A Lagrangian Approach for Computational Acoustics with Meshfree Method

Transcription

1 A Lagrangan Approach for Computatonal Acoustcs wth Meshfree Method Yong Ou Zhang,, Stefan G. Llewellyn Smth, Tao Zhang,*, Tan Yun L School of aval Archtecture and Ocean Engneerng, Huazhong Unversty of Scence and Technology, Wuhan 4374, Chna; zhangyo989@gmal.com (Y. O. Zhang), sgls@ucsd.edu (S. G. Llewellyn Smth), ltyz8@hust.edu.cn (T. Y. L) Department of Mechancal and Aerospace Engneerng, Unversty of Calforna, San Dego, La Jolla, CA 993, USA * Correspondence: zhangt7666@hust.edu.cn; Tel: Abstract: Although Euleran approaches are standard n computatonal acoustcs, they are less effectve for certan classes of problems lke bubble acoustcs and combuston nose. A dfferent approach for solvng acoustc problems s to compute wth ndvdual partcles followng partcle moton. In ths paper, a Lagrangan approach to model sound propagaton n movng flud s presented and mplemented numercally, usng three meshfree methods to solve the Lagrangan acoustc perturbaton equatons (LAPE) n the tme doman. The LAPE splt the flud dynamc equatons nto a set of hydrodynamc equatons for the moton of flud partcles and perturbaton equatons for the acoustc quanttes correspondng to each flud partcle. Then, three meshfree methods, the smoothed partcle hydrodynamcs (SPH) method, the correctve smoothed partcle (CSP) method, and the generalzed fnte dfference (GFD) method, are ntroduced to solve the LAPE and the lnearzed LAPE (LLAPE). The SPH and CSP methods are wdely used meshfree methods, whle the GFD method based on the Taylor seres expanson can be easly extended to hgher orders. Applcatons to modelng sound propagaton n steady or unsteady fluds n moton are outlned, treatng a number of dfferent cases n one and two space dmensons. A comparson of the LAPE and the LLAPE usng the three meshfree methods s also presented. The Lagrangan approach shows good agreement wth exact solutons. The comparson ndcates that the CSP and GFD method exhbt convergence n cases wth dfferent background flow. The GFD method s more accurate, whle the CSP method can handle hgher Courant numbers. Keywords: Lagrangan approach; Lagrangan acoustc perturbaton equatons; computatonal acoustcs; meshfree method; smoothed partcle hydrodynamcs; generalzed fnte dfference method

2 . Introducton Computatonal acoustcs s wdely used n ndustral applcatons to model devces such as fans, automoble manufactures, underwater vehcles, room acoustcs, and so on. Many dfferent models have been proposed and used n these applcatons, such as drect numercal methods [], the lnearzed Euler equatons (LEE) [], acoustc analogy methods (Lghthll s acoustc analogy theory [3], Ffowcs-Wllams and Hawkngs (FW-H) method [4], etc ), and hydrodynamc/acoustc splttng methods (expanson around an ncompressble flow (EIF) model [5, 6], acoustc perturbaton equatons (APE) [7, 8], perturbed compressble equatons (PCE) [9, ], etc...). Euleran approaches have manly been used to mplement numercal approxmatons to these models, and dfferent numercal methods nclude the fnte dfference method (FDM) [], the fnte element method (FEM) [], the boundary element method (BEM) [3], and other modfed or coupled methods [4-6]. These methods have shown ther power n solvng varous acoustc problems so far. However, they are stll not deal for solvng certan acoustc problems, such as transent acoustc problems wth free surfaces, complex materal nterfaces, nhomogeneous meda, movng or deformable boundares, and multphase systems. Some specfc examples are bubble acoustcs [7], combuston nose [8], sound propagaton n multphase flows [9], etc Compared to the Euleran approach whch solves for quanttes at fxed locatons n space, the Lagrangan approach uses ndvdual partcles whch move through both space and tme, and have ther own physcal propertes, such as densty, velocty, and pressure, to represent the dynamcally evolvng flud flow. The flow s descrbed by recordng the tme hstory of each flud partcle. A comparson of two approaches s outlned n Fg.. Fg. Euleran and Lagrangan approaches n numercal computaton. (U stands for computatonal varables, t stands for tme, and + are computatonal ponts) It should be noted that even for a Lagrangan approach, Lagrangan quanttes are

3 usually represented n an Euleran framework, snce equatons wrtten n the Lagrangan descrpton are very awkward for most purposes, and are seldom used []. A comparson between Euleran and Lagrangan descrptons can also be found n []. Consequently, a Lagrangan approach wth Lagrangan quanttes represented by the Euleran descrpton s an alternatve way to deal wth flud dynamcs problems or acoustc problems. The applcaton of Lagrangan approaches n CFD has shown the advantages of such methods n solvng problems wth multphase systems [-3]. However, n the feld of computatonal acoustcs, there have been almost no applcatons of Lagrangan approaches. It s possble to solve the aver-stokes (-S) equatons drectly over all spatal and temporal scales n a Lagrangan approach [4, 5], whch can be vewed as a Lagrangan drect numercal smulaton (DS), but ths Lagrangan DS s as costly as an Euleran DS. Moreover, t s worth notng that a partal Lagrangan method, whch s also a hydrodynamc/acoustc splttng method, has been obtaned prevously: ths partal Lagrangan method, whch consders flud flow wth Euleran quanttes and represents acoustc varables wth a Lagrangan descrpton, had been used n Galbrun s equatons [6, 7] for aeroacoustcs. The perturbaton of the partcle poston due to sound waves s the only Lagrangan varable used n ths partal Lagrangan method. The flow feld s computed based on fxed grds n ths approach, and hence the advantages of the Lagrangan approach mentoned before cannot be used n the flud flow computaton. The purpose of our work s to devse a Lagrangan approach for computatonal acoustcs, whch computes the flud partcle moton and the sound waves based on the acoustc perturbaton assumpton [8, 9]. In ths way, ths approach can consder sound generaton and propagaton wth complex background flow or multple meda, and can be more effcent than the Lagrangan DS method by gnorng scale dspartes snce the flow and acoustcs are treated separately [, 8] as n hydrodynamc/acoustc splttng methods. In addton, we choose meshfree method, nstead of tradtonal mesh based methods to realze the computaton numercally, because dependent varables correspond flud partcles, whch can move wth the flud, nstead of fxed ponts. Meshfree methods has been used to solve the acoustc wave equaton n quescent meda n the tme doman or the Helmholtz equaton n the frequency doman, as n the multplescale reproducng kernel partcle method (RKPM) [9], the element-free Galerkn method (EFGM) [3], the method of fundamental solutons (MFS) [3], etc [3, 33]. When usng a meshfree method, the Lagrangan approach proposed n ths paper not only has the advantages of the Lagrangan property mentoned before, but also has the benefts of the meshfree method tself, such as avodng mesh generaton. 3

