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Observations of Reynolds number sensitivity in the separated flow region on a bluff body
Journal of Wind Engineering and Industrial Aerodynamics 73 (1998) 231—249
Observations of Reynolds number sensitivity in the separated flow region on a bluff body R.P. Hoxey*, A.M. Reynolds, G.M. Richardson, A.P. Robertson, J.L. Short Silsoe Research Institute, Wrest Park, Silsoe, Bedford MK45 4HS, UK Received 6 June 1996; accepted 10 September 1997
Abstract Measurements have been made at both full scale and model scale of the surface pressures on a conventional low-rise, pitch-roofed building. The full-scale measurements were made under natural strong wind conditions in a near-equilibrium atmospheric boundary layer. Replicated model-scale experiments were made in simulated atmospheric boundary-layer flows at 1 : 100 scale in two wind tunnels. The transverse pressure distribution at the mid-length section of the building produced by a transverse wind has been examined for sensitivity to Reynolds number. The influence of changes in mean wind speed at full scale, and of changes in length scale between full scale and model scale is presented. Both sets of results support the hypothesis that the separated flow at the windward eaves of the building is Reynolds number dependent. Corroborating evidence was obtained from further full-scale and model-scale experiments on the same building by changing the conventional sharp-line eaves to a curved eaves: this produced predominately attached flow over the roof, for which no Reynolds number sensitivity was observed. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Wind loads; Full-scale; Model-scale; Reynolds number; Bluff Body; Low-rise; Atmospheric boundary layer
1. Introduction It is a widely held view that the aerodynamics of bluff bodies are Reynolds number, Re, insensitive, and hence that tests on small-scale models in simulated atmospheric boundary layers are an acceptable method for obtaining information which can be applied to full-scale structures. This view was expressed by Jensen [1] in presenting his ideas regarding the model-law. He focused on the location of the start of the vortex layer at the sharp corners of bluff-bodies. He, like others to follow, made the
* Corresponding author. 0167-6105/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 7 - 6 1 0 5 ( 9 7 ) 0 0 2 8 7 - 0
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assumption that provided the separation point is predetermined by geometry, “all essential relations in the flow must also be rather independent of Re”. Since the 1950s, several studies have been made on low-rise buildings at both full scale and model scale which can be used to test the modelling assumption. Retrospective examination shows that consistent discrepancies exist between full-scale and model-scale results in separated flow regions. The work described here on the Silsoe Structures Building, and, for example, that by Surry and Johnson [2], illustrate this discrepancy. It has been proposed that one possible cause is an inadequate boundary layer simulation, in particular, one not matching turbulence length scales, and this has been the subject of an investigation by Li and Melbourne [3]. However, the full-scale studies at Silsoe, complemented by two wind-tunnel comparisons where the boundary layer was well simulated, cast doubt on the basic assumption that bluff-body flows are insensitive to Reynolds number. Full-scale surface pressure measurements have been made under natural wind conditions on a range of low-rise buildings for the purpose of defining design loads (see Refs. [4—6]). To improve the understanding of wind loading and wind load transfer, an experimental portal-framed building was constructed at Silsoe (the Silsoe Structures Building) on which detailed surface pressure measurements were made. These fullscale measurements provided definitive benchmark data with which to compare and assess model-scale and computational fluid dynamic (CFD) data. Following improvements to wind-tunnel experiments made by Richardson and Surry [7], differences emerged between full-scale and model-scale data which exceeded the expected error, and which showed distinct patterns that suggested an Re sensitivity. The full-scale data were thus re-examined and a statistical analysis conducted to explore in greater detail the existence of trends between mean pressure coefficient values and wind speed for a common wind direction. This exercise was repeated for a second set of aerodynamically distinct data obtained from the Silsoe Structures Building when its configuration had been changed by replacing the conventional sharp-line eaves with a modern curved eaves detail. The curved eaves geometry induced attached flow around the eaves and the windward roof slope which was in direct contrast to the localised separated flow produced by the conventional sharp-line eaves (see Refs. [8—10]). This second set of data fortuitously provided extremely valuable supplementary information for establishing Reynolds number sensitivity (fortuitous since the curved eaves were selected as being representative of modern building practice to investigate consequential wind loading distributions, but it was not known a priori that this eaves detail would produce a contrasting attached flow regime). To test further the Re sensitivity hypothesis, a systematic comparison between the full-scale data and the results from two independent 1 : 100 scale wind-tunnel tests was undertaken. This provided additional evidence of the existence of a significant dependency. 2. Experimental arrangements Similar experimental techniques were used in the full-scale and model-scale tests, employing conventional surface pressure tappings, pressure transmission tubing and differential pressure transducers. Reference wind speed, direction and static pressure
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were sensed upstream at building ridge height at a point where the pressure field generated by the building was not significant (see Ref. [11]). The wind-tunnel modelling work was conducted at 1 : 100 linear scale, and particular care was taken to scale correctly the boundary-layer surface roughness parameter (z ) and turbulence length 0 scales (¸ , ¸ and ¸ ). Instrumentation response was also matched for both pressure x y z measurements and wind speed, although a different type of wind-speed sensor was used at model scale. Details of the experimental arrangements have been published previously and so only a brief re´sume´ containing references which give further details is presented here. 2.1. Full-scale measurements The Silsoe structures building was designed and constructed specifically to undertake wind-load and building response measurements. Full details are given in Ref. [12]. The building, measuring 24.1 m long by 12.9 m span by 5.3 m ridge height, with a 10° duo-pitched roof, was constructed of modern cold-formed steel portal frames clad in a single skin of coated, cold-formed, profiled steel sheeting. Pre-cast concrete pad foundations were used which incorporated special optional frame fixings for structural testing purposes. Pressure tappings were positioned at the centre of aluminium sheets measuring 600 mm square which were fixed to the cladding to provide a locally smooth surface spanning three cladding corrugations. A tapping hole of 9 mm diameter was used, as this was sufficiently large to prevent rain-water blocking the hole. An internal porous ceramic drain removed unwanted water from the pressure tubing. A solid-state pressure transducer with a range of $1.2 kPa was connected to each tap by a 1 m length of flexible plastic tube. A total of 32 pressure taps distributed over one-half of the building were monitored at a scan frequency of 5 Hz. There was no measurable attenuation of the pressure system below 50 Hz. Initially, the Silsoe Structures Building had a conventional sharp-line eaves detail formed by the roof sheets overhanging the side-wall by 50 mm; no gutter was present. Following measurements for this configuration, which extended over one year, the eaves detail was changed to one of a modern curved eaves of radius 635 mm, and wind-load measurements were repeated for a second year. The ridge detail consisted of a simple sharp-line folded ridge cap throughout. The basic full-scale flow patterns developed over the two eaves details and the ridge are shown in Fig. 1. Smoke from a canister was introduced locally into the flow to illustrate the separated flow over the sharp eaves, the attached flow over the curved eaves, and, for both curved and sharp eaves, attached flow over the ridge. Some divergence of the flow can be seen on the leeward roof slope, towards the downwind eaves, which indicates that separation takes place before the leeward eaves. A three-component sonic anemometer was positioned upstream at the ridge height of the building to sense the undisturbed wind velocity. At an adjacent upstream position, the total wind pressure and the static pressure were sensed using a directional Pitot tube and a static probe, respectively. The static probe was designed to be insensitive to the horizontal flow direction and was calibrated at full scale against the pressure from a tapping hole set flush into level ground. The provision of an accurate
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Fig. 1. Flow visualisation for a transverse (perpendicular to ridge) wind: (a) sharp eaves; (b) 635 mm radius curved eaves; (c) ridge.
