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Abstract When a fountain is injected into a linearly stratified fluid, its behavior will be governed by the stratification of the ambient fluid, represented by the dimensionless temperature stratification parameter (s), in addition to the Reynolds number (Re) and the Froude number (Fr). In this study, a series of three-dimensional direct numerical simulations (DNS) were carried out using ANSYS Fluent for transitional plane fountains in linearly-stratified fluids with Re , Fr and s over the ranges 28 ≤ Re ≤ 300, 3 ≤ Fr ≤ 10, and 0.1 ≤ s ≤ 0.5 to study the effects of these governing parameters on the characteristics of such unsteadiness. Empirical correlations to quantify the effects of Fr , Re and s over the studied ranges on the characteristics of the unsteadiness are also obtained and compared to the relevant scaling relations developed for weak plane fountains. [ABSTRACT FROM AUTHOR]

The transition to asymmetry of transitional fountains is further complicated by the stratification of the ambient fluid. In this study, a series of three-dimensional direct numerical simulations were carried out for transitional round fountains in linearly-stratified fluids over the ranges of 100 ≤ Re ≤ 400, 1 ≤ Fr ≤ 8 and 0.0 ≤ s ≤ 0.3, where Fr , Re and s are the Froude, Reynolds, and dimensionless temperature stratification parameters, respectively, to examine, both qualitatively and quantitatively, the effect of these parameters on their axisymmetric transition and their maximum penetration heights. The results show that when Fr or Re are small enough, a fountain remains axisymmetric for all time; however, when Fr or Re are increased sufficiently, the fountain will be axisymmetric initially, before becoming asymmetric and unsteady, ultimately reaching a fully developed quasi-steady stage when the maximum fountain penetration height fluctuates over a constant, time-average, value. The stratification is found to play a positive role to stabilize the flow and to reduce or even to eliminate the asymmetric behavior. The numerical results were also used to develop a series of scaling relations, in conjunction with the scalings obtained with dimensional analysis, for the initial and time-averaged fountain penetration heights, and the time for the fountain to attain the initial penetration height. It is further found that in general these characteristic parameters are strongly dependent on Fr and s , while only weakly dependent on Re . [ABSTRACT FROM AUTHOR]

In this study, the PIV and flow visualization techniques and three-dimensional direct numerical simulation are used to investigate the behavior of triple transitional round fountains, which are aligned along a straight line, interacting with each other in a homogeneous fluid over the ranges of 1 ≤ Fr ≤ 10 and 100 ≤ Re ≤ 1000, at a fixed spacing of D / X 0 = 6 , where Re and Fr are the Reynolds and Froude numbers, D is the spacing between the two neighbouring fountain sources, and X 0 is the radius of orifices at the fountain source, respectively. The results show that the interaction behavior of triple transitional round fountains, over the ranges of Re and Fr studied, is dominated by the bobbing and flapping motions, and is either steady, or unsteady weakly multi-modal, or unsteady strongly multi-modal. In a steady interaction, the bobbing-flapping motions are only present in its initial development stage and the interaction will attain a steady state in the later stage in which the bobbing-flapping motions are no longer present. In contrast, in an unsteady interaction, the interaction remains unsteady all the time and the bobbing-flapping motions are present at all stages. Among all cases considered, a steady interaction occurs at Re ≤ 300 with F r = 1 and at R e = 100 with Fr ≤ 5; an unsteady weakly multi-modal interaction occurs at R e = 100 with 6 ≤ Fr ≤ 10, at 200 ≤ Re ≤ 300 with 2 ≤ Fr ≤ 10, and at R e = 400 with F r = 1 ; and an unsteady strongly multi-modal interaction occurs at 600 ≤ Re ≤ 1000 with F r = 1 and at 400 ≤ Re ≤ 1000 with 2 ≤ Fr ≤ 10. Such interaction behavior is found to be similar to that of twin transitional round fountains over comparable ranges of Re and Fr , as discovered by us in an earlier study. It is also found that the two interaction regions formed by the three fountains have essentially symmetrical behavior about the middle fountain for all cases considered. Furthermore, the maximum fountain penetration heights and the maximum thicknesses of the interaction regions of the triple transitional round fountains are quantified with the experimental and numerical results and compared to the available scalings for single fountains at comparable Fr and Re values. Several scaling relations are then developed for these parameters over the ranges of Fr and Re considered. [ABSTRACT FROM AUTHOR]

