Phase front analysis of vortex streetsMarcel Lefrançois and Boye Ahlborn Citation: Physics of Fluids (1994-present) 6, 2021 (1994); doi: 10.1063/1.868209 View online: http://dx.doi.org/10.1063/1.868209 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/6/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Von Kármán vortex streets on the sphere Phys. Fluids 21, 116603 (2009); 10.1063/1.3258066 Modification of a vortex street by a polymer additive Phys. Fluids 13, 867 (2001); 10.1063/1.1347962 A criterion for vortex street breakdown Phys. Fluids A 3, 588 (1991); 10.1063/1.858120 Interaction of Vortex Streets J. Appl. Phys. 35, 3458 (1964); 10.1063/1.1713250 Formation of Vortex Streets J. Appl. Phys. 24, 98 (1953); 10.1063/1.1721143

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Phase front analysis of vortex streets Marcel Lefrangois and Boye Ahlborn Department of Physics, The University of British Columbia, Vancouver V6T lZl, Canada

(Received 19 August 1993; accepted 14 February 1994)

The continuous formation and development of a laminar vortex street behind a circular cylinder of radius D in a flow of velocity U 00 has been modeled as a Huygens-type wave process, where the transverse velocity U and the vorticity w in the near wake oscillate at the vortex shedding frequency f. Starting ~rom the Biot-Savart law for fluids a phase front propagation integral is derived. This formalism is used to calculate for each point along the span the. phase of vortex shedding as a function of the phase of the previously shed vortex generation and the shedding frequency. The amplitude is determined by a simple renormalization c~1culation. In good a~ree~ent with experiments, the model predicts the propagation of spanwls~ phase perturbations. In~O

subsequent vortex generations for two-dimensional (2-D) flow geometnes and the cell formation In

three-dimensional flows around tapered cylinders.

I. INTRODUCTION

Laminar vortex streets are produced when cylindrical objects are immersed into a flow with Reynolds numbers of less than about 300. Initially it was assumed that these "Von Kanmln vortex streets" with their alternating vortices are two-dimensional flows. 1- 3 However, it is now clear that they are three dimensional at all but the lowest Reynolds numbers.4 Gaster5 (1969) proposed that the three­dimensional (3-D) wake of straight and tapered cylinders can be modeled using van der Pol oscillators. Only recently have theoretical predictions of the three-dimensional wake forma­tion patterns been advanced.6 These computational models require the input of the shedding frequency, I, the cylinder geometry, and a number of experimentally determined coef­ficients which are not directly observable in the wake. We present here a phase propagation model which can predict the cell structure of a newly formed wake. The input param­eters of our model are only the shedding frequency and the geometry of the object. The model is based on the Biot­Savart law, and it adds to some previously used ideas, the new concept of diffraction-type phase superposition.

A. Background

Early investigations of the vortex street revealed a double row of parallel vortices [as shown in Fig. l(a)], with a vortex shedding cycle of frequency, I, which is propor­tional to the flow speed, U 00, and inversely proportional to the diameter, D, of the shedding objece

SU.., I=D' (1)

The Strouhal number, S, is an empirical value which depends on the Reynolds number, R~= U ooD/v. For circular cylinders at high Reynolds numbers S = 0.22 and for the range 40<R <150 the value of S has been found empirically by (i

Roshk03,8 to be

4.5 "-0 "1"--,.,J- .;&.J - Re' (2)

Using the Strouhal equation for shedding frequency (1), and von Karman's stability analysis,! an adequate model of the two-dimensional vortex street can be formulated.9 However, the simple wake of parallel organized 2-D motion is rare or difficult to maintain.1o The initial fluid motion may be simple and parallel but it usually breaks up into smaller structures which are no longer parallel to the main flow. For example, in shedding from a straight 2-D cylinder, oblique shedding modes11,12 appear in the wake above4 R,,=65. A comprehen­sive model should describe both the parallel and oblique shedding modes and also be able to predict the three­dimensional structure of the wake of a gradually tapered cyl­inder. In both geometries the shedding frequency is locally determined by the local cylinder diameter, but the shedding pattern is globally determined by the coupling between adja­cent shedding elements.5

,l3 As a result, over short sections of the span, vortic~s shed coherently at the same frequency and angle. These shedding sections are bound by dislocations to form shedding cells.14

Shedding cells arise from a variation in vorticity along the span, causing the vortices to link to adjacent vortex gen­erations to preserve the continuity of vorticity.11 Vortex dis­locations are due to phase variations along the span of the cylinder, and displacements of vortex ends. IS Linkages form between the vortex ends, marking the boundaries between shedding cells. Obviously, any computational model must show the formation of these shedding cells with both the connections between the vortices and the dislocated sections.

Recent attempts have been made to model the wake as a set of van der Pol oscillators

A + €(A 2-1),4 +A =0, (3)

where A is the strength of the oscillator and € is the nonlinear growth rate parameter. Gaster13 was the first to propose that the wake of a cylinder could be modeled by a string of van der Pol oscillators positioned along its axis, but he made no attempt to derive Eq. (3) from first principles or from the Navier-Stokes equation. Each oscillator emits at the locally defined frequency, but coupling to adjacent oscillators yields a globally determined vortex structure. The state (phase and amplitude information) of the oscillator is convected down-

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(a)

(b)

(c)

FIG. 1. Vortex street in the lab reference frame with cylinder moving from left toright at U~. (a) Schematic top view; (b) side view produced by strip marked dye technique; (c) transverse velocity Uy and vorticity was a func­tion of position.

stre~m, its condition reflecting the phase of vortex shedding (a vortex is assumed to be shed at a maximum or minimum ill the oscillator's cycle).

