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Surface pressure on a cubic building exerted by conical vortices
Journal of Fluids and Structures 92 (2020) 102801
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Surface pressure on a cubic building exerted by conical vortices Hrvoje Kozmar Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lučića 5, 10000 Zagreb, Croatia
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Article history: Received 7 March 2019 Received in revised form 24 September 2019 Accepted 8 November 2019 Available online xxxx Keywords: Cubic building Turbulent boundary layer Turbulence Conical vortices Surface pressure distribution and fluctuations Pressure spectra Wind-tunnel experiments
a b s t r a c t Experiments were performed to study surface pressure on a cubic building underlying conical vortices, which are known to cause severe structural damage and failure. The focus is on the effects of turbulence in the incident flow. Three turbulent boundary layers were created in a boundary layer wind tunnel. A wall-mounted cube, i.e. a cube situated on the horizontal ground floor surface of the wind-tunnel test section, was used as an experimental model. The cube was subjected to the incidence flow at 40◦ . Steady and unsteady pressure measurements were performed on the cube surface. The analysis suggests that conical vortices developed above the top surface of the wallmounted cube. A larger mean suction was observed on the top cube surface in the less turbulent boundary layer. With an increase in turbulence in the incoming flow, the strong suction zones decreased in size. The fluctuating pressure coefficient profiles retained their shape when the turbulence in the upstream flow of the cube increased. The fluctuating pressure coefficient was observed to be larger in more turbulent flows. The pressure fluctuations were larger on the cube surface underlying outer boundaries of the conical vortex. The fluctuating pressure coefficient under the conical vortex was three to four times larger than in the weak suction zone on the central area of the top cube surface. Close to the leading cube corner, the pressure spectra were dominated by a single low frequency peak. As the conical vortex developed, this primary peak weakened and a secondary peak emerged at a higher reduced frequency. There is a general trend of shifting the pressure spectra towards higher reduced frequencies when the turbulence in the undisturbed incident flow increases. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Conical vortices develop on the top surface of cubic buildings subjected to the oblique wind flow, as shown in Fig. 1, adapted from Banks et al. (2000). Extreme suction occurs on the body surface underlying such vortices, e.g. Kind (1986) and Richards and Hoxey (2008). These aerodynamic loads may cause severe structural damage and failure, e.g. Ginger and Letchford (1993), which may be mitigated by using various engineering solutions, e.g. Baskaran and Stathopoulos (1988), Bienkiewicz and Sun (1992), Tieleman et al. (1994), Hoxey et al. (1998) and Banks (2013). Conical vortices are highly relevant for a wide range of aerospace, civil, mechanical, and wind engineering applications. Many studies have comprehensively tackled this complex issue, while there are still some open questions including further aspects of the boundary layer turbulence effects on pressure distribution and particularly fluctuations on a flat top body surface in the oblique fluid flow. E-mail address: [email protected]. https://doi.org/10.1016/j.jfluidstructs.2019.102801 0889-9746/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Schematic view of conical vortices developing on the top surface of a bluff body subjected to the oblique fluid flow. Source: Adapted from Banks et al. (2000).
Previous studies have thoroughly addressed flow characteristics of conical vortices and the surface pressure on bluff bodies underlying these vortices. In these studies, field measurements, wind-tunnel experiments, Computational Fluid Dynamics (CFD), and semi-empirical methods have been used. It is generally accepted that the large suction on the top surface of bluff bodies near their corner is caused by the conical (corner, delta-wing) vortex that develops in the oblique fluid flow, as given in Kawai and Nishimura (1996) and Kawai (1997, 2002). These authors report that a pair of conical vortices has rotating spiral axes and that they rotate opposite to each other. The centerlines of the vortices are along the line inclined at 13◦ in the smooth flow and at 10◦ in the turbulent flow relative to the eaves. These lines correspond to the maximum suction on the top body surface directly underneath the conical vortices. A strong effect of turbulence on the conical vortices and the underlying surface pressure has been observed. In particular, the vortices are generally stronger in the smooth flow than in the turbulent flow, i.e. a larger mean suction acts on the top body surface in the smooth flow, which was reported by Lin et al. (1995) as well. In the turbulent flow, vortices are flatter and placed closer to the top body surface. The maximum mean suction occurs a little downstream of the corner of the top body surface and it is larger in the smooth flow than in the turbulent flow. The suction fluctuations result from the combined flapping and rotation of the conical vortices. Vortex flapping is the intermittent up and down vertical movement of the conical vortex at some characteristic frequency, which is similar to the movement of flying birds’ wings. Each conical vortex grows and decays alternately to each other, yielding a mismatch in the suction fluctuations relative to the diagonal line of the top body surface. The fluctuating pressure coefficient is three times larger in the turbulent shear flow than in the smooth flow, Kawai and Nishimura (1996). Banks et al. (2000) confirmed that the peak suction lies beneath the vortex core and it moves with the vortex. The vortex becomes larger with an increase in the flow incidence angle. The increasing turbulence moves the mean vortex core position closer to the leading edge. The faster the vortex spins, the lower the core pressure and the lower the surface pressure, Banks and Meroney (2001a,b), so the vortex acts as an amplifier of the local pressure drop. The pressure coefficient decreases with the inverse of the square root of the distance from the corner, Tryggeson and Lyberg (2010). The incident flow velocity predominantly controls the strength of conical vortices, and it has rather marginal effect on the overall structure of conical vortices (Wu et al., 2001a,b). The horizontal flow incidence angle determines the formation and configuration of vortices and the underlying surface pressure fluctuations, as substantial discrepancies in the suction on the top body surface were observed when the horizontal flow incidence angle is changed, Mehta et al. (1992). The vertical flow incidence angle modifies the configuration and movement of the vortices, whereas large and quick excursions of the vertical angle can significantly change the configuration of the vortices and yield extremely high, instantaneous peak suctions. Characteristics of the conical vortex vary in the vertical and the horizontal direction, Marwood and Wood (1997). The surface suction under the conical vortex is maximal when the vortex axis is located farthest away from the body edge and highest above the surface of the body. Variations in position are smaller in the low turbulence flow thus indicating that the vortex movement is driven by incident turbulence. The shape of the vortex cross section is nearly elliptical, whereas the angle between the vortex axis and the leading edge is smaller in the turbulent flow, Sun and Ye (2016). Detailed flow and surface pressure characteristics were obtained by Ginger and Letchford (1992) and Kim et al. (2001, 2003). The maximum turbulence kinetic energy was found in the upper boundary of the separation bubble near the leading edge of the top bluff body surface, whereas the maximum energy magnitude is substantially larger in that area than in the wake region. Analytical and semi-empirical approaches have been used to account for the flow and underlying surface pressure characteristics caused by conical vortices. A semi-empirical model was proposed for the pressure field beneath corner vortices, Cook (1990), whereas the flow can be considered as a conical vortex ‘growth region’ near the corner of the top
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body surface, and as a cylindrical vortex ‘mature region’ further downstream of the corner. Williams and Baker (1997) confirmed the existence of the ‘growth region’, while they did not observe the ‘mature region’ in the form suggested by Cook (1990). Banks and Meroney (2001c) advised caution when using the quasi-steady theory to predict peak and fluctuating pressure coefficients under conical vortices. The incorporation of the vertical turbulence component improved this approach, but discrepancies remained as the quasi-steady theory proved its inadequacy in regard to flow distortion and building-generated turbulence, Letchford and Marwood (1997). Vortex movement was driven by upstream turbulence and generally behaved in a quasi-steady manner; however, the peak pressures are underestimated by the quasi-steady theory, Marwood and Wood (1997). Ye and Dong (2014) improved the quasi-steady theory by including the effect of vortices, where it was possible to correctly predict the mean and peak suction under conical vortices. Wu and Sarkar (2006) proposed a bivariate quasi-steady model, which accounted for the influence of the horizontal and vertical flow components on the surface pressure underlying conical vortices. The quasi-steady theory in this form proved to be applicable in the separated flow region where vortices were present. Zhao et al. (2002) developed a simple analytical model to predict fluctuating pressure coefficients on the leading corner of a wall-mounted bluff body. CFD has become an increasingly important tool to analyze flow characteristics of conical vortices and underlying surface pressure distribution and fluctuations on bluff bodies, e.g. He and Song (1997), Thomas and Williams (1999a,b), He et al. (2007) and Ono et al. (2008). This literature survey reveals many important previous studies that focused on the flow characteristics of conical vortices and pressure characteristics on the top surface of underlying bluff bodies; nevertheless, there are still some open questions. In particular, Tieleman (1993), Saathoff and Melbourne (1997), Wu et al. (2001a,b), Letchford and Marwood (1997) and Marwood and Wood (1997) point out that it is important to accurately model turbulence characteristics to fully account for relevant aspects of conical vortices and bluff body aerodynamics. Furthermore, it is still not sufficiently understood how the fluctuating pressures under conical vortices compare to the fluctuating pressures in the center of the top bluff body surface, i.e. outside of the conical vortex. It is therefore particularly important to study the power spectral density of pressure fluctuations on the top body surface to account for the fluctuating pressure energy distribution in the frequency domain. This will enable a more thorough understanding of dynamic loads on bluff bodies exerted by conical vortices. Given these facts, the scope of the present study is the pressure distribution and fluctuations on the top surface of a wall-mounted cube, i.e. a cube situated on the horizontal ground floor surface of the wind-tunnel test section, subjected to oblique turbulent boundary layers (TBLs). A particular focus is on the power spectral density of surface pressure fluctuations on the top cube surface underlying conical vortices and outside of them for various levels of turbulence in the incoming 3D boundary layer flows. 2. Experimental procedure Experiments were performed in a boundary layer wind tunnel (BLWT) at the Technical University of Munich (TUM), Germany. The rectangular test section is 1.80 m high, 2.70 m wide and 21 m long. The turbulence intensity at the inlet of the test section is less than 0.5%. Across the cross section of the test section inlet, the mean velocities in the main flow direction differ by less than 1%. These results were analyzed away from the walls, i.e. outside of the near-wall region that represents 10% of the respective width and height dimensions of the wind-tunnel test section. The test section is equipped with a ceiling adjustable in height that enables a zero pressure gradient in the main flow direction. There is a turntable centered 11.3 m downstream of the nozzle that enables airflows on structural models to be developed at various incidence angles. A blower powered by a 210 kW electric motor generates airflows in the velocity range of 1 m/s – 30 m/s. Three TBLs were created using the Counihan (1969) approach, i.e. the castellated barrier wall, vortex generators and surface roughness elements. The fundamental difference between these three TBLs was in their turbulence levels, i.e. they were either low-turbulent (LT), mid-turbulent (MT), or high-turbulent (HT) boundary layers. While the Counihan approach has been commonly employed to generate the entire TBL thickness using the full-size Counihan vortex generators, in the present study the truncated Counihan vortex generators were employed. These vortex generators were truncated from the 2.0 m high ‘classical’ Counihan vortex generators to 1.7 m, e.g. Kozmar (2011a,b, 2012). The TBL thickness δ is 2.0 m; however, in the wind tunnel only a lower TBL portion from the ground surface up to 0.7δ (1.4 m) was reproduced. This approach generally allows achieving thicker turbulent boundary layers, Kozmar and Laschka (2019a). Surface roughness was created by using LEGO bricks. In each of the three TBLs, the LEGO bricks were of the same height and arranged in a symmetrical and staggered pattern. Preliminary arrangement of the spacing (packing) density, which is the test section area covered with surface roughness elements with respect to the total test section area, and the height of the LEGO bricks was based on recommendations given by Gartshore and De Croos (1977). The final arrangements were made through trial and error and reported in Table 1. In all the experiments, the basic height of the castellated barrier wall was 292 mm with the castellation height of 84 mm on top of it. An example of the experimental arrangement in the wind-tunnel test section, the HT setup, is shown in Fig. 2. The two other configurations, LT and MT boundary layers, are basically the same, but with smaller spacing density and a lower height of the surface roughness elements (LEGO bricks).
