(A Primitive Approach to) Vortex Sound Theory and Application to Vortex Leapfrogging Christophe Schram von Karman Institute for Fluid Dynamics Aeronautics and Aerospace & Environmental and Applied Fluid Dynamics depts ERCOFTAC course Computational Aeroacoustics
Plan Lighthill s analogy The quadrupolar character of low-mach turbulence noise (M 8 law) is not an accident! There has been premeditation! Vortex Sound Theory, choice of the source term Powell s analogy, Mohring s analogy, conservative formulation Application to vortex pairing described by wrong flow models If your flow model is wrong just tell your analogy that it s correct!
Lighthill s aeroacoustical analogy : concept The problem of sound produced by a turbulent flow is, from the listener s point of view, analogous toa problem of propagation in a uniform medium at rest in which equivalent sources are placed. Wave propagation region: linear wave operator applies No source source region S y V x observer in uniform stagnant fluid propagation region uniform fluid at rest Turbulent region: fluid mechanics equations apply mass momentum No external forces no external dipole!
Lighthill s analogy: formal derivation
Lighthill s aeroacoustical analogy: how to make it useful? Reformulation of fluid mechanics equations, and use of arbitrary speed c 0 : Definition of a reference state: Aeroacoustical analogy : source region S y V observer position x propagation region uniform fluid at rest with Lighthill s tensor Exact and perfectly useless!
Sound produced by free isothermal turbulent flows at low Mach number Solution using Green s fct sound scattering at boundaries No boundary no external forces no external dipole! integral solution Purpose: simplify the RHS High Reynolds number Isentropic Low Mach number No monopole! Using free field Green s fct Quadrupolar source
Lighthill s M 8 law Integral solution: Scaling law: Acoustic scale: D U 0 Flow time scale: Spatial derivative: λ Acoustical power:
Vortex Sound Theory Conservation principles Powell s analogy Mohring s analogy Conservative formulation Application to vortex pairing
Lighthill s analogy: some issues for free subsonic flows Spatial extent of source term for a localized distribution of vorticity (Oseen vortex): 1 / r Alternative formulation of the analogy : Vortex Sound Theory Yields a more localised source term Allows reinforcing the quadrupolar character of free turbulence
Invariants of incompressible, inviscid vortex flows in absence of external forces (Saffman, 1992) Circulation: where C is a closed material line. Impulse (momentum): vanishes if the force f derives of a single-valued potential, and for inviscid flows. vanishes in absence of non-conservative body forces. Kinetic energy: conserved quantity using the same assumptions.
Vortex Sound Theory: Powell s analogy for free flows Vectorial identity: Momentum equation becomes: Similar manipulation as for Lighthill s analogy: low Mach isentropic Retaining leading order terms in M 2 : M 2 conservation of kinetic energy Integral solution using free field Green s function and first order Taylor expansion of the retarded time: conservation of impulse Powell s integral formulation:
Vortex Sound Theory: Möhring s analogy for free flows Starting from Powell s integral formulation: Using vectorial s identity: By substitution: conservation of kinetic energy Using Helmholtz s vorticity transport equation: Möhring s integral formulation:
Vortex Sound Theory: 2 solutions for the same problem We have derived two (formally) equivalent formulations of the Vortex Sound Theory: Powell s analogy: Mohring s analogy: Although formally equivalent, these two formulations do not yield the same numerical robustness! The choice of a source term affects the numerical performance of the prediction!
Vortex Sound Theory for axisymmetrical flows Coordinate of a vortex element: General form of velocity and vorticity: Powell s analogy becomes: Möhring s analogy becomes:
Application: vortex ring pairing Vortex pairing = inviscid interaction (Biot-Savart) Vortex leapfrogging: periodic motion Vortex merging : requires core deformation Can be easily stabilized and studied at laboratory scale One of the mechanisms of sound production in subsonic jets Generic interaction showing how the reciprocal exchange of impulse b two vortex elements produces a quadrupole in far field for each vortex ring
Vortex pairing: U 0 = 5.0 m/s
2D and 3D models of vortex ring leapfrogging 2D model (σ << d << R 0 ): locally planar interaction, neglects vortex stretching: 3D model (σ << d = O(R 0 ) ): accounts for vortex stretching:
2D model: vortex trajectories and flow invariants Two cases considered: d / R 0 = 0.1 and 0.3. Locus of the vortex cores: 2D d / R 0 = 0.1 d / R 0 = 0.3 2D 3D d / R 0 = 0.1 d / R 0 = 0.3 2D Flow invariants: secular term
2D model: sound prediction Powell s analogy: secular term Möhring s analogy: secular term Conclusion: failure of both Powell s and Möhring s analogies when applied to a flow model that does not respect the conservation of momentum and kinetic energy.
