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Phase front analysis of laminar vortex streets Lefrançois, Marcel 1992

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PHASE FRONT ANALYSISOFLAMINAR VORTEX STREETSbyMarcel LefrancoisB.A.Sc., The University of British Columbia, 1985M.A.Sc., The University of British Columbia, 1987A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust, 1992© Marcel Lefrancois 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(SignatureDepartment of  PA/ 5 ;c_ The University of British ColumbiaVancouver, CanadaDate^(--)c-71- / 3 l 2-DE-6 (2/88)ABSTRACTThe continuous formation and development of a laminar vortex street behind a circular cylinderhas been modelled as a Huygens-type wave process. The phase of a point next to the cylinderis defined by the amplitude and phase of secondary wavelet centres along the preceding vortexin the near wake. Vortices are located at places where the phase in each cycle is 0 or n. Thephase is defined from a Kirchhoff-type phase-amplitude integral which is derived from the Biot-Savart law. Model predictions agree with reported experimental results on cylinders with stepsand tapers, and with our own tapered cylinder experiments. To test the predictive power of themodel, with regards to preset phase, experiments were performed with new experimentaltechniques that allow us to set the start-up phase of vortex shedding. The experiments yieldknown starting conditions, which, when introduced into the phase front model, yield predictionsof the street in good agreement with the experiments with regard to the shape and position of theshed vortices in the wake.fiTABLE OF CONTENTSABSTRACT^  iiLIST OF TABLES  viLIST OF FIGURES ^  viiACKNOWLEDGEMENTS  xiii1 - INTRODUCTION^  11.1 Related Works  101.2 Aim and Synopsis of Thesis ^  121.3 Conventions ^  142 - VORTEX STREET MODELS ^  162.1 Two-Dimensional Vortex Models ^  172.1.1 Rankine Vortex ^  172.1.2 Oseen Vortex  192.2 The Karman Vortex Street ^  212.3 The Frequency and Phase of Vortex Shedding ^  262.4 Tapered Vortices ^  302.5 Kelvin's Circulation Theorem ^  332.6 Vortex Street Instabilities  372.7 Van Der Pol Oscillator Model of Vortex Shedding ^  401113 - PHASE FRONT ANALYSIS OF LAMINAR VORTEX STREETS ^ 463.1 A Huygens Construction of the Vortex Street ^  483.2 Biot-Savart Law For Fluids ^  533.3 Phase Coupling Equation (Diffraction Integral For Fluids ^ 563.4 The Vorticity, Emitters Phase, and Amplitude In A Vortex Street ^ 603.5 Emitter Plane Location In A Vortex Street ^  643.6 Diffraction Model of the Vortex Street  663.7 Oblique Vortex Shedding ^  713.8 Modulation Zones From Step Changes In Cylinder Diameter^ 763.9 Small Increase In Diameter ^  803.10 Shedding Cells On A Tapered Cylinder ^  813.11 Summary^  864 - APPARATUS  874.1 Set-up Overview ^  874.2 Test Basin and Cart  894.3 Trigger Plates ^  914.4 Vortex Shedding Cylinders and Cylinder Holder ^  934.5 Vortex Street Visualization ^  944.6 Data Logging and Analysis  954.7 Velocity Sensor ^  974.8 Viscosity  1004.9 Experimental Procedure ^  101iv5 - TRIGGERED EXPERIMENTS ^  1025.1 Setting the Initial Phase of the Vortex Street ^  1045.2 End Conditions of the Cylinder ^  1105.2.1 Condition at The Bottom of the Tank ^  1105.2.2 End Condition at the Free Surface  1145.3 Inclined Wave Fronts ^  1165.4 Sudden Phase Jump of Al2 Along the Span ^  1225.5 Small Phase Jumps Over Short Spanwise Sections  1255.6 Large Phase Jumps Over Short Spanwise Sections ^  1305.7 Frequency Variation Along The Phase Fronts  1366 - CONCLUSIONS ^  147BIBLIOGRAPHY 152A - LIST OF VARIABLES ^  162B - THE FLUID DYNAMICS EQUATIONS ^  164C - BURGERS' VORTEX ^ 166D - EXAMPLE CALCULATION  167E - CALCULATION OVERVIEW ^ 170F - HUYGENS.0 ^  174vLIST OF TABLESI^Wake structure variation with R. for flow past a cylinder ^ 4II^Input parameters for modelling oblique vortex shedding.  72III^Input parameters for modelling a step change in diameter. ^ 76IV^Input parameters for modelling a step change in diameter, the repairing process.^79V^Input parameters for modelling a collar. ^  80VI^Input parameters for modelling a tapered cylinder  81VII Viscosity of pure water at various temperatures. ^  99VIII Experimental procedure. ^  101IX^Input parameters for modelling an inclined triggered experiment. ^ 116X^Input parameters for a phase jump along the span^  123XI^Input parameters for a small phase jump over a short spanwise section. ^ 126XII Input parameters for a large phase jump over a short spanwise section. ^ 131XIII Input parameters for a tapered cylinder. ^  137viLIST OF FIGURES1.1 Side view of two-dimensional vortex street^ - 51.2 Definition of 3D vortices. ^  71.3 Vortex street (a) vortex cores marked with dye (b) tracer particles showing flow^81.4 Streamwise vortices at R e=326. ^  91.5 Coordinate systems used  152.1 Rankine vortex velocity profile. ^  182.2 Oseen vortex velocity profile at different vt values normalized to I'/2it. ^ 212.3 Von Karman vortex street formed by the superposition of potential vortices. ^ 222.4 Recirculation bubble behind a 10 cm wide plate moving at 9.45cm/s in water. Thecamera was moving with the plate. ^  252.5 C.H.K. Williamson's vortex linkage model. The relative vortex strengths areindicated. ^  292.6 Vortex street produced by a tapered cylinder (length 410mm). D=8mm (R e=92.6) atthe surface, and D=2mm at the bottom (R e=23.1). ^  352.7 Blow-up of area indicated on Figure 2.6 showing interconnections between thevortices. ^  372.8 Van der Pol Oscillator model. Slices of fluid close to the cylinder are displaced adistance q from equilibrium. ^  42vii3.1 Huygens principle in the wake of a circular cylinder. A line of emitters forms asecondary wave front at the cylinder. The resulting diffraction pattern is sweptbackward, forming a new set of emitters and, eventually, the vortex wake^ 493.2 Vortices pass through the emitter plane which is positioned in the wake, behind theshedding cylinder. ^  493.3 Phase information propagation in the two reference frames^  513.4 Fresnel-Kirchhoff diffraction^  523.5 Definition of angles for the diffraction theory of vortex streets. ^ 563.6 "Streak photo" of vortex position behind a cylinder (R e=107) composed from slicesof 14 consecutive side views (top half labelled (a)). The bottom half (b) is a widersection of the last frame. ^  623.7 Contribution to Uy from vorticity elements at different distances behind thecylinder. ^  653.8 Phase fronts used to find the input conditions for oblique shedding. The startingposition of the cylinder is dashed on the left. ^  713.9 Frozen phase picture of oblique shedding at 30°, the vortex lines from one side of thecylinder are shown more solidly than the striped lines from the other side. Thephase front ray propagates away from the parallel shedding region. ^ 743.10 Progression of Uy calculated at the cylinder. The phase fronts are at an angle to thecylinder (zIL axis) and are parallel to the waves in the Uy, surface. The top edgeis on the right. ^  753.11 Indirect mode of vortex shedding from a step change in diameter. ^ 76VII'3.12 Input phase fronts for step change in diameter to produce growing modulation zoneseen in Figure 3.11. ^  773.13 Repair process of Lewis and Gharib wake. ^  783.14 Input phase front for step diameter change to return to original sheddingconditions. ^  793.15 Kink instability production maintained by a collar on the shedding cylinder. ^ 803.16 Shedding frequency versus position along a tapered cylinder. ^ 833.17 Noack et. al. tapered shedding simulation. ^  854.1 Overview of the experimental setting.  884.2 The test basin. ^  894.3 Velocity sensor mounted on cart. ^  904.4 Cart velocity versus applied voltage  914.5 Trigger plates. (a) is a top view of a cylinder passing between trigger plates. (b) isa side view of a number of single plates. In the split plate, the cylinder passesbetween the two plates. ^  924.6 Striped dye marking of vortices. The cylinder is painted solidly on one side and ina striped pattern on the other. The dye comes off, rolling up into striped and solidvortices. ^  954.7 Camera timing circuit and arrangement ^  964.8 Camera timing signals for calibration purposes.  964.9 Velocity sensor circuit (G.E. emitter-receiver assembly H21B5). ^ 97^4.10 Velocity sensor signal    97ix^4.11 Position and velocity comparison    984.12 Detail of velocity sensor signal. The arrival of an edge is known within 1/2f. Thesampling rate used is much higher than indicated here. ^ 994.13 Viscosity calibration curve. ^  995.1 Vortex street development in time with no trigger plate (random start). (R e=98) ^ 1055.2 Wake development for single trigger plate (position indicated by arrow) whencylinder is started many shedding periods from plate. The plate is on the side ofthe wake away from the camera. (Rp96) ^  1065.3 Vortex wake with one trigger plate on the side away from the camera at the arrow.One side is triggered (solid) while the other side is not (striped). (Rp110) ^ 1075.4 Vortex street with two trigger plates placed X/2 apart on opposite sides of the wake.The perspective causes the edge of both plates to appear at the arrow. (R e-107) . 1075.5 Street triggering with different plate separations. The plate on this side of the wakeis at the left arrow and on the opposite side at the right arrow. ^ 1085.6 Vortex street at the bottom of the tank Rp200. (a) spacing=4.4mm, (b) 2.7mm (c)1.0mm^  1105.7 View of the bottom structure (Re=76.7). The front (F) and back (B) vortices aremarked in the sketch. The arrows indicate the direction of the vorticity vector. ^ 1125.8 Bottom view of vortex street production^  1135.9 The effect of a contaminated (dirty) water surface on vortex shedding. (Rp84) ^ 1155.10 Desired contaminant free (clean) water surface condition. (R em 126) ^ 1155.11 Vertical velocity component. ^  1175.12 Slanted vortex street at 32° to the vertical. Water depth 365mm. (R e=96.3) ^ 1185.13 Poorly triggered street by an inclined plate at 32° to the vertical, water depth371mm. (R e=97) ^  1205.14 Poorly triggered street by plate inclined 32° to the vertical, water depth 371mm.Loops appear in wake. (Re=97.4) ^  1205.15 Slanted plate at 19.3° to the vertical. Water depth 410 mm. (R e=89) ^ 1215.16 Split plate set up for producing a sudden phase jump. The cylinder moves out of thepage, passing between the two plates. ^  1225.17 Phase jump of 180° (X./2). (Uz=14.02 mm/sec, Re=108) ^  1245.18 Cylinder moves past a stepped plate to produce a small phase jump. ^ 125^5.19 Small phase jump of 118°. Step plate width 53mm, R e=123, U.,=1.5538 cm/sec.   1275.20 Jump in phase. (Re=111, step 116°, U0,1.4738 cm/sec) 15.75 sec betweenpictures. ^  1285.21 Simulation of Figure 5.20 (drawn to same scale). ^  1295.22 Simulation of a 210° phase step at Re=116, U.,=1.4669 cm/sec. ^ 1305.23 Experiment with a phase jump of 210°, 13.11 sec between frames. (R e=116) ^ 1345.24 Large phase jump of 199°. (R e=128, U.=1.5362 cm/sec) ^  1355.25 Taper ratio 137, D=8mm at the surface, 5mm at the bottom, L=410mm. ^ 1405.26 Taper ratio 82, D=8mm at the surface, 3mm at the bottom, L=410mm ^ 1415.27 Taper ratio 68, D=8mm at the top, 2mm at the bottom, L=410mm  1425.28 Part 1. ^  1435.28 Part 2.  144xi5.29 Re=92.6 at the top of the taper (D=8mm) for taper ratio 68. A blow up of the boxedarea in frame #7 is given showing loops. ^  1455.30 Taper ratio 68, initial wave front angle of 15°.  146C.1 Tangential velocity profile for Burgers vortex compared to Rankine. ^ 166D.1 Input to model. The emitter plane is on the left and the cylinder is on the right. . . . ^ 167D.2 The initial vorticity distribution is assumed to be sinusoidal without axialcoupling. ^  168D.3 Diffraction integral is used to sum the contributions from the emitter plane in thewake. ^  168D.4 The phase information convects back towards the emitter plane. ^ 169xiiACKNOWLEDGEMENTSThere are many people I would like to thank for their patience and assistance in preparing thisthesis. Fraser Duncan a fellow graduate student and friend who has persevered through the sameeducational process as me at UBC. Many coffees have been shared where I would bounce ideasoff him or describe what I was doing, which helped me to firm up concepts and the direction ofmy thesis. Al Cheuck (who I called "Super-Tech") was designer, builder and advisor inelectronics as well as other technical support. My wife Mae Lefrancois who I met in the labduring her masters degree. She helped maintain and run the lab and proved the value of triggerplates in setting the wake. She also introduced me to the graduate students in the MechanicalEngineering department of UBC, which allowed me to see a different approach to fluid research.Dr. W. Shuter lent me a gray-scale scanner to introduce the photos electronically into the thesis.This was invaluable in quickly preparing and revising the thesis. He also asked me to be ateaching assistant in the Engineering Physics undergraduate optics course. It was because of this,I was reminded of the Kirchhoff diffraction integral and Huygens' principle, without which Iwould not have been able to build a phase front model of vortex streets. By far the greatestinfluence on the work came from my supervisor Dr. Boye Ahlborn who gave me ongoing supportand guidance. He treated me as a colleague and friend, advising me on matters of careerdevelopment and life while keeping me in touch with other research areas through discussions,papers and student projects. Thank you Boye for believing in my capabilities to do a Ph.D. thesisand especially for giving me the opportunity.Marcel LefrancoisOctober 5, 19921 - INTRODUCTIONThe deflection of a moving fluid by an obstacle immersed in the flow will often result in theformation of a double row of vortices in the wake. These vortex streets may exist downstreamfrom chimneys, towers, struts or cables on bridges, pilings in rivers, or around ocean platforms,possibly generating significant lift and drag forces. Controlling the organization of the vortexshedding process may be necessary in order to manage these forces (eg. surface protuberances,called stakes, are installed on towers to prevent coherent shedding). At low flow speeds, thewake behind a two-dimensional object (such as a plate or circular cylinder) is periodic and thevelocity profiles are smooth and continuous. This regular pattern, smoothed by viscous friction,is known as a laminar vortex street. Surprisingly, the regular two-dimensional wake behind anaxisymmetric object is difficult to produce and maintain. Discontinuities, inhomogeneities, andinstabilities conspire to create three-dimensional vortex structures. The appearance of these 3Dstructures could be started by random processes but once established they affect the overallorganization and development of the wake.Several types of three-dimensional structures were studied and have been repeatedly producedin controlled experiments on a circular cylinder. The structures were investigated both1I - Introduction 2theoretically and experimentally in the laminar vortex street, the results of which are reported inthis thesis. On the theoretical side, a Huygens type phase front propagation integral has beenderived starting from the generalized Biot-Savart Law for fluids. This analysis is used to predictthe phase of newly shed vortices as a function of the phase of the previous vortex generation andthe shedding frequency. On the experimental side, a new method has been developed to set theinitial vortex phase. In other words, the phase of the first vortex generation is set and knownthereby allowing comparisons to be drawn between the experiments and the phase front integralcalculations. In most cases, the calculated vortex street pattern agrees with that observed in theexperiment.During the course of this work, the concepts used in the diffraction theory were found to havea direct bearing on the analysis of the vortex shedding pattern of laminar vortex streets. Bothdiffraction patterns and vortex fields can be described by phase fronts with an associatedwavelength and frequency, and show constructive and destructive interference. This treatment isintuitively appealing; the power and elegance of the Huygens wavelet construction and theKirchhoff diffraction integral can be used to predict the temporal development of laminar vortexstreets. This model is capable of predicting the propagation of spanwise phase perturbations intosubsequent vortex generations of 2D flow geometries, and it has been extended to the three-dimensional flows encountered around tapered cylinders, reliably predicting the onset of cellformation (dislocations) in the wake.2I - Introduction 3The wake of a bluff body may be a simple or complicated conglomeration of coherentstructures [Hussain, 1986] (a turbulent fluid mass connected by a phase-correlated vorticity[Hussain, 1981]) with the most prevalent being eddies or vortices. An eddy or vortex is a massof swirling fluid possessing angular momentum about a point. It is marked by a low pressurecore about which the surrounding fluid rotates.Through extensive experiments [Gerrard, 1978; Roshko,' 1954; Blevins, 1984: p.313 & p.339] ithas become clear that the wake structure for flow past a cylinder varies significantly withReynolds number:U D°where U., is the free stream velocity, D is a characteristic dimension of the bluff body, and v isthe kinematic viscosity. In effect, Re is a dimensionless size or it can be considered adimensionless speed. Table I gives a survey of the wake structure-Reynolds number relation forcircular cylinders (Re is based on cylinder diameter). At very low Reynolds numbers no eddiesform, the streamlines close smoothly behind the cylinder. Increasing the Reynolds number causesvortices to form that remain attached to the cylinder. As Re passes 40, the vortices start to shed,forming a periodic wake of alternating vortices possessing smooth velocity profiles, belowRe =300 this is a laminar vortex street known as the Von KArman Vortex Street [Von KanIlanand Rubach, 1912; Schlichting, 1979: p.31-32].1.0.131 - Introduction 4c/.R,<5: Regime of unseparated orpotential flow.5<Re<40: a pair of FOppl vorticesin the wake and attached to thecylinder.---------‹ ''7"-■---........,..„ .........4,^, 4:,... ,....---0.-0040<Re<150:^alternating^vortexstreet forms where the vortices andstreamlines are laminar. (known asthe Von Kaman vortex street),,,,,,,,^.,4.„5.Z■i^0-----,.:,^/000150<k<300: Transition range.300<k<300,000: vortex street iscompletely turbulent.?0---------.,< , ,--300,000eSe<3,500,000:^laminarboundary layer undergoes turbulenttransition and wake is narrowerand disorganized. (completelyturbulent)^ ( , ///---------- -/„..,/z__.--'.----- ,"-4:'40Re>3,500,000:^turbulent^vortexstreet^is^reestablished^but^isnarrower than in previous cases.,Table I Wake structure variation with Re for flow past a cylinder [Blevins, 1984: p.340]. Thestandard model of the wake behind a circular cylinder is the two dimensional vortex street[Roshko, 1953]. At Reynolds numbers around 40<R e<150 a periodic wake of laminar vorticesis observed, known as the Von Karman vortex street [Schlichting, 1979: p.31-32].41 - Introduction 5Figure 1.1 Side view of two-dimensional vortex street.A side view of a vortex street shows parallel rows of eddies (Figure 1.1); those on one side ofthe wake rotate in the opposite direction from those on the other side. Since the process isperiodic, the shedding state can be described by a phase, 0, with one shedding cycle occurringover /10=360°. In one shedding cycle two eddies are shed 180° apart. The eddies are associatedwith an arbitrarily chosen phase value such as (1)=0° on one side of the wake, and 43=180° onthe other side of the wake. In the laminar vortex street (13(z) is uniform along the length of thecylinder (z). In general, (130(z) can vary which results in three-dimensional vortex structuresappearing in the wake.As the Reynolds number increases further, past 300, the velocity profiles start to fluctuate andthe vortices become turbulent. In a range 300,000R e53x106 the alternating wake is suppressed,5I - Introduction 6yielding a turbulent wake without large scale structures. The periodic wake is reestablished atmuch higher Reynolds numbers.An examination of the wake reveals that vortices are the important feature at all but the lowestof Reynolds numbers. Even in the completely turbulent wake vortices exist but at very smallscales and in three dimensions. Therefore, a study of vortex behaviour is a study of the processeswhich create drag and lift forces and set the overall structure of the wake.In practice, the simple wake of parallel organized 2D motion is rare or difficult to maintaint[Lewis and Gharib, 1992]. The initial fluid motion may be simple and parallel; however it usuallybreaks up into smaller structures no longer parallel to the main flow. If the processes which leadto three-dimensional structures can be identified and modelled, better engineering predictions canbe made. One of the aims of this study was to isolate the individual processes that might possiblylead to a model of the three-dimensional wake.In the work reported here, a novel method utilizing trigger plates [Seto et. al., 1991] was usedto modify the starting phase of the laminar vortex shedding cycle from a circular cylinder. Theplate shapes were varied in order to introduce three-dimensional features in the two-dimensionalf In fact, the inspiration for this work was due in part to Stuart Loewen's Ph.D. thesis[1987] [see also Loewen et. al., 1986]. He observed the surface eddies of the wakecreated behind a grid. The eddies were assumed to be two-dimensional but on furtherinvestigation they were shown to be quite three-dimensional below the surface.61 - Introduction 7wake. The resulting three-dimensional structures could then be treated as variations of the two-dimensional vortex shedding along the span. In a complicated flow field the distinction between2D and 3D features is not obvious; however, in the vortex street the span direction serves as aunique reference axis. For this study, a two-dimensional eddy is assumed axisymmetric and hasno spanwise velocity component (along the vortex line, parallel to the cylinder) and uniform sizealong the vortex line. A three-dimensional eddy may have spanwise flow, loss of axial symmetryor varying radius along the span direction (see Figure 1.2).Figure 1.2 Definition of 3D vortices.71 - Introduction 8Figure 1.3 Vortex street (a) vortex cores marked with dye (b) tracer particles showing flowFigure 1.3 shows the street under study as visualized by 2 different techniques: In (a) the vorticesare marked by dye and in (b) the fluid motion is made visible through the motion of tracerparticles on the surface of the water. These are visualizations of cross sections of the wake,which, in general, is three-dimensional. However, assuming the vortices extend into and out ofthe page to infinity, like straight rods, and that the flow between the vortices is the same at every8p 7'^(Figure 1.4 Streamwise vortices at Re=326.I - Introduction 9spanwise location along the rod, a two-dimensional vortex street is formed as indicated inFigure 1.1. A sheet of fluid (called the "wake river") meanders between the two rows ofoppositely rotating vortices. As the cylinder moves relative to the fluid, the flow separates infront of the cylinder and convects around the two sides, meeting at the rear of the cylinder tobecome part of the continuously emerging wake river.Unfortunately, this simple picture breaks down at higher Reynolds numbers, where complicated3D effects can be observed in the wake as shown in Figure 1.4. For this reason, this study isrestricted to an investigation of 3D effects in laminar vortex streets at 40<R e<150. With the9I - Introduction 10results of these studies, it is hoped that the more complicated 3D flow field at higher Reynoldsnumbers may be interpreted.1.1 Related WorksAt low Reynolds numbers, the vortex street behind a circular cylinder is two-dimensional but iteasily breaks into 3D structures such as vortex loops, inclined sections, undulations, dislocationsand interconnections. The reason for the appearance of the 3D structures has been a focus ofcontroversy with some adherents to the nonuniformity of the flow being the cause [Gaster, 1971]and others contending that the structures arise from the end conditions [Tritton, 1959]. Evidenceindicates that the cause of many effects is the end conditions but one cannot ignore that theeffects can be enhanced through applying 3D variations in the oncoming flow.Many researchers (J.H. Gerrard [1966], D.J.Tritton [1959], and others) have observed obliquevortex shedding and experiments by C.H.K. Williamson [1988a] indicate the wake changes fromvortices shedding parallel to the cylinder (2D) to shedding obliquely (3D) around R e=65. In manycases the vortex ends bowed towards the end of the cylinder [Gerrard, 1978; Gerich andEckelmann, 1982] giving the impression that the end conditions are of importance. Furtherinvestigation by Slaouti and Gerrard [1981] verified this suspicion by showing that theobliqueness could be enhanced or suppressed by changing the end conditions. In later studies,Williamson [1989c] showed that by attaching endplates to the cylinder the obliqueness could be101 - Introduction 11suppressed altogether. In a similar way, Hammache and Gharib [1989] forced parallel sheddingby placing 'control' cylinders close to the ends of the main shedding cylinder (smaller cylinderswere positioned transverse to the shedding cylinder).In order to acquire more information, some experimenters have modified the straight 2D sheddingcylinder experiment to observe the 3D shedding effects. One of the simplest is to observeshedding from an inclined cylinder in hope of forming inclined vortices. Ramberg [1983] triedthis and observed that for small angles of the rod to the flow the vortices shed obliquely