Generalised Abel transform for the analysis of fluid vibrations

Abstract

Development of hybrid numerical-experimental techniques is an important method of analysis used for interpretation and validation of experimental results. Laser holography is a powerful experimental technique for analysis of high frequency vibrations of the fluid, especially when the amplitudes of those vibrations are relatively small. It is assumed that the angular frequency of excitation coincides with the eigenfrequency of the appropriate eigenmode and thus the eigenproblem is solved. The problem of fluid eigen-vibrations in axi-symmetric geometry according to the first harmonic in the circumferential direction is analyzed. This problem has a practical application in the analysis of transverse vibrations of the fluid in an axi-symmetric pipe. The numerical model is developed on the basis of the finite element model of the fluid and the method of analysis of the problems with axi-symmetric geometry. The irrotational motions of the ideal compressible fluid are analyzed. The numerical model of the system is based on the approximation of nodal displacements via the shape functions. Thus the field of the amplitudes of the circumferential variation of the volumetric strain is calculated using the procedure of conjugate approximation. By using the procedure of conjugate approximation the amplitudes of circumferential variation of nodal volumetric strains for the eigenmodes are determined. The obtained field of volumetric strain is used in the numerical procedure for the construction of the digital holographic image. Thus the calculation of the amplitudes of circumferential variation of nodal volumetric strains consists of two stages: 1) calculations using the displacement formulation for the first harmonic in the circumferential direction by the method of finite elements; 2) determination of the field of the amplitudes of the circumferential variation of the volumetric strain by using the procedure of conjugate approximation. The obtained nodal values of the amplitudes of the circumferential variation of the volumetric strain are used in the construction of the digital holographic images. For this purpose the Abel transform which is usually used in axi-symmetric problems is generalized for the problems with axi-symmetric geometry with the variation of displacements according to the first harmonic in the circumferential direction. On the basis of the generalized Abel transform the digital holographic images are constructed. The obtained digital holographic images are used in the hybrid experimental-numerical procedure for the determination of the correlation with the experimental holographic images.