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H. Kozmar / Journal of Fluids and Structures 92 (2020) 102801 Table 1 Spacing density and height of surface roughness elements in turbulent boundary layers. Turbulent boundary layer
Roughness spacing density
Roughness height
LT MT HT
0.5% 0.9% 6.3%
68 mm 68 mm 78 mm
Fig. 2. Photograph of the experimental arrangement in the wind-tunnel test section for the high-turbulent boundary layer.
Instantaneous velocities were first measured in the empty wind-tunnel test section, i.e. without the cube. This was performed simultaneously in three coordinate directions at each measuring point using a triple hot-wire probe DANTEC 55P91. The sampling rate was 1.25 kHz and the record length was 150 s. The preliminary tests proved that these parameters were sufficient to successfully acquire all relevant flow phenomena. Measurements were carried out at 16 measuring points placed along a vertical line at the center of the turntable. Previous experiments proved that the TBLs created using this approach were fully developed and uniform at the turntable center in the empty test section, Kozmar (2011c). This is exactly the position where the cube was situated in the subsequent experiments focused on the surface pressure on a wall-mounted cube exerted by conical vortices. The TBLs were characterized by the mean velocity in the longitudinal (x), lateral (y), and vertical (z) directions, i.e. u, v , w, respectively. The turbulence intensities Iu (z), Iv (z), Iw (z) in these three directions were calculated by using the respective fluctuating velocities u’, v’, w’ and the local mean flow velocity u in the main (x) flow direction. This characterization of flow characteristics based on simultaneous measurements in all three directions is important as the turbulence in each of these three directions is relevant for bluff body aerodynamics including conical vortices, e.g. Tieleman (1993, 2003), Letchford and Marwood (1997), Marwood and Wood (1997), Saathoff and Melbourne (1997) and Wu et al. (2001a,b). The Reynolds shear stress τ = −ρ u′ w ′ profile was provided because it was previously proven that it influences roof pressure fluctuations, Sharma and Richards (1999), so it was anticipated that it would also affect the conical vortices and the underlying pressure fluctuations. The turbulence length scale Lu,x was calculated using the autocorrelation functions and assuming the Taylor’s frozen turbulence hypothesis. One cube with 0.2 m long edges was used as the experimental model. It created the test section blockage of 0.84% that is considerably smaller than the critical blockage of 6% (West and Apelt, 1982), so the results were not corrected. The predicted thickness of the TBLs that would be equal to the full size of 2 m high non-truncated Counihan vortex generators, in agreement with Balendra et al. (2002), was in all experiments ten times larger than the cube height, which is comparable to the Castro and Robins (1977) experiments. The pressure measurements were simultaneously performed on all cube surfaces except the bottom surface that was attached to the wall. For the purposes of general understanding and qualitative validation, the results were reported on all the cube surfaces equipped with pressure measuring points, while they were studied in detail on the top surface underlying conical vortices only as this was the focus of the present study. The flow incidence angle was at 40◦ in all experiments. This setup allows for two different conical vortex structures to be achieved above the top cube surface rather than two identical conical vortices that would appear at 45◦ and thus this setup yields a larger variety in the results. In general, the conical vortex is larger downstream of the leading edge of the top cube surface that is oriented at a smaller angle to the main flow direction (the flow perpendicular to the leading edge is the zero incidence angle), e.g. Banks et al. (2000), whereas it is not that clear at the moment what the minimum/maximum flow incidence angle is that would result in the
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formation of conical vortices. The conical vortex more perpendicular to the positive x-axis is larger because the incident flow angle in the case of this vortex is smaller to the respective leading edge of the top cube surface, in agreement with Banks et al. (2000). On each of the vertical cube surfaces there were 35 pressure taps and on the top surface there were 49 pressure taps used for performing steady pressure measurements. Close to the cube edges, the pressure taps were placed more densely to make the capturing of strong pressure gradients that were anticipated in these regions possible. The steady surface pressure measurements on the cube and the reference Prandtl–Pitot static pressure measurements in the freestream flow were simultaneously performed by using PDCR-22 sensors and the Natec Schultheiss Scanivalve System. The pressure taps on the cube surface were connected to the steady pressure sensors via short plastic tubes. The sampling rate was 100 Hz during 15 s because the mean pressure and the pressure standard deviation changed negligibly for longer time records. The steady surface pressure coefficients cp on the cube were calculated using the mean flow velocity uH at the height that corresponds to the height of the top cube surface (H = 0.2 m). uH was measured in the empty test section at the height of the top cube surface before the pressure measurements on the cube actually took place. The unsteady surface pressure was measured at 15 points on the top cube surface by using Kulite XCS-093-0.35bar-D miniature pressure sensors. The pressure taps needed for the unsteady pressure measurements were designed as described in Kozmar and Laschka (2019b). The unsteady pressure coefficient cp,rms was calculated using the mean flow velocity uH , whereas uH was determined in the same way as for the steady pressure measurements. The sampling rate and the recording time were 2 kHz and 20 s, respectively. The dynamic Kulite pressure sensors were placed directly in the measuring points at the cube surface, i.e. plastic tubes were not used to connect the pressure taps to the pressure sensors, so there was no need to perform pressure corrections to account for the tubing effects. During preliminary measurements, this experimental setup proved accuracy with respect to the intended experiments. The power spectral density of pressure fluctuations on the cube surface Sp (f ) was provided in the pre-multiplied form f · Sp /σp2 , which identifies frequencies that carry the majority of the energy in the observed phenomenon, e.g. Hearst et al. (2016); this way of presenting the spectra is common in the engineering community. However, it needs to be noted that plotting the spectra in this pre-multiplied f · Sp /σp2 form always leads to a peak, whether or not the signal contains any dominating periodicity, while on the other hand, the peaks observed in the non-pre-multiplied form Sp are more genuinely caused by the periodicity in the signal and commonly used to identify characteristic flow phenomena. The distribution of the measuring points on the top surface of the wall-mounted cube for steady and unsteady pressure measurements is given in Fig. 3. 3. Results and discussion Turbulent boundary layers Basic characteristics of the TBLs created in the empty test section (without the wall-mounted cube) are outlined in Table 2; α is the exponent of the empirical power-law approximation of the mean velocity profile, d is the displacement height of the mean velocity profile, z0 is the aerodynamic surface roughness length, uτ is the friction velocity, ReR is the roughness Reynolds number calculated as ReR = (uτ · z0 )/ν , where the air viscosity ν is 1.5 · 10−5 m2 /s. The mean velocity results were fitted to the power law to estimate α and d, and to the logarithmic law to determine z0 and uτ . The profiles of the mean u velocity in the main (x) flow direction, u′ w ′ Reynolds shear stress, Lu,x turbulence length scale, and Iu , Iv and Iw turbulence intensity in the x-, y- and z-directions, respectively, are given in Fig. 4. The mean reference velocity uH was recorded in the empty test section (without the wall-mounted cube) at the height H = 0.2 m, which corresponds to the height H of the wall-mounted cube in the subsequent experiments. While this work may be understood as a fundamental fluid mechanics study (wall-mounted cube subjected to TBL), it may also represent an experimental small-scale model of a cubic building subjected to the rural, suburban and urban atmospheric boundary layer (ABL) flows, characterized by the power-law exponent of the mean velocity profiles α = 0.17, 0.21 and 0.35, respectively, in agreement with Dyrbye and Hansen (1997) for the suggested terrain types. In the latter case, the length scale factor of the ABL wind-tunnel simulation calculated as suggested in Cook (1978) is S = 91.3 (z − d)0.491 /Lu1,.403 z00.088 = 210, which is then to be applied to the cubic building model as well; z is the vertical x distance from the wind-tunnel floor, d is the displacement height, Lu,x is the integral length scale of turbulence, z0 is the aerodynamic surface roughness length. In that case, a full-scale cubic building has a height of 42 m; the characteristic full-scale ABL parameters in the LT, MT and HT boundary layer cases are d = 0.42 m, 5.04 m and 5.88 m, whereas
Table 2 Basic characteristics of the turbulent boundary layers. Case
α
d, mm
z0 , mm
uτ , m/s
ReR
uH , m/s
LT MT HT
0.17 0.21 0.35
2 24 28
0.80 1.59 9.34
1.11 1.18 1.43
59 125 890
14.97 13.48 10.14
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Fig. 3. Schematic view of (a) wall-mounted cube with indicated measuring points on the top (number 5) cube surface for (b) steady and (c) unsteady pressure measurements.