at twodifferent angles, both at a shallower angle than the rod. At even greater rod angles, both sheddingangles still existed but at different frequencies. Slight bends were also applied to the cylinder byTritton [1959] who noted that kinks easily form at the bend elevation in the wake. Many havealso varied the diameter, producing 3D wake structures. The direct effect in the wake of avarying diameter cylinder is a local variation in vortex shedding frequency but the observedresults are much more complicated. In one experiment, Williamson [1989b] added a small collarto increase the cylinder diameter over a small local area of the span. He observed A vorticesforming in the wake much larger than the collar and interconnections between the vortexelements. In a similar vein, local diameter reductions have been introduced. For sudden stepchanges in diameter, Yagita et. al. [1984] have shown that vortex breaking and reconnection donot occur unless the reduced diameter is less than 0.8D. In a more recent study, Lewis andGharib [1992] identified three different shedding modes in the region of the step; 1. periods ofinterconnecting loops form in the wake; 2. loops are seen in the wake but are inclined along a111 - Introduction 12modulation zone; and 3. loops form on only the laminar side of the wake. A more gradual axialvariation is to use a tapered cylinder. This has been investigated by Gaster [1969; 1971] whofound the shedding frequency was locally determined but that the vortices shed in cells along thespan. The results are the same for cylinders mounted in shear flows [Tavoularis et. al., 1987;Kiya et. al, 1980] (the experiments are similar since the shedding frequency is proportional tovelocity).The observed cell structures, linkages and kinks prevent loose vortex ends and arise from avariation in voracity along the vortex. Gerrard [1966] proposed that linkages form as arequirement to close the vortex ends and may loop to an adjacent vortex if there is no other freevortex end available. A similar but more complete model was put forth by Williamson [1989c].He proposed that the interconnections (or 'vortex divisions') are due to relative differences in thetime the vortices are shed along the span of the cylinder. Linkages form between vortices whenthe ends are displaced form each other and their strengths differ in the spanwise direction.1.2 Aim and Synopsis of ThesisThe aforementioned work of other researchers provides details on individual phenomena, but donot provide a comprehensive theory of vortex shedding. The work here advances the theory ofvortex shedding towards a single framework in which the structure of the vortex street can bedetermined. This is done through a novel experimental technique and the introduction of a phase12I - Introduction 13front analysis which is inspired by the Huygens principle. The difference between theexperimental work here and previous studies is the ability to manipulate the 2D street byintroducing plates to locally set the phase of the vortex shedding cycle. This new technique hasthe capability to set experimental configurations which are repeatable over many runs. This hasnot been available to previous studies of the vortex street. Because of this new technique,predictions of where, when , and what effects will be seen in the wake can be checked ratherthan coaxing random chance to exhibit the sought after phenomena. That is, instead of draggingthe cylinder through the fluid, observing the resulting structures and then sorting them out fromthe other effects and structures, the desired structure is cleanly produced without the confusionof random effects through setting the starting conditions. In this way, the nonuniformity of thebackground flow has a very limited influence. By setting the starting conditions, the wake candevelop in the same manner from run to run so that a structure is observed at the same locationin the tank with each test run. Small modifications to the starting condition will cause smallvariations in the structure so that transition points (to new structures) can be defined.On the conceptual side, the ability to trigger spanwise variations led to the development of aphase front model that is based on a Huygens construction but uses the Biot-Savart Law in theform of a Kirchhoff integral. Using this analysis, some transitions and behaviours have beenidentified, modelled and confirmed experimentally.131 - Introduction 14The thesis is composed as follows: Chapter 2 presents the simple, isolated vortex models anddiscusses the expected behaviours when a vortex is distorted into a third dimension. These vortexmodels and associated theories are used as a basis for the established wake models. Thefollowing section, chapter 3, introduces the Biot-Savart law for fluids, from which a diffraction-type integral for fluids is derived. The diffraction equation is then used to build a comprehensivephase front model of the wake which is used to explain other researchers' published results.Chapter 4 contains a description of the experimental apparatus used to verify the phase fronttheory, through setting known initial conditions. This is followed by the experimental results inchapter 5. The fluid equations and additional background information is collected in theappendices.1.3 ConventionsIn the hope that the casual reader will find this work easy to follow and refer to, the conventionsbelow have been adopted throughout this thesis except where explicitly noted.The coordinate system is as defined in Figure 1.5. Notice that the cylinder motion is in the +xdirection and its axis is parallel to the z axis. In the case of two-dimensional vortex shedding thiswould result in the voracity vectors also being parallel to the z axis. The shed vortices aredescribed locally using cylindrical coordinates as indicated.14U.(Cylinder)Motiona • ElVortexr45.._' ._...--/—Y1 - Introduction 15The list of variables commonly used is given in Appendix A (page 162), and a summary of thefluid dynamic equations is given in Appendix B (page 164).Figure 1.5 Coordinate systems used.Most of the photos are side views of the vortex street. They were always taken in the samemanner with the same settings. The shutter speed was 1/60th second at an f stop of 22. The same50 gallon tank was used throughout and the working fluid is tap water. In each case an 8mmdiameter rod was used (in the tapered cases the cylinder starts at 8mm at the water's surface andreduces downward in the picture). The submerged length of the cylinder is 410mm except in afew noted instances. The cylinder always travels from left to right at Re.--100. The eddies aremarked by a solid dye line on the side of the cylinder away from the camera and are marked bya dashed line on the side facing the camera. In the background is a 2 cm ruled grid.152 - VORTEX STREET MODELSThe wake behind an object is an aggregate of individual vortices linked by a ribbon of flow.Therefore, the appearance and development of the wake is a reflection of the evolution of itsvortices. The vortices themselves evolve in response to nearby structures or undergo self-induceddevelopment (see for instance Hama [1962]). The structure of the wake, and the vortices areinextricably tied to one another; the wake structure cannot be described without reference to itsvortices. Hence, before proceeding to the discussion of vortex street models, it is necessary toreview the standard isolated vortex models and how they develop when subjected to a three-dimensional distortion.This chapter begins with a review of the standard vortex models, which are then used to buildthe two-dimensional von Karman vortex street model. This is followed by a discussion of theknown frequency of vortex shedding. Three dimensional effects on a single vortex are thenconsidered for the case of a tapered vortex, where axial flow is expected. A simple model isdeveloped to determine the axial flow velocity as a function of angular velocity, radius variation,and viscosity. The implications of tapered vortices on the structure of the wake are considered162 - Vortex Street Models 17and the role of instabilities is discussed. Finally, the chapter concludes with a model of the vortexstreet based on van der Pol oscillators.2.1 Two-Dimensional Vortex ModelsThere are two well known two-dimensional vortex models, the Rankine vortex and the Oseenvortext. Both models have no axial or radial velocities, are uniform in the axial direction andhave an axisymmetric tangential velocity profile. The Rankine vortex is a steady state solutionin an inviscid fluid whereas the Oseen velocity profile develops in time in a viscous fluid.Although the Oseen solution is inherently more realistic, the Rankine model is the most widelyused owing to its mathematical simplicity while yielding reasonable approximations to real flows.The Oseen solution itself is of interest because it shows the vorticity diffusing away from thecore area as the rotation speed is reduced by viscous action.2.1.1 Rankine VortexThe Rankine vortex's [1921: Rankine calls his vortex a 'combined vortex' section 633, p.576]velocity profile is found by solving Euler's equation:t The Oseen solution is occasionally referred to as Lamb's vortex due to its inclusion inLamb's book [1932, section 334a, p.590-592] on hydrodynamics, but Oseen [1911] solvedthe problem earlier.172 - Vortex Street Models 18al rrirr-.^VP= --at2.1.1where t is time, P is the pressure, p is the density, and Y is the velocity vector. Assuming noradial or axial velocities and axial symmetry, the swirl velocity for the Rankine vortex is:if OsrsR""c = {0R2ir if mRwhere S2 is the angular velocity, R is the size of the vortex, and r is the radial coordinate. Thistwo dimensional velocity field is plotted in Figure 2.1. Physically, it describes a core of fluidrotating like a solid cylindrical body, surrounded by a slower rotating halo with a velocity profilethat falls off ast lfr.This vortex model for R=0 is valid only inpotential flows (very high Re or inviscid orvery low Re). This model has shown to be agood approximation of vortices in real flowseven though the sharp peak in the velocitydistribution and its steadiness in time are notreflected in nature. For example, PiercyFigure 2.1 Rankine vortex velocity profile.In Rankine's [1921, p.576], he named the vortex which rotates solely as a rotatingcylinder (R=0.) a 'forced vortex'. He called a vortex which only exhibits the haloproperties (i.e. R=0) a 'free spiral vortex', and any combination thereof he termed a`combined vortex'.2.1.2182 - Vortex Street Models 19[1923] found trailing vortices from aircraft wings have nearly this velocity profile except forsmoothing of the peak in the velocity profile where the core meets the halo. SuperimposingRankine vortices to form more complex flows such as vortex streets has also proven successful[Von Kalman and Rubach, 1912].2.1.2 Oseen VortexA model which has both time development and a smooth velocity profile is the Oseen vortex.Oseen [1911] [Lamb, 1932: section 334a, p.590-592] found a viscous eddy solution to theNavier-Stokes equation by noting the similarity between the heat equation and the vorticityequation:ac)at =vx(Oxa)A-vv2a^2.1.3Assuming axisymmetric velocity with no axial or radial components, the vorticity can only bechanged by diffusion, and the above equation is reduced to a form similar to the heat equation:(;)^vV2o3^ 2.1.4Changing variables to t and =r2/4vt and identifying coz as the new function, h(,t)lt, equation(2.1.4) is transformed to:th =h+(1+01/ / +(li ll^ 2.1.5192 - Vortex Street Models 20where the dot is the derivative with respect to time and the primes indicate derivatives withrespect to C. A solution for tit.° is obviously:0 = h l +h^h= Ce -cThe vorticity and velocity are obtained by solving:h^a= — =(vxu) = —(rU )t z r ar^opgiving the Oseen solution for the viscous vortex (plotted in Figure 2.2):U^__e —r2/412nrwhere F is the circulation at r=0.. This viscous vortex is very similar to the Rankine model. Thecentral core rotates almost like a solid cylinder and the halo's velocity profile falls off as lfr.Unlike the Rankine solution, the Oseen model has a rounded peak in the velocity profile wherethe core meets the halo and the velocity decays with time. As the Oseen vortex's rotation slowsthe size increases as R5_Nl5vt.Even though this vortex is physically more appealing than the Rankine model it too cannot existin nature because, first the vortex has a singularity for t=0 and second, as in the Rankine vortex,the lfr halo contains an infinite amount of energy. However, both models closely match knownvelocity distributions.2.1.62.1.72.1.8202 - Vortex Street Models 21Superimposing vortices of opposite rotationsto form a flow structure improves thesituation, the velocity field at large r falls offmore rapidly as the influence of the twovortices cancel each other like in a dipolefield (1/r2). A viscous solution for isolatedvortices which places bounds on the energycould be obtained by retaining the th termFigure 2.2 Oseen vortex velocity profile atdifferent vt values normalized to 1727c.when solving equation (2.1.5) to obtain a new velocity profile. A more complete and accuratesolution is not required for the work here, the simpler Rankine solution provides reasonableapproximations to most flows while the Oseen solution serves as a good model for timedependent applications. This will be demonstrated by building vortex street models based on thestandard vortex models.2.2 The 'Carman Vortex StreetThe Kaman vortex street exists at low Reynolds numbers (40<k<150) in the wake of a circularcylinder. Von Kaman [Von Karm6n and Rubach, 1912] constructed a mathematical model ofthe street by superimposing Rankine vortices, as shown in Figure 2.3, to form an infinitely longwake. Though presented as an infinite wake, Figure 2.3 is similar to the stable portion of a vortexstreet well behind a cylinder moving to the right. The street moves to the right at the drift21Figure 2.3 Von Karman vortex street formed by2- Vortex Street Models 22velocity, U drip and the configuration is stableonly when a/b=0.2805 (from cosh(r,a/b)=4 -i).Lamb [1932: section 156, p.224-229] includesthe proof of stability in his book, showing avortex perturbed from its equilibrium positionthe superposition of potential vortices.will experience a velocity which brings it backto equilibrium. The drift velocity of the stable configuration can be found using potential flowtheory and the coordinate system defined in Figure 1.5, such that [Batchelor, 1967: PotentialTheory is covered in Chapter 2, specific equations are on p.106-107]:Ux = A =^and^Uy = =-ax ay ax2.2.1where x is the velocity potential, and is the stream function. Using the imaginary plane [Saffand Snider, 1976]:Z=x+iythe velocity potential and the stream function form a complex potential:from which the velocity field can be found by a simple derivative:2.2.22.2.3222 - Vortex Street Models 23^U s- Uy az^ 2.2.4The complex potential for a Rankine vortex with R= 0 centred at Z=Zo and having a circulation,F, is:w = --27c In(Z-Z0)^ 2.2.5Summing the potentials for vortices centred at Z0=Zi , Zo= ±b-Z,, Zo= ±2b-Z,, etc., gives thepotential function for a line of vortices:w^n(Z-Zi)in{sin( 271^bThe potential of the infinite 'Carman vortex street is obtained by adding the potential of a lineof vortices with circulation F at Zi =(ial2) to a line with circulation -F at Zi .(b-ia)12. Thevortices move at the local fluid velocity, hence, the drift velocity of the street is found byapplying (2.2.4) at the centre of any vortex (eg. Z=(ial2) or Z=(b-la)12) to obtain:U2b(btanh na)drift 2.2.7This is the velocity at which the street follows the shedding cylinder.The Von Karman model of the vortex street assumes an infinitely long wake, however, realvortex streets are finite. The absolute maximum length of a real vortex street is from the vortexshedding cylinder to the point in the fluid where shedding first began. The length of the street2.2.6232 - Vortex Street Models 24is further reduced by the drift velocity and by diffusive processes causing real vortices to decayand diffuse outward from the wake. The effect of the drift velocity can be ascertained throughvon Karman's treatment (equation 2.2.7). The viscous processes can be modelled by replacingthe Rankine vortices with Oseen eddies as demonstrated by Schaefer and Eskinazi [1959].Schaefer and Eskinazi's [1959] Oseen based model of the vortex street has the character ofvortex growth and velocity decay as the eddies move downstream from the shedding cylinder.Each Oseen vortex distribution starts at time t= 0 at the cylinder and develops in time as it movesdownstream. Beginning directly behind the cylinder and extending downstream, Schaefer andEskinazi observed a region of vortex street formation. This region extends to x0=260DIR, forRe<150. It is a site of lower pressure where the fluid is accelerated into a circulating patternsimilar to the region visible in Figure 2.4. This is where the boundary layer rolls up into thevortices that become part of the wake. As the vortices are forming, they move along with thecylinder. Once a vortex is fully formed its velocity with respect to the cylinder increases and itmoves out of the formation region and into the developed wake. The developed wake follows thecylinder, moving forward at about the drift velocity with a vortex spacing ratio as predicted byvon Karman. After some time, the vortices start to diffuse into one another. At this point, thewake becomes unstable, its behaviour is irregular and eventually undergoes a transition toturbulence [Schaefer and Eskinazi, 1959].242 - Vortex Street Models 25Figure 2.4 Recirculation bubble behind a 10 cm wide plate movingat 9.45cm/s in water. The camera was moving with the plate.Schaefer and Eskinazi developed an Oseen vortex based model of the wake in conjunction withexperimental observations of the vortex street. This is a phenomenological model; the vortexmotion and spacing, and their associated circulation was found from experiments. In other words,the velocity field was found by placing Oseen eddy models at the locations observed in theexperiment and then finding the circulation for the eddy models by matching the predictedvelocity with that in the experiment.This analysis of the wake accounts for the viscous effects and demonstrates a mechanism for thebreakdown of the wake but it does not in itself predict the structure of the wake as Von252 - Vortex Street Models 26Karman's model does. Von Karman achieved his result without reference to experimental results,appealing only to a stability analysis. This stability analysis gives the vortex spacing ratio for thevortices but not the absolute distance between them. The vortex spacing, b, is given by:U -Ub -^fThe shedding frequency, f, is obtained from observations. Various studies over the last 100 yearshave shown that a well defined shedding frequency exists, as discussed in the following section.2.3 The Frequency and Phase of Vortex SheddingThe first measurements of the vortex shedding frequency of a circular cylinder were done byStrouhal [1878] while investigating the Aeolian tones heard from telegraph wires. Strouhalobserved that the shedding frequency depends on the fluid velocity and on the diameter of thewire such that:sucef = D2.3.1where the proportionality, S, is called the Strouhal Number. It is often treated as adimensionless frequency. Measurements show that the Strouhal number is relatively constant athigh R„ as theorized by Rayleigh [1879] [1896: p.413-414], and changes with the cross-sectionalshape of the vortex shedding cylinder [Blevins, 1990: p.50]. For a circular cylinder, S isapproximately 0.21 to 0.22, but at lower speeds it has been found to vary as [Roshko, 1953]:2.2.8262 - Vortex Street Models 27S = 0.212 - R4.5^/2for^50<.<150^2.3.2ReS = 0.212 - 2'7 for 300</V2000R,2.3.3The transition in S between the two ranges given above is generally smooth, but there is adiscontinuity [Williamson, 1989a] around Re=180 and a second one between R e=230 and 260 dueto changes in the shedding modes. The above results are for a two-dimensional flow of parallelshedding, but when the vortices shed obliquely (a three-dimensional mode) the observedfrequency is lower than expected.For straight cylinders, oblique shedding modes [Gerrard, 1966; Tritton, 1959] (tilted in the planeof the shedding cylinder) are observed many diameters downstream from the start of theexperimental run. Williamson [1989c] reports that for experiments where /2,45 and L/D=140,even after towing the cylinder 100D the shedding pattern is still parallel to the cylinder withoblique shedding beginning to appear close to the cylinder ends. The oblique mode diffuses infrom the ends until after 600D oblique shedding is observed over the entire span in a 'chevron'pattern. Measurements of the shedding frequency of an oblique mode at angle 0 to the cylinderhave shown the frequency is lowered to [Williamson, 1988a]:Se = S cose or fe =fcose 2.3.4For a fixed convection speed of the wake, U drift, the above result implies that the spacingmeasured perpendicular to the vortex lines is fixed at DIS. If the drift velocity does vary with272 - Vortex Street Models 28shedding angle, the deviation will not be large, therefore, the vortex spacing should be close toconstant.The shedding frequency is uniform over most of the span, except within a few diameters of theend (6-15 D) [Gerich and Eckelmann, 1982] where it is lower [Stager and Eckelmann, 1991] by10-15%. The lower frequency is associated with the shedding angle being greater at the cylinderends than over the rest of the span [KOnig et. al., 1992]. Since there exists more than onefrequency along the span, the phase difference in the shedding cycle between adjacent sectionswill increase as:= 2n(fi-f2)t 2.3.5Once the phase difference becomes too large, the vortices form interconnections and eventuallybreak in order to realign the phases (dislocations form). Along the span, the advance in the phaseof shedding is slowed by the lower frequency shedding at the ends of the cylinder, causing thewake to incline or bow towards the ends. After a sufficient number of shedding cycles, the phasedifferences at the ends of the cylinder are large enough to cause the vortex generations to formdislocations and interlink (in a manner similar to the model proposed by Williamson [1989c]shown in Figure 2.5). These dislocations are indicative of a phase jump between the two sheddingsections on either side of the break. The formation of the dislocated region allows the vortexstreet to break so that the shedding sections may interlink with a phase difference of 360°(shedding cycles are in harmony once again). This is accomplished by one extra vortex sheddingcycle occurring in one section than in its neighbouring section with interconnections to 'tie' the282 - Vortex Street Models 29vortex ends that would result. This dislocation and interconnection process occurs at the interfacebetween two shedding regions, allowing fixed vortex shedding 'cells' to be defined along thespan [Konig et. al., 1990]. The interface between the cells is indicative of the difference inshedding modes between the two cells. In other words, the shedding angle [KOnig et. al., 1992]and frequency [Stager and Eckelmann, 1991] change slowly over a cell and abruptly changebetween cellst. For a straight cylinder, the cells remain fixed around the ends of the cylinder,allowing the wake to link to the lower frequency ends while maintaining a stable obliqueshedding mode over the rest of the span.Figure 2.5 C.H.K. Williamson's vortex linkage model. The relative vortex strengths areindicated.I A similar variation in phase is known for the minima in optical diffraction patterns.292 - Vortex Street Models 30The vortex shedding mode discussed above pertains to the wake of a circular cylinder where theresults are highly sensitive to the end conditions. A simple geometry which exhibits similarbehaviour and is less sensitive to end conditions is vortex shedding from a tapered cylinder.Examining equation (2.3.1), the shedding frequency varies inversely with diameter, therefore thevortices should be inclined to the cylinder. The inclination is seen in Figure 2.6, the diameter ofthe shedding cylinder decreases with depth, therefore, the vortex shedding frequency is higherat the bottom than at the top of the picture and varies smoothly over the span. The wake inclines,increasing the phase differences until the vortices break and, in addition, an axial flow develops.The inclination and breaking behaviour can be predicted by a model based on phase fronts, tocome later, but first, one ought to have a simple model that shows axial flow, and the presentstate of understanding of the vortex shedding process must be reviewed.2.4 Tapered VorticesOne aim of this thesis is to model the flow field in the wake of a conical cylinder as shown inFigure 2.6. The resulting vortex street is composed of tapered vortices with axial flow. For thepurpose of this investigation, a new vortex model must be found that can predict such axial flow.A vortex model which includes axial flow is Burgers solution [Burgers, 1948] [Odgaard, 1986](Appendix C) which assumes an axial velocity proportional to z and a radial velocity proportionalto r. These assumptions do not seem to be applicable here. For that reason a new model has beendeveloped for these studies. If a tapered vortex is considered to be a swirling flow on the inside302 - Vortex Street Models 31of a gently tapered pipe a simple model can be formulated. The pipe is the size of the vortex,its diameter is the local vortex size, R. Spinning the pipe at the fluid velocity where the coremeets the halo will transfer no viscous stresses to the tangential fluid motion, but the viscousstress will be felt in the axial direction. The effects of the tapered vortex are then