Entrainment and dilution in a fountain top

Turbulent fountains are common phenomena in geophysical and engineering applications. They are formed when a less dense fluid is injected downwards into a heavy ambient fluid or when a dense fluid is released from a localized source vertically upwards into a lighter ambient fluid. The opposing buoyancy due to the difference in the densities of the ambient fluid and the source slows down the flow until zero mean momentum is achieved. Thereafter, the flow reverses and collapses around the inner flow. While returning, however, the flow is diluted by entraining the ambient fluid into the fountain.

There are several theoretical models for describing the time-averaged behavior of forced fountains, which consist of inner upflow and outer counterflow. Unfortunately, these models fail to account for the entrainment of ambient fluid into the fountain top (also called the ‘cap’ region). Previous studies have revealed that entrainment into the cap region is not negligible as was originally thought. Thus, it must be considered when estimating volume flux bulk entrainment in turbulent fountains because neglecting it might also induce an inevitable imbalance between the momentum and buoyancy fluxes. Additionally, a clear distinction between scalar concentration dilution and entrainment of volume flux is still lacking, mostly due to the absence of well-resolved experimental data on scalar concentration and velocity.

Herein, Dr. Krishna Talluru (currently at UNSW Canberra), Dr.  Nicholas Williamson and Professor S.W. Armfield from The University of Sydney experimentally investigated the entrainment of ambient fluid and the scalar concentration dilution in the fountain top and their dependence on source Froude number at larger Reynolds numbers. In their approach, the source Froude number was varied between 10 – 30 while the Reynolds number was set at a minimum value of 3000 to guarantee fully turbulent flow at the source. High-fidelity spatial measurements of velocity and concentration at a fountain top were conducted at varying source Froude numbers ranging from 30 – 40. Their work is currently published in the journal, Journal of Fluid Mechanics.

From the mean concentration fields, the authors showed that the cap was approximately hemispherical and its base scaled well with the local Froude number ≈ 1.5 over the range of the Froude number tested in this study. The entrainment of the ambient fluid into the fountain top was estimated to be the difference in volume fluxes between the upflow and return flow of the fountain top. The ratio of the entrained volume flux at the fountain top and the supplied volume flux at the base varied from 1.5 to 3.5, and it exhibited a non-monotonic variation with the Froude number. The entrained volume flux values measured here were greater than those reported in previous studies.

The research team developed a robust metric to estimate the dilution of scalar in the fountain top. It was compared with the entrained volumetric ratio and the two quantities were found to have different mean values, which varied with Froude number. Nevertheless, the proposed metric proved more effective and accurate in characterizing scalar dilution at the fountain top, considering the induced downward flow. Although the scalar concentration dilution was not equal to the entrainment ratio, the variation of the dilution ratio was associated with the local Reynolds number at the base of the fountain top or the cap region.

In summary, the study provided detailed concentration and velocity measurements on the turbulent fountain top for source Froude number in the range 30 – 40. Importantly, the volume flux entrainment and the scalar concentration dilution at the fountain top were governed by the local and source Froude and Reynolds numbers Reynolds number. The presented experimental results were in good agreement with the numerical simulation results. In a statement to Advances in Engineering, Dr. Krishna Talluru said their study provided valuable insights that would contribute to a thorough understanding of the entrainment and dilution in a fountain top.

Dr. Krishna Talluru is a Lecturer in the School of Engineering and Information Technology at UNSW Canberra. His past research focused on low-speed turbulent flows that include turbulent boundary layers, wakes, neutral and buoyant jets (plumes and fountains), scalar dispersion, and wind engineering. His current research focusses on high-speed aerodynamics studying fluid-structure interactions (in particular, the shock-wave-boundary-layer-interactions) and wakes in supersonic and hypersonic flow regimes.

Dr Nicholas Williamson is a Senior Lecturer in the School of Aerospace, Mechanical and Mechatronic Engineering at the University of Sydney. His research focuses on buoyant turbulent flows, in particular natural convection boundary layers and free shear flows such as negatively buoyant jets. He also works collaboratively in riverine science towards understanding the physical mechanisms which cause algal blooms and how stable thermal stratification aids bloom formation.

Professor Steve Armfield investigates the fluid mechanics of a range of environmental and industrial flows using computational, theoretical and experimental approaches, leading to applications as diverse as improved river management and the design of more efficient building ventilation systems. His research focuses on the development of computational models and algorithms to allow the prediction of highly unsteady, buoyancy-driven and -dominated flows, such as the natural convection boundary layers that develop adjacent to vertical heated surfaces, the two-layer mixing flow that occurs when a lighter fluid passes over a denser fluid, and thermal fountains and plumes. Such flows occur in many environmental and industrial settings, such as in rivers, estuaries and atmospheric boundary layers, and in building heating, cooling and ventilation. His research has made a substantial contribution to the understanding of the mechanism of flow initiation and transition, which has led to improved computational models for these flows that will inform the development of river management strategies, large-scale fluid dynamics models and the design of heating, ventilation and cooling systems.


Talluru, K. M., Williamson, N., & Armfield, S. (2022). Entrainment and dilution in a fountain topJournal of Fluid Mechanics, 941, 941.


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