Scattering of Acoustic Duct Modes by Axial Liner Splices

Transcription

1 Scattering of Acoustic Duct Moes by Axial Liner Splices Christopher K.W. Tam * an Hongbin Ju Floria State University, Tallahassee, Floria , USA an Eugene W. Chien Goorich Aerostructures Group, Chula Vista, California , USA Recent engine test ata an results of computational analysis show that the engine inlet acoustic liner splices coul have a significant impact on aircraft certification flight noise an cabin noise levels. The phenomenon of scattering of acoustic uct moes by axial liner splices is investigate. Previous stuies, invariably, follow the frequency omain approach. The present stuy, however, uses the time omain approach. It is emonstrate that time omain computation yiels results that are in close agreement with frequency omain results. The scattering phenomenon uner consieration is very complex. This stuy concentrates on the effects of four parameters. They are the with of the splices, the frequency of the incient uct moe, the number of splices an the length of splices. Base on the compute results, the conitions uner which scattere wave moes woul significantly increase the intensity of transmitte waves are ientifie. It is also foun that surface scattering by liner splices has the tenency to istribute energy equally to all the cuton scattere azimuthal moes. On the other han, for each scattere azimuthal moe, the high orer cut-on raial moe, generally, has the highest intensity. More over, scattering by liner splices is a local phenomenon. It is confine primarily to an area of the uct ajacent to the junction between the har wall near the fan face an the splice liner. S I. Introuction CATTERING takes place when an acoustic uct moe propagates into a region of a circular uct line with soun absorbing material an har wall splices. An incient uct moe is often scattere into a multitue of other azimuthal an raial moes epening on the number an with of the splices. If the frequency of the incient uct moe is slightly higher than the cut-on frequency, the energy flux of the scattere acoustic moes may be many ecibels higher than that of the incient acoustic moe at the exit en of the splice liner. This is a most unesirable situation. Recently, the phenomenon of uct moe scattering by axial liner splices has attracte a goo eal of interest. In an early work, Fuller 1,2 use uct moe expansions to investigate the scattering effect. More recently, Regan an Eaton 3 stuie the problem using finite element formulation in the frequency omain. The finite element metho leas immeiately to a large matrix system as in the case of Fuller. To obtain an accurate solution of a very large matrix system, as it turns out, is non-trivial. Elnay, Boen an Glav 4 evise a point matching metho to moel the splice liner surface. This metho again results in the nee to solve a large matrix system. In a follow-up work, Elnay an Boen 5 use the point matching metho to stuy the effect of har wall splices for both locally reacting an non-locally reacting liners. They conclue that non-locally reacting liners were less affecte by the presence of har strips than locally reacting liners. Bi, Pagneux an Lafarge 6 stuie soun propagation in ucts with varying cross-section an with nonuniform impeance. They expane their solutions using the propagating uct moes as basis functions. As in most eigenfunction expansion methos, the proper choice of a set of basis functions is crucial. However, there is no obvious an simple choice. Bi et al. consiere the use of rigi wall uct moes. But they are not the natural moes * Department of Mathematics, Robert O. Lawton Distinguishe Professor, Fellow AIAA. Department of Mathematics, Research Associate, Senior Member AIAA. Staff engineer, Acoustics. 1

