258 Lecture 14 
cording to a -—*h power law (Kolmogorov) [8] for high frequencies and according 
to a -7 power law (Heisenberg law) [9] for very high frequencies. The curves 
show that, apart from relatively unimportant deviations at very low frequencies 
(or space wave numbers), the velocity fluctuations and the Reynolds stresses 
(that is, the velocity products uv’, etc.) have approximately the same spectral 
distribution. Similar results have been obtained for the power spectrum of the 
flow noise by M. Harrison [10], by W.W. Willmarth [11], and by the Ordnance 
Research Laboratory [12]. 
The spectral density of the turbulence noise may thus be considered to be 
constant up to a frequency uo/d determined by the ratio of the free-stream ve- 
locity to the thickness of the boundary layer. From there on, it may be expected 
to decrease inversely proportional to a certain power m of the scale of the 
turbulence or the space wave number; a different power law may then be ex- 
pected to apply in the viscous range. This power law may be deduced from the 
known laws of turbulence or deduced from the experimental results. 
The power spectrum of the pressure referred to unit frequency interval is 
thus known by 
Pte) = p3 (22) for @2W (7) 
2 
Sor os c soe (8) 
In Eq. (8), the frequency #0 at which the noise spectrum starts to decrease is 
expressed by the ratio of the free-stream velocity to a quantity 6 which is of 
the same order of magnitude as the boundary-layer thickness. The relation 
6 =56*, where 6* is the displacement thickness of the boundary layer, seems to 
lead to very good agreement with the experimental results. This relation has 
therefore been assumed in all the following computations. 
The preceding considerations in conjunction with the pressure, Eq. (5), lead 
to the prediction of the spectral distribution of the noise pressure. The constant 
ps, however, remains unknown. This constant can be determined by computing 
the rms noise pressure (neglecting the viscous range): 
fea} wo Tas} a 
2 2.2 da _ 2 dw 2(@o0\ dw 
p =ar -{ po) d= f cate |) p3(22) = (9) 
ty) 0 0 
and by equating the result to the value given by the Kraichnan equation: 
2 _ 0.75.10-° a2p2ug 5* |3(m—— (10) 
pPo=vV. . P uo 9} m ™m 
If this value is substituted above, the power spectrum P(w) of the noise becomes 
P(@) = 0.75-1075 a®p*uad*[3(m-2)] for @<@o (11) 
