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Research Papers: Fundamental Issues and Canonical Flows

Nonlinear Control of Axisymmetric Swirling Flows in a Long Finite-Length Pipe

[+] Author and Article Information
Lei Xu

Department of Mechanical, Aerospace,
and Nuclear Engineering,
Rensselaer Polytechnic Institute,
Troy, NY 12180-3590
e-mail: lei.joe.raven@gmail.com

Zvi Rusak

Department of Mechanical, Aerospace,
and Nuclear Engineering,
Rensselaer Polytechnic Institute,
Troy, NY 12180-3590
e-mail: rusakz@rpi.edu

Shixiao Wang

Department of Mathematics,
University of Auckland,
38 Princes Street,
Auckland 1142, New Zealand
e-mail: wang@math.auckland.ac.nz

Steve Taylor

Department of Mathematics,
University of Auckland,
38 Princes Street,
Auckland 1142, New Zealand
e-mail: taylor@math.auckland.ac.nz

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received September 10, 2014; final manuscript received June 10, 2015; published online August 31, 2015. Assoc. Editor: D. Keith Walters.

J. Fluids Eng 138(2), 021201 (Aug 31, 2015) (16 pages) Paper No: FE-14-1503; doi: 10.1115/1.4031255 History: Received September 10, 2014; Revised June 10, 2015; Accepted June 13, 2015

Feedback stabilization of inviscid and high Reynolds number, axisymmetric, swirling flows in a long finite-length circular pipe using active variations of pipe geometry as a function of the evolving inlet radial velocity is studied. The complicated dynamics of the natural flow requires that any theoretical model that attempts to control vortex stability must include the essential nonlinear dynamics of the perturbation modes. In addition, the control methodology must establish a stable desired state with a wide basin of attraction. The present approach is built on a weakly nonlinear model problem for the analysis of perturbation dynamics on near-critical swirling flows in a slightly area-varying, long, circular pipe with unsteady changes of wall geometry. In the natural case with no control, flows with incoming swirl ratio above a critical level are unstable and rapidly evolve to either vortex breakdown states or accelerated flow states. Following an integration of the model equation, a perturbation kinetic-energy identity is derived, and an active feedback control methodology to suppress perturbations from a desired columnar state is proposed. The stabilization of both inviscid and high-Re flows is demonstrated for a wide range of swirl ratios above the critical swirl for vortex breakdown and for large-amplitude initial perturbations. The control gain for the fastest decay of perturbations is found to be a function of the swirl level. Large gain values are required at near-critical swirl ratios while lower gains provide a successful control at swirl levels away from critical. This feedback control technique cuts the feed-forward mechanism between the inlet radial velocity and the growth of perturbation's kinetic energy in the bulk and thereby enforces the decay of perturbations and eliminates the natural explosive evolution of the vortex breakdown process. The application of this proposed robust active feedback control method establishes a branch of columnar states with a wide basin of attraction for swirl ratios up to at least 50% above the critical swirl. This study provides guidelines for future flow control simulations and experiments. However, the present methodology is limited to the control of high-Reynolds number (nearly inviscid), axisymmetric, weakly nonparallel flows in long pipes.

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Figures

Grahic Jump Location
Fig. 1

Failed feedback stabilization of a Lamb–Oseen vortex with ω=0.895, δ=−36, and γ = 2: (a) the time-history of perturbation's maximum (dashed-dotted line), minimum (solid line), and outlet (dashed line) values and (b) pipe contraction ratio λ as a function of t*

Grahic Jump Location
Fig. 9

The time-history of the perturbation's energy-production rate terms and the total energy production rate; ω=0.87, δ=−36, and γ=−10

Grahic Jump Location
Fig. 10

Feedback stabilization of a Lamb–Oseen vortex with ω=0.8, δ=−36, and γ=−0.75: (a) the time-history of perturbation's maximum (dashed-dotted line), minimum (solid line), and outlet (dashed line) values and (b) pipe contraction ratio λ as a function of t*

Grahic Jump Location
Fig. 11

The time-history of the perturbation's energy-production rate terms and the total energy production rate; ω=0.8, δ=−36, and γ=−0.75

Grahic Jump Location
Fig. 13

Feedback stabilization of a Lamb–Oseen vortex with ω = 1, Re = 16,000, A(X,0)=−36 sin(πX/2), γ = 2, kν=0.9: (a) the time-history of perturbation maximum (dashed-dotted line), minimum (solid line), and outlet (dashed line), (b) the pipe contraction ratio λ as a function of time t*, and (c) controlled time-asymptotic state

Grahic Jump Location
Fig. 14

Feedback stabilization of a Lamb–Oseen vortex with ω = 1, Re = 16,000, A(X,0)=−36 sin(πX/2), γ = 2, kν=1.1 : (a) the time-history of perturbation maximum (dashed-dotted line), minimum (solid line), and outlet (dashed line), (b) the pipe contraction ratio λ as a function of time t* and (c) controlled time-asymptotic state

Grahic Jump Location
Fig. 2

Feedback stabilization of a Lamb–Oseen vortex with ω=0.895, δ=−36, and γ = 10: (a) the time-history of perturbation's maximum (dashed-dotted line), minimum (solid line), and outlet (dashed line) values and (b) pipe contraction ratio λ as a function of t*

Grahic Jump Location
Fig. 3

The time-history of the perturbation's energy-production terms and the total energy production rate; ω=0.895, δ=−36, and γ = 10

Grahic Jump Location
Fig. 4

Feedback stabilization of a Lamb–Oseen vortex with ω=1.3, δ=−36, and γ = 2: (a) the time-history of perturbation's maximum (dashed-dotted line), minimum (solid line), and outlet (dashed line) values and (b) pipe contraction ratio λ as a function of t*

Grahic Jump Location
Fig. 5

The time-history of the perturbation's energy-production rate terms and the total energy production rate; ω=1.3, δ=−36, and γ = 2

Grahic Jump Location
Fig. 6

Feedback stabilization of a Lamb–Oseen vortex with ω=1.3, δ=−36, and γ = 10: (a) the time-history of perturbation maximum (dashed-dotted line), minimum (solid line), and outlet (dashed line) values and (b) pipe contraction ratio λ as a function of t*

Grahic Jump Location
Fig. 7

The time-history of the perturbation's energy-production rate terms and the total energy production rate; ω=1.3, δ=−36, and γ = 10

Grahic Jump Location
Fig. 8

Feedback stabilization of a Lamb–Oseen vortex with ω=0.87, δ=−36, and γ=−10: (a) the time-history of perturbation's maximum (dashed-dotted line), minimum (solid line), and outlet (dashed line) values and (b) pipe contraction ratio λ as a function of t*

Grahic Jump Location
Fig. 12

(a) The swirling flow steady-state As(X) at ω=0.8 and Re = 16,000 and (b) the time-history of perturbation's maximum (dashed-dotted line), minimum (solid line), and outlet (dashed line) values when ω=0.8, Re = 16,000, γ = 0, and λ(t*)=−0.0031

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