Method of characteristics for transient, planar, cylindrical flows

Abstract

Cylindrical radiation of individual pulses is studied and strong differences from spherical radiation are highlighted. Prior to this study, the class of available analytical solutions for cylindrical radiation was rather limited. It included (i) a well-known solution for a steady-state, time-harmonic line source and (ii) the solution for a delta-pulse line source. The first of these has no analytical counterpart for individual pulses. By means of the second (actually the Green's function) a more general class of line source solutions can be formulated, but each one is in the form of an integral that has to be evaluated numerically and requires tedious preparation of a singular integrand, making it difficult to achieve perfect accuracy. In the present paper, use is made instead of a particular family of analytically-exact (linear) solutions that has been developed recently. It is used to assess the accuracy of solutions obtained numerically by the method of characteristics (MoC), which is then applied to more general sources than those amenable to analytical treatment. The MoC analysis is specially formulated to reduce the influence of terms involving the reciprocal of the radius (such terms are numerically unmanageable close to the origin). It is found that the qualitative behaviour of waves radiating from the surface of a cylinder of finite radius can depend strongly on the duration of the initiating disturbance and that this can be interpreted in a convolution-like manner. After validation, the numerical method is coupled with a conventional MoC analysis of planar wave propagation and is used to simulate the reflection and radiation of an initially planar wavefront arriving at a flanged duct exit. The numerical representation is one-dimensional in both domains – (x,t) inside the duct and (r,t) outside it. Although its use implies identical behaviour in all radial directions in the external domain, the solution is found to be in close agreement with circumferential-average values from a CFD solution that simulates azimuthal variations as well as radial variations. Whilst being less comprehensive than the CFD simulation, the MoC analysis has the important practical benefit of making highly efficient use of human and computational resources. By comparing the MoC and CFD solutions, it is found that the nominally-cylindrical radiation from the duct approximates closely to radiation from a line source a small distance inside the duct (not at the exit plane). An approximate location of the effective source is quantified.