Numerical Computation of the Eigenstructure of Cylindrical Acoustic Waveguides with Heated (or Cooled) Walls

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1 Alied Mathematical Sciences, Vol. 3, 29, no. 7, Numerical Comutation of the Eigenstructure of Clindrical Acoustic Waveguides with Heated (or Cooled) Walls Brian J. M c Cartin Alied Mathematics, Kettering Universit 7 West Universit Avenue, Flint, MI , USA bmccarti@kettering.edu Abstract A erturbation rocedure for the modes and cut-off frequencies of clindrical acoustic waveguides of arbitrar cross-section with a small temerature gradient along the walls was resented in [, 2]. In the resent aer, that work is etended to arbitrar wall temerature distributions ( warm waveguides ) as outlined in [3]. We utilize a hsical formulation due to DeSanto [4] which incororates both an inhomogeneous sound seed and a densit gradient. This hsical model is altered to include an aial velocit comonent []. The associated eigenvalue roblem [6] is then discretized via the Control Region Aroimation [7, 8] which is a finite difference rocedure alicable on arbitrar comutational grids [9]. The resulting sarse generalized matri eigenvalue roblem [] is then solved numericall [] using MATLAB c. This general comutational rocedure is then used to stud the acoustical effect of heating (or cooling) the walls of rectangular waveguides [2]. Mathematics Subject Classification: 3P99, 3Q99, 6F, 6M6, 76M2, 76Q, 8A2, 8M2 Kewords: acoustic waveguides, temerature gradient, Control Region Aroimation, generalized eigenvalue roblem, inhomogeneous waveguide Introduction Modal roagation characteristics of clindrical acoustic waveguides at constant temerature have been investigated analticall for canonical duct crosssections such as rectangular and circular [6] and numericall for general simlconnected cross-sections [7]. However, if there is a temerature variation across

2 826 B. J. McCartin the duct, then these analses are inadequate. This is due to the inhomogeneous sound seed and densit gradient induced b such a temerature distribution. Such temerature variations occur in the ehaust sstems of vehicles [3] where the bottom ortion of the waveguide is in direct contact with the ambient atmoshere while the uer ortion is either near or in contact with the bod of the vehicle which is, of course, at a different temerature. As we shall show b our subsequent numerical analsis, such an induced temerature variation across the waveguide can significantl alter the modal characteristics of the ehaust sstem with a consequent modification in its overall effectiveness at suressing selected acoustic frequencies. In ocean acoustics [4], both deth and range deendent sound seeds are ticall considered. Densit gradients are usuall ignored since the satial scale of such variations is much larger than the wavelength of the acoustic disturbance. The same situation obtains in atmosheric roagation []. Unlike such ocean and atmosheric waveguides, the ducts under consideration in this aer are full enclosed and hence densit gradients must be accounted for. For small erturbations of an isothermal ambient state, such an analsis was erformed in [, 2]. In [3], a numerical rocedure was outlined for the case of arbitrar temerature distribution within the cross-section of the waveguide. Full details of the analsis and aroimations as well as a suorting numerical eamle are rovided in this aer. The resent stud will commence with the formulation of a hsical model which includes both an inhomogeneous sound seed and a densit gradient across the duct. Uniform aial flow is also accomodated. The corresonding mathematical model will involve a generalized Helmholtz oerator whose eigenvalues (related to the cut-off frequencies) and eigenfunctions (modal shaes) are to be determined. This continuous roblem is discretized via the Control Region Aroimation [7, 8, 9] (a finite difference rocedure alicable to general geometries) resulting in a generalized matri eigenvalue roblem. The aforementioned generalized eigenvalues and eigenvectors are then aroimated numericall. In turn, this ermits the determination of the cut-off frequenc, disersion curve and ressure distribution for each of the modes. This rocedure is illustrated for the case of a rectangular duct with a heated wall. All numerical comutations were erformed using MATLAB c. 2 Phsical Model With reference to Figure, we consider the acoustic ressure field within an infinitel long, hard-walled, clindrical duct of general cross-section, W. The analsis of the modal roagation characteristics of such an acoustic waveguide ticall roceeds b assuming a constant temerature throughout the undisturbed fluid.

