flow separation behind ellipses at reynolds numbers less than 10

Applied Mathematical Modelling 33 (2009) 1633–1643

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Applied Mathematical Modelling

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Flow separation behind ellipses at Reynolds numbers less than 10

David Stack a,*, Hector R. Bravo b

a University Information Technology Services, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USAb College of Engineering and Applied Science, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 10 April 2007Received in revised form 9 February 2008Accepted 21 February 2008Available online 13 March 2008

Keywords:Laminar flowSeparationCellular automataCylinderEllipsePlate

S0307-904X/$ - see front matter � 2008 Elsevier Indoi:10.1016/j.apm.2008.02.016

* Corresponding author. Tel.: +1 414 229 5371; faE-mail address: [email protected] (D. Stack).

Flow separation behind two-dimensional ellipses with aspect ratios ranging from 0, a flatplate, to 1, a circular cylinder, were investigated for Reynolds numbers less than 10 usingboth a cellular automata model and a commercial computational fluid dynamics softwareprogram. The relationship between the critical aspect ratio for flow separation and Rey-nolds number was determined to be linear for Reynolds numbers greater than one. Atslower velocities, the critical aspect ratio decreases more quickly as the Reynolds numberapproaches zero. The critical Reynolds numbers estimated for flow separation behind a flatplate and circular cylinder agree with extrapolations from experimental observations. Fluc-tuations in the values of the stream function for laminar flow behind the ellipses werefound at combinations of Reynolds number and aspect ratio near the critical values forseparation.

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1. Introduction

For Reynolds numbers between 1 and 10, smooth flows around circular cylinders are likely to remain attached whereasflows around flat plates oriented normal to the flow are separated and exhibit recirculating downstream vortices. It is there-fore reasonable to expect that flows around nearly circular ellipses are likely to remain attached, whereas flows around nar-rower ellipses will exhibit separation and recirculating vortices. By implication, for a given Reynolds number, there should bea critical aspect ratio, defined as the length of the ellipse axis aligned parallel to the mean flow divided by the length of thecrosswise axis, at which flow separation occurs. The critical aspect ratio is expected to vary as a function of Reynolds number.Similarly, for an ellipse of given aspect ratio there should be a critical Reynolds number for the onset of flow separation.

These investigations examined the onset of flow separation behind two-dimensional ellipses with aspect ratios rangingfrom 0, a flat plate, to 1, a circular cylinder at Reynolds numbers less than 10. For Reynolds numbers greater than 1, a linearrelationship was found between the Reynolds number and the critical aspect ratio for separation. For slower flows, the crit-ical aspect ratio decreases more quickly as the Reynolds number goes to 0. The critical Reynolds numbers for separation be-hind a cylinder and a flat plate agree with extrapolations from experimental observations. Fluctuations in both the attachedand separated laminar flow behind ellipses were found at combinations of Reynolds number and aspect ratio that were nearthe critical values for separation.

In this paper we review the literature pertaining to low Reynolds number flow around two-dimensional ellipses and theprevious applications of the FHP-I cellular automata model to such flows. Next, we present our application of the model toellipses of various aspect ratios and compare the determinations of attached and separated flow to those obtained with theFLUENT computational fluid dynamics software program. Finally, the results of both models are compared to experimentaldata.

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1634 D. Stack, H.R. Bravo / Applied Mathematical Modelling 33 (2009) 1633–1643

2. Slow flow around elliptical cylinders

The flow around circular cylinders has been studied with experimental, analytical and numerical techniques [1]. How-ever, there is a paucity of experimental and computational data for flow around ellipses at low Reynolds numbers that isdue, at least in part, to the economic imperatives for studying higher Reynolds number flows that are common for vehiclesand machinery. However, microfluidics research is now becoming recognized as a field in its own right [2].

When using inviscid models, the flow pattern is symmetrical not only above and below the cylinder, but also upstreamand downstream. However, for viscous flows above Reynolds numbers of approximately one, dynamic forces override thefore and aft symmetry. As the flow rate increases, a separation appears on the downstream side that contains two recircu-lating eddies that grow laterally as the Reynolds number increases.

