International audienceThe description of coherent features of fluid flow is essential to our understanding of fluid-dynamical and transport processes. A method is introduced that is able to extract dynamic information from flow fields that are either generated by a (direct) numerical simulation or visualized/measured in a physical experiment. The extracted dynamic modes, which can be interpreted as a generalization of global stability modes, can be used to describe the underlying physical mechanisms captured in the data sequence or to project large-scale problems onto a dynamical system of significantly fewer degrees of freedom. The concentration on subdomains of the flow field where relevant dynamics is expected allows the dissection of a complex flow into regions of localized instability phenomena and further illustrates the flexibility of the method, as does the description of the dynamics within a spatial framework. Demonstrations of the method are presented consisting of a plane channel flow, flow over a two-dimensional cavity, wake flow behind a flexible membrane and a jet passing between two cylinders. © 2010 Cambridge University Press

Hydrodynamic stability theory has recently seen a great deal of development. After being dominated by modal (eigenvalue) analysis for many decades, a different perspective has emerged that allows the quantitative description of short-term disturbance behavior. A general formulation based on the linear initial-value problem, thus circumventing the normal-mode approach, yields an efficient framework for stability calculations that is easily extendable to incorporate time-dependent flows, spatially varying configurations, stochastic influences, nonlinear effects, and flows in complex geometries.129 Annu. Rev. Fluid Mech. 2007.39:129-162. Downloaded from www.annualreviews.org by University of Virginia on 10/09/12. For personal use only.

Dynamic mode decomposition (DMD) represents an effective means for capturing the essential features of numerically or experimentally generated flow fields. In order to achieve a desirable tradeoff between the quality of approximation and the number of modes that are used to approximate the given fields, we develop a sparsity-promoting variant of the standard DMD algorithm. In our method, sparsity is induced by regularizing the least-squares deviation between the matrix of snapshots and the linear combination of DMD modes with an additional term that penalizes the 1-norm of the vector of DMD amplitudes. The globally optimal solution of the resulting regularized convex optimization problem is computed using the alternating direction method of multipliers, an algorithm well-suited for large problems. Several examples of flow fields resulting from numerical simulations and physical experiments are used to illustrate the effectiveness of the developed method. *

Streak breakdown caused by a spanwise inflectional instability is one phase of the following transition scenarios, which occur in plane Poiseuille and Couette flow. The streamwise vortex scenario is described by (SV) streamwise vortices =⇒ streamwise streaks =⇒ streak breakdown =⇒ transition.The oblique wave scenario is described by (OW) oblique waves =⇒ streamwise vortices =⇒ streamwise streaks =⇒ streak breakdown =⇒ transition.The purpose of this paper is to investigate the streak breakdown phase of the above scenarios by a linear stability analysis and compare threshold energies for transition for the above scenarios with those for transition initiated by Tollmien-Schlichting waves (TS), two-dimensional optimals (2DOPT), and random noise (N) at subcritical Reynolds numbers. We find that if instability occurs, it is confined to disturbances with streamwise wavenumbers α 0 satisfying 0 < α min < |α 0 | < α max . In these unstable cases, the least stable mode is located near the centre of the channel with a phase velocity approximately equal to the centreline velocity of the mean flow. Growth rates for instability increase with streak amplitude. For Couette flow streak breakdown is inhibited by mean shear. Using the linear stability analysis, we determine lower bounds on threshold amplitude for transition for scenario (SV) that are consistent with thresholds determined by direct numerical simulations.In the final part of the paper we show that the threshold energies for transition in Poiseuille flow at subcritical Reynolds numbers for scenarios (SV) and (OW) are two orders of magnitude lower than the threshold for transition initiated by Tollmien-Schlichting waves (TS) and an order of magnitude lower than that for (2DOPT). Scenarios (SV) and (OW) occur on a viscous time scale. However, even when transition times are taken into account, the threshold energy required for transition at a given time for (SV) and (OW) is lower than that for the (TS) and (2DOPT) scenarios at Reynolds number 1500.

This paper investigates the pseudospectra and the numerical range of the Orr-Sommerfeld operator for plane Poiseuille flow. A number z E C is in the E-pseudospectrum of a matrix or operator A if Il(zI-A)-'II 2 E-l, or, equivalently, if z is in the spectrum of A + E for some perturbation E satisfying IlEll 5 E. The numerical range of A is the set of numbers of the form (Au, u), where (., .) is the inner product and u is a vector or function with llull = 1. The spectrum of the Orr-Sommerfeld operator consists of three branches. It is shown that the eigenvalues at the intersection of the branches are highly sensitive to perturbations and that the sensitivity increases dramatically with the Reynolds number. The associated eigenfunctions are nearly linearly dependent, even though they form a complete set. To understand the high sensitivity of the eigenvalues, a model operator is considered, related to the Airy equation that also has highly sensitive eigenvalues. It is shown that the sensitivity of the eigenvalues can be related qualitatively to solutions of the Airy equation that satisfy boundary conditions to within an exponentially small factor. As an application, the growth of initial perturbations is considered. The near-linear dependence of the eigenfunctions implies that there is potential for energy growth, even when all the eigenmodes decay exponentially. Necessary and sufficient conditions for no energy growth can be stated in terms of both the pseudospectra and the numerical range. Bounds on the growth can be obtained using the pseudospectra.

International audienceThe decomposition of experimental data into dynamic modes using a data-based algorithm is applied to Schlieren snapshots of a helium jet and to time-resolved PIV-measurements of an unforced and harmonically forced jet. The algorithm relies on the reconstruction of a low-dimensional inter-snapshot map from the available flow field data. The spectral decomposition of this map results in an eigenvalue and eigenvector representation (referred to as dynamic modes) of the underlying fluid behavior contained in the processed flow fields. This dynamic mode decomposition allows the breakdown of a fluid process into dynamically revelant and coherent structures and thus aids in the characterization and quantification of physical mechanisms in fluid flow. © 2010 Springer-Verlag

A framework for the input-output analysis, model reduction and control design of spatially developing shear flows is presented using the Blasius boundary-layer flow as an example. An input-output formulation of the governing equations yields a flexible formulation for treating stability problems and for developing control strategies that optimize given objectives. Model reduction plays an important role in this process since the dynamical systems that describe most flows are discretized partial differential equations with a very large number of degrees of freedom. Moreover, as system theoretical tools, such as controllability, observability and balancing has become computationally tractable for large-scale systems, a systematic approach to model reduction is presented.

Linear stability of incompressible flow in a circular pipe is considered. Use is made of a vector function formulation involving the radial velocity and radial vorticity only. Asymptotic as well as transient stability are investigated using eigenvalues and ε-pseudoeigenvalues, respectively. Energy stability is probed by establishing a link to the numerical range of the linear stability operator. Substantial transient growth followed by exponential decay has been found and parameter studies revealed that the maximum amplification of initial energy density is experienced by disturbances with no streamwise dependence and azimuthal wavenumber n = 1. It has also been found that the maximum in energy scales with the Reynolds number squared, as for other shear flows. The flow field of the optimal disturbance, exploiting the transient growth mechanism maximally, has been determined and followed in time. Optimal disturbances are in general characterized by a strong shear layer in the centre of the pipe and their overall structure has been found not to change significantly as time evolves. The presented linear transient growth mechanism which has its origin in the non-normality of the linearized Navier–Stokes operator, may provide a viable process for triggering finite-amplitude effects.