STROUHAL NUMBERS OF RECTANGULAR CYLINDERS AT INCIDENCE: A REVIEW AND NEW DATA

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1 Journal of Fluids and Structures (199) 4, STROUHAL NUMBERS OF RECTANGULAR CYLINDERS AT INCIDENCE: A REVIEW AND NEW DATA C. W. KNISELY Mechanical Engineering Department, Bucknell University, Lewisburg, PA 17837, U.S.A. (Received 5 May 1987 and in revised form 7 December 1989) Experimentally determined Strouhal numbers for a family of rectangular cylinders with side ratios (B/D) ranging from.4 to 1. and with angles of attack from to 9 are presented. Tests were conducted both in a water channel with Reynolds numbers between 7-2 x 12 and and in a wind tunnel with Reynolds numbers between 8.8 x 13 and 8-1 x 14, based on the projected cross stream dimension and the mean velocity. In addition, the mean drag as well as the mean and fluctuating lift were measured as functions of the angle of attack in the wind tunnel tests for BID =.25,.5 and 1. The Strouhal number of all except the thinnest plate showed qualitatively similar behavior with changing angle of attack. The general tendency is for a rapid rise in the Strouhal number to occur at relatively small angles of attack. This rapid rise is associated with reattacliment of the separated shear layer. The angle of attack where reattachment and, hence, the rapid rise in Strouhal number occurs is dependent on the BID ratio. After the sudden rise, the Strouhal number levels off with a further increase in angle of attack. As the angle of attack approaches 9, there is a sudden decline in the value of the Strouhal number, again associated with shear layer-corner interaction (i.e. detachment). For the thinnest plate, the Strouhal number was found to be essentially independent of the angle of attack over a wide range of angles. Measured force coefficients show a strong dependence on the angle of attack, suggesting that for -5 -< BID <- 2. there is an optimum incidence that makes most efficient use of the load-bearing capacity of the structure. The present data agree well with available results from previously published papers which are reviewed. 1. INTRODUCTION AND REVIEW 1.1. MOTIVATION Most ENGINEERS CONCERNED WITH THE design of structures to be exposed to fluid flow are well aware of the hydrodynamic disadvantages of sharp-edged rectangular cross-sectional shapes, particularly their susceptibility to flow-induced vibration. However, fluid dynamic considerations are often only one aspect of the design process and may play a subsidiary role to other aspects such as economics, ease of construction or structural requirements, and frequently these nonideal rectangular geometries are employed. Conceivably, situations may arise in which the designer has sufficient leeway to choose the orientation of the rectangular cross-sectional member relative to the mean flow direction. In such cases, knowledge of the expected Strouhal frequencies and flow-induced loadihg due to vortex shedding might be useful in deciding the inclination of the rectangular member. Alternatively, one can imagine a situation in which a /9/ $3./ 199 Academic Press Limited

2 372 c.w. KNISELY completed structure is experiencing flow-induced vibration. Knowledge of expected Strouhal numbers permits a very quick cursory check on whether the vibrations are vortex-induced. Structures that typically have rectangular cross-sections include architectural features on buildings, the buildings themselves, and occasionally stays and supports in internal flow geometries. In addition, the component bars in trashracks at hydraulic intakes frequently have rectangular cross sections (Syamalaras 1989). For the reader unfamiliar with trashracks, they are structures which usually consist of a number of bars or rods, relatively closely spaced and stiffened with some type of cross bars; they are placed at hydraulic intakes to prevent large pieces of debris from entering the flow system. In several cases, the vibrations were severe enough to lead to structural failure (Crandall et al. 1975, Behring & Yeh 1979, James & Katakara 1971, Neilson & Picket 198). One prevalent trashrack design employs an array of rectangular bars with the side ratios of the individual bars ranging from about 1:4 to 1 : 1. Some of the reported failures have occurred when the oncoming flow was at an angle of attack relative to the bars (Syamalarao 1989). In a recent laboratory investigation of simulated trashrack bars (Callander 1987), very large amplitude in-line vibrations were observed for certain geometries where the flow was at an angle of attack relative to these bars. Flow visualization studies indicated that these vibrations may well be associated with vortex shedding in a way analogous to that proposed by Wootton et al. (1974) for circular cylinders. It was then apparent that information concerning vortex shedding frequencies of these arrays would be necessary before any conclusions could be drawn. Obviously, the spacing ratio will affect the shedding frequency, especially at small spacing ratios. It is expected, however, that for moderate to large spacing ratios, the difference between the frequency of vortex shedding from the trashrack and that of a single cylinder at the same angle of attack would be slight. Unfortunately, a literature search, the results of which will be discussed below, revealed less extensive data for single rectangular cylinders at an angle of attack than had been hoped. To fill the gap and provide information for the trashrack vibration problem, a laboratory study of vortex shedding from rectangular cylinders at incidence was undertakert (Knisely 1985). The results of this study, along with those of a more recent complementary study, will be presented following the review of the available literature REVIEW The earliest work concerned with vortex shedding from sharp-edged rectangular structures was that of Fage & Johansen (1927), dealing with a flat plate (side ratio- 3:1). For later comparison it is important to note that this geometry was not truly rectangular, but rather almost triangular as shown in Figure 1. A similar triangular geometry was employed in Abernathy's (1962) study. Abernathy found that a rounded leading edge resulted in a significant increase in the Strouhal number in. 1 I.,1685 in. - I Figure 1. Geometry of cylinder of Fage & Johansen (1927); 1 in = 25-4 mm.

