EACWE 5

Florence, Italy 19th – 23rd July 2009

Flying Sphere image © Museo Ideale L. Da Vinci

Keywords: Reynolds number, bluff body, BARC, rectangular cylinder, flow separation.

ABSTRACT

The paper reports on experiments carried out over a wide range of Reynolds numbers (2·104 < Re < 2·106) in the high pressure windtunnel in Göttingen. The model was a rectangular cylinder with aspect ratio 1:5. It has a sharp-edged generic cross section, which was selected as benchmark problem for bluff bodies in general (www.aniv-iawe.org/barc/BARC). High Reynolds numbers were reached by increasing the pressure in the flow medium up to 70 bar. Steady and unsteady forces were measured with a piezoelectric balance whose major characteristic is its stiffness, which makes the instrument particularly useful for investigating unsteady phenomena. Thus apart from 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.

Considering the dependence of the lift coefficient upon the Reynolds number, in particular for α = 4° a significant Reynolds number effect is obvious in the curve, which is double bent and the values are ranging from 0.3 to 0.6. In addition, the trend to lower or to higher Reynolds numbers indicates a further decrease and increase of the lift coefficient, respectively. Also for α = 6° and α = 2° there is a significant increase of the lift in the range Re < 105, at which the rise is shifted to lower for α = 6° and to higher Reynolds numbers in case of 2°.

Probably the shape of separation bubbles, that depends on the Reynolds number is responsible for these effects, which are stronger in asymmetric flow (α ≠ 0°) than in the symmetric case(α = 0°), where the effects on the upper and the lower side can compensate each other.

Concerning vortex resonance phenomenon, a response diagram was taken for the natural bending

Reynolds-Number-Effects in Flow around a rectangular Cylinder with Aspect Ratio 1:5

Günter Schewe DLR – Institut für Aeroelastik - guenter.schewe@dlr.de - 37073 Göttingen, Germany

motion of the rather slender model, which was fixed at both sides in the stiff balance. Finally, as a supplement the section was also investigated with the bluff i.e. much longer side normal to the flow, a case which can be seen as a test case for the absence of Reynolds number effects.

1. INTRODUCTION A rectangular cylinder with aspect ratio 1:5 is a generic cross section for a sharp-edged bluff body,

which often is seen as a benchmark test case for bridge aerodynamics and -aeroelasticity. The flow around such a body is determined by flow separations that results in a vortex street in the wake. A characteristic feature of wake flows is unsteadiness caused by the flow separation process. If a rectangular element of the structure is elastic itself or elastically suspended, the interplay between the fluid and the structure can lead to flow-induced vibrations, which in the worst case can be destructive. Thus the propensity of bluff bodies to vibrate in fluid flow is one of the limiting factors when designing and constructing large engineering structures in natural wind.

Concerning the fluid dynamics part of the problem the Reynolds number Re can play an important role here because flow separations are often Reynolds number dependent, even if the bodies have sharp edges. This has already been observed and explained for the approach span cross section of the Great Belt Bridge (Schewe & Larsen 1998). The reason for this dependence is that, the state of the boundary layer has a far-reaching influence on the entire flow field about a body. Both the state of the boundary layer and the location of transition are often responsible for the formation, length, and shape of separation bubbles.

In particular when the symmetry is broken, i.e. the cross section is asymmetric or the angle of incidence α is not zero, the behaviour of the separated flow that depends on the Reynolds number, can be different on the upper- and the lower side of the section. Thus global values, like the lift coefficient Cl, for example, can be affected by the Reynolds number. This in turn can have a large influence on the derivatives, which are differential values and thus sensitive to small variation of the underlying Cl(α) curve, the latter are, in addition, typically nonlinear in case of bluff bodies. In general the derivatives and the nonlinearities are determining the type and strength of possible flow induced vibrations.

The experiments were carried out over a very large range of Reynolds numbers (104 < Re < 107) in the high-pressure wind tunnel in Göttingen. Very high Reynolds numbers can be reached by increasing the pressure in the flow medium by a factor of 100. Steady and unsteady forces were measured with a piezoelectric balance whose major characteristic is its stiffness, which makes the instrument particularly useful for investigating unsteady phenomena (Schewe 1983, 2007). With the same model and balance, we can vary the Reynolds number over almost three orders of magnitude by controlling the flow velocity u∞ and the pressure p.

Concerning vortex resonance phenomena a response diagram was taken for the natural bending motion of the rather slender model which was fixed at both sides in the stiff balance. The paper is focused on the case, where the section is oriented lengthwise, but as a supplement it was also investigated when the longer side of the rectangular cross section is normal to the flow, a case which can be seen as test case for the absence of Reynolds number effects.

