Ffowcs Williams & Hawkings’ Formulation of the Acoustic Analogy
1.2 The Wave Equation
Lighthill’s equation and alternative forms such as Powell-Howe’s equation are reformulations of general gas dynamics equations which do not address specifically the question of physical boundaries. Yet aerodynamic noise from wall-bounded flows can be predicted from this general background by solving the wave equation together with relevant boundary conditions imposed on the wall surfaces. The needed material can be any code or software solving the wave equation, or the Helmholtz equation provided that a Fourier transform is performed to investigate single frequencies. This makes the sources of sound interpreted as distributed quadrupoles in Lighthill’s analogy, and their radiation understood as just scattering by the surfaces. Such a view can be inconvenient if the character of the sources is fundamentally modified by the scattering. Furthermore the formal simplicity of the formalism, brought by the homogeneity of the propagation space, is partly lost because of the needed account of boundaries. Another interpretation is obtained when replacing the surfaces by additional equivalent sources supposed to radiate in free space, thus extending the original idea of the acoustic analogy. This is the essence of Ffowcs Williams & Hawkings’ formulation (1969) presented now (Curle’s analysis introduced in chapter 1 for a stationary surface can be considered included in this more general one).
The principle is as follows. The physical surfaces are removed and replaced by mathematical surfaces (Fig. 2). The corresponding inner volume is assumed to contain the same fluid at rest as in the distant propagating medium, whereas the surrounding flow is kept as such. In order to maintain the discontinuity between the inner volume and the real flow outside, additional sources of mass and momentum must be distributed on the surfaces. This is achieved by writing the equations in the sense of generalized functions. The continuity equation becomes
chapter 1 and involving the stress tensor aj. Furthermore if the function f is properly scaled the normal unit vector on the surfaces is just Vf = n.
Equation (3) is exact, as a reformulation of the general equations of fluid dynamics. p and Tij are understood in the sense of generalized functions: they are zero inside the mathematical surfaces and equal, respectively, to the density fluctuations and Lighthill’s tensor of the flow outside. According to the new statement of the analogy the density fluctuations in the real fluid, in the presence of flow and rigid bodies, are those which would exist in an equivalent acoustic medium perfectly at rest and forced by three source distributions. The first term is responsible for the noise produced by virtue of flow mixing and distortions around the solid bodies. It is just the continuation of the quadrupole sources recognized by Lighthill. The second term is a surface distribution of dipoles (divergence of a vector field); it generates what is referred to as loading noise by reference to the aerodynamic loading of a surface in a flow. The third source term involving the time derivative of a scalar quantity is a distribution of monopoles. The resulting acoustic field will be called thickness noise.
The practical use of the formal result is subjected to the same need for simplifications of the source terms as for Lighthill’s equation, if explicit solutions are expected from the general background of linear acoustics. The simplifications are summarized in next section.