Archives of Acoustics, 45, 4, pp. 633–645, 2020

Finite Element Modelling of a Flow-Acoustic Coupling in Unbounded Domains

AGH – University of Science and Technology

Ireneusz CZAJKA
AGH – University of Science and Technology

Andrzej GOŁAŚ
AGH – University of Science and Technology

One of the main issues of design process of HVAC systems and ventilation ducts in particular is correct modelling of coupling of the flow field and acoustic field of the air flowing in such systems. Such a coupling can be modelled in many ways, one of them is using linearised Euler equations (LEE). In this paper, the method of solving these equations using finite element method and open source tools is decribed. Equations were transformed into functional and solved using Python language and FEniCS software. The non-reflective boundary condition called buffer layer was also implemented into equations, which allowed modelling of unbounded domains. The issue, influence of flow on wave propagation, could be adressed using LEE equations, as they take non-uniform mean flow into account. The developed tool was verified and results of simulations were compared with analytical solutions, both in one- and two-dimensional cases. The obtained numerical results are very consistent with analytical ones. Furthermore, this paper describes the use of the developed tool for analysing a more complex model. Acoustic wave propagation for the backward-facing step in the presence of flow calculated using Navier-Stokes equations was studied.
Keywords: linearised Euler equations (LEE); FEniCS; finite element method; non-reflective boundary conditions; open source
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Åbom M. (2006), An Introduction to Flow Acoustics, Stockholm, Sweden: KTH.

Ahrens J., Geveci B., Law C. (2005), Paraview: An end-user tool for large data visualization, [in:] C.D. Hansen, C.R. Johnson (Eds), Visualization Handbook, Butterworth-Heinemann, pp. 717–731, doi: 10.1016/B978-012387582-2/50038-1.

Alnaes M.S. et al. (2015), The Fenics project version 1.5, Archive of Numerical Software, 3(100): 9–23, doi: 10.11588/ans.2015.100.20553.

Armaly B.F., Durst F., Pereira J.C.F., Schonung B. (1983), Experimental and theoretical investigation for backward-facing step flow, Journal of Fluid Mechanics, 127: 473–496, doi: 10.1017/S0022112083002839.

Atkins H., Casper J. (1994), Nonreflective boundary conditions for high-order methods, AIAA Journal, 32(4): 512–518, doi: 10.2514/3.12015.

Bailly C., Juve D. (2000), Numerical solution of acoustic propagation problems using linearized euler equations, AIAA Journal, 38(1): 22–29, doi: 10.2514/2.949

Bendat J.S., Piersol A.G. (2010), Random Data. Analysis and Measurement Procedures, Hoboken, US: John Wiley & Sons, Inc.

Berenger J.-P. (1994), A perfectly matched layer for the absorption of electromagnetic waves, Journal of Computational Physics, 114(2): 185–200, doi: 10.1006/jcph.1994.1159

Bermudez, A., Hervella-Nieto, A., Prieto, A., & Rodriguez, R. (2008), Perfectly matched layers, [in:] Computational Acoustics of Noise Propagation in Fluids – Finite and Boundary Element Methods, pp. 167–196, Springer Berlin Heidelberg, doi: 10.1007/978-3-540-77448-8_7.

Biswas G., Breuer M., Durst F. (2004), Backward-facing step flows for various expansion ratios at low and moderate reynolds numbers, Journal of Fluids Engineering, 126(3): 362–374, doi: 10.1115/1.1760532.

Butcher J.C. (2016), Numerical Methods for Ordinary Differential Equations, Chichester, England: John Wiley & Sons Inc., doi: 10.1002/9781119121534.

Colonius T. (1997), Lectures on Computational Aeroacoustics, Pasadena, United States: California Institute of Technology.

Czajka I., Gołas A. (2017), Engineering Methods of Numerical Analysis and Design of Experiment [in Polish], Kraków, Poland: Wydawnictwa AGH.

Dykas S., Wroblewski W. (2006), Method of Modeling Aerodynamic Noise in Transonic Flows [in Polish], Gliwice, Poland: Wydawnictwo Politechniki Śląskiej.

Dykas S., Wroblewski W., Rulik S., Chmielniak T. (2010), Numerical method for modeling of acoustic waves propagation, Archives of Acoustics, 35(1): 35–48, doi: 10.2478/v10168-010-0003-7.

Epikhin A., Evdokimov I., Kraposhin M., Kalugin M., Strijhak S. (2015), Development of a dynamic library for computational aeroacoustics applications using the OpenFOAM open source package, Procedia Computer Science, 66: 150–157, doi: 10.1016/j.procs.2015.11.018.

