3D efficiency of some road barrier crownings using a 2D½ BEM

Transcription

1 3D efficiency of some road barrier crownings using a 2D½ BEM paper ID: 244 /p.1 J. Defrance, L. Bouilloud, E. Premat, Ph. Jean CTB (Centre cientifique et Technique du Bâtiment) 24, rue Joseph Fourier 384 aint-martin-d Hères, France. j.defrance@cstb.fr This paper presents a methodology for the assessment of 3D acoustical efficiency of road noise barrier crownings (or s). The studied protections are the T and the cylinder of two different sizes covered with an absorbing material. The 3 rd octave band mid frequency band efficiency of these s has been calculated with reference to a reflective straight barrier of same overall height (4 m) considered as the reference. The approach used is numerical taking advantage of recent developments in the 2D½ Boundary Element Method. Original results are presented. The post-processing of these calculations allows the creation of a efficiency database as a function of geometrical and frequency parameters. From this database have been determined the analytical expressions function of the Fresnel number which can be straight used in a classical geometrical 3D approach. 1. INTRODUCTION At the time being there is no operational method to take into account complex shape anti-noise barriers in environmental noise predictions. This lack leaves the engineer no possibility to precisely integrate in a road project the effect of devices such as barrier crownings. This paper proposes an original methodology for the integration of the 3D effect of diffracting devices in a geometrical acoustical approach such as ray tracing method. From numerical results, simple analytical laws function of geometrical parameters and frequency are proposed. These approximate expressions can be used directly in the calculation of diffraction along an acoustic path. 2. PREENTATION OF MICADO-BEM CODE 2.1 The 2D Boundary Element Method MICADO (Integral Method for the Acoustical Calculation of Diffraction by Obstacles) is a calculation code based on a variational approach of the Boundary Element Method [1]. The geometry of the 2D problem ((O,x,z) plane) where all the elements remain unchanged in the 3 rd dimension (y-axis) parallel to the linear coherent source, is shown in Figure 1. O z y x M z= Figure 1. Geometry of the 2D problem defined in the (O,x,z) plane

2 paper ID: 244 /p.2 The ground (z=) as well as any obstacle surface are rigid or can be characterised by their own acoustical admittance. The theoretical formalism [1] uses an integral representation of the pressure at any point ( z ) as a function a the pressure on the boundaries, the admittances as well as the elementary Green s solution G (solution for a point source N and for a receiver M above the impedant ground defined by z=) which can be written as the sum of three different terms: i i, (1) 4 4 M N H kr H kr' P M N G, where r is the distance between M and N, r between M and the image of N in the plane z= and H the Hankel function of first kind and zero order. The second term in equation 1 represents the contribution of the reflection of the cylindrical wave on a perfectly rigid ground and the last term P is a corrective factor taking into account the ground admittance [2]. 2.2 From 2D to 2D½ After solving the linear system of the 2D problem, one can calculate the acoustic field at any point in the vertical plane (O,x,z), z. Using then a representation of the point source as an integral of Hankel functions [3, 4] it has been shown that the 2D pressure field (y=) and the 2D½ one (same configuration with the receiver at any z ) are related. Pressure P can thus be written at the receiver as: P 1 x, z px, z, k, Z k 2 where p is the 2D pressure at point (x,z) for a value k of the wavenumber, P is the 2D½ 2 2 pressure at point (x,z) for a value K of the wavenumber, with k K a, Y y y being the ordinate of the point source. y, Z k represents for the wavenumber account an infinitely extended ground. K k e iay da (2) k the varying boundaries impedances taking into One has to notice that k may become imaginary when a>k. However in practice the imaginary part of the pressure decreases very quickly with frequency and can be neglected in many configurations of road traffic noise [5]. 3. REULT OF MICADO CALCULATION 3.1 Presentation of the studied crownings Four different crownings have been studied: two T-shape and two cylindrical (Figure 2). All these complex barriers have an overall height of 4 m as well as the rigid straight barrier considered as the reference one (thickness: 1 cm). The crownings are made of a rigid material

3 paper ID: 244 /p.3 covered with a 5 cm thick layer of glasswool. Acoustic impedance of glasswool is determined through Delany and Bazley s semi-empirical model [6] considering a flow resistivity of 3 kpa s m -2.,85 1,5,6 1,1 m 4 m Figure 2. Geometry of the studied crownings. From left to right: small T, large T, small cylinder, large cylinder and straight reference barrier (absorbing material location in dashed lines). All dimensions in meter 3.2 Calculations hypotheses In this work the barrier lateral diffractions are ignored since they can be neglected in the case of long barriers along road traffic sources. In order to isolate only the top edge diffraction, reflections on the ground have to be cancelled on both sides of the barrier in the BEM calculations (since this phenomenon is of no interest in the present purpose). This is done by setting the point source and receiver at zero height on the reflecting horizontal ground. In the following the angle of diffraction is defined as the angle between a diffracted ray segment (O- R for instance) and a segment perpendicular to the straight barrier top edge, the two segments and the barrier edge being in the same plane (Figure 3). z ground Q x R Q y x R Figure 3. Geometry of a diffracted ray from source to receiver R (side and top views) The aim here is to calculate for a given 3 rd octave band mid frequency (1 to 5 Hz) the value of the attenuation A due to the crowning compare to the one obtained in the case of the straight reference barrier. imulations are carried out making vary the diffraction angle as well as the path difference defined as (Figure 3): Q QR R (3)

