# Sound Radiation of Aerodynamically Excited Flat Plates into Cavities
## 2 Theory  
Before analyzing the sound radiation of the plate, a determination of the boundary layer parameters is required to achieve a starting position with well-defined initial and boundary conditions. For illustrative purposes, a scheme of the theoretical setup is depicted in Figure 1.  
To specify whether the boundary layer is laminar or turbulent, an analytical calculation of the dimensionless velocity profiles is required. Therefore, different approximate solutions for the wall shear stress \(\tau_{w}\) are examined in detail. The first one has resulted from empirical measurements of a pipe flow and can be transferred to smooth plate walls due to similarity. The analogous approach for a turbulent boundary layer [6; 7]  
\[\frac{u_{\infty}}{u_{\tau}}=8.75\left(\frac{u_{\tau}\,\delta}{\
u}\right)^{ \frac{1}{7}} \tag{1}\]is solved for the friction velocity with the main flow velocity \(u_{\infty}\), the wall shear stress velocity \(u_{\tau}\), the boundary layer thickness \(\delta\) and the kinematic viscosity \(\
u\). Using the following equation \(\tau_{w}=\rho u_{\tau}^{2}\) with the density \(\rho\) yields:  
\[\tau_{w_{1}}=0.0225\cdot\rho\,u_{\infty}^{2}\bigg{(}\frac{\
u}{u_{\infty}\cdot \delta}\bigg{)}^{\frac{1}{4}} \tag{2}\]  
The boundary layer thickness \(\delta\) of the turbulent flow as a function of the running length \(x\) is developed through substitution in conjunction with the \(\
icefrac{{1}}{{7}}\) power law, rearrangement and integration from 0 to \(x\)[6, 8]:  
\[\delta(x)=0.371\cdot x\cdot Re_{x}{}^{-\frac{1}{5}}. \tag{3}\]  
with the Reynolds number based on the running length in flow direction \(Re_{x}\). For Reynolds numbers ranging from \(5\times 10^{5}\) to \(10^{7}\), the local friction coefficient for a turbulent flow over an isothermal plate is correlated by [9, 10]:  
\[C_{f,x}=0.0592\cdot Re_{x}{}^{-\frac{1}{5}} \tag{4}\]  
The friction coefficient as the dimensionless shear stress is defined as [9]:  
\[C_{f,x}=\frac{\tau_{w}}{\frac{\rho}{2}\cdot{u_{\infty}}^{2}} \tag{5}\]  
Equalizing the two Equations (4) and (5) results in the following wall shear stress \(\tau_{w_{2}}\) as second equation:  
\[\tau_{w_{2}}=0.0296\cdot\rho\,u_{\infty}^{2}\,\,Re_{x}{}^{-\frac{1}{5}} \tag{6}\]  
We tried to approach the velocity distribution in the boundary layer to the real course of the experimental measurement data. Indirect turbulence models are used as a basis of the direct velocity modeling:  
* 1/7 Power Law[6, 9] \[u(z)=u_{\infty}\cdot\bigg{(}\frac{z}{\delta(x)}\bigg{)}^{\frac{1}{7}}\] (7)  
with the distance perpendicular to the plate \(z\). The dimensionless velocity is calculated using the following equation \(u^{+}=\frac{u(z)}{u_{\tau}}\) with \(u_{\tau}=\sqrt{\frac{\tau_{w}}{\rho}}\). * **Logarithmic Law after [PERSON] and Adapted by [PERSON][11, 12]**  
The dimensionless velocity which follows the outer-layer corrected logarithmic law after [PERSON] is given by:  
\[u^{+}=\frac{u(z-z_{0})}{u_{\tau}}=\frac{1}{\kappa}\ln\bigg{(}Re_{\delta}\frac {u_{\tau}}{u_{\infty}}z\bigg{)}+B+\frac{\Pi}{\kappa}\omega\left(\xi\right)\quad (0\leq(z-z_{0})\leq\delta) \tag{8}\]  
Figure 1: Scheme of the theoretical setup. This indirect turbulence model uses a Reynolds number which is based on the boundary layer thickness \(\delta\):  
\[\frac{u_{\infty}}{u_{\tau}}=8.75\left(\frac{u_{\tau}\,\delta}{\
u}\right)^{ \frac{1}{7}} \tag{1}\]0  
The dimensionless wall distance \(\xi\) is defined as  
\[\frac{u_{\infty}}{u_{\tau}}=8.75\left(\frac{u_{\tau}\,\delta}{\
u}\right)^{ \frac{1}{7}} \tag{1}\]1  
with the position of the wall \(z_{0}\). The von [PERSON] constant is \(\kappa=0.4\)[11] and the boundary layer constant is \(B=5.1\). [PERSON][13] fitted the data analytically from an empirical table after [PERSON] and derived the following relation for the wake-function:  
\[\frac{u_{\infty}}{u_{\tau}}=8.75\left(\frac{u_{\tau}\,\delta}{\
u}\right)^{ \frac{1}{7}} \tag{1}\]2  
The profile parameter \(\Pi\) is dependent on the pressure gradient and results in Equation (8) for \(\xi=1\).