4 In ths paper, we have chosen three dfferent meshfree methods to mplement and valdate the Lagrangan acoustc perturbaton equatons (LAPE): smoothed partcle hydrodynamc (SPH), the correctve smoothed partcle method (CSP), and the generalzed fnte dfference (GFD) method, a meshfree method derved from tradtonal fnte dfference methods. These three methods are representatve of meshfree methods n general. The SPH method s a Lagrangan meshfree method frst poneered ndependently by Lucy [34] and Gngold and Monaghan [35] to solve astrophyscal problems n 977. It has been wdely used n dfferent felds as outlned n recent revews [-3, 36]. Detals about the standard SPH computaton can be found n Lu and Lu s book [37], and the CSP method can be found n [38, 39]. To date, ths Lagrangan method has shown ts advantages n flud dynamcs smulatons, and the Lagrangan partcle trackng method has been wdely used n modellng bubble motons [4], free surface [4], dust [4], and other problems. In the prelmnary work, we have gven a compact dscusson concerns the numercal performance of the SPH method n solvng acoustc wave equatons n quescent meda [43-45]. The GFD method, also called the meshless fnte dfference method, was frst dscussed for fully arbtrary meshes by Jensen [46] n 97. Perrone and Kao [47] also contrbuted to the development of ths method at that tme. Subsequently, a varaton of the GFD method usng the movng least squares approxmaton was proposed by Lzska and Orksz [48]. Recently, Bento, Urena and Gavete gave a dscusson about the nfluence of several factors n the GFD method, and a comparson between the GFD method and the EFGM n solvng Laplace equaton s also presented n [49-5]. The GFD method shows more accurate than the EFGM. In addton, the GFD method was appled to solve acoustc wave equatons n quescent meda recently [5]. The present paper s organzed as follows. In secton, the LAPE are obtaned. In secton 3, the SPH and CSP formulatons for smulatng acoustc waves n a movng flow s gven based on the LAPE. In secton 4, the GFD formulaton for computng LAPE s shown. Several test cases are gven n order to valdate the numercal method n secton 4, whle secton 5 summarzes the results of ths work.. Lagrangan acoustc perturbaton equatons The fundamental assumptons and equatons are presented frst, and then the LAPE for computatonal acoustcs are obtaned.. Fundamental assumptons and equatons The basc dea of the hydrodynamc/acoustc splttng method s to splt the background flow and acoustc perturbaton based on dfferences n ampltude. Detals of ths approach 4

5 can be found n [8, 9]. Followng ths dea, consder a flud partcle whch has physcal propertes: P= p + δ p, δ p << p, () ρ = ρ + δρ, δρ << ρ, () v = v + δv, δv << v, (3) S = S + δs, δs << S, (4) where P, ρ, v, and S are the pressure, densty, velocty, and entropy of the flud partcle respectvely; p, ρ, v, and S are the components n the absence of sound, whle δ p, δρ, δv, and δ S are the perturbatons of correspondng parameters due to acoustc waves. All these varables are supposed to be the physcal propertes of a flud partcle whch can move through both space and tme, rather than at a fxed poston. evertheless, as mentoned n the ntroducton, all these parameters are represented n Euleran coordnates. The Lagrangan form of the flud dynamcs equatons s gven n Chapter 4 of Lu and Lu s book [37]. The flud dynamcs equatons ncludng gravty can be wrtten as: the contnuty equaton Dρ = ρ v, (5) the momentum equaton Dv = P + g, (6) ρ the equaton of state P = P( ρ, S), (7) where the materal or Lagrangan dervatve s wrtten as D = + v. (8) t ow we consder the sound waves n an nhomogeneous medum wthout the effects of vscosty. Sound propagaton can be modelled as an adabatc process, and vscosty, dffuson and thermal conductvty can be neglected. Let c be the sound velocty; then the equaton of state, accordng to Chapter 4 n Brekhovskkh and Godn s book [8], becomes DP Dρ = c, (9) where 5

6 P c = c( ρ, S) = ( ) S ρ. () For adabatc processes we have DS =. () Eq. (5), Eq. (6), and Eqs. (9)-() are a closed system for equatons of flud dynamcs. For an deal gas, Eq. () can be replaced by c γ P where γ s the rato of specfc heats (γ =.4 for ar). =, () ρ. Equatons for background flud moton ow we consder the flow wthout sound waves. The usual hydrodynamc equatons nclude gravty can be wrtten as Dρ = ρ v, (3) D v = p + g, (4) ρ where the materal dervatve s D p = p ( ρ, S ), (5) = + v. (6) t For an adabatc process neglectng vscosty, dffuson, and thermal conductvty, we have the equaton of state where Dp Dρ = c, (7) p = ( ρ, ) = ( ) S ρ c c S. (8) For an adabatc process we have DS =, (9) where c s the sound speed not nfluenced by the acoustc wave, wth c = c + δ c. For an deal gas, Eq. (8) can be replaced by 6

7 c γ p =. () ρ.3 Lagrangan acoustc perturbaton equatons Consderng acoustc perturbatons generated by sound, and substtutng Eqs. ()-(3) nto the contnuty equaton (Eq. (5)), the momentum equatons (Eq. (6)), and the equaton of state (Eq. (9)) leads to D( ρ + δρ) = ( ρ + δρ) ( v + δ v ), () D( v + δ v) = ( p + δ p) + g, () ( ρ + δρ) D ( p + δ p ) ( ) ( c c ) D ρ + = + δ δρ. (3) Rearrangng these equatons usng the hydrodynamc equatons (Eqs. (3), (4), and (7)), we obtan the LAPE as Dδρ = ρ δv δρ ( v + δv) ( δv )( ρ + δρ), (4) Dδ v δρ = δ p ( δv )( v + δv ) + ( p + δ p), (5) ρ ρ ρ δρ ( + ) Dδ p Dδρ c = δv ( p + δ p) + c( δv )( ρ + δρ), (6) Dρ Dδρ + δc + + ( δv )( ρ + δρ) where the materal dervatve used s Eq. (6). These equatons are exact. Ignorng the second-order terms, leads to lnear equatons for sound waves. In order to take nto account sound sources, two terms are added to the rght hand sde of equatons, and the lnearzed LAPE (LLAPE) are Dδρ = ρ δv δρ v ( δ v ) ρ + δρa, (7) D δ v δ p δ δρ p δ f = v v + +, (8) ρ ρ ρ ( ) where D δ p D D c δρ = c( δv ) ρ ( δv ) p + δc ρ, (9) δρ a represents a volume velocty generated by a source, and δ f ρ s a force 7