In parallel the rectangular cylinder has been numerically investigated partly at the DLR - Göttingen using the CFD code TAU, which is an inhouse development with unstructured grid. The results herein have been obtained through 2D Reynolds Averaged Navier-Stokes simulations (Mannini et al., 2008) and detached eddy simulations (Mannini et al., 2009) and are described and discussed ibidem.

2. EXPERIMENTAL SETUP AND THE MODEL The model itself was originally used as a generic Tacoma section applied in an investigation,

described in Schewe(1989). At this model the gaps were filled with plastics forming a rectangular cylinder (Figure 1). Consequently all details concerning the experimental arrangement are the same or similar as described ibidem. General points concerning the high pressure windtunnel, the

piezobalance and the entire environment can be found in Schewe(1983, 2007). The main feature of the piezo balance is its inherent high stiffness, leading to high natural frequencies. This property is particularly important for the measurement of values, which are the result of time series analysis, like frequencies, RMS values, higher statistical moments etc.

Figure 1: Dimensions of the rectangular cylinder and definition of the coordinate system ( L: lift; D: drag; M: moment; Cpb : base pressure coefficient ). The dashed lines show the relation to the Tacoma section.

The model in Figure 1 with width l = 55.4mm, height h = 11 mm and a span of 600mm was fixed at both ends in the rigid piezobalance such that the lowest frequency of 99 Hz (at 1 bar) was determined purely by the bending of the rather slender model (height/span = 1/55). This value decreased to 94 Hz at a pressure of 70 bar. In addition to the measurement of the steady and unsteady forces and the moments, we also measured the time averaged base pressure coefficient Cpb at the backside of the model. At angle of incidence α = 0° the blockage of the model amounts to only 1.83%. When the section is lengthwise oriented, i.e. 0° < α < 6° then as usual for bluff bodies the crosswind dimension i.e. height h is the reference length for drag coefficient and the Strouhal number. For the special case of α = 90° the width l is the characteristic length for Cd and Strouhal number. For the Reynolds number, the steady lift, the RMS of lift and the moment width l is taken as reference length in all cases.

3. RESULTS

3.1 Section lengthwise oriented Since the model, fixed at both ends in the stiff piezobalance, is allowed to vibrate in its natural

bending mode, a response diagram depending on flow speed can be taken. In Figure 2 the RMS-value of the lift fluctuations is plotted upon the flow speed for different pressures up to 70 bar. In this case the section is lengthwise oriented at an angle of incidence of α = 0°. As the force measurement is a global one, in particular near the resonances the signals have in addition to the aerodynamic forces also contributions of inertia and elastic forces. The main resonance for vortex resonance is obvious around u∞ = 10 m/s. For safety reasons the vortex resonance range was not crossed for pressures higher than 5 bar. Due to the low damping of the steel model the amplitudes in the resonance case would become too large even when the crossing is transient. Thus only for p = 1 bar measurements were taken within the main resonance.

Figure 2: RMS-value of lift depending of flow speed and pressure in the wind tunnel (α = 0°). The curve corresponds to a response diagram of bending motion, since the unsteady forces contain apart from the aerodynamic- also inertia- and elastic contributions.

It is interesting to compare the present results with corresponding measurements using the generic Tacoma section in the same experimental environment with similar structural properties (Schewe 1989). At first glance the peak value of the main resonance is much less than in the case of the Tacoma section and secondly the occurrence of superharmonic resonances is not obvious as in the case of the Tacoma section. Nevertheless we can estimate that a typical RMS-value of lift below and far from resonance will have a value around Cl rms = 0.08.

Figure 3: Time function of lift force, when the main resonance is crossed by a flow speed sweep, left figure: increasing, right-: decreasing

Figure 3 gives an impression about 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 angle of incidence is α = 4°, which can be seen in the small offset of the averaged lift force at about t > 25 s in the left - and for about t ≥ 0 in the beginning of the right time function. The left figure shows the time function of the lift for increasing- and the right for decreasing flow speed. The different shapes reflect a small asymmetry of the resonance curve, which probably is bended to the right side. In addition hysteresis seems to be responsible for the difference in the maximal amplitudes.

Figure 4: Strouhal number, force/moment coefficients and base pressure depending on Reynolds number and angle of incidence α for the rectangular section.

