Viscous vs. inviscid interaction of a vorticity structure with a circular cylinder

The interaction of a given two-dimensional vorticity distribution with a circular cylinder is analyzed by comparing the numerical solutions provided by an inviscid and by a viscous approach. While the vorticity dynamics of high Reynolds flows in free space shows an almost inviscid behavior, at least in the starting phase, this is not the case in the presence of a solid wall where a considerable effect of viscosity is experienced since the initial stage of the evolution. In fact, the vorticity generation process at the wall may significantly influence the overall flow field even in the case of a weak interaction.A multilevel contour dynamics technique plus a vortex sheet at the body surface are introduced to study the inviscid evolution, while a viscous vortex method has been adopted for the solution of the complete Navier-Stokes equations. An energy-like relation involving forces and other global quantities of the flow is proposed together with its use as a way to control the accuracy of the numerical solution.The numerical simulation of a vorticity patch orbiting around a circular cylinder gives an interesting source of information for the study of unsteady separation providing, at the same time, a proper test to devise a simplified model within the limit of vanishing viscosity.

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Sommario In questo lavoro si studia l'interazione di una distribuzione bidimensionale di vorticità con un cilindro circolare confrontando tra loro la soluzione numerica ottenuta dalle equazioni di Navier-Stokes con quella relativa ad un modello basato sull'ipotesi di fluido non viscoso. La dinamica di strutture vorticose nello spazio libero ad alti Reynolds ha un carattere prevalentemente non viscoso, per lo meno nella fase iniziale. Invece la presenza di una parete solida introduce nel campo nuova vorticità e di conseguenza rende importanti gli effetti viscosi già dai primi istanti dell'evoluzione anche nel caso di interazione debole.Per ottenere la soluzione in assenza di viscosità è stata utilizzata una metodologia di soluzione numerica basata sullaContour Dynamics insieme ad una discontinuità della velocità tangenziale sulla parete. La soluzione delle equazioni di Navier-Stokes è invece ottenuta con un modello viscoso a vortici. Si ricava una relazione di tipo energetico tra le forze agenti sul corpo ed altre grandezze globali del campo fluidodinamico che viene utilizzata per il controllo dell'accuratezza della soluzione numerica.La simulazione numerica del moto di una distribuzione di vorticità, inizialmente uniforme, in prossimità di un cilindro circolare, mentre permette di studiare più approfonditamente i fenomeni connessi alla separazione non stazionaria dello strato limite, offre, nel contempo, uno strumento appropriato per individuare un modello semplificato per viscosità tendente a zero.

     

An inviscid/viscous coupling approach for vortex flow field calculations

A new computational approach is developed for the analysis of vortex-dominated flow fields around highly swept wings at high angles of attack. In this approach an inviscid Euler technology is coupled with viscous models, similar to inviscid/boundary layer coupling. The viscous nature of the vortex core is represented by an algebraic model derived from the Navier–Stokes equations. The approach also accounts for the effects of the viscous shear layer near a wing surface through a modified surface boundary condition. The inviscid/viscous coupling consistently provides improved predictions of leading edge separation, vortex bursting and secondary vortex formation at relatively low computational cost. Results for several cases are compared with wind tunnel tests and other Euler and Navier-Stokes solutions.     

Suppression of vortex shedding from a rectangular cylinder at low Reynolds numbers

Small elements of circular, square, triangular and thin-strip cross-sections are used to suppress vortex shedding from a rectangular cylinder of stream-wise to transverse scale ratio L/B=3.0 at Reynolds numbers in the range of Re=V_∞B/ν=75–130, where V_∞ is the on-coming velocity of the stream, and ν is the kinematic viscosity. The relative transverse dimension of the small element b/B is fixed at 0.2. The results of numerical simulation and visualization experiment show that, vortex shedding from both sides of the cylinder can be suppressed and the fluctuating drag and lift of the cylinder can be greatly reduced, if the element is placed in a certain region referred to as the effective zone. Comparisons at a specific Reynolds number indicate that the square element produces the largest size of the effective zone, whereas the triangular element yields the smallest. Results also show that the effective zone for the square element shrinks with increasing Re and disappears at Re>130. Independent of element cross-section shape and Reynolds number, the center of the effective zone is always at X/B=2.5–3.0 and Y/B≈1.0. The mechanism of the suppression is discussed from the view points of velocity profile stability and stress distribution.     

