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Low-dimensional modelling of a transient cylinder wake using double proper orthogonal decomposition

Published online by Cambridge University Press:  08 August 2008

STEFAN G. SIEGEL ,
JÜRGEN SEIDEL ,
CASEY FAGLEY ,
D. M. LUCHTENBURG ,
KELLY COHEN  and
THOMAS MCLAUGHLIN
STEFAN G. SIEGEL
Affiliation:
Department of Aeronautics, US Air Force Academy, HQ USAFA/DFAN, Colorado Springs, CO 80840, USA
JÜRGEN SEIDEL
Affiliation:
Department of Aeronautics, US Air Force Academy, HQ USAFA/DFAN, Colorado Springs, CO 80840, USA
CASEY FAGLEY
Affiliation:
Department of Aeronautics, US Air Force Academy, HQ USAFA/DFAN, Colorado Springs, CO 80840, USA
D. M. LUCHTENBURG
Affiliation:
Institute of Fluid Dynamics and Technical Acoustics, Berlin University of Technology MB1, Straße des 17 Juni 135, D-10623 Berlin, Germany
KELLY COHEN
Affiliation:
Department of Aeronautics, US Air Force Academy, HQ USAFA/DFAN, Colorado Springs, CO 80840, USA
THOMAS MCLAUGHLIN
Affiliation:
Department of Aeronautics, US Air Force Academy, HQ USAFA/DFAN, Colorado Springs, CO 80840, USA
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Abstract

For the systematic development of feedback flow controllers, a numerical model that captures the dynamic behaviour of the flow field to be controlled is required. This poses a particular challenge for flow fields where the dynamic behaviour is nonlinear, and the governing equations cannot easily be solved in closed form. This has led to many versions of low-dimensional modelling techniques, which we extend in this work to represent better the impact of actuation on the flow. For the benchmark problem of a circular cylinder wake in the laminar regime, we introduce a novel extension to the proper orthogonal decomposition (POD) procedure that facilitates mode construction from transient data sets. We demonstrate the performance of this new decomposition by applying it to a data set from the development of the limit cycle oscillation of a circular cylinder wake simulation as well as an ensemble of transient forced simulation results. The modes obtained from this decomposition, which we refer to as the double POD (DPOD) method, correctly track the changes of the spatial modes both during the evolution of the limit cycle and when forcing is applied by transverse translation of the cylinder. The mode amplitudes, which are obtained by projecting the original data sets onto the truncated DPOD modes, can be used to construct a dynamic mathematical model of the wake that accurately predicts the wake flow dynamics within the lock-in region at low forcing amplitudes. This low-dimensional model, derived using nonlinear artificial neural network based system identification methods, is robust and accurate and can be used to simulate the dynamic behaviour of the wake flow. We demonstrate this ability not just for unforced and open-loop forced data, but also for a feedback-controlled simulation that leads to a 90% reduction in lift fluctuations. This indicates the possibility of constructing accurate dynamic low-dimensional models for feedback control by using unforced and transient forced data only.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 610 , 10 September 2008 , pp. 1 - 42
Copyright
Copyright © Cambridge University Press 2008

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References

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