In the mid 19th century, Stokes developed an analytical description of flow past a motionless sphere at Reynolds numbersless than 1. Oseen extended Stokes’s work to cylinders using an analysis based upon slight deviations from a known flow [3].

Schlichting reported on the work by Blasius and others in the early 1900s to develop exact solutions for flow around sim-ple shapes [4]. The velocity of the potential flow and the velocity profile of the boundary layer were expressed as power ser-ies in x, the distance from the stagnation point measured along the object’s contour. Lack of sufficient computationalresources to include an adequate number of terms limited the accuracy of the calculations, especially for slender body shapessuch as streamlined ellipses.

Taneda experimented with circular cylinders and flat plates aligned parallel to the flow for Reynolds numbers in the range1–2000 [5]. He observed the formation of twin rear vortices behind a circular cylinder at a Reynolds number of 7. He laterextended his work by taking detailed measurements of the recirculating eddies behind flat plates oriented normal to the flow[6]. He found measurable eddies at a Reynolds number of 0.92. He also measured the relationship between the eddy size andReynolds number.

Dennis and Chang presented finite difference solutions of the equations of motion for steady, incompressible flow arounda circular cylinder for Reynolds numbers in the range 5–100 [7]. Like Taneda [5], they found a linear growth in eddy lengthwith increasing Reynolds number behind a circular cylinder. They calculated that flow separation begins at a critical Rey-nolds number of 6.2.

Nieuwstadt and Keller modeled viscous flow around a circular cylinder for Reynolds numbers in the range 1–40 using thesemi-analytical method of series truncation to express the stream function and vorticity in a Fourier series that was substi-tuted into the Navier–Stokes equation to yield a finite system of nonlinear ordinary differential equations [8]. Their resultscompared favorably with Dennis and Chang [7] for Reynolds numbers less than 40, and were computationally more efficient.

Coutanceau and Bouard photographed features of wakes behind circular cylinders for Reynolds numbers in the range of5–40 [1]. They noted that the maximum recirculating velocity on the axis between the eddies increased linearly with Rey-nolds number.

Van Dyke published experimental visualizations of flow around circular cylinders at Reynolds numbers of 0.16 and 1.54[9]. In the former case, the flow was almost completely symmetrical upstream and downstream of the object. In the lattercase, the streamlines downstream of the cylinder were elongated, but the flow was not separated. He also published Taneda’sphotograph [6] of flow past a flat plate normal to the flow at a Reynolds number of 0.334. Although Taneda did not claim flowseparation for this case, Van Dyke entertained the possibility.

Shintani, Umemura and Takano asymptotically matched the Stokes and Oseen solutions of the Navier–Stokes equationsfor two overlapping regions near elliptic cylinders at a Reynolds number of 0.1 [10]. For a flat plate, two symmetrical recir-culating vortices formed on the downstream side. They reported qualitative agreement with Taneda’s illustration [6] of flowat a Reynolds number of 0.44.

Nakayama et al. presented visualizations of flows around a circular cylinder at Reynolds numbers of 0.038 and 1.1 [11]. Inboth cases the flow was attached but did not exhibit fore and aft symmetry.

Wu and Lee presented experimental data and mathematical calculations using the FIDAP computational fluid dynamicssoftware program for the free settling of solid and porous ellipsoids of revolution for Reynolds numbers in the range 0.1–40[12]. For a solid ellipsoid of revolution of aspect ratio 0.7 with the major axis aligned parallel to the flow, the upstream anddownstream streamlines were symmetrical at a Reynolds number of 0.1. At a Reynolds number of 40, recirculating eddieswere visible on the downstream side.

3. Previous cellular automata models and applications

Flow systems are usually modeled mathematically in terms of continuous functions in an Eulerian system. Fluids, how-ever, are composed of discrete molecules. Although, it is not feasible to track every molecule, the field of cellular automata(CA) has demonstrated that it is possible to compute a large, yet tractable, number of automata that emulate fluid particles.The system as a whole can be considered a continuum so long as the length scale of the macroscopic motions is much largerthan the length scale of the automata [13].