Figure 4 shows the dependence on Reynolds number and angle of incidence α of the time averaged force / moment coefficients Cd, Cl , Cm, the Strouhal number St and the base pressure Cpb . For the two angles of incidence α = 0° and 4° the Reynolds Number was varied in the range 3 x 104 < Re < 2 x 106 by increasing the pressure from 1 to 70 bars and flow speeds ranging between 3 and 8m/s. Restriction to this low flow speeds was necessary to hold possible resonance effects in reasonable limits. For the two angles of incidence α = 2° and 6° the Reynolds Number was extended only up to Re = 6 x 105 for time and economic reasons.

The Strouhal numbers were determined from the power spectra of the lift forces, examples will be shown later. Concerning the drag coefficient CD and the Strouhal number there are no drastic Reynolds Number effects obvious regarding the angle of incidence α = 0°. Zooming the ordinate, we see in both lines Cd(Re) and St(Re) a slight curvature with opposite sign. This inversed behaviour was to be expected.

The values of the moment are very small and thus difficult to measure, in particular at low Reynolds numbers. Thus the moment coefficients Cm are included only for Reynolds numbers larger than about 105. For α = 4° the moment coefficients are negative and nearly constant, while for α = 2° and 6° there is a small increase in negative direction, when Reynolds number is increasing.

For angles of incidence α ≠ 0° ( except for the special case α = 90°) a significant Reynolds number effect is obvious in particular in the curve of the lift coefficient. In the case of α = 4° the curve is twice bent and the values are ranging from 0.3 to 0.6. In addition the trend to lower or to higher Reynolds numbers indicates a further decrease and increase of the lift coefficient, respectively. For all α there is a significant increase of the lift in the lower range Re < 105, at which the range of rise is shifted to lower Reynolds number, when α is increased.

Figure 5: Force and moment coefficients depending on the angle of incidence α with the Reynolds number as parameter. Cl α and Cm α are the derivatives at α = 0° , which are the key parameter for the aeroelastic stability.

In order to illuminate the Reynolds number effects from a different point of view, Figure 5 shows the characteristic diagrams for Cd(α), Cl(α) and Cm(α) with Reynolds number as parameter. Since the measurements for different α were not taken at the same Reynolds numbers, partly we extracted the data from the compensating curves displayed in figure 4. Even if the data are of reduced accuracy and the number of supporting points is rather low, the curves show significant Reynolds number effects in particular 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 around α ≈ 6°, and the slopes ∂Cl/∂α at α = 0° become steeper from 5.7 rad-1 at a Re = 3 ⋅ 104, 7.8 rad-1 at a Re = 6 ⋅ 104 to 9.3 rad-1 at a Re = 3 ⋅ 105. The mentioned maximum of the lift seems to be shifted to lower α, when Re is increasing. The moment characteristic should not be over-interpreted, but considering all measuring points up to and including α = 4°, we get for the slope ∂Cm/∂α = -0.66 rad-1. Concerning a Reynolds number effect we can say that in both cases the curves are bent down to lower absolute values, namely more for the lower Reynolds number. Additionally also the dependence of the drag depending on α is shown, although the Reynolds number effects are not so obvious. The curve is single bent at low and double bent at the higher Reynolds number. The knowledge of the derivatives allows a rough estimate concerning the streamvise location of the center of pressure Δx using ∂Cm/∂Cl = - Δx/l = 0.66 / 7.8 = 0.09. That means the lift attacks roughly 10% of the width downstream the midpoint of the section. This value is nearly the same as in the case of a Tacoma section (Schewe 1989).

Figure 6: Power spectra of lift for rectangular section for two angles of incidence α

Figure 6 displays two representative power spectra of lift fluctuations for two angles of incidence. In both cases there are rather narrow peaks due to the vortex shedding, which have nearly the same Strouhal number St = 0.11. The bending frequency of the model is visible at f ≈ 94 Hz, its power spectral density is more than 20 db below the Strouhal peak. Further, for the asymmetric case (α = 4°, pressure in the windtunnel p = 70 bar, flow speed u∞ = 5.7 m/s) the low frequency part is more pronounced and the occurrence of a significant second harmonic is obvious. It should be remarked here that, no parasitic resonances of the balance are present. For α = 0° the flow parameter were as follows: p = 50 bar, u∞ = 3.9 m/s. The Reynolds numbers, noted in the figures, are rather high, around 106.

