Low-Reynolds-number wakes of elliptical cylinders: from the circular cylinder to the normal flat plate
TL;DR: In this article, the Strouhal number and drag coefficient variations with Reynolds number are documented for the two-dimensional shedding regime for elliptic cylinders, and different three-dimensional transition modes are also examined using Floquet stability analysis based on computed 2D periodic base flows.
Abstract: While the wake of a circular cylinder and, to a lesser extent, the normal flat plate have been studied in considerable detail, the wakes of elliptic cylinders have not received similar attention. However, the wakes from the first two bodies have considerably different characteristics, in terms of three-dimensional transition modes, and near- and far-wake structure. This paper focuses on elliptic cylinders, which span these two disparate cases. The Strouhal number and drag coefficient variations with Reynolds number are documented for the two-dimensional shedding regime. There are considerable differences from the standard circular cylinder curve. The different three-dimensional transition modes are also examined using Floquet stability analysis based on computed two-dimensional periodic base flows. As the cylinder aspect ratio (major to minor axis) is decreased, mode A is no longer unstable for aspect ratios below 0.25, as the wake deviates further from the standard Benard–von Karman state. For still smaller aspect ratios, another three-dimensional quasi-periodic mode becomes unstable, leading to a different transition scenario. Interestingly, for the 0.25 aspect ratio case, mode A restabilises above a Reynolds number of approximately 125, allowing the wake to return to a two-dimensional state, at least in the near wake. For the flat plate, three-dimensional simulations show that the shift in the Strouhal number from the two-dimensional value is gradual with Reynolds number, unlike the situation for the circular cylinder wake once mode A shedding develops. Dynamic mode decomposition is used to characterise the spatially evolving character of the wake as it undergoes transition from the primary Benard–von Karman-like near wake into a two-layered wake, through to a secondary Benard–von Karman-like wake further downstream, which in turn develops an even longer wavelength unsteadiness. It is also used to examine the differences in the two- and three-dimensional near-wake state, showing the increasing distortion of the two-dimensional rollers as the Reynolds number is increased.
Citations
More filters
TL;DR: In this paper, a review of techniques for analyzing fluid flow data is presented, with the aim of extracting simplified models that capture the essential features of these flows, in order to gain insight into the flow physics, and potentially identify mechanisms for controlling these flows.
Abstract: Advances in experimental techniques and the ever-increasing fidelity of numerical simulations have led to an abundance of data describing fluid flows. This review discusses a range of techniques for analyzing such data, with the aim of extracting simplified models that capture the essential features of these flows, in order to gain insight into the flow physics, and potentially identify mechanisms for controlling these flows. We review well-developed techniques, such as proper orthogonal decomposition and Galerkin projection, and discuss more recent techniques developed for linear systems, such as balanced truncation and dynamic mode decomposition (DMD). We then discuss some of the methods available for nonlinear systems, with particular attention to the Koopman operator, an infinite-dimensional linear operator that completely characterizes the dynamics of a nonlinear system and provides an extension of DMD to nonlinear systems.
567 citations
Cites methods from "Low-Reynolds-number wakes of ellipt..."
...…et al. (2014) Transition of Mach 6 boundary layer with roughness element (DNS) Thompson et al. (2014) Flow past elliptic cylinders (DNS) Tissot et al. (2014) Flow past a cylinder (experiment), mode extraction for reduced-order modeling Dunne & McKeon (2015) Dynamic stall on a…...
[...]
Posted Content•
TL;DR: The proposed CNN-based approximation procedure has a profound impact on the parametric design of bluff bodies and the feedback control of separated flows.
