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The flow behavior of weak symmetric plane fountains in linearly stratified fluids is studied numerically with three-dimensional simulations over a range of the Froude ( F r ), Reynolds ( R e ), and stratification numbers (s). The two main parameters describing the fountain characterization are the dimensionless maximum fountain penetration height ( z m ) and intrusion velocity ( u i n t ), which differ significantly at different flow development stages. It was found that the stratification stabilizes the symmetry of the weak fountains, which makes the fountain become asymmetric at a larger F r value, and z m at the fully developed stage continues to increase as a result of the intrusion, which continually changes the ambient fluid stratification features, thus the buoyant force. The evolution of intrusion experiences three distinct stages. Both F r and s have effects on z m and u i n t , with the effect of F r usually larger than that of s. The overall impacts of F r and s can be quantified in terms of F r a s b , with a and b varying for different parameters. With numerical results, empirical correlations are produced in terms of F r a s b for each relevant parameter, which generally predict the results very well. [ABSTRACT FROM AUTHOR]

A "fountain filling box" flow produced by discharging a weak laminar plane fountain in a confined open channel is studied numerically. Two‐dimensional direct numerical simulations were performed for weak plane fountains. The development of the fountain flow experiences five stages; the initial upflow and the subsequent downflow after the fountain penetrates to the maximum height, followed by the outward movement of the intrusion of the fallen fountain fluid on the channel bottom, and then the wall fountain formed by the impingement of the intrusion on the vertical sidewall, which results in the reversed flow, and finally the gradual stratification of the fluid. The behavior of the intrusion can be approximately described with the plane gravity current theory. The period for the intrusion to reach the bounded side wall increases with increasing Re or decreasing Fr. Three regimes are found for the wall fountain behavior; "no‐falling," "slumping down," and "rolling down" behavior. Convection, mixing, conduction, and filling all contribute to the formation and development of stratification, but their effects vary at different stages. For the initial stages, convection and mixing play a key role, resulting in an increasing bulk entrainment rate, while conduction and filling are dominant after quasi‐steady stratification is created, presenting a decreasing bulk entrainment rate. [ABSTRACT FROM AUTHOR]

In this study, a series of three-dimensional direct numerical simulations were carried out for transitional round fountains in a linearly-stratified fluid 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 transition to asymmetry and the asymmetric behavior. It is found that the transition to asymmetry are well represented and quantified by azimuthal velocity, with non-zero or noticeable azimuthal velocity indicating asymmetry. The results show that when Fr or Re are small, a fountain remains axisymmetric for all time; however, when Fr or Re are increased to be sufficiently large, the fountain will be axisymmetric initially, before becoming asymmetric and unsteady, ultimately reaching a fully developed quasi-steady stage when each quantity 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 the scaling for the time for transition to asymmetry, which is found to be strongly dependent on Fr and s , while only weakly dependent on Re . [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, 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]

The interaction of multiple sourced fountains is very common in many applications such as computers and electronic instruments, discharge of waste water from power plants to marine environments, displacement ventilation and air conditioning of large building spaces. But the understanding of such an interaction is currently scarce. This paper is the first part of the study on the behavior of the interaction between twin transitional round fountains with equal power in a homogeneous fluid. In this paper, the interaction behavior is investigated experimentally using a noninvasive PIV technique and flow visualization over the ranges 25 ⩽ Re ⩽ 400 and 1 ⩽ Fr ⩽ 5 at the fixed spacing of D / X 0 = 10 , where Re and Fr are the Reynolds and Froude numbers, D is the spacing between the two fountain sources, and X 0 is the radius of orifices at the fountain source. The interaction behavior is observed to be dominated by bobbing and flapping motions and is either steady, or unsteady and weakly multi-modal or strongly multi-modal, depending on the specific values of Re and Fr . 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 development stage in which the bobbing–flapping motions are no longer present and the maximum interaction height, z i , becomes constant. In an unsteady interaction, the interaction remains unsteady all the time and the bobbing–flapping motions are present at all development stages. The unsteady interaction is characterized by a finite number of discrete modes, with the time averaged z i approximately constant. Among all cases considered, the steady interaction behavior is observed for the cases of Re ⩽ 100 with all Fr values considered and of Fr ⩽ 1 with all Re values considered, except Re = 400 ; the unsteady weakly multi-modal interaction behavior is observed for 150 ⩽ Re ⩽ 300 with 1.5 ⩽ Fr ⩽ 5 and Fr = 1 with Re = 400 ; and the unsteady strongly multi-modal interaction behavior is found for Re = 400 with 1 < Fr ⩽ 5 . Dimensional analysis and the experimental results are also used to develop an empirical scaling relation between z ¯ i , which is the time-averaged dimensionless maximum interaction height at full development, and Re and Fr , i.e., z ¯ i = 0.194 Fr 7 / 4 Re 1 / 4 over the ranges 25 ⩽ Re ⩽ 400 and 1 ⩽ Fr ⩽ 5 , at the fixed spacing of D / X 0 = 10 . [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]

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: The behaviour of unsteady planar strong fountains, impinging on a ceiling with an opposing heat flux on the ceiling, is investigated numerically for at and . The height of the ceiling is varied in , where is the half-width of the fountain source, and the non-dimensional gradient of the temperature difference between the fountain source and the ceiling is varied for . It is found that the fountain does not hit the ceiling, but instead stagnates and spreads at some distance from the ceiling due to the stratification of the fluids in the immediate vicinity of the ceiling. The scaling and direct numerical simulation results show that the augmented spreading distance has the scaling of in the range studied, where is the maximum fountain height, is the spreading distance measured at , respectively. [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]