The first fully deterministic lattice gas model, known as HPP, is based upon unit-mass, unit-speed fluid particles movingeither horizontally or vertically between nodes along a square lattice [14]. When only two particles experience a head-oncollision at a node, they leave at right angles along the two previously unoccupied directions. Viscosity is anisotropic because

D. Stack, H.R. Bravo / Applied Mathematical Modelling 33 (2009) 1633–1643 1635

the total momentum in each row of the grid that is parallel to the flow is conserved. Therefore, there is no convection ofmomentum in the direction orthogonal to the flow [15]. Nevertheless, Succi used an HPP model to simulate flow arounda flat plate as an exercise in implementing a CA program on a computer with vector processing architecture [16].

The HPP model was superseded by a series of FHP models that are based upon a lattice of equilateral triangles. Each nodeis connected to its six nearest neighbors. The hexagonal symmetry of the lattice allowed Frisch and his colleagues to modelthe Navier–Stokes equations using the version that they named FHP-I [17]. The FHP-I model is inherently viscous and there-fore restricted to low Reynolds numbers. It also lacks Galilean invariance, which is the principle that Newton’s laws are validin all inertial reference frames [18]. Consequently, the convective term in the CA form of the Navier–Stokes equations, u � $u,is scaled by the factor g(q) = (3 � q)/(6 � q) where q is the average number of particles per node. The scaling factor arisesbecause all particles have unit-speed and unit-mass and the particle motions are constrained to the links of the hexagonallattice. Therefore, the flow is compressible because regions of higher flow rate also have higher densities. During the latterhalf of the1980s, successively more complex CA models were developed to mitigate these limitations.

Wolfram used a CA model to study eddies and a vortex street in the wake of a circular cylinder at a Reynolds number ofapproximately 100 to demonstrate the capabilities of the massively parallel Connection Machine Computer [19].

Duarte and Brosa used a two-dimensional CA model to investigate viscous drag on both a circular cylinder and a cylinderwith hexagonal cross section in the center of a channel with Poiseuille stream flow [20]. They determined drag coefficients atReynolds numbers in the range 1–100 that were within 10% of experimental values. For Reynolds numbers in the range of10–30, they obtained close correlation with experimental data for the location of the turning point downstream of the cyl-inder where the backflow between the recirculating vortices ends.

During the 1990s, attention shifted toward Lattice Boltzmann (LB) models that employ statistical distributions of the par-ticle velocities at the lattice nodes rather than particle tracking. Although LB models are free from Galilean invariance and theinherent noise of CA models, microdynamical features of the flow are ignored [21]. LB models may also be numerically unsta-ble, which can be exacerbated by the use of floating point computations in contrast to the integer operations that underlie CAmodels [22].

This study investigates the laminar flow downstream of elliptical cylinders with aspect ratios ranging from cylinders toflat plates. Previous experimental research has been largely confined to flat plates and cylinders and overlooked intermediateaspect ratios. Prior analytical and numerical solutions have focused on specific Reynolds numbers or aspect ratios. The pres-ent investigations determine critical values and the overall relationships between flow separation, Reynolds number and as-pect ratio that, for example, are important in the settling of particles in atmospheric and benthic environments.

We confirm the aforementioned linear relationships between the dimensions of the recirculating vortices and Reynoldsnumbers greater than 1. We also determine that the relationship between the critical aspect ratio for flow separation andReynolds number is linear for Reynolds numbers greater than one. The critical Reynolds numbers determined for flow sep-aration behind a flat plate and circular cylinder compare favorably with extrapolations from experimental observations.Unexpected fluctuations in the values of the stream function for laminar flow behind the ellipses were found at combina-tions of Reynolds number and aspect ratio near the critical values for separation.

