A VIRTUAL SOURCE METHOD FOR THE PREDICTION OF THE SOUND FIELD AROUND COMPLEX GEOMETRIES

Transcription

1 23 rd International Congress on Sound & Vibration Athens, Greece July 2016 ICSV23 A VIRTUAL SOURCE METHOD FOR THE PREDICTION OF THE SOUND FIELD AROUND COMPLEX GEOMETRIES Penelope Menounou and Christos Klagkos University of Patras Department of Mechanical and Aeronautical Engineering, Rion 26504, Greece menounou@mech.upatras.gr The prediction of the acoustic field produced by a knon sound source around a complex geometry is investigated. On the triangulated surface of a solid object sound is reflected from the planar surfaces of the triangles and is diffracted by edges of to adjacent triangles that are not co-planar. The present study proposes (i) the use of existing frequency-domain solutions to describe the contribution of each part of the geometry and (ii) an algorithm, suited for complex geometries ith large number of triangles, that replaces each part of the geometry ith its corresponding virtual source. Specifically, planar surfaces are replaced ith virtual sources according to the image source method (to account for reflections), hile the edges of adjacent non-planar surfaces by virtual sources according to the directive line source model (to account for edge diffraction). In the cases examined the surrounding fluid as considered quiescent and homogenous, the sound source an omni-directional spherical or hemispherical source, hile the surfaces of the objects rigid or partially absorbing. The geometries considered span from noise barriers on the ground and desks configurations inside an office, to airfoil geometries. Results of the method agree favorably ith experimental data and/or results from other numerical methods. As opposed to the traditional computational aeroacoustics methods, the proposed method is considerably faster and requires a coarser geometry discretization. Moreover, the computational cost remains the same irrespective of the sound source frequency or the propagation distance. It is, thus, ell suited for high frequencies and very long propagation distances. As opposed to ray methods, the proposed virtual source method does not give rise to caustics, nor does it rely on diffraction ray models that are applicable to infinitely long diffracting edges and thus miscalculate the sound field, particularly in the shado zones. 1. Introduction The acoustic field produced by a knon source around a complex geometry is investigated. The problem is usually addressed ith computational aeroacoustics methods (and less often) ith ray tracing methods. In the former category belong the methods of acoustical analogies [1], the Kirchhoff method or other boundary element methods [2], as ell as computational fluid dynamics methods [3] or hybrid computational fluid dynamics / computational aeroacoustics methods [4], just to name a fe. These approaches have been used both extensively and successfully. Their main disadvantage is the computational cost, hich increases dramatically ith increased propagation distances and increased source frequencies. As a result, they cannot be employed for long propagation distances and high frequency noise sources. Ray methods have been used as an alternative (see, for example, Ref. [5] regarding airplane geometries). Hoever, ray methods quite often either ignore diffraction effects completely, or overestimate their contribution by employing the solution of infi- 1

