Three-dimensional Floquet stability analysis of the wake of a circular cylinder
TL;DR: In this paper, a global numerical stability analysis of the periodic wake of a circular cylinder for Reynolds numbers between 140 and 300 is presented, showing that the two-dimensional wake becomes (absolutely) linearly unstable to three-dimensional perturbations at a critical Reynolds number of 1885±10.
Abstract: Results are reported from a highly accurate, global numerical stability analysis of the periodic wake of a circular cylinder for Reynolds numbers between 140 and 300 The analysis shows that the two-dimensional wake becomes (absolutely) linearly unstable to three-dimensional perturbations at a critical Reynolds number of 1885±10 The critical spanwise wavelength is 396 ± 002 diameters and the critical Floquet mode corresponds to a ‘Mode A’ instability At Reynolds number 259 the two-dimensional wake becomes linearly unstable to a second branch of modes with wavelength 0822 diameters at onset Stability spectra and corresponding neutral stability curves are presented for Reynolds numbers up to 300
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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: The intent of this document is to provide an introduction to modal analysis that is accessible to the larger fluid dynamics community and presents a brief overview of several of the well-established techniques.
Abstract: Simple aerodynamic configurations under even modest conditions can exhibit complex flows with a wide range of temporal and spatial features. It has become common practice in the analysis of these flows to look for and extract physically important features, or modes, as a first step in the analysis. This step typically starts with a modal decomposition of an experimental or numerical dataset of the flowfield, or of an operator relevant to the system. We describe herein some of the dominant techniques for accomplishing these modal decompositions and analyses that have seen a surge of activity in recent decades [1–8]. For a nonexpert, keeping track of recent developments can be daunting, and the intent of this document is to provide an introduction to modal analysis that is accessible to the larger fluid dynamics community. In particular, we present a brief overview of several of the well-established techniques and clearly lay the framework of these methods using familiar linear algebra. The modal analysis techniques covered in this paper include the proper orthogonal decomposition (POD), balanced proper orthogonal decomposition (balanced POD), dynamic mode decomposition (DMD), Koopman analysis, global linear stability analysis, and resolvent analysis.
1,110 citations
Cites methods from "Three-dimensional Floquet stability..."
...We note that time-periodic flow can also be utilized as the base flow [148]; as we will see in the first illustrative example below [149]....
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TL;DR: An extensive review of the literature in V&V in computational fluid dynamics (CFD) is presented, methods and procedures for assessing V &V are discussed, and a relatively new procedure for estimating experimental uncertainty is given that has proven more effective at estimating random and correlated bias errors in wind-tunnel experiments than traditional methods.
Abstract: Verification and validation (V&V) are the primary means to assess accuracy and reliability in computational simulations. This paper presents an extensive review of the literature in V&V in computational fluid dynamics (CFD), discusses methods and procedures for assessing V&V, and develops a number of extensions to existing ideas. The review of the development of V&V terminology and methodology points out the contributions from members of the operations research, statistics, and CFD communities. Fundamental issues in V&V are addressed, such as code verification versus solution verification, model validation versus solution validation, the distinction between error and uncertainty, conceptual sources of error and uncertainty, and the relationship between validation and prediction. The fundamental strategy of verification is the identification and quantification of errors in the computational model and its solution. In verification activities, the accuracy of a computational solution is primarily measured relative to two types of highly accurate solutions: analytical solutions and highly accurate numerical solutions. Methods for determining the accuracy of numerical solutions are presented and the importance of software testing during verification activities is emphasized. The fundamental strategy of validation is to assess how accurately the computational results compare with the experimental data, with quantified error and uncertainty estimates for both. This strategy employs a hierarchical methodology that segregates and simplifies the physical and coupling phenomena involved in the complex engineering system of interest. A hypersonic cruise missile is used as an example of how this hierarchical structure is formulated. The discussion of validation assessment also encompasses a number of other important topics. A set of guidelines is proposed for designing and conducting validation experiments, supported by an explanation of how validation experiments are different from traditional experiments and testing. A description is given of a relatively new procedure for estimating experimental uncertainty that has proven more effective at estimating random and correlated bias errors in wind-tunnel experiments than traditional methods. Consistent with the authors’ contention that nondeterministic simulations are needed in many validation comparisons, a three-step statistical approach is offered for incorporating experimental uncertainties into the computational analysis. The discussion of validation assessment ends with the topic of validation metrics, where two sample problems are used to demonstrate how such metrics should be constructed. In the spirit of advancing the state of the art in V&V, the paper concludes with recommendations of topics for future research and with suggestions for needed changes in the implementation of V&V in production and commercial software.