An inviscid model for vortex shedding from a deforming body

An inviscid vortex sheet model is developed in order to study the unsteady separated flow past a two-dimensional deforming body which moves with a prescribed motion in an otherwise quiescent fluid. Following Jones (J Fluid Mech 496, 405–441, 2003) the flow is assumed to comprise of a bound vortex sheet attached to the body and two separate vortex sheets originating at the edges. The complex conjugate velocity potential is expressed explicitly in terms of the bound vortex sheet strength and the edge circulations through a boundary integral representation. It is shown that Kelvin’s circulation theorem, along with the conditions of continuity of the normal velocity across the body and the boundedness of the velocity field, yields a coupled system of equations for the unknown bound vortex sheet strength and the edge circulations. A general numerical treatment is developed for the singular principal value integrals arising in the solution procedure. The model is validated against the results of Jones (J Fluid Mech 496, 405–441, 2003) for computations involving a rigid flat plate and is subsequently applied to the flapping foil experiments of Heathcote et al. (AIAA J, 42, 2196–2204, 2004) in order to predict the thrust coefficient. The utility of the model in simulating aquatic locomotion is also demonstrated, with vortex shedding suppressed at the leading edge of the swimming body.      

Numerical prediction of vortex shedding behind a square cylinder

It is common knowledge that flow around bluff bodies exhibits oscillatory behaviour. The aim of the present study is to compute the steady two-dimensional flow around a square cylinder at different Reynolds numbers and to determine the onset of unsteadiness through a linear stability analysis of the steady flow. Stability of the steady flow to small two-dimensional perturbations is analysed by computing the evolution of these perturbations. An analysis of various time-stepping techniques is carried out to select the most appropriate technique for predicting the growth of the perturbations and hence the stability of the flow. The critical Reynolds number is determined from the growth rate of the perturbations. Computations are then made for periodic unsteady flow at a Reynolds number above the critical value. The predicted Strouhal number agrees well with experimental data. Heat transfer from the cylinder is also studied for the unsteady laminar flow.     

Effect of vortex shedding in unsteady aerodynamic forces for a low Reynolds number stationary wing at low angle of attack

This study elucidates the relation between wake vortex shedding and aerodynamic force fluctuations for a low Reynolds number wing from time resolved particle image velocimetry (TR-PIV) experimental measurements. The results reveal a periodic lift and drag variation within the shedding cycle and resolve the frequencies of those fluctuations from a proper orthogonal decomposition (POD) and power spectral density (PSD) analysis. To show the effect of vortex shedding on the body force fluctuations, the evolution of instantaneous aerodynamic forces is compared to the pressure field of the fluid flow and to the vortical structures in the wake of the airfoil. A six step model describing the vortex-force relation is proposed. It shows that changes in lift such as maximum lift and minimum lift are associated with the detachment of a vortex. It also shows that the minimum or local minimum drag value is obtained at the onset formation of a vortex on the airfoil wake. Similarly, the maximum or local maximum drag is obtained at the onset formation of the saddle on the airfoil wake. The model further explains the asymmetry observed in the unsteady drag force evolution. The model can be used to optimize flow control and fluid-structure interaction applications.     

Control of vortex breakdown in a closed cylinder with a small rotating rod

Effective control of vortex breakdown in a cylinder with a rotating lid was achieved with small rotating rods positioned on the stationary lid. After validation with accurate measurements using a novel stereoscopic particle image velocimetry (SPIV) technique, analysis of numerical simulations using a high-order spectral element method has been undertaken. The effect of a finite length rod creates additional source terms of vorticity as the rod rotates. These additional source terms and their spatial locations influence the occurrence of the vortex breakdown.     

The variation in the maximum loading of a circular cylinder impacted by a 2D vortex with time of impact

Direct numerical simulation is used to study the loading of a rigid, circular cylinder impacted by a 2D vortex. The vortex travels within a stream of fluid characterized by Reynolds number of 150. Vortex impact occurs at twenty-five different times within one vortex shedding cycle. Substantial variation is observed in the maximum values of the drag and lift force coefficients. This variation is due to interaction between the impinging vortex and those attached to the cylinder. As the radius of the impinging vortex is increased from one to three times the cylinder’s diameter, the variation in maximum force coefficients with time of impact decreases. The variation decreases because the larger vortex alters the flow field and vortex shedding cycle prior to impacting the cylinder. For structures impacted by a vortex similar in size, significant under-prediction of the maximum loading may occur if variation in loading with vortex impact time is not considered.     