confined solelyto the core area contained in the pipe. Provided the vortex taper is shallow the axial velocity ofthe fluid is closely approximated by the Hagen-Poiseuille equation[Streeter and Wylie, 1981:p.192]:d// - -^ (P+Pgz)4vp dz2.4.1where g is the gravitational constant. The pgz term is neglected since the vortex is surroundedby fluid of the same density (no buoyancy forces are felt by the fluid). The P+pgz term isactually the energy per unit volume in the fluid available to the axial flow, a component of whichis the energy in the swirling motion. The energy in the rotation is not a component of the usualpipe flow, but since it, like the pressure, may change as the fluid travels along the vortex it mustbe included. The kinetic energy due to rotation in a Rankine vortex core has a density:p 2r 2 2.4.2 2The pressure term is found by balancing the centrifugal force with the pressure gradient suchthat:312 - Vortex Street Models 322aP PUT^2=^- pa rar^rAssuming the conditions outside the Rankine vortex core are uniform, the solution becomes:P PO20.240^2.4.42Combining equations (2.4.2) and (2.4.4) yields the total energy density:p 02(r2 ...R2) p Q2r2^p 02R2^ 2.4.52^2^2This is the energy which binds the vortex together [Ahlborn et. al., 1991a]. Notice that the energydistribution is uniform across the vortex core like the conditions assumed in pipe flow.Substituting back into equation (2.4.1) gives:R2-r2 d p 2R2)  R(z)2-r2 d (Q(z)2R(z)2)=  4v p dz 2^8v dz2.4.6Notice that Uz=0 if OR=constant along the vortex (this condition is met if the eddy's peripheralvelocity equals the free stream velocity UJ. However, a constant state could not persist for long,the action of viscosity would redistribute the rotational energy in the spanwise direction so thatthe vortex (at least locally) rotates as a solid cone. Assuming da/dz =0 locally, (2.4.6) istransformed into the viscous pipe flow velocity for tapered vortices:^R 2 -r2^dR=^ (QzRzN^4vR^dz2.4.32.4.7322 - Vortex Street Models 33This relation for the axial velocity implies an important instability mechanism: a minor initialconstriction of the vortex radius leads to an axial transport of mass out of the reduced area,resulting in a further reduction in the radius. As the radius shrinks, viscous dissipation increasesuntil the vortex breaks up or pinches off. This can be thought of as a hydrodynamic sausageinstability. This motion and eventual breaking is evident in experimental runs like that ofFigure 2.6, the results of which will be discussed later.2.5 Kelvin's Circulation TheoremThough the above analysis of a tapered vortex implies that a vortex can be easily broken, thereare some limitations to the manner in which the break occurs. The breaking process is modifiedby the conservation of circulation of the vortex in an inviscid fluid. This law was presented byLord Kelvin [1869] who stated that the circulation around a closed contour enclosing a vortexin an inviscid fluid is constant in time.The conservation of circulation is easily proven [Tritton, 1988: p.113] through taking thecomplete time derivative of the circulation:dt = i0•aidt 2.5.1where F is the circulation and the integral is around the closed contour, 7, surrounding the vortex332 - Vortex Street Models 34line. The enclosing contour moves with the fluid as the vortex changes with time, therefore,carrying out the differentiation yields:dr= dO al xo. daidt j dt^J dt^ 2.5.2The final integral in the above equation can be evaluated since the contour moves with the localfluid velocity:r" •74. dai _ 1,,17 _^_dt I"^2 2.5.3The result of the above integral equals 0 since the integration is around a closed contour (thestarting and ending points have the same value). Substituting Euler's equation (2.1.1) into theremaining term of (2.5.2) gives:dr = -p JIvPai = odt^ 2.5.4For a pressure field without discontinuities, the integral on the right hand side equals 0 since theintegration is around a closed contour. Combining equations (2.5.2), (2.5.3), and (2.5.4) yieldsKelvin's Circulation Theorem:r = constant = fe•ai = f w .aA^2.5.5Awhere the integration is over the area bounded by the contour 1 enclosing the vortex and A isnormal to the described surface. The final step in the above equation is an application of Green'sTheorem and the definition of voracity (65=Vx0.342 - Vortex Street Models 35Therefore, in an inviscid fluid, the circulation of a vortex line is constant both in time and alongthe vortex line. Even when the vortex tube is stretched, the circulation remains unchanged. Asa vortex tube is stretched, its diameter will decrease to conserve mass. The reduction in diameternecessitates an increase in the vorticity to conserve angular momentum. Thus, a vortex line inan inviscid fluid cannot be broken. This is further evident from taking the divergence of thevorticity:V•6 = V•Vx CI= 0^ 2.5.6Therefore, the vorticity lines must form closed loops in the fluid, much like magnetic field lines.Figure 2.6 Vortex street produced by a tapered cylinder (length 410mm). D=8mm (R e=92.6) atthe surface, and D=2mm at the bottom (R e=23.1).352 - Vortex Street Models 36Equation (2.5.6) holds for viscous as well as inviscid fluids, however, in a real fluid, viscosityexerts a drag on the smaller vortex sections, taking away the rotational energy at a higher ratethan for the larger, unstretched sections. This would allow the vortex to break at the reducedsection. Even though the divergence of the vorticity (2.5.6) apparently indicates that the vortexmust remain unbroken, it is still possible to observe broken vortices in a viscous fluid. There isa subtle point as to why: though a vortex contains vorticity it is the circulation that is observedand considered to be the vortex. Consider a Rankine vortex with R=0. The vorticity isconcentrated at the centre with the halo containing no vorticity. The tangential velocity in thehalo is given by rarcR, this is the area of observable circulation. The circulation can spread farinto the fluid with a low tangential velocity, thus an apparent break may be observed. Further,in cases of flows containing vortices of opposite rotation, the circulations may overlap,superimposing to form regions where no circulation is observable, and hence no vortices exist.Just such a case can seen in Figure 2.6 where the vortices terminate in a region of no vortexshedding (Re is too low) about 1/4 of the way up from the bottom of the photo.Vortex terminations are seemingly more difficult to form in a region of vortex shedding owingto the concentrated circulations (vortices) already existent in the flow field. The wake may appearbroken at places but close inspection shows interconnections between the vortices like those inFigure 2.7. These dislocated regions are most likely locations of reduced circulation due tomixing of oppositely rotating vortices. The connections form in an effort to conserve circulationbut dissipate rapidly owing to their smaller size and interactions with each other. It is even362 - Vortex Street Models 37possible that their rotations had ceased by thetime the photo was taken, leaving the dyelines as a mark of their passing and giving theimpression that the vortex lines are stillsolidly connected.Therefore Kelvin's circulation theorem doesnot hold in a viscous fluid but does have aFigure 2.7 Blow-up of area indicated onFigure 2.6 showing interconnections between thevortices.limited effect, mediating what will happen between vortices. In general, the vortices will beinterlinked but the circulations are allowed to diffuse outward and mix with opposite rotations,yielding weaker vortex sections at these interconnections than over the rest of the vortex street.2.6 Vortex Street InstabilitiesFor this study, an instability is considered to be a perturbation, or wave in the fluid flow or ona coherent structure which grows or develops in time into a coherent structure of different shape.The presence of instabilities usually leads to turbulence or a breakdown of structure. The steadyvortex shedding itself may be explained as a self-limiting instability [Saffman and Baker, 1979]:An eddy grows on one side of the shedding object until it reaches a state where further growthdoes not occur causing another eddy to begin growing on the other side of the object. Thisgrowth cycle alternates from side to side forming new eddies and casting off older ones at regular372 - Vortex Street Models 38intervals forming a periodic vortex street. This process is said to be absolutely unstable [Oertel,1990]; effects of the downstream flow (the wake) reach back to the obstacle affecting the growthand shedding of vortices, maintaining the unstable condition at the vortex shedding obstacle. Inthis model the shedding "clockwork" is maintained through absolute instabilities. The wake alsocontains elements of convectively unstable phenomena, which are convected downstreamgrowing as they move away from the shedding object but having no influence on the state of theflow upstream!At the beginning of an experiment the vortex shedding process is initiated by small perturbationsin the oncoming flow as evidenced from computer simulations by Anderson et. al. [1990] Theseperturbations are of convective nature since their initial effect diminishes rapidly after thebeginning of the experimental run, giving way to an absolute instability which maintains thevortex street. In our work, the effect of the irregularities in the oncoming flow is overwhelmedfirst by starting the experiments in quiescent water and further by using trigger plates to applya strong perturbation when the vortex street first forms. This is a convective event, the initialperturbation is left at the starting point of the wake and is eventually so far downstream that itcannot affect the shedding conditions at the object.t Huerre and Monkewitz [1990] describe an absolutely unstable flow as one in which thedisturbances spread both upstream and downstream, contaminating the entire flow. Theydescribe a convectively unstable flow as one in which the disturbances are convectedaway from the source.382 - Vortex Street Models 39Absolute instabilities may play a roll in maintaining the production of three-dimensional vortices,an example is the vortex loops seen in Figure 1.4. These are the same type of loops observed byWilliamson [1989a] who describes the mechanism as an absolute instability: "The process ofloop generation is self sustaining in that there is a feedback from one loop to the next so that awhole string of vortex loops form at the same spanwise position."The production of an eddy and its influence on the vortex in a subsequent generation takes placeclose to the shedding object. As the eddy is swept downstream into the far wake it can no longerinfluence the vortex shedding mechanism nor can it feel the change in conditions at the sheddingobject. Any changes that the eddy undergoes in the far wake are due to the instabilities carriedwithin itself or from the nearby fluid. These are the convective instabilities which get carrieddownstream with the eddy, affecting the wake structure in its vicinity. The imposed instabilitiesmay cause a two-dimensional eddy to undergo a transition to a three-dimensional structure. Forexample, Inoue [1991] has shown numerically that introducing a small wave on a vortex ringleads to large scale instabilities. In the case of a straight vortex, a small wave or undulation leadsto soliton behaviour and may develop into helical twists on the vortex [Aref and Flinchem, 1984].The waves may even grow to a point where the vortex can no longer support them and is tornapart. Vortex breakdown is a route to turbulence, transferring energy directly into the dissipationrange rather than through a cascade of eddy sizes [Pierrehumbert, 1986].392 - Vortex Street Models 40In the study undertaken here, the experimental technique, using trigger plates, provides certaintypes of 3D instabilities and thereby allows a study of their effect on the transition to three-dimensionality through observing the subsequent development of the vortices and the wake.The concept of 3D instabilities yields a qualitative understanding of the shedding cycle, but doesnot address the propagation of wake structures. Two different models have recently been reportedwhich look at details of the wake formation.2.7 Van Der Pol Oscillator Model of Vortex SheddingThe vortex shedding cycle is a periodic phenomenon with a natural frequency which depends onU.., as well as on the shape, and size of the shedding object. The periodic cycle is maintainedby a process which may be described as a self-excited, self-limited absolute instability originatingwithin the near wake. Once started, the wake stabilizes in a steady vortex production process[Seto et. al, 1991]. These are also the properties of the van der Pol oscillator [Boyce andDiPrima, 1977]:+ e(q 2 -1)4 +q = 0 2.7.1where q is the strength of the oscillator and c is the nonlinear growth rate parameter. The solutionis periodic, having a single closed limit cycle with an amplitude around 2. This equation is usedwidely in mechanics to model vibrations or frequency dependent phenomena [Rao, 1990; VonKarman, 1940].402 - Vortex Street Models 41The use of the van der Pol oscillator in modelling a vortex wake was first considered by Gaster[1969]. He noted that the shedding frequency of a conical cylinder was locally determined butthat the shedding pattern was not. Based on this, he proposed that the wake can be modelled bya string of van der Pol oscillators positioned along the axis of the conical cylinder. Eachoscillator emits at the locally defined frequency but coupling to the adjacent oscillators causesthe vortex formation to be globally determined. The state (phase and amplitude information) ofthe oscillator is convected downstream, its condition reflecting the phase of vortex shedding (avortex is assumed to be shed at a maximum or minimum in the oscillator's cycle). While the vander Pol oscillator has properties as observed in the wake, Gaster made no attempt to derive(2.7.1) from first principles or from the Navier-Stokes equation.This proposal was pursued by Noack et al. [1991] in a computer simulation of vortex sheddingfrom a tapered cylinder. In their model, the oscillating components are planes or slices of fluid,close to the cylinder, undergoing periodic motion as indicated in Figure 2.8. The slices of fluidprefer to oscillate at the locally determined frequency and are coupled by viscous frictionbetween adjacent slices. The strength or amplitude of the motion is assumed to be proportionalto the local diameterA0 - CID 2.7.2and the axial coupling due to viscosity is:412 - Vortex Street Models 42a2u^aq c2v ---z - c2vaz at2.7.3where the constants C1 and C2 are both oforder unity. The motion of an oscillatingslab of fluid cannot be too different inmagnitude from the cylinder size, henceC1-l. The viscous coupling is derivedfrom the viscous force between two layersof fluid. The only reason for C2#1 is toinclude other coupling effects, but here itis assumed that the oscillators do notinteract with the surrounding fluid. Afurther simplification is to choose thegrowth parameter, e, small so that themotion is approximately sinusoidal:q - A0 sin(2rft)^2.7.4Modifying the van der Pol equation (2.7.1)Figure 2.8 Van der Pol Oscillator model. Slices offluid close to the cylinder are displaced a distanceq from equilibrium.for these conditions yields the differential equation for the model:2a24 + eartf) „^-21 + (27cf)2q = C2v a3q01,2^AN^at^az2at 2.7.5422 - Vortex Street Models 43The input parameters required for the model are: the local frequency f; the proportionality forstrength C1 , taken to be 1; the coupling coefficient C2 between oscillators, also taken to be 1; andthe excitation term e. The local frequency is known from the Strouhal number (equation 2.3.2)(Noack et. al.'s work was at Re=60 to 180). The growth rate parameter, e, was assumed small inorder to keep the solution close to sinusoidal, however, e should be large enough to keep theamplitude of the oscillation close to the strength of shedding from a straight cylinder (A 0). Areasonable value of e yielding credible agreement between theory, model, and experiment eludedNoack et. al. They finally settled on a compromise of e=0.2. The model produced the correctqualities, breaking and linking of vortices, but the values for local frequency, oscillator strength,and number of shedding cells, did not agree with experiment. First, the model frequencies werehigher than in the experiment, but this may have been due to not taking into account thereduction in shedding frequency when the vortices shed obliquely (equation 2.3.4). Second thenumber of shedding cells predicted by the model was in the 100's while the experiment yieldedonly 6. A good match could be achieved by setting C 2=324 which, as the authors note, is"surprisingly high and indicates other physical processes contribute to the coupling strength."An improvement is to lower to 0.1, but C2 is still quite high at 80. Lowering c cannot be donead infinitum since this would appreciably affect the amplitude. It may be possible to improve thismodel however a more detailed understanding of the coupling is required.In the above model, the coupling was introduced as an additional term in an oscillator equation.An improved model may be realized by using a model equation based on stability arguments432 - Vortex Street Models 44which includes wave-type coupling. One such model is based on a Ginzburg-Landau equations,as investigated by Albarede and Monkewitz [1992] and more recently by Park and Redekopp[1992] * :aA--a—t = (a ,+iadA +^—82A^+az 2 2.7.6The above equation reduces to the van der Pol oscillator for a constant diameter cylinder at theonset of vortex shedding. Equation (2.7.6) describes the growth of the oscillator strength, A. Thefirst term on the right hand side describes the growth, the last term limits the amplitude, and thecentral term is a wavelike coupling. Albarede and Monkewitz did not derive the Ginzburg-Landauequation from first principles, it is, in their words, "validated by experimental data for all of itscoefficients." There are a number of coefficients to be found (6,p,1 including imaginary parts),the values of which are not intuitively obvious but obtained from experimental data. The modelcoefficients used by Albarede and Monkewitz were found at the onset of vortex shedding, which,as they point out, differ by more than 50% from the values found at R e= 100. The model also didnot account for the existence of cells close to the end of the cylinder; nevertheless, the keyfeatures of vortex shedding from a constant diameter cylinder started from rest are seen in the-I. The time dependent Ginzburg-Landau equation also appears in the field of statisticalphysics where it is used to describe the approach of systems to equilibrium (it is a routeto handling problems dependent on fluctuations and inhomogenieties). One such exampleis the condensation of electrons (forming Cooper pairs) close to the transition from aconductor to a superconductor where IA 1 2 is the superconducting electron density[Lifshitz and Pitaevskii, 1980: p.181].Park and Redekopp's treatment is similar to that of Albarede and Monkewitz except thatthe calculation is extended to include streamwise disturbances as well as the spanwisedisturbances (2 dimensions instead of 1).442 - Vortex Street Models 45model (the development of a chevron pattern or oblique shedding is seen). Despite the model'ssuccess in predicting the shedding patterns of finite spans and the angle of shedding, the resultsrely on the values chosen for the coefficients which, in turn, may depend on boundary conditions.As already indicated, these values are not obvious, making it difficult to apply the model to newflow geometries without some experimental input.In the next chapter, a new model of vortex shedding is proposed, where the parameters relatedirectly to the known properties of the Von Kaman vortex street, as already described in thischapter. The proposed model also utilizes wavelike coupling, as in Albarede and Monkewitz'sGinzburg-Landau model, but it uses a simple oscillator approach similar to that of Noack et. al.453 - PHASE FRONT ANALYSIS OFLAMINAR VORTEX STREETSVortex shedding is a periodic phenomenon which can be likened to an oscillator in which theflow parameters (velocity U, pressure P, and vorticity co) are described by the frequency, f, thewave number, k=27crk, and an initial phase, O s of the oscillator:U(1) = F(keleb Fobe ir21`h-il +..(1" .^ 3.0.1w (g)F is a function of space describing the amplitude of the given parameter, and cto is the completephase. The phase front model of the wake, proposed here, associates the formation of a vortexwith a particular complete phase value, such as c1:0=ir or 0. In other words, entire vortices aredescribed by the crests or troughs of a series of waves associated with the vortex street. Thisapproach is in correspondence with the concepts of diffraction but incorporates the knownproperties of the Von Karman vortex street described in the previous chapter, the important pointsof which are:1.^There is a locally determined frequency of shedding which varies with cylinder diameterand the angle of shedding.463 - Phase Front Analysis of Laminar Vortex Streets 472. The spacing between vortices is fixed regardless of shedding angle (equation (2.3.4)).3. There is a feedback region (an absolute instability) which apparently helps set the vortexshedding clockwork.4. Von Karman's treatment of the wake utilizes potential flow theory to linearly superimposesolutions to form a 2D vortex street. It may be possible to formulate a 3D model whichemploys linear superposition to calculate shedding patterns in a simple straightforwardmanner.5. The process may be treated as a series of emitters or as an oscillation in the wake whichis independent of the outside influences (like vortex shape, background flows, or wakesfrom other bodies).6.^A phase can be associated with the shedding cycle. The in-phase shedding regions arebounded by fractures, across which there is a phase jump. However, the regions do joinsmoothly where the phase difference is a multiple of 27r.This behaviour is similar to interference in diffraction patterns. Waves destructively interferewhen 180° out of phase and constructively interfere at 360° phase difference. Further, 'diffractionrelies on emitters of known frequency and fixed wavelength, the emissions of which superimposelinearly. This leads to a Huygens phase front construction of the wake.473 - Phase Front Analysis of Laminar Vortex Streets 483.1 A Huygens Construction of the Vortex StreetThe process of vortex shedding is periodic and has an associated spacing. Further to this, areduction in shedding frequency with inclination of the wake, fe=f cogs (the empirical relation(2.3.4)), indicates that the spacing between vortices is fixed for a specific flow speed and cylindersize. This is similar to optics where the electro-magnetic field oscillates with a fixed wavelength.In addition, electro-magnetic fields superimpose linearly, simplifying calculations. The possibilityof bringing such a simple yet powerful analytic tool into a vortex street model is highlyattractive. Could, then, the vortices be associated with wave fronts emanating from an emittingplane or slit in the fluid? Is it, therefore, possible to build a model of vortex shedding based onan analogy with optics?The vortex shedding process is maintained by a mechanism where the information to form thenext vortex generation comes from downstream in the wake. Something like an absoluteinstability affects the upstream conditions at the cylinder to produce the next vortex generation,which, in turn, is swept downstream into the wake, growing into a new structure. At some point,this new structure will be in a position and have enough strength to affect upstream conditionsand thus, become the source that influences the form of a following vortex generation.484-1vortexwakewave cylinderfrontemittersEmitter PlaneSheddingCylinderFigure 3.2 Vortices pass through the emitter plane which is positioned in the wakebehind the shedding cylinder.3 - Phase Front Analysis of Laminar Vortex Streets 49Figure 3.1 Huygens principle in the wake of a circular cylinder. A line of emitters formsa secondary wave front at the cylinder. The resulting diffraction pattern is swept backward,forming a new set of emitters and, eventually, the vortex wake.493 - Phase Front Analysis of Laminar Vortex Streets 50This process may be modelled by a line of emitters placed in the fluid behind the cylinder(Figure 3.1). The emitters are placed at a fixed place behind the cylinder, their state is incorrespondence to the vortices which convect through the location (Figure 3.2). Isolating theemitters from influences exterior to the wake, their emissions will reflect the periodicity of theshedding process. The emitters transmit phase information upstream forming an interferencepattern at the cylinder (see Figure 3.1). This pattern is then swept downstream, passing throughthe emitting plane, and, thus, the phase and amplitude of the emitter is redefined by this newgeneration.In the object frame, the phase information seemingly forms a closed loop (Figure 3.3). The phaseinformation propagates forward to the cylinder, interferes, then is swept back with the fluidthrough the emitter plane to form the secondary wave front. This continuous formation of newwake vortices appears much more wavelike when viewed in the fluid reference frame. The wavescontinually travel forward, following the cylinder, in the fluid frame (Figure 3.3). The primaryphase front interferes at the cylinder. This interference pattern is swept forward with the fluid atUthiftU.,-Useporation into the secondary emitter plane. By Huygens' principle: "every point on awave front may be considered as a centre of a secondary disturbance which gives rise tospherical wavelets" [Born and Wolf, 1980: p.370-371]. In the wake, the wave fronts interfereat the cylinder to form a secondary wave front which travels forward forming a third wave front.50object frameEmitter PlaneV CylinderUseparationWave InformationSecondaryEmitterPlanePrimaryEmitterPlane/000.1.1°."