Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken. We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator. This generalizes DMD to a larger class of datasets, including nonsequential time series. We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively. We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples. Such computations are not considered in the existing literature, but can be understood using our more general framework. In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory. It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science. We show that under certain conditions, DMD is equivalent to LIM.

On Dynamic Mode Decomposition: Theory and Applications

The Structure of Turbulent Shear Flow By Dr. A. A. Townsend. Pp. xii + 315. 8¾ in. × 5½ in. (Cambridge: At the University Press.) 40s.

/pdf/the-structure-of-turbulent-shear-flow-2yf3szhaqb.pdf

The Structure of Turbulent Shear Flow

Fluid dynamic turbulence is one of the most challenging computational physics problems because of the extremely wide range of time and space scales involved, the strong nonlinearity of the governing equations, and the many practical and important applications. While most linear fluid instabilities are well understood, the nonlinear interactions among them makes even the relatively simple limit of homogeneous isotropic turbulence difficult to treat physically, mathematically, and computationally. Turbulence is modeled computationally by a two-stage bootstrap process. The first stage, direct numerical simulation, attempts to resolve the relevant physical time and space scales but its application is limited to diffusive flows with a relatively small Reynolds number (Re). Using direct numerical simulation to provide a database, in turn, allows calibration of phenomenological turbulence models for engineering applications. Large eddy simulation incorporates a form of turbulence modeling applicable when the large-scale flows of interest are intrinsically time dependent, thus throwing common statistical models into question. A promising approach to large eddy simulation involves the use of high-resolution monotone computational fluid dynamics algorithms such as flux-corrected transport or the piecewise parabolic method which have intrinsic subgrid turbulence models coupled naturally to the resolved scales in the computed flow. The physical considerations underlying and evidence supporting this monotone integrated large eddy simulation approach are discussed.

New insights into large eddy simulation

Turbulence, Coherent Structures, Dynamical Systems and SymmetryP. Holmes, J. L. Lumley, G. Berkooz, and C. W. Rowley, 2nd ed., Cambridge University Press, Cambridge, England, U.K., 2012, 386 pp., $90

Turbulent jet noise is a controversial fluid mechanical puzzle that has amused and bewildered researchers for more than half a century. Whereas numerical simulations are now capable of simultaneously predicting turbulence and its radiated sound, the theoretical framework that would guide noise-control efforts is incomplete. Wave packets are intermittent, advecting disturbances that are correlated over distances far exceeding the integral scales of turbulence. Their signatures are readily distinguished in the vortical, turbulent region; the irrotational, evanescent near field; and the propagating far field. We review evidence of the existence, energetics, dynamics, and acoustic efficiency of wave packets. We highlight how extensive data available from simulations and modern measurement techniques can be used to distill acoustically relevant turbulent motions. The evidence supports theories that seek to represent wave packets as instability waves, or more general modal solutions of the governing equations, and confirms the acoustic importance of these structures in the aft-angle radiation of high subsonic and supersonic jets. The resulting unified view of wave packets provides insights that can help guide control strategies.

/pdf/wave-packets-and-turbulent-jet-noise-1j77wgcl8y.pdf

Wave Packets and Turbulent Jet Noise

Noise prediction for a turbulent jet using different hybrid methods

We propose a generalization of proper orthogonal decomposition (POD) for optimal
flow resolution of linearly related observables. This Galerkin expansion, termed
‘observable inferred decomposition’ (OID), addresses a need in aerodynamic and
aeroacoustic applications by identifying the modes contributing most to these
observables. Thus, OID constitutes a building block for physical understanding, leastbiased
conditional sampling, state estimation and control design. From a continuum of
OID versions, two variants are tailored for purposes of observer and control design,
respectively. Firstly, the most probable flow state consistent with the observable is
constructed by a ‘least-residual’ variant. This version constitutes a simple, easily
generalizable reconstruction of the most probable hydrodynamic state to preprocess
efficient observer design. Secondly, the ‘least-energetic’ variant identifies modes
with the largest gain for the observable. This version is a building block for
Lyapunov control design. The efficient dimension reduction of OID as compared
to POD is demonstrated for several shear flows. In particular, three aerodynamic
and aeroacoustic goal functionals are studied: (i) lift and drag fluctuation of a
two-dimensional cylinder wake flow; (ii) aeroacoustic density fluctuations measured
by a sensor array and emitted from a two-dimensional compressible mixing layer; and (iii) aeroacoustic pressure monitored by a sensor array and emitted from a
three-dimensional compressible jet. The most ‘drag-related’, ‘lift-related’ and ‘loud’
structures are distilled and interpreted in terms of known physical processes.

/pdf/on-least-order-flow-representations-for-aerodynamics-and-3npla2js2z.pdf

On least-order flow representations for aerodynamics and aeroacoustics

The flow field and the acoustic field of various realistic coaxi al jet configurations are numerically analyzed. The flow data, which are computed via large-eddy simulations (LES), provide the distributions being plugged in the source terms of the acoustic perturbation equations (APE) to compute the acoustic near field. The investigation emphasizes the core flow to have a major impact on the radiated jet noise. In particular the effect of heating the inner stream generates substantial noise to the sideline of the jet, whereas the vorticity-velocity interaction is the dominant source for the downstream noise. Furthermore, two additional jet configurations are investi gated to study a noise reduction approach. A silencing high density fluid is injected between the outer jet strea m and the ambient flow and between the inner and the outer stream to modify the vortex dynamics. The acoustic results show the injection into the shear layer between the inner and outer stream to reduce the noise at the acute angles from the jet axis, while the noise at obtuse angles is enhanced. That is, the high frequency noise downstream of the jet is reduced.

Towards Noise Reduction of Coaxial Jets

In the present study, subsonic turbulent jet noise is investigated employing reduced-order representations of the flow field and its noise source targeting 'least-order' approximations of the key processes. These representations utilize LES data for a compressible jet at Mach number 0.9 and Reynolds number 3600. The fluctuations of the velocity field and of the Lamb vector as noise source are investigated with three methods. Firstly, the streamwise development is characterised by a statistical analysis. Thus, the most active region of the flow field and the Lamb vector are observed at 11 and 8 jet diameters downstream, respectively. Secondly, an azimuthal mode decomposition is carried out. The first five azimuthal modes resolve most of the flow field and Lamb vector fluctuation. Thirdly, the dimension of the dynamics phase space is estimated by the proper orthogonal decomposition (POD). About 350 modes are necessary to resolve at least 50% of the fluctuation level of the hydrodynamics and even more modes are required for the noise source. As expected, the end of the potential core correlates with the location of a distinct peak in the noise source magnitude, thus indicating a highly active acoustical region. Intriguingly, the noise source efficiency per unit energy increases with higher azimuthal modes. The comparison of the compressible jet results with the incompressible LES at the same Reynolds number reveals a significant smaller energy concentration in the first azimuthal modes, i.e. the incompressible flow is dynamically more complex. The current results are part of an ongoing effort to predict far-field jet noise by reduced- order modelling of its hydrodynamic source.

/pdf/reduced-order-representation-of-turbulent-jet-flow-and-its-3aplkq07u3.pdf

E. Gröschel

Papers

Noise prediction for a turbulent jet using different hybrid methods

On least-order flow representations for aerodynamics and aeroacoustics

Towards Noise Reduction of Coaxial Jets

Reduced-order representation of turbulent jet flow and its noise source

Noise Generation Mechanisms in Single and Coaxial Jets