Hostname: page-component-546b4f848f-q5mmw Total loading time: 0 Render date: 2023-06-01T01:43:02.329Z Has data issue: false Feature Flags: { "useRatesEcommerce": true } hasContentIssue false

Frequency–wavenumber mapping in turbulent shear flows

Published online by Cambridge University Press:  15 October 2015

Roeland de Kat*
Engineering and the Environment, University of Southampton, Highfield, Southampton SO17 1BJ, UK
Bharathram Ganapathisubramani
Engineering and the Environment, University of Southampton, Highfield, Southampton SO17 1BJ, UK
Email address for correspondence:


Spatial turbulence spectra for high-Reynolds-number shear flows are usually obtained by mapping experimental frequency spectra into wavenumber space using Taylor’s hypothesis, but this is known to be less than ideal. In this paper, we propose a cross-spectral approach that allows us to determine the entire wavenumber–frequency spectrum using two-point measurements. The method uses cross-spectral phase differences between two points – equivalent to wave velocities – to reconstruct the wavenumber–frequency plane, which can then be integrated to obtain the spatial spectrum. We verify the technique on a particle image velocimetry (PIV) data set of a turbulent boundary layer. To show the potential influence of the different mappings, the transfer functions that we obtained from our PIV data are applied to hot-wire data at approximately the same Reynolds number. Comparison of the newly proposed technique with the classic approach based on Taylor’s hypothesis shows that – as expected – Taylor’s hypothesis holds for larger wavenumbers (small spatial scales), but there are significant differences for smaller wavenumbers (large spatial scales). In the range of Reynolds number examined in this study, double-peaked spectra in the outer region of a turbulent wall flow are thought to be the result of using Taylor’s hypothesis. This is consistent with previous studies that focused on examining the limitations of Taylor’s hypothesis (del Álamo & Jiménez, J. Fluid Mech., vol. 640, 2009, pp. 5–26). The newly proposed mapping method provides a data-driven approach to map frequency spectra into wavenumber spectra from two-point measurements and will allow the experimental exploration of spatial spectra in high-Reynolds-number turbulent shear flows.

© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


del Álamo, J. C. & Jiménez, J. 2009 Estimation of turbulent convection velocities and corrections to Taylor’s approximation. J. Fluid Mech. 640, 526.CrossRefGoogle Scholar
Buxton, O. H. R., de Kat, R. & Ganapathisubramani, B. 2013 The convection of large and intermediate scale fluctuations in a turbulent mixing layer. Phys. Fluids 25, 125105.CrossRefGoogle Scholar
Cenedese, A., Romano, G. P. & Di Felice, F. 1991 Experimental testing of Taylor’s hypothesis by L.D.A. in highly turbulent flow. Exp. Fluids 11, 351358.CrossRefGoogle Scholar
Davies, P. O. A. L. & Fisher, M. J. 1963 Statistical properties of the turbulent velocity fluctuations in the mixing region of a round subsonic jet. AASU Report 233. University of Southampton.Google Scholar
Davies, P. O. A. L., Fisher, M. J. & Barrat, M. J. 1963 The characterisics of the turbulence in the mixing region of a round jet. J. Fluid Mech. 15, 337367.CrossRefGoogle Scholar
Dennis, D. J. & Nickels, T. B. 2008 On the limitations of Taylor’s hypothesis in constructing long structures in a turbulent boundary layer. J. Fluid Mech. 614, 197206.CrossRefGoogle Scholar
Dennis, D. J. C. & Nickels, T. B. 2011 Experimental measurement of large-scale three-dimensional structures in a turbulent boudnary layer. Part 1. Vortex packets. J. Fluid Mech. 673, 180217.CrossRefGoogle Scholar
Elsinga, G. E., Poelma, C., Schröder, A., Geisler, R., Scarano, F. & Westerweel, J. 2012 Tracking of vortices in a turbulent boundary layer. J. Fluid Mech. 697, 273295.CrossRefGoogle Scholar
Fisher, M. J. & Davies, P. O. A. L. 1964 Correlation measurements in a non-frozen pattern of turbulence. J. Fluid Mech. 18, 97116.CrossRefGoogle Scholar
Freegarde, T. 2013 Introduction to the Physics of Waves. Cambridge University Press.Google Scholar
Geng, C., He, G., Wang, Y., Xu, C., Lozano-Durán, A. & Wallace, J. M. 2015 Taylor’s hypothesis in turbulent channel flow considered using a transport equation analysis. Phys. Fluids 27, 025111.CrossRefGoogle Scholar
Goldschmidt, V. W., Young, M. F. & Ott, E. S. 1981 Turbulent convective velocities (broadband and wavenumber dependent) in a plane jet. J. Fluid Mech. 105, 327345.CrossRefGoogle Scholar
Harrison, M.1958 Pressure fluctuations on the wall adjacent to a turbulent boundary layer. Tech. Rep. 1260, David Taylor Model Basin.Google Scholar
Herpin, S., Stanislas, M. & Soria, J. 2010 The organization of near-wall turbulence: a comparison between boundary layer SPIV data and channel flow DNS data. J. Turbul. 11 (47), 130.CrossRefGoogle Scholar
Hutchins, N. & Marusic, I. 2007 Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 128.CrossRefGoogle Scholar
Hutchins, N., Nickels, T. B., Marusic, I. & Chong, M. S. 2009 Hot-wire spatial resolution issues in wall-bounded turbulence. J. Fluid Mech. 635, 103136.CrossRefGoogle Scholar
de Kat, R. & Ganapathisubramani, B. 2013 Characteristics of Reynolds stresses in a turbulent boundary layer. In Proceedings of the International Symposium on Turbulence and Shear Flow Phenomena, TSFP-8, 28–30 August 2013, Poitiers, France. Paper 4A-4.Google Scholar
de Kat, R. & Ganapathisubramani, B. 2014 Convection of momentum transport events in a turbulent boundary layer. Bull. Am. Phys. Soc. II 59 (20), 140.Google Scholar
Krogstad, P.-Å., Kaspersen, J. H. & Rimestad, S. 1998 Convection velocities in a turbulent boundary layer. Phys. Fluids 10 (4), 949957.CrossRefGoogle Scholar
Lee, J., Lee, J. H., Choi, J.-I. & Sung, H. J. 2014 Spatial organization of large- and very-large-scale motions in a turbulent channel flow. J. Fluid Mech. 749, 818840.CrossRefGoogle Scholar
Lee, M. & Moser, R. D. 2015 Direct numerical simulation of turbulent channel flow up to $Re_{{\it\tau}}\approx 5200$ . J. Fluid Mech. 774, 395415.CrossRefGoogle Scholar
LeHew, J., Guala, M. & McKeon, B. J. 2011 A study of the three-dimensional spectral energy distribution in a zero pressure gradient turbulent boundary layer. Exp. Fluids 51, 9971012.CrossRefGoogle Scholar
LeHew, J., Guala, M. & McKeon, B. J. 2013 Time-resolved measurements of coherent structures in the turbulent boundary layer. Exp. Fluids 54, 1508.CrossRefGoogle Scholar
Lin, C. C. 1953 On Taylor’s hypothesis and the acceleration terms in the Navier–Stokes equations. Q. Appl. Maths 10 (4), 295306.CrossRefGoogle Scholar
Mathis, R., Hutchins, N. & Marusic, I. 2011 A predictive inner–outer model for streamwise turbulence statistics in wall-bounded flows. J. Fluid Mech. 681, 537566.CrossRefGoogle Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336382.CrossRefGoogle Scholar
Moin, P. 2009 Revisiting Taylor’s hypothesis. J. Fluid Mech. 640, 14.CrossRefGoogle Scholar
Monty, J. P. & Chong, M. S. 2009 Turbulent channel flow: comparison of streamwise velocity data from experiments and direct numerical simulation. J. Fluid Mech. 633, 461474.CrossRefGoogle Scholar
Nickels, T. B., Marusic, I., Hafez, S. & Chong, M. S. 2005 Evidence of the $k_{1}^{-1}$ law in a high-Reynolds-number turbulent boundary layer. Phys. Rev. Lett. 95 (7), 074501.CrossRefGoogle Scholar
Perry, A. E. & Abell, C. J. 1975 Scaling laws for pipe-flow turbulence. J. Fluid Mech. 67 (2), 257271.CrossRefGoogle Scholar
Renard, N. & Deck, S. 2015 On the scale-dependent turbulent convection velocity in a spatially developing flat plate turbulent boundary layer at Reynolds number $R_{{\it\theta}}=13000$ . J. Fluid Mech. 775, 105148.CrossRefGoogle Scholar
Romano, G. P. 1995 Analysis of two-point velocity measurements in near-wall flows. Exp. Fluids 20, 6883.CrossRefGoogle Scholar
Rosenberg, B. J., Hultmark, M., Vallikivi, M., Bailey, S. C. C. & Smits, A. J. 2013 Turbulence spectra in smooth- and rough-wall pipe flow at extreme Reynolds numbers. J. Fluid Mech. 731, 4663.CrossRefGoogle Scholar
Taylor, G. I. 1938 The spectrum of turbulence. Proc. R. Soc. Lond. 164 (919), 476490.CrossRefGoogle Scholar
Vallikivi, M., Ganapathisubramani, B. & Smits, A. J. 2015 Spectral scaling in boundary layers and pipes at very high Reynolds numbers. J. Fluid Mech. 771, 303326.CrossRefGoogle Scholar
Wilczek, M., Stevens, R. J. A. M. & Meneveau, C. 2015 Spatio-temporal spectra in the logarithmic layer of wall turbulence: large-eddy simulations and simple models. J. Fluid Mech. 769, R1.CrossRefGoogle Scholar
Wills, J. A. B. 1964 Convection velocities in turbulent shear flows. J. Fluid Mech. 20, 417432.CrossRefGoogle Scholar