Gaster's proposal was pursued by Noack et al. 6 in a computer simulation of vortex shedding from a tapered cyl­inder at low Reynolds numbers. In their mode, the oscillating componeJ?ts are planes or slices of fluid, close to the cylin­der, undergoing periodic motion. The slices of fluid oscillate at the locally determined frequency and are coupled laterally by viscous friction. This model jJroduced the correct quali­ties, breaking and linking of vortices, but the values for local frequency, oscillator strength, and number of shedding cells, did not agree with the experimental results obtained from a 180 mm tapered rod (3 mm diameter at one end, 1 mm at the other) at R ,,= 179 at the 3 mm diameter end. First, the model frequencies were higher than in the experiment. Second, the number of shedding cells predicted by the model was in the 100's while the experiment yielded only 6. A good match could be achieved by increasing the viscous coupling by more than a factor bf 300 until, as Noack et al. note, it is "surprisingly high and indicates other physical processes contribute to the coupling strength." A possible mechanism is wave-type coupling. One such model is based on a Ginzburg-Landau equation, as investigated by A11:Jarede and Monkewitzl6 (and recently extended to include streamwise

. 17 disturbances by Park and Redekopp )

aA a2A 2 - = ((Tr+ i(T;)A + (/Lr+i/Li) -:;:Ta -(lr+ili)iA/ A. at z

(4)

This equation for the oscillator strength, A, reduces to the van der Pol oscillator for a constant diameter cylinder at the onset of vortex shedding. The first term on the right-hand side describes the growth, the last term limits the amplitude, and the central term is a wave-like coupling. Albarede and

2022 Phys. Fluids, Vol. 6, No.6, June 1994

Monkewitz did not derive the Ginzburg-Landau equation from first principles. It is, in their words, "validated by ex­perimental data for all of its coefficients." There are a num­ber of coefficients (T,/L,f (including imaginary parts) which are not intuitively obvious but must be found from experi­ments. Unfortunately, these coefficients appear not to be unique since their values depend on the Reynolds number at which they are determined. The model is capable of describ­ing the key features of vortex shedding starting from rest: development of chevron patterns and oblique shedding. However, since the model requires the input of empirical coefficients (T,/L,f it is necessary to measure these parameters before extending the model to new flow geometries.

We have developed a model that can predict the structure of the vortex street, using only the Strouhal frequency, f, the flow velocity U 00, and the object geometry as input param­eters. Our model adds to some previously used ideas the concept of diffraction-type superposition. This principle of­fered itself as a physical explanation of the processes ill a vortex street when we noted that the structures in a wake have considerable similarity with optical diffraction patterns. These similarities are observed in laminar vortex shedding experiments.

B. Phase propagation in vortex streets

The similarity between the cell structure in the wake and optical diffraction patterns becomes apparent in flow visual­ization experiments. Consider, for instance, the wake of a circular cylinder, in the fluid reference frame (coordinates x' ,y' ,Z') as shown in Fig. l(b). The street was generated by dragging" a rod of D = 8 mm diameter at R e = 100 in water. The vortices are marked by dye (continuous dye line on the side away froin the camera and dashed dye line on the side facing the camera) as described by Lefran<;ois and Ahlborn.18

In this experiment the street was quickly established by using a trigger plate to fix the illitial shedding phase uniformly along the span. 19 Figure l(c) shows the idealized vorticity and transverse wake velocity along the x' axis. The wake itself moves at a slow velocity, U w with respect to the back­ground fluid. In a reference frame that moves with the wake at the velocity U w (coordinates x I ,y I ,z ') the magnitude of the vorticity w may be approximated by

w=Ae-i21Tx'/A=Ae-ikx' , (5)

where k=2n"/A. is the wave number and A. is the vortex spac­ing shown in Fig. 1. For convenience we associate the vortex centers with the location of maximum and minimum vortic­ity. In this reference frame, the wake appears as a stationary wave left in the fluid by the passing of the cylinder. By a simple transformation [x=x'+(Uoo-Uw)t, y'=y, and z' =z], the phenomenon is transferred into the reference frame of the cylinder (coordinates x,y,z). Since Uw is gen­erally small compared to U co, we replace U 00 - U w by U co'

Then the vorticity (5) as a function of space and time be­comes

(6)

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Noting that the vortex spacing is the distance the cylinder travels in one vortex shedding cycle

21T U"" "-=T=/' (7)

Eq. (6) can be rewritten to reveal that the wake in this refer­ence frame appears to be a traveling wave appearing behind the cylinder and traveling downstream (+ x direction) at U 00

w = A ei (27rf t - kx) • (8)

Similarly, the transverse velocity in the wake can be repre­sented by a function

U = U ei(27rft-kx+/lcp) y yO (9)

which is out of phase by /lrp= 1T/2 relative to the vorticity. The velocity and the vorticity wave fields are, of course, inseparable parts of the same phenomenon. This arises since the vorticity is the curl of the velocity field. Equation (6) indicates that the vorticity-velocity field travels in the x di­rection at the phase velocity U 00 •

When superposition of signals is possible, wave func­tions like Eq. (8) or Eq. (9) must by necessity lead to inter­ference and diffraction phenomena. This arises from the fact that a particular point in space may be reached by wave elements with different phases having traveled through dif­ferent distances from their various points of origin. It is therefore not surprising that the velocity-vorticity wave field in the wake is similar to electromagnetic diffraction patterns: The vortices look like the field of maxima and minima of a diffraction pattern, where the vortices are located at the maxima and the minima of the w(x) curve. The pattern ex­pands in the x direction with a phase velocity V= U",=21Tf/k as new maxima and minima are steadily added to the field. This addition happens in a manner where the phase of the vortex just being born behind the cylinder [shaded area in Fig. l(a)] is a smooth continuation of the phase of the previous vortex.