8 densty [8]. Ths corresponds to a sngular source n the flud, whch can be useful for modellng purposes. Here δ c be the perturbaton of the square of sound velocty caused by the sound wave. In order to obtan δ c, we expand Eq. (8) as c ( ρ, S) c ( ρ, S) c = c( ρ, S) + δρ+ δs+ ρ S. (3) Snce c s known from Eq. (8), ths equaton becomes c ( ρ, S) c ( ρ, S) δc = δρ + δ S ρ S. (3) For an deal gas, Eq. (3) becomes c γ ( p + δ p) + δ c =, (3) ( + δρ) ρ and δc γ( ρ δ p pδρ) γδ p c δρ = =, (33) ( ρ + δρ) ρ ρ where the frst fracton s an exact result and the last s lnearzed. It can be seen that the left hand sde of Eqs. (3), (4), (4), and (5) are the materal dervatves of the varables ρ, v, δρ, δv whle the rght hand sde terms only have spatal dervatves. So, f the spatal dervatve at each moment n tme can be computed wth the meshfree method, we can solve these varable values correspondng to flud partcles by temporal ntegraton of the materal dervatve. Followng ths, the background moton of the flud partcle can be obtaned from Eqs. (3), (4), and (7)-(9), and the acoustc varables of each partcle can be obtaned from Eqs. (7)-(3). In ths way, we have a Lagrangan approach to solve the LLAPE..4 Equatons for dfferent cases.4. Sound propagaton n steady unform flow The LAPE are now appled to determne sound propagaton n steady unform flow. A general unform flow gnorng gravty can be represented as ρ = constant, v = constant, p =. (34) Substtutng nto Eqs. (7)-(9), the LLAPE for steady unform flow become Dδρ = ρ δv + δρa, (35) 8

9 Dδ v δ = δ p + f, (36) ρ ρ Dδ p Dδρ = c. (37) In quescent meda, Eqs. (35)-(37) leads back to acoustc wave equatons..4. Sound propagaton n ncompressble steady flow Let sound waves travel through ncompressble steady flow wth constant densty. Solutons of Eqs. (3), (4), and (7) gnorng gravty are obtaned from ρ =constant, v =, v v = p. (38) ρ The correspondng lnearzed Lagrangan acoustc wave equatons for ncompressble steady flow are Dδρ = ρ δv + δρa, (39) D δ v δ p δρ p δ f = + +, (4) ρ ρ ρ ( ) D δ p D δρ + δ v p = c. (4) 3. SPH and CSP Equatons Snce the SPH method s a famous and wdely used meshfree method n computatonal flud dynamcs, the LAPE and LLAPE are solved usng SPH and a correcton to t, CSP, n ths secton. 3. Basc concepts of SPH SPH s a meshfree Lagrangan partcle method. Functons n the SPH method are represented by a partcle approxmaton. Some basc concepts are presented n ths secton. A functon f (r) can be represented as < f() r >= f( r ) W( r r', h)dr, (4) Ω where f s a functon of the vector r, Ω s the volume of the ntegral, W s the smoothng kernel, and h s the smoothng length whch represents the sze of the support doman. The kernel approxmaton operator s denoted by the angle bracket <>. The partcle approxmaton for the functon f (r) at partcle can be wrtten as 9

10 m < f ( r) >= f( r ) W, (43) = ρ where r and r are the postons of partcles and, s the number of partcles n the computatonal doman, m s the mass of partcle, W = W( r, h) and r s the dstance between partcle and partcle as shown n Fg.. Fg. Flud partcles, support doman and the kernel functon. The spatal dervatve f ( r ) s smlarly obtaned as m < f ( r) >= f( r ) W, (44) = ρ where r r W = r W r and r = r r. Ths can also be derved for the dvergence of a vector feld. The cubc splne functon gven by Monaghan and Lattanzo [53], whch s wdely used n the computatonal flud dynamcs, s chosen as the smoothng kernel n the present paper. 3. SPH equatons for background flow Partcle approxmaton for flud dynamc equatons (Eqs. (5), (6), and (9)) n the absence of forces are gven n [37]. The partcle approxmaton used for the hydrodynamc equatons ncludng gravty (Eqs. (3), (4), and (7)) can be obtaned n a smlar way. The results are D m ρ = ρ v, (45) W = ρ

11 Dv p + p = + g, (46) m W = ρρ Dp Dρ = c, (47) where p, ρ, and v are the pressure, densty, and velocty of the flud partcle n the absence of sound, correspondng to p, ρ, and v n secton, v = v v, and denotes the gradent of the kernel taken wth respect to the coordnate of partcle. Other forms of (45) and (46) are mw = ρ =, (48) Dv p p = m + W + g. (49) = ( ρ) ( ρ) 3.3 LLAPE n SPH form Applyng the SPH partcle approxmaton equaton (Eq. (44)) to the LLAPE (Eqs. (7)- (9)) yelds D m m δρ = ρ δ δρ v W v W = ρ = ρ m δv ρ + δρ W a = ρ, (5) Dδ v m m = ρ ρ ρ δ p W δv v W = = δρ m δ f + p + ( ρ ) ρ ρ W =, (5) D δ p D δρ v m v m D ρ,(5) c = cδ ρ W δ p W + δc = ρ = ρ where δ p, δρ, and δ v are the perturbatons of pressure, densty, and velocty of the flud partcle due to acoustc waves. Consderng the partcle approxmaton of the gradent of the unty and the transformaton method presented n [37], the partcle approxmaton equatons of the contnuty and momentum equatons of acoustc waves (Eqs. (5) and (5)) can be rewrtten

12 as D m m δρ = ρ δ + δρ v W v W = ρ = ρ m δv ρ + δρ W a = ρ, (53) Dδ v δ p + δ p m = + m W δ v v W = ρρ = ρ δρ p + p δ f + + ρ ρ ρ ρ m W =. (54) For sound propagaton n steady unform flow, the SPH form of the LLAPE s D m δρ = ρ δv + δρ, (55) Dδ W a = ρ δ p + δ p v = m W + f, (56) = ρρ ρ Dδ p D δρ δ = c. (57) 3.4 CSP method The CSP method mprove the partcle approxmaton n the standard SPH method usng Taylor seres. The Taylor seres expanson of a functon f (r) at a poston ξ can be represented as α α α α β β f( r) = f( ξ) + ( r ξ ) fα( ξ) + ( r ξ )( r ξ ) fαβ( ξ ) +..., (58)! where α, β =,, 3 are the dmenson and f α f ( ξ ) ( ξ ) =, (59) r α ( ) ( ξ f ξ ) =. (6) r r f αβ α β Multplyng both sdes of Eq. (58) by the smoothng functon W (r - ξ) yelds α α f ( r) W( r-ξ, h) dr= f( ξ) W( r-ξ, h) dr+ ( r ξ ) f ( ξ) W( r-ξ, h) dr Ω Ω Ω ( α α )( β β + r ξ r ξ ) fαβ ( ξ ) W ( r -ξ, h ) dr+...! Ω Ignorng all dervatve terms, f (ξ) can be represented as α.(6)