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Fig. 4. Characteristic profiles of the turbulent boundary layers: (a) mean u velocity in the main (x) flow direction, (b) u′ w ′ Reynolds shear stress, (c) Lu,x turbulence length scale, (d)–(f) Iu , Iv , Iw turbulence intensity in the longitudinal (x), lateral (y) and vertical (z) directions, respectively; the dotted line represents the height H of the top surface of the wall-mounted cube used in subsequent experiments.
z0 = 0.168 m, 0.334 m and 1.961 m, respectively. More details on the wind-tunnel ABL simulation procedure of this type may be found in Kozmar (2011a) along with a justification of the studied parameters. The gradient in the mean velocity profile is larger on rougher surfaces for z ≤H, Fig. 4a. In this same flow region, the absolute values of the Reynolds shear stress and the turbulence intensity are higher on the rougher wall (Fig. 4b, d–f), while the turbulence length scale is smaller (Fig. 4c). The observed flow behavior was expected and previously confirmed by many relevant studies, e.g. Nikuradse (1933). The constant shear stress close to the wall is present as well, which phenomenon is in agreement with Schlichting and Gersten (1997). The flow characteristics recorded in the empty test section at the height H, which is the height of the top surface of the wall-mounted cube used in the subsequent experiments, are given in Table 3. The power spectral density of the longitudinal velocity fluctuations at z /δ = 0.20 (z /H = 2) in the TBLs upstream of the wall-mounted cube is shown in Fig. 5. The trends observed at other heights were nearly the same and thus not
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H. Kozmar / Journal of Fluids and Structures 92 (2020) 102801 Table 3 Characteristics of turbulent boundary layers recorded in the empty test section at a height which corresponds to the height H of the top surface of the wall-mounted cube used in subsequent experiments. Case
Iu
Iv
Iw
u′ w ′ / uH
LT MT HT
0.17 0.22 0.33
0.14 0.17 0.24
0.11 0.13 0.20
−0.00688 −0.01000 −0.02108
2
Lu,x /H 4.03 3.78 2.42
Fig. 5. Power spectral density of longitudinal velocity fluctuations at z /H = 2 in turbulent boundary layers upstream of the wall-mounted cube in comparison with the Kolmogorov (1941) and Von Kármán (1948) models.
presented. The velocity power spectra in all three TBLs agree well with the Kolmogorov (1941) and Von Kármán (1948) models. The flow characteristics recorded simultaneously in all three directions allowed a reliable analysis of pressure characteristics under conical vortices on the top surface of the wall-mounted cube in three TBLs of various turbulence characteristics. Steady pressure distribution The characteristics of the steady pressure distribution on the wall-mounted cube surface were analyzed at Reynolds (Re) numbers greater than 5·104 , which value previously proved to be critical for bluff bodies, Tieleman (2003). The Re numbers obtained in the presented experiments are given in Table 4. However, it needs to be noted that, according to Lim et al. (2007), the Re effects are more likely to occur if the flow around the bluff body contains strong concentrated vortex motions (similar to the conical vortices studied in the present work). Even though the present study focuses on the conical vortices on the top cube surface only, the cp distribution on all cube surfaces (except the bottom surface) is given first in Fig. 6 in the cube-based coordinate system so as to analyze whether the characteristic surface pressure phenomena were present. The steady surface pressure on the top cube surface was further analyzed in more detail in Fig. 7. On the vertical cube surfaces subjected to the incoming TBLs, the cp distribution is in agreement with Castro and Robins (1977). The flow generally ‘preferred’ the surface that was more perpendicularly oriented to the incoming flow, where the stagnation area was larger and the respective cp values higher. The vertical leeward surfaces that are oriented toward the negative x- and y-coordinate axes are both characterized by nearly uniform suction, which phenomenon is in agreement with Castro and Robins (1977) as the differences in cp on these surfaces, as observed in Fig. 6, are rather negligible.
Table 4 Reynolds number in steady pressure measurements on the wall-mounted cube. Case
Re = (uH · H)/ν
LT MT HT
2.0·105 1.8·105 1.4·105
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Fig. 6. Distribution of cp coefficients on the wall-mounted cube subjected to (a) low-turbulent, (b) mid-turbulent, (c) high-turbulent boundary layers.
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Fig. 7. Distribution of cp coefficients on the top surface of the wall-mounted cube subjected to (a) low-turbulent, (b) mid-turbulent, (c) high-turbulent boundary layers.
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The conical vortices develop above the top surface of the wall-mounted cube as the flow separates from the leading edges of the top cube surface, Fig. 1. These flow characteristics are exhibited in the cp distribution on the top cube surface, Figs. 6 and 7. Two zones immediately downstream of the leading edges of the top cube surface characterized by strong suctions can be clearly observed. These two suction zones are underlying the cores of the two conical vortices, which causality is inferred from Kind (1986), Kawai and Nishimura (1996), Kawai (1997, 2002) and Banks et al. (2000). The conical vortex perpendicular to the positive x-axis of the cube-based coordinate system is larger because the incident flow for this vortex is closer to the perpendicular flow with respect to the leading edge of the top cube surface. This characteristic is in agreement with Banks et al. (2000). Along the diagonal line across the top cube surface the suction is considerably weaker than close to the leading edges, thus indicating that this area is outside of the conical vortices. The larger mean suction acts on the top body surface in less turbulent flows (e.g. the LT boundary layer compared to the MT boundary layer and more so to the HT boundary layer), thus indicating that the conical vortices are generally stronger in the smooth flow than in the turbulent flow, which is in agreement with Lin et al. (1995). In particular, the maximum LT suction (cp = −2.0) is almost double the maximum HT suction (cp = −1.2). At the same time, as the turbulence in the incoming flow increases, the strong suction zones decrease in size with a simultaneous increase in the size of the weak suction zone on the central area of the top cube surface. This indicates that the conical vortices are generally larger in less turbulent (smoother) flows than in more turbulent flows. This is in line with previously observed shortening of the separation bubble along with an increase in the turbulence of the incident flow, Hillier and Cherry (1981). The conical vortices are situated closer to the respective edges of the top cube surfaces in the more turbulent flow (HT boundary layer), which result is in agreement with Sun and Ye (2016) as the angles between the vortex axes and the respective leading edges of the top cube surface are smaller in the HT boundary layer than in the two other cases, Figs. 6 and 7. The conical vortices spin faster in less turbulent flows, as exhibited in the stronger underlying surface suction on the top cube surface, which is inferred from Banks and Meroney (2001a,b), i.e. the conical vortices spin faster in the LT than in the MT boundary layer and even more so than in the HT boundary layer, as cp on the top cube surface underlying conical vortices are minimum (strongest suction) in the LT boundary layer, Figs. 6 and 7. Unsteady pressure characteristics Two conical vortices rotating opposite to each other along with their flapping substantially influence pressure fluctuations on the top cube surface, as reported by Kawai and Nishimura (1996). They grow and decay alternately, which yields a mismatch in the suction fluctuations relative to the diagonal line of the top cube surface. The fluctuating pressure coefficient was previously observed to be three times larger in the turbulent shear flow than in the smooth flow, Kawai and Nishimura (1996). The present results for the cp,rms fluctuating pressure coefficient are given in Fig. 8. Based on these cp,rms results along with the cp distributions presented in Figs. 6 and 7, which approximately mirror the boundaries of the conical vortices, it is possible to analyze the pressure fluctuations (cp,rms ) underlying conical vortices in comparison with the respective values obtained in the weak suction area in the central part of the top cube surface. A particular emphasis is put on the effects of turbulence in the incoming TBLs on the cp,rms characteristics on the top cube surface inside and outside the conical vortices. It can be observed that the cp,rms profiles retain their general shape when adding turbulence to the undisturbed upstream TBL along with a simultaneous increase in pressure fluctuations. Accordingly, at each particular measuring point on the top cube surface, regardless of whether it is inside or outside the conical vortex, the HT cp,rms values are the highest and the LT cp,rms values are the lowest. In this context, the added turbulence acts as an amplifier of the pressure fluctuations on the top cube surface. In all experiments, the cp,rms profiles substantially change their shape as they leave the surface area underlying conical vortices and progress toward the weak suction area in the central part of the top cube surface. The peak values are observed at a/H = 0.25 (Fig. 8a), at a/H = 0.10 (Fig. 8b), and at a/H = 0.90 (Fig. 8c). The first two points correspond to the boundaries between the conical vortices and the weak suction area close to the sharp top cube surface edge, which is documented in Figs. 6 and 7. This indicates that the pressure fluctuations reach their maximum at the surface underlying outer boundaries of the rotating conical vortex near the flow separation point. Further downstream of the conical vortex boundary, but still under the conical vortex, cp,rms decreases in magnitude (Fig. 8a), while the exhibited minimum in Fig. 8b corresponds to the surface area characterized by a weak suction between two conical vortices in the central part of the top cube surface. Fig. 8c serves as evidence for the cp,rms characteristics on the top cube surface underlying the conical vortex in comparison with the weak suction area. This is because this profile starts outside the conical vortex in the weak suction area, progresses towards the center of the top cube surface and finishes underneath the vortex. This is unlike the cp,rms profiles presented in Fig. 8a and b that start and finish under the conical vortices while their central part is in the weak suction area. In Fig. 8c, a substantial difference in the cp,rms values can be observed in the weak suction area in comparison with the area underlying the conical vortex, as for each of the profiles the cp,rms values observed under the conical vortex are three to four times higher than in the weak suction area of the respective profile. This is the same order of magnitude difference that Kawai and Nishimura (1996) observed for the fluctuating pressure coefficient between the turbulent and smooth incident flows.
