Investigation of the elliptical resonant vibration of high-rise buildings induced by the oblique-downwind interference effects

ABSTRACT The interference effect is one of the difficult topics in wind engineering research themes. 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 effects of the neighboring building located downstream. This study intends to investigate several mentioned downstream interference effects based on the high-frequency-force-balance tests. Four different cross-sections are selected to understand how the adjacent building’s appearance interferes with the square principal building. Results show that the 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. Graphical Abstract


Introduction
High-rise buildings are the most remarkable structures in highly developed urban terrain.It is commonly seen that high-rise buildings locate in condensed areas and cause interference effects to each other.Figure 1 shows two closely located square cross-sectional high-rise buildings in Taipei City.The line of their geometric centers forms an almost 45° inclination arrangement.No matter what the most unfavorable wind direction is for these two highrise buildings, the interference effects from the upstream or the downstream potentially alter the design wind loads on them.According to the current codes, AIJ 2015 or GB 50009-2012, the interference effects have been mentioned to remind structural engineers to pay attention to the significant amplified wind loads.However, structural engineers might sometimes ignore the consideration of the resonant buffeting motion caused by the downstream buildings under low reduced velocities among those various interference mechanisms.
The phenomenon of resonant buffeting between two buildings was first mentioned by Cooper and Wardlaw (1971).When the vortex-shedding frequency from the upstream building meets the fundamental frequency of the downstream building located in the former's wake area, a strong resonant buffeting motion is expected to occur at lower reduced velocities, say 6.8 in the case of Cooper and Wardlaw (1971).The downstream interference effect was then pointed out by Bailey and Kwok (1985), indicating that when the interfering building locates at a specific downstream area and under a certain velocity range, a rhythmic elliptical oscillation for the channel space between the two buildings can be expected.The elliptical shape is in line with the two diagonally arranged buildings, so the specific area is also recognized as the oblique-(diagonal-) downwind location.In the case of Bailey and Kwok (1985), the critical location for the interfering building is (x/b, y/b) = (−1.5,1.22), where b is the building breadth, x and y are the along-wind and the across-wind distances to the principal building, respectively.The negative sign indicates the downstream area.From the conclusion of Bailey and Kwok (1985), with different building shapes and sizes, such an elliptical resonant vibration between two buildings shall still be observable at other critical locations and perhaps under different reduced velocities.Yahyai et al. (1992) later utilized the vibration test of an aeroelastic model to investigate the downstream interference effects.From their work, a distance of two to three times the building breadth seems to be the critical distance to the principal building in the downstream area to induce the same phenomenon indicated by Bailey and Kwok (1985).In addition to that, from the results of Yahyai et al. (1992), the downstream location right behind the principal building, (x/b, y/b) = (−2, 0), also seems 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 number has a minor effect on the downstream interference than the upstream interference.To enhance the interference 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 discussed by Lo, Kim, and Li (2016) and ( 2020) shows a significant amplification in the acrosswind response instead of the elliptical vibration in both along-wind and across-wind responses.Various interference mechanisms, like the theme in this study, have also been explored in other research works, such as Li and Ishihara (2021) and Wang et al. (2022).
By summarizing the above most related works, the downstream interference effects may be categorized into two mechanisms, one is the elliptical resonant motion at oblique-downwind locations by Bailey and Kwok (1985), and the other is the amplified across-wind response at downