JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING https://doi.org/10.1080/13467581.2023.2172343 BUILDING STRUCTURES AND MATERIALS Investigation of the elliptical resonant vibration of high-rise buildings induced by the oblique-downwind interference effects a b c c Yi-Chao Li , Yuan-Lung Lo , Cheng-Wei Chen and Cheng-Hsin Chang a b Taiwan Building Technology Center, National Taiwan University of Science and Technology, Taipei, Taiwan; Department of Civil Engineering, National Taipei University of Technology, Taipei, Taiwan; Department of Civil Engineering, Tamkang University, Taipei, Taiwan ABSTRACT ARTICLE HISTORY Received 10 September 2022 The interference effect is one of the difficult topics in wind engineering research themes. Accepted 19 January 2023 Enormous publications have been made to understand its whole picture qualitatively and quantitatively in the past several decades. However, rare results focused on the interference KEYWORDS effects of the neighboring building located downstream. This study intends to investigate Interference effect; several mentioned downstream interference effects based on the high-frequency-force- downstream interference; balance tests. Four different cross-sections are selected to understand how the adjacent elliptical resonant vibration; building’s appearance interferes with the square principal building. Results show that the High-rise building; High- frequency-force-balance test interference effects induced by the downstream neighboring buildings can be categorized into two locations – the oblique-downwind location and the downwind location. The oblique- downwind interference effect generates two different motion shapes of the principal building, including the inclined hollow elliptical motion from the two-directional resonant vibration and the standing elliptical motion from the one-directional resonance. The downwind interference effect generates a similar one-directional resonance. Several force spectrum examples are given in this study to illustrate how the oblique-downwind interference effect mechanism forms at a specific distance and reduced velocity. In addition, the results from the augmented experiment with more combinations of two buildings suggest more efforts are necessary for inspiring the phenomenon of an inclined hollow elliptical motion. 1. Introduction most unfavorable wind direction is for these two high- rise buildings, the interference effects from the upstream High-rise buildings are the most remarkable structures in highly developed urban terrain. It is commonly seen that or the downstream potentially alter the design wind loads on them. According to the current codes, AIJ 2015 high-rise buildings locate in condensed areas and cause or GB 50009–2012, the interference effects have been interference effects to each other. Figure 1 shows two closely located square cross-sectional high-rise buildings mentioned to remind structural engineers to pay atten- tion to the significant amplified wind loads. However, in Taipei City. The line of their geometric centers forms an structural engineers might sometimes ignore the almost 45° inclination arrangement. No matter what the CONTACT Yi-Chao Li firstname.lastname@example.org; email@example.com Department of Civil Engineering, National Taipei University of Technology, Taipei, Taiwan © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the Architectural Institute of Japan, Architectural Institute of Korea and Architectural Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 Y.-L. LO ET AL. significant to amplify the across-wind response of the principal building. However, the discussions regarding these downstream locations farther than x/b < −2 were not made in detail. Lo, Kim, and Li (2016) and (2020) then continued investigating the downstream interference effects through the high-frequency-force balance and vibration tests. In their results, downstream interference effects are mainly velocity-dependent. The Scruton num- ber has a minor effect on the downstream interference than the upstream interference. To enhance the interfer- ence mechanism of the downstream interference effect, the CFD simulation technique was also utilized to explain how the flow interaction within the narrow space between the two buildings. Compared to Bailey and Kwok (1985), the downstream interference effect dis- cussed by Lo, Kim, and Li (2016) and (2020) shows a significant amplification in the across-wind response Figure 1. Two closely located high-rise buildings in Taipei City. instead of the elliptical vibration in both along-wind and across-wind responses. Various interference mechan- consideration of the resonant buffeting motion caused isms, like the theme in this study, have also been by the downstream buildings under low reduced velo- explored in other research works, such as Li and cities among those various interference mechanisms. Ishihara (2021) and Wang et al. (2022). The phenomenon of resonant buffeting between two By summarizing the above most related works, the buildings was first mentioned by Cooper and Wardlaw downstream interference effects may be categorized into (1971). When the vortex-shedding frequency from the two mechanisms, one is the elliptical resonant motion at upstream building meets the fundamental frequency of oblique-downwind locations by Bailey and Kwok (1985), the downstream building located in the former’s wake and the other is the amplified across-wind response at area, a strong resonant buffeting motion is expected to downwind locations by Lo, Kim, and Li (2016) and (2020). occur at lower reduced velocities, say 6.8 in the case of Unfortunately, so far, there is no reference given to exam- Cooper and Wardlaw (1971). The downstream interfer- ine how different these interference mechanisms are. ence effect was then pointed out by Bailey and Kwok This study aims to distinguish the mechanisms of differ - (1985), indicating that when the interfering building ent downstream interference effects from previous locates at a specific downstream area and under works, including the elliptical resonant vibration and the a certain velocity range, a rhythmic elliptical oscillation amplified across-wind motion, and validate the rare for the channel space between the two buildings can be occurrence of the former motion through a series of well- expected. The elliptical shape is in line with the two designed experiments. Five specific interference loca- diagonally arranged buildings, so the specific area is tions are selected for interfering buildings in four differ - also recognized as the oblique- (diagonal-) downwind ent shapes. By changing the reduced velocity, wind- location. In the case of Bailey and Kwok (1985), the critical induced responses are estimated from the measured location for the interfering building is (x/b, y/b) = (−1.5, overturning moments of the principal building based 1.22), where b is the building breadth, x and y are the on the high-frequency-force balance test. Besides the along-wind and the across-wind distances to the princi- downstream interference effects, the upstream interfer- pal building, respectively. The negative sign indicates the ence effects are also investigated for the interfering build- downstream area. From the conclusion of Bailey and ings in different shapes. This study discusses the Kwok (1985), with different building shapes and sizes, optimum oblique-downwind locations for interfering such an elliptical resonant vibration between two build- buildings in different shapes in terms of response spectra ings shall still be observable at other critical locations and and resonant-component interference factors. The con- perhaps under different reduced velocities. clusion is then given for the occurring mechanisms of Yahyai et al. (1992) later utilized the vibration test of an different downstream interference effects. aeroelastic model to investigate the downstream inter- ference effects. From their work, a distance of two to three times the building breadth seems to be the critical 2. Experimental setting and response distance to the principal building in the downstream area estimation to induce the same phenomenon indicated by Bailey and 2.1. Experimental setting Kwok (1985). In addition to that, from the results of Yahyai et al. (1992), the downstream location right behind the The approaching flow is simulated in an atmospheric principal building, (x/b, y/b) = (−2, 0), also seems turbulent boundary layer wind tunnel at Tamkang JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 3 University in Taiwan. The wind tunnel has a testing manufactured to have the same height of 60 cm, section of 2.2 m in width, 1.8 m in height, and 12.0 m which is normalized to 0.46 with the simulated bound- in length, enabling the passive development of an ary layer height of 130 cm, Z , inside the wind tunnel. ideal turbulent boundary layer flow with a length Figure 4 shows the reduced wind speed spectrum at scale of 1/400. Wooden spires and roughness blocks the model height, showing the consistency between are adequately equipped to simulate eddies in differ - the generated turbulences inside the wind tunnel and ent sizes and good vertical wind profiles specified by the Karman spectrum in the field. This study adopts the power law. Figure 2 shows the photo of the experi- different cross-section shapes for the interfering build- mental setting of the principal and interfering build- ings and remains the principal building in the square ings in square cross-sections inside the wind tunnel. In cross-section. To ensure the subsequent comparisons the photo, the interfering building is located at one of meet this study’s purpose, all interfering models are the selected oblique-upwind locations to generate the manufactured to have the same volume of 6,000 cm upstream interferences. and model height of 60 cm. The geometric information Figure 3 shows the vertical profiles of the simulated of all the models is listed in Table 1. turbulent boundary layer flow, including the normal- The mean wind speed at the model height is 9.2 m/s ized mean wind speeds, turbulent intensities, and the for all the measurements. The turbulent intensity estimated turbulent length scales. The simulated ter- ranges from 5% to 15% over the model height. The rain is open country terrain with a power law index of turbulent length scale is about 60 cm at the model 0.13. The models adopted in this study are all height, about 1/3 of the height of the wind tunnel, as commonly seen in a conventional suck-in type wind tunnel facility. The overturning moments of the princi- pal building model are measured through the Figure 4. Reduced wind speed spectrum at the model height. Figure 2. Photo of the experimental setting. Figure 3. Vertical profiles of the simulated turbulent boundary layer flow. 4 Y.-L. LO ET AL. Table 1. Geometric information of all building models in this study. Interfering Principal Square Square Circular Rectangular Rectangular Unit: cm (SQ) (SQ) (CIR) (R0.5) (R2.0) Breadth, B 10 10 11.3 7.1 14.1 Depth, D 10 10 (Diameter) 14.1 7.1 Height, H 60 60 60 60 60 installation of the JR3 Universal Force-Moment Sensor (2006, 2012). The target building’s fundamental fre- System from Nitta Co. Figure 5 shows the coordinate quency of 0.2 Hz is then converted based on the system of the six base reactions, including three base estimated time-scale factors to be 11.5 Hz for U = 8 shears and three overturning moments. The sampling and 22.9 Hz for U = 4. The converted frequency range rate is 1,000 Hz for each measurement at any interfer- is relatively lower than the identified frequency of ence location. The sampling length is assumed to be 62 Hz of the principal building model, which allows 180 seconds, long enough to meet the stable ensem- the correct calculation of resonant-component ble averaging requirement. To apply the high- responses under all reduced velocities. The estimated frequency-force-balance (HFFB) tests, all models are time-scale factors also divide the continuous 180- rigid and light enough to avoid the high-frequency second measurements into at least 17 records of 10- noise signals and not disturb the estimation of struc- min samples in the field scale, ensuring ensemble tural responses based on the spectral analysis stability in calculating force spectra and aerodynamic approaches. Table 2 ensures that the experimental coefficients. The Reynolds number of the model in the setting meets the similarity rules of the wind tunnel wind tunnel is estimated to be 6.3 × 10 , fulfilling the test and avoids the experimental limitation due to the essential requirement in AWES(2019) for a standard model manufacture. square building model. The Pitot tube is set up at the According to the Taiwan Wind Code(2015), the 50- elevation of the model height, providing the reference year-return-period design wind speeds for a 240- velocity pressure for force coefficient normalization. It m-high building range from 32.5 m/s to 65 m/s, is worth noting that when the interfering building is in which can be ideally transferred to reduced velocities a circular cross-section, the flow field around it shall be from 4 to 8 in a 0.5 interval with the assumption of recognized as in a sub-critical Reynolds number con- fundamental frequency equals 0.2 Hz based on Tamura dition. The Reynolds number estimated by actual sizes 7 9 should be in the 10 –10 range in the field scale, which is rarely achievable in a conventional wind tunnel simulation. This study attempts to increase the Reynolds number by covering the model’s appearance with a delicate rough skin layer. In addition, a mediate level of approaching turbulence could help stabilize the flow pattern around the curved geometric model in wind tunnel tests (Cheng and Fu 2010). The simu- lated turbulent flow in this study provides a 12–15% turbulence intensity over the model height range, enhancing the stable flow field generated from the circular interfering building. In this study, five interference location series are Figure 5. Coordinate system of base forces and overturning selected to install interfering buildings in four different moments. Table 2. Scale factors under different assumed reduced velocities. Design wind speed at building height U (m/s), H = 240 m 64 60 56 52 48 44 40 36 32 H, field Fundamental frequency n (Hz) 0.2 0, field Building breadth, B (m) 40 Reduced velocity U 8 7.5 7 6.5 6 5.5 5 4.5 4 Mean wind speed at the model height, U (m/s) 9.2 H, lab Velocity scale factor λ 0.144 0.153 0.164 0.177 0.192 0.209 0.230 0.256 0.288 Length scale factor λ 0.0025 −2 Time scale factor λ (x10 ) 1.739 1.630 1.522 1.413 1.304 1.196 1.087 0.978 0.870 Converted frequency n (Hz) 11.5 12.3 13.1 14.2 15.3 16.7 18.4 20.4 22.9 0, lab Model frequency n (Hz) 62 S, lab Sampling length (seconds) 180 Segment number of 10-min sample 17 18 19 21 23 25 27 30 34 JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 5 shapes, while the principal building remains