Figure 7 shows the probability density functions (PDF) of lift fluctuations for two angles of incidence α and the same flow conditions as in figure 6. For α = 4° (solid line), the asymmetric case, a significant skewness of the PDF is obvious, the probability for the occurrence of positive high amplitudes up to 3 RMS-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 amounts to 2.1. The corresponding values for the symmetric case α = 0° are skewness = 0.01 and flatness = 1.85, the dotted curve is nearly symmetric regarding zero amplitude.

Figure 7: Probability density functions of lift fluctuations for two angles of incidence

3.2 Section with bluff side normal to the flow At an angle of incidence of α = 90° we have the possibility to study a special case of bluff body

aerodynamics, which is quite near to a two-dimensional flat plate normal to the flow i.e. the crosswind dimension l is much larger than the lengthwise dimension h. In such flows around sharp edged cross sections, there is no possibility for the formation of larger separation bubbles at the side faces and thus Reynolds number effects should not to be expected and if they occur, then they should be very small.

Figure 8: Drag coefficient, Strouhal number and the RMS-value of the lift fluctuations depending on the Reynolds number

Figure 8 displays the drag coefficient, the Strouhal number and the RMS-value of the lift fluctuations for the range 2.7 x 104 < Re < 6.4 x 105. For this special case, as already mentioned the characteristic length also for Cd and St is always the present crosswind dimension l. The pressure in the windtunnel was increased from 1 up to 12 bar. The results show that, there is indeed no systematic variation depending on the Reynolds number.

The mean drag coefficient with Cd = 2.4 is roughly 10% higher than for a square cylinder, which was investigated in the same test setup (Schewe 1984). The mean Strouhal number is St = 0.152 and the mean RMS- value of the lift Cl_rms = 0.14, which is significantly higher than for lengthwise

orientation but small, about 20% of the value, compared with a square cylinder. Since the wind tunnel blockage in this position is no longer small (9.15%), an estimate concerning

the effect on drag and flow speed is made using the method of Allen Vincenti (see e.g. Roshko 1961). The corrected values are denoted by an asterisk (u* and Cd

*). The correction would be for the flow velocity u*/u = 1.06 and for the drag coefficient Cd

*/Cd = 0.87 leading to Cd* = 2.08 and St* = 0.143,

respectively.

Figure 9: Power spectra of the lift-, drag- and the moment fluctuations

Figure 9 shows power spectra Φ(f) of the lift-(index l), drag- (index d) and the moment fluctuations (index m) taken at Re = 3.3 x 105 , a pressure of 12 bar and the relatively low flow velocity of u∞ = 7.25 m/s, such that there is a large distance between the vortex shedding frequency fV at about 20 Hz and the bending frequency of the model at about 97 Hz. As the model is now turned to α = 90° the lowest bending mode of the model appears in the drag component, as indicated in the spectrum. For the lift, there is a rather narrow peak at St = 0.155, which arises more than four orders of magnitude from the background, also the threefold of the vortex shedding frequency is visible around 60 Hz.

Concerning the piezobalance it should be remarked that in the lift spectrum, there is no crosstalk visible due to the bending in drag direction. In the drag spectrum we see the vortex shedding at 2 x fV and broadband low frequency contributions. Also here, there is no crosstalk from the lift to the drag obvious. The spectrum of the moment fluctuation has nearly the same appearance as the lift spectrum with the exception that here also the bending of the model is present. Higher statistical moment were calculated from the same time functions of the lift, as they were used for the power spectra. In Figure 10, there is an extract of a typical time function of the lift, normalized by the RMS-value.

The symmetry of the signal is reflected in the third moment, the skewness factor, which amounts to -0.009, which is about zero, as it was to be expected in the symmetric flow conditions. The fourth

moment i.e. the flatness factor amounts to 2.65, which is near the flatness of 3 for a Gaussian process. This value reflects that there is a higher degree of irregularity in the time history than in case of lengthwise orientation of the section (α = 0°), where the flatness was lower than 2. In this case the envelope of the time function is more stochastic with many abrupt changes from one period to the next one. Probably the coherence of the separated flow in spanwise direction is rather low, compared to, for example, a square cylinder.