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Fig. 8. Profiles of cp,rms coefficients on the top surface of the wall-mounted cube at (a) x/H = 0.10, (b) x/H = 0.25, (c) y/H = 0.50.
Power spectral density of surface pressure fluctuations The power spectral density of pressure fluctuations on the top cube surface underlying conical vortices is given in Figs. 9–11. The pressure spectra for square prisms oriented perpendicularly to the incident flow commonly exhibit a distinct peak at the characteristic Strouhal frequency St = f ·H /uH ∼ 0.1 (f is frequency) due to the vortex shedding on the sharp body edges. This vortex-shedding peak was previously observed to weaken while approaching the trailing edge, while another peak emerged at higher reduced frequencies, Kareem and Cermak (1984). The present results (Figs. 9–11) show that close to the leading corner (point 1), the pressure spectra are dominated by a single low-frequency peak at f ·H / uH ∼ 0.2 for the LT and MT boundary layers, and at ∼ 0.3 for the HT boundary layer. Its magnitude is the largest for the LT boundary layer, Fig. 9. As the conical vortex develops from point 1 to 5, this primary peak weakens and a secondary peak at a higher reduced frequency at ∼ 1.0 emerges. This trend is comparable to what was observed by Kareem and Cermak (1984) for a square prism normal to the incident flow. They argued that the reduction in the vortex-shedding peak amplitude was partly caused by the redistribution of energy to a wider bandwidth and a concentration of energy at a frequency higher than the Strouhal frequency. This feature was more pronounced in
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Fig. 9. Power spectral density of surface pressure fluctuations at measuring points placed on the top cube surface along the line at x/H = 0.10; x-axis is reduced frequency f · H /uH , y-axis is pre-multiplied reduced pressure power spectrum f · Sp /σp2 .
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Fig. 10. Power spectral density of surface pressure fluctuations at measuring points placed on the top cube surface along the line at x/H = 0.25; x-axis is reduced frequency f · H /uH , y-axis is pre-multiplied reduced pressure power spectrum f · Sp /σp2 .
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Fig. 11. Power spectral density of surface pressure fluctuations at measuring points placed on the top cube surface along the line at y/H = 0.50; x-axis is reduced frequency f · H /uH , y-axis is pre-multiplied reduced pressure power spectrum f · Sp /σp2 .
more turbulent flows suggesting a possible re-attachment of the separated shear layer on the downstream portion of the square prism surfaces. At similar portions of the square prism subjected to two different incident flows there was a concomitant decrease in the spectral peaks with an increase in the incident flow turbulence. These results of Kareem and Cermak (1984) partly explain the characteristic trends in surface pressure fluctuations given in Figs. 9–11, whereas in addition to the flow separation and re-attachment phenomena analyzed by Kareem and Cermak (1984), in Figs. 9–11 there were conical vortices acting as well. In comparison to the HT boundary layer, the magnitude of the secondary peak at point 5 is larger for less turbulent incident flows (LT and MT boundary layers). This is likely so because of a complex interplay between three dominant
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phenomena that are present in that portion of the top cube surface and the changes that occur in their contribution to the fluctuating pressure when adding turbulence to the oncoming flow. These flow phenomena are (a) the conical vortex that weakens farther away from the leading corner, (b) the flow separation at the leading edge of the top cube surface between the cube surfaces 1 and 5, (c) the secondary flow separation at the trailing edge of the top cube surface between the cube surfaces 2 and 5. There is a general trend of shifting the pressure spectra towards higher reduced frequencies in more turbulent incident flows. The trends observed at points 1–5 are generally present at points 6–10 as well, whereas the secondary highfrequency peak is not that well exhibited as it is for points 1–5. This indicates that the effects of the conical vortex are stronger for the surface area closer to the leading edge (points 1–5). The pressure spectra given in Fig. 11 are particularly important as they compare the characteristics of the pressure spectra inside (points 3 and 8) and outside (points 11 and 12) the conical vortex. At points 3 and 8, there is a strong low-frequency peak and a less exhibited high-frequency peak. At the same time, adding turbulence to the incident flow causes the surface pressure spectra to shift towards higher reduced frequencies. At central point 11, the high-frequency peak increases in magnitude and becomes fully dominant at point 12 close to the trailing edge. 4. Conclusions This study focuses on pressure distribution and fluctuations, including power spectral density of pressure fluctuations, on the top surface of a cubic building subjected to oblique turbulent boundary layers. In this configuration, conical vortices develop and substantially influence pressure characteristics on the top building surface. The study particularly focuses on the effects of turbulence in the incident flow. Three turbulent boundary layers of various turbulence characteristics were created experimentally in a boundary layer wind tunnel. The flow was characterized by the mean velocity, turbulence intensity, Reynolds shear stress, turbulence length scales, and power spectral density of velocity fluctuations, which were simultaneously recorded in all three directions. One cube with 0.2 m long edges was used as an experimental model. In all conducted experiments, the flow incidence angle was at 40◦ so as to create conical vortices above the top cube surface. The thickness of the turbulent boundary layers was ten times the cube height. For the purpose of taking steady pressure measurements, on each of the vertical cube surfaces there were 35 pressure taps and on the top surface there were 49 pressure taps. Unsteady surface pressures were measured at 15 points on the top cube surface. The conical vortices developed above the top surface of the wall-mounted cube are clearly observed as two zones characterized by strong suction. These two suction zones are underlying the cores of the two conical vortices. The conical vortex is larger downstream of the leading edge of the top cube surface that is oriented at a smaller angle to the main flow direction, i.e. closer to the case when the flow is perpendicular to the leading cube edge. The larger mean suction acts on the top cube surface in the less turbulent boundary layer, thus indicating that the conical vortices are stronger in the smooth flow than in the turbulent flow. With an increase in the turbulence in the incoming flow, the strong suction zones decrease in size with a simultaneous increase in the size of the weak suction zone on the central area of the top cube surface. This indicates that the conical vortices are generally larger in less turbulent (smoother) flows than in more turbulent flows. The conical vortices are situated closer to the respective edges of the top cube surface when subjected to a more turbulent incident flow. The profiles of the fluctuating pressure coefficient retain their general shape when turbulence is added to the flow upstream of the cube. However, the entire profile shifts so the fluctuating pressure coefficient is larger in a more turbulent flow. In fact, the added turbulence acts as an amplifier of pressure fluctuations on the top cube surface. The pressure fluctuations are larger at the surface underlying outer boundaries of the rotating conical vortex near the flow separation point. Further downstream of the conical vortex boundary, but still underneath the conical vortex, the fluctuating pressure coefficient decreases in magnitude, while the exhibited minimum corresponds to the surface area characterized by weak suction between two conical vortices in the central part of the top cube surface. In the same experiment, the fluctuating pressure coefficient underneath the conical vortex is three to four times larger than in the weak suction zone on the central area of the top cube surface. Close to the leading cube corner, the pressure spectra are dominated by a single low-frequency peak. This peak is larger than the one commonly observed on square prisms oriented perpendicularly to the incident flow, while its magnitude is larger in less turbulent incident flows. As the conical vortex develops, this primary peak weakens and a secondary peak at a higher reduced frequency emerges. The magnitude of the secondary peak is larger for less turbulent incident flows. There is a general trend of shifting the pressure spectra towards higher reduced frequencies in more turbulent incident flows. The effects of the conical vortex are stronger in the case of the surface area closer to the leading cube edge. Declaration of competing interest The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Acknowledgments Experiments were carried out at the former Chair of Fluid Mechanics, Department of Mechanical Engineering, Technical University of Munich (TUM), which was headed by Professor Emeritus Boris Laschka, whose kind support and advice along with his associates’ help are greatly appreciated. The study was funded by the Croatian Ministry of Science and Education, German Academic Exchange Service (DAAD) and the Croatian Academy of Sciences and Arts (HAZU), which support is gratefully acknowledged. References Balendra, T., Shah, D.A., Tey, K.L., Kong, S.K., 2002. Evaluation of flow characteristics in the NUS-HDB wind tunnel. J. Wind Eng. Ind. Aerodyn. 90, 675–688. Banks, D., 2013. The role of corner vortices in dictating peak wind loads on tilted flat solar panels mounted on large, flat roofs. J. Wind Eng. Ind. Aerodyn. 123, 192–201. Banks, D., Meroney, R.N., 2001a. A model of roof-top surface pressures produced by conical vortices: Model development. 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Surface pressure on a cubic building exerted by conical vortices Hrvoje Kozmar Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lučića 5, 10000 Zagreb, Croatia
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Article history: Received 7 March 2019 Received in revised form 24 September 2019 Accepted 8 November 2019 Available online xxxx Keywords: Cubic building Turbulent boundary layer Turbulence Conical vortices Surface pressure distribution and fluctuations Pressure spectra Wind-tunnel experiments
a b s t r a c t Experiments were performed to study surface pressure on a cubic building underlying conical vortices, which are known to cause severe structural damage and failure. The focus is on the effects of turbulence in the incident flow. Three turbulent boundary layers were created in a boundary layer wind tunnel. A wall-mounted cube, i.e. a cube situated on the horizontal ground floor surface of the wind-tunnel test section, was used as an experimental model. The cube was subjected to the incidence flow at 40◦ . Steady and unsteady pressure measurements were performed on the cube surface. The analysis suggests that conical vortices developed above the top surface of the wallmounted cube. A larger mean suction was observed on the top cube surface in the less turbulent boundary layer. With an increase in turbulence in the incoming flow, the strong suction zones decreased in size. The fluctuating pressure coefficient profiles retained their shape when the turbulence in the upstream flow of the cube increased. The fluctuating pressure coefficient was observed to be larger in more turbulent flows. The pressure fluctuations were larger on the cube surface underlying outer boundaries of the conical vortex. The fluctuating pressure coefficient under the conical vortex was three to four times larger than in the weak suction zone on the central area of the top cube surface. Close to the leading cube corner, the pressure spectra were dominated by a single low frequency peak. As the conical vortex developed, this primary peak weakened and a secondary peak emerged at a higher reduced frequency. There is a general trend of shifting the pressure spectra towards higher reduced frequencies when the turbulence in the undisturbed incident flow increases. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Conical vortices develop on the top surface of cubic buildings subjected to the oblique wind flow, as shown in Fig. 1, adapted from Banks et al. (2000). Extreme suction occurs on the body surface underlying such vortices, e.g. Kind (1986) and Richards and Hoxey (2008). These aerodynamic loads may cause severe structural damage and failure, e.g. Ginger and Letchford (1993), which may be mitigated by using various engineering solutions, e.g. Baskaran and Stathopoulos (1988), Bienkiewicz and Sun (1992), Tieleman et al. (1994), Hoxey et al. (1998) and Banks (2013). Conical vortices are highly relevant for a wide range of aerospace, civil, mechanical, and wind engineering applications. Many studies have comprehensively tackled this complex issue, while there are still some open questions including further aspects of the boundary layer turbulence effects on pressure distribution and particularly fluctuations on a flat top body surface in the oblique fluid flow. E-mail address: [email protected]. https://doi.org/10.1016/j.jfluidstructs.2019.102801 0889-9746/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Schematic view of conical vortices developing on the top surface of a bluff body subjected to the oblique fluid flow. Source: Adapted from Banks et al. (2000).