."\04,44:xVWaveFrontWaveFrontUoo— Useparation Uoo— Useparationfluid frame3 - Phase Front Analysis of Laminar Vortex Streets 51Figure 3.3 Phase information propagation in the two reference frames.513 - Phase Front Analysis of Laminar Vortex Streets 52This parallels the principle of interference inoptics. Moreover, it may be possible to writean interaction integral for laminar vortexstreets similar to the Fresnel-Kirchhoffdiffraction integral in optics which relates the Figure 3.4 Fresnel-Kirchhoff diffraction.electric field I at a point Ito the contributions from elementary waves in the aperture plane[Hecht and Zajac, 1974: p.388-392; Born and Wolf, 1980: p.378-380]:E = -A ife i(21eft4.1) [COSP + COSP lidi111^2 3.1.1The integral is over the surface of the aperture as indicated in Figure 3.4. The angle between theaperture plane and xis (3, the angle between the forward direction of the wave and the apertureplane is 13", and r is the wave vector (k=2ic/X where X is the wavelength). In this case, thecontributions of the secondary emitters in the aperture plane are assumed to be due to a singleemitter. If more than one emitter exists, then the integration can be extended over all space soas to linearly superimpose their contributions. In general wave energy flows along the rays whichare perpendicular to the wave fronts.As already seen, a reasonable approximation of the far wake in the von Karman vortex street canbe made by superimposing Rankine vortex solutions. A similar process is now used for the nearwake: Superposition of line elements of the first fully developed downstream vortex to find thecondition at the cylinder that will, in turn, form the next vortex generation. The Von K inan523 - Phase Front Analysis of Laminar Vortex Streets 53vortex street was formulated using potential flow theory in two-dimensions. The extension ofthese concepts to three dimensions is embodied in the Biot-Savart law for fluids, from which adiffraction type equation (or phase coupling equation) for fluid dynamics can be found.3.2 Biot-Savart Law For FluidsThe Biot-Savart law for fluids is an induction equation much like the Biot-Savart law forelectromagnetic theory. It is a long range interaction equation where the velocity at a point is thesummation over all of the vortex elements in the fluid.Consider the equations for a magnetic field with no electric field present:= 0 and Vx/Y =^ 3.2.1where pn, is the permeability. This leads to the Biot-Savart Law for magnetic fields [Jordan andBalmain, 1950: p.87]:*Ito) = 41k r ottri)xf(i)^dXII0 -11 33.2.2t In this case, the displacement current term: pmemormo has been omitted from the secondequation. This is valid since the Biot-Savart law is a summation of the contributions ofthe current elements only.533 - Phase Front Analysis of Laminar Vortex Streets 54where ro is the location in space where the magnetic field is being calculated and the integrationis over all of space (a=axayaz). Compare (3.2.1) against the fluid equations; continuity in anincompressible flow, and the definition of vorticity:V47= 0 and Vx C/ =^ 3.2.3The similarity to the electromagnetic equations (3.2.1) is evidentt, therefore there must be aBiot-Savart Law for fluids [Batchelor, 1967: p.86] of the form:Cad = - 41w iff  (10-i-oxism IiO413^3.2.4A complete derivation is given in Karamcheti's book [1980: p.524-528 and p.532-534] but is alsoavailable in other references [Leonard, 1985; Lamb, 1932: p.211-212]. Assuming the flow isincompressible (V -0+=0) a solution can be found by using a vector potential, ir, such that:3.2.5Substituting into the definition for vorticity yields:- a = -VxVx = v26 - wv•b.)^3.2.6Assuming V•g-.0 (to be verified below) this reduces to- iv = v26^ 3.2.7t This similarity was also noted by Batchelor [1967: p.86].543 - Phase Front Analysis of Laminar Vortex Streets 55This is Poisson's equationt for IC. The solution to which is given by:lir  6(1-)Ild6(10) ^JJ 110-X '3.2.8The assumption of V•g=0 can be tested by taking the divergence of the above expression:V•6(7 4) = Idi4„^14 _11which, since V.65=V.Vx/7=0, is the same as110 -11_^4-4)  )cd _ - 1^c().rriiidA .04n- rr vi' li_i^4o 7c n il03.2.10The final integral is over the surface of the volume which contains all of the vorticity, iris thenormal to the surface. The result of (3.2.10) is zero since the volume contains all of the vorticity,and V• (754, no vorticity line can be normal to the surface, therefore vir=o holds and (3.2.8)is valid. One important new aspect must be mentioned: We assume that w may be a function ofspace and time. Substituting (3.2.8) into (3.2.5) and carrying out the differentiation yields theBiot-Savart Law for fluids:t The electromagnetic equation is V2Vp=-p/c„., where Vp is the voltage potential, p is thecharge density, and En, is the permeability [Hayt, 1981: p.206].3.2.9553 - Phase Front Analysis of Laminar Vortex Streets 56Ciao) - _Lilf Oto 750 x WI) di47t^144133.2.11This equation relates the velocity field to thevorticity field. The integration is extended overthe volume containing the elements of vorticity.Similar to the Kirchhoff integral, the Biot-Savartlaw assumes that the elementary contributionsfrom the localized vorticity elements linearlysuperimpose to give a resultant velocity at ro, asindicated in Figure 3.5. Figure 3.5 Definition of angles for thediffraction theory of vortex streets.3.3 Phase Coupling Equation (Diffraction Integral For Fluids)When the Biot-Savart law is used to describe time varying parameters co and U, a subtle pointarises: The Biot-Savart law does not specify a travel time for information emanating from anemitter at f, however, the contributions from the emitting elements cannot spread with infinitevelocity, but will propagate at some finite velocity, V. This "phase" velocity characterizes thepropagation of "information" from the source point to a new point. The "information" compelsthe new point to acquire the velocity C. In analogy to the Huygens construction, the flow of563 - Phase Front Analysis of Laminar Vortex Streets 57information from the emitter to a new point is considered a "wave" process and, hence, V iscalled the "phase velocity". In the situation considered here, the emitting element has anoscillating vorticity:e 12nfts 3.3.1Of course, the induced velocity is a superposition of information which arrives simultaneouslyat 4 hence, the emitter's vorticity is evaluated at the retarded timer, t':-X It i - t - '^k-(1 -1)°^- t^° 3.3.2I PI 2tfThe phase velocity of a periodic vibration implies a wavelength, X=V1f, which, as usual, can beexpressed as a wave number, lc:=27tfIV. Introducing the oscillating vorticity (3.3.1) into the Biot-Sayan law (3.2.4), forms an expression for radiating the flow information:^üor. ) = _ 1^(10 -1) x 6 (1) e^di^47c^ii; -1'1 3i 1^ i[2.fooii.t _ . (o _11)+0,0) ,^3.3.3The initial phase factor Os has been introduced to allow for position dependent phase variationsin the oscillating vorticity field. The local vorticity vector, (.715=co k, and the position vector,gl-r--4x0-xii+(zo-z)/c, can be broken into components such that:t Retarded times are also applied in electromagnetic radiation, but the result is much morecomplicated than here. This is due to the time dependence of the magnetic field on theelectric field [Jordan and Balmain, 1950: p.316-320]. There is no such dependence for thefluid equations used to derive the Biot-Savart law.573 - Phase Front Analysis of Laminar Vortex Streets 58-1 fief  x(zo^GqX0 - x)^ fMlii[27ct--(z0 -1)+0,(?)]Uy (x0 y0' zo) = —47c dX4-113^eThe vorticity of strength A(X) can be reduced to:wx = A(16 cosa and w = A(I) sina^3.3.5where a is the angle the vorticity vector makes to the x axis as shown in Figure 3.5. Similarly,identifying: R as the angle ro-fmakes with the x axis yields:zo -z^xo-xsinP -^  and cosy - lio3.3.6Equation (3.3.4) can be rewritten as:^1^A(1)(cosa sinP -sina cosP) e i[brf(l)t-E.(10 -1)+0,(1)] ID? 3.3.7U (x0 ,y0 ,z0) = -^47c if° _g 12which can be transformed to the final form of the fluid diffraction equation:1^A(l)sfigoe — 13) i[27:foot-E.010 -1)+0 objUy(X0 ,y0 ,z0) - e^s dX47r —1123.3.8The above equation looks very similar to the Fresnel-Kirchhoff diffraction equation (3.1.1), anddescribes a wave-like propagation of the transverse velocity field Uy which moves at the speedV=27rf/k in a direction orthogonal to vy and 6. The major difference between the diffraction3.3.4583 - Phase Front Analysis of Laminar Vortex Streets 59equation for a fluid and the diffraction equation in optics is the obliquity factor, sin(oc-P), yet,this modified factor still accounts for the angular difference between the wave front and thedirection to the calculation point.Using the fluid diffraction equation (3.3.8), the velocity close to the cylinder, Uy, can be deducedfrom the distribution of vorticity in the fluid downstream. The state associated with the velocityis then swept downstream developing into the vorticity distribution which, in turn, forms a newdiffraction pattern for the velocity at the cylinder. In other words, given a vorticity distributionat one time t1 , one calculates a velocity distribution, Uy(4) as an amplitude and a phase. Thevelocity profile is swept back into the near wake during the time At, developing into a vorticitydistribution at the time to-At, which is used for the next calculation of Uy(to-At).Using an oscillating vorticity distribution at some emitter plane (the parent state) at a location;, to be discussed later, the above equation can be used to calculate the magnitude and phase ofthe transverse velocity at the cylinder (offspring state). In the vortex street, the vorticity at a fixedplane relative to the cylinder oscillates as vortices convect through it as indicated in Figure 3.2.By considering the wake in this manner, the above diffraction equation can be utilized to predictthe vortex street pattern as a function of space and time which results from a given flowgeometry and initial conditions.593 - Phase Front Analysis of Laminar Vortex Streets 60Calculations of the vortex street using the diffraction integral are presented later in this chapter.It will be seen that this method accurately predicts the development of vortex streets and offersa new intuitive method of deducing the vortex street development of new flow geometries.3.4 The Vorticity, Emitters Phase, and Amplitude In A Vortex StreetGiven a flow geometry where D(z), U.„ and R. are known, the resulting vortex street can bepredicted providing the frequency, placement, wavelength, phase, and strength of the emitters areknown. Though it is difficult to completely characterize the emitters, it is possible to judge theirproperties well enough to predict the pattern of a vortex street.The characteristic frequency of the emitter,f(z) is that of the wake convecting through the emitterlocation. This is simply described by the shedding frequency (2.3.1):suf- DThe frequency is assumed to be fixed by a process that is independent of the wave frontpropagation mechanism, so that f can be treated as a known input parameter. For a circularcylinder below Re=40, no shedding occurs, no wake forms and hence, f=0. In the range40<Re<150, as used in the models below and in the experiments of chapter 5, the Strouhalnumber has been found empirically by Roshko [1953] (2.3.2) for 2D vortex shedding:603 - Phase Front Analysis of Laminar Vortex Streets 61S = 0.212 - 4 '5The Strouhal number relationship varies with cylinder shape and with Reynolds number, valuesfor which can be found in the literature [Blevins, 1990: p.50]. Locally, one can assume that theemitter frequency is determined only by Re, as has been verified experimentally on taperedcylinders [Gaster, 1969].Frequency alone does not describe the emitters' effects, a phase velocity, V, or wavelength, X,is also required. Close to the cylinder inside the recirculation region, the fluid remains practicallyattached to the cylinder. Therefore, the mean velocity is close to the cylinder speed, U,,. As thevortices detach from the cylinder, their speed drops [Kovasznay, 1949; Schaefer and Eskinazi,1959]. Within the near wake the speed in the lab reference frame was measured from the slopeof the line in Figure 3.6, giving approximately 0.4U.. This is higher than the velocity in thedeveloped wake, where the vortices move at the drift velocity 'drift. The drift velocity wasmeasured from Figure 3.6 and found to be Udrfr-(0.15±30%)U.. The place where the velocitychanges separates the near wake from the far wake. The near wake is the region of vortex rollup and growth. All of the vortices within this developing region should have an influence on thestate of vortex shedding at the cylinder. Therefore, the waves within this region must havesufficient velocity to reach the cylinder. This condition requires a phase speed of at least V=U.,-U drVt • Using this, the wavelength, X=V/f, becomes the vortex spacing b (2.2.8). This wavelengthis the same in all directions as implied by Williamson's cosine law (2.3.4), f=focos0. It is613 - Phase Front Analysis of Laminar Vortex Streets 62conceivable that the wavelength could be altered by an additional velocity component, in asimilar manner as the wave field of sound waves is Doppler distorted when the source is inmotion.Near Wake1Frame #123456789101112131412_04 sec(a)^16_16 sec (b)► 1121 mmFigure 3.6 "Streak photo" of vortex position behind a cylinder (R e=107) composed from slicesof 14 consecutive side views (top half labelled (a)). The bottom half (b) is a wider section of thelast frame.The frequency and wavelength are fixed by the vortex shedding process, but the oscillatorstrength, A, and its phase, O s , could have arbitrary initial values. In the experiments describedlater, the phase is set using trigger plates. In most work performed by other researchers Os israndom at the beginning of the experiment.623 - Phase Front Analysis of Laminar Vortex Streets 63The initial strength of the oscillators is assumed proportional to the diameter of the cylinder. Thisis justifiable since the strength of the vorticity is proportional to cylinder diameter, giving acirculation 1-•=70. A similar assumption is made by Noack et. al. [1991] The strengths and phasesof the vorticity emitters, at time t+At are deduced from the transverse velocity Uy calculated atthe cylinder at time t. The transverse velocity contains the flow information which convects backinto the emitter plane in a time At to form the new voracity, hence, there is a growth function,G, which describes the transformation of Uy into vorticity strength:(1, t +A t) = d(Uy(lo,t)) 3.4.1The growth function relates the amplitude and phase of the velocity Uy created at the cylinderto the vorticity in the wake. In the work presented here, it is assumed there is a one to onecorrespondence between the velocity calculated at the cylinder and the vorticity which developsin the same fluid element after it has convected downstream in the wake. In other words, thegrowth function is a multiplying factor, Cr, which transforms the velocity into a vorticity value:A(1, t + A = CrUy(10, t) 3.4.2This factor is chosen in a manner which maintains the vorticity strength at the cylinders.Therefore, at a fixed distance behind the cylinder the total emitter strength is the same as theinitial strength:t This is akin to renormalizing the problem. This also maintains the process in a mannersimilar to self-excitation of the vortex shedding process but allows for suppression ofvortex shedding through the phase coupling processes.633 - Phase Front Analysis of Laminar Vortex Streets 64fA(x,z,t+At)dz = fD(zo )dzo = constant^3.4.3where L is the length of the cylinder. The renormalization is then simply:fD(zo )dzoA(x,z,t+At) = CrUy(x0 ,zo ,t) - L °^Uy(xo,zo,t)^3.4.4f Uy(xo , zo , t)ItizoThe above renormalization gives a vorticity which is in phase with Uy at the cylinder. Therenormalization implies that the associated emitters develop their full strength within a fewdiameters from the cylinder after the vortices at time t+At have fully grown in the near wake.This renormalization may seem overly simple unless the location of the emitters can be chosenat a location far enough downstream for the eddies to be strongly developed but still closeenough to be the major influence on the vortex shedding process.3.5 Emitter Plane Location In A Vortex StreetFor simplicity, all of the emitters are considered to be in one plane in the wake as indicated inFigure 3.2. In the general formulation of the diffraction equation (3.3.8) the location x of thesecondary wavelets is still arbitrary, however since the determination of U(t+At) requires that U(t)turns into vorticity with strength A(t+At) by drifting downstream, the location of the sources mustbe chosen to indicate this. The index xs is used to describe the location of the emitters in the643 - Phase Front Analysis of Laminar Vortex Streets 65vortex wake. The choice of location of this source plane, is made as follows: The emitters arein the wake, close to the cylinder, where they have the strongest influence on vortex formation.This is where recirculation brings flow information back to the cylinder, the vortices aredeveloping and gaining strength, and are moving with the cylinder. The vortices are much morestrongly developed towards the rear of the recirculation region, which, in general, extends a fewdiameters downstream [Kovasznay, 1949; Schaefer and Eskinazi, 1959] and, from Figure 3.6,seems to be limited to the near wake region of about 3D. Therefore, the emitters must bepositioned far enough from the cylinder to allow the vorticity emitters to gain enough strengthto be of consequence in the vortex shedding process and near enough to interfere effectively atthe cylinder (contribution falls off as 1/x2). A qualitative depiction of the growth of the emittersand the drop off of their contribution is given in Figure 3.7.In actual fact, all of the vorticity in the wakeis produced close to the cylinder but does notappear as an oscillating contribution to thewake close to the cylinder. The vorticity isproduced in relatively equal amounts on bothsides of the cylinder but the opposingrotations are collected at different locations,hence the combined effect of the emitters inthe y-z plane directly behind the cylinder is xsFigure 3.7 Contribution to Uy from vorticityelements at different distances behind thecylinder.653 - Phase Front Analysis of Laminar Vortex Streets 66small. As the vorticity is concentrated in the growing eddies, a separation occurs which placesmore vorticity of one rotation at x than in the other rotation. This separation process increasesthe emitter strength and, hence, intensifies the contribution to UrWithout appreciably affecting the results, the emitter distribution can be collapsed to a singleplane across the wake in the fluid as shown in Figure 3.2. The set of emitters in front of theplane are too weak to affect the shedding cycle by comparison to the influence of those behindthe emitter plane. The wave fronts approaching the cylinder from behind the plane can besuperimposed to form a secondary wave front at the plane, xs. By Huygens' principle, thesecondary wave fronts produce the same results, therefore, the emitters can be considered to existin one plane of the wake within a few diameters of the cylinder. In the work here, x, is assumedto be within the near wake region identified in Figure 3.6 and, as will be shown later, x 8=1.0Dproduces reasonable results.3.6 Diffraction Model of the Vortex StreetHaving established the strength, phase, frequency, wave speed, wavelength, and the location ofthe emitter plane, the diffraction integral (3.3.8) can be applied to a vortex street.The emitters are assumed to be in one plane of the fluid located a distance ; behind the cylinderin the vortex street. We assume that the phase at one point at a spanwise location, z, is equal to663 - Phase Front Analysis of Laminar Vortex Streets 67the phase at all other points across the wake with the same z and x. Therefore, the combinedeffect of the emitters at z can be represented by a single emitter centred at y=() in the wake atthe source plane. The vortex street model is then a two-dimensional diffraction problem from aline source (the emitter plane) in the wake. The contributions from the line source superimposeat the cylinder forming a transverse velocity Uy , which (in the cylinder reference frame) isconvected downstream into the source plane to form a new set of emitters at a later time. These,in turn, form a diffraction pattern at the cylinder. Under these conditions the diffraction equation,(3.3.8), for this propagation mechanism becomes a wake propagation equation:Uy(Zo ,t) — 1 fA(z)sill(a - 13) exp{i 2n f (z) it + 1+ 41) s(z ,tt.)471  2xs +Z VX2 2 Z 3.6.1The strengths, and frequencies are found as described in the previous section. The phase, 0.,(z,t.),is taken at a retarded time t* such that all of the contributions to Uy from the emitters arrivesimultaneously at zo. The time t>, is found by using the propagation time:\Xs +Z 2^3.6.2VThe angle of the vortices, a defined in Figure 3.5, is difficult to solve for as the computer modelprogresses. For the most part, a was set to 90° but other values were used when the angle isobvious (such as in oblique vortex shedding). The angle, f3, of the placement vector, 4-.4, to theforward motion of the cylinder (L/j) is given by:673 - Phase Front Analysis of Laminar Vortex Streets 68tang= Zo -z 3.6.3 xsEquation (3.6.1) has been written using a phase velocity, V= IX, rather than wave numbers. Thenegative sign is introduced in order to maintain the forward phase progression of the wake (theconvection velocity Uo. is in the opposite direction to the wave velocity V). In principle V=U,„.-Udrift. However, in the numerical work we used V=L1.„ first because the measured drift velocitywas found to be small (Udrifi=(0.15±30%)UJ in our experiments, and second, it is not alwaysstraightforward to estimate a drift velocity from the experiments performed by others. As willbe seen, this choice of V does produce reasonable agreement between the calculations andexperiment. The placement of the emitter plane gave the best results when set at xs=1D, otherlocations are considered in the following investigation.Using equation (3.6.1), and the results summarized above, the progression of the shedding phasecan be calculated as follows: The initial information for the problem is the cylinder geometry(inclined, tapered, D), R„ U,,,, and v, and the initial phase distribution, O s(z). From the initialinformation the shedding frequency, f, can be found from the Strouhal frequency. Initially, thereis insufficient vortex strength or phase information to begin using the phase coupling equation.Therefore, for the start-up condition, the wake is assumed to develop sinusoidally in time withoutcoupling, its phase is given by:(1),(z , = 0.