in the coefficient, the moment coefficient C is calculated square cross-section. The red intersectional dots in by normalizing the measured overturning moment Figure 6 are the chosen interference locations. This by the reference moment, q BH . study ignores the direction effects caused by the dif- For a high-rise building under the assumption of ferent incidental winds. Both the windward faces of the a continuously mass-distributed linear system, the buildings are perpendicular to the wind direction. In governing equation of the motion can be described some references, for example, Kim, Tamura, and by Equation (2) in terms of the space variable z and the Yoshida (2011), the directional effect could be crucial time variable t. for estimating local peak loading designs. mðzÞu €ðz; tÞþ cu _ðz; tÞþ kðzÞuðz; tÞ ¼ fðz; tÞ (2) where m, c, k the systematic mass, damping, and stiff - 2.2. Response estimation based on the HFFB test ness; uðz; tÞ the lateral displacement against height z at time instant t, and fðz; tÞ the external wind loading. From the measured base forces and overturning Based on the orthogonal vibration mode shapes esti- moments of the principal building model in the wind mated from the eigenvalue analysis, Equation (3) can tunnel tests, aerodynamic coefficients can be defined be further transformed to Equation (4), the generalized in a non-dimensional format for further investigations governing equation of the jth mode: on wind loadings or wind-induced responses of build- ings in the field scale. Equations (1) define the force € _ M UðtÞþ C UðtÞþ K UðtÞ ¼ F j j j j j j j and moment coefficients. where FðtÞ C ðtÞ ¼ i ¼ x; y; z (1) F T T T T M ¼ ϕ mϕ C ¼ ϕ cϕ K ¼ ϕ kϕ F ¼ ϕ f (3) q BH j j j j H j j j j j j j In Equation (3), ϕ is the jth vibration mode and UðtÞ is MðtÞ C ðtÞ ¼ i ¼ x; y; z M the jth generalized coordinate. Normally for a high-rise q BH building, the first model dominates its oscillation beha- In Equations (1), FðtÞ and MðtÞ are measured instan- i i vior when excited by earthquakes or winds. The high- taneous base forces and moments as indicated in frequency-force-balance test provides a convenient Figure 4. For a high-rise building, F is usually ignored, z calculation of Equation (3) for response estimations. and M is the twisting loading along the vertical z-axes z Since the building model is made in a rigid format, of a high-rise building. The overturning moment M is y the first mode shape can be assumed to be linear, linearly related to the base force F , as well as M to F . x x y which is ϕ ¼ z=H. The generalized force F in The force coefficient of C is calculated by normalizing F Equation (3) is then written in the format of the over- the measured base force by the reference force, q BH, H turning moment in Equation (4). In this study, MðtÞ is where the velocity pressure q ¼ 0:5ρU (pa) with ρ the measured overturning moment M for the estima- H y the air density in kg/m and U the mean wind speed tion of the along-wind response x and is M for the H x at the model height in m/s. BH is the projected area estimation of the across-wind response y. H is the in m of the windward face. Same as the force model height. Figure 6. Interference locations of interest in this study. 6 Y.-L. LO ET AL. z MðtÞ estimated by the narrow-band assumption of F ¼ fðz; tÞ ¼ (4) a Gaussian distributed variable. H H 2� � � � As a result, the governing equation of the motion for 2 q BH πn H 2 0 2 0 σ ¼ C þ S ðn Þ M C 0 a high-rise building based on the HFFB test can be x y M 2 y 4� ð2πn Þ M H 1 0 1 provided as Equation (5) in conjunction with � � � � q BH πn Equation (2). H 2 0 2 0 σ ¼ ðC Þ þ S ðn Þ (7) M C 0 y x Mx 4� ð2πn Þ M H 0 1 MðtÞ q BH € _ UðtÞþ 2� ω UðtÞþ ω UðtÞ ¼ ¼ C ðtÞ (5) 1 1 M 0 0 M H M where C and C the fluctuating force coefficients of 1 1 M M y x the along-wind and the across-wind directions; S ðn Þ C 0 where � the damping ratio is assumed for the princi- and S ðn Þ the spectrum values at the fundamental C 0 pal building, � ¼ 1% in this study, ω the circular 1 1 frequency n from force coefficient spectra. Table 3 is frequency of the first mode, ω ¼ 2πn in this study, 1 0 given to verify the acceptable precision of the response M the generalized mass of the first mode. This study estimation method by Equation (7). In this table, the assumes that the mass density is 150 kgf/m . Equation results from Equation (7) and the direct integration (5) can be derived by the spectral analysis method method of the first generalized mode under different through Equation (6) for the along-wind and across- chosen reduced velocities are compared to confirm wind responses. a good agreement between the two methods. The error q BH percentage between the two methods decreases as the S ðnÞ ¼ jHðnÞj S ðnÞ x C M reduced velocity increases since a higher reduced velocity leads to a better frequency resolution, so the estimation of the power spectra improves. Equation (7) shall be q BH S ðnÞ ¼ jHðnÞj S ðnÞ y C M considered acceptable for the subsequent response esti- mation in this study. where 2 3. Results and discussions jHðnÞj ¼ (6) � � � � � � 2 2 n n 3.1. Interference factor definition 1 þ 2� n n 0 0 The discussions are divided into two parts. In the first The mechanical function jHðnÞj is a function to indicate part, the interference effects on aerodynamic forces how amplified or reduced the dynamic response is when and estimated responses at five location series are a structural system is excited by a dynamic loading. It is investigated in Sections 3.2 and 3.3. In the second assumed to be identical in the along-wind and the across- part, the downstream interference effect focuses on wind directions in this study because the principal build- those chosen locations to examine the optimum ing has a square cross-section. Same as Equation (5), n areas to cause resonant vibration for different building represents the fundamental frequency of the first mode in shapes Section 3.4. In both parts, the principal building both directions. � represents the damping ratio of the is in a square cross-section, while the interfering build- first mode in both directions. In Equation (6), S ðnÞ and ing is in four cross-sections, as indicated in Table 1 and S ðnÞ are response spectra of the along-wind and the Figure 6. The interference effects on aerodynamic across-wind responses. S ðnÞ and S ðnÞ are moment C C My Mx forces are based on the normalized interference factors coefficient spectra corresponding to S ðnÞ and S ðnÞ. of measured force/moment coefficients; meanwhile, x y Based on the Davenport Chain, the response variance the interference effects on the estimated responses obtained by integrating the response spectra can be are based on reduced velocities ranging from 4 to 8 considered to consist of two components – the back- and Equation (7). ground component and the resonant component, as Interference factors for the aerodynamic forces and Equation (7). The background component can be derived the estimated responses are determined by Equations from the approaching fluctuating wind based on the (8) ~ (10). In Equation (10), an enveloped interference quasi-static assumption; the resonant component is factor is defined to identify each location’s most Table 3. Comparison of response estimations based on Equation (7) and the direct integration method. Reduced velocity U 8 7 6 5 4 σ Eq. (7) 0.0608 0.0412 0.0282 0.0195 0.0085 Direct Integration 0.0603 0.0407 0.0287 0.0176 0.0077 Error (%) 0.81 1.39 1.55 9.42 10.60 