# Wavelet analysis of wall turbulence to study large-scale modulation of small scales

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## Abstract

Wavelet analysis is employed to examine amplitude and frequency modulations in broadband signals. Of particular interest are the streamwise velocity fluctuations encountered in wall-bounded turbulent flows. Recent studies have shown that an important feature of the near-wall dynamics is the modulation of small scales by large-scale motions. Small- and large-scale components of the velocity time series are constructed by employing a spectral separation scale. Wavelet analysis of the small-scale component decomposes the energy in joint time–frequency space. The concept is to construct a low-dimensional representation of the small-scale time-varying spectrum via two new time series: the instantaneous amplitude of the small-scale energy and the instantaneous frequency. Having the latter in a time-continuous representation allows a more thorough analysis of frequency modulation. By correlating the large-scale velocity with the concurrent small-scale amplitude and frequency realizations, both amplitude and frequency modulations are studied. In addition, conditional averages of the small-scale amplitude and frequency realizations depict unique features of the scale interaction. For both modulation phenomena, the much studied time shifts, associated with peak correlations between the large-scale velocity and small-scale amplitude and frequency traces, are addressed. We confirm that the small-scale amplitude signal leads the large-scale fluctuation close to the wall. It is revealed that the time shift in frequency modulation is smaller than that in amplitude modulation. The current findings are described in the context of a conceptual mechanism of the near-wall modulation phenomena.

## Keywords

Amplitude Modulation Time Shift Instantaneous Frequency Wavelet Power Spectrum Streamwise Velocity Fluctuation## Notes

### Acknowledgments

The authors wish to gratefully acknowledge the Australian Research Council for financial support. Furthermore, we would like to give special thanks to Dr. Daniel Chung for insightful discussions.

