# On the Outer Layer Controversy for a Turbulent Boundary Layer over a Rough Wall

## Abstract

An impressive number of experimental and numerical studies of turbulent flows over rough walls has appeared over the last 20 years. Much useful information has been obtained in terms of the turbulence structure both near the roughness canopy and in the outer flow region. However, the issue of whether or not the outer region of the boundary layer is affected by the nature of the wall has yet to be resolved satisfactorily. While the available data, mostly at sufficiently large values of the Reynolds numbers and δ∕*k* (δ and *k* are the boundary layer thickness and characteristic roughness height, respectively), seem to suggest that 3D and transverse 2D rough surfaces may affect the outer layer differently, this suggestion can only be tested rigorously once the measurement of the wall-normal velocity fluctuation over the transverse 2D roughness is improved. With the benefit of the channel flow DNS data, it is argued that, for this latter surface type, the wall shear stress, as inferred from the form drag or drag balance, has actually been measured reasonably accurately in the past.

## Keywords

Wall Shear Stress Reynolds Stress Outer Region Roughness Element Reynolds Shear Stress## Notes

### Acknowledgements

RAA is most grateful to P-A. Krogstad, S. Leonardi and P. Burattini for useful discussions on several issues addressed on in this paper. The past support from the ARC and the collaboration with P. Orlandi, R. Smalley and H. Shafi are warmly acknowledged.

