Aeroakustische Analyse mittels natürlicher Helmholtz-Hodge-Zerlegung

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1 Fachtagung Eperimentelle Strömungsmechanik September 6, Cottbus Aeroakustische Analse mittels natürlicher Helmholtz-Hodge-Zerlegung Aeroacoustic analsis using natural Helmholtz-Hodge decomposition Daniel Haufe, Johannes Gürtler, Anita Schulz, Friedrich Bake, Lars Enghardt,3, Andreas Fischer, Jürgen Czarske Technische Universität Dresden, Fakultät Elektrotechnik und Informationstechnik Institut für Grundlagen der Elektrotechnik und Elektronik, Professur für Mess- und Sensorsstemtechnik Helmholtzstraße 8, 69 Dresden Deutsches Zentrum für Luft- und Raumfahrt Berlin Institut für Antriebstechnik, Abteilung Triebwerksakustik Müller-Breslau-Straße 8, 63 Berlin 3 Technische Universität Berlin, Fakultät Verkehrs- und Maschinenssteme, Institut für Strömungsmechanik und Technische Akustik, Fachgebiet Turbomaschinen- und Thermoakustik Müller-Breslau-Straße 8, 63 Berlin Strömungsmessung, Schallschnelle, Aeroakustik, Schalldämpfung flow measurement; acoustic particle velocit; aeroacoustics; sound absorption Abstract We present the first application of the Helmholtz-Hodge decomposition HHD) to measured velocit data in aeroacoustics in order to analze damping phenomena at an aeroacoustic sound absorber. As an eample, a generic bias flow liner is investigated, i.e. a perforated acoustic liner whose damping is resulting from an interaction of sound and flow but provides an increased damping efficienc due to an additional injected flow through the perforation. For the aeroacoustic investigation, three-dimensional optical measurements of the velocit vector are performed at 496 locations with a spatial resolution of 66 µm and a measurement rate of khz. Appling the natural HHD to the phase-resolved oscillation velocit regarding the acoustic ecitation frequenc of Hz ields an irrotational velocit term, the acoustic particle velocit, and a solenoidal velocit term, the aerodnamic flow velocit. As a result, the interaction between the sound and the flow results in a coherent flow vortices dominating the velocit oscillation. In addition, a local acoustic source near the perforation is supposed, having a local acoustic particle velocit which is three times higher compared to the ecitation sound wave. The HHD result shows slight artifacts, whose cause is identified as the spatial discretization of the data and, thus, can be suppressed b using a finer data grid size in the future. As a consequence, the HHD will be a valuable tool for analzing and eventuall optimizing bias flow liners for enhanced sound absorption in jet engines and stationar gas turbines. Introduction For the investigation of aeroacoustic phenomena, for eample, at sound absorbers liners), the measured velocit field is an attractive data set, because it is composed of both the aerod- Copright 6 and published b German Association for Laser Anemometr GALA e.v., Karlsruhe, German, ISBN

