Application of the Ffowcs Williams/Hawkings equation to two-dimensional problems

This paper discusses the application of the Ffowcs Williams/Hawkings equation to two-dimensional problems. A two-dimensional version of this equation is derived, which not only provides a very efficient way for numerical implementation, but also reveals explicitly the features of the source mechanisms and the characteristics of the far-field noise associated with two-dimensional problems. It is shown that the sources can be interpreted, similarly to those in three-dimensional spaces, as quadrupoles from turbulent flows, dipoles due to surface pressure fluctuations on the bodies in the flow and monopoles from non-vanishing normal accelerations of the body surfaces. The cylindrical spreading of the two-dimensional waves and their far-field directivity become apparent in this new version. It also explicitly brings out the functional dependence of the radiated sound on parameters such as the flow Mach number and the Doppler factor due to source motions. This dependence is shown to be quite different from those in three-dimensional problems. The two-dimensional version is numerically very efficient because the domains of the integration are reduced by one from the three-dimensional version. The quadrupole integrals are now in a planar domain and the dipole and monopole integrals are along the contours of the two-dimensional bodies. The calculations of the retarded-time iraterpolation of the integrands, a time-consuming but necessary step in the three-dimensional version, are completely avoided by making use of fast Fourier transform. To demonstrate the application of this,a vortex/airfoil interaction problem is discussed, which has many practical applications and involves important issues such as vortex shedding from the trailing edge.