2 nitely long diffracting edges for all edges, irrespective of their length. The inaccuracies introduced by both approaches can be important in the shado regions, i.e. areas that are not illuminated by the source. The present study proposes the combination of elementary solutions that describe the contribution of the incident, reflected, and diffracted aves (including edges of finite length) ith an algorithm that replaces parts of the geometry that cause reflections and /or diffraction ith appropriate virtual sources that contribute to the acoustic field in accordance ith the aforementioned elementary solutions. Specifically: an acoustically rigid solid is considered ith its outer surface triangulated. On the triangulated surface sound is reflected from the planar surfaces of the triangles and is diffracted by edges of to non- planar adjacent triangles. Accordingly, planar surfaces are replaced ith virtual sources according to the image source method (to account for reflections), hile the edges of adjacent non-planar surfaces by virtual sources according to the Directive Line Source Model (see Ref. [6], [7]) (to account for diffractions by edges). Virtual sources of mixed type are also required. The physical source is considered an omni-directional point source and the surrounding fluid quiescent and homogenous. The method proposed here cannot (at its current stage) handle complex propagation environments nor flo-sound interaction. Hoever, as opposed to the traditional computational aeroacoustics methods, it is considerably faster and requires a coarser geometry discretization. Moreover, the computational cost remains the same irrespective of the sound source frequency or the propagation distance. It is, thus, ell suited for high frequencies and very long propagation distances. As opposed to ray methods, the proposed virtual source method does not give rise to caustics, nor does it rely on diffraction ray models that are applicable to infinitely long diffracting edges and thus miscalculate the sound field, particularly in the shado zones. 2. Types of virtual sources In the present ork three types of virtual sources are employed: (i) virtual point sources (VPS), (ii) virtual edge sources (VES), and (iii) virtual sources of mixed type (VEpS). In the folloing their analytical expressions are presented along ith their spreading type (point or line sources) and their radiation region (conical or edge). 2.1 Virtual point sources Virtual point sources (VPS) are the knon image sources that describe reflections on the surfaces of the triangles. Consider a triangle on the surface of the geometry that is illuminated by the source S and thus gives rise to reflection. The triangle can be removed from the geometry and replaced by its corresponding VPS (a point source positioned at the mirror location ith respect to the source S and the triangle). The contribution of this VPS at the receiver location R rec is as follos: P VPS ikr e inc (R ), (1) rec P0 r inc here P 0 the amplitude of the source, r inc the distance beteen VPS and the receiver R rec, i the imaginary unit, and k the ave number. As opposed to the physical source, hich is omni-directional, reflections from the triangle, and thus contributions from its corresponding VPS, reach only receivers that lie ithin the semi-infinite region defined by the VPS and the three semi-straight lines starting from the VPS and passing through the three vertices of the triangles [see Fig.1(a)]. This semiinfinite region ill be called in the folloing conical radiation region. VPS are, therefore, point sources ith a conical region of radiation. A triangle that is subsequently illuminated by this VPS gives rise to a ne VPS (2 nd order reflection), and so on. Symbolically, the sequence of propagation effects considered is R (1) R (2) R (N), here R denotes the effect considered (reflection) and the superscript the order of reflection. It is noted the VPS are strong sources contributing significantly to the acoustic field. 2 ICSV23, Athens (Greece), July 2016

3 0 Figure 1: The three types of virtual sources and their radiation regions: virtual point sources (VPS) [(a)], virtual edge sources (VES) [(b)], virtual sources of mixed type (VEpS) [(c)]. 2.2 Virtual edge sources (VES) Virtual edge sources (VES) describe diffractions on the edges of non-planar adjacent triangles. Consider an edge shared by to non-planar adjacent triangles in the geometry that is illuminated by the source S and thus gives rise to diffraction. According to the Directive Line Source Model [7], the diffraction field is equivalent to radiation from a line source coinciding ith the diffracting edge and having a specified directivity D glob. The edge can therefore be replaced by its corresponding VES, hose contribution at the receiver location R rec is: 2 2 ik L l 1 1 e PVES (R rec ) P0 D 2 2 globaldl 4 rr0 edge L l i u1 1 i u D 2 2u e F if u 2 u e F if u global 0 sin 0 sin 2 cos ( ) cos 2 cos (2 ) cos 1, 2 here F the Fresnel integral, ( r0, 0, z0) and (, r, z) the coordinates of source S and receiver R rec, respectively (based on a cylindrical coordinate system having the edge as its z-axis), 2 2 u krr ( L R ), L r r 2 z z ( R) r r 2rr cos z z 2, 1,2 0 1,2 0 0, ( R ) r r 2rr cos z z, 2 2, and 2Ω the inner angle of the edge formed by the planes of the to triangles that share the diffracting edge. Contributions from the VES reach only receivers that lie ithin the semi-infinite region defined by the outer edge (angle 2π-2Ω) [see Fig. 1(b)]. The region ill be called in the folloing edge radiation region. VES, therefore, are line sources ith a edge region of radiation. It is noted that an edge is similarly replaced by its corresponding VES, if it is illuminated by any VPS instead by the source directly (recall that VPS are also point sources). Symbolically, the sequence of propagation effects considered is D (1) and/or R (1) R (2) R (N) D (1). It is noted that in the present ork diffraction is considered only up to the first order [D (1) ], as the contributions of higher order diffraction are very eak compared to the rest of the contributions. In general, VES are much eaker sources than VPS. 2.3 Virtual sources of mixed type (VEpS) Virtual sources of mixed type (VEpS) are virtual sources that are line sources, like VES, but have a conical region of radiation, like VPS. They describe reflections of a VES. An edge that is illuminated by a VES is not considered, as higher order diffractions are ignored. Hoever, a triangle that is illuminated by the VES gives rise to reflections, hich are as strong as the VES and are, (2) ICSV23, Athens (Greece), July

