333 



for wall pressure due to turbulent boundary layer 

 flow. However, in many ways less is known about 

 the turbulent pressure field for boundary layers 

 than for free turbulent shear flows . Not only is 

 the theoretical problem made more difficult 

 (impossible to the present) by the presence of the 

 wall, the experimental problem is considerably 

 complicated by the dynamical significance of the 

 small scales near the wall. 



Thus, in spite of over two decades of concentrated 

 attention we cannot say with confidence even what 

 the rms wall pressure level is, although recent 

 evidence points to a value of [Willmarth (1975) ] : 



p' = c p u^^ 



c = 2 to 3 



The basic problem is that the most interesting part 

 of a turbulent boundary layer appears to the region 

 near the wall where intense dynamical activity 

 apparently gives rise to the overall botmdary layer 

 activity. While the details of the process are 

 debatable, most investigators concur on the importance 

 of the wall region on overall boundary layer 

 development. Unfortunately, under most experimental 

 conditions, the scales of primary activity are 

 smaller than standard wall pressure probes can 

 resolve [Willmarth (1975) ] . Thus we have virtually 

 no information concerning the contribution of the 

 small scales to the pressure field, although we 

 suspect that the small scales are significant or 

 even dominant. 



Pressure Spectra in Boundary Layers 



Our knowledge of the pressure spectra may be 

 summarized as follows : 



1) Pressure fluctuations arising from motions 



in the main part of the boundary layer (y/5 > 0.1) 

 scale with the outer parameters u and 6. 



2) Pressure fluctuations arising from the inner 

 part of the boundary layer scale with the inner 

 parameters : 



a) hydraulically smooth, u , V 



b) hydraulically rough, u^ , h ; where h is 

 roughness height 



3) Pressure fluctuations arising from the 

 inertial sublayer (logarithmic layer) scale only 

 with u^ and y, the distance from the wall. 



4) The wall pressure spectrum is a composite 

 of all these factors and has a distinct region 

 corresponding to each factor. 



A composite picture of the wall pressure spectrum 

 is shown in Figures 7a and 7b. The 1/k range is 

 evident in both the inner and outer scalings and 

 arises from the inertial sublayer contribution 

 [Bradshaw (1967)]. 



The pressure spectrum within the near wall region 

 should closely resemble the wall spectrum (although 

 this has never been confirmed) . The spectrum in 

 the main part of the boundary layer, should, however, 

 resemble that obtained for a free shear flow at 

 high Reynolds numbers. Again there is no information 

 available to either prove or disprove this conjecture. 



The Lagrangian model developed in the preceding 

 section depends in part on the assumption that a 

 material point is in a stationary random field. 

 As long as the Eulerian field is homogeneous, there 

 is no problem. This is approximately true in many 

 shear flows, but is never true in a turbulent 

 boundary layer. Thus our Lagrangian spectral picture 

 must be abandoned entirely (or used with great 

 restraint) . 



However, a number of features of the Lagrangian 

 model can be applied to this problem. In particular, 

 the "spectral peaks" in the outer flow can be 

 identified with the Lagrangian integral scale, 

 ^ ~ i/u' . The highest frequencies in the flow will 



Table 1. Noirmalized correlation functions with pressure probes 

 arranged as shown in Figure 6 (corresponding velocity 

 correlations in brackets) . 



