independent of r at least along the path flown by the vehicle. At high altitudes, turbulence appears to occur in large patches, each of which can reasonably be taken to be homogeneous—but with differences from patch to patch. At low altitudes, near the ground, there are fairly rapid changes in the turbulence with altitude. However, for airplanes in nearly horizontal flight, homogeneity along the flight path is a reasonable approximation.

In general, the functions R(j and du depend on the directions of the axes of Fa. This is especially so in the ground boundary layer. When this de­pendence is absent, and the evidence is that this is the case at high altitudes, then the turbulence is isotropic, i. e. all the statistical properties at a point are independent of the orientation of the axes. In this case it follows that the three mean-square velocity components are equal, i. e. the intensity is

<r2 = <m12> = <m22> = <w32> (13.2,5)

When the turbulence is stationary and homogeneous it is also ergodic, so that time averages can replace ensemble averages—a matter of no small importance for experimental work.

Finally, the last simplifying assumption relates not so much to the turbulence itself but to the nature of the present problem. Airplanes fly for the most part at speeds large compared to the turbulent velocities and to their rates of change. Thus the vehicle can traverse a relatively large patch of turbulence in a time so short that the turbulent velocities have not had time to change very much. This amounts to neglecting t in the argument of u(r, t), i. e. to treating the turbulenceas a frozen pattern in space. This
assumption is known as “Taylor’s hypothesis.” Its consequence is that

r) -> B{j(%) and 0„(&, m) -> 0y(fl)

and the Fourier integrals of (13.2,3) are triple rather than quadruple. The problem of computing aerodynamic forces and vehicle responses is corre­spondingly simplified.

Finally, then, the simplest model we can obtain is of homogeneous, isotropic, Gaussian, frozen turbulence. This is the model most commonly used for analysis of flight outside the ground boundary layer. Unfortunately, the strong anisotropy of boundary layer turbulence makes it unsuitable for landing and take-off; and for hovering flight the assumption of frozen turbulence is clearly also invalid.

Batchelor (13.2) has shown that in isotropic turbulence B(i(%) can be expressed in terms of two fundamental correlations, f(£) and g(£), viz.

M = [/(f) – s'(f)] Ці + g№,

о £

where £ = |5|, 6i} is the Kronecker delta, and o’2 is given by (13.2,5). It should be observed that B{j is zero whenever і ф j and either £t or £} vanishes, so that Bfj(0) = 0 for і Фд. Other situations are illustrated in Fig. 13.2, a wing-fin system; the correlation of иг at A with either u2 or u3 at В vanishes because £t and £3 are both zero, but that of % at A with u2 at G is not zero because £[ and £2 are both nonzero. Furthermore, the equation of continuity for an incompressible fluid imposes the condition

9 =/+ iff’

f(£) is known as the longitudinal correlation, typified by Bn(£1,0, 0) and is associated with the condition illustrated in Fig. 13.3a. g{£) is the lateral

(c)

Pig. 13.3 Correlations in isotropic turbulence, (a) Longitudinal correlation, /(f) = (uu). (b) Lateral correlation, g(f) = {uu’). (c) Typical forms of/and g.

correlation, typified by Bu(0, f2, 0) and is associated with the condition illustrated in Fig. 13.36. The typical forms of these correlations are shown in Fig. 13.3c, when normalized to unity at f = 0.

The spectrum function in isotropic turbulence is expressible in terms of the basic energy spectrum function E(Q), i. e.

W) = ® (ПЧі – OA) (13.2,8)

47rLr

jEJ(Q) is a scalar function that describes the turbulent energy density as a function of wave number magnitude, Q = |£2| such that

As with Ri}, the spectral density 0i5 is zero whenever і ф j and £2г or Q3- vanishes. Thus 0Й(О) = 0 for і ф j, and for many special values of the wave number vector.

The mean product of two velocity components at one point in frozen turbulence is Лй(0), which is from (13.2,4) (for frozen turbulence, со and г do not appear, and it is a triple integral)

00

2, 03) сШх сШ2 dQ3

—00

Integration successively w. r.t. 03 and 02 yields the two-dimensional (T“) and one-dimensional (Ф) spectrum functions, i. e.