2 namic) flow velocit and the acoustic particle velocit. However, the separation of both quantities is challenging, since both coincide with each other at the measured velocit signal and in the temporal spectrum of the velocit as well, because flow oscillations are induced at the acoustic frequenc b the interaction of sound and flow. For this reason, a separation in the spatial domain is suggested in the following. Previous approaches like proper orthogonal decomposition POD, see Rupp et al. ) suffer from ambiguities regarding the interpretation, because the mathematical modes resulting from the POD are not phsicall related to the sound and flow field, respectivel. In contrast, the Helmholtz-Hodge decomposition HHD) allows distinguishing between the aerodnamic flow velocit and the acoustic particle velocit b their characteristic properties of being irrotational and solenoidal, respectivel, see De Roeck et al. 7. Using the HHD requires considering of boundar conditions for the given velocit field, according to Denaro 3. In case of measured data, these conditions are usuall not or not eactl known, which precludes this approach. Though, the natural HHD was introduced b Bhatia et al. 4, which allows the decomposition to be independent from boundar conditions, but the natural HHD has not been applied to aeroacoustics so far. Therefore, the natural HHD is applied to an aeroacoustic sound absorber here, as an eample. In this paper, the natural HHD algorithm and its implementation in MATLAB R is eplained at first. Subsequentl, a verification of the implementation is presented showing its suitabilit for the aeroacoustic application. Then, the application to measured velocit data, acquired at a bias flow liner, is demonstrated. Finall, a conclusion and an outlook on future work is given. Algorithm Assuming a given simpl connected vector field v that is spatiall unbounded, the natural Helmholtz-Hodge decomposition NHHD) can be performed, such that v = d + r + h ) is decomposed into a irrotational term d, a solenoidal term r, and a remaining term h that is both irrotational and solenoidal. The terms d s) = G s, s ) ) v s d s and Ω r s) = G s, s ) ) v s d s, ) are calculated in the spatial region Ω with s as the position vector. The Green s function Ω G s, s ) = π ln s s ) or G s, s ) = 4π s s 3) has to be emploed in Eqs. ), for two-dimensional or three-dimensional problems, respectivel. Finall, h is obtained as a remainder using Eq. ). The algorithm is implemented in MATLAB b approimating the integrals in Eqs. ) b central finite differences corresponding to a Riemann sum on a regular grid at a Cartesian coordinate sstem,, z). Moreover, all differential operators are replaced b their discrete counterparts, i.e., using difference quotients. Copright 6 and published b German Association for Laser Anemometr GALA e.v., Karlsruhe, German, ISBN

3 vsource v vorte v shift.5 vsource v vorte vshift.5.5 vsource v vorte v shift.5 Fig. : Input vector field data top) used for the snthetic test of the implementation of the natural Helmholtz-Hodge decomposition, as well as the corresponding divergence middle) and curl bottom). Verification In order to verif the implementation of the NHHD algorithm described in the previous section, a snthetic test is accomplished using eemplar data, which is shown for the two-dimensional case, for the sake of simplicit. The input data reads v = v source + v vorte + v shift with v source =,) ep [.5 + )], v vorte =,) ep [.5 + )], and v shift =.,.) 4) and is depicted in Fig. top). As also depicted in Fig. middle and bottom), the source term is irrotational v source = ), the vorte term is solenoidal v vorte = ) and the shift term is both irrotational and solenoidal v shift = v shift = ). Consequentl, for the epected output field, resulting from the NHHD, d = v source, r = v vorte and h = v shift hold. Copright 6 and published b German Association for Laser Anemometr GALA e.v., Karlsruhe, German, ISBN

4 d r h.5 d r h.5.5 d r h.5 Fig. : Output vector field data top) from the natural Helmholtz-Hodge decomposition of the snthetic test data from Fig., as well as the corresponding divergence middle) and curl bottom). The actual output field is depicted in Fig. top). The decomposed vector terms fields qualitativel agree with the epected fields, cf. Fig.. However, there are slight deviations from the epectation, which can especiall be identified in the term h which does not eactl equal v shift. As a result, the divergence v shift and the curl v shift do not complete vanish, unlike epected. Both have a root mean square ) value of less than %, regarding the value of v and v. This behavior is resulting from the discretization of the vector field data and can be suppressed when choosing a smaller data grid size not shown here), which corresponds to a finer spatial resolution at the measurement. Eperimental setup and results For the aeroacoustic analsis using the NHHD, a generic bias flow liner based on Heuwinkel et al. is chosen as an eample. A bias flow liner is an aeroacoustic sound absorber and consists of a perforated sheet with a cavit underneath, where an additional bias) flow is fed into the cavit to pass through the perforation, which increases the damping efficienc. The aeroacoustic damping is based on an interaction between the sound and the flow and needs to be understand further. For this purpose, an optical velocit measurement using Doppler global velocimetr with frequenc modulation FM-DGV, see Haufe et al. 3, 4b) emploing Copright 6 and published b German Association for Laser Anemometr GALA e.v., Karlsruhe, German, ISBN

