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JIM Jk AIAA 2002-2477 Trailing Edge Noise Prediction Based on a New Acoustic Formulation J. Casper and F. Farassat NASA Langley Research Center Hampton, VA 8th AIAA/CEAS Aeroacoustics Conference and Exhibit June 17-19, 2002 / Breckenridge, CO For permission to copy or republish, contact the copyright owner named on the first page. For AlAA-held copyright, write to AIAA Permissions Department, 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344. AIAA 2002-2477 TRAILING EDGE NOISE PREDICTION BASED ON A NEW ACOUSTIC FORMULATION 1 J. Casper* and F. Farassat 1 NASA Langley Research Center Hampton, VA 23681 Abstract A new analytic result in acoustics called “Formula- tion IB,” proposed by Farassat, is used to compute broadband trailing edge noise from an unsteady surface pressure distribution on a thin airfoil in the time do- main. This formulation is a new solution of the Ffowcs Williams-Hawkings equation with the loading source term, and has been shown in previous research to pro- vide time domain predictions of broadband noise that are in excellent agreement with experiment. Furthermore, this formulation lends itself readily to rotating reference frames and statistical analysis of broadband trailing edge noise. Formulation IB is used to calculate the far field noise radiated from the trailing edge of a NACA 0012 airfoil in low Mach number flows, using both analyti- cal and experimental data on the airfoil surface. The results are compared to analytical results and experi- mental measurements that are available in the literature. Good agreement between predictions and measurements is obtained. Nomenclature b = airfoil semi-span (m) C = airfoil chord (m) Co = ambient sound speed (m/sec) / = frequency (Hz) / = geometry function for airfoil surface (Fig. 1) E* — combination of Fresnel integrals (Eq. (2c)) g = surface pressure transfer function k c = u>/U c , convective wave number (m _1 ) £2 = spanwise correlation length (m) M = V/co, Mach number vector M r = M ■ rjr Mach number in radiation direction M v = M ■ v Mach number in direction of v AP = unsteady airfoil surface pressure jump (Pa) p = unsteady airfoil surface pressure (Pa) p' = sound pressure radiated to observer (Pa) dp/ds = surface pressure gradient in the direction of V qo = poll 2 /2, dynamic head (kg-m/s 2 ) Copyright © 2002 by the American Institute of Aero- nautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Gov- ernment has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental Purposes. All other rights are reserved by the copyright owner. 1 This manuscript has been revised since its original pre- sentation. Last revision July 19, 2002. * Research Scientist, Computational Modeling and Simula- tion Branch, AIAA Senior Member 1 Senior Research Scientist, Aeroacoustics Branch, AIAA Assoc. Fellow r = x — y. sound radiation vector (m) S qq = surface pressure correlation function T =1//, acoustic period (sec) t = observer time (sec) U = uniform freestream speed (m/sec) u = unsteady streamwise velocity (m/sec) V = airfoil velocity vector x = [aq, X 2 , * 3 ] t , observer position S = [l/i 1 y~‘ 0]L surface source position 0 = v/1 -M 2 A = Co//, acoustic wave-length (m) /r = Mu/0 2 U ip = directivity angle (Fig. 4) v = unit inward facing normal on surface edge (Fig. 1) 9 = angle between surface normal and f (Fig. 1) pa = ambient density (kg/m 3 ) t = t — r/co , source time (sec) = power spectral density of surface pressure q> = random phase variable (radians) uj = 2nf, circular frequency (radians/sec) Subscripts 1,2,3 = Cartesian coordinate directions (Fig. 2) ret = evaluated at source time r 1. Introduction Trailing edge (TE) noise has been the subject of ex- tensive research within the aeroacoustic community for decades, both experimentally and analytically. Areas of current research include the prediction of TE noise from rotating machinery and airframes. Research in the area of TE noise prediction has, in large part, been mo- tivated by the desire to incorporate the results of TE noise analysis into a design methodology. The present work is similarly motivated, and the resulting formula- tion should lend itself well to an engineering design tool suite when aeroacoustics plays a role in the design. The literature abounds with various theoretical ap- proaches to the prediction of TE noise. Howe 1 catego- rized the various theories of TE noise into three groups: (i) Theories based on the Lighthill 2 acoustic analogy, e.g., Ffowcs Williams and Hall 3 . (n) Theories based on the solution of special problems approximated by the linearized hydrodynamics equa- tions, e.g., Amiet 4 ’ 3 and Goldstein 6 . (iii) Ad hoc models, involving postulated source dis- tributions whose strengths and types are empirically de- termined. The present work falls into the first category. This 1 American Institute of Aeronautics and Astronautics new result, “Formulation IB,” is a solution of the loading source term of the Ffowcs Williams-Hawkings equation.' Such time domain methods provide for a total decou- pling of the acoustic signal from the aerodynamics. As such, these methods readily avail themselves to acoustic predictions that are based on input from experimental measurements or computational fluid dynamics (CFD) solutions. For example, Singer et alf used a solution of the Ffowcs Williams-Hawkings equation to predict TE noise from sources that were modeled with CFD simula- tions. The acoustic formulation in their work 8 is known as “Formulation 1A.” 9 What distinguishes Formulation IB from this prior formulation is its relative simplicity, which makes it highly suitable for rotational reference frames and statistical analysis of broadband TE noise. In Ref. 10, Formulation IB was applied to the pre- diction of far field noise due to incident turbulence on a NACA 0012 airfoil at tunnel speeds ranging from 40 m/s to 165 m/s, and compared to the experimental results of Paterson and Amiet. 