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A new type of bistable flow around circular cylinders withspanwise stiffening rings

F. Lupi a,n, C. Borri a, L. Facchini a, H.-J. Niemann b, U. Peil c

a Department of Civil and Environmental Engineering, Università degli Studi di Firenze, Via S. Marta 3, 50139 Firenze, Italyb Windingenieurwesen und Strömungsmechanik, Ruhr-Universität Bochum, Universitätsstraße 150, 44801 Bochum, Germanyc Institut für Stahlbau, Technische Universität Carolo-Wilhelmina Braunschweig, Beethovenstraße 51, 38106 Braunschweig, Germany

a r t i c l e i n f o

Available online 8 November 2013

Keywords:Circular cylindersBistable flowReynolds numberWind tunnel testsNon-linear conservative systemsTwo-wells systems

a b s t r a c t

The bistable flow condition around a single circular cylinders is a well-known fluid dynamic pheno-menon in the critical range of Re. It is sensitive not only to small variations of the Reynolds number, butalso to turbulence of the incoming flow and to surface roughness of the cylinder. Bistable flows are alsocommon for side-by-side cylinders, depending on their centre-to-centre transverse pitch ratio.

The paper reveals – through wind tunnel tests – the existence of a new type of bistable flow, inducedaround a single circular cylinder with a free-end by the presence of spanwise rings. There are someanalogies with the aforementioned bistable phenomena, but the conditions of occurrence are profoundlydifferent. The peculiarity of this phenomenon is that it does not disappear at moderately high Re. Itsexistence is confirmed by a cross-check of results in two different wind tunnels.

In order to characterize such a bistable pressure field, the pressure is modeled in the paper as theoutput of a non-linear conservative systemwith asymmetric potential wells forced by a proper stochasticprocess. The identification of the system parameters is performed by the comparison between thetheoretical distribution of the oscillator response and the histogram of the recorded pressure.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction: bistable flows in literature

Bistable flows are fluid-dynamic phenomena characterizedby two stable configurations of equilibrium. Typically, they arenot-symmetric configurations. It is a matter of fact that a not-symmetric state of flow can occur and stabilize around symmetricstructures like single circular cylinders and circular cylinders inpairs. However, the type of bistable flow addressed in this paperhas never been reported as far as the authors know, and it does notdisappear even at moderately high Reynolds numbers. The aim ofthe paper is to prove, by means of wind tunnel experiments, theexistence and the novelty of such a state of flow. In view of that,the background and the physical interpretation of the well-knownbistable flow phenomena around circular cylinders and cylinder inpairs are introduced at first.

The bistable flow condition around an isolated circular cylinder– as reported in literature – is an effect of the Reynolds number.The Reynolds number governs the transition from laminar toturbulent flow and this may generate, in certain conditions, anasymmetric and bistable flow even around a symmetric structure,

which is sensitive to Reynolds effects. The discovery of theexistence of a stable not-symmetric pressure distribution aroundcircular cylinders is very far in time. It was at first observed byEisner (1925). He measured mean pressure distributions aroundboth sides of a circular cylinder and discovered a stable asym-metric pressure distribution in the critical range of the Reynoldsnumber. The low and high pressure on the two sides of thecylinder interchanged in different runs. Because of that, this flowstate was named bi-stable. Eisner concluded, correctly, that thekey of the phenomenon was in the transition from laminar toturbulent boundary layer, which occurred on one side only ofthe cylinder. This produced, on one side only of the cylinder, theformation of a laminar separation bubble.

Symmetric laminar separation bubbles on two sides of thecylinder characterize the state of the flow at Re¼Recrit. Accordingto Roshko's (1961) classification, the presence of two laminarseparation bubbles persists in the first stable state of the super-critical regime, associated to the horizontal plateau at the mini-mum drag coefficient. Bearman (1969) identified the bistability ofthe flow in a sub-range of Re before symmetric twin bubblesappeared, by experimental evidence of discontinuous changes inCpb and St at Re¼3.4�105 and 3.8�105. He also found the cause inthe formation of a laminar separation bubble on one side only ofthe cylinder. The Cpb distribution along the height showed that the

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0167-6105/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jweia.2013.09.004

n Corresponding author. Tel.: þ39 55 4796276.E-mail addresses: francesca.lupi@dicea.unifi.it, francescalupi@alice.it (F. Lupi).

J. Wind Eng. Ind. Aerodyn. 123 (2013) 281–290

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bubble took place along the complete length of the cylinder. Healso noted that the asymmetric pressure distribution was accom-panied by the displacement of the stagnation point away from thebubble.

Further studies on the formation on only one bubble arereported by Farell and Blessmann (1983). Contrary to the resultsof Bearman (1969), who found the single-bubble regime to occuralways with the bubble consistently on the same side of thecylinder, in Farell and Blessmann's experiments no preference wasdetected for the bubble to form on either side of the cylinder.

