206 Lecture 12 
Figure 12.6 shows an extension of the rms temperature fluctuation meas- 
urements for long distances. As the distance is increased, the slope increases 
above the value given by the one-third power law. However, when the distance 
between the two points becomes equal to the depth of the measurements, the 
curves become practically horizontal. It is relatively easy to show that this dis- 
continuity in the slope of the measured curve is due to a similar discontinuity 
in the slope of the power-spectrum frequency curve (or that it is even due to a 
cutoff of the power-spectrum) at a wavelength equal to one-fourth the depth of 
the measurement. Figure 12.7 shows a curve that is computed on the assumption 
of a Kolmogorov-type power spectrum and cutoff. This curve is similar to the 
experimental curves shown in the same figure. The discontinuity in the slope 
of the power spectrum also can be found directly from a Fourier analysis of 
the temperature fluctuations (Fig. 12.8), or from a Fourier analysis of the cor- 
relation function of the temperature fluctuations (Fig. 12.9). The results shown 
in Fig. 12.8 are of an orientative nature only, and the measurements are still 
affected by the time constants of the recording equipment. The frequency spec- 
trum shown in Fig. 12.9 is practically constant up to a wave number equal to 
the depth, and decreases with increasing frequency. Figure 12.10 shows similar 
results obtained for the turbulent velocity fluctuations in a boundary layer. 
The last result makes it possible to prove that the approximate constancy 
of the thermal patch diameters in all the temperature recordings (see, for in- 
stance, Fig. 12.2) is not due to a predominant spectral space component of the 
temperature fluctuation, but is due to a cutoff of the spectral distribution. For 
instance, let us consider a Fourier distribution with slowly varying amplitudes 
A(t) and slowly varying phase angles w(t). (The introduction of A(t) and y(t) elimi- 
22:1073¢ 
THEORETICAL CURVE 
22:10°4 
EFFECT OF TIME 
CONSTANT OF 
THERMISTOR 
e |28-IN SPACING ° °,, > 
© 16 - IN. SPACING +++ °+ 9 eh 
+ 64-IN. SPACING 
22-1075 ++ Or 
% \e 
ek 
° 
TEMPERATURE FLUCTUATION 
(deg C/unit wave -number interval) 
Fig. 12.8. Space spectrum of 
the temperature fluctuations. 
22-1076 w 
TIME CONSTANT~6  * 
OF FILTERING + 
CONDENSER 
760 76 7.6 076 0.076 
SPACE-WAVE LENGTH (cm) 
