Acoustic Field inside a Rigid Cylinder with a Point Source

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Acoustic Field iside a Rigid Cylider with a Poit Source 1 Itroductio The ai objectives of this Deo Model are to Deostrate the ability of Coustyx to odel a rigid cylider with a poit source usig

Validate Coustyx software by coparig the results fro Coustyx to the aalytical solutios i the presece of acoustic sources. 2 Model descriptio We odel a cylider of radius a = 1 ad legth L = 6.

1 Acoustic Field iside a Rigid Cylider with a Poit Source 1 Itroductio The ai objectives of this Deo Model are to Deostrate the ability of Coustyx to odel a rigid cylider with a poit source usig Coustyx MultiDoai odel ad solve for the acoustic field distributio iside the cylider. Derive aalytical solutio usig odal expasio ethod. Validate Coustyx software by coparig the results fro Coustyx to the aalytical solutios i the presece of acoustic sources. 2 Model descriptio We odel a cylider of radius a = 1 ad legth L = 6. The cylider axis is parallel to the z-axis ad the ceter of the botto ed of the cylider coicides with the origi (as show i Figure 1. The fluid ediu iside the cylider is air with ea desity ρ o = 1.21 kg/ 3 ad soud speed c = 343 i 1 /s. A coplex speed of soud itroduces dapig i the syste. The iagiary part of the speed of soud should always be egative for a decayig soud wave. The waveuber at a frequecy ω is give as k = oega/c. A oopole source of uit volue velocity is itroduced at (-.2,-.35,4 to siulate the poit source i the cylider. The cylider is assued to be rigid. The BE esh of the cylider is show i Figure 1. S z O x φ r y (a (b Figure 1: Rigid cylider with a poit source. (a Acoustic proble, Note S source, O observatio poit; (b Boudary eleet esh. 3 Boudary coditios I the Coustyx MultiDoai odel, the rigid boudary coditio is siulated by applyig the boudary coditio o the cylider as Uifor Noral Velocity type with zero aplitude. That is, v =, where v is the particle oral velocity o the surface of the cylider i the Doai Noral directio. Note that all boudary coditios i a MultiDoai odel are defied with respect to the Doai Noral, which always poits away fro the doai of iterest. For this 1

2 2 exaple, the iterior doai is the doai of iterest; hece, doai oral is poitig away fro the cylider iterior. 4 Aalytical solutio We first copute odes for a rigid cylider (of radius a ad legth L. These odes are the used i odal expasio to evaluate field poit pressure at ay poit iside the cylider. 4.1 Eigevalue proble Table 1: Eigevalues of a rigid cylider fro the roots of J (k r, =, k r, ( = = 1 = 2 = 3 = 4 = The eige-fuctio Ψ(r, satisfies the Helholtz equatio at ay poit r(r, φ, z iside the cylider 2 + k 2 ( ] Ψ(r, = (1 where k 2 ( is the eigevalue. The eige-fuctio should also satisfy the rigid boudary coditios o the surface of the cylider Ψ(r, ˆ where ˆ is the surface oral. I cylidrical coordiate syste 2 is give by = (2 2 = 2 r r r r 2 φ z 2 (3 To solve the eigevalue proble, we assue that the eige-fuctio ca be factored ito a for Ψ(r, = R(re iφ Z(z. The Helholtz equatio is reduced to 1 2 R(r R(r r r ] R(r 2 r r Z(z Z(z z 2 + k 2 = (4 Applyig separatio of variables, we obtai idepedet equatios for the axial factor Z(z ad the radial factor R(r. The axial factor Z(z satisfies ad also the rigid boudary coditios o both eds of the cylider, 2 Z(z z 2 + k 2 zz(z = (5 Z z =, z =, ad L (6 Therefore, the axial factor Z(z is the solutio to above equatios, which is Z(z = cos(k z z, ad the eige values k z = z π/l for z =, 1, 2...