Both vortex fields and diffraction patterns can be de­scribed by phase fronts with wavelength and frequency, ex­hibiting constructive and destructive interference effects. We therefore suspected that the powerful and elegant tools of diffraction theory, Huygens wavelet analysis and the Fresnel-Kirchhoff superposition integral, could possibly be used to describe pattern formation in vortex streets. It turned out that such an analysis can indeed predict the temporal development of laminar vortex streets.

With these similarities in mind we have developed the model described in the next section.

c. Phase propagation model

The cornerstone of our phase propagation model is Huy­gens principle which sets the phase and amplitude of physi­cal parameters in a new volume element by superposition of information from nearby wave front elements of the next older vortex generation. We utilize Noack's idea6 that each point along the span acts as a simple oscillator with natural frequency described by the local Strouhal number. We also note that Albarede and Monkewitz's idea16 that adjacent

Phys. Fluids, Vol. 6, No.6, June 1994

points are coupled in a wave-like process is taken care of automatically if we use the superposition principle.

The phase propagation model is based on the following points and assumptions; which are summarized below and explained or derived later.

(1) The phase and magnitude of the transverse velocity Uix=O,z) of a fluid element that has just entered the wake at the rear of the cylinder is determined by induc­tion of the nearby vorticity elements. It can be calculated with the help of the Biot-Savart law, in which vorticity w oscillates periodically with the vortex shedding fre­quency f.

(2) The local frequency f depends only on the local diam­eter D, the velocity U 00, and the Strouhal number S.

(3) The physical quantities which induce the velocity U v at the rear of the cylinder travel at the phase velocity V = 21Tfl k= U 00 relative to the fluid.

(4) The wavelength "- of a vortex street pattern is indepen­dent of the alignment of the wave fronts. This implies that the phase velocity is also independent of the direc­tion of travel.

(5) Retarded times must be used when summing up the con­tributions of different vorticity elements.

(6) The phase of Uy of a fluid element acquired by this process does· not change significantly as the fluid ele­ment convects downstream through the near wake re­gion.

To describe the expanding vortex field in a wake we will first derive a fluid diffraction equation to determine the am­plitude and phase of newly "enrolled" fluid elements in the near wake. Then we will discuss all the pertinent parameters and describe a recursive algorithm by which the phases and amplitudes are continuously calculated to determine the evo­lution of the vortex shedding pattern. Three examples are given at the end to test this formalism.

II. FLUID DIFFRACTION EQUATION·

The diffraction theory, which provides the basic con­cepts of this work is well documented in optics textbooks (see, for instance, Born and Wolf2o). However, for conve­nience, some details of the optical theory are summarized before we derive an equivalent relation for wake flows. Fig­ure 2(a) shows a series of spherical wave fronts emanating from a line of wavelet centers. According to Huygens prin­ciple, the common tangent of these wave fronts forms the new wave front. Each point on the new wave front can again be considered a wavelet source, as the primary front was, forming a unique relation between all wave fronts in the expanding wave field. Thus all of the wave fronts in the field can be described by only one of the wave fronts (even if this wave front was found from some other wave front in the field). Huygens principle therefore allows us to calculate the amplitude and phase of a point Xo on any secondary wave front from the contributions from elementary waves in any source window we choose. This is indicated in Fig. 2(b). The calculation of the new wave amplitude is accomplished by using the Fresnel-Kirchhoff diffraction integraeO,21

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(a)

(b)

Xs

_Secondary - Source

,_Secondary / Source

Aperture Plane ~ (Source Window)

FIG. 2. (a) mustration of Huygens principle: Tangents of secondary wave fronts form a new wave front. (b) Illustration of Fresnel-Kirchhoff's inte­gration: Secondary wavelets in the source window are superimposed at ob­servation point xo.

_ iff e i(27T!t-k.X) (cos /3+ cos /3')

E-y;A IX'IIXI 2 dX, (10)

which relates the electric field E at a point Xo to the wavelet contributions from the aperture window. The angle between the aperture plane normal and X is /3, the angle between the forward direction, X', of the wave and the aperture plane is /3', and k is the wave vector (k=2n/A, where A is the wave­length). In Fig. 2(b) the wavelet sources in the aperture plane are assumed to be due to a single primary source. If more than one primary source exists, the integration can be ex­tended over all space so as to linearly superpose their con­tributions.

The idea of using superposition to calculate vortex streets is not new. Using potential flow theory in two dimen­sions, von Karman1

,9 and Schaefer and Eskinazi22 demon­strated that the vortex street could be modeled as a superpo­sition of vortices. New in our model is the incorporation of an oscillating vorticity, which must lead to a fluid diffraction equation useful for describing three-dimensional vortex streets. Our model begins with the Biot~Savart law for fluids23- 26

- 1 J J' J (Xo~ X) Xw(X) U(Xo)--4'1T !X

o-X!3 dX (11)

which relates the velocity field U(xo) to the vorticity field w(x). The integration is extended over the volume contain­ing the elements of vorticity. The Biot-Savart law deals with sources of vorticity, which may be arbitrarily distributed in

2024 Phys. Fluids, Vol. 6, No.6, June 1994

(a)

(b)

<1>=0 <1>=1t <1>=21t <1>=31t

Source Window

Vortex Line

x ~

FIG. 3. Superposition of vorticity information. (a) Illustration of the Biot­Savar! law, where velocity at X 0 is calculated. (b) Vortex lines in the wake, and source window used for calculation of Uy(Xo}.

space, similar to the radiation phase fronts on which Huy­gens secondary wave centers are arranged. Similar to the Fresnel~Kirchhoff integral, the integration of the Biot~

Savart law implies that the elementary contributions from the localized vorticity elements superimpose linearly to give a resultant velocity at Xo, as indicated in Fig. 3(a).