13 < f ( ξ ) >= Multplyng both sdes of Eq. (58) wth W (r - ξ) yelds Ω f() r W( r ξ, hd ) r = f() ξ W( r ξ, hd ) r Ω Ω f ( r) W( r -ξ, h) dr W( r-ξ, h) dr. (6) Ω α α + ( r ξ ) f ( ξ) W( r ξ, h) dr Ω α ( α α )( β β + r ξ r ξ ) fαβ ( ξ ) W ( r ξ, h ) dr! Ω, (63) +... where W( r ξ, h) W( r ξ, h) =. (64) r eglectng second and hgher dervatve terms, fα (ξ) can be represented as < f ( ξ ) >= [ ξ ] Ω α α α f () r f() W( r ξ, h) dr. (65) ( r ξ ) W( r ξ, h) dr Ω Followng a smlar dervaton to that n the standard SPH method, the partcle approxmatons for Eqs. (6) and (65) at partcle become m f ( r) W = ρ f ( r ) =, (66) m W ρ = f α ( r ) = = m f ( ) f ( ) W ρ r r m α α ( r r ) W ρ =. (67) 3.5 Partcle moton and tme ntegraton In our Lagrangan approach, calculaton of the partcle moton and tme ntegraton s performed wth a second-order leapfrog ntegraton [54]. In ths scheme, the equatons for updatng the poston of partcles and the velocty are D v () t v( t+ Δ t) = v ( t Δ t) +Δt, (68) 3

14 r( t+δ t) = r( t) +Δ tv ( t+ Δt), (69) where v ( t+ Δt) s the velocty of flud partcle at tme t+ Δ t, t s the tme of prevous tmestep, and Δ t s the tme step. Eq. (68) starts wth the ntal velocty offset gven by D v () v( Δ t) = v () + Δt. (7) 4. GFD Equatons The consstency problem always lmt the accuracy of SPH method [, 55], and the gradent of kernel functon have some specal requrements on selectng a kernel functon [37]. So, n ths secton, we show another meshfree method, namely the GFD method, whch only use the kernel tself. Ths method reduces to standard fnte dfference methods (FDM) on an orthogonal grd, and can be easly extended to hgher orders [56]. 4. Spatal dervatve approxmaton by GFD Consder a partcle surrounded by partcles =,,,, wth all + of them are n a compact support doman. The value of an nfntely dfferentable functon F at the poston of partcles and are defned as F and F. Let us expand n the Taylor seres of F at the partcle : F h F F = F + h + +, (7) x x F F h F k F F F = F + h + k h k +, (7) x y x y x y where Eqs. (7) and (7) are for D and D, respectvely, wth h = x - x, and k = y - y. Ignorng hgher order terms, the approxmaton of F s denoted by f : f f h f x = + + h f x, (73) f f h f k f f f = f + h + k h k x y x y x y. (74) After rearrangng these equatons, the sum of these expressons for all partcles wth multplyng weght functons are obtaned: 4

15 f h f ( f f + h + ) W( h, ) h = x x, (75) = f f f f + h + k x y W( h,, ) k h =, (76) = h f k f f hk x y x y where W(h, h) and W(h, k, h) are weght functons n D and D respectvely, and h represents the sze of the support doman, whch s always called smoothng length n the SPH method (see secton 3.). For a crcular shape doman, h stands for the radus of the support doman. In order to compare wth the SPH method, the closest nodes to the partcle are selected as partcles, and these partcles should be n the support doman at the same tme. In addton, cubc splnes are selected n both SPH, CSP, and GFD methods as the weghtng functon. A comparson of dfferent crterons of partcle selecton and dfferent weghtng functons n the GFD method s gven n [47]. In the followng equatons, we use W for the weghtng functon. Defnng functons B n D and B5 n D as and f h f B ( f) = ( f f + h + ) W, (77) = x x f f f f + h + k x y B5 ( f) = W. (78) = h f k f f hk x y x y Accordng to Eqs. (75) and (76), the norms B and B5 equal to, so we obtan: B ( f) = ΦhW = f, (79) = ( ) x B ( f) = ΦhW =, (8) f = ( ) x B ( f) 5 = Φ5hW = f, (8) = ( ) x B5 ( f) = Φ5kW = f, (8) = ( ) y 5

16 B ( f) 5 = Φ5hW =, (83) f = ( ) x B ( f) 5 = Φ5kW =, (84) f = ( ) y B ( f) 5 = Φ5hkW =, (85) f = ( ) xy where Φ f f = f f + h + x h x, (86) where Φ h k x y x y x y. (87) f f f f f 5 = f f + h + k h k Eqs. (79)-(85) gve us the followng equatons: A 5 = AD u = b, (88) AD 5 u5 = b, 5 (89) 3 hw hw = = A =, (9) 3 4 hw hw = = 4 3 hw hkw hw hkw hkw = = = = = 3 hkw kw hkw kw hkw = = = = = hw hkw hw hkw hkw = = = 4 = 4 = hkw kw hkw kw hkw = = = 4 = 4 = 3 3 hkw hkw hkw hkw hkw = = = = = 6 T,(9) f f D u =,, (9) x x

17 f f f f f D u5 =,,,,, (93) x y x y x y ( f f) hw = b =, (94) h ( f f) W = ( f f) hw = ( f f) kw = h b ( ) 5 = f f W. (95) = k ( f f) W = ( f f) hkw = Snce the matrces A and A5 are symmetrcal, Eqs. (88) and (89) can be solved usng Cholesky factorzaton. In ths way, the spatal dervatve can be obtaned and the materal dervatve n LAPE can be ntegrated usng the tme ntegraton scheme. T 4. GFD equatons for background flow The coeffcents of D and b are denoted by Dm(f) and bm(f) wth m =,, and the coeffcents of D5 and b5 are denoted by D5m(f) and b5m(f) wth m =,, 5. For example, D f ( f ) =, x b h ( f ) = ( ) f f W, ( ) D5 f = y f =, and b 5 ( f) = ( f f) kw. = Besdes, the matrces A and A5 has a decomposton n upper and lower trangular matrces T T A = LL and A 5 = 5 5 L L. The coeffcents of the matrces L and L5 are denoted by L(m, n) wth m, n =, and L5(m, n) wth m, n =,, 3, 4, 5. For example, the matrces A s wrtten as L(,) L(,) L(,) A =. (96) L(,) L(, ) L(, ) By usng the GFD method and Cholesky factorzaton to solve Eqs. (88) and (89), we obtan the solutons for Eqs. (3) and (4) n one dmenson from 7

18 Du In two dmensons, we have Dρ = ρ ( ) D u, (97) = D( p ) +g. (98) ρ Dρ = ρd5( u) ρd5( v), (99) Du = D5( p ), () ρ Dv = D5( p ) +g, () ρ where u and v are the velocty of partcle along two drectons, and where D D 5 Y [ b ( f) L (,) L (,) D( f) ] = L (,) ( f) =, () L(,) b( f) b ( f) = L(,) L(,) ( f) 5 Y5 f L5 D5 f = L5 (,) =+ = Y55( f) = L5 (5,5) 5 5 ( f) = ( ) (, ) ( ),,3,4 5, (3) b5 ( f) = L (,). (4) b5( f) L5(, ) Y5( f) =,3,4,5 L5 (,) = For an deal gas, the equaton of state s Eq. (). 4.3 LLAPE n GFD form The GFD form of the LLAPE (Eqs. (7) and (8)) gnorng the force densty s gven n ths secton as an example. The GFD formulaton n one dmenson s Dδρ = ρd( δu) δρd( u) δud( ρ) + δρa, (5) Dδu = δρ D ( δ p ) δu D ( u ) D ( p ) ρ +, (6) ρ 8

