Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T12:46:39.276Z Has data issue: false hasContentIssue false
  • English
  • Français

Initial development of a free-surface wall jet at moderate Reynolds number

Published online by Cambridge University Press:  03 August 2017

Roger E. Khayat Open the ORCID record for Roger E. Khayat [Opens in a new window]
Roger E. Khayat*
Affiliation:
Department of Mechanical and Materials Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The steady laminar flow of a moderately inertial wall jet is examined theoretically near the exit of a channel. The free-surface jet emerges asymmetrically from the channel as it adheres to an infinite (upper) wall subject to a pressure gradient. The problem is solved using the method of matched asymptotic expansions. The small parameter involved in the expansions is the inverse cubic power of the Reynolds number. The flow field is obtained by matching the inviscid rotational core flow separately with the free-surface and the two wall layers. The upstream influence is examined as well as the break in the symmetry between the two wall layers. The wall jet exhibits a contraction near the channel exit that is independent of inertia, and eventually expands for any Reynolds number. Unlike the flow of a wall jet emerging into the same ambient fluid, the free-surface jet experiences a limited weakening in shear stress along the infinite wall, suggesting the possibility of separation for a jet with relatively low inertia. Significant shearing and elongation ensue at the exit, accompanied by flattening of the velocity profile near the upper wall.

JFM classification

Boundary Layers: Boundary Layers Materials Processing Flows: Coating Interfacial Flows (free surface): Interfacial Flows (free surface)
Type
Papers
Information
Journal of Fluid Mechanics , Volume 826 , 10 September 2017 , pp. 235 - 269
Check for updates
Copyright
© 2017 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