In analogy with optical diffraction we will determine the amplitude and phase of the transverse velocity in the near wake as induced by the nearby vortices. Due to the cyclic nature of the wake field, one immediately sees the similarity between optical radiation phase planes (locations of the phases 0, 2'1T, 4'1T, ... , of the wave field) and the location of the vortex axes (locations of phase 0, 2'1T, 4'1T, ... , of the vorticity function). Similar to the Fresnel~Kirchhoff formalism, the contribution from the nearby vortices will be summarized in a "source window" located a distance Xs behind the cylinder [Fig. 3(b)]. This yields the same result as calculating contri­butions from the vortex lines as long as the phases of the secondary source elements in the window are correctly as­signed.

The Biot-Savart law converts into a diffraction equation when the time varying vorticity, w(x) Eq. (8), is inserted. This causes a certain problem: The Biot-Savart law does not specify a travel time for information emanating from a source at X. It is improbable that the contributioI1s spread with infinite velocity. They must propagate at some finite velocity, V=2'1TJlk. This velocity characterizes the propaga­tion of "information" from all the source points of a new

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point where by superposition of effects the velocity U is induced. In analogy to the Huygens construction, the flow of information from the source to a new point is considered a "wave" process and, hence, V is called the "phase velocity." More will be said about the magnitude of V later.

By Eq. (8) the source element at a fixed x coordinate measured in the object reference frame has an oscillating vorticity

(12)

The induced velocity is a superposition of information which arrives simultaneously at Xo traveling at V, hence, the source's vorticity must be evaluated at the retarded time, t *

IXo-XI t*=t IVI (13)

This assures that the contributions from various source points are added with their correct relative phases. The phase velocity of a periodic vibration implies a wavelength, A= VI f, or, alternately, a wave number, k=2'TrfIV. Introducing the oscillating vorticity (12) into the Biot-Savart law (11), yields an expression for the induced velocity

U(V)= -1 f r f (Xo-X)xws(X) ,.<1(J 4'Tr. IXo-XI 3

X ei[21Tf{X)t-k'lXo-X) + <l>s(X)] dX. (14)

The initial phase factor <1>, has been introduced to allow for position dependent phase variations in the oscillating vortic­ity field. Equation (14) yields all the induced velocity com­ponents. The hallmark of the wake vortex field is the oscil­lating transverse velocity, Uy • It is expected that the value of Uy in the center of the wake (x-z plane) is indicative of the flow velocities and vorticity observed in the entire flow field since the street is phase correlated. In order to simplify the calculations, only the components of vorticity in the x-z plane will be considered. Under this condition the local vor­ticity vector, ?;>= w) + w);, and the position vector, Xo-X=(xo-x)i+(zo-z)k, can be broken into components such that

U ( )- -1 f f J" wx(zo-z)-wz(xo-x) y xo,Yo,zo - 4'Tr IXo-XI3

Xe i [21Tf{X)t-k'(Xo-XH<l>s(X)] dX. (15)

The vorticity of strength A (X) can also be broken into com­ponents

wx=A(X)cos a and wz=A(X)sin a, (16)

where a is the angle of the vorticity vector to the x axis as shown in Fig. 3(a). Similarly, {3 is defined as the angle which the position vector Xo-X makes with the x axis

(17)

Equation (15) can therefore be transformed into the fluid diffraction equation

Phys. Fluids, Vol. 6, No.6, June 1994

Xe i [21T!lX)t-k'(Xo- Xl+<l>s(Xl] dX. (18)

The integration is extended over the entire space which con­tains sources of vorticity but due to the dropoff with l/(xo-.x)2 only the nearby sources will be considered. Simi­lar to Fresnel integral calculations we assign a calculation window (the source window), Fig. 3(b), at some distance Xs

to carry out the calculations. Equation (18) allows us to cal­culate the magnitude and phase of the transverse velocity of each fluid volume element that enters the wake. In addition it assigns a phase <1>0 that is correlated to the phase structure downstream. This phase provides an identity for the volume element which wiII give it a fixed position relative to the older and younger vortices. The phase marked blob of fluid subsequently drifts away from the vortex shedding cylinder becoming a part of the wake structure.

The model assumes that the phase of the vorticity of this fluid blob does not change quickly so that it is still approxi­mately equal to <1>0 as it drifts through the calculation win­dow. The field moves relative to this window at the speed V = 2 'Tr f I k in a direction orthogonal to U y and w. Equation (18) describes a wave-like propagation of the transverse ve­locity field U y driven by an oscillating vorticity field. The oscillating elements of vorticity emit information about the flow field which induce the transverse velocity of the emerg­ing wake continuously just as the Huygens secondary wave­lets superimpose to produce the diffraction pattern of spread­ing light wave fronts. Equation (18) looks very similar to the Fresnel-Kirchhoff diffraction equation (10). The major dif­ference between Eq. (18) and Eq. (10) is the geometric fac­tor, sin(a-{3), which accounts for the vector nature of the vorticity source element in the fluid diffraction equation.

Using the fluid diffraction equation (18), the velocity close to the cylinder, U y , can be deduced from the distribu­tion of vorticity in the fluid downstream. The phase im­printed onto each volume element along with the amplitude may be used to predict the subsequent phases of the younger parts of the street.

III. THE MODEL PARAMETERS

In order to use the fluid diffraction equation for a par­ticular experiment one must know the source parameters: Frequency f, phase velocity, V, window position x s ' wave­length A, oscillation strength A, and the start-up phase <1>$' These parameters are now discussed individually.

A. Frequency f

The characteristic frequency of the sources, fCz) is the shedding frequency given by Eqs. (1) and (2). Implicitly, we assume the Strouhal frequency is set by the local geometry of the cylinder and that the observed wave field structure may be modified by the interaction of the phase fronts from different portions of the shedding cylinder. (Gaster13 has ob­served this behavior in tapered cylinder experiments.) The model does not preclude the use of a modified frequency.