static pressure for differential pressure measurement is essential, and considerable effort went into the calibration and operation of the static pressure probe. However, the experimental results presented here suggest the existence of a small Reynolds number dependency in the sensed static pressure, which is quantified later. Pressure transducer instabilities were invariably small, but nonetheless were corrected by means of a computer-controlled sequence which applied a zero pressure difference, followed by the total pressure from the Pitot tube, to all the pressure transducers at the beginning and end of each 1 h recording period. The 1 h records were processed as six 10 min records to give mean and fluctuating properties. The analysis in this report is based on the 10 min mean pressure coefficient, C , and the p corresponding mean wind speed and mean direction at ridge height. Wind speed, as sensed by the sonic anemometer, was monitored continuously and recordings were initiated when the 5 min mean wind speed exceeded 8 m s~1. The fetch was predominantly cut grass extending over 600 m. Measurements were made over the winter months during which the condition of the fetch remained constant. Extensive measurements were made to classify the approach flow by using anemometers positioned at heights from 0.1 to 100 m. The mean velocity ratios, referred to the
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Fig. 2. Full-scale velocity profile: (a) mean velocity; (b) turbulence intensity.
velocity measured by a fixed anemometer at 10 m, and the turbulence intensities are presented in Fig. 2. The mean velocity profile was well represented by a log-law with a roughness length of z "0.010 m (Fig. 2a). The frictional velocity, º , derived from 0 q u—w measurements in the lower 10 m of the boundary layer was 0.0607º — a value 10 consistent with the velocity profile and a von Karman constant of i"0.42. Although building surface pressure measurements were made over a wide range of wind direction, the results presented here are restricted to those where the 10 min mean wind direction was within one degree of perpendicular to the long side of the
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building. This narrow band of mean wind direction was selected to remove the possibility of changes in C with wind direction influencing the result. (Experience has p shown that the method used here to test for Re sensitivity can be applied only when other parameters that can produce variability in C are minimised, for example by p selecting a wind direction, h, for which dC /dh"0.) p 2.2. Model-scale measurements 2.2.1. Boundary-layer wind tunnels and building models The two wind tunnels used in the present work were the University of Western Ontario (UWO) Boundary Layer Wind Tunnel No. 1 [7,13] and the Building Research Establishment (BRE) Boundary Layer Wind Tunnel No. 3 [14]. Two 1 : 100 geometrical scale models of the Silsoe Structures Building (one for the sharp eaves geometry, the other for the curved eaves geometry) were carefully constructed from acrylic sheet and pressure tappings were incorporated at positions corresponding to the full-scale tapping positions. 2.2.2. UWO experiments The boundary layer in the UWO No. 1 open-return wind tunnel was developed over a distance of 20 m as the air was sucked down the 2.5 m wide by 2 m high parallel sided tunnel by an axial flow fan positioned downstream of the working section. Three large spires near the inlet were used to increase the boundary layer thickness and to increase the turbulence levels. These were followed by sets of 25 mm cubes positioned over the floor to within 2 m of the model. Randomly spaced, 7 mm high machine nuts were placed over the remaining 2 m of the approach. The resulting velocity profile at the model position (with the model removed) was measured using two hot wires mounted on a vertical traverse. The velocity profile is presented in Fig. 3. The mean velocity profile (Fig. 3a) shows a region close to the ground where the velocity exceeds the desired log-law. This is associated with the change in roughness 2 m upstream of the model position. Also, the upper part of the boundary layer is not fully developed owing to the finite length of the tunnel. The mid-region of the boundary layer, extending from the height of the model to approximately 50 m full-scale equivalent height, has a roughness length of 0.1 mm, giving a mean velocity profile scale of 1 : 100 as required. Examination of the spectrum of turbulence also shows a good 1 : 100 scale fit to the full-scale spectrum. Total and static pressures were sensed by a conventional pitot-static probe positioned near the roof of the working section of the tunnel. Correction factors were thus required to derive pressure coefficients consistent with those derived at full scale. Reference static pressure was corrected to give agreement between full- and model-scale pressure coefficients at the mid-length of the building for flow parallel to the building ridge. A Scanivalve system incorporating a Statham 2.5 psi transducer, which gave a flat response up to 100 Hz, was used to measure surface pressures over the model building. 2.2.3. BRE experiments The BRE No. 3 open-return wind tunnel has a cross-section measuring 2 m wide by 1.5 m high, and an overall length of 20 m. The “blow-down” tunnel has a large
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Fig. 3. UWO boundary-layer wind tunnel velocity profiles: (a) mean velocity; (b) turbulence intensity.
centrifugal fan at the inlet. The working section was fitted with permeable aerofoil slats in the walls and roof to reduce the influence of blockage. A turbulence grid and a deeply saw-toothed barrier were placed at the tunnel inlet. Between the inlet and the working section, 10 mm high roughness elements were distributed over the tunnel floor. Roughness was added close to the model to maintain a near-equilibrium boundary layer. The mean velocity profile measured by a cross-wire is shown in
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Fig. 4. BRE boundary-layer wind tunnel velocity profiles: (a) mean velocity; (b) turbulence intensity.
Fig. 4a, and the turbulence intensity is shown in Fig. 4b. The influence of insufficient roughness close to the model is again apparent in the mean velocity profile, but the extent of the linear log-law region of the boundary layer is greater than in the UWO simulation (Fig. 3a), probably as a result of the combined influence of the centrifugal fan and the saw-toothed barrier giving a non-uniform inlet velocity. The scaling of the mean velocity profile and of the turbulence was again approximately 1 : 100.
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In the BRE experiments, a pitot-static probe was positioned at the ridge height of the model (53 mm), upstream and to one side of the model, in a position equivalent to that used at full scale. Corrections were applied for static pressure error caused by turbulence effects on the probe by making comparisons with a surface pressure tapping at the centre of the turntable when the model had been removed. The uncertainty in pressure coefficient values resulting from the use of continuous local flow monitoring (BRE approach) is significantly less than that resulting from the use of a profile calibration undertaken prior to model testing (UWO approach).
