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Inverse Problems in Unsteady Aerodynamics and Aeroacoustics

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Inverse Problems in Unsteady Aerodynamics and Aeroacoustics INVERSE PROBLEMS IN UNSTEADY AERODYNAMICS AND AEROACOUSTICS y z Sheryl M. Patrick and Ha z M. Atassi Aerospace and Mechanical Engineering Department University of Notre Dame Notre Dame, Indiana and x William K. Blake David Taylor Mo del Basin C.D./N.S.W.C. Bethesda, Maryland ABSTRACT duce uctuating pressure forces on the structural com- p onents which leads to vibration and noise. The feasibility of determining an unknown vortical disturbance in an approach ow to a streamlined solid In order to either reduce or eliminate suchunwanted b o dy from the radiated sound is studied. The problem e ects, the aero dynamics and aeroacoustics of the prob- is treated in two separate parts. First, the inverse aero- lem need to be understo o d. This entails determining dynamic problem, wherein the upstream disturbance which are the key parameters resulting in ecient or is determined from unsteady surface pressure, is con- nonecient sources of vibration and noise and optimiz- sidered. It is shown that the transverse comp onentof ing them for vibration and noise control. an incoming vortical disturbance can b e uniquely pre- To this end, the study of the aero dynamic prob- dicted from the surface unsteady pressure. This result lem asso ciated with impinging ow nonuniformities on can b e extended for incoming turbulence in terms of the a streamlined b o dy such as wings or blades, known as surface pressure density sp ectrum. The second problem the gust problem, has b een the fo cus of considerable in- treated is the inverse acoustic problem, wherein the un- terest in unsteady aero dynamics. The direct gust prob- steady surface pressure is determined in terms of the lem is to calculate the unsteady pressure distribution radiated sound. It is shown that for a single frequency, along the airfoil/blade surface knowing the upstream the governing equation reduces to the Helmholtz equa- disturbances and the shap e of the b o dy. The rst treat- tion, and the solution to the inverse acoustic problem is ment of this problem in incompressible owwas given unique. However, the problem remains ill-p osed and ex- by Sears 1941. A recent review on the sub ject was hibits extreme sensitivity to noise in the far- eld data. given byAtassi 1994. The singular value decomp osition metho d and a simple Another e ect asso ciated with the interaction of a regularization metho d are used to solve the discretized nonuniform ow and a solid body is the radiation of system of equations resulting from the inverse acoustic acoustic sound. Atlow Machnumb er, the direct acous- problem. The simple regularization metho d is, however, tic radiation problem can be treated using Lighthill's more accurate. acoustic analogy Lighthill, 1952 and the Ffowcs Wil- liams-Hawkings equations Ffowcs Williams and Hawk- INTRODUCTION ings, 1969. In this treatment the unsteady pressure Aircraft wings and propulsive system structural com- along the airfoil/blade surface is equivalent to a dip ole p onents often op erate in nonuniform ow conditions. distribution. More recent analyses by Amiet 1976 and The nonuniformities arise from atmospheric and inlet Atassi et al. 1990 derive the far- eld sound directly turbulence, viscous wakes, secondary ows, and rotor/ from the unsteady aero dynamic solutions. These treat- stator interaction. Upstream ow nonuniformities pro- ments consider dip ole and quadrup ole e ects. The lat- ter b ecome signi cant as the mean- ow Machnumber y Fellow, Center for Applied Mathematics and the reduced frequency increase and as the mean z Professor airfoil/blade loading increases Atassi et al., 1993. x Research Scientist 1 ularization scheme. The Tikhonovscheme can b e used Another imp ortant asp ect of active control of vibra- alone to obtain a solution or in conjunction with the tion and noise is sensing and analyzing data to predict single value decomp osition to alleviate the dep endence their sources. This is the fo cus of the work presented on very small singular values. in this pap er. Here, the feasibility of inverse aero dy- namic and aeroacoustic problems is considered. The The inverse problem discussed in this pap er, dif- inverse problem allows for new metho ds of determining fers from the inverse scattering problem investigated the origins of unwanted vibration and noise. For the by those mentioned ab ove. Here the incident eld is aero dynamic problem, the pap er studies the feasibil- unknown, although it is assumed to b e either vortical ity of determining the upstream disturbances from the or acoustic. The shap e of the b o dy is known, however uctuating unsteady pressure on a moving streamlined the unsteady pressure along the b o dy is not given. De- body. The aeroacoustic