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Internal hydraulic jump in plane Poiseuille two-layer flow: theoretical, numerical and experimental study

Published online by Cambridge University Press:  16 February 2021

Mrinmoy Dhar ,
Subhabrata Ray ,
Gargi Das Open the ORCID record for Gargi Das [Opens in a new window]  and
Prasanta Kumar Das Open the ORCID record for Prasanta Kumar Das [Opens in a new window]
Mrinmoy Dhar
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur721302, India
Subhabrata Ray
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kharagpur721302, India
Gargi Das*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Kharagpur721302, India
Prasanta Kumar Das
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur721302, India
*
† Email address for correspondence: [email protected]
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Abstract

This paper discusses a hitherto unexplored flow phenomenon, namely the internal hydraulic jump in thin films during co-current and counter-current two-layer flow between parallel plates. The problem corresponds to a special case of plane Poiseuille flow where the velocity profile changes continuously in the streamwise distance. Since an exact solution of Navier–Stokes equations is not possible, we reformulate the approximate shallow water theory, conventionally adopted to analyse viscous jumps in single-layer laminar flow. The standalone theory has been extensively validated with experimental data for coflow of the two phases and numerical simulations for both co- and counter-current flow. In the limit of zero viscosity ratio, the theoretical results reduce to the expression proposed by Dhar et al. (J. Fluid Mech., vol. 884, no. A11, 2020, pp. 1–26) for single-layer viscous jumps. For a holistic understanding, numerical simulations are used to unravel the physics at the jump, where the analysis displays singularity. The theory in conjunction with simulation reveals recirculation zones even in co-current laminar flow and delineates wavy, smooth and submerged jumps, displayed as a phase diagram. We thus demonstrate the efficacy of shallow water theory which, despite the approximations involved, can be used as a reliable tool for a priori prediction of viscous jumps in two-layer flow with much less effort and resources compared to numerical simulations. Use of an approximate analysis to obtain multifaceted results for a complex flow phenomenon has rarely been explored previously. This paper is also the first study reporting experiments on viscous jumps for two-layer flow in a shallow water analogue.