Three-dimensional phase dynamics in a cylinder wake

Recently there has been a new surge of interest in three-dimensional wake patterns, from both an experimental and analytical standpoint. One of the central discoveries is that the patterns of vortex shedding are dependent on the specific end conditions of a long cylinder span. However, a number of outstanding questions have remained unanswered, in part because techniques had not existed to control such patterns in a continuous fashion and from outside a test facility. In the present work, we have devised a method to control the end conditions of a cylinder span by non-mechanical and continuously-variable means, namely by the use of end suction. The technique allows a continuous variation of end conditions and admits transient or impulsive control. With the method, the classical steady-state patterns, such as parallel or oblique shedding or the chevron patterns are simply induced. These experiments demonstrate that the wake, at a given Reynolds number, is receptive to a continuous (but limited) range of oblique shedding angles (), rather than to discrete angles. There is excellent agreement in these results with the cos formula for collapsing oblique-shedding frequencies onto a single universal frequency curve. The use of suction has avoided the grossly unsteady motions at the ends of the cylinder span brought about by the wakes of mechanical end manipulators, and we show that the laminar shedding regime exists up to Reynolds numbers (Re) of 205. The surprisingly large disparity among reported measurements of criticalRe for wake transition (Re=140–200), over the last forty years, can now be explained in terms of spanwise end contamination.The control technique has also allowed experiments to be performed, which have resulted in the discovery of new phenomena such as phase shocks and phase expansions. A major difference between these phenomena is that phase shocks (involving regions of straight vortices) translate spanwise at constant speed, crossing the complete span in a finite time, whereas a phase expansion (involving curved vortices) requires an infinite time to complete its development across the span. These transient wake patterns are well illustrated using a simple model, based on experimental measurements, that thenormal wavelength for oblique or parallel vortices remains constant. However, a detailed and close comparison between our experimental results and those results from analytical modelling of the wake using Ginzburg-Landau modelling (in collaboration with Peter Monkewitz at Lausanne) is presently underway. These equations yield a Burger's equation for the spanwise wavenumber (or phase gradient), from which both (phase) shocks and expansions are well-known solutions.

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Sommario Recentemente è rinato l'interesse per i modelli tridimensionali di scie, sia da un punto di vista sperimentale che da uno analitico. Una delle scoperte centrali è che le modalità di produzione dei vortici sono dipendenti dalle condizioni agli estremi di un lungo cilindro. Nel presente lavoro, è stato formulato un metodo per controllare le condizioni finali agli estremi di un cilindro per mezzo di variabili continue di significato non-meccanico, cioè per mezzo dell'aspirazione agli estremi. La tecnica permette una variazione continua delle condizioni agli estremi ed ammette controllo transitorio od impulsivo. Con questo metodo sono semplicemente indotti i classici modelli a stato fissato, come anche quelli che prevedono produzioni di vortici parallele od oblique o quellichevron. Questi esperimenti dimostrano che la scia, ad un dato numero di Reynolds, ammette una distribuzione continua (ma limitata) di angoli per distribuzioni oblique, piuttosto che una discreta. In questi risultati c'è un eccellente accordo con la formula del coseno per frequenze di produzioni oblique e collassanti su di una curva di frequenzauniversale. L'uso dell'aspirazione ha evitato i moti largamente instabili alle estremità del cilindro, provocati dalle scie delle parti terminali dei manipolatori, e si osserva che il regime laminare diffondente esiste oltre un numero di Reynolds pari a 205. La sorprendentemente larga disparità tra le misure di numeri di Reynolds critici, riportati per transizioni di scia (Re=140÷200) durante gli ultimi quattro anni, può essere ora spiegata in termini di contaminazione della estensione della lunghezza del cilindro.

     

Modes of vortex formation and transition to three-dimensionality in the wake of a freely vibrating cylinder

The present work is aimed to give some insight into the relation between vortex shedding modes and transition to three-dimensionality in the wake of a freely vibrating cylinder by establishing a numerical model and analyzing the relevant results of two- and three-dimensional simulations. The compressible flow past an elastically-mounted cylinder is solved by using the immersed boundary method (IB method). The cylinder is free to vibrate in the transverse direction with zero structure damping. The response of displacement amplitude is studied with the variation of reduced velocity. Whether P+S mode exists in three-dimensional flow and the occurrence of 2P mode is caused by flow transition from two-dimensional to three-dimensional are problems of concern. Both 2P and P+S wake modes are observed in two- and three-dimensional simulations. The numerical results indicate that the flow transition from two-dimensional to three-dimensional is coupled with the cylinder vibration in the synchronization/lock-in regime. The wake formation given by three-dimensional simulations suggests that the P+S mode might exist in reality when the flow is reverted to two-dimensional by vortex induced vibration (VIV) at Re=300–350. When Reynolds number increases to 425, the wake formation undergoes transition to three-dimensionality and 2P mode is observed. The effect of mass ratio on the flow transition to three-dimensionality is studied. The relationship between wake modes and aerodynamic forces is discussed.     