5 a high-power diode laser W optical output power) is performed at a measurement rate of khz. The eperiments are conducted at the duct acoustic test rig with a rectangular cross section DUCT R) with a length of about 3 m and a cross-sectional area of 6 mm 8 mm. The optical access is provided b glass windows and light scattering particles of diethlhel sebacate having a diameter of about µm) are provided b a particle generator from the compan PIVTEC GmbH. For the first time, a velocit measurement is accomplished in a three-dimensional region of interest at the vicinit of the central orifice, which is visualized in Fig. 3. Measurement data is acquired at 496 locations using multipoint detection and traversing stages. A grazing flow with a velocit of about 34 m/s is provided b a radial compressor, 8.5 mm grazing flow + acoustic wave region of interest.5 mm z glass window Fig. 3: Eperimental setup for the velocit measurement at a generic bias flow liner for the analsis of aeroacoustic phenomena near the central orifice of the perforated sheet, where its center is the point of origin. The bias flow is fed into the cavit underneath the perforated sheet and interacts with the sound. the acoustic ecitation is accomplished b a Monacor R speaker at a frequenc of Hz with a maimum sound pressure level of about 8 db in the duct. A bias flow with a controlled mass flow of 5 kg/h is fed into the cuboid shaped cavit underneath the mm thick perforated sheet. The decomposed velocit terms resulting from the NHHD applied to the measured data are depicted for z = in Fig. 4 as values of the phase-averaged oscillation velocit regarding the acoustic ecitation frequenc. The spatial resolution of the measurements is 66 µm and an interpolation using cubic splines with an interpolation factor of 6 is applied in order to suppress discretization effects for the calculation of the NHHD. In Fig. 4, the highest oscillation velocit is at a low distance of mm to the perforated sheet, which agrees well with previous investigations b Heuwinkel et al.. Note that the solenoidal term r, which is related to flow vortices, is dominating compared to the irrotational term d, which is related to the acoustic particle velocit. However, d has a maimum value of.3 m/s which is more than three times larger compared to the estimated acoustic particle velocit of.4 m/s of the ecitation sound wave assuming plane waves in air at C). This is an indication of a local acoustic source, which results from the interaction between the sound wave and the flow. Future investigations will show, whether and where the sound wave propagates and at which intensit. As alread observed in Fig., the harmonic term is not completel irrotational, which is again due to discretization effects. In order to further reduce these effects, a finer spatial resolution, e.g. using the laser Doppler velocit profile sensor from Haufe et al. 4a), or a finer interpolation is needed in the future. However, the calculation effort for the NHHD will rise with the sith power of the number of data points per dimension for the investigation of the full volume, according to Eqs. ). To face this challenge, an accelerated signal processing, e.g. using graphics processing units like in Kirk and Hwu ), will be a perspective. Conclusion and Outlook The aeroacoustic analsis of measured velocit data using the natural Helmholtz-Hodge decomposition NHHD) was performed for the first time. For this purpose, the three-dimensional vector field of the velocit has been measured using Doppler global velocimetr with sinusoidal Copright 6 and published b German Association for Laser Anemometr GALA e.v., Karlsruhe, German, ISBN