Abstract: We present an efficient deep learning technique for the model reduction of the Navier-Stokes equations for unsteady flow problems. The proposed technique relies on the Convolutional Neural Network (CNN) and the stochastic gradient descent method. Of particular interest is to predict the unsteady fluid forces for different bluff body shapes at low Reynolds number. The discrete convolution process with a nonlinear rectification is employed to approximate the mapping between the bluff-body shape and the fluid forces. The deep neural network is fed by the Euclidean distance function as the input and the target data generated by the full-order Navier-Stokes computations for primitive bluff body shapes. The convolutional networks are iteratively trained using the stochastic gradient descent method with the momentum term to predict the fluid force coefficients of different geometries and the results are compared with the full-order computations. We attempt to provide a physical analogy of the stochastic gradient method with the momentum term with the simplified form of the incompressible Navier-Stokes momentum equation. We also construct a direct relationship between the CNN-based deep learning and the Mori-Zwanzig formalism for the model reduction of a fluid dynamical system. A systematic convergence and sensitivity study is performed to identify the effective dimensions of the deep-learned CNN process such as the convolution kernel size, the number of kernels and the convolution layers. Within the error threshold, the prediction based on our deep convolutional network has a speed-up nearly four orders of magnitude compared to the full-order results and consumes an insignificant fraction of computational resources. The proposed CNN-based approximation procedure has a profound impact on the parametric design of bluff bodies and the feedback control of separated flows.
82 citations
Cites result from "Low-Reynolds-number wakes of ellipt..."
...This behavior is also observed for Re = 100 in the recent study by [32]....
[...]
TL;DR: In this paper, a reduced-order model (ROM) was proposed to couple an incompressible flow with a transversely vibrating bluff body in a state-space format.
Abstract: We present an effective reduced-order model (ROM) technique to couple an incompressible flow with a transversely vibrating bluff body in a state-space format. The ROM of the unsteady wake flow is based on the Navier–Stokes equations and is constructed by means of an eigensystem realization algorithm (ERA). We investigate the underlying mechanism of vortex-induced vibration (VIV) of a circular cylinder at low Reynolds number via linear stability analysis. To understand the frequency lock-in mechanism and self-sustained VIV phenomenon, a systematic analysis is performed by examining the eigenvalue trajectories of the ERA-based ROM for a range of reduced oscillation frequency , while maintaining fixed values of the Reynolds number ( ) and mass ratio ( ). The effects of the Reynolds number , the mass ratio and the rounding of a square cylinder are examined to generalize the proposed ERA-based ROM for the VIV lock-in analysis. The considered cylinder configurations are a basic square with sharp corners, a circle and three intermediate rounded squares, which are created by varying a single rounding parameter. The results show that the two frequency lock-in regimes, the so-called resonance and flutter, only exist when certain conditions are satisfied, and the regimes have a strong dependence on the shape of the bluff body, the Reynolds number and the mass ratio. In addition, the frequency lock-in during VIV of a square cylinder is found to be dominated by the resonance regime, without any coupled-mode flutter at low Reynolds number. To further discern the influence of geometry on the VIV lock-in mechanism, we consider the smooth curve geometry of an ellipse and two sharp corner geometries of forward triangle and diamond-shaped bluff bodies. While the ellipse and diamond geometries exhibit the flutter and mixed resonance–flutter regimes, the forward triangle undergoes only the flutter-induced lock-in for at . In the case of the forward triangle configuration, the ERA-based ROM accurately predicts the low-frequency galloping instability. We observe a kink in the amplitude response associated with 1:3 synchronization, whereby the forward triangular body oscillates at a single dominant frequency but the lift force has a frequency component at three times the body oscillation frequency. Finally, we present a stability phase diagram to summarize the VIV lock-in regimes of the five smooth-curve- and sharp-corner-based bluff bodies. These findings attempt to generalize our understanding of the VIV lock-in mechanism for bluff bodies at low Reynolds number. The proposed ERA-based ROM is found to be accurate, efficient and easy to use for the linear stability analysis of VIV, and it can have a profound impact on the development of control strategies for nonlinear vortex shedding and VIV.
61 citations
Cites methods from "Low-Reynolds-number wakes of ellipt..."