Schewe (1983) described the physical phenomenon of bistableflow in detail. A discontinuity in the drag coefficient and a suddenjump of St (up to 0.33) mark the onset of a bistable flow condition.The two stable states corresponds to the two possible signs ofthe lift force. The asymmetric stable state persists for a very smallrange of Re. Then, a second discontinuity in the drag, as well asanother sudden jump in St (up to 0.48, that is the highest value)mark the abrupt disappearance of the steady lift and the attain-ment of the minimum drag, corresponding to the critical Re. Thebistable flow condition is preceded and followed by two unstableranges. The explanation for the phenomena lies in the behaviourof the boundary layer. The asymmetric flow and thus the steadylift in the critical range are caused by the fact that the boundarylayer transition from laminar to turbulent has occurred on one sideonly of the cylinder. Thus a laminar separation bubble is formed asfollow: the transition from laminar to turbulent flow occurs in thedetached boundary layer just downstream from the laminarseparation point. After reattachment of the boundary layer onthe back of the cylinder, the separation is turbulent. The reason forwhich transition in the detached boundary layer is initiated on oneside only of the cylinder (and not simultaneously on both sides) isthe occurrence – only on that side – of perturbations or fluctua-tions, which are inherent in the boundary layer and in the freestream. Then, once transition and then reattachment haveoccurred on one side, there is an acceleration of the fluid on thatside and deceleration on the other side. Since deceleration delaystransition in the free shear layer (it reduces Re), the formation ofthe bubble also on the other side is delayed. According to Schewe(1983), this is the reasonwhich stabilizes and fixes the asymmetricflow state. Of course, such a bistable flow condition is extremelysensitive to Re and it is only possible if there is a very lowprobability for simultaneous occurrence of perturbations on bothsides. In fact, the bistable flow disappears on rough cylinders aswell as in turbulent flow. As soon as the flow reattaches also onthe other side of the cylinder, symmetric conditions are achievedwith two bubbles on the two sides of the cylinder: the drag is thenminimum and that is the critical Re.

The bistable flow condition around isolated circular cylinders,as previously described, represent a well-known and interestingfluid-dynamic phenomenon, but without any relevant applicationin the design of structures. The bistable regime occurs in a verysmall range of Re, just before Recrit. Most of the structures are farbeyond this regime. Moreover, the bistable phenomenon is extre-mely sensitive not only to Re, but also to any flow disturbance suchas turbulence of the incoming flow and cylinder surface rough-ness. So, practically, conditions of occurrence of such a bistableflow are rarely fulfilled.

Bistable flow conditions around circular cylinders are muchmore common in side-by-side configurations of cylinders in pair.In this case, the bi-stability is regarded as a phenomenon ofinteraction between cylinders. The fluid behaviour is primarily afunction of the centre-to-centre transverse pitch ratio (T/D).Reynolds effects for the side-by-side configuration exist, but theyare less prominent than, for example, the tandem configuration.Depending on T/D, different flow patterns are possible for the side-by-side configuration (Zdravkovich, 2003). At small pitch ratios

(approximately 1.0oT/Do1.1–1.2) a single eddy street is formedbehind both cylinders, which appear as a single bluff body with aweak flow through the gap. At intermediate pitch ratios (approxi-mately 1.1–1.2oT/Do2.0–2.2) a biased flow pattern develops.Narrow and wide wakes are formed behind two identical cylin-ders. The gap flow forms a jet biased towards the narrow wake.The biased gap flow is bistable, and it may intermittently switch toeither side. The origin of the bistable biased flow between side-by-side cylinders has been attributed to various causes (e.g. Coandaeffect), but it still remains unsolved. At sufficiently high pitchratios (approximately T/D42.0–2.5) both wakes are equal in sizeand eddy shedding is synchronized in frequency and phase. Thepredominant out-of-phase coupling produces two eddy streets,which mirror each other relative to the gap axis.

A review paper by Sumner (2010) describes further theasymmetrical or biased flow pattern at intermediate values ofT/D for two side-by-side circular cylinders. The cylinder towardswhich the gap flow is biased has a narrow near-wake, higher-frequency vortex shedding, and a higher drag coefficient, while theother cylinder has a wider near-wake, lower-frequency vortexshedding, and a lower drag coefficient. Because of that, the twomodes “narrow wake” and “wide wake” can be identified. Thedeflection of the biased gap flow varies with T/D. The trend istoward a smaller degree of deflection with increasing T/D. In somecases, the biased flow pattern switches intermittently from beingdirected towards one cylinder to the other, and the flow patternis termed bistable. This “flip-flopping” of the gap flow directionand wake sizes occurs spontaneously and irregularly, but betweenswitchovers the flow remains stably biased to one of the cylindersfor long durations (perhaps a few orders of magnitude larger thanthe vortex shedding period). The bistable characteristic is notcaused by misalignment of the cylinders or other extraneousinfluences, but is an intrinsic property of the flow.

Mahbub Alam et al. (2003) applied a wavelet analysis in orderto detect the frequency of vortex shedding in the two modes(narrow wake, wide wake) and the switching phenomenon. Threemodes of flow, associated with wider wake, symmetric wake andnarrow wake are described by Mahbub Alam and Meyer (2011).The formation and burst of a separation bubble when the gap flowis biased towards one of the two cylinder can even produce aquadri-stable flip-flopping flow regime.

The study addressed in the paper could develop only on thebasis of the aforementioned similar known flow phenomena.Several analogies were found with them and are presented inthe following. However, the conditions of occurrence and thephysical reason are profoundly different.