3. Analysis and results Mean pressure coefficients for both the sharp and curved eaves configurations at building mid-length for a transverse wind are shown in Fig. 5. The full-scale coefficients, Fig. 5a, were derived from curve fitting to over 1000 data points of 10 min mean values obtained over a range of mean wind directions. An example of this method is shown in Fig. 6 for tap 28 on the windward roof slope of the building with the sharp eaves detail. The transverse, or perpendicular to ridge, wind direction is denoted by 180°. The uncertainty in full-scale coefficients for the transverse wind direction is small (of order 0.01) due to the large number of data points. The coefficients from the wind tunnels, Fig. 5b and Fig. 5c, were obtained with the transverse axis of the model aligned with the tunnel axis. The uncertainty in the results from both wind tunnels is greater than that at full scale as there were no repetitions of measurements within each wind tunnel. The UWO reference pressures were recorded at a position remote from the model, giving an overall uncertainty of the order of 0.1 in coefficients. The use of local reference pressures in the BRE tunnel reduced the uncertainty to 0.05. The arithmetic differences between the full-scale coefficients and the average of the two wind-tunnel results are shown in Fig. 7. Good agreement exists on the windward wall which is to be expected for a well-simulated boundary-layer flow. A marked difference is observed on the windward roof slope for the sharp eaves configuration which is associated with the flow separation over the eaves and subsequent reattachment. A less significant difference is observed on the windward roof for the curved eaves configuration, but examination of the two sets of wind-tunnel results for this case, Fig. 5b and Fig. 5c, shows a significant difference between the model-scale data sets, with the coefficients from the BRE tunnel being in closer agreement with the full-scale data. There was evidence from flow visualisation using smoke, that at model scale there was separation at the ridge with early reattachment; this was not the case at full scale where the flow remained attached (Fig. 1c). This difference may, in part, be caused by the changes in flow on the windward roof at model scale. The suggestion from these measurements is of a change in flow pattern over the windward eaves which is possibly Reynolds number dependent; this observation encouraged further analysis of the full-scale data. The full-scale pressure coefficients were derived by curve fitting to the entire data set, as illustrated in Fig. 6. The number of data points with a mean flow direction
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Fig. 5. Mean pressure coefficients at building mid length for a transverse wind: (a) Full-scale (showing also the full-scale tapping point numbers); (b) UWO wind tunnel; (c) BRE wind tunnel.
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Fig. 5. (Continued).
Fig. 6. Full-scale pressure coefficients for tap-28 measured over a range of wind directions.
within one degree of perpendicular to the side wall was sufficient, however, to test the hypothesis that the flow was Reynolds number sensitive since they relate to a range of wind speeds of some 7—15 m s~1. For this narrow band of direction, 43 data points were available for the sharp eaves case, and 41 for the curved eaves case. The pressure
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Fig. 7. Arithmetic difference between full-scale and averaged wind-tunnel pressure coefficients.
coefficients were plotted against Reynolds number based on the building ridge height dimension (5.3 m) and the measured mean velocity at the reference position for the same recording period. An example is presented for tap 28 in Fig. 8a for the sharp eaves configuration, and in Fig. 8b for the curved eaves configuration. The scatter of the data points in Fig. 8 is associated with the separation of the anemometer from the tapping point. This scatter was quantified by using two anemometers on a similar open site with no building present and varying the horizontal distance between the anemometers. Analysis of the measurements shows that for a spacing of 20 m and an averaging period of 10 min, the standard deviation of the difference in dynamic pressure was approximately 3%. This difference was found to be normally distributed and independent of wind speed. The scatter of points in Fig. 8 is consistent with this observation. For the data presented in Fig. 8, linear regression analysis and evaluation of Student t values shows that for the sharp eaves case the slope is highly significant (t"!4.26), with a value of !0.281 per decade of Re, but that for the curved eaves case the slope was not significant (t"!0.96). The analysis for all the tapping points shown in Figs. 5 and 7 is summarised in Fig. 9. The Student t values are given in Fig. 9a, and the slopes of the regression lines in Fig. 9b (which have units of C per p decade of Re).
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Fig. 8. Pressure coefficients for tap-28 showing: (a) significant regression with Reynolds number for the sharp eaves; (b) non-significant regression with Reynolds number for the curved eaves.
4. Discussion and analytical indicators Before comparing wind-tunnel and full-scale coefficients to investigate any similarity in Reynolds number sensitivity, examination of Fig. 9b shows that negative regression occurs for all tappings on both the sharp and curved eaves building configurations. For the curved eaves case, only the lower tap on the leeward roof has a significant slope (t"!3.7). The probability of all the slopes being negative (mean value !0.0775, standard deviation 0.0384 for all non-significant regression slopes) by chance is very low ((1%) and a physical reason must be assumed. The most likely
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Fig. 9. Analysis of full-scale data: (a) Student t values; (b) slope of regression lines.
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cause is considered to be a Reynolds number dependency in the instrumentation, specifically the omni-directional static pressure probe. This instrument comprises a vertical cylindrical tube (38 mm in diameter) with an annular set of holes (each 3 mm in diameter) that are covered by an adjustable shroud. It can be expected to have some Reynolds number dependency, but previous calibrations have not quantified this. Combining all non-significant regression coefficients (DtD)2), indicates a mean regression of !0.0775C per decade of Re. Over the entire measurement p range (which equates to 0.3 of a decade), the pressure change is small, i.e. 0.02C , and p this does not increase the uncertainty in full-scale values which are of the order 0.01 since the probe was calibrated at wind velocities close to the overall mean velocity of 9 m s~1. Another possible cause of Re sensitivity was the 9 mm diameter tapping hole size. Applying the methods described by Shaw [15], indicates that the expected sensitivity to Reynolds number of these large tapping holes is of order 0.003C per decade of Re, p i.e. more than an order of magnitude smaller than the measured value. According to Bryer and Pankhurst [16], directional Pitot tube readings are not Reynolds number dependent, other than at very low Re, nor are any Mach number effects significant at these low velocities. With the elimination of these possible causes, the hypothesis remains that the flow over this bluff body is Reynolds number sensitive. Assuming that the hypothesis is true, it is of interest to extrapolate from the full-scale data, using a linear model, to compare trends with the model-scale data (two decades of Re lower), whilst acknowledging that the application of the linear-regression model for Reynolds number outside the full-scale measurement range is experimentally unjustified. To do this, the slope of the regression line was corrected by subtracting the !0.0775 static probe factor. The result of this correction was to reduce the number of tappings at which a significant Re sensitivity remained to those on the lower part of the windward roof of the sharp eaves building. The results given in Table 1 are for the three tapping points near the windward sharp eaves. These show good agreement in the welldeveloped central region of the separated flow (taps 26 and 27), but indicate a breakdown of the linear interpolation at tap 28 which is in the vicinity of reattachment (estimated from full-scale flow visualisation in Fig. 1b to be at approximately 3 m from the leading edge).
Table 1 Full-scale and model-scale pressure coefficients over lower part of windward roof slope of building with sharp eaves Tapping point
Distance from leading edge (m)
Full scale C p
UWO model-scale C p
BRE model-scale C p
Linear-regression estimate of model-scale C from full-scale C p p
26 27 28
0.965 1.965 3.465
!1.389 !1.148 !0.697
!1.098 !0.697 !0.444
!1.109 !0.633 !0.498
!1.035 !0.650 !0.289
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5. Theoretical understanding An intuitive, conceptual understanding of the observed Reynolds number effects can be gained by focusing attention on the dynamics of vortices. It is proposed that the observed Reynolds number effects can be attributed to the formation and subsequent instability of the vortex sheet formed from vortices generated at the eaves. To satisfy the constraint of having no flow normal to the surface of the roof, vortices can be thought of as belonging to contrarotating vortex-image pairs (see Fig. 10a). Once generated, these vortex pairs would, in the absence of a superimposed advecting flow, be transported, by their self-induced motion, upstream in a direction normal to the line joining their centres. However, at least initially, the surrounding flow will overcome this self-induced motion and the vortex pair will be advected downstream.
