Abstract
In 2000, a non-field analytical method for solving various problems of energy and information transport has been developed by Kulish and Lage. Based on the Laplace transform technique, this elegant method yields closed-form solutions written in the form of integral equations, which relate local values of an intensive properties such as, for instance, velocity, mass concentration, temperature with the corresponding derivative, that is, shear stress, mass flux, temperature gradient. Over the past 20 years, applied to solving numerous problems of energy and information transport, the method—now known as the method of Kulish—proved to be very efficient. In this paper—for the first time—the method is applied to problems in aeroacoustic. As a result, an integral relation between the local values of the acoustic pressure and the corresponding velocity perturbation has been derived. The said relation is valid for axisymmetric cases of planar, cylindrical and spherical geometries.
Highlights
In 2000, a non-field analytical method for solving various problems of energy and information transport has been developed by Kulish and Lage
Based on the Laplace transform technique, this elegant method yields closed-form solutions written in the form of integral equations, which relate local values of an intensive properties such as, for instance, velocity, mass concentration, temperature with the corresponding derivative, that is, shear stress, mass flux, temperature gradient
The method provides a unified view on how the domain geometry and boundary conditions influence the transient behaviour of the acoustic pressure field
Summary
The condition on the boundary, r = 0, is deliberately not imposed This will become clear from the following solution procedure. When applied to partial differential equations, this technique leads to the appearance of extra terms in the transforms of the time derivatives, unless the relevant initial conditions are zero. In the present case, introducing the excess pressure p(r) does not change the differential equations involved, but eliminates the necessity to carry on extra terms in the Laplace space. Once the solution has been obtained, the original variables are restored and the reference pressure p0 is added to the final result. Where α and β are arbitrary constants, while Iγ (z) and Kγ (z) denote modified Bessel functions of order γ.
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