4. Cellular automata model methods and results

The FHP-I cellular automata model used in these investigations is based upon Rothman’s investigation of flow in porousmedia [23]. It is the simplest lattice gas formulation that has sufficient symmetry to ensure isotropy in the equations offluid mechanics [24]. At a given time step, up to six particles may arrive at a node, at most one per link. When onlytwo particles collide head-on at a node they both change direction by p/3, either clockwise or counter-clockwise dependingon whether the timestep is odd or even. When only three particles spaced at intervals of 2p/3 collide at a node they bouncebackwards along the incident links. In all other collisions the particles appear to pass through each other withoutinteracting.

The model was comprised of 1380 rows and 1600 columns. Because the two-dimensional flow above and below the ellip-ses was symmetrical, only the upper halves were modeled. Particles that encountered either the top or bottom slip bound-aries were reflected similar to light striking a mirror, i.e., the angle of reflection equaled the angle of incidence. Half ellipsesmade of non-slip material with vertical semi-major axes of 40 lattice units were placed on the bottom boundary and cen-tered on column 640 because their effects on the flow were more prominent downstream than upstream. Particles encoun-tering the ellipses were reflected back along the incoming links.

Particles exiting either end of the model were reintroduced at the opposite end via periodic boundary conditions, thusconserving the total number of particles and the total momentum. At each time step, the particle configurations in eachof the first 5 columns at the upstream end were randomly re-arranged to prevent any downstream effects of the ellipsesupon the flow from wrapping around to the input. Momentum was added as necessary by randomly changing the particleconfigurations at the nodes in the first column such that the average horizontal velocity in that column was set as near aspossible to the desired uniform flow velocity without exceeding it. The model thus started with a surge of flow proceedingtoward the downstream end at the Mach number of

p2/2 lattice units per time step [17]. Finer details of the flow behind the

elliptical objects took longer to develop. Experience indicated that a spin-up of 60,000 timesteps was sufficient for Reynoldsnumbers of 2.76 and above. For lower Reynolds numbers, the spin-up was lengthened proportionately.


The model density was 2.47 particles per node. A lower density would have enabled the model to run at higher Reynoldsnumbers and somewhat mitigated the effects of Galilean invariance. However, large spatial averages would have been re-quired, which would have masked the structure of the flow.

Seven flow rates were modeled across a Reynolds number range of 1.38–5.53, which corresponded to flow rates of 0.1–0.4lattice units per timestep. To mitigate Galilean invariance and compressibility, an FHP-I model should not be run near itsMach number [20]. The maximum flow rate used in these investigations was slightly higher than the values of 0.35 recom-mend by Frisch et al. [14] and 0.3 recommended by Kohring [25]. Forty-one aspect ratios were modeled for each flow rate,ranging from 1, a circular cylinder, to 0, a vertical flat plate.

For each combination of Reynolds number and aspect ratio, the model was spun-up and then four data samples were ta-ken. Each sample consisted of temporally averaging the particle velocities at each node across at least 1000 timesteps. Sincethe particles moved in a quasi-random fashion, the four samples were taken to guard against the peculiarities of a single rununduly influencing the interpretation of the results. The model showed limited sensitivity with respect to the number of runsbeyond four. Each set of temporal averages was smoothed spatially by averaging horizontally and vertically across regions of16 lattice units per side. To increase the resolution, the spatial averages were successively overlapped by four lattice units inboth the horizontal and vertical directions. Table 1 summarizes the combinations of parameters that were investigated withthe cellular automata model.

The computed vector fields were visually inspected and categorized as either separated or attached depending uponwhether there was evidence of a recirculating eddy. An eddy was recorded if the field clearly showed a complete loop includ-ing at least one vector pointed downstream at the top of the eddy, one vector pointed downward at the far extent of theeddy, one vector pointed upstream at the bottom of the eddy and one vector pointed upward close behind the object. Thismethod is similar to that employed by Agrawal and Prasad [26] who also recap a number of vortex definitions. In caseswhere one or more portions of an eddy were missing due to the inability to compute vectors within 8 lattice units of eitherthe line of symmetry or the object, the flow was also classified as separated. Fig. 1 shows a clearly resolved eddy.