11 The time dependent surface pressure required as input to Formulation IB was gener- ated by stochastic modeling of the incident turbulence and approximation of the airfoil response with a re- sult from thin airfoil theory. Formulation IB was then used to predict the acoustic pressure as a function in time at a prescribed microphone location. The time do- main results were then Fourier analyzed to determine the spectral density of the far field noise. The far field spectra were found to be in excellent agreement with the frequency domain predictions and experimental mea- surements of Paterson and Amiet 11 . The time domain approach that is described in Ref. 10 is used in the current work to predict far field radia- tion from the trailing edge of an airfoil. In the following section, Formulation IB is briefly reviewed for the case of a flat surface in a general non-uniform motion. (For a formal derivation, see Ref. 10.) Some advantages of this new formulation relative to other solutions of the Ffowcs Williams-Hawkings equation are described. In Section 3, a model problem is considered in which an unsteady surface pressure that is comprised of a single frequency induces an acoustic source at the trailing edge of a flat plate in uniform motion. The unsteady surface pressure is an analytical result from thin airfoil theory that is taken from the work of Amiet. 4,5 * 12 Two simple test cases are presented for validation purposes. The directivity of the tone induced by this surface pressure is examined for qualitative correctness. The results of a velocity scaling exercise are shown to be consistent with the results of Ffowcs Williams and Hall. 3 In Section 4, the surface pressure formulation intro- duced in Section 3 is used as the basis function of a linear superposition that provides an analytic source model for broadband TE noise. This stochastically modeled surface pressure is used as input to Formulation IB to predict broadband TE noise from a NACA 0012 airfoil in a low-turbulence uniform mean flow. The surface pres- sure correlations that are required in the aerodynamic model are taken from two sources: an empirical flat plate formulation 13,14 and experimental data. 15 The resulting calculations are compared to the acoustic predictions of Schlinker and Amiet 16 and the experimental measure- ments of Brooks and Hodgson. 17 2. Acoustic Formulation Consider a flat, finite surface moving in the plane Xu = 0 along a velocity vector V . The velocity vector and the plate’s geometry are related to the coordinate axes as pictured in Fig. 1. Let f(xi,X2,t) denote a geometric function that is so defined that / = 0 on the surface edge i and / > 0 on the interior of the surface. Let i> = V / denote the unit inward geodesic normal that lies in the plane of the surface. Let x = [xi,X 2 ,xs] t denote the po- sition of an observer, and by y = [yi,y 2 , 0] T the position of a source point on the plate’s surface. The unsteady perturbation pressure p(y, r) on the surface gives rise to sound that radiates along r = x—y to the observer. This sound is described by p'(x, t), the perturbation pressure that arrives at the point (xi,X2,X3) at time t. The derivation of Formulation IB can be found in Ref. 10. However, for derivation purposes, both x and y frames of reference are considered fixed relative to the medium at rest. The resulting formulation contains a time derivative p that is evaluated relative to an ob- server that is fixed with respect to the medium at rest, e.g., as measured by a transducer just above the surface that remains stationary as the surface passes by it. This quantity p can be related to dp/dr, the time derivative of pressure in the reference frame of the moving surface, e.g., as measured by a transducer attached to the surface. This relation is where dp/ ds is the gradient of p in the direction of V, and V is the local magnitude of V. Here, s is in the direction of the velocity V of the surface in the reference frame fixed to the undisturbed medium. The final expression for the sound radiated to the observer is = j + L -L ( dp /dr - V dp/ds ) cos 8 cor (1 p cos 9 /> o [r 2 {\-M r ) Mr p cos 9 r ( 1 — M r ) J - M r ) dS ret dl. dS ( 1 ) where co is the ambient sound speed, r is the magnitude of the radiation direction vector f from a point on the surface to the observer, M r is the Mach number in the direction of r, M„ is the Mach number in the direction of the inward-facing geodesic normal 0, and 9 is the an- gle subtended by the surface normal and the radiation vector r (See Fig. 1.). The subscript “ret” denotes eval- uation at retarded time r = t — r/c o- This is the source time at which a surface pressure fluctuation at the point (y i ,3/2,0) made its contribution to the signal detected by the observer at time t. Note that y , r, and 9 are pictured in observer time in Fig. 1. For a far field observer in a low Mach number flow, the first integral in Eq. (1) dominates the acoustic signal. 2 American Institute of Aeronautics and Astronautics This is because the second integral in Bq. (1) is propor- tional to 1/r 2 and the third integral is proportional to M. Perhaps most significant in regard to the form of Eq. (1) is that it is valid, as is, for rotating surfaces. Its predeces- sor, Formulation lAf is significantly more complicated in its rotational form, and cannot be approximated by only one surface integral in the far field for low Mach number flows, as can the present formulation. Such a significant simplification for far field calculations makes Formulation IB more suitable for statistical analysis of broadband noise for rotating surfaces. A statistical for- mulation based on Eq. (1) was derived in Ref. 10. 3. Model Problem - Trailin g Edge Tone Any noise prediction made with Eq. (1) will be only as good as the input surface pressure p(y , r). The cur- rent thinking is that such time-dependent pressure data would result from experimental measurement or a com- putational fluid dynamics (CFD) calculation. However, in this section, a simplified analytic expression is used for p(y, t ) to serve as a model problem. A result from thin airfoil theory 4 will be used to describe the unsteady sur- face pressure that is produced by the passage of a single frequency disturbance past the trailing edge of a slender airfoil. This simple surface pressure formulation will be extended to a broadband source model in the following section. 