4 taken into account. The illuminated triangle is replaced by its corresponding VEpS: a VES (line source) ith the same amplitude as the illuminating VES located at the mirror location ith respect to the triangle and ith a conical region of radiation (like a VPS). The conical radiation region is defined by the midpoint of the VEpS and the three semi-straight lines starting from the midpoint and passing through the three vertices of the triangles [see Fig.1(c)]. Similarly to VPS, successive reflections can be handled. Symbolically, D (1) R (1) R (2) R (N) and/or R (1) R (2) R (N) D (1) R (1) R (2) R (N). In general, VEpS sources are as eak as VES sources. 3. Virtual source algorithm In the folloing the basic elements of the virtual source algorithm are outlined. 3.1 Geometry representation and pre-processing The geometry of the solid object is represented in STL (StereoLithography) format. The outer surface of the object is triangulated and for each triangle on the surface its unit normal vector (facing outards) is provided along ith the coordinates of the three vertices of the triangle. The source S is an omni-directional point source located at any point outside the object, and can be either monochromatic (containing energy in a single frequency) or polychromatic (in hich case its spectrum is considered knon). The receiver R rec can be also any point outside the solid object. The presented method provides the acoustic pressure on a grid of receiver locations, hich is normal to either one of the x, y or z axis. The grid spatial extension and density are specified by the user. At the pre-processing stage the edges that give rise to diffraction are separated from those that do not cause diffraction. Common edges shared by adjacent triangles that are co-planar or almost coplanar (i.e. their unit normal vectors are parallel or deviate from being parallel by a user specified angle) do not give rise to diffraction and are appropriately flagged. In cases of polychromatic sound sources, the virtual sources are firstly created based on the location of the source and independent of the frequency. At a later stage, hen the contribution of each virtual source is computed, the calculations are done separately for each frequency in the source spectrum. 3.2 Illuminated triangles The virtual source algorithm presented here relies on determining hich triangles/edges are illuminated either by the original (physical) source or by any of the virtual sources. A triangle (and thus its edges) is considered illuminated by the original source, if the ray connecting the source to the centroid of the triangle: (i) does not intersect any other triangle of the geometry, or (ii) intersects other triangles of the geometry, but all such triangles are behind the triangle in consideration. To determine hether a ray intersects a triangle, the Moeller algorithm [8] is employed, hile to determine hether a triangle is behind another triangle (ith respect to the source) the length and the direction of the rays from the source to the centroids of the corresponding triangles are checked. The above described algorithm is slightly modified in the cases of virtual sources, here, unlike the omni-directional original source, regions of radiation are specified. Firstly, it is checked hether a triangular (and its edges) lies ithin the radiation region of the virtual source, and subsequently hether the triangle is also illuminated by that source. It should be noted that parts of the geometry, hich led to the creation of a virtual source are not considered in the illumination check for this specific virtual source (i.e. the triangle that gave rise to creation of a VPS or a VEpS, or the triangles that are attached to a VES). 3.3 Creation of virtual sources Creation of VPS: Identify all triangles that are illuminated by the source S. For each illuminated triangle m create the VPS ith respect to that triangle, R m (1), to account for sound being reflected on triangle m. It is noted that the location of R m (1) is the same for all co-planar triangles, each one, 4 ICSV23, Athens (Greece), July 2016