B. Wavelength A.

The wavelength of a wave field is the closest distance in space between two locations of phase difference 2'Tr as de-

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scribed by Eq. (7) and shown in Fig. 3(b). In many experi­ments the streamwise wavelength X. x = U 001 Ix is measured, namely, the distance that the cylinder moves during one vor­tex shedding cycle. When the shedding occurs at an oblique angle () the downstream wavelength increases '

x. x. =-­x cos()' (19)

One important result of Williamson's4 research is that the Strouhal frequency decreases in oblique shedding, namely,

Sx=S cos 0 which implies Ix= I cos (). (20)

Combining this empirical result with (19) yields

Uoo Uoo Uoo x.=x. cos ()=- cos ()=--- cos ()=-. (21) x Ix I cos () I

Equation (21) implies that the spacing X. (or wave number k=27T1X.) measured perpendicular to the vortex lines is not a function of shedding angle e but is fixed at DIS [Eq. (1)].

C. Phase velocity V

The phase velocity of a wave phenomenon is in general found as the product of wavelength and frequency. For vor­tices shed at the angle e. These values are

A X.x=--() and Ix= I cos B.

cos

The phase velocity is therefore

(22)

(23)

The shedding angle drops out. It was shown above that the spacing X. is independent of the shedding angle and the fre­quency I is the local Strouhal frequency. Therefore, the phase velocity of the vorticity-velocity field is V = U 00 •

It should be acknowledged that this result is not intu­itively obvious. One could argue that the velocity field deter­mined by the Biot-Savart induction equation must be cre­ated instantaneously, so that the characteristic expansion velocity ought to be 00. This does not make physical sense since no information travels faster than the speed of light. If the velocity field of a vorticity source was created by pres­sure and flow fields it could spread at the speed of sound which is still very large compared to U 00' If the transverse velocity field was created by viscous shear forces it would expand at a diffusion speed of -[vlt, which is typically much smaller than U", at Reynolds numbers around 100.

The velocity calculated in Eq. (23) concurs with a pre­vious result where we derived the phase velocity V cp for azi­muthal waves on eddies as a function of the local swirl ve­locity U cp' finding V cp = ± U cp (Ahlborn et at. 27). This result was derived from a very simple physical model which only assumes that a segment of fluid of velocity U cp is forced onto a curved path. Such a situation is also found in the meander­ing wake river of the vortex street. Assumption (23) also agrees with Kim and Hussain's assertion that the velocity, vorticity, and pressure fluctuations in a turbulent field propa­gate at the mean velocity of the flow.28 Now since X. is inde-

2026 Phys. Fluids, Vol. 6, No.6, June 1994

pendent of the inclination of the vortex field by Eq. (21) and 1= S U 001 D is a constant for given values of D, U 00 ,S the phase velocity is independent of the direction. If a random or systematic velocity field is superimposed onto the fluid in which the vortex street is formed, frequency and wavelength will be modified like in the acoustical Doppler effect.

D. Location of the oscillating vorticity source

In optical diffraction theory the location of the source window is important because it affects the relative phase of adjacent points. In the general formulation of the diffraction equation (18) the location Xs of the source window is still arbitrary. However, since the phase <I> of U v of a blob of fluid is set when it enters the wake, and since we assume that <P changes only very slowly the actual position of the source window is not very important. For simplicity, all of the sources are considered to be in one plane in the wake. By definition (6) the vorticity is known only as a function of x and z, so that the source distribution can be collapsed to a single line in x-z plane, a distance Xs behind the cylinder. Then the area integration reduces to a line integral. This line is located far enough downstream for the eddies, and hence the oscillators, to be strongly developed but still close enough to be the major influence on the vortex shedding process.

The choice of location for this line is made as follows: The sources are in 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 are developing and gaining strength, and are moving with the cylinder. The vortices are much more strongly developed towards the rear of the recirculation re­gion, which, in general, extends a few diameters downstream22

,29 (about 3D). Therefore, the sources must be positioned far enough from the cylinder to allow the vorticity sources to gain enough strength to be of consequence in the vortex shedding process and near enough to interfere effec­tively at the cylinder.

For reasons discussed later we assume x s = 1. OD. At this location the vortices are about fully formed. Their amplitude 11 has reached its full value and the inverse square law of distance dependence (XO-X)2 in the denominator of Eq. (18) has not yet reduced their influence on the new transverse velocity too much. If one vortex happens to be found at xo then the next older vortex with opposite vorticity is about X./2=2.3D downstream, so that its contribution to the trans­verse velocity is about a factor 5 smaller and may hence be neglected.

E. Oscillator strength A and phase <l>s

The frequency and wavelength and wave speed are fixed by the vortex shedding process, but the oscillator strength, A, and its phase, <Ps ' could have arbitrary initial values.

The initial strength of the oscillators is assumed to be the circulation of the vortices, which in turn are proportional to the diameter of the cylinder. This is reasonable since the strength of the vorticity is proportional to cylinder diameter, giving a circulation fIXD. A similar assumption is made by

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Noack et al. 6 The strengths and phases of the· vorticity sources, at time t+ Ilt are deduced from the transverse ve­locity Uy calculated at the cylinder at time t. The transverse velocity contains the phase information which convects into the source plane in a time Ilt to determine the phase of the new vorticity, hence, there is a "growth function," G, which describes the transformation of U y into vorticity

It)(X,t+ 6.t) =G[Uy{Xo ,t)]. (24)

Transverse velocity is 90° out of phase with the vorticity and the growth function is a multiplying factor, C r' which trans­forms the velocity into a vorticity value, namely,

A(X,t+ 6.t)exp[i<Ps(X,t+ 6.t)] = iCrUy(Xo,t). (25)

The multiplying factor is chosen in a manner which main­tains the vorticity strength at the cylinder. Therefore, at a fixed distance behind the cylinder the total source strength is assumed to be the same as the initial strength

JoLA(X,z,t+ Ilt)dz= J: D(zo)dzo= cortst, (26)

where L is the length of the cylinder. Therefore, the multi­plying factor of Eq. (25) can be rewritten as

f~D(zo)dzo (27)

The renormalization equation (25) yields the new vorticity strength for a fluid element as it passes through the source plane. Its phase is already known because we assume that it has not changed since the fluid element was "enrolled" into the wake. The new vorticity strength and phase are the input parameters for the secondary phase front to be used for the calculation at the next time step.

