MODEL OF HIGH-ALTITUDE TURBULENCE

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.