The determinations of which combinations of Reynolds number and aspect ratio resulted in recirculating eddies, andwhich did not, for each of the four runs are displayed in Fig. 2. The four quadrants of each circle in the figure representthe four runs of the model. The first temporal average is represented by the upper-right quadrant of the circle, the secondby the upper-left quadrant, and so forth. A black quadrant indicates that a recirculating eddy was observed downstream ofthe object. A completely filled circle indicates that all four runs of the CA model yielded a recirculating eddy for the givenReynolds number and aspect ratio.

Generally speaking, the figure has blacker regions that indicate a predominance of flow separation toward the lower righthand where both the flow rates are higher and the curvatures of the ellipses are greater. However, for many combinations ofReynolds number and aspect ratio the four samples yielded both separated and attached flows, e.g., flow around a plate at aReynolds number of 1.38. In other cases, as the aspect ratio increased for a given Reynolds number the runs began to spo-radically indicate attached flow instead of switching completely from separated to attached. This can be seen by traversingthe figure from bottom to top. These inconsistencies are the result of the inherent limitations of the FHP-I CA model and theunderlying physics of the flow that are discussed below.

The FHP-I model is known to be noisy [27] which is the result of the lack of Galilean invariance and the quasi-randomnessof the collisions of thousands of particles at each timestep. The size of the spatial averages were a compromise designed to beboth larger than the mean free path of approximately 12 lattice units for these parameters [23] and smaller than the char-acteristic size of the ellipses, which is 80 lattice units.

The temporal averages were negatively impacted by variations in the mean flows, which are due to the randomness of theparticle motions and the mechanism for setting the upstream boundary condition described above. Fig. 3 shows a sample ofthe average downstream velocity of the entire model for 5000 timesteps at a Reynolds number of 2.76. Although the rootmean square signal to noise ratio of the short term fluctuations in the mean flow rate is less than 1E-4 across an averageof 64 timesteps, the RMS ratio is larger for the longer term fluctuations. During typical temporal averages of 1000–2000timesteps, the overall flow rate may vary by a tenth of a percent. The experimental observations of Taneda [6] indicate thatthe sizes and lengths of recirculating eddies are proportional to the magnitude of the mean flow. Consequently, because ofthe temporal averaging, there is a smearing of the averaged flow pattern that could lead to a blurring of one or more of the

Table 1Comparison of the Reynolds numbers, flow rates, spin-up times and durations of temporal averages of the CA and FLUENT models

Reynolds number

1.38 2.07 2.76 3.46 4.15 4.84 5.53

Comparison of parameters of the CA and fluent modelsFLUENT model flow rate (m/sec) 2E�4 3E�4 4E�4 5E�4 6E�4 7E�4 8E�4CA model flow rate (lattice units per timestep) 0.1 0.15 0.2 0.25 0.3 0.35 0.4Length of CA model spin-up (timesteps) 120,000 80,000 60,000 60,000 60,000 60,000 60,000Length of CA temporal averages (timesteps) 2000 1500 1000 1000 1000 1000 1000Size of CA spatial averages (lattice units) 16 � 16 16 � 16 16 � 16 16 � 16 16 � 16 16 � 16 16 � 16Overlaps of spatial averages (lattice units) 4 4 4 4 4 4 4

615 625 650 700

0

650 675

0

50

LATTICE UNITS

LA

TT

ICE

UN

ITS

= 0.05 lattice units per timestep

Fig. 1. Example of a clearly resolved eddy rendered by the cellular automata model behind an ellipse with an aspect ratio of 0.25 at a Reynolds number of5.53.


four vectors that form the distinctive signature of an eddy described above. In such cases, the modeled flow may be errone-ously categorized as attached.

