The 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.
The dynamic mode decomposition (DMD) is a data-decomposition technique that allows the extraction of dynamically relevant flow features from time-resolved experimental (or numerical) data. It is ...based on a sequence of snapshots from measurements that are subsequently processed by an iterative Krylov technique. The eigenvalues and eigenvectors of a low-dimensional representation of an approximate inter-snapshot map then produce flow information that describes the dynamic processes contained in the data sequence. This decomposition technique applies equally to particle-image velocimetry data and image-based flow visualizations and is demonstrated on data from a numerical simulation of a flame based on a variable-density jet and on experimental data from a laminar axisymmetric water jet. In both cases, the dominant frequencies are detected and the associated spatial structures are identified.
Dynamic mode decomposition (DMD) is a factorization and dimensionality reduction technique for data sequences. In its most common form, it processes high-dimensional sequential measurements, extracts ...coherent structures, isolates dynamic behavior, and reduces complex evolution processes to their dominant features and essential components. The decomposition is intimately related to Koopman analysis and, since its introduction, has spawned various extensions, generalizations, and improvements. It has been applied to numerical and experimental data sequences taken from simple to complex fluid systems and has also had an impact beyond fluid dynamics in, for example, video surveillance, epidemiology, neurobiology, and financial engineering. This review focuses on the practical aspects of DMD and its variants, as well as on its usage and characteristics as a quantitative tool for the analysis of complex fluid processes.
Rotating Rayleigh–Bénard convection is typified by a variety of regimes with very distinct flow morphologies that originate from several instability mechanisms. Here we present results from direct ...numerical simulations of three representative set-ups: first, a fluid with Prandtl number
$Pr=6.4$
, corresponding to water, in a cylinder with a diameter-to-height aspect ratio of
$\unicodeSTIX{x1D6E4}=2$
; second, a fluid with
$Pr=0.8$
, corresponding to
$\text{SF}_{6}$
or air, confined in a slender cylinder with
$\unicodeSTIX{x1D6E4}=0.5$
; and third, the main focus of this paper, a fluid with
$Pr=0.025$
, corresponding to a liquid metal, in a cylinder with
$\unicodeSTIX{x1D6E4}=1.87$
. The obtained flow fields are analysed using the sparsity-promoting variant of the dynamic mode decomposition (DMD). By means of this technique, we extract the coherent structures that govern the dynamics of the flow, as well as their associated frequencies. In addition, we follow the temporal evolution of single modes and present a criterion to identify their direction of travel, i.e. whether they are precessing prograde or retrograde. We show that for moderate
$Pr$
a few dynamic modes suffice to accurately describe the flow. For large aspect ratios, these are wall-localised waves that travel retrograde along the periphery of the cylinder. Their DMD frequencies agree with the predictions of linear stability theory. With increasing Rayleigh number
$Ra$
, the interior gradually fills with columnar vortices, and eventually a regular pattern of convective Taylor columns prevails. For small aspect ratios and close enough to onset, the dominant flow structures are body modes that can precess either prograde or retrograde. For
$Pr=0.8$
, DMD additionally unveiled the existence of so far unobserved low-amplitude oscillatory modes. Furthermore, we elucidate the multi-modal character of oscillatory convection in low-
$Pr$
fluids. Generally, more dynamic modes must be retained to accurately approximate the flow. Close to onset, the flow is purely oscillatory and the DMD reveals that these high-frequency modes are a superposition of oscillatory columns and cylinder-scale inertial waves. We find that there are coexisting prograde and retrograde modes, as well as quasi-axisymmetric torsional modes. For higher
$Ra$
, the flow also becomes unstable to wall modes. These low-frequency modes can both coexist with the oscillatory modes, and also couple to them. However, the typical flow feature of rotating convection at moderate
$Pr$
, the quasi-steady Taylor vortices, is entirely absent in low-
$Pr$
flows.
