Stability of Developing Flow in a Pipe—Nonparallel Effects

[+] Author and Article Information
V. K. Garg, S. C. Gupta

Department of Mechanical Engineering, Indian Institute of Technology Kanpur, P.O. IIT Kanpur 208016, India

J. Appl. Mech 48(2), 243-248 (Jun 01, 1981) (6 pages) doi:10.1115/1.3157604 History: Received March 01, 1980; Revised August 01, 1980; Online July 21, 2009


A theoretical investigation into the linear, spatial instability of the developing flow in a rigid circular pipe, incorporating the effects of nonparallelism of the main flow, has been made at several axial locations. The velocity profile in the developing flow region is obtained by a finite-difference method assuming uniform flow at the entry to the pipe. For the stability analysis, the continuity and momentum equations have been integrated separately using fourth-order Runge-Kutta integration scheme and applying selectively the Gram-Schmidt orthonormalization procedure to circumvent the parasitic error-growth problem. It is found that the critical frequency, obtained from different growth rates, decreases first sharply and then gradually with increasing X , where X = x/aR = X/R; x being the streamwise distance measured from the pipe inlet, a being the radius of the pipe, and R the Reynolds number based on a and average velocity of flow. However, the critical Reynolds number versus X curves pass through a minima. The minimum critical Reynolds number corresponding to gψ(X , O), the growth rate of stream function at the pipe axis, to gE (X ), the growth rate of energy density, and to the parallel flow theory are 9700 at X = 0.00325, 11,000 at X = 0.0035, and 11,700 at X = 0.0035, respectively. It is found that the actual developing flow remains unstable over a larger inlet length of the pipe than its parallel-flow approximate. The first instability of the flow on the basis of gψ(X , O), gE (X ) and the parallel flow theory, is found to occur in the range 30 ≤ X ≤ 36, 35 ≤ X ≤ 43, and 36 ≤ X ≤ 45, respectively. The critical Reynolds numbers obtained on the basis of gψ(X , O) are closest to the experimental values.

Copyright © 1981 by ASME
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