## References

- Addison PS (2002) The illustrated wavelet transform handbook. Taylor & Francis, New YorkzbMATHCrossRefGoogle Scholar
- Baars WJ, Tinney CE (2013) Transient wall pressures in an overexpanded and large area ratio nozzle. Exp Fluids 54:1468CrossRefGoogle Scholar
- Baars WJ, Ruf JH, Tinney CE (2015) Non-stationary shock motion unsteadiness in an axisymmetric geometry with pressure gradient. Exp Fluids 56(92):1Google Scholar
- Bandyopadhyay PR, Hussain AKMF (1984) The coupling between scales in shear flows. Phys Fluids 27(9):2221–2228CrossRefGoogle Scholar
- Bernardini M, Pirozzoli S (2011) Inner/outer layer interactions in turbulent boundary layers: a refined measure for the large-scale amplitude modulation mechanism. Phys Fluids 23(6):061701CrossRefGoogle Scholar
- Blackwelder RF, Kovasznay LSG (1972) Time scales and correlations in a turbulent boundary layer. Phys Fluids 15(9):1545–1554CrossRefGoogle Scholar
- Boashash B (1992) Estimating and interpreting the instantaneous frequency of a signal—part 1: fundamentals. Proc IEEE 80(4):520–538CrossRefGoogle Scholar
- Brown GL, Thomas ASW (1977) Large structure in a turbulent boundary layer. Phys Fluids 20(10):S243–S252CrossRefGoogle Scholar
- Chauhan K, Philip J, de Silva CM, Hutchins N, Marusic I (2014) The turbulent/non-turbulent interface and entrainment in a boundary layer. J Fluid Mech 742:119–151CrossRefGoogle Scholar
- Chauhan KA, Monkewitz PA, Nagib HM (2009) Criteria for assessing experiments in zero pressure gradient boundary layers. Fluid Dyn Res 41:021404CrossRefGoogle Scholar
- Chung D, McKeon BJ (2010) Large-eddy simulation of large-scale structures in long channel flow. J Fluid Mech 661:341–364zbMATHCrossRefGoogle Scholar
- Cohen L (1989) Time-frequency distributions—a review. Proc IEEE 77(7):941–981CrossRefGoogle Scholar
- Cohen L (1995) Time-frequency analysis. Prentice-Hall Inc, Upper Saddle RiverGoogle Scholar
- Daubechies I (1992) Ten lectures on wavelets. Society for Industrial and Applied Mathematics, PhiladelphiazbMATHCrossRefGoogle Scholar
- Dimotakis PE (2005) Turbulent mixing. Annu Rev Fluid Mech 37:329–356MathSciNetCrossRefGoogle Scholar
- Duvvuri S, McKeon BJ (2015) Triadic scale interactions in a turbulent boundary layer. J Fluid Mech 767:R4CrossRefGoogle Scholar
- Farge M (1992) Wavelet transforms and their application to turbulence. Annu Rev Fluid Mech 24:395–457MathSciNetCrossRefGoogle Scholar
- Favre AJ, Gaviglio JJ, Dumas R (1967) Structure of velocity space–time correlations in a boundary layer. Phys Fluids 10(9):S138–S145CrossRefGoogle Scholar
- Ganapathisubramani B, Longmire EK, Marusic I (2003) Characteristics of vortex packets in turbulent boundary layers. J Fluid Mech 478:35–46zbMATHCrossRefGoogle Scholar
- Ganapathisubramani B, Hutchins N, Monty JP, Chung D, Marusic I (2012) Amplitude and frequency modulation in wall turbulence. J Fluid Mech 712:61–91zbMATHMathSciNetCrossRefGoogle Scholar
- Guala M, Metzger M, McKeon BJ (2011) Interactions within the turbulent boundary layer at high Reynolds number. J Fluid Mech 666:573–604zbMATHCrossRefGoogle Scholar
- Hutchins N (2014) Large-scale structures in high Reynolds number wall-bounded turbulence. In: Talamelli A et al (eds) Progress in turbulence V, vol 149. Springer International Publishing, pp 75–83Google Scholar
- Hutchins N, Marusic I (2007a) Evidence of very long meandering structures in the logarithmic region of turbulent boundary layers. J Fluid Mech 579:1–28zbMATHCrossRefGoogle Scholar
- Hutchins N, Marusic I (2007b) Large-scale influences in near-wall turbulence. Philos Trans R Soc A 365:647–664zbMATHCrossRefGoogle Scholar
- Hutchins N, Nickels TB, Marusic I, Chong MS (2009) Hot-wire spatial resolution issues in wall-bounded turbulence. J Fluid Mech 635:103–136zbMATHCrossRefGoogle Scholar
- Hutchins N, Monty JP, Ganapathisubramani B, Ng HCH, Marusic I (2011) Three-dimensional conditional structure of a high-Reynolds-number turbulent boundary layer. J Fluid Mech 673:255–285zbMATHCrossRefGoogle Scholar
- Jacobi I, McKeon BJ (2013) Phase relationships between large and small scales in the turbulent boundary layer. Exp Fluids 54:1481CrossRefGoogle Scholar
- Klewicki JC (2013) A description of turbulent wall-flow vorticity consistent with mean dynamics. J Fluid Mech 737:176–204zbMATHMathSciNetCrossRefGoogle Scholar
- Kulandaivelu V (2011) Evolution and structure of zero pressure gradient turbulent boundary layer. Ph.D. thesis, The University of Melbourne, Dept. of Mechanical Engineering, Melbourne, AustraliaGoogle Scholar
- Larsson C, Öhlund O (2014) Amplitude modulation of sound from wind turbines under various meteorological conditions. J Acoust Soc Am 135(1):67–73CrossRefGoogle Scholar
- Ligrani PM, Bradshaw P (1987) Spatial resolution and measurement of turbulence in the viscous sublayer using subminiature hot-wire probes. Exp Fluids 5(6):407–417CrossRefGoogle Scholar
- Marusic I, Heuer WD (2007) Reynolds number invariance of the structure inclination angle in wall turbulence. Phys Rev Lett 99:114504CrossRefGoogle Scholar
- Marusic I, Mathis R, Hutchins N (2010) Predictive model for wall-bounded turbulent flow. Science 329(5988):193–196zbMATHMathSciNetCrossRefGoogle Scholar
- Mathis R, Hutchins N, Marusic I (2009) Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J Fluid Mech 628:311–337zbMATHCrossRefGoogle Scholar
- Mathis R, Marusic I, Chernyshenko SI, Hutchins N (2013) Estimating wall-shear-stress fluctuations given an outer region input. J Fluid Mech 715:163–180zbMATHMathSciNetCrossRefGoogle Scholar
- Neuberg J (2000) External modulation of volcanic activity. Geophys J Int 142:232–240CrossRefGoogle Scholar
- Nickels TB, Marusic I, Hafez S, Chong MS (2005) Evidence of the \(k_{1}^{-1}\) law in a high-Reynolds-number turbulent boundary layer. Phys Rev Lett 95:074501CrossRefGoogle Scholar
- Pope SB (2000) Turbulent flows. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
- Rao KN, Narasimha R, Narayanan MAB (1971) The ‘bursting’ phenomenon in a turbulent boundary layer. J Fluid Mech 48:339–352CrossRefGoogle Scholar
- Schlatter P, Örlü R (2010) Quantifying the interaction between large and small scales in wall-bounded turbulent flows: a note of caution. Phys Fluids 22Google Scholar
- Sreenivasan KR, Prabhu A, Narasimha R (1983) Zero-crossings in turbulent signals. J Fluid Mech 137:251–272CrossRefGoogle Scholar
- Talluru KM, Baidya R, Hutchins N, Marusic I (2014a) Amplitude modulation of all three velocity components in turbulent boundary layers. J Fluid Mech 746:R1CrossRefGoogle Scholar
- Talluru KM, Kulandaivelu V, Hutchins N, Marusic I (2014b) A calibration technique to correct sensor drift issues in hot-wire anemometry. Meas Sci Technol 105304:1–6Google Scholar
- Tomkins CD, Adrian RJ (2003) Spanwise structure and scale growth in turbulent boundary layers. J Fluid Mech 490:37–74zbMATHCrossRefGoogle Scholar
- Wark CE, Nagib HM (1991) Experimental investigation of coherent structures in turbulent boundary layers. J Fluid Mech 230:183–208CrossRefGoogle Scholar