σ Eq. (7) 0.1970 0.1165 0.0663 0.0377 0.0183 Direct Integration 0.1932 0.1136 0.0641 0.0345 0.0168 Error (%) 1.99 2.45 3.29 8.47 9.01 JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 7 8 9 � � significant interfered case under all reduced velocities < = y;interfered (Yu, Xie, and Gu 2018). EIF ¼ max: (10) : ; y;isolated V ¼4 8 M ;interfered IF ¼ (8) mean;x M ;isolated 3.2. Interference effects on aerodynamic forces M ;interfered IF ¼ rms;x M ;isolated Figures 7–9 indicate how interference effects vary with different interfering buildings based on Equations (8) and (9). It is worth noting in these figures that, for some M ;interfered IF ¼ (9) rms;y close locations, say (x/B, y/B) = (1.5, 0) or (0, 1.5), inter- M ;isolated fering buildings in rectangular (R2.0 and R0.5) shapes 8 9 are not installable since their model plate sizes will � � < = collide with the installation of the high-frequency- x;interfered EIF ¼ max: : σ ; force-balance sensor. ;isolated V ¼4 8 Upwind Oblique upwind 2 2 SQ-SQ SQ-SQ CIR-SQ CIR-SQ 1.5 R2.0-SQ R2.0-SQ 1.5 R0.5-SQ R0.5-SQ 0.5 0.5 0 2 4 6 8 10 12 0 2 4 6 x/B (y/B = 0) x/B = y/B Side SQ-SQ CIR-SQ R2.0-SQ 1.5 R0.5-SQ 0.5 0 2 4 6 y/B (x/B = 0) Oblique downwind Downwind 2 2 SQ-SQ SQ-SQ CIR-SQ CIR-SQ R2.0-SQ R2.0-SQ 1.5 1.5 R0.5-SQ R0.5-SQ 1 1 0.5 0.5 0 0 -4 -3 -2 -1 0 -4 -3 -2 -1 0 x/B = -y/B x/B (y/B = 0) Figure 7. Interference factors of the along-wind mean force coefficients. IF IF mean,x mean,x IF mean,x IF IF mean,x mean,x 8 Y.-L. LO ET AL. Upwind Oblique upwind 2 2 SQ-SQ SQ-SQ CIR-SQ CIR-SQ R2.0-SQ R2.0-SQ 1.5 1.5 R0.5-SQ R0.5-SQ 1 1 0.5 0.5 0 0 0 2 4 6 8 10 12 0 2 4 6 x/B (y/B = 0) x/B = y/B Side SQ-SQ CIR-SQ R2.0-SQ 1.5 R0.5-SQ 0.5 0 2 4 6 y/B (x/B = 0) Oblique downwind Downwind 2 2 SQ-SQ SQ-SQ CIR-SQ CIR-SQ R2.0-SQ R2.0-SQ 1.5 1.5 R0.5-SQ R0.5-SQ 1 1 0.5 0.5 0 0 -4 -3 -2 -1 0 -4 -3 -2 -1 0 x/B = -y/B x/B (y/B = 0) Figure 8. Interference factors of the along-wind fluctuating force coefficients. Figure 9. Power spectra of force coefficient of along-wind fluctuating forces for the SQ-interfering building cases. IF IF rms,x rms,x IF rms,x IF IF rms,x rms,x JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 9 between the two buildings. However, when it changes to 3.2.1. Static aerodynamic force the suburban terrain, with a higher approaching turbu- Figure 7 shows the interference factors of the mean along-wind forces. Among five location series, the lent intensity, such a spectral peak is no longer seen, which means the wake structure behind the interfering interference effect on the mean along-wind force has building is also changed. For the interfering buildings in a noticeable reduction effect at upwind locations, a circular or rectangular shape, such a vortex-induced while in other series, nearly no effect is identified. The interfering building in the rectangular shape with amplification in the along-wind fluctuating forces is hardly seen, perhaps due to weak wake structures or a wider windward face and narrower side faces, the different spaces for development. R2.0 case, essentially blocks the approaching wind force in the along-wind direction. At very close loca- The oblique-upwind location series also show an apparent amplification effect at (x/B, y/B) = (2.5 ~ tions, for example, x/B < 4.0, the positive drag force 4, 0) for the SQ-interfering building. The vortex shed turns negative since the principal building is sub- merged in the wake of the R2.0-interfering building. from the upstream SQ-interfering building hit or par- tially hit the windward face of the principal building to The negative force acting on the windward face due to slightly amplify the along-wind fluctuating force. the wake of the R2.0-interfering building is even more significant than the negative force acting on the lee- However, other interfering buildings do not have the ward face of the principal building itself. The same same effect at the oblique-upwind location series. As for the side, oblique-downwind, and downwind loca- reduction and the sign switch can also be indicated tion series, nearly no interference effect is found from in the interfering building in the square shape; how- ever, it is not as significant as in the R2.0 case. The CIR- the interfering buildings in all four shapes. and the R0.5-interfering buildings can reduce the mean along-wind force to nearly zero. 3.2.3. Across-wind fluctuating aerodynamic forces Figure 10 shows the interference factors of the across- 3.2.2. Along-wind fluctuating aerodynamic forces wind fluctuating force coefficients. Different from Figure 8 shows the interference factors of the fluctuating Figures 7 and 8, the interfering buildings in four shapes along-wind forces coefficients. Like Figure 7, the upwind produce various interference mechanisms. A reduction location series has the most significant effect on the force effect is generally seen at the upwind locations when coefficients; however, it is not just the reduction effect. the four interfering buildings are close to the principal When the interfering building is square and is located building, say at x/B < 3.0. As the interfering building upstream of the principal building, the amplification moves farther from the principal building, the SQ- and effect is observed at (x/B, y/B) = (3.5 ~ 5, 0). To explain R2.0-interfering buildings show amplified across-wind this amplification effect, Figure 9 shows the power spec- fluctuating forces. In contrast, the CIR- and tra of the wind force coefficient of the along-wind fluc - R0.5-interfering buildings maintain almost the same tuating forces at some selected locations under the open reduction effect as when they are close to the principal country and suburban terrains. According to the authors’ building. The distance of x/B = 12 seems insufficient to previous conclusion (Chen, Li, and Lo 2022), the wind- recover the zero-interference effect. At the oblique- induced response due to the open terrain is more appar- upwind locations, the SQ- and R2.0-interfering build- ent than the suburban and urban ones, especially in ings again produce apparent amplification effects explaining the interference effect mechanisms. To con- when x/B = y/B ≥ 3.0 but do not last longer than x/ centrate on the subsequent discussion of the resonant B = y/B = 6.0. For the CIR- and R0.5-interfering build- buffeting, the open country terrain is the main simulated ings, when the interfering building moves farther than flow condition. The results of the suburban and urban x/B = y/B = 3.0, the across-wind fluctuating wind force terrains can be referred to Chen, Li, and Lo (2022). It is is almost no different from the situation of the isolated interesting to indicate that when the principal building is principal building. Comparing the upwind and the submerged in the wake of the interfering building, the oblique-upwind locations, the amplified interference windward face of the principal building is directly effects at the oblique-upwind locations show affected by the vortices in the wake. In the case of a narrower area than at the upwind locations, indicat- isolated square building case, the approaching turbu- ing that the wake from the upstream interfering build- lence will generally lower the Strouhal number. When ing strongly depends on the interference location as the terrain changes from the open country, the suburban, well as the building shapes. to the urban terrain, the isolated square principal building The side locations have shown a reduction effect in has a Strouhal number changing from 0.096, 0.089, to the across-wind fluctuating wind force when the inter- 0.086 gradually (Chen, Li, and Lo 2022). The power spec- fering building, no matter which shape, is close to the tra of the wind force coefficient at (x/B, y/B) = (3.5 ~ 5, 0) principal building, for example when x/B < 3.0. The show small spectral peaks near the reduced vortex- R2.0-interfering has a longer distance for reduction induced frequency of 0.082, which is a little biased from effect than the other three because of its longer geo- 0.096 and is supposed to be the increased turbulence in metric shape in the across-wind direction. The oblique- 10 Y.