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.

https://hal-polytechnique.archives-ouvertes.fr/hal-01020654/document

Dynamic mode decomposition of numerical and experimental data

The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

: This thesis applies neural network feature selection techniques to multivariate time series data to improve prediction of a target time series. Two approaches to feature selection are used. First, a subset enumeration method is used to determine which financial indicators are most useful for aiding in prediction of the S&P 500 futures daily price. The candidate indicators evaluated include RSI, Stochastics and several moving averages. Results indicate that the Stochastics and RSI indicators result in better prediction results than the moving averages. The second approach to feature selection is calculation of individual saliency metrics. A new decision boundary-based individual saliency metric, and a classifier independent saliency metric are developed and tested. Ruck's saliency metric, the decision boundary based saliency metric, and the classifier independent saliency metric are compared for a data set consisting of the RSI and Stochastics indicators as well as delayed closing price values. The decision based metric and the Ruck metric results are similar, but the classifier independent metric agrees with neither of the other metrics. The nine most salient features, determined by the decision boundary based metric, are used to train a neural network and the results are presented and compared to other published results. (AN)

/pdf/nonlinear-time-series-analysis-1j3a35n4y3.pdf

Nonlinear Time Series Analysis.

Advanced Mathematical Methods for Scientists and Engineers

The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems In this manuscript, we present a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a “black box” integrator We will show that this approach is, in effect, an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the “stochastic Koopman operator” (Mezic in Nonlinear Dynamics 41(1–3): 309–325, 2005) Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data and two that show potential applications of the Koopman eigenfunctions

A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

The 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.

Applications of the dynamic mode decomposition

Nine decades of rocket engine and gas turbine development have shown that thermoacoustic oscillations are difficult to predict but can usually be eliminated with relatively small ad hoc design changes These changes can, however, be ruinously expensive to devise This review explains why linear and nonlinear thermoacoustic behavior is so sensitive to parameters such as operating point, fuel composition, and injector geometry It shows how nonperiodic behavior arises in experiments and simulations and discusses how fluctuations in thermoacoustic systems with turbulent reacting flow, which are usually filtered or averaged out as noise, can reveal useful information Finally, it proposes tools to exploit this sensitivity in the future: adjoint-based sensitivity analysis to optimize passive control designs and complex systems theory to warn of impending thermoacoustic oscillations and to identify the most sensitive elements of a thermoacoustic system

Sensitivity and Nonlinearity of Thermoacoustic Oscillations

With a sufficiently large impulse, a thermoacoustic system can reach self-sustained oscillations even when it is linearly stable, a process known as triggering. In this paper, a procedure is developed to find the lowest initial energy that can trigger self-sustained oscillations, as well as the corresponding initial state. This is known as the ‘most dangerous’ initial state. The procedure is based on adjoint looping of the nonlinear governing equations, combined with an optimization routine. It is developed for a simple model of a thermoacoustic system, the horizontal Rijke tube, and can be extended to more sophisticated thermoacoustic models. It is observed that the most dangerous initial state grows transiently towards an unstable periodic solution before growing to a stable periodic solution. The initial energy required to trigger these selfsustained oscillations is much lower than the energy of the oscillations themselves and slightly lower than the lowest energy on the unstable periodic solution. It is shown that this transient growth arises due to non-normality of the governing equations. This is analogous to the sequence of events observed in bypass transition to turbulence in fluid mechanical systems and has the same underlying cause. The most dangerous initial state is calculated as a function of the heat-release parameter. It is found that self-sustained oscillations can be reached over approximately half the linearly stable domain. Transient growth in real thermoacoustic systems is 10 5 –10 6 times greater than that in this simple model. One practical conclusion is that, even in the linearly stable regime, it may take very little initial energy for a real thermoacoustic system to trigger to high-amplitude self-sustained oscillations through the mechanism described in this paper.

http://www2.eng.cam.ac.uk/~mpj1001/papers/JFM_Juniper_2011_final.pdf

Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition

A detailed understanding of liquid propellant combustion is necessary for the development of improved and more reliable propulsion systems. This article describes experimental investigations aimed at providing such a fundamental basis for design and engineering of combustion components. It reports recent applications of imaging techniques to cryogenic combustion at high pressure. The flame structure is investigated in the transcritical range where the pressure exceeds the critical pressure of oxygen (p > p c (O2 = 5.04MPa)) but the temperature of the injected liquid oxygen is below its critical value . Data obtained from imaging of OH* radicals emission, CH* radicals emission in the case of LOx/GCH4 flames and backlighting provide a detailed view of the flame structure for a set of injection conditions. The data may be used to guide numerical modelling of transcritical flames and the theoretical and numerical analysis of the stabilization process. Calculations of the flame edge are used to illust...

http://www2.eng.cam.ac.uk/~mpj1001/papers/CST_Candel_Juniper_2006.pdf

Structure and dynamics of cryogenic flames at supercritical pressure

The wake characteristics of bluff-body-stabilized flames are a strong function of the density ratio across the flame and the relative offset between the flame and shear layer. This paper describes systematic experimental measurements and stability calculations of the dependence of the flow field characteristics and flame sheet dynamics upon flame density ratio, , over the Reynolds number range of 1000–3300. We show that two fundamentally different flame/flow behaviours are observed at high and low values: a stable, noise-driven fixed point and limit-cycle oscillations, respectively. These results are interpreted as a transition from convective to global instability and are captured well by stability calculations that used the measured velocity and density profiles as inputs. However, in this high-Reynolds-number flow, the measurements show that no abrupt bifurcation in flow/flame behaviour occurs at a given value. Rather, the flow field is highly intermittent in a transitional range, with the relative fraction of the two different flow/flame behaviours monotonically varying with . This intermittent behaviour is a result of parametric excitation of the global mode growth rate in the vicinity of a supercritical Hopf bifurcation. It is shown that this parametric excitation is due to random fluctuations in relative locations of the flame and shear layer.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112012002480

Matthew P. Juniper

Papers

Applications of the dynamic mode decomposition

Sensitivity and Nonlinearity of Thermoacoustic Oscillations

Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition

Structure and dynamics of cryogenic flames at supercritical pressure

Density ratio effects on reacting bluff-body flow field characteristics