3.1 Surface Pressure from Thin Airfoil Theor y The airfoil for this model problem is a rectangular flat plate in the plane £3 = 0, undergoing a uniform rectilinear motion, as in Fig. 2. The velocity vector V = [— U, 0,0] r , where U is a constant subsonic speed. The plate’s surface and its boundary, f > 0, are defined by the rectangle { — C < x\ <0} x { — b < xz < 6}, with the trailing edge at £i = 0. An unsteady pressure distribution is assumed on this surface, and is analyt- ically prescribed from thin airfoil theory, as discussed below. Amiet 4 has proposed a formulation to model the re- sponse of an airfoil to the passage of a pressure distur- bance over its trailing edge. This formulation, formally derived in Ref. 12, is based on the theory of a thin airfoil of infinite span and models the moving disturbance as stationary in the variable Xi — U c t, where U c is the con- vection speed of the disturbance. The induced pressure jump on the airfoil surface can be written AP(ai,t) = 2Pog(xi,kc)e~ rkc( ' xl ~ v ‘ e> (2a) where k c = oj/U c is the streamwise convective wave num- ber, and Po is the amplitude of the disturbance. The factor of two in Eq. (2a) indicates that the pressure is assumed to be antisymmetric between the upper and lower surfaces, and this expression thereby accounts for the pressure on both sides of the airfoil, i.e., the pres- sure jump. Note that Eq. (2a) differs from the general form for the pressure jump in Ref. 10 because the explicit term e ~ lkcX1 [ n Eq. (2a) was incorporated into the trans- fer function g in Ref. 10. The formulation in Eq. (2a) is used here for consistency with the TE noise research of Schlinker and Amiet. 16 The transfer function g(xi,k c ) is g(xi,k c ) = -1 + (1 + i)E~[— xi( k c +//(1 + A/))] (2b) where p = Mu>//3 2 U, = \/1 — A/ 2 , and the function E* is given by E*(0 = r —— du = C(0 - iS(0 (2c) Jo (2 tvu)2 The quantities C(£) and 5(f) are the Fresnel cosine and sine integrals, and will be evaluated numerically by the formulas derived by Boersma. 18 The final representation for the unsteady surface pressure p{yi, r), assumed to be a real quantity, is P(y i, t) = »{— AP(i/i, r)} (2d) The pressure jump is negative in Eq. (2d) because the acoustic formulation in Eq. (1) is derived from a form of the Ffowcs Williams-Hawkings equation in which the unit surface normal <33 is assumed to point into the fluid, i.e., in the positive *3 direction on the upper surface, and in the negative *3 direction on the lower surface. There- fore, using the same positive surface normal on both sides of the airfoil, the sum of the pressure on both sides is P = Pupper — Power, and this expression is the negative of the conventional notion of a pressure jump. Note that the transfer function in Eq. (2b) represents the effect of the induced surface pressure only, and ne- glects the effect of the incident pressure. The neglect of the incident pressure field effect is not of concern here, as this model problem is presented for illustrative purposes only. After the initial derivation of this induced pres- sure formulation 4,12 Amiet later altered the formulation to include the effect of the incident pressure field.' 7 The effects of both induced and incident surface pressure will be employed in the broadband formulation in Section 4. 3.2 Directivity Calculation Using Eqs. (2a)-(2d) as the input surface pressure in Eq. (1), the directivity of a single frequency source is now examined. The flat plate has a chord length C = 0.5 meter, and a span 26 = 2.0 meters. The flow speed U is determined by a free stream Mach number M = 0.2, with co = 343 m/s. The disturbance amplitude Po is taken as one percent of the dynamic head qo = pou / 2, with po = 1.23 kg-m/s , and the convection speed is taken to be U c = 0.8 U. The initial surface pressure p(x i,0), — C < xi < 0, is shown in Fig. 3. This pressure profile represents the surface pressure over the entire span at observer time t = 0. Note, again, that the formulation in Eqs. (2a)-(2d) represents the induced surface pressure only. The radiated noise p'(x, t) is calculated at 360 equally spaced locations on a circular arc in the plane £2 = 0. The radius of this arc emanates from the mid-span lo- cation on the trailing edge, as shown in Fig. 4. The arc trajectory (r, «6) is determined by r = 2 meters and 0 < V 7 < 27T. The surface discretization is a uniform grid of 100 x 400 surface elements. The directivity is deter- mined by the peak pressure amplitude calculated at each 3 American Institute of Aeronautics and Astronautics position, during one period T = 1// for a frequency of 2.5 kHz, with 128 timesteps in a period. Fig. 5 shows the results, in polar form, where the notation ||p'|| is interpreted as The results are Fourier analyzed and compared with ex- perimental TE noise spectra. 1 ' 4.1 Ex p eriment Descri p tion lb'll = 0 “ t ^ r b'OM)l The upstream directivity of the major lobes is consistent with the research of previous authors, e.g., Singer, et al [ 8 ], 3.3 Velocit y Scalin g Law Attention is now turned to the way in which the in- tensity of the far field noise, as predicted by Eq. (1), will scale as a function of velocity, when the surface pressure is described by Eqs. (2a)-(2d). A scaling law' will be de- termined under the assumption that the acoustic source is noncompact, i.e. , A < C. Furthermore, the observer is assumed to be in the acoustic and geometric far field, i.e., r A and r ~S> C, respectivey. Because scaling laws are typically determined for low Mach number flow's, 3 ’ 19 the Mach number range of inter- est is 0.01 < M < 0.2. The surface pressure amplitude Po is one percent of the dynamic head. The plate’s phys- ical dimensions are the same as in the above directivity- problem. The observer is chosen at a distance of 10 me- ters, directly above the trailing edge, i.e., x = [0,0, 10] 7 in meters. The calculations are performed on a 100 x 400 uniform surface grid. The surface pressure in Eqs. (2a)-(2d), with a fre- quency of 2.5 kHz, is used as input to equation to Eq. (1) to predict the far field sound p'(x,t) to the observer. A separate calculation is run for each of 50 equally spaced Mach numbers between 0.01 and 0.2. Each calculation is performed for one period with 128 timesteps. The aver- age intensity I(x) of the acoustic signal at the observer x, assuming spherical spreading, is then calculated by [p'{x, £)] 2 dt poco The average acoustic intensities for this test case, as a function of Mach number, are represented as circles in Fig. 6. The slope of these results on a log-log plot can be visually determined by observing their proximity to the dotted line w'hose slope is exactly five. This U° proportionality is consistent with the result of Ffowcs Williams and Hall, 3 as expected from the idealized con- ditions placed upon the calculations. 