A numerical and experimental study on the divergence instability of an inverted cantilevered plate in wall effect

Abstract

We present a numerical and experimental study on the static-divergence instability of an inverted cantilevered plate in an uniform axial subsonic airflow. The flow is assumed ideal and is confined by a rigid channel wall. The inverted cantilevered plate, unlike a conventional one, is with a clamped trailing edge and a free leading edge. The equation of the plate motion is solved by the finite-difference method, and the linearized boundary element method is applied for the fluid equations. A new trigonometric form vortex panel model is developed for the solution of fluid force. An equivalent test method is developed for the divergence instability and is applied for the experimental study. The effect of the rigid wall is evaluated for various distances between the plate and the rigid channel wall, and the critical airflow velocity increases with the increase in the distance. The numerical calculations and predication show good agreement with the experimental results. When the distance is large enough, the numerical results are in accordance with the existing theory for an unconfined airflow. Finally, an approximated solution of the critical airflow velocity in terms of the distance is suggested.

Appendix

The influence coefficients of the normal flow-induced velocity at the control points are calculated as follows.

$$\begin{aligned} \begin{aligned}&\left( {{}^\mathrm{v}_\mathrm{{n}}{} \mathbf{{C}}_{\mathrm{p}/\mathrm{bw}/\mathrm{cw}}^\mathrm{{p}}}\right) _{i,j}= \left\{ \begin{aligned}&-\frac{{{\vec n_i}\cdot {\vec n_j}}}{{2\pi }}\int _{- \frac{{ \Delta }}{2}}^{\frac{{\Delta }}{2}} {\frac{\left( {\frac{{\Delta }}{2} - \xi } \right) }{\Delta }\frac{{{{\xi _j} - \xi }}}{{{{\left( {{\xi _j} - \xi } \right) }^2} + \eta _j^2}}} d\xi ,\quad \quad (i = 1)\\&{}\begin{aligned}&- \frac{{{\vec n_i}\cdot {\vec n_j}}}{{2\pi }}\int _{ -\frac{{\Delta }}{2}}^{\frac{{\Delta }}{2}} {\frac{\left( {\xi - \frac{{\Delta }}{2}} \right) }{\Delta }\frac{{{{\xi _j} - \xi }}}{{{{\left( {{\xi _j} - \xi } \right) }^2} + \eta _j^2}}} d\xi \\&- \frac{{{\vec n_i}\cdot {\vec n_j}}}{{2\pi }}\int _{ - \frac{{\Delta }}{2}}^{\frac{{\Delta }}{2}} {\frac{\left( {\frac{{\Delta }}{2} - \xi } \right) }{\Delta }\frac{{ {{\xi _j} - \Delta - \xi }}}{{{{\left( {{\xi _j} - \Delta - \xi } \right) }^2} + \eta _j^2}}}d\xi \end{aligned},(i = 2\sim {N_\mathrm{p}}) \end{aligned}\right. \\&\left( {{}^\mathrm{v}_\mathrm{{n}}{} \mathbf{{C}}_{\mathrm{p}/\mathrm{bw}/\mathrm{cw}}^{\mathrm{bw}/\mathrm{cw}}}\right) _{i,j}=\frac{{{\vec n_i}\cdot {\vec n_j}}}{{2\pi }}\int _{\frac{{ - \Delta }}{2}}^{\frac{{\Delta }}{2}} {\frac{{{\xi _i-\xi }}}{{{{\left( {{\xi _j} - \xi } \right) }^2} + \eta _j^2}}} d\xi ,(i >N_{p}) \end{aligned} \end{aligned}$$

(A.1)

The influence coefficients of the flow-induced potential at the control points of plate are calculated as follows.