Bakke, P. 1957 An experimental investigation of a wall jet. J. Fluid Mech. 2, 467.Google Scholar
Benilov, E. S., Benilov, M. S. & Kopteva, N. 2008 Steady rimming flows with surface tension. J. Fluid Mech. 597, 91.CrossRefGoogle Scholar
Elliotis, M., Georgiou, G. & Xenophontos, C. 2005 Solution of the planar Newtonian stick–slip problem with the singular function boundary integral method. Intl J. Numer. Meth. Fluids 48, 1001.CrossRefGoogle Scholar
Glauert, M. B. 1956 The wall jet. J. Fluid Mech. 1, 625.Google Scholar
Goren, S. L. & Wronski, S. 1966 The shape of low-speed capillary jets of Newtonian liquids. J. Fluid Mech. 25, 185.Google Scholar
Higgins, B. G. 1982 Downstream development of two-dimensional viscocapillary film flow. Ind. Engng Chem. Res. 21, 168.Google Scholar
Iliopoulos, I. & Scriven, L. E. 2005 A blade-coating study using a finite-element simulation. Phys. Fluids. 17, 127101.CrossRefGoogle Scholar
Khayat, R. E. 2014 Free-surface jet flow of a shear-thinning power-law fluid near the channel exit. J. Fluid Mech. 748, 580.Google Scholar
Khayat, R. E. 2016a Slipping free-jet flow near channel exit at moderate Reynolds number for large slip length. J. Fluid Mech. 793, 667.Google Scholar
Khayat, R. E. 2016b Impinging planar jet flow on a horizontal surface with slip. J. Fluid Mech. 808, 258.Google Scholar
Khayat, R. E. & Kim, K. 2002 Influence of initial conditions on transient two-dimensional thin-film flow. Phys. Fluids 14, 4448.CrossRefGoogle Scholar
Khayat, R. E. & Welke, S. 2001 Influence of inertia, gravity, and substrate topography on the two-dimensional transient coating flow of a thin Newtonian fluid film. Phys. Fluids 13, 355.CrossRefGoogle Scholar
Levin, O., Chernoray, V. G., Fdahl, L. L. & Henningson, D. S. 2005 A study of the Blasius jet. J. Fluid Mech. 539, 313.Google Scholar
Maki, H. 1983 Experimental studies on the combined flow field formed by a moving wall and a wall jet running parallel to it. Bull. JSME 26, 2100.Google Scholar
Miyake, Y., Mukai, E. & Iemoto, Y. 1979 On a two-dimensional laminar liquid jet. Bull. JSME 22, 1382.Google Scholar
Muhammad, T. & Khayat, R. E. 2004 Effect of substrate movement on shock formation in pressure-driven coating flow. Phys. Fluids 13, 355.Google Scholar
Pasquali, M. & Scriven, L. E. 2002 Free surface flows of polymer solutions with models based on the conformation tensor. J. Non-Newtonian Fluid Mech. 108, 363.CrossRefGoogle Scholar
Peters, F., Rupel, C., Javili, A. & Kunkel, T. 2008 The two-dimensional wall jet. Velocity measurements compared with similarity theory. Forsch. Ing. Wes. 72, 18.Google Scholar
Phan-Thien, N., Jin, H. & Tanner, R. I. 1989 On the hydrodynamic braking flow of a viscoelastic fluid. Wear 133, 323.Google Scholar
Philippe, C. & Dumargue, P. 1991 Étude de l’établissement d’un jet liquide laminaire émergeant d’une conduite cylindrique verticale semi-infinie et soumis à l’influence de la gravité. Z. Angew. Math. Phys. J. Appl. Math. Phys. 42, 227.CrossRefGoogle Scholar
Ruschak, K. J. & Scriven, L. E. 1977 Developing flow on a vertical wall. J. Fluid Mech. 81, 305.Google Scholar
Saffari, A. & Khayat, R. E. 2009 Flow of viscoelastic jet with moderate inertia near channel exit. J. Fluid Mech. 639, 65.Google Scholar
Schmidt, M., Schlosser, U. & Schollmeyer, E. 2009 Computational fluid dynamics investigation of the static pressure at the blade in a blade coating. Textile Res. J. 79, 579.CrossRefGoogle Scholar
Shi, J. M., Breuer, M. & Durst, F. 2004 A combined analytical–numerical method for treating corner singularities in viscous flow predictions. Intl J. Numer. Meth. Fluids 45, 659.Google Scholar
Smith, F. T. 1976a Flow through constricted or dilated pipes and channels: part 1. Q. J. Mech. Appl. Maths 29, 343.CrossRefGoogle Scholar
Smith, F. T. 1976b Flow through constricted or dilated pipes and channels: part 2. Q. J. Mech. Appl. Maths 29, 365.Google Scholar
Smith, F. T. 1977 Upstream interactions in channel flow. J. Fluid Mech. 79, 631.CrossRefGoogle Scholar
Smith, F. T. 1979 The separating flow through a severely constricted symmetric tube. J. Fluid Mech. 90, 725.CrossRefGoogle Scholar
Sobey, I. J. 2000 Interactive Boundary Layer Theory. Oxford University Press.Google Scholar
Soederberg, L. D. 2003 Absolute and convective instability of a relaxational plane liquid jet. J. Fluid Mech. 493, 89.Google Scholar
Talke, F. E. & Berger, S. A. 1970 The flat plate trailing edge problem. J. Fluid Mech. 40, 161.Google Scholar
Tanner, R. I. 2000 Engineering Rheology. Oxford University Press.Google Scholar
Tillett, J. P. K. 1968 On the laminar flow in a free jet of liquid at high Reynolds numbers. J. Fluid Mech. 32, 273.Google Scholar
Van Dyke, M. D. 1964 Perturbation Methods in Fluid Mechanics. Academic.Google Scholar
Watson, E. 1964 The spread of a liquid jet over a horizontal plane. J. Fluid Mech. 20, 481.CrossRefGoogle Scholar
Wilson, D. E. 1986 A similarity solution for axisymmetric viscous-gravity jet. Phys. Fluids 29 (3), 632.Google Scholar