2. Experimental set-up

2.1. The two wind tunnels and the model

The experimental investigation was performed in two windtunnels, in order to cross-check the results: the WiSt laboratory atRuhr-University Bochum (Germany) and the Criaciv laboratory atUniversity of Florence (Italy). The comparison of results allowedto exclude any dependency of the phenomenon on some localconditions or flow distortions of a given laboratory.

WiSt laboratory at Ruhr-University Bochum is an open circuitwind tunnel with a total length of about 17 m. The tunnel itself hasa length of 9.3 m and the test section is 1.8 m in width and 1.6 m inheight. A honeycomb grid is located at the inlet of the tunnel. Theturbulent atmospheric boundary layer develops over a length ofabout 7.6 m. After a castellated barrier having a maximum heightof 425 mm, there are three turbulent generators of 1.5 m in height.They are built according to Counihan's (1969) specifications. The

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roughness field consists of six panels with 36�36�36 mm3 cubesalternated to 36�36�18 mm3 square prisms. The tests areperformed in both atmospheric boundary layer and uniform flow(empty tunnel). The diffuser and the centrifugal fan are placedat the end of the wind tunnel. The engine allows to attain amaximum wind speed of about 28–30 m/s with 1500 rounds perminute. A Prandtl tube, placed at 1.3 m in height, allows tomeasure the dynamic pressure of the incoming flow. Temperaturesensors acquire temperature during the measurements. The pres-sure sensors are four-active-element piezoresistive bridges. Twodifferent pressure sensors are used: the type n.1 (Honeywell 170PC) has a measurement range735 mbar and the type 2 (AMSYS5812-0001-D-B) has a measurement range710.34 mbar. The A/Dconverter scans the pressures in a sample-and-hold modus, whichproduces simultaneous sampling of the measurements. A sam-pling frequency of 2 kHz is used for the measurements.

The Criaciv laboratory is an open-circuit wind tunnel located atPolo Universitario Città di Prato, which is a branch of the University ofFlorence. The total length is about 22 m. The tunnel itself has a lengthof 11 m, with a slightly divergent shape from the inlet (in order toguarantee a constant pressure along the x-axis) and a test section of2.4 m in width and 1.6 m in height. The atmospheric boundary layerprofile which is used in these experiments is produced by threebigger turbulence generators of Counihan type and four smallerspires, followed by roughness panels with wooden cubes. The motor– with a nominal power of 160 kW – and the fan are placed at theend of the wind tunnel, followed by a T-shaped symmetric diffuser.The engine allows to attain a maximumwind speed of about 30 m/s.The pressures on the tower are measured with two different types ofpressure scanners: PSI 8400, scanning at a frequency of 250 Hz; DTC-Initium, scanning at a sampling frequency of 500 Hz.

The model for wind tunnel tests (Fig. 1) is a circular cylinderwith a free-end and aspect ratio H/D¼6.7. The cylinder is a rigidbody of 1 m in height and 15 cm in diameter. Pressure taps areplaced at 17 levels along the height and at an angular spacing of201 in the circumferential direction. It is also possible to createan efflux inside the cylinder like a chimney. Both external andinternal pressures can be measured at each level.

The peculiarity of the model is the presence of a certainnumber of circular ring beams along the height (spanwise rings).The rings are removable, so that two situations were investi-gated in parallel: circular cylinder with/without rings. The mostinteresting results, which are presented in this paper, are obtainedin the following configuration, namely “reference configuration”:

� 10 ring beams, equally spaced along the height at a distance of10 cm in the wind tunnel scale, i.e. 2/3 of the diameter D;

� width of the rings w¼7 mm in the wind tunnel scale, i.e.w/D¼7/150¼4.67�10�2;

� no-efflux out of the chimney;� atmospheric boundary layer flow;� rough cylinder (ks/D¼0.25/150¼1.67�10�3);� Re¼2.5�105 at WiSt (Re¼2.8�105 at Criaciv).

However, several other configurations (not shown in the paper)had to be tested, by changing position and/or size of rings andeffective Re. The whole workplan consisted of all possible combi-nations of the following conditions (Lupi, 2013):

� None/10/7/5 ring beams, equally spaced along the height;� w/D¼7/150¼4.67�10�2 and 3.5/150¼2.33�10�2;� efflux/no-efflux;� atmospheric boundary layer flow/uniform flow;� Re from 1.1�105 until 2.5�105 at WiSt (from 0.4�105 in case

of smooth cylinder) and Re¼2.8�105 at Criaciv;� smooth cylinder/rough cylinder with ribs: ks/D¼0.25/150¼

1.67�10�3; 0.375/150¼2.50�10�3; 0. 50/150¼3.33�10�3.

All these tests were done first at WiSt wind tunnel, then themost interesting conditions (including the “reference configura-tion”) were tested also at Criaciv. Expectedly, the bistable flowtends to disappear as the distance between rings increases and/ortheir size decreases.

The ring beams along the height of the model representstiffening rings, that are usually applied in the design of thinand slender shells. For cooling towers, one ring on top is usuallysufficient. However, in super-tall structures like solar updrafttowers, described in Section 5, several ring beams are necessaryalong the height in order to guarantee a beam-like behaviour andreduce ovalling deformations. The experiments within this workaim at investigating the aerodynamic of the flow in such designconditions.

