Method of characteristics for transient, spherical flows

Abstract

• Spherically-symmetric unsteady flow is solved numerically by the method of characteristics. • Close agreement is demonstrated with an analytical solution for a single pulse propagating spherically. • Interaction of waves propagating internally in a duct with acoustic waves radiating from the duct's exit. • Flows containing both uni-axial and spherically-symmetric domains are calculated with a highly efficient use of resources. • Two-way coupling is demonstrated to be a significant advantage over conventional one-way coupling. The equations of motion for spherically-symmetric, unsteady, inviscid, compressible flow are expressed in a form that enables accurate numerical solutions to be obtained using a one-dimensional formulation based on the method of characteristics. Unlike the corresponding equations for uniaxial unsteady flows, the spherically-symmetric equations necessarily include terms involving the reciprocal of the radius and, close to the radial origin, numerical integration is unreliable. Nevertheless, good accuracy is obtained over a wide range of radii, including regions inside the range where the pressure and acceleration are approximately in phase. The range of validity of the method is assessed by comparison with an analytical solution for a single pulse and the method is then used to predict the radiation of acoustic waves from the exit of a duct in which a pulse is propagating internally. A method of obtaining efficient solutions of flows containing both uniaxial and spherically-symmetric domains is then obtained. Interfaces between the domains are solved in a manner that ensures continuity of pressure and flowrate. The use of spherically-symmetric assumptions limits the range of three-dimensional domains that can be approximated, but the combination of the two forms of one-dimensional analysis makes highly efficient use of resources. Furthermore, the two-way coupling is a significant advantage over two-step methods that use independent solutions of the internal, uniaxial domain to provide prescribed boundary conditions for solutions of the external domain.