To better distinguish the predominantly blacker and whiter regions in Fig. 2, a clarification scheme was employed thatsmoothed the data across adjacent aspect ratios. The flow was deemed unattached for a particular combination of Reynoldsnumber and aspect ratio if the sum of the black quadrants of the corresponding circle in Fig. 2 as well as one half times thenumber of black quadrants in the circles immediately above and below it, exceeded a threshold of seven. Using this scheme,the flow was deemed separated for all aspect ratios below the stair-stepped line in Fig. 2.

5. Steady state FLUENT model

Because of the paucity of relevant experimental and analytical data, the FLUENT [28] computational fluid dynamics soft-ware was employed as a tool to complement the CA model and improve the understanding of the flows. Similar to the CAmodel, symmetry was assumed and flows were only modeled over the upper halves of objects that were placed in the middleof the flat boundary of a semi-circular domain that was five meters in radius. The major axes of the objects were all 0.1 m andthey protruded 0.05 m above the line of symmetry.

A flat plate was modeled in the same semi-circular domain. To enable comparisons with Taneda’s experimental observa-tions [6], the same flat plate aspect ratio was used, 0.0154. Table 1 shows the free stream velocities of both the CA and FLU-ENT models in their native units for the primary Reynolds numbers under investigation.

Visual examination of the contours of stream function computed by the steady state FLUENT model provided an unam-biguous indication of whether or not the flow was attached. The critical aspect ratios for separation are illustrated by filledsquares in Fig. 4. Lesser values for the critical aspect ratio as calculated by the CA model are represented by filled circles anddescribed below.

6. Comparing the cellular automata and FLUENT models

The CA model has a tendency to switch back and forth between attached and separated flow, especially for ellipses thathave near-critical aspect ratios. The clarification scheme described above conservatively errs on the side of separated flowrather than attached. Hence, as seen in Fig. 4, the critical aspect ratios are systematically underestimated by the CA model.What is notable is the striking similarity between the results of the CA and FLUENT models as demonstrated by the linearslopes of the relationships between the critical aspect ratios and Reynolds numbers greater than 1.

To investigate whether there exists an inherent unsteadiness in the underlying physics of the flow that affected the CAmodel, the FLUENT model was also run in the unsteady state mode and the maximum values of the stream function inthe cell immediately downstream of the object and nearest the line of symmetry were monitored.

Although the shape of the flows did not change over time, Fig. 5 shows fluctuations in the values of the stream functionfor laminar flow downstream of ellipses of various aspect ratios at a Reynolds number of 5.53, the highest velocity testedwith the CA model. The streamlines behind the cylinder were attached whereas the streamlines for the other three cases

Plate

0.75

0.25

0.5

Cylinder

Asp

ect

Rat

io

1.38

2.07

2.76

3.46

4.15

4.84

5.53

Reynolds Number

Fig. 2. Quadrants of circles indicate the presence (black) and lack (white) of recirculating eddies in four temporal averages of the cellular automata model atthe indicated combinations of Reynolds number and aspect ratio.


indicated flow separation. The magnitudes of the fluctuations are the greatest downstream of the cylinder and the leastdownstream of the plate. This is counterintuitive because the cylinder bends the incident streamlines less than the plate

0.1994

0.1996

0.1998

0.2000

30,000 30,500 31,000 31,500 32,000 32,500 33,000 33,500 34,000 34,500 35,000

Timesteps

Vel

oci

ty (

latt

ice

un

its/

tim

este

p)

Fig. 3. Sample of the average downstream velocity of the entire cellular automata model at a Reynolds number of 2.76.

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6 7 8Reynolds Number

Asp

ect R

atio

CA FLUENTTaneda (observed) Taneda (extrapolated)Dennis and Chang (extrapolated) Dennis and Chang (calculated)

Fig. 4. Critical aspect ratios for separation as a function of Reynolds number for the CA model, the steady state FLUENT model and findings from otherinvestigators.


and it is reasonable to expect the flow around the cylinder to be smoother in all respects. Bearing in mind that the criticalaspect ratio for separation at this Reynolds number is 0.95 (see Fig. 4), it is clear that the magnitudes of the fluctuations arelarger for aspect ratios near the critical value for separation, than for aspect ratios further removed.

