Regelung aerodynamischer Strömungen am Beispiel einer rückwärts gewandten Stufe (Robust Control of a Backward-Facing Step Flow)

AbstractMethods for investigating and approximating the linear dynamics of amplifier flows are examined in this paper. The procedures are derived for incompressible flow over a two-dimensional backward-facing step. First, the singular value decomposition of the resolvent is performed over a frequency range in order to identify the optimal and suboptimal harmonic forcing and responses of the flow. These forcing/responses are shown to be organized into two categories: the first accounting for the Orr and Kelvin–Helmholtz instabilities in the shear layer and the second for the advection and diffusion of perturbations in the free stream. Next, we investigate the dynamics of the flow when excited by a white in space and time noise. We compute the predominant patterns of the random flow which optimally account for the sustained variance, the empirical orthogonal functions (EOFs), as well as the predominant forcing structures which optimally contribute to the sustained variance, the stochastic optimals (SOs). The leading EOFs and SOs are expressed as a linear combination of the suboptimal forcing and responses of the flow and are related to particular instability mechanisms and/or frequency intervals. Finally, we use the leading EOFs, SOs and balanced modes (obtained from balanced truncation) to build low-order models of the flow dynamics. These models are shown to accurately recover the time propagator and resolvent of the original dynamical system. In other words, such models capture the entire flow response from any forcing and may be used in the design of efficient closed-loop controllers for amplifier flows.

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Time-Dependent Laminar Backward-Facing Step Flow in a Two-Dimensional Duct

Journal of Fluids Engineering ◽

10.1115/1.3243547 ◽

1988 ◽

Vol 110 (3) ◽

pp. 289-296 ◽

Cited By ~ 42

Author(s):

F. Durst ◽

J. C. F. Pereira

Keyword(s):

Reynolds Number ◽

Steady State ◽

Separated Flow ◽

Flow Region ◽

Reynolds Numbers ◽

Steady State Flow ◽

Backward Facing Step ◽

Step Flow ◽

Recirculating Flow ◽

State Flow

This paper presents results of numerical studies of the impulsively starting backward-facing step flow with the step being mounted in a plane, two-dimensional duct. Results are presented for Reynolds numbers of Re = 10; 368 and 648 and for the last two Reynolds numbers comparisons are given between experimental and numerical results obtained for the final steady state flow conditions. In the computational scheme, the convective terms in the momentum equations are approximated by a 13-point quadratic upstream weighted finite-difference scheme and a fully implicit first order forward differencing scheme is used to discretize the temporal derivatives. The computations show that for the higher Reynolds numbers, the flow starts to separate on the lower and upper corners of the step yielding two disconnected recirculating flow regions for some time after the flow has been impulsively started. As time progresses, these two separated flow regions connect up and a single recirculating flow region emerges. This separated flow region stays attached to the step, grows in size and approaches, for the time t → ∞, the dimensions measured and predicted for the separation region for steady laminar backward-facing flow. For the Reynolds number Re = 10 the separation starts at the bottom of the backward-facing step and the separation region enlarges with time until the steady state flow pattern is reached. At the channel wall opposite to the step and for Reynolds number Re = 368, a separated flow region is observed and it is shown to occur for some finite time period of the developing, impulsively started backward-facing step flow. Its dimensions change with time and reduce to zero before the steady state flow pattern is reached. For the higher Reynolds number Re = 648, the secondary separated flow region opposite to the wall is also present and it is shown to remain present for t → ∞. Two kinds of the inlet conditions were considered; the inlet mean flow was assumed to be constant in a first study and was assumed to increase with time in a second one. The predicted flow field for t → ∞ turned out to be identical for both cases. They were also identical to the flow field predicted for steady, backward-facing step flow using the same numerical grid as for the time-dependent predictions.

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