The study of turbulence has been of interest to scientists and engineers for many years. Because of the random nature of turbulence, it is subject to the same types of analysis as random noise, such as is produced in electronic circuits by the random discharge of individual electrons or in a rarefied gas by the random collisions of molecules. The ran-dom motions of small particles in a liquid that result from bombardment of the particles by the molecules of the liquid is called Brownian motion. This phenomenon was analyzed by Albert Einstein in a famous paper written in 1905 (ref. 12.1). Turbulence became of interest to aerodynamicists first through studies of the boundary layer-formed by air or liquid flowing over a solid bound-ary. The boundary layer could be either lam-inar or turbulent. The turbulent boundary layer was found to produce skin friction drag many times greater than that found in the
laminar boundary layer. The tendency for the boundary layer to become turbulent depends strongly on the Reynolds number, a quantity that increases with the size of the model being tested, the speed, and the density of the test medium, and varies inversely with the viscosity of the medium. Early tests of small models in low-speed wind tunnels were found to give erroneous results because of the greater tendency of the boundary layer to remain laminar on the model as compared to that on a full-scale airplane. The boundary layer was studied extensively by Hugh L. Dryden, later a director of the NACA and by a famous British scientist, G. 1. Taylor, as well as many other noted scientists (ref. 12.2). I first became aware of these studies when I wrote my bachelor's thesis on boundary layers in 1939. Soon the subject of turbulence became a discipline of interest in many fields and could easily become the focus of a lifetime career.
The turbulence in the atmosphere is a random three-dimensional turbulence field that can extend for large distances compared to the size of an airplane penetrating it. Early applications of turbulence theory to atmospheric turbulence were made by T. von Karman in 1934 (ref. 12.3) and by G. 1. Taylor in 1935. These reports were written before I started work, but I made an effort to review them after I had worked at Langley for some years.
When an airplane flies through turbulence, large loads are imposed on the structure that may be critical for the design strength of the airplane or repeated smaller loads may lead to fatigue failure of structural components. Also, the violent motions of the airplane may cause airsickness or fatigue of the crew and passengers. As a result of these effects, atmospheric turbulence was recognized as an important subject for research.
The type of turbulence most frequently studied in the theoretical analyses is  called homogeneous isotropic turbulence. Homogeneous means that the turbulence is the same at any location, and isotropic means that the turbulence field is the same in any direction. In such turbulence, an airplane would experience the same type of disturbances flying in any direction. Not all atmospheric turbulence has these characteristics, but this theory appears to be useful in many practical applications. Two terms associated with this turbulence are the autocorrelation function and the power spectrum. The autocorrelation function shows how, on the average, the turbulent velocity at some point is related to the turbulent velocity at some other point. Thus, if a gust hits an airplane, experience has shown that the velocity is approximately the same at the wing tips as at the centerline. The velocity at some point far away from the airplane would, on the average, be unrelated to the velocity at the airplane, which shows that the correlation function approaches zero at large distances. The power spectrum considers the turbulence in the atmosphere to be composed of superimposed sinusoidal waves of different wavelengths. The spectrum shows how the amplitude of a wave, on the average, varies with the wavelength. It can be shown mathematically that the correlation function is related to the power spectrum, so that if one is known, the other can be calculated
The shape of the spectrum of vertical gusts as predicted by von Karman on a plot of the gust power (that is, the square of the gust velocity in each increment of frequency) as a function of frequency shows that the power is constant at very low frequencies. Then at some frequency, the power starts to decrease, and falls off as the frequency to the -5/3 power. This region of the spectrum is called the inertial subrange, indicating that the mass effects are important. This range is usually most important in influencing the loads and motions of an airplane. At still higher frequencies, the spectrum was shown by W. Heisenberg, the famous physicist, to fall off very rapidly and vary as the frequency to the -7 power (ref. 12.4). He called
this part of the spectrum the viscous subrange, indicating that viscosity effects in the air are important. When I became aware of Heisenberg's report, I had it translated and published as a NACA Technical Memorandum. I thought that this part of the spectrum might be of interest for airplanes, but it turns out that the wavelength is so short in this region that it has a negligible effect on full-scale airplanes, though it may be of some interest in wind-tunnel studies of turbulence.
The work on atmospheric turbulence made it apparent that the accuracy of turbulence measurements made in flight would be important and that studies to show how well the assumption of homogeneous, isotropic turbulence would apply in flight would be of interest. As a result, studies made of these subjects are presented in the following sections.
Analytical Studies of the Accuracy of Turbulence Data from Flight Records
In the time period of the early 1950's, numerous applications occurred in flight research for the analysis of random data. Records requiring this type of analysis were obtained, for example, in studies of pilot tracking error, in the response of airplanes to turbulence, and in records of human response to arbitrary inputs. The approach used was to analyze the record by fitting it with a Fourier series, which showed the amplitude of the harmonics at various frequencies. The earliest attempts at this type of analysis used a Coradi rolling sphere analyzer, a delicate instrument that mechanically read the amplitude of a component at a given frequency when the record was carefully traced by manually moving a stylus along the record. Later, when electronic computers became available, the records were digitized on punch cards and power spectral data were obtained, but the process was quite lengthy....
.....with the capabilities of computers available at that time. Now, of course, the availability of data recording in digitized form, the use of the fast Fourier transform, and the high speed of computation makes the analysis of such data a routine procedure.