2 of a line uct. The use of locally parallel uct moes with liners might seem to be a better choice. Unfortunately, for eigenmoes of a line uct, there is no guarantee that the compute moes form a complete set of basis functions. In aition, the computation of these moes is by no means straightforwar. Moreover, this expansion metho will once more lea to a large matrix that may be ense an not easy to solve. In a relate work by Bi, Pagneux, Lafarge an Auregan 7, this uct moe expansion metho was use to analyze the scattering of uct moes by rigi splices in line ucts. Preliminary results for an incient moe at slightly above the cut-on frequency were reporte. The influence of liner length, the azimuthal moe orer of the incient uct moe, as well as axial nonuniformities were briefly consiere. McAlpine et al. 8 investigate the effect of splice intake liner at supersonic fan tip spee. They employe a numerical coe name ACTRAN for their computation. The ACTRAN coe is base on the finite element metho. For incient uct moes at typical blae passage frequency (BPF), their three-imensional finite element simulation require consierable amount of computing resources. To reuce computing requirements to a manageable level, they stuie the problem at half BPF of a typical commercial turbofan engine. The number of splices was also reuce to half while the liner length increase to twice the typical lengths. McAlpine et al. also use the ACTI3S Airbus France bounary element coe in their stuy. They reporte that the finite element ACTRAN coe an the bounary element ACTI3S coe yiele similar results for certain cases in their investigation. A major conclusion of their stuy is that fan spee is a critical factor of the splice scattering phenomenon an that thinner splices woul reuce the level of scattere tones. Recently, Tester et al. 9 performe a valiation test of the Cargill analytical metho for uct moe scattering by liner splices. The Cargill metho is base on the Kirchhoff approximation. Its main avantage is simplicity. The valiation test was carrie out by comparing the results obtaine by the Cargill metho with those of the ACTRAN finite element coe. Tester et al. reporte that the ACTRAN an Cargill results agree well for the no-flow case except for some minor iscrepancies in the back-scattere moes. The back-scattere moes are of seconary importance compare to the forwar scattere moes. When a mean flow up to Mach 0.4 was inclue, the agreement between the ACTRAN an Cargill results was still acceptable except again for the fiel ominate by back-scattering. Encourage by the agreement, Tester et al. plan to improve the Cargill metho so as to provie a rapi methoology for evaluating splices scattering effects. In this work, a specially esigne time-omain coe base on the 7-point stencil time marching Dispersion- Relation-Preserving (DRP) scheme is use to compute the scattere an transmitte acoustic uct moes. Previous works on this subject invariably use the frequency-omain approach. This stuy intens to emonstrate that the accuracy of the time-omain solution is, at least, comparable to that obtaine by frequency omain coes. One avantage of time omain approach over frequency omain metho is that multiple frequencies an broaban input can be calculate in a single computation. Further, nonlinear effects may also be inclue. In this paper, however, the primary objective is to stuy the axial splice scattering phenomenon. These other computational issues will not be consiere an will be left to a future work. In aition to the DRP scheme, avance computational aeroacoustics (CAA) bounary conitions are use in the computation. They inclue time-omain impeance bounary conitions, ghost-point metho (for enforcing bounary conitions) as well as the Perfectly Matche Layer (PML). The results of a parametric stuy are reporte in this paper. The stuy focuses on four parameters of the acoustic scattering phenomenon. First, is the with of the axial splices. Our interest is to examine its effect on the energy flux of the scattere an transmitte wave moes. Our stuy suggests that if the with of the splices is not too narrow, the scattere moes essentially comprise of all the cut-on azimuthal moes an that the soun power levels (PWLs) of the scattere moes are nearly the same. This fining is consistent with the equal energy assumption wiely use in uct aeroacoustics. However, if the splice with is narrow, PWL of the scattere uct moes is negligible. The secon parameter investigate is the frequency of the incient uct moe. The results of our stuy show that the total PWL of all the scattere moes woul be significant relative to the transmitte incient moe if the frequency is slightly above the cut-on frequency. On the other han, if the frequency is well above cut-on frequency, then the increase in the total PWL of the scattere moes is relatively negligible. The thir parameter examine in this stuy is the number of splices. The results of our investigation reveal that a larger number of splices woul result in a reuction in the PWL of the scattere acoustic wave moes. This is important from the point of view of minimizing noise raiation. An explanation for the reuction is provie. The fourth parameter investigate is the length of the axial splices an the location at which most of the scattering takes place. Extensive compute results show that acoustic scattering by axial liner splices is a local phenomenon. Most of the scattering takes place primarily near the junction of the uct where the incient wave moe first encounters the splice liner. Thus a long axial splice an a meium length splice have practically the same effect. 2

3 The rest of the paper is organize as follows. Section II presents the computational moel an numerical methos use in this work. Since a time omain approach is followe, the formulation of the time omain impeance bounary conition an its implementation for splice liner are escribe. To ensure the numerical simulation is accurate an of high quality, the numerical coe is valiate by comparing the compute results with those of Tester et al9. Tester et al. s results are calculate by the ACTRAN coe. Section III reports the results of a parametric stuy. This stuy focuses on the effects of splice with, the frequency of the incient uct moe, the number an length of the splices on the scattering phenomenon. Section IV summarizes the overall results of this work. II. Computational Moel an Numerical Methos In this section, we will introuce the physical an computational moel use in this investigation. This is followe by a escription of the computational methos. The present computer coe is valiate by comparing compute results with those of Tester et al9. Tester et al. employe the ACTRAN coe that is base on the finite element metho. A. Computational Moel Figure 1. Computation moel. Figure 2. Splice liner moel. 3