4 paper ID: 244 /p.4 For a point source-receiver couple and a given frequenc the attenuation A due to the crowning is given in db by: A P( x, z) 2 ref 1 log1 (4) P( x, z) where P ) ( x, z ref and P x, z) ( are the calculated pressures at receiver R(x,z) for the case of the reference straight barrier and the crowned barrier, respectively. 2D½ calculations have been carried out for a very important number of source and receiver positions. The distance (in the x-axis direction) between the source (or receiver) and the barrier can vary between 4 and 512 m. The angle of diffraction is between and Results For given frequency and diffraction angle, the results show a strong correlation between the value of A and the Fresnel number defined as N 2 where is the wavelength. The acoustical effect of the crowning appears very complex at lowest frequencies (in the range 1-5 Hz). A point of interest is that a negative value of A can be observed around 4 Hz for all studied crownings when the angle of diffraction is less than 55. It means that in this range of frequencies (2-5 Hz) and diffraction angles, the crowned barrier is less efficient than the reference one. This behaviour has been confirmed by many other calculations as well as outdoor measurements on a test-wall. An example of results at 25 Hz for the case of a large T-shape barrier (1.5 m wide) is given in Figure 4a. The range of N when the values of A are negative is clearly visible. Large T 25 Hz Large cylinder 4 Hz Cap Attenuation (db) = Fresnel Number N Cap Attenuation (db) Fresnel Number N Figure 4. A as a function of N for different values of. Cases of the large T at 25 Hz (left) and large cylinder at 4 Hz (right) = 5

5 paper ID: 244 /p.5 From 1 Hz and up, attenuation A becomes proportional to cos that is to sa in the case of the T, proportional to the distance covered by the creeping wave on the top of the crowning. An example of result at 4 Hz is given in Figure 4b for the case of the large cylinder (1 m in diameter). 4. INTEGRATION IN A GEOMETRICAL MODEL 4.1 Determination of attenuation laws In order to take into account the effect of the crownings in a 3D geometrical approach, a general approximate expression of the attenuation A has been developed as a function of N, with and frequency fixed: E N A A N B C D e (5) F where A, B, C, D, E and F are real parameters to be determined by a mean least squares method. In Figure 5 is shown the comparison of A calculated with BEM (equations 1 and 2) and evaluated from the approximate expression (eq. 5) for the cases of the small T at 2 Hz and the small cylinder at 125 Hz. Cap Attenuation - db mall T - 2 Hz = Cap Attenuation - db mall cylinder Hz = Fresnel Number Fresnel Number Figure 5. A as a function of N at for different values of. Comparison between MICADO calculations and approximate laws. Cases of the small T at 2 Hz (left) and small cylinder at 125 Hz (right) (caution: vertical axis is the opposite compared to Figure 4) 4.2 Integration in a geometrical model. Creation of a crowning diffraction database All the determined values of A, B, C, D, E and F (eq. 5) are reported in a database as a function of the following parameters: type of crowning, size of crowning, 3 rd octave band mid frequency and diffraction angle (9 different angles of diffraction as reported in Figure 5). In a geometrical approach as the one used in France [9, 1], the integration of the crowning effect is achieved by applying for any acoustical diffracted path the term A as a correction on any diffraction attenuation A dif due to a straight barrier: A dif, total Adif A (5)

6 paper ID: 244 /p.6 A dif can be related to a source-to-receiver diffracted path, but also to an image source-to-receiver or a source-to-image receiver or an image source-to-image receiver diffracted path. For each of these four paths a specific value of A has to be considered and added to each of the four A dif terms. As the database contains results for 9 diffraction angles onl the values of A have to be interpolated according to the real value of. 5. CONCLUION A methodology for the integration of the diffraction effect due to barrier crownings in a 3D acoustical geometrical approach has been presented. This work has been carried out in the case of an homogeneous atmosphere. Research on 3D models where meteorological effects and crownings diffraction are coupled is in progress. A first solution is the integration of the atmospheric refraction into the Green s function of the problem. A BEM calculation code based on this approach is being developed [11,12]. Another way is the use of a hybrid BEM-PE (Parabolic Equation) method [13, 14] where the field in the vicinity of the complex barrier calculated with BEM in an homogeneous atmosphere is used as a starter for the long range propagation with varying meteorological profiles carried out with a GFPE (Green s Function Parabolic Equation) method. Acknowledgements The authors would like to thank the French Institution ADEME (Agence de L Environnement et de la Maîtrise de l Energie) for its financial support. References 1. P. Jean, Journal of ound and Vibration 212, pp (1998) 2. D.C. Hothersall,.N. Chandler-Wilde and N.N. Hajmirzae, J. ound and Vib. 146, pp (1991) 3. D. Duhamel, Journal of ound and Vibration 197, pp (1996) 4. D. Duhamel and P. ergent, Journal of ound and Vibration 218(5), pp (1998) 5. P. Jean, J. Defrance and Y. Gabillet, Journal of ound and Vibration 212, pp (1998) 6. M.E. Delany and E.N. Bazle Applied Acoustics 3, pp (197) 7. J. Defrance, P. Jean and Y. Gabillet, Proceedings of EuroNoise 1998, München, pp (1998) 8. J. Defrance, Y. Gabillet and P. Jean, Proceedings of the 6th ICV, Copenhagen, pp (1999) 9. J. Defrance and Y. Gabillet, Applied Acoustics 57(2), pp (1999) 1. J. Defrance, M. Bérengier and J.F. Rondeau, Proceedings of InterNoise 21, La Haye (21) 11. E. Premat and Y. Gabillet, Journal Acoust. oc. Am. 18(6), pp (2) 12. E. Premat, Y. Gabillet and J. Defrance, Proceedings of the 9th LRP ymposium, Delft, pp (2) 13. N. Barrière and Y. Gabillet, Acta Acustica 85, pp (1999) 14. E. Premat, J. Defrance, F. Aballéa and M. Priour, Proceedings of the LRP ymposium, Grenoble (22)