5 hoever, has a different conical region of radiation determined by the corresponding triangle. For (1) each receiver that is illuminated by R m (i.e. the ray beteen R (1) m and R rec (i) lies ithin the VPS radiation region and (ii) is not intersected by any part of the geometry) the contribution of R (1) m is computed by Eq. (1). Subsequently, for each illuminated triangle n in the radiation region of R (1) m, create the VPS for the illuminated triangle n, R (1) m R (2) n. The VPS R (1) (2) m R n describes a second order reflection. Higher order VPS [ R (1) (2) m R n R (N) k ] are similarly created until no more illuminated triangles can be found ithin the radiation region of the preceding order VPS, or until a user-defined maximum order of reflection is reached. Creation of VES: Identify all edges that are illuminated by the source S or by any VPS [R (1) (2) m R n R (N) k ]. For each illuminated edge J create the edge source VES ith respect to that edge, (D (1) J or R (1) (2) m R n R (N) k D (1) J, the former, if illuminated directly by the source and the latter, if illuminated by an N th order VPS]. For each receiver that is illuminated by the VES (i.e. the ray beteen the midpoint of VES and R rec (i) lies ithin the VES radiation region and (ii) is not intersected by any part of the geometry) the contribution of VES is computed by Eq. (2). After all edges illuminated by the source and by all VPS are considered, no more VES are created, as second order diffraction is ignored in the present ork. Creation of VEpS: Identify all triangles that are illuminated by any VES [D (1) J or R (1) (2) m R n R (N) k D (1) J ]. For each so illuminated triangle l create the mixed type source VEpS:D (1) (1) J R l or R m (1) R n (2) R k (N) D J (1) R l (1). For each receiver that is illuminated by the VEpS (i.e. the ray beteen the midpoint of VEpS and R rec lies ithin the VEpS radiation region and is not intersected by any part of the geometry) the contribution of the VEpS is computed by Eq. (2). For each created VEpS all triangles illuminated by that VEpS are identified (note that illuminated edges are not considered as second order diffraction is ignored). For each so illuminated triangle q a ne VEpS is created ith respect to triangle q, D J (1) R l (1) R q (2) or R m (1) R n (2) R k (N) D J (1) R l (1) R q (2) and its contribution is computed. The conditions for terminating the creation of VEpS are the same as the corresponding conditions for VPS. In the results presented in the current ork, only first order reflections of a VES have been considered. 4. Results Discussion In the present section three case studies are presented: noise barrier on the ground, open desk configuration and multi-element airfoil. The results for each case are discussed and their reasonably good agreement ith experimental data or other numerical methods is demonstrated. 4.1 Noise barrier on the ground Comparison ith experiments A finite length noise barrier on rigid ground is considered. Figure 2 depicts the comparison beteen numerical results from the presented method and experimental data taken from Ref. [9]. The results are presented as insertion loss (i.e. sound pressure level difference ith and ithout the barrier present) at a receiver location in the shado zone behind the barrier. The reasonably good agreement can be observed for a ide range of frequencies. The presented case constitutes a challenging benchmark, as it tests the accuracy of the method in cases here diffraction by finite length edges, hich is often miscalculated, is the sole contributor to the acoustic field at the receiver location. The top edge and the to side edges of the barrier are all of finite length. It should be emphasized that all frequencies have been computed simultaneously and ith the same geometry discretization. The computational cost remains the same for higher frequencies and for longer propagation distances. ICSV23, Athens (Greece), July

6 Figure 2: Finite length barrier on rigid ground (left); comparison beteen insertion loss calculated by the virtual source method ( ) and measurements ( ) taken from Ref. [9] (right). 4.2 Open desk configuration Comparison ith ray tracing methods / experiments The case of an open office is considered next. The geometry consists of a double desk, acoustically hard ground and partially absorbing ceiling made of plaster board (see Fig. 3). Besides being geometrically more complex than the previous case, the case at hand presents to additional challenges. Firstly, results are presented for a hemispherical source, in addition to the omni-directional point source, providing, therefore, evidence that the method can handle directional sources. Secondly, the reflection coefficient of the ceiling is a function of frequency, as opposed to the perfectly rigid surfaces considered as far. This provides an evidence that reflections from partially absorbing surfaces can also be taken into account by the presented method. Figure 3: Open office configuration. Comparison of insertion loss calculated by the virtual source method ith results by the Chevret-Chatillon model (a ray-tracing approach) and measurements, both taken from Ref. [10]. Figure 3 shos the comparisons beteen results (insertion loss) obtained by the virtual source method and experiments taken from Ref. [10] for various geometrical parameters of the open office 6 ICSV23, Athens (Greece), July 2016