2.2. Preliminary investigation on the cylinder without rings

Due to the scale of the model, the Reynolds number plays a keyrole in the wind tunnel tests. A preliminary investigation of Re onthe smooth and rough cylinder without rings in turbulent flowwas performed (Lupi, 2013). The maximum Reynolds numberachievable in the WiSt wind tunnel, calculated by referring tothe velocity at z¼H, is Re¼2.5�105. Surface roughness, consistingof ribs, is applied on the model in order to attain a much highereffective Re (ESDU 80025).

The target full-scale condition is that of a smooth cylinder attranscritical Re. Codified data for smooth and rough surfaces attranscritical Re are available in the VGB guideline (2010). Accord-ing to them, the drag coefficient for a smooth surface is in therange 0.46–0.49. The value of the minimum lateral suction (1.5–1.6for curves K1.5 and K1.6 in the VGB, respectively) should not beconsidered alone, rather, the difference Cp,b–Cp,min is a much moresignificant parameter, since the base pressure depends on theslenderness while the pressure recovery is not so influenced bythe aspect ratio. In the target configuration, it should be expectedCp,b–Cp,min in the range 1.0–1.1.

The investigation about the effect of surface roughness on thecircular cylinder (without rings) was performed at WiSt tunnel inturbulent boundary layer flow. The main results are reported inFigs. 2 and 3. They refer to z/H¼0.65. Tests were performed at firston the smooth cylinder (namely R0) at several wind tunnelvelocities in the range Re¼4�104�2.5�105. The critical Re is1.9�105. Then, different types of surface roughness were tested(namely R1–R5), with ks ranging from 0.25 to 0.5 mm (being ks the

Fig. 1. Wind tunnel model.

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thickness of the ribs) and at an angular spacing of 101 or 201. Inparticular:

� R1: ks/D¼1.67�10�3 with ribs at angular spacing of 201;� R2: ks/D¼1.67�10�3 with ribs at angular spacing of 101;� R3: ks/D¼2.50�10�3 with ribs at angular spacing of 201;� R4: ks/D¼2.50�10�3 with ribs at angular spacing of 101;� R5: ks/D¼3.33�10�3 with ribs at angular spacing of 201.

The graphs show that state of the flow on the rough cylinder isalready beyond the critical range and it approaches transcriticalconditions.

The target full-scale condition (as previously described, CD¼0.46–0.49, Cp,b–Cp,min¼1.0–1.1) is well fulfilled by the surfaceroughness R1 (red curve) at Re¼2.5�105. This configuration isthen selected for further investigation.

3. A new type of bistable flow

3.1. Investigation on the cylinder with rings

Wind tunnel tests on the circular cylinder with spanwise ringsrevealed the existence of a new type of bistable flow. It is createdby the presence of rings along the height and it does not disappearon the rough cylinder even at moderately high Reynolds number(up to Re¼2.5�105 at WiSt, Re¼2.8�105 at Criaciv). The phe-nomenon is at first measured at WiSt tunnel in both uniform andboundary layer flow and then its existence is confirmed at Criacivwind tunnel. In time, the side pressures jump between two states.This is particularly evident at φ¼1001 (φ¼2601) and at the highestlevels close to the tip of the cylinder. It is reminded that the resultspresented here are those obtained in the “reference configura-tion”, as defined in Section 2.1.

A time history of the pressure coefficient at 950 mm, 1001,calculated by using the local velocity pressure (0.5ρVm(z)2) isplotted in Fig. 4 (WiSt wind tunnel) and Fig. 5 (Criaciv wind

tunnel), to show a direct comparison between results in the twowind tunnels. The jumps occur in the same manner in the twowind tunnels, therefore it can be excluded any responsibility of alocal distortion of the flow, which might be typical of a givenlaboratory. The phenomenon is then somewhat fundamental, anintrinsic property. The standard deviations are of course differentin the two plots, because they depend on the turbulence intensityin the two wind tunnels (at the height of the pressure measure-ments shown in the figures, Iv¼7.94% at WiSt and 2.80% atCriaciv).

3.2. Literature and novelty

It is important to discuss the conditions of occurrence of thephenomenon, which make this phenomenon original and physi-cally unique.

In the conditions of these experiments, the incoming flow isturbulent and the cylinder is rough. In these conditions, laminarseparation can be absolutely excluded, as turbulent transition ofthe boundary layer occurs close to stagnation. The correspondingstate of the flow in the selected surface roughness and Reconditions was shown in Figs. 2 and 3 (reference configuration:R1-curve, Re¼2.5�105). In fact, on the rough cylinder, for anyroughness R1–R5 and at any Re in the range of the tests, the stateof the flow is always beyond the critical drop and it tends toapproach the transcritical state.