6 d [m/s] r [m/s] h [m/s] liner distance [mm] liner distance [mm] liner distance [mm] d [/s] d [/s] 4 6 aial position [mm] r [/s] r [/s] 4 6 aial position [mm] h [/s] h [/s] 4 6 aial position [mm] Fig. 4: Resulting velocit fields top) of the oscillation at the acoustic ecitation frequenc of Hz, using natural Helmholtz-Hodge decomposition of the velocit data measured according to Fig. 3, as well as corresponding divergence middle) and curl bottom). All values are given as the root mean square ) of the l norm of the velocit vectors. frequenc modulation FM-DGV) at a generic bias flow liner. To this aim, volumetric FM-DGV measurements at 496 locations have been accomplished, providing three-dimensional phaseresolved oscillation velocit data regarding the acoustic ecitation frequenc. As a result of the decomposition, the measured oscillation velocit is dominated b flow vortices, which are induced b the interaction of sound and flow and, thus, being coherent to the acoustic ecitation. The results of this work prove the suppositions in previous investigations, like in Heuwinkel et al.. Moreover, there is an indication of a local acoustic source with a comparativel high acoustic particle velocit three times higher compared to the acoustic ecitation) which results from the interaction of sound and flow. The implementation of the NHHD algorithm in MATLAB has been verified using snthetic test data. The verification revealed that a smaller grid size finer spatial resolution) is necessar in the future to further suppress decomposition artifacts resulting from the numerical discretization. In conclusion, the aeroacoustic analsis using NHHD represents a valuable tool for the investigation of the aeroacoustic damping phenomena at bias flow liners, which enables the design of optimized liners for more efficient sound absorption in jet engine and stationar gas turbines. Copright 6 and published b German Association for Laser Anemometr GALA e.v., Karlsruhe, German, ISBN

7 Acknowledgements The authors thank the German Research Foundation DFG) for their financial support of the research projects CZ 55/5-3 and EN 797/-3. We kindl thank the compan Toptica Photonics for continuous support concerning the modulatable high power diode laser. Special thanks are given to Dr. Harsh Bhatia from the Lawrence Livermore National Laborator for fruitful discussions and Heiko Scholz for his contribution to the NHHD implementation in MATLAB. References Bhatia, H., Pascucci, V., Bremer, P.T., 4: The Natural Helmholtz-Hodge Decomposition for Open- Boundar Flow Analsis, IEEE Transactions on Visualization and Computer Graphics, Vol., No., pp De Roeck, W., Baelmans, M., Desmet, W., 7: An Aerodnamic / Acoustic Splitting Technique for Hbrid CAA Applications, in: 3 th AIAA/CEAS Aeroacoustics Conference, 7-376, Rome, Ital Denaro, F.M., 3: On the application of the Helmholtz Hodge decomposition in projection methods for incompressible flows with general boundar conditions, International Journal for Numerical Methods in Fluids, Vol. 43, No., pp Haufe, D., Fischer, A., Czarske, J., Schulz, A., Bake, F., Enghardt, L., 3: Multi-scale measurement of acoustic particle velocit and flow velocit for liner investigations, Eperiments in Fluids, Vol. 54, No. 7, pp. 7 Haufe, D., Pietzonka, S., Fischer, A., Schulz, A., Bake, F., Enghardt, L., Czarske, J., 4a: Aeroacoustic near-field measurements with microscale resolution, Measurement Science and Technolog, Vol. 5, No., pp Haufe, D., Schulz, A., Bake, F., Enghardt, L., Czarske, J., Fischer, A., 4b: Spectral analsis of the flow sound interaction at a bias flow liner, Applied Acoustics, Vol. 8, pp Heuwinkel, C., Fischer, A., Röhle, I., Enghardt, L., Bake, F., Piot, E., Micheli, F., : Characterization of a Perforated Liner b Acoustic and Optical Measurements, in: 6 th AIAA/CEAS Aeroacoustics Conference, -3765, Stockholm, Sweden Kirk, D.B., Hwu, W.W., : Programming Massivel Parallel Processors: A Hands-on Approach, Morgan Kaufmann Rupp, J., Carrotte, J., Spencer, A., : Interaction Between the Acoustic Pressure Fluctuations and the Unstead Flow Field Through Circular Holes, Journal of Engineering for Gas Turbines and Power, Vol. 3, No. 6, pp Copright 6 and published b German Association for Laser Anemometr GALA e.v., Karlsruhe, German, ISBN