A high-order super-grid-scale absorbing layer and its application to linear hyperbolic systems

We continue the development of the super-grid-scale model initiated in [T. Colonius, H. Ran, A super-grid-scale model for simulating compressible flow on unbounded domains, J. Comput. Phys. 182 (1) (2002) 191–212] and consider its application to linear hyperbolic systems. The super-grid-scale model consists of two parts: reduction of an unbounded to a bounded domain by a smooth coordinate transformation and a damping of those scales. For linear problems the super-grid scales are analogous to spurious numerical waves. We damp these waves by high-order undivided differences. We compute reflection coefficients for different orders of the damping and find that significant improvements are obtained when high-order damping is used.     

Special issue on global flow instability and control

This special issue is the second on the topic of “Global Flow Instability and Control,” following the first in 2011. As with the previous special issue, the participants of the last two symposia on Global Flow Instability and Control, held in Crete, Greece, were invited to submit publications. These papers were peer reviewed according to the standards of the journal, and this issue represents a snapshot of the progress since 2011. In this preface, a sampling of important developments in the field since the first issue is discussed. A synopsis of the papers in this issue is given in that context.     

An algebraic characterization of closed small attainability subspaces of delay systems

If small attainability subspaces of linear time delay systems are closed in a certain Sobolev space, the existence of Lagrange multipliers for optimal control to small solutions is guaranteed. This paper characterizes the required closedness property using an algebraic approach due to B. Jakubczyk. As a main result it turns out that closedness is—in an algebraic sense—generic in the variety of system matrices (A₀,A₁, B₀) with rank A₁ not greater than the dimension of the control space. This is in contrast to known results on closedness of attainability subspaces playing an analogous role for optimal control to fixed final states instead of small solutions.     

A splitting theorem for affine skew‐product flows

Topological conjugacy for continuous skew product flows on topological group bundles and semidirect products is studied resulting in a splitting theorem. The results are used to discuss fiber entropy.     

Flow around a NACA0018 airfoil with a cavity and its dynamical response to acoustic forcing

Trapping of vortices in a cavity has been explored in recent years as a drag reduction measure for thick airfoils. If, however, trapping fails, then oscillation of the cavity flow may couple with elastic vibration modes of the airfoil. To examine this scenario, the effect of small amplitude vertical motion on the oscillation of the shear layer above the cavity is studied by acoustic forcing simulating a vertical translation of a modified NACA0018 profile. At low Reynolds numbers based on the chord (O(10⁴)), natural instability modes of this shear layer are observed for Strouhal numbers based on the cavity width of order unity. Acoustic forcing sufficiently close to the natural instability frequency induces a strong non-linear response due to lock-in of the shear layer. At higher Reynolds numbers (above 10⁵) for Strouhal number 0.6 or lower, no natural instabilities of the shear layer and only a linear response to forcing were observed. The dynamical pressure difference across the airfoil is then dominated by added mass effects, as was confirmed by numerical simulations.     

A note on the existence of Lagrange multipliers

This note extends a Lagrange multiplier theorem due to J. Zowe and S. Kurcyusz. Under some additional assumptions the required regularity condition can be weakened. The cone corresponding to the explicit constraint is replaced by its closure.     

Controllability Properties and Invariance Pressure for Linear Discrete-Time Systems

For linear control systems in discrete time controllability properties are characterized. In particular, a unique control set with nonvoid interior exists and it is bounded in the hyperbolic case. Then a formula for the invariance pressure of this control set is proved.

     

Linear control semigroups acting on projective space

Linear control semigroupsL?=Gl(d,R) are associated with semilinear control systems of the form whereA:R ^m → gl(d,R) is continuous in some open set containingU. The semigroupL then corresponds to the solutions with piecewise constant controls, i.e., L acts in a natural way onR ^d {0}, on the sphereS ^d?1, and on the projective spaceP ^d?1. Under the assumption that the group generated byL in Gl(d,R) acts transitively onP ^d?1, we analyze the control structure of the action ofL onP ^d?1: We characterize the sets inP ^d?1, where the system is controllable (the control sets) using perturbation theory of eigenvalues and (generalized) eigenspaces of the matrices g εL For nonlinear control systems on finitedimensional manifoldsM, we study the linearization on the tangent bundleTM and the projective bundleP M via the theory of Morse decompositions, to obtain a characterization of the chain-recurrent components of the control flow onU×PM. These components correspond uniquely to the chain control sets onP M, and they induce a subbundle decomposition ofU×TM. These results are used to characterize the chain control sets ofL acting onP ^d?1 and to compare the control sets and chain control sets.     

Special issue on global flow instability and control

This special issue is intended to provide a snapshot of current research in the area of ??Global Flow Instability and Control??. The original papers, and to a certain extent the topic itself, are intimately linked with the series of symposia by the same name that were held in Crete, Greece, between 2001 and 2009. As members of the organizing committees of the Crete symposia, we invited all past participants to contribute, and all papers were reviewed following the strict standards of the journal. This preface gives a brief historical account of events that have shaped ideas in the field over the past decade, followed by a synopsis of the papers published herein.