Fig. 10. Vortex dynamics: (a) vortex-image vortex pair; (b)—(d) time evolution of the motion of two vortex-image vortex pairs ((———) motion due to advecting flow and corresponding image vortex, (— — —) motion due to interaction between the two vortex-image vortex pairs) where the trailing vortex “separates” from the surface as it is advected downstream whilst the leading vortex “reattaches” to the surface and then “recirculates”; (e) predicted evolution of a vortex in a system of many vortex-image vortex pairs (the vortex “separates” from the surface at the leading edge before eventually “reattaching” and “recirculating”).
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In practice, this process of vortex pair generation and advection will be continuous and will lead to the formation of a vortex sheet. For clarity of understanding, it is, however, beneficial to consider a system of discrete vortex pairs. Vortex pairs will, by virtue of the flow fields they induce, interact with one another. The effects of this interaction on the motion of two vortex pairs is shown in Fig. 10b—10d. The flow field induced by the leading vortex pair causes the trailing vortex pair to move apart. As the trailing vortex and image vortex move apart, their self-induced back-flow will decrease and, consequently, their forward motion due, in part, to their induced backflow and the superimposed advecting flow, will increase. Conversely, the flow induced by the trailing vortex pair on the leading vortex pair will cause the separation and forward motion between the leading vortex and image vortex to decrease. Eventually, the leading vortex pair will come close together, their self-induced back flow will exceed the advection velocity, and they will be transported back upstream towards the eaves. Vortices generated at the eaves can thus be expected to initially lift-off from the surface of the roof as they are swept downstream before returning to the roof ’s surface to be transported upstream towards the eaves. In other words, as shown in Fig. 10b—10d, vortices generated at the eaves are expected to initially “separate” from the roof ’s surface as they are transported downstream before eventually “reattaching” and “recirculating”. This was indeed found to be the case in numerical studies of a simple system of such vortex pairs. The system studied consisted of pairs of point vortices, of strength $C, confined to move in two-dimensions. Pairs of vortices were generated at the eaves every C/º2 seconds, the natural time scale of the system, where º is the superimposed advection velocity. The dynamics of these vortex pairs was studied by numerical integration of the corresponding equations of motion. The equations of motion and the numerical method can be found in Ref. [17]. In the numerical studies, the effects of viscosity, which will cause the strength of real vortices to decay in time and impose a no-slip boundary condition at the surface of the roof, was neglected. The vortex sheet formed by such vortices was found to lift off from the surface of the roof at the eaves and then reattach further downstream before eventually rolling-up (see Fig. 10e). In accordance with the experimental observations, the average location of the “reattachment” point, r, was found to scale with Reynolds number, r&C/ º¸&1/Re, ¸ being an arbitrary length scale. Moreover, the height of the separation bubble was also found to scale as 1/Re. Thus, intriguingly, the length-to-height ratio of the separation bubble is predicted to be Reynolds number independent. The shape of the leading edge of the bubble is predicted to be stationary and Reynolds number independent, whereas the trailing edge is predicted to be transient and Reynolds number dependent. That is, for a given Reynolds number, the location of the “reattachment” point fluctuates, in time, about its mean position. This particular form of transience is intrinsic to the dynamics of the vortices. In practice, transience can also result from turbulent velocity fluctuations and fluctuations in the mean wind speed and direction. The apparent stability of the leading edge of the separation bubble, and the transient nature of the trailing edge and location of the reattachment are supported by observations of smoke plumes around the building.
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For the curved eaves case, the flow was found, experimentally, to remain attached. Within the vortex picture, this can be attributed to the suppression of vortex generation at the rounded eaves.
6. Concluding comments The reported experimental observations of Reynolds number sensitivity in bluffbody aerodynamics have been explained by considering the motion of a theoretical vortex pair which indicated a Reynolds number dependency in the separated flow region, particularly, in the vicinity of reattachment. This concept predicts no Reynolds number effect near the leading edge which is consistent with the experimental data, in so much as the effect is reduced for the tapping point closest to the leading edge. There is some indication of Reynolds number effect associated with separation of the flow off the leeward roof for both the sharp and curved eaves details, but the magnitude of pressure coefficients in this region is small and the dependency not therefore fully established. It is likely, however, that a strong dependency exists near the gable ends where pronounced flow separations are generated. The findings presented here are statistically highly significant and provide strong evidence that a Reynolds number effect is present in the separated flows around bluff bodies. Some indication of the magnitude of the effect has been estimated (approximately 0.25C per decade of Re), but more detailed work is required to produce p generalised correction terms for extrapolation from model-scale to full-scale values, as is required to obtain reliable design information. The implication for high-rise building design, where smaller scale models are used, is likely to be more significant than for low-rise buildings, and this should provide the impetus for further work to be done.
Acknowledgements This programme of work was funded by the UK Ministry of Agriculture, Fisheries and Food, and the Building Research Establishment. Financial assistance was also received from the Natural Sciences and Engineering Research Council of Canada through operating grants made available to D. Surry and A.G. Davenport at the University of Western Ontario.
References [1] M. Jensen, The model-law for phenomena in natural wind, Ingenioren - Int. Edition 2 (1954) 4. [2] D. Surry, G.L. Johnson, Comparison between wind-tunnel and full-scale estimates of wind loads on a mobile home, J. Wind Eng. Ind. Aerodyn. 23 (1986) 165—180. [3] Q.S. Li, W.H. Melbourne, An experimental investigation of the effects of free-stream turbulence on streamwise surface pressures in separated and reattaching flows, 3rd Asia-Pacific Symp. on Wind Engineering, 13—15 December 1993, Hong Kong, J. Wind Eng. Ind. Aerodyn. 54/55 (1993) 313—323.
