A dynamic mode decomposition of the saturation process in the open cavity flow

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Abstract

The flow over the incompressible open cavity has been studied both numerically and experimentally in previous works by the authors [6], [16]. In these studies the in-depth linear stability analysis of the problem provided a detailed description of the flow features in the linear regime, where span-wise periodic perturbations over a steady two-dimensional flow were studied. The experiments conducted in said work, with span-wise walls, show the flow in a saturated state. The main structures that appear in the flow were identified as three-dimensional centrifugal modes. Even if the flow conditions for these two approaches are different, common features appear in both: the spatial structure of the modes in the saturated regime according to the experiments is the one predicted by linear analysis, but there is a shift in frequencies for the oscillating modes. The present work uses a three-dimensional non-steady direct numerical simulation (DNS) with periodic span-wise conditions to study the effect of saturation. In order to analyze the DNS results a Dynamic Mode Decomposition (DMD) was used.

Introduction

There are numerous studies in the literature focusing on the flow over an open cavity. The appearance of this problem in numerous industrial applications, such as landing gears or weapon bays, and the richness of the physics involved, make it a well studied problem. The majority of the early work focused on the two-dimensional resonance that produces self-sustained oscillations in the shear layer, commonly known as the Rossiter modes [24], [21], [23]. Later studies reported a modulation of the shear layer modes by far smaller frequencies [22], [19]. This modulation is the result of the presence of three-dimensional structures that coil around the main recirculating vortex inside the cavity. The first linear computations of these three-dimensional instabilities were presented in [28], and they were proven to be dominant under certain flow conditions by [4], as well as independent of the two-dimensional shear layer modes. Recent experimental studies have focused on the characterization and visualization of these centrifugal modes, such as [9], [3].

A complete parameter analysis of the three-dimensional incompressible cavity flow was presented in [16]. The behavior of the linear modes with the variation of the different parameters of the problem (length to depth aspect ratio of the cavity (L/D), Reynolds number, incoming boundary layer thickness, and span-wise wavenumber β) was documented, as well as their morphological structure, characteristic frequencies, neutral stability curves, and laws of dependence among the aforementioned parameters. An in-depth comparative study between the flow described by linear analysis and the one observed in an experimental campaign was presented in [6], where two different setups with ReD=1500 (Case A) and ReD=2400 (Case B) were studied, for the L/D=2 cavity. The main structures present in the saturated and wall-bounded regime were found to match the ones from linear analysis, given there was a difference in flow conditions. One of the main results obtained in the aforementioned paper was the apparent reduction of the characteristic frequencies of the most energetic eigenmode from the theoretical value predicted by the linear analysis. The authors postulated that the possibility of said reduction could be consequence of the presence of the span-wise walls, which had the effect of slowing down the main centrifugal recirculation of the cavity, thus reducing the characteristic Strouhal number of this structures. Other possible sources for this phenomenon could be the saturation of the flow, or the non-linear interactions among several unstable eigenmodes.

A preliminary study was presented in [15], trying to separate these three effects. A three-dimensional DNS computation was performed for the same flow parameters of cases A and B of [6], but with periodic boundary conditions. Thus, the effect of the presence of end-walls was neglected, and a restriction on the span-wise wavenumber β was set, limiting the number of eigenmodes interacting and leaving the saturation as the main mechanism present in the study. The DNS results revealed that the reduction of characteristic Strouhal number reported in [6] occurs also in absence of span-wise walls. The work of [15] relied on the study of instantaneous flow-fields (or snapshots), as well as on the evolution of the flow variables at one control point. A deeper study of the data is however required to fully understand the physics behind it, as no relevant conclusions were reached due to the high complexity of the flow.

Data-sequences of snapshots collected from numerical simulations or experiments can be used to approximate the inherent fluid flow into dynamic modes, allowing then the identification of the relevant coherent structures in the flow. The most commonly used data-based techniques so far are the Fourier Transform analysis, the Proper Orthogonal Decomposition (POD) and, more recently, the Dynamic Mode Decomposition (DMD). The first approach is particularly efficient when dealing with periodic sampled data-fields. However, it loses accuracy when dealing with more complex and time-dependent fluid flows. With the use of POD one can identify the relevant structures in the flow ranked by their energy content but, since the different POD modes are orthogonal in space, their temporal behavior is characterized by the presence of multiple frequencies. For a detailed discussion about the use of Fourier Transform analysis, POD, and alternative data-based decomposition techniques on the identification of coherent structure in the fluid flow, the reader is directed to [17] and [2].

