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Journal of Fluids and Structures

Journal of Fluids and Structures 39 (2013) 15–26

0889-97

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n Tel.:

E-m

journal homepage: www.elsevier.com/locate/jfs

Reynolds-number-effects in flow around a rectangularcylinder with aspect ratio 1:5

Gunter Schewe n

Deutsches Zentrum fur Luft- und Raumfahrt (DLR) – Institut fur Aeroelastik, Bunsenstrasse 10, 37073 Gottingen, Germany

a r t i c l e i n f o

Article history:

Received 21 March 2011

Accepted 11 February 2013Available online 18 March 2013

Keywords:

Reynolds number

Bluff body

Rectangular cylinder

Flow separation

Vortex resonance

46/$ - see front matter & 2013 Elsevier Ltd.

x.doi.org/10.1016/j.jfluidstructs.2013.02.013

þ49 551 709 2423.

ail address: guenter.schewe@dlr.de

a b s t r a c t

The paper reports on experiments carried out over a wide range of Reynolds numbers in

a high pressure wind tunnel. The model was a sharp-edged rectangular cylinder with

aspect ratio height/width 1:5 (width/span ratio 1:10.8), which was investigated in both

basic orientations, lengthwise (4�103oReo4�105) and perpendicular to the flow

(2.7�104oReo6.4�105). The Reynolds number is based on the height of the model

normal to the flow. Steady and unsteady forces were measured with a piezoelectric

balance. Thus along with steady (i.e. time averaged values) including the base pressure

coefficient, also power spectra and probability density functions were measured yielding

for example Strouhal numbers, higher statistical moments, etc. A response diagram for

the vortex resonance phenomenon was taken for the natural bending motion of the

slender model. If lift coefficient for constant angle of attack is plotted against Reynolds

number, a significant Reynolds number effect is seen. For a¼41, the curve shows an

inflection point and the lift varies between 0.3 and 0.6. For a¼61 and 21 there are similar

variations shifted to lower and higher values of Re, respectively. Probably the shapes of

separation bubbles that depend on the Reynolds number are responsible for these

effects. No Reynolds number effects were observed when the long side was normal to the

flow, an orientation where reattachment at the side walls is not possible. Comparing

both basic cases (a¼01 and 901), the interpretation of the probability distributions of lift

force leads to the conclusion that the possibility of reattachment (a¼01) seems to

enhance the degree of order in the vortex shedding process.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The flow around two-dimensional bluff bodies is determined by flow separations that result in a vortex street in the wake.The flow topology and the time dependent behaviour of the vortex shedding process are thus strongly dependent on the cross-section of the bodies. The circular cylinder is the prime example of bluff bodies, representing rounded geometries, which havethe highest degree of symmetry. The rectangular cylinder, including the special case of the square section, is the prime examplefor a sharp-edged bluff body with a high degree of symmetry. For the circular cylinder, it is well known that the Reynoldsnumber dependence plays a dominant role because the location of the separation varies due to the continuous, rounded shape.The flow around a rectangular cylinder is often seen as an example where the Reynolds number plays a minor role because thelocation of the separation is fixed at the sharp edges. This is no longer the case, however, when the separated flow has the

All rights reserved.

G. Schewe / Journal of Fluids and Structures 39 (2013) 15–2616

possibility of reattachment. In this case the Reynolds number Re can play an important role regarding the formation and shapeof separation bubbles. This has already been observed and explained for the clean trapezoidal ‘‘Belt Bridge Deck Section’’(Schewe and Larsen, 1998). The reason for this dependence is that the behaviour of the flow around a cylinder with an inclinedfront plate is similar to the case of a circular cylinder, where the laminar/turbulent transition occurs in the free shear layers inthe wake. With increasing Reynolds number the location of transition moves upstream. If, at critical Reynolds numbers, thetransition in the free shear layers is close to the surface, reattachment is possible, forming a separation bubble (Fig. 1). Thelength, shape and stability of such separation bubbles is again dependent on the Reynolds number with consequences forglobal values like force coefficients or Strouhal numbers. Up to a certain degree this scenario seems to be universal as it can beobserved in flow over bodies having quite different geometries (Schewe, 2001).

In particular when the symmetry is broken, i.e. the cross-section is asymmetric as for the Belt section or the angle ofincidence a of a symmetric body is not zero, the behaviour of the separated flow that depends on the Reynolds number canbe different on the upper- and the lower-side of the section. Thus global values, like the lift coefficient Cl, for example, canbe affected by the Reynolds number.

