Nonuniqueness of solutions to extended Kirchhoff integral formulations

Abstract

This paper examines the nonuniqueness of solutions to an extended Kirchhoff integral formulation at certain discrete frequencies corresponding to the eigenfrequencies of the related interior boundary value problem. In particular, effects of surface motion and interaction between turbulence and vibrating surface in motion on nonuniqueness difficulties are investigated. It is shown that surface motion affects nonuniqueness difficulties in two ways: (1) it shifts the eigenfrequencies of the related interior boundary value problem and (2) it excites more eigenfrequencies at which nonuniqueness difficulties occur than the corresponding stationary case. The interaction between turbulence and vibrating surface in motion further shifts the eigenfrequencies. Although changes in these eigenfrequencies are small at low Mach numbers, they increase with the Mach number. It is also demonstrated that although an extended Kirchhoff integral equation fails to yield a unique solution at certain discrete frequencies, a combined extended surface Kirchhoff integral equation and an extended interior Kirchhoff integral equation always yield a unique solution for all frequencies. This is because there is only one set of solutions that satisfy both an extended surface Kirchhoff integral equation and an extended interior Kirchhoff integral equation.