An investigation of Kirchhoff's methodologies for computational aeroacoustics

This paper presents a comprehensive and comparative study of the various moving Kirchhoff formulations, including the recent S-Surface Kirchhoff formulation by employing a simple moving source test case for which an analytical solution exists. The formulations are tested in a wide range of Mach numbers in order to assess the accuracy of the prediction of the acoustic signal as well as their robustness, accuracy and feasibility. Special care is given for Mach numbers in the vicinity of the sonic point, where under certain circumstances a Doppler singularity can occur. Introduction In recent years, Computational Aeroacoustics (CAA) has emerged as a new discipline, dealing with the numerical calculation of the aerodynamically induced sound. The numerical prediction tools used in modern computational aeroacoustics work in conjunction with aerodynamic data provided by CFD solvers. The advances in CFD technology during the two last decades, made CFD solutions accurate, robust and reliable. However, since in aeroacoustics we are mainly interested in the acoustic wave propagation in the linear far field, CFD codes cannot be realistically utilized for acoustic predictions. Navier-Stokes as well as Euler solvers become expensive tools and the memory and CPU requirements of extending the computational grid in distances several length scales away from the source region, are prohibiting with current computational resources. However, the combination of a reliable CFD near field solution and an acoustic methodology for the the extension of the CFD results to the linear far-field is becoming a very popular method today. Kirchhoff *AIAA paper 97-0487 presented at the 35th Aerospace Science Meeting and Exhibit, Reno, January 1997 ^Graduate Research Assistant, Student Member AIAA * Associate Professor, Associate Fellow AIAA Copyright ©1997 by E.K.Koutsavdia and A.S. Lyrintzis. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. methods fall in this category, where the CFD solver provides aerodynamic data of the non-linear near field on a specified stationary or non-stationary surface. These data, mainly pressure and temporal pressure derivatives are integrated providing acoustical information for multiple observers in the far field. Kirchhoff methods are integral solutions of the linear wave equation in the unbounded space and they have been applied in several areas; helicopter noise, fan noise propeller noise are some of them. A review of the various uses of the methods was given by Lyrintzis [1]. Kirchhoff's formula was first published in 1882 and used initialy for light diffraction problems. Morgans [2] in (1930) derived a Kirchhoff formula for a moving surface using Green's function approach. Farassat and Myers [3] (1988) published a retarded time Kirchhoff formulation for surfaces in arbitrary subsonic motion using generalized function theory. Recently, (1995) Farassat and Myers [4] derived a new acoustic planform Kirchhoff formulation for moving surfaces that allows supersonic motion. The major limitation of the retarded time formulation [3] is that restricts the surface motion to subsonic speeds due to the presence of a Doppler singularity. The new acoustic planform Kirchhoff formulation can utilize the CFD input provided on a Kirchhoff surface but the integration is carried out on the acoustic platform, which is the locus of the points of intersection of the Kirchhoff surface with the collapsing sphere which has as center the observer position. By performing the integration on the £ surface we avoid the Doppler singularity. The formula requires a great deal of aerodynamic data processing as well as advanced geometry handling. Due to possible singularities, the Kirchhoff surface should have a special shape for supersonic cases. The main motivation behind the derivation of the acoustic planform formulation is the fact that for high speed rotors, the flow around the tip becomes delocalized. Delocalization is the phenomenon in which pockets of non-linear flow are extended beyond the tip of the blade towards the sonic cylinder. That, from the computational point of view, means that one has to extend the Kirchhoff surface away from the tip in order to include all those non-linearities. When the tip Mach number is high enough, portions of the Kirchhoff surface may move with nearly sonic or even supersonic speeds. The presence of the Doppler singularity in the retarded time formulation does not allow for the projection of the Mach number in the radiation direction to become one and thus, this formulation poses significant limitations for those high speed cases. A possible supersonic application of the acoustic planform formula is the evaluation of the impulsive noise in the propeller noise of tilt-rotor aircraft, where the tip Mach number is in the high transonic regime. It could also be applied to High-Speed Civil Transport (HSCT) aeroacoustic predictions. The following relation transforms the integrands between the various domains [6] dS cdrdT A |l-A/r| sine (6) As we can see from the first of the above formulations, the term (1 — AfP) can produce a singularity in the case where the Mach number in the radiation direction reaches the sonic point. That is a major limitation of the retarded time formulation. The collapsing sphere formulation also contains a singularity the sin & = 0, which however can be overcome [5] . In the E surface integration, the singularity A = 0 is harder to occur and can be avoided by specially constructing the / = 0 surface. More details of the above formulations can be found hi [5]. Various Formulations for the Acoustic Integrals For the purpose of illustrating the key differences between the various forms of the acoustic integrals, we briefly consider the solution to the following inhomogeneous wave equation [5] A retarded time formulation of the solution of the above equation can be written as ',«) = .-Mr\ }T.dS (2) where dS is the surface element of the physical Kirchhoff Surface, y denotes the source location and Mr = M • r, the radiation Mach number. The same solution written in a collapsing sphere formulation is = /' / •/—oo Jf=0,g=0 Q(y,T) rsinfl cdTdr (3) where cosfl = n • r. As a collapsing sphere, we define the sphere with center at the observer point and radius c(t T). Finally, the E surface form of the solution is (4) F=O r In the above formulation, / = 0 is the physical source surface, F = 0 is the acoustic planform or E surface, mathematically defined as F = f ( y , t r/c), that is the locus of the points from which the acoustic signal, that arrives at the observer at time t, was emitted. The F curve is the curve that is produced from the intersection of the collapsing sphere with the / = 0 surface. Finally we define the quantity A as (5) Retarded Time and Acoustic Planform Kirchhoff Formulations The retarded time and the acoustic planform Kirchhoff formulations are the integral representations of the solution to the wave equation. In the retarded time Kirchhoff formulation of Farassat and Myers [3] , the integration is carried out on the original Kirchhoff surface. The formulation requires that pressure and its spatial and temporal derivatives are known on the original Kirchhoff surface for an observer at a point (x, t) the acoustic pressure is where the E\ and E% are [3] (7) -[(nP Mn nM)p + (cos 9 Afn)p]