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[4] G.M. Richardson, R.P. Hoxey, A.P. Robertson, J.L. Short, The Silsoe Structures Building: the completed experiment Part 1. Proc. 9th Int. Conf. on Wind Engineering, New Delhi, India, 9—13 January 1995. [5] R.P. Hoxey, P.J. Richards, G.M. Richardson, A.P. Robertson, J.L. Short, The Silsoe Structures Building: the completed experiment, Part 2, Proc. 9th Int. Conf. on Wind Engineering, New Delhi, India, 9—13 January 1995. [6] R.P. Hoxey, A.P. Robertson, Pressure coefficients for low-rise building envelopes derived from full-scale experiments, Presented at Invitational Seminar on Wind, Rain and the Building Envelope, London, Ontario, Canada, 16—18 May 1994, J. Wind Eng. Ind. Aerodyn. 53 (1,2) (1994) 283—297. [7] G.M. Richardson, D. Surry, The Silsoe Building: comparison between full-scale and wind-tunnel data, J. Wind Eng. Ind. Aerodyn. 51 (1994) 157—176. [8] G.M. Richardson, D. Surry, Silsoe Building: comparison of pressure coefficients and spectra at model and full-scale, J. Wind Eng. Ind. Aerodyn. 41—44 (1992) 1653—1664. [9] G.M. Richardson, P.A. Blackmore, The Silsoe structures building: comparison of 1:100 model-scale data with full-scale data, J. Wind Eng. Ind. Aerodyn. 57 (1995) 191—201. [10] A.P. Robertson, Effect of eaves detail on wind pressures over an industrial building, J. Wind Eng. Ind. Aerodyn. 38 (1991) 325—333. [11] R.P. Hoxey, P.J. Richards, Flow patterns and pressure field around a full-scale building, J. Wind Eng. Ind. Aerodyn. 50 (1993) 203—212. [12] A.P. Robertson, A.G. Glass, The Silsoe Structures Building — its design, instrumentation and research facilities, Div. Note DN 1482, AFRC Inst. Eng. Res., Silsoe, October 1988. [13] A.G. Davenport, N. Isyumov, The application of the boundary- layer wind tunnel to the prediction of wind loading, Proc. Int. Research Seminar on Wind Effects on Buildings and Structures, Ottawa, University of Toronto Press, Toronto, 1967. [14] G.V. Parkinson, N.J. Cook, Blockage tolerance of a boundary-layer wind tunnel, J. Wind Eng. Ind. Aerodyn. 41—44 (1992) 873—884. [15] R. Shaw, The influence of hole dimensions on static pressure measurements, J. Fluid Mech. 7 (1960) 550—564. [16] D.W. Bryer, R.C. Pankhurst, Pressure-probe methods for determining wind speed and flow direction, NPL, HMSO, 1971. [17] A.J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973) 785—796.
Observations of Reynolds number sensitivity in the separated flow region on a bluff body R.P. Hoxey*, A.M. Reynolds, G.M. Richardson, A.P. Robertson, J.L. Short Silsoe Research Institute, Wrest Park, Silsoe, Bedford MK45 4HS, UK Received 6 June 1996; accepted 10 September 1997
Abstract Measurements have been made at both full scale and model scale of the surface pressures on a conventional low-rise, pitch-roofed building. The full-scale measurements were made under natural strong wind conditions in a near-equilibrium atmospheric boundary layer. Replicated model-scale experiments were made in simulated atmospheric boundary-layer flows at 1 : 100 scale in two wind tunnels. The transverse pressure distribution at the mid-length section of the building produced by a transverse wind has been examined for sensitivity to Reynolds number. The influence of changes in mean wind speed at full scale, and of changes in length scale between full scale and model scale is presented. Both sets of results support the hypothesis that the separated flow at the windward eaves of the building is Reynolds number dependent. Corroborating evidence was obtained from further full-scale and model-scale experiments on the same building by changing the conventional sharp-line eaves to a curved eaves: this produced predominately attached flow over the roof, for which no Reynolds number sensitivity was observed. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Wind loads; Full-scale; Model-scale; Reynolds number; Bluff Body; Low-rise; Atmospheric boundary layer
1. Introduction It is a widely held view that the aerodynamics of bluff bodies are Reynolds number, Re, insensitive, and hence that tests on small-scale models in simulated atmospheric boundary layers are an acceptable method for obtaining information which can be applied to full-scale structures. This view was expressed by Jensen [1] in presenting his ideas regarding the model-law. He focused on the location of the start of the vortex layer at the sharp corners of bluff-bodies. He, like others to follow, made the
* Corresponding author. 0167-6105/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 7 - 6 1 0 5 ( 9 7 ) 0 0 2 8 7 - 0
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assumption that provided the separation point is predetermined by geometry, “all essential relations in the flow must also be rather independent of Re”. Since the 1950s, several studies have been made on low-rise buildings at both full scale and model scale which can be used to test the modelling assumption. Retrospective examination shows that consistent discrepancies exist between full-scale and model-scale results in separated flow regions. The work described here on the Silsoe Structures Building, and, for example, that by Surry and Johnson [2], illustrate this discrepancy. It has been proposed that one possible cause is an inadequate boundary layer simulation, in particular, one not matching turbulence length scales, and this has been the subject of an investigation by Li and Melbourne [3]. However, the full-scale studies at Silsoe, complemented by two wind-tunnel comparisons where the boundary layer was well simulated, cast doubt on the basic assumption that bluff-body flows are insensitive to Reynolds number. Full-scale surface pressure measurements have been made under natural wind conditions on a range of low-rise buildings for the purpose of defining design loads (see Refs. [4—6]). To improve the understanding of wind loading and wind load transfer, an experimental portal-framed building was constructed at Silsoe (the Silsoe Structures Building) on which detailed surface pressure measurements were made. These fullscale measurements provided definitive benchmark data with which to compare and assess model-scale and computational fluid dynamic (CFD) data. Following improvements to wind-tunnel experiments made by Richardson and Surry [7], differences emerged between full-scale and model-scale data which exceeded the expected error, and which showed distinct patterns that suggested an Re sensitivity. The full-scale data were thus re-examined and a statistical analysis conducted to explore in greater detail the existence of trends between mean pressure coefficient values and wind speed for a common wind direction. This exercise was repeated for a second set of aerodynamically distinct data obtained from the Silsoe Structures Building when its configuration had been changed by replacing the conventional sharp-line eaves with a modern curved eaves detail. The curved eaves geometry induced attached flow around the eaves and the windward roof slope which was in direct contrast to the localised separated flow produced by the conventional sharp-line eaves (see Refs. [8—10]). This second set of data fortuitously provided extremely valuable supplementary information for establishing Reynolds number sensitivity (fortuitous since the curved eaves were selected as being representative of modern building practice to investigate consequential wind loading distributions, but it was not known a priori that this eaves detail would produce a contrasting attached flow regime). To test further the Re sensitivity hypothesis, a systematic comparison between the full-scale data and the results from two independent 1 : 100 scale wind-tunnel tests was undertaken. This provided additional evidence of the existence of a significant dependency. 2. Experimental arrangements Similar experimental techniques were used in the full-scale and model-scale tests, employing conventional surface pressure tappings, pressure transmission tubing and differential pressure transducers. Reference wind speed, direction and static pressure