3 STROUHAL NUMBERS AT INCIDENCE 373 A later paper by Schramm (1966) gave the Strouhal number for many structural configurations at incidence including sharp-edged rectangular cylinders with side ratios of 3: 4 and 1 : 1. Further data on square cross section cylinders can be found in papers by Vickery (1966), Lee (1975), Rockwell (1977), Obasaju (1983) and Bokaian & Geoola (1981). Additional information on structural shapes at incidence can be found in Grant & Barnes (1981) and Modi & Slater (1983). After the present investigation was completed, the author became aware of the articles by Otsuki et al. (1978) and Ohya et al. (198), in which Strouhal number data for rectangular cylinders with side ratios 1 : 1, 2: 1, 3 : 1 and 4 : 1 at angles of incidence from to 5 may be found. Several authors (Nakaguchi et al. 1968, Bearman & Trueman 1971, Parkinson 1971, Novak 1974, Washizu et al. 1978, Parker & Welsh 1983, Bokaian & Geoola 1983) present Strouhal number data for rectangular cylinders at zero incidence as a function of the side ratio Reynolds number effects Okajima (1982) showed that, at zero angle of attack, cylinders with side ratios of 1 : 1, 2:1 and 3:1 have a significant Reynolds number dependence for Re In addition, Bauer (1961) has shown that the shedding frequency of thin flat plates (with side ratios of 48 and 3) parallel to the freestream increases with increasing Reynolds number over the range 14<Re~<3 15. Thus, one must expect a significant Reynolds number effect of the type described by Okajima for Re for B/D < 4 and of the type described by Bauer for larger BID ratios for Re - 3 l s Turbulence effects Another factor that affects the Strouhal frequency of rectangular bodies is the turbulence level in the freestream. A comparison of Novak's (1972) data for rectangular cylinders at zero angle of attack and for a thin flat plate (BID =.1) at varying angles of incidence, respectively, with other available data, suggests that a small amount (2-5%) of turbulence increases the shedding frequency, but a higher level of turbulence (15%) tends to decrease the Strouhal frequency to a value near its smooth flow value. Lee (1975) examined the effects of freestream grid turbulence on the pressure field of a square cross section cylinder. For turbulence levels of 4-4% and 6.5%, Strouhal numbers were observed to increase, relative to smooth flow values, for angles of incidence less than about 2. With higher levels of turbulence, 8.% and 12-5%, Strouhal numbers returned to near their smooth flow values. For angles of attack greater than approximately 2, the effect of all levels of grid turbulence was to reduce the Strouhal number. In a more recent study of turbulence effects on square cylinders at zero incidence, Huot et al. (1986) showed for turbulence intensities of the order of 1%, the separated shear layer can reattach, resulting in a significant increase in the Strouhal number. The large uncertainty in the data of Huot et al. prohibited any definite conclusion regarding the effect of turbulence scale on the shedding frequency. Additional data on scale and intensity effects on flow past square prisms at zero incidence can be found in Petty (1979). The studies of Laneville et al. (1976), Courchesne & Laneville (1982) and Nakamura & Ohya (1984), although containing no direct data on shedding frequencies, indicate that the change in the mean drag coefficient with increasing turbulence is a function of the afterbody length. In addition, Nakamura & Ohya (1984) examined the effect of the

4 374 c.w. KNISELY turbulence scale on the strength of vortex shedding. For small scale turbulence, an increase in the turbulence level results in a decrease in the drag coefficient for bodies with BID >.62, while for shorter bodies the drag coefficient initially increases, and then subsequently decreases with a further increase in intensity. If one accepts that the product of drag coefficient and Strouhal number is approximately constant, one must suspect a similar, though inverted, afterbody length dependence for the change in Strouhal number with increasing freestream turbulence. Large-scale turbulence was found to reduce to spanwise correlation, resulting in weaker vortex shedding (Nakamura & Ohya 1984) but there was no information concerning its effect on the frequency of shedding. To the author's knowledge, other than studies of the square cylinder by Vickery (1966) and Lee (1975), for example), the simultaneous effects of turbulence and incidence on the mean and fluctuating flow about rectangular cylinders with various BID ratios has not yet been examined. If one attempts such a study, a large aspect ratio should be used since recently published data by Lee (1988) suggest that end effects in turbulent flow can affect the base pressure over the entire cylinder length to a greater extent than previously realized. A more thorough overview of the effects of turbulence intensity and scale can be found in Laneville (1988). Since turbulence can play a significant role in determining the Strouhal frequency, caution is advised when extrapolating laboratory data for smooth flow cases to field situations where significant turbulence levels are encountered Effects of rounded corners A further area of concern when extrapolating laboratory data is the effect of rounded corners on the Strouhal frequency. The data concerning this effect are somewhat limited and scattered, but from the data of Schramm (1966) (radius of corner not specified), Parker & Welsh (1983) (various side ratios with semicircular leading edges), Bokaian & Geoola (1982) (side ratio 1:2), and Bokaian & Geoola (1984) (side ratio 1 : 1, both cases with specified rounding) one can conclude that the general tendency is for the Strouhal number to increase with increasing rounding radius. The rate of increase with rounding appears to depend upon the afterbody length as well, as shown in the collected data in Figure 2. In this figure, the Strouhal number with rounding divided by that with sharp corners is plotted as a function of the dimensionless corner radius. Note that for some bodies (1.4 < B/D < 3-2) this increase can exceed a factor of 3 (Parker & Welsh 1983). The review of the literature revealed no complete data set for variation of the angle of attack from to 9, except for the square cylinder. Further, there is no data that come from a single test facility for a family of cylinders with various B/D ratios. To fill this gap, the experiments which are described below were conducted. 2. APPARATUS AND EQUIPMENT Measurements of vortex shedding frequency for various cylinder geometries and angles of attack were carried out in a 6 cm wide water channel. Later experiments were conducted in a 1- x.7 m closed wind tunnel. Figure 3(a,b) shows sketches of the water channel and the wind tunnel, respectively. For the water channel data, blockage effects were accounted for by geometrical corrections, that is, by defining the mean velocity as the volumetric flow rate divided by the geometrical flow area. This

5 STROUHAL NUMBERS AT INCIDENCE 5.O 375 /k ~ m3. 4. /k <> Symbol BIB v o [] Reference 3.16 t 1. Parker & Welsh (1983) O.5 Bokaian & 1-O Geoola (1984).5 tsokaian & Geoola (1982) 2"..ET.L3 / s/ =O'5 y. ~ }.-~ I ~O'BIo=I.O I in ~.~----~,,,,,,,, I - O.1 "2 '3.4 "5 RIB Figure 2. Effects of rounding of the leading edge on the Strouhal number, as given by the ratio of Strouha] number with rounded edge, StR, to Strouhal number with square edge, Stso. Data taken from the literature. Honeycomb Screen Glass floor Depth control plate Side view (a) H =q> Flow 163 4o- Top view All dimensions in cm Figure 3. (a) Schematic diagram of water channel.