(z,0) +27rft^ 3.6.4683 - Phase Front Analysis of Laminar Vortex Streets 69The wake propagation equation (3.6.1) can be applied. Calculation of the vorticity strength andphase at each time is done in two steps: First, the earlier phase and strength values, at the emitterplane, are superimposed using the diffraction equation to give Uy at the cylinder. Second, thetransverse velocity information is convected back towards the emitter plane at a velocity U.,while being transformed into a new vorticity strength and phase. The transformation of Uy (z,t)to A and Os at time t+xjU, is assumed to be a simple growth without axial coupling. We haveassumed the new vorticity value at z to have the same phase as Uy and for A to be proportionalto the magnitude (equation (3.4.4)). These new values of A and O s are then used in the nextcalculation of the vortex street formation. A simple example of this calculation process is givenin appendix D.The above calculation of the transverse velocity oscillates in an approximately sinusoidal fashion,forming a phase front which moves away from the cylinder. In parallel shedding, the phase frontsremain parallel to the cylinder and the wave front normals, or rays, travel perpendicular to thefronts. The vortices are imbedded in the moving phase fronts; hence a vortex can be identifiedwith a specific phase value. For computational simplicityt, it is assumed that a vortex is shedon one side of the cylinder at 0=0 and on the other at 0=n, other values are possible but at thisstage of the analysis there is little to be gained from using other phase values for their location.Since the phase fronts are connected with the vortex lines, the phase fronts travel in a directionA vortex shedding corresponds to a sign change in the phase calculation: from 41)=-0° to+0° on one side and +ir to -7C on the other side of the cylinder.693 - Phase Front Analysis of Laminar Vortex Streets 70perpendicular to the front, similar to wave fronts in optics. In analogy to optics we assume thatthe wave influence flows along the trajectories or rays of the vortex field, Figure 3.8.This principle of phase front formation and propagation can be used to explain many structuresseen in laminar vortex streets. It is also possible to predict the structure of the vortex streetquantitatively using the wake propagation equation (3.6.1). A computer program was developedfor this task, a listing is given in Appendix F and the general calculation procedure is given inAppendix E. This computer program was used to analyze the examples of oblique shedding,cylinders with a step change in diameter, and tapered cylinders, the results of which are in goodagreement with experimental observations.703 - Phase Front Analysis of Laminar Vortex Streets 713.7 Oblique Vortex SheddingSurface\ \Image360040ansFronts0Ray AImage\\One of the simplest configurations to test isoblique shedding from a circular cylinder asreported by Williamson [1989a}. The vorticesshed at an angle 0 to the cylinder (a to the xaxis). In phase front terms, this mode is a wavefront which is at an angle a=90°-0 to the originalflow direction U.,. An oblique mode can beformed by an initial phase distribution of41:),(z) .27c sin(a _ a) zS(z)^3.7.12^DFigure 3.8 Phase fronts used to find theinput conditions for oblique shedding. Thestarting position of the cylinder is dashed onthe left.The above result comes from projecting a fixedwavelength from a phase front set at an angle 0onto the cylinder when it is in the startingposition as indicated in Figure 3.8. This initial phase distribution starts the model in an obliqueshedding mode immediately, as if the cylinder were already producing a steady, slanted vortexfield.This configuration was used to calculate vortex shedding at an angle of 0=30°, R e=85, L/D=140,and cylinder diameter 1.04 mm. The input parameters are given in Table II. The resulting vortex713 - Phase Front Analysis of Laminar Vortex Streets 72lines are shown in Figure 3.9. Thetransverse velocity, Ur in the portionindicated in the top left box of the figureis shown in Figure 3.10. In thiscalculation, the method of images wasused to describe the boundary conditions.The phase front angle to the direction ofDiameter D 1.04 mmReynolds Number R. 85Emitter Plane x3 1.0DStarting Phase 0, eqn (3.7.1)Length L 140DEnd Conditions z=0z=Lmethod ofimagesPhase Front Ray Angle 0 -30°motion was set to a=90°-30°=60°. The vortex shedding.method of images is commonly employedin fluid dynamics calculations to maintain a straight, flat boundary [Granger, 1985] (such as atthe interface between two fluids with no waves distorting the surface). By reflecting the flowparameters about the interface, the fluid at the boundary flows parallel to and with the boundary,unable to cross it. When comparing numerical results such as Figure 3.9 to actual side-on photos,one must bear in mind that the vortex positions are shown immediately after their production (inFigure 3.10 Uy is shown immediately after its production). Any subsequent motion of the vortexfield is due to drift and internal instabilities and is not considered. Hence, in experiments,sections far to the left of the vortex shedding cylinder may actually look somewhat different thanthat predicted by the computer calculations. From the computer simulation, it is evident that thevortices are formed at a 30° angle to the cylinder, however, this shedding mode is transformedinto a parallel mode by the influence of the end conditions. A reflected condition forces thevortices to meet the boundary orthogonally. This is the seed which grows into the parallelTable II Input parameters for modelling oblique723 - Phase Front Analysis of Laminar Vortex Streets 73shedding "cell". In tow tank experiments this boundary condition is achieved with clean watersurfaces [Slaouti and Gerrard, 1981] as will be seen in chapter 5.The opposite effect, transforming a parallel mode into an oblique mode is possible using thismodel but it requires a detailed knowledge of the end conditions [Gerich and Eckelmann, 1982].For example, in steady wake production, experimentalists have observed a variation in sheddingfrequency coupled with, and possibly due to, a spanwise flow close to the ends of the cylinder[Ktinig et. al., 1990; 1992]. If this frequency difference were included in the diffraction basedtheory, oblique shedding would be predicted. The emitters at the ends of the cylinder emit at alower frequency than those at the centre of the span. This frequency difference causes the phasefronts to bow towards the ends of the cylinder. As the ends bow inwards, the "rays" divergefrom the ends and focus the wave energy inward towards the centre of the span, increasing thestrength of the emitters at centre span at the expense of those towards the ends. The strongeremitters at the centre counteract the inward bending by producing an outward travelling wavefront. Eventually, the inward and outward waves are in balance, producing a stable obliqueshedding mode.The knowledge gained from the use of the wake propagation equation can be summarized asfollows: Oblique shedding with a reflected boundary condition eliminates itself as the area ofparallel shedding grows from the perpendicular "seed" at the boundary. Stable oblique shedding730mmCylinderMotionUco145.63 - Phase Front Analysis of Laminar Vortex Streets 74may be the result of a variation of shedding frequency along the span. This would establish"cells" of oblique shedding. A case in point is a cylinder with a step change in diameter.Figure 3.9 Frozen phase picture of oblique shedding at 30°, the vortex lines from one side of thecylinder are shown more solidly than the striped lines from the other side. The phase front raypropagates away from the parallel shedding region.74InstantaneousLocationof CylinderUycz.cA?C,3 - Phase Front Analysis of Laminar Vortex Streets 75csFigure 3.10 Progression of Ely calculated at the cylinder. The phase fronts are at an angle to thecylinder (zIL axis) and are parallel to the waves in the Uy surface. The top edge is on the right.753 - Phase Front Analysis of Laminar Vortex Streets 761.575mmCylinderMotion2.108_1mm Nip U378mm^ 0mmFigure 3.11 Indirect mode of vortex shedding from a step change in diameter.3.8 Modulation Zones From Step Changes In Cylinder DiameterA sudden spanwise change in sheddingfrequency can be achieved by a stepchange in cylinder diameter. One suchexperiment was done by Lewis andGharib [19921 who observed a slantedinterface along which the two sheddingzones interacted as shown in Figure 3.11.This situation occurred with cylinderdiameters D=2.108/1.575 mmcorresponding to Re=76/56. A simulationwas run with the input conditions given inDiameter D1D21.575 mm2.108 mmReynolds Number Re,Reg5776Emitter Plane .x., LOD/Starting Phase Cosi(Ds20°eqn (3 .7 . 1 )Length L1L250D1112D2End Conditions Z=0z=Lmethod ofimagesPhase Front Ray Angle 0,020°-20°Table at Input parameters for modelling a stepchange in diameter.Table III and in Figure 3.12, yielding the results seen in Figure 3.11.761.575 mmFigure 3.12 Input phase fronts for stepchange in diameter to produce growingmodulation zone seen in Figure 3.11.3 - Phase Front Analysis of Laminar Vortex Streets 77Though not stated in Lewis and Gharib's work,the growth of the interface at an angle a=9° ismost likely due to an oblique shedding modeover the entire region where the diameter of thevortex is larger. The picture included in theirwork shows an oblique mode at 0=20° to thevertical (Figure 3.12), this was introduced intothe program using equation (3.7.1). The bottomportion of Figure 3.11 does not show the oblique shedding angle since it is a limited view of theinteraction region, however, the lower section was initially set at an angle in order to producea growth boundary. The oblique shedding mode is a necessary condition to forming a slantedinteraction zone; the associated wave fronts direct energy away from the interaction region,weakening the oscillators below the step in diameter. It is evident from Figure 3.11 that after aperiod of growth, the opened region's wave front normals are directed back towards the steps,hence the growth of the modulation region should cease and the region should close. The processmay then be repeated once again. The repeated growth and collapse of the interaction region wasobserved by Lewis and Gharib.t The lower frequency at the bottom causes the vortex lines to connect at an oblique angle.The phase advance at the bottom is slower than in the smaller cylinder region, hence theray inclination as shown at 30°.771 1.575MM2.108MMCylinderMotion11+ Uee3 - Phase Front Analysis of Laminar Vortex Streets 78In order to simulate the entire process the wave front angle, a, must be calculated at each stepfrom the previous conditions. Such an improved calculation of wave front adjustment goesbeyond the scope of this thesis and it can be included in future work. An indication that wavefront readjustment should yield the desired results is shown in Figure 3.13, with inputs given inTable IV. In this figure, the final conditions on the right of Figure 3.11 were introduced into theprogram but the wave angle, 9, was set to 30° in the modulation zone (as measured offFigure 3.11 and shown pictorially in Figure 3.14). The modulation zone immediately collapses,forcing the interaction zone back to the level of the step discontinuity. This would be a returnto conditions similar to the start of the run in Figure 3.11.189mm^0mmFigure 3.13 Repair process of Lewis and Gharib wake.The physics learned from modelling a step with the phase propagation equation: The diffractiontheory is capable of modelling the characteristics of the wake at a step change in diameter. Thegrowth and collapse of the interaction region has been demonstrated and is due to varying phasefront angles. This process is dependent on the end conditions which form a constant sheddingcondition that keeps reimposing itself onto the interaction zone.781.575 mm2.108 mm3 - Phase Front Analysis of Laminar Vortex Streets 79Figure 3.14 Input phase front for stepdiameter change to return to originalshedding conditions.Diameter D1D21.575 mm2.108 mmReynolds Number RelReg5776Emitter Plane .x:, 1.0DiStarting Phase Oa(1)2.(Dib0°eqn (3.7.1)(Dia (B ottom)Length L1LeaLib50D112D21 00D2End Conditions z=0z=Lmethod ofimagesPhase Front Ray Angle 0102.02b0°30°0°Table IV Input parameters for modelling a stepchange in diameter, the repairing process.793 - Phase Front Analysis of Laminar Vortex Streets 80 I CylinderMotionII. Ue.,• :^•120D^ 60D^ ODFigure 3.15 Kink instability production maintained by a collar on the shedding cylinder.3.9 Small Increase In DiameterDiameter D 1.00 mmReynolds Number Re 100Emitter Plane x, 1DStarting Phase Os 0°Length L 100DEnd Conditions z=0z=Lmethod ofimagesWave Front Ray Angle 0 0°Collar at centre span Diameter 1.5D.Width 0.5DTable V Input parameters for modelling a collar.The frequency variation along the spancan also be limited to a small region ofthe cylinder's span. This was done byWilliamson [1989b] when he added asmall collar of 0.5D width and 1.5Ddiameter. The resulting wake at R e=100contains A vortices. A calculation with theinput conditions in Table V yields theresult shown in Figure 3.15. As in theoblique shedding calculation, these resultsshow the vortices immediately after their production. Any subsequent development of the vorticesin the wake has been neglected. Identified in Figure 3.15 is the steady production of kinkedvortices, which in turn are expected to develop into the lambda vortices observed by Williamson803 - Phase Front Analysis of Laminar Vortex Streets 81as they move downstream (this would be akin to the development of a soliton into a largerstructure [Aref and Flinchem, 1984; Hasimoto, 1972].Therefore, the diffraction model of the vortex street is capable of maintaining a stable sheddingmode which introduces an instability into every vortex that is shed. In the case of a small collar,small kinks are produced on each vortex, which have the potential to develop into A vortices.3.10 Shedding Cells On A Tapered CylinderSimilar to the collar experiment above, astable shedding mode is maintained invortex shedding from tapered cylinders. Inthis case, variations of the cylinderdiameter are modelled with a set ofemitters at different frequencies, whichform a stable diffraction pattern at thecylinder. Stable cells of different sheddingfrequencies were observed in experimentsby Noack et. al. [1991]. In theirDiameter Dm,DBottom3.00 mm1.00 mmReynolds Number R. TopR. Bott.17959.7Emitter Plane x3 1.0DBotto.Starting Phase ct's 0°Length L 180DEnd Conditions z=0z=Lf=0 hzf=0 hzWave Front Ray Angle 0 0°Table VI Input parameters for modelling a taperedcylinder.experiment, a tapered cylinder of length 180mm ranging from 3mm diameter at one end to lmm813 - Phase Front Analysis of Laminar Vortex Streets 82diameter at the other in a wind tunnel at v=14.9x10 -6 m2/s U.,=0.89 m/s produced 6 distinctshedding cells.These conditions were introduced into the model, the results of which are shown in Figure 3.16and Figure 3.17. The ends were assumed fixed, f=0 hz, which is the same condition used byNoack et. al. in their model. The experiment was carried out in a wind tunnel, with the cylinderterminating at the walls inside the boundary layer, hence no vortex shedding should occur at theends. It is evident from the figures that the model produces 3 shedding cells. In Noack et. al.'scomputer prediction, a random start-up was assumed. This had little consequence on their worksince their work was focused on the stable shedding condition. In the work here, initialrandomness was not introduced instead, a constant phase of (I)=0° was used. This had little effecton the results since the predicted shedding patterns developed within about 350 diameters fromthe start of the run and the entire computer run lasted for about 6000 diameters.These results are an improvement over the van der Pol oscillator model used by Noack et. al.which predicted 100's of shedding cells. The discrepancy between the diffraction model, usedhere, and experiment is most likely a result of improperly defining the end conditions of thecylinder.82130 ^--Roshko^0.2D.0D^1.4Da-0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.912503 - Phase Front Analysis of Laminar Vortex Streets 833mm zIL 1mmend endFigure 3.16 Shedding frequency versus position along a tapered cylinder.The results of placing the emitter plane elsewhere in the wake was also investigated, the resultsof which are shown in Figure 3.16. The closer the emitter plane is to the cylinder, the more cellsare observed, but the less well defined they are. As the plane moves closer, the sheddingfrequency curve seems to approach the local shedding frequency predicted by Roshko (2.3.2). Atdistances further out, fewer cells are observed but the transition between them is sharper.Therefore, the position of the emitting plane is bounded as evidenced by the number of cells itproduces. The best results are for xs approximately 1D bc,„, but in the tapered case this is a bit833 - Phase Front Analysis of Laminar Vortex Streets 84ambiguous. This placement can be more fully investigated and varied over the span in futurework, however, since the calculation is not overly sensitive reasonable results are obtained bychoosing a single distance, x0-x, =Dbottom•The diffraction model of the street of a tapered cylinder produces stable shedding cells asobserved in experiments. The number of shedding cells observed in the model is dependent onthe position of xs and on the end conditions, which if more fully defined, have the potential toproduce the experimental shedding pattern. This is an improvement after other models, which failto show the stability of the cell formation.84Figure 3.17 Noack et. al. tapered shedding simulation.853 - Phase Front Analysis of Laminar Vortex Streets 863.11 SummaryThe phase front analysis illustrated by these examples is well able to explain experiments fromother labs where the shedding conditions vary as a function of z and have unknown or randominitial conditions. The treatment is simple with easily identified input parameters. The behaviourparallels optics, having a ray analogy, making this a powerful principle for understanding vortexstreets. The analysis reveals that the cylinder end conditions were the primary influence on thewake structure. However it is clear that the initial conditions at t=0 also play an important rolefor the development of the wake. No experiments have been reported where (1:1 s (z) is set in acontrolled manner. It was felt that such experiments would give additional conclusive evidenceof the usefulness of the newly derived phase coupling equation. Hence further to the resultspresented here, experiments were performed in which the starting phase of the shedding cyclewas set and where straight as well as tapered cylinders could be investigated. The procedure,experimental methods, and results are presented in the next two chapters.864 - APPARATUSParallel to the development of the diffraction theory, a set of experiments was performed inwhich the starting phase of the vortex street was set and the subsequent development observed.The test basin, vortex shedding cylinders, associated apparatus, and experimental methods aredescribed below.4.1 Set-up OverviewThe experimental setting is shown in Figure 4.1. The wake was produced in a 50 gallon fish tankfilled with tap water. An 8mm diameter vortex shedding cylinder was towed through the tank bya cart which travels on rails along the top of the tank. During a run, a pictorial record lookingat the wake from the side was taken using a 35mm camera. Simultaneously, the camera shuttersignal and the cart position were recorded by a computer. After the run the photos weredeveloped and printed and the computer data was analyzed to obtain the shutter timing and cartvelocity.8735mm Camera400 ISO FilmV6o Sec. Shutterf stop 22PhotosCylinder-„,..4„,/ 50 Gal. AquariumData Storage1.4 Meg Disc/UM°Computer 113M ATWith Data TranslationDT2805 A/D Carda04 - Apparatus 88Figure 4.1 Overview of the experimental setting.88Trigg er PlateCylinde ^1EE450 Gal. AquariumI ^/AO895mmails^0-■^HolderCart4 - Apparatus 89Figure 4.2 The test basin.4.2 Test Basin and CartThe experiments were performed at Re--100 with an 8mm diameter cylinder. The cylinder sizewas chosen small enough to avoid blockage effects and large enough to provide visualization ofthe vortex structure while limiting the cart speed (U 00=1.25 cm/sec) so that a short tank could beused. The 50 gallon fish tank with height 488mm, width 390mm and length 895mm gives a 70second run, plus has the advantage of glass on all sides, making 3D viewing easy. The tank waslevelled within 1mm over the length of the run so that the depth of water is constant at 410mm,894 - Apparatus 90and hence, the submerged lengthof the cylinder is fixed atL=410mm and the clearancebetween the bottom of thecylinder and the bottom of thetank is maintained at less thanlmm. The tank is filled with tapwater which has passed through a5 pm filter to remove sedimentand is allowed to stand for 1 to 3hours to minimize the backgroundturbulence before a run is made.Figure 4.3 Velocity sensor mounted on cart.Just prior to the run, the cylinder is slipped into a holder on the cart which runs along the topof the tank as shown in Figure 4.2. The cart drags the cylinder through the water as it rides onnickel plated HO scale model rail-road trace where the joints have been soldered and filedsmooth to avoid unnecessary bumps. The cart wheels are nickel plated brass HO scale electricThe track is made by Atlas Tool Co. Inc., Hillside NJ, 07205. It comes in 36 inch lengths,known as flex track, and cost $2.85 Canadian each.9032109IN Data Points87Volts Applied to Motor1.1.0 1.0.a o.o.0.4 - Apparatus 91model train wheels t with a 3/32 inch diameter axle mounted in teflon bearing blocks as shownin Figure 4.3.The cart is pulled by a wire cable connectedthrough a gearbox to a 16V DC electricmotor, normally run at 16V. The velocity isrelatively constant above 10 volts as shown inFigure 4.4. The velocity range and cylinderdiameter limits the Reynolds number to64<k<120 at 20° C..Figure 4.4 Cart velocity versus applied voltage.4.3 Trigger PlatesThe starting phase of the vortex street is set through the use of trigger plates [Seto et. al., 1991][Lefrancois and Ahlborn, 1992]. These are plexiglass plates positioned in the tank as indicatedin Figure 4.2 and Figure 4.5.The plates span the entire length of the cylinder, are 3/16 inches thick and have a 45° bevel alongthe active edge. As the cylinder passes the plate, a vortex is shed on the side opposite the plate,t The wheels are available from hobby stores in packs of 12 for $11.60 Canadian. They aredescribed as "42" replacement wheels for powered Athearn Diesels." The supplier isNorthWest Short Line, Box 423, Seattle WA, 98111.914 - Apparatus 92 ► Straight Slant Step SplitFigure 4.5 Trigger plates. (a) is a top view of a cylinder passing between trigger plates. (b) isa side view of a number of single plates. In the split plate, the cylinder passes between the twoplates.locking the phase of the street in relation to the plate, and, hence, setting the phase at the startof the run [Seto et. al., 1991]. A single plate may be used but the street is more reliably triggeredby using two plates, one on each side of the wake half a shedding cycle apart [Lefrancois andAhlborn, 1992]. The actual placement of the plates is not critical. In the experiments here thecylinder cleared the plates by roughly 0.5 to lmm and the phase between plates was in general160° to 200° but trials as low as 45° and as high as 360° produced the standard two-dimensionalvortex street. By modifying the shape of the plates, other vortex shapes can be produced asindicated in Figure 4.5. Their effects will be described in the next chapter.924 - Apparatus 934.4 Vortex Shedding Cylinders and Cylinder HolderThe vortex shedding cylinders are 8mm diameter s brass rods. Seven different rods were made,each with a 410mm working length, one a straight parallel cylinder and the other six tapered.Each taper measures D=8mm at the water surface and tapers down to diameters: 7, 6, 5, 4, 3, and2 mm at the bottom of the tank, giving taper ratios (length/AD): 410, 205, 136.7, 102.5, 82, and68.3 respectively. The working section of each cylinder was lightly sandblasted so that the dyeused in the visualization would wet the cylinder when it was applied (stick to it while drying).This roughness is very small and was not expected to appreciably affect the results [Votaw andGriffin, 1971; Granger, 1985: p.783-786].Just prior to a run, a cylinder is slipped into the holder on the cart (Figure 4.2) which is a 2 1/4inch diameter nylon rod, 3'/8 inch long with a 5/16 inch diameter hole drilled down the centre. Thehole is the same size as the cylinder so that the tight fit minimizes cylinder vibration [Couder andBasdevant, 1986].the nominal size is 5/16 inch diameter which is equal to 7.94mm.934 - Apparatus 944.5 Vortex Street VisualizationFluid flow visualization is an art of many possible techniques and methods (A good review isgiven by Merzkirch [1987]). The choice in this study was Ethyl Violet organic stain painted onthe cylinder and allowed to dry. The dye dissolves off the cylinder into the boundary layer whichrolls up into the vortices seen in the wake.The dye is neutrally buoyant, easy to apply, optically dense, and does not spread rapidly in thefluid (long after a run, streaks of dye could still be seen in the test basin). A 2% solution waspainted on the cylinder and allowed to dry before it was used in an experiment. The dye wasapplied completely to one side of the cylinder and in a striped pattern on the other side(approximately lcm apart and 0.5cm wide strips) as indicated in Figure 4.6. The intention of thispainting scheme was to have the vortices on one side of the cylinder show up as solid lines andthose on the other side as dashed lines. In this way, axial velocity information could also beobtained by following the movement of the strips between photos taken a known time apart.The photos were taken using a 35mm Minolta X700 camera with auto-winder. The Film usedwas Ilford XP1 400, a high-resolution 400 ISO black and white film which is developed usinga C41 process. In all of the photos, a 2cm square grid is in the background. This grid was drawnf Ethyl Violet is C311-142N3C1, name hexa-ethyl pararosanilin. It was obtained from Gurrmicroscopy materials, BDH chemicals. [Gurr, 1960; Conn, 1961; Society of Dyers andColourists..., 1956: p.1634 of Volume 1]94on vellum and taped to the back of the tank.Vellum is translucent so lighting the tank witheight 75 Watt flood lamps from the backprovided enough illumination. Since the vortexstreet is in the middle of the tank, a 2cmobject at the location of the street appears tobe bigger than the grid spacing. The grid isintended for relating placements of vorticesand objects between photos. The scalingchanges with camera-wake distance, however,a scaling factor can be determined fromknown sizes in the picture such as the waterdepth or the cylinder diameter.4 - Apparatus 95Figure 4.6 Striped dye marking of vortices. Thecylinder is painted solidly on one side and in astriped pattern on the other. The dye comes off,rolling up into striped and solid vortices.4.6 Data Logging and AnalysisThe data are recorded using an AT computer with a Data Translation DT2805 analog to digitalconverter. The converter has 8 double ended inputs and is capable of 6000 samples per second(750 per channel if all 8 channels are being used). Typically, data were collected at 250 samplesper second on two channels: a velocity channel and a camera channel.95lkflPhotoResistorView finderenters.xko.:,f •^:•••The mirror reflectslight up through theview finder when theshutter is closed.Vs i gn al9V —7—aperture shuts dowIView finderisignalI irror swings to closed 1-- -4---I II aperture opens upI I^II_ 1^11 ! irror swings to open1I^T----Signal through opent -41111 back of camera0:1^0 2^0'3^0 4^o.s^0:6^0'7Time (sec)3.5310.50The camera timing is accomplished bylooking through the viewfinder with a photo-resistor as shown in Figure 4.7. Light istransmitted through the viewfinder when theshutter is closed and is blocked when theshutter is open. The output of the viewfindercircuit is shown in Figure 4.8. The viewfindersignal was checked against the signal received4 - Apparatus 96Figure 4.7 Camera timing circuit andarrangement.through the back of the camera in order to find the relative liming between the mirror blockinglight from the viewfinder and the shutter opening. From results like that presented in Figure 4.8the time of the shutter opening can be found as a time difference from the minimum in the viewfinder signal.Figure 4.8 Camera timing signals for calibration purposes.964 - Apparatus 974.7 Velocity SensorThe velocity sensor measures the time of flight between points spaced a known distance apart.It consists of a photodiode-phototransistor pair which shines through a series of 5/32" diameterholes drilled 0.762 cm apart (0.3 inches). The sensor [General Electric, 1984] (shown inFigure 4.9) is attached to the cart, straddling the drilled 3/16 inch aluminium angle which isattached to the tank at the base of the rails as shown in Figure 4.3. As the cart moves along, thephotodiode shines through the holes giving a step output from the photo-resistor as shown inFigure 4.10. Each pulse corresponds to one hole, thus, by counting pulses one knows how far thecart has moved from the beginning of the run. Given the position and time data, the velocity canbe found through numeric differentiation in the computer.Figure 4.10 Velocity sensor signal.Figure 4.9 Velocity sensor circuit (G.E. emitter-receiver assembly H21B5). 