Figure 10: Typical time function of the lift, normalized by the RMS-value

4. DISCUSSION AND CONCLUSIONS A main result of the experiments is that, for angles of incidence α ≠ 0° a significant Reynolds

number effect is obvious, in particular in the curve of the lift coefficient (figure 4). For α = 4° the curve is double bent and Cl is ranging from 0.3 to 0.6. Probably the Reynolds number dependent shape of separation bubbles on the sides is responsible for this behaviour (figure 11). Numerical simulations of Mannini et al. (2009) in the same Reynolds number range show that the length of the separation bubble on the lower side of the cylinder decreases with increasing Reynolds number. Thus it can be expected that, the variation of the size of the bubble will probably have an effect on the lift coefficient. Even if in the numerical simulation the variation of the corresponding force coefficient is small compared to the experiments, one can say that, the tendency has the same direction. Considering figure 4, one can say that the Reynolds number effects are obviously stronger in asymmetric flow (α ≠ 0°) than in the symmetric case(α = 0°). In the latter case, where the time averaged flow is symmetric, the effects on the upper and the lower side probably can compensate each other, apart from a small effect on the drag coefficient.

Because of the same aspect ratio it is worth, to compare the results with the corresponding values of the Tacoma section, which amount to Cd = 1.2 and St = 0.11, respectively. Thus the drag coefficient of the Tacoma section is a little bit higher and the Strouhal number correspondingly lower by a few percent than the present ones. However the RMS-value of the Tacoma section is roughly 3 times the corresponding rectangular cross section, leading to the conclusion that the vortex shedding process is much more intense in the first case. This can also be concluded considering the response diagram (figure 4), where the peak value in the main resonance is roughly one order of magnitude smaller than in case of the Tacoma section (figure 4,5 in Schewe 1989). As mentioned, these measurements cannot be compared directly, because of the fact that the structural properties are not the same, nevertheless the modifications are not that much and the trend is very strong in the right direction.

Figure 11: Simplified sketch of flow topology. The lift force attacks downstream the midpoint. The Reynolds number effects are different on both sides. The separation bubbles can be seen as modification of the shape of the body, here leading to two additional bumps of different length (if α ≠ 0°). The curvature can lead to different underpressure at the surface of each bump. If the lower bubble shortens ( Re↑), as consequence, the resultant lift force increases.

Now the question arises, which mechanism can be responsible for the large difference in the vortex shedding intensity? In Schewe(1989) a particular feature of the unsteady wake flow of the Tacoma section was discussed as follows. The flow along the Tacoma section, between the two small vertical bars (figure 1), is similar to the flow about a cavity or gap, where the shear layer separating from the sharp leading edge is impinging upon the sharp trailing edge. Thus we have an impinging shear layer instability leading to self-generation of organized oscillations, which are coupled with vortex-like structures traveling downstream. It is conceivable that the unsteady wake flow is the result of two fluid oscillators, the first is the Karman vortex shedding mechanism and the second is the vortical motion in the shear layers, which are probably synchronized and are oscillating with the same frequency. Since we have investigated a rectangular section and a Tacoma section with the same aspect ratio, the first one can be seen as Tacoma section with filled gaps and thus up to a certain degree we can separate the effect of both mechanism on the intensity of the vortex shedding. That means, in case of the rectangular section the impinging shear layer instability should not be present or very weak leading to a lower RMS-value of the lift fluctuations. On the other hand, the relative high intensity of the vortex shedding process in case of the Tacoma bridge section can be attributed to the impinging shear layer instability.

Concerning aeroelastic stability the derivatives of Cl(α) and Cm(α) are the key parameters. As ∂CL/∂α > 0 the section is stable for α = 0° concerning galloping instability of the heaving mode, but the fact that ∂Cm/∂α = -0.66 rad-1 < 0 indicates torsional instability, that means both derivatives show the same trend as in case of a Tacoma section (∂Cm/∂α = -0.77 rad-1, Schewe 1989). In a simplified approach (Blevins 1997, Larsen 2002) one can show that the critical velocity for the onset of the torsional galloping oscillations is proportional to the inverse of the derivative ∂Cm/∂α. Thus in the case of the rectangular cylinder the smaller absolute value of the derivative leads to a corresponding higher critical velocity of about 15% compared to the Tacoma section. In figure 5 we have seen that the curves for Cl(α) and Cm(α) are highly nonlinear, also similar as for the Tacoma section. The down bending of the curves and the formation of local extrema are the reason for the development of limit cycle oscillations, that means in particular the limit cycle amplitude is strongly dependent on the location of such nonlinearities. The latter in turn are strongly dependent on the Reynolds number, thus we can conclude the Reynolds number can have a significant effect on flow induced vibrations. Finally it should be mentioned that numerical investigations of flow around a rectangular cylinder and a comparison with our experiments can be found in Mannini et al. (2008, 2009).

ACKNOWLEDGEMENT

The author would like to thank the colleague M. Rippl from the DNW-windtunnel-department for his continuous support and valuable advise. With him the experiments at the high pressure windtunnel 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.

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