4 The computational moel, consisting of a circular uct encase by splice liner an har walls, is as shown in Figure 1. A three-imensional sketch of the splice liner section is given in Figure 2. The moel was use previously by Regan an Eaton 3, McAlpine et al 8 an Tester et al. 9. The har wall on the right represents the space between the fan blae an the splice liner. The har wall on the left represents the har wall segment at the inlet of the intake uct of a jet engine. The uct is assume to have a constant iameter D (raius a = D/2). The uct carries a uniform mean flow at Mach number M. A spinning uct moe of azimuthal moe number m an frequency W (blae passage frequency or interaction tone frequency) propagates upstream from the fan face on the right of Figure 1. The length of the liner is taken to be L. The resistance an reactance of the liner are R an X, respectively. It will be assume that the number of har wall splices is N seg. A spinning uct moe propagates upstream from the har wall region of the uct on the right to the splice liner segment. Acoustic scattering takes place as soon as the incient uct moe reaches the junction between the har wall an the splice liner. The N seg rigi splices scatter the incient uct moe with azimuthal moe number m into spinning uct moes of the same frequency but with azimuthal moe number equal to m ± jn seg (j = 0,1,2,3, ). In aition to generating upstream propagating uct moes, there is back scattering of rigi wall uct moes. Of interest is the intensity of the transmitte acoustic waves in the har wall region upstream of the splice liner. B. Governing Equations Dimensionless variables are use in the computation. The scales use for non-imensionalization purpose are, D (iameter of the uct) = length scale, a 0 (spee of soun of gas) = velocity scale, D a 0 r 0 (gas ensity) 2 r 0 a 0 r 0 a 0 = time scale, = ensity scale, = pressure scale, = impeance scale. The uct is assume to carry a uniform mean flow (in the positive x irection) at a Mach number M. The acoustic isturbances insie the uct are governe by the linearize Euler equations. In cylinrical coorinates (r,f,x) an velocity components (v,w,u), these equations are, r t + M r x + v r + v r + 1 w r f + u x = 0, (1) v t + M v x = - p r, (2) w t + M w x = - 1 p r f, (3) u t + M u x = - p x, (4) p t + M p x + v r + v r + 1 w r f + u x = 0. (5) For a spinning uct moe with azimuthal moe number n, the solution may be written in the form, 4

5 È r Ï È ˆ r n (r, x,t) Í Ô Í Í u Ô Í u ˆ n (r, x,t) Í Ô Í Í v = ReÌ Í v ˆ n (r, x,t) Í Ô Í Í w Ô Í w ˆ n (r,x,t) Í Ô Í Î Í p Ó Ô Î Í p ˆ n (r, x,t) e inf Ô Ô Ô Ô Ô Ô Ô (6) where Re{ } is the real part of { }. Substitution of (6) into (1) to (5) gives the governing equations for ( ˆ r n, u ˆ n, v ˆ n, w ˆ n, p ˆ n ) ˆ r n t ˆ r + M n x + v ˆ n r + v ˆ n + in w r r ˆ n + u ˆ n x = 0 (7) ˆ v n t + M ˆ v n x = - ˆ p n r (8) w ˆ n t w ˆ + M n x = - in p r ˆ n (9) ˆ p n t u ˆ n t ˆ p + M n u ˆ + M n x = - ˆ p n x x + v ˆ n r + v ˆ n + in r r w ˆ + u ˆ n n x (10) = 0. (11) C. Time Domain Bounary Conitions at the Splice Liner Surface At the splice liner surface, part of the surface is liner material an part of the surface is har wall. On the part of the surface that is har wall, the bounary conition is, r = a, v = 0 (a = D 2). (12) On the part of the surface with liner materials, time-omain impeance bounary conition is use. Let the surface impeance be (e iwt time epenence assume) Z = R -ix (13) where R an X are the imensionless resistance an reactance. Here the three parameters time omain impeance bounary conition of Tam an Auriault 10 will be use. The reactance which may be positive or negative is parametrize by, X = X -1 W + X 1 W (14) where W is the angular frequency of the incient soun. Numerical stability requires that R > 0, X -1 < 0 an X 1 > 0. Here for a given impeance value Z an frequency W, the two parameters of the moel X -1 an X 1 are compute by the following formulas, 5

6 1 X 1 = È 2 Z - X Í Î Í Z 2 W Ê - 1- X ˆ Á W Ë Z, X -1 = È 2 Z - X Í Î Í Z 2 2 (15) It is easy to check that (15) satisfies (14) an the constraints on X -1 an X 1. Upon incorporating the Myers 11 convective term, the time omain impeance bounary conition on the liner surface in a grazing flow at Mach M is, p t + M p x = R v t - X 2 v -1v + X 1 t 2 (16) Now (12) an (16) will be combine into a single splice liner bounary conition. Let N seg be the number of axial splices. Then the angle f HW subtene by a har wall splice is equal to 2W/D where W is the with of a splice. Azimuthally, the splice liner surface is a perioic function with perio 2p/N seg. Let G(f) be a perioic function with perio 2p/N seg as shown in Figure 3. G(f) takes the value of 1 for 0 < f < (2p/N seg f HW ) an zero for (2p/N seg f HW ) < f < 2p/N seg. The graph of G(f) may be expane in a Fourier series in f, where G(f) = c s e isn segf  (17) s=- Ï i Ê e -is ( 2p -f HW N seg ) ˆ Ô Á 2ps -1, s not equal to zero c s = Ì Ë Ô 1- f HW N seg Ó, s = 0 2p. (18) Figure 3. Graph of function G(f). 6