4. Broadband Predictions The analytic surface pressure in the previous section is extended to model a broadband trailing edge source on a slender airfoil at zero angle of attack. Following the approach of Schlinker and Amiet., 16 the surface pres- sure correlations required as input are evaluated by flat plate theory and by experimental measurements. This broadband surface pressure is used as input to Formula- tion IB to predict far field radiation in the time domain. The experiment that is modeled in this section is re- ported by Brooks and Hodgson. 1 ' A NACA 0012 airfoil is placed between two plates at zero angle of attack in the test section of an open jet wind tunnel. A schematic of this experimental setup is shown in Fig. 7. Noise prop- agates from the test section into an anechoic chamber that is instrumented with microphones. The airfoil has a chord length of 0.6096 m and a span of 0.46 m. The tunnel speeds of interest here are 38.6 m/s and 69.5 m/s. The chord-based Reynolds numbers are 1.57 million and 2.82 million, respectively. Boundary layer tripping was applied at 15 percent chord downstream of the leading edge to ensure a spanwise uni- form transition location and a fully developed turbulent boundary layer at the trailing edge. For radiated noise measurements, eight microphones are located in the plane perpendicular to the airfoil midspan. The presence of extraneous noise sources pre- cludes direct mearurement of TE noise by a single mi- crophone. Therefore, to evaluate the TE noise, a cross- spectral analysis of pairs of microphones w'as employed in a manner consistent with the coherent output power method. 20,21 The microphone pictured in Fig. 7 repre- sents the location for which the current predictions are made, at a distance of 1.22 m directly above the airfoil trailing edge. Note that a shear layer forms dowmstream of the nozzle lip, between the airfoil and the microphone. Although both the directivity and the amplitude of the TE noise are affected by refraction through this shear layer, the corrections for the microphone at this location are small enough to ignore (see Ref. 17). 4.2 Broadband Anal y sis For prediction purposes, the airfoil is modeled as a flat plate in order to evaluate the unsteady surface pres- sure w'it.h a broadband extension of the analytic formu- lation in Section 3. The airfoil geometry is oriented with respect to the coordinate axes as in Fig. 2, with { — C < *i < 0 } x { — b < X 2 < b}, where C = 0.6096 m and 2b = 0.46 m. The surface pressure arises from boundary layer tur- bulence that is assumed to convect in a frozen pattern along the airfoil surface towards the trailing edge. Unlike the single frequency source in Section 3, the broadband nature of the surface pressure in the present case requires consideration of both chordw-ise and spanw'ise wave num- bers, k i and fe, respectively. Each Fourier component of this broadband surface pressure jump is associated with a wave number pair (k\, k-i) and can be written AF(/ci,fe; n,X2,l) = (3) 2P(k u k2)g(x 1 M,k2)e- l[kl{xi - Uct)+k * X2] where P(ki, fe) is the amplitude of the pressure jump as- sociated with the wave number pair (ki,ko)- Because of 4 American Institute of Aeronautics and Astronautics the assumed convective nature of the turbulence, the no- tation for the chordwise wave number fo will be replaced by fo to emphasize its dependence on the convection speed U c and to avoid confusion with the conventional notion of fo = ui/U. In an exact sense, there are in- finitely many combinations of frequency and convection speed whose ratio uj/U c yields a given value of fo . How- ever, it is assumed here that the acoustically relevant structures in the turbulent boundary layer are frozen with respect to a single convection speed that is taken as U c = 0.8 U. The complete broadband spectrum for the surface pressure jump AP(xi, x 2 ,t) is obtained by summing all Fourier components in Eq. (3): AP(xi,X 2 ,t)= (4) poo poo 2 / / P(kc,k 2 )9(x 1 ,k c ,k2)e- ilk ^ 1 - Uct)+k2X2] dkcd^ J — ooJ—oo A straightforward approach for predicting the desired broadband far field measurements is to use the real part of Eq. (4) as input to Formulation IB. This approach requires knowledge of a two-component surface pres- sure spectrum and a dual wave number transfer function g(xi, k € , fo). However, because one of the objectives of the current work is to reproduce the results of Schlinker and Amiet 16 from a time domain perspective, an ap- proach similar to that taken in Ref. 16 will be used to model the surface pressure. The analysis for the general formulation in Ref. 16 comes from previous work 22 in which Amiet argues that, within certain limitations, integration over all spanwise wave numbers is not required. His conclusion, derived mathematically in the frequency domain, is that only one spanwise wave number contributes to the sound de- tected by an observer in a given location. In particular, Amiet focuses on an observer in a spanwise symmetric location, for which only the zero spanwise wave num- ber needs to be considered. This result is argued to be exact in the limit of infinite span and a good approx- imation for an airfoil of finite span that responds to a high frequency disturbance. Although Amiet.’s analysis was initially presented to derive an acoustic formulation for incidenct turbulence noise, the result pertaining to spanwise wave numbers is sufficiently general to apply to the present trailing edge problem. The derivation of Amiet’s analytical result can be gen- erally described as follows. First, Eq. (4) is transformed into Fourier space. Then, a two-point cross-correlation function is formed and related to the far field power spec- trum through Kirchhoff ’s formula 23 and Curie’s result.. 