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Interaction between the Blasius boundary layer and a free surface

Published online by Cambridge University Press:  25 January 2018

Jonathan Michael Foonlan Tsang Open the ORCID record for Jonathan Michael Foonlan Tsang [Opens in a new window] ,
Stuart B. Dalziel Open the ORCID record for Stuart B. Dalziel [Opens in a new window]  and
N. M. Vriend Open the ORCID record for N. M. Vriend [Opens in a new window]
Jonathan Michael Foonlan Tsang*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Stuart B. Dalziel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
N. M. Vriend
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: jmft2@cam.ac.uk
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Abstract

We consider the steady supercritical flow of a fluid layer. The layer is bounded above by a free surface and below by a rigid no-slip base. The base is in two parts: the downstream part of the base is stationary, while the upstream part translates in the streamwise direction with a uniform speed; there is an abrupt transition. At high Reynolds number, a boundary layer forms in the fluid above the base downstream of the transition point. The displacement due to this boundary layer creates a perturbation to the outer flow and therefore to the free surface. We show that the Blasius boundary layer solution, which applies in an infinitely deep fluid, also applies at high Froude numbers. The Blasius solution no longer applies for flows that are just supercritical, as the outer flow is strongly affected by the presence of the boundary layer. We outline possible applications of this work to depth-averaged models of gravity currents.

JFM classification

Boundary Layers: Boundary Layers Geophysical and Geological Flows: Shallow water flows Geophysical and Geological Flows: Topographic effects
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 839 , 25 March 2018 , R1
Copyright
© 2018 Cambridge University Press 

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References

Ablowitz, M. J. & Fokas, A. S. 2003 Complex Variables. Cambridge University Press.CrossRefGoogle Scholar
Abramowitz, M. & Stegun, I. A. 1970 Handbook of Mathematical Functions: with Formulas, Graphs and Mathematical Tables. Dover.Google Scholar
Acheson, D. J. 1989 Elementary Fluid Dynamics. Oxford University Press.Google Scholar
Baker, J. L., Johnson, C. G. & Gray, J. M. N. T. 2016 Segregation-induced finger formation in granular free-surface flows. J. Fluid Mech. 809, 168–212.CrossRefGoogle Scholar
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Billingham, J. & King, A. C. 2000 Wave Motion. Cambridge University Press.Google Scholar
Chow, V. T. 1959 Open-Channel Hydraulics. The Blackburn Press.Google Scholar
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.CrossRefGoogle Scholar
Nelson, J. J., Alving, A. E. & Joseph, D. D. 1995 Boundary layer flow of air over water on a flat plate. J. Fluid Mech. 284, 159–169.CrossRefGoogle Scholar
Petley, D. 2012 Global patterns of loss of life from landslides. Geology 40 (10), 927–930.CrossRefGoogle Scholar
Prandtl, L. 1905 Über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandlungen des III Internationalen Mathematiker-Kongresses, pp. 484–491. B. G. Teubner.Google Scholar
Schlichting, H. & Gersten, K. 2003 Boundary-Layer Theory, 8th edn. Springer.Google Scholar