JFM classification

Interfacial Flows (free surface): Thin films Geophysical and Geological Flows: Shallow water flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 912 , 10 April 2021 , A45
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Arakeri, J.H. & Rao, K.P.A. 1996 On radial film flow on a horizontal surface and the circular hydraulic jump. J. Indian Inst. Sci. 76, 73–91.Google Scholar
Baines, P.G. 1995 Topographic Effects in Stratified Flows, p. 482. Cambridge University Press.Google Scholar
Bohr, T., Dimon, P. & Putkaradze, V. 1993 Shallow-water approach to the circular hydraulic jump. J. Fluid Mech. 254, 635–548.CrossRefGoogle Scholar
Bohr, T., Putkaradze, V. & Watanabe, S. 1997 Averaging theory for the structure of hydraulic jumps and separation in laminar free-surface flows. Phys. Rev. Lett. 79, 1038–1041.CrossRefGoogle Scholar
Bonn, D., Andersen, A. & Bohr, T. 2009 Hydraulic jumps in a channel. J. Fluid Mech. 618, 71–87.CrossRefGoogle Scholar
Brauner, N., Rovinsky, J. & Maron, D.M. 1996 Analytical solution for laminar-laminar two-phase stratified flow in circular conduits. Chem. Engng Commun. 141–142, 103–143.CrossRefGoogle Scholar
Chanson, H. 2009 Development of the Bélanger equation and backwater equation by Jean-Baptiste Bélanger (1828). J. Hydraul. Engng ASCE 135 (3), 159–163.CrossRefGoogle Scholar
Chow, V.T. 1959 Open-Channel Hydraulics. McGraw-Hill.Google Scholar
Dasgupta, R., Tomar, G. & Govindarajan, R. 2015 Numerical study of laminar, standing hydraulic jumps in a planar geometry. Eur. Phys. J. E 38, 1–14.CrossRefGoogle Scholar
Dhar, M., Das, G. & Das, P.K. 2020 Planar hydraulic jumps in thin film flow. J. Fluid Mech. 884 (A11), 1–26.CrossRefGoogle Scholar
Higuera, F.J. 1994 The hydraulic jump in a viscous laminar flow. J. Fluid Mech. 274, 69–92.CrossRefGoogle Scholar
Higuera, F.J. 1997 The circular hydraulic jump. Phys. Fluids 9, 1476–1478.CrossRefGoogle Scholar
Holland, D., Rosales, R., Stefanica, D. & Tabak, E. 2002 Internal hydraulic jumps and mixing in two-layer flows. J. Fluid Mech. 470, 63–83.CrossRefGoogle Scholar
Kate, R.P., Das, P.K. & Chakraborty, S. 2007 Hydraulic jumps due to oblique impingement of circular liquid jets on a flat horizontal surface. J. Fluid Mech. 573, 247–263.CrossRefGoogle Scholar
Kate, R.P., Das, P.K. & Chakraborty, S. 2008 An investigation on non-circular hydraulic jumps formed due to obliquely impinging circular liquid jets. Exp. Therm. Fluid Sci. 32, 1429–1439.CrossRefGoogle Scholar
Mejean, S., Faug, T. & Einav, I. 2017 A general relation for standing normal jumps in both hydraulic and dry granular flows. J. Fluid Mech. 816, 331–351.CrossRefGoogle Scholar
Ogden, K.A. & Helfrich, K.R. 2016 Internal hydraulic jumps in two-layer flows with upstream shear. J. Fluid Mech. 789, 64–92.CrossRefGoogle Scholar
Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190, 572–600.CrossRefGoogle Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228, 5838–66.CrossRefGoogle Scholar
Rayleigh, Lord 1914 On the theory of long waves and bores. Proc. R. Soc. Lond. A 90, 324–328.Google Scholar
Roberts, P.A. & Hibberd, S. 1996 Internal hydraulic jumps at T-junctions. J. Fluid Mech. 314, 331–347.CrossRefGoogle Scholar
Singha, S.B., Bhattacharjee, J.K. & Ray, A.K. 2005 Hydraulic jump in one-dimensional flow. Eur. Phys. J. B 426, 417–426.CrossRefGoogle Scholar
Sonim, A.A. 2002 Criteria for locally fully developed viscous flow. Massachusetts Institute of Technology. Course material. http://ocw.mit.edu.Google Scholar
Tani, I. 1949 Water jump in the boundary layer. J. Phys. Soc. Japan 4, 212–215.CrossRefGoogle Scholar
Thorpe, S.A. & Kavčič, I. 2008 The circular internal hydraulic jump. J. Fluid Mech. 610, 99–129.CrossRefGoogle Scholar
Thorpe, S.A., Malarkey, J., Voet, G., Alford, M.H., Girton, J.B. & Carter, G.S. 2018 Application of a model of internal hydraulic jumps. J. Fluid Mech. 834, 125–148.CrossRefGoogle Scholar
Vishwanath, K.P., Dasgupta, R., Govindarajan, R. & Sreenivas, K.R. 2016 The effect of initial momentum flux on the circular hydraulic jump. Trans. ASME J. Fluids Engng 137, 1–7.Google Scholar
Watanabe, S., Putkaradze, V. & Bohr, T. 2003 Integral methods for shallow free-surface flows with separation. J. Fluid Mech. 480, 233–265.CrossRefGoogle Scholar
Watson, E.J. 1964 The radial spread of a liquid jet over a horizontal plane. J. Fluid Mech. 20, 481–499.CrossRefGoogle Scholar
Wilkinson, D.L. & Wood, I.R. 1971 A rapidly varied flow phenomenon in a two-layer flow. J. Fluid Mech. 47, 241–256.CrossRefGoogle Scholar
Yeh, H. 1991 Vorticity generation mechanisms in bores. Proc. R. Soc. Lond. A 432, 215–231.Google Scholar