The model investigated was a rectangular cylinder with aspect ratio 5. This sharp-edged generic cross-section wasselected, since often it is seen as a test case for bluff bodies and also for bridge aerodynamics and bridge aeroelasticity.Such a benchmark test case is called BARC, which stands for Benchmark on the Aerodynamics of a Rectangular 5:1 Cylinder(http://www.aniv-iawe.org/barc/; Paıdoussis, 2009). In addition in Schewe (1989) non-linear resonances of a genericsection of the Tacoma bridge were investigated in the same experimental environment and similar structural properties.Thus because of the same outer dimensions, the changes of aerodynamic or aeroelastic behaviour between an H-shape anda corresponding rectangular cylinder (due to filled gaps) can be studied. For this reason a response diagram was also takenfor the natural bending motion of the slender rectangular cylinder, which was fixed at both sides in the stiff balance. Thusa comparison of the vortex resonance behaviour can be made to the Tacoma section.

For rectangular cylinders at zero incidence, the chord-to-thickness ratio c/t is an important parameter, where c is thelength in flow direction and t the crosswind dimension. This ratio influences the type of the wake topology and the kind ofvortex shedding. According to Parker and Welsh (1983), Stokes and Welsh (1986) and Nakamura et al. (1991) the selectedratio c/t¼5 belongs to the regime of longer plates, where the flow separation occurs at the leading edge corner. The freeshear layer then reattaches to the trailing edge more or less periodically in time and then forms a regular vortex street in thewake. For values c/to3.2 there is no reattachment to the plate surface and for c/t47.6 the shear layer is always reattached.

The measurements are focused on a first case, where the section is oriented lengthwise c/t¼5, but in addition the casewhen the longer side of the rectangular cross-section is normal to the flow c/t¼0.2 was also investigated. This is a casewithout reattachment and thus an example for the absence of Reynolds number effects.

The measurements were carried out over a wide range of Reynolds numbers (4�103oReo6�105) in the high-pressure wind tunnel in Gottingen. Very high Reynolds numbers can be achieved by increasing the pressure in the windtunnel tube up to 100 bar. Steady and unsteady forces were measured with a piezoelectric balance whose majorcharacteristic is its high stiffness, thus the force measuring system is particularly useful for investigating unsteadyphenomena (Schewe, 1983, 2007). With the same model and balance, variation of the Reynolds number is possible overalmost three orders of magnitude by merely changing the flow parameters flow speed uN and pressure p. The results willbe interpreted, inspired by the numerical simulations of Mannini et al. (2010), which showed a Reynolds numberdependent behaviour of a separation bubble.

2. Experimental set-up and the model

The same model was originally used as a generic Tacoma section applied in an investigation, described in Schewe(1989), but here the gaps were filled with plastic forming a rectangular cylinder (Fig. 2). Consequently all detailsconcerning the experimental arrangement are the same or similar as described ibidem.

The wind tunnel was built with the purpose of achieving very high Reynolds numbers in incompressible flow. Theentire wind tunnel tube, which is of the closed return type, can be pressurised up to 100 bar, which allows maximumReynolds numbers of 107. The maximum flow speed is U¼38 m/s, the size of square test section is 0.6�0.6 m2 and the

laminar BLtransition

turb.reattachment

recirculationregion

laminarseparation

Flow

Fig. 1. Formation of a separation bubble. The separated boundary layer BL is a free shear layer, in which the laminar/turbulent transition can occur. If the

location of transition is close to the surface, the now turbulent BL can reattach, forming a separation bubble.

l = 5h

h

cpb

L

DMαU 8

Fig. 2. Dimensions of the rectangular cylinder and definition of the coordinate system (L, lift; D, drag; M, moment; Cpb, base pressure coefficient ). Width l

and height h are fixed at the model. The dashed lines show the relation to the Tacoma section.

G. Schewe / Journal of Fluids and Structures 39 (2013) 15–26 17

contraction ratio is 5.6. For the purpose of achieving good flow quality, there are guide vanes, a honeycomb and screens.The turbulence intensity of the flow increases slightly with increasing flow speed and pressure, respectively, but remainsless than 0.4% for the range investigated in this study.

General points concerning the high pressure wind tunnel, the piezo balance and the entire environment can be found inSchewe (1983, 2007). The main feature of the piezo balance is its inherent high stiffness, leading to high naturalfrequencies. This property is particularly important for the measurement of values, which are the result of time seriesanalysis, like frequencies, rms values, higher statistical moments, etc.

The sharp edged model in Fig. 2 has a width of l¼55.4 mm, height h¼11 mm and a span of 600 mm, thus the aspectratio span/width is 10.8. As l and h are fixed for the model, in the case of a¼01 width l corresponds to the chord length c