As a matter of fact, early laminar separation typically occurs inthe sub-critical range of Re. In the turbulent flow which charac-terizes these experiments, it can be seen in Fig. 2 that the sub-critical range (high plateau of mean drag coefficient) is not fullyachieved even at the lowest Re (4�104) on the smooth cylinder(curve R0). In the critical range, i.e. within the fall of drag beforethe critical minimum, it is known that transition from laminarto turbulent boundary layer is around the separation point(Zdravkovich, 1997). As Re increases, transition moves upstream,so that separation is always turbulent. This is actually the state ofthe flow which occurs on the rough cylinder (curves R1–R5), at anyRe in the range of the tests. Therefore, in the reference condition aswell (R1-curve, Re¼2.5�105), it can be stated that transition hasalready occurred when the flow separates: laminar separation isexcluded.

Consequently, even though the higher suction on one side ofthe cylinder may suggest the existence of a separation bubble, likein the well-known bistable flow in the critical range of Re, itcannot be a bubble in the classical sense, i.e. a laminar separationbubble. A laminar separation bubble occurs when the transitionpoint from laminar to turbulent boundary layer is around separa-tion and laminar separation is followed by turbulent transition,reattachment and turbulent separation. Moreover, its occurrenceis very sensitive to Re, to the surface roughness of the cylinder andto the turbulence of the incoming flow.

Moreover, a laminar separation bubble is caused by a physicalreason, that is transition to turbulent conditions in the free-shearlayer after separation and consequent reattachment. As the for-mation of the laminar separation is excluded in the conditions ofthe experiments, the corresponding physical explanation fails.

Within each stable state before and after a jump, higher suctionon one side of the cylinder implies a non-zero mean lift coefficient.The dependency of the phenomenon on Re was investigated inmore detail, in order to see whether the bistable steady lift is goingto disappear as Reynolds increases, or a certain trend appears. Thisstudy is synthesized in Fig. 6. In the figure, the intervals of timebefore and after a jump are considered separately, so that two statesof the flow can be identified. They are marked with blue andmagenta colors. The lift force is obtained by integration of pressurecomponents in the cross-wind direction. The mean values of the lift

0.100.200.300.400.500.600.700.800.901.00

CD

,m

R0

R1

R2

R3

R4

R5

0 50000 100000 150000 200000 250000 300000

Re

Fig. 2. Drag coefficient vs Re; z/H¼0.65, smooth (R0) and rough (R1–R5)cylinder.

2.0

1.41.61.8

R0

0.81.01.2

0.6

0.00.20.4

0 50000 100000 150000 200000 250000 300000

Re

Cp,

b - C

p,m

in

R1

R2

R3

R4

R5

Fig. 3. Pressure recovery vs Re; z/H¼0.65, smooth (R0) and rough (R1–R5)cylinder.

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forces, within each state, are plotted at different Re in Fig. 6.The first result to be observed is that the mean lift force withineach stable state (either 1 or 2, i.e. blue or magenta in the figure) isnot zero. The result is astonishing, being the structure symmetric.However, Section 1 explained that this is a physically possible resultand it has been already observed in other conditions of occurrence(e.g. Eisner, 1925). The further interesting feature shown by thefigure is that such a result – i.e. a non-zero mean lift – is achievedon a relatively wide range of Re, i.e. on the whole range which waspossible to investigate in the wind tunnel during these experiments.Even though the range of Re which was possible to test in the twowind tunnels was limited, it is a much larger ΔRe than what wasfound in literature (e.g. Schewe, 1983) on the smooth cylinder in thecritical regime. Of course, extension of the Re range with furthertests in high pressurized wind tunnels Re would be a ratherimportant proof. Within this work, they could not be performed.

What is interesting, moreover, is the stable nature of the flowwithin each state. After being initiated, a certain state establishes.The time histories of pressures in the tip region of the cylinder(Figs. 4 and 5) show that not-symmetric conditions persist for arelatively long time, before a jump takes place.

From the experiments, it was not straightforward to infer thatsymmetric conditions can be achieved by averaging on an infi-nitely long period. Any characteristic periodical feature has not

been found in the jumps. But again, it should be reminded that thestable nature of the not-symmetric condition is also a peculiarfeature of the well-known bistable flow of literature, as explainedby Schewe (1983) and mentioned in Section 1. It motivates alsothe name itself, bi-stable.

The fact that the occurrence of jumps between the two statesdoes not present regularity reflects the random nature of theturbulent flow. Also the bistable flow in the critical range of Represents the random characteristic: a random perturbation in theflow, on either side of the cylinder, may initiate the phenomenon.In any case, the random nature should not be surprising. In fact,it is also typical, for example, of side-by-side cylinders andconstitutes one of the paradoxes cited in Zdravkovich (2003)(pp. 1027): “an entirely symmetrical oncoming flow leads to theasymmetric narrow and wide wakes behind two identical side-by-side cylinders and a uniform and stable flow induces a non-uniform and random bistable flow. The origin of bistable biasedflow has been attributed to various causes, but still remainsunsolved”.

Figs. 4 and 5 showed the most significant results. However,many more conditions were tested, as described in Section 2.1.By using different wind tunnels, different atmospheric boundarylayers and turbulence conditions of the incoming flow (bothturbulence intensity and length scales) were applied. Uniform

Fig. 4. Time history of Cp at z¼950 mm and φ¼1001, 10 rings (WiSt wind tunnel, surface roughness R1, Re¼2.5�105).

Fig. 5. Time history of Cp at z¼950 mm and φ¼1001, 10 rings (Criaciv wind tunnel, surface roughness R1, Re¼2.8�105).