F. Retarded times

In all wave-like superposition problems retarded times must be used in order to add contributions that arrive at the same time from different source elements.

As already discussed, the transverse velocity Uy is in­duced by phase information arriving simultaneously at the calculation point "0. Equation (18) has been written with the factork·(xo-x) to account for the phase difference between the vorticity source at (x,z) and the calculation point at the cylinder. However, for coding the model the fluid diffraction equation is recast using distances and velocities rather than with retarded times, t * , using Eq. (13)

~x;+z2 k(Xo-X)=21TI(t-t*)=21TI V' (28)

For infinitely large phase velocities, the wavelength becomes infinite and no wave phenomena are possible. Retarded times allow us to superimpose the traveling phase fronts using the information from the vorticity sources at an earlier time. The final step in the above equation assumes that the oscillating vorticity elements are located a distance x s behind the cylin­der and are a distance z above the calculation point. It is als'o assumed there is no significant background velocity: If so,

Phys. Fluids, Vol. 6, No.6, June 1994

the phase velocity, V, has to be modified similarly as in the acoustical Doppler effect. Detailed discussions of such ef­fects go beyond the framework of this study.

All the parameters; I, V, x s , A, <Po must be combined to apply the model to a given experiment.

IV. CALCULATIONS OF CONTINUOUS WAVE FRONTS

The fluid diffraction equation. (18) becomes a wave propagation equation when all the parameters described in the previous section are substituted. We use the phase veloc­ity V= AI rather than wave numbers. The negative sign di­rects the wave towards the cylinder so that the forward phase progression of the wake is maintained. This yields

1 J A(z)sin(a-f3) {.[ Uy(zo,t) =-4 OJ 2 exp l 21TI(z)

1T x;+z

( ~) ]} X t+ V + <PsCz,t*) dz. (29)

We note that the model parameters are all functions of Ueo , D, and SeRe) so that no measurements are needed be­fore this model can be applied to a new experiment. We do, however, need the angle of the wave fronts (vortices), a, and the angle of the placement vector, {3, defined in Fig. 3(a). The angle, {3, of the placement vector, xo-xs to the forward mo­tion of the cylinder (U 00£) is given by

zo-Z tan {3=--.

Xs (30)

The vortex line angle a is difficult to solve for as the com­puter model progresses. The angle of the wave fronts could be calculated as the street emerges but this goes beyond the work presented here. For simplicity, a was set equal to the initial shedding angle (for parallel shedding a=900).

With the help of Eq. (29), the progression of the shed­ding phase can be calculated as follows: The initial informa­tion for the problem is the cylinder geometry (inclined, ta­pered, D), R e , U co, v, and the initial phase distribution, <P s(z). From the initial information the shedding frequericy, t, can be found by Eq. (1). Initially, there is insufficient vortex strength or phase information to begin the calculation using the phase coupling equation. Therefore, we assume a start-up value <PsCz).

To apply the wake propagation equation (29) the vortic­ity strength and phase at each time is calculated in tWo steps: First, the earlier phase and strength values, at the source plane, are superimposed using the diffraction equation (29) to give Uy with magnitude I Uyl and phase <Pl at the cylinder. Second, the transverse velocity information is convected back towards the source plane while being transformed into a new vorticity strength with the phase <Pl' The transformation of Uy(z,t) toA and <Ps at time t+xsIUeo is calculated by the renormalization equatiori (25). These new values of A and <Ps are then used in the next mesh point of the growing vortex street.

In the phase propagation model the transverse velocity oscillates in an approximately sinusoidal fashion. Lines of constant phase of the transverse velocity form a phase front which moves away from the cylinder. In parallel shedding,

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the phase fronts remain parallel to the cylinder and the wave front normals, or rays, travel perpendicular to the fronts. The vortices are imbedded in the moving phase fronts; hence, a vortex can be identified with a specific phase value. For com­putational simplicity, it is assumed that a vortex is shed on one side of the cylinder at $=0 and on the other at $=7T. The phase fronts may be thought to be localized at the posi­tion of vortex centers. They travel in the direction of their normals, similar to wave fronts in optics. In analogy to optics we assume that the wave influence flows along the trajecto­ries or rays of the vortex field.

The principle of phase front formation and propagation can be used to explain many structures seen in laminar vor­tex streets. In order to make full use of the phase front inte­gral (29) a computer program was developed3o to analyze the examples of oblique shedding and tapered cylinders.

v. PREDICTION OF SHEDDING CELLS OF TAPERED CYLINDERS

The anticipated shedding pattern of a tapered cylinder is formed by a continuous frequency variation due to the de­pendence of / on D shown by Eq. (1). This simple prediction is only half true. Experimental evidence from Gaster13 and Noack et at. 6 suggest that the frequency is locally deter­mined, but the observed shedding pattern is governed by the coupling between shedding sections to form stable shedding cells along short segments of the span at different frequen­cies. In our model the shedding cells of the tapered cylinder take the form of a stable diffraction pattern at the cylinder created by the interference of vorticity sources of different frequencies.

