Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-01T01:10:45.306Z Has data issue: false hasContentIssue false
  • English
  • Français

On the near-wall instability of oblique flow over a wavy wall

Published online by Cambridge University Press:  01 July 2022

Philip Hall Open the ORCID record for Philip Hall [Opens in a new window]
Philip Hall*
Affiliation:
School of Mathematics, Monash University, VIC 3800, Australia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The instability of a shear flow passing over a wavy wall with wave crests not perpendicular to the flow direction is investigated. The friction Reynolds number for the flow is large and the wave amplitude scaled on the wavelength is small compared with the viscous wall layer. The instability takes the form of a streamwise vortex of wavelength comparable to the viscous wall layer in which the basic flow adjusts to the presence of the wall. For a given wall amplitude, the instability considered is the first one to arise as the Reynolds number is increased and modes of wavelength comparable to the viscous layer grow much faster than modes of wavelength comparable to the wall wavelength. The instability is not driven by centrifugal or viscous effects but is a novel kind of cross-flow vortex instability associated with a spatially periodic flow; the existence of the instability is associated with the orientation of the wave crests. The instability is investigated for wavelengths comparable to the depth of the viscous wall layer; the limiting cases of large and small wavelengths are investigated asymptotically. At small wavenumbers and roughness heights the instability connects with disturbances of wavelength comparable to the wall wavelength. At high wavenumbers and roughness heights a new structure emerges and the disturbance moves away from the wall, that structure takes on a self-similar form with progressively faster variations in the streamwise and spanwise directions as the roughness height increases.

JFM classification

Instability: Transition to turbulence Instability: Shear-flow instability Turbulent Flows: Shear layer turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 944 , 10 August 2022 , A37
Check for updates
Copyright
© The Author(s), 2022. Published by 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.)

References

Bassom, A.P. & Hall, P. 1991 Vortex instabilities in three-dimensional boundary layers: the relationship between Görtler and crossflow vortices. J. Fluid Mech. 232, 647–680.CrossRefGoogle Scholar
Charru, F., Andreotti, B. & Claudin, P. 2013 Sand ripples and dunes. Annu. Rev. Fluid Mech. 45, 469–493.CrossRefGoogle Scholar
Cotrell, D.L., McFadden, G.B. & Alder, B.J. 2008 Instability in pipe flow. Proc. Natl Acad. Sci. USA 105, 428–430.CrossRefGoogle ScholarPubMed
Choi, K.S. 2002 Near-wall structure of turbulent boundary layer with spanwise-wall oscillation. Phys. Fluids 14, 2530–2542.CrossRefGoogle Scholar
Darcy, H. 1857 Recherches Experimentelle Relatives au Mouvement de l'eau dans les Tuyaux. Mallet-Bachelier.Google Scholar
Davey, A., DiPrima, R.C. & Stuart, J.T. 1968 On the instability of Taylor vortices. J. Fluid Mech. 31, 17–52.CrossRefGoogle Scholar
Fage, A. 1943 The smallest size of a spanwise surface corrugation which affects the drag of a laminar flow aerofoil. ARC Reports and Memoranda No. 2120. HM Stationery Office.Google Scholar
Floryan, J.M. 2002 Centrifugal instability of flow over a wavy wall. Phys. Fluids 14, 301–322.CrossRefGoogle Scholar
Floryan, J. 2003 Wall-transpiration-induced instabilities in plane Couette flow. J. Fluid Mech. 488, 151–188.CrossRefGoogle Scholar
Floryan, J.M. 2015 Flow in a meandering channel. J. Fluid Mech. 770, 52–84.CrossRefGoogle Scholar
Ghebali, S., Chernyshenko, S. & Lechziner, M.A. 2017 Can large-scale oblique undulations on a solid wall reduce the turbulent drag? Phys. Fluids 29, 105102.CrossRefGoogle Scholar
Gregory, N., Stuart, J.T. & Walker, W.S. 1955 On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk. Phil. Trans. R. Soc. Lond. A 248, 155–159.Google Scholar
Goldstein, M.E. 1983 The evolution of Tollmien–Schlichting waves near a leading edge. J. Fluid Mech. 127, 59–81.CrossRefGoogle Scholar
Hagen, G. 1854 Über den Einfluss der Temperatur auf die Bewegung des Wassers in Röhren. Königliche Akademie der Wissenschaften.Google Scholar
Hall, P. 1982 Taylor–Görtler vortices in fully developed or boundary layer flows: linear theory. J. Fluid Mech. 124, 475–494.CrossRefGoogle Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58.CrossRefGoogle Scholar
Hall, P. 1986 An asymptotic investigation of the stationary modes of instability of the 3D-boundary layer on a rotating disc. Proc. R. Soc. Lond. A 406, 93–106.Google Scholar
Hall, P. 2020 An instability mechanism for channel flows in the presence of wall roughness. J. Fluid Mech. 899, R2.CrossRefGoogle Scholar
Hall, P. 2021 a Long wavelength streamwise vortices caused by curvature or wall roughness. J. Engng Maths 128, 2.CrossRefGoogle Scholar
Hall, P. 2021 b On the roughness instability of growing boundary layers. J. Fluid Mech. 922, A28.CrossRefGoogle Scholar
Hall, P. 2022 A vortex-wave interaction theory describing the effect of boundary forcing on shear flows. J. Fluid Mech. 932, A54.CrossRefGoogle Scholar
Hall, P. & Horseman, N.J. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357–375.CrossRefGoogle Scholar
Hall, P. & Ozcakir, O. 2021 Poiseuille flow in rough pipes: linear instability induced by vortex-wave interactions. J. Fluid Mech. 913, A43.CrossRefGoogle Scholar
Hall, P. & Smith, F.T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641–666.CrossRefGoogle Scholar
Kadivar, M., Tormey, D. & McGranaghan, G. 2021 A review on turbulent flow over rough surfaces: fundamentals and theories. Intl J. Thermofluids 10, 100077.CrossRefGoogle Scholar
Kandlikar, S.G. 2008 Exploring roughness effect on laminar internal flow-are we ready for change? Nanoscale Microscale Thermophys. Engng 12, 61–82.CrossRefGoogle Scholar
Loh, S.A. & Blackburn, H.M. 2011 Stability of steady flow through an axially corrugated pipe. Phys. Fluids 23, 111703.CrossRefGoogle Scholar
Phillips, W.R.C. 2002 Langmuir circulations beneath growing or decaying surface waves. J. Fluid Mech. 469, 317–342.CrossRefGoogle Scholar
Ruban, A.I. 1984 On Tollmien–Schlichting wave generation by sound. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 5, 44–52.Google Scholar
Seminara, G. & Hall, P. 1976 Centrifugal instability of a Stokes layer: linear theory. Proc. R. Soc. Lond. A 350, 299–316.Google Scholar
Smith, F.T. 1982 On the high Reynolds number theory of laminar flows. J. Appl. Maths 20, 207–281.CrossRefGoogle Scholar