3 Warm waveguides 827 Figure : Acoustic Waveguide Cross-Section However, in the resent stud, we ermit a stead-state temerature variation throughout the waveguide cross-section (Figure 2). This temerature variation is due to an alied temerature distribution along the walls of the duct governed b the boundar value roblem ΔT =in W; T = T alied on W () where, here as well as in the ensuing analsis, all differential oerators are transverse to the longitudinal ais of the waveguide. This temerature variation across the duct will roduce inhomogeneities in the background fluid densit ˆ ˆρ(, ) = (2) R T (, ) and sound seed c 2 (, ) =γr T (, ), (3) where γ is the ratio of secific heats, R is the universal gas constant and ˆ is the ambient ressure (which is constant). The self-consistent governing equations for the hard-walled acoustic ressure wave with frequenc ω and roagation constant β, e j(ωt βz) (, ), are [4] (n is the direction normal to the waveguide wall): ˆρ ( ˆρ [ ) ( ˆρ ] ˆρ) +( ω2 c 2 β2 ) =in W; =on W, (4) n

4 828 B. J. McCartin T Figure 2: Temerature Profile assuming that the fluid is at rest (the cut-off frequencies, corresonding to β =, are unaltered b uniform flow down the duct []). However, a uniform aial flow with velocit V will alter the disersion curves for the waveguide modes. In this event, the DeSanto model, Equation (4) ma be modified as follows []: ˆρ ( ˆρ [ ) ( ˆρ ] ˆρ) + [ ] (ω V β) 2 β 2 =in W. () c 2 We net nondimensionalize the temerature so that the minimum scaled temerature is unit: t(, ) := T (, ) T min ˆρ(, ) = ρ ma t(, ),c2 (, ) =c 2 min t(, ). (6) We then utilize Equation (6) to eliminate the densit from Equation () which results in the generalized Helmholtz boundar value roblem for (, ): [ ( ( ))] (t ) t t β 2 t = Ω 2 inw; t where Ω 2 := n =on W, (7) ( ω c min M ma β ) 2 ; Mma := V c min. (8)

5 Warm waveguides 829 Figure 3: Dirichlet/Delauna Tessellations 3 Control Region Aroimation The Control Region Aroimation [7, 8] is a generalized finite difference rocedure that accomodates arbitrar geometries [9]. It involves discretization of conservation form eressions on Dirichlet/Delauna tessellations (see below). This ermits a straightforward alication of relevant boundar conditions. The first stage in the Control Region Aroimation is the tessellation of the solution domain b Dirichlet regions associated with a re-defined et arbitrar distribution of grid oints. Denoting a generic grid oint b P i,we define its Dirichlet region as D i := {P : P P i < P P j, j i}. (9) This is seen to be the conve olgon formed b the intersection of the half-saces defined b the erendicular bisectors of the straight line segments connecting P i to P j, j i. It is the natural control region to associate with P i since it contains those and onl those oints which are closer to P i than to an other grid oint P j. If we construct the Dirichlet region surrounding each of the grid oints, we obtain the Dirichlet tessellation of the lane which is shown dashed in Figure 3. There we have also connected b solid lines neighboring grid oints which share an edge of their resective Dirichlet regions. This construction tessellates the conve hull of the grid oints b so-called Delauna triangles. The union of these triangles is referred to as the Delauna tessellation. The grid oint distribution is tailored so that the Delauna triangle edges conform to W. It is essential to note that these two tessellations are dual to one another in the sense that corresonding edges of each are orthogonal.

6 83 B. J. McCartin Figure 4: Control Region Aroimation With reference to Figure 4, we will eloit this dualit in order to aroimate the Dirichlet roblem, Equation (), for the temerature distribution at the oint P. We first reformulate the roblem b integrating over the control region, D, and aling the divergence theorem, resulting in: T dσ =, () D ν where (ν, σ) are normal and tangential coordinates, resectivel, around the eriher of D. The normal temerature flu, T, ma now be aroimated ν b straightforward central differences [6] thereb ielding the Control Region Aroimation: τ m (T m T )=, () m where the inde m ranges over the sides of D and τ m := τm +τ m +. Equation () for each interior grid oint ma be assembled in a (sarse) matri equation which can then be solved for T (, ) (with secified boundar values) and iso facto for t(, ). It remains to discretize the homogeneous Neumann boundar value roblem, Equation (7), for the acoustic ressure wave b the Control Region Aroimation. For this urose we first rewrite it in the integral form: t [ ( ( ))] D ν D dσ t t da β 2 t da = Ω 2 da. t D D (2)