Previous studies have thoroughly addressed flow characteristics of conical vortices and the surface pressure on bluff bodies underlying these vortices. In these studies, field measurements, wind-tunnel experiments, Computational Fluid Dynamics (CFD), and semi-empirical methods have been used. It is generally accepted that the large suction on the top surface of bluff bodies near their corner is caused by the conical (corner, delta-wing) vortex that develops in the oblique fluid flow, as given in Kawai and Nishimura (1996) and Kawai (1997, 2002). These authors report that a pair of conical vortices has rotating spiral axes and that they rotate opposite to each other. The centerlines of the vortices are along the line inclined at 13◦ in the smooth flow and at 10◦ in the turbulent flow relative to the eaves. These lines correspond to the maximum suction on the top body surface directly underneath the conical vortices. A strong effect of turbulence on the conical vortices and the underlying surface pressure has been observed. In particular, the vortices are generally stronger in the smooth flow than in the turbulent flow, i.e. a larger mean suction acts on the top body surface in the smooth flow, which was reported by Lin et al. (1995) as well. In the turbulent flow, vortices are flatter and placed closer to the top body surface. The maximum mean suction occurs a little downstream of the corner of the top body surface and it is larger in the smooth flow than in the turbulent flow. The suction fluctuations result from the combined flapping and rotation of the conical vortices. Vortex flapping is the intermittent up and down vertical movement of the conical vortex at some characteristic frequency, which is similar to the movement of flying birds’ wings. Each conical vortex grows and decays alternately to each other, yielding a mismatch in the suction fluctuations relative to the diagonal line of the top body surface. The fluctuating pressure coefficient is three times larger in the turbulent shear flow than in the smooth flow, Kawai and Nishimura (1996). Banks et al. (2000) confirmed that the peak suction lies beneath the vortex core and it moves with the vortex. The vortex becomes larger with an increase in the flow incidence angle. The increasing turbulence moves the mean vortex core position closer to the leading edge. The faster the vortex spins, the lower the core pressure and the lower the surface pressure, Banks and Meroney (2001a,b), so the vortex acts as an amplifier of the local pressure drop. The pressure coefficient decreases with the inverse of the square root of the distance from the corner, Tryggeson and Lyberg (2010). The incident flow velocity predominantly controls the strength of conical vortices, and it has rather marginal effect on the overall structure of conical vortices (Wu et al., 2001a,b). The horizontal flow incidence angle determines the formation and configuration of vortices and the underlying surface pressure fluctuations, as substantial discrepancies in the suction on the top body surface were observed when the horizontal flow incidence angle is changed, Mehta et al. (1992). The vertical flow incidence angle modifies the configuration and movement of the vortices, whereas large and quick excursions of the vertical angle can significantly change the configuration of the vortices and yield extremely high, instantaneous peak suctions. Characteristics of the conical vortex vary in the vertical and the horizontal direction, Marwood and Wood (1997). The surface suction under the conical vortex is maximal when the vortex axis is located farthest away from the body edge and highest above the surface of the body. Variations in position are smaller in the low turbulence flow thus indicating that the vortex movement is driven by incident turbulence. The shape of the vortex cross section is nearly elliptical, whereas the angle between the vortex axis and the leading edge is smaller in the turbulent flow, Sun and Ye (2016). Detailed flow and surface pressure characteristics were obtained by Ginger and Letchford (1992) and Kim et al. (2001, 2003). The maximum turbulence kinetic energy was found in the upper boundary of the separation bubble near the leading edge of the top bluff body surface, whereas the maximum energy magnitude is substantially larger in that area than in the wake region. Analytical and semi-empirical approaches have been used to account for the flow and underlying surface pressure characteristics caused by conical vortices. A semi-empirical model was proposed for the pressure field beneath corner vortices, Cook (1990), whereas the flow can be considered as a conical vortex ‘growth region’ near the corner of the top
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body surface, and as a cylindrical vortex ‘mature region’ further downstream of the corner. Williams and Baker (1997) confirmed the existence of the ‘growth region’, while they did not observe the ‘mature region’ in the form suggested by Cook (1990). Banks and Meroney (2001c) advised caution when using the quasi-steady theory to predict peak and fluctuating pressure coefficients under conical vortices. The incorporation of the vertical turbulence component improved this approach, but discrepancies remained as the quasi-steady theory proved its inadequacy in regard to flow distortion and building-generated turbulence, Letchford and Marwood (1997). Vortex movement was driven by upstream turbulence and generally behaved in a quasi-steady manner; however, the peak pressures are underestimated by the quasi-steady theory, Marwood and Wood (1997). Ye and Dong (2014) improved the quasi-steady theory by including the effect of vortices, where it was possible to correctly predict the mean and peak suction under conical vortices. Wu and Sarkar (2006) proposed a bivariate quasi-steady model, which accounted for the influence of the horizontal and vertical flow components on the surface pressure underlying conical vortices. The quasi-steady theory in this form proved to be applicable in the separated flow region where vortices were present. Zhao et al. (2002) developed a simple analytical model to predict fluctuating pressure coefficients on the leading corner of a wall-mounted bluff body. CFD has become an increasingly important tool to analyze flow characteristics of conical vortices and underlying surface pressure distribution and fluctuations on bluff bodies, e.g. He and Song (1997), Thomas and Williams (1999a,b), He et al. (2007) and Ono et al. (2008). This literature survey reveals many important previous studies that focused on the flow characteristics of conical vortices and pressure characteristics on the top surface of underlying bluff bodies; nevertheless, there are still some open questions. In particular, Tieleman (1993), Saathoff and Melbourne (1997), Wu et al. (2001a,b), Letchford and Marwood (1997) and Marwood and Wood (1997) point out that it is important to accurately model turbulence characteristics to fully account for relevant aspects of conical vortices and bluff body aerodynamics. Furthermore, it is still not sufficiently understood how the fluctuating pressures under conical vortices compare to the fluctuating pressures in the center of the top bluff body surface, i.e. outside of the conical vortex. It is therefore particularly important to study the power spectral density of pressure fluctuations on the top body surface to account for the fluctuating pressure energy distribution in the frequency domain. This will enable a more thorough understanding of dynamic loads on bluff bodies exerted by conical vortices. Given these facts, the scope of the present study is the pressure distribution and fluctuations on the top surface of a wall-mounted cube, i.e. a cube situated on the horizontal ground floor surface of the wind-tunnel test section, subjected to oblique turbulent boundary layers (TBLs). A particular focus is on the power spectral density of surface pressure fluctuations on the top cube surface underlying conical vortices and outside of them for various levels of turbulence in the incoming 3D boundary layer flows. 2. Experimental procedure Experiments were performed in a boundary layer wind tunnel (BLWT) at the Technical University of Munich (TUM), Germany. The rectangular test section is 1.80 m high, 2.70 m wide and 21 m long. The turbulence intensity at the inlet of the test section is less than 0.5%. Across the cross section of the test section inlet, the mean velocities in the main flow direction differ by less than 1%. These results were analyzed away from the walls, i.e. outside of the near-wall region that represents 10% of the respective width and height dimensions of the wind-tunnel test section. The test section is equipped with a ceiling adjustable in height that enables a zero pressure gradient in the main flow direction. There is a turntable centered 11.3 m downstream of the nozzle that enables airflows on structural models to be developed at various incidence angles. A blower powered by a 210 kW electric motor generates airflows in the velocity range of 1 m/s – 30 m/s. Three TBLs were created using the Counihan (1969) approach, i.e. the castellated barrier wall, vortex generators and surface roughness elements. The fundamental difference between these three TBLs was in their turbulence levels, i.e. they were either low-turbulent (LT), mid-turbulent (MT), or high-turbulent (HT) boundary layers. While the Counihan approach has been commonly employed to generate the entire TBL thickness using the full-size Counihan vortex generators, in the present study the truncated Counihan vortex generators were employed. These vortex generators were truncated from the 2.0 m high ‘classical’ Counihan vortex generators to 1.7 m, e.g. Kozmar (2011a,b, 2012). The TBL thickness δ is 2.0 m; however, in the wind tunnel only a lower TBL portion from the ground surface up to 0.7δ (1.4 m) was reproduced. This approach generally allows achieving thicker turbulent boundary layers, Kozmar and Laschka (2019a). Surface roughness was created by using LEGO bricks. In each of the three TBLs, the LEGO bricks were of the same height and arranged in a symmetrical and staggered pattern. Preliminary arrangement of the spacing (packing) density, which is the test section area covered with surface roughness elements with respect to the total test section area, and the height of the LEGO bricks was based on recommendations given by Gartshore and De Croos (1977). The final arrangements were made through trial and error and reported in Table 1. In all the experiments, the basic height of the castellated barrier wall was 292 mm with the castellation height of 84 mm on top of it. An example of the experimental arrangement in the wind-tunnel test section, the HT setup, is shown in Fig. 2. The two other configurations, LT and MT boundary layers, are basically the same, but with smaller spacing density and a lower height of the surface roughness elements (LEGO bricks).