00

UiU< = I [ M'<3(Q1, II2) dQ1 <Ш2 = J dOi (13.2,10)

—00

^іДОіФг) — f @ij(Q 1) Оз) d03

J—00

Лео

0^.(0^= Т,.Д01; Q2)dQ2 (13.2,11)

Note that the mean-square value of any velocity component is [cf. (2.6,11)]

^=Гфй(Ш1 (13.2,12)

J-co

There is a more direct physical interpretation of the one-dimensional spec­trum functions than the formal one given above. In homogeneous frozen turbulence consider the measurement of u{ and u} along the x1 axis (corre­sponding to measurement in flight along a straight line, or at a fixed point on a tower when the frozen field sweeps by it with the speed of the mean wind). The corresponding correlation is – R#(£i, 0, 0) and its one-dimensional transform is ФІЗ(Г21) i. e.

Ф«(йг) = ^ Г Вді, 0, 0)e-“ib di, (13.2,13)

J—СО

Furthermore, if the хг axis is traversed at speed V (or the wind past the tower has speed U), then = Ur, where r is the time interval associated with the separation

Corresponding to the two basic correlations /(£) and gr(|) for isotropic turbulence, are their two Fourier integrals, the longitudinal and lateral one-dimensional spectra, i. e. Ф11(01) and Ф^О,), respectively. By virtue of the
relation between / and g, Batchelor shows that

Фзз(Оі) = ІФп^) – іо/фп(0і) (13.2,14)

ЙІ!

The isotropy, of course, requires the symmetry relations

Фгг(^і) = Ф»(£іі) if * Фэ

= Ф11(£І1) if i=j (13.2,15)

Most of the experimental information collected about atmospheric turbulence, on towers and by aircraft, is in the form of the above two one-dimensional spectra.

SPECTRAL COMPONENT OF TURBULENCE

We showed in Sec. 2.6 that a one-dimensional random function could be represented as a superposition of sinusoids (2.6,4). The analogous relation for three-dimensional turbulence is

which indicates that the individual spectral component is a velocity field of the form exp + Q,2×2 -+- Q3*3) and amplitude dC. The triple integral

signifies that integration is over — oo to + oo in each of the wave number com­ponents ; or to put it another way, individual sinusoidal waves of all possible wave numbers are superimposed to make up the turbulent field. The in­dividual spectral component has been shown by Ribner (ref. 13.3) to be an

inclined shear wave as illustrated in Fig. 13.4. The velocity vector is per­pendicular to the wave number vector, and is constant in planes normal to it. It is no more surprising that a superposition of waves like that shown can represent turbulence than that an infinite Fourier series can represent an arbitrary random function of time.

The spectral component in two dimensions, say Qx and Qa, has the form exp г (LI A + 02ж2); and is the sum (more properly integral) of all the three­dimensional waves having the given values Qx, 02, but differing Q3. It can be pictured as in Fig. 13.5, which shows the node lines and the distribution

of uz through a section of the wave. The two-dimensional wave number vector is Й’ = [Qx, Q2]T, and is seen to lie at an angle в to the x1 axis. The wave­length is A = 2t7/Q’ and associated with the components of S2′ are the wave­lengths along the coordinate axes, Ax = 27r/£Ix and A2 – 27t/Q2-

Finally, the one-dimensional spectral component is a sinusoid on one axis, e. g. егПл, and is the sum of all two-dimensional components having the same Qx or Ax. This is the familiar spectral component of one-dimensional Fourier analysis.

The total velocity field of the atmosphere is variable in both space and time, composed of a “mean” value and variations from it. The mean wind is a problem primarily for navigation and guidance and is not of interest here. We eliminate it by choosing as our reference frame the atmosphere – fixed frame Fa (see Sec. 4.2.4) relative to which the mean motion is zero. Let the velocity of the air relative to FA at position r = [x1x2x3]T and time t be

u(r, t) = [игиги^т (13.2,1)

Then ufv, t) are random functions of space and time, i. e. we have to deal with the statistics of a random vector function of four variables (x1; x2, x3, t).

