Keywords: vortex wave, reactor, oscillatory flow, low-Reynolds number, numerical simulation, two-dimensional reactor.

In environmental engineering, the performance of membrane reactors is limited by concentration polarization and membrane fouling. Turbulent flow can reduce these effects but results in damage to microbial cells in the reactor. Low-Reynolds-number flows provide a good form to overcome these problems. Vortex wave flow works at low Reynolds numbers, which can be generated either by an unsteady motion of a channel wall [1] or by unsteady flow through an asymmetric channel expansion [2]. The vortex wave, generated at low speed, enhances the mass transfer of the boundary layer and has a significant impact on the hydrodynamics and the mass transfer rates. Superimposing vortex wave flow on membrane systems is a novel method that combines oscillatory flow and vortex formation to prolong the membrane's working life.

Sobey [2] showed that a vortex wave would form downstream of a fixed channel expansion during oscillatory flow by both experimental observations and calculations using finite-difference discretization of the momentum equations. Other calculations of unsteady flow past a fixed channel expansion, using the finite-element method, can be found in Deblois & Sobey [4]. The combination of vortex wave flow and membrane process has found an application in plasma filtration by Millward et al. [6,7] and their work shows that the vortex wave can generate impressive convective mixing in relatively wide channels, and consequently the wall shear rates are low enough for shear sensitive mediums.

In this paper we consider calculations of some characteristics of a vortex wave to find the potential use of a vortex wave mechanism in membrane processes [3]. We present our calculations using the finite control volume method and primitive variables, based on the SIMPLE method [5]. This numerical method has proven credible and produced results which compare closely to the experimental results obtained by particle image velocimetry. It can be showed that the numerical simulations of the unsteady Navier-Stokes equations with simple computation region grids were reliable to the calculations of vortex wave flow field. Details of the flow patterns are presented, which give additional insight into the physical phenomena. The calculation results showed that a vortex wave can be present in the reactor when the Reynolds number (Re) and the Strouhal number (St) are moderate, and the vortex strength enhances as Re increases but the streamwise wavelength increases as St decreases. When the Reynolds number is too small or the Strouhal number is too large to form into vortex waves owing to the domination of the viscosity in the field. When the Reynolds number is too large, it becomes a three-dimensional field. As the Strouhal number is small, the flow can be treated as a quasi-steady field and cannot form a wave of vortices either. The main feature of the vortex wave is that it is a two-dimensional standing wave formed during the deceleration period and the core flow follows a curving path with a sequence of vortices forming alternately on each wall between the core flow and the walls of the channel.

In summary, a substantial increase in hydrodynamic and mass transfer of the vortex wave represents at relatively low Reynolds number, which is an absorbing characteristic in environmental engineering.

1

T.J. Pedley, K.D. Stephanoff, "Flow along a channel with a time-dependent indentation in one wall: the generation of vorticity waves", J Fluid Mech, 1985, 160:337 367 doi:10.1017/S0022112085003512

2

I.J. Sobey, "Observation of waves during oscillatory channel flow", J. Fluid Mech., 1985, 151:395-426 doi:10.1017/S0022112085001021

3

F. Li, "A numerical and experimental study of vortex wave flow", Master's degree Dissertation, DaLian University of Technology,2004.

4

B.M. Deblois, I.J. Sobey and S. Alani, "Characteristics of the vortex wave", J. Fluid Mech. 1993, 253:27-43 doi:10.1017/S0022112093001703

5

S.V. Pantankar, D.B. Spalding, "A calculation procedure for heat, mass and momentum transfer in thre-dimensional parabolic flows", Int. J. Heat Mass Transfer, 1972,15:1787-1806 doi:10.1016/0017-9310(72)90054-3

6

H.R. Millward, B.J. Bellhouse, I.J. Sobey and R.W.H. Lewis, "Enhancement of plasma filtration using the concept of the vortex wave", J. Membr. Sci., 1995,100:121-129 doi:10.1016/0376-7388(94)00257-Y

7

H.R. Millward, B.J. Bellhouse, I.J. Sobey, "The vortex wave membrane bioreactor: hydrodynamics and mass transfer", The biochemical engineering journal, 1996,62:175-181 doi:10.1016/0923-0467(96)03087-4

8

W.Q. Tao, "Numerical method for heat transfer", Xi'an Jiaotong University publishing company, Xi'an, 1995.

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