7 configuration and for a ide range of frequencies. The reasonably good agreement can be observed in almost all cases. Finally, Figure 4 depicts the A-eighted overall sound pressure level on a crosssection of the open office as calculated by the virtual source method and by the Chevret-Chatillon model (based on a ray-tracing method) taken from Ref. [10] for a point source and for a hemispherical source. Figure 4: Open office configuration. Comparison of A-eighted overall sound pressure level at a crosssection of the configuration calculated by the virtual source method (left column) ith results by the Chevret-Chatillon model (a ray-tracing approach) taken from Ref. [10] (right column) for an omni-directional point source (upper ro) and a hemi-spherical source (loer ro). 4.3 Airfoil Qualitative comparison ith computational aeroacoustics methods Finally, a multi-element airfoil is considered. A point source is positioned in the space beteen an airfoil and its slat as shon in Fig. 5. The case is even more challenging than the previous cases, as the geometry contains curved surfaces and narro gaps. The method presented here is employed to compute the acoustic field produced by the point source around the airfoil on a plane perpendicular to it and in the absence of mean flo. The results depicted in Fig. 5 are instantaneous acoustic pressures around the multi-element airfoil obtained by the virtual source method and by the ONERA computational aeroacoustics softare sabrina, as taken from Ref. [11]. It is observed that the acoustic patterns predicted by the numerical solution of the Euler equations (sabrina) are reasonably similar to the patterns predicted by the virtual source method. This first qualitative result seems encouraging for the possibility of employing the virtual source method to complex geometries ith curved surfaces and narro gaps. It should be emphasized that the presented method, at its current stage, does not take into account second order diffraction nor creeping aves on curved surfaces. Note, for example, the area right above the airfoil, hich, as a result, is predicted as a complete shado zone (no penetration of sound). The inclusion of these effects ould improve the comparisons in this challenging case study. ICSV23, Athens (Greece), July

8 Figure 5: Multi-element airfoil geometry (left) and instantaneous acoustic pressure from the computational aeroacoustics code sabrina (center) taken from Ref. [11]; results from the virtual source method (right). 5. Summary and future ork A virtual source method has been presented for predicting the acoustic field around complex geometries. Parts of the geometry are replaced by appropriate elementary virtual sources (point or line sources ith specified amplitudes and radiation regions). The virtual sources have a contribution to the acoustic field that is equal to the contribution of any reflections and /or diffractions that ould have been caused by the replaced parts of the geometry. The method seems to yield results in reasonably good agreement ith experiments and other numerical methods. It is ell suited for all frequencies (including high frequencies) and for very long propagation distances. Future ork shall extend the method to incorporate high order diffraction, diffraction by absorbing edges, creeping aves, the existence of mean flo and the spatial variation of acoustic parameters in the propagation medium. REFERENCES 1 Goldstein, M. E. A generalized acoustic analogy, Journal of Fluid Mechanics, 455, , (2003). 2 Lyrintzis, A. S. Revie of the use of Kirchhoff s method in computational aero-acoustics, Journal of Fluids Engineering, 116, , (1994). 3 Kurbatskii, K. A. and Mankbadi, R. R. Revie of computational aeroacoustics algorithms, International Journal of Computational Fluid Dynamics, 18, , (2004). 4 Farshchi, M., Siamak, K., and Mohammad, E. Linearized and non-linear acoustic/viscous splitting techniques for lo Mach number flos, International Journal of Numerical Methods in Fluids, 42, , (2003). 5 Agaral A., Doling A.P., Shin Ho-Chul, Graham W., and Sefi S. A Ray Tracing Approach to Calculate Acoustic Shielding by the Silent Airframe, 27th AIAA Aeroacoustics Conference, Cambridge, Massachusetts, USA, 8-10 May, (2006). 6 Menounou, P., Busch-Vishniac, I.J., and Blackstock, D.T. Directive Line Source Model: a ne model for sound diffraction by half planes and edges, The Journal of the Acoustical Society of America, 107, , (2000). 7 Menounou, P. and Nikolaou, P. An Extension to the Directive Line Source Model for Diffraction by Half Planes and Wedges, Acta Acustica united ith Acustica, 102, , (2016). 8 Moller, T. and Trumbore, B. Fast, Minimum Storage Ray/Triangle Intersection, Journal of Graphics Tools, 2, 21 28, (1997). 9 Lam, Y. W. and Roberts, S. C. A simple method for accurate prediction of finite barrier insertion loss, The Journal of the Acoustical Society of America, 93, , (1993). 10 Chevret, P. and Chatillon, J. Implementation of diffraction in a ray tracing model for the prediction of noise in open-plan offices, The Journal of the Acoustical Society of America, 132, , (2012). 11 Manoha, E., Juvingy, X., Redonnet, S. and Guenanff, R. Numerical simulation of acoustic effects of engine installation for ne concepts of aircrafts, CEAS- X 2 NOISE orkshop, Budapest, Hungary, November, (2004). 8 ICSV23, Athens (Greece), July 2016