The flow of a two-dimensional liquid laminar jet near the exit of a channel.
INTRODUCTION:
A high Reynolds number Newtonian liquid jet, exiting a two-dimensional channel, is examined in this study. Fully developed Poiseuille flow, driven by an applied pressure gradient, is assumed to prevail upstream of the channel exit. Flow is assumed to be laminar, steady and incompressible.
A free surface develops as the fluid exits the channel. The resistance of the air on the free surface of the liquid jet can be neglected so that, the shear stress at the channel wall disappears and becomes zero at the free surface. This stress singularity is a major obstacle in calculating the flow near the channel exit both numerically and analytically. No exact analytical solution is available for the problem. However, the flow is predicted near channel exit in literature using some approximate analytical methods.
High Reynolds number Newtonian jet contracts downstream of the channel due to the emergence of the normal stress as soon as the fluid detaches itself from the channel wall. The effect of inertia on the free surface height and free surface velocity and the flow downstream has been explored. Results obtained are compared with the analytical work of the relevant problem.
The computational domain includes the upstream and downstream near the channel exit. Fully developed Poiseuille flow is used as the boundary condition at the inlet and zero traction on the free surface at the outlet.
MATHEMATICAL MODEL:
The problem is a two phase flow where there are distinct regions for each of the phases. Multiphase model VOF is the most suitable for this specific flow problem. In the VOF model, a single set of momentum equations is shared by the fluids, and the volume fraction of each of the fluids in each computational cell is tracked throughout the domain.
The tracking of the interface between the phases is accomplished by the solution of a continuity equation for the volume fraction of one (or more) of the phases. For the ‘q’th phase, this
A high Reynolds number Newtonian liquid jet, exiting a two-dimensional channel, is examined in this study. Fully developed Poiseuille flow, driven by an applied pressure gradient, is assumed to prevail upstream of the channel exit. Flow is assumed to be laminar, steady and incompressible.
A free surface develops as the fluid exits the channel. The resistance of the air on the free surface of the liquid jet can be neglected so that, the shear stress at the channel wall disappears and becomes zero at the free surface. This stress singularity is a major obstacle in calculating the flow near the channel exit both numerically and analytically. No exact analytical solution is available for the problem. However, the flow is predicted near channel exit in literature using some approximate analytical methods.
High Reynolds number Newtonian jet contracts downstream of the channel due to the emergence of the normal stress as soon as the fluid detaches itself from the channel wall. The effect of inertia on the free surface height and free surface velocity and the flow downstream has been explored. Results obtained are compared with the analytical work of the relevant problem.
The computational domain includes the upstream and downstream near the channel exit. Fully developed Poiseuille flow is used as the boundary condition at the inlet and zero traction on the free surface at the outlet.
MATHEMATICAL MODEL:
The problem is a two phase flow where there are distinct regions for each of the phases. Multiphase model VOF is the most suitable for this specific flow problem. In the VOF model, a single set of momentum equations is shared by the fluids, and the volume fraction of each of the fluids in each computational cell is tracked throughout the domain.
The tracking of the interface between the phases is accomplished by the solution of a continuity equation for the volume fraction of one (or more) of the phases. For the ‘q’th phase, this
References: 1. J. P. K. Tillett. On the laminar flow in a free jet of liquid at high Reynolds numbers. J. Fluid Mech. 32 (1968) 273. Figure 1: Contours of the volume fraction of water as it comes out of the channel as a jet.(Re = 1000) Figure 2: Free surface height vs. distance from the exit for the present numerical result and the approximate analytical result obtained by Tillett (1968) (Re =1000) Figure 4 : Free surface velocity vs. distance from the exit for the present numerical result and the approximate analytical result obtained by Tillett (1968) (Re = 1000) Figure 6 : Velocity profile colored by static pressure at different upstream and downstream position of the channel exit (Re = 1000) Figure 8 : Axial velocity at the centerline inside the channel(Re = 1000)