E. J. Skudrzyk and G. P. Haddle 257 
The shear velocity v* turns out to be very nearly equal to 4% of the free-stream 
velocity. This result is in good agreement with, for instance, Laufer channel 
measurements [2]. The shear velocity is 4% at the outside of the laminar sub- 
layer and then decreases linearly with the distance until it becomes Zero at 
the outer limit of the boundary layer. 
The shear velocity v* is connected with the nearfield noise pressure by an 
equation very similar to the well-known Bernoulli equation, the only difference 
being a modified constant: 
p=apv*? (5) 
Computations have been performed by Batchelor [4] fora region of homogeneous 
turbulence, and by Kraichnan [5] for a simplified model of a boundary layer. 
For the boundary layer, the factor '4 in the Bernoulli equation turns out to be 
the factor a of the order of magnitude 7. The constant a may be expected to de- 
pend slightly on the Reynolds number. The nearfield sound pressure that is 
generated by the velocity fluctuations is similar to the acoustic radiation pres - 
sure, or the stagnation pressure on a moving body, except that stagnation now 
is generated not by solid bodies but by the velocity gradients of the moving 
eddies. This pressure is proportional to the square of the velocity fluctuations 
and is, therefore, a second-order phenomenon in terms of the velocities. 
The spectral distribution of the nearfield pressure can be estimated in a 
very simple manner. The turbulent eddies are correlated over distances of the 
same order of magnitude as the boundary-layer thickness. Each one of the 
turbulent eddies that moves over the sensitive area of the hydrophone represents 
a pressure pulse whose magnitude is constant as long as the pulse is in full 
contact with the hydrophone and then decreases rapidly to zero when contact 
is lost. The spectrum of such a pulse is well known. It is practically constant 
up to a frequency » whose semiperiod is equal to the pulse duration t;. From 
then on, it decreases as 
sin(@t;/2) sin (f/f) 
aE (6) 
where t, = 5/uy = 1/f), uy being the flow velocity, 5 the space length of the pressure 
pulse, and f) the repetition frequency of the pulses. The low-frequency part of 
the spectrum is practically independent of the details of the pressure distribu- 
tion inside the pulse. It is proportional to the average value of the pressure 
during this interval. In contrast, the high-frequency part of the spectrum de- 
pends on the details of the turbulence. It is practically impossible therefore to 
predict the high-frequency spectrum with adequate accuracy. However, the 
experimental results show that the spectrum decreases inversely proportional 
to the second or third power of the frequency. 
Figure 12.10* shows the power spectra of the longitudinal velocity fluctua- 
tions u’ of the turbulence and of the Reynolds stress u'v'as given in [6] and [7] 
(for a point close to the wall, a flow velocity of 49 ft/sec, and a displacement 
thickness of the boundary layer of approximately 0.24 in.). The frequency f, is 
therefore 200 cps. As predicted, the spectral amplitudes are practically con- 
stant at low frequencies (particularly those of v’ andu'v’); they decrease ac- 
*See page 208 of this volume. 