(length in flow direction) and h corresponds to thickness t (crosswind dimension). For the case a¼901 the allocation is c¼h

and t¼ l. The radius of curvature of the corners was measured at different positions and had an average value of about20 mm. The model was fixed at both ends in the rigid piezo balance such that the lowest frequency of 99 Hz (at 1 bar) wasdetermined purely by the bending of the slender model (height/span¼1/55). This value decreased to 94 Hz at a pressure of70 bar. In the lengthwise orientation the time averaged base pressure coefficient Cpb was measured on the back side of themodel in addition to the measurement of the steady and unsteady forces and moments. At angle of incidence a¼01 theblockage of the model is only 1.83%, thus in case of lengthwise orientation no correction for blockage effects was applied.In this orientation, i.e. 01oao61 as usual for bluff bodies the crosswind dimension, i.e. height h, is the reference length forthe drag coefficient Cd, the Strouhal number St and the Reynolds number Re. For the lift and moment coefficient Cl, Cm andthe rms of lift Cl rms, the longer side of the section l is the reference length. For the special case of a¼901 the longer side, i.e.width l (Fig. 2) is the characteristic length for Cd, the Strouhal number and the Reynolds number Re. For this orientation therms of lift Cl rms is non-dimensionalized by the width h of the side wall. For the case a¼901 the blockage is 9.15%. Estimatesof the effect on the drag coefficient and the flow speed were made as described in Section 3.2.1, but all results ofcoefficients presented in the figures are uncorrected.

3. Results

3.1. Section lengthwise oriented (chord/thickness ratio¼5)

3.1.1. Vortex resonance phenomena

Since the model, fixed at both ends in the stiff piezo balance, is allowed to vibrate in its natural bending mode, aresponse diagram depending on flow speed can be taken. In Fig. 3 the rms-value of the lift fluctuations is plotted againstthe flow speed for different pressures up to 70 bar. In this case the section is oriented lengthwise at an angle of incidence ofa¼01. As the force measurement is a global one the signals include contributions of inertia forces in addition to theaerodynamic forces, in particular near the resonances. The main response for vortex resonance can be seen arounduN¼10 m/s. For safety reasons the vortex resonance range was not crossed for pressures higher than 5 bar. Due to the lowdamping of the steel model the amplitudes in the resonance case would become too large even when the crossing istransient. Thus only for p¼1 bar measurements were taken within the main resonance.

It is interesting to compare the present results with corresponding measurements using the generic Tacoma section inthe same experimental environment with similar structural properties (Schewe, 1989). At first glance the peak value of themain resonance is much less pronounced and secondly no superharmonic resonances occur as in the case of the Tacomasection. Nevertheless it is estimated that a typical rms-value of lift far below resonance will have a value aroundCl rms¼0.08. The numerical results reported in Mannini et al. (2011) gave values in close agreement with this estimate,using an advanced method of turbulence modelling (Detached Eddy Simulation, DES), with a spanwise extension of thecomputational domain, i.e. aspect ratio span/width of at least 2.

10 20 30t [s]

U 8 increasing

-200

-100

0

100

200

L [N]

0 10 20t [s]

U 8 decreasing

Fig. 4. Time function of lift force, when the main resonance is crossed by a flow speed sweep. Left, increasing; right, decreasing.

0 10 20 300.01

0.10

1.00

Uoo[m/s]

1 bar510203040506070

Clrms

Fig. 3. The rms-value of lift depending on flow speed and pressure in the wind tunnel (a¼01). The curve corresponds to a response diagram of bending

motion, since the unsteady forces contain inertia contributions in addition to the aerodynamic contributions.

G. Schewe / Journal of Fluids and Structures 39 (2013) 15–2618

In general the unsteady forces measured by the balance contain inertia contributions in addition to the aerodynamicforces. In particular the increase in the resonance is mainly caused by the inertia forces of the oscillating bending motion.A further aerodynamic contribution could be produced by motion induced effects.

Fig. 4 gives an impression of the vortex resonance effects, when the main resonance is crossed by a flow speed sweep,i.e. a transient excitation by a vortex shedding frequency sweep. The left figure shows the time function of the lift forincreasing- and the right for decreasing-flow speed. The angle of incidence is a¼41, which can be seen in the small offset ofthe averaged lift force at about t425 s in the left and for about tZ0 in the beginning of the right time function.Unfortunately the time function of the instantaneous flow velocity during the sweep was not measured. Nevertheless thedifferent shapes reflect a small asymmetry of the resonance curve, which is bent to the right side. In addition hysteresisseems to be responsible for the difference in the maximal amplitudes. It should be remarked that the describedphenomena were caused by vortex resonance. Self-excited torsional oscillations of rectangular cylinders, occurring atlower frequencies are investigated and described by Robertson et al. (2003) and Matsumoto et al. (2005).

3.1.2. Steady force coefficients and Strouhal number

Fig. 5 shows the dependence on Reynolds number and angle of incidence a of the time averaged force/momentcoefficients Cd, Cl, Cm, the Strouhal number St and the base pressure Cpb. For the two angles of incidence a¼01 and 41 theReynolds number was varied in the range 6�103oReo4�105 by increasing the pressure from 1 to 70 bar and flowspeeds ranging between 3 and 8 m/s. Restriction to this low flow speed was necessary to keep possible resonance effectswithin reasonable limits. For the two angles of incidence a¼21 and 61 the Reynolds number was extended only up toRe¼1.2�105 for time and economic reasons.

