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Research Papers: Techniques and Procedures

A Numerical Simulation Algorithm of the Inviscid Dynamics of Axisymmetric Swirling Flows in a Pipe

[+] Author and Article Information
J. Granata, L. Xu, Z. Rusak

Department of Mechanical,
Aerospace, and Nuclear Engineering,
Rensselaer Polytechnic Institute,
Troy, NY 12180-3590

S. Wang

Department of Mathematics,
University of Auckland,
38 Princes Street,
Auckland 1142, New Zealand

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received September 22, 2015; final manuscript received March 16, 2016; published online June 6, 2016. Assoc. Editor: Praveen Ramaprabhu.

J. Fluids Eng 138(9), 091402 (Jun 06, 2016) (14 pages) Paper No: FE-15-1682; doi: 10.1115/1.4033321 History: Received September 22, 2015; Revised March 16, 2016

Current simulations of swirling flows in pipes are limited to relatively low Reynolds number flows (Re < 6000); however, the characteristic Reynolds number is much higher (Re > 20,000) in most of engineering applications. To address this difficulty, this paper presents a numerical simulation algorithm of the dynamics of incompressible, inviscid-limit, axisymmetric swirling flows in a pipe, including the vortex breakdown process. It is based on an explicit, first-order difference scheme in time and an upwind, second-order difference scheme in space for the time integration of the circulation and azimuthal vorticity. A second-order Poisson equation solver for the spatial integration of the stream function in terms of azimuthal vorticity is used. In addition, when reversed flow zones appear, an averaging step of properties is applied at designated time steps. This adds slight artificial viscosity to the algorithm and prevents growth of localized high-frequency numerical noise inside the breakdown zone that is related to the expected singularity that must appear in any flow simulation based on the Euler equations. Mesh refinement studies show agreement of computations for various mesh sizes. Computed examples of flow dynamics demonstrate agreement with linear and nonlinear stability theories of vortex flows in a finite-length pipe. Agreement is also found with theoretically predicted steady axisymmetric breakdown states in a pipe as flow evolves to a time-asymptotic state. These findings indicate that the present algorithm provides an accurate prediction of the inviscid-limit, axisymmetric breakdown process. Also, the numerical results support the theoretical predictions and shed light on vortex dynamics at high Re.

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Figures

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Fig. 3

A snapshot of the flow evolution for ω = 0.95 and δ = −4 at (a) t = 50, (b) t ∼ 90, and at (c) time-asymptotic state, for meshes 100 × 100, 150 × 150, 200 × 200, and 300 × 300. The results from all meshes are essentially the same for all time.

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Fig. 2

Time history of Amin (solid line), Aout (dashed line), and Amax (dotted line) computed from the meshes 100 × 100, 150 × 150, 200 × 200, and 300 × 300; for ω = 0.95 and δ = −4

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Fig. 4

Comparison of the profile of stream function at the outlet from four meshes 100 × 100, 150 × 150, 200 × 200, 300 × 300 and the theoretical prediction of Wang and Rusak [17] (heavy line); for ω = 0.95

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Fig. 5

Evolution of σ* as a function of t* for meshes 100 × 100 and 200 × 200. Also shown for reference are the linear stability growth rate according to Wang and Rusak [16] (dashed–dotted line) and the growth rate according to the weakly nonlinear model of Rusak et al. [21] (dashed line); for ω = 1.1 and δ = −4.

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Fig. 9

Stream function contour-line snapshots of flow evolution from a perturbed columnar state to a breakdown state when ω = 0.8 and δ = −15 at (a) t = 0, (b) t = 66, (c) t = 85, (d) t = 105, (e) t = 134, (f) t = 155 (time-asymptotic state)

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Fig. 10

Stream function contour-lines of time-asymptotic breakdown states at (a) ω = 0.8, (b) ω = 0.78, (c) ω = 0.76, and (d) ω = 0.74

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Fig. 11

Stream function contour-line snapshots of flow evolution from a breakdown state at ω = 0.74 to a columnar state at ω = 0.735 at (a) t = 0, (b) t = 80, (c) t = 180, and at (d) time-asymptotic state

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Fig. 1

Evolution of σ* as a function of t* for meshes 100 × 100, 150 × 150, 200 × 200, and 300 × 300. Also shown for reference are the linear stability growth rate according to Wang and Rusak [16] (dashed–dotted line) and the growth rate according to the weakly nonlinear model of Rusak et al. [21] (dashed line); for ω = 0.95 and δ = −4.

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Fig. 12

Stream function contour-line snapshots of flow evolution from a near-columnar state to a breakdown state when ω = 1.2 and δ = −4 at (a) t = 0, (b) t = 10, (c) t = 25, (d) t = 40, (e) t = 60 and at (f) time-asymptotic state

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Fig. 13

Comparison between the numerical results and predictions according to the theory of vortex breakdown

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Fig. 6

Time history of Amin (solid line), Aout (dashed line), and Amax (dotted line) computed from the meshes 100 × 100 and 200 × 200; for ω = 1.1 and δ = −4

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Fig. 7

Snapshots of the flow evolution for ω = 1.1 and δ = −4 at (a) t = 15 and at (b) time-asymptotic state, for meshes 100 × 100 and 200 × 200. The results from all meshes are essentially the same for all time.

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Fig. 8

Comparison of the profile of stream function at the outlet from two meshes 100 × 100 (circles), 200 × 200 (triangles) and the theoretical prediction of Wang and Rusak [17] (solid line); for ω = 1.1

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