...…physical features of interest, a linear model based on the model reduction provides a way to perform stability analysis for the flow past a bluff body and to design active control strategies (Marquet, Sipp & Jacquin 2008; Mettot, Renac & Sipp 2014; Thompson et al. 2014; Flinois & Morgans 2016)....
[...]
TL;DR: In this paper, the transition from a circular cylinder to a two-layered and secondary vortex street is quantified by a new method based on the time-averaged transverse velocity field.
Abstract: Instabilities and flow characteristics in the far wake of a circular cylinder are examined through direct numerical simulations. The transitions to the two-layered and secondary vortex streets are quantified by a new method based on the time-averaged transverse velocity field. Two processes for the transition to the secondary vortex street are observed: (i) the merging of two same-sign vortices over a range of low Reynolds numbers ( ) between 200 and 300, and (ii) the pairing of two opposite-sign vortices, followed by the merging of the paired vortices into subsequent vortices, over a range of between 400 and 1000. Single vortices may be generated between the merging cycles due to mismatch of the vortices. The irregular merging process results in flow irregularity and an additional frequency signal (in addition to the primary vortex shedding frequency ) in the two-layered and secondary vortex streets. In particular, a gradual energy transfer from to with distance downstream is observed in the two-layered vortex street prior to the merging. The frequency spectra of are broad-band for –300 but become increasingly sharp-peaked with increasing because the vortex merging process becomes increasingly regular. The ratio of the sharp-peaked frequencies and is equal to the ratio of the numbers of vortices observed after and before the merging. The general conclusions drawn from a circular cylinder are expected to be applicable to other bluff bodies.
40 citations
Cites background from "Low-Reynolds-number wakes of ellipt..."
...In contrast, Thompson et al. (2014) found that at Re = 150 the secondary vortices in the wake of an elliptical cylinder (with an aspect ratio of 0.25) were formed from merging of the two-layered vortices....
[...]
...…case at higher Re values but much smaller than the circular cylinder case at moderate Re values of 150 and 200), such as an elliptical cylinder (Thompson et al. 2014), a rectangular cylinder, a flat plate and some multi-cylinder arrangements (e.g. two circular cylinders in tandem reported by…...
[...]
...…total of 46 091 macro-elements, of which approximately 44 000 macro-elements are placed in the region of x/D > 2 because the transition to the secondary vortex street (especially at low Re values such as Re= 200) is extremely disturbance-dependent (Thompson et al. 2014) and therefore meshdependent....
[...]
...…numerical simulations, Vorobieff et al. (2002) and Kumar & Mittal (2012) observed the second transition for Re= 200 at approximately x/D= 60 and 100, respectively, while Thompson et al. (2014) did not observe the secondary vortex street at Re= 200 up to a wake domain length of x/D = 280....
[...]
TL;DR: In this article, the authors present the results of numerical stability analysis of the wake of an elliptical cylinder and show that the three modes A, B and QP are all stabilized by increasing the aspect ratio of the ellipse.
Abstract: This paper presents the results of numerical stability analysis of the wake of an elliptical cylinder. Aspect ratios where the ellipse is longer in the streamwise direction than in the transverse direction are considered. The focus is on the dependence on the aspect ratio of the ellipse of the various bifurcations to three-dimensional flow from the two-dimensional Karman vortex street. It is shown that the three modes present in the wake of a circular cylinder (modes A, B and QP) are present in the ellipse wake, and that in general they are all stabilized by increasing the aspect ratio of the ellipse. Two new pertinent modes are found: one long-wavelength mode with similarities to mode A, and a second that is only unstable for aspect ratios greater than approximately 1.75, which has similar spatiotemporal symmetries to mode B but has a distinct spatial structure. Results from fully three-dimensional simulations are also presented confirming the existence and growth of these two new modes in the saturated wakes.
35 citations
References
More filters
TL;DR: In this article, the authors propose a definition of vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor, which captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers.