## References

- 1.J.C. Rotta, Turbulent boundary layers in incompressible flow. Prog. Aero. Sci.
**2**, 1–220 (1962)CrossRefGoogle Scholar - 2.I. Tani, Some equilibrium turbulent boundary layers. Fluid Dyn. Res.,
**1**, 49–58 (1968)CrossRefGoogle Scholar - 3.R.J. Smalley, R.A. Antonia, L. Djenidi, Self-preservation of rough-wall turbulent boundary layers. Eur. J. Mech. B-Fluids
**20**, 591–602 (2001)zbMATHCrossRefGoogle Scholar - 4.M.R. Raupach, R.A. Antonia, S. Rajagopalan, Rough-wall turbulent boundary layers. Appl. Mech. Rev.
**44**, 1–25 (1991)CrossRefGoogle Scholar - 5.P.-A. Krogstad, R.A. Antonia, L.W.B. Browne, Comparison between rough- and smooth-wall turbulent boundary layers. J. Fluid Mech.
**245**, 599–617 (1992)CrossRefGoogle Scholar - 6.P.-A. Krogstad, R.A. Antonia, Structure of turbulent boundary layers on smooth and rough walls. J. Fluid Mech.
**277**, 1–21 (1994)CrossRefGoogle Scholar - 7.P.-A. Krogstad, R.A. Antonia, Surface roughness effects in turbulent boundary layers. Exp. Fluids
**27**, 450–460 (1999)CrossRefGoogle Scholar - 8.J. Jimenez, Turbulent flows over rough walls. Ann. Rev. Fluid Mech.
**36**, 173–196 (2004)MathSciNetCrossRefGoogle Scholar - 9.S. Leonardi, P. Orlandi, R.J. Smalley, L. Djenidi, R.A. Antonia, Direct numerical simulations of turbulent channel flow with transverse square bars on the wall. J. Fluid Mech.
**491**, 229–238 (2003)zbMATHCrossRefGoogle Scholar - 10.J. Cui, V.C. Patel, C.-L. Lin, Large eddy simulation of turbulent flow in a channel with rib roughness. Int. J. Heat, Fluid Flow
**24**, 372–388 (2003)CrossRefGoogle Scholar - 11.A. Ashrafian, H.I. Andersson, M. Manhart, DNS of turbulent flow in a rod-roughened channel. Int. J. Heat, Fluid Flow
**25**, 373–383 (2004)CrossRefGoogle Scholar - 12.T. Ikeda, P.A. Durbin, Direct simulations of a rough wall channel flow. J. Fluid Mech.
**571**, 235–263 (2007)zbMATHCrossRefGoogle Scholar - 13.M.F. Tachie, D.J. Bergstrom, R. Balachandar, Rough wall turbulent boundary layers in shallow open channel flow. J. Fluids Eng.
**122**, 533–541 (2000)CrossRefGoogle Scholar - 14.M.F. Tachie, D.J. Bergstrom, R. Balachandar, Roughness effects in low-Re number open-channel turbulent boundary layers. Exp. Fluids
**35**, 338–346 (2002)CrossRefGoogle Scholar - 15.L. Keirsbulck, L. Labraga, A. Mazouz, C. Tournier, Surface roughness effects on turbulent boundary layer structures. J. Fluids Eng.
**124**, 127–135 (2002)CrossRefGoogle Scholar - 16.K.A. Flack, M.P. Schultz, J.S. Connelly, Examination of a critical roughness height for outer layer similarity. Phys. Fluids
**19**, 095104 (2007)CrossRefGoogle Scholar - 17.K.A. Flack, M.P. Schultz, T.A. Shapiro, Experimental support for Townsend’s Reynolds number similarity hypothesis on rough walls. Phys. Fluids
**17**, 035102 (2005)CrossRefGoogle Scholar - 18.M.P. Schultz, K.A. Flack, Outer layer similarity in fully rough turbulent boundary layers. Exp. Fluids
**38**, 328–340 (2005)CrossRefGoogle Scholar - 19.Y. Wu, K.T. Christensen, Outer-layer similarity in the presence of a practical rough-wall topography. Phys. Fluids
**19**, 085108 (2007)CrossRefGoogle Scholar - 20.P. Orlandi, S. Leonardi, R.A. Antonia, Turbulent channel flow with either transverse or longitudinal roughness elements on one wall. J. Fluid Mech.
**561**, 279–305 (2006)zbMATHCrossRefGoogle Scholar - 21.P. Orlandi, S. Leonardi, DNS of turbulent channel flows with two and three-dimensional roughness. J. Turb
**7**, no 53 (2006)Google Scholar - 22.P. Orlandi, S. Leonardi, Direct numerical simulation of three-dimensional turbulent rough channels: parameterization and flow physics. J. Fluid Mech.
**606**, 399–415 (2008)zbMATHCrossRefGoogle Scholar - 23.K. Bhaganagar, J. Kim, G. Coleman, Effect of roughness on wall-bounded turbulence. Flow Turb. Comb.
**72**, 463–492 (2004)zbMATHCrossRefGoogle Scholar - 24.R.J. Volino, M.P. Schultz, K.A. Flack, Turbulence structure in rough- and smooth-wall boundary layers. J. Fluid Mech.
**592**, 263–293 (2007)zbMATHCrossRefGoogle Scholar - 25.P.-A. Krogstad, H.I. Andersson, O.M. Bakken, A. Ashrafian, An experimental and numerical study of channel flow with rough walls. J. Fluid Mech.
**530**, 327–352 (2005)zbMATHCrossRefGoogle Scholar - 26.P. Burattini, S. Leonardi, P. Orlandi, R.A. Antonia, Comparison between experiments and direct numerical simulations in a channel flow with roughness on one wall. J. Fluid Mech.