948 citations
Cites background from "Three-dimensional Floquet stability..."
...Our review of the literature has identified a number of authors who have contributed to the verification of CFD solutions [11, 28, 33, 34, 3942, 45, 47, 49, 57, 60, 65, 74, 87-89, 92, 97, 110, 116, 117, 120, 127, 129-131, 137-139, 141, 142, 144, 147, 165, 171, 177, 209, 218, 219, 221, 228, 229, 273, 274, 277, 278, 281, 298, 306-309, 318, 325, 327, 329, 333-337, 340, 347, 350, 351, 353]....
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...Examples of benchmark PDE solutions in fluid dynamics are the following: incompressible laminar flow over a semi-infinite flat plate [87, 329, 347]; incompressible laminar flow over a parabolic plate [49, 88, 92]; incompressible laminar flow in a square cavity driven by a moving wall [34 , 130, 131, 141, 147, 228, 298, 318]; laminar nat convection in a square cavity [89, 165], incompressible laminar flow over an infinite-length circular cylinder [28, 33, 177, 218]; and incompressible laminar flow over a backward-facing step, with and without heat transfer [40, 127, 138, 142]....
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TL;DR: In this article, the effects of Reynolds number in the nominal case of an infinitely long and non-confined cylinder in a smooth oncoming flow are discussed, from about Re = 47 to 2 x 10(5), i.e., from the onset of vortex shedding up to the end of the subcritical regime.
Abstract: Apart from providing some new experimental data the paper reviews previous investigations concerning fluctuating lift acting on a stationary circular cylinder in cross-flow. In particular, effects of Reynolds number in the nominal case of an infinitely long and nonconfined cylinder in a smooth oncoming flow are discussed. The Reynolds number range covered is from about Re = 47 to 2 x 10(5), i.e., from the onset of vortex shedding up to the end of the subcritical regime. At the beginning of the subcritical regime (Reesimilar or equal to0.3 x 10(3)) a spanwise correlation length of about 30 cylinder diameters is indicated, the correlation function being based on near-cylinder velocity fluctuations in outer parts of the separated shear layer. In between Reynolds numbers 1.6 x 10(3) and 20 x 10(3), an approximate 10-fold increase in the sectional r.m.s. lift coefficient is indicated. This range contains a fundamental change-over from one flow state to another, starting off at Re similar or equal to 5 x 103 and seemingly fully developed at Re similar or equal to 8 x 10(3). (C) 2002 Elsevier Science Ltd. All rights reserved. (Less)
939 citations
Cites background or methods from "Three-dimensional Floquet stability..."
...The constants have been adjusted towards the results that were considered the most reliable in terms in terms of accuracy and large enough calculation domain sizes, in particular those of Rosenfeld (1994), Beaudan and Moin (1994), Barkley and Henderson (1996), He and Doolen (1997), Park et al....
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...The present measurements indicated a spanwise correlation length of about 7 diameters at the lowest attained Reynolds number, Re 1⁄4 230; which is about twice this wavelength (Barkley and Henderson, 1996). As shown in Norberg (1994), there is, within ReC165 to ReC230; a relatively weak influence of the necessary aspect ratio to obtain independent global results....
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...The spanwise correlation length found at Re 1⁄4 0:24 10 is of the same order as twice the wavelength of the most unstable mode A instability (Barkley and Henderson, 1996)....
