**Turbulence **– a chaotic state of fluid motion that is commonly observed in the atmosphere, oceans and rivers. This physical phenomenon is best described by its characters because a mathematically precise definition is still lacking. One such character is the greatly enhanced mixing of mass, heat and momentum when compared to a laminar (smooth) flow at the same Reynolds number. Another defining character is its multiscale nature in both time and space and the existence of an energy cascade.

**Energy cascade** – the nonlinear transfer of kinetic energy across different turbulent scales/eddies. The cascade is termed forward (inverse) when energy is transferred from large to small (small to large).

**The von Karman-Howarth-Monin (K-H-M) equation** – the evolution equation of the second-order velocity structure function, also known as the scale energy, that represents the kinetic energy content of a particular size class of turbulent eddies. The equation tracks the changes in both physical space and scale space such that one can study all pathways of energy transfer in turbulence – a complete description not offered by the single-point turbulent kinetic energy equation.

**Atwood number ***At* – is a dimensionless number between 0 and 1 used to quantify the magnitude of density gradients in a flow. It is zero when there is no density difference. The value of 0.05 is commonly taken as the upper bound for Boussinesq flows.

**Near-far field transition** – is used in the context of jet/plume discharges in the environment, and it refers to the transition from jet-induced mixing to ambient turbulence mixing.

**Production P_B by bubbles** – refers to the creation of liquid-phase turbulent kinetic energy at the bubble-liquid interface due to the hydrodynamic forces exerted on the bubble.

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