6 376 c.w. ~'~lsely Side I,,,u...on.,...,,oz,. T! *Lon Load cell detail end view 7 "i I M r Load view ~X"~'- \ ~ " ~ : :I,, (b) Figure 3. contd. (b) Schematic diagram of wind tunnel. 1 ~I rl area is in turn defined as the channel width minus the projected cross stream dimension of the body times the depth. At the flow velocities employed in this investigation, there was no significant drawdown of the water surface in the vicinity of the cylinder. The depth was therefore regarded as remaining constant. From earlier measurements in the same channel (Knisely et al. 1984, Knisely, Matsumoto & Menacher 1986, Knisely, Matsumoto & Shiraishi 1986), it is known that the mean flow was uniform to within 2% over the central 75% of the channel. Freestream fluctuations were less than.5% of the freestream velocity when measured with a hot film anemometer at 23 cm/s. Detailed velocity profiles as well as turbulence intensity distributions can be found in Knisely et al. (1984). The second set of experiments were conducted in the wind tunnel shown schematically in Figure 3(b). Experiments were conducted at freestream velocities of 7 m/s and 14 m/s. The freestream turbulence was about.5%. Blockage ranged from 2% to about 9.2%. Since base pressure was not recorded, any correlation method relying on correcting the base pressure could not be employed. Instead, the method of Ranga Raju & Singh (1975), as reviewed by Courchesne & Laneville (1979), was employed to correct the mean drag coefficient: i.e., Coo~Co = (1 - AM~As) N, (1) where Ara is the model area, As is the wind tunnel cross-sectional area and N is an empirical function of the B/D ratio and is given by Courchesne & Laneville (1979). Based on arguments in Vickery (1966) and Awbi (1983), the correction factor for the Strouhal number to account for blockage is taken to be the square root of the drag correction given by equation (1). Also from Vickery (1966), it is known that for small blockage ratios, the ratio of fluctuating lift to mean drag is essentially independent of the blockage ratio, and so the correction factor for the fluctuating lift was taken to be the same as for the mean drag. The correction for the mean lift coefficient was accomplished using the observation from Keshavan (1977) that, for airfoils, the fluctuating lift is proportional to the product of square of the mean lift times the drag coefficient. For angles of attack at which there is significant mean lift this relationship is

7 STROUHAL NUMBERS AT INCIDENCE 377 TABLE 1 Geometry of cylinders employed B D B'/D' End plate (crn) (cm) BID (at tr = 9 ) H/B Rem,x Retain (cm) Water channel experiments x x " " x x x x x x x 14 Wind tunnel experiments " x x (o~-- 15 ) (o~_< 2 ) x approximately true for rectangular cylinders as well. The correction of the mean lift coefficient for blockage is then assumed to be equal to the square root of the correction for drag. For the water channel study, all of the model cylinders, except the thinnest, were manufactured from plexiglas. For reasons of rigidity, the thinnest cylinder was made from aluminum. All cylinders had machined sharp edges. In the wind tunnel tests, the cylinders were constructed from light weight wood and reinforced with a rib construction. Table 1 lists the geometric properties of the cylinders investigated, along with the maximum and minimum Reynolds numbers. The Reynolds number changed with angle of attack and from cylinder to cylinder. All Reynolds numbers are based on the projected cross stream dimension and the mean velocity as previously defined. Figure 4 defines the notation used in Table 1 and throughout this report. In the water channel tests, the cylinder model was hung from a turntable above the channel. The gap between the cylinder and the floor of the channel was maintained at approximately 1. to 1-5 mm. A wider gap resulted in a significantly more threedimensional wake, as visualized with injected dye. Nominal water depth was 24 cm, yielding the aspect ratios shown in Table 1. In the wind tunnel tests, the cylinders were supported on both sides by three-component load cells (Nissho Electric Works Co., Ltd., Model LMC-351-2). During the experiments, however, the measurement of the moment proved unreliable, so that only drag and lift data are presented. The total lift and drag were determined by adding the signals from the two load cells. End plates, typically 2- to 3-times the cross-sectional area of the cylinder and with 2 cnl radius rounded corners, were employed. The Strouhal frequency in the water channel tests was determined by spectral

8 378 C. W. KNISELY B O Figure 4. Cross section of cylinder model, defining notation employed in report. analysis of the output signal of a DISA (now DANTEC) conical hot film probe located approximately one body width, D', downstream of the body and near the edge of the wake. Dye visualization was used to assist in positioning the probe. The probe output was fed into a DISA 55M1 anemometer. Spectral analysis was carried out on a Hewlett Packard 5451C Fourier Analyzer. In the wind tunnel tests, the Strouhal frequency was determined by spectral analysis of the fluctuating lift signal and the fluctuating velocity signal originating from a single element Kanomax hot wire probe located near the edge of the shear layer approximately one body length downstream and about one half to three quarters of a body length away from the centerline. The signal from the probe was processed using a Nihon Kagaku Kogyo Co., Ltd CTA Anemometer, Model 111. Spectral analysis of both the fluctuating lift and fluctuating velocity was done using an NEC San-ei Instruments Co., Ltd Signal Processor, Model 7T16 with a typical frequency resolution of.25 Hz. 3. RESULTS 3.1. STROUHAL NUMBERS The most frequently investigated rectangular cylinder is the square cross section cylinder. To validate the blockage corrections discussed in the preceding section, the present corrected results for the square cross-section cylinder are compared with those of Obasaju (1983) in Figure 5. The data of Obasaju were chosen for comparison and are considered representative for the following reasons: (i) the turbulence level in his wind tunnel was very low, -.4%; (ii) end plates were employed, giving results more representative of two-dimensional conditions; (iii) his aspect ratio of about 17 was relatively high; (iv) his data were corrected for wind tunnel blockage, which in his case was 7.78%; and (v) Obasaju's (1983) data lie in the middle of the scatter range (approximately + 9% of his values) of reported results. If average values are calculated from the collected data, they are found to correspond extremely well with Obasaju's values.

