Aeroacoustics: Flow induced sound (Part I)

Transcription

1 VIENNA UNIVERSITY OF TECHNOLOGY Aeroacoustics: Flow induced sound (Part I) Manfred Kaltenbacher Summer School and Workshop Waves in Flows Prague, August 27 - August 31

2 Flow induced sound 2 Flow induced sound is called aeroacoustics

3 Simple Example 3 Flow around an obstacle Obstacle: infinite cylinder Low Mach number Generation of Kár á s vortex street Tonal sound at vortex shedding frequency Shedding frequency Constant over a large Reynolds number-range

4 Scientific Challenge 4 An eddy of size l with typical velocity U generates sound at frequency: with wavelength: Main challenge of CAA (Computational Aeroacoustics) Disparity of length and energy scales Confined flow and large acoustic domains Frequency f(hz) Infra Hearing Ultrasound Hypersound Turbulence Length scale L(m) Turbulence Hearing Acoustic far field Pressure scale p(pa) Hearing Flow

5 Understanding Aeroacoustics 5 Scientist in fluid dynamics Scientist in acoustics Scientist in aeroacoustics

6 Content 6 Fluid dynamics Conservation & constitutive equations Vorticity Towards acoustics Acoustics Wave equations as a perturbation to compressible flow equations Solution approaches to wave equation ( Helmholtz equation) Near- and far-field, compactness Aeroacoustics Lighthill s & Curle s analogy Vortex sound Perturbation equations for low Mach number flows General aeroacoustic approach Applications Axial fan Human phonation

7 Literature 7 G. K. Batchelor. An introduction to fluid dynamics. Cambridge University Press, 1967 M. E. Goldstein. Aeroacoustics. McGraw-Hill, 1976 D. G. Crighton, A. P. Dowling, J. E. Ffowcs-Williams, M. Heckl, and F. G. Leppington. Modern methods in analytical acoustics. Springer Lecture Notes, M. S. Howe. Theory of Vortex Sound. Cambridge Texts in Applied Mathematics, 2003 S. Glegg and W. Devenport. Aeroacoustics of low Mach number flows. Academic Press, 2017 M. Kaltenbacher (editor): Computational Acoustics. Springer International Publishing, 2018 M. Kaltenbacher: Flow induced sound - Course notes, 2018

8 Fluid dynamics 8 Continuum mechanics Total change

9 Fluid dynamics 9 Reynolds' transport theorem (applied to conservation of mass) We consider Total change Divergence (Gauss) integral theorem

10 Fluid dynamics 10 Conservation of momentum (no volume forces) May be rewritten by using mass conservation as

11 Fluid dynamics 11 Conservation of energy Change of total inner energy (no gravity) The change of energy is balanced by volume and surface terms heat production surface pressure heat exchange over surface shear force on surface

12 Fluid dynamics 12 Constitutive equations First law of thermodynamics Enthalpy Thermodynamic equilibrium and homogeneous fluid: two intrinsic state variables determine the state of the fluid; i.e. for the pressure Isentropic case (ds=0): isentropic speed of sound

13 Fluid dynamics 13 Constitutive equations Newtonian fluid Hypothesis of Stoke:

14 Fluid dynamics 14 Vorticity Motion of a fluid particle is a superposition of Translational motion Rotation Distortion of shape (strain) Rate of strain tensor Vorticity A fluid particle with angular velocity has vorticity about the origin, i.e. velocity

15 Fluid dynamics 15 Vorticity Vorticity lines are always tangential to the vorticity vector and form closed field lines pure solenoidal Divide momentum conservation by density and assume Stokesian, homentropic fluid Crocco s form of momentum conservation Lamb vector

16 Fluid dynamics 16 Vorticity Assume incompressible fluid results in the vortex equation This term is only important in regions of high shear, in particular near boundaries. Near walls the velocity becomes very small, so we may use In the absence of viscosity, vortex lines move with the fluid. They are rotated and stretched in a manner determined by the flow. When a vortex tube is stretched, the crosssectional area is decreased and therefore the amplitude of vorticity has to increase in order to preserve the strength of the tube. Vorticity is generated at solid boundaries and the viscosity is responsible for its diffusion into the fluid domain, where it may subsequently be convected by the flow.