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were sensed upstream at building ridge height at a point where the pressure field generated by the building was not significant (see Ref. [11]). The wind-tunnel modelling work was conducted at 1 : 100 linear scale, and particular care was taken to scale correctly the boundary-layer surface roughness parameter (z ) and turbulence length 0 scales (¸ , ¸ and ¸ ). Instrumentation response was also matched for both pressure x y z measurements and wind speed, although a different type of wind-speed sensor was used at model scale. Details of the experimental arrangements have been published previously and so only a brief re´sume´ containing references which give further details is presented here. 2.1. Full-scale measurements The Silsoe structures building was designed and constructed specifically to undertake wind-load and building response measurements. Full details are given in Ref. [12]. The building, measuring 24.1 m long by 12.9 m span by 5.3 m ridge height, with a 10° duo-pitched roof, was constructed of modern cold-formed steel portal frames clad in a single skin of coated, cold-formed, profiled steel sheeting. Pre-cast concrete pad foundations were used which incorporated special optional frame fixings for structural testing purposes. Pressure tappings were positioned at the centre of aluminium sheets measuring 600 mm square which were fixed to the cladding to provide a locally smooth surface spanning three cladding corrugations. A tapping hole of 9 mm diameter was used, as this was sufficiently large to prevent rain-water blocking the hole. An internal porous ceramic drain removed unwanted water from the pressure tubing. A solid-state pressure transducer with a range of $1.2 kPa was connected to each tap by a 1 m length of flexible plastic tube. A total of 32 pressure taps distributed over one-half of the building were monitored at a scan frequency of 5 Hz. There was no measurable attenuation of the pressure system below 50 Hz. Initially, the Silsoe Structures Building had a conventional sharp-line eaves detail formed by the roof sheets overhanging the side-wall by 50 mm; no gutter was present. Following measurements for this configuration, which extended over one year, the eaves detail was changed to one of a modern curved eaves of radius 635 mm, and wind-load measurements were repeated for a second year. The ridge detail consisted of a simple sharp-line folded ridge cap throughout. The basic full-scale flow patterns developed over the two eaves details and the ridge are shown in Fig. 1. Smoke from a canister was introduced locally into the flow to illustrate the separated flow over the sharp eaves, the attached flow over the curved eaves, and, for both curved and sharp eaves, attached flow over the ridge. Some divergence of the flow can be seen on the leeward roof slope, towards the downwind eaves, which indicates that separation takes place before the leeward eaves. A three-component sonic anemometer was positioned upstream at the ridge height of the building to sense the undisturbed wind velocity. At an adjacent upstream position, the total wind pressure and the static pressure were sensed using a directional Pitot tube and a static probe, respectively. The static probe was designed to be insensitive to the horizontal flow direction and was calibrated at full scale against the pressure from a tapping hole set flush into level ground. The provision of an accurate
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Fig. 1. Flow visualisation for a transverse (perpendicular to ridge) wind: (a) sharp eaves; (b) 635 mm radius curved eaves; (c) ridge.
static pressure for differential pressure measurement is essential, and considerable effort went into the calibration and operation of the static pressure probe. However, the experimental results presented here suggest the existence of a small Reynolds number dependency in the sensed static pressure, which is quantified later. Pressure transducer instabilities were invariably small, but nonetheless were corrected by means of a computer-controlled sequence which applied a zero pressure difference, followed by the total pressure from the Pitot tube, to all the pressure transducers at the beginning and end of each 1 h recording period. The 1 h records were processed as six 10 min records to give mean and fluctuating properties. The analysis in this report is based on the 10 min mean pressure coefficient, C , and the p corresponding mean wind speed and mean direction at ridge height. Wind speed, as sensed by the sonic anemometer, was monitored continuously and recordings were initiated when the 5 min mean wind speed exceeded 8 m s~1. The fetch was predominantly cut grass extending over 600 m. Measurements were made over the winter months during which the condition of the fetch remained constant. Extensive measurements were made to classify the approach flow by using anemometers positioned at heights from 0.1 to 100 m. The mean velocity ratios, referred to the
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Fig. 2. Full-scale velocity profile: (a) mean velocity; (b) turbulence intensity.
velocity measured by a fixed anemometer at 10 m, and the turbulence intensities are presented in Fig. 2. The mean velocity profile was well represented by a log-law with a roughness length of z "0.010 m (Fig. 2a). The frictional velocity, º , derived from 0 q u—w measurements in the lower 10 m of the boundary layer was 0.0607º — a value 10 consistent with the velocity profile and a von Karman constant of i"0.42. Although building surface pressure measurements were made over a wide range of wind direction, the results presented here are restricted to those where the 10 min mean wind direction was within one degree of perpendicular to the long side of the
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building. This narrow band of mean wind direction was selected to remove the possibility of changes in C with wind direction influencing the result. (Experience has p shown that the method used here to test for Re sensitivity can be applied only when other parameters that can produce variability in C are minimised, for example by p selecting a wind direction, h, for which dC /dh"0.) p 2.2. Model-scale measurements 2.2.1. Boundary-layer wind tunnels and building models The two wind tunnels used in the present work were the University of Western Ontario (UWO) Boundary Layer Wind Tunnel No. 1 [7,13] and the Building Research Establishment (BRE) Boundary Layer Wind Tunnel No. 3 [14]. Two 1 : 100 geometrical scale models of the Silsoe Structures Building (one for the sharp eaves geometry, the other for the curved eaves geometry) were carefully constructed from acrylic sheet and pressure tappings were incorporated at positions corresponding to the full-scale tapping positions. 2.2.2. UWO experiments The boundary layer in the UWO No. 1 open-return wind tunnel was developed over a distance of 20 m as the air was sucked down the 2.5 m wide by 2 m high parallel sided tunnel by an axial flow fan positioned downstream of the working section. Three large spires near the inlet were used to increase the boundary layer thickness and to increase the turbulence levels. These were followed by sets of 25 mm cubes positioned over the floor to within 2 m of the model. Randomly spaced, 7 mm high machine nuts were placed over the remaining 2 m of the approach. The resulting velocity profile at the model position (with the model removed) was measured using two hot wires mounted on a vertical traverse. The velocity profile is presented in Fig. 3. The mean velocity profile (Fig. 3a) shows a region close to the ground where the velocity exceeds the desired log-law. This is associated with the change in roughness 2 m upstream of the model position. Also, the upper part of the boundary layer is not fully developed owing to the finite length of the tunnel. The mid-region of the boundary layer, extending from the height of the model to approximately 50 m full-scale equivalent height, has a roughness length of 0.1 mm, giving a mean velocity profile scale of 1 : 100 as required. Examination of the spectrum of turbulence also shows a good 1 : 100 scale fit to the full-scale spectrum. Total and static pressures were sensed by a conventional pitot-static probe positioned near the roof of the working section of the tunnel. Correction factors were thus required to derive pressure coefficients consistent with those derived at full scale. Reference static pressure was corrected to give agreement between full- and model-scale pressure coefficients at the mid-length of the building for flow parallel to the building ridge. A Scanivalve system incorporating a Statham 2.5 psi transducer, which gave a flat response up to 100 Hz, was used to measure surface pressures over the model building. 2.2.3. BRE experiments The BRE No. 3 open-return wind tunnel has a cross-section measuring 2 m wide by 1.5 m high, and an overall length of 20 m. The “blow-down” tunnel has a large
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Fig. 3. UWO boundary-layer wind tunnel velocity profiles: (a) mean velocity; (b) turbulence intensity.
centrifugal fan at the inlet. The working section was fitted with permeable aerofoil slats in the walls and roof to reduce the influence of blockage. A turbulence grid and a deeply saw-toothed barrier were placed at the tunnel inlet. Between the inlet and the working section, 10 mm high roughness elements were distributed over the tunnel floor. Roughness was added close to the model to maintain a near-equilibrium boundary layer. The mean velocity profile measured by a cross-wire is shown in
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Fig. 4. BRE boundary-layer wind tunnel velocity profiles: (a) mean velocity; (b) turbulence intensity.
