Quantitative analysis of the kinematics and induced aerodynamic loading of individual vortices in vortex-dominated flows: A computation and data-driven approach
Introduction
The dynamical influence of vortex structures is key to a wide range of fluid-flow phenomena [1], [2], [3]. This is particularly true in vortex-dominated flows, where coherent vortex structures and their interactions exert a dominant influence on the force/moment production on immersed surfaces. In the case of fluid-structure interaction problems in particular, these vortex structures can drive the motion of immersed bodies, which in turn leads to the generation of additional vortices, and gives rise to complex non-linearities in the structural response. Such behavior is relevant in a number of arenas including bluff-body oscillation, wing flutter, biological propulsion, and physiological flows [4], [5], [6], [7], [8].
A prototypical problem that manifests much of the complexity associated with such flows – vortex dominated behavior and complex vortex interactions – is the flow around a pitching airfoil. To illustrate this, Fig. 1(a) shows a data set consisting of 165 two-dimensional Navier-Stokes simulations of flow past a sinusoidally pitching airfoil. This ensemble of cases represents a parameter sweep through the pitching frequency, , and amplitude, (both defined in the caption of Fig. 1). The complex vortex dynamics inherent in such a problem is highlighted by the snapshots of the vorticity field shown for some select cases in Figs. 1(b)-(e). These snapshots show that this problem is characterized by a variety of vortex patterns that are quite sensitive to changes in oscillation kinematics. The snapshots in Figs. 1(b)-(d), corresponding to the same oscillation amplitude and relatively small differences in oscillation frequency, all show the growth of a strong leading-edge vortex (LEV), along with several other distinct, interacting vortices. However, we see that the phase of the LEV-growth is slightly different in each case, although the snapshots are at the same phase in the oscillation.
One way to assess the overall aeroelastic interaction of the flow with the airfoil is to evaluate the energy that could potentially be extracted by the airfoil from the surrounding flow as a function of oscillation kinematics. The contours in Fig. 1(a) show this energy extraction, which is defined as, where is coefficient of moment about mid-chord, and is the angular velocity (see ref. [9] for details). This energy is known to determine the flow-induced oscillation response as well as energy-harvesting potential of the airfoil [9], [10]. We see in Fig. 1(a) that kinematic states with positive as well as negative energy extraction are possible, which is primarily dictated by the phase difference between the dominant aerodynamic loading mechanisms and the oscillation kinematics [9], [11]. As evidenced by these energy contours, the slight variation in oscillation kinematics for the cases shown in Figs. 1(b)-(d), which results in subtle changes in the phase of the LEV as well as different vortex interactions, has consequences for the sign of energy extraction and therefore the dynamics of flow-induced motion. Additionally, all these cases exhibit very different vortex interactions close to the leading and trailing-edges, and this can influence the force production, propulsion, and fluid-structure interaction [12], [13], [14] associated with such configurations.
Thus we see that these flows are generally characterized by several interacting vortex structures. The kinematics and the dynamical influence of these vortices on an immersed/control surface are governed by their inception, phase, location, as well as their interactions with each other and the immersed surface. While prior studies have highlighted the dynamical importance of specific vortex structures such as the LEV [8], as well as distinct patterns in vortex shedding [4], they have largely been limited to qualitatively correlating the occurrence and evolution of these structures to the observed dynamics of the problem. The question of precisely quantifying their dynamical influence, in terms of force and moment production on an immersed body for instance, has not been adequately addressed. Moreover, a systematic and rigorous way to partition the influence of the various vortex structures generally present in such problems, and to identify those that are the most dynamically important to that specific problem, do not currently exist.