17 Fluid dynamics 17 No vorticity in an incompressible flow: Potential flow Note: is divergence- and curl-free! Helmholtz decomposition for incompressible flow Taking the curl results in vortical flow Gree s function for Laplace operator

18 Fluid dynamics 18 Towards acoustic Consider mass conservation in the following form Apply an Helmholtz decomposition to the compressible flow velocity Triple orthogonal decomposition Includes compressible effects and so wave propagation Irrotational deformation without volume change A rigid-body rotation described by vorticity Isotropic expansion

19 Fluid dynamics 19 Towards acoustic Helmholtz-Hodge decomposition, which also considers the topology of a domain, and in special considers the decomposition on a bounded domain Orthogonality requires Vortical field being divergence-free and parallel to boudary PDEs for the potentials

20 Fluid dynamics 20 Towards acoustic We can substract from the scalar potential a harmonic field, which is divergence- and curl-free (flow potential field) Triple decomposition (all orthogonal) divergence- and curl-free PDE for the vortical field PDE for the compressible, radiating (acoustic) field PDE for the potential flow field

21 VIENNA UNIVERSITY OF TECHNOLOGY Aeroacoustics: Flow induced sound (Part II) Manfred Kaltenbacher Summer School and Workshop Waves in Flows Prague, August 27 - August 31

22 Acoustics: Basics 2 Thermodynamic relation: entropy constant and linear pressure-density relation via speed of sound c 0 Perturbation ansatz Substitution into mass and momentum conservation (neglecting viscosity) Linearization

23 Acoustics: Basics 3 Acoustic pressure potential relation Using this relation, acoustic pressure-density relation and linearized mass conservation leads to Same is obtained for acoustic pressure General solution for plane waves (d Alembert) Backward propagating wave General solution for spherical waves (d Alembert Forward propagating wave

24 Acoustics: Basics 4 Sound waves

25 Acoustics: Basics 5 Waves in air, water, solid material Acoustic pressure wavelength Sound pressure level Threshold of hearing Typical SPLs

26 Acoustics: Basics 6 Acoustic Intensity Acoustic power Typical sound power levels

27 Acoustics: Green s function 7 Free field Green s function Solution of wave equation for general source distribution Idea: source is regarded as a superposition of impulsive point sources For each constituent source strength the solution is given by

28 Acoustics: Green s function 8 Overall solution Free field Green s function in frequency domain

29 Acoustics: Far Field Approximation 9 Search approximation for Apply following approximations Fraunhofer approximation

30 Acoustics: Compactness 10 Consider an oscillating sphere with velocity amplitude Model as a point dipole According to free field Green s function we get Explore the relations

31 Acoustics: Compactness 11 Near field term is dominant at sufficient small distances In this case the retarded time can be neglected and the near-filed solution coincide with the incompressible potential flow (hydrodynamic near-field) Compact regions and sources: dimensionless form of wave equation In this case the wave equation reduces to the Laplace equation and the acoustic field coincide with the incompressible potential flow!

32 Acoustics: General solution of wave equation 12 Wave equation Green s function fulfilling Now, we multiply (2) by and (1) by and subtract the so obtained equations

33 Acoustics: General solution of wave equation 13 Now we integrate over source space and time The integrand is zero at the lower limit, when we specify and to be zero at. Causality condition: sound heard at time t must be generated at

34 Acoustics: General solution of wave equation 14 Similar expansion for the second term Using the divergence theorem, we obtain For the solution, we need the pressure as well as normal derivative of on the surfaces; furthermore, we need a Green s function satisfying the inhomogeneous wave equation and causality condition (free field Green s function can be used). For radiating surfaces, we may use the acoustic momentum equation