Associated with any given point r and time t there is а З X 3 correlation matrix (second-order tensor)

r) = <u{(r, t)u}(v + t + r)> (13.2,2)|

As indicated, it is the ensemble average of the product of ut at r and t with Uj at the different point r + and the later time t + r. The associated four­dimensional Fourier integral is the 3×3 matrix of four-dimensional spectrum functions

The inverse relation for Fourier integrals gives oo

, со)еі(П-?+0>г> dOx d0.2 dQз dm

— co

The functions Bu and di} serve (together with the assumption of normality) to describe the needed statistics of the turbulence. From them all the pertinent results can be derived (see Sec. 2.6); a principle objective of re­search into atmospheric turbulence is to ascertain their forms, and how their parameters depend on meteorological conditions, terrain, etc.

f 0, r) should not be confused with the time-delayed correlation measured by a fixed instrument in a flow passing it at a mean speed U.

Of those obstacles with which nature confronts man in his use of the air as a medium of transportation, two are transcendent in importance—poor visibility that prevents him from seeing where he is going, and turbulent movement of the surrounding air that disturbs his vehicle and its flight path. To overcome these obstacles has always been and continues to be a major challenge to aviation. Poor visibility is associated with both darkness and weather, turbulence with weather alone. The former of these obstacles has to a great extent been overcome—modem navigation techniques permit blind flying with adequate safety for all but the critical phases of landing and take-off", and there is hope that the safety margins for these too will ultimately be acceptable.

The subject of this chapter is the second obstacle, turbulence. The motion of an aircraft in turbulence is akin to that of a ship on a rough sea, or an automobile on a rough road. It is subjected to buffeting by random external forces and as a result the attitude angles and trajectory experience random variations with time. The time scale and intensity of these responses are governed by the scale and intensity of the turbulence, as well as the speed and characteristics of the vehicle. Their effect is to produce fatigue in both the pilot and the structure, to endanger the structural integrity of the air­craft, to produce an uncomfortable, possibly even unacceptable, ride for

the passengers and cargo, and to impair the precise control of flight path needed for collision avoidance and safe landing.

To understand and analyze these responses, which is to provide the basis for ameliorating them, we dissect the total phenomenon into several parts, as illustrated in Fig. 13.1. The first is to describe the turbulence itself, the “output” of this description being the velocity field in which the airplane is immersed. Next, it is necessary to determine how these velocities result in aerodynamic forces and moments; these in turn become inputs to the mechanical/structural system whose mathematical modelling was the subject of Chapter 5. Finally, the motions and stresses that result serve to define the problems faced by the structure and the pilot. The diagram indicates that the pilot feeds back into the dynamic system via the controls—a feature that cannot be overlooked for realistic analysis. A study of all the problems embraced by the figure clearly spans the disciplines of meterology, aero­dynamics, vehicle and structural dynamics, metal fatigue, and human factors. We make no attempt here to go in depth into all of these! The aim of the following is to extend the mathematical models previously given to embrace a description of the turbulence and the inputs provided by it. This model then provides the tool for calculating the responses of interest for any design or operational problem.

Since turbulence is a random process that cannot be described by explicit functions of time, only a statistical, probabilistic approach can be taken. The basic random-process theory needed was presented in Secs. 2.6 and 3.4, and the following relies heavily on that material. In particular the role of input spectra in computing output spectra should be recalled at this point [see (3.4, 48 to 61)], and the role of output spectra in calculating response probabilities (Sec. 2.6).

The term control power is used to describe the efficacy of a control in producing a range of steady equilibrium or maneuvering states. For example, an elevator control which by taking positions between full up and full down can hold the airplane in equilibrium at all speeds in its speed range, for all configurations and C. G. positions, is a powerful control. On the other hand a rudder that is not capable at full deflection of maintaining equilibrium of yawing moments in a condition of one engine out and negligible sideslip is not powerful enough. The flying qualities requirements normally specify the specific speed ranges that must be achievable with full elevator deflection in the various important configurations, and the asymmetric power condition that the rudder must balance. They may also contain references to the elevator angles required to achieve positive load factors, as in steady turns and pull-up maneuvers (“elevator angle per g,” Sec. 6.10).

CONTROL FORCES

The requirements invariably specify limits on the control forces that must be exerted by the pilot in order to effect specific changes from a given trimmed condition, or to maintain the trim speed following a sudden change in configuration or throttle setting. They frequently also include requirements on the control forces in pull-up maneuvers (“stick force per g,” Sec. 6.10).