problem concerns the feasibil- termining the b oundary condition on the b o dy, i.e. the ity of determining the surface unsteady pressure from unsteady pressure, from far- eld measurements will b e the radiated far eld. Ultimately b oth inverse problems referred to as the inverse acoustic problem, while de- can b e combined to analyze the upstream disturbances termining the incident disturbance from the b oundary from the radiated sound. conditions on the b o dy will b e termed the inverse gust problem. In spite of the imp ortance of these inverse problems to applications and, in particular, to control and sens- The problem of determining unknown b oundary con- ing, they have received much less attention than the ditions for a given ob ject from far- eld measurements direct problem. Like most inverse problems, these are has b een extensively investigated in conjunction with ill-p osed in the Hadamard sense; that is, a problem is acoustic holographyVeronesi and Maynard, 1989, Kim ill-p osed if one of the following criteria fails: a solution and Lee, 1990, Sarkissian et al., 1993. These problems exists, it is unique, it depends continuously on the data di er from the present problem in that they do not con- Hadamard, 1923. These issues will be considered in sider a mean owvelo city. regard to the present application. Rayleigh, in 1896, showed that a source of sound THE DIRECT PROBLEM could not be determined uniquely from the far- eld In order to gain a fundamental understanding of is- sound Rayleigh, 1896. This statement is true for a sues involved with our inverse problems, we restrict our- general sound source whose propagation is governed by selves to a at-plate airfoil in subsonic ow with vortical the wave equation. For sound which is pro duced by disturbances imp osed upstream. For the at-plate air- a disturbance which is harmonic in time, however, the foil, the mean ow remains uniform. The formulation governing equation reduces to the Helmholtz equation for the direct problem of a at-plate airfoil in a subsonic and inverse solutions to the Helmholtz equation may ow with convected vortical gusts is given in Atassi et b e unique. The inversion of problems governed by the al., 1990, where the unsteady pressure jump across the Helmholtz equation has b een studied quite extensively at plate is calculated using an integral equation and in recent years. This recent interest can be traced to then the scattered eld is calculated using Green's the- an article by Kac 1966 entitled, \Can One Hear the orem. The formulation is summarized here for clarity Shap e of a Drum?". The emphasis in inverse acous- and for reference. tic scattering problems has b een on surface reconstruc- tion. Many researchers have attempted to use the far- The Gust Direct Problem eld acoustics, pro duced by the interaction of a known Assume the owisinviscid, and non-heat conduct- acoustic wave and an unknown body, to reconstruct the ing. When the upstream disturbance is small, i.e. the b o dy surface This research has b een motivated by ap- owisweakly rotational, the velo city, pressure and den- plications in tomography, geophysics, optics and sonar. sity can b e linearized ab out their mean values Atassi, It has led to the establishmentofseveral theorems p er- 1994. taining to the existence and uniqueness of solutions to b oth direct and inverse scattering problems governed by the Helmholtz equation. Researchers studying inverse ~ ~ acoustic scattering have proven that a radiating solu- U ~x; t = U + ~u~x; t 1 1 tion to the Helmholtz equation can b e obtained uniquely 0 p~x; t = p ~x+p ~x; t 2 from the far- eld data, provided a radiating solution ex- 0 ists Colton and Kress, 1992. 0 ~x; t = ~x+ ~x; t 3 0 Even though the theorems guarantee uniqueness when a solution exists, the diculty remains in prac- In the absence of upstream incident acoustic waves, tice to nd this solution. In many cases, the formulation the upstream ow can b e written as leads to a Fredholm integral equation of the rst kind. To solve this integral equation, usually, one has to invert ~ ~ ~ an ill-conditioned matrix. Hence, existing metho ds for U ~x; t=U i + ~u ~x i U t 4 1 1 1 1 1 1 solving very ill-conditioned matrices have b een further where ~u represents the upstream rotational distur- explored and new techniques have b een found Kress, 1 ~ 1989. The most widely used metho d found in the bance in the ow and i is a unit vector. The axis and 1 literature is the singular value decomp osition metho d. ow direction are shown in Figure 1. Another metho d commonly used is the Tikhonov reg- 2 a longitudinal 1 ~ ik tk ~x 1 x ; 0;x =a e for 1 <x < 1: 9 2 u a2 1 3 2 1 k 2 Along the vortex sheet extending downstream from the trailing edge, u is discontinuous, but 1 x a 2 2 0 p = 0 k 4 1 x a 1 u = 0 x > 1 and x =0 10 2 4 a2 1 2 x 3 transverse 0 0 0 where 4p signi es p p .
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