19 Dδ p Dδρ c = cδud( ρ) δud( p) δc ρd( u). (7) In two dmensons, we have Dδρ = ρ δ + δ δρ + δud ( ρ ) δvd ( ρ ) + δρa [ D ( u ) D ( v )] [ D ( u ) D ( v )] , (8) Dδu = δρ D ( δ p ) δu D ( u ) δvd ( u ) D ( p ) ρ +, (9) ρ Dδv = δρ D ( δ p ) δu D ( v ) δvd ( v ) D ( p ) ρ +, () ρ Dδ p Dδρ c = c u D + vd δud ( p) δvd ( p) [ δ ( ρ ) δ ( ρ )] [ ( ) ( )] δ ρ + c D5 u D5 v, () where δ u and v δ are the velocty perturbaton of partcle along two drectons. For an deal gas, the equaton of state s Eq. (33). 5. umercal Tests and Dscusson In ths secton, dfferent test cases wth applcatons to modelng sound propagaton n movng fluds are outlned. 5. D Sound propagaton n steady unform flow Sound propagaton n steady unform background flow s examned n ths secton. The test example s orgnally provded n the Fourth Computatonal Aeroacoustcs Workshop on Benchmark Problems [57]. Consder sound propagaton n a D homogeneous, steady unform movng medum. The ntal condtons are [ ] δu( x, t = ) = αc + cos( kx) exp α( kx), () [ ] δ p( x, t = ) = αρc + cos( kx) exp α( kx), (3) where [ ] δρ( x, t = ) = αρ + cos( kx) exp α ( kx), (4) δ u s the velocty perturbaton along the x-axs, t represents tme, x s the geometrc poston, k s the wave number, and α = 4/34, α = ln /. 9

20 The flud s an deal gas wth sound speed c 34 m/s and densty. kg/m 3. The exact soluton for the LLAPE at tme t s [ ] δu( x, t) = αc + cos( kx wt) exp α( kx wt), (5) [ ] δ p( x, t) = αρc + cos( kx wt) exp α( kx wt), (6) [ ] δρ( x, t) = αρ + cos( kx wt) exp α ( kx wt). (7) In the model, sound waves propagate along the x-axs over a tme nterval of.5 s, w s the angular frequency of 34 rad/s, and k s set as. rad/m. The computatonal doman spans -5 m to 5 m. Snce the computatonal regon s far longer than the sound propagaton dstance, Drchlet boundary condtons are appled to both ends of the computatonal regon. The sound pressure and perturbatons of densty and velocty for the last three partcles at each end are set as, whch are ther ntal value. In the computaton, the partcle spacng ( x) s set to be.5 m, the Courant-Fredrchs- Lewy (CFL) number s.5, and the smoothng length (h) for the smulaton s.45 x. The CFL number s defned as Δx Δ t = CCFL, (8) c where x s the tme step and CCFL s the CFL number. CSP results at tme t are compared wth exact solutons n Fg. 3, and three test cases wth the movng medum at dfferent Mach numbers are shown. Computatonal results are plotted wth ponts at ntervals of 5 grd ponts, whle exact solutons are plotted wth sold lnes. The Mach number of the background flow changes from -. to.. Results from the SPH and GFD methods are not gven here because they are so smlar to the CSP method that the dfference cannot be dstngushed n the fgure.

21 Fg. 3 Sound pressure comparson between CSP and exact solutons for the LLAPE wth the movng medum at dfferent Mach numbers. The fgure shows that sound pressure ampltude and tendency obtaned by the CSP method agree well wth exact solutons, and the CSP method s valdated for ths problem. To evaluate the convergence of the three meshfree methods, a non-dmensonal sound pressure error (εpre) s used to dscuss the effects of x and CCFL. εpre s defned as: ε = c pre 3αρ p p( x), (9) = where s the number of partcles n the computatonal doman, p s the numercal solutons of sound pressure for partcle, p(x) s the theoretcal sound pressure at the poston x, and x s the poston of partcle. Convergence and CPU tme (tcpu) curves for SPH, CSP, and GFD solutons of LLAPE are shown n Fg. 4 and Fg. 5. The exact solutons for LLAPE come from Eqs. (5)-(7). All computatonal parameters stay the same as n the model for Fg. 3 except x. All computaton n the present paper were performed wth processor Intel(R) Core(TM) 7-47MQ CPU.5 GHz and RAM 8. GB. Accordng to our computatons, the Mach number has lttle effects on the error, so only one Mach number s used n the followng comparson.

22 Fg. 4 Convergence curves for SPH, CSP, and GFD solutons of LLAPE at Ma =.. Fg. 5 CPU tme curves for SPH, CSP, and GFD solutons of LLAPE at Ma =. at dfferent x. As shown n Fg. 4, all three meshfree methods are convergent for ths problem. The GFD method has better accuracy for dfferent x compared wth the other two methods, whle the SPH and CSP methods are smlar. As x ncreases from.3 to.9 m, εpre for the GFD method ncreases from around. -5 to.5-4, and εpre for the SPH and CSP methods grow from about. -5 to over.5-4. However, the decrease of x not only brngs accuracy mprovements, but also ncreases the tme of computaton accordng to Fg. 5. The GFD method has the hghest tme cost, whle the SPH method has the lowest cost. The graph shows a fvefold ncreases for all three methods n tcpu when x decreases from.9 to.3 m. The effects of CCFL on εpre and tcpu for the three meshfree methods solvng the LLAPE at Ma =. are shown n Fg. 6 and Fg. 7. All computatonal parameters are the same as n

23 the computaton for Fg. 3 wth only CCFL changng. Fg. 6 ε pre for SPH, CSP, and GFD solutons of LLAPE at Ma =. wth dfferent C CFL. Fg. 7 CPU tme curves for SPH, CSP, and GFD solutons of LLAPE at Ma =. wth dfferent C CFL. Fg. 6 shows that accurate numercal results depend on an approprate choce of CCFL. When CCFL s small enough, εpre remans small. As CCFL ncreases, there s a rapd growth n εpre. The value of CCFL to keep the computaton stable s dfferent for the three methods. Accordng to the graph, the maxmum CCFL value for the SPH, CSP, and GFD methods s about.8,.35, and.3 respectvely, and for larger CCFL convergence s lost. Fg. 7 shows that small CCFL leasds to notceable growth n tcpu, and the GFD method requres more tcpu than other two methods. For the three mehfree methods, as CCFL changes from.5 to.4, tcpu decreases from around 3 s to 5 s. When CCFL s small, ths tendency s more obvous then at large CCFL. The effects of h/ x on numercal solutons of LLAPE at Ma =. are shown n Fg. 8. 3

