Hostname: page-component-7c8c6479df-hgkh8 Total loading time: 0 Render date: 2024-03-19T10:27:13.833Z Has data issue: false hasContentIssue false

Unsteady draining of reservoirs over weirs and through constrictions

Published online by Cambridge University Press:  06 November 2019

Edward W. G. Skevington
Affiliation:
School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, UK
Andrew J. Hogg*
Affiliation:
School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, UK
*
Email address for correspondence: a.j.hogg@bris.ac.uk

Abstract

The gravitationally driven flow of fluid from a reservoir following the partial collapse of its confining dam, or the partial opening of its confining lock, is modelled using the nonlinear shallow water equations, coupled to outflow conditions, in which the drainage is modelled as flow over a constricted, broad-crested weir. The resulting unsteady motion reveals systematic draining, on which strong and relatively rapid oscillations are imposed. The oscillations propagate between the outflow and the impermeable back wall of the reservoir. This dynamics is investigated utilising three methods: hodograph techniques to yield quasi-analytical solutions, asymptotic analysis at relatively late times after initiation and numerical integration of the governing equations. The hodograph transformation is used to find exact solutions at early times, revealing that from initially quiescent conditions the fluid drains and yet repeatedly generates intervals during which there are regions of constant depth and velocity adjacent to the boundaries. A novel modified multiscale asymptotic analysis designed for late times is employed to determine the limiting rate of draining and wave structure. It is shown that the excess height drains as $t^{-2}$ and, when the obstacle has finite height, the velocity field decays as $t^{-3}$, and exhibits a wave structure that tends towards a constant and relatively rapid phase speed. In the case of a pure constriction, for which all the fluid ultimately drains out of the reservoir, the motion adjusts to a self-similar state in which the velocity field decays as $t^{-1}$. Oscillations around this state have an exponentially increasing period. Numerical simulations with a novel implementation of boundary conditions are performed; they confirm the hodograph solution and provide data for the asymptotic results.

Type
JFM Papers
Copyright
© 2019 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