-L. LO ET AL. downwind locations have a similar tendency as the building moves farther. It reaches the maximum reduc- side locations. The CIR-interfering building shows tion effect at (x/B, y/B) = (−3, 0). After this location, the a less reduced effect in a shorter distance between interference effect again turns to an amplification effect two buildings among four interfering buildings. at (x/B, y/B) = (−4, 0). Interestingly, the R2.0-interfering The downwind locations have noticeable variations building has the same variation as the CIR-interfering when the location of the interfering building is close or building; however, all the interference factors shift far from the principal building for four cases. For the SQ- upward and show only the amplification effect. The interfering building, a general reduction effect is found location of (−2, 0) has the largest amplification effect. except for the location of (x/B, y/B) = (−2, 0), which As for the R0.5-interfering building, no regularity is shows the exact downstream interference mechanism found to vary with the downwind locations. mentioned in Lo, Kim, and Li (2016) and (2020). For the CIR-interfering building, the location of (x/B, y/ 3.3. Interference effects on estimated responses B) = (−1.5, 0) has an amplified effect, which may be the same phenomenon as the SQ-interfering building at Wind-induced responses are estimated based on (−2, 0). However, a further examination based on Equation (7) and the structural information mentioned a more detailed experiment is necessary. This amplifica - in Section 2.2. In Table 2, those design wind speeds in tion effect gradually reduces when the CIR-interfering the first row are referred to Taiwan Code for a 240 m Upwind Oblique upwind 2 2 SQ-SQ SQ-SQ CIR-SQ CIR-SQ R2.0-SQ R2.0-SQ 1.5 1.5 R0.5-SQ R0.5-SQ 1 1 0.5 0.5 0 0 0 2 4 6 8 10 12 0 2 4 6 x/B (y/B = 0) x/B = y/B Side SQ-SQ CIR-SQ R2.0-SQ 1.5 R0.5-SQ 0.5 0 2 4 6 y/B (x/B = 0) Oblique downwind Downwind 2 2 SQ-SQ SQ-SQ CIR-SQ CIR-SQ R2.0-SQ R2.0-SQ 1.5 1.5 R0.5-SQ R0.5-SQ 1 1 0.5 0.5 0 0 -4 -3 -2 -1 0 -4 -3 -2 -1 0 x/B = -y/B x/B (y/B = 0) Figure 10. Interference factors of the across-wind fluctuating force coefficients. IF IF rms,y rms,y IF rms,y IF IF rms,y rms,y JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 11 high building. The magnitude range is similar to other 3.3.1. Along-wind estimated responses international codes. Different velocity and time-scale At the upwind locations, the SQ-interfering building factors are obtained by choosing different design wind and the R2.0-interfering building produce amplifica - speeds. Therefore, the fundamental structural fre- tion effects to the principal building’s along-wind quency of the building can vary to show various responses at all observed upwind locations. The max- reduced velocity conditions. The converted frequency, imum amplification is observed at (x/B, y/B) = (2, 0) for n_0, in the ninth row of Table 2, is then substituted to the SQ-interfering building under the reduced velocity Equation (7) for response calculations. By doing so, of U = 4.5. Compared to the same SQ-interfering interference factors are calculated, and the maximum building in Figure 8, the maximum effects occur at is picked up for discussion. Figures 11 and 12 show the different locations, indicating that the resonant com- enveloped interference factor of estimated responses ponent significantly increases the along-wind vibration at five location series in the along-wind and across- of the principal building when the SQ-interfering wind directions, respectively. Figure 13 shows results building is getting closer to it. In Figure 13(a), Bailey from other references to validate the observed inter- and Kwok (1985) and Huang and Gu (2005) show ference effects in this study; however, only the two a similar tendency for the SQ-interfering building at commonly compared locations are shown. the upwind locations under the reduced velocity of U Upwind Oblique upwind 3 3 SQ-SQ SQ-SQ CIR-SQ CIR-SQ 2.5 2.5 R2.0-SQ R2.0-SQ R0.5-SQ R0.5-SQ 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 2 4 6 8 10 12 0 2 4 6 x/B (y/B = 0) x/B = y/B Side SQ-SQ CIR-SQ 2.5 R2.0-SQ R0.5-SQ 1.5 0.5 0 2 4 6 y/B (x/B = 0) Oblique downwind Downwind 3 3 SQ-SQ SQ-SQ CIR-SQ CIR-SQ 2.5 2.5 R2.0-SQ R2.0-SQ R0.5-SQ R0.5-SQ 2 2 1.5 1.5 1 1 0.5 0.5 0 0 -4 -3 -2 -1 0 -4 -3 -2 -1 0 x/B = -y/B x/B (y/B = 0) Figure 11. Interference factors of the along-wind fluctuating displacement coefficients. EIF EIF x x EIF EIF EIF x x 12 Y.-L. LO ET AL. Upwind Oblique upwind 3 3 SQ-SQ SQ-SQ CIR-SQ CIR-SQ 2.5 2.5 R2.0-SQ R2.0-SQ R0.5-SQ R0.5-SQ 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 2 4 6 8 10 12 0 2 4 6 x/B (y/B = 0) x/B = y/B Side SQ-SQ CIR-SQ 2.5 R2.0-SQ R0.5-SQ 1.5 0.5 0 2 4 6 y/B (x/B = 0) Oblique downwind Downwind 3 3 SQ-SQ SQ-SQ CIR-SQ CIR-SQ 2.5 2.5 R2.0-SQ R2.0-SQ R0.5-SQ R0.5-SQ 2 2 1.5 1.5 1 1 0.5 0.5 0 0 -4 -3 -2 -1 0 -4 -3 -2 -1 0 x/B = -y/B x/B (y/B = 0) Figure 12. Interference factors of the across-wind fluctuating displacement coefficients. = 6, even their aspect ratios of testing building models interference factors under all reduced velocities for are different. Unlike the SQ-interfering building, the four interfering buildings indicates that the estimated maximum amplification for the R2.0-interfering build- responses at upwind locations are not quite depen- ing is observed at a more upstream location, (x/B, y/ dent on the reduced velocity, confirming the applic- B) = (8, 0), under the reduced velocity of U = 7. This ability of the enveloped interference factor in this location does not show a relatively larger value. study. Instead, the amplification effects at almost all upwind For the interfering buildings located at the oblique- locations for the R2.0-interfering building are pretty upwind locations, the estimated responses of the prin- consistent. The same observation point as the SQ- cipal building have a consistent enveloped interfer- interfering building is that the resonant component ence factor variation with the interference factors in contribution is significantly large when the interfering Figure 8. The SQ-interfering building and the building is close to the principal building. The CIR- R2.0-interfering building show apparent amplified interfering building only has a least limited amplifica - along-wind fluctuating responses in the range of (x/B, tion effect at (x/B, y/B) = (1.5 ~ 2, 0), while the y/B) = (2, 2) ~ (4,4). The location of (x/B, y/B) = (3, 3) R0.5-interfering building is challenging to find any indicates the maximum amplified effects for both amplified or reduced effects. Examining the interfering buildings under a reduced velocity of 5.5. EIF EIF y y EIF EIF EIF y y JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 13 Upwind Oblique upwind 1.6 1.4 1.4 