Stability of Developing Flow in a Pipe—Nonparallel Effects

Abstract

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.