Fig. 4a, and the turbulence intensity is shown in Fig. 4b. The influence of insufficient roughness close to the model is again apparent in the mean velocity profile, but the extent of the linear log-law region of the boundary layer is greater than in the UWO simulation (Fig. 3a), probably as a result of the combined influence of the centrifugal fan and the saw-toothed barrier giving a non-uniform inlet velocity. The scaling of the mean velocity profile and of the turbulence was again approximately 1 : 100.
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In the BRE experiments, a pitot-static probe was positioned at the ridge height of the model (53 mm), upstream and to one side of the model, in a position equivalent to that used at full scale. Corrections were applied for static pressure error caused by turbulence effects on the probe by making comparisons with a surface pressure tapping at the centre of the turntable when the model had been removed. The uncertainty in pressure coefficient values resulting from the use of continuous local flow monitoring (BRE approach) is significantly less than that resulting from the use of a profile calibration undertaken prior to model testing (UWO approach).
3. Analysis and results Mean pressure coefficients for both the sharp and curved eaves configurations at building mid-length for a transverse wind are shown in Fig. 5. The full-scale coefficients, Fig. 5a, were derived from curve fitting to over 1000 data points of 10 min mean values obtained over a range of mean wind directions. An example of this method is shown in Fig. 6 for tap 28 on the windward roof slope of the building with the sharp eaves detail. The transverse, or perpendicular to ridge, wind direction is denoted by 180°. The uncertainty in full-scale coefficients for the transverse wind direction is small (of order 0.01) due to the large number of data points. The coefficients from the wind tunnels, Fig. 5b and Fig. 5c, were obtained with the transverse axis of the model aligned with the tunnel axis. The uncertainty in the results from both wind tunnels is greater than that at full scale as there were no repetitions of measurements within each wind tunnel. The UWO reference pressures were recorded at a position remote from the model, giving an overall uncertainty of the order of 0.1 in coefficients. The use of local reference pressures in the BRE tunnel reduced the uncertainty to 0.05. The arithmetic differences between the full-scale coefficients and the average of the two wind-tunnel results are shown in Fig. 7. Good agreement exists on the windward wall which is to be expected for a well-simulated boundary-layer flow. A marked difference is observed on the windward roof slope for the sharp eaves configuration which is associated with the flow separation over the eaves and subsequent reattachment. A less significant difference is observed on the windward roof for the curved eaves configuration, but examination of the two sets of wind-tunnel results for this case, Fig. 5b and Fig. 5c, shows a significant difference between the model-scale data sets, with the coefficients from the BRE tunnel being in closer agreement with the full-scale data. There was evidence from flow visualisation using smoke, that at model scale there was separation at the ridge with early reattachment; this was not the case at full scale where the flow remained attached (Fig. 1c). This difference may, in part, be caused by the changes in flow on the windward roof at model scale. The suggestion from these measurements is of a change in flow pattern over the windward eaves which is possibly Reynolds number dependent; this observation encouraged further analysis of the full-scale data. The full-scale pressure coefficients were derived by curve fitting to the entire data set, as illustrated in Fig. 6. The number of data points with a mean flow direction
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Fig. 5. Mean pressure coefficients at building mid length for a transverse wind: (a) Full-scale (showing also the full-scale tapping point numbers); (b) UWO wind tunnel; (c) BRE wind tunnel.
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Fig. 5. (Continued).
Fig. 6. Full-scale pressure coefficients for tap-28 measured over a range of wind directions.
within one degree of perpendicular to the side wall was sufficient, however, to test the hypothesis that the flow was Reynolds number sensitive since they relate to a range of wind speeds of some 7—15 m s~1. For this narrow band of direction, 43 data points were available for the sharp eaves case, and 41 for the curved eaves case. The pressure
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Fig. 7. Arithmetic difference between full-scale and averaged wind-tunnel pressure coefficients.
coefficients were plotted against Reynolds number based on the building ridge height dimension (5.3 m) and the measured mean velocity at the reference position for the same recording period. An example is presented for tap 28 in Fig. 8a for the sharp eaves configuration, and in Fig. 8b for the curved eaves configuration. The scatter of the data points in Fig. 8 is associated with the separation of the anemometer from the tapping point. This scatter was quantified by using two anemometers on a similar open site with no building present and varying the horizontal distance between the anemometers. Analysis of the measurements shows that for a spacing of 20 m and an averaging period of 10 min, the standard deviation of the difference in dynamic pressure was approximately 3%. This difference was found to be normally distributed and independent of wind speed. The scatter of points in Fig. 8 is consistent with this observation. For the data presented in Fig. 8, linear regression analysis and evaluation of Student t values shows that for the sharp eaves case the slope is highly significant (t"!4.26), with a value of !0.281 per decade of Re, but that for the curved eaves case the slope was not significant (t"!0.96). The analysis for all the tapping points shown in Figs. 5 and 7 is summarised in Fig. 9. The Student t values are given in Fig. 9a, and the slopes of the regression lines in Fig. 9b (which have units of C per p decade of Re).
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Fig. 8. Pressure coefficients for tap-28 showing: (a) significant regression with Reynolds number for the sharp eaves; (b) non-significant regression with Reynolds number for the curved eaves.
4. Discussion and analytical indicators Before comparing wind-tunnel and full-scale coefficients to investigate any similarity in Reynolds number sensitivity, examination of Fig. 9b shows that negative regression occurs for all tappings on both the sharp and curved eaves building configurations. For the curved eaves case, only the lower tap on the leeward roof has a significant slope (t"!3.7). The probability of all the slopes being negative (mean value !0.0775, standard deviation 0.0384 for all non-significant regression slopes) by chance is very low ((1%) and a physical reason must be assumed. The most likely
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Fig. 9. Analysis of full-scale data: (a) Student t values; (b) slope of regression lines.
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cause is considered to be a Reynolds number dependency in the instrumentation, specifically the omni-directional static pressure probe. This instrument comprises a vertical cylindrical tube (38 mm in diameter) with an annular set of holes (each 3 mm in diameter) that are covered by an adjustable shroud. It can be expected to have some Reynolds number dependency, but previous calibrations have not quantified this. Combining all non-significant regression coefficients (DtD)2), indicates a mean regression of !0.0775C per decade of Re. Over the entire measurement p range (which equates to 0.3 of a decade), the pressure change is small, i.e. 0.02C , and p this does not increase the uncertainty in full-scale values which are of the order 0.01 since the probe was calibrated at wind velocities close to the overall mean velocity of 9 m s~1. Another possible cause of Re sensitivity was the 9 mm diameter tapping hole size. Applying the methods described by Shaw [15], indicates that the expected sensitivity to Reynolds number of these large tapping holes is of order 0.003C per decade of Re, p i.e. more than an order of magnitude smaller than the measured value. According to Bryer and Pankhurst [16], directional Pitot tube readings are not Reynolds number dependent, other than at very low Re, nor are any Mach number effects significant at these low velocities. With the elimination of these possible causes, the hypothesis remains that the flow over this bluff body is Reynolds number sensitive. Assuming that the hypothesis is true, it is of interest to extrapolate from the full-scale data, using a linear model, to compare trends with the model-scale data (two decades of Re lower), whilst acknowledging that the application of the linear-regression model for Reynolds number outside the full-scale measurement range is experimentally unjustified. To do this, the slope of the regression line was corrected by subtracting the !0.0775 static probe factor. The result of this correction was to reduce the number of tappings at which a significant Re sensitivity remained to those on the lower part of the windward roof of the sharp eaves building. The results given in Table 1 are for the three tapping points near the windward sharp eaves. These show good agreement in the welldeveloped central region of the separated flow (taps 26 and 27), but indicate a breakdown of the linear interpolation at tap 28 which is in the vicinity of reattachment (estimated from full-scale flow visualisation in Fig. 1b to be at approximately 3 m from the leading edge).