The DMD allows the extraction of spatial modal structures from a flow field. Each identified dynamic mode is associated with a single and unique frequency, consequence of the orthogonalization in time of the decomposition. The present technique is based on the Koopman analysis of a dynamical system [25], aiming to approximate the Koopman modes and eigenvalues of a linear infinite dimensional operator that describes that system, even if its dynamic behavior is nonlinear. In this case, DMD retrieves the structures of a linear tangent approximation to the underlying fluid flow [26]. For a system behaving linearly, the extracted DMD modes are expected to match the global stability modes. Moreover, if one performs a DMD over periodic solutions, [25] analytically demonstrated that the decomposition is identical to a discrete temporal Fourier Transform. Contrary to the POD, the DMD does not rank the extracted coherent structures in terms of energy content. However, their amplitudes provide a feedback about the individual contribution of a specific mode to the original data-set [27], enabling models of lower complexity [12], as it already happens with the POD.

This last decomposition technique is becoming more and more attractive in post-processing of numerical and experimental results, mainly due to its easiness of implementation, efficient data analysis, inherent low computational cost, and possibility of application to large data-sets or to sub-domains of a flow region. Some implementations of this tool for cavity problems can be found in the literature, like [8], [10]. The method has also demonstrated superior performance over other traditional data-based decomposition techniques for oscillatory dominated problems [27], and for fluid flows exhibiting strong peaks in the spectrum [17]. Nonetheless, the original DMD still has some relevant limitations, as recognized in [2]. According to this last reference, there is yet no validation between Koopman and DMD modes for chaotic and noisy high-Reynolds number flow-data and, based on the work of [7], the Dynamic Decomposition can be sensitive to the presence of noise in the data-field. Furthermore, [18] observed no particular differences between the POD and the DMD modes, in a flow characterized by a broad frequency spectrum and no dominant spectral peaks. There are also cases reported in literature where this decomposition may not guarantee the best approximation of the flow field, opening new ways for improved variants of the original algorithm based on optimization techniques, like the ones proposed by [12] and [5].

In the present paper, the original DMD algorithm [26] is applied to the incompressible fluid flow over a rectangular open cavity, from the linear to the saturated regime (as detailed in [15]), in order to understand the evolution of span-wise instabilities of the flow and the interactions between different dynamic modes. The numerical solutions required to construct the data-sequences of snapshots were obtained by means of a three-dimensional unsteady DNS solver. The most relevant DMD modes and associated oscillation frequencies are then compared to the ones obtained using linear stability analysis, allowing us to assess the accuracy of the aforementioned snapshot-based decomposition. The numerical details of the different methods used for the present investigation are explained in section 2. The BiGlobal and DNS results, and the DMD analysis performed are presented in section 3. Finally, the most significant conclusions are summarized in section 4.

Section snippets

Problem description

A schematic representation of the flow configuration is depicted in Fig. 1. The parameters defining the flow are the Reynolds number based on cavity depth (ReD), and the incoming boundary layer displacement thickness (θ0/D). The geometrical parameters of the problem are the length-to-depth aspect ratio (L/D), and the wavelength in the span-wise direction (Lz), related with the corresponding wavenumber β=2π/Lz. The case studied here corresponds to the experimental Case B of [16], with ReD=2400

Regime I

The preliminary study conducted in [15] shows that the behavior of the DNS in the linear growth phase (region I of Fig. 4) matches perfectly with the results predicted by linear analysis, with the most unstable eigenmode appearing and growing exponentially with the predicted σ. Those results will not be repeated here. The aforementioned study has also served as a validation for the DNS code.

Regime II

In regime II, after the linear growth phase, saturation of the leading mode occurs (t=500) and the

Conclusions

This work shows the power of the DMD tool to analyze a complex problem. The separation of structures provided by this technique allows to identify the two most relevant linear modes well outside the linear regime of the DNS computation. The description of the dominant structures in the five different flow regimes allows to better understand how the process of saturation affects the morphology of the modes, as well as their characteristic frequencies, without having to deal with the effect of

Conflict of interest statement

The authors have no conflict of interest to declare.

Acknowledgements

The authors would like to thank J. Basley and E. Ferrer for the many fruitful discussions. The support of the Marie Curie Grants PIRSES-GA-2009-247651 “FP7-PEOPLE-IRSES: ICOMASEF Instability and Control of Massively Separated Flows” and PITN-GA-2013-608087 “FP7-PEOPLE-ITN: AIRUP Airbus-UPM European Industrial Doctorate in mathematical methods applied to aircraft design” are gratefully acknowledged.

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