Characterising line fountains

Abstract

We present analytical solutions for the initial rise height zm of two-dimensional turbulent fountains issuing from a horizontal linear source of width 2b0 into a quiescent environment of uniform density. Using the initial rise height prediction as a measure we classify line fountains into three types depending on their source conditions. For source Froude numbers Fr0 ≫ 1, the near-source flow of the ‘forced’ fountain is dominated by source momentum flux and behaves like a jet; the asymptotic solution to the fountain equations yields, in agreement with previous studies, zm/b0 ~ Fr04/3. For Fr0 = O(1) the fountain is ‘weak’ and fluid is projected vertically to a height that is consistent with an energy-conserving flow – the sensitivity of the rise height with Fr0 increases as zm/b0 ~ Fr02. For Fr0 ≪ 1, the fountain is ‘very weak’ and we find that zm/b0 ~ Fr02/3. As the local value of the Froude number decreases with height, all three forms of fountain behaviour identified are expected above a highly forced source and we provide scalings for the three lengths that contribute to the total rise height. Comparisons between our predicted rise heights and the previous experimental results show good agreement across a wide range of Fr0. The collated data highlights that experiments have focused in the majority on fountains above sources with intermediate Fr0. Notably there is a lack of measurements on very weak line fountains and of independent experimental confirmation of the initial rise heights across the range of Fr0.