Möhring s solution: reinforcement of conservation assumptions Using Lamb (1932) identities: d / R 0 = 0.1 2D 3D Imposing further conservation of impulse: We obtain: d / R 0 = 0.3 2D 3D Imposing further conservation of kinetic energy:
Generalization of Möhring s solution Using Lamb (1932) identities: will disappear in subsequent derivations Second correction: subtraction of the vortex centroid axial coord. from the axial coord. of each vortex element. Doesn t harm if impulse is conserved, since Improves numerical stability. Imposing further conservation of impulse and kinetic energy: conservative formulation
Robustness of different formulations of Vortex Sound Theory Investigation of the robustness of Powell s form, Möhring s form and of the conservative form when the flow data is perturbed. Perturbation = addition of random noise to coordinates, circulation. Effects on conservation of impulse and kinetic energy? Effects on sound prediction? Relation b both?
Reference case vortex trajectory d / R 0 = 1 impulse sound production kinetic energy
Effect of perturbation on flow invariants Perturbation of coordinates impulse Perturbation of circulation impulse kinetic energy kinetic energy
Effect on sound prediction perturb. of coordinates Powell Möhring perturb. of circulation Powell Möhring conservative form perturbation of coordinates perturbation of circulation
Application to PIV data
PIV results: flow invariants Low frequency fluctuations of about 10%. Increasing scatter due to growing instabilities. 5.0 m/s 5.0 m/s 5.0 m/s 34.2 m/s 34.2 m/s 34.2 m/s
PIV results: acoustical source terms 5.0 m/s 5.0 m/s 5.0 m/s 34.2 m/s 34.2 m/s 34.2 m/s
Acoustic predictions 2nd time derivative: 4th order polynomial fit over moving interval acoustical source term. Case U 0 = 5.0 m/s: good agreement between predictions obtained from PIV data and 3D leapfrogging model. Case U 0 = 34.2 m/s: order of magnitude OK, but quite different frequency content. 5.0 m/s PIV 3D model 34.2 m/s PIV 3D model
Acoustic measurements SPL (db Re 20µPa) 80 70 60 50 40 30 20 10 0-10 background noise -20 0 2.5 5 7.5 10 12.5 f (khz) x θ = = 0.9 m o 90
Surprise: double pairing SPL (db Re 20µPa) 80 70 60 50 40 30 20 1P+2P 2P 2P EXC+1P+2P 2P 1P+2P EXC+1P+2P 2P 1P+2P 2P EXC+1P+2P unexcited excited 10 0 0 2.5 5 7.5 10 12.5 f (khz)
Intermittence of double pairing and comparison prediction - measurement SPL (db Re 20µPa) 80 70 60 50 40 30 20 1P+2P 2P 2P 2P 1P+2P 2P EXC+1P+2P EXC+1P+2P 2P 1P+2P average spectrum EXC+1P+2P SPL (db Re 20µPa) 80 70 60 50 40 30 20 1P EXC+1P 1P EXC+1P 1P EXC+1P single spectrum 10 10 0 0 2.5 5 7.5 10 12.5 f (khz) 0 0 2.5 5 7.5 10 12.5 f (khz) For some acquisitions: absence of double pairing. Quite good agreement between prediction and measurement, for first and third pairing harmonics.
Summary Aeroacoustical analogies allow extracting a maximum of acoustical information from a given description of the flow field Assuming a decoupling between the sound production and propagation, the analogies provide an explicit integral solution for the acoustical field at the listener position Improves numerical robustness Permits drawing scaling laws Some formulations make the dominant character of the source appear more explicitly, and allow making useful approximations. Without approximations, the analogy is useless!
A few references A. Pierce, Acoustics: an Introduction to its Physical Principles and Applications, McGraw-Hill Book Company Inc., New York, 1981. S.W. Rienstra and A. Hirschberg, An Introduction To Acoustics(corrections), Report IWDE 01-03 May 2001, revision every year or so M.E. Goldstein, Aeroacoustics, McGraw-Hill International Book Company, 1976. A.P. Dowling and J.E. Ffowcs Williams, Sound and Sources of Sound,Ellis Horwood-Publishers, 1983. D.G. Crighton, A.P. Dowling, J.E. Ffowcs Williams, M. Heckl and F.G. Leppington, Modern Methods in Analytical Acoustics, Springer-Verlag London, 1992. And of course: the VKI Lecture Series
VKI Lecture Series - 2-4 Dec 2014 Fundamentals of Aeroengine Noise Tue 2 Wed 3 Thu 4 9:00 Innovative architectures Combustion noise Subsonic jet noise prediction 10:15 N. Tantot (SNECMA) M. Heckl (Univ. Keele) C. Bailly (ECL) 10:45 12:00 Aeroengine airframe integration and aeroacoustic installation effets T. Node-Langlois(Airbus) Aeroengine nacelle liner design and optimization G. Gabard (ISVR) An introduction to supersonic jet noise C. Bailly 14:00 15:15 Fundamentals of aeroacoustic analogies C. Schram (VKI) Analytical methods for turbomachinery noise prediction (con td) M. Roger Experimental methods applied to jet noise M. Felli (INSEAN) 15:45 17:00 Analytical methods for turbomachinery noise prediction M. Roger (ECL) Visit VKI Laboratories Advanced analysis techniques (wavelets, LSE, POD) for noise sources identification R. Camussi (Univ. Roma3) www.vki.ac.be