CLC number: O35

On-line Access: 2009-11-30

Received: 2008-11-11

Revision Accepted: 2009-03-06

Crosschecked: 2009-11-05

Cited: 1

Clicked: 5491

Zhao-sheng YU, Xue-ming SHAO, Jian-zhong LIN. Numerical computations of the flow in a finite diverging channel[J]. Journal of Zhejiang University Science A, 2010, 11(1): 50-60.

@article{title="Numerical computations of the flow in a finite diverging channel",

author="Zhao-sheng YU, Xue-ming SHAO, Jian-zhong LIN",

journal="Journal of Zhejiang University Science A",

volume="11",

number="1",

pages="50-60",

year="2010",

publisher="Zhejiang University Press & Springer",

doi="10.1631/jzus.A0800782"

}

%0 Journal Article

%T Numerical computations of the flow in a finite diverging channel

%A Zhao-sheng YU

%A Xue-ming SHAO

%A Jian-zhong LIN

%J Journal of Zhejiang University SCIENCE A

%V 11

%N 1

%P 50-60

%@ 1673-565X

%D 2010

%I Zhejiang University Press & Springer

%DOI 10.1631/jzus.A0800782

TY - JOUR

T1 - Numerical computations of the flow in a finite diverging channel

A1 - Zhao-sheng YU

A1 - Xue-ming SHAO

A1 - Jian-zhong LIN

J0 - Journal of Zhejiang University Science A

VL - 11

IS - 1

SP - 50

EP - 60

%@ 1673-565X

Y1 - 2010

PB - Zhejiang University Press & Springer

ER -

DOI - 10.1631/jzus.A0800782

**Abstract: **The flow in a finite diverging channel opening into a large space and resembling the experimental prototype of Putkaradze and Vorobieff (2006) was numerically investigated. The effects of the Reynolds number, initial condition, intersection angle, length of the wedge edges, and the outer boundary condition were examined. The numerical results showed that the flow in the wedge undergoes a change from symmetrical flow to unsymmetrical flow with a weak backflow, then a vortical (circulation) flow and finally an unsteady jet flow as the Reynolds number is increased for an intersection angle of 32( and a wedge edge of length 30 times the width of the inlet slit. For the unsteady flow, the jet attached to one side of the wedge constantly loses stability and rolls up into a mushroom-shaped vortex-pair near the outlet of the wedge. As the intersection angle is increased to 50(, a stable jet flow is observed as a new regime between the vortex and unsteady regimes. Both the intersection angle and the wedge length have negative effects on the stability of the flow, although the effect of the wedge length on the critical Reynolds number for the symmetry-breaking instability is not pronounced. The outer boundary condition was found not to affect the flow patterns inside the wedge significantly. At a certain

**
**

[1] Allmen, M.J., Eagles, P.M., 1984. Stability of divergent channel flows: a numerical approach. *Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences*, **392**(1803):359-372.

[2] Banks, W.H.H., Drazin, P.G., Zaturska, M.B., 1988. On perturbations of Jeffery-Hamel flow. *Journal of Fluid Mechanics*, **186**:559-581.

[3] Boutros, Y.Z., Abd-el-Malek, M.B., Badran, N.A., Hassan, H.S., 2007. Lie-group method solution for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. *Applied Mathematical Modelling*, **31**(6):1092-1108.

[4] Burde, G.I., Nasibullayev, I.S., Zhalij, A., 2007. Stability analysis of a class of unsteady nonparallel incompressible flows via separation of variables. *Physics of Fluids*, **19**(11):114110.

[5] Dennis, S.C.R., Banks, W.H.H., Drazin, P.G., Zaturska, M.B., 1997. Flow along a diverging channel. *Journal of Fluid Mechanics*, **336**:183-202.

[6] Eagles, P.M., 1988. Jeffery-Hamel boundary-layer flows over curved beds. *Journal of Fluid Mechanics*, **186**:583-597.

[7] Fraenkel, L.E., 1962. Laminar flow in symmetrical channels with slightly curved walls. I. On the Jeffery-Hamel solutions for flow between plane walls. *Proceedings of the Royal Society London A*, **267**:119-138.

[8] Georgiou, G.A., Eagles, P.M., 1985. The stability of flows in channels with small wall curvature. *Journal of Fluid Mechanics*, **159**:259-287.

[9] Glowinski, R., Pan, T.W., Hesla, T.I., Joseph, D.D., 1999. A distributed Lagrange multiplier/fictitious domain method for particulate flows. *International Journal of Multiphase Flow*, **25**(5):755-794.

[10] Glowinski, R., Pan, T.W., Hesla, T.I., Joseph, D.D., Periaux, J., 2001. A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: Application to particulate flow. *Journal of Computational Physics*, **169**(2):363-426.

[11] Goldshtik, M., Hussain, F., Shtern, V., 1991. Symmetry breaking in vortex-source and Jeffery-Hamel flows. *Journal of Fluid Mechanics*, **232**:521-566.