9 STROUHAL NUMBERS AT INCIDENCE 379 "2 2 2 o o ~ o o ~l ~ -1 II V3 O Re Tu Aspect ratio Present 2.2 to 4-4x14(Air) -.5 /~ 12 (with end plates) Present 1.Sx14(Water) -.2 /, 3,2 /X Obasaju 47x14(Air) ~.4 /o 17(with end plates) (1983) (All sets of data corrected for blockage).5 I I I I l I I I Angle of attack, a Figure 5. Strouhal number of square cross section cylinder as a function of the angle of attack. (The data set given on the figure are corrected for blockage). In Figure 5 the sudden jump in the Strouhal number value in the vicinity of 12 to 13 can be taken to be indicative of reattachment of the separated shear layer (e.g. Rockwell 1977, Obasaju 1983). Again, the exact angle where reattachment occurs depends upon the freestream turbulence (Lee 1975). The data from the water channel and the data from the wind tunnel differ slightly. This difference is believed to be attributable primarily to the use of geometric blockage to correct the water channel data. In the wind tunnel data, the geometrical blockage correction was found to be approximately equal to the square of the correction as determined by the method of Ranga Raju & Singh (1975). The water channel data in the remainder of this paper were corrected using the geometrical blockage and are likely to be two to four percent too low. An additional source of the difference between the two sets of data is difference in the aspect ratios, as given in Table 1. In Figure 6, a comparison of the present results from the water channel experiments with those of Parker & Welsh (1983) for the dependence of the Strouhal number on the side ratio, B/D, at zero angle of attack is made. As indicated in the introduction, there are many sets of data (Nakaguchi et al. 1968, Bearman & Trueman Washizu et al. 1978, Bokaian & Geoola 1983) available for comparison. In all of these reported experiments, the general trends are quite similar, although there are some crucial differences in detail.

10 38 c. W, KNISELY I I rllll I i I T'T~III I I l I IIIII Present:square leading edg, f.o Parker & Welsh (1983): StT square leading edg, "3 - Parker& Welsh (1983): - semi-circular leading edge E.25.2 ~ 1.15 o.ic.5 "\ "\i C ct=o.in,. St becomes ~~. increasingly I ~ ~ Re-dependent ~x~ ~. i "- ~, :NNo regular \, l F//~ vortex ~. ~ street & Welsh I I I I I IIII I I I llllll I I I llll 1' 1 1 Slenderness ratio, BID Figure 6. Strouhal number as a function of side ratio, B/D, at zero angle of attack. The data of Parker & Welsh (1983) indicate four distinctive regions, as can be seen in Figure 5, which they discuss in detail in their paper. The data of Washizu et al. (1978) suggest the presence of an additional characteristic region, which lies between the first and second regions of the Parker & Welsh (1983) data. Washizu et al. found two distinct Strouhal numbers for side ratios in the range 2 < BID < 2-5 and somewhat sporadic shedding in the range 2.5<B/D <3. Beyond B/D =3 and up to their maximum value of B/D = 4, the results of Washizu et al. are in agreement with those of Parker & Welsh. It can be seen from Figure 6 that the present data follow quite closely the results from Parker & Welsh (1983). The discrepancy at the right hand side of the plot, i.e. at larger BID values, is most likely due to the Reynolds number dependence previously discussed in the Introduction, but may also be influenced to a small extent by the geometrical blockage correction and the aspect ratio. As will be shown subsequently, the limited data from the wind tunnel experiments at zero incidence agree fairly well with the water channel results. In Figures 7 through 16, Strouhal numbers for the various side ratios from the present tests are plotted as functions of the incidence. It is to be noted that all BID ratios listed are less than one and the angle of attack ranges from to 9. To find Strouhal numbers for bodies with BID greater than one, i.e. the reciprocal values, the lower abscissa scale must be employed. For a very thin plate with B/D = -4 (or B'/D' =25 at o:=9 ), the Strouhal number, as shown in Figure 7, is essentially independent of the angle of attack for -< o: -< 7. For angles between 7 and 9, no organized shedding could be detected in the present tests. Slight disagreement is seen between the present data and that of Fage & Johansen (1927), also plotted in Figure 7. Their data show a more significant

11 STROUHAL NUMBERS AT INCIDENCE 381 r ] i! I.2 o.15~ O ~.1 slo =.4 B "5 O Present (Water) z~ Fage & Johansen (1927) i I i I i I i I i I i I i ] i I v o Angle of attack,~ I I ~ I I [ I ] I I (for B'Io'= 25') Figure 7. Strouhal number as a function of the angle of attack for B/D =.4. rising tendency as the angle of attack exceeds -75. The possible source of this difference is the non-rectangular geometry shown in Figure 1, the effect of which becomes more significant as the angle of attack approaches 9. The present results are in agreement with the experiments of Abernathy (1962); his Strouhal numbers varied only between.147 and.153 over the range of bluff body type flow ( --- :-< 65 ). In Figure 8 similar data are presented for the case BID = -1. There is excellent agreement between the present smooth flow results and Novak's (1972) results with 15% turbulence, but Novak's 2-5% turbulence data are significantly higher than the other two cases. Based on the previous discussion of the effects of turbulence, one must regard this agreement as somewhat fortuitous. The results for B/D =.2 (water channel) and B/D =.25 (wind tunnel) are shown "2, i, ~, i, i,-:, ~, ~, I,.15, ~,,@ -1 BID ='1 B O Present (O-5 /o Tu-Water) Nov&k (2-5 /otu) A Nov.4k (15 O/o Tu) 5 n i I r ] I I i I i I I I ~ I ~ J J ~o Angle of attack, I t J I I I I / I J (for B'Io'=IO.O) Figure 8. Strouhal number as a function of the angle of attack for B/D = -1.