1.2 1.2 1 1 0.8 along-wind(P.A. BAILEY & K.C.S. KWOK, 1985) along-wind(P.A. BAILEY & K.C.S. KWOK, 1985) along-wind(P. Huang & M. Gu, 2005) along-wind(this study) along-wind(this study) 0.8 0.6 0 2 4 6 8 10 1.5 2 2.5 3 3.5 4 x/B(y/B = 0) x/B = y/B (a) Along-wind Upwind Oblique upwind across-wind(P.A. BAILEY & K.C.S. KWOK, 1985) 1.6 1.4 across-wind(P. Huang & M. Gu, 2005) across-wind(this study) 1.4 1.2 1.2 1 1 0.8 across-wind(P.A. BAILEY & K.C.S. KWOK, 1985) across-wind(this study) 0.8 0.6 0 2 4 6 8 10 1.5 2 2.5 3 3.5 4 x/B(y/B = 0) x/B = y/B (b) Across-wind Figure 13. Comparison of interference factors of the estimated responses for the SQ-SQ arrangement. The amplified effects caused by the CIR-interfering discussed later with the across-wind estimated building and the R0.5-interfering building are not as responses for a general understanding of the ellipti- apparent as the SQ- and the R2.0-interfering buildings. cal resonant vibration induced by the oblique- Their maximum amplified effects occur at the location downwind interference effects. of (x/B, y/B) = (2.5, 2.5) under the reduced velocity of 6 Nearly no apparent amplified estimated response of and 4, respectively. The reduced velocity dependency the principal building is found at the downwind loca- is less significant for the estimated responses at all the tions by all the four interfering buildings under the oblique-upwind locations. In Figure 13, the compari- investigated range of reduced velocity in this study. sons at the oblique-upwind locations show similar tendencies to those in Bailey and Kwok (1985). 3.3.2. Across-wind estimated responses All the interfering buildings at the side locations do The across-wind responses of the principal building not have essential contributions from the resonant due to the presence of the interfering building in four component of the along-wind estimated responses different shapes are normalized with that of the iso- under all reduced velocities. Therefore, the variations lated principal building and then plotted in Figure 12 of the enveloped interference factors are all around by the enveloped interference factors. unity and look almost the same as the variations of the When the interfering building is located at the interference factors by simple aerodynamic forces in upwind locations, the interfered across-wind Figure 8. responses of the principal building show different pat- In the figure of the oblique-downwind location, terns for four shapes. The apparent amplification the SQ-interfering building and the R2.0-interfering effects are indicated when the CIR-interfering building building produce the maximum amplified responses is located at (x/B, y/B) > 2.5. The most significant one is of the principal building when they are at the loca- (x/B, y/B) = (6, 0) under a reduced velocity of U = 7.5. tions of (x/B, y/B) = (−1.5, 1.5) and (−2, 2), respec- The R0.5-interfering building has a consistent trend at tively. The former case occurs under a reduced similar locations but with smaller amplification effects. velocity of U = 5.5, while the latter occurs under The location of (x/B, y/B) = (6, 0) is still the most U = 8. The along-wind estimated responses shall be prominent; however, the reduced velocity is decreased IF IF V =6 V =6 r r IF IF V =6 V =6 r r 14 Y.-L. LO ET AL. to U = 6.5. When the interfering building is in a square side locations are mostly due to the channel effect shape, the principal building has a general amplifica - between two side-by-side buildings. tion effect of EIF = 1.4 in the across-wind responses at The along-wind response of the principal building is all observed upwind locations, which is slightly higher amplified when the interfering building is at very close than the general amplification effect of EIF = 1.3 in the oblique-downwind locations, no matter which shape along-wind responses. When the shape changes to the the interfering building is. The across-wind response is R2.0-interfering building, the enveloped interference amplified differently and significantly depends on the factor varies more or less around unity, with no appar- interfering building’s shape. The maximum amplifica - ent interference effect. Compared to the interfered tion occurs at (x/B, y/B) = (−2, 2) for the SQ- and the along-wind responses in Figure 11, the interference CIR-interfering buildings. The former has a larger mechanisms for four interfering buildings are very dif- amplification effect than the latter. The ferent. When the SQ-interfering building is at upwind R2.0-interfering building produces a similar but smaller locations, it enlarges the principal building’s responses amplification effect than the SQ-interfering building. in both the along- and the across-wind directions. The maximum amplification occurs at (x/B, y/B) = (−2.5, When the interfering building turns to the R2.0 2.5), a bit farther from the SQ-interfering building. The shape, only the along-wind responses are increased R0.5-interfering building produces its maximum ampli- at all the upwind locations. When the interfering build- fication effect at (x/B, y/B) = (−2, 2); however, from its ing is in the CIR or the R0.5 shape, the amplification tendency and compared with the other three shapes, it effect is identified in the across-wind responses, just seems the R0.5-interfering building has not yet the other way around. reached its maximum amplification effect. The obser- For the interfering building at the oblique-upwind vations at the oblique-downwind locations imply that locations, the SQ- and the R2.0-interfering buildings it may be interesting to dig further into the interfer- produce amplified across-wind response of the princi- ence effect at those regions in the distance smaller pal building by 50%. The maximum amplification of than two times the building breadth. the former occurs at (x/B, y/B) = (2.5, 2.5) and (x/B, y/ As for the downwind locations, no interference B) = (3, 3) for the latter. Combining the interfered effect is identified in the across-wind responses for all along-wind and the across-wind responses from four interfering buildings under the chosen reduced Figures 11 and 12, the two interfering buildings gen- velocities. It is worth mentioning that, in Lo, Kim, and Li erate the commonly mentioned upstream interference (2016) and (2020), the downstream interference effects effect due to the wake shed from the upstream build- due to the presence of the interfering building at the ing. However, when the wake structure of the downwind locations are indicated under higher upstream is weaker or narrower, for example, the reduced velocities and the assumption that the wakes from the R0.5- and the CIR-interfering buildings, Scruton number of the principal building is low. In such an amplified effect only occurs in the along-wind this study, the estimated responses of the principal responses with a much smaller amount. building are not expected to be significantly interfered Compared to the reference shown in Figure 13, the with by the downwind interfering building. observations in this study at the upwind and oblique- It is interesting to point out that, except for the upwind locations under the reduced velocity of U = 6 oblique-downwind locations, the interference factors are validated with a good agreement. of fluctuating responses in Figures 11 and 12 generally Unlike the almost un-interfered along-wind follow the interference factors of fluctuating forces in responses, the across-wind responses of the