Bernd R. Noack

A hierarchy of low-dimensional models for the transient and post-transient cylinder wake

A hierarchy of low-dimensional Galerkin models is proposed for the viscous, incompressible flow around a circular cylinder building on the pioneering works of Stuart (1958), Deane et al . (1991), and Ma & Karniadakis (2002). The empirical Galerkin model is based on an eight-dimensional Karhunen–Loeve decomposition of a numerical simulation and incorporates a new ‘shift-mode’ representing the mean-field correction. The inclusion of the shift-mode significantly improves the resolution of the transient dynamics from the onset of vortex shedding to the periodic von Karman vortex street. In addition, the Reynolds-number dependence of the flow can be described with good accuracy. The inclusion of stability eigenmodes further enhances the accuracy of fluctuation dynamics. Mathematical and physical system reduction approaches lead to invariant-manifold and to mean-field models, respectively. The corresponding two-dimensional dynamical systems are further reduced to the Landau amplitude equation.

On the transition of the cylinder wake

The transition of the cylinder wake is investigated experimentally in a water channel and is computed numerically using a finite‐difference scheme. Four physically different instabilities are observed: a local ‘‘vortex‐adhesion mode,’’ and three near‐wake instabilities, which are associated with three different spanwise wavelengths of approximately 1, 2, and 4 diam. All four instability processes can originate in a narrow Reynolds‐number interval between 160 and 230, and may give rise to different transition scenarios. Thus, Williamson’s [Phys. Fluids 31, 3165 (1988)] experimental observation of a hard transition is for the first time numerically reproduced, and is found to be induced by the vortex‐adhesion mode. Without vortex adhesion, a soft onset of three‐dimensionality is numerically and experimentally obtained. A control‐wire technique is proposed, which suppresses transition up to a Reynolds number of 230.

The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows

Low-dimensional empirical Galerkin models are developed for spatially evolving laminar and transitional shear layers, based on a Karhunen–Loeve decomposition of incompressible two- and three-dimensional Navier–Stokes simulations. It is shown that the key to an accurate Galerkin model is a novel analytical pressure-term representation. The effect of the pressure term is elucidated by a modal energy-flow analysis in a mixing layer, which generalizes the framework developed by Rempfer (1991). In convectively unstable shear layers, it is shown in particular that neglecting small energy terms leads to large amplitude errors in the Galerkin model. The effect of the pressure term and small neglected energy flows is very important for a two-dimensional mixing layer, is less pronounced for the three-dimensional analogue, and can be considered as small in an absolutely unstable wake flow.

Feedback shear layer control for bluff body drag reduction

Drag reduction strategies for the turbulent flow around a D-shaped body are examined experimentally and theoretically. A reduced-order vortex model describes the interaction between the shear layer and wake dynamics and guides a path to an efficient feedback control design. The derived feedback controller desynchronizes shear-layer and wake dynamics, thus postponing vortex formation. This actuation is tested in a wind tunnel. The Reynolds number based on the height of the body ranges from 23 000 to 70 000. We achieve a 40% increase in base pressure associated with a 15% drag reduction employing zero-net-mass-flux actuation. Our controller outperforms other approaches based on open-loop forcing and extremum-seeking feedback strategies in terms of drag reduction, adaptivity, and the required actuation energy.

A global stability analysis of the steady and periodic cylinder wake

A global, three-dimensional stability analysis of the steady and the periodic cylinder wake is carried out employing a low-dimensional Galerkin method. The steady flow is found to be asymptotically stable with respect to all perturbations for Re less than 54. The onset of periodicity is confirmed to be a supercritical Hopf bifurcation which can be modeled by the Landau equations. The periodic solution is observed to be only neutrally stable for 54 less than Re less than 170. While two-dimensional perturbations of the vortex street rapidly decay, three-dimensional perturbations with long spanwise wavelengths neither grow nor decay. The periodic solution becomes unstable at Re = 170 by a perturbation with the spanwise wavelength of 1.8 diameters. This instability is shown to be a supercritical Hopf bifurcation in the spanwise coordinate and leads to a three-dimensional periodic flow. Finally the transition scenario for higher Reynolds numbers is discussed.

Reduced-Order Modelling for Flow Control

The book focuses on the physical and mathematical foundations of model-based turbulence control: reduced-order modelling and control design in simulations and experiments. Leading experts provide elementary self-consistent descriptions of the main methods and outline the state of the art. Covered areas include optimization techniques, stability analysis, nonlinear reduced-order modelling, model-based control design as well as model-free and neural network approaches. The wake stabilization serves as unifying benchmark control problem.

Model-based Control of Vortex Shedding Using Low-dimensional Galerkin Models

A model-based flow control strategy is proposed for the suppression of vortex shedding behind a circular cylinder. The control design is based on a hierarchy of low-dimensional Galerkin models of the cylinder wake. These models are constructed from a Karhunen-Loeve decomposition of a simulation without actuation. The key enablers are an additional physical mode in the Karhunen-Loeve approximation and an energy-based control which respects the regime of model validity. The developed control strategy is successfully tested in direct numerical simulations. Copyright  2003 by J. Gerhard, M. Pastoor, R. King, B.R. Noack, A. Dillmann, M. Morzynski and G. Tadmor. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. ∗Research engineer. Corresponding author: phone: ++49-30-314.79574, x24100; fax: x21129; e-mail: johannes.gerhard@tu-berlin.de †Research engineer ‡Professor §Research engineer ¶Professor ‖Professor ∗∗Associate Professor

Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier-Stokes equation

MACIEJ J. BALAJEWICZ1†, EARL H. DOWELL2 AND BERND R. NOACK3 Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA Institut PPRIME, CNRS – Université de Poitiers – ENSMA, UPR 3346, Départment Fluides, Thermique, Combustion, CEAT, 43 rue de l’Aérodrome, F-86036 POITIERS Cedex, France

A low-dimensional Galerkin method for the three-dimensional flow around a circular cylinder

A low‐dimensional Galerkin method for the three‐dimensional flow around a circular cylinder is constructed. The investigation of the wake solutions for a variety of basic modes, Hilbert spaces, and expansion modes reveals general mathematical and physical aspects which may strongly effect the success of low‐dimensional simulations. Besides the cylinder wake, detailed information about the construction of similar low‐dimensional Galerkin methods for the sphere wake, the boundary‐layer, the flow in a channel or pipe, the Taylor–Couette problem, and a variety of other flows is given.

On cell formation in vortex streets

A simple, phenomenological model is proposed for the formation of spanwise cells behind slender bodies of revolution in crosswise, uniform or non-uniform oncoming flow. The model yields estimates for the position of the cells, their frequencies, their amplitudes of oscillation along the span, and the local shedding angle. The qualitative features of the solutions of this theory agree well with experiments. A quantitative comparison with experiments for a slender cone is presented.