$$\begin{aligned} \begin{aligned}&\left( {{}^{\varphi }{\mathbf{C}}_{\mathrm{p}}^{\mathrm{p}}}\right) _{i,j}= \left\{ \begin{aligned}&- \frac{{1}}{{2\pi }}\int _{-\frac{{ \Delta }}{2}}^{\frac{{\Delta }}{2}} {\frac{{\left( {\frac{{\Delta }}{2} - \xi } \right) }}{{\Delta }}{{\tan }^{ - 1}}\left( {\frac{{{\eta _j}}}{{{\xi _j} - \xi }}} \right) } d\xi ,\quad (i = 1)\\&{}\begin{aligned}&- {\frac{{ 1}}{{2\pi }}\int _{- \frac{{\Delta }}{2}}^{\frac{{\Delta }}{2}} {\frac{{\left( {\xi + \frac{{\Delta }}{2}} \right) }}{{\Delta }}{{\tan }^{ - 1}}\left( {\frac{{{\eta _j}}}{{{\xi _j} - \xi }}} \right) } d\xi }\\&{ - \frac{{ 1}}{{2\pi }}\int _{ - \frac{{\Delta }}{2}}^{\frac{{\Delta }}{2}} {\frac{{\left( {\frac{{\Delta }}{2} - \xi } \right) }}{{\Delta }}{{\tan }^{ - 1}}\left( {\frac{{{\eta _j}}}{{{\xi _j} - \Delta - \xi }}} \right) } \xi } \end{aligned},(i = 2\sim {N_\mathrm{p}}) \end{aligned}\right. \\&\left( {{}^{\varphi }{\mathbf{C}}_{\mathrm{p}}^{\mathrm{bw}/\mathrm{cw}}}\right) _{i,j}=\frac{1}{{4\pi }}\int _{ -\frac{{\Delta }}{2}}^{\frac{{\Delta }}{2}} {\ln \left[ {{{\left( {\xi - {\xi _j}} \right) }^2} + \eta _j^2} \right] } d\xi ,(i >N_{p}) \end{aligned} \end{aligned}$$

(A.2)

The influence coefficients of the tangential flow-induced velocity at the control points of plate are calculated as follows.

$$\begin{aligned} \begin{aligned}&\left( {{}^\mathrm{v}_{\mathrm{t}}{\mathbf{C}}_{\mathrm{p}}^{\mathrm{p}}}\right) _{i,j}= \left\{ \begin{aligned}&\frac{{ {1}}}{{2\pi }}\int _{- \frac{{\Delta }}{2}}^{\frac{{\Delta }}{2}} {\frac{\left( {\frac{{\Delta }}{2} - \xi } \right) }{\Delta }\frac{{\eta _j}}{{{{\left( {{\xi _j} - \xi } \right) }^2} + \eta _j^2}}} d\xi ,\quad \quad (i = 1)\\&{}\begin{aligned}&\frac{{{1}}}{{2\pi }}\int _{ -\frac{{\Delta }}{2}}^{\frac{{\Delta }}{2}} {\frac{\left( {\xi - \frac{{\Delta }}{2}} \right) }{\Delta }\frac{{\eta _j}}{{{{\left( {{\xi _j} - \xi } \right) }^2} + \eta _j^2}}} d\xi \\&\frac{{{1}}}{{2\pi }}\int _{ - \frac{{\Delta }}{2}}^{\frac{{\Delta }}{2}} {\frac{\left( {\frac{{\Delta }}{2} - \xi } \right) }{\Delta }\frac{{\eta _j}}{{{{\left( {{\xi _j} - \Delta - \xi } \right) }^2} + \eta _j^2}}}d\xi \end{aligned},(i = 2\sim {N_\mathrm{p}}) \end{aligned}\right. \\&\left( {{}^\mathrm{v}_{\mathrm{t}}{\mathbf{C}}_{\mathrm{p}}^{\mathrm{bw}/\mathrm{cw}}}\right) _{i,j}=\frac{{{1}}}{{2\pi }}\int _{- \frac{{ \Delta }}{2}}^{\frac{{\Delta }}{2}} {\frac{{{\eta _j}}}{{{{\left( {{\xi _j} - \xi } \right) }^2} + \eta _j^2}}} d\xi ,(i >N_{p}) \end{aligned} \end{aligned}$$

(A.3)

Note that the above equations are all derived by using the local coordinates as shown in Fig. 3a, and are calculated by using the suggested methods in Ref. [35].