principal Figures 8 and 10. The interference factors of fluctuating building are significantly amplified by the SQ- and the responses are those picked-up enveloped maximum R2.0-interfering building at the side locations. The values under various reduced velocities, which means reduction effect is first identified for both interfering that except for the oblique-downwind locations, all buildings closer to the principal building. As the inter- other locations show a less velocity-dependent inter- fering building moves farther from the principal build- ference mechanism. As the reduced velocity changes, ing, the interference effect dramatically changes from the response estimation based on the resonant com- reduction to amplification and soon reaches the max- ponents varies with the fundamental frequency. The imum amplification effect at (x/B, y/B) = (0, 2.5) for the response might be amplified even with a reduced SQ-interfering building and at (x/B, y/B) = (0, 4) for the spectral area for fluctuating wind force in Figure 10. R2.0-interfering building, respectively. The CIR- and The resonant motion of the principal building contri- R0.5-interfering buildings have a slight amplification butes significantly to the amplified responses as long effect near (x/B, y/B) = (0, 2 ~ 2.5); however, the ampli- as the vortices from the two buildings merge to form fication effect soon decreases to zero when (x/B, y/B) > a more prominent spectral peak. The following section (0, 4). Generally speaking, the observed interference details the downstream interference effects under dif- effects when the interfering building is located at the ferent reduced velocities. JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 15 straight downstream building does not necessarily 3.4. Downstream interference effects reduce the wake behind the principal building. With Comparing the oblique-downwind figure and the the contribution from the resonance induced by the downwind figure in Figure 10 implies two different fundamental structural characteristics under different downstream interference effects. For the oblique- reduced velocities, the across-wind responses at the downwind locations, the across-wind aerodynamic oblique-downwind locations are potentially amplified. forces are reduced due to the close existence of the This section focuses on the interfered responses at the interfering building, which effectively disturbs or col- oblique-downwind locations. Section 1 explains the lapses the vortex shed from the upstream building or difference between the two downstream mechanisms the wake structure formation. For the downwind loca- from the results by Bailey and Kwok (1985) and Lo, Kim, tions, the across-wind aerodynamic forces are poten- and Li (2016) and (2020). This section continues to tially amplified at certain distances for different discuss how to identify the mechanism of the former, buildings, indicating that the disturbance from the the elliptical motion induced from the rhythmic (a) SQ-interfering building at (-1.5, 1.22) under U = 6.2 from Bailey and Kwok (1985) Along-wind U = 5.5@(-1.5,1.5) Across-wind U = 5.5@(-1.5,1.5) r r -2 -1 10 10 isolated isolated interfered interfered -2 -3 -3 -4 -4 10 10 -3 -2 -1 -3 -2 -1 10 10 10 10 10 10 n*B/U nB/U (b) SQ-interfering building at (-1.5, 1.5) under U = 5.5 (from this study) Figure 14. Examples of an inclined hollow elliptical motion due to dominant two-directional resonant responses. 16 Y.-L. LO ET AL. vibration of two buildings, in terms of force spectra different because of their different portion ratios of and resonance-derived interference factor. the background and the resonant responses. When As indicated in Bailey and Kwok (1985), when the the SQ-interfering building moves farther to (x/B, y/ force spectra in the along-wind and the across-wind B) = (−2.5, 2.5), the resonant response is still dominant directions both show spectral humps or peaks at the in the across-wind direction; however, there is no inter- same vortex-induced frequency, the resonant ference-induced spectral peak in the along-wind force responses in the two directions will be simultaneously spectrum for the possibility of any resonant response amplified under the same reduced velocity. If drawn in to occur. Figure 16 shows the hand-drawn schemes of terms of the along-wind and the across-wind displace- an inclined hollow elliptical motion and a standing ments, the interfered responses will behave like an elliptical motion. inclined hollow elliptical trajectory motion due to the Examining the other three interfering buildings high correlation between the responses in two direc- shows that all the along-wind force spectra do not tions. Figure 14 shows two examples of force spectra, indicate any resonance-induced spectral peak at the one is the extracted case from Bailey and Kwok (1985), oblique-downwind locations, which means no inclined and the other one is from the location of (x/B, y/ hollow elliptical motion is expected. However, in some B) = (−1.5, 1.5) under the reduced velocity U = 5.5. cases, as long as the spectral peak caused by the Although these two cases are slightly different regard- interfering building is inspired, a standing elliptical ing the aspect ratio of buildings, the observed location, motion in the across-wind response is still possible. and the reduced velocity, it is supposed to indicate the To simplify the identification of the inclined hollow same interference effect mechanism. The vertical red elliptical motion, Equation (11) is given to show the line in Figure 14(b) represents the reduced velocity of estimated resonant-component response. Equation U = 5.5. From the interfered along-wind and across- (12) is then given as the resonant-component interfer- wind force spectra, it is reasonable to conclude a more ence factor based on Equation (11). significant portion of resonant responses compared to �� � � q BH πn H 0 the background responses if multiplied by the σ ¼ S ðn Þ C 0 xR M 2 y 4� ð2πn Þ M H 1 mechanical functions. The elliptical motion shall 0 1 behave like an inclined hollow elliptical trajectory – 2�� � � a combination of two harmonic movements with two 2 q BH πn H 0 σ ¼ S ðn Þ (11) C 0 yR M different amplitudes but in the same frequency. 2 x 4� ð2πn Þ M H 1 0 1 Two other interesting examples occur at the loca- tions of (x/B, y/B) = (−2, 2) under the reduced velocity 0:5 S ðn Þ C ;interfered of U = 6.5 and (x/B, y/B) = (−2.5, 2.5) under the reduced IF ¼ rms;x;R 0:5 S ðn Þ velocity of U = 8 for the SQ-interfering building. 