7 Warm waveguides 83 We then aroimate each of the integral oerators aearing in Equation (2) as follows: D t ν dσ m τ m t + t m 2 [ ( ( ))] [ t t da D t m β 2 D D ( m ), (3) (t m t ) 2 ] τ m, (4) 2t m t da β 2 t A, () da A, (6) where A is the area of D (restricted to W, if necessar) and we have utilized in Equation (4) the fact that m τ m (t m t ) = imlies that t m τ m t + t m 2 ( ) = t m t m τ m (t m t ) 2 2t m. (7) The hard boundar condition, =, is enforced b siml modifing an ν τ m in Equation (3) corresonding to boundar edges. Because Δt =, the aroimation of Equation (4) remains valid for boundar oints rovided that the summation ecludes the ortion of D ling on W. 4 Generalized Eigenvalue Problem Substitution of Equations (3-6) into Equation (2) ields the matri generalized eigenvalue roblem [] A = λb, (8) with λ = Ω 2 and where A is sarse and smmetric while B is ositive and diagonal (its elements are recisel the areas of the Dirichlet olgons restricted to W ). The realit of Ω is guaranteed b the following result. Theorem The matri A aearing in Equation (8) is nonositive semidefinite. Proof: As A is comrised of the discrete oerators of Equations (3-), we consider searatel the effect of each comonent uon the sectrum of A. B Gershgorin s Circle Theorem [7], the matri corresonding to Equation (3) is nonositive semidefinite (since the τ m are nonnegative). B the Discrete Maimum Princile [8], t is ositive throughout W so long as it is so along W (which we have imlicitl assumed throughout). Thus, adding the nonositive

8 832 B. J. McCartin diagonal matrices corresonding to Equations (4) and () onl serves to ush the sectrum further to the left. Thus, A is seen to be nonositive semidefinite. In comuting the generalized eigenvalues of Equation (8), λ, we benefit greatl from the sarsit of A and the diagonalit of B []. We ma efficientl swee out the modal disersion curves b the following numerical rocedure. Fi β and find Ω = λ from Equation (8). Then, obtain ω =Ω+M ma β (9) c min from Equation (8) which ma then be lotted versus β for an desired mode. Numerical Eamle b a Figure : Rectangular Waveguide Cross-Section With reference to Figure, we net al the above numerical rocedure to an analsis of the modal characteristics of the warmed rectangular duct with cross-section W =[,a] [,b]. The eact eigenvalue corresonding to the (, q)-mode with constant temerature is [2]: ( ) π 2 ( ) qπ 2 Ω 2,q = β (2) a b

9 Warm waveguides 833 Figure 6: (, )-Mode (Hard Wall) Figure 7: (, )-Mode (Hard Wall)

10 834 B. J. McCartin Figure 8: (2, )-Mode (Hard Wall) Figure 9: (, )-Mode (Hard Wall)

11 Warm waveguides 83.4 Disersion Relation.2 Ω.8.6 (,) (2,).4 (,).2 (,) β Figure : Sectral Structure (Warm Rectangular Waveguide) For this geometr and mesh, the Dirichlet regions are rectangles and the Control Region Aroimation reduces to the familiar central difference aroimation [9]. Also, eact eressions are available for the eigenvalues and eigenvectors of the discrete oerator in the constant temerature case [2]. The lower wall is subjected to the arabolic temerature rofile T alied =+ 4(T ma ) a 2 (a ). (2) while the other three walls are set to T alied =. We then solve ΔT = b the rocedure described above subject to this boundar condition for the cross-sectional temerature rofile, T (, ), which is dislaed in Figure 2. Secificall, we set a =, b =,T ma = 2 and V = so that Ω = ω/c min. Utilizing 33 7 and 7 9 comutational meshes followed b Richardson etraolation [6], we disla the first four comuted modes (corresonding to the four smallest generalized eigenvalues) in Figures (6-9). In each figure, the lot in the uer left corresonds to constant temerature, T =, while that in the uer right corresonds to variable temerature with β =. The lots in the lower left/right corresond to variable temerature with β =. /., resectivel. Figure resents the disersion relation corresonding to each of these four modes. The dashed curves corresond to constant temerature while the solid curves corresond to variable temerature. Of course, the cutoff frequenc for each mode corresonds to β =. Collectivel, these figures tell an intriguing tale. Firstl, the (, )-mode is no longer a lane wave in the resence of temerature variation. In addition, all variable-temerature modes ossess a cut-off frequenc so that, unlike the case of constant temerature, the duct cannot suort an roagating modes at

