# Streamwise velocity statistics in turbulent boundary layers that spatially develop to high Reynolds number

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## Abstract

Well-resolved measurements of the streamwise velocity in zero pressure gradient turbulent boundary layers are presented for friction Reynolds numbers up to 19,670. Distinct from most studies, the present boundary layers undergo nearly a decade increase in Reynolds number solely owing to streamwise development. The profiles of the mean and variance of the streamwise velocity exhibit logarithmic behavior in accord with other recently reported findings at high Reynolds number. The inner and mid-layer peaks of the variance profile are evidenced to increase at different rates with increasing Reynolds number. A number of statistical features are shown to correlate with the position where the viscous force in the mean momentum equation loses leading order importance, or similarly, where the mean effect of turbulent inertia changes sign from positive to negative. The near-wall peak region in the 2-D spectrogram of the fluctuations is captured down to wall-normal positions near the edge of the viscous sublayer at all Reynolds numbers. The spatial extent of this near-wall peak region is approximately invariant under inner normalization, while its large wavelength portion is seen to increase in scale in accord with the position of the mid-layer peak, which resides at a streamwise wavelength that scales with the boundary layer thickness.

## Keywords

Boundary Layer Reynolds Number High Reynolds Number Streamwise Velocity Reynolds Number Range## Notes

### Acknowledgments

This work was partially supported by the National Science Foundation and by the Office of Naval Research. The authors are grateful to Dr. V. Kulandaivelu for making his data available.