Factors influencing non-circular ring vortex motion

Non-circular ring vortices are innately unstable, giving rise to a range of new phenomena. Here we report on our and Heertsch's [1] experiments in which vortices were generated at rectangular holes and nozzles with aspect ratios 2<<20. Different piston histories were also used. For forestrokes alone we were able to confirm the typical non-splitting motion of the primary vortex. On introducing a backstroke following the forestroke even for values of as low as 2 — values which should not give rise to splitting vortices — vortices could be made to split into 2, 3 or 4 secondary vortices. For cases where they rejoined the process was significantly different to that predicted by theory [2]. For 3 for a nozzle geometry the splitting angle is extremely sensitive to the stroke (length) so long as splitting takes place, whereas for 9>>5 the splitting angle tends to become independent of the stroke. This sensitivity on the stroke is reduced for vortices generated at a hole geometry. For all cases investigated here the splitting angle seems to be relatively insensitive to the Reynolds number. Vortices generated at hole geometries also tend to be less stable than those generated at tube geometries. Finally, the dependence of the splitting angle on the stroke length only scales with the nozzle breadth for 7>>5.

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Sommario Vortici ad anello non circolari sono intrinsecamente instabili e danno luogo ad una gamma di nuovi fenomeni. In questo articolo vengono riportati gli esperimenti degli Autori e di Heertsch in cui sono generati vortici in ugelli e fori rettangolari con rapporti geometrici =2÷20. Sono state anche usate differenti storie del moto del pistone. Nel caso in cui si usi solo la corsa in avanti si è stati capaci di confermare il moto tipico del vortice primario senza divisione del vortice stesso. Introducendo una corsa inversa, subito dopo la corsa in avanti, persino per pari circa a 2 — valore in cui il vortice non si dovrebbe dividere — i vortici si potevano dividere in due, tre o quattro vortici secondari. Nei casi in cui si verificava la riconnessione, l'evoluzione del processo era molto differente rispetto alla teoria. Per <3, per una data geometria dell'ugello, l'angolo di separazione è estremamente sensibile alla lunghezza della corsa, mentre per =5÷9 l'angolo di separazione tende a diventare indipendente dalla corsa. Questa sensibilità è ridotta per vortici generati in fori. In tutti i casi l'angolo di separazione sembra abbastanza indipendente dal numero di Reynolds. Vortici generati in corrispondenza di fori tendono ad essere meno stabili di quelli generati in ugelli. Infine, la dipendenza dall'angolo di separazione sulla lunghezza della corsa scala con l'ampiezza dell'ugello solamente per =5÷7.

     

Formation and interaction of intense vortex sheets in 3-dimensional,incompressible, hydrodynamics

Various questions related to the physics of inviscid flows are reviewed. The emergence of strong vortex sheets has repeatedly been observed in the simulation of the 3-dimensional equations, with a variety of initial conditions. In the case of axisymmetric Euler flows, the origin of these sheets can be easily understood with the help of an analogy with thermally driven flows. A more general mechanism to explain these sheets is proposed. Questions of singularities are briefly reviewed. Lastly, preliminary results on the connection between the sheets forming in inviscid flows, and the vortex tubes observed in high Reynolds number flows are presented.

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Sommario Sono considerate varie questioni correlate con la fisica dei flussi non-viscosi. La nascita di strati di forti vortici è stata ripetutamente osservata nella simulazione delle equazioni tridimensionali, per diverse condizioni iniziali. Nel caso di flussi di Eulero assialsimmetrici, l'origine di questi piani può essere facilmente compresa con l'aiuto di una analogia con i flussi guidati termicamente. Viene inoltre proposto un più generale meccanismo per giustificare questi strati e si passano in rassegna brevemente questioni riguardanti le singolarità. Infine, vengono presentati alcuni risultati preliminari sulla connessione tra i piani formantisi in flussi non viscosi ed i vortici tubolari osservati nei flussi ad alti numeri di Reynolds.