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...Strouhal number versus Reynolds number: X; Bearman (1969); Norberg (1987a, 1994): J; laminar shedding; &; wake transition; n; turbulent shedding; - - -, Barkley and Henderson (1996), 2-D; ; Kwon and Choi (1996), 2-D; þ; Posdziech and Grundmann (2000), 2-D; ——, formulas in Appendix A....
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...The present measurements indicated a spanwise correlation length of about 7 diameters at the lowest attained Reynolds number, Re 1⁄4 230; which is about twice this wavelength (Barkley and Henderson, 1996)....
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TL;DR: In this paper, the dynamics of open flows are considered as a superposition of linear or nonlinear instability waves that behave at each streamwise station as if the flow were homogeneous in the streamwise direction.
Abstract: The objective of this review is to critically assess the different approaches developed in recent years to understand the dynamics of open flows such as mixing layers, jets, wakes, separation bubbles, boundary layers, and so on. These complex flows develop in extended domains in which fluid particles are continuously advected downstream. They behave either as noise amplifiers or as oscillators, both of which exhibit strong nonlinearities (Huerre & Monkewitz 1990). The local approach is inherently weakly nonparallel and it assumes that the basic flow varies on a long length scale compared to the wavelength of the instability waves. The dynamics of the flow is then considered as a superposition of linear or nonlinear instability waves that, at leading order, behave at each streamwise station as if the flow were homogeneous in the streamwise direction. In the fully global context, the basic flow and the instabilities do not have to be characterized by widely separated length scales, and the dynamics is then viewed as the result of the interactions between Global modes living in the entire physical domain with the streamwise direction as an eigendirection. This second approach is more and more resorted to as a result of increased computational capability. The earlier review of Huerre & Monkewitz (1990) emphasized how local linear theory can account for the noise amplifier behavior as well as for the onset of a Global mode. The present survey first adopts the opposite point of view by demonstrating how fully global theory accounts for the noise amplifier behavior of open flows. From such a perspective, there is strong emphasis on the very peculiar nonorthogonality of linear Global modes, which in turn allows a novel interpretation of recent numerical simulations and experimental observations. The nonorthogonality of linear Global modes also imposes severe constraints on the extension of linear global theory to the fully nonlinear regime. When the flow is weakly nonparallel, this limitation is so severe that the linear Global mode theory is of little help. It is then much more appropriate to develop a fully nonlinear formulation involving the presence of a front separating the base state region from the bifurcated state region.
725 citations
Cites background from "Three-dimensional Floquet stability..."
...For the cylinder wake, the primary and secondary instabilities were studied by Jackson (1987), Zebib (1987), Noack & Eckelmann (1994), and Barkley & Henderson (1996), among others....
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References
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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 review of recent developments in the hydro- dynamic stability theory of spatially developing flows pertaining to absolute/convective and local/global instability concepts is presented.
Abstract: The goal of this survey is to review recent developments in the hydro dynamic stability theory of spatially developing flows pertaining to absolute/convective and local/global instability concepts. We wish to dem onstrate how these notions can be used effectively to obtain a qualitative and quantitative description of the spatio-temporal dynamics of open shear flows, such as mixing layers, jets, wakes, boundary layers, plane Poiseuille flow, etc. In this review, we only consider open flows where fluid particles do not remain within the physical domain of interest but are advected through downstream flow boundaries. Thus, for the most part, flows in "boxes" (Rayleigh-Benard convection in finite-size cells, Taylor-Couette flow between concentric rotating cylinders, etc.) are not discussed. Further more, the implications of local/global and absolute/convective instability concepts for geophysical flows are only alluded to briefly. In many of the flows of interest here, the mean-velocity profile is non-
1,988 citations
TL;DR: Improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods, and a new family of stiffly stable schemes is employed in mixed explicit/implicit time-intgration rules.