The analysis of these general vortex-dominated flows, which are usually characterized by several distinct, interacting vortices therefore requires two key elements: (1) the isolation, tracking and segmentation (i.e. determination of size and shape) of multiple individual vortex structures that are generated in the unsteady flow; (2) the rigorous quantification of the aerodynamic forces and moments induced on an immersed body by each of these vortex structures. In this work, we propose a computational framework to perform such an analysis of high-dimensional, time-resolved flow-fields at the level of individual vortex structures. In particular, we combine a rigorous force and moment partitioning method (FMPM) which enables the precise estimation of the aerodynamic loads induced by individual vortices, with data-driven techniques that facilitate the efficient use of this method in complex vortex-dominated flows. The result of this combined physics-based data-driven approach is a versatile and largely automated framework that can decode the vortex kinematics and dynamics of such problems by isolating each vortex structure in the flow, and evaluating its dynamical effect on an immersed body through its entire spatio-temporal evolution.
A central piece in this analysis framework is a mathematically rigorous method for partitioning fluid dynamic forces and moments on an immersed body into contributions from individual vortices, as well as other viscous and inviscid forcing mechanisms. The method used here, which is based on an exact analytical formulation derived from the Navier-Stokes equations, follows from the work of Quartappelle and Napolitano [15], with extensions by Chang [16] and Zhang et al. [17] for the partitioning of flow-induced forces on an immersed body into physically relevant mechanisms. Here we extend the formulation that has been applied to flow-induced forces, to also include the partitioning of flow-induced moments. This is particularly relevant for problems with rotational/pitch degrees-of-freedom. Furthermore, the specific formulation developed in this work results in force/moment components that have clear physical interpretations, thereby allowing us to relate, using first principles, the mechanisms behind force/moment generation in incompressible flows to the local kinematics of the flow. We note that the partitioning of flow-induced pressure forces used here is not unique, and there have been other mathematically rigorous force partitioning formulations [18], [19], as well as extensions to the formulation used here [20], [21], [22], [23].
In the context of analyzing force production in vortex dominated flows, these partitioning methods have proven to be very useful in delineating the overall contribution of vortex-induced effects, as well as other physically relevant forcing mechanisms, in various unsteady aerodynamics and fluid structure interaction problems [24], [17], [14], [25], [26]. While there have also been other efforts to quantify the force production from vortex-induced mechanisms, these have primarily used inviscid vortex models and vortex impulse formulations [27], [28], or simple inviscid theory to separate added-mass from so-called “vortex-flow force” [29], [30], [31]. However, conceptual validity of these latter methods has been questioned [32], and the extension of these methods to general viscous flows is unclear. Further, these prior efforts have mostly focused on analyzing the global effect of vorticity in the flow. Estimations of force/moment production due to local vortical regions have thus far been limited to determining vortex-induced forces within static spatial volumes of the flow-field [24], [26], rather than individual vortices. This is primarily due to the computational complexity associated with accurately isolating and tracking several interacting vortical regions that are evolving in highly complicated, time-varying flow fields. Therefore, as mentioned earlier, a critical aspect in deploying these rigorous force/moment estimation methods for the analysis of highly dynamic vortex-dominated flow-fields is a systematic way to individually isolate and track these vortical regions.
Here we leverage data-driven clustering techniques to perform this task of isolating and tracking several vortices in high-dimensional, unsteady flow-fields. These techniques are a class of unsupervised statistical inference methods [33], [34] which attempt to find clusters of data that share similar characteristics within a large set of data. In previous studies, this has been used in identifying coherent structures in a flow-field, by clustering regions that share similar dynamic or Lagrangian behavior [35], [36], [37], [38], [39], [40]. In the context of this work, clustering provides an automated way to isolate an arbitrary number of spatial regions corresponding to individual vortex structures from high-dimensional flow-fields. Furthermore, this data-driven approach also facilitates the spatio-temporal tracking of these vortex structures, as well as the grouping of vortices in distinct categories (such as LEVs or TEVs) based on various attributes. The automated isolation of these distinct vortical regions therefore provides an efficient way to employ the aforementioned force and moment partitioning methods in precisely quantifying their contributions to force/moment production. Furthermore, the spatio-temporal tracking of each of these structures also allows us to relate their evolution and interactions to the dynamics of the problem, which has been shown to be very insightful in past work [41], [42], [43]. Hence this combined physics-based and data driven approach provides an automated and rigorous method to analyze vortex kinematics as well as the force and moment production in complex vortex dominated flows.