     

Interaction of an obliquely rising vortex ring with a free surface in a viscous fluid

A vortex ring, which approaches a free surface under various initial conditions and different values of flow parameters, is investigated by means of numerical solutions of the Navier-Stokes equations. The study focuses on the connection process of the vortex ring with the free surface at low Reynolds number. Discrepancies between a numerical solution and experimental observations found recently with regard to the shape of the reconnected vorticity tubes have been resolved. The crucial parameter, which determines the essential differences at the low Reynolds number selected, is the angle of inclination at which the vortex ring rises toward the free surface. At an angle of inclination of 20° circular vorticity tubes at the free surface evolve, while at an angle of 45° the reconnection takes place in the form of an almost circular sheet. At higher Reynolds number this form of reconnection may change to the other form through instability. The possibility of vortex-ring reflection at the free surface is discussed.

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Sommario Usando soluzioni numeriche delle equazioni di Navier-Stokes, viene analizzato il comportamento di un vortice and anello che si avvicina ad una superficie libera, per diverse condizioni iniziali e valori dei parametri del flusso. Lo studio è incentrato sul processo di connessione del vortice adnanello con la superficie libera per bassi numeri di Reynolds. con riguardo alla forma dei tubi di vorticità riconnessi con la superficie, tutte le differenze tra osservazioni sperimentali e simulazioni numeriche sono state risolte. Per bassi numeri di Reynolds, il parametro cruciale dell' interazione è l'angolo di inclinazione tra vortice e superficie. Per un angolo di 20° si formano tubi circolari di vorticità alla superficie, mentre a 45° la riconnessione ha luogo in forma di strato sottile di vorticità. Per altri numeri di Reynolds la forma della riconnessione può cambiare in altre forme a causa dell'instabilità. Infine viene discussa la possibilità di riflessione del vortice ad anello alla superficie.

     

Two-degree-of-freedom vortex induced vibration of low-mass horizontal circular cylinder near a free surface at low Reynolds number

Two-degree-of-freedom vortex induced vibration (VIV) of a low-mass zero-damping circular cylinder horizontally placed near a free surface at Re = 100 was numerically studied with an adaptive Cartesian cut-cell/level-set method. Two Froude numbers and various normalized submergence depths were considered. The results reveal that the Froude number affects the critical normalized submergence depth and possible physical mechanisms are proposed. The in-line vibration amplitude cannot be neglected. Proximity to a free surface strengthens and suppresses the VIV for low and high Froude numbers, respectively; increases the occurrence of amplitude modulation; and in general enhances the magnitude of the time-averaged lift coefficient, which is always negative. The phase lag of the transverse displacement behind the lift coefficient jumps at some reduced velocity, which strongly depends on the Froude number and normalized submergence depth. Regular trajectories exist only in cases with a small vibration amplitude or a large normalized submergence depth. The vortex structures in any case with large transverse amplitude basically originate from the alternative vortex shedding with the negative vortex weaker than the positive one. For the higher Froude number, an extra free surface positive vortex interacts with the vortices from the cylinder surface. The vibration frequency deviates from the natural structure frequency in fluids in the large-amplitude regime.     

Numerical simulation of the noise generated by a low Mach number, low Reynolds number jet

In this paper we study the sound field produced by a turbulent round jet with a Mach number of 0.6 based on the centerline velocity and the ambient speed of sound c_∞. The turbulent flow field is found by solving the fully compressible Navier–Stokes equations with help of high-order compact finite difference schemes. It is shown that the simulated flow field is in good agreement with experiments. The corresponding sound field has been obtained with help of the Lighthill equation using two different formulations for the Lighthill stress tensor T_ij. In the first formulation of T_ij the fluctuating density is taken into account. In the second formulation the density is assumed to be constant. As an additional check we have also performed an acoustic calculation using a formulation in which a homogeneous wave equation is solved. The boundary conditions for this homogeneous wave equation are obtained from the numerical simulation of the Navier–Stokes equation. The results obtained with both formulations of the Lighthill stress tensor are nearly identical. This implies that an incompressible formulation of the conservations laws could be used to predict jet noise at low Mach numbers.     

低马赫数下斜爆轰波的结构

利用Euler方程和两步化学反应模型,对低马赫数入流时的驻定斜爆轰波进行了数值模拟,并重点研究了斜爆轰波的驻定过程和结构。数值结果显示,当入流马赫数较低时,即使其本身是附体的,在诱导区后侧的高压区的作用下,斜爆轰波也会从其起始位置向来流方向运动。在这种情况下,斜爆轰波会驻定在靠近斜面前缘的位置,诱导区的长度仅有1 mm左右。通过设置初始条件,让斜爆轰波在斜面前缘附近被触发,则其将一直维持在靠近斜面前缘的位置。