24 In the test, h/ x changes from. to.7 correspondng to 4 partcles n the support doman, whch s suggested for the SPH method [37]. Fg. 8 ε pre for SPH, CSP, and GFD solutons of LLAPE wth dfferent h at Ma =.. Accordng to the graph, t can be seen that h/ x affects the SPH results a lot, whle t has a mnmal effect on the accuracy of the CSP and GFD results. For SPH, the mnmum εpre s obtaned when h/ x equals.45. For other cases, εpre rapdly changes to around On the other hand, the CSP and GFD solutons are accurate and stable, and the GFD method s more accurate than the CSP method. Both results have εpre below. -4. Snce dfferent cases cost smlar tcpu, a comparson of tcpu of dfferent methods s not gven here. For further evaluaton, Table compares the accuracy and effcency of the three meshfree methods n solvng LAPE and LLAPE. Snce there s no theoretcal soluton for LAPE, the exact solutons come from DS. In the DS computaton, we use a 4th order fnte dfference scheme for specal dervatve and a 4th order Runge-Kutta tme ntegraton scheme. In order to valdate the DS computaton, t s used to solve LLAPE, and the results are compared wth theoretcal solutons (Eqs. (5)-(7)). The data show that εpre of the DS method s lower than. -8, whch s close enough to be consdered exact. Consderng dfference between the DS mesh and meshfree grds, there s maybe no DS soluton correspondng to a flud partcle n the meshfree computaton. The blnear nterpolaton s used to nterpolate the DS data to a value correspondng to a flud partcle. Table Comparson of SPH, CSP, and GFD methods n solvng LAPE and LLAPE at dfferent Mach numbers. Mach umber Equatons SPH CSP GFD εpre tcpu εpre tcpu εpre tcpu ( -5 ) (s) ( -5 ) (s) ( -5 ) (s) 4

25 LAPE Ma = -. LLAPE LAPE Ma =. LLAPE LAPE Ma = +. LLAPE As s shown n the table, both LAPE and LLAPE are valdated to be useful n solvng problems wth acoustc propagaton n steady unform flow. Compared wth LAPE, LLAPE can save some computatonal tme wth acceptable loss n the accuracy. The table also ndcate that three meshfree methods are successful n numercal mplementaton of the Lagrangan approach. In solvng LAPE, smlarly to the prevous results of solvng LLAPE n Fg. 4 and Fg. 5, the GFD method show a hgher accuracy, and the SPH method cost less computatonal tme. In addton, Dfferent Mach numbers of the unform flow do lttle effects on εpre and tcpu. 5. D Sound propagaton n unsteady flow After analyzng sound propagaton n steady unform flow, the same analyss s used to evaluate the Lagrangan approach n modellng sound propagaton n unsteady flow. Theoretcal solutons for a D unsteady background flow s gven n Chapter 3 n [58]. Consder sound propagaton n a D unsteady flow wth u = x C γ + t +, () x C ( γ ) + C + + γ ( γ + ) t ( γ ) = + t + γ + ( γ + ) t p C3 C 3γ γ C ( γ 3) γ + x ρ = x C ( γ + ) C + + ( γ ) ( γ ) t + ( γ ) C 3γ C ( γ 3) ( γ ) / ( γ+ ) x + t + γ + ( γ+) t γ x 3 γ c = C ( γ ) Ct γ + t γ + ( γ ) ( γ ) / + γ/ ( γ ) ( γ ) /, (), (), (3) 5

26 where u s the ntal velocty of background flow, C = 3., C = -. 6, C3 = 857., γ =.4, and the ntal tme ( t ) s at.5 s. By selectng these values, the ntal velocty of background flow s about 4 m/s, and the sound speed s around 34 m/s. Both depend on poston. Sound waves propagate along the x-axs over a tme nterval of.5 s, and the ntal sound waves are set as Eqs. ()-(4). The computatonal doman spans -5 m to 5 m. The same Drchlet boundary condtons to secton 5. are appled to both ends of the computatonal regon. In the computaton, Δx s set to be.5 m, CCFL s.5, and h s selected as.45δx. A comparson of pressure obtaned from CSP results n solvng LLAPE and exact solutons s gven n Fg. 9. Fg. 9(b) gves a detaled vew of the dashed box n Fg. 9(a). The CSP results are plotted wth ponts at ntervals of 3 grd ponts n fgure (a) and 5 grd ponts n fgure (b), and the exact solutons are plotted wth sold lnes. The exact solutons come from DS results, and data for the CSP results n the graph s the sum of CSP solutons of sound pressure and p from Eq. (). The graph llustrates that CSP results get good agreement wth exact solutons n solvng LLAPE. (a) pressure at dfferent tmes 6

27 (b) detaled vew Fg. 9 Pressure of CSP solutons of LLAPE versus exact solutons. (a: pressure at dfferent tmes; b: detaled vew of dashed box n (a)) For further evaluaton, convergence and CPU tme curves for SPH, CSP, and GFD solutons of LLAPE are shown n Fg. and Fg.. All computatonal parameters stay the same as n the model for Fg. 9 except x. Fg. shows that the SPH method does not converge, whle the CSP and GFD methods do. Ths results consstent wth the analyss n [59], whch gves an error estmaton for four schemes ncludng both the standard SPH method and the CSP method. For rregular partcle arrangement, [59] shows that the CSP method converges as x decreases, whle convergence for the standard SPH method requres both x and h / x. Ths concluson can also be found n [6]. On the other hand, for the CSP and GFD methods, εpre decreases along wth x. The GFD method has a better accuracy for dfferent x compare to the other methods. 7

28 Fg. Convergence curves for SPH, CSP, and GFD solutons of LLAPE. Fg. CPU tme curves for SPH, CSP, and GFD solutons of LLAPE at dfferent x. A decrease of x ncreases the tme cost as can be seen from Fg.. The GFD method has the hghest tcpu cost, whle the SPH method has the lowest. evertheless, the dfference s small between the three methods. tcpu ncreases from less than s to about 6s when x decreases from.9 m to.3 m. To evaluate the effects of CCFL, a comparson of εpre from dfferent CCFL computaton s shown n Fg.. When CCFL s small enough, εpre stays stable for the three methods. When CCFL ncreases, εpre dverges. The value of CCFL to mantan a small and stable εpre for the SPH, CSP, and GFD methods s around.3,.3, and. respectvely. When CCFL s smaller than these values, the GFD method s more accurate than other two methods. A comparson of tcpu from dfferent CCFL computaton s shown n Fg. 3. Large CCFL reduces tcpu, and the rate of change slows down as CCFL ncreases. The GFD method has the 8

