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On turbulent particle fountains


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Authors

Woods, AW 

Abstract

jats:pWe describe new experiments in which particle-laden turbulent fountains with source Froude numbers jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline1" />jats:tex-mathMisplaced &20&gt;Fr_{0}&gt;620&gt;Fr_{0}&gt;6</jats:tex-math></jats:alternatives></jats:inline-formula> are produced when particle-laden fresh water is injected upwards into a reservoir filled with fresh water. We find that the ratio jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline2" />jats:tex-mathU</jats:tex-math></jats:alternatives></jats:inline-formula> of the particle fall speed to the characteristic speed of the fountain determines whether the flow is analogous to a single-phase fountain (jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline3" />jats:tex-mathU≪1</jats:tex-math></jats:alternatives></jats:inline-formula>) or becomes a fully separated flow (jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline4" />jats:tex-mathU⩾1</jats:tex-math></jats:alternatives></jats:inline-formula>). In the single-phase limit, a fountain with momentum flux jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline5" />jats:tex-mathM</jats:tex-math></jats:alternatives></jats:inline-formula> and buoyancy flux jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline6" />jats:tex-mathB</jats:tex-math></jats:alternatives></jats:inline-formula> oscillates about the mean height, jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline7" />jats:tex-mathhm=(1.56±0.04)M3/4B−1/2</jats:tex-math></jats:alternatives></jats:inline-formula>, as fluid periodically cascades from the maximum height, jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline8" />jats:tex-mathht=hm+Δh</jats:tex-math></jats:alternatives></jats:inline-formula>, to the base of the tank. Experimental measurements of the speed jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline9" />jats:tex-mathu</jats:tex-math></jats:alternatives></jats:inline-formula> and radius jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline10" />jats:tex-mathr</jats:tex-math></jats:alternatives></jats:inline-formula> of the fountain at the mean height jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline11" />jats:tex-mathhm</jats:tex-math></jats:alternatives></jats:inline-formula>, combined with the conservation of buoyancy, suggest that jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline12" />jats:tex-mathFr(hm)=u(gr)−1/2≈1</jats:tex-math></jats:alternatives></jats:inline-formula>. Using these values, we find that the classical scaling for the frequency of the oscillations, jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline13" />jats:tex-mathωBM−1</jats:tex-math></jats:alternatives></jats:inline-formula>, is equivalent to the scaling jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline14" />jats:tex-mathu(hm)/r(hm)</jats:tex-math></jats:alternatives></jats:inline-formula> for a fountain supplied at jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline15" />jats:tex-mathz=hm</jats:tex-math></jats:alternatives></jats:inline-formula> with jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline16" />jats:tex-mathFr=1</jats:tex-math></jats:alternatives></jats:inline-formula> (Burridge & Hunt, jats:italicJ. Fluid Mech.</jats:italic>, vol. 728, 2013, pp. 91–119). This suggests that the oscillations are controlled in the upper part of the fountain where jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline17" />jats:tex-mathFr⩽1</jats:tex-math></jats:alternatives></jats:inline-formula>, and that they may be understood in terms of a balance between the upward supply of a growing dense particle cloud, at the height where jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline18" />jats:tex-mathFr=1</jats:tex-math></jats:alternatives></jats:inline-formula>, and the downward flow of this cloud. In contrast, in the separated flow regime, we find that particles do not reach the height at which jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016001671_inline19" />jats:tex-mathFr=1</jats:tex-math></jats:alternatives></jats:inline-formula>: instead, they are transported to the level at which the upward speed of the fountain fluid equals their fall speed. The particles then continuously sediment while the particle-free fountain fluid continues to rise slowly above the height of particle fallout, carried by its momentum.</jats:p>

Description

Keywords

complex fluids, plumes/thermals, suspensions

Journal Title

Journal of Fluid Mechanics

Conference Name

Journal ISSN

0022-1120
1469-7645

Volume Title

793

Publisher

Cambridge University Press (CUP)
Sponsorship
This work has been funded through the BP Institute, EPSRC and Hughes Hall, Cambridge. We gratefully acknowledge the technical assistance of A. Pluck, and the constructive comments of three anonymous referees.