19 In order to follow a similar line of reasoning in the time domain, Eq. (4) itself must be related to the far field acoustic pressure through Formulation IB. In the case of a distant observer directly overhead of a finite-span airfoil, the terms r, M r , and 9 in Eq. (1) are weak functions of y\ and y 2 on the airfoil surface, and there- fore will be considered constants. Furthermore, for the observer position considered here, the differences in re- tarded time, as a function of airfoil surface location, can be neglected. These assumptions are consistent with the acoustic model employed by Amiet. 22 For the present problem, including the above assumptions, Eq. (1) is ap- proximated by 4 np(x,t)~ (5) cos# Cor (l -M r ) a b L §=p(y,f) + u-^-p(y,f) dy where the over-bars on 9, r, and M r denote mean values over the airfoil surface, and therefore the retarded time t = t — f/co is constant for fixed t. Recall that only the first integral in Eq. (1) is significant under the present assumptions of a far field observer in a low Mach number flow. Before the surface pressure p(y, t) is specified, Eq. (5) is further simplified. For convenience, the terms 9 and M r will be neglected, as they are small (M r ss 0 and cos 9 ~ 1) for a distant observer directly above the airfoil. With these additional simplifications, if — AP in Eq. (4) is substituted for p(y, t) in Eq. (5), the far field acoustic pressure can be approximated in the form 2 f'C r-b poo poo 4np’(x,t )& — z / / / / F(yi,k c , ki) c 'o r J 0 J_ b J_ oa J_ ao x e -'^-Ucf)+k 2 y 2 ] dk2dkcdy2dyi ( 6a ) where T{y\ , fo, fo) = (6b) -P(k c . fo) d ik c (U c - U)g(yi,k c ,k 2 )+U—g(yi,k c ,k 2 ) Sufficient conditions 24 on AP and its derivatives have been assumed for the commutation of integration and differentiation in Eqs. (6a) and 6(b). The y 2 integration in Eq. (6a) can now be explicitly evaluated, yielding 4np(x, t) : co o „ . ’ J o J — OG J — O 2 sin (fob) fo T{yi, fo, fo) x e- ikAyi - u ‘ f ) dk 2 dkcdyi (7) Integrating with respect to fo, the term sin(fob) /fo acts like a Dirac delta function when integrating over an un- bounded domain, and the result is 9 /'C poo 4irp'(x, t.) ss — = / / 2nJ r (yi,kc,0)e~' kc(vi ~ UcT) dk c dyi co r J 0 J-oo ( 8 ) Eq. (8) indicates that only the zero spanwise wave number contributes to the noise detected by the far field observer. Eqs. (7) and (8) are time domain analogies to Eqs. (15) and (17) in Ref. 22. Furthermore, Eq. (8) suggests that the acoustic source p(y, r) in Eq. (1) can be evaluated as the real part of a simplified pressure jump: / OO P(k c , 0)g( Xl ,k c ,0)e~ ik ‘ (xi ~ Uct) dk c p(y,r ) = S{-AP(t/i, r)} ( 9 ) 5 American Institute of Aeronautics and Astronautics Note that, when performing the actual calculation, the y-i integration will be explicitly performed when Eq. (9) is input to Eq. (6). Only the k 2 integration will be neglected. Furthermore, the evaluation of the surface pressure terms at retarded time r = t — r/co will be ex- ecuted in an exact fashion, as prescribed by Eq. (1). All three integrals in Eq. (1) will be evaluated for the predic- tions that follow, although the first integral is expected to dominate the signal. The evaluation of the surface pressure in Eq. (9) is accomplished by first recognizing the turbulent fluctu- ations as a stochastic process. This process can be approximated by a truncated series whose limit exhibits the required relationship between the autocorrelation and the power spectrum of that process (e.g., Ref. 25). This relationship is achieved by evaluating the pressure amplitudes P(k c ,k 2 ) as a function of 'hpp, the power spectral density (PSD) of the surface pressure. To this end, the infinite wave number domain, —00 < k c < 00 , in Eq. (9) is integrally discretized and truncated such that k c ,-N < k c , n < k c ,N . The largest convective wave number fe c ,jv represents an “upper cutoff” wave number, beyond which the surface pressure amplitude P(k c , 0) is considered negligible or is out of range of experimental measurement. The unsteady surface pressure jump in Eq. (9) is then approximated by N AP(xi,t) A u ,oe i ' l ’ n g(xi,kc,n,0)e~ ikc n(xi ~ Uct) n=-N (10a) kc,n = n A k c , n = 0, ±1, ±2, . . . , ±N A k c = k c ,N/N The discrete surface pressure amplitudes {.4„,o} are evaluated by A n , 0 = [$pp (&.,„, 0) Afc t ]i (10b) where <&pp(k c ,k2) is the two-component PSD of the sur- face pressure. Amiet 16 argues that the required single wave number spectrum <&p.p(fc t , 0) can be evaluated by $pp(fe, 0) = — 4 2 (w)S<, g (w,0) (10c) 7T where t X2 (w) is the spanwise correlation length and (u,Ax 2 ) is the spanwise surface pressure correlation function. The phase angles {<j>n} are independent random vari- ables uniformly distributed on [0, ’2tv]. The transfer function in Eq. (2b) can be used for g(x\,k c , n , 0) with the following modification. As previously noted, Eqs. (2a) and (2b) represent the induced pressure jump. Amiet. 5 has suggested that the incident pressure, i.e., that which results from turblent eddies that contact the trailing edge, can be accounted for by the addition of an ex- ponential convergence factor of the form e ekcXl , where £ is a positive parameter. For —C < xi < 0, this ad- ditional term will be significantly larger than zero only in the immediate vicinity of the trailing edge, provided that ek c C is large. Therefore, to include the effect of the incident pressure, the transfer function to be used in Eq. (10a) is the two-component function g(x i,k c ,ko) in Ref. 16, with k -2 = 0. g (a i,fc c ,0) = e ekaX1 - 1 + (1 + i)E*[-x\{k c +g.