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H. Kozmar / Journal of Fluids and Structures 92 (2020) 102801 Table 1 Spacing density and height of surface roughness elements in turbulent boundary layers. Turbulent boundary layer
Roughness spacing density
Roughness height
LT MT HT
0.5% 0.9% 6.3%
68 mm 68 mm 78 mm
Fig. 2. Photograph of the experimental arrangement in the wind-tunnel test section for the high-turbulent boundary layer.
Instantaneous velocities were first measured in the empty wind-tunnel test section, i.e. without the cube. This was performed simultaneously in three coordinate directions at each measuring point using a triple hot-wire probe DANTEC 55P91. The sampling rate was 1.25 kHz and the record length was 150 s. The preliminary tests proved that these parameters were sufficient to successfully acquire all relevant flow phenomena. Measurements were carried out at 16 measuring points placed along a vertical line at the center of the turntable. Previous experiments proved that the TBLs created using this approach were fully developed and uniform at the turntable center in the empty test section, Kozmar (2011c). This is exactly the position where the cube was situated in the subsequent experiments focused on the surface pressure on a wall-mounted cube exerted by conical vortices. The TBLs were characterized by the mean velocity in the longitudinal (x), lateral (y), and vertical (z) directions, i.e. u, v , w, respectively. The turbulence intensities Iu (z), Iv (z), Iw (z) in these three directions were calculated by using the respective fluctuating velocities u’, v’, w’ and the local mean flow velocity u in the main (x) flow direction. This characterization of flow characteristics based on simultaneous measurements in all three directions is important as the turbulence in each of these three directions is relevant for bluff body aerodynamics including conical vortices, e.g. Tieleman (1993, 2003), Letchford and Marwood (1997), Marwood and Wood (1997), Saathoff and Melbourne (1997) and Wu et al. (2001a,b). The Reynolds shear stress τ = −ρ u′ w ′ profile was provided because it was previously proven that it influences roof pressure fluctuations, Sharma and Richards (1999), so it was anticipated that it would also affect the conical vortices and the underlying pressure fluctuations. The turbulence length scale Lu,x was calculated using the autocorrelation functions and assuming the Taylor’s frozen turbulence hypothesis. One cube with 0.2 m long edges was used as the experimental model. It created the test section blockage of 0.84% that is considerably smaller than the critical blockage of 6% (West and Apelt, 1982), so the results were not corrected. The predicted thickness of the TBLs that would be equal to the full size of 2 m high non-truncated Counihan vortex generators, in agreement with Balendra et al. (2002), was in all experiments ten times larger than the cube height, which is comparable to the Castro and Robins (1977) experiments. The pressure measurements were simultaneously performed on all cube surfaces except the bottom surface that was attached to the wall. For the purposes of general understanding and qualitative validation, the results were reported on all the cube surfaces equipped with pressure measuring points, while they were studied in detail on the top surface underlying conical vortices only as this was the focus of the present study. The flow incidence angle was at 40◦ in all experiments. This setup allows for two different conical vortex structures to be achieved above the top cube surface rather than two identical conical vortices that would appear at 45◦ and thus this setup yields a larger variety in the results. In general, the conical vortex is larger downstream of the leading edge of the top cube surface that is oriented at a smaller angle to the main flow direction (the flow perpendicular to the leading edge is the zero incidence angle), e.g. Banks et al. (2000), whereas it is not that clear at the moment what the minimum/maximum flow incidence angle is that would result in the
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formation of conical vortices. The conical vortex more perpendicular to the positive x-axis is larger because the incident flow angle in the case of this vortex is smaller to the respective leading edge of the top cube surface, in agreement with Banks et al. (2000). On each of the vertical cube surfaces there were 35 pressure taps and on the top surface there were 49 pressure taps used for performing steady pressure measurements. Close to the cube edges, the pressure taps were placed more densely to make the capturing of strong pressure gradients that were anticipated in these regions possible. The steady surface pressure measurements on the cube and the reference Prandtl–Pitot static pressure measurements in the freestream flow were simultaneously performed by using PDCR-22 sensors and the Natec Schultheiss Scanivalve System. The pressure taps on the cube surface were connected to the steady pressure sensors via short plastic tubes. The sampling rate was 100 Hz during 15 s because the mean pressure and the pressure standard deviation changed negligibly for longer time records. The steady surface pressure coefficients cp on the cube were calculated using the mean flow velocity uH at the height that corresponds to the height of the top cube surface (H = 0.2 m). uH was measured in the empty test section at the height of the top cube surface before the pressure measurements on the cube actually took place. The unsteady surface pressure was measured at 15 points on the top cube surface by using Kulite XCS-093-0.35bar-D miniature pressure sensors. The pressure taps needed for the unsteady pressure measurements were designed as described in Kozmar and Laschka (2019b). The unsteady pressure coefficient cp,rms was calculated using the mean flow velocity uH , whereas uH was determined in the same way as for the steady pressure measurements. The sampling rate and the recording time were 2 kHz and 20 s, respectively. The dynamic Kulite pressure sensors were placed directly in the measuring points at the cube surface, i.e. plastic tubes were not used to connect the pressure taps to the pressure sensors, so there was no need to perform pressure corrections to account for the tubing effects. During preliminary measurements, this experimental setup proved accuracy with respect to the intended experiments. The power spectral density of pressure fluctuations on the cube surface Sp (f ) was provided in the pre-multiplied form f · Sp /σp2 , which identifies frequencies that carry the majority of the energy in the observed phenomenon, e.g. Hearst et al. (2016); this way of presenting the spectra is common in the engineering community. However, it needs to be noted that plotting the spectra in this pre-multiplied f · Sp /σp2 form always leads to a peak, whether or not the signal contains any dominating periodicity, while on the other hand, the peaks observed in the non-pre-multiplied form Sp are more genuinely caused by the periodicity in the signal and commonly used to identify characteristic flow phenomena. The distribution of the measuring points on the top surface of the wall-mounted cube for steady and unsteady pressure measurements is given in Fig. 3. 3. Results and discussion Turbulent boundary layers Basic characteristics of the TBLs created in the empty test section (without the wall-mounted cube) are outlined in Table 2; α is the exponent of the empirical power-law approximation of the mean velocity profile, d is the displacement height of the mean velocity profile, z0 is the aerodynamic surface roughness length, uτ is the friction velocity, ReR is the roughness Reynolds number calculated as ReR = (uτ · z0 )/ν , where the air viscosity ν is 1.5 · 10−5 m2 /s. The mean velocity results were fitted to the power law to estimate α and d, and to the logarithmic law to determine z0 and uτ . The profiles of the mean u velocity in the main (x) flow direction, u′ w ′ Reynolds shear stress, Lu,x turbulence length scale, and Iu , Iv and Iw turbulence intensity in the x-, y- and z-directions, respectively, are given in Fig. 4. The mean reference velocity uH was recorded in the empty test section (without the wall-mounted cube) at the height H = 0.2 m, which corresponds to the height H of the wall-mounted cube in the subsequent experiments. While this work may be understood as a fundamental fluid mechanics study (wall-mounted cube subjected to TBL), it may also represent an experimental small-scale model of a cubic building subjected to the rural, suburban and urban atmospheric boundary layer (ABL) flows, characterized by the power-law exponent of the mean velocity profiles α = 0.17, 0.21 and 0.35, respectively, in agreement with Dyrbye and Hansen (1997) for the suggested terrain types. In the latter case, the length scale factor of the ABL wind-tunnel simulation calculated as suggested in Cook (1978) is S = 91.3 (z − d)0.491 /Lu1,.403 z00.088 = 210, which is then to be applied to the cubic building model as well; z is the vertical x distance from the wind-tunnel floor, d is the displacement height, Lu,x is the integral length scale of turbulence, z0 is the aerodynamic surface roughness length. In that case, a full-scale cubic building has a height of 42 m; the characteristic full-scale ABL parameters in the LT, MT and HT boundary layer cases are d = 0.42 m, 5.04 m and 5.88 m, whereas
Table 2 Basic characteristics of the turbulent boundary layers. Case
α
d, mm
z0 , mm
uτ , m/s
ReR
uH , m/s
LT MT HT
0.17 0.21 0.35
2 24 28
0.80 1.59 9.34
1.11 1.18 1.43
59 125 890
14.97 13.48 10.14
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Fig. 3. Schematic view of (a) wall-mounted cube with indicated measuring points on the top (number 5) cube surface for (b) steady and (c) unsteady pressure measurements.