7 Now the combine bounary conition of (12) an (16) for the splice liner may be written as 2 v X 1 t 2 + R v t - X -1 v = G ( f È ) p t + M p Î Í x. (19) It is easy to verify that on the liner surface, where G(f) = 1.0, (19) reuces to (16). On the surface of the rigi splices, where G(f) = 0, (19) reuces to It is straightforwar to show, if the initial conitions X 1 2 v t 2 + R v t - X -1 v = 0. (20) v = 0, v = 0 at t = 0 (21) t are impose, the solution of (20) is v = 0, which is the rigi wall bounary conition. For convenience of enforcing bounary conition (19) at r = D/2, we will rewrite this equation as a system of first orer equation in t by introucing an auxiliary variable q efine by (19) may now be rewritten as v t = q (22) q t = - R q + X -1 v - 1 È G( f) v X 1 X 1 X 1 r + v r + 1 w r f + u Í. (23) Î x In eriving (23), use has been mae of (5). Finally, by replacing the time erivative of v in (2) at r = D/2 by q accoring to (22), a time erivative free bounary conition is erive. p r = -q - M v x. (24) In summary, the splice liner surface bounary conitions at r = D/2 are (22), (23) an (24). D. Computational Algorithm Due to scattering by the liner splices, the soun fiel insie the uct woul consist of many azimuthal moes. The moe numbers are space by N seg apart. The full solution insie the uct may be expresse in the form, È u( r,f, x,t) Ï È u ˆ m + j ( r, x,t) Í Ô Í Ô Í v( r,f, x,t) Í v ˆ m + j ( r, x,t) Ô Í w( r,f, x,t) = Re Â Í w ˆ m + j ( r, x,t) e i(m + jn seg )f Ô Ì Í Í p( r,f, x,t Ô Í ) j Í p ˆ m + j ( r, x,t Ô Ô ) Ô Î Í q( f, x,t) Î Í q ˆ m + j ( x,t) Ó Ô Ô (25) where m is the azimuthal moe number of the incient uct moe. The incient an scattere moes are couple by the splice liner surface bounary conitions (22), (23) an (24). It is easy to see that only surface bounary conition (23) causes moe coupling. The amplitue functions of the incient an scattere uct moes of (25), i.e., (ˆ u m+ j, v ˆ m+ j, w ˆ m+ j, p ˆ m+ j ) are calculate by (8) to (11) with n replace by (m + jn seg ). The values of v ˆ m+ j at r = D/2 are, however, not compute this way, since at this r location 7

8 (8) has been effectively replace by bounary conition (24). Each value of ˆ v m+ j is compute by a time marching metho accoring to (22). Upon equating the azimuthal Fourier components in f, (22) becomes ˆ v m+ j = q ˆ t m+ j. (26) To enforce bounary conition (24), the ghost point metho 12 is employe. For this purpose, a row of ghost value of p at k = M + 1 is introuce. k is the mesh inex in the r-irection. k = M correspons to r = D/2 as shown in Figure 4. In the bounary region, at or ajacent to k = M, 7-point backwar ifference stencils are use to compute the r-erivative. The r-erivative stencils for all the variables, with the exception for p, terminate at k = M (see Figure 4). Those for p terminate at the ghost point k = M + 1. The iscretize form of (24) for each azimuthal moe, is (l is the mesh inex in the x-irection; superscript n is the time level). Figure 4. Backwar ifference stencils an ghost points in the upper bounary region of the computational omain 1 Dr 1 Â s=-5 a s 15 (n) ( p ˆ m + j ) l,m +s (n) Thus on solving for the ghost value, ( p ˆ m + j ) l,m +1, it is foun (n) = -( q ˆ m + j ) - M l,m Dx 3 a v (n) Â s ( ˆ ). m + j l +s,m s=-3 (n) ( p ˆ m + j ) = Dr l,m a 1 È Í - ˆ Î Í q (n) ( m + j ) l,m - M Dx 3 Â s=-3 a s (n) ( v ˆ m + j ) l +s,m - 1 Dr 0 Â s=-5 a s 15 (n) ( p ˆ m + j ) l,m +s. (27) This is the formula by which the ghost values are foun. To implement the remaining bounary conition (23), the first step is to substitute (25) into (23) an equate terms accoring to the azimuthal moe number. This leas to an infinite system of equations. A finite system may be erive by ropping the azimuthal moes outsie the range of importance. Suppose the scattere moes are limite to J j K, then (23) leas to 8