The predictive power of our model can be demonstrated using the results from Noack et al. 'SD experiments. In their experiment. a tapered cylinder of length 180 mm ranging from 3 mm diameter at one end to 1 mm diameter at the other in a wind tunnel at v= 14.9X 10-6 m2/s and U ",,=0.89 mis produced six distinct shedding cells. Phase front calcu­lations were carried out with these conditions and with the vorticity sources positioned at x. = 1 mm downstream and the shedding frequency at the ends of the cylinder fixed at /=0 Hz. The end conditions are those used by Noack et al. in their model and are reasonable since the experiment was car­ried out in a wind tunnel with the cylinder terminating at the walls inside the boundary layer where little or no vortex shedding should occur. The pattern calculated from our model is shown in Fig. 4. It is evident that the model pro­duces three shedding cells. In Noack et at. 's computer pre­diction, a random start-up was assumed. This had little con­sequence on their work since their focus was the stable shedding condition. In the work here, initial randomness was not introduced, instead a constant phase of $=00 was used to produce a start-up vortex parallel to the cylinder. The computed shedding patterns developed within about 350 mm from the start of the run and the entire run lasted for about 6000 mm. Figure 4 shows a section of this pattern at a dis­tance x =720 to 960 mm from start-up.

These results are an improvement over the van der Pol oscillator model used by Noack et at. which predicted hun­dreds of shedding cells. The discrepancy between the diffrac-

2028 Phys. Fluids, Vol. 6, No.6, June 1994

3mm

Imm I~

240mm

FIG. 4. Simulation of Noack et al.'s tapered cylinder experiment (Ref. 6). The vortices shedding from one side of the cylinder are dashed, from the other are solid, shown for a distance of x=720 mm after start-up.

tion model (three cells), used here, and the experiment (six cells) is most likely a result of improperly defined end con­ditions at the cylinder.

The results of placing the vorticity source plane at vari­ous locations in the wake was also investigated, the results of which are shown in Fig. 5 for curves for xs=0.2D, 0.6D, 1.0D, and l.4D. The closer the source plane is to the cyl­inder, the more cells are observed, but the less well defined they are. As the plane moves very close, the shedding fre­quency curve seems to approach the local shedding fre­quency predicted by Roshko [Eq. (2)]. At distances farther out, fewer cells are observed but the transition between them is sharper. Therefore, the position of the source plane is bounded as evidenced by the number of cells it produces. The placement of the calculation window can be more fully investigated in future work. In particular one might consider varying x s as a function of z. However, since the calculation is not overly sensitive to the streamwise position of the source window, x s can be chosen to produce a reasonable compromise between number of cells and the sharpness of transition between the cells. It is our assertion that the model will produce satisfactory results using x s = 1. OD.

130r------.------.------,~----,-----_,

soL 0.0 0.2

~l _____ ~ __ _

Q4 o~ o~ 1~

3mmend z/L 1 mmend

FIG. 5. Shedding frequency versus position along tapered cylinder for vari­ous placements, x" of the source window, for Noack et al.'s experiment (Ref. 6).

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28

FIG. 6. Side view of vortex pattern produced by a tapered cylinder (8 mm diameter at the top of the photo, 2 mm diameter at the bottom). Cylinder moves from left to right. A simulation of the experiment is shown below the photo.

VI. COMPARISON TO START-UP EXPERIMENTAL RESULTS

Having demonstrated the ability of our model to predict shedding cells and having chosen the final parameter in the diffraction integral (18) as xs=D, the model can be further tested by checking its predictions against other experiments. For this, we compared the model results against experimental shedding patterns produced in the start-up phase of vortex shedding from tapered cylinders (before cells are evident) and against the shedding patterns produced in oblique shed­ding.

In our first experiment, a tapered rod 410 mm long, rang­ing from 8 mm diameter at the top to 2 mm at the bottom was pulled through a tank of water. It produced results as shown in Fig. 6. The cylinder was towed from left to right at Re=112 at the 8 mm diameter end. The dark bar seen on the left is the support structure for the trigger plate, and the cylinder is still partly seen at the right edge. The vortices were marked by a striped dye technique so that vortices shedding on the side of the cylinder away from the camera appear as solid lines while those that shed on the side facing the camera are dashed. This was achieved by painting the cylinder solidly on one side and dashed on the other, then allowing the dye to dry before inserting the tapered cylinder in the water just prior to a run.

Control of the start-up conditions were essential for meaningful comparisons between the experiments and the computer simulations. The initial experimental conditions were set by employing trigger plates18•19 to force the first vortex to shed parallel to the cylinder. In the computer simu­lation this corresponds to a starting condition of <1'>=const along the cylinder (<1'>=0° was chosen). The end conditions

Phys. Fluids, Vol. 6, No.6, June 1994

used were f=O Hz at the bottom of the tank and the method of images31 was used at the top surface of· the water. The source plane was placed at xs=2 mm behind the cylinder.

The computed output is shown in the lower portion of Fig. 6. The vortex lines are dashed coming off one side of the tapered cylinder and more solid coming off the other side, similar to their appearance in the photo. In the lower portions of both the photo and the simulation no vortices shed since the Reynolds number is not high enough. From the simula­tion, each vortex sheds more obliquely than the previously shed vortex until the inclination is too great causing disloca­tions and the joining of vortex ends. This is also seen in the experiment with the breakdown occurring in approximately the same location as predicted in the simulation.

In the above simulation the vortex angle was taken to be fixed at £1'=90° (parallel to the cylinder). This was done for ease in coding yet it is obvious that a does not remain fixed. A better prediction may be obtained by computing the vortex shedding angle as the simulation progresses to redefine a. However, reasonable results have been obtained for the ta­pered cylinder with a constant.