We present a conditional space–time proper orthogonal decomposition (POD) formulation that is tailored to the eduction of the average, rare or intermittent events from an ensemble of realizations of ...a fluid process. By construction, the resulting spatio-temporal modes are coherent in space and over a predefined finite time horizon, and optimally capture the variance, or energy of the ensemble. For the example of intermittent acoustic radiation from a turbulent jet, we introduce a conditional expectation operator that focuses on the loudest events, as measured by a pressure probe in the far field and contained in the tail of the pressure signal’s probability distribution. Applied to high-fidelity simulation data, the method identifies a statistically significant ‘prototype’, or average acoustic burst event that is tracked over time. Most notably, the burst event can be traced back to its precursor, which opens up the possibility of prediction of an imminent burst. We furthermore investigate the mechanism underlying the prototypical burst event using linear stability theory and find that its structure and evolution are accurately predicted by optimal transient growth theory. The jet-noise problem demonstrates that the conditional space–time POD formulation applies even for systems with probability distributions that are not heavy-tailed, i.e. for systems in which events overlap and occur in rapid succession.
A novel data-driven modal decomposition of fluid flow is proposed, comprising key features of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD). The first mode is the ...normalized real or imaginary part of the DMD mode that minimizes the time-averaged residual. The
$N$
th mode is defined recursively in an analogous manner based on the residual of an expansion using the first
$N-1$
modes. The resulting recursive DMD (RDMD) modes are orthogonal by construction, retain pure frequency content and aim at low residual. Recursive DMD is applied to transient cylinder wake data and is benchmarked against POD and optimized DMD (Chen et al., J. Nonlinear Sci., vol. 22, 2012, pp. 887–915) for the same snapshot sequence. Unlike POD modes, RDMD structures are shown to have purer frequency content while retaining a residual of comparable order to POD. In contrast to DMD, with exponentially growing or decaying oscillatory amplitudes, RDMD clearly identifies initial, maximum and final fluctuation levels. Intriguingly, RDMD outperforms both POD and DMD in the limit-cycle resolution from the same snapshots. Robustness of these observations is demonstrated for other parameters of the cylinder wake and for a more complex wake behind three rotating cylinders. Recursive DMD is proposed as an attractive alternative to POD and DMD for empirical Galerkin models, in particular for nonlinear transient dynamics.
The generation of discrete acoustic tones in the compressible flow around an aerofoil is addressed in this work by means of nonlinear numerical simulations and global stability analyses. The ...nonlinear simulations confirm the appearance of discrete tones in the acoustic spectrum and, for the chosen flow case, the global stability analyses of the mean-flow dynamics reveal that the linearized operator is stable. However, the flow response to incoming disturbances exhibits important transient growth effects that culminate in the onset of aeroacoustic feedback loops, involving instability processes on the suction- and pressure-surface boundary layers together with their cross-interaction by acoustic radiation at the trailing edge. The features of the aeroacoustic feedback loops and the appearance of discrete tones are then related to the features of the least-stable modes in the global spectrum: the spatial structure of the direct modes displays the coupled dynamics of hydrodynamic instabilities on the suction surface and in the near wake. Finally, different families of global modes will be identified and the dynamics that they represent will be discussed.
Pulsatile channel and pipe flows constitute a fundamental flow configuration with significant bearing on many applications in the engineering and medical sciences. Rotating machinery, hydraulic pumps ...or cardiovascular systems are dominated by time-periodic flows, and their stability characteristics play an important role in their efficient and proper operation. While previous work has mainly concentrated on the modal, harmonic response to an oscillatory or pulsatile base flow, this study employs a direct–adjoint optimisation technique to assess short-term instabilities, identify transient energy-amplification mechanisms and determine their prevalence within a wide parameter space. At low pulsation amplitudes, the transient dynamics is found to be similar to that resulting from the equivalent steady parabolic flow profile, and the oscillating flow component appears to have only a weak effect. After a critical pulsation amplitude is surpassed, linear transient growth is shown to increase exponentially with the pulsation amplitude and to occur mainly during the slow part of the pulsation cycle. In this latter regime, a detailed analysis of the energy transfer mechanisms demonstrates that the huge linear transient growth factors are the result of an optimal combination of Orr mechanism and intracyclic normal-mode growth during half a pulsation cycle. Two-dimensional sinuous perturbations are favoured in channel flow, while pipe flow is dominated by helical perturbations. An extensive parameter study is presented that tracks these flow features across variations in the pulsation amplitude, Reynolds and Womersley numbers, perturbation wavenumbers and imposed time horizon.
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