97Figure 4.11 Position and velocity comparison.4 - Apparatus 98Figure 4.11 shows a typical test run with ahole reached at every cross slash. A simpledifference between holes was performed toevaluate the velocity. This simple schemeshows a fair bit of noise in the velocity yetthe position data form a straight, smooth line.The actual velocity error is much smaller thanindicated, it is the digitization noise whichintroduces the fluctuation. This noise is a placement error of when the hole is reached(Figure 4.12) and has a maximum given by:Ax= U 7+Ad 4.7.1where f is the data sampling frequency, Ad is the error in the placement of the holes, and U.,/fis the distance travelled between samples. A more reasonable estimate of the velocity is obtainedby fitting a curve to the position data and differentiating. The order of the fit is chosen such thatthe maximum error from any data point is less than AL In most cases, the sampling frequencywas 250 samples/sec and the error in machining the holes was estimated to be 0.001 inches.When the cart is moving at 1.36 cm/sec, Ax=0.08mm. For the run shown in Figure 4.11, apolynomial fit of order 4 has a maximum error of 0.08mm from the data points. This yielded anaverage velocity of 1.3613 cm/s with a maximum deviation of 0.0064 cm/s. This indicates thatthe velocity is steady to less than 0.5% over the run. The absolute maximum error on the mean982f1 f•Ta■s/ d •^Mean LevelDataSamplesVoltage Signal• Data Points I6^42^0Fitted Curve to8^5 n i^n^1C^nn^•IrFigure 4.13 Viscosity calibration curve.4 - Apparatus 99velocity is approximately 2(&)/t---2(0.08)/50=0.0032 mm/sec which is negligible by comparisonto the deviation over the run.Figure 4.12 Detail of velocity sensor signal. The arrival of an edge is known within 1/2f.The sampling rate used is much higher than indicated here.Temperature(°C)Viscosity (v)(m2/sec x107)0 17.875 15.1910 13.0715 11.4020 10.0425 8.93030 8.00935 7.237Table VII Viscosity of pure water atvarious temperatures.994 - Apparatus 1004.8 ViscosityThe viscosity can vary by 10% over the temperature range of the water. The water comes in aslow as 9° C and can be as high as room temperature (approximately 22° C). Using the viscosityof pure water given by Blevins [1984: p.523], reproduced in Table VII, a least squares fit wasdone from 0° C to 35° C in steps of 5° C. This yielded the expression for the viscosity:v = (-0.000110T 3 +0.0119T2 -0.586T+17.86)xle m 2/sec 4.8.1Where T is the temperature in °C. The resulting curve and data values are plotted in Figure 4.13.The maximum error of the formula from the tabulated values is 0.036x10 7 m2/sec. The accuracyin measuring temperature is 0.1° C. The error is given by the derivative of the above equation:A v = ( -0.000330T2+0.0238 T-0.586) x10 -7A T m2/sec 4.8.2which works out to be: 0.024x10 -7 m2/sec. This gives a maximum overall error of 0.06x10 -7m2/sec or about 0.6%. Combining this with the error in velocity, 0.5%, the overall error in theReynolds number, Re=t1,.,DIv, is about 1%.1004 - Apparatus 1014.9 Experimental ProcedureThe experimental procedure is outlined in Table VDT.1 Fill the tank with water and let it sit for 1 to 3 hours.2 While waiting for the water to become still, paint the rod with dye and allow to dry.3 Just prior to the data run, gently slip the cylinder into the holder on the cart.4 Turn the lights on for picture taking.5 Start the run, recording the position and camera signals while taking pictures.6 Record the water temperature.7 Process the data and films.Table VIII Experimental procedure.1015 - TRIGGERED EXPERIMENTSThe original aim of the experimental program undertaken by the author was to study the three-dimensional nature of the vortex street. Early on, it was recognized that the nature of the studyrequired a method of introducing perturbations into the vortex street which would grow intothree-dimensional vortex structures. Particularly, a method which would yield reproducible resultswas sought. The perfect instrument for setting the phase in this study is a set of trigger platesclose to the beginning of the experimental run, as described in the previous chapter.After spending some period developing the "art" of flow visualization and perfecting theexperimental procedure, a strong similarity was noted between the vortex street patterns seen inthe photos and the alternating light and dark bands observed in diffraction patterns. This in turnlead to the development of the Huygens wavelet analysis of the vortex street. This treatmentyields the phase and amplitude of a new vortex generation as a function of the phase, frequency,and strength of the parent generation. The ensuing experimental work was undertaken to verifythe validity of the phase front analysis by performing experiments where the phase and frequencycan be set as a function of z.1025 - Triggered Experiments 103All of the experiments were performed using the facilities described in the previous chapter. Thephotographic results presented here were all taken in the same manner, at 1160th second exposureand f stop 22. The shedding cylinder moves from left to right, shedding striped vortices on theside toward the camera and solidly marked vortices on the side away from the camera. Forreference, a 2cm grid is in the background but due to the camera's perspective the scale does notcorrespond to 2cm. The scale is found from an object in the picture such as the cylinder whichis 410mm long (submerged length).Simulations are also included in this chapter, reproduced here at the same scale as the associatedphoto. In the simulations, the total phase is set to 1=0 at the location of the edge of the plate.Once the phase has been set at a spanwise location, z, that section is included in the diffractionintegral. This assumes that the wake developed before the trigger plate was encountered has noinfluence on the street produced after passing the trigger plate. The emitter plane is located 1Dbehind the cylinder in the calculations. In the case of the tapered cylinders, the emitter plane isplaced at the average location,1035 - Triggered Experiments 1045.1 Setting the Initial Phase of the Vortex StreetThe initial phase of a vortex street is usually random as shown in Figure 5.1. In these cases, thefirst vortices are not parallel to the cylinder but have many discontinuities and are bent. Eachsuccessive vortex generation is smoothed, eventually yielding a wake of parallel vortices(Figure 5.1 (c)). The initially random phase of the street complicates experimental studies of thewake and shortens the useful length of the tank. A long length of travel may be required beforea parallel shedding pattern emerges so that a desired shedding mode may not be achieved or theresearcher must wait for a random event to produce the desired effect. The initial phase variationalong the cylinder must be quantified in order to understand the results. However, if trigger platesare employed the effective length of the tank is longer, the initial phase distribution known, andthe street can be coaxed into producing the sought for vortex structures.The result of a single straight plate (Figure 4.5) introduced in the tank and positioned far fromthe start of the run is shown in Figure 5.2. The discontinuities which existed in the street beforethe shedding cylinder encountered the plate disappear within a few vortex shedding periods afterthe plate [Seto et. al., 1991]. The plate sets the phase of one side of the street, forcing the otherto lock in its phase within a few shedding cycles [Lefrangois and Ahlborn, 1992].1045 - Triggered Experiments 105(a) t=9.68 sec.^(b) t=25.11 sec.^(c) t=27.86 sec.Figure 5.1 Vortex street development in time with no trigger plate (random start). (Re=98)This process is clearly seen in Figure 5.3. The striped vortices are on the side of the wakeopposite the plate and the solid ones are on the plate side. The solid vortices are parallel to thecylinder immediately after the plate is encountered while the dashed vortices meander andbecome parallel within a few shedding cycles.105(a) t=22.83 sec. (d) t=43.83 sec.5 - Triggered Experiments 106(b) t=28.99 sec.^ (e) t=54.63 sec.Figure 5.2 Wake development for single triggerplate (position indicated by arrow) whencylinder is started many shedding periods fromplate. The plate is on the side of the wake awayfrom the camera. (R 8-96) (c) t=35.77 sec.1065 - Triggered Experiments 107Figure 5.3 Vortex wake with one trigger plateon the side away from the camera at the arrow.One side is triggered (solid) while the otherside is not (striped). (Re--110)Figure 5.4 Vortex street with two triggerplates placed X/2 apart on opposite sides of thewake. The perspective causes the edge of bothplates to appear at the arrow. (Re=A single plate does not always produce a street of parallel vortices; it offers a strong perturbationwhich phase locks one side of the street but the perturbation may not be enough to quell therandomness of the other side of the street. The effectiveness of the method can be enhanced byintroducing a second plate on the opposite side of the wake. Placing the second plate half ashedding period (A/2) from the first plate (as indicated in Figure 4.5 (a)) triggers both sides ofthe street immediately as seen in Figure 5.4.1075 - Triggered Experiments 108AcI)=88.3° Re=109^Acto=271° R e=113^0do=357° Re=114(a) (b) (c)Figure 5.5 Street triggering with different plate separations. The plate on this side of the wakeis at the left arrow and on the opposite side at the right arrow.The plate separation is not crucial in producing a parallel wake as seen in Figure 5.5. Theimportant factor is that a plate should be placed on each side of the cylinder (street). The firstplate promotes parallel shedding while the last plate locks the shedding phase. A plate separationof approximately M3=90° (A.14) was tried producing the result in Figure 5.5 (a). The vortices areparallel and properly spaced immediately after the plate. Separations of Ado=270° (3X/4) andAcI)=360° (X) (Figure 5.5 (b) and (c)) produced similar results. In the 270° and 360° cases thefirst vortex formed was trapped behind the second plate while the rest of the street followed thecylinder at the drift velocity. In the 90° situation no vortex was trapped since there was nochance of shedding an eddy before the second plate was reached. The 'fingering' or spanwise1085 - Triggered Experiments 109stretching of the vortex evident in Figure 5.5 (c) appears only in the first vortex generation asa possible consequence of suddenly flipping the phase of the street by 180°.Based on these results, the experiments were performed (in most cases) using two trigger plates,one on each side of the wake, spaced approximately X/2 apart so that the phase difference isapproximately 180°. This new triggering technology is a necessary tool to investigate details ofthe three-dimensional structure of laminar vortex streets.1095 - Triggered Experiments 1105.2 End Conditions of the CylinderThe end conditions of the shedding cylindercan govern the development and appearanceof the wake. It is known that obliqueshedding modes can be induced or suppressedby altering the ends of the cylinder (endplates [Williamson, 1989a; Ramberg, 1983] orcontrol cylinders can be added [Hammacheand Gharib, 1989]). Therefore, a knowledgeof the end conditions is necessary in order tointerpret the experimental results.5.2.1 Condition at The Bottom of the TankThe spacing between the bottom end of thecylinder and the bottom of the tank must bechosen such that the three dimensional effectscaused by the cylinder end are minimized. Ifthe cylinder end is far from the bottom, a freeend condition is in effect. In this case, the(a)(b)(c)Figure 5.6 Vortex street at the bottom of thetank Re=200. (a) spacing=4.4mm, (b) 2.7mm (c)1.0mm.1105 - Triggered Experiments 111vortex street shrivels up towards the centre of the span (the vortex cores are regions of lowerpressure representing an axial force that causes an acceleration inward from the ends). Positioningthe cylinder end close to the tank bottom can reduce this effect as seen in Figure 5.6. At spacingsof 4.4 and 2.7 mm the bottom point of the street moves rapidly away from the bottom (Figure 5.6(a) and (b)). A gap between cylinder end and tank bottom of ltnm slows this effect, producinga reasonably stable street structure along the bottom of the tank. As a result, in all the followingexperiments, the cylinder position was adjusted so that the bottom end just cleared the bottomof the tank (less than 1.0mm).Close inspection of the bottom structure with a small gap reveals a complex loop structureinterconnecting the vortices as shown in Figure 5.7. There are two different lines of dye, one atthe very bottom of the tank and the other linking the vortices. The bottom dye line is due to adownwash at the back of the cylinder, bringing dye downward from the region close to the endof the cylinder. Photos taken from the bottom of the tank confirm this (Figure 5.8). When dyeis applied over the bottom 2cm of the cylinder, both lines are evident and seem to cross overeach other (Figure 5.8 (a)). With no dye directly at the bottom, only one dye line is seen(Figure 5.8 (b) and (c)). The vortices are simply connected to nearest neighbours but a loopedline forms, connecting to the dye line along the bottom of the tank.1115 - Triggered Experiments 112(a)(b)Figure 5.7 View of the bottom structure (R e=76.7). The front (F) and back (B) vortices aremarked in the sketch. The arrows indicate the direction of the vorticity vector.1125 - Triggered Experiments 113(a) R e=89.5 Dye on the bottom 2cm of the cylinder.(b) k=95.9 Dye painted from 4mm to 2cm from bottom of cylinder.(c) Re=97.5 Dye painted from 1cm to 2cm from bottom of cylinder.Figure 5.8 Bottom view of vortex street production.1135 - Triggered Experiments 1145.2.2 End Condition at the Free SurfaceThe end condition where the cylinder intersects the surface of the water can dramatically affectthe development of the wake as shown in Figure 5.9 and Figure 5.10. A contaminated, or dirty,water surface completely distorts a parallel vortex street. This is caused by a surfactant of organicmolecules forming a solid mat on the surface. The molecular mat prevents the surface fromrotating, a condition necessary for a vortex to meet the surface. Since the vortices are unable topierce the surface, they bend away, following the cylinder as seen in Figure 5.9.In order to produce a parallel street one must rid the free surface of contaminants prior to eachrun. To this end, a siphon was fitted in a corner of the tank with the opening just below thesurface of the water as done by Slaouti and Gerrard [1981]. The siphon cleans the surface byforming an air column (like in a bathtub drain) which pulls the fluid surface down the siphontube. The contaminated surface layer is removed, exposing a clean fluid surface. The cleaningaction is enhanced by adding a drop of dish soap to push all of the contaminants towards thesiphon.The final step in maintaining a clean surface is to add a drop of soap just before inserting theshedding cylinder into the tank. This prevents the dye painted on the cylinder from immediatelydissolving and spreading over the surface when the cylinder is inserted into the tank. Dye is thecontaminant which caused the inclination of the street in Figure 5.9.1145 - Triggered Experiments 115Figure 5.9 The effect of a contaminated (dirty) Figure 5.10 Desired contaminant free (clean)water surface on vortex shedding. (R e,--84)^water surface condition. (Re-126)With this experimental technique well in hand a number of experiments were performed toproduce vortex streets with several different start up phase functions, (1)(z).1155 - Triggered Experiments 1165.3 Inclined Wave FrontsIn many reported experiments on laminar vortex streets inclined vortices are observed. In suchexperiments the phase of shedding varies continuously in the axial direction. Trigger platetechnology is used to generate such structures. This shedding mode can be produced byintroducing an inclined trigger plate (Figure 4.5) as shown in Figure 5.12 (a) with a plate angleof 0=32°. As can be seen, the wake is inclined parallel to the plate. Close to the free surface,at the top of the picture is a zone where the vortices straighten; a region that grows at an angle6=13°, becoming parallel with the cylinder. The simulation of oblique shedding in the lastchapter, Figure 3.9, has the same features however, the parallel region's growth rate is too large.Including an upward axial velocity slows this growth as was done in the simulation ofFigure 5.12 (b). The input conditions are listed in Table IX, the initial phase distribution isdescribed by (3.7.1):Diameter D 8 mmReynolds Number  Re  96.3Emitter Plane x., 1.0DStarting Phase b, eqn (3.7.1)Length L 365 mmEnd Conditions z=0z=Lreflectedf=0 hzPhase Front Ray Angle 9 -32°Axial velocity Uz 6.35 mm/secTable IX Input parameters for modelling an inclinedtriggered experiment.clos(z) = 2,t sin (9) zSD(z)This forces the wave fronts topropagate a little more strongly in theupward direction and, since the entirewave front experiences the samevelocity, the wave front angle ismaintained.116The axial velocity chosen for the simulation wasthe vertical component of the oncoming flowparallel to the vortices (Figure 5.11). This is thevelocity necessary to produce an inclinedvorticity distribution as implied by Hammacheand Gharib [1991]. The velocity is given by:5 - Triggered Experiments 117Vortex lineor wave front.Figure 5.11 Vertical velocity component.UZ = U.,cose sin@ = (14.14 mm/sec) cos32° sin32° = 6.35 mm/sec^5.3.11175 - Triggered Experiments 118(a)E^572 mm(b)Figure 5.12 Slanted vortex street at 32° to the vertical. Water depth 365mm. (R g=96.3)1185 - Triggered Experiments 119The calculated value, U3=6.35 mm/sec, yields an angle a of approximately 13° for the interfacebetween the oblique section and the growing parallel section. This is very roughly the same angleobserved in the experiment. Surprisingly, this seems to be a consistent angle for disturbances toappear between the vortex generations for a plate at 0=32°. The interface between parallel andinclined sections is downward at approximately 13° as seen in Figure 5.13. A straight sectiongrows at the top of the picture and the straight region at the bottom shrinks while the inclinedsection between them maintains its size. A similar angle is observed for the series ofinterconnecting loops between generations in Figure 5.14.The findings from the above experiments and application of the phase front model are as follows:An inclined trigger plate can be used to produce oblique vortex shedding modes. A zone ofparallel shedding grows from the surface, at the expense of the oblique section, with thespreading angle a. For the numerical prediction, a velocity U, was introduced parallel to thecylinder and the phase fronts were assumed at the plate angle over the entire simulation. Thephase fronts are projected away from the interface, opening a section of parallel shedding. Theangle, a=13°, observed in the model matches that in the experiment.The above analysis is consistent. A simulation at 0=19.3° (Figure 5.15) also reproduced theexperimental findings with reasonable accuracy. The vertical velocity used in this simulation waschosen in the same manner as before: Uz=13.24 cos 19.3° sin 19.3° = 4.13 mm/sec.1195 - Triggered Experiments 120Figure 5.13 Poorly triggered street by an inclined plate at 32° to thevertical, water depth 371mm. (R e=97)529 mma=12.5°Figure 5.14 Poorly triggered street by plate inclined 32° to the vertical,water depth 371mm. Loops appear in wake. (Re=97.4)1205 - Triggered Experiments 121(a)672 mm^1 ^.• •^ :^I^•^.•.• •^'^i •r:^•'^...,.^t .• a'^•^r^•^;^•^i^•^t^•^•II :^•^r^•^: ^^r^•^a^•^i^• •^r^•r^•^ir i^•^1•• .1.r^. '^I^•^I^•^i• .f ' 1 : 1      ^: r. r. 1 :^:I : i : / : / • i • 1 • f•i•1-1.:•:•:...:.1•1•1-1 : II^1:^.:• 41•1•R• :^a ^•^a^•^r^•^l•^I^•I•e•I:1;1:1,1: I : I „. 1 • I • I - Ir ^.•^r^:^r^r^r^•^•^i^•^:^•^rf •^a^:^r^:^1^..•^r^•^.r^•^r^•^i^•^:^•^r^•^:^ ^.^.1. ^I r 1•1•I•I•I•1•11.I.I.:•1^•^•1•I•I^I'l e^r^r^'^r^.I^'^:^•^r^•^r^•^ '^r^•^:^•^t^'^r : ^•' : j : I : 1: / :1:1:1:1. • : /^.•; • / • r • i • ri ' / ' .^1^1^11 • / ' / • i : II^r^1r •^•^• 1^ ••11^ •1•1•••^•I•1• 1 •: / '1•1•1•/•/•!•/./.:1:fil.../ . ./../ ..../.../.../.../.: 1 : 1 :1^r ^i•i'^1•41*/•i•l:ti:l/../..i...1:::: . : .,...i ^r.^r. r^r^1^1^i^;^;r i •i•rt •i•I - 1•1 -1 • 1 •^r ;^ 1 1::1:1: 1^ 1:1 1 • 1 *l•^ I•a•I• :•^:• :.I r^I^1 i ^i^.^I^I^I^1^E^I^'1 (b)Figure 5.15 Slanted plate at 19.3° to the vertical. Water depth 410 mm. (R e=89)6=10°121Figure 5.16 Split plate set up for producinga sudden phase jump. The cylinder movesout of the page, passing between the twoplates.5 - Triggered Experiments 1225.4 Sudden Phase Jump of X12 Along the SpanBesides a continual smooth phase variation over the span, a sudden phase jump at a point z j canbe introduced. This is accomplished through the use of a split trigger plate as indicated inFigure 4.5. The cylinder passes between two half-plates which are at the same x position in thetank but on opposite sides of the cylinder, as shown in Figure 5.16, producing a phase jump ofAci:•=180° (A/2) between adjacent points above and below zj . The resulting phase jump can beseen in Figure 5.17 (a), the vortices on one side of the street are in line with the vortices on theother side of the wake (this is most evident close to the cylinder on the right side of the photo).The connection never occurs, the vortices bendsharply away connecting to another vortex on thesame side of the phase jump. This creates a seriesof loops interconnecting the vortices with thestreet opening up (tearing apart) along the planeZ=Z-J .A computer simulation with the input given inTable X was attemptedt, giving the output seenin Figure 5.17 (b). Recall that this is a productionNotice that the phase jump is actually set at 0.997t. This was done in order to force abreak from symmetry that would produce connections, and eventually lead to loops.1225 - Triggered Experiments 123Diameter D 8 mmReynolds Number Re 108Emitter Plane ; 1.0DStarting Phase^z<ziz>zjI0,20°0.99xLength L1L215D15DEnd Conditions z=0z=Lmethod ofimagesPhase Front Ray Angle 81020°0°model showing the location of all vorticesat the place where they are produced. Nosubsequent motion of the wake field isshown. In reality the wake drifts in thedirection of the cylinder at the velocityUthift and significant axial motion takesplace near Loops form which are notevident in the model. However, thestraining of the vortices, forming short Table X Input parameters for a phase jump alonghorizontal connections indicated by the the span.simulation is seen in the experiment, close to the cylinder. These horizontal sections develop intothe connecting loops downstream from the cylinder, thus the phase front model accuratelyportrays the vortex street production but it would have to be expanded if one wanted to describethe time dependence of the vortex field downstream of the cylinder.The results of the split plate experiment are: A split trigger plate produces sudden phase jumpsin the spanwise direction on the wake. Since the vortices are highly strained, the street tears apartforming a dislocation with loops interconnecting the broken vortices. The computer simulationreproduces the vortices at the time of production but not their subsequent development.1235•MI6••It•••••0a•••••0•••••0•El•If000•0•0••00a0a0••00•000^. 0•0••etII • '• 0• 0• •• 00• 0• ••0^ 00^•^00••••00000•a•00•0•••■•0••••••••000•0•a0••0a•0aa0e•••••^TriggerPlate53mmZlosFigure 5.18 Cylinder moves past a steppedplate to produce a small phase jump.5 - Triggered Experiments 1255.5 Small Phase Jumps Over Short Spanwise SectionsIn this work, a small phase jump is considered tobe a sudden phase change between sections of thetrigger plate of AO less than 180° (X/2). Thesephase jumps were applied over zup-zio„,=53mm atthe centre of the span as indicated in Figure 5.18.The initial shedding phase is set when thecylinder moves past the first section of the plate,the phase jump is set when the cylinder movespast the 53mm section. The phase jump lags therest of the span by:360°x^360° x,S(Re)^ 5.5.1DAn experimental run in which the step was set at 118° (15mm at U.,=15.54 mm/sec, R e=123) isshown in Figure 5.19. The first feature of note is the bulging of the vortex toward the cylinderin both the model and the experiment. The vortex fingers in the photo are most likely due toeffects from the top and bottom edges of the stepped section of the plate. The extent of the bulgealso lessens in later generations but spreads outward in the spanwise direction. An experimentalresult showing the initial vortices and the later generations is Figure 5.20 and the simulation isgiven in Figure 5.21, with settings as indicated in Table XI.1255 - Triggered Experiments 126Diameter D 8 mmReynolds Number Re 123Emitter Plane x, 1.0DStarting Phase^z<zio,zio„,<z<zupZ>ZupcI43,2(13 s 30°eqn (5.5.1)0°Length L1L2L39.25D6.62D9.25DEnd Conditions z=0z=Lmethod ofimagesPhase Front Ray Angle 0 0°Table XI Input parameters for a small phase jumpover a short spanwise section.1265 - Triggered Experiments 127183 mm(b)Figure 5.19 Small phase jump of 118°. Step plate width 53mm, Re=123, U..