12 382 c.w. KNISELY.2-15 O<'z'v o o o o O <> o o o o o <, ~".1( -5 B Pr'esent, BID = "25 (Air') OO Present, ~1o = -2 (Water) i i I i t i I i I I i I i I, I Angle of attack,(~ 19O I I t I I I I 1 I I (for B'ID' =4- or 5-) Figure 9. Strouhal number as a function of the angle of attack for B/D =.2 and.25. in Figure 9. The trends for both sets of data are similar. From Figure 6, one would expect the BID =.25 data to fall slightly below the B/D =.2 data. However, as was the case in Figure 5, the wind tunnel results are consistently higher than the water channel results, again most likely due to the differences in Reynolds number, blockage correction and aspect ratio. In Figure 1 the results for the BID =.3 rectangular cylinder are plotted. For angles above about 5, the available data from Ohya et al. (198) are also included. The two sets of data agree quite well, although that of Ohya et al. is consistently above that of the present test. The difference is most likely due to the blockage correction, smaller aspect ratio and the lower Reynolds number of the present experiment. For rectangular cylinders with B/D = -4 and B/D =.5, in Figures 11 and 12, respectively, there are also two sets of data, that from the present experiment and,.2 l I ~ I I i i ' I I I i I l -15 O O a Oz~Oz~ A ~AZ~AZ~A.- -1C ii BIo="3 Present (Water) A Ohya etal, (198) B 5 u~ U~ O i I ~ I l I I ~ i L~_I t I ~ I l Angle of attack, 1 I I I I 1 _~ J ] J (for B'ID'=3'33) Figure 1. Strouhal number as a function of the angle of attack for B/D =.3.

13 STROUHAL NUMBERS AT INCIDENCE I ' I ' I ~ I J I ' I 7 I ~ I '-- OA o B % o o o o o o Q.1( BID =.4 B O Present (Water)] Ohya eta,(198o) / %.5 ~, I t I I i I I I I I i I, I i I i Angle of attack,a I ~ i_ t i I L I _ I I Q -2-1 (for B' ID'=2"5 ) Figure 11. Strouhal number as a function of the angle of attack for B/D =.4. again, the data from Ohya et al. (198). Again the agreement between the two sets of data is good, with the Ohya et al. data lying slightly above the values from the present test. Also included in Figure 12 are the results from the wind tunnel tests on a BID =.5 cylinder. For values of a~ less than about 2, the agreement is excellent. At higher angles of attack the wind tunnel data are again consistently above the water channel results. The probable reasons for this discrepancy are, as before, differences in blockage correction, Reynolds number and aspect ratios. As o: approaches 9 for the wind tunnel data in Figure 12, two Strouhal numbers are plotted. The small squares represent a weak higher frequency peak that appears to be an extension of vortex shedding mode at lower angles of attack. This high frequency mode is consistent with the previously mentioned observation of Washizu et al. of dual mode shedding for B/D < The dominant peak in the spectra, represented by.2 o.15 B/D ~,5-1 -~ a ~ ~ 5 o, I Angle of attack,a I I I I I I_ I I I (fop B'/D'= 2.) Figure 12, Strouhal number as a function of the angle of attack for BID =.5.

14 384 o W. KNISELY tt~.2.15 (.1 ' I ' I r I I I i I ~ I I I ~ I o BID =.6 Present (Water)].5 u o' Oo I 1 i I i I i t i t i I I I I I t Angle of attack, I I I I I I I I 1 I (for S'ID'=1"67) Figure 13. Strouhal number as a function of the angle of attack for BID =.6. the large diamonds, makes a sudden jump to a much lower frequency. A detailed discussion of this multiple model behavior follows the discussion of Strouhal numbers. For'larger BID ratios,.6 -< B / D -<.9, Strouhal number data can be found in Figures 13 through 16. The general trends are the same for all B/D ratios considered. There is a relatively quick rise in the Strouhal number at small angles of attack and then a subsequent plateau region for a wide range of angles of attack before a very steep decrease close to 9. In Figure 14, the data from Schramm (1966) for B/D = -75 is plotted to provide a comparison. In this figure there is considerable disagreement between the data from the literature and the present results. One possible explanation may be that Schramm's unspecified turbulence level was sufficiently high to cause the increase in Strouhal numbers shown in Figure 14. The sudden rise in Strouhal number with a slight increase in the angle of attack has i i I i I ~ I i i i I i I ' z~ i I ' '2O A A A A A a A A A A O O.15j~- A A o BID =.7 Present (Water) I ~".1 ASchramm (1966) I ii I (BID="75) l o I r I I I I J I 61 I 7[ l 81 9 o~ 11 ~ I Angle of attack, a I I I I I I t I I I (for B'ID'=1.43) Figure 14. Strouhal number as a function of the angle of attack for B/D = -7.

15 STROUHAL NUMBERS AT INCIDENCE 385.2, l ~ l....,..... i. ~ l ~ l ~ l ~ l, i,.15 B/O=-8 a in.1c I opresent (Water) I -5 u O I I J I I I ~ I, I i I, I, I t Angle of attack,a L I I I I I I I I I (for B'ID'= 1"25) Figure 15. Strouhal number as a function of the angle of attack for B/D = -8. been interpreted as signaling the reattachment of the separated shear layer. By examining the data in Figures 5 and 7 through 16, one can see that the angle at which reattachment occurs is dependent on the afterbody length. For B/D=.1, reattachment occurs near 35 (:1:5 ), while for the square cross section cylinder it occurs near 12 to 13. For a longer body, reattachment occurs at even smaller angles. For example, for-the B/D = 5 body, reattachment appears to occur near an angle of attack of MULTIPLE MODE BEHAVIOR FOR B/D = 2 Spectra of the fluctuating lift for small angles of attack are shown in Figure 17. Note that Figure 17 is drawn for BID = 2, and hence the angle of attack is the complement of the angle for BID =.5 (Figure 12). For o: =, there is a very dominant peak in.2 l * l l l l l l l l l l l l l l "15 BID =-9 n.1( -- O Present (Water) ] "5 o~ ' I -9 t I 1 J I I I i I I I I r I i Angle of attack, I I I I i I i j (for B'IB'=1.11 ) Figure 16. Strouhal number as a function of the angle of attack for B/D = -9.