Ewert R., Schröder W. (2003), Acoustic perturbation equations based on flow decomposition via source filtering, Journal of Computational Physics, 188(2): 365–398, doi: 10.1016/s0021-9991(03)00168-2.

Giles M.B. (1990), Nonreflecting boundary conditions for euler equation calculations, AIAA Journal, 28(12): 2050–2058, doi: 10.2514/3.10521.

Gill J., Fattah R., Zhang X. (2017), Towards an effective non-reflective boundary condition for computational aeroacoustics, Journal of Sound and Vibration, 392: 217–231, doi: 10.1016/j.jsv.2016.11.036

Givoli D. (2008), Computational absorbing boundaries, [in:] Computational Acoustics of Noise Propagation in Fluids – Finite and Boundary Element Methods, Berlin–Heidelberg: Springer, doi: 10.1007/978-3-540-77448-86.

Hagstrom T., Goodrich J. (2003), Accurate radiation boundary conditions for the linearized euler equations in cartesian domains, SIAM Journal on Scientific Computing, 24(3): 770–795, doi: 10.1137/s1064827501395914.

Kaltenbacher M. (2017), Fundamental equations of acoustics, [in:] Kaltenbacher M. (Ed.), Computational Acoustics. CISM International Centre for Mechanical Sciences (Courses and Lectures), Vol. 579, pp. 1–33, Springer, Cham.

Koloszár L., Villedieu N., Deconinck H., Rambaud P., Anthoine J. (2019), Improved characteristic non-reflecting boundary conditions for the linearized euler equations, [in:] 16th AIAA/CEAS Aeroacoustics Conference, doi: 10.2514/6.2010-3984.

Kosloff R., Kosloff D. (1986), Absorbing boundaries for wave propagation problems, Journal of Computational Physics, 63(2): 363–376, doi: 10.1016/0021-9991(86)90199-3.

Kuttruff H. (2007), Acoustics. An introduction, Abingdon, England: Taylor & Francis.

Langtangen H.P., Logg A. (2017), Solving PDEs in Python – The FEniCS tutorial. Vol. I. Cham, Switzerland: Springer, doi: 10.1007/978-3-319-52462-7.

Le H., Moin P., Kim J. (1997), Direct numerical simulation of turbulent flow over a backwardfacing step, Journal of Fluid Mechanics, 330: 349–374, doi: 10.1017/S0022112096003941.

Lighthill M.J. (1952), On sound generated aerodynamically. I. General theory, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 211(1107): 564–587, doi: 10.1098/rspa.1952.0060.

Logg A., Mardal K.A., Wells G. (Eds), (2012), Automated solution of differential equations by the finite element method. The FEniCS book, Lecture Notes in Computational Science and Engineering, Berlin–Heidelberg: Springer, doi: 10.1007/978-3-642-23099-8.

Lyrintzis A.S., George A.R. (1989), Use of the Kirchhoff method in acoustics, AIAA Journal, 27(10): 1451–1453, doi: 10.2514/3.10285.

Łojek P., Czajka I. (2019), Scalable finite element implementation of linearized Euler equations, [in:] XXIII Conference on Acoustic and Biomedical Engineering.

Mechel F.P. (Ed.) (2008), Formulas of Acoustics, Heidelberg, Germany: Springer-Verlag GmbH, doi: 10.1007/978-3-662-07296-7.

Povitsky A. (2000), Numerical study of wave propagation in a non-uniform flow, Technical Report, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center.

Ribes A., Caremoli C. (2007), Salomé platform component model for numerical simulation, [in:] Compsac 07: Proceeding of the 31st Annual International Computer Software and Applications Conference (COMPSAC 2007), Beijing, 2007, pp. 553–564, doi: 10.1109/COMPSAC.2007.185.

Richards S.K., Zhang X., Chen X.X., Nelson P.A. (2004), The evaluation of non-reflecting boundary conditions for duct acoustic computation, Journal of Sound and Vibration, 270(3): 539–557, doi: 10.1016/j.jsv.2003.09.042.

Rienstra S.W., Hirschberg A. (2004), An Introduction to Acoustics, Eindhoven University of Technology, Eindhoven, Netherlands.

Suder-Debska K., Gołas A., Filipek R. (2018), An Introduction to Applied Acoustics [in Polish], Kraków, Poland: Wydawnictwa AGH.

Wagner C.A., Huttl T., Sagaut P. (Eds) (2006), Large-Eddy Simulation for Acoustics, New York, US: Cambridge University Press, doi: 10.1017/CBO9780511546143.

DOI: 10.24425/aoa.2020.135251