Similarly, Fig. 6 shows values of the stream function for the laminar flows immediately behind a cylinder at three Rey-nolds numbers for which the flow is attached. The three traces indicate fluctuations in the values of the stream function thatincrease with respect to Reynolds number as the separation Reynolds number is approached. The FLUENT model predictsseparation at a Reynolds number of approximately 5.8 as shown in Fig. 4.

We believe that these fluctuations contribute to the CA model switching back and forth between the attached and unat-tached states, especially near-critical aspect ratios, as evidenced in Fig. 2.

7. Comparing model results to experiments

Taneda investigated the relationship between the horizontal and vertical locations of the center of the recirculating eddywith respect to Reynolds number for the case of a flat plate [6]. The data were presented in non-dimensional ratios, s*/d and

1.E-13

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-0630,000 30,500 31,000 31,500 32,000 32,500 33,000 33,500 34,000 34,500 35,000

Time (seconds)

Str

eam

Fu

nct

ion

Cylinder Aspect 0.95 Aspect 0.875 Plate

Fig. 5. Fluctuations in the values of the stream function in the laminar flow immediately downstream of ellipses of various aspect ratios at a Reynoldsnumber of 5.53. The flow behind the cylinder was attached. The flow behind the other three objects was separated.

0.E+00

2.E-10

4.E-10

6.E-10

8.E-10

1.E-09

30,000 30,500 31,000 31,500 32,000 32,500 33,000 33,500 34,000 34,500 35,000

Time (seconds)

Str

eam

Fu

nct

ion

Re = 1.38 Re = 3.46 Re = 5.53

Fig. 6. Fluctuations in the values of the stream function in the laminar flow immediately downstream of a cylinder at three Reynolds numbers. All threeflows were attached.

d/2

d*/2

s *s

Fig. 7. Taneda’s [6] nomenclature for measurements of the location of the center of an eddy downstream of a flat plate at a Reynolds number of 3.46.



d*/d, where d is the length of the plate. These definitions are illustrated in Fig. 7, which shows the output of the steady stateFLUENT model at a Reynolds number of 3.46.

The steady state FLUENT model was used to generate a set of coordinates for the center of the eddy, i.e., the coordinates ofthe center of the FLUENT cell downstream of the plate that had the lowest value of stream function. Data published by Tan-eda [6], and the corresponding FLUENT data, are graphed in Figs. 8 and 9. In both figures, Taneda’s data show a stair-steppattern owing to the challenges of experimental observation. The FLUENT data are also slightly stair-stepped because smallchanges in the overall size or location of the eddy may not move the location of the minimum stream function into a differ-ent cell of the model, in which case there would be no change to s* or d*/2.

In both Figs. 8 and 9, the calculations agree very well with Taneda’s experimental measurements [6]. Taneda estimated acritical Reynolds number of 0.4 for the onset of eddies behind a flat plate. Since the FLUENT model can be run at lower Rey-nolds numbers than is practical for experimental investigations, it yielded a critical Reynolds number of 0.08 for the forma-tion of eddies.

Fig. 8 shows a steep drop in the values of d*/d nearer to the origin than Taneda was able to measure [6], which illustratesthe increasing influence of viscosity upon the flow as the Reynolds number approaches 0. Moving to the right of the origin,

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30

Reynolds Number

d*/

d

FLUENT Taneda

Fig. 8. Comparison of the values of d*/d, the vertical distance between the centers of eddies downstream of a flat plate, with respect to Reynolds number asobserved by Taneda and computed using FLUENT.