The Strouhal numbers were determined from the power spectra of the lift forces, and examples will be shown later. Thereare no visible Reynolds number effects at the angle of incidence a¼01 for the drag coefficient Cd and the Strouhal number.Zooming in, a slight curvature with opposite sign is visible in both lines Cd(Re) and St(Re). This inversed behaviour was to beexpected. Comparing the drag coefficient for zero incidence (Cd¼1.03, a¼01) at corresponding Reynolds numbers, it agreeswith the results of Nakamura and Mizota (1975) and also with the numerical simulation of Shimada and Ishihara (2002).A corresponding comparison regarding the Strouhal number (St¼0.111) with the two last investigations shows agreement, alsoOkajima, cited in Shimada and Ishihara (2002), found experimentally St¼0.106, but Matsumoto et al. (2003) obtained a higher

104

Reh

0.81.01.21.41.6

Cd

0.105

0.110

0.115

0.120

St

-0.10

-0.05

0.00

0.05

Cm

0.0

0.2

0.4

0.6

Cl

-0.6

-0.4

-0.2

Cpb

α [°]0246

105 106

Fig. 5. Strouhal number, force/moment coefficients and base pressure depending on Reynolds number and angle of incidence a for the rectangular

section. No blockage correction was applied.

G. Schewe / Journal of Fluids and Structures 39 (2013) 15–26 19

value of St¼0.132. One reason could be the different aspect ratio of the models (span/width), which was 3:1 and in the presentcase more than 10:1. Comparing with the 2D numerical URANS simulations of Mannini et al. (2010), in which two differentturbulence models were applied, it can be seen that both turbulence models give Strouhal numbers slightly smaller than theexperimental results, while the corresponding drag coefficient is in one case about 14% higher and in the other case 5% smallerthan the experimental value. In Mannini et al. (2011) the more advanced 3D DES approach was applied to the same test case,giving practically the same drag coefficient as in the present experiments and a Strouhal number 7% lowers the valuemeasured here.

The values of the moment are very small and thus difficult to measure, in particular at low Reynolds numbers. Thus themoment coefficients Cm are included only for Reynolds numbers larger than about 2�104. For a¼41 the momentcoefficients are negative and nearly constant, while for a¼21 and 61 there is a small increase in the negative direction,when Reynolds number is increasing.

For angles of incidence aa01 (except for the special case a¼901) a significant Reynolds number effect can be seen, inparticular in the curve of the lift coefficient. In the case of a¼41 the curve shows an inflection point and the lift coefficientvaries between 0.3 and 0.6. In addition the trend to lower or to higher Reynolds numbers indicates a further decrease orincrease of the lift coefficient, respectively. For all a there is a significant increase of the lift in the lower range Reo2�104,at which the range of rise is shifted to lower Reynolds number, when a is increased. Considering these measurements, it isnot yet clear at which angle of incidence the Reynolds number effects would cease.

For angle of incidence a¼41 the numerical simulations of Mannini et al. (2010) show the following: for all turbulence modelsthe deviation in Strouhal number and drag coefficient from experiments is below 13% and in most cases less than 10%.

G. Schewe / Journal of Fluids and Structures 39 (2013) 15–2620

The numerical simulations show that applying three different eddy-viscosity turbulence models, the lift coefficients at a¼41 arevirtually independent of Reynolds number, and some agreement with experiments can be found only at low Reynolds numbers.Only if the EARSM-LEA turbulence model (Explicit Algebraic Reynolds Stress Model, Rung et al., 1999) is used, a lift increase of 26%with Reynolds number can be observed, although the increase is not as strong as in the experiments.

In order to illuminate the Reynolds number effects from a different point of view, Fig. 6 shows the characteristicdiagrams for Cd(a), Cl(a) and Cm(a) against Reynolds number. Since the measurements for different a were not taken at thesame Reynolds numbers, the data were interpolated from the compensating curves displayed in Fig. 5. Even if the data areof reduced accuracy and the number of supporting points is low, the curves show significant Reynolds number effects inparticular in the case of the lift coefficient. With increasing Reynolds number the curves are more and more bent,indicating the formation of a (local) maximum a little bit before aE61, and the slopes qCl/qa at a¼01 increase from5.7 rad�1 at Re¼6�103, 7.8 rad�1 at Re¼1.2�104 to 9.3 rad�1 at Re¼6�104. The maximum of the lift seems to beshifted to lower a, when Re is increasing. The moment characteristic should not be over-interpreted, but considering allmeasuring points up to a¼41, the slope is qCm/qa¼�0.66 rad�1. In both cases the curves Cm(a) are bent down for thelower Reynolds number. Additionally the dependence of the drag on a is shown, although the Reynolds number effects arenot so strong. The curve is single bent at low and shows an inflection point at the higher Reynolds number.

The knowledge of the derivatives allows a rough estimate concerning the streamwise location of the centre of pressureDx using qCm/qCl¼�Dx/l¼0.66/7.8¼0.09. That means the lift force attacks roughly 10% of the chord length l downstreamof the midpoint of the section. This value is nearly the same as in the case of a Tacoma section (Schewe, 1989).