Abstract: A new pressure formulation for splitting methods is developed that results in high-order time-accurate schemes for the solution of the incompressible Navier-Stokes equations. In particular, improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods. A new family of stiffly stable schemes is employed in mixed explicit/implicit time-intgration rules. These schemes exhibit much broader stability regions as compared to Adams-family schemes, typically used in splitting methods. Their stability properties remain almost constant as the accuracy of the integration increases, so that robust third- or higher-order time-accurate schemes can readily be constructed that remain stable at relatively large CFL number. The new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation. A model Stokes problem is studied in detail, and several examples of Navier-Stokes solutions of flows in complex geometries are reported. Comparison is made with the previously used first-order in time spectral element splitting and non-splitting (e.g., Uzawa) schemes. High-order splitting/spectral element methods combine accuracy in space and time, and flexibility in geometry, and thus can be very efficient in direct simulations of turbulent flows in complex geometries.
1,341 citations
TL;DR: In this article, it was shown that the Strouhal discontinuity is not due to any of the previously proposed mechanisms, but instead is caused by a transition from one oblique shedding mode to another oblique mode.
Abstract: Two fundamental characteristics of the low-Reynolds-number cylinder wake, which have involved considerable debate, are first the existence of discontinuities in the Strouhal-Reynolds number relationship, and secondly the phenomenon of oblique vortex shedding. The present paper shows that both of these characteristics of the wake are directly related to each other, and that both are influenced by the boundary conditions at the ends of the cylinder, even for spans of hundreds of diameters in length. It is found that a Strouhal discontinuity exists, which is not due to any of the previously proposed mechanisms, but instead is caused by a transition from one oblique shedding mode to another oblique mode. This transition is explained by a change from one mode where the central flow over the span matches the end boundary conditions to one where the central flow is unable to match the end conditions. In the latter case, quasi-periodic spectra of the velocity fluctuations appear; these are due to the presence of spanwise cells of different frequency. During periods when vortices in neighbouring cells move out of phase with each other, ‘vortex dislocations’ are observed, and are associated with rather complex vortex linking between the cells. However, by manipulating the end boundary conditions, parallel shedding can be induced, which then results in a completely continuous Strouhal curve. It is also universal in the sense that the oblique-shedding Strouhal data (S_θ) can be collapsed onto the parallel-shedding Strouhal curve (S_0) by the transformation, S_0 = S_θ/cosθ, where θ is the angle of oblique shedding. Close agreement between measurements in two distinctly different facilities confirms the continuous and universal nature of this Strouhal curve. It is believed that the case of parallel shedding represents truly two-dimensional shedding, and a comparison of Strouhal frequency data is made with several two-dimensional numerical simulations, yielding a large disparity which is not clearly understood. The oblique and parallel modes of vortex shedding are both intrinsic to the flow over a cylinder, and are simply solutions to different problems, because the boundary conditions are different in each case.
976 citations
TL;DR: In this article, the Strouhal number and the mean base suction coefficient were measured at the mid-span position Reynolds numbers from about 50 to 4 × 104 were investigated.
Abstract: The investigation is concentrated on two important quantities – the Strouhal number and the mean base suction coefficient, both measured at the mid-span position Reynolds numbers from about 50 to 4 × 104 were investigated Different aspect ratios, at low blockage ratios, were achieved by varying the distance between circular end plates (end plate diameter ratios between 10 and 30) It was not possible, by using these end plates in uniform flow and at very large aspect ratios, to produce parallel shedding all over the laminar shedding regime However, parallel shedding at around mid-span was observed throughout this regime in cases when there was a slight but symmetrical increase in the free-stream velocity towards both ends of the cylinder At higher Re, the results at different aspect ratios were compared with those of a ‘quasi-infinite cylinder’ and the required aspect ratio to reach conditions independent of this parameter, within the experimental uncertainties, are given For instance, aspect ratios as large as L/D = 60–70 were needed in the range Re ≈ 4 × 103–104 With the smallest relative end plate diameter and for aspect ratios smaller than 7, a bi-stable flow switching between regular vortex shedding and ‘irregular flow’ was found at intermediate Reynolds number ranges in the subcritical regime (Re ≈ 2 × 103)
620 citations