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30Reynolds Number

s*/d

FLUENT Taneda

Fig. 9. Comparison of the values of s*/d, the horizontal distance of the centers of eddies from the flat plate, with respect to Reynolds number as observed byTaneda and computed using FLUENT.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9s*/d

d*/d

23.5

18.0

5.533.46

1.380.346

8.98 13.1

22.8

0.086

0.022

Fig. 10. Relationship between the horizontal (s*/d) and vertical (d*/d) coordinates of the eddy center behind a flat plate as Reynolds number increases usingthe steady state FLUENT model. Reynolds numbers are indicated above selected data points.


the eddy center is almost immediately 40% as tall as the plate. As the Reynolds number increases above 1, the center getsfarther from the line of symmetry as a consequence of the dynamic forces. These observations are confirmed by examinationof Fig. 10, which illustrates the relationship between the horizontal and vertical measurements of the eddy center forincreasing Reynolds number.

Taneda observed that the flow around a circular cylinder exhibits twin vortices at a Reynolds number of 7 [5]. He did notobserve twin vortices at a Reynolds number of 6, but noted the difficulty of making observations at such low Reynolds num-bers because the dimensions of the vortices and the flow velocities are both small immediately adjacent to the rear stagna-tion point. He estimated that the critical Reynolds number for eddy formation is 5.

Coutanceau and Bouard deduced several critical Reynolds numbers for separation depending on the ratio of the cylinderdiameter to the diameter of the experimental chamber [1]. They calculated that the separation value for unbounded flow, i.e.,a tank of infinite diameter, would lie between Reynolds numbers of 5 and 7.

In their computations, Dennis and Chang also observed twin vortices behind a circular cylinder at a Reynolds number of 7but not at a value of 5 [6]. By linear interpolation, they posited a critical Reynolds number of 6.2.

The above findings are included in Fig. 4, which also shows the Reynolds numbers corresponding to the critical aspectratios for separation as determined with the steady state FLUENT model. The critical Reynolds number for a flat plate is seento be 0.08, and the value for a cylinder is 5.8. The figure illustrates that the latter value is very close those determined by theother investigators and within the range of 5–7 measured by Coutanceau and Bouard [1].

8. Conclusions

Both the cellular automata and FLUENT models clearly demonstrate that the relationship between the Reynolds numberand the critical aspect ratio for separation is linear for Reynolds numbers greater than 1. For Reynolds numbers less than 1,the critical aspect ratio decreases more quickly the closer the Reynolds number is to 0. The computed critical Reynolds num-bers for separation behind a flat plate and a cylinder are in close agreement with extrapolations made from experimentalobservations.

Taneda’s horizontal and vertical measurements of the location of the eddy center behind a flat plate with respect to Rey-nolds number [6] were verified and extended. The relationships between the Reynolds number and both of these coordinatesare linear for Reynolds numbers greater than 1. For Reynolds numbers less than 1, the vertical coordinate of the eddy centerdecreases more quickly than the horizontal coordinate as the Reynolds number approaches 0.

Regardless of whether the flow is attached or separated, the unsteady state FLUENT model predicts fluctuations in thevalues of the stream function in the laminar flow downstream of the ellipses for combinations of aspect ratios and Reynoldsnumbers that are near the critical combination for separation. The fluctuations, and the inherent characteristics of the CAmodel, resulted in a tendency for the CA model to switch back and forth between attached and separated flow and thereforeunderestimate the critical aspect ratio for separation.

Although the cellular automata model exhibited a certain amount of uncertainty in regard to the values of the criticalaspect ratios, it may provide valuable insight into how flow at such low Reynolds numbers occurs in the natural world. Tan-eda notes the care he took to create a quiescent chamber in which to move a plate and a cylinder as smoothly as possible[5,6]. As valuable as these experiments are, they don’t represent naturally occurring flow. Steady state mathematical models


of laminar flow are even more idealized. A cellular automata representation, with its alternating flow separations and attach-ments, could be a more realistic representation of the physical processes that are at work in the case of a human hair fallingthrough air at a Reynolds number of approximately 2 than either a steady state computational fluid dynamics model or anexperiment conducted under ideal laboratory conditions.

Acknowledgements

The authors gratefully acknowledge the comments of two anonymous reviewers. This work was partially supported bythe National Center for Supercomputing Applications under Grant CTS040031N and utilized the IBM P690.

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flow separation behind ellipses at reynolds numbers less than 10

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