0 C ;isolated r M Although the spectral peak is still indicatable in the 0:5 former example, the one in the along-wind force spec- S ðn Þ C ;interfered Mx IF ¼ (12) trum occupies a relatively smaller portion than the rms;y;R 0:5 S ðn Þ C ;isolated background. On the other hand, the spectral peak in the across-wind spectrum is the dominant feature. Figure 17 shows the resonant-component interference When the reduced velocity approaches 6.5 at this loca- factor varying with the reduced velocity at all the tion, a resonant movement is expected in the across- oblique-downwind locations for the SQ-interfering wind response, while the along-wind movement is building cases. Figure 17 correctly identifies the ampli- expected to be a buffeting response due to the back- fied resonant responses in the along-wind and the ground fluctuating force. Consequently, the overall across-wind directions for the case with the SQ- motion of the principal building is supposed to be interfering building located at (−1.5, 1.5) under a standing elliptical trajectory without a hollow core Ur = 5.5. The other two examples of the standing part. The same motion was shown at the location elliptical motion are also identified in Figure 16. (−2, 2) under the reduced velocity of U = 6.8 in Lo, Unfortunately, at the oblique-downwind locations in Kim, and Li (2016). The two elliptical motions in Bailey this study, the CIR-, R2.0-, or R0.5-interfering building and Kwok (1985) and Lo, Kim, and Li (2016) are cannot interfere with the principal building to Table 4. Augmented experiments for elliptical resonant motion searching. Principal building Combination Name (Interfering – Principal) SQ CIR R2.0 R0.5 Interfering building SQ SQ-SQ SQ-CIR SQ-R2.0 SQ-R0.5 CIR CIR-SQ CIR-CIR × × R2.0 R2.0-SQ × × × R0.5 R0.5-SQ × × × JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 17 Along-wind U = 6.5@(-2.0,2.0) Across-wind U = 6.5@(-2.0,2.0) r r -2 -1 10 10 isolated isolated interfered interfered -2 -3 -3 -4 -4 10 10 -3 -2 -1 -3 -2 -1 10 10 10 10 10 10 n*B/U nB/U (a) SQ-interfering building at (-2, 2) under U = 6.5 Along-wind U = 8@(-2.5,2.5) Across-wind U = 8@(-2.5,2.5) r r -2 -1 10 10 isolated isolated interfered interfered -2 -3 -3 -4 -4 10 10 -3 -2 -1 -3 -2 -1 10 10 10 10 10 10 n*B/U nB/U (b) SQ-interfering building at (-2.5, 2.5) under U = 8 Figure 15. Examples of a standing elliptical motion due to a dominant one-direction resonant response. generate the elliptical resonant motion. Moreover, the locations. Instead, a larger oblique-downwind area identified location and reduced velocity for the case between the side locations and the downwind loca- with the SQ-interfering building to occur in such tions is required for future searching for their optimum a motion is not the exact location and reduced velo- locations. city, as mentioned in Bailey and Kwok (1985). The Table 4 lists the augmented experiments to exam- optimum location of (−1.5, 1.22) under U = 6.2 in ine the elliptical resonant vibration by switching the Bailey and Kwok (1985) suggests that the optimum buildings’ principal and interfering roles. The first label locations for the other three interfering buildings of of the combination name means the interfering build- different shapes may not be at the oblique-downwind ing’s cross-section shape, and the second label means 18 Y.-L. LO ET AL. Figure 16. Hand-drawn schemes of elliptical motions induced by oblique-downwind interference effects. Alongwind - Oblique downwind Acrosswind - Oblique downwind 2.5 2.5 (-1.5,1.5) (-1.5,1.5) (-2.0,2.0) (-2.0,2.0) 2 2 (-2.5,2.5) (-2.5,2.5) (-3.0,3.0) (-3.0,3.0) (-4.0,4.0) (-4.0,4.0) 1.5 1.5 1 1 0.5 0.5 4 5 6 7 8 4 5 6 7 8 Ur Ur Figure 17. Resonant-component interference factor for the SQ-interfering building cases. the principal building’s cross-section shape. Among all inclinational angle of the align line of the two build- these combinations, only the SQ-R2.0 combination, the ings. In this study, the geometric centers assign the principal building in an R2.0 shape with the interfering relative position of two buildings. However, due to the building in a square shape shows a standing elliptical different cross-sections, the actual space for the flow motion at the (x/B, y/B) = (−1.5, 1.5) location under the between the two buildings may be slightly different. reduced velocity of U = 6, similar to that in Figure 15 Unfortunately, this study cannot cover all cross- (a). Other combinations at the oblique-downwind loca- sections to conclude an empirical formula for the tions do not show any features in their force spectra abovementioned variables. The novelty of this study that fulfill the conditions required to inspire an inclined lies in providing a suggestive explanation for distin- hollow elliptical or standing elliptical motion. guishing two different kinds of downstream interfer- From the above observations, the standing elliptical ence effects. motion is found to occur a bit easier than the inclined hollow elliptical motion since the latter requires the 4. Conclusions same shedding vortex frequencies from both buildings and proper space between the two buildings for This study discussed the effects of four different cross- motion development. The shedding vortex in the sectional interfering buildings at typical locations. The along wind force spectra is the crucial element to first part gives the variations of the interference factors validate the existence of the inclined hollow elliptical for fluctuating forces and estimated responses in the motion. Therefore, it may not be surprising that, with along-wind and the across-wind directions. By examin- very different flow separation phenomena, the combi- ing these variations, the discrepancies in the oblique- nations of the SQ and CIR buildings have little possibi- downwind locations were found to be different from lity of inspiring such a motion. While for the varieties of other locations. In the second part, the downstream the SQ and the two rectangular buildings, it might be interferences induced by the interfering buildings at worth examining further the relative distance and the the oblique-downwind locations were examined in JOURNAL OF ASIAN ARCHITECTURE AND BUILDING ENGINEERING 19 detail. A schematic diagram and a necessary condition reduced velocity but the relative distance and the for the rhythmic resonant motion, i.e., the inclined inclinational angle of the aligned line of the two build- hollow elliptical motion, were given, along with poten- ings may play essential roles in inspiring the incline tial reasons for those interfering buildings not being hollow elliptical motion. able to inspire such a motion. Here are some conclu- sions given as follows. Acknowledgments 1. Cooper and Wardlaw (1971) first mentioned the downstream interference effect. Bailey and Kwok The authors would like to acknowledge Mr. Min-Wei Hsu’s (1985) then pointed out a specific rhythmic vibration assistance in the execution work of wind tunnel tests. The happening at one relative distance and under science research project 105-2221-E-032 −004-MY2 finan - cially supported the model manufacturing work in this a specific low reduced velocity. Recently Lo, Kim, and research. Li (2016) and (2020) pointed out the existence of another downstream interference effect. This study conducted a series of wind tunnel experiments to Disclosure statement point out the velocity-dependent features of the downstream interferences. In contrast, the upstream No potential conflict of interest was reported by the authors. interference effects show a more location-dependent feature. 2. When both the principal building’s along-wind ORCID and across-wind force spectra show a similar magni- Yi-Chao Li http://orcid.org/0000-0001-9520-8952 tude of vortex-induced spectral peaks at the same Yuan-Lung Lo http://orcid.org/0000-0003-1138-7506 frequency, the first downstream interference mechan- ism, an inclined hollow elliptical motion, is expected to occur as long as the reduced velocity happens to References inspire the resonance with a larger contribution than AWES (Australasian Wind Engineering Society). 2019. Quality the background. 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Journal of Asian Architecture and Building Engineering
– Taylor & Francis
Published: Feb 12, 2023
Keywords: Interference effect; downstream interference; elliptical resonant vibration; High-rise building; High-frequency-force-balance test