While the framework described above allows the detailed dynamical analysis of a single flow, the automated nature of the method makes it particularly well suited for examining a large ensemble of flows. We demonstrate this by applying our method to the data set generated from the 165 pitching airfoil simulations represented in Fig. 1. In the current study, application of the method to this large ensemble is preceded by a data-driven reduction in the “rank” of the data set via identification of distinct vortex-dynamic regimes in this flow. This identification of distinct regimes has similarities to previous studies on the wake of oscillating cylinders [44], [45], [46], [4], [47] as well as in biomimetic propulsion [48], [49], [50]. However, the visual identification employed in these past studies is impractical for more complex flows; a fact that has motivated data-driven approaches to this problem [51], [52], [53]. However, these prior data-driven efforts have focused on idealized wakes, using point vortices for example, and have mostly assumed a-priori knowledge of the possible wake patterns in order to classify observed wakes into these known categories. Here we demonstrate a more robust method to identify distinct flow regimes from high-dimensional, time-resolved flow-fields by using dimensionality reduction and clustering techniques to identify patterns in large ensembles of these flow-fields. This approach presented here is robust and also obviates the need for a-priori definition of wake-patterns.
Hence, the framework developed here allows the analysis of a large ensemble of flows at the resolution of individual vortices, and includes a quantification of the vortex kinematics as well as and their dynamical influence. In the following sections, we first describe the methods developed to enable this analysis, i.e. the force and moment partitioning method, the vortex isolation, tracking and dynamical analysis methodology, and lastly the vortex-regime identification procedure. This description of the methods is then followed by an application of these methods to the pitching airfoil data set mentioned above, followed by concluding remarks.
Section snippets
Force and moment partitioning method
We first provide an overview of the force and moment partitioning methods (FMPM) used in this work, since this method drives much of the data-driven methodology described in the current paper. The partitioning is described with reference to the configuration shown schematically in Fig. 2, where we have a body with its surface represented as B, immersed in a fluid domain of volume . The unit normal vector points into the surface B at every point along it, and the fluid flow around this body
Automated tracking of vortices and estimation of aerodynamic loads
While the force and moment partitioning method (FMPM) can determine the aerodynamic loading associated with any vorticity-containing region of the flow, complex vortex-dominated flows (such as in Fig. 1, Fig. 3) typically contain multiple vortices which interact, deform as well as change their volume and location as they are advected with the flow. Thus, to effectively apply FMPM to such flows, appropriate methods are required to isolate, track, as well as determine the time-varying volumes
Application to pitching airfoils
We now present an application of the methods described in the previous sections to the configuration which was introduced earlier in Fig. 1: the two-dimensional flow past an airfoil which is undergoing prescribed sinusoidal pitch oscillations over a range of amplitudes and frequencies of oscillation. As we will show, this canonical problem exhibits numerous distinct vortex-dynamic regimes, which have been shown to be more complex [63] than those behind the more well-studied problem of an
Summary
In this work, we have presented a data-driven and physics-based computational framework for the analysis of vortex-dominated flows. The main focus is a flexible and automated method to accurately evaluate kinematic quantities and the aerodynamic loading of individual vortex structures in complex vortex-dominated flows. This method uses a novel force and moment partitioning formulation which breaks down the aerodynamic loading on an immersed body into physically insightful components. In the
CRediT authorship contribution statement
Karthik Menon: Conceptualization, Methodology, Software, Writing – original draft. Rajat Mittal: Conceptualization, Methodology, Supervision, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work is supported by the Air Force Office of Scientific Research Grant Number FA 9550-16-1-0404, monitored by Dr. Gregg Abate. This work also benefited from the computational resources at Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562, through allocation number TG-CTS100002, and at the Maryland Advanced Research Computing Center (MARCC).
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