# STATISTICAL AVERAGES

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

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

 —- T 1 П+Т u{t) “ 2T. u(t) dt It —T 1 і ft + T ut) = 2rJ ut) dt l – T

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 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).  Fig. 8.8. A truncated function.

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

 p(m) – lim (ZWoo 1 (9) C oo u2(t) — p(m) dm Jo (10)

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(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

 J* 00 p(co) 0

 cos сот dco

 (18)

 yj(r)

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.

 The constants by and L2 in the above equations are quantities known as scales of turbulence. They are proportional to the areas under the normalized correlation curves

 (27)