# Research on Panel Flutter Considering the Effect of Convective Active Cooling
## 2 Theoretical Analysis  
Consider a two-dimensional isotropic simply-supported panel, as shown in Figure 1, where the panel length is \(a\), the thickness is \(h\), and the transverse vibration displacement is \(w(x)\). The upper surface of the panel is acted on by supersonic airflow, and the Mach number, pressure, and density of the freestream are \(Ma_{\infty},p_{\infty},\rho_{\infty}\), respectively. Below the panel is a convective cooling channel filled with coolant driven by a constant pressure gradient \(-G\), ignoring the viscous shear. Based on [PERSON]'s large deformation thin plate theory and the Kirchhoff-Love hypothesis, the equation of motion for the panel is  
\[\rho h\frac{\partial^{2}w}{\partial t^{2}}+D\frac{\partial^{4}w}{\partial x^{4 }}-(N_{x}+N_{0})\frac{\partial^{2}w}{\partial x^{2}}+(p-p_{\infty})=f_{c}-p_{\infty} \tag{1}\]  
The boundary conditions are  
\[\left\{ \begin{array}{l}w(0,t)=w(1,t)=0\\ \frac{\partial^{2}w}{\partial x^{2}}(0,t)=\frac{\partial^{2}w}{\partial x^{2} }(1,t)=0\end{array} \right. \tag{2}\]  
where \(w\) is transverse vibration displacement, \(\rho\) is material density, \(D=Eh^{3}/\left[12(1-v^{2})\right]\) is bending stiffness, \(E\) is elastic modulus, \(v\) is [PERSON]'s ratio. \(N_{0}\) is in-plane load caused by assembly or heating, and \(N_{x}\) is nonlinear membrane load caused by panel deformation. \[N_{x}=\frac{Eh}{2a}\int_{0}^{a}\bigg{(}\frac{\partial w}{\partial x}\bigg{)}^ {2}dx \tag{3}\]  
In order to predict the aerodynamic load under supersonic/hypersonic conditions accurately, the nonlinear modified piston theory [13] is used. \[p-p_{\infty}=\frac{2q}{\sqrt{{M_{\infty}}^{2}-1}}\Bigg{[}\frac{1}{U_{\infty}} \bigg{(}\frac{{M_{\infty}}^{2}-2}{{M_{\infty}}^{2}-1}\bigg{)}\frac{\partial w }{\partial t}+\frac{\partial w}{\partial x}+\frac{\gamma+1}{4}M_{\infty} \bigg{(}\frac{\partial w}{\partial x}\bigg{)}^{2}\Bigg{]} \tag{4}\]  
\(U_{\infty}\) is the velocity of freestream and \(q=\rho_{\infty}{U_{\infty}}^{2}/2\) is dynamic pressure. Figure 1: Geometric model of panel with cooling channel. The coolant load on the panel is  
\[f_{c}=p_{c}+f_{in}+f_{vis} \tag{5}\]  
The three items on the right side of Equation (5) are coolant pressure, inertial resistance, and viscous resistance. The flow of the coolant in the cooling channel is approximated by the Poiseuille flow. It is called forward flow when the coolant and air flow are in the same direction; otherwise, it is the reverse flow. Take forward flow as an example  
\[p_{c}=p_{ch}-Gx \tag{6}\]  
\(p_{ch}\) is the head-panel pressure. The inertial resistance and viscous resistance are proportional to the second derivative and the first derivative of the deformation to time, respectively. \[f_{in}=c_{c1}\frac{\partial^{2}w}{\partial t^{2}} \tag{7}\]  
\[f_{vis}=c_{c2}\frac{\partial w}{\partial t} \tag{8}\]  
Through dimensional analysis  
\[c_{c1}\sim\frac{\rho_{c}h^{2}}{h_{c}},\ c_{c2}\sim\frac{\
u_{c}}{\delta_{c}}\]  
where \(\rho_{c},\
u_{c},h_{c},\delta_{c}\) are the coolant density, the coolant viscosity, the cooling channel thickness, and the thickness of the oscillating boundary layer describing the friction effect, respectively. The dimensionless variables are defined as  
\[W=w/h,\ \zeta=x/a,\ \tau=t/\sqrt{\rho ha^{4}/D},\ R_{0}=\frac{N_{0}\mu^{2}}{D},\ \mu=\frac{\rho_{\infty}a}{\rho h},\ \lambda=\frac{2\mu^{3}}{D \sqrt{{M_{\infty}}^{2}-1}}\]  
\[P_{ch}=\frac{\mu^{4}}{Dh}p_{ch},\ P_{Gr}=\frac{p_{ch}}{Ga},\ C_{c1}=\frac{c_{ c1}}{\rho h},\ C_{c2}=\left(\frac{a^{4}}{D\rho h}\right)^{1/2}c_{c2},\ P_{\infty}=\frac{a^{4}}{Dh}p_{\infty}\]  
where \(R_{0},\mu,\lambda,P_{ch},P_{Gr},c_{c1},C_{c2}\) denote the in-plane load, air/plate mass ratio, freestream dynamic pressure, head-panel pressure, pressure drop ratio, inertial resistance, and viscous resistance, respectively. \(P_{ch}=aP_{\infty}\) is valued near the atmospheric pressure of the incoming flow, where \(\alpha\) is coefficient of head-panel pressure. \(P_{Gr}\) cannot be less than 1.0.