1. Prediction of jet mixing noise in flight from static tests
Ulf Michel
CFD Software Entwicklungs- und Forschungsgesellschaft mbH Berlin
22nd AIAA/CEAS Aeroacoustics Conference Lyon, France, 30 May – 1 June 2016

2. AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016
Introduction
Influence of flight velocity on jet noise very important
Example: Engine noise tests performed on static test beds
Take-off certification noise levels must be predicted correctly for Mf = 0.26 … 0.28
Cruise noise must be known (noise in rear cabin dominated by jet noise) for Mf = 0.78 … 0.85
Flight effect on jet mixing noise has surprising properties
Noise is much louder in flight than to be expected from reduced relative velocity.
Frequency is not reduced in flight as to be expected from smaller relative velocity.
Frequency is Doppler-shifted for observers on the ground, indicating a source that is stationary relative to the nozzle.
Noise level is almost unchanged in forward arc for take-off flight Mach numbers.
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

3. Industry standard for flight effect prediction
Industry standard for modeling the effect of flight speed is using a relative velocity exponent m(θe)
m(θe) determined with flight simulations of model jets in free-jet tunnels
re is wave-normal distance (flyover: observer distance at emission time)
Predicted noise levels for cruise Mach numbers much too low
Relative velocity exponent approach too simple, does not describe physics properly.
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

4. Theoretical analysis of flight effect
Acoustic analogy in a uniform flight stream best described with convective wave equation for the pressure (Michalke & Michel 1979)
Nozzle fixed coordinate system:
Source field stationary random
Emitted acoustic field also stationary random
Integration boundaries stationary
Source term q can be approximated by quadrupole and dipole sources (Morfey 1973)
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

5. Integral solution of convective wave equation
Green function for radiation into free space (differs in Doppler factor Df from normal wave equation, re is wave-normal distance)
Solution in acoustic far field
Qq and Qd are quadrupole and dipole source terms for the acoustic far field (see paper)
Both integrals yield forward arc amplification (Df-n)
Amplification larger for quadrupole than for dipole noise.
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

6. Power-spectral density in acoustic far field
Time averaged quantities are of interest
Jet noise sources non-compact
Source interference effects dominate directivity (Michel 2007, 2009)
Source interference can only be studied in frequency domain
Power-spectral density in acoustic far field consists of four terms
conjugate complex
Cross-spectral densities between quadrupole and dipole contributions cannot be neglected since they are both dominated by the same instability wave motions in the jet.
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

7. Power-spectral density in geometric far field
Power-spectral density of quadrupole noise
Power-spectral density of sources
All interference effects with neighboring source positions are contained in interference integral Gqq
Ψs: Phase of source cross spectral density (wavelike motion)
Ψr: Phase due to retarded time difference
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

8. Influence of interference integral
Source interference explains
Directivity of jet mixing noise (Michel 2007, 2009, Michel & Ahuja 2014)
Deviation of Mach number exponent from theoretical values 8 (quadrupole noise) and 6 (dipole noise) at 90° to jet axis (see paper)
Directivity of broadband shock noise (Michel 1995)
Peak frequency and width of peak of broadband shock noise (Michel 1995)
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

9. Decomposition into azimuthal components
Source field and far field can be decomposed into azimuthal components
Each component m in the far field depends only on the same component in the source field (for jets with axisymmetric mean flow, Michalke 1972)
Power-spectral density in far field composed of azimuthal components
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

10. Stretching of flow field in flight
Flow field stretched by influence of flight stream
Same normalized flow quantities located further downstream in a flight stream
Coherence field also stretched
Stretching factor
Take-off σ ≈ 1.3
Cruise σ ≈3.0 … 4.0
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

11. Influence of jet stretching
Stretching factor result of instability analysis (Michalke & Hermann 1982)
Normalizations (indicated by star)
Normalized axial source position
Normalized axial source length scale
Normalized axial coherence length scale
Normalized radial source length scale unaffected by stretching
Normalized frequency (result of instability analysis)
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

12. Normalization of source strength
Normalized quadrupole source strength
Normalized dipole source strength
Assumptions:
Normalized source quantities independent of jet and flight Mach numbers
Normalized turbulence quantities independent of jet and flight Mach numbers
Normalized mean velocity profiles independent of jet and flight Mach numbers
Decomposition into azimuthal components independent of jet and flight Mach numbers
All sources are located on circular cylinder with radius rs= Dj/2
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

13. AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016
Azimuthal components
Solution for one component m of mean square sound pressure due to quadrupole sources in geometric far field:
Interference function
Azimuthal Interference
Axial Interference integral
Interference effects are described by
azimuthal interference function Jm and
axial interference integral Fqqm
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

14. Mean square sound pressure
Quadrupole sources
Quadrupole-dipole sources
Interference integrals, (azimuthal and axial interference),
assumed to be identical for all three sources (wave-like sources)
Dipole sources
Main difference: power of (Mj-Mf)/Df
Michalke & Michel proposed to carry out static experiments for Mach number Me= (Mj-Mf)/Df and multiply results with σ3 Df2
Condition: interference integrals only functions of Me (not completely satisfied)
Prediction valid for any combination of quadrupole and dipole sources.
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

15. Comparison with Aérotrain data
Predictions (Michalke & Michel 1979) in comparison with experimental data
Dotted lines indicates range with extrapolated static data.
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

16. Comparison with Aérotrain data
Predictions (Michalke & Michel 1979) in comparison with experimental data
Dotted lines indicates range with extrapolated static data.
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

17. Extrapolations of static Aérotrain data
Interpolation of static directivities
Available range Mj=1.09 to 1.49
Extrapolation to smaller and larger Mach numbers by use of Mach number exponent
Extrapolation to small Mach numbers obviously non-physical
Peak in rear arc outcome of Helmholtz number scaling through wave refraction (Michel & Ahuja 2014)
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

18. Prediction of cruise noise
Cruise noise louder than static noise (reduced ambient pressure not considered)
Cause: jet stretching
Predictions with relative velocity exponents m yield noise reduction for 90 degrees. (m=3 yields noise level reduction by 10.8 dB for cruise Mach number 0.84, m=4 yields reduction by 14.4 dB)
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

19. Interference integral
Assumption was that interference integral G depends only on Me
Definition
Azimuthal Interference
Azimuthal interference depends on
Axial interference depends on
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

20. Axial source interference
Axial interference function
Rear arc: all sources almost in phase (“super directivity”)
Other angles: radiation decreased due to oscillating sign of integrand
Oscillations get stronger in the forward arc and for higher jet speeds
This behavior has led to the proposal that rear arc radiation is caused by large scales, while all other angles are results of fine scale) + -
Integrand as function of normalized separation ξc
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

21. Scaling for axial source interference
Condition for identical axial source interference
Mach number of static jet must be larger by stretching factor σ.
Higher levels in rear arc
Lower levels in forward arc.
Ms=σ Me
Applied to dipole noise only
Ms=Me
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

22. AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016
Conclusion
Jet mixing noise in a flight stream can be predicted well with the method of Michalke and Michel (1979).
Method only based on Lighthill theory with no arbitrary constant.
Small errors in forward and rear arc are results of incorrect modelling of the axial and azimuthal source interference.
Levels in forward arc slightly overpredicted, in rear arc slightly underpredicted.
Jet mixing noise in cruise is much louder than predicted with relative velocity relations.
Increases in the order of 20 dB are caused in cruise by stretching of the jet flow field by the flight stream.
Stretching factor σ
Take-off: σ ≈ 1.3
Cruise: σ ≈ 3, on future engines σ ≈ 4
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016

23. AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016
Explanation of surprising properties of flight effect on jet mixing noise
Jet noise much louder than originally expected from reduced relative velocity
Increase of source length, coherence length and frequency range due to jet stretching
Frequency not reduced in flight as to be expected from smaller relative velocities
Strouhal numbers of most unstable instability waves increase with flight stream
Jet noise is result of growing and decaying instability waves in jet shear layer
Frequency is Doppler-shifted for observers on the ground
Sources are stationary random relative to the nozzle.
Moving source model of original jet noise theories incorrect.
Wave model of Michalke must be used.
Noise level surprisingly high in forward arc
Forward arc amplification is outcome of theory for stationary sources in a flight stream
AIAA/CEAS Aeroacoustics Conference, Lyon, 30 June 2016