Entrainment is a mechanism leading to the growth of the jet radius and volume flux with distance from the point of discharge through the capture of ambient fluid ( Hunt et al., 2011). At low discharge velocities Selleckchem CAL101 the jet becomes laminar, the consequence of this is that mixing with ambient fluid is significantly reduced due to the dominance of viscous forces ( Batchelor, 2001). Entrainment models for laminar jets are discussed by Morton (1967). In order to obtain optimal dilution through turbulent mixing we introduce a constraint equation(5) Re=2b0u0ν>Rec,where RecRec is a critical Reynolds number and νν is the kinematic viscosity of water. Certainly Rec=3000Rec=3000 is

sufficient for the jet to be turbulent ( McNaughton and Sinclair, 1966). We describe a mathematical model of a buoyant jet discharged horizontally and tangentially into a uniform unstratified stream in order to calculate Palbociclib mw the jet trajectory and dilution. An unstratified ambient is considered because the draught depth of merchant vessels is at most 20 m and in this range the effects of stratification are not significant. It is assumed that the issuing fluid is perfectly mixed across the width of the jet and that the dilution processes have a far longer timescale than the chemical processes that happen very rapidly (Ülpre

et al., 2013). In the ‘top-hat’ model (Morton et al., 1956), the jet is characterized by a radius b  , average

centre line velocity u   and a density contrast of ρ-ρaρ-ρa compared to the ambient ρaρa. These variables are combined about to form the volume flux Q  , specific momentum flux M   and specific buoyancy flux B  , which are defined as equation(6a,b,c) Q=πb2u,M=πb2u2,B=πb2ugρa-ρρa.The initial values of Q,M and B   at the point of discharge are Q0,M0 and B0B0. The conservation of mass and momentum are expressed in terms of how Q   and M   vary with distance s   along the jet trajectory. The jet is directed along the y  -axis, rises due to buoyancy along the z  -axis and is swept by an ambient flow along the x  -axis. Two forces act on the buoyant jet in the presence of an ambient flow U∞U∞, the Lamb force and buoyancy. In conclusion this gives equation(7) dQds=2πuEb,ddsMdxds=2πuEU∞b,ddsMdyds=0,ddsMdzds=πb2gρa-ρρa,where uEuE is the entrainment velocity that must be closed by an empirical relationship between the mean jet velocity and the ambient flow ( da Silva et al., 2014). We use the closure relationship applied by Woodhouse et al. (2013) equation(8) uE=αudzds+udxds-U∞+udyds,but others have also been proposed e.g.   Jirka (2004). Since the discharges are likely to be in the form of jets we can assume the empirically determined entrainment coefficient to be α=0.08α=0.08 ( Turner, 1969).