**600**, 403–426 (2008)zbMATHCrossRefGoogle Scholar - 27.K. Bhaganagar, G. Coleman, J. Kim, Effect of roughness on turbulent fluctuations in a turbulent channel flow. Phys Fluids
**19**, 028103 (2007)CrossRefGoogle Scholar - 28.P.R. Spalart, Direct simulation of a turbulent boundary layer up to Re?=1410. J. Fluid Mech.
**187**, 61–98 (1988)zbMATHCrossRefGoogle Scholar - 29.S.G. Saddoughi, S.V. Veeravalli, Local isotropy of turbulent boundary layers at high Reynolds number. J. Fluid Mech.
**268**, 333–372 (1994)CrossRefGoogle Scholar - 30.M.P. Schultz, K.A. Flack, The rough-wall turbulent boundary layer from the hydraulically smooth to the fully rough regime. J. Fluid Mech.
**580**, 381–405 (2007)zbMATHCrossRefGoogle Scholar - 31.P.M. Ligrani, R.J. Moffat, Structure of transitionally rough and fully rough turbulent boundary layers. J Fluid Mech.
**162**, 69–98 (1986)MathSciNetCrossRefGoogle Scholar - 32.S.-H. Lee, H.J. Sung, Direct numerical simulation of the turbulent boundary layer over a rod-roughened wall. J. Fluid Mech.
**584**, 125–146 (2007)zbMATHCrossRefGoogle Scholar - 33.M. Acharya, M.P. Escudier, Turbulent flow over mesh roughness, in
*Turbulent Shear Flows*, vol 5 (Springer, 1987), pp. 177–185Google Scholar - 34.A.E. Perry, K.L. Lim, S.M. Henbest, An experimental study of turbulence structure in smooth- and rough-wall turbulent boundary layers. J. Fluid Mech.
**177**, 437–466 (1987)CrossRefGoogle Scholar - 35.T. Kameda, S. Mochizuki, H. Osaka, LDA measurement in roughness sub-layer beneath turbulent layer developed over two-dimensional square rough surface, in
*Proceedings of 12th International Symposium on Application of Laser Techniques to Fluid Mechanics (CDROM-paper 28.3)*, Lisbon, July 12–14 (2004)Google Scholar - 36.L. Djenidi, R.A. Antonia, M. Amielh, F. Anselmet, A turbulent boundary-layer over a two-dimensional rough wall. Exp. Fluids
**44**, 37–47 (2008)CrossRefGoogle Scholar - 37.R. Antonia, R.E. Luxton, The response of a turbulent boundary layer to a step change in surface roughness part 1. Smooth to rough. J. Fluid Mech.
**48**, 721–761 (1971)CrossRefGoogle Scholar - 38.P. Mulhearn, Turbulent flow over a periodic rough surface. Phys Fluids
**21**, 1113–1115 (1978)CrossRefGoogle Scholar - 39.T. Kameda, H. Osaka, S. Mochizuki, Mean flow quantities for the turbulent boundary layer over a k-type rough wall, in
*Proceedings of 13th AFMC*, Monash Univerity, December 13–18, 1998, pp. 357–360Google Scholar - 40.S. Leonardi, P. Orlandi, L. Djenidi, R.A. Antonia, Structure of turbulent channel flow with square bars on one wall. Int. J. Heat Fluid Flow
**25**, 384–392 (2004)CrossRefGoogle Scholar - 41.J.D. Li, A.E. Perry, Shear stress profiles in zero pressure-gradient turbulent boundary layers, in
*Proceedings of 10th AFMC*, University of Melbourne, December, 11–15, 7.9–7.12 (1989)Google Scholar - 42.P.J. Mulhearn, J.J. Finnigan, Turbulent flow over a very rough random surface. Boundary-Layer Met.
**15**, 109–132 (1978)CrossRefGoogle Scholar - 43.M.R. Raupach, A.S. Thom, I. Edwards, A wind-tunnel study of turbulent flow close to regularly arrayed rough surfaces. Boundary-Layer Met.
**18**, 373–397 (1980)CrossRefGoogle Scholar - 44.A.E. Perry, W.H. Schofield, P. N. Joubert, Rough wall turbulent boundary layers. J. Fluid Mech.
**37**, 383–413 (1969)CrossRefGoogle Scholar - 45.T. Kameda, H. Osaka, S. Mochizuki, Turbulent structure in the vicinity of a roughness element for boundary layer over a k-type rough wall. Trans. JSME Series B
**50-458**, 2299–2306 (2000)Google Scholar - 46.S. Leonardi, P. Orlandi, R.A. Antonia, Properties of d- and k-type roughness in a turbulent channel flow. Phys. Fluids
**19**, 125101 (2007)CrossRefGoogle Scholar - 47.F. Pineau, V.D. Nguyen, J. Dickinson, J. Belanger, Study of flow over a rough surface with passive boundary layer manipulators and direct wall drag measurements. AIAA-87-0357 (1987)Google Scholar
- 48.Y. Furuya, M. Miyata, H. Fujita, Turbulent boundary layer and flow resistance on plates roughened by wires. J. Fluids Eng.
**98**, 635–644 (1976)CrossRefGoogle Scholar - 49.B.G. Brzek, R.B. Cal, G. Johansson, L. Castillo, Transitionally rough zero pressure gradient turbulent boundary layers. Exp. Fluids
**44**, 115–124 (2008)CrossRefGoogle Scholar