35 Acoustics: General solution of wave equation 15 Frequency domain formulation Often not known! E.g., mechanical vibration velocity Solve by a numerical method, i.e. boundary element method Taylored Green s function, e.g., fulfilling

36 VIENNA UNIVERSITY OF TECHNOLOGY Aeroacoustics: Flow induced sound (Part III) Manfred Kaltenbacher Summer School and Workshop Waves in Flows Prague, August 27 - August 31

37 Aeroacoustics: Lighthill s analogy 2 Lighthill s wave equation Conservation of momentum Momentum flux tensor Case of now flow Constant speed of sound (isentropic case) Conservation of mass

38 Aeroacoustics: Lighthill s analogy 3 Lighthill s wave equation Case of flow Lighthill s stress tensor Momentum equation Again using mass conservation Source term

39 Aeroacoustics: Lighthill s analogy 4 Lighthill s integral formulation (free radiation) Lighthill s analogy is an exact reformulation of the fluid dynamic equations; so the source term not only accounts for generation of sound but also for acoustic self modulation caused by acoustic nonlinearity convection and refraction of sound waves by the flow attenuation due to thermal and viscous actions. Main challenges for practical application: In order to compute the sound, one needs Lighthill s tensor for which the full set of compressible flow dynamics equations have to be solved! No separation between sources, interaction and propagation!

40 Aeroacoustics: Lighthill s analogy 5 Practical application at low Mach numbers Sound emission from an eddy involves three scales Eddy size: l Wavelength of the sound: λ Dimension of the region: L Problem is solved for Ma << 1 and L/l~1 and by matching the compressible eddy core scaled by l to the surrounding acoustic field scaled by λ (method of asymptotic expansion). Leading order term for Lighthill s tensor is incompressible flow velocity

41 Aeroacoustics: Lighthill s analogy 6 Practical application at low Mach numbers Hybrid approach First perform a incompressible flow simulation Compute source term and acoustic propagation For an incompressible flow, we may rewrite the reduced Lightill source by Furthermore, the divergence of the momentum conservation (neglecting viscous terms) leads to incompressible flow pressure

42 Aeroacoustics: Lighthill s analogy 7 Lighthill s investigation: jet engine noise

43 Aeroacoustics: Lighthill s analogy 8 Solving Lighthill s wave equation including solid bodies (scatterer): Use generalized functions theory Use Green s function method Curle s equation General integral formulation of Lighthill s wave equation not only taking into account stationary scattering objects but also moving surfaces as occurring for propellers and helicopter rotor noise results in Ffowcs Williams and Hawkings equation

44 Aeroacoustics: Vortex sound 9 An eddy of size l with typical velocity U generates sound with frequencies of O(U/l). The wavelength of the sound, λ= O(l/Ma), greatly exceeds the eddy size l of a low Mach number flow. Leading order term of Lighthill s source at low Mach number By vector identities, one obtains small contribution to far field acoustic pressure

45 Aeroacoustics: Vortex sound 10 Using the relation Integrating over τ For far-field approximation, we may use

46 Aeroacoustics: Vortex sound 11 Resulting integral Example of a spinning vortex pair

47 Aeroacoustics: Equation of vortex sound 12 Total enthalpy irrotational flow Medium at rest Assuming homentropic flow (no viscosity), the equation of vortex sound is Considers sound propagation through non-uniform flow! compressible flow velocity Flow can generate sound only, if moving vorticity is present

48 Aeroacoustics: Perturbation equations at low Mach numbers 13 Perturbation ansatz Splitting in temporal mean and fluctuating quantities Additionally splitting of velocity field in solenoidal (incompressible flow) and irrotational (acoustic) components Mean flow Irrotational part of flow velocity = acoustic particle velocity Solenoidal part of flow velocity Acoustic Perturbation Equations (APE) A. Hüppe et al.: Construction and Analysis of an Adapted Spectral Finite ElementMethod to Convective Acoustic Equations; Communications in Computational Physics, 2016

