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.

Green's Functions in Computational Aeroacoustics

“…An obvious example of non-uniform flow of great interest in aeroacoustics is the cylindrical shear flow that forms the basis of Lilley's equation [7]. Semi-analytic approaches to finding Green's functions for Lilley's equation are reviewed in [8].…”

“…2, we obtain 7where (8) Equation 7is an inhomogeneous wave equation, valid for all (x,t), that describes the windowed variable ψ -. In the source distribution (8), it is seen that the first two terms are equivalent to the initial conditions for ψ, the next two terms are equivalent to the boundary conditions and the final term is due to the volume sources. Now let V ′ be an extended region, V ′ ⊃ V, bounded by a fixed closed surface Σ.…”

“…Examination of source distributions (8) and (11) shows, for example, that the homogeneous wave equation with either stationary or moving boundaries can be cast in this form.…”

“…We therefore have Note that the reciprocity relation (43) holds only for a uniform flow: in a nonuniform flow, the relationship between the appropriate wave operator and its adjoint may be rather more complicated. An important example in aeroacoustics arises in the conditions for Lilley's equation; Green's functions for Lilley's equation are discussed in [8].…”