Transient Solution of an Incompressible Viscous Flow in a Channel with Sudden Expansion/Contraction

In this paper, a numerical study has been made to analyze the transient 2-D flows of a viscous incompressible fluid through channels with forward or backward constriction. Problems addressed include flow through sudden contraction and sudden expansion channel geometries with rounded and increasingly sharp reentrant corner. In both the cases, numerical results are presented for the separation and reattachment points, streamlines, vorticity and flow patterns. A fourth order accurate compact scheme has been employed to efficiently capture steady state solutions of the governing equations. It appears from our study that sharpness of the throat in the channel is one of the important parameters to control the strength and size of the separation zone without modifying the general flow patterns. The comparison between the two cases shows that the upstream geometry plays a significant role on vortex growth dynamics.

Authors:

• Durga C. Dalal
• Swapan K. Pandit

Keywords:

• Forward and backward constriction
• HOC scheme
• Incompressible viscous flows
• Separation and reattachment points.

References:

[1] B. F. Armaly, F. Durst, J. C. F. Pereira and B. Schonung, "Experimental
and theoretical investigation of backward-facing step flow," Journal of
Fluid Mechanics, vol. 127, pp. 473-496, Feb. 1983.
[2] F. Durst, A. Melling and J. H. Whitelaw, "Low Reynolds number flow
over a plane symmetrical sudden expansion," Journal of Fluid
Mechanics, vol. 64, no. 1, pp. 111-128, June 1974.
[3] O. R. Tutty and T. J. Pedley, "Oscillatory flow in a stepped channel,
Journal of Fluid Mechanics," vol. 247, pp. 179-204, Feb. 1993.
[4] I. J. Sobey, "Observation of waves during oscillatory channel flow,"
Journal of Fluid Mechanics, vol. 151, pp. 395-426, Feb. 1985.
[5] W. Cherdron, F. Durst and J. H. Whitelaw, "Asymmetric flows and
instabilities in symmetric ducts with sudden expansions," Journal of
Fluid Mechanics, vol. 84, no. 1, pp. 13-31, Jan. 1978.
[6] F. Durst and J. C. F. Pereira, "Time dependent laminar backward-facing
step flow in a two dimensional duct," Trans. ASME: Journal of Fluids
Engineering, vol. 110, no. 3, pp. 289-296, Sept. 1988.
[7] F. Durst, J. C. F. Pereira and C. Tropea, "The plane symmetric sudden
expansion flow at low Reynolds numbers," Journal of Fluid Mechanics,
vol. 248, pp. 567-581, March 1993.
[8] I. J. Sobey and P. G. Drazin, "Bifurcations of two dimensional channel
flows," Journal of Fluid Mechanics, vol. 171, pp. 263-287, Oct. 1986.
[9] R. Wille, and H. Fernholz, "Report on the first European Mechanics
Colloquium on the Coanda effect," Journal of Fluid Mechanics, vol. 23,
no. 4, pp. 801-819, Dec. 1965.
[10] M. S. Borgas and T. J. Pedley, "Non-uniqueness and bifurcation in
annular and planar channel flows," Journal of Fluid Mechanics, vol.
214, pp. 229-250, May 1990.
[11] M. Shapira, D. Degani and D. Weihs, "Stability and existence of
multiple solutions for viscous flow in suddenly enlarged channels,"
Computers and Fluids, vol. 18, no. 3, pp. 239-258, 1990.
[12] G. Pedrizetti, "Unsteady tube flow over an expansion," Journal of Fluid
Mechanics, vol. 310, no. 5, pp. 89-111, March 1996.
[13] P. F. A. Mancera and R. Hunt, "Fourth order method for solving the
navier-stokes equations in a constricted channel," International Journal
of Numerical Methods in Fluids, vol 25, no. 11, pp. 1119-1135, June
1997.
[14] P. F. A. Mancera, "A study of numerical solution of the steady two
dimensional navier-stokes equations in a constricted channel problem by
a compact fourth order method," Applied Mathematics and
Computation, vol. 146, no. 2-3, pp. 771-790, Dec. 2003.
[15] S. K. Pandit, J. C. Kalita and D. C. Dalal, "A transient higher order
compact scheme for incompressible viscous flows on geometries beyond
rectangular," Journal of Computational Physics, vol. 225, no. 1 pp.
1100-1124, July 2007.
[16] J. C. Tannehill, D. A. Anderson and R. H. Pletcher, Computational Fluid
Mechanics and Heat Transfer, Hemisphere Publishing Corporation,
New York, 1984.
[17] C. T. Kelley, Iterative methods for Linear and Nonlinear Equations,
SIAM Publications, Philadelphia, 1995.
[18] Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing
Company, 1996.
[19] H. V. D. Vorst, "BiCGSTAB: A fast and smoothly converging variant of
BiCG for the solution of nonsymmetric linear systems," SIAM J. Sci.
Comput., vol. 13, no. 2, pp. 631-644, 1992.