29 hghest tme cost, but the dfference among three methods s small. Increasng CCFL from.5 to.4 decreases tcpu from around 6 s to less than s. Fg. ε pre for SPH, CSP, and GFD solutons of LLAPE wth dfferent C CFL. Fg. 3 CPU tme curves for SPH, CSP, and GFD solutons of LLAPE wth dfferent C CFL. The effects of h/ x on numercal solutons of LLAPE are shown n Fg. 4. The lne graph shows that εpre from the SPH method s larger than other two methods, and h/ x affects the SPH results sgnfcantly. The most accurate case corresponds to h/ x of.45. The CSP and GFD methods mantan ther accuracy well wth dfferent h/ x. The GFD method s more accurate than the CSP method, but the dfference s small. Snce a sutable h s unknown before the computaton, the CSP and GFD methods appear more practcal than the SPH method. 9

30 Fg. 4 ε pre for SPH, CSP, and GFD solutons of LLAPE wth dfferent h. For further evaluaton, Table compares εpre and tcpu of the SPH, CSP, and GFD methods n solvng LAPE and LLAPE wth unsteady background flow. Table Comparson of SPH, CSP, and GFD methods n solvng LAPE and LLAPE wth unsteady flow. SPH CSP GFD Equatons εpre tcpu εpre tcpu εpre tcpu ( -5 ) (s) ( -5 ) (s) ( -5 ) (s) LAPE LLAPE The table ndcates that both LAPE and LLAPE are able to solve acoustc problems wth unsteady background flow. In addton, LLAPE save much computatonal tme wth acceptable loss n the accuracy comparng wth LAPE. In solvng LAPE, smlarly to the prevous results of solvng LLAPE n Fg. and Fg., the GFD method s more accurate than the other two methods, whle the SPH method cost the least computatonal tme. 5.3 D sound propagaton n steady unform flow In ths case, a D acoustc model s selected to valdate the Lagrangan approach. Ths model s smlar to the problem gven n the ICASE/LaRC workshop concernng benchmark problems n computatonal aeroacoustcs (CAA) [6]. In addton, the standard SPH method s not dscussed n D problems, because t has more lmtaton to mantan convergence and the selecton of h s also more dffcult than the CSP and GFD methods n the applcaton as shown n Secton 5. and 5.. Consder sound propagaton n steady unform flow wth Ma =.. Let α = 34, 3

31 α = ln 9, 3 ln 7 ntal condtons are α =, ( x y ) η = +, where x and y are the poston coordnates. The 3 ( ) δ p = α ρc exp α x + y, (4) ( ) (( ) ) ( ) ( ) ( ) δρ = αexp α x + y +.αexp α3 x + y, (5) δu =.4αcyexp α3 x + y, (6) ( ) ( ) δv =.4αc x exp α3 x + y, (7) where δu and δv are the velocty perturbaton of partcles along the x-axs and y-axs respectvely. The flud s an deal gas wth sound speed c 34 m/s and ρ. kg/m 3. The exact solutons for the sound waves when tme equals t are αρc δ p = ξt J ξη ξdξ ξ exp cos( ) ( ) α 4α, (8) +.αexp α3 + α ξ δρ = exp ξ ξη ξdξ α 4α cos( t) J ( ) (( x ) y ), (9) αcx ξ δu = exp sn ( ξt) J ( ξη ) ξdξ αη 4 α, (3) +.4αcy exp α3( ( x ) + y) αcy ξ δv = exp sn ( ξt) J ( ξη) ξdξ αη 4 α.4αc( x ) exp α3 ( x ) + y ( ), (3) where J( ) and J( ) are Bessel functons of order and respectvely. In the smulaton, x s set as. m, CCFL s selected as., and h s. x. The boundary partcles around the computatonal doman are taken to be Drchlet boundary wth perturbaton varables set to. A model wth larger computatonal doman s mplemented to show the process of sound propagaton n the unform flow at frst, then a smaller model s used for the evaluaton n order to save the tme. The CSP results for the change n densty at dfferent tme steps are shown n Fg. 5. In ths smulaton, the computatonal doman s -8 m < x < 8 m, -8 m < y < 8 m. Durng

32 the computaton, all computatonal partcles are movng along wth the background flow, and sound waves propagate n the meda at the same tme. The drecton of the background flow s represented by an arrow. Computatonal results show that the CSP method can smulate the propagaton process of Gaussan mpulses. (a).5 s (b).5 s (c).75 s 3

33 (d). s Fg. 5 Contour of densty change obtaned by the CSP method at dfferent tme. To evaluate the computatonal results, Fg. 6 gves a comparson between CSP results and DS solutons along the x-axs. In ths smulaton, the computatonal doman s -4 m < x < 4 m, -4 m < y < 4 m, and the tme for sound propagaton s 5 ms. The ponts n the fgure represent the numercal values at dfferent postons, and the lnes denote the exact solutons. In order to clearly dentfy numercal results, the ponts are plotted at ntervals of 5 grd ponts. Snce the GFD results are smlar to Fg. 6, they are not shown n here. As can be seen from the lne graph, numercal results of all perturbaton varables agrees well wth exact solutons. The comparson suggests that the CSP results are accurate. (a) densty change 33

34 (b) sound pressure (c) velocty perturbaton Fg. 6 Comparson between CSP results and exact solutons along the x-axs at.5s. Convergence and CPU tme curves for CSP and GFD solutons of LLAPE are shown n Fg. 7 and Fg. 8. As can be seen from the fgure, both the CSP and GFD methods converge when solvng LLAPE. The GFD method has better accuracy for dfferent x compared to the CSP method. εpre for the two meshfree methods ncreases from about. -6 to more than. -4 along wth x ncreasng from. to.4 m. However, tcpu decreases sgnfcantly along as x ncrease, and the GFD method costs more tcpu than the CSP method. For the two methods, tcpu changes from more than s to around s when x changes from. m to.4 m. 34

35 Fg. 7 Convergence curves for CSP and GFD solutons of LLAPE at Ma =.. Fg. 8 CPU tme curves for CSP and GFD solutons of LLAPE at Ma =. at dfferent x. The effects of CCFL on the CSP and GFD solutons of LLAPE at Ma =. are shown n Fg. 9 and Fg.. Fg. 9 ndcates that an approprate CCFL s necessary for accurate numercal results. When CCFL s small enough, εpre stays low value and the method remans stable. An ncrease of CCFL leads to dvergence n εpre. The methods reman stable when CCFL <.. After that, εpre fluctuates slghtly but stll acceptable snce εpre s under Fnally, εpre rses sharply when CCFL s larger than.7 and.55 for the CSP and GFD method respectvely. For most acceptable choces of CCFL, the GFD method has lower εpre than the CSP method. 35