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Fig. 4. Characteristic profiles of the turbulent boundary layers: (a) mean u velocity in the main (x) flow direction, (b) u′ w ′ Reynolds shear stress, (c) Lu,x turbulence length scale, (d)–(f) Iu , Iv , Iw turbulence intensity in the longitudinal (x), lateral (y) and vertical (z) directions, respectively; the dotted line represents the height H of the top surface of the wall-mounted cube used in subsequent experiments.
z0 = 0.168 m, 0.334 m and 1.961 m, respectively. More details on the wind-tunnel ABL simulation procedure of this type may be found in Kozmar (2011a) along with a justification of the studied parameters. The gradient in the mean velocity profile is larger on rougher surfaces for z ≤H, Fig. 4a. In this same flow region, the absolute values of the Reynolds shear stress and the turbulence intensity are higher on the rougher wall (Fig. 4b, d–f), while the turbulence length scale is smaller (Fig. 4c). The observed flow behavior was expected and previously confirmed by many relevant studies, e.g. Nikuradse (1933). The constant shear stress close to the wall is present as well, which phenomenon is in agreement with Schlichting and Gersten (1997). The flow characteristics recorded in the empty test section at the height H, which is the height of the top surface of the wall-mounted cube used in the subsequent experiments, are given in Table 3. The power spectral density of the longitudinal velocity fluctuations at z /δ = 0.20 (z /H = 2) in the TBLs upstream of the wall-mounted cube is shown in Fig. 5. The trends observed at other heights were nearly the same and thus not
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H. Kozmar / Journal of Fluids and Structures 92 (2020) 102801 Table 3 Characteristics of turbulent boundary layers recorded in the empty test section at a height which corresponds to the height H of the top surface of the wall-mounted cube used in subsequent experiments. Case
Iu
Iv
Iw
u′ w ′ / uH
LT MT HT
0.17 0.22 0.33
0.14 0.17 0.24
0.11 0.13 0.20
−0.00688 −0.01000 −0.02108
2
Lu,x /H 4.03 3.78 2.42
Fig. 5. Power spectral density of longitudinal velocity fluctuations at z /H = 2 in turbulent boundary layers upstream of the wall-mounted cube in comparison with the Kolmogorov (1941) and Von Kármán (1948) models.
presented. The velocity power spectra in all three TBLs agree well with the Kolmogorov (1941) and Von Kármán (1948) models. The flow characteristics recorded simultaneously in all three directions allowed a reliable analysis of pressure characteristics under conical vortices on the top surface of the wall-mounted cube in three TBLs of various turbulence characteristics. Steady pressure distribution The characteristics of the steady pressure distribution on the wall-mounted cube surface were analyzed at Reynolds (Re) numbers greater than 5·104 , which value previously proved to be critical for bluff bodies, Tieleman (2003). The Re numbers obtained in the presented experiments are given in Table 4. However, it needs to be noted that, according to Lim et al. (2007), the Re effects are more likely to occur if the flow around the bluff body contains strong concentrated vortex motions (similar to the conical vortices studied in the present work). Even though the present study focuses on the conical vortices on the top cube surface only, the cp distribution on all cube surfaces (except the bottom surface) is given first in Fig. 6 in the cube-based coordinate system so as to analyze whether the characteristic surface pressure phenomena were present. The steady surface pressure on the top cube surface was further analyzed in more detail in Fig. 7. On the vertical cube surfaces subjected to the incoming TBLs, the cp distribution is in agreement with Castro and Robins (1977). The flow generally ‘preferred’ the surface that was more perpendicularly oriented to the incoming flow, where the stagnation area was larger and the respective cp values higher. The vertical leeward surfaces that are oriented toward the negative x- and y-coordinate axes are both characterized by nearly uniform suction, which phenomenon is in agreement with Castro and Robins (1977) as the differences in cp on these surfaces, as observed in Fig. 6, are rather negligible.
Table 4 Reynolds number in steady pressure measurements on the wall-mounted cube. Case
Re = (uH · H)/ν
LT MT HT
2.0·105 1.8·105 1.4·105
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Fig. 6. Distribution of cp coefficients on the wall-mounted cube subjected to (a) low-turbulent, (b) mid-turbulent, (c) high-turbulent boundary layers.
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Fig. 7. Distribution of cp coefficients on the top surface of the wall-mounted cube subjected to (a) low-turbulent, (b) mid-turbulent, (c) high-turbulent boundary layers.
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The conical vortices develop above the top surface of the wall-mounted cube as the flow separates from the leading edges of the top cube surface, Fig. 1. These flow characteristics are exhibited in the cp distribution on the top cube surface, Figs. 6 and 7. Two zones immediately downstream of the leading edges of the top cube surface characterized by strong suctions can be clearly observed. These two suction zones are underlying the cores of the two conical vortices, which causality is inferred from Kind (1986), Kawai and Nishimura (1996), Kawai (1997, 2002) and Banks et al. (2000). The conical vortex perpendicular to the positive x-axis of the cube-based coordinate system is larger because the incident flow for this vortex is closer to the perpendicular flow with respect to the leading edge of the top cube surface. This characteristic is in agreement with Banks et al. (2000). Along the diagonal line across the top cube surface the suction is considerably weaker than close to the leading edges, thus indicating that this area is outside of the conical vortices. The larger mean suction acts on the top body surface in less turbulent flows (e.g. the LT boundary layer compared to the MT boundary layer and more so to the HT boundary layer), thus indicating that the conical vortices are generally stronger in the smooth flow than in the turbulent flow, which is in agreement with Lin et al. (1995). In particular, the maximum LT suction (cp = −2.0) is almost double the maximum HT suction (cp = −1.2). At the same time, as the turbulence in the incoming flow increases, the strong suction zones decrease in size with a simultaneous increase in the size of the weak suction zone on the central area of the top cube surface. This indicates that the conical vortices are generally larger in less turbulent (smoother) flows than in more turbulent flows. This is in line with previously observed shortening of the separation bubble along with an increase in the turbulence of the incident flow, Hillier and Cherry (1981). The conical vortices are situated closer to the respective edges of the top cube surfaces in the more turbulent flow (HT boundary layer), which result is in agreement with Sun and Ye (2016) as the angles between the vortex axes and the respective leading edges of the top cube surface are smaller in the HT boundary layer than in the two other cases, Figs. 6 and 7. The conical vortices spin faster in less turbulent flows, as exhibited in the stronger underlying surface suction on the top cube surface, which is inferred from Banks and Meroney (2001a,b), i.e. the conical vortices spin faster in the LT than in the MT boundary layer and even more so than in the HT boundary layer, as cp on the top cube surface underlying conical vortices are minimum (strongest suction) in the LT boundary layer, Figs. 6 and 7. Unsteady pressure characteristics Two conical vortices rotating opposite to each other along with their flapping substantially influence pressure fluctuations on the top cube surface, as reported by Kawai and Nishimura (1996). They grow and decay alternately, which yields a mismatch in the suction fluctuations relative to the diagonal line of the top cube surface. The fluctuating pressure coefficient was previously observed to be three times larger in the turbulent shear flow than in the smooth flow, Kawai and Nishimura (1996). The present results for the cp,rms fluctuating pressure coefficient are given in Fig. 8. Based on these cp,rms results along with the cp distributions presented in Figs. 6 and 7, which approximately mirror the boundaries of the conical vortices, it is possible to analyze the pressure fluctuations (cp,rms ) underlying conical vortices in comparison with the respective values obtained in the weak suction area in the central part of the top cube surface. A particular emphasis is put on the effects of turbulence in the incoming TBLs on the cp,rms characteristics on the top cube surface inside and outside the conical vortices. It can be observed that the cp,rms profiles retain their general shape when adding turbulence to the undisturbed upstream TBL along with a simultaneous increase in pressure fluctuations. Accordingly, at each particular measuring point on the top cube surface, regardless of whether it is inside or outside the conical vortex, the HT cp,rms values are the highest and the LT cp,rms values are the lowest. In this context, the added turbulence acts as an amplifier of the pressure fluctuations on the top cube surface. In all experiments, the cp,rms profiles substantially change their shape as they leave the surface area underlying conical vortices and progress toward the weak suction area in the central part of the top cube surface. The peak values are observed at a/H = 0.25 (Fig. 8a), at a/H = 0.10 (Fig. 8b), and at a/H = 0.90 (Fig. 8c). The first two points correspond to the boundaries between the conical vortices and the weak suction area close to the sharp top cube surface edge, which is documented in Figs. 6 and 7. This indicates that the pressure fluctuations reach their maximum at the surface underlying outer boundaries of the rotating conical vortex near the flow separation point. Further downstream of the conical vortex boundary, but still under the conical vortex, cp,rms decreases in magnitude (Fig. 8a), while the exhibited minimum in Fig. 8b corresponds to the surface area characterized by a weak suction between two conical vortices in the central part of the top cube surface. Fig. 8c serves as evidence for the cp,rms characteristics on the top cube surface underlying the conical vortex in comparison with the weak suction area. This is because this profile starts outside the conical vortex in the weak suction area, progresses towards the center of the top cube surface and finishes underneath the vortex. This is unlike the cp,rms profiles presented in Fig. 8a and b that start and finish under the conical vortices while their central part is in the weak suction area. In Fig. 8c, a substantial difference in the cp,rms values can be observed in the weak suction area in comparison with the area underlying the conical vortex, as for each of the profiles the cp,rms values observed under the conical vortex are three to four times higher than in the weak suction area of the respective profile. This is the same order of magnitude difference that Kawai and Nishimura (1996) observed for the fluctuating pressure coefficient between the turbulent and smooth incident flows.
