aa r X i v : . [ phy s i c s . f l u - dyn ] A p r Jets with Reversing Buoyancy
R. V. R. Pandya ∗ and P. Stansell † Abstract
A jet of heavy fluid is injected upwards, at time t = 0 , into a lighterfluid and reaches a maximum height at time t = t i and then flows backaround the upward flow. A similar flow situation occurs for a light fluidinjected downward into a heavy one. In this paper an exact analyticalexpression for t i is derived. The expression remains valid for laminarand turbulent buoyant jets with or without swirl. The study of jets with reversing buoyancy has found many applications inengineering. A heavy fluid jet injected upward into a lighter fluid is referredto as a negatively buoyant jet or fountain. This type of flow situation oftenoccurs in disposal of industrial effluent. Typical examples are brine andgypsum discharged into the ocean through multi-port diffusers.Similarly, a positively buoyant jet occurs when a light fluid is injecteddownwards into a heavier one. This jet reaches a maximum depth and thenturns upward and rises around the downward flow. These types of jets arealso referred to as inverted fountains. Typical examples are heating of a largeopen structure by fan-driven heaters on the ceiling and mixing of a two-layerwater reservoir with propellers [1]. In both cases the jets also possess someswirl.One of the first experimental studies on negatively buoyant jets was con-ducted by Turner [2]. He used a nozzle to inject a salt solution of density ρ upward into a tank of stationary fresh water of density ρ a < ρ . He measuredthe maximum height, z i , attained by the jet and the mean height, z m < z i , ∗ [email protected] † [email protected] i z ρ a Contol volumeArea, A g u z z = 0 Figure 1: A schematic diagram of a negative buoyancy jet at t = t i .at which the jet finally settles down. He related the two parameters z i /D and z m /D as proportional to the densimetric Froude number F r = U ρ g D ( ρ − ρ a ) ! by using a dimensional analysis. Here, U is the uniform velocity at thenozzle outlet, g is the acceleration due to gravity and D is diameter of thenozzle.Later, Abraham [3] studied the problem theoretically and confirmed theresult of Turner. Recently Baines et al. [1, 4] made an extensive study, bothexperimentally and theoretically on negatively buoyant jets. None of thesestudies provide an expression for t i . Here a simple analysis is presented toobtain an exact analytical expression for t i . A schematic diagram of the jet and control volume at time t = t i is shownin Figure 1. 2onsider a flow situation in which a fluid of density ρ is injected, attime t = 0 in the upward direction at a constant flow rate, from a nozzle at z = 0 into an ambient fluid of density ρ a < ρ . Let the cross sectional areaof the nozzle be A and the fluid velocity distribution at the nozzle to be u .The jet continuously looses its momentum due to the reverse buoyancy forceand reaches a maximum height z i at time t = t i after which the jet turnsdownwards.For times t < t i the z -component of the momentum in the control volumeincreases continuously until time t = t i from which time it decreases due tothe downward flow. Therefore, at t = t i , the rate at which the momentumis changing inside the control volume is equal to zero. Application of themomentum theorem to the flow situation at time t = t i implies that the netforce F b (in this case the buoyancy force in the z -direction) acting on thecontrol volume is equal to the rate at which the net momentum M is leavingthe control volume, that is to say, F b = M (1)If we take the control volume to be sufficiently large then the net rate atwhich the momentum is leaving the control volume can be written as M = − ρ Z A u dA ′ k , (2)where k is the unit vector in the z -direction.The total volume of the fluid jet of density ρ entering the control volumein time t i is V = t i Z A u dA ′ . This is also equal to the volume of fluid of density ρ a displaced by the jetin time t i . Therefore, the net buoyancy force acting on the fluid inside thecontrol volume is F b = − t i ( ρ − ρ a ) g Z A u dA ′ k . (3)From equations (1), (2) and (3) we obtain an analytic expression for t i as t i = ρ R A u dA ′ ( ρ − ρ a ) g R A u dA ′ , (4)which reduces to t i = U ρ | ρ − ρ a | g , (5)for u = U = constant over the area A . The modulus is introduced so thatequation (5) is valid for positively and negatively buoyant jets. It is worth3entioning that we have not made any assumptions about the regime offlow. Therefore the expression for t i remains valid for laminar and turbulentbuoyant jets with reversing buoyancy. Also, as any swirl added to the fluid atthe nozzle does not affect the z -momentum of the fluid, equation (4) remainsvalid for this case too. The following symbols are used in this paper: t time t i time at which the jet reaches its maximum height ρ density of the fluid in jet ρ a density of the ambient fluid z i maximum height attained by the jet z m mean height at which the jets settles F r Froude number U uniform fluid velocity as the nozzle outlet D diameter of the nozzle outlet g acceleration due to gravity A cross sectional area of nozzle outlet u fluid velocity distribution at the nozzle outlet F b buoyancy force M b rate at which momentum is leaving the control volume M b rate at which momentum is leaving the control volume k unit vector in the z -direction V total volume of jet fluid References [1] W. D. Baines, A. F. Corriveau and T. J. Reedman, “Turbulent fountainsin a closed chamber,” J. Fluid Mech.,212