Abstract: Considerable confusion surrounds the longstanding question of what constitutes a vortex, especially in a turbulent flow. This question, frequently misunderstood as academic, has recently acquired particular significance since coherent structures (CS) in turbulent flows are now commonly regarded as vortices. An objective definition of a vortex should permit the use of vortex dynamics concepts to educe CS, to explain formation and evolutionary dynamics of CS, to explore the role of CS in turbulence phenomena, and to develop viable turbulence models and control strategies for turbulence phenomena. We propose a definition of a vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor ${\bm {\cal S}}^2 + {\bm \Omega}^2$ are respectively the symmetric and antisymmetric parts of the velocity gradient tensor ${\bm \Delta}{\bm u}$. This definition captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers, unlike a pressure-minimum criterion. We compare our definition with prior schemes/definitions using exact and numerical solutions of the Euler and Navier–Stokes equations for a variety of laminar and turbulent flows. In contrast to definitions based on the positive second invariant of ${\bm \Delta}{\bm u}$ or the complex eigenvalues of ${\bm \Delta}{\bm u}$, our definition accurately identifies the vortex core in flows where the vortex geometry is intuitively clear.
5,837 citations
TL;DR: In this paper, a finite-difference method for solving the time-dependent Navier-Stokes equations for an incompressible fluid is introduced, which is equally applicable to problems in two and three space dimensions.
Abstract: A finite-difference method for solving the time-dependent Navier- Stokes equations for an incompressible fluid is introduced. This method uses the primitive variables, i.e. the velocities and the pressure, and is equally applicable to problems in two and three space dimensions. Test problems are solved, and an ap- plication to a three-dimensional convection problem is presented.
4,991 citations
TL;DR: In this article, 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.
Abstract: 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.
4,150 citations
TL;DR: A review of wake vortex dynamics can be found in this article, with a focus on the three-dimensional aspects of nominally two-dimensional wake flows, as well as the discovery of several new phenomena in wakes.
Abstract: Since the review of periodic flow phenomena by Berger & Wille (1972) in this journal, over twenty years ago, there has been a surge of activity regarding bluff body wakes. Many of the questions regarding wake vortex dynamics from the earlier review have now been answered in the literature, and perhaps an essential key to our new understandings (and indeed to new questions) has been the recent focus, over the past eight years, on the three-dimensional aspects of nominally two-dimensional wake flows. New techniques in experiment, using laser-induced fluorescence and PIV (Particle-Image-Velocimetry), are vigorously being applied to wakes, but interestingly, several of the new discoveries have come from careful use of classical methods. There is no question that strides forward in understanding of the wake problem are being made possible by ongoing three- dimensional direct numerical simulations, as well as by the surprisingly successful use of analytical modeling in these flows, and by secondary stability analyses. These new developments, and the discoveries of several new phenomena in wakes, are presented in this review.
3,206 citations
TL;DR: In this article, a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an infinite-dimensional linear operator associated with the full nonlinear system, is presented.
Abstract: We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an infinite-dimensional linear operator associated with the full nonlinear system. These modes, referred to as Koopman modes, are associated with a particular observable, and may be determined directly from data (either numerical or experimental) using a variant of a standard Arnoldi method. They have an associated temporal frequency and growth rate and may be viewed as a nonlinear generalization of global eigenmodes of a linearized system. They provide an alternative to proper orthogonal decomposition, and in the case of periodic data the Koopman modes reduce to a discrete temporal Fourier transform. The Arnoldi method used for computations is identical to the dynamic mode decomposition recently proposed by Schmid & Sesterhenn (Sixty-First Annual Meeting of the APS Division of Fluid Dynamics, 2008), so dynamic mode decomposition can be thought of as an algorithm for finding Koopman modes. We illustrate the method on an example of a jet in crossflow, and show that the method captures the dominant frequencies and elucidates the associated spatial structures.
1,591 citations