3 3 The radial factor R(r satisfies 2 R(r r r R(r r + (k 2r 2 where k 2 r = k 2 k 2 z, ad the rigid boudary coditios are R r r 2 R(r = (7 =, r = a (8 The solutio to the above equatio is the cylidrical Bessel fuctio J (k r r, where is the order ad k r the eigevalue. The eigevalues (k r, ( are the roots of the derivative of the cylidrical Bessel fuctio at r = a, that is, J (k r, (a = (9 The above equatio is solved uerically usig Maple to fid the eigevalues. Table 4.1 shows the first te eigevalues for =,..., 5. Therefore, the eige-fuctio for a rigid cylider is give by Ψ z,( = J (k r, (re iφ cos(k z z (1 ad the eigevalues are k z = z π/2 for z =, 1, 2..., ad k r, ( give i Table Modal expasio The acoustic field pressure p at ay poit r(r, φ, z (iside the cylider due to the presece of a poit source at r s (r s, φ s, z s satisfies the Helholtz equatio 2 p + k 2 p = ϖδ(r r s (11 where the source stregth ϖ = ikρ o cβ o, β o is the volue velocity; δ(r r s is the Dirac delta fuctio. The odal eige-fuctios derived above for a coplete set. Hece, the acoustic solutio p iside the cylider ca be approxiated as a liear cobiatio of these eige-fuctios. p = C z,,ψ z,(r, z = (12 J (k r, (r cos(k z z A z,, cos φ + B z,, si φ] z where A z,,, B z,,, ad C z,, are ode participatio coefficiets. We copute the ode participatio coefficiets by substitutig Equatio 12 ito Equatio 11, that is, k 2 ko 2 ] J (k r, (r cos(k z z A z,, cos φ + B z,, si φ] = ϖδ(r r s (13 z We use the followig properties to copute A z,, ad B z,,. Noralizatio itegral for cylidrical Bessel fuctios (refer 1], or, ad also, N l = a 1 J(α 2 rrdr = 1 2α α 2 2 2] J(α 2, for α = 1 2, for α = J 2 (k r, (rrdr = a 2 2(k r, a 2 (k r, a 2 2 ] J 2 (k r, a, for k r, a L = 1 2, for k r,a = cos 2 { zπ L, L zdz = z = L/2, z

4 4 2π 2π cos 2 φdφ = si 2 φdφ = { 2π, = π, {, = π, To copute A z,,, ultiply both sides of Equatio 13 with J (k r, (r cos(k z z cos(φ ad itegrate over the etire volue. Usig the properties of various fuctios described above, alog with the properties of Dirac delta fuctio, the coefficiet A z,, is derived. A z,, = ϖ J (k r, (r s cos(k z z s cos φ s k 2 k 2 o] N l 2πεLη (14 where ε = 1 for =, ad ε = 1/2 for ; η = 1 for z = ad η = 1/2 for z. Siilarly, B z,, is coputed by ultiplyig both sides of Equatio 13 with J (k r, (r cos(k z z si(φ. Therefore, B z,, = ϖζ J (k r, (r s cos(k z z s si φ s k 2 k 2 o] N l 2πLη where ζ = for =, ad ζ = 2 for. Thus the odal expasio for the pressure at a observatio poit r(r, φ, z iside the rigid cylider i the presece of a poit source at r s (r s, φ s, z s is, p(r, φ, z = z ϖ J (k r, (r s cos ( z π L z s k 2 ko] 2 N l 2πεLη (15 ( z π J (k r, (r cos L z cos ((φ φ s (16 where z =, 1, 2,..., =, 1, 2,... ad = 1, 2,... If the positio is i Cartesia coordiates, it ca be trasfored ito cylidrical coordiates usig the trasforatios give below: r = x 2 + y 2 φ = arcta(y/x, < φ 2π z = z (17 5 Results ad validatio Acoustic aalysis is carried out by ruig oe of the Aalysis Sequeces defied i the Coustyx MultiDoai odel. A Aalysis Sequece stores all the paraeters required to carry out a aalysis, such as frequecy of aalysis, solutio ethod to be used, etc. I the deo odel, the aalysis is perfored at a frequecy f = 2Hz usig the Fast Multipole Method (FMM by ruig Ru Validatio - FMM. Coustyx aalysis results, alog with the aalytical solutios, are writte to the output file validatio results f.txt. The results ca be plotted usig the atlab file PlotResults.. Coustyx MultiDoai odel uses Direct BE ethod to solve the acoustic proble. I Direct BE ethod, the priary variables are the pressure ad the pressure gradiet o the boudary. Field poit solutios are the coputed fro the surface solutios. Figure 2 shows coparisos of field poit pressures coputed fro both Coustyx ad aalytical ethods. The specified field poits are located at (r, φ, z, where r =.8485, φ = π/4 ad z = 2 + i.1, i =,..., 3. The coparisos show very good agreeet betwee the solutios coputed fro Coustyx ad aalytical expressios.

http://www.asol.co 5.4.35 Coustyx - FMM Aalytical.3.25 P / ϖ.2.15.1.5 2 2.5 3 3.5 4 4.

5 Coustyx - FMM Aalytical.3.25 P / ϖ z ( Figure 2: Field poit pressure coparisos iside a rigid cylider with a poit source fro Coustyx ad aalytical ethods. Note that P is the field poit pressure ad ϖ = ikρ o cβ o, where β o is the volue velocity of the oopole source. Refereces 1] M. Abraowitz ad I. E. Stegu. Hadbook of Matheatical Fuctios, AMS55. U.S. Departet of Coerce, Natioal Bureau of Stadards, 1972.

COMPARISON OF LOW WAVENUMBER MODELS FOR TURBULENT BOUNDARY LAYER EXCITATION

Fluid Dyaics ad Acoustics Office COMPARISON OF LOW WAVENUMBER MODELS FOR TURBULENT BOUNDARY LAYER EXCITATION Peter D. Lysa, Willia K. Boess, ad Joh B. Fahlie Alied Research Laboratory, Pe State Uiversity