Fig. 6. Mean lift coefficient at z¼950 mm vs Re, 10 rings (WiSt).

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flow was also tested at WiSt wind tunnel, as well as theinfluence of Re and surface roughness R0–R5 (including smoothsurface) on the cylinder with rings. The phenomenon is influ-enced by the surface roughness conditions, but the intrinsicphysical cause is neither created nor destroyed by the presenceof ribs along the height. Therefore, the whole investigation ofthe phenomenon in different flow conditions has suggested thatnothing but the presence of rings – at a proper distance – is themain cause of bistable flow. This is also proved by the progres-sive disappearance of the phenomenon as the distance betweenrings increases. It has experimental evidence: with only fiverings instead of 10, the bi-stability is lost (Fig. 7). In the limitingcase (no rings), the phenomenon is completely absent. For adetailed description of the full investigation the reader isreferred to Lupi (2013).

4. Characterization of the time histories of thepressure coefficients

Characterization and modeling of bistable time histories areimportant in view of a dynamic calculation of the structure (seeSection 5 for practical design application).

The time histories of the recorded pressure coefficients aremodeled as the response of proper non-linear conservative sys-tems to a white noise; the potential energy function of eachsystem is characterized by one or two wells.

Such time histories are computed from the recorded pressuresby the relation:

CpðtÞ ¼pðtÞ�p02ρv2m

ð1Þ

where p(t) is the recorded pressure, p0 the atmospheric pressure inabsence of air flow, ρ is the air density and eventually vm the meanflow velocity at the level of pressure measurement.

Thus, a single degree of freedom system is determined for eachcoefficient. Such approach can subsequently be used in order tocompute the statistics of the tower response by means, for instance,of equations or similar approaches.

The equation of motion of the considered system will thereforebe of the type:

€xþHðEÞ_xþgðxÞ ¼ f ðtÞ ð2ÞThe total energy of the system is given by the expression

Eðx; _xÞ ¼ _x2=2þΦðxÞ where Φ(x) is the system potential energygiven by ΦðxÞ ¼ R x

0 gðuÞ du. If the forcing process can be modeled

as a stationary white noise of intensity S0, a solution can be foundin literature due to Kramers (1940) successively expanded byCaughey (1964):

px;_xðx; _xÞ ¼ Cexp � 1πS0

Z E

0HðuÞ du

� �ð3Þ

where E is the total energy of the system, as above, and C anormalizing constant.

This solution was subsequently applied to two-wells systemsby several Authors, such as Alaoui Ismaili and Bernardt (1997),who considered a system endowed with linear viscous damping:

HðEÞ ¼ 2νω; px;_xðx; _xÞ ¼ Cexp �2νωπS0

Eðx; _xÞ� �

; Eðx; _xÞ ¼ _x2

2þΦðxÞ

ð4ÞIn this case, the system velocity and displacement are inde-

pendent as the their joint distribution can be written as

px;_xðx; _xÞ ¼ Cexp �2νωπS0

_x2

2

!exp �2νω

πS0ΦðxÞ

� �ð5Þ

Thus obtaining a distribution for the system displacement inthe form

pxðxÞ ¼ Cxexp �2νωπS0

ΦðxÞ� �

ð6Þ

and for the system velocity as

p_xð_xÞ ¼ C _xexp � _x2

2s2_x

!; s2_x ¼

πS02νω ð7Þ

It is interesting to notice that the velocity is characterized by aGaussian distribution, but cannot be considered a Gaussian pro-cess, as its integral, the displacement, is not Gaussian.

In the following, it will be assumed that any pressure coeffi-cient can be modeled as the response of a system characterized byan equation as (4), namely

€cpþ2νω _cpþgðcpÞ ¼ f ðtÞ ð8Þ

The pressure coefficient taken into consideration for the illus-tration of the method is the one derived from a tap placed inproximity of a stiffening ring, and identified by an angle ϑ¼ 1001with respect to the incoming flow direction. An example of thecomputed cp for such tap is shown in Fig. 8. This sample case,measured close to a ring beam, is selected for the purpose ofvalidating the model because it shows – within the time window ofthe measurement – an interesting and articulated series of jumps.

In order to identify the system parameters, the histograms ofthe pressure coefficient and its time derivative will be compared tothe theoretical distributions (6) and (7).

The first comparison carried out was the one concerningthe derivative with respect to time; as it is clear from Fig. 9, thedistribution of the derivative of the pressure coefficient can bewell considered zero-mean Gaussian Fig. 10.

This means that system is affected by linear viscous damping andthe pressure coefficient is statistically independent on its derivative.

The histogram of cp(t) has therefore to be compared to theprobability distribution given by (6).

Several expressions have been considered for such distribution,namely:

1. A polynomial approximation:

pcðcÞ ¼ Ccexp⌊�Pð8ÞðcÞ⌋ ð9Þwhere P(8) stands for an 8th degree polynomial and Cc is anormalizing constant.Fig. 7. Histograms of lift coefficient at z¼950 mm, 10 and 5 rings (WiSt).