## References

- Coles DE (1956) The law of the wake in the turbulent boundary layer. J Fluid Mech 1:191–226MathSciNetCrossRefzbMATHGoogle Scholar
- DeGraaff DB, Eaton JK (2000) Reynolds number scaling of the flat plate turbulent boundary layer. J Fluid Mech 422:319–346CrossRefzbMATHGoogle Scholar
- Fife P, Wei T, Klewicki J, McMurtry P (2005) Stress gradient balance layers and scale hierarchies in wall-bounded turbulent flows. J Fluid Mech 532:165–189MathSciNetCrossRefzbMATHGoogle Scholar
- Fife P, Klewicki J, Wei T (2009) Time averaging in turbulence settings may reveal an infinite hierarchy of length scales. J Discrete Continuous Dyn Syst 24:781–807MathSciNetCrossRefzbMATHGoogle Scholar
- Gad-el-Hak M, Bandyopadhyay P (1994) Reynolds number effects in wall-bounded turbulent flows. Appl Mech Rev 47:307–365CrossRefGoogle Scholar
- Hultmark M, Vallikivi M, Bailey SCC (2012) Turbulent pipe flow at extreme Reynolds numbers. Phys Rev Lett 108:094501CrossRefGoogle Scholar
- Hutchins N (2012) Caution: tripping hazards. J Fluid Mech 710:1–4MathSciNetCrossRefzbMATHGoogle Scholar
- Hutchins N, Marusic I (2007) Large-scale influences in near-wall turbulence. Phil Trans R Soc Lond A 365:647–664CrossRefzbMATHGoogle Scholar
- Hutchins N, Marusic I (2007) Evidence of very long meandering streamwise structures in the logarithmic region of turbulent boundary layers. J Fluid Mech 579:1–28CrossRefzbMATHGoogle Scholar
- Hutchins N, Nickels TB, Marusic I, Chong MS (2009) Hot-wire spatial resolution issues in wall-bounded turbulence. J Fluid Mech 635:103–136CrossRefzbMATHGoogle Scholar
- Johansson AV, Alfredsson PH (1983) Effects of imperfect spatial resolution on measurements of wall-bounded shear flows. J Fluid Mech 137:409–421CrossRefGoogle Scholar
- Klewicki J (2010) Reynolds number dependence, scaling and dynamics of turbulent boundary layers. J Fluids Eng 132:094001CrossRefGoogle Scholar
- Klewicki JC (2013a) Self-similar mean dynamics in turbulent wall-flows. J Fluid Mech 718:596–621MathSciNetCrossRefGoogle Scholar
- Klewicki JC (2013b) A description of turbulent wall-flow vorticity consistent with mean dynamics. J Fluid Mech (in press)Google Scholar
- Klewicki J, Falco R (1990) On accurately measuring statistics associated with small-scale structure in turbulent boundary layers using hot-wire probes. J Fluid Mech 219:119–142CrossRefGoogle Scholar
- Klewicki J, Fife P, Wei T, McMurtry P (2007) A physical model of the turbulent boundary layer consonant with mean momentum balance structure. Phil Trans R Soc A 365:823–839MathSciNetCrossRefzbMATHGoogle Scholar
- Klewicki J, Fife P, Wei T (2009) On the logarithmic mean profile. J Fluid Mech 638:73–93CrossRefzbMATHGoogle Scholar
- Kulandaivelu V (2012) Evolution of zero pressure gradient turbulent boundary layers from different initial conditions. PhD Dissertation, University of MelbourneGoogle Scholar
- Ligrani PM, Bradshaw P (1987) Spatial resolution and measurement of turbulence in the viscous sublayer using subminiature hot-wire probes. Exp Fluids 5:407–417CrossRefGoogle Scholar
- Marusic I, McKeon BJ, Monkewitz PA, Nagib HM, Smits AJ, Sreenivasan KR (2010) Wall-bounded turbulent flows: recent advances and key issues. Phys Fluids 22:065103CrossRefGoogle Scholar
- Marusic I, Monty JP, Hultmark M, Smits AJ (2013) On the logarithmic region in wall turbulence. J Fluid Mech 716:R3MathSciNetCrossRefGoogle Scholar
- Mathis R, Hutchins N, Marusic I (2009) Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J Fluid Mech 628:311–337CrossRefzbMATHGoogle Scholar
- Mathis R, Marusic I, Hutchins N, Sreenivasan KR (2012) The relationship between the velocity skewness and the amplitude modulation of the small scale by the large scale in turbulent boundary layers. Phys Fluids 23:121702CrossRefGoogle Scholar
- Metzger MM, Klewicki JC (2001) A comparative study of near-wall turbulence in high and low Reynolds number boundary layers. Phys Fluids 13:692–701CrossRefGoogle Scholar
- Mochizuki S, Nieuwstadt FTM (1996) Reynolds-number-dependence of the maximum in the streamwise velocity fluctuations in wall turbulence. Exp Fluids 21:218–226CrossRefGoogle Scholar
- Monkewitz PA, Chauhan KA, Nagib HM (2008) Approach to an asymptotic state for zero pressure gradient turbulent boundary layers. Phil Trans R Soc A 365:755–770Google Scholar
- Morrill-Winter C, Klewicki J (2013) Scale separation effects on the mean vorticity transport mechanism of wall turbulence. Phys Fluids 25:015108CrossRefGoogle Scholar
- Nagib HM, Chauhan KA, Monkewitz PA (2007) Comparisons of mean flow similarity laws in zero pressure gradient turbulent boundary layers. Phys Fluids 20:105102Google Scholar
- Patel VC (1965) Calibration of the Preston tube and limitations on its use in pressure gradients. J Fluid Mech 23:185–208CrossRefGoogle Scholar
- Priyadarshana PJA, Klewicki JC, Treat S, Foss JF (2007) Statistical structure of turbulent-boundary layer velocity–vorticity products at high and low Reynolds numbers. J Fluid Mech 570:307–346CrossRefGoogle Scholar
- Pullin D, Inoue M, Saito N (2013) On the asymptotic state of high Reynolds number, smooth-wall turbulent flows. Phys Fluids 25:015116CrossRefGoogle Scholar
- Schlatter P, Orlu R (2012) Turbulent boundary layers at moderate Reynolds numbers: inflow length and tripping effects. J Fluid Mech 710:5–34CrossRefzbMATHGoogle Scholar
- Smits AJ, McKeon BJ, Marusic I (2011) High Reynolds number wall turbulence. Ann Rev Fluid Mech 43:353–375CrossRefGoogle Scholar
- Wei T, Fife P, Klewicki J, McMurtry P (2005) Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows. J Fluid Mech 522:303–327MathSciNetCrossRefzbMATHGoogle Scholar