3.1.3. Power spectra and probability density distributions

Fig. 7 displays two representative power spectra of lift fluctuations for two angles of incidence. In both cases there arenarrow peaks due to the vortex shedding, which have nearly the same Strouhal number St¼0.11. The bending frequency ofthe model is visible at fE94 Hz, and its power spectral density is more than 20 db below the Strouhal peak. Furthermore,for the asymmetric case (a¼41, pressure in the wind tunnel p¼70 bar, flow speed uN¼5.7 m/s) the low frequency part ismore pronounced and a significant superharmonic occurs. No parasitic resonances of the balance are present. For a¼01 theflow parameters were as follows: p¼50 bar, uN¼3.9 m/s. The Reynolds numbers, noted in the figures, are around 2�105.

Fig. 8 shows the probability density functions (PDF) of lift fluctuations L(t) for two angles of incidence a and the sameflow conditions as in Fig. 7. For comparison, also a Gaussian distribution is included. For a¼41 (solid line), the asymmetriccase, a significant skewness of the PDF can be seen. The probability for the occurrence of positive high amplitudes up to 3rms-values is significantly higher than for corresponding negative ones. This is reflected in a positive third moment,the skewness factor with a value of 0.2. The fourth moment, the flatness factor, is 2.1. The corresponding values for

0 1 2 3 4 5 6 70.8

1

1.2

1.4

1.6

1.8

α [°]

Cd

00.10.20.30.40.50.6

Cl6e3 5.7

1.2e4 7.86e4 9.3

00.010.020.030.040.050.06

-Cm

Reh Cl α

Cmα= -0.66

Fig. 6. Force and moment coefficients depending on the angle of incidence a with the Reynolds number as parameter. Cla and Cma are the derivatives at

a¼01, which are the key parameters for the aeroelastic stability.

0 40 80 12010-2

100

102

104

0 40 80 120

100

102

104

Φl

[N2 /H

z]

f [Hz]

2 fV

bending

f [Hz]

Φl

[N2 /H

z]

St = 0.114 α = 4°α = 0°

Reh = 2.6 x 105Reh = 1.3 x 10 5

bending

St = 0.113

Fig. 7. Power spectra of lift for rectangular section for two angles of incidence a.

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

L/Lrms

Pro

babi

lityd

ensi

ty

Gaussian

α=0°

α=4°

Fig. 8. Probability density functions of lift fluctuations for two angles of incidence.

104 105 1060.0

0.5

1.0

1.5

2.0

2.5

3.0

Cd

Cd

Rel

St St

0.30

0.25

0.20

0.15

0.10

0.05

0.00

α=90°

Cl

1.5

1.0

0.5

0.0

l

C l

U 8

Fig. 9. Drag coefficient, Strouhal number and the rms-value of the lift fluctuations depending on the Reynolds number. No blockage correction was

applied.

G. Schewe / Journal of Fluids and Structures 39 (2013) 15–26 21

the symmetric case a¼01 are skewness¼0.01 and flatness¼1.85, the dotted curve is nearly symmetric around zeroamplitude.

3.2. Section with the long side normal to the flow (chord/thickness ratio¼0.2 )

3.2.1. Analysis of global values

The incidence angle of a¼901 is a special case of bluff body aerodynamics, which is quite near to a two-dimensional flatplate normal to the flow, i.e. the crosswind dimension is much larger than the lengthwise dimension, the chord length.In such flows around sharp edged cross-sections, there is no possibility for the formation of larger separation bubbles atthe side faces and thus minimal Reynolds number effects should be expected.

Fig. 9 displays the drag coefficient, the Strouhal number and the rms-value of the lift fluctuations for the range2.7�104oReo6.4�105. For this special case, the characteristic length for Cd and St is always width l, which is thecrosswind dimension, when a¼901. The pressure in the wind tunnel was increased from 1 up to 12 bar. The results showthat, there is no systematic variation depending on the Reynolds number.

The mean drag coefficient Cd¼2.4 is roughly 14% higher than for a square cylinder, which was investigated in the sametest set-up (Schewe, 1984). The mean Strouhal number St¼0.152 and the mean rms-value of the lift Cl rms¼0.70. Thatmeans the rms-value, here normalised with height h is much higher than for the lengthwise orientation. Even normalised

G. Schewe / Journal of Fluids and Structures 39 (2013) 15–2622

by the longer side l, the rms of the lift fluctuations would be significantly higher than in case of lengthwise orientation.Comparing with a square cylinder (Schewe, 1984), then the rms of lift Cl rms of the rectangular cylinder is about 12% lower.