{l+M))\ (lOd) where E* is the same complex combination of Fresnel integrals as in Eq. (2c). Amiet 5 was able to avoid the di- rect use of the parameter e because of the manner in which the transfer function in Eq. (lOd) was used in his analysis. Amiet used the transfer function to de- fine an unsteady lift response function that involved the chordwise integration of the transfer function with other terms. The result of this integration yields an expres- sion that, upon clever manipulation of limits, does not contain e but still provides an additional term to the lift response function that accounts for the incident pres- sure. In the present case, the transfer function in Eq. (lOd) must be explicitly used and therefore a value for t must be specified. This value e = 1.5 is chosen for reasons that are discussed in the following subsection. To illustrate the effect of including this incident pres- sure term in the surface pressure formulation, Fig. 8 shows the same single frequency surface pressure case in Section 3, with and without the the incident pres- sure term. Clearly, the incident pressure term has a significant effect only near the trailing edge, as expected. However, for a given frequency, differing values of e will result in differing amounts of upstream chordlength to be so affected. Note that the addition of this term causes the pressure jump to vanish at the trailing edge for all time, i.e., the Kutta condition is satisfied. Note also t.he increase in spatial oscillation that is caused near the trailing edge when the effect of this incident pressure term is included. Eqs. (10a)-(10d) represent the the complex- valued broadband surface pressure formulation to be used for the present TE noise predictions. The final representa- tion for the unsteady broadband pressure on the airfoil surface is then given by the real part of —A P in Eq. (10a). Using symmetry arguments and algebraic ma- nipulation, the indicial bounds for the surface pressure’s spectral representation are altered so that the domain in- cludes only positive wave numbers. As input to Eq. (1), the resulting real-valued surface pressure can be written p(lJU t) = -47T ^ -4»,o{ B n COs[fc c ,„(f/l — U C T ) + (j>n + D n sin[fc c , n (j/i - U c t ) + <j>„] } (11a) where A, t ,o — — 4(tu„) S qq (uj n , 0) A k c 7 r (11b) Bn = c «*‘.«» 1-1+C(£ b ) +$(£„) (11c) Dn = C(4)-«S(4) (lid) £,n = -yi[kc,n +Mn(l + M)\ (lie) and C(4) and S(£ i „) are the Fresnel cosine and sine inte- grals in Eq. (2c). Specific evaluations for the correlation 6 American Institute of Aeronautics and Astronautics lengths ti and surface pressure correlations S m will be discussed in the following section. 4.3 Time Domain Predictions The lower and upper frequency bounds for both cal- culations are 25 Hz and 10 kHz, respectively. Therefore, / = 25 Hz also serves as the fundamental frequency and the numerical bandwidth A/. Each calculation is per- formed for one period of the lowest frequency, T = 0.04 s. The numerical solution is sampled at the Nyquist fre- quency, i.e. , Af = T/2N. The calculation is performed on a 500 x 100 surface grid with grid-point clustering near the trailing edge, as shown in Fig. 9. This trailing edge clustering allows for better resolution to account for the effect of the incident pressure term in Eq. (lOd), as previously shown in Fig. 8. As in the constant frequency case, because the surface pressure in Eqs. (lla)-(lle) is cast in only one spatial variable y%, and the observer location is symmetric relative to the airfoil span, the acoustic predictions are found to be relatively insensitive to the discretization in y 2 , and the primary concern for grid resolution is in the streamwise direction. With 500 points in the streamwise direction and clustering near the trailing edge, a sufficient resolution of at least 10 points per wavelength was obtained for the entire length of the chord. This conclusion was reached by inspection of surface pressure profiles for the highest frequency of 10 kHz. The coordinate system for the calculation is such that the 212 -axis coincides with the center span line, so that the microphone position is in the plane X 2 = 0. The experimental microphone position for which comparisons are made is at a distance of 1.22 m from the model, and at an angle of 90 degrees relative to the chord and directly above the trailing edge. The measured observer position for the prediction is, then, x = [0, 0, 1.22 ] T in meters. The baseline prediction case under consideration is for a tunnel speed of U = 69.5 m/s. This flow condition, the above observer location, and airfoil geometry are incor- porated into an acoustic prediction using Eq. (1) with the surface pressure defined by Eqs. (lla)-(lle). Following the example of Schlinker and Amiet, 16 a first-cut pre- diction is performed using flat plate theory to evaluate the surface pressure correlations S m (ca,0) and spanwise corelation lengths £o(u) in Eq. (lib). The authors used empirical formulations for these quantities that they de- rived from previous analysis and boundary layer mea- surements of Corcos 26 and Willmarth and Roos. 2 ' The surface pressure correlations are approximated by x* 2 x 10" 5 5w ( W, 0) ~ q ° U 1+CC + 0.217th 2 + 0.00562 th 4 (12a) where go = poV 2 /2, 5 * is the trailing edge displacement thickness, and th = a >5*/U. The displacement thickness is also taken from a flat plate approximation for turbu- lent boundary layer thickness 5 on a flat plate, based on the chord Reynolds number Rec, i.e., 5 0.37 c 5 * C C (12b) For the experiment of Brooks and Hodgson, 1 ' Schlinker and Amiet 16 used Eq. (12b) to compute the boundary layer thickness 5 and accounted for boundary layer tripping by taking the 15 percent chord station as the initial point of the calculation. Surface curvature was also accounted for in the downstream distance used in the calculation. The ratio 5/C used by Schlinker and Amiet for this experiment was reported as 0.0166 for U = 69.5 m/s and 0.0187 for U = 38.6 m/s. The displacement thickness was then taken as 1/8 of the boundary layer thickness. The expression that Schlinker and Amiet) 6 suggest for the spanwise correlation length is £a(u) » (12c) U) Fig. 10 shows the far field signal p'(x,t) that is pre- dicted by Formulation IB at the experimental micro- phone location, for a tunnel speed of 69.5 m/s. The surface pressure is modeled with Eqs. (lla)-(lle) and (12a)-(12c). The time signal p'(x,t) is Fourier ana- lyzed to determine a discrete set of spectral amplitudes {Pn}n=i- The far field sound pressure level (SPL) spec- trum is calculated by SPL(/„) = 20 log Pn_' Prrf . 71 = 1,2,..., A (13) where the reference pressure is P re f = 20 yuPa. The SPLs are converted to a 1.0 Hz bandwidth by reducing the values in Eq. (13) by 10 log(A/). The resulting narrowband SPLs are compared with the prediction of Schlinker and Amiet 16 in Fig. 11. Also on this plot are the narrowband SPLs that were ex- perimentally measured by Brooks and Hodgson. 