One question that came up, particularly in the analysis of atmospheric turbulence, was how long a record was required to obtain sufficiently accurate data on the low-frequency components of turbulence. In the case of the early studies, a problem apparent to the researcher was that the time history record obtained as a trace on oscillograph film had a certain width. Frequently, engine vibration picked up by the instrument made this trace considerably wider than that of the trace under smooth conditions. The width of the trace could be observed on a physical record or could be considered as indicative of an error band that depended on the accuracy of the instrumentation involved. The actual value of the recorded quantity at any time could be anywhere within the width of the error band. A study was made to determine how large a variation in the amplitude and phase of a given harmonic component could occur without going outside the boundaries of the trace over the length of the recorded data. In particular, it should be noted that attempts were made to obtain data on harmonics with wavelengths considerably....
.....longer than the length of the time history available. This analysis involved relatively straightforward trigonometric manipulations. One reason for showing the results here is that the resulting plot of amplitude versus phase angle of the harmonics at various wavelengths has an unusually artistic form.
First, in figure 12.1 (a) are shown several examples of harmonic components of waves that just fit within a rectangle with length equal to the length of the record and width equal to the error band. In practice, the record is many times longer than its width, but the figure shows a much shorter and wider rectangle for clarity. As shown, so long as the wavelength is less than the record length, the maximum error of the component is the same as the width of the error band. When the wavelength of the harmonic component is larger than the record length, however, the error of the component exceeds the width of the error band by an amount depending on the phase angle of the component. Figure 12.1(b) shows the ratio of error to line width as a function of ratio of record length to wavelength for different conditions of phase angle of the harmonic. The conditions shown are those giving the maximum error, those giving the minimum error, and the error obtained by averaging over all phase angles. Inasmuch as the phase angles in power spectral analysis are assumed to be uniformly distributed, the latter curve is probably more representative of what could be obtained with a statistical analysis.
Figure 12.2 shows the polar plot of amplitude versus phase angle for harmonics of different ratios of wave length to record length. The nesting of the various lenticular curves presents an unusual artistic effect not often seen in theoretical plots.
This analysis was never published, mainly because more sophisticated approaches were being developed by mathematicians with much more knowledge of statistical analyses than I had. The next section describes a flight investigation to make accurate measurements....
...of the spectrum of atmospheric turbulence that uses the most advanced statistical techniques available at the time.
Measurements of the Spectrum of Atmospheric Turbulence Over a Large Range of Wavelengths
A much more important source of error than the error introduced by the instrumentation is the variability of the power spectral data caused by the random nature of the process being analyzed. Thus, various samples of a given process taken at different times will show different harmonic content, particularly at the longer wavelengths. To obtain an accurate measure of the power spectrum, many records should be analyzed and the results averaged or one very long record taken under uniform conditions should be analyzed. As a result of these considerations, a rule of thumb is that approximately ten to twenty wavelengths of the lowest frequency under consideration should be contained in the record. Obviously, in considering atmospheric turbulence that contains much energy at wavelengths of several thousand feet, a very long record in uniform conditions of turbulence is required.
In my position as head of the Stability and Control Branch of the Flight Research Division in the early 1950's, I was able to plan programs of flight research and had available test airplanes and instrumentation to perform research studies. One of my engineers, Robert G. Chilton, a recent MIT graduate, was familiar with the latest mathematical techniques to analyze records of turbulence. The measurement of atmospheric turbulence was officially assigned to the Aircraft Loads Division, but the results of such studies were of immediate use in my branch to study response of airplanes to turbulence. Inasmuch as accurate measurements over a wide range of wavelengths....
....were not then available, I obtained approval to conduct a flight test to obtain these data. These tests used instrumentation with greater accuracy than had been used previously, such as a sun camera (rather than a gyroscope) to determine airplane pitch angle and a high-frequency vane to measure angle of attack.
The airplane and some of the instrumentation are shown in figure 12.3.
With the aid of this equipment, a flight was made at an altitude of 1700 feet over a distance of 170 miles between Norfolk and Baltimore with a Navy F9F-3 Cougar fighter  airplane. The weather conditions were favorable in that a cold front had just passed through the entire area, which gave a strong ground wind and consistent clear-air turbulence over the entire path. The results of this test are presented in a NACA Technical Note (ref. 12.5).
The measured spectrum of turbulence is shown in figure 12.4. To obtain maximum accuracy in different frequency ranges, different portions of the record were analyzed with different spacing of the recorded points. Thus, for high frequencies, a shorter portion of the record with more closely spaced points was analyzed. As a result, the plotted curve is broken into sections. Since these tests were made, many studies have been made by NASA, the Air Force, and other organizations to obtain turbulence measurements at various altitudes and meteorological conditions. It is believed, however, that the study presented in reference 12.5 is still one of the most useful for comparing the spectrum of atmospheric turbulence with the theoretical predictions over a wide range of wavelengths.
It is evident that my early analysis has little relation to the more important problems of obtaining accurate power spectral data. The artistic results of the analysis, therefore, may be considered more a mathematical oddity than a useful analytical result. Subsequent developments of turbulence theory by experts in this field, however, were followed closely and allowed useful data to be obtained on the spectrum of turbulence or in other studies involving random data.