Wave packet analysis and break–up length calculations for an accelerating planar liquid jet

This paper examines the process of transition to turbulence within an accelerating planar liquid jet. By calculating the propagation and spatial evolution of disturbance wave packets generated at a nozzle where the jet emerges, we are able to estimate break–up lengths and break–up times for different magnitudes of acceleration and different liquid to air density ratios. This study uses a basic jet velocity profile which has shear layers in both the air and the liquid either side of the fluid interface. The shear layers are constructed as functions of velocity which behave in line with our CFD simulations of injecting Diesel jets. The non–dimensional velocity of the jet along the jet centre–line axis is assumed to take the form V (t) = tanh(at) where the parameter a determines the magnitude of the acceleration. We compare the fully unsteady results obtained by solving the unsteady Rayleigh equation, to those of a quasi–steady jet to determine when the unsteady effects are significant, and if the jet can be regarded as quasi–steady in typical operating conditions for Diesel engines. For a heavy fluid injecting into a lighter fluid (density ratio ρair/ρjet = q < 1) it is found that unsteady effects are mainly significant at early injection times where the jet velocity profile is changing fastest. When the shear layers in the jet thin with time, the unsteady effects cause the growth rate of the wave packet to be smaller than the corresponding quasi–steady jet, while for thickening shear layers the unsteady growth rate is larger than that of the quasi–steady jet. For large accelerations (large a) the unsteady effect remains at later times but its effect on 1 Corresponding author: [email protected] 2 Present address: Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH UK 3 Present address: ANSYS UK, Ltd., 97 Milton Park, Abingdon, Oxfordshire OX14 4RY UK