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2. A Gram–Charlier III distribution (GC3 in the following):

pcðcÞ ¼ Ccexp½Ψ ðcÞ�; Ψ ðcÞ ¼ ∑N

k ¼ 1

1kγkH

ðkÞðcÞ ; c¼ c�μcsc

ð10Þ

where H(k) is the kth order Hermite polynomial; mc and sc are,respectively, the mean and standard deviation of the consid-ered coefficient. γk are coefficients to be determined accordingto Muscolino et al. (2003).

3. A generalized regression function (GRF in the following, see f.i.Specht 1995):

pcðcÞ ¼∑N

h ¼ 1phexp½�12ððc�chÞ=sÞ2�

∑Nh ¼ 1exp½�1

2ððc�chÞ=sÞ2�ð11Þ

where ch and ph are, respectively, the centers and the asso-ciated probabilities of the histogram of the computed pre-ssure coefficients. N is the number of classes of the histogram.

Fig. 8. Time history of the pressure coefficient.

Fig. 9. Comparison between the histogram and the theoretical distribution of the derivative of the pressure coefficient.

Fig. 10. The histogram and the results obtained for the various expressions used to approximate the distribution of the pressure coefficient.

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4. A superimposition of Gaussian bells in the form (see Nelles2001):

pðcÞ ¼ 1Ns

ffiffiffiffiffiffi2π

p ∑N

h ¼ 1exp �1

2c�chs

� �2" #

ð12Þ

where ch is the hth value of the coefficient time history. N is therecord length and s a decay coefficient to be tuned by the user.

The distribution of the pressure coefficient, combining togetherEqs. (6) and (7), can be written as

pcðcÞ ¼ Ccexp � 1s2_c

ΦðcÞ !

ð13Þ

and therefore, by inversion,

ΦðcÞ ¼ �s2_c f ln ½pcðcÞ�� lnðCcÞg ð14Þ

Eventually, on derivation, it can be obtained that

gðcÞ ¼Φ′ðcÞ ¼ �s2_cp′cðcÞpcðcÞ

ð15Þ

Once the probability distribution of the pressure coefficient isdetermined, the restoring force can be derived by derivation as inEq. (15); as the best fit was obtained for the superimposition of theGaussian bells, the restoring force has been determined in thiscase, and is shown in Fig. 11.

Once the restoring force is obtained, from the equation ofmotion (8) the forcing process can be obtained in the form

f ðtÞ ¼ €cþπS0s2_c

_cþgðcÞ ð16Þ

Its time history is shown in Fig. 12; in order to cross-test theprocedure, the histogram and power spectral density of the forcingprocess were computed, obtaining that the forcing process canactually be modeled as a Gaussian white noise, as shown in Figs. 13and 14.

5. Structural application

Even though the results presented in the paper are of generalvalidity, the investigation was performed for specific application toultra-high cylindrical towers, such as Solar Updraft Towers. In thiscase, the presence of rings along the height at a distance com-parable to the diameter is a necessary design feature. So, theexperiments presented in this paper aimed at investigating theaerodynamic of the flow in such a design condition. The full-scaleRe of such ultra-high towers is in the order of 108. In the windtunnel, the concept of effective Re was applied by adding surfaceroughness (Section 2.2). However, tests at higher Re would berecommended in future research.

The Solar Updraft Power Plants Technology (SUPPs) is a newsustainable natural resource for electric power generation. It

Fig. 11. The restoring function obtained for the superimposition of Gaussian bells.

Fig. 12. The time history of the identified forcing process.

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produces renewable energy by sun-wind energy harvesting.The working principle and the main features of the power plantare described in Lupi et al. (2011). In particular, a SUPP consistsof three elements: the collector, the turbine(s) with coupledgenerators as power conversion unit and the solar tower. Thecollector is a large glass-covered area, where solar radiationheats the ground and consequently warms up the air under theroof. The heated, less dense buoyant air rises up into thechimney of the plant. The stream of warm air turns the turbinesat the chimney foot and in the power conversion unit thekinetic energy of the flow is then transformed into electricpower. The production of energy is proportional to the volumeof the cylinder having the height of the tower and the diameterof the collector (Schlaich et al., 2005). For this reason, provideda sufficiently high solar radiation input (e.g. 2000 kW h/m2 oreven more), a very good efficiency of the power plant can bereached with extra-large dimensions of the tower and/or thecollector.

Because of that, Solar Updraft Towers are slender and extre-mely thin shells, up to 1 km or even 1.5 km in height, usually madeof reinforced concrete. Their shape is cylindrical and it may turninto a hyperboloid in the lowest part, in order to apply the benefitsof shape strengthening (Krätzig et al., 2008, 2009). The wallthickness of high-performance reinforced concrete (C70/85) variesfrom 25 cm until 60 cm. In order to reduce ovalizations of thecross-section and guarantee a beam-like behaviour, stiffeningrings must be applied along the height of the tower.

As shown by the previous results, the stiffening rings along theheight might be responsible for creating the bistable flow condi-tion, depending on their position and size. Therefore, despite theimproved structural behaviour, this may induce an even moresevere load condition.