16 386 c.w. KNISELY o -1-2 ~ -3o u Iu -~u JU ~u :) Frequency, Hz Figure 17. Spectra of fluctuating lift as a function of angle of attack, showing multiple mode behavior with decay of low frequency component and growth of high frequency component for BID = 2; U = 7 m/s; B = 5 cm; D = 2.5 cm; ReD = 1.1 x 14; db re 9-81N. the spectrum at about 24 Hz and a weak peak at about 39 Hz. These two peaks correspond to the low and high frequency modes in Figure 12. As the angle of attack is increased, the low frequency peak quickly diminishes and by tr = 5 is weaker than the high frequency peak. Above ~ = 6 there is no trace of the low frequency component. Since the spectra in Figure 17 are time averaged, it is not possible to say whether the weak high frequency peak is due to an intermittent switching of the shedding mode (due perhaps to transitory reattachment of the shear layer) or whether the two modes of shedding coexist simultaneously. Interestingly, the spectrum of velocity fluctuations taken at the edge of the shear layer (about 1 cm downstream and about 7-5 cm away from the wake centerline) indicate a reversed predominance of the two frequency components at angles near 3 to to 4. Comparing the fluctuating lift spectrum in Figure 18(a) with the simultaneously recorded velocity fluctuation spectrum in Figure 18(b), one is struck by the disappearance of the low frequency peak in the velocity spectrum. Other than during this occurrence of multiple mode shedding for the B/D = 2 cylinder, the frequency of the peak in the fluctuating velocity spectra always agreed with that of the fluctuating lift spectra. One conjectural explanation of the multiple mode behavior is that it represents the reorganization of the shear layer vortices downstream of the body. At zero incidence,

17 STROUHAL NUMBERS AT INCIDENCE 387 (a) -2 " I I I I I (b) d. -2 E ro b -4 N -6 "1" "o -81 I 1 I I I I Frequency, Hz Figure 18. (a) Fluctuating lift spectra (db re 9.81 N), and (b) fluctuating velocity spectra (arbitrary amplitude) recorded near the edge of the shear layer about 1 cm downstream of the trailing edge of the body and about 7.5cm away from the wake centerline for B/D=2; B=1cm, D=5cm; o:=4; Re D = 2-2 x 14. the reorganization occurs far enough downstream that only a small lift fluctuation due to the reorganized vortex street is detected. As the angle of attack increases, the location of reorganization moves closer to the body, as is evidenced by the dominant high frequency peak in the velocity spectrum. Finally, at large enough incidence, the re-organization occurs before the vortices detach from the body and there is no longer any dual mode behavior. More detailed measurements are needed to confirm or refute this speculative explanation FORCE COEFFICIENTS For BID ratios of 1, -5 and.25 the wind tunnel experiments yielded mean drag and lift coefficients as well as fluctuating lift coefficients as a function of the angle of attack. Figure 19 shows the measured data for B/D = 1 and comparison with available published results. In Figure 19(a), the quality factor is plotted. The quality factor Q is defined as the frequency of the peak in the spectrum divided by the difference in frequencies of the points which are 3 db down relative to the peak (the half-power points). The parameter plotted in Figure 19(a) has been normalized by the maximum value of Q. This was necessary since the bandwidth of the spectral peak did not change significantly with a doubling of the velocity; the frequency of the peak doubled, but the bandwidth remained almost unchanged, resulting in a doubling of the Q-values: The Q values measured with a freestream velocity of 7 m/s were approximately twice those reported by Obasaju (1983), and those for 14m/s were about four times his values. When normalized with the maximum Q values, the present results support Obasaju's observation of a peak near 5 incidence and a second peak near 13. The only significant difference in the two data sets is that the sudden drop in the Q value

18 1. /~ "8 E O.6 I.4,.2-2 o.16.12!- -- I [] [] -ID---~ D (a) (b) 2.2 ~.. I 3 llol o28 I ^ I.2[- I ~7-, V V V \7 I V -o.5 - ~/A~~ -lo - I A _ = Turbulent I I l I I I I I I Angle of attack, degrees Figure 19. Wind tunnel results for B/D = 1 as a function of angle of attack: (a) normalized quality factor; A, present results; O, Obasaju's (1983) data; (b) Strouhal number; (c) mean drag coefficient; O, present results; [], range of data compiled by Obasaju (1983); (d) Strouhal number-drag coefficient product; (e) mean lift; ~7, present data; A, Parkinson & Brooks (1961); (f) fluctuating lift; O, present rms value; V, present spectral peak value; *, Vickery (1966).

19 STROUHAL NUMBERS AT INCIDENCE 389 between the first and second peaks in the present data was much less severe than in Obasaju's data. The Strouhal number is re-plotted in Figure 19(b) for comparison. The measured drag from the present experiment is seen to skirt the lower edge of the range of previously reported results in Figure 19(c). The inverse behavior of St and CD with OE.5 2 U- U... 2:I1 I -8 o F O.1 "5 (e.2) -.5 _ ~ ~ (f.1) (f.2) & o.5' Angle of attack, degrees Figure 2. Wind tunnel results for B/D =.5 and.25 as a function of angle of attack: (a) quality factor Q; (b) Strouhal number; (c) mean drag coefficient; (d) Strouhal number-drag coefficient product; (e) mean lift coefficient; and (f) fluctuating lift coefficient;, rms value; O, spectral peak value.