Since the wind tunnel blockage in this position is no longer small (9.15%), an estimate of the effect on drag and flowspeed is made using the correction method of Allen and Vincenti (1944), following the formula derived by Roshko (1961).The corrected values are denoted by an asterisk (un and Cd

n). The correction would be for the flow velocity un/u¼1.06 and

for the drag coefficient Cdn/Cd¼0.87 leading to Cd

n¼2.08 and Stn¼0.143, respectively. The corrected value of the drag

coefficient is higher than the measurement of Nakaguchi et al. (1968) (Cd¼2.0). The results of Bostock and Mair (1972) alsoshow a drag of Cd¼2.0, if their measurements are extrapolated to chord/thickness¼0.2. Bearman and Trueman (1972) gotnearly the same results with Cd¼2.06 for the drag coefficient and St¼0.141 for the Strouhal number. In both latterinvestigations the Reynolds number was in the same range and also a blockage correction was applied.

Switching between orientations at a¼901 and 01, the span/chord ratio is changed by a factor of 5. This means thatbecause of the different crosswind dimensions the vortex shedding frequency and the typical size of the vortices shedchange drastically. A further consequence could be that, if there is a three-dimensional span wise cell-like flow structure,then in case of lengthwise orientation (a¼01) the number of cells would be much higher than for a¼901.

3.2.2. Power spectra and probability density distributions

Fig. 10 shows power spectra F(f) of the lift- (index l), drag- (index d) and the moment-fluctuations (index m) taken atRe¼3.3�105, a pressure of 12 bar and the relatively low flow velocity of uN¼7.25 m/s, such that there is a large distancebetween the vortex shedding frequency fV at about 20 Hz and the bending frequency of the model at about 97 Hz. As themodel is now turned to a¼901 the lowest bending mode of the model appears in the drag component, as indicated inthe spectrum. For the lift, there is a narrow peak at St¼0.155, which rises more than four orders of magnitude from thebackground. In addition, the second superharmonic of the vortex shedding frequency is visible around 60 Hz.

In the lift spectrum, there is no crosstalk in the piezo balance visible due to the bending in drag direction. In the dragspectrum the vortex shedding is at 2fV and there are broadband low frequency contributions. There is also no crosstalkfrom the lift to the drag. The spectrum of the moment fluctuation has nearly the same appearance as the lift spectrum withthe exception that here the bending of the model is also present. Higher statistical moments were also calculated fromthe time functions of the lift. In Fig. 11, there is an extract of a typical time function of the lift L(t), normalised by the

10-2

100

102

104

100

102

104

0 20 40 60 80 100 120

10-4

10-2

100

Φl

[N2 /H

z]Φ

d

[N2 /H

z]Φ

m

[N2 m

2 /Hz] 3fV

f [Hz]

fV

2fV

3fV

bending

St=0.155

Fig. 10. Power spectra of the lift-, drag- and the moment-fluctuations.

0 0.5 1 1.5 2 2.5 3

-3

-2

-1

0

1

2

3

timet [s]

LLrms

α=90°

Fig. 11. Typical time function of the lift L, normalised by its rms-value (long side normal to flow, a¼901).

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

L/Lrms

Pro

babi

lityd

ensi

ty

Gaussian

α=0°

α=90°

-3 -2 -1 0 1 2 3

10-3

10-2

10-1

L/Lrms

Pro

babi

lityd

ensi

ty

Gaussian

α=0°

α=90°

Fig. 12. (a): Probability density functions of lift fluctuations for two angles of incidence, i.e. lengthwise oriented (a¼01, same as in Fig. 8) and with long

side normal to flow (a¼901), respectively. (b) Same as in Fig. 12(a) but in logarithmic representation in order to emphasis the tails of the probability

density functions.

G. Schewe / Journal of Fluids and Structures 39 (2013) 15–26 23

rms-value. The symmetry of the signal is reflected in the third moment, the skewness factor (0.009), which is about zero,as was to be expected in the symmetric flow conditions. The fourth moment, i.e. the flatness factor is 2.65, which is nearthe flatness of 3 for a Gaussian process. This value reflects that there is a higher degree of irregularity in the time historythan in case of lengthwise orientation of the section (a¼01), where the flatness was lower than 2. In this case the envelopeof the time function is more stochastic with many abrupt changes from one period to the next one.

These properties can also be seen in Fig. 12, where the probability density functions (PDF) of lift fluctuations for the two basicorientations of the cylinder are displayed. The dotted curve is for a¼01 and taken from Fig. 8. The solid line represents the casea¼901 based on the same data as in Fig. 10. Here a Gaussian distribution is included for comparison. The most obvious differenceis that for lengthwise orientation (a¼01) the distribution is bimodal, as for a sinusoidal process with random contributions.

G. Schewe / Journal of Fluids and Structures 39 (2013) 15–2624

In contrast, for a¼901, when the long side is normal to the flow, the probability distribution tends to a Gaussian process, whichwas already concluded considering the time function and the higher statistical moments. This point will be discussed further inthe following section.