STATIC STABILITY

The requirement for static longitudinal stability (see Chapter 6) is usually stated in terms of the neutral point (defined in Sec. 6.3). It is usually required that the relevant neutral point (stick-free or stick-fixed) shall lie some distance (e. g. 5 % of the mean aerodynamic chord) behind the most aft position of the C. G. This ensures that the airplane will tend to fly at a constant speed and angle of attack as long as the controls are not moved.

The requirement on static lateral stability is usually mild. It is simply that the spiral mode (see Chapter 9) if divergent shall have a time to double greater than some stated minimum (e. g. 4 see).

DYNAMIC STABILITY

Generally the requirement on dynamic stability takes the form of a specifi­cation on the time to damp to half amplitude. The damping required for good flying qualities varies with the period.

STALLING AND SPINNING

Finally, most requirements specify that the airplane’s behavior following a stall or in a spin shall not include any dangerous characteristics, and that the controls must retain enough effectiveness to ensure a safe recovery to normal flight

As a result of inability to carry out completely rational design of the man-machine combination, it is customary for the government agencies that are responsible for the procurement of military airplanes, or for licensing civil airplanes, to specify compliance with certain handling qualities require­ments (e. g. refs. 12.26 to 12.28).

These requirements have been developed from extensive and continuing flight research. In the final analysis they are based on the opinions of research test pilots, substantiated by careful instrumentation. They vary from country to country and from agency to agency, and, of course, are different for different types of aircraft. They are subject to continuous study and modifi­cation in order to keep them abreast of the latest research and design infor­mation.

The purpose of these regulations is to ensure the safety of operation of new aircraft. If the rules are too lenient or incomplete the result can be degraded performance, poor flight safety, and perhaps an inability to complete the intended mission. On the other hand, if the rules are too stringent the penalties can be degraded performance, added complexity, and reduced economic efficiency. When a new aircraft is designed with novel features and per­formance characteristics, the old regulations are not always sufficient to cover the situation, and subsequent prolonged vehicle flight testing is then required before it can be certified. In the past, regulations have merely specified minima for the various aspects of handling qualities. It is anticipated that ongoing research in this field will lead to the specification of optimum values for the various handling qualities parameters and the definition of acceptable ranges for these parameters.

The following is intended to show the nature, not the detail, of typical handling qualities requirements. Most of the specific requirements can be classified under one of the following headings.

The ratio а>фІ(оа is a significant parameter when studying lateral-directional handling qualities. If [соф, = [<wd, £d] then the quadratic factors in the numerator and denominator of (12.9,1) cancel. Or, to put it another way, the associated poles and zeros exactly cancel. The major consequence of such an occurrence is that the ф response to aileron becomes non-oscfflatory, a very desirable circumstance. Another consequence would be the disappearance of the valley-peak sequence in the frequency response for ф/да, as illustrated in Fig. 10.12c. When this special circumstance is not the case, then aileron inputs produce oscillatory responses. The cancellation of the quadratic

factors depends mainly on the values of а>ф and cod and less on 1,ф and ‘(d. Hence the importance of (0фІ(0а as a parameter.

Detailed analysis shows that for (оф/соа > 1 favorable yaw is generated, the opposite being true for (oJcod < 1. The yaw that occurs determines the amount and direction of rudder deflection needed to execute a coordinated turn. In addition, closed-loop bank angle control is difficult when софІсоа > 1. The general trend of pilot rating with софІ(ол is shown in Fig. 12.21. The general and marked preference for ojJo)d = 1 is apparent. Figure 12.22 gives typical pilot iso-opinion curves for a range of (о)ф1<оа)2 and £й. These curves indicate that, depending upon the value of £й, the optimum value of o>Jojd may differ from unity.

ROLL CONTROL SYSTEM CHARACTERISTICS (“FEEL”)

The roll control system dynamics are important in establishing the handling qualities of an aircraft. From the small amount of research performed in this area the following general remarks apply:

1. Full aileron deflection with a wheel-type control should not require a rotation exceeding 90°.

2. Control sensitivities as high as 0.5 deg/sec rate of roll per degree of wheel displacement can be satisfactory.

3. The force required to apply full control should be about 40 lb.