3 results
Confined turbulent entrainment across density interfaces
- Ajay B. Shrinivas, Gary R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 779 / 25 September 2015
- Published online by Cambridge University Press:
- 14 August 2015, pp. 116-143
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In pursuit of a universal law for the rate of entrainment across a density interface driven by the impingement of a localised turbulent flow, the role of the confinement, wherein the environment is within the confines of a box, has to date been overlooked. Seeking to unravel the effects of confinement, we develop a phenomenological model describing the quasi-steady rate at which buoyant fluid is turbulently entrained across a density interface separating two uniform layers within the confines of a box. The upper layer is maintained by a turbulent plume, and the localised impingement of a turbulent fountain with the interface drives entrainment of fluid from the upper layer into the lower layer. The plume and fountain rise from sources at the base of the box and are non-interacting. Guided by previous observations, our model characterises the dynamics of fountain–interface interaction and the steady secondary flow in the environment that is induced by the perpetual cycle of vertical excursions of the interface. We reveal that the dimensionless entrainment flux across the interface $E_{i}$ is governed not only by an interfacial Froude number $\mathit{Fr}_{i}$ but also by a ‘confinement’ parameter ${\it\lambda}_{i}$, which characterises the length scale of interfacial turbulence relative to the depth of the upper layer. By deducing the range of ${\it\lambda}_{i}$ that may be regarded as ‘small’ and ‘large’, we shed new light on the effects of confinement on interfacial entrainment. We establish that for small ${\it\lambda}_{i}$, a weak secondary flow has little influence on $E_{i}$, which follows a quadratic power law $E_{i}\propto \mathit{Fr}_{i}^{2}$. For large ${\it\lambda}_{i}$, a strong secondary flow significantly influences $E_{i}$, which then follows a cubic power law $E_{i}\propto \mathit{Fr}_{i}^{3}$. Drawing on these results, and showing that for previous experimental studies ${\it\lambda}_{i}$ exhibits wide variation, we highlight underlying physical reasons for the significant scatter in the existing measurements of the rate of interfacial entrainment. Finally, we explore the implications of our results for guiding appropriate choices of box geometry for experimentally and numerically examining interfacial entrainment.
Unconfined turbulent entrainment across density interfaces
- Ajay B. Shrinivas, Gary R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 757 / 25 October 2014
- Published online by Cambridge University Press:
- 23 September 2014, pp. 573-598
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We present theoretical models describing the quasi-steady downward transport of buoyant fluid across a gravitationally stable density interface separating two unbounded quiescent fluid masses. The primary transport mechanism is turbulent entrainment resulting from the localised impingement of a vertically forced high-Reynolds-number axisymmetric jet with steady source conditions. The entrainment across the interface is examined in the large-time asymptotic state, wherein the interfacial gravity current, formed by the fluid entrained from the upper layer and the jet, becomes infinitesimally thin and a two-layer stratification persists. Characterising flows with small interfacial Froude numbers $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}({{\mathrm{Fr}}}_i)$ as an axisymmetric semi-ellipsoidal impingement dome, we combine conservation equations with a mechanistic model of entrainment and reveal that, in this regime, the dimensionless entrainment flux $E_i$ across the interface follows the power law $E_i = 0.24{{\mathrm{Fr}}}_i^2$. For large-${{\mathrm{Fr}}}_i$ impingements, modelled as a fully penetrating turbulent fountain, we show that $E_i$ no longer scales with ${{\mathrm{Fr}}}_i^2$, but linearly on ${{\mathrm{Fr}}}_i$, following $E_i = 0.42{{\mathrm{Fr}}}_i$. We establish the intermediate range of ${{\mathrm{Fr}}}_i$ over which there is a transition between these quadratic and linear power laws, thus enabling us to classify the dynamics of entrainment across the interface into three distinct regimes. Finally, the close agreement of our solutions with existing experimental results is illustrated.
Transient ventilation dynamics induced by heat sources of unequal strength
- Ajay B. Shrinivas, Gary R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 738 / 10 January 2014
- Published online by Cambridge University Press:
- 02 December 2013, pp. 34-64
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We examine theoretically the transient displacement flow and density stratification that develops within a ventilated box after two localized floor-level heat sources of unequal strengths are activated. The heat input is represented by two non-interacting turbulent axisymmetric plumes of constant buoyancy fluxes ${B}_{1} $ and ${B}_{2} \gt {B}_{1} $. The box connects to an unbounded quiescent external environment of uniform density via openings at the top and base. A theoretical model is developed to predict the time evolution of the dimensionless depths ${\lambda }_{j} $ and mean buoyancies ${\delta }_{j} $ of the ‘intermediate’ $(j= 1)$ and ‘top’ $(j= 2)$ layers leading to steady state. The flow behaviour is classified in terms of a stratification parameter , a dimensionless measure of the relative forcing strengths of the two buoyant layers that drive the flow. We find that $\mathrm{d} {\delta }_{1} / \mathrm{d} \tau \propto 1/ {\lambda }_{1} $ and $\mathrm{d} {\delta }_{2} / \mathrm{d} \tau \propto 1/ {\lambda }_{2} $, where $\tau $ is a dimensionless time. When $\hspace{0.167em} \hspace{0.167em} \ll \hspace{0.167em} \hspace{0.167em} $1, the intermediate layer is shallow (small ${\lambda }_{1} $), whereas the top layer is relatively deep (large ${\lambda }_{2} $) and, in this limit, ${\delta }_{1} $ and ${\delta }_{2} $ evolve on two characteristically different time scales. This produces a time lag and gives rise to a ‘thermal overshoot’, during which ${\delta }_{1} $ exceeds its steady value and attains a maximum during the transients; a flow feature we refer to, in the context of a ventilated room, as ‘localized overheating’. For a given source strength ratio $\psi = {B}_{1} / {B}_{2} $, we show that thermal overshoots are realized for dimensionless opening areas $A\lt {A}_{oh} $ and are strongly dependent on the time history of the flow. We establish the region of $\{ A, \psi \} $ space where rapid development of ${\delta }_{1} $ results in ${\delta }_{1} \gt {\delta }_{2} $, giving rise to a bulk overturning of the buoyant layers. Finally, some implications of these results, specifically to the ventilation of a room, are discussed.