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Critical regime of gravity currents flowing in non-rectangular channels with density stratification

Published online by Cambridge University Press:  14 February 2018

L. Chiapponi*
Affiliation:
Dipartimento di Ingegneria e Architettura (DIA), Università di Parma, Parco Area delle Scienze, 181/A, 43124 Parma, Italy
M. Ungarish
Affiliation:
Department of Computer Science, Technion, Israel Institute of Technology, Haifa 32000, Israel
S. Longo
Affiliation:
Dipartimento di Ingegneria e Architettura (DIA), Università di Parma, Parco Area delle Scienze, 181/A, 43124 Parma, Italy
V. Di Federico
Affiliation:
Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali (DICAM), Università di Bologna, Viale Risorgimento, 2, 40136 Bologna, Italy
F. Addona
Affiliation:
Dipartimento di Ingegneria e Architettura (DIA), Università di Parma, Parco Area delle Scienze, 181/A, 43124 Parma, Italy
*
Email address for correspondence: [email protected]

Abstract

We present theoretical and experimental analyses of the critical condition where the inertial–buoyancy or viscous–buoyancy regime is preserved in a uniform-density gravity current (which propagates over a horizontal plane) of time-variable volume ${\mathcal{V}}=qt^{\unicode[STIX]{x1D6FF}}$ in a power-law cross-section (with width described by $f(z)=bz^{\unicode[STIX]{x1D6FC}}$, where $z$ is the vertical coordinate, $b$ and $q$ are positive real numbers, and $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FF}$ are non-negative real numbers) occupied by homogeneous or linearly stratified ambient fluid. The magnitude of the ambient stratification is represented by the parameter $S$, with $S=0$ and $S=1$ describing the homogeneous and maximum stratification cases respectively. Earlier theoretical and experimental results valid for a rectangular cross-section ($\unicode[STIX]{x1D6FC}=0$) and uniform ambient fluid are generalized here to a power-law cross-section and stratified ambient. Novel time scalings, obtained for inertial and viscous regimes, allow a derivation of the critical flow parameter $\unicode[STIX]{x1D6FF}_{c}$ and the corresponding propagation rate as $Kt^{\unicode[STIX]{x1D6FD}_{c}}$ as a function of the problem parameters. Estimates of the transition length between the inertial and viscous regimes are also derived. A series of experiments conducted in a semicircular cross-section ($\unicode[STIX]{x1D6FC}=1/2$) validate the critical values $\unicode[STIX]{x1D6FF}_{c}=2$ and $\unicode[STIX]{x1D6FF}_{c}=9/4$ for the two cases $S=0$ and $1$. The ratio between the inertial and viscous forces is determined by an effective Reynolds number proportional to $q$ at some power. The threshold value of this number, which enables a determination of the regime of the current (inertial–buoyancy or viscous–buoyancy) in critical conditions, is determined experimentally for both $S=0$ and $S=1$. We conclude that a very significant generalization of the insights and results from two-dimensional (rectangular cross-section channel) gravity currents to power-law cross-sections is available.

Type
JFM Papers
Copyright
© 2018 Cambridge University Press 

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References

Didden, N. & Maxworthy, T. 1982 The viscous spreading of plane and axisymmetric gravity currents. J. Fluid Mech. 121, 2742.CrossRefGoogle Scholar
Huppert, H. E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 4358.CrossRefGoogle Scholar
Longo, S., Di Federico, V., Archetti, R., Chiapponi, L., Ciriello, V. & Ungarish, M. 2013 On the axisymmetric spreading of non-Newtonian power-law gravity currents of time-dependent volume: an experimental and theoretical investigation focused on the inference of rheological parameters. J. Non-Newtonian Fluid Mech. 201, 6979.Google Scholar
Longo, S., Di Federico, V. & Chiapponi, L. 2015a Non-Newtonian power-law gravity currents propagating in confining boundaries. Environ. Fluid Mech. 15 (3), 515535.Google Scholar
Longo, S., Ungarish, M., Di Federico, V., Chiapponi, L. & Addona, F. 2016a Gravity currents in a linearly stratified ambient fluid created by lock release and influx in semi-circular and rectangular channels. Phys. Fluids 28, 125.CrossRefGoogle Scholar
Longo, S., Ungarish, M., Di Federico, V., Chiapponi, L. & Addona, F. 2016b Gravity currents produced by constant and time varying inflow in a circular cross-section channel: experiments and theory. Adv. Water Resour. 90, 1023.Google Scholar
Longo, S., Ungarish, M., Di Federico, V., Chiapponi, L. & Maranzoni, A. 2015b The propagation of gravity currents in a circular cross-section channel: experiments and theory. J. Fluid Mech. 764, 513537.Google Scholar
Marino, B. M. & Thomas, L. P. 2011 Dam-break release of a gravity current in a power-law channel section. J. Phys.: Conf. Ser. 296, 012008.Google Scholar
Maxworthy, T. 1983 Gravity currents with variable inflow. J. Fluid Mech. 128, 247257.CrossRefGoogle Scholar
Monaghan, J., Mériaux, C., Huppert, H. & Monaghan, J. 2009 High Reynolds number gravity currents along v-shaped valleys. Eur. J. Mech. (B/Fluids) 28 (5), 651659.CrossRefGoogle Scholar
Shringarpure, M., Lee, H., Ungarish, M. & Balachandar, S. 2013 Front conditions of high-Re gravity currents produced by constant and time-dependent influx: an analytical and numerical study. Eur. J. Mech. (B/Fluids) 41 (0), 109122.CrossRefGoogle Scholar
Simpson, J. E. 1997 Gravity Currents in the Environment and the Laboratory. Cambridge University Press.Google Scholar
Takagi, D. & Huppert, H. E. 2007 The effect of confining boundaries on viscous gravity currents. J. Fluid Mech. 577, 495505.Google Scholar
Takagi, D. & Huppert, H. E. 2008 Viscous gravity currents inside confining channels and fractures. Phys. Fluids 20, 023104.Google Scholar
Ungarish, M. 2009 An Introduction to Gravity Currents and Intrusions. Chapman & Hall/CRC Press.Google Scholar
Ungarish, M. 2012 A general solution of Benjamin-type gravity current in a channel of non-rectangular cross-section. Environ. Fluid Mech. 12 (3), 251263.CrossRefGoogle Scholar
Ungarish, M. 2015 Shallow-water solutions for gravity currents in non-rectangular cross-area channels with stratified ambient. Environ. Fluid Mech. 15 (4), 793820.Google Scholar
Ungarish, M. 2018 Thin-layer models for gravity currents in channels of general cross-section area, a review. Environ. Fluid Mech. 18, 283333.CrossRefGoogle Scholar
Ungarish, M., Mériaux, C. A. & Kurz-Besson, C. B. 2014 The propagation of gravity currents in a V-shaped triangular cross-section channel: experiments and theory. J. Fluid Mech. 754, 232249.Google Scholar
Zemach, T. & Ungarish, M. 2013 Gravity currents in non-rectangular cross-section channels: analytical and numerical solutions of the one-layer shallow-water model for high-Reynolds-number propagation. Phys. Fluids 25, 026601.Google Scholar