9 ˆ q m + j t This is the equation for ˆ q m+ j. = - R q ˆ m + j + X -1 v ˆ m + j X 1 X 1-1 È ˆ c s X 1 Â Í Î r -J ( j-s) K sum over s v m + j-s + v ˆ m + j-s r + i[m + ( j - s)n seg ] r u m + j-s w ˆ m + j-s + ˆ x. (28) In this work, governing equations (8) to (11) for each retaine azimuthal moe (m J n m + K) as well as (26) an (28) are solve computationally by iscretizing the system of equations accoring to the 7-point stencil DRP scheme 13. To suppress possible numerical instability an to remove spurious short waves, artificial selective amping 14 is ae to the iscretize equations. It is known that time omain impeance bounary conition with Myers 11 convective term inclue is subjecte to Kelvin-Helmholtz instability 10. This instability is suppresse by artificial selective amping. The computation is carrie out on a Cartesian mesh in the x r plane within the computation omain as shown in Figure 5. For each azimuthal moe, the axis bounary treatment evelope by Shen an Tam 15 is impose. This bounary treatment eliminates the apparent singularity of the system of equations (8) to (11) at the axis (r = 0). At the uct wall (r = D/2) outsie the splice liner region, bounary conition (12) is use. This bounary conition is again enforce by the ghost point metho. At the two open ens of the computation omain (see Figure 5), Perfectly Matche Layers (PML), as propose by Hu 16 are ae. The PML absorb all outgoing waves. On the right bounary, split variables are use to allow the incient uct moe to enter the computation omain from the PML region. Figure 5. The computational Domain To start the computation, all variables are set equal to zero insie the computation omain. The incient acoustic wave moe propagates into the computation omain from the right. This begins the transient perio of the computation. The computation continues an the numerical solution is monitore at a number of well-chosen locations. A stopping criterion is set base on how close the solution is to its former value at the monitoring stations after a perio accoring to the incient wave. Numerical results are measure only after further computation over a few more perios. E. Coe Valiation For the purpose of valiating the present time-omain coe, the recently publishe results of Tester et al 9 are use. Tester et al compute their results by means of the ACTRAN coe. ACTRAN is base on a finite element formulation. In the work of Tester et al., there is no iscussion of mesh esign an numerical resolution. It will, therefore, be assume that a very fine mesh an long computing time were use to obtain their results. The resolution of the present time-omain coe was teste by comparing the numerical results with exact solution for the case of a spinning acoustic moe propagating own a har wall uct. Comparisons of the moe 9

10 shape incluing more than one raial moe were mae. Excellent agreements were foun. The tests were conucte over a range of frequencies an azimuthal moe numbers. In aition to irect comparisons with exact analytical solution for har wall uct moes, gri refinement tests were also carrie out for ucts with splice liners. The final mesh size aopte for the coe yiele numerical solutions that were quite insensitive to mesh refinement. Two cases are consiere in the paper by Tester et al 9. One is at zero flow Mach number. The other case is at M = 0.4. The incient uct moe has a frequency equal to 1.1 cut-on frequency. Other etails of their moel problem are as follows, Raius of uct (enote by a) = 50 inches Impeance of liner, Z = 2 + i (time factor = e iwt ) Length of liner (L) = 24 inches (L/a = 0.48) Number of splices = 2 With of splices = 3 inches Azimuthal moe number of incient wave = No Mean Flow Case In this case, the frequency of the incient wave is Hz. Figure 6 shows a comparison of the results of the present time omain coe an those of ACTRAN. This figure shows the axial istribution of Soun Power Level (PWL) in B (Re: PWL of incient wave moe at x/a = 0.48.) The splice liner extens from x/a = 0.0 to In this figure, the full lines are the results of time omain computation an the ashe lines are the results of Tester et al. It is evient that the numerical results are in close agreement. This is true for the PWL of the total transmitte soun as well as the incient an the iniviual scattere azimuthal moes. At the en of the splice liner region (x/a = 0.0), the incient wave moe is significantly ampe. Its PWL is about 4 B below the PWL of the scattere wave moes. In the present computation, a gri refinement has been performe. The results reporte here are regare as gri converge. Figure 6. Axial istribution of transmitte wave energy. Flow Mach number = 0. Soli lines present results; Dashe lines ACTRAN results by Tester et al. 2. Mach Number 0.4 Case In this case, the frequency of the incient uct moe is Hz. Figure 7 shows a comparison of the present compute axial istributions of PWL an those of Tester et al. There are goo agreements with respect to the total transmitte acoustic power as well as the PWL of iniviual scattere azimuthal moes. This is true over the entire 10

11 length of the acoustic liner. However, in the absence of error estimates from the ACTRAN coe computation, it is not possible to ascertain the reason for the slight ifferences in the two sets of compute results. Figure 7. Axial istribution of transmitte wave energy. Flow Mach number = 0.4 Soli lines present results; Dashe lines ACTRAN results by Tester et al. III. A Parametric Stuy A parametric stuy focuse on four parameters of the splice liner acoustic scattering phenomenon has been carrie out. The stuy concentrates on the following effects an questions. A. Effect of Splice With Figure 8. Comparison of the transmitte wave energy flux (PWL) for splices with with = 3, 1, 0.5, 0. 11

12 In this stuy, the effect of the with of the splices on the effectiveness of acoustic scattering is examine. Figure 8 shows a comparison of the axial istribution of the total transmitte acoustic power (PWL) for the test case specifie in section IIE (the same test conitions as Tester et al.) at a mean flow Mach number 0.4 an ifferent splice withs. In aition to the case with splice with equal to 3 (shown in Figure 7), the cases of 1, 0.5 an 0 (uniform liner) are also shown. Clearly, as anticipate, the intensity of scattere acoustic waves increases with splice with, all other variables are hel fixe. For splice with 0.5 an narrower, there is little acoustic scattering. Figure 9. Distribution of total transmitte PWL in azimuthal moes upstream of the splice liner. Two 3 splices. Figure 10. Distribution of total transmitte PWL in azimuthal moes upstream of the splice liner. Two 1 splices. 12