4. Discussion and conclusions

4.1. Influence of the Reynolds number

A main result of the experiments is that for lengthwise orientation and angles of incidence aa01 a significant Reynoldsnumber effect is seen, in particular in the curve of the lift coefficient (Fig. 5). For a¼41 the curve shows an inflection pointand the lift Cl ranges from 0.3 to 0.6. The Reynolds number dependent topology of the separated flow is probablyresponsible for this behaviour.

Fig. 13 shows a simplified sketch of flow topology for a¼01 and aa01. As found in the force measurements the lift forceattacks downstream of the midpoint. In general the separation bubbles can be seen as a modification of the shape of thebody, here leading to two additional bumps of equal length in the symmetric case. For aa01 the bubbles have differentlengths. At higher incidence, as in Fig. 13, reattachment on the upper side is no longer possible. That means in theasymmetric flow case, the different shape and curvature of the bubbles can lead to different surface pressure resulting intraverse or lift force. If the bubble at the lower side shortens due to a variation of Reynolds number, this results in anincrease in the resultant lift force. That means the Reynolds number effects can be different on both sides, withconsequences for the lift.

Numerical simulations of Mannini et al. (2010) in the same Reynolds number range and for a¼41 show that the lengthof the separation bubble on the lower side of the cylinder decreases with increasing Reynolds number. Thus it can beexpected that the variation of the size of the bubble has an effect on the lift coefficient. Even if in the numerical simulationthe variation of the corresponding force coefficient is small compared to the experiments, the tendency has the samedirection as already mentioned in Section 3.1.2. From Fig. 5, it can be seen that the Reynolds number effects are stronger inasymmetric flow (aa01) than in the symmetric case (a¼01). In the latter case, where the time averaged flow is symmetric,the effects on the upper and the lower side can probably compensate each other, apart from a small effect on the dragcoefficient.

In this context it should be remarked that a very high Reynolds number effect was reported by Delany and Sorensen(1953) who found a significant dip in the drag coefficient of a rectangular cylinder with aspect ratio 2:1 for Reynoldsnumbers based on chord length Re42�106. But it is an open question whether the effect was caused by the finitesharpness of the corners. The radius of the curvature of the edges of the rectangular cylinder was 4% of the depth and wasgiven as a cause for concern by Larose and D’Auteuil (2006). In any case, if the Reynolds number is based on the chordlength, then there is no tendency for a dip in the drag coefficient up to Re¼2�106 in these experiments.

In the previous discussion, the effect of the elongated body (chord/thickness c/t¼5) was observed, which allowsreattachment at the side walls. From the results of the short rectangular cylinder c/t¼0.2 (a¼901), it can be concluded thatthe lack of reattachment seems to be the main reason for the absence of significant Reynolds number effects. Anotherinteresting difference was seen in the probability density function of the lift fluctuations. For lengthwise orientation(a¼01) the distribution is bimodal, as for a sinusoidal process with random contributions. In contrast, for a¼901, when thelong side is normal to the flow, the probability distribution tends to a Gaussian process.

Schewe (1983) noted that the lift fluctuations of a circular cylinder look like sine waves, which are randomly modulatedin amplitude and frequency. This view can be applied also considering other bluff cylinders. Then the non-randomprocesses are represented by the Strouhal frequency acting as a carrier frequency, thus their probability density will tend

Fig. 13. Simplified sketch of flow topology. The formation of a separation bubble on the lower side and their changes of the shape can act as a

modification of the geometry of the body. The reattachment point s moves upstream with Re increasing.

G. Schewe / Journal of Fluids and Structures 39 (2013) 15–26 25

to be as for a sinusoidal process. The random processes are wideband and at low frequency and are mainly reflected in theamplitude modulation, thus they tend to be Gaussian distributed.

In conclusion, the non-random contributions representing the regular vortex shedding process are more dominantwhen the cylinder is lengthwise oriented, while for the case with the long side normal to flow, the influence of irregularityincreases. Probably the coherence of the separated flow in the span wise direction is higher in the first case, resulting inhigher degree of regularity concerning the vortex shedding process. Finally, when the cylinder is oriented lengthwise,the separated shear layers can reattach. This seems to enhance the span wise coherence and thus the degree of order inthe vortex shedding process.

As mentioned in the introduction the chord-to-thickness ratio c/t is an important parameter concerning flow aroundrectangular cylinders. For the three cases studied, there is a collection of characteristic values in Table 1, including theolder results for a square cylinder taken from Schewe (1984). The data are uncorrected, but the blockage is included inorder to give the possibility for application of different correction methods.

4.2. Comparison with the Tacoma section

The results of the lengthwise orientation of the cylinder were compared with the corresponding values of the Tacomasection of the same aspect ratio (Cd¼1.2 and St¼0.115). Thus the drag coefficient of the rectangular section is at least 20%lower. The Strouhal number in the measured Reynolds number range is nearly equal at low and high values of Re, but has aweak dip around ReE2�104 (Fig. 5). However the rms-value of the Tacoma section is roughly three times thecorresponding rectangular cross-section, leading to the conclusion that the vortex shedding process is much more intensein the first case. This can also be concluded from the response diagram (Fig. 3), where the peak value in the main resonanceis roughly one order of magnitude lower than in case of the Tacoma section (Figs. 4 and 5 in Schewe, 1989). Thesemeasurements cannot be compared directly because the structural properties are not the same, nevertheless themodifications are minor and the trend is very strong in the right direction.