1 ' The predicted results of Schlinker and Amiet and the mea- surements of Brooks and Hodgson were obtained by digitizing the appropriate plots in Figure 34 of Ref. 16. Various values of the paramenter e in Eq. (lOd) were as- sessed in this comparison stage of the research. With an arbitrary parameter in the formulation, the value e = 1.5 was chosen for its agreement with the flat plate correlation results of Schlinker and Amiet. 16 This value of e is held fixed at 1.5 for all remaining calculations. Clearly, Fig. 11 shows that significant error exists be- tween the predictions and the measurements when flat plate formulations are used for the required surface pres- sure correlations. Fig. 12 sheds light on this error with a comparison of the flat plate formula in Eq. (12a) and the measured surface pressure correlations of Yu and Joshi. The notation S m denotes that the surface pressure cor- relations are normalized by go 5* [U. The measured data in Fig. 12 were obtained by digitizing the “average” plot in Figure 35(a) of Ref. 16. The normalized flat plate surface pressure correlations are significantly lower than the measured data, by as much as 7 dB. The reason that the flat plate approximation is so much in error is only in small part because of the the lack of pressure gradient. The most significant error made in the approximation in Eq. (12a) is the lack of a trailing edge; this empir- ical formulation is based on experimental measurement and analysis in which the flat plate is assumed to be infi- nite. Clearly, surface pressure correlations that are based 7 American Institute of Aeronautics and Astronautics on flat plate theory are inappropriate for predicting TE noise in this case. The experimental surface pressure correlations 28 in Fig. 12 will now be used in the surface pressure formu- lation to predict the TE noise associated with the two tunnel speeds of interest and compared with experimen- tal measurements. The modified formula for the surface pressure correlations is S qq (u,Q) « go jj S vq {u, 0) (14) where S M (w.O) denotes the normalized measured data in Fig. 12. The tabulated data obtained from digitiz- ing this information from Ref. 16 is stored in a file that is accessed and interpolated to obtain S qv (io, 0) for any frequency. Having altered the surface pressure correla- tion function by experimental data, the evaluation of the spanwise correlation length is now brought into question. However, it was concluded by Brooks and Hodgson 17 that the function £a(w) for a flat plate and a thin airfoil are identical under suitable normalization. Therefore, the use of Eq. (12c) for £2 (w) will be retained for the remaining calculations. The predicted and measured far field SPLs for the two tunnel speeds are shown in Fig. 13. The experi- mental data in Fig. 13 were obtained by digitizing the measurements plotted in Figure 34 of Reference 16. The agreement with the measured data is significantly im- proved when the calculation includes surface pressure correlations that account for the trailing edge of an air- foil. In fact, it was concluded by Schlinker and Amiet 16 that airfoil surface pressure correlations were absolutely necessary for realistic TE noise predictions. Concludin g Remarks The prediction of broadband trailing edge noise from rotating machinery and airframes is currently the sub- ject of intense research in aeroacoustics. The physics of broadband noise generation are well understood as the result of the pioneering research of Howe, 14 ’ 29 ' 30 Amiet and coworkers, 4 ’ 5 ’ 12 ' 10 ’ 22 and Brooks and coworkers. 17 ’ 31 The previous work of these aeroacoust.icians, and many others, has clearly demonstrated that any successful broadband loading noise prediction requires an under- standing of two physical processes: the character of the time-dependent surface pressure that provides the acous- tic source, and the manner in which that source gives rise to an acoustic signal. Obtaining the fluctuating surface pressure distribu- tion analytically, numerically, or experimentally is itself a difficult problem. For this reason, past researchers have most often resorted to modeling the surface pressure, us- ing guidance from experiments to aid in the development of these models. Today, high resolution surface pressure fluctuations can be obtained from turbulence simula- tions in realistic situations where the airfoil geometry and kinematics are accurately modeled. Therefore, the improvement of the acoustic radiation model becomes an important research topic. In the past, acoustic radiation models were most often developed for airfoils in uniform rectilinear motion. In addition, other restrictive assump- tions, such as far field positioning of the observer, were often used to simplify the acoustic analysis. The present work further develops a simple and gen- eral acoustic result in the time domain, based on the solution of the loading noise term of the Ffowcs Williams- Hawkings equation. This new: solution, called Formula- tion IB, is, to date, the simplest analytical result for the prediction of loading noise and is suitable for statistical analysis of broadband noise for a surface in general mo- tion. The new formulation has been validated with time domain calculations that predict trailing edge noise on a NACA 0012 airfoil in a low r Mach number flow. The time domain predictions are found to be in excellent agree- ment with the frequency domain predictions of Schlinker and Amiet 16 as well as with the experimental measure- ments of Brooks and Hodgson. 17 These results are, to the authors’ knowledge, the first successful broadband trailing edge noise predictions in the time domain. The authors advocate the use of time domain meth- ods in the prediction of broadband noise. Because of the decoupling of the aerodynamics from the acoustics, the chief advantage of time domain methods is their po- tential for direct use of time-dependent surface pressure statistics from experiments or computer simulations. Acknowled g ements The authors would like to express their gratitude to Dr. Roy K. Amiet, whose input from his past experience was invaluable to this research. The authors are also grateful to Dr. Meelan Choudhari of NASA Langley Re- search Center for several enlightening discussions on the subject of stochastic modeling. References 1. Howe, M. S., “A Review: of the Theory of Trailing Edge Noise,” Journal of Sound and Vibration, Vol. 61, 1978, pp. 437-465. 