=1.5538 cm/sec.1275 - Triggered Experiments 128(a)E•^360 mm(b)Figure 5.20 Jump in phase. (R e-111, step 116°, U.,=1.4738 cm/sec) 15.75 sec between pictures.1285 - Triggered Experiments 129360 mmFigure 5.21 Simulation of Figure 5.20 (drawn to same scale).The results are consistent: the wake contains vortices which bulge towards the cylinder, the bulgediminishes and spreads in the axial direction with every vortex generation. The spanwisespreading could be altered in the model by accounting for the angular change of the phase frontin the bulged area, but since the extent of the bulge is small in both the simulations and theexperiments, the effect would be only a small increase in the spreading rate. The salient featuresare seen here without accounting for the direction of the phase front normal. Thus, the phasefront model, as applied here, adequately predicts the behaviour of the wake with a small phasejump.1295 - Triggered Experiments 1305.6 Large Phase Jumps Over Short Spanwise SectionsTo further test the predictive power of the phase coupling equation, experiments and calculationswere made with large phase jumps. As in the previous section, experiments were made with thesame 53mm step plate length, but at separations over 180° (X/2). There is a marked differencebetween the small and large phase jumps: For AO>180° the first vortex formed after the cylinderpasses the edge of the stepped section is connected to the later generations.I- . 1360 mmFigure 5.22 Simulation of a 210° phase step at Re=116, U.,=1.4669 cm/sec.1305 - Triggered Experiments 131The simulation of one such run is given in Figure 5.22, with input conditions similar to those inthe previous section except for Re=116 and the phase jump has been increased to 210°. Theexperimental results are shown in Figure 5.23. In the experiment, the aforementioned vortexconnects to later vortex generations whereas in the model it connects to the vortex assumed tobe created at the edge of the step (the vortex is at the edge of the plate in the simulation image).No evidence of a connection between generations is seen in the model. Even the gradualreduction of the phase jump in each generation indicated by the model does not show up in theexperiment. The experiment shows a rapid reduction of the phase jump within a few vortices.Diameter D 8 mmReynolds Number Re 128Emitter Plane xs 1.0DStarting Phase^z<zio,zlow<Z <Zupz>z4,0.1CDs 21:10,30°eqn (5.5.1)0°Length L1L2L39.25D6.6W9.25DEnd Conditions z=0z=Lmethod ofimagesPhase Front Ray Angle 9 0°Initial Vortex Strength^z<z,o,,„Ziow<Z<ZupZ>2upAlA2A3D0DTable XII Input parameters for a large phase jump over a shortspanwise section.1315 - Triggered Experiments 132This development is distinctly different from the small phase jump experiments. The differenceis due to the formation of a vortex that extend from above and below the step before the cylinderhas reached the stepped portion of the plate. Hence, an established wake exists before the phasejump is applied. Further to this, no vortex is observed at the edge of the step, which was assumedin the model. In order to model the process adequately, a knowledge of the interaction betweenthe plate and a formed vortex street is required. It seems as if the step annihilates the formingvortex section at the location of the step rather than forcing a new vortex section out of step withthe rest of the wake. This condition, with no vortex at the edge of the step was input into themodel yielding the result of Figure 5.24 (the input conditions are in Table XII). The strength ofthe vortex, A(z), is proportional to the dot size in the simulation. As can be seen, the street closesrapidly as observed in the photo, gaining strength in the later generations. The strength of thevortex line is difficult to infer from the photos (or at least should be interpreted with caution),however, it would seem from the concentration of dye that the two vortices directly after the stepshow a weakness in the phase jumped region.The failure to predict the wake for a large phase jump is not due to a weakness in the phase frontmodel but in not being able to quantify the start up conditions and axial dynamics in the vicinityof the trigger plate on a developed street. What was attempted here was an interaction betweenthe formed street and an exterior influence (the step). At this point, the interaction behaviour isnot defined well enough to allow its introduction into the model. The model is limited, for now,1325 - Triggered Experiments 133to describing the wake given all of the initial conditions and no external perturbations beingapplied later in the wake.The phase coupling equation gives the vortex shedding phase of the youngest vortex generationas a function of the phase and frequency of the previous generation. The first set of experimentstested the model for spanwise variation of the phase. It remains to prove that spanwise frequencyvariations are also correctly described by the phase coupling equation.133(a)360 mmINNri mommmeiimihmummmummEnnommom momre I I•^ii ir A ":!.^.%44 Art INI 11 ill 1I1Ian!Ir ii Hi on I imii IF 1 11 1111mnaseitric 1^1111Willi60111111r9E.11 111.1111111111iiil1 111l'irjt. .N5,47 :121 1 IIIill• 11111,11"111111210i^• '10;10,1041k1v9,1P.F.1111.).! I Ifir =love^t .S - Triggered Experiments 134(b)Figure 5.23 Experiment with a phase jump of 210°, 13.11 sec between frames. (R e=116)134I000O00000•1000O000000••••*00000000O000O00OOOo001 0OO05 - Triggered Experiments 135183 mm(b)Figure 5.24 Large phase jump of 199°. (R e=128, U.=1.5362 cm/sec)1355 - Triggered Experiments 1365.7 Frequency Variation Along The Phase FrontsSince the vortex shedding frequency is inversely proportional to the diameter of the sheddingcylinder, a frequency variation along the phase front is easily accomplished through using atapered rod. Vortex shedding from tapered cylinders has been investigated by others. Gaster[1969] found that the shedding frequency is locally determined by the diameter but that vortexformation is not. He also concluded that the circulation is not constant along the span andobserved a strong axial flow but did not measure it. Using hot-wire anemometers, Gerrard [1966]measured the shedding frequency along the span, observing frequency cells (shedding cells). Thework of Noack et. al. [1991], discussed earlier, was concerned with modelling the wake of atapered cylinder in the hopes of observing the frequency cells.All the above mentioned works deal with the developed wake, with no concern for the initialconditions and without generating predictions of where (x,z position) the dislocations will occur.In the work here, a trigger plate is used to set the initial conditions. Knowing the initial phase,the diffraction theory of the wake can be applied to predict both x and z where the dislocationwill appear in the wake.In the following runs, the initial phase was set by towing a cylinder past a straight edged plate,setting the initial phase to cbs=0 along the entire span. The taper is 8mm at the surface of thewater (top of the pictures) and tapers down to the bottom of the tank as indicated. The tapered136In the first experiment, a cylinder with ataper ratio of 137 was used. The resultingphotos and computer simulation(Figure 5.25) show each vortex leaningover more than their predecessor (inputparameters are in Table XIII) Thisinclination is expected due to the shedding5 - Triggered Experiments 137Diameter DtopDbottom8 mm5 mmReynolds Number RetopRibott.10968Emitter Plane x3 6.5 mmStarting Phase Os 0°Length L 410 mmEnd Conditions z top; atomreflectedf =0 hzPhase Front Ray Angle 8 0°Initial Vortex Strength A D (z)Table XIII Input parameters for a tapered cylinder.rods are described by a taper ratio:ID, -D2 /2L.frequency being greater at the bottom of the cylinder (smaller diameter) than at the top. Theinclination angles, 0, seem to match but it should be noted that the drift velocity in the wakereduces the experimental inclination angle, making the angle observed in the simulation appearslightly greater than that in the photo. The wake is compressed towards the right edge of thephoto since it shows all vortices at the time of exposure, whereas the model is the location ofthe vortex at the time they were created.Though the above result shows good agreement between the simulation and the experiment, thesought after vortex dislocations were not observed. By increasing the degree of taper it washoped that dislocations would appear in the wake. To this end a cylinder with a taper ratio of 82was used (Figure 5.26). The observed wake did show signs of forming dislocations but the run1375 - Triggered Experiments 138was not long enough to observe the full development. The beginnings are seen in the lower rightcorner of the photo and at the arrow on the right edge. The simulation also indicated that adislocation would occur late in the run in the lower right corner.Increasing the degree of taper (ratio 68) did result in a dislocation appearing in the run. Thisappears in the photo of Figure 5.27 (at the point marked by the two arrows). This is more clearlyseen in the time series of Figure 5.28. The development of the dislocation starts in frame #42 asa sharply kinked vortex being shed. Following strips of dye from frame to frame one can observean axial motion along the vortex, this was expected and predicted in chapter 2. The axial velocityalong the vortex was measured and found to be U ‘i=0.81 mm/sec upward on the vortex just beforethe dislocation. This is quite close to the maximum allowed viscous pipe flow velocity at centrespan, predicted by equation 2.4.7:Uz = -L? (Q2R2) dR -  (8+2)/2 inni (143 mm/sec)2(8-2)/2 min -0.92 mm/sec 5.7.18v dz (8)(1.02 mm 2/sec) 410 mmThe mechanism for the dislocation may, therefore, be similar to the phenomena of vortexbreakdown [Hall, 1972] (the vortex breaks down when a critical axial velocity is reached).Though the phase front model does not consider this, it shows a vortex dislocation in the correctplace in the run.In a similar run, the initial angle of the street was at 15° (Figure 5.29), causing the dislocationsto occur earlier in the wake. Two dislocations are clearly visible, one at the centre and one1385 - Triggered Experiments 139towards the top right (it is blown up in Figure 5.29). The simulation of this also shows the firstdislocations in approximately the same location (Figure 5.30).As before, the changing angle of the vortices has not been accounted for in the phase coupling(diffraction) integral. Around the dislocations, the phase front angle varies dramatically. Thevariation of the wave front normals may affect how quickly the street repairs itself and shift thelocation of the dislocations occurring thereafter. By including a changing phase front angle, abetter model may emerge but the simple assumption of all of the phase fronts being projectedparallel to the cylinder does yield a reasonable approximation to the observed triggered streets.The locations of the dislocations are accurately portrayed, and may be sights of vortex breakdownphenomena.1395 - Triggered Experiments 140621mm:I:•I•i•l-^1^•^1^•^1^•I•E:1^-^I^•i-:I:^:I :1 : i :I :I :I :I : 1 •I:i: 1 :• :---•-:-:•:•:-:-•-:•: I:I:1:^-. 1:1:1: 1 - 1.-^/^"i /^/•...• :-^8•I• •I•1^•;•1• •e-s•s- if •.:I:I:I:^:I:1^l•• 1.x.i.l.i.a.r.ff.it. 1s. 1r.^I .^t• I •^1^•^•^i^•^i •^1^•^i •^IV^11•• •^I• 1^•^I• •I•I•f•I•11•1•••1••• s•s•^•^•^•:•-• ..s...• 1^ i•i•i••i•i•i•i•I•i/•/*/^i:i: i :/:1:1 :I :1:i:1:1:1:1/:/. . °: I : f : i : 1 : I '' ." : f : I : I : I : I : I : 1 : 1 : I :• 1.1•1:1:1:1;i:i:1:1:1:1:/:!:/../       ,1::1.1 •^•1:1:1:,:/^/i:i:/:/:/:/:/....1.1.z.z.z.• f•s•f•i• :• r • 1•1•1• : •1•/•//•/•1:^:I:I:I 'il:1:1:I:1:1:1:1 .: 1:/:F:f:1:^•!•!•1•i•I1•111:14 :1•11:1•:1•1•:^:1:1:1:i:1::1:I:1:i:i:t'i`iLi• 1•1• 81 . 1 . 1*1 : 1••1•1"["1 : 1 : 1 : 1 :11 :(b)Figure 5.25 Taper ratio 137, /3--.8mm at the surface, 5mm at the bottom, L=410mm.140109— 98.8Re—88.6—78.468A 5 - Triggered Experiments 141(a)621mm• I^•^1^•^I^•^1^:^1^• ^I^•^:^6^:^•^I' 1 • i :  : i :1:1:1-^:^-^:^•^i^•..^1^:^1^: i^•^1^•.; •=•1•1*•/•,-/•/•,^••If "-i- 11•1•O'i• 1 •:•.=•/•/././•/•/•:- •^8*/°:• 1 d^•••^•.•^•• 11•11.^•1•1•J• ^•t•^IP^•^e•^i^•^.•^•^.•^•^.: 1 : 1 : I : 1 : 1 : 1 : I • • ^:' ^ • ^ • / •• .•••• • • /. •I 11:1:1:1/:/••/••/••• 1 • 11 • 1 • • •1• 1 • 1 •1•1• •r• 1 • •1•1• 1 • ,.i . t:i.1.1.1.!.• 1.ii ./..•• . • /././. • .••! •f•i•1•1•/•:•/•/•/•.(•/•/•/•/•^•^:. I o " . /. /.//1/1: ./. ." ./1...: 1 : I : 1 : I : I : 1 :^•I .• i .• i . • it . • / . • I . •,/ . • I . • I . • ./ .104— 87_8Re—71.5• ./e •/*^ :"„' •^:^'^:^:^:^.61 ^•":^I if: I:.^: 1 :/:/ • I *• j •j: 1 •1• 1 •1•1: 1 :1 . /.: /:/:/::1 ::/ • :1 • I. I:• :?:/. 1.:1 • I: 1^ •i•^:• I • I'''' • : • • . 1 . 1 *I •/:I• ! ^:^•I• .^iiiiiiiiiiiiiiiii:• I ; •I.1 • • • l• =*1—t:k:!.k.t.1.•1.•t•k . .... . .... .... .— 55-339(b)Figure 5.26 Taper ratio 82, D=8mm at the surface, 3mm at the bottom, L=410mm.141. •^•• 1-f•i•••1-E•I•1°!•:•: '1":•1":•^• I^°^•^•11•1•)•1)1.1.1.1•1•1 : 1.1•1•1•1 : 1•1•1•; t . 1• /• ^/• ; / ; ;^:11f ^ . / • if • •1?• ./ • ••••• • • ./. • :.V. • /::1 :i: :1 :11 :^:^I: I :-* " e" . • •••• •^• 1 ' 1 • :1 ' 1 • 1 • i • 1 • 1 •^• :^:• .* •^•^• •^1 • r .•^•^°^• 1:1:1:1:11.' " -** • -**^• -•• •^• 1. '^• 1. • 1 • I ; • •^•• ••• ••••••.r•r•r•I•r•r•!•:•■• i • l•^: ^• I °^▪ 1"^• -I • °^• I • I :^:^: • I / •^.°^•^0". •^• er •^ • •^•^•^• i •^•^•^i•!•!• !•1•1•I' r„^f„^„• :^•^I^• •^:^:^-^:^- - ,tr^r ;1^•• •^i^•^•^i^• I^•^i^•MON•—04—a?II 6 —ZI^ II Mk^.^',it %/... 46,Z-17Iluw0I -17=7 tuopoq alp n uauz `dol aqi le wwg.a '89 our/ iadvj, Lz•s aarblu(q)uruu I Z9nuatuyadrg padanyi -5 - Triggered Experiments 143Frame #25 t=6.422 secFrame #26 t=8.150 secFrame #27 t=9.930 secFrame #28 t=11.918 secFrame #29 t=13.710 secFigure 5.28 Part 1.Frame #33 t=20.714 sec Frame #38 t=29.814 secFrame #34 t=22.614 sec Frame #39 t=31.402 sec143Frame #30 t=15.410 sec Frame #35 t=24.374 secFrame #31 t=17.210 sec Frame #36 t=26.182 secFrame #32 t=18.986 sec Frame #37 t=27.818 sec5 - Triggered Experiments 144Figure 5.28 Part 2.Frame #40 t=33.202 secFrame #41 t=34.890 secFrame #42 t=36.646 secFrame #0 t=38.230 secFrame #1 t=39.790 secFrame #5 t=44.554 secFrame #6 t=45.894 secFrame #10 t=48.142 secFrame #11 t=48.702 secFrame #2 t=41.166 sec Frame #7 t=46.466 secFrame #8 t=47.026 secFrame #4 t=43.982 sec Frame #9 t=47.582 sec144Frame #7 t=44.7 secFrame #33 t=21.5 sec.621 mm240 mmFrame #38 t=31.1 sec Frame #7 blow-up of boxed area5 - Triggered Experiments 145Frame #0 t=38.9 secFigure 5.29 Re 92.6 at the top of the taper(D=8mm) for taper ratio 68. A blow up of theboxed area in frame #7 is given showingloops.145(a)621 mm4 r- -^)^I ^"-fFJ^Lip, 7r v5 - Triggered Experiments 146! 1!).^• ,^•^ r• • ^• ^•^• ./• / • /,^•: /  ^ 1• .^/r^.^:^: . ... . ..• . ..• . e• . .• . I . 1 .: i • i : I : i? .• .4' .• I .• i . • ., . ..• . ...* . ..: . e .. 1: .. I .. I .. if: / :. i .- I •• i .• if . • / . • i . - .0- . • „r" . •i^... . - •• . .,- •• •• , • 1 • 1 • , • f • I • i - i • 1 • 1 •^• 1 •^•• ^• i • if;T• ^•' I • • ' •a 'I^••'' " I • ' e •^•• e •,-. ".—% - z - I - z • I - : - i -.i ' ' • • I^I•I ' i^I • I:if :f:I.:.!.1:I:^:•:^:• (•(^•t•^t^i^•t•i•I•tel•k^..^ .•..^ •%....^• ••""^• 92.6—75.2Re—57_7— 40.523_2(b)Figure 5.30 Taper ratio 68, initial wave front angle of 15°.1466 - CONCLUSIONSThis thesis has proposed a new near-wake model of the vortex street and has shown it to be anaccurate predictor of the development of laminar vortex streets. The model is based on thesimilarity between the vortex street and optical interference patterns. The spreading of phasefronts of both optical patterns and vortex streets may be described by Huygens-type phase frontconstruction and by ray type propagation directions. The new experimental technique of settingthe initial phase of the street was utilized to substantiate the model's utility.The phase front model of the vortex street is based on the Biot-Savart law for fluids, yieldinga phase coupling equation for the vortex street (3.3.8):1 ffiuy(xo ,y0 ,z0 )^A (1)sin(a^expfi [27cf(l)t -E. (10 -1) + c•,(1)1} di= 110^12This relates the velocity Uy to the amplitude, A, and phase, O s of the oscillating vorticity in theflow field, and in doing so, becomes a tool for studying local influences in fluid flows throughwave coupling and superposition. Analogous to optics, the coupling can be described byHuygens' secondary wave fronts and by rays. Since both vorticity and velocity are functions oftime in the phase coupling equation a phase velocity V must be defined.1476 - Conclusions 148In the laminar vortex street produced by a given cylinder and flow speed, U.,„ the vortices areassociated with the phase, (10=0 on one side of the wake, and 0.7c on the other and are spaceda constant distance apart, X=V/f. The shedding frequency is not determined in this model butobtained from the empirical Strouhal relation The question of what sets the sheddingfrequency is still unresolved.The phase velocity used was the free stream velocity 17..U.„. In principle any background and driftvelocity should be accounted for in the model. Including the drift velocity will reduce the phasevelocity but not by much. The drift could be included in later work, providing it is known, butas a first order approximation, the results obtained here without U drift are reasonable portrayalsof the experimental results. It was assumed that V was the same in all directions, however, it maywell be that V varies with direction and may be affected by background velocities, in which casethis may be likened to a doppler shift.It was demonstrated that the vortex street was adequately modelled by a line of emitters placedxs=1D. The emitters (primary wave front) were coupled through superposition at the cylinder,yielding Uy. As seen from the reference frame of the cylinder, the information of the velocitystate is swept back into the wake, developing into a new set of emitters which replaces the onesthat created them (secondary wave front). In the work presented here, Uy is assumed to have thesame phase as the vorticity emitter it creates and is proportional to the amplitude. The single lineof emitters is an approximation that works well; however, an improved model may be realized1486 - Conclusions 149by using two lines of emitters, one for each side of the wake. This may, in some way, allow forthe modelling of the trigger plate-cylinder interaction, which was beyond the scope of the workhere.The model was applied to a number of flow configurations, with various boundary conditions.Using simple boundary conditions (eg. f=0, or the method of images), the phase propagationmodel accurately portrays the development of the vortex street. The concept of a phase front alsooffers an intuitive qualitative approach to the phenomenon of wake formation by considering the"ray" in which vortex influences are propagated in the vortex street: The rays focus or defocusthe oscillator energy allowing the phase fronts to interact more strongly in one direction than inanother.In the experimental program, a new technique of setting the start-up phase of the vortex streetwas explored. End conditions are known to produce oblique shedding modes, however it wasdemonstrated, by using slanted trigger plates, that the end condition can produce a parallel modewhich grows at the expense of the oblique mode. The trigger plate technique is quite useful sinceit allows the introduction of sudden phase changes, so that flow structures can be"manufactured" in the wake. At this point we have managed to form step discontinuities, tornvortex streets containing interconnecting loops, and undulated vortices. In tapered cylinderexperiments, the plates provided a known initial starting condition, which, in turn, allowed thetesting of the phase model in predicting the formation of shedding cells. Sectional dye markings1496 - Conclusions 150facilitated obtaining the axial flow velocity. The trigger plate technique will be useful for furtherstudies. This technique could be applied to experiments on vortex shedding from irregularlyshaped objects. Plates could be shaped to set the starting conditions of vortex shedding from asphere or bent rod for example.The limits of the applicability of the model and its extensions could be pursued in later studies.Several questions appear immediately:1. Would different growth functions: A(t+At)=G(Uy(t)) yield better results and allow forinteractions with other influences?.2. What sets the phase velocity, V?3. Does its value vary with direction and is it a type of "structure" wave?4. If V does vary, can the transfer of energy and cell formation be described by ray opticsusing a refractive index: V=c/n?There are some possibilities that the phase front treatment could lead to new insight into thedynamics of other flows: It may be possible to predict the wake pattern resulting from theinteraction of two or more streets (multi-bar arrangements). If so, then geometries for vortexsuppression and creation could be found. This would be an application of the diffraction integralas a design tool for managing vortices and their production (eg. make a better stake or predictflow oscillations on power lines and bridges).1506 - Conclusions 151Based on the experimental results presented and the calculations, it is, therefore, possible toformulate a model of the vortex street based on the concepts of superposition. In this model, avortex generation's shape is in direct correspondence to the superposition pattern created by thenear wake.Coupling the power of diffraction theory with input parameters based on simple and reasonableassumptions and well established observations of the wake could yield, for further research, ananalytical tool for studying vortex streets.In summary, a model which incorporates the simplicity of diffraction theory and easilyidentifiable wake parameters has been successfully used to predict the experimentally observedpatterns of vortex shedding from cylindrical objects in this thesis. 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Free stream velocityUr ^ Radial velocityUz  Axial velocityUe ^ Circumferential velocityU   Drift velocity of the vortex street relative to the fluid^ Wave speedw  Complex potential (w=x+iT)x ^ Lab frame coordinate in direction of cylinder motionxs  Position of emitter planeX ^ Position coordinate (x ,y,z), or as a vectory  Lab frame coordinate the y axis is perpendicular to both the x and z axisz ^ Coordinate in direction parallel to the vortex axis (or cylinder axis)Z  Complex position coordinate (x+iy)a ^ Angle a vortex makes with U.,R  Angle to U.,F ^ Circulation: (I ga :f)0  Angle to the cylinder (z axis)O Plate angle to the cylinder15 ^ Vector potential such that (7--Vxd#X Wavelength (vortex spacing)v ^ Kinematic viscosity162A - List of Variables 163P ^ Densitya  Disturbance growth angle4) ^ Azimuthal coordinate4:13  Total phasecp. ^ Initial phaseX  Velocity potential'P ^ Stream functionco  VorticityS2^ Angular velocity163B - THE FLUID DYNAMICS EQUATIONSThe following is the set of basic fluid flow equations which are covered in any elementary fluidmechanics text [Batchelor, 1967; Streeter and Wylie, 1981]. In most cases they cannot be solvedanalytically due to the nonlinear terms.In most cases, a fluid flow is described mathematically by a momentum equation and byconservation of mass. The momentum balance is described by the Navier-Stokes equation:Tr +(ü•V1 = -°13 +v020p^ B.1where U is the velocity vector, t is time, P is the pressure, p is the mass density of the fluid, andv is the kinematic viscosity. This is a set of 3 equations containing 4 unknowns. The set ofequations is closed by conserving mass through the continuity equation:aP -F(Cm)p ÷p(7.0) =0at^ B.2This assumes the fluid properties are fixed, however, the fluid properties may vary with pressureand temperature, transferring energy to the atomic state and/or heating the fluid. Therefore, onemay need to include the equation of state for the fluid. Fortunately, in this study, as in manycases, the flow is incompressible and there is little or no heating, therefore a good approximationof the flow can be obtained using only Navier-Stokes and continuity.It is often more convenient to solve dimensionless equations since similar flow situations maybe described by solving only one set of equations. The Navier-Stokes equation is transformed bytaking a length scale for the flow L and free stream velocity U.„ then multiplying equation B.1by DU!. Identifying the dimensionless time scale as 7=tU.,IL, position as T=X/L, velocity asV= U/U.„ and pressure as 71=P1pUO2., then:80 ^v 2+ ((IV) = -VP+V^ B.3at^LU.From which we see the same solutions for the same Reynolds number:LU^ B.4This allows us to scale the solutions to larger diameter cylinders with slower moving mean flows.164B - The Fluid Dynamics Equations 165In the inviscid case (v=0) or large Re equation B.1 reduces to Euler's equation:au .