36 Fg. 9 ε pre for CSP and GFD solutons of LLAPE at Ma =. wth dfferent C CFL. Fg. CPU tme curves for CSP and GFD solutons of LLAPE at Ma =. wth dfferent C CFL. Fg. shows that for small CCFL, tcpu grows rapdly, and the GFD method costs more than the CSP method. For the CSP method, CCFL changes from.5 to.8 decreases tcpu from about 5 s to 5 s. Table 3 compares the accuracy and effcency of the CSP and GFD methods when solvng LAPE and LLAPE. Table 3 Comparson of CSP and GFD methods n solvng LAPE and LLAPE wth unform flow. CSP GFD Equatons εpre tcpu εpre tcpu ( -5 ) (s) ( -5 ) (s) LAPE LLAPE

37 The table shows that both LAPE and LLAPE are able to solve D acoustc problems n steady unform flow, and LLAPE can save some computatonal tme wth acceptable loss n the accuracy comparng wth LAPE. Moreover, the GFD method shows more accurate than the CSP method. 5.4 D sound propagaton n rotatonal vortex D sound propagaton n a rotatonal vortex s consdered n ths case. The vortex s descrbed by u = ωy, (3) v = ωx, (33) p = ω x + ω y + Cpre, (34) where ω s the angular velocty of the rotatonal vortex, x and y are the poston coordnates, and u and v are the velocty of partcles along the x-axs and y-axs respectvely. A D acoustc model smlar to secton 5.3 s selected to valdate the Lagrangan ln ln approach. Let α =, α =, α 3 = The ntal condtons are { } δ p = α ρc exp α ( x ) + ( y ), (35) { x y } { α x y } δρ = α exp α ( ) + ( ) +.αexp 3( ) + ( ) { } { }, (36) δu =.4 αc( y )exp α 3( x ) + ( y ), (37) δv =.4 αc( x )exp α 3( x ) + ( y ). (38) A model wth larger computatonal doman s mplemented to show the process of sound propagaton n the rotatonal flow at frst, then a smaller model s used for the evaluaton n order to save the tme. The contour of densty change at dfferent tme s shown n Fg.. In ths smulaton, the computatonal doman s -3 m < x < 3 m, -3 m < y < 3 m, and ω s set as 8 rad/s. As shown n the contour, the red lne at poston (, ) s the center of the rotatonal vortex. All partcles are movng durng the computaton, and sound waves propagate n the meda at the same tme. 37

38 (a) s (b).5 s (c).5 s 38

39 (d).75 s (c). s Fg. Contour of CSP solutons of densty change at dfferent tme. For further evaluaton, a model close to the actual stuaton s consdered. The computatonal doman s -3 m < x < 5 m, -3 m < y < 5 m, and ω s set as rad/s. Fg. gves a comparson of densty change between the CSP method and exact solutons at y = m. The lne represents exact solutons, and ponts n the fgure represent the CSP results. As can be seen from the lne graph, the CSP results show good agreement wth the exact solutons. Fg. Comparson of densty change between CSP results and exact solutons along y = m. Convergence and CPU tme curves for CSP and GFD solutons of LLAPE are shown n Fg. 3 and Fg. 4. All computatonal parameters stay the same as n secton 5.3 except x. Fg. 3 shows that both the CSP and GFD method are convergent n ths case, and the GFD method has better accuracy for dfferent x compared to the CSP method. εpre for two meshfree methods ncreases from less than. -5 to over. -4 along wth x changng from. to.4 m. However, a decrease of x ncreases the tme cost as can be seen from Fg. 4. The GFD method has a hgher tme cost than the CSP method. For both 39

40 meshfree methods, t CPU ncreases over tmes when x decreases from.4 to. m. Fg. 3 Convergence curves for CSP and GFD solutons of LLAPE. Fg. 4 CPU tme curves for CSP and GFD solutons of LLAPE at dfferent x. To evaluate the effects of CCFL, a comparson of εpre from the computaton wth dfferent CCFL s shown n Fg. 5. The lne graph shows that εpre stays stable when CCFL s small enough. There s then a clear ncrease n εpre when CCFL ncreases and gets hgher. The stablty condton for CCFL s dfferent for the two methods. Accordng to the graph, the hghest CCFL value for the CSP and GFD methods s about.7 and.5 separately. A comparson of tcpu from dfferent CCFL computatons s shown n Fg. 6. The fgure shows the GFD method has a hgher tcpu compared to the CSP method. For the two methods, changng CCFL from.5 to.8 decreases tcpu from over 5 s to about 5 s. 4

41 Fg. 5 ε pre for CSP and GFD solutons of LLAPE wth dfferent CFL numbers. Fg. 6 CPU tme curves for CSP and GFD solutons of LLAPE wth dfferent CFL numbers. For further evaluaton, Table 4 compares εpre and tcpu of two meshfree methods n solvng LAPE and LLAPE. Table 4 Comparson of CSP, and GFD methods n solvng LAPE and LLAPE wth rotatonal flow. CSP GFD Equatons εpre tcpu εpre tcpu ( -5 ) (s) ( -5 ) (s) LAPE LLAPE The table reveals that both LAPE and LLAPE are able to solve D acoustc problems n rotatonal background flow. LLAPE can save some computatonal tme wth acceptable loss n the accuracy compared wth LAPE. Smlar to the prevous three cases, the GFD method 4

42 shows better accuracy than the CSP method. 6. Conclusons A Lagrangan approach for computatonal acoustcs s presented and mplemented numercally, usng the SPH, CSP, and GFD methods to solve LAPE and LLAPE. In ths Lagrangan approach, both acoustc and flud varables are supposed to be the physcal propertes of a flud partcle whch can move through both space and tme, and LAPE and LLAPE are derved based on the hydrodynamc/acoustc splttng method. Applcatons to modelng sound propagaton n movng fluds are outlned wth four examples, and a comparson of the LAPE and LLAPE usng the three meshfree methods s presented. Effects of partcle spacng, smoothng length, and Courant number on SPH, CSP and GFD methods are evaluated. The error analyss show that the present Lagrangan approach gves good agreement wth exact solutons, and LLAPE can save some computatonal tme wth acceptable accuracy loss compared wth LAPE. For four test cases, three meshfree methods are accurate for the tme doman smulaton of acoustc waves. The CSP and GFD method are convergent for all cases consdered, whle the standard SPH method s not. The GFD method s more accurate, but need a smaller CFL number to keep low numercal error, whch ncreases computatonal tme. As llustrated n ths paper, the Lagrangan approach s supposed to have potental for acoustc problems lke bubble acoustcs and combuston nose. evertheless, before engneerng applcatons, varety of boundary condtons for the present Lagrangan approach stll need to be analyzed wth more numercal experments. Ths ssue s stll under study and wll be reported n a subsequent paper. Acknowledgments Ths study was supported by the atonal atural Scence Foundaton of Chna (o and ), and the Independent Innovaton Foundaton of Huazhong Unversty of Scence and Technology (o ). Y. O. Zhang was supported by the Chna Scholarshp Councl (4663). References [] P. Mon, K. Mahesh, Drect numercal smulaton: a tool n turbulence research. Annual Revew of Flud Mechancs 3 (998) [] C. Bally, D. Juve, umercal soluton of acoustc propagaton problems usng 4

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