CALCULATION OF STEADY AND OSCILLATING FLOWS IN TUBES USING A VORTICITY TRANSPORT ALGORITHM

Abstract

SUMMARY Steady and oscillating axisymmetric tube flows are modelled using a vorticity transport algorithm. The axisymmetric convectivediffisive Navier-Stokes equations are solved using a splitting technique. Axisymmetric ring vortex filaments are introduced on the walls and subsequently convected and diffused throughout the flow field. An axisymmetric equation similar to the Oseen diffusion equation is used to diffuse the ring vortex filaments. Vorticity is reflected from the tube walls using two techniques. Results are presented for the developing Poiseuille flow and for the developed flow in the form of the entrance length and the axial velocity and vorticity profiles. Good agreement is achieved with a finite difference method in the developing region of Poiseuille flow. The developed flow results are compared with the analytical solutions. The developed profiles of velocity and vorticity have errors of less than 0.3 per cent for both methods of dealing with reflection of difkion at the bounding surfaces and similar accuracy is obtained for the velocity profiles in oscillating flow except at the wall. Oscillating flow is produced with a discretized sinusoidal piston motion. Velocity profiles, boundary layer thickness and entrance length are presented for oscillating flow. Good agreement is achieved for lowWomersley-number non-dimensional frequency. At higher values of this parameter, flows are inaccurately simulated, because the number of piston positions used to discretize the piston motion is inversely proportional to the non-dimensional frequency. 1. NTRODUCTION The extension of Lagrangian vortex element methods to the computation of unsteady viscous flows from two to three dimensions has evolved in recent years.’.’ A description of a two-dimensional computational model which was applied to the flow past a circular cylinder has been published by Benson et and is the basis of the present work. For axisymmetric flows the non-dimensional convective-diffusive NavierStokes equations are solved by the introduction of ring vortex filaments to represent a shear layer on the solid surfaces and by the subsequent convection and difision of these vortex filaments. The convection routine involves a cloud- in-cell type of calculation as given by Christiansen4 and the diffusion routine involves the redistribution of circulation on a mesh. The diffusion is computed from the axisymmetric solution which is derived in the Appendix. To ensure the conservation of vorticity near the solid surfaces, vorticity must be ‘treated’ in these regions. Two approximate methods of wall treatment are compared: vorticity is either specularly reflected from the solid surfaces or is allowed to diffuse beyond the solid surfaces, reallocated to mesh points and rediffised into the flow field. Results are presented using both methods. The flows are produced by the motion of a piston. Wagner’ discusses the difficulties associated with specification of the inlet