H/sub ∞/ control for non-periodic planar channel flows

Baramov, L.; Tutty, O.R.; Rogers, Eric

doi:10.1109/.2001.980993

Cited by 3 publications

(8 citation statements)

References 4 publications

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“…Substituting the flow decomposition (4) and (6) into the vorticity vector definition ω i (i = x,y,z) and simplifying by taking into account that k 1 1, it is found that…”

Section: Mathematical Formulation Of the Physical Problemmentioning

confidence: 99%

“…The equations for these variables can be obtained by substituting (4) and (6) into the Navier-Stokes equations. The equations can be linearized about the undisturbed Blasius solution when r t 1.…”

Section: Mathematical Formulation Of the Physical Problemmentioning

confidence: 99%

“…The matrix L and the linear operator M can be obtained by inspection. In summary, we have a deterministic parabolic system (11) from which the variables that contain the flow randomness {ū 0 ,v 0 ,w 0 ,p 0 } are obtained using the multiplicative transformation (6) and not additively as usual. Of course, the objective function to be minimized by the control action must be expressed in terms of the original random variables {ū 0 ,v 0 ,w 0 ,p 0 }.…”

Section: Mathematical Formulation Of the Physical Problemmentioning

confidence: 99%

“…Along the homogeneous directions, the variation can be represented by Fourier series expansions, and the property of linearity allows the decoupling of the evolution equations for each wave number pair, thus greatly simplifying the control design [6][7][8]. Bamieh et al [9] proved that, using this approach in spatially invariant systems with distributed actuation and sensing, spatially localized convolution kernels with exponential decay rates can be obtained.…”

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confidence: 99%

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Closed-loop control of boundary layer streaks induced by free-stream turbulence

Papadakis

Riccó

2016

Phys. Rev. Fluids

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The central aim of the paper is to carry out a theoretical and numerical study of active wall transpiration control of streaks generated within an incompressible boundary layer by free-stream turbulence. The disturbance flow model is based on the linearized unsteady boundary-region (LUBR) equations, studied by Leib, Wundrow, and Goldstein [J. Fluid Mech. 380, 169 (1999)], which are the rigorous asymptotic limit of the NavierStokes equations for low-frequency and long-streamwise wavelength. The mathematical formulation of the problem directly incorporates the random forcing into the equations in a consistent way. Due to linearity, this forcing is factored out and appears as a multiplicative factor. It is shown that the cost function (integral of kinetic energy in the domain) is properly defined as the expectation of a random quadratic function only after integration in wave number space. This operation naturally introduces the free-stream turbulence spectral tensor into the cost function. The controller gains for each wave number are independent of the spectral tensor and, in that sense, universal. Asymptotic matching of the LUBR equations with the free-stream conditions results in an additional forcing term in the state-space system whose presence necessitates the reformulation of the control problem and the rederivation of its solution. It is proved that the solution can be obtained analytically using an extension of the sweep method used in control theory to obtain the standard Riccati equation. The control signal consists of two components, a feedback part and a feed-forward part (that depends explicitly on the forcing term). Explicit recursive equations that provide these two components are derived. It is shown that the feed-forward part makes a negligible contribution to the control signal. We also derive an explicit expression that a priori (i.e., before solving the control problem) leads to the minimum of the objective cost function (i.e., the fundamental performance limit), based only on the system matrices and the initial and free-stream boundary conditions. The adjoint equations admit a self-similar solution for large spanwise wave numbers with a scaling which is different from that of the LUBR equations. The controlled flow field also has a self-similar solution if the weighting matrices of the objective function are chosen appropriately. The code developed to implement this algorithm is efficient and has modest memory requirements. Computations show the significant reduction of energy for each wave number. The control of the full spectrum streaks, for conditions corresponding to a realistic experimental case, shows that the root-mean-square of the streamwise velocity is strongly suppressed in the whole domain and for all the frequency ranges examined.

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“…Substituting the flow decomposition (4) and (6) into the vorticity vector definition ω i (i = x,y,z) and simplifying by taking into account that k 1 1, it is found that…”

Section: Mathematical Formulation Of the Physical Problemmentioning

confidence: 99%

Section: Mathematical Formulation Of the Physical Problemmentioning

confidence: 99%

Section: Mathematical Formulation Of the Physical Problemmentioning

confidence: 99%

mentioning

confidence: 99%

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Closed-loop control of boundary layer streaks induced by free-stream turbulence

Papadakis

Riccó

2016

Phys. Rev. Fluids

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“…The different formulations of the governing equations are discretised in space on computational meshes with grid spacing δ in both the x and y directions. Second-order accurate centred finite difference schemes are used for spatial derivatives as this yields simple interconnections between neighbouring node subsystems, which enables efficient evaluation of system frequency response by chaining nodes together, as demonstrated in [17,15]. Such finite difference schemes are of the form…”

Section: Individual Computational Node Subsystemsmentioning

confidence: 99%

Dynamically correct formulations of the linearised Navier–Stokes equations

Dellar

Jones

2017

Numerical Methods in Fluids

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SUMMARYMotivated by the need to efficiently obtain low-order models of fluid flows around complex geometries for the purpose of feedback control system design, this paper considers the effect on system dynamics of basing plant models on different formulations of the linearised Navier-Stokes equations. We consider the dynamics of a single computational node formed by spatial discretisation of the governing equations in both primitive variables (momentum equation & continuity equation) and pressure Poisson equation (PPE) formulations. This reveals fundamental numerical differences at the nodal level, whose effects on the system dynamics at the full system level are exemplified by considering the corresponding formulations of a two-dimensional (2D) channel flow, subjected to a variety of different boundary conditions.

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Linear and non‐linear simulations of feedback control in plane Poiseuille flow

McKernan

Papadakis

Whidborne

2008

Numerical Methods in Fluids

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SUMMARYThis paper examines the performance of optimal linear quadratic state and output feedback controllers in stabilizing two-dimensional perturbations in a plane Poiseuille flow. The synthesis of the controllers is based on a linearized model of the flow using a new set of interpolating polynomials in the wall-normal direction, which automatically satisfy the homogeneous Dirichlet and Neumann boundary conditions at the walls and eliminate spurious eigenvalues. The controllers are implemented into a non-linear NavierStokes solver, which is modified to compute the evolution of the flow perturbations. Two cases are examined, one with small initial disturbances that do not violate the linearity assumptions and the other with much larger disturbances that trigger the non-linear convection terms. For the smallest disturbances, the solver accurately reproduced the results of the linear simulations of open-and closed-loop systems. The simulations for the larger disturbances without control showed a rapid initial growth but the flow soon reached a saturated state in agreement with previous findings in the literature. The large initial growth is a consequence of the non-normal nature of the system dynamics. The state feedback and output feedback controllers were able to reduce significantly the perturbation energy. For the larger disturbances, the energy calculated from the state variables is well below the energy evaluated by direct integration of the velocity field. This is probably due to the non-linear terms transferring energy to harmonics of the considered wavenumber, which are not sensed by the linear controller.

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H/sub ∞/ control for non-periodic planar channel flows

Cited by 3 publications

References 4 publications

Closed-loop control of boundary layer streaks induced by free-stream turbulence

Closed-loop control of boundary layer streaks induced by free-stream turbulence

Dynamically correct formulations of the linearised Navier–Stokes equations

Linear and non‐linear simulations of feedback control in plane Poiseuille flow

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