Solving the Lilley Equation with Quadrupole and Dipole Sources

Abstract

The literature contains various methods for solving the Lilley equation with different types of quadrupole and dipole sources to represent the mixing noise radiated into the far-field by isothermal and heated jets. These include two basic numerical solution methods, the ‘direct’ and the ‘adjoint’, and a number of asymptotic, analytic solutions. The direct and adjoint equations are reviewed and it is shown that their solutions are not only related through the adjoint property: the radial ODE for the adjoint displacement Green's function is the same as that governing the direct displacement Green's function because this particular Green's function obeys classical reciprocity with respect to its radial dependence. Further, by comparing the two numerical solution methods within the context of the parallel flow assumption of the Lilley equation, it is shown that the numerical effort for the two methods is equivalent. The numerical solutions are compared with analytic low frequency ‘thin shear layer’ solutions and WKB solutions, both outside and inside the cone of silence. It is concluded that the former should be used with caution at all angles, while the WKB has some limitations inside the cone of silence. Although numerical solutions can be obtained with little computational effort and are the preferred route for jet mixing noise predictions, the analytic solutions still offer important physical insights as well as verification of numeric results.