Nonreflecting boundary conditions for the time-dependent convective wave equation in a duct

Abstract

Solutions for the three-dimensional wave equation in an annular duct are represented in terms of Fourier–Bessel modes, each obeying a one-dimensional dispersive wave equation. An exact nonreflecting boundary condition, nonlocal in time but local in space, is derived for each mode. Since, in most applications, the number of propagating modes is a small finite subset of all the modes, the present condition yields a computationally efficient scheme. Convergence of the solution depends on the radial eigenvalues which characterize the dispersion of the propagating duct modes. For periodic forcing, convergence analysis shows that the solution tends toward its asymptotic limit as the reciprocal of the square root of time. Near a duct mode cut-on, the computational time required for convergence is proportional to the reciprocal of the square of the group velocity. The boundary condition is implemented numerically and tested by computing the propagation of dispersive waves with various frequencies and comparing the results with the analytic solution. Two local boundary conditions are also implemented and their performance relative to the nonlocal boundary condition is studied. Significant improvements in accuracy are observed by using the present boundary condition near cut-on. Numerical results also show that the rate of convergence of the solution to a time-periodic solution significantly decreases as the group velocity of the incident waves becomes small, consistent with the analytic results.