2. Lighthill, M. J., “On Sound Generated Aerodynam- icallv. I. General Theory,” Proceedings of the Royal Society of London, A 211, 1952, pp. 564-587. 3. Ffowcs Williams, J. E. and Hall, L. H., “Aerodynamic Sound Generation by Turbulent Flow in the Vicinity of a Scattering Half-Plane,” Journal of Fluid Mechanics, Vol. 40, 1970, pp. 657-670. 4. Amiet, R. K., “Noise Due to a Turbulent Flow Past a Trailing Edge,” Journal of Sound and Vibration, Vol. 47, No. 3, 1976, pp. 387-393. 5. Amiet, R. K., “Effect of the Incident Surface Pressure Field on Noise Due to a Turbulent Flow Past a Trailing Edge,” Journal of Sound and Vibration, Vol. 57, No. 2, 1978, pp. 305-306. 6. Goldstein, M. E., “Scattering and Distortion of the Unsteady Motion on Transversely Sheared Mean Flows,” Journal of Fluid Mechanics, Vol. 91, No. 4, 1979, pp. 601-632. 7. Ffowcs Williams, J. E. and Hawkings, D. L., “Sound Generation by Turbulence and Surfaces in Arbitrary 8 American Institute of Aeronautics and Astronautics Motion,” Philosophical Transactions of the Royal So- ciety , A 264, 1969, pp. 321-342. 8. Singer, B. A., Brentner, K. S., Lockard, D. P., and Lilley, G. M., “Simulation of Acoustic Scattering from a Trailing Edge,” Journal of sound and Vibration, Vol. 230, No. 3, 2000, pp. 541-560. 9. Farassat, F. and Succi, G. P., “The Prediction of Heli- copter Rotor Discrete Frequency Noise,” Vertica, Vol. 7, No. 4, 1983, pp. 309-320. 10. Casper, J. and Farassat, F., “ Broadband Noise Pre- dictions Based on a New Aerocoustic Formulation,” AIAA Paper No. 2002-0802. 11. Paterson, R. W., and Amiet, R. K., “Noise and Surface Pressure Response of an Airfoil to Incident Turbu- lence,” AIAA Journal of Aircraft, Vol. 14, No. 8, 1977, pp. 729-736. 12. Amiet, R. K., “High Frequency Thin-Airfoil Theory for Subsonic Flow,” AIAA Journal, Vol. 14, No. 8, 1976, pp. 1076-1082. 13. Willmarth, W. W. and Roos, F. W., “Reslution and Structure of the Wall Pressure Field Beneath a Tur- bulent Boundary Layer,” Journal of Fluid Mechanics, Vol. 22, 1965, pp. 81-94. 14. Corcos, G. M., “The Structure of the Turbulent Pres- sure Field in Boundary Layer Flows,” Journal of Fluid Mechanics, Vol. 18, 1964, pp. 353-378. 15. Yu, J. C. and Joshi, M. C., “On Sound Radiation from the Trailing Edge of an Isolated Airfoil in a Uniform Flow,” AIAA Paper No. 79-0603, 1979. 16. Schlinker, R. H., and Amiet, R. K., “Helicopter Rotor Trailing Edge Noise,” NASA Contractor Report No. 3470, 1981. 17. Brooks, T. F., and Hodgson, T. H., “Trailing Edge Noise Prediction from Measured Surface Pressures,” Journal of Sound and Vibration, Vol. 78, No. 1, 1981, pp. 69-117. 18. Boersma, J., “Computation of Fresnel Integrals,” Mathematics of Computation, Vol. 14, 1960, p. 380. 19. Curie, N., “The Influence of Solid Boundaries on Aero- dynamic Sound,” Proceedings of the Royal Society of London, A 231, 1954, pp. 505-514. 20. Halvorsen, W. G. and Bendat, J. S., “Noise Source Identification Using Coherent Output Power Spectra,” Journal of Sound and Vibration, Vol. 9, 1975, pp. 15- 24. 21. Piersol, A. G., “Use of Coherence and Phase Data Between Two Receivers in Evaluation of Noise Envi- ronments,” Journal of Sound and Vibration, Vol. 56, pp. 215-228. 22. Amiet, R. K., “Acoustic Radiation from an Airfoil in a Turbulent Stream,” Journal of Sound and Vibration, Vol. 41, 1975, pp. 407-420. 23. Lamb, H., Hydrodynamics, Dover Publications, New York, 6th edition, p. 501. 24. Courant, R. and John, F. Introduction to Calculus and Analysis, Vol. 2, Springer-Verlag, New York, 1989. 25. Shinozuka, M. and Deodatis, G., “Simulation of Stochastic Processes by Spectral Representation,” Ap- plied Mechanics Review, Vol. 44, No. 4, 1991, pp. 191-204. 26. Corcos, G. M., “The Structure of the Turbulent Pres- sure in Boundary Layer Flows,” Journal of Fluid Me- chanics, Vol. 18, 1964, pp. 353-378. 27. Willmarth, W. W. and Roos, F. W., “Resolution and Structure of the Wall Pressure Field Beneath a Tur- bulent Boundary Layer,” Journal of Fluid Mechanics, Vol. 22, 1965, pp. 81-94. 28. Yu, J. C. and Joshi, M. C., “On Sound Radiation from the Trailing Edge of an Isolated Airfoil in a Uniform Flow,” AIAA Paper No. 79-0603, 1979. 29. Howe, M. S., “Trailing Edge Noise at Low Mach Num- bers,” Journal of Sound and Vibration, Vol. 225, No. 2, 1999, pp. 211-238. 30. Howe, M. S., “Trailing Edge Noise at Low Mach Num- bers, Part 2: Attached and Separated Flows,” Journal of Sound and Vibration, Vol. 234, No. 5, 2000, pp. 761— 775. 31. Brooks, T. F., Pope, D. S., and Marcolini, A. M., “Air- foil Self-Noise and Prediction,” NASA RP-1218, July, 1989. 9 American Institute of Aeronautics and Astronautics Bd ‘(o ,l K)d Fig. 4 Schematic for directivity calculation. Observer on circular path in plane X 2 = 0. Fig. 2 Schematic for the constant-frequency trailing edge noise problem in Section 3. 90° Fig. 5 Directivity for a constant frequency source of 2.5 kHz; observer distance 2 m; spacing between concentric circles on grid represents 0.25 Pa. Fig. 3 Initial surface pressure profile using Eqs. (2a)- 2(d). 10 American Institute of Aeronautics and Astronautics Acoustic Intensity Fig. 6 Velocity scaling properties as determined by Formulation IB and the surface pressure in Eqs. (2a)- (2d). Fig. 8 Effect of incident pressure term on initial surface pressure profile using Eqs. (2a)-2(d), and modified with Eq. (lOd). microphone | Fig. 7 Schematic for trailing edge noise experiment of Brooks and Hodgson. 1 ' Fig. 9 Surface grid for prediction of experiment of Brooks and Hodgson; 1 7 every fourth point of in each di- rection is shown. 11 American Institute of Aeronautics and Astronautics Fig. 10 Predicted far-field signal, U = 69.5 m/s; mi- crophone at 90 u , 1.22 m above trailing edge. Fig. 11 Predicted and measured far field noise spectra, using surface pressure correlations from flat plate theory; U = 69.5 m/s; frequency domain prediction from Ref. 16; experimental data from Ref. 17. Fig. 12 Normalized surface pressure correlations; flat plate theory from Ref. 16; experimental measurements from Ref. 28. Fig. 13 Predicted and measured far-field noise spec- tra; predictions obtained with measured surface pressure correlations (Ref. 28); experimental SPLs from Ref. 17. 12 American Institute of Aeronautics and Astronautics