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Research Papers: Fundamental Issues and Canonical Flows

The Linear Stability Analysis of the Lamb–Oseen Vortex in a Finite-Length Pipe

[+] Author and Article Information
S. Wang1

Department of Mathematics, University of Auckland, Auckland 1142, New Zealandwang@math.auckland.ac.nz

S. Taylor

Department of Mathematics, University of Auckland, Auckland 1142, New Zealandtaylor@math.auckland.ac.nz

K. Ku Akil

Department of Mathematics and Statistics, University of Teknologi MARA, Arau Perlis o2600, Malaysia

1

Corresponding author.

J. Fluids Eng 132(3), 031201 (Mar 17, 2010) (13 pages) doi:10.1115/1.4001106 History: Received August 03, 2009; Revised December 15, 2009; Published March 17, 2010; Online March 17, 2010

This article extends the perturbation method introduced by Wang (2008, “A Novel Method for Analyzing the Global Stability of Inviscid Columnar Swirling Flow in a Finite Pipe,” Phys. Fluids, 20(7), p. 074101) to determine the global stability of a swirling flow in a straight circular pipe with specified inlet and outlet boundary conditions. To accurately compute the flow stability characteristics, a general procedure to treat the complexity arising from high-order terms is developed. It extends the previous fourth-order method to an eighth-order method. The technique is first applied to the benchmark case of a solid-body rotation flow with a uniform axial speed. It is demonstrated that the eighth-order method is sufficient to construct the growth rate curve between the first and second critical swirls of this flow. Note that this range of swirl is crucial for the study of the vortex breakdown phenomenon since the base flow is unstable and starts the initial stage of transition to a breakdown state. The method is then applied to the Lamb–Oseen vortex to construct the growth rate curve between the first and second critical swirls of this flow. Calculated results are compared with the growth rate curve computed from direct numerical simulations and an overall agreement between the two computations is found. This demonstrates that the Wang and Rusak (1996, “On the Stability of an Axisymmetric Rotating Flow in a Pipe,” Phys. Fluids, 8(4), pp. 1007–1076) instability mechanism captures quantitatively the initial growth of disturbance, which eventually evolves into a breakdown state.

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Figures

Grahic Jump Location
Figure 2

The convergence history of the approximations from the fifth-order method to the eighth-order method at σ≈0.0003

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Figure 3

The comparison of Δ1,1Ω(σ) (dashed line − − denotes eighth-order (a)), Δ1,2Ω(σ) (dashed line − − denotes eighth-order (a)), σ1(Δ1Ω) (plus sign + denotes eighth-order (b1)), and σ2(Δ2Ω) (diamond ◇ denotes eighth-order (b2)). The solid line indicates the exact solution.

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Figure 4

The comparison of the approximation by using the original eighth-order Taylor expansion Δ1,2Ω(σ) (dashed line − − denotes eighth-order (a)) and the approximations by using the new approach with a=300,150,80. The solid line indicates the exact solution.

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Figure 5

The construction of the growth rate curve by jointly using the improved first branch (○) and second branch (⋅) of approximations. The solid line indicates the exact solution.

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Figure 1

The first-order to eighth-order approximations of the growth rate curves between first and second critical swirls. The smooth solid black curve indicates the exact solution. (a): first-order to fourth-order approximations. (b): fifth-order to eighth-order approximations. The asterisk is the branching point (Ω∗,σ∗).

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Figure 6

The first-order to eighth-order approximations of the growth rate curves between first and second critical swirls. (a): first-order to fourth-order approximations. (b): fifth-order to eighth-order approximations.

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Figure 7

The eighth-order approximations of the growth rate curve by using different approaches: Δ1,1Ω(σ) (solid line denotes eighth-order (a)), Δ1,2Ω(σ) (solid line denotes eighth-order (a)), σ1(Δ1Ω) (plus sign + denotes eighth-order (b1)), and σ2(Δ2Ω) (diamond ◇ denotes eighth-order (b2))

Grahic Jump Location
Figure 8

The comparison of the approximation by using the original eighth-order Taylor expansion Δ1,2Ω(σ) (dashed line − − denotes eighth-order (a)) and the approximations by using the new approach with a=10,4,2.57. The first branch approximation σ1(Δ1Ω) (solid line denotes eighth-order (b)) is used in the matching process to optimize the second branch approximation.

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Figure 9

Reconstruction of the whole growth rate curve: the solid line is the approximated growth rate curve and the diamond (◇) is the direct numerical simulation result. The asterisk is the branching point.

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