In this study, a series of three-dimensional direct numerical simulations (DNS) were carried out using ANSYS Fluent for transitional plane fountains in linearly stratified fluids with the Reynolds number ( Re ), Froude number ( Fr ) and dimensionless temperature stratification parameter ( s ) over 28 ≤ Re ≤ 300, 3 ≤ Fr ≤ 10, and 0.1 ≤ s ≤ 0.5, to study and quantify the effects of these governing parameters on the maximum fountain penetration height, including the initial one during the early developing stage and the time-averaged one at the quasi-steady state, as well as the time to reach the initial maximum penetration height. The results show that both the initial and time-averaged maximum fountain heights as well as the time to attain the initial maximum fountain height increase with Fr but decrease with s , whereas the effect of Re is negligible, and the fluctuations of the maximum fountain penetration height at some specific locations at the quasi-steady state also follow the similar trends. Empirical correlations to quantify the effects of Fr , s and Re on these bulk fountain behavior parameters were obtained from the DNS results over the ranges of Fr , s and Re considered. [ABSTRACT FROM AUTHOR]

Fountains injected into stratified fluids are widely found in environmental and industrial settings. The onset of asymmetry and entrainment that occurs in transitional fountains is the key to understanding turbulence generation and entrainment mechanisms in fountains. In addition to the Reynolds number ( Re ) and the Froude number ( Fr ), the stratification of the ambient fluid, represented by the dimensionless temperature stratification parameter ( s ), also has a significant effect on the onset of asymmetry, unsteadiness, and entrainment in a fountain, and on the maximum height that the fountain penetrates in the ambient fluid. In this study, a series of three-dimensional direct numerical simulations (DNS) were carried out using ANSYS Fluent for transitional plane fountains in linearly-stratified fluids with Re and s in the ranges of 25 ⩽ Re ⩽ 300 and 0 ⩽ s ⩽ 0.5 , all at Fr = 10 . The transient behavior of the fountains, in particular the effects of Re and s on the asymmetric transition and the maximum fountain penetration heights, is analysed and quantified using the DNS results. It is found that fountains are generally symmetric in the early developing stage, but become asymmetric and unsteady subsequently. The stratification of the ambient fluid is shown to stabilize the fountain flow and to reduce its asymmetry and unsteadiness. However, the effect of s on the asymmetric behavior of a fountain is found to be weaker than that of Re . Empirical correlations were developed, using the numerical results, to quantify such effects on the time for the asymmetric transition. The effects of Re and s on the initial and time-averaged maximum fountain heights ( z m , i and z m , a , respectively), and the time to attain z m , i are also analysed. Quantified correlations are developed using the DNS results, which demonstrate that both z m , i and z m , a increase with Re , but decrease with s , apparently due to the increasing negative buoyancy that the fountain must overcome to penetrate the ambient fluid. The results further show that the effect of s on z m , i and z m , a is much stronger than that of Re . [ABSTRACT FROM AUTHOR]

This paper is the second part of our study on the behavior of the interaction between twin transitional round fountains with equal power in a homogeneous fluid. In the first part (Mahmud et al., 2015), an experimental study using PIV techniques and flow visualization was carried out over the ranges 1 ⩽ Fr ⩽ 5 and 25 ⩽ Re ⩽ 400 , where Re and Fr are the Reynolds and Froude numbers, respectively. In the current paper, this work is extended to wider ranges, i.e. , 1 ⩽ Fr ⩽ 10 and 25 ⩽ Re ⩽ 1000 , by a series of three-dimensional direct numerical simulation (DNS). In addition, a preliminary study on the effect of the distance between the two fountain sources is also carried out for three representative cases. The DNS results are found to capture all the major features of the steady, and unsteady weakly multi-modal and strongly multi-modal interaction behavior, which is dominated by bobbing and flapping motions, as found by the experimental study (Mahmud et al., 2015), but over wider ranges of Fr and Re . Furthermore, the maximum fountain penetration height and the maximum thickness of the interaction region, obtained with the DNS, together with the experimental results of (Mahmud et al., 2015), are quantified and compared to the available scalings for single fountains in a homogeneous fluid at comparable Fr and Re values, and several scaling relations are then developed for these parameters over the ranges of Fr and Re considered in the current study. [ABSTRACT FROM AUTHOR]

Numerical simulations of turbulent fountain flow are used to investigate the important energy and mass transfer mechanisms present in the forced fountain flow regime, which has been reported to exist at Froude numbers (Fr) greater than 3. The flow is equivalent to a negatively buoyant jet with three flow streams, the inner upflow (IF), the outer downflow (OF) and the surrounding ambient fluid (AF). Simulation results are presented for Fr = 4 and 7 at Reynolds number Re = 3350. The mean fountain penetration height scales with the previously reported relation Zm/R0 = 2.46Fr, where R0 is the source radius, but the assumptions behind analytical derivations of the relation are not supported by the present results. The results suggest that the OF may be relatively well described by the dynamics of a pure line plume surrounding the IF but with higher entrainment owing to the unsteady pulsing behaviour of the flow entering the OF from the IF. The length scale for a pure plume appears to apply at Fr = 7 in the OF and a degree of self-similarity exists. Comparisons with previous results suggest the IF is not fully developed at Fr = 7 and entrainment into the IF from the OF may not occur until Fr > 15. [ABSTRACT FROM AUTHOR]

