Viscous starting jets

Abstract

This paper is concerned with the transient motion produced when a viscous incompressible fluid is forced from an initial state of rest. The applied force is time dependent in the form of an impulse, step and ramp function applied at a point and along a line. These cases have been chosen because they form a logical progression for investigating the connection between the flow Reynolds number and the sequence of events leading to the creation of a starting vortex. Much of the structure of the starting process can be revealed through a study of boundary conditions, integrals of the motion and the invariance properties of the governing equations prior to the consideration of a particular solution. The method used to bring out the flow structure is applicable to flows that can be treated as self-similar over some interval in time. The equations for unsteady particle paths are written in terms of similarity variables and then analysed as a quasi-autonomous system with the, usually time-dependent, Reynolds number treated as a parameter. The structure of the flow is examined by finding and classifying critical points in the phase portrait of this system. Bifurcations in the phase portrait are found to occur at specific values of the Reynolds number of the flow in question. When exact solutions of the Stokes equations for the low-Reynolds-number limit are examined they are found to contain two critical Reynolds numbers and three distinct states of motion which culminate in the onset of a vortex roll-up. An interesting feature of the Stokes solutions for planar unsteady jets is that they are uniformly valid over 0 < r < ∞.