Now the question arises, which mechanism is responsible for the large difference in the vortex shedding intensity? InNakamura and Nakashima (1986) the behaviour of the ‘‘impinging shear layer instability’’ occurring in flow around prismswith elongated rectangular and H cross-sections are described and flow visualisations are presented. Based on theirfindings a particular feature of the unsteady wake flow of the Tacoma section was discussed in Schewe (1989) as follows:at zero incidence the flow along the Tacoma section, between the two small vertical bars (Fig. 2), is similar to the flowabout a cavity or gap, where the shear layer separating from the sharp leading edge is impinging upon the sharp trailingedge. Thus impinging shear layer instability leads to self-generation of organised oscillations, which are coupled withvortex-like structures travelling downstream. It is conceivable that the unsteady wake flow is the result of two fluidoscillators, the first would be the Karman vortex shedding mechanism and the second would be the vortical motion in thefree shear layers. Nakamura and Nakashima (1986) found that for chord-to-thickness ratios c/t44 including the Tacomasection both oscillation mechanism are probably synchronized and are oscillating with the same frequency.

Since the rectangular section and a Tacoma section had the same aspect ratio, the rectangular section can be seen as aTacoma section with filled gaps. Thus up to a certain degree the effect of both mechanisms on the intensity of the vortexshedding can be investigated. That means, in case of the rectangular section the impinging shear layer instability should bepresent (Nakamura et al., 1991), but only weak, leading to a base rms-value of lift due to vortex shedding plus a smallcontribution of the impinging shear layer instability. On the other hand, the relative high intensity of the lift fluctuations incase of the Tacoma bridge section can be attributed to a strong additional contribution of the impinging shear layerinstability.

The Reynolds number dependence of the assumed two oscillation mechanisms is difficult to qualify in the absence ofinformation on the location of the laminar/turbulent transition in the free shear layer, which can play a prominent role inthis context. It is assumed that effects due to different Reynolds number dependencies would be small for the rangeinvestigated. Strong effects should be reflected in the drag coefficient and the Strouhal number. Since for a¼01 nosignificant Reynolds number were observed, it can be assumed that because of the symmetry of the time averaged flow,the effects on the upper and the lower side compensate each other.

Table 1Collection of characteristic values of rectangular cylinders with different chord to thickness ratio c/t measured in the same test set-up at zero incidence.

The results concerning the square cylinder (c/t¼1) are from Schewe (1984), the ranges for Cd and Cl rms reflect the small dependence on Re. In all cases the

reference length for the rms-value Cl rms is the chord, i.e. the width of the side wall of the cylinder. For Reynolds number Re the crosswind dimension is

the reference length. No blockage correction was applied.

c/t Cd St Cl rms Re Blockage %

0.2 2.4 0.152 0.70 2�104–7�105 9.15

1 2.07–2.17 0.121 0.7–0.9 2�105–4�106 6.92

5 1.03 0.111 E0.08 2�104 1.83

G. Schewe / Journal of Fluids and Structures 39 (2013) 15–2626

The derivatives of Cl(a) and Cm(a) are the key parameters for aeroelastic stability. As qCL/qa40 the section is stable fora¼01 concerning galloping instability of the heaving mode, but the fact that qCm/qa¼�0.66 rad�1o0 indicates possibilityof torsional instability, that means both derivatives show the same trend as in case of a Tacoma section (qCm/qa¼�0.77 rad�1, Schewe, 1989). In a simplified quasi-steady approach (Blevins, 1997; Larsen, 2002) it can be shown that thecritical velocity for the onset of the torsional galloping oscillations is proportional to the inverse of the derivative qCm/qa.Thus in case of the rectangular cylinder the smaller absolute value of the derivative leads to a corresponding higher criticalvelocity compared to the Tacoma section. In Fig. 6 the curves for Cl(a) and Cm(a) are highly non-linear, similar tothe Tacoma section. The downward bending of the curves and the formation of local extrema are the reason for thedevelopment of limit cycle oscillations meaning in particular that the limit cycle amplitude is strongly dependent on thelocation of such non-linearities. The latter in turn are strongly dependent on the Reynolds number, thus the Reynoldsnumber can have a significant effect on flow induced vibrations.

Acknowledgement

The author would like to thank the colleague M. Rippl for his continuous support and valuable advice. With himthe experiments in the high pressure wind tunnel were performed. Further thanks go to Dr. C. Mannini for the fruitful,long-time cooperation and stimulating discussions concerning our common theme, the rectangular cylinder. Furtherthanks are extended to Dr. T. Gardner for helpful discussions and suggestions concerning the manuscript.

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