Abstract: It is of fundamental interest to understand the behavior of transitional fountains with intermediate Froude and Reynolds numbers, together with the associated entrainment and turbulence. In this work, the transient behavior of axisymmetric fountains with 1⩽ Fr ⩽8 and 200⩽ Re ⩽800 is studied by direct numerical simulation. It is found that at Re ⩽200, there is little entrainment present at the upflow–downflow interface and at the downflow–ambient interface, even for a value of Fr as high as 8; however, at Re >200, entrainment is present at these interfaces and the extent increases with Re, which clearly demonstrates that entrainment is strongly dependent on Re whereas the contribution from the Fr effect is relatively much smaller. The DNS results also show that z m, which is the maximum fountain penetration height, fluctuates, even when the flow reaches full development, due to the entrainment at the upflow–downflow and the downflow–ambient interfaces, and the averaged z m scales with for 1< Fr ⩽8 and 100⩽ Re ⩽800. [Copyright &y& Elsevier]

• Behavior of round fountains in a cylindrical container is studied experimentally. • Transition of flow behavior and secondary flows increase with Froude number. • Secondary flows change from buoyancy-inertial regime to buoyancy-viscous regime. • Stratification rate increases with Froude number, but decreases with Reynolds number. In this paper, high-speed cameras and flow visualization techniques are used to investigate the behavior of the 'fountain filling box' flow resulted from releasing a round fountain in a homogeneous quiescent fluid in a cylindrical container over the ranges of 1.0 ⩽ Fr ⩽ 20.0 , 102 ⩽ Re ⩽ 1502 , and 27.9 ⩽ λ ⩽ 48.75 , where Fr , Re and λ are the Froude number, the Reynolds number, and the dimensionless radius of the container, respectively, with λ non-dimensionalized by the fountain source radius. The results show the transition of the flow behavior of the fountain and its secondary flows (i.e., the intrusion, reversed flow and stratification) from laminar to turbulent with increasing Fr , and turbulence of the flow strengthened with increasing Re. For intermediate (e.g., Fr = 3.0) and forced turbulent fountains (e.g., Fr = 5.0 , 8.0, 15.0) with a specific λ , the non-dimensionalized time-scale for the intrusion front to impinge upon the sidewall, τ w , is nearly constant for Re ≳ 500. This is because the secondary intrusion flow is dominated by the wall-jet and buoyancy-inertial regimes where the non-dimensionalized intrusion front velocity, v i , is only time-dependent ( v i ~ τ - 1 / 2 ) or time-dependent but also under the influence of Fr ( v i ~ Fr - 1 / 2 τ - 1 / 4 ). However, τ w for the fountains of Re ≲ 204 is significantly different, which may result from the change in the dominant regime for the intrusion or the interaction between the upflow, downflow of the fountain and the ambient fluids. Furthermore, it is found that the non-dimensionalized quasi-steady development rate of the stratification, v s , increases with Fr , but decreases with Re , since the diffusion effect is suppressed with decreasing Fr or increasing Re. [ABSTRACT FROM AUTHOR]

Abstract: The behaviour of plane laminar fountains with parabolic velocity inlet profile is studied using numerical simulation over the parametric range 0.25 ≤ Fr ≤ 10.0, 50 ≤ Re ≤ 150 and Pr = 7, 300, 700. The behaviour of the flow is most strongly affected by the Froude number and to a lesser extent by the Reynolds number, particularly for weak fountains at low Reynolds numbers. Behaviour is independent of the Prandtl number over the parametric range investigated. Three distinct regimes are observed: a steady symmetric pattern at low Froude numbers (Fr < 1.25), an unsteady periodic flapping characterised by lateral oscillations at intermediate Froude numbers (1.25 ≤ Fr ≤ 2.25) and an unsteady aperiodic flapping at higher Froude numbers. Based on dimensional analysis, the rise height of the fountain is shown to follow the correlation z m∼Fr 4/3−2/3(c+d) Re −(c+d) Pr −d . The constants c and d are determined from numerical results for each regime. [Copyright &y& Elsevier]

Abstract: In this paper, we present line fountain behavior at low-Reynolds numbers obtained by experiments. The experiments are conducted over the range of Reynolds number and Froude number . It is observed that the fountain behavior can be categorized broadly into four regimes: the steady; flapping; laminar mixing; and jet-type mixing behavior, at full development. The critical Froude number for transition from a steady to unsteady flow varies with the Reynolds number. For , the transition is independent of Re and is well described by the line. Over the range , the transition can be approximated by a constant line. For , there is a higher dependency on the Reynolds number with a very sharp increase in the critical Froude number and it is hypothesized that the demarcation line follows , where . The fountain exhibits flapping behavior in the range . These observed fountain behaviors are mapped on to a plot. In addition, the observed, non-dimensionalised fountain height is found to be in reasonable agreement with previous results on laminar line fountains. In particular, the experimental results confirm the scaling in the steady regime which was obtained previously by scaling analysis. [Copyright &y& Elsevier]