Ackers, P., White, W. R., Perkins, J. A. & Harrison, A. J. M. 1978 Weirs and Flumes for Flow Measurement. Wiley.Google Scholar
Antuono, M. & Hogg, A. J. 2009 Run-up and backwash bore formation from dam-break flow on an inclined plane. J. Fluid Mech. 640, 151164.Google Scholar
Antuono, M., Hogg, A. J. & Brocchini, M. 2009 The early stages of shallow flows in an inclined flume. J. Fluid Mech. 633, 285309.Google Scholar
Ascher, U. M. & Petzold, L. R. 1998 Computer Methods for Ordinary Differential Equations and Differential Algebraic Equations. SIAM.Google Scholar
Azimi, A. H., Rajaratnam, N. & Zhu, D. Z. 2013 Discharge characteristics of weirs of finite crest length with upstream and downstream ramps. J. Hydraul. Engng 139 (1), 7583.Google Scholar
Bender, C. M. & Orszag, S. A. 1999 Asymptotic Methods and Perturbation Theory. Springer.Google Scholar
Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209248.Google Scholar
Chow, V. T. 2009 Open-Channel Hydraulics. The Blackburn Press.Google Scholar
Dake, J. M. K. 1983 Essentials of Engineering Hydraulics. Macmillan Press.Google Scholar
Fannelop, T. K. & Waldman, G. D. 1972 Dynamics of oil slicks. J. AIAA 10, 506510.Google Scholar
Garabedian, P. R. 1986 Partial Differential Equations. Chelsea Publishing.Google Scholar
Goater, A. J. N. & Hogg, A. J. 2011 Bounded dam-break flows with tailwaters. J. Fluid Mech. 686, 160186.Google Scholar
Grundy, R. E. & Rottman, J. W. 1985 The approach to self-simiilarity of the solutions representing gravity current releases. J. Fluid Mech. 156, 3953.Google Scholar
Hinch, E. J. 1992 Perturbation Methods. Cambridge University Press.Google Scholar
Hogg, A. J. 2006 Lock-release gravity currents and dam-break flows. J. Fluid Mech. 569, 6187.Google Scholar
Hogg, A. J., Baldock, T. E. & Pritchard, D. 2011 Overtopping a truncated planar beach. J. Fluid Mech. 666, 521553.Google Scholar
Horton, R. E.1907 Weir experiments, coefficients, and formulas. Water Supply and Irrigation Paper 200 (Series M, General Hydrographic Investigations). Government Printing Office.Google Scholar
Hoult, D. P. 1972 Oil spreading on the sea. Annu. Rev. Fluid Mech. 2, 341368.Google Scholar
Karelskii, K. V. & Petrosyan, A. S. 2006 Problem of steady-state flow over a step in the shallow-water approximation. Fluid Dyn. 41 (1), 1220.Google Scholar
von Kármán, T. 1940 The engineer grapples with nonlinear problems. Bull. Am. Math. Soc. 46, 615683.Google Scholar
Kevorkian, J. & Cole, J. D. 1996 Multiple Scale and Singular Perturbation Methods, Applied Mathematical Sciences. Springer.Google Scholar
Kunkel, P. & Mehrmann, V. 2006 Differential-Algebraic Equations. European Mathematics Society.Google Scholar
Kurganov, A., Noelle, S. & Petrova, G. 2001 Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations. SIAM J. Sci. Comput. 23, 707740.Google Scholar
Lane-Serff, G. F., Beal, L. M. & Hadfield, T. D. 1995 Gravity current flow over obstacles. J. Fluid Mech. 292, 3953.Google Scholar
Leveque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge.Google Scholar
Mathunjwa, J. S. & Hogg, A. J. 2006 Stability of gravity currents generated by finite volume releases. J. Fluid Mech. 562, 261278.Google Scholar
Momen, M., Zheng, Z., Bou-Zeid, E. & Stone, H. A. 2017 Inertial gravity currents produced by fluid drainage from an edge. J. Fluid Mech. 827, 640663.Google Scholar
Nabi, S. & Flynn, M. R. 2013 The hydraulics of exchnage flow between adjacent confined building zones. Build. Environ. 59, 7690.Google Scholar
Peregrine, D. H. 1972 Equations for water waves and the approximations behind them. In Waves on Beaches and Resulting Sediment Transport (ed. Meyer, R.), chap. 3, pp. 95121. Academic Press.Google Scholar
Rottman, J. W. & Simpson, J. E. 1983 Gravity currents produced by instantaneous releases of heavy fluid in a rectangular channel. J. Fluid Mech. 135, 95110.Google Scholar
Shu, C. & Osher, S. 1988 Efficient implementation of essentially non-oscillatory schemes. J. Comput. Phys. 77, 439471.Google Scholar
Simpson, J. E. 1982 Gravity currents in the environment and the laboratory. Annu. Rev. Fluid Mech. 14, 213234.Google Scholar
Simpson, J. E. 1997 Gravity Currents in the Environment and the Laboratory. Cambridge University Press.Google Scholar
Ungarish, M. 2009 An Introduction to Gravity Currents and Intrusions. CRC Press.Google Scholar
Ungarish, M. & Hogg, A. J. 2018 Models of internal jumps and the fronts of gravity currents: unifying two-layer theories and deriving new results. J. Fluid Mech. 846, 654685.Google Scholar
Valiani, A. & Caleffi, V. 2017 Momentum balance in the shallow water equations on bottom discontinuities. Adv. Water Resour. 100, 113.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Wilson, R. I., Friedrich, H. & Stevens, C. 2017 Turbulent entrainment in sediment-laden flows interacting with an obstacle. Phys. Fluids 29, 036603.Google Scholar
Zachoval, Z., Knéblová, M., Rous̆ar, L., Rumann, J. & S̆ulc, J. 2014 Discharge coefficient of a rectangular sharp-edged broad-crested weir. J. Hydrol. Hydromech. 62, 145149.Google Scholar