SIMPLIFYING ASSUMPTIONS

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.