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Fig. 8. Profiles of cp,rms coefficients on the top surface of the wall-mounted cube at (a) x/H = 0.10, (b) x/H = 0.25, (c) y/H = 0.50.
Power spectral density of surface pressure fluctuations The power spectral density of pressure fluctuations on the top cube surface underlying conical vortices is given in Figs. 9–11. The pressure spectra for square prisms oriented perpendicularly to the incident flow commonly exhibit a distinct peak at the characteristic Strouhal frequency St = f ·H /uH ∼ 0.1 (f is frequency) due to the vortex shedding on the sharp body edges. This vortex-shedding peak was previously observed to weaken while approaching the trailing edge, while another peak emerged at higher reduced frequencies, Kareem and Cermak (1984). The present results (Figs. 9–11) show that close to the leading corner (point 1), the pressure spectra are dominated by a single low-frequency peak at f ·H / uH ∼ 0.2 for the LT and MT boundary layers, and at ∼ 0.3 for the HT boundary layer. Its magnitude is the largest for the LT boundary layer, Fig. 9. As the conical vortex develops from point 1 to 5, this primary peak weakens and a secondary peak at a higher reduced frequency at ∼ 1.0 emerges. This trend is comparable to what was observed by Kareem and Cermak (1984) for a square prism normal to the incident flow. They argued that the reduction in the vortex-shedding peak amplitude was partly caused by the redistribution of energy to a wider bandwidth and a concentration of energy at a frequency higher than the Strouhal frequency. This feature was more pronounced in
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Fig. 9. Power spectral density of surface pressure fluctuations at measuring points placed on the top cube surface along the line at x/H = 0.10; x-axis is reduced frequency f · H /uH , y-axis is pre-multiplied reduced pressure power spectrum f · Sp /σp2 .
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Fig. 10. Power spectral density of surface pressure fluctuations at measuring points placed on the top cube surface along the line at x/H = 0.25; x-axis is reduced frequency f · H /uH , y-axis is pre-multiplied reduced pressure power spectrum f · Sp /σp2 .
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Fig. 11. Power spectral density of surface pressure fluctuations at measuring points placed on the top cube surface along the line at y/H = 0.50; x-axis is reduced frequency f · H /uH , y-axis is pre-multiplied reduced pressure power spectrum f · Sp /σp2 .
more turbulent flows suggesting a possible re-attachment of the separated shear layer on the downstream portion of the square prism surfaces. At similar portions of the square prism subjected to two different incident flows there was a concomitant decrease in the spectral peaks with an increase in the incident flow turbulence. These results of Kareem and Cermak (1984) partly explain the characteristic trends in surface pressure fluctuations given in Figs. 9–11, whereas in addition to the flow separation and re-attachment phenomena analyzed by Kareem and Cermak (1984), in Figs. 9–11 there were conical vortices acting as well. In comparison to the HT boundary layer, the magnitude of the secondary peak at point 5 is larger for less turbulent incident flows (LT and MT boundary layers). This is likely so because of a complex interplay between three dominant
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phenomena that are present in that portion of the top cube surface and the changes that occur in their contribution to the fluctuating pressure when adding turbulence to the oncoming flow. These flow phenomena are (a) the conical vortex that weakens farther away from the leading corner, (b) the flow separation at the leading edge of the top cube surface between the cube surfaces 1 and 5, (c) the secondary flow separation at the trailing edge of the top cube surface between the cube surfaces 2 and 5. There is a general trend of shifting the pressure spectra towards higher reduced frequencies in more turbulent incident flows. The trends observed at points 1–5 are generally present at points 6–10 as well, whereas the secondary highfrequency peak is not that well exhibited as it is for points 1–5. This indicates that the effects of the conical vortex are stronger for the surface area closer to the leading edge (points 1–5). The pressure spectra given in Fig. 11 are particularly important as they compare the characteristics of the pressure spectra inside (points 3 and 8) and outside (points 11 and 12) the conical vortex. At points 3 and 8, there is a strong low-frequency peak and a less exhibited high-frequency peak. At the same time, adding turbulence to the incident flow causes the surface pressure spectra to shift towards higher reduced frequencies. At central point 11, the high-frequency peak increases in magnitude and becomes fully dominant at point 12 close to the trailing edge. 4. Conclusions This study focuses on pressure distribution and fluctuations, including power spectral density of pressure fluctuations, on the top surface of a cubic building subjected to oblique turbulent boundary layers. In this configuration, conical vortices develop and substantially influence pressure characteristics on the top building surface. The study particularly focuses on the effects of turbulence in the incident flow. Three turbulent boundary layers of various turbulence characteristics were created experimentally in a boundary layer wind tunnel. The flow was characterized by the mean velocity, turbulence intensity, Reynolds shear stress, turbulence length scales, and power spectral density of velocity fluctuations, which were simultaneously recorded in all three directions. One cube with 0.2 m long edges was used as an experimental model. In all conducted experiments, the flow incidence angle was at 40◦ so as to create conical vortices above the top cube surface. The thickness of the turbulent boundary layers was ten times the cube height. For the purpose of taking steady pressure measurements, on each of the vertical cube surfaces there were 35 pressure taps and on the top surface there were 49 pressure taps. Unsteady surface pressures were measured at 15 points on the top cube surface. The conical vortices developed above the top surface of the wall-mounted cube are clearly observed as two zones characterized by strong suction. These two suction zones are underlying the cores of the two conical vortices. The conical vortex is larger downstream of the leading edge of the top cube surface that is oriented at a smaller angle to the main flow direction, i.e. closer to the case when the flow is perpendicular to the leading cube edge. The larger mean suction acts on the top cube surface in the less turbulent boundary layer, thus indicating that the conical vortices are stronger in the smooth flow than in the turbulent flow. With an increase in the turbulence in the incoming flow, the strong suction zones decrease in size with a simultaneous increase in the size of the weak suction zone on the central area of the top cube surface. This indicates that the conical vortices are generally larger in less turbulent (smoother) flows than in more turbulent flows. The conical vortices are situated closer to the respective edges of the top cube surface when subjected to a more turbulent incident flow. The profiles of the fluctuating pressure coefficient retain their general shape when turbulence is added to the flow upstream of the cube. However, the entire profile shifts so the fluctuating pressure coefficient is larger in a more turbulent flow. In fact, the added turbulence acts as an amplifier of pressure fluctuations on the top cube surface. The pressure fluctuations are larger at the surface underlying outer boundaries of the rotating conical vortex near the flow separation point. Further downstream of the conical vortex boundary, but still underneath the conical vortex, the fluctuating pressure coefficient decreases in magnitude, while the exhibited minimum corresponds to the surface area characterized by weak suction between two conical vortices in the central part of the top cube surface. In the same experiment, the fluctuating pressure coefficient underneath the conical vortex is three to four times larger than in the weak suction zone on the central area of the top cube surface. Close to the leading cube corner, the pressure spectra are dominated by a single low-frequency peak. This peak is larger than the one commonly observed on square prisms oriented perpendicularly to the incident flow, while its magnitude is larger in less turbulent incident flows. As the conical vortex develops, this primary peak weakens and a secondary peak at a higher reduced frequency emerges. The magnitude of the secondary peak is larger for less turbulent incident flows. There is a general trend of shifting the pressure spectra towards higher reduced frequencies in more turbulent incident flows. The effects of the conical vortex are stronger in the case of the surface area closer to the leading cube edge. Declaration of competing interest The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Acknowledgments Experiments were carried out at the former Chair of Fluid Mechanics, Department of Mechanical Engineering, Technical University of Munich (TUM), which was headed by Professor Emeritus Boris Laschka, whose kind support and advice along with his associates’ help are greatly appreciated. The study was funded by the Croatian Ministry of Science and Education, German Academic Exchange Service (DAAD) and the Croatian Academy of Sciences and Arts (HAZU), which support is gratefully acknowledged. References Balendra, T., Shah, D.A., Tey, K.L., Kong, S.K., 2002. Evaluation of flow characteristics in the NUS-HDB wind tunnel. J. Wind Eng. Ind. Aerodyn. 90, 675–688. Banks, D., 2013. The role of corner vortices in dictating peak wind loads on tilted flat solar panels mounted on large, flat roofs. J. Wind Eng. Ind. Aerodyn. 123, 192–201. Banks, D., Meroney, R.N., 2001a. A model of roof-top surface pressures produced by conical vortices: Model development. 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