[12] Hamadiche, M., Scott, J., Jeandel, D., 1994. Temporal stability of Jeffery-Hamel flow. *Journal of Fluid Mechanics*, **268**:71-88.

[13] Hamel, G., 1916. Spiralförmige Bewegungen zäher Flüssigkeiten. Jahrber. *Deutsch. Math. Ver*., **25**:34-60.

[14] Hooper, A.P., Duffy, B.R., Moffatt, H.K., 1982. Flow of fluid of non-uniform viscosity in converging and diverging channels. *Journal of Fluid Mechanics*, **117**:283-304.

[15] Hwang, W.R., Hulsen, M.A., 2006. Direct numerical simulations of hard particle suspensions in planar elongational flow. *Journal of Non-Newtonian Fluid Mechanics*, **136**(2-3):167-178.

[16] Jeffery, G.B., 1915. The two-dimensional steady motion of a viscous fluid. *Philosophical Magazine Series 6*, **29**:455.

[17] Kerswell, R.R., Tutty, O.R., Drazin, P.G., 2004. Steady nonlinear waves in diverging channel flow. *Journal of Fluid Mechanics*, **501**:231-250.

[18] Landau, L.D., Lifshitz, E.M., 1987. Fluid Mechanics. Pergamon Press, Oxford.

[19] Majdalani, J., Zhou, C., 2003. Moderate-to-large injection and suction driven channel flows with expanding or contracting walls. *ZAMM*, **83**(3):181-196.

[20] McAlpine, A., Drazin, P.G., 1998. On the spatio-temporal development of small perturbations of Jeffery-Hamel flows. *Fluid Dynamics Research*, **22**(3):123-138.

[21] Nakayama, Y. (Ed.), 1988. Visualized Flow. Pergamon Press, Oxford.

[22] Pan, T.W., Glowinski, Joseph, D.D., 2005. Simulating the dynamics of fluid-cylinder interactions. *Journal of Zhejiang University SCIENCE A*, **6**:97-109.

[23] Pan, T.W., Glowinski, R., Hou, S.C., 2007. Direct numerical simulation of pattern formation in a rotating suspension of non-Brownian settling particles in a fully filled cylinder. *Computers and Structures*, **85**(11-14):955-969.

[24] Pan, T.W., Chang, C.C., Glowinski, R., 2008. On the motion of a neutrally buoyant ellipsoid in a three-dimensional Poiseuille flow. *Computer Methods in Applied Mechanics and Engineering*, **197**(25-28):2198-2209.

[25] Putkaradze, V., Vorobieff, P., 2006. Instabilities, bifurcations, and multiple solutions in expanding channel flows. *Physical Review Letters*, **97**(14):144502.

[26] Schlichting, H., Gersten, K., 2000. Boundary Layer Theory (8th Ed.). Springer.

[27] Shao, X., Yu, Z., Sun, B., 2008. Inertial migration of spherical particles in circular Poiseuille flow at moderately high Reynolds numbers. *Physics of Fluids*, **20**(10):103307.

[28] Sobey, I.J., Drazin, P.G., 1986. Bifurcations of two-dimensional channel flows. *Journal of Fluid Mechanics*, **171**:263-287.

[29] Stow, S.R., Duck, P.W., Hewitt, R.E., 2001. Three-dimensional extensions to Jeffery-Hamel flow. *Fluid Dynamics Research*, **29**(1):25-46.

[30] Tutty, O.R., 1996. Nonlinear development of flow in channels with non-parallel walls. *Journal of Fluid Mechanics*, **326**:263-284.

[31] Uribe, F.J., Daz-Herrera, E., Bravo, A., Peralta-Fabi, R., 1997. On the stability of the Jeffery-Hamel flow. *Physics of Fluids*, **9**(9):2798-2800.

[32] Veeramani, C., Minev, P.D., Nandakumar, K., 2007. A fictitious domain formulation for flows with rigid particles: A non-Lagrange multiplier version. *Journal of Computational Physics*, **224**(2):867-879.

[33] Yu, Z., Shao, X., 2007. A direct-forcing fictitious domain method for particulate flows. *Journal of Computational Physics*, **227**(1):292-314.

[34] Yu, Z., Shao, X., Wachs, A., 2006a. A fictitious domain method for particulate flows with heat transfer. *Journal of Computational Physics*, **217**(2):424-452.

[35] Yu, Z., Wachs, A., Peysson, Y., 2006b. Numerical simulation of particle sedimentation in shear-thinning fluids with a fictitious domain method. *Journal of Non-Newtonian Fluid Mechanics*, **136**(2-3):126-139.

Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou
310027, China

Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn

Copyright © 2000 - 2022 Journal of Zhejiang University-SCIENCE

Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn

Copyright © 2000 - 2022 Journal of Zhejiang University-SCIENCE

Open peer comments: Debate/Discuss/Question/Opinion

<1>