6. Conclusions

The paper reveals the existence of a new type of bistable flowaround circular cylinders with a free-end. It is induced by ringbeams along the height. Although some analogies can be foundwith the well-known bistable flows described in literature, theconditions of occurrence are completely different. The debatebetween literature and novelty, which characterizes this newphenomenon, has been addressed in the paper. Issues likeReynolds number and surface roughness are discussed. Theauthors would recommend, for future research, wind tunnel testsin pressurized wind tunnels at transcritical Re on a smoothcylinder.

In the second part of the paper, the time history of a genericpressure coefficient has been modeled as the response of a non-linear system endowed with linear viscous damping and a propernon-linear restoring force. The expression of the restoring forcewas modeled by means of the closed form solution given byKramers and Caughey. The forcing process was modeled as aGaussian white noise: this fact is particularly important as it

Fig. 13. The histogram, together with the pdf, of the identified forcing process.

Fig. 14. The power spectral density of the identified forcing process.

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enables to use the equations or similar approaches to compute astatistical approximation of the response of the tower.

A structural application for this investigation is illustrated inSection 5.

References

Alaoui Ismaili, M., Bernardt, P., 1997. Asymptotic analysis and linearization of therandomly perturbed two-wells Duffing oscillator. Probabilistic EngineeringMechanics 12 (3), 171–178.

Caughey, T.K., 1964. On the response of a class of nonlinear oscillators to stochasticexcitation. Proc. Coll. Int. du Centre Nat. de la Rechercher Scient. 148, 393–402.(Marseille, France).

Bearman, P.W., 1969. On vortex shedding from a circular cylinder in the criticalReynolds number regime. Journal of Fluid Mechanics 37 (3), 577–585.

Counihan, J., 1969. An improved method of simulating an atmospheric boundarylayer in a wind tunnel. Atmospheric Environment 3 (2), 197–214.

Eisner, F., 1925. Pressure measurements on cylinder surrounded by flowing fluid (inGerman). Zeitschrift Fuer Angewandte Mathematik und Mechanik 5, 486–489.

ESDU 80025 1986, Mean forces, pressures and flow field velocities for circularcylindrical structures: single cylinder with two-dimensional flow. EngineeringSciences Data Unit, London.

Farell, C., Blessmann, J., 1983. On critical flow around smooth circular cylinders.Journal of Fluid Mechanics 136, 375–391.

Kramers, H.A., 1940. Brownian motion in a field of force and the diffusion model ofchemical reactions. Physica 7, 284–304.

Krätzig, W.B., Harte, R., Wörmann, R., 2008. Large shell structures for powergeneration technologies. In: Proceedings of the Sixth International Conferenceon IASS-IACM 2008. Cornell University, Ithaca, NY, USA.

Krätzig, W.B., Harte, R., Montag, U., Wörmann, R., 2009. From large natural draftcooling tower shells to chimneys of solar upwind power plants. In: Proceedingsof the International Association for Shell and Spatial Structures (IASS). Valencia,Spain.

Lupi, F., Borri, C., Krätzig, W.B., Niemann, H.-J., 2011. Solar Updraft Power Planttechnology: basic concepts and structural design. In: Encyclopedia Online ofLife Support Systems (EOLSS) Developed Under the Auspices of the UNESCO,Eolss Publishers, Oxford, UK.

Lupi, F., 2013. A New Aerodynamic Phenomenon and its Effects on the Design ofUltra-high Cylindrical Towers. Ph.D. dissertation. University of Florence, TUBraunschweig.

Mahbub Alam, Md., Sakamoto, H., Moriya, M., 2003. Reduction of fluid forces actingon a single circular cylinder and two circular cylinders by using tripping rods.Journal of Fluids and Structures 18 (3–4), 347–366.

Mahbub Alam, Md., Meyer, J.P., 2011. Two interacting cylinders in cross flow.Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 84, 5.

Muscolino, G., Ricciardi, G., Cacciola, P., 2003. Monte Carlo simulation in the stochasticanalysis of non-linear systems under external stationary Poisson white noiseinput. International Journal of Non-Linear Mechanics 38, 1269–1283.

Nelles, O., 2001. Nonlinear System Identification. Springer, Berlin.Roshko, A., 1961. Experiments on the flow past a circular cylinder at very high

Reynolds number. Journal of Fluid Mechanics 10 (3), 345–356.Schewe, G., 1983. On the forces acting on a circular cylinder in cross-flow from

subcritical up to transcritical Reynolds numbers. Journal Fluid Mechanics 133,265–285.

Schlaich, J., Bergermann, R., Schiel, W., Weinrebe, G., 2005. Design of commercialsolar updraft tower systems—utilization of solar induced convective flows forpower generation. Journal of Solar Energy Engineering 127, 117–124.

Specht, D.F., 1995. General regression neural network. IEEE Transactions on NeuralNetworks 2 (6), 568–576.

Sumner, D., 2010. Two circular cylinders in cross-flow: a review. Journal of Fluidsand Structures 26 (6), 849–899.

VGB, 2010. Structural Design of Cooling Towers. Guideline VGB R 610e, VGBPowerTech, Essen.

Zdravkovich, M.M, 1997. Flow Around Circular Cylinders, vol. 1. Oxford UniversityPress (Fundamentals).

Zdravkovich, M.M, 2003. Flow Around Circular Cylinders, vol. 2. Oxford UniversityPress (Applications).

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