Table 1 Full-scale and model-scale pressure coefficients over lower part of windward roof slope of building with sharp eaves Tapping point
Distance from leading edge (m)
Full scale C p
UWO model-scale C p
BRE model-scale C p
Linear-regression estimate of model-scale C from full-scale C p p
26 27 28
0.965 1.965 3.465
!1.389 !1.148 !0.697
!1.098 !0.697 !0.444
!1.109 !0.633 !0.498
!1.035 !0.650 !0.289
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5. Theoretical understanding An intuitive, conceptual understanding of the observed Reynolds number effects can be gained by focusing attention on the dynamics of vortices. It is proposed that the observed Reynolds number effects can be attributed to the formation and subsequent instability of the vortex sheet formed from vortices generated at the eaves. To satisfy the constraint of having no flow normal to the surface of the roof, vortices can be thought of as belonging to contrarotating vortex-image pairs (see Fig. 10a). Once generated, these vortex pairs would, in the absence of a superimposed advecting flow, be transported, by their self-induced motion, upstream in a direction normal to the line joining their centres. However, at least initially, the surrounding flow will overcome this self-induced motion and the vortex pair will be advected downstream.
Fig. 10. Vortex dynamics: (a) vortex-image vortex pair; (b)—(d) time evolution of the motion of two vortex-image vortex pairs ((———) motion due to advecting flow and corresponding image vortex, (— — —) motion due to interaction between the two vortex-image vortex pairs) where the trailing vortex “separates” from the surface as it is advected downstream whilst the leading vortex “reattaches” to the surface and then “recirculates”; (e) predicted evolution of a vortex in a system of many vortex-image vortex pairs (the vortex “separates” from the surface at the leading edge before eventually “reattaching” and “recirculating”).
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In practice, this process of vortex pair generation and advection will be continuous and will lead to the formation of a vortex sheet. For clarity of understanding, it is, however, beneficial to consider a system of discrete vortex pairs. Vortex pairs will, by virtue of the flow fields they induce, interact with one another. The effects of this interaction on the motion of two vortex pairs is shown in Fig. 10b—10d. The flow field induced by the leading vortex pair causes the trailing vortex pair to move apart. As the trailing vortex and image vortex move apart, their self-induced back-flow will decrease and, consequently, their forward motion due, in part, to their induced backflow and the superimposed advecting flow, will increase. Conversely, the flow induced by the trailing vortex pair on the leading vortex pair will cause the separation and forward motion between the leading vortex and image vortex to decrease. Eventually, the leading vortex pair will come close together, their self-induced back flow will exceed the advection velocity, and they will be transported back upstream towards the eaves. Vortices generated at the eaves can thus be expected to initially lift-off from the surface of the roof as they are swept downstream before returning to the roof ’s surface to be transported upstream towards the eaves. In other words, as shown in Fig. 10b—10d, vortices generated at the eaves are expected to initially “separate” from the roof ’s surface as they are transported downstream before eventually “reattaching” and “recirculating”. This was indeed found to be the case in numerical studies of a simple system of such vortex pairs. The system studied consisted of pairs of point vortices, of strength $C, confined to move in two-dimensions. Pairs of vortices were generated at the eaves every C/º2 seconds, the natural time scale of the system, where º is the superimposed advection velocity. The dynamics of these vortex pairs was studied by numerical integration of the corresponding equations of motion. The equations of motion and the numerical method can be found in Ref. [17]. In the numerical studies, the effects of viscosity, which will cause the strength of real vortices to decay in time and impose a no-slip boundary condition at the surface of the roof, was neglected. The vortex sheet formed by such vortices was found to lift off from the surface of the roof at the eaves and then reattach further downstream before eventually rolling-up (see Fig. 10e). In accordance with the experimental observations, the average location of the “reattachment” point, r, was found to scale with Reynolds number, r&C/ º¸&1/Re, ¸ being an arbitrary length scale. Moreover, the height of the separation bubble was also found to scale as 1/Re. Thus, intriguingly, the length-to-height ratio of the separation bubble is predicted to be Reynolds number independent. The shape of the leading edge of the bubble is predicted to be stationary and Reynolds number independent, whereas the trailing edge is predicted to be transient and Reynolds number dependent. That is, for a given Reynolds number, the location of the “reattachment” point fluctuates, in time, about its mean position. This particular form of transience is intrinsic to the dynamics of the vortices. In practice, transience can also result from turbulent velocity fluctuations and fluctuations in the mean wind speed and direction. The apparent stability of the leading edge of the separation bubble, and the transient nature of the trailing edge and location of the reattachment are supported by observations of smoke plumes around the building.
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For the curved eaves case, the flow was found, experimentally, to remain attached. Within the vortex picture, this can be attributed to the suppression of vortex generation at the rounded eaves.
6. Concluding comments The reported experimental observations of Reynolds number sensitivity in bluffbody aerodynamics have been explained by considering the motion of a theoretical vortex pair which indicated a Reynolds number dependency in the separated flow region, particularly, in the vicinity of reattachment. This concept predicts no Reynolds number effect near the leading edge which is consistent with the experimental data, in so much as the effect is reduced for the tapping point closest to the leading edge. There is some indication of Reynolds number effect associated with separation of the flow off the leeward roof for both the sharp and curved eaves details, but the magnitude of pressure coefficients in this region is small and the dependency not therefore fully established. It is likely, however, that a strong dependency exists near the gable ends where pronounced flow separations are generated. The findings presented here are statistically highly significant and provide strong evidence that a Reynolds number effect is present in the separated flows around bluff bodies. Some indication of the magnitude of the effect has been estimated (approximately 0.25C per decade of Re), but more detailed work is required to produce p generalised correction terms for extrapolation from model-scale to full-scale values, as is required to obtain reliable design information. The implication for high-rise building design, where smaller scale models are used, is likely to be more significant than for low-rise buildings, and this should provide the impetus for further work to be done.
Acknowledgements This programme of work was funded by the UK Ministry of Agriculture, Fisheries and Food, and the Building Research Establishment. Financial assistance was also received from the Natural Sciences and Engineering Research Council of Canada through operating grants made available to D. Surry and A.G. Davenport at the University of Western Ontario.
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