The structure of a turbulent line fountain

Abstract

Line fountains form when heavy miscible fluid is ejected steadily upwards as a jet from a high-aspect-ratio rectangular slot, of length $L$ and half-width $b_{0}$, into lighter quiescent surroundings. Viewed along the slot from one end, previous observations reveal that the ejected fluid mixes with the environment and reaches a peak height before partially collapsing back downward under gravity to form a fountain whose top thereafter fluctuates vertically about a mean height. While the motion as perceived from this single view has provided insights that have successfully guided theoretical predictions for the initial rise height, until now a wider understanding of line fountains, and corresponding predictive capability, has been limited to this single prediction due to a lack of any other observational data. Indeed, the general behaviour of line fountains, including the structure internally and along the spanwise length $L$ of the slot, has not been reported previously. To address this, flow visualisations and comprehensive measurements of saline fountains in an aqueous environment are presented here that reveal their complex overall structure and behaviours. After establishing the uniformity of the source conditions from slots of aspect ratio $600:1$ and $300:1$, we first show that double-averaged (spanwise and time) rise heights $\overline{\overline{z}}_{v}/b_{0}$ scale on $Fr_{0}^{4/3}$, $Fr_{0}$ being the source Froude number, with vertical fluctuations being circa 20 % of these heights. Then, simultaneously interrogating the flow as viewed from above and from the side onto the spanwise dimension, we identify three distinct patterns of behaviour. Instrumental to distinguishing these behaviours were the contrasting signatures we observed in the time series of rise height departures from the mean which led us to the following classification: (i) non-uniform flapping for $0.05\lesssim \overline{\overline{z}}_{v}/L\lesssim 0.30$, in which the lateral motion of the fountain takes the form of an oscillatory wave with a wavelength of $2L/3$ (approx.); (ii) uniform flapping for $0.30\lesssim \overline{\overline{z}}_{v}/L\lesssim 0.45$, in which the entire fountain sways to the left and then to the right side of the slot; and (iii) disorganised flapping for $\overline{\overline{z}}_{v}/L\gtrsim 0.45$. Regarding the internal structure, we show that unlike a classic round fountain, eddying structures comparable in scale with the rise height form towards the top of the fountain, and the counterflow forms predominantly to one side of the jet. We then identify the single dominant mechanism driving the flapping motions, successfully linking the wave-like behaviour observed along the span to the internal structure and vertical oscillations. Quantifying the oscillatory motions, both the vertical and flapping frequencies scale as $Fr_{0}^{-2}$, and we demonstrate and explain a robust coupling between these frequencies that follows a ratio of 2:1.

Vertical Plane Buoyant Jets and Fountains in a Uniform Stagnant Ambient Revisited

“…There one can see that our data compare well with those by Papakonstantis and Mylonakou (2021) who have used a different 2-D jet generating arrangement in a much bigger dispersion tank. Our data deviate from those of Baines et al (1990) that are 15% higher, and are quite different from the measurements by Zhang and Baddour (1997) and Hunt et al (2019). This is more evident in the present data when Froude number is greater than 50.…”

“…This is more evident in the present data when Froude number is greater than 50. In the same figure the results by Hunt et al (2019) regarding terminal penetration height are bigger when the slot length to width ratio is doubled from L/d=150 to 300. In the present experiment the slot length to width ratio was variable because we kept the length L=20 cm while the width varied from 0.05 cm to 0.25 cm, thus obtaining L/d= 80 to 400.…”

“…The initial conditions and measured parameters are depicted in Table 1. Flapping due to the instability of the fountain was observed as in Hunt (2019) and the flow would return to the ejection elevation on either side. Two instantaneous photos of the flow, one from shadowgraph and one from PLIF imaging, are depicted in Figure 8.…”

“…Thus, the vertical jet of negative buoyancy does not return symmetrically to the outflow elevation due to instability, but like a fountain, it will sometimes fall on the one side of the rising vein and sometimes on the other side. This instability was recently studied by Hunt et al (2019) who have shown that part of the reversing flow may fall to one side and simultaneously part the rest of the reversing flow to the other side of ejected vain.…”

“…There is disagreement regarding the structure of the vein and the measurement of the penetration height, that is the inconsistency in the parameterization among the different researchers, making it difficult to compare the results. This inconsistency stems from the fact that Baines, et al (1990), Zhang andBaddour, (1997), Srinarayana, et al (2010) and Hunt, et al (2019) used the half-width of the slot to calculate the dimensionless parameters. This methodology, however, goes against the physical characteristic scales derived by dimensional analysis and used by earlier researchers like Kotsovinos (1975) and Fischer et al (1979) and the most recent one by Papakonstantis and Mylonakou (2021).…”