# Sound Radiation of Aerodynamically Excited Flat Plates into Cavities
## 4 Experimental Results
### Flow Characterization
#### 4.1.2 Boundary Layer  
The dimensionless velocity profiles of the measurement, of the three indirect turbulence models and of the approach after Blasius (refer to Section 2) for a distance to the nozzle outlet of \(x=350\,\mathrm{mm}\) are shown in Figure 8. The turbulence models correlate in exponential curve characteristics compared with the measurement in contrast to the approach after Blasius which shows a linearly increasing course at the beginning changing to an exponential growth till the maximum value of \(u^{+}\). This result suggests that the boundary layer is turbulent. To evaluate the quality of the indirect turbulent models more clearly, another depiction of the dimensionless velocity profiles of the turbulent presumed boundary layer is presented in Figure 9. Besides this, the graphs of the logarithmic overlap law and of the laminar sublayer are added. The \(\
icefrac{{1}}{{7}}\) power law reproduces the curve shape of the measurement most closely at the beginning, whereas the logarithmic law after Coles overlaps the measurement between a dimensionless distance  
Figure 8: Dimensionless velocity profiles at \(45\,\mathrm{m}\,\mathrm{s}^{-1}\) with a distance of \(x=350\,\mathrm{mm}\) from the nozzle outlet. (a) Turbulent presumed boundary layer in case of shear stress \(1\); (b) Laminar presumed boundary layer. Figure 7: Turbulence intensity at \(45\,\mathrm{m}\,\mathrm{s}^{-1}\) with a distance \(x=100\), \(350\) and \(520\,\mathrm{mm}\) from the nozzle outlet. from \(z^{+}=90\) to 200. Towards the end, a plateau with constant dimensionless velocity is to be noticed which means that this section is already outside of the boundary layer. The curve progressions of the logarithmic law and the logarithmic overlap law are the same till reaching the outer part of the boundary layer. There, the law of the wake as the third summand added to the logarithmic overlap law takes effect and describes the outer part in the corrected equation of [PERSON] (Equation (8)). The laminar sublayer completely deviates from the other curves. The shape factor \(H_{12}\) as the ratio of the displacement thickness \(\delta_{1}\) to the momentum loss thickness \(\delta_{2}\) is used to characterize the boundary layer developing on a flat plate. The integral calculation of the displacement thickness \(\delta_{1}\) and the momentum loss thickness \(\delta_{2}\) is done by a spline interpolation with a following exact numerical integration using global adaptive quadrature and preset fault tolerance [16]. The values of \(H_{12}\) converge towards 1.3 despite varying values for \(\delta_{1}\) and \(\delta_{2}\) for different \(x\)-positions. For laminar flows on a flat plate, \(H_{12}\) is typically about 2.59. At the transition point it falls to a considerably smaller number range between 1.4 and 1.0 [9; 17]. [PERSON] [18] concluded that \(H_{12}\) is dependent on the Reynolds number \(Re_{x}\) based on the running length \(x\) and on the skin friction coefficient \(C_{f}\) for equilibrium boundary layers. In this case of constant pressure profiles, for a range of \(10^{6}<Re_{x}<10^{7}\), the values of \(H_{12}\) are expected to be between 1.27 and 1.35 with \(C_{f}\) between 0.025 and 0.036. According to [PERSON], [PERSON] and [PERSON] and [PERSON] [19]\(H_{12}\) assumes higher values between 1.3 and 1.42 in the same range of \(Re_{x}\). [PERSON] [20] concluded after measuring the boundary layer for a pipe flow that \(H_{12}=1.3\) is constant in the measurement range of \(1.7\times 10^{6}\leq Re_{x}\leq 1.8\times 10^{7}\). All boundary layer parameters are shown in Table 1. The resulted values of \(H_{12}\) clearly prove that there is a turbulent flow in the entire area of the flexible flow-excited plate. Moreover, the calculated \(x\)-distance \(x_{\text{new}}\) from Equation (15) is nearly twice as high as the real \(x\)-position. The boundary layer thickness \(\delta_{100}\) increases with the distance to the nozzle.