12 836 B. J. McCartin the lowest frequencies. Moreover, the resence of a temerature gradient alters all of the cut-off frequencies. In shar contrast to the constant temerature case, the variable-temerature modal shaes are frequenc deendent. Lastl, the resence of a temerature gradient removes the modal degenerac between the (2, )- and (, )-modes that is rominent for constant temerature. 6 Conclusion The receding sections have resented a hsical model as well as a numerical rocedure for studing the modal characteristics of clindrical acoustic waveguides in the resence of temerature gradients induced b an alied temerature distribution along the walls of the duct. Rather than siml making the sound seed satiall varing, as is done in atmosheric roagation and underwater acoustics, we have been careful to utilize a self-consistent hsical model (due to DeSanto [4]) which also includes densit gradients since we have dealt here with a fluid full confined to a narrow region. Also, we have etended DeSanto s model to allow for uniform aial flow down the duct. While revious work [, 2] was restricted to small erturbations of a constant temerature rofile, the resent aer allows arbitrar cross-sectional temerature rofiles. Moreover, the Control Region Aroimation develoed herein ermits waveguide cross-sections of arbitrar shae. We have resented a detailed numerical eamle (utilizing MATLAB c ) intended to intimate the broad ossibilities offered b the resulting numericall comuted cut-off frequencies, disersion curves and modal shaes. 7 Acknowledgement I would like to thank m devoted wife Mrs. Barbara A. McCartin for her indisensable aid in constructing the figures. References [] B. J. McCartin, A Perturbation Method for the Modes of Clindrical Acoustic Waveguides in the Presence of Temerature Gradients, Journal of Acoustical Societ of America, 2 () (997), [2] B. J. McCartin, Erratum, Journal of Acoustical Societ of America, 3 () (23), [3] B. J. McCartin, A Numerical Procedure for 2D Acoustic Waveguides with Heated Walls, Bulletin of American Phsical Societ, 44 (3) (999), 38.

13 Warm waveguides 837 [4] J. A. DeSanto, Scalar Wave Theor, Sringer-Verlag, Berlin, 992. [] P. M. Morse and K. U. Ingard, Linear Acoustic Theor, Handbuch der Phsik, Vol. XI/, Acoustics I, S. Flügge (ed.), Sringer-Verlag, Berlin, 96, -28. [6] D. S. Jones, Acoustic and Electromagnetic Waves, Oford, 986. [7] B. J. McCartin, Numerical Comutation of Guided Electromagnetic Waves, Proceedings of the 2th Annual Conference on Alied Mathematics, U. Central Oklahoma, Edmond, OK, 996, [8] B. J. McCartin, Control Region Aroimation for Electromagnetic Scattering Comutations, Comutational Wave Proagation, B. Engquist and G. A. Kriegsmann (eds.), Sringer-Verlag, New York, 996, [9] B. J. McCartin, Seven Deadl Sins of Numerical Comutation, American Mathematical Monthl, () (998), [] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Oford, 96. [] Y. Saad, Numerical Methods for Large Eigenvalue Problems, Halsted, New York, 992. [2] H. R. L. Lamont, Waveguides, Third Edition, Methuen, London, 9. [3] M. L. Munjal, Acoustics of Ducts and Mufflers, Wile, New York, 987. [4] L. M. Brekhovskikh and Yu. P. Lasov, Fundamentals of Ocean Acoustics, Second Edition, Sringer-Verlag, Berlin, 99. [] L. Brekhovskikh and V. Goncharov, Mechanics of Continua and Wave Dnamics, Sringer-Verlag, Berlin, 98. [6] S. D. Conte and C. de Boor, Elementar Numerical Analsis: An Algorithmic Aroach, Third Edition, McGraw-Hill, New York, 98. [7] D. S. Bernstein, Matri Mathematics, Princeton Universit Press, Princeton, 2. [8] K. W. Morton and D. F. Mers, Numerical Solution of Partial Differential Equations, Second Edition, Cambridge Universit Press, Cambridge, 2. [9] A. Thom and C. J. Aelt, Field Comutations in Engineering & Phsics, Van Nostrand, London, 96. [2] F. B. Hildebrand, Finite-Difference Equations and Simulations, Prentice- Hall, Englewood Cliffs, NJ, 968. Received: October, 28