13 Figure 11. Distribution of total transmitte PWL in azimuthal moes upstream of the splice liner. Two 0.5 splices. Figure 12. Distribution of transmitte PWL in raial moes for azimuthal moe m = 0 in the region upstream of the splice liner. Two 3 splices. To obtain a better unerstaning of the scattering phenomenon, the istribution of total transmitte wave energy (PWL) for all azimuthal moes at the har wall region upstream of the acoustic liner is plotte in Figure 9 for the splices with 3 with. It is easy to see that the scattere wave energy is concentrate in the propagating azimuthal moes (base on an equivalent soli wall uct consieration). The PWL of the propagating moes of the scattere waves is almost the same. In this case, the level of each scattere moe is higher than that of the incient wave moe. Figures 10 an 11 show similar results for splices with a with of 1 an 1/2, respectively. Again, most of the transmitte energy of the scattere moes is concentrate on the propagating moes. The level of the propagating moes is again nearly constant. In both of these cases, the incient wave moe has amplitue higher than that of the scattere moes. 13

14 Figures 12, 13 an 14 show the istribution of PWL in raial moes for the m = 0, 8 an 16 azimuthal moes of Figure 9. These figures reveal that most of the acoustic energy of an azimuthal moe is concentrate in the high orer propagating raial moes. This energy concentration in the high orer raial moes turns out to be true in all the cases compute in the present parametric stuy. Since the energy of low orer moes is concentrate near the uct wall whereas that of the high orer moes is sprea out more evenly over the cross section of the uct, the scattering phenomenon has the tenency to equalize energy istribution in a uct. Figure 13. Distribution of transmitte PWL in raial moes for azimuthal moe m = 8 in the region upstream of the splice liner. Two 3 splices. Figure 14. Distribution of transmitte PWL in raial moes for azimuthal moe m = 16 in the region upstream of the splice liner. Two 3 splices. B. Effect of Frequency How important is frequency as a parameter of the splice liner scattering phenomenon? To examine the effect of frequency, 3 cases are stuie using the Tester et al. problem but at 1.5, 1.1 an 0.9 cut-on frequency. Figure 15 14

15 shows the axial istribution of the PWL of the incient wave moe, the sum of the PWL of all waves an the PWL of a few selecte scattere waves at a 1.5 cut-on frequency. It is clear from the results shown that at high frequency the amplitue of the total acoustic wave is nearly the same as that of the incient wave moe (m = 26). Thus acoustic scattering is not important. Figure 16 shows the istribution of PWL of all the transmitte waves in azimuthal moes. The PWL of the scattere wave moes is over 30 B below that of the incient moe. Figure 15. Axial istribution of transmitte wave energy. Flow Mach number = 0.4. Two 3 splices. Frequency = 1664 Hz ; frequency = 1.5 cut-on frequency Figure 16. Distribution of total transmitte PWL in azimuthal moes upstream of the splice liner. Frequency = 1664 Hz ; frequency = 1.5 cut-on frequency The case of incient wave at a slightly above cut-on frequency is shown in Figure 17. It is to be note that at this frequency the total PWL of the transmitte scattere waves is 19 B higher than that of the incient moe. Figure 18 15

16 shows similar axial istribution at 0.9 cut-on frequency. In this case the incient wave moe is heavily ampe. At the en of the liner region, only the scattere waves remain. Figure 19 shows the istribution of PWL of the total transmitte waves in azimuthal moes. The intensity of the scattere waves is comparable to the case at 1.1 cut-on frequency. Acoustic scattering is, without question, important for incient moe at below cut-on frequency. Figure 17. Axial istribution of transmitte wave energy. Flow Mach number = 0.4. Two 3 splices. Frequency = 1221 Hz ; frequency = 1.1 cut-on frequency Figure 18. Axial istribution of transmitte wave energy. Flow Mach number = 0.4. Two 3 splices. Frequency = Hz ; frequency = 0.9 cut-on frequency 16

17 Figure 19. Distribution of total transmitte PWL in azimuthal moes upstream of the splice liner. Frequency = Hz ; frequency = 0.9 cut-on frequency C. Effect of Number of Splices Suppose the total with of har wall splices is fixe. Is it avantageous to use a larger number of narrower splices? For instance, in the problem consiere by Tester et al., the total splice with is 6. By keeping the total with at 6, one can use two 3 splices or four 1.5 splices. Figure 20 shows the axial istributions of the PWL of the total transmitte waves for both cases. It is evient that there is more than 3 B less transmitte soun when four splices are use. In other wors, there is a efinite avantage in using a larger number of splices for a fixe total with. As another example, Figure 21 shows the results of the same problem but for a total splice with of 12. In this figure, the axial istributions of the PWL of the transmitte waves for two 6 splices an four 3 splices are provie. It is clear that the use of more splices has an avantage of slightly over 3 B. Figure 20. Axial istribution of transmitte wave energy. Flow Mach number = 0.4. Total with of splices = 6. Frequency = 1221 Hz.: frequency = 1.1 cut-on frequency 17