20 39 c.w. KNISELY increasing t~ leads onto suspect that their product will be constant. Figure 19(d), however, shows that the StC~ product is not constant and decreases up to reattachment and thereafter returns to and then exceeds its original value at zero incidence. The mean lift coefficient shown in Figure 19(e) compares well with the uncorrected data from Parkinson & Brooks (1961). The blockage correction would reduce the magnitude of Parkinson & Brooks' data, making the agreement even better. The fluctuating lift coefficient, on the other hand, exhibits noticeable deviations from Vickery's (1966) published data. The dip in the fluctuating lift coefficient occurring near reattachment (o:= 12 to 13 ) was not detected in Vickery's study because of his coarse increment in angle of attack. At small angles of attack, the present data lie believably between Vickery's smooth flow and turbulent flow cases. At angles of attack above 15, the present data lie inexplicably above Vickery's data. The fluctuating lift coefficient determined from the spectral peak value agrees quite well with the measured rms value, indicating essentially all the fluctuating lift is due to coherent narrow band vortex shedding. Analogous data are presented for the B/D = -5 and.25 cylinders in Figure 2. In both cases the normalized Q factor and the Strouhal number are seen to peak first just after reattachment occurs. Near c~ = 9 a second peak occurs in the Q factor curve, indicating the reattachment points for the B/D = 2 and B/D = 4 cylinders. The force coefficients all have local minima just before reattachment for the B/D = -5 cylinder. Those of the B/D =.25 cylinder exhibit similar behavior, except for the fluctuating lift coefficient which exhibits a local maximum at reattachment. For both cases shown in this figure, the spectral peak value of fluctuating lift exceeds the measured rms value as approaches 9, indicating that there was a slight error either in measuring the rms value or in calculating the spectra. The trend, however, is qualitatively the same for both the rms value and the spectral peak values. In general, for -5 - B/D --< 2 geometries one can maximize the load bearing capacity of a support strut by specifying an incidence near the angle at which reattachment occurs. The advantages are reduced mean drag and reduced fluctuating lift, but the disadvantage is an increased mean lift. For a square cylinder, this combination represents a more efficient use of the available material. 4. CONCLUSION Experimentally determined Strouhal numbers have been presented for a family of rectangular cylinders, ranging from a flat plate to a square cross section cylinder, at varying angles of attack. The Strouhal number of all except the thinnest plate showed qualitatively similar behaviour with changing angle of attack. The general tendency is for a rapid rise in the Strouhal number to occur at relatively small angles of attack. This rapid rise is associated with reattachment of the separated shear layer. The angle of attack where reattachment and, hence, the rapid rise in Strouhal numbers occurs is dependent on the BID ratio. After the sudden rise, the Strouhal number levels off with a further increase in angle of attack. As the angle of attack approaches 9, there is a sudden decline in the value of the Strouhal number. For the thinnest plate, the Strouhal number was found to be essentially independent of the angle of attack over a wide range of angles. The bandwidth of the spectral peak in the lift spectrum was found to be essentially independent of the velocity, resulting in Q factors that increased with increasing velocity. For the B/D = 2. cylinder, dual mode behavior was found for angles of attack

21 STROUHAL NUMBERS AT INCIDENCE 391 ranging from to about 6. More extensive measurements are needed to clarify definitively the nature of this dual mode shedding. Measured force coefficients show a strong dependence on the angle of attack, suggesting that for.5 <-BID <-2. there is an optimum incidence that makes most efficient use of the load bearing capacity of the structure. ACKNOWLEDGEMENTS The author wishes to acknowledge the financial support of the SFB 21, University of Karlsruhe, and its sponsor, the German Science Foundation (DFG) for the water channel experiments. The encouragement of Professor Eduard Naudascher and the staff of the Institute for Hydromechanics is also acknowledged. The wind tunnel tests were conducted in a wind tunnel belonging to the Bridge Engineering Laboratory at Kyoto University, Japan. The assistance of Professor M. Matsumoto in arranging for the use of this tunnel is most gratefully acknowledged. REFERENCES ABERNATHY, F. H Flow over an inclined plate. ASME Journal of Basic Engineering 84, AwBI, H. B Effect of blockage on the Strouhal number of two-dimensional bluff bodies. Journal of Wind Engineering and Industrial Aerodynamics 12, BAUER, A. B Vortex shedding from thin flat plates parallel to the free stream. Journal of the Aerospace Sciences 28, BEARMAN, P. W. & TRUEMAN, O. M An investigation of the flow around rectangular cylinders. Aeronautical Quarterly 23, BEHRING, A. G. YEH, C. H Flow-induced trashrack vibration. In Proceedings of Joint ASME-CSME Applied Mechanics, Fluids Engineering and Bioengineering Conference, Niagara Falls, NY, pp BOKAIAN, A. K. t~ GEOOLA, F Hydrodynamic galloping of rectangular cylinders. In Proceedings of the International Conference on Flow-Induced Vibrations in Fluids Engineering, Reading, England (eds H. S. Stephens and G. B. Warren), pp , BHRA Fluid Engineering. BOKAIAN, A. K. & GEOOLA, F On the cross flow response of cylindrical structures. Proceedings of the Institution of Civil Engineering, Part 2, 75, BOKAIAN, A. K. & GEOOLA, F Hydroelastic instabilities of square cylinders. Journal of Sound and Vibration 92, CALLANDER, S. J Str6mungsbedingte Schwingungen von Rechteckst~iben (Flow-induced vibration of rectangular bars). Ph.D. Dissertation, Institute of Hydromechanics, University of Karlsruhe, Karlsruhe, West Germany. COURCHESNE, J. (~ LANEVILLE, A A comparison of correction methods used in the evaluation of drag coefficient measurements for two-dimensional rectangular cylinders. ASME Journal of Fluids Engineering 11, COURCHESNE, J. & LANEVILLE, A An experimental evaluation of drag coefficient for rectangular cylinders exposed to grid turbulence. ASME Journal of Fluids Engineering 14, CRANDALL, S. H., VIGANDER, S. & MARCH, P. A Destructive vibration of trashracks due to fluid-structure interaction. ASME Journal of Engineering for Industry, 97, FAGE, A. & JOHANSEN, F. C On the flow of air behind an inclined flat plate of infinite span. Proceedings of the Royal Society of London, Series A 116, GRANT, I. & BARNES, F. H The vortex shedding and drag associated with structural angles. Journal of Wind Engineering and Industrial Aerodynamics 8, HuoT, J. P., REY, C. & ARBEY, n Experimental analysis of the pressure field induced on a square cylinder by a turbulent flow. Journal of Fluid Mechanics 162, JAMES, E. C. & KATAKARA, F. T Oroville intake vibration resolved. In Proceedings of the ASCE National Water Resources Engineering Meeting, Phoenix, AZ, Reprint 1323.