It is not always possible to compute the shedding pattern accurately with a=90° especially in regards to the proper shedding frequency. Such is the case in oblique vortex shed­ding as reported by Williamson32 and othersP·12 Iri this con­figuration the vortices shed at an angle () to the cylinder (a to the x axis). In phase front terms, this mode is a wave front inclined at an angle £1'=90° - () to the original flow direction U 00 (x axis) and it has the phase distribution at the cylinder

(31)

This distribution implies a fixed wavelength and a phase front set at an angle e in the starting position.

We produced an oblique starting mode experimentally by mounting a trigger plate at ()= - 32°. The edge of the trigger plate is emphasized by a white broken line in the photo of Fig. 7. In this run, Re'-'96.3, L/D=45.6, and cylin­der diameter D =8 mm. As can be seen in the experiment the oblique shedding turns into parallel shedding near the sur­face. The parallel shedding spreads at an angle of 12° which is similar to that predicted in the calculations (bottom of Fig. 7).

When comparing numerical results such as Fig. 7 to the actual side-on photo, one must bear in mind that the vortex positions of the calculation are shown immediately after their production. Any subsequent motion of the vortex field due to drift and internal instabilities and is not considered in our model. From the simulation, it is evident that the vortices are formed at a 32° angle to the cylinder, however, this shedding mode is transformed into a parallel mode by the influence of the end conditions. A reflected condition forces the vortices to meet the boundary orthogonally. This is the seed which grows into the parallel shedding "cell." In tow tank experi­ments this boundary condition is achieved with clean water surface.33 In this simulation, the same end conditions were used as in the tapered cylinder experiments ([=0 at the bot­tom of the tank and the method of images at the water sur­face). An axial velocity of

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Ie

572mm

FIG. 7. Oblique shedding at 32°. The edge of the inclined trigger plate is dashed in white.

Uz=U", cos 0 sin 0=(14.14 mm/s)cos 32° sin 32°

=6.35 mmls (32)

was also introduced due to the vertical component of the oncoming flow paraIIel to the vortices as shown in Fig. 8. This axial velocity is associated with oblique shedding and has been studied experimentally by Hammache and Gharib.34

We were able to measure this axial velocity by tracking the dye strips from one photo to the next. By this method we found Uz =2.7±O.1 mmls, and indications are that the veloc­ity close to the cylinder is much higher (at times reaching approximately 1 crn/s). Unlike in optics, the phase fronts in the fluid are somewhat attached to the fluid elements, hence, this vertical velocity translates the wave pattern upward without changing the shedding angle. The consequence of the upward velocity is a squeezing of the structure at the top surface of the photo: The section of parallel shedding near

Vortex: line or wave front

FIG. 8. Vertical velocity component.

2030 Phys. Fluids, Vol. 6, No.6, June 1994

the surface grows at a slower rate than it would otherwise. When axial flow of Uz=6.35 mrn/s is included the growth angle was found to be 13 0, without the vertical velocity the predicted angle was 23°.

The above results confirm the applicability of this wave front model of vortex streets. We therefore feel justified in using the model to qualitatively describe the development of oblique shedding from a paraIIel mode. This requires a de­tailed knowledge of the end conditions.35 For example, in steady wake production, experimentalists have observed a variation in shedding frequency coupled with, and possibly due to, a spanwise flow close to the ends of the cylinder.14,36

The vorticity sources at the ends of the cylinder oscillate at a lower frequency than those at the center of the span. Since the phase velocity U 00 is the same along the span, the stream­wise wavelength near the end must be larger. The wave fronts attain a concave curvature with the ends bent towards the bar, explaining qualitatively the chevron shaped topology that has been observed.

VII. CONCLUSIONS

We have developed a phase propagation integral, Eq. (29), from the Biot-Savart law for oscillating vorticity sources. The near wake of a cylinder. is considered as an oscillating vorticity field where this model can be applied. The frequency of this field is the Strouhal frequency, f = S U ",I D, the phase velocity is the free stream velocity, V = U 00 and the wavelength of the phase front field is fixed, A=DIS. In oblique shedding the streamwise wavelength Ax is larger and the local shedding frequency is smaIIer than f. The integral equation (29) yields the phase of the newly en­roled fluid elements as they enter the wake behind a vortex shedding object. The amplitude can be calculated by a simple renormalization procedure, Eq. (25).

We have assumed that the vorticity in a source region in the near wake of the object varies as w=woei271Jt and that this oscillation becomes a wave function w=woXexp [i21Tf(t- Uoox)] when transferred into the fluid reference frame that has the relative velocity U 00 to the object frame. By this transformation U 00 becomes the phase velocity of the wave. It was also noted that even for oblique shedding the phase velocity remains equal to U 00' .

The fluid diffraction equation (18) has been used to pre­dict the phase structure of the continuously expanding vortex street. The model has successfuIIy predicted the inclination and onset of vortex dislocations in the wake of tapered cyl­inders and has demonstrated the ability to produce shedding cells in accord with experimental results. It is evident from this work that the wave front coupling modifies the observed local frequency from the Strouhal frequency. We have also shown that the phase front integral can produce the shedding pattern observed in oblique shedding where the initial phase distribution, <f>(z), has been arbitrarily set.

In each case the treatment was simple with easily iden­tified input parameters making this a powerful tool for study­ing local influences in fluid flows through wave coupling and superposition. The model was applied to experimental situa­tions with a small spanwise flow component (oblique shed­ding). In principle, it could be extended to cover fields with

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additional background flow components. Then one would have to use a direction dependent phase velocity analogous to the formulation of the acoustical doppler shift with a mov­ing source. Also it should be investigated how source win­dows placed at different lateral y positions would affect the results so that one could analyze the wakes of multiple cy­lindrical structures.

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