The values of ΔT0ΔT0 result in the density contrast at the nozzle ((ρa-ρ0)/ρa(ρa-ρ0)/ρa) of 0.0006, 0.0015, 0.0040 Ganetespib supplier and 0.0075 kg/m3 for Standard Mean Ocean Water (SMOW) ( Tanaka et al., 2001). The system of equations in (7) is solved using the Euler method in Matlab R2013b for b0=0.05m, the results are plotted in Fig. 3 and discussed in the following section. For large initial jet velocities (i.e. u0≫gb0(ρa-ρ0)/ρa,U∞) the influence of buoyancy and ambient flow is negligible in the near field. The benefit of a high discharge velocity is that it also results in a more coherent jet within 4 m from the nozzle.
In this limit, from (7), the following linear relationships can be established equation(9a,b) bb0=1+Djet,Djet=2αyb0.Jet
dilution and volume flux increase ( Morton et al., 1956) along the centre line are related to each other through the following relationship equation(10) QQ0=1+Djet.The comparison with the full model in Section 2.2 and the estimates in (9a,b) are Selleck Roxadustat plotted in Fig. 3a and b. The jet forms a conical shape with an angle tan-1(4α)=17.74°tan-1(4α)=17.74°. Over a distance of y = 4 m, the jet fluid has been diluted by a factor of equation(11) Djet=0.64b0.The decay in u and ΔTΔT of the jet with distance y due to entrainment of ambient fluid (dilution) can be estimated as equation(12a,b) uu0=11+2αyb0,ΔTΔT0=11+D.By inserting the terms in (9a) and (12a) into (5) it can be shown that the local Reynolds number within a momentum dominated jet cone will stay constant, so if the jet is initially turbulent at the outlet it will be turbulent along its path. When measuring the location of the jet centre line at 4 m it is important to make a correction due to the effect of U∞U∞ and ΔTΔT. The influence of ΔTΔT causes the jet to rise above the point of discharge. This rise can be estimated from equation(13) M0d2zdy2≃πb2gρa-ρ0ρa.Since the buoyancy flux is conserved, we can integrate (13) to obtain equation(14) z≃gy2u0212+αy3b0ρa-ρ0ρa,where
Lenvatinib research buy the distance z is the amount the jet has risen. Similarly the jet trajectory deflection due to a weak cross flow is estimated from equation(15) M0d2xdy2≃2πuEU∞b0≃2παu0U∞b0,where entrainment (uEuE) is simplified to αu0αu0. Integrating (15) results in an approximation for the jet deflection downstream equation(16) x≃αU∞y2u0b0.A comparison between the full numerical model in Section 2.2, (14) and (16) is shown in Fig. 3c and d respectively. The agreement is good for |ΔT0|<20°C and U∞/u0<0.01U∞/u0<0.01. The previous discussion gives practical estimates of the centre line dilution. Additional information is required to understand how average dilution varies across the jet width. To examine this effect we analysed the dilution of a jet containing passive dye as it is gradually diluted.