49 Aeroacoustics: Perturbation equations at low Mach numbers 14 Use of acoustic scalar potential Use of mass conservation form APE and substitute acoustic pressure Perturbed Convective Wave Equation (PCWE)

50 Aeroacoustics: Perturbation equations at low Mach numbers 15 Cylinder in cross-flow Pseudo-2D computation Cylinder diameter D = 1m Reynolds-Number 250 Lighthill s equation Perturbed Convective Wave Equation

51 Towards general aeroacoustics 16 Cavity with a lip Vorticity Third Computational Aeroacoustics (CAA) Workshop on Benchmark Problems Category 6., NASA, vol. CP , Cleveland, Ohio, Nov

52 Towards general aeroacoustics 17 Generalization Consider acoustic-flow feedback No restriction on Mach number Solve compressible flow equations on an restricted domain Aeroacoustic formulation Perform compressible flow simulation on a restricted domain Split flow variables in a base flow (non-radiating) and a remaining component (acoustic, radiating fluctuations) using Helmholtz-Hodge decomposition Compute the radiating fluctuations (acoustic field) Radiating pressure Base flow quantities (non-radiating)

53 Towards general aeroacoustics 18 Non-radiating base flow The incompressibility condition leads to the Helmholtz-Hodge decomposition of the flow field Vortical (solenoidal) part of flow velocity (vorticity field) Harmonic part (divergence- and curl-free; potential flow) Irrotational part of flow velocity = acoustic particle velocity

54 Towards general aeroacoustics 19 Vortex sound equation (low Mach number) Door cavity B. Henderson. Automobile noise involving feedback- sound generation by low speed cavity flows. Technical report, In: Third Computational Aeroacoustic(CAA) Workshop on Benchmark Problems, 2000.

55 Application: Cavity with a lip 20 Rate of expansion: Vorticity: 0 50

56 Application: Cavity with a lip 21 Lamb-vector (source term): Fourier transformed at first shear layer mode x-component of Lamb vector y-component of Lamb vector x-component of corrected Lamb vector y-component of corrected Lamb vector

57 Application: Cavity with a lip 22 Field of total enthalpy fluctuation Comparison of acoustic pressure at 1 st shear layer mode

58 VIENNA UNIVERSITY OF TECHNOLOGY Aeroacoustics: Flow induced sound (Part IV) Manfred Kaltenbacher Summer School and Workshop Waves in Flows Prague, August 27 - August 31

59 Application: Axial Fan 2 Experimental setup Stefan Becker University of Erlangen-Nuremberg, Germany Provided as a benchmark at EAA ( Kaltenbacher et al.: Computational Aeroacoustics for Rotating Systems with Application to an Axial Fan; AIAA 2017

60 Application: Axial Fan 3 Experimental setup (ipat, University of Erlangen) (1500 rpm) Microphone positions

61 Computational Aeroacoustics: Workflow 4

62 Computational Aeroacoustics: Low Mach number 5 Extension to rotating systems Euler formulation: Fluid mechanics and acoustics Lagrange formulation: Structural mechanics (rotation of rotor) ALE (Arbitrary Lagrangian Eulerian) Mechanical velocity of rotating parts

63 FE Formulation for stationary and rotating domains 6 Computational setup PDEs in the sub-domains Interface conditions Boundary conditions

64 FE - Formulation for stationary and rotating domains 7 Weak formulation Addition of the two equations

65 FE - Formulation 8 Retain symmetry by adding Retain stability by adding Needs intersection operations Nitsche-type mortaring (IP-DG on interface)

66 Computational Aeroacoustics: Interpolation 9 Source term interpolation Apply a grid intersection approach (conservative interpolation) CFD grid cells are small compared to acoustic ones CFD grid cells are even larger compared to acoustic ones Needs conservative interpolation with cell intersection algorithm CAA CFD

