"An investigation is made of the linear stability of the developing flow of an incompressible fluid in the entrance region of annular ducts, circular tubes, and parallel-plate channels. Small axisymmetric disturbances for annular duct and tube flows and small two-dimensional disturbances for channel flow are considered in the analysis. In formulating the stability problems, account is taken of the transverse velocity component of the mainflow. This results in the modified Orr-Sommerfeld equations, one for annular duct and tube flows and the other for channel flow. The mainflow velocity fields utilized in the stability analysis are those from the solutions of the linearized momentum equations.
The governing equation for the disturbances and the boundary conditions for each of the flow configurations constitute an eigenvalue problem. The eigenvalue problems for the annular duct and circular tube flows are solved by a fourth order Runge-Kutta integration scheme along with a differential correction iteration technique. An orthonormalization process is used to remove the "parasitic error" inherent in the numerical integration of the disturbance equations. For flow in the parallel-plate channels, the eigenvalue problem is solved by a finite difference method and the differential correction iteration scheme is employed to obtain the eigenvalues.
Neutral stability characteristics and critical Reynolds numbers at various axial locations are obtained for the developing flow in the annular ducts with radius ratios of 2.0 and 3.33, in the circular tubes, and in the parallel-plate channels, using the modified Orr-Sommerfeld equations. These stability results for the annular duct flow are also computed using the conventional Orr-Sommerfeld equation. Representative eigenfunctions are also presented for the annular duct flow. Comparisons of the results from the modified Orr-Sommerfeld equations are made with those from the conventional equations for all three flow configurations.
The main findings of the present study are: (1) laminar flow in the entrance region of annular ducts, circular tubes, and parallel-plate channels is unstable to small axisymmetric or two-dimensional disturbances; (2) the critical Reynolds number for the developing flow in the annular ducts and parallel-plate channels decreases monotonically as the axial distance increases; (3) the flow in the annular ducts becomes more stable as the ratio of the outer to inner radius increases; (4) the minimum critical Reynolds numbers for annular duct flow occur in the fully developed flow region and have the values of 9720 and 40530, respectively, for radius ratios of 2.0 and 3.33; (5) the minimum critical Reynolds number for tube flow is about 19780 and occurs in the entrance region; (6) the modified Orr-Sommerfeld equation provides critical Reynolds numbers that differ somewhat from those obtained from the conventional equation; and (7) the effect of non-parallelism of the mainflow (that is, the effect of the mainflow transverse velocity) on the stability characteristics of the developing flow in ducts is of significance only in the range of small Reynolds numbers"--Abstract, pages ii-iv.
Chen, T. S.
Faucett, T. R.
Rhea, L. G.
Ho, C. Y. (Chung You), 1933-1988
Penico, Anthony J., 1923-2011
Howell, Ronald H. (Ronald Hunter), 1935-
Mechanical and Aerospace Engineering
Ph. D. in Mechanical Engineering
University of Missouri--Rolla
xv, 128 pages
© 1973 Francis Chung-ti Shen, All rights reserved.
Dissertation - Open Access
Flows (Differentiable dynamical systems)
Laminar flow -- Mathematical models
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Shen, Francis Chung-ti, "Linear stability of non-parallel flow in the entrance region of ducts" (1973). Doctoral Dissertations. 238.