^_ VPat^" p^ B.5The 2D flow solutions to B.5 can be superimposed using potential flow theory[Landau andLifshitz, 1987: §9]. In this way, vortex solutions can be superimposed to form the Karman vortexstreet (see page 21) observed behind a circular cylinder.In a fluid flow, the vortices are easily identified as a closed circulation pattern. Mathematically,a vortex is a mass of fluid possessing circulation:r =^ B.6where s is the length along a closed loop surrounding a vortex. The circulation is a difficultquantity to solve for since it does not relate easily to either Navier-Stokes or continuity andleaves a pressure term to be found. In vortex solutions it is much simpler to use the local spinin a fluid [Shapiro, 1972] known as the vorticity:(7) = Vx(1^ B.7Integrating the vorticity over the area of the vortex gives the circulation (Green's Theorem). Thevorticity is also directly related to the angular velocity, 51:=2L2^ B.8The Navier-Stokes equation is easily transformed into the vorticity equation by taking the curlof equation B.1 giving:'36 =vx(Ox3)A-vv2aatNotice that there is no pressure term to solve for but by using equation B.7 to find the velocitycomponents then substituting into Navier-Stokes equation an equation for pressure is obtained.B.9165e 50.8a4 0.6_s..s..— 0.402r^,Burger's-r,rIRC - BURGERS' VORTEXThree dimensional vortices as might appear in wake flows must have some axial velocities. Asteady state vortex which features axial and radial velocities as well as viscosity is Burgersvortex. Burgers formulated this vortex [Burgers, 1948] to fit into a turbulence model he wasinvestigatingt. He assumed a radial velocity:U,=Cir^C.1and axial velocity (z direction)Uz = 2C1z^C.2C1 is a constant. By solving Navier-Stokes hefound the azimuthal velocityC^-c 212vU = —1(1 e 11. )9 2nrC.3.^Figure C.1 Tangential velocity profile forwhere C2 is a constant. The circumferential Burgers vortex compared to Rankine.velocity of Burgers' and Rankine's models areshown in Figure C.1. For both vortices thetails of the distributions fall off as 1/r, approaching each other asymptotically. Burgers'distribution actually bears stronger resemblance to a steady state Oseen eddy profile with its peakvelocity at R-45v/2C 1 . It is unlikely that a real flow would exhibit a large radial velocity far fromthe vortex core and an associated axial velocity, therefore, the Burgers model is applied onlyclose to the vortex core itself. This limits the utility of Burgers' model to steady vortices ofuniform size. An example of a steady vortex to which this model applies is a driven vortex withsuction from below such as a bathtub vortex [Odgaard, 1986]. Burgers' model serves as thesimplest axial flow vortex, but it is unsuitable to describe the vortices in the wake of a conicalcylinder.t The paper Burgers wrote is more widely known for the vortex model than for theturbulence model he proposed.166Figure D.1 Input to model. The emitterplane is on the left and the cylinder is on theright.D - EXAMPLE CALCULATIONThis is an example calculation of parallel vortex shedding from a 1 mm diameter circularcylinder at Re=100 in water. The diffraction integral is applied in the same manner as used in themodel calculations. The input to the model are the cylinder diameter D=lmm, Re=100, viscosityv=10-6 m2/sec, and the initial phase distribution at the emitter plane, 4:13s . The flow velocity canbe found from the Reynolds number: U„.=R ev/D=100mm/sec. The wave speed, V is assumed tobe U.,. The shedding frequency is found by using the Strouhal number:f=SUID=(0.212-4.51ROUJD=16.7 hzThe emitters are assumed to be a distance x3=D from the cylinder and spaced 0.2D apart asshown in Figure D.1. In this example only 3 emitters are considered but in the calculations ofchapters 3 and 4 up to 31 emitters contributed to the result at zo. The distance between thecalculation point and the emitters can be determined from the geometry, yielding the travel timetn=r,I17.rn.1 =1.02 mmrn=1.00 mmr„.,4=1.02 mm4,4=0.0102 sec4,4=0.0100 sec4,„4=0.0102 sec(a-f3)„./=90°+arctan(0.2)=101.31°(a- (3).-1=90°(a-P)„„=90°-arctan(0.2)=78.69°A=D=1 mmCoe=0 Radians at the emitter plane.167The calculation requires a developed wake beforecoupling through diffraction from the emitterplane can be considered. Initially, starting thecylinder from rest, there is no vorticity in theemitter plane. During this start-up phase, thevortex street develops simply with no axialcoupling. The phase of the street emerging fromthe cylinder is assumed to be:Os(z,t)=27rf t+ck(z,0).This phase is convected at a velocity U., towardsthe emitter plane. Therefore, the phase of theemitters at time t=0 sec is taken to be 0 and attime 0.0002 sec is 0.02.1 radians (1.202°).Once the vorticity distribution of the wake at theemitter plane is known, the diffraction integralcan be applied:1 A sin(tc - 13) e iantv+t,,)+04,toidzU (zo,t) - 47 r 2The contributions from the emitters arrivesimultaneously at point zo at t=0.0000 sec. Thesecontributions leave the emitters at different times.The part from z„ 1 leaves the emitter 0.0002 secbefore the part from zn. These contributions to thediffraction integral are then:D - Example Calculation 168 000Figure D.2 The initial voracity distributionis assumed to be sinusoidal without axialcoupling.Figure D.3 Diffraction integral is used tosum the contributions from the emitter planein the wake.A U^_ (1)sk(101.31°) e if2It(16 •7)0.000040.0102)+M =0.0258 +0.0471iyA-1 47c(1.02)2A U _ (1)x(90° ) a a6 7)(0 0000+0 0100)+0 021] =0.0382 +0.0698ie' • •^•^•" 4.tc (1.00)2AU ^- (1)sin(78.69°) Libras mo.0000.omoz)+0] 0 0258 0 0471 'e^•^= .^+ .^14n(1.02)2Superimposing the results yields the transverse velocity at the cylinder at t=0.0000 sec:Uy(zo , t=0.0000 sec) = (0.0258 +0.0382+0.0258) + (0.0471 +0.0698 +0.0471)i =0.0898 +0.1640i168 000Figure D.4 The phase information convectsback towards the emitter plane.D - Example Calculation 169The phase information from Uy convects backtowards the emitter plane at the flow velocity,reaching the emitter plane in x/U.,=0.01 sec. Asthe information moves, the velocity informationgrows into the new vorticity distribution seen atthe emitter plane. The vorticity has the samephase as U calculated at the cylinder: 1.070radians (61.30 0). The strength is renormalized toits original value: A=Constant*1 Uy I=lmm.This new emitter distribution (A, and do s) canthen be used in the diffraction equation toevaluate the form of the emerging vortex street atthe next time step.This method is not limited to time steps of 0.01sec. Time steps smaller than the cycle time for the phase information between the emitter planeand the cylinder. These smaller time steps can be used along with the corresponding phaseinformation. The time steps are not independent, the phase velocity couples the time stepinformation through the phase (13 8(z,t.) used in each calculation.169E - CALCULATION OVERVIEWThe calculation of the development of the vortex street is performed by the program Huygens.c,listed in Appendix F. A flow chart of the program is given on the next 3 pages.The inputs to the program are: the local diameter of the cylinder (D(z)), the free stream velocity(U..), the initial shedding phase distribution (4)(z)), the location of the emitters (x,), and thenumber of time steps (n) for the cylinder to travel a distance x3. The first portion of the programsets up the data areas with these input values plus calculates the parameters to be used in thesimulation. The directly calculated parameters are the shedding frequency (f(z)), the oscillationstrength (A(z)), and the total oscillation strength (A s.. The distance (r), the obliquity factor(sin(a-(3)/r2) (from eqn (3.3.8)), and the propagation time (t') could be calculated during thesimulation; however, in order to speed up the calculation look-up tables are formed for eachvalue at every emitter (at za) which contributes to the Uy(zo) calculated at the cylinder.The initial oscillation strength (vortex strength) is assumed to be proportional to the cylinderdiameter (A(z,t=0)=D(z)). The shedding frequency is found through the definition of the Strouhalnumber (S=fDIU,.; eqn (2.3.1)) and empirical data for the Strouhal number (in our case, for acircular cylinder in the range of 50<R e<150 Roshko (1953) has found S.212-4.5/Re ; eqn(2.3.2)). Below Re=40 no shedding occurs and, therefore f=0.The values for the distance and obliquity are calculated from the distances and angles describedby the calculation point (0,0,z 0) and the emitters (x„0,z, i). The travel time (retarded time) t' isfound by assuming that the wave speed is that of the free-stream velocity, yielding (=r/U.., (eqn(3.3.2)).Once the input data values are set and the look-up tables calculated, the time for the initial phasevalues to be convected from the cylinder to the emitter plane is found. This is the time beforethe coupling from the street can appreciably affect the shedding phase at the cylinder. During thistime the street is assumed to develop simply, its phase progressing sinusoidally without couplingto the neighbouring emitters. This calculation is given on the second page of the flow chart.When the initial phase has reached the emitter plane, the shedding phase at the cylinder isdetermined through the diffraction equation (wave coupling). The calculation flow is shown onthe third page of the flow chart. The transverse velocity (U y) is calculated at the cylinder fromthe contributions of each emitter (the data of each are in look-up tables) (using eqn 3.3.8). Thephase of the vorticity in the street is assumed to be that of Uy and is convected back towards theemitter plane with the wake. The vorticity strength (A) is assumed to be proportional to U andits total value is also assumed to be fixed (The strength of the vortex street is unvarying, alwaysproducing the same size eddies in parallel shedding from a uniform diameter cylinder). Thevorticity strength is then A(z,t)=Uy(z)k.„/Us„„, (eqn (3.4.4)). The results of the calculation areretained since they reenter the calculation at later times. The results take time to convect awayfrom the cylinder, and reenter the calculation when they pass through the emitter plane. Thephase information takes a time At=x1U., to reach the emitter plane and a time t' for its effect tobe felt at the cylinder. Therefore, the calculation looks back to earlier time results to find thevalue of Ur170E - Calculation Overview 171StartCalculate time step sizet„ep=x)(nUjOpen output filesOUTPUTFILEO -- for 0=0 crossingsOUTPUTFILEPI -- for CD=It crossingsSet local diameterD(z)Set up oscillation strengthtableA (z ,t=0)=D(z)A(z,M0)=0Calculate total oscillation strengthAei,„,=EA(z,t=0)Set the initial phase00(z,t=0)171Calculate the local shedding frequencyf(z)=S (1? e)U eiD(z)whereS(Re)=0.212-4.5/ReandRe=U,,,,D(z)/vNote that f(z)=0 below R erit=40Make look-up tables for sin(a-(3)/r2 andretarded time, t', for the emitter points at(xe,0,z,z) associated with the calculationpoint zo .r=sqrt((zo-z02+x32)tf(zo,zn)=r1VV=U,,,, (assumed)f3=asin((zo-z„)/r)a is a user input = initialsinrr(zo ,z„)=sin(cc-(3)/r2find (min in t'(zo,z,i)The wake developsfor the time beforefeedback startsno whilet<emin E - Calculation Overview 172yesif 4:10(z,t)*0(z,t-ts„p)<0^noa 41)=0 or it transition(vortex shed)yesif (13(z,t) has 0transitionyesOutput t, z, and A(z,t)to OUTFILEO(record shedding location)Advance timet----t+ts„pAdvance shedding phase0:13(z,t)=0(z,t-t„ep)+27cf tstepcos(a)(f varies with shedding angle oc)(1)(z,t) has it transitionOutput t, z, and A(z,t)to OUTFILEPI(record shedding location)no172E - Calculation Overview 173Advance timet=t+cpCalculationwith feedbackTransform Uy into voracity(Renormalize the vortex strength)Usum=E Uy(z)A(z,t)=IUy(z)I*As.,,,JUsu„,if 43(z,t)*cD(z,t-t„ep)<0^no((=0 or It transition)(a vortex shed)yes<if (13(z,t) has 0transitionno1.^Compute new shedding phasePhase of z„ contribution iscion(z,t)=4)(z,,,t-f)+27rfSuperimpose the contributions at z(apply diffraction equation)Uy(z)=real+i imag=Esinrr(zo,z„)*exp(idon)Amplitude isUy(z)=sqrt(reaf+imag2)New shedding phase is013(z,t)=atan(imag/real)(from -7C to +7C)(13(z,t) has it transitionOutput t, z, and A(z,t)to OUTFILEPI(record shedding location)yesOutput t, z, and A(z,t)to OUTFILEO(record shedding location)173F - HUYGENS.0The program used to simulate the laminar vortex street using wave front analysis based on aKirchhoff diffraction type formula is given below. The code is written in Microsoft C 5.1. Thecode has been written referencing all angles to the cylinder axis rather than the normal to thecylinder given in the thesis./************************************ **************** Program: Huygens.c* Programmer: Marcel Lefrancois** This C program is a simulation of the wake of* a circular cylinder based on the Kirchhoff* integral, or Huygens wavefronts.***************************************************/#include <stdio.h>#include <math.h>*include <conio.h>#include <graph.h>#include <errno.h>#include <stdlib.h>#include <fontl.h>#include <sys\types.h>#include <sys\stat.h>#include <io.h>#define OUTFILEO#define OUTFILEPI"VO.MDL""VPI.MDL"/* OUTPUT FILE OF 0 PHASE TRANSITION *//* OUTPUT FILE OF PI PHASE TRANSITION */#define TWOPI 6.28318530717958#define PI 3.14159265358979#define HALFPI 1.57079632679490*define Z_SLICES 410 /* NUMBER OF SLICES IN ROD */#define ROD_LENGTH 410.0 /* LENGTH OF ROD (mm) */#define TIME_TOTAL 1000 /* TIME STEPS TO RUN FOR */#define V_FACTOR 1.0 /* THE RATIO OF WAVE VELOCITYTO THE FREE STREAM VELOCITY */#define SPAN FACTOR 3.00 /* HOW MANY X_EMIT LENGTHS TO USE INSUMMING PHASE OVER SPAN */174#define#define#define#define#define#define#define#define#define#defineD_TOP^8.000D_BOTTOM^2.000MAX_INTENSITY 256ZD_LOW^0ZD_UP 0DSTEP_SIZE^1.00ZP_LOW^0ZP_UP 0PSTEP_SIZE^1.0PLATE_ANGLE 00.0F - Huygens.c 175#define U_INF#define NU#define R_e(a)#define R_CRIT#define S(a)#define N_t_D#define X_SLICES#define Z_LEVELS#define X_EMIT#define TOP_END^0#define BOTTOM END^2#define RANDOM^0#define AXIAL_V^0.0/* FREE STREAM VELOCITY (mm/sec) *//* THE KINEMATIC VISCOSITY (*10e6) *//* REYNOLDS NUMBER (a) IS DIAMETER *//* R_e FOR NO VORTEX SHEDDING *//* STROUHAL NUMBER (a) is R_e *//* NUMBER OF TIME STEPS IN MOVINGROD A DISTANCE X_EMIT*D_BOTTOM *//* NUMBER OF TIME SLICES TO KEEPTHIS MUST BE LARGER THAN N_t_D/* MAXIMUM NUMBER OF SLICES NEEDEDFOR SUMMING WAVE INFO AT A POINT *//* LOCATION OF EMITTER IND_BOTTOM DIAMETERS *//* CYLINDER DIAM (mm) AT z=0 *//* CYLINDER DIAM (mm) z=Z_SLICE-1 *//* OUTPUT NUM FOR NORMAL STRENGTH *//* z LOWER FOR SUDDEN DIAM CHANGE *//* z UPPER FOR SUDDEN DIAM CHANGE *//* RATIO OF DIAMETER CHANGE *//* z LOWER FOR SUDDEN PHASE CHANGE */* z UPPER FOR SUDDEN PHASE CHANGE */* RATIO OF PHASE CHANGE *//* ANGLE TRIGGER PLATE MAKES TO THEVERTICAL FOR CONTINUAL CHANGE OFPHASE. POSITIVE FOR TOP OF PLATELEANING INTO FLOW (IN DEGREES) *//* 2 IF TOP END IS FIXED1 IF TOP END (z=0) IS A FREE END0 IF CYLINDER IS REFLECTED *//* 2 IF TOP END IS FIXED1 IF BOTTOM END IS A FREE END0 IF CYLINDER IS REFLECTED *//* 1 FOR A RANDOM START0 OTHERWISE *//* AXIAL VELOCITY IN mm/s14.21.23(U_INF*(a)/NU)40.0(0.212-(4.5)/(a))414252.000main()POSITIVE IS UPWARD TOWARDz=0 */static float huge phase[X_SLICES][Z_SLICES]; /* PHASE OF SHEDDING */static float far D[Z_SLICES];^ /* DIAMETER OF ROD */static float huge A[X_SLICES][Z_SLICES];^/* EMITTER STRENGTH */static float far A_temp[Z_SLICES];static float far omega[Z_SLICES];^/* ANGULAR FREQUENCY */static int huge zcoordz(Z_LEVELS][Z_SLICES]; /* LOOKUP TABLE - z'S */static int huge zcoordx[Z_LEVELS][Z_SLICES]; /* LOOKUP TABLE - x'S */static float huge ztime[Z_LEVELS][Z_SLICES]; /* LOOKUP TABLE-delayed time */static float huge zcosrrr(Z_LEVELS][Z_SLICES]; /* LOOKUP TABLE - cos/rr */175F - Huygens.c 176static int nz[Z_SLICES];^ /* LOOKUP TABLE - NUMBER OFSLICES FOR CALCULATION */int i,j;^ /* COUNTERS */int t,xn,zn; /* TEMPORARY VARIABLES FORTIME AND COORDINATES */int maxz,minz;^ /* MAX-MIN z COORDINATES */int maxx,minx; /* MAX-MIN x COORDINATES */int maxnz; /* MAX X_SLICES NEEDED */int x,y;^ /* x,z COUNTERS */double t_step;^ /* sec FOR A TIME STEP */double real,imag; /* COMPLEX PARTS OF CALC */double Aoverrrr; /* A/r*r*r */double tphase;^ /* PHASE IN EXPONENT */double normal_A_sum;^ /* TOTAL EMITTER STRENGTHTO MAINTAIN */double sum A;^ /* SUM OF STRENGTHS */double A_factor; /* MULTIPLIER TO RESTOREEMITTER STRENGTH AFTERCALCULATION */int handle0,handlepi;^ /* FILE HANDLES */int open_ftle(char *,int);int intens;^ /* STRENGTH VALUE OUTPUTTO FILES */double theta, time, r, z;^ /* FOR CALCULATING LOOK-UP */long int duml;int dummy,flag;double dumf;/*^CALCULATE THE TIME IT TAKES FOR ONE TIME STEP*/t_step=(double)X_EMIT*D_BOTTOM/(double)U_INF/(double)N_t_D;/*^OPEN OUTPUT FILES*/if((handle0=open_file(OUTFILE0,0_BINARYIO_CREATIO_TRUNCIO_WRONLY))==-1)printf("COULD NOT OPEN FILE\n");return(0);if( (handlepi=open_füe(OUTFILEPI,O_BINARYIO_CREATIO_TRUNCIO_WRONLY) )==-l)printf("COULD NOT OPEN FILE\n");return(0);/*^SET THE SIZE OF THE CYLINDER (IN mm) AND THE VORTEX STRENGTHSprintf("SETTING CYLINDER SIZE\n");for(i=0;i<Z_SLICES;++i)176F - Huygens.c 177D[i]=(float)(D_TOP-(D_TOP-D_BOTTOM)*(float)i/((float)Z_SLICES-1.0));/* VORTEX SHEDDING HAS NOT STARTED SO SET PREVIOUS TIME STRENGTHS TO 0for(j=1;j<X_SLICES;++j)A[j] [i]=(float)0.0;}/* IF THERE IS A STEP */for(i=ZD_LOW;i<=ZD_UP;++i)D[i]=(float)(DSTEP_SIZE*D[i]);/* THE INITIAL STRENGTH IS THE CYLINDER DIAMETERif(RANDOM==1)for(i=0;i<Z_SLICES;++i)A[0][i]=(float)(rand())/32767.0*D[i];elsefor(i=0;i<Z_SLICES;++i)A[0][i]=D[i];normal_A_sum=0.0;for(i=0;i<Z_SLICES;++i)normal_A_sum+=(double)D[i];*SET THE INITIAL PHASE AREA TO A RANDOM PHASE SINCE NO VORTEX SHEDDING^ */printf("SETTING CYLINDER PHASE\n");for(i=0;i<Z_SLICES;++i)for(j=0;j<X_SLICES;++j)phase[j][i]=(float)(((double)(rand())/16383.5-1.0)*PI);/*^SET THE INITIAL PHASE OF THE CYLINDER*/* PHASE WITH A PLATE ANGLE */if(RANDOM!=1)for(i=0;i<Z_SLICES;++i)phase[0][i]=(float)(TWOPI*sin(PLATE_ANGLE*PI/180.0)*(float)i*S(R_e(D[i]))*ROD_LENGTH/D[i]/(float)(Z_SLICES-1));/* PHASE IF THERE IS A STEP */for(i=ZP_LOW;i<ZP_UP;++i)phase[0][i]=(float)(PSTEP_SIZE*PI);/* PHASE OVER IS IN RANGE -PI TO PI */for(i=0;i<Z_SLICES;++i)while((float)phase[0] [i]>(float)PI)phase[0][i]-=(float)(TWOPI);for(i=0;i<Z_SLICES;++1)while((float)phase[0][i]<(float)-PI)phase[0][i]+=(float)(TWOPI);/*^COMPUTE THE LOCAL SHEDDING FREQUENCY177F - Huygens.c 178printf("SETTING SHEDDING FREQUENCY\n");for(i=0;i<Z_SLICES;++i)/* THE R_e MUST BE HIGH ENOUGH FOR SHEDDING */if((float)R_e(D[i])>(float)(R_CRIT))omega[i]=(float)(TWOPI*S(R_e(D[i]))*U_INF/D[i]);elseomega[i]=(float)0.0;phase[0][i]=(float)(((double)(rand())/16383.5-1.0)*PI);/* IF END IS FIXED THEN NO VORTEX SHEDDING */if(TOP_END==2)omega[0]=(float)0.0;if(BOTTOMEND==2)omega[Z_SLICES-1]=(float)0.0;/*^COMPUTE time AND radius VALUES FOR EACH POINT*printf("CREATING LOOKUP TABLE\n");for(i=0;i<Z_SLICES;++i)/* SPAN TO SUM OVER IS BASED ON (X_EMIT*D_BOTTOM) LENGTHS *//* FIRST FIND EXTENT OF SUMMATION AND SEE IF THERE IS ENOUGH *//* DATA AREA FOR PROGRAM TO PROCEED *//* GET MINIMUM AND MAXIMUM SLICES TO BOUND CENTER POINT */minz=i-(int)(X_EMIT*D_BOTTOWSPAN_FACTOR*Z_SLICES/ROD_LENGTH+0.5);maxz=i+(int)(X_EMIT*D_BOTTOM*SPAN_FACTOR*Z_SLICES/ROD_LENGTH+0.5);/* HOW MANY Z_SLICES ARE NEEDED TO FORM RESULT */nz[i]=maxz-minz+1;/* CHECK FOR ENOUGH DATA AREA */if(nz[i]>Z_LEVELS)printf("Z_LEVELS TOO SMALL: SET AT %i, NEED %i\n",Z_LEVELS,nz[i]);return(0);/* NOW CREATE LOOK-UP DATA OF time, distance, location */for(j=0;j<nz[i];++j)theta=atan((double)(i-minz-j)*ROD_LENGTH/Z_SLICES/X_EMIT/D_BOTTOM);time=X_EMIT*D_BOTTOM/V_FACTOR/U_INF/cos(theta);/* SET z COORDINATE FOR TABLE *//* AN AXIAL VELOCITY CARRIES THE INFORMATION FURTHER */zcoordz[j][i]=abs(minz+j+(int)(AXIAL_V*time*Z_SLICES/ROD_LENGTH+0.5));if(zcoordz[j][i]>=Z_SLICES)zcoordz[j][i]=2*Z_SLICES-zcoordz[j][i]-2;/* COMPUTE APPARANT z COORD */z=(float)(i-minz-j-(int)(AXIAL_V*time+0.5))178F - Huygens.c 179*(float)ROD_LENGTH/(float)Z_SLICES;/* COMPUTE DISTANCE */r=sqrt((double)z*(double)z+(double)(X_EMIT*D_BOTTOM)*(double)(X_EMIT*D_BOTTOM));/* FIND THE ANGLE BETWEEN THE CALC POINT AND THE VORTICITY VECTOR*/if(minz+j<0)theta+=(float)PLATE_ANGLE*PI/180.0;else if(minz+j>Z_SLICES-1)theta+=(float)PLATE_ANGLE*PI/180.0;else if((minz+j!=0)&&(minz+ji=Z_SLICES-1))theta-=PLATE_ANGLE*PI/180.0;/* PLACE VALUES INTO LOOKUP TABLE */zcosrrr[j][i]=(float)(cos(theta)/r/r);zcoordx[j][i]=(int)(time/t_step+0.5);ztime[j][i]=(float)(zcoordx[j][i])*t_step;/* IF THE ANGLE IS GREATER THAN PI THEN THIS IS NOT A FORWARD WAVE *//* AND THE DATA SHOULD BE WRITTEN OVER *//* CHECK FOR END CONDITION IF NOT INFINITE CYLINDER */flag=1;if(TOP_ENDi=0)if(zcoordz[j][i]<0)nz[i]-=1;- - j;flag=0;)if(BOTTOMEND!=0)if(zcoordz[j][i]>=Z_SLICES)nz[i] - =1;- - j;flag=0;}if(((float) (fabs(theta))>(float)PI)&&(flag==1))nz[i]-=1;-- j;))}/*^DO A SEARCH FOR MINIMUM AND MAXIMUM TIME STEPS/* minx IS x WHERE FEEDBACK STARTS *//* maxx IS x WHERE FEEDBACK IS CUTOFF */minx=X_SLICES;maxx=0;maxnz=0;for(i=0;i<Z_SLICES;++i)179F - Huygens.c 180if(nz[i]>maxnz)maxnz=nz[i];for(j=0;j<nz[i];++j)if(minx>zcoordx[j][i])minx=zcoordx[j][i];if(maxx<zcoordx[j][i])maxx=zcoordx[j][i];})if(maxx>=X_SLICES)printf("PROGRAM HALTED --- NOT ENOUGH X_SLICES FOR TASK\n");printf("REQUIRED %i X_SLICES\n",maxx+1);return(0);)printf("MAXIMUM NUMBER OF CUTS PER LEVEL: %i\n",maxnz);printf("FEEDBACK STARTS AT TIME STEP %i AND ENDS AT %i\n",minx,maxx);/*^START COMPUTING THE STREET UP TO WHEN FEEDBACK STARTS*//* THIS IS JUST A NATURAL PERIODIC PROGRESSION OF THE OSCILATOR */for(t=1;(t<minx)&&(t<TIME_TOTAL);++t)if((t%25)==0)printf("%5i",t);/* CHECK FOR PHASE TRANSITIONS AND WRITE TO FILE */for(x=0;x<Z_SLICES;++x)/* TRANSITION WHEN GO FROM POSITIVE TO NEGATIVE PHASE *//* EITHER CROSSES FROM -0 TO +0 OR PI TO -PI */if((float)(phase[0][x]*phase[1][x])<(float)0.0)intens=(int)(A[0][x]/D[x]*(float)MAX_INTENSITY);if((float)(fabs((double)phase[0][x]))<(float)(HALFPI))y=write(handle0,(char *)&t,2);y=write(handle0,(char *)&x,2);y=write(handle0,(char *)&intens,2);)elsey=write(handlepi,(char *)&t,2);y=write(handlepi,(char *)&x,2);y=write(handlepi,(char *)&intens,2);)))180F - Huygens.c 181/* SHIFT THE PHASE DATA OVER FOR NEXT TIME STEP */for(i=0;i<Z_SLICES;++i)for(j=X_SLICES-1;j>0;--j)A[j)[i]=A(j-1)[i];phase[j][i]=phase[j-l][i];)/* UNTIL FEEDBACK STARTS, PHASE PROGRESSES LINEARLY *//* IF INCLINED IT FOLLOWS WILLIAMSON'S CRITERION */for(i=0;i<Z_SLICES;++i)phase[0][i]+=omega[i]*t_step*cos(PLATE_ANGLE*PI/180.0);if((float)phase[0][i]>(float)PI)phase[0](i)-=(float)(2.0*PI);)/*^COMPUTE THE STREET FROM WHEN FEEDBACK STARTS TO COMPLETE FEEDBACKfor(t=minx;(t<TIME_TOTAL)&&(t<maxx);++t)if((t%25)==0)printf("%5i",t);/* CHECK FOR PHASE TRANSITIONS AND WRITE TO FILE */for(x=0;x<Z_SLICES;++x)/* TRANSITION WHEN GO FROM POSITIVE TO NEGATIVE PHASE *//* EITHER CROSSES FROM -0 TO +0 OR PI TO -PI */if((float)(phase[0][x]*phase(1)[x])<(float)0.0)intens=(int)(A[0][x)/D[x)*(float)MAX_INTENSITY);if((float)(fabs((double)phase[0](xj))<(float)(HALFPI))y=write(handle0,(char *)&t,2);y=write(handle0,(char *)&x,2);y=write(handle0,(char *)&intens,2);)elsey=write(handlepi,(char *)&t,2);y=write(handlepi,(char *)&x,2);y=write(handlepi,(char *)&intens,2);)))/* SHIFT THE PHASE DATA OVER FOR NEXT TIME STEPfor(i=0;i<Z_SLICES;++i)for(j=X_SLICES-1;j>0;--j)181F - Huygens.c 182A[j][i]=A[j - l][i];phase[j][i]=phase[j-l][i];)/* SUM THE NEW PHASE INFORMATION FOR NEW WAVEFRONT */for(i=0;i<Z_SLICES;++i)imag=(double)0.0;real=(double)0.0;for(j=0;j<nz[i];++j)xn=zcoordx[j][i];zn=zcoordz[j][1];if(xn<=t)Aoverrrr=(double)A[xn][zn]*zcosrrr[j][i];tphase=(double)(phase[xn][zn]+omega[zn]*ztime[j][1]);real+=Aoverrrr*sin(tphase);imag+=Aoverrrr*cos(tphase);if(((double)real!=(double)0.0)&&((double)imag!=(double)0.0))/* THERE IS A BUG IN MICROSOFT C 5.1 OR AN ERROR IN THEDOCUMENTATION THE FOLLOWING CALCULATES arctan(real/imag)THE FUNCTION RETURNS AN ANGLE FROM -PI TO PI */phase[0][i]=(float)atan2(real,imag);/* LEAVE PHASE AS IS IF NO AMPLITUDE */)else/* FEEDBACK HAS NOT YET STARTED SO PROGRESSES LINEARLY */if(zn==i)phase[0][i]+=omega[i]*t_step*cos(PLATE_ANGLE*PI/180.0);if((float)phase[0][i]>(float)PI)phase[0][i]-=(float)(2.0*PI);}})A_temp[i]=(float)(sqrt(real*real+imag*imag));)/* RENORMALIZE THE STRENGTH TO ORIGINAL POWER OF ROD */sum_A=0.0;for(i=0;i<Z_SLICES;++i)sum_A+=(double)A_temp[i];A_factor=normal_A_sum/sum_A;for(i=0;i<Z_SLICES;++i)A[0][i]=(float)(A_factor*(double)A_temp[i]);/*^COMPUTE THE REST OF THE STREET WITH COMPLETE FEEDBACK182F - Huygens.c 183/* ENDS WHEN TIME STEPS ARE REACHED OR KEYBOARD IS HIT */for(t=maxx;(t<TIME_TOTAL)&&(!kbhit());++t)if((t%25)==0)printf("%5i",t);/* CHECK FOR PHASE TRANSITIONS AND WRITE TO FILE */for(x=0;x<Z_SLICES;++x)/* TRANSITION WHEN GO FROM POSITIVE TO NEGATIVE PHASE *//* EITHER CROSSES FROM -0 TO +0 OR PI TO -PI */if((float)(phase[0][x]*phase[1][x])<(float)0.0)intens=(int)(A[0][x]/D[x]*(float)MAX_INTENSITY);if((float)(fabs((double)phase[0][x]))<(float)(HALFPI))y=write(handle0,(char *)&t,2);y=write(handle0,(char *)&x,2);y=write(handle0,(char *)&intens,2);)elsey=write(handlepi,(char *)&t,2);y=write(handlepi,(char *)&x,2);y=write(handlepi,(char *)&intens,2);))}/* SHIFT THE PHASE DATA OVER FOR NEXT TIME STEP *//* SUM THE NEW PHASE INFORMATION FOR NEW WAVEFRONT */for(i=0;i<Z_SLICES;++i)for(j=X_SLICES-1;j>0;--j)A[j][i]=A[j-1][i];Phase[j][i]=Phase[j - 1][1];)/* SUM THE NEW PHASE INFORMATION FOR NEW WAVEFRONT */for(i=0;i<Z_SLICES;++i)imag=(double)0.0;real=(double)0.0;for(j=0;j<nz[i];++j)xn=zcoordx[j][i];zn=zcoordz[j][i];Aoverrrr=(double)A[xn][zn]*zcosrrr[j][i];tphase=(double)(phase[xn][zn]+omega[zn]*ztime[j][i]);real+=Aoverrrr*sin(tphase);imag+=Aoverrrr*cos(tphase);)183F - Huygens.c 184if(((double)real!=(double)0.0)&&((double)imag!=(double)0.0))/* THERE IS A BUG IN MICROSOFT C 5.1 OR AN ERROR IN THEDOCUMENTATION THE FOLLOWING CALCULATES arctan(real/imag)THE FUNCTION RETURNS AN ANGLE FROM -PI TO PI */phase[0][i]=(float)atan2(real,imag);/* LEAVE PHASE AS IS IF NO AMPLITUDE */A_temp[i]=(float)(sgrt(real*real+imag*imag));/* RENORMALIZE THE STRENGTH TO ORIGINAL POWER OF ROD */sum A=0.0;for(i=0;i<Z_SLICES;++i)sumA+=(double)A_temp[i];A_factor=normal_A_sum/sum_A;for(i=0;i<Z_SLICES;++i)A[0][i)=(float)(A_factor*(double)A_temp[i]);while(kbhit())dummy=getch();close(handle0);close(handlepi);184

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