67 Application: Axial Fan 10 CFD mesh CFD-grid: 29.8 million volume cells meshing with Numeca HEXPRESS /Hybrid Solver: OpenFOAM with adapted solver Version of pimpledymfoam Detached Eddy Simulation (DES) Use of AMI (Arbitrary Mesh Interface) for coupling stationary and rotating domains; Time step: Dt = 10µs 256 Cores on VSC (Vienna Scientific Cluster)

68 Application: Axial Fan 11 LDA measurement vs. simulation U xmean [m/s] Measurement

69 Application: Axial Fan 12 Instationary pressure (Kulite sensors) Measurement (0.28s) Measurement (30s) OpenFoam (0.28s)

70 Application: Axial Fan 13 CFD: flow field

71 Application: Axial Fan 14 Acoustic source terms: frequency domain 254Hz 1004Hz

72 Application: Axial Fan 15 Acoustic computation Nitsche-type mortaring between rotating and stationary mesh Grid: finite elements nodes Time step: Dt = 20µs PML (Perfectly matched Layer) Axial fan Motor Propagation: FRONT PML Propagation : BACK Rotating domain Hybrid mesh Kaltenbacher et al.: A modifed and stable version of a perfectly matched layer technique for the 3-d second order wave equation in time domain with an application to aeroacoustics. Journal of Computational Physics, 2013

73 Application: Axial Fan 16 Acoustic Field

74 Application: Axial Fan 17 Acoustic result PSD: Power spectral density of microphone 3

75 Human Phonation 18 In-vivo measurements are very restricted A computer model is more appropriate to analyze the human phonation Sound sources: Modulated air stream Pressure gradient Turbulent eddies Vibrating solid

76 Human Phonation 19 Modelling approach I: Full coupled simulation (restrict to 2D models) Flow Pressure- & Viscous Forces Mechanical displacement Mechanics Acoustics Cooperation with J. Valasek, P. Svacek; CTU Prague, Czech Republic J. Valasek et al.: On the Application of Acoustic Analogies in the Numerical Simulation of Human Phonation Process, Flow Turbulence Combust, 2018

77 Human Phonation 20 Modelling approach II: Prescribed mechanical vibration of vocal folds Flow Prescribed movement Mechanics Acoustics 3D Geomety 1-3 Mio. cells Prescribed vibration of vocal folds Open Foam & CFS++ Cooperation with Petr Šidlof, Technical University of Liberec, Czech Republic Zörner et al.: Flow and Acoustic Effects in the Larynx for Varying Geometries. Acta Acustica united with Acustica, 2016

78 Human Phonation 21 Flow field at characteristic time step Velocity (m/s)

79 Human Phonation 22 Acoustic simulation domain (just glottis)

80 Human Phonation 23 Vocal tract Cross section Distance

81 Human Phonation 24 Comparison between perturbation equation and Lighthill s analogy Frequencies of tonal components are the same Lighthill s analogy: pressure fluctuation PCWE: acoustic pressure PCWE Lighthill Frequency

82 Human Phonation 25 SPL outside mouth /i/ N /i/ S F F F /u/ N /u/ S

83 Human Phonation 26 Influence of false vocal folds Vocal folds 12mm False vocal folds 50mm Harmonics are stronger deveoped in case of considering false vocal folds. Amplitudes of non-harmonics are reduced.

84 Human Phonation 27 Influence of shape of glottis opening and closing Rectangular shape, symmetric opening-closing. Rectangular shape, lower vocal Fold is stationary, simulating an unilateral paralysis.

85 Human Phonation 28 Velocity magnitude and jet contours (iso-surface at 5 m/s) at maximum opening Symmetric moving Unilateral paralysis

86 Human Phonation 29 Acoustic source terms (Fourier-transformed) Symmetric moving 100Hz 2665Hz Unilateral paralysis 100Hz 2665Hz

87 Human Phonation 30 Acoustic spectra Symmetric Paralysis

88 31 WAVES 2019 Conference Vienna, Austria: August 25 30, th International Conference on Mathematical and Numerical Aspects of Wave Propagation