22 392 c.w. KNISELY KESHAVAN, N. R Vortex shedding noise of aerofoils in smooth flow. Journal of Sound and Vibration 52, KMSELY, C. W Strouhal numbers of rectangular cylinders at incidence. Report No. SFB 21/E/13, Sonderforschungsbereich 21, University of Karlsruhe, Karlsruhe, West Germany. KNISELY, C. W., MATSUMOTO, M. & MENACHER, F Experimental investigations of prismatic cylinders in flow with harmonic pulsations. Report SFB 21/E/4, Sonderforschungsbereich 21, University of Karlsruhe, Karlsruhe, West Germany. IC~ISELV, C. W., MATSUMOTO, M. & MENACHER, F Rectangular cylinders in flows with harmonic perturbations. ASCE Journal of the Hydraulics Division, 112, No. 8, I~ISEL, C. W., MATSUMOTO, M. & SHIRAISHI, N Cylinders in a mean flow with pulsations--a review. In Proceedings of ASCE Conference on Advancements in Aerodynamics, Fluid Mechanics and Hydraulics, Minneapolis, MN (eds R. E. A. Arndt, H. G. Stefan, C. Farrel and S. M. Peterson), pp LANEVlLLE, A Turbulence and blockage effects on two-dimensional rectangular cylinders In Proceedings of the International Colloquium on Bluff Body Aerodynamics and its Applications (ed. Japan Assoc. Wind Engineers), Kyoto, Japan, LANEVILLE, A., GARTSHORE, I. S. & PARKINSON, G. V An explanation of some effects of turbulence on bluff bodies. In Proceedings of the 4th International Conference on Wind Effects on Buildings and Structures, Heathrow, England (ed. K. J. Eaton), pp , Cambridge Univ. Press. LEE, B. E The effect of turbulence on the surface pressure field of a square prism. Journal of Fluid Mechanics 69, LEE, B. E Some observations of the effect of aspect ratio on the influence of turbulence on the drag of rectangular cylinders. In Proceedings of the International Colloquium on Bluff Body Aerodynamics and its Applications (ed. Japan Assoc. Wind Engineers), Kyoto, Japan, Mooi, V. J. & SEATER, J. E Unsteady aerodynamics and vortex induced aeroelastic instability of a structural angle section. Journal of Wind Engineering and Industrial Aerodynamics 11, NAKAGUCHI, H., HASHIMOTO, K. & MUTO, S An experimental study on aerodynamic drag of rectangular cylinders. Journal of the Japan Society of Aeronautical and Space Sciences 16, 1-5. NAKAMURA, Y. & OHYA, Y The effects of turbulence on the mean flow past two-dimensional rectangular cylinders. Journal of Fluid Mechanics 149, NEILSON, F. M. & PICKET, E. B. 198 Corps of Engineers experience with flow-induced vibrations. In Proceedings of the Symposium on Practical Experience with Flow-Induced Vibrations, Karlsruhe (eds E. Naudascher and D. Rockwell), pp , Berlin: Springer Verlag. NOVAK, J Strouhal number and flat plate oscillation in an air stream. Acta Technica CSAV (Czechoslovakia), NOVAK, J Strouhal number of a quadrangular prism, angle iron, and two circular cylinders arranged in tandem. Acta Technica CSAV (Czechoslovakia), OBASAJU, E. D An investigation of the effects of incidence on the flow around a square section cylinder. Aeronautical Quarterly 34, OHYA, m., WASHIZU, K., FuJII, K. & OTSUKI, Y. 198 Wind tunnel experiments on aerodynamic forces and pressure distributions of rectangular cylinders in a uniform flow (Part 2) (in Japanese). In Proceedings of the 6th National Symposium on Wind Engineering (Japan), pp OKAJIMA, A Strouhal numbers of rectangular cylinders. Journal of Fluid Mechanics 123, OTSUKI, Y., FuJII, K., WASHIZU, K. & OHYA, A Wind tunnel experiments on aerodynamic forces and pressure distributions of rectangular cylinders in a uniform flow (in Japanese). In Proceedings of the Japan Symposium on Wind-Structure Interaction, pp PARKER, R. & WELSH, M. C Effects of sound on flow separation from blunt flat plates. International Journal of Heat and Flud Flow 4, PARKINSON, G. V Wind-induced instability of structures. Philosophical Transactions of the Royal Society of London, Series A 269,

23 STROUHAL NUMBERS AT INCIDENCE 393 PARKINSON, G. V. & BROOKS, N. P. H On the aeroelastic instability of bluff cylinders. ASME Journal of Applied Mechanics 28, PEarY, D. G The effects of turbulence intensity and scale on the flow past square prisms. Journal of Wind Engineering and Industrial Aerodynamics 4, RANOA RAGU, K. G. & SINGH, V. 1975/76 Blockage effects on drag of sharp-edged bodies. Journal of Wind Engineering and Industrial Aerodynamics 1, ROCKWELL, D. O Organized fluctuations due to flow past a square cross section cylinder. ASME Journal of Fluids Engineering 99, SCHRAMM, W Wirbelfrequenzmessungen an umstr6mten Bauteilen. IfL-Mitteilungen (Institut fiir Leichtbau und Okonomische Verwendung yon Werkstoffen Mitteilungen) 5, SVAMALARAO, B. C A review of trashrack failures and related investigations. Water Power and Dam Construction 41, No. 1, VICKERY, B. J Fluctuating lift and drag on a long cylinder of square cross-section in a smooth and turbulent stream. Journal of Fluid Mechanics 25, WASHIZU, K., OHYA, A., OTSUKI, Y. & FuJII, K Aeroelastic instability of rectangular cylinders in a heaving mode. Journal of Sound and Vibration 59, WooTroN, L. R., WARNER, M. H. & COOPER, D. H Some aspects of the oscillation of full-scale pile. In Proceedings of the Flow-Induced Structural Vibrations Symposium, Karlsruhe (ed. E. Naudascher), pp , Berlin: Springer Verlag. APPENDIX: NOMENCLATURE A AM As B B' Co CL CL D f D' f H N Q R U Re St Tu ol v Flow Area of Water Channel (A = (Channel Width - D') Depth) Wind Tunnel Model Area Wind Tunnel Cross Sectional Area Cylinder Dimension in Streamwise Direction (cm) Projected Streamwise Dimension (B' = B cos o: + D sin o:) Mean Drag Coefficient Mean Lift Coefficient RMS Fluctuating Lift Coefficient Cylinder Dimension in Transverse Direction (cm) Projected Transverse Dimension (D' = B sin : + D cos o:) Frequency of Vortex Shedding (Hz) Cylinder Length (cm) Empirical Function of B/D Used in Blockage Correction Quality Factor of Spectral Peak Corner Radius (cm) Mean Velocity (Q/A) (cm/s) Reynolds number (ud'/v) Strouhal Number (fd'/u) Turbulence Intensity in Percent of Freestream Velocity Angle of Attack (Angle between flow direction and normal to upstream face of cylinder) Kinematic Viscosity (m2/s)