18 Figure 21. Axial istribution of transmitte wave energy. Flow Mach number = 0.4. Total with of splices = 12. Frequency = 1221 Hz ; frequency = 1.1 cut-on frequency Figure 22. Distribution of total transmitte PWL in azimuthal moes upstream of the splice liner. Two 3 splices. Frequency = 1221 Hz. 18

19 Figure 23. Distribution of total transmitte PWL in azimuthal moes upstream of the splice liner. Four 1.5 splices. Frequency = 1221 Hz. Figure 24. Distribution of total transmitte PWL in azimuthal moes upstream of the splice liner. Two 6 splices. Frequency = 1221 Hz. 19

20 Figure 25. Distribution of total transmitte PWL in azimuthal moes upstream of the splice liner. Four 3 splices. Frequency = 1221 Hz. The reason why there is an avantage in using a larger number of splices can be foun by examining the istribution of PWL of the total transmitte waves in azimuthal moes. Figures 22 an 23 show the istribution for the case of two 3 splice an four 1.5 splices. For both cases, the transmitte acoustic energy is associate mainly with the propagating moes as mentione before. It is to be note that the levels of the propagating azimuthal moes are about the same for both cases. Now for the four splices configuration, the scattere azimuthal moes have moe numbers separate by 4 (i.e. m = 26 ± 4j, j = 0,1,2, ) while those for the 2 splices configuration have moe numbers separate by 2 (i.e. m = 26 ± 2j, j = 0,1,2 ). Thus there are nearly twice as many propagation azimuthal moes for the two splices configuration as the four splices configuration. This ifference accounts for 3 B less transmitte wave energy for the four splices configuration. Figures 24 an 25 show similar istribution of PWL in azimuthal moes for the two 6 splices an the four 3 splices configurations. Again, the two splice configuration has nearly twice as many propagating azimuthal moes. This results in 3 B more in the total transmitte acoustic energy. D. Effect of Splice Length Extensive computation of the splice liner acoustic scattering phenomenon suggests that scattering is confine mainly to the region close to the junction between the har wall an the splice liner near the fan face. To emonstrate that scattering is, inee, a local phenomenon, the scattering problem of Tester et al. is reconsiere with the moification that the splices have a length shorter than the acoustic liner. Figure 26 shows a threeimensional view of the moifie configuration of the splice liner. Figure 27 shows the computation omain of the moifie configuration. Figure 28 shows the axial istributions of PWL of the total transmitte acoustic waves for the four cases with har wall splice length equal to 24 (same length as the liner), 12, 6 an 0 (no splice). It is reaily seen that the transmitte acoustic energy is nearly ientical for splice length of 12 an 24. The axial istribution iffers only slightly for splice length of 6. This result strongly inicates that most of the acoustic scattering takes place over a length of approximately 6 to 8. 20

21 Figure 26. Moifie splice liner moel. Figure 27. Moifie computational moel. Figure 29 shows the istribution of the PWL of the transmitte soun in azimuthal moes at the upstream har wall region of the uct for splice length equal to 6, 12 an 24. Again, it is reaily seen the intensities of the scattere propagating wave moes are nearly the same regarless whether the splice length is 12 or 24. The intensities are slightly reuce for the 6 length splices. The fact that there is almost no ifference between the 12 an 24 long splices further confirms the belief that acoustic scattering is quite local. There is little scattering beyon the first 12 measure from the ownstream en of the acoustic liner. 21

22 Figure 28. Axial istribution of transmitte wave Energy. Flow Mach number = 0.4. Two 3 wie splices. Length of splices = 24, 12, 6, 0. Frequency = 1221 Hz. Figure 29. Distribution of total transmitte PWL in azimuthal moes upstream of the splice liner. Two 3 wie splices. Length of splices = 24, 12, 6. Frequency = 1221 Hz. IV. Summary The scattering of acoustic uct moes by axial liner splices is a rather complex phenomenon. It is influence by a host of parameters. This work concentrates on the stuy of four aspects of the phenomenon. The effect of splice with is first examine. It is foun that scattering increases with splice with. However, for a typical moern ay jet engine, scattering is not important if the splice with is less than 0.5 inches (1.77 cm). Extensive computations 22

23 inicate that scattering by liner splices tens to result in relatively uniform istribution of scattere wave energy insie the uct. It is observe that all the scattere cut-on azimthal moes, generally, have nearly equal amplitue. In aition, for each cut-on azimuthal moe, the high orer cut-on raial moe consistently has the high amplitue in all the compute results. These high orer raial moes have a more even energy istribution across the crosssection of the uct whereas low orer raial moes concentrate their energy an fluctuations close to the uct wall. Whether the scattering phenomenon uner stuy significantly increases the transmitte wave energy epens strongly on the incient moe frequency. For uct moes at a frequency much higher than the cut-on frequency, the amplitue of the incient moe is, generally, not ampe by the liner. Because of this, the total energy of the scattere moes is small compare to that of the incient moe at the inlet of the uct. Thus scattering by liner splices have little effect on the raiate noise. At slightly above cut-on frequency, an incient uct moe woul be heavily ampe unless the liner is very short. On the other han, the cut-on scattere moes woul not experience significant amping by the liner. In this case, the scattere wave moes coul cause a large increase in the transmitte acoustic wave energy when compare with that of the liner without splices. This is extremely unesirable. At below cut-on frequency, the energy of the scattere cut-on moes at the uct inlet is usually larger than that of the incient moe. To minimize the effect of scattering, it is recommene that liner splices be installe far away from the fan face. If the surface area of liner splices is kept fixe, then it is avantageous to use a large number of narrow splices. This is because the azimuthal moe number of the scattere moe are space further apart when there is a larger number of splices. In other wors, when there are more splices, although they are narrower, there are fewer scattere cut-on azimuthal moes. Since the magnitues of the energy of the scattere cut-on azimuthal moes ten to be more or less the same, the result is that the transmitte wave energy is less when there are more splices. The present stuy establishes that acoustic scattering by liner splices is a local phenomenon. For moern commercial jet engines, most scattering takes place over a short length of the splices (approximately 12 inches in length). The remaining length of the splices oes not seem to cause further increase in the energy of the scattere waves. Finally, a search of the literature reveals that most previous works on scattering by liner splices use the frequency omain approach in their computation. The present investigation is one of the first to perform all computations in the time omain. It is emonstrate that time omain results are as accurate as those of high quality frequency omain computation. Time omain computation has the intrinsic avantages that nonlinear effects as well as broaban soun may easily be inclue in the calculation. However, ue to the limite scope of this work, these avantages have yet to be emonstrate. Acknowlegments The authors wish to thank the Floria State University for permission to use their SP-4 supercomputer. References 1 Fuller, C.R., Propagation an Raiation of Soun from Flange Circular Ducts with Circumferentially Varying Wall Amittances, I.: Semi-Infinite Ducts, Journal of Soun an Vibration, Vol. 93, No. 3, 1984, pp Fuller, C.R., Propagation an Raiation of Soun from Flange Circular Ducts with Circumferentially Varying Wall Amittances, II: Finite Duct with Sources, Journal of Soun an Vibration, Vol. 93, No. 3, 1984, pp Regan, B. an Eaton, J., Moelling the Influence of Acoustic Liner Non-Uniformities on Duct Moes, Journal of Soun an Vibration, Vol. 219, No. 5, 1999, pp Elnay, T., Boen, H. an Glav, R., Application of the Point Matching Metho to Moel Circumferentially Segmente Non-Locally Reacting Liners, AIAA Paper , Elnay, T an Boen, H., Har Strips in Line Ducts, AIAA Paper , Bi, W.P., Pagneux, V. an Lafarge, D., Soun Propagation in Varying Cross-Section Ducts Line with Non-Uniform Impeance by Multimoe Propagation Metho, AIAA Paper , Bi, W.P., Pagneux, V., Lafarge, D. an Auregan, Y. Characteristics of Penalty Moe Scattering by Rigi Splices in Line Ducts, AIAA Paper , McAlpine, A., Wright, M.C.M., Batar, H. an Thezelais, S., Finite/Bounary Element Assessment of a Turbofan Splice Intake Liner at Supersonic Fan Operating Conitions, AIAA Paper , Tester, B.J., Baker, N.J., Kempton, A.J. an Wright, M.C., Valiation of an Analytical Moel for Scattering by Intake Liner Splices, AIAA Paper , Tam, C.K.W. an Auriault, L., Time-Domain Impeance Bounary Conitions for Computational Aeroacoustics, AIAA Journal. Vol. 34, No. 5, 1996, pp Myers, M., On the Acoustic Bounary Conition in the Presence of Flow, Journal of Soun an Vibration, Vol. 71, No. 3, 1980, pp

24 12 Tam, C.K.W. an Dong, Z., Wall Bounary Conitions for High Orer Finite Difference Schemes in Computational Aeroacoustics, Journal of Theoretical an Computational Flui Dynamics, Vol. 6, No. 6, 1994, pp Tam, C.K.W. an Webb, J.C., Dispersion-Relation-Preserving Finite Difference Scheme for Computational Acoustics, Journal of Computational Physics, Vol. 107, No. 2, 1993, pp Tam, C.K.W., Webb, J.C. an Dong, Z., A Stuy of the Short Wave Components in Computational Acoustics, Journal of Computational Acoustics, Vol. 1, No. 1, 1993, pp Shen, H. an Tam, C.K.W., Three-Dimensional Simulation of the Screech Phenomenon, AIAA Journal, Vol. 40, No. 1, 2002, pp Hu, F.Q., A Stable Perfectly Matche Layer for Linearize Euler Equations in Unsplit Physical Variables, Journal of Computational Physics, Vol. 173, No. 3, 2001, pp