Exact theory of unsteady separation for two-dimensional flows

Haller, George

doi:10.1017/s0022112004009929

Cited by 100 publications

(120 citation statements)

References 23 publications

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“…far from the wall) shrinks exponentially down to the diffusion or measurement scale; yet (ii) wide strips of unmixed fluid of width ∆(t) ∝ t −2 are periodically interweaved with these fine structures. Both protocols have in common a chaotic region that spans the entire domain, which imposes the presence of parabolic separation points on the boundary [35,37,38]. In the next section, we generalize the baker's map model to include such a parabolic point at the boundary, and reproduce the dominant features observed experimentally and numerically.…”

Section: B Hydrodynamics Near the Wallmentioning

confidence: 92%

Slow decay of concentration variance due to no-slip walls in chaotic mixing

et al. 2008

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Chaotic mixing in a closed vessel is studied experimentally and numerically in different 2-D flow configurations. For a purely hyperbolic phase space, it is well-known that concentration fluctuations converge to an eigenmode of the advection-diffusion operator and decay exponentially with time. We illustrate how the unstable manifold of hyperbolic periodic points dominates the resulting persistent pattern. We show for different physical viscous flows that, in the case of a fully chaotic Poincaré section, parabolic periodic points at the walls lead to slower (algebraic) decay. A persistent pattern, the backbone of which is the unstable manifold of parabolic points, can be observed. However, slow stretching at the wall forbids the rapid propagation of stretched filaments throughout the whole domain, and hence delays the formation of an eigenmode until it is no longer experimentally observable. Inspired by the baker's map, we introduce a 1-D model with a parabolic point that gives a good account of the slow decay observed in experiments. We derive a universal decay law for such systems parametrized by the rate at which a particle approaches the no-slip wall.

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Section: B Hydrodynamics Near the Wallmentioning

confidence: 92%

Slow decay of concentration variance due to no-slip walls in chaotic mixing

et al. 2008

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“…In that approach, the solution of the skin-friction equation (7) was controlled via two-point boundary actuation to satisfy the kinematic separation conditions of Haller [6]. The velocity derivative σ, however, was obtained from observations rather than from a model.…”

Section: Discussionmentioning

confidence: 99%

Reduced Navier-Stokes Equations Near a Flow Boundary

Kılıç¹,

Jacobs²,

Haller³

2005

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We derive a hierarchy of PDEs for the leading-order evolution of wall-based quantities, such as the skin-friction and the wall-pressure gradient, in two-dimensional fluid flows. The resulting Reduced Navier-Stokes (RNS ) equations are defined on the boundary of the flow, and hence have reduced spatial dimensionality compared to the Navier-Stokes equations. This spatial reduction speeds up numerical computations and makes the equations attractive candidates for flow-control design. We prove that members of the RNS hierarchy are well-posed if appended with boundary-conditions obtained from wall-based sensors. We also derive the lowest-order RNS equations for three-dimensional flows. For several benchmark problems, our numerical simulations show close finite-time agreement between the solutions of RNS and those of the full Navier-Stokes equations.

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“…Of course, similar combination of models appear with stochastic systems (see, for example, [Elbeyli et al, 2005]) or with distributed parameter systems (see, for example, [Haller, 2004]), too, but the present study concentrates on delay and time-periodicity together. Such problems appeared in remote periodic motion and haptic control [Insperger & Stepan, 2004d], or in control of periodic flows, but one of the most transparent engineering problems is high-speed milling in this respect.…”

Section: Introductionmentioning

confidence: 95%

Delay, Parametric Excitation, and the Nonlinear Dynamics of Cutting Processes

Stépàn

Insperger

Szalai

2005

Int. J. Bifurcation Chaos

100

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It is a rule of thumb that time delay tends to destabilize any dynamical system. This is not true, however, in the case of delayed oscillators, which serve as mechanical models for several surprising physical phenomena. Parametric excitation of oscillatory systems also exhibits stability properties sometimes defying our physical sense. The combination of the two effects leads to challenging tasks when nonlinear dynamic behaviors in these systems are to be predicted or explained as well. This paper gives a brief historical review of the development of stability analysis in these systems, induced by newer and newer models in several fields of engineering. Local and global nonlinear behavior is also discussed in the case of the most typical parametrically excited delayed oscillator, a recent model of cutting applied to the study of high-speed milling processes.

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Exact theory of unsteady separation for two-dimensional flows

Cited by 100 publications

References 23 publications

Slow decay of concentration variance due to no-slip walls in chaotic mixing

Slow decay of concentration variance due to no-slip walls in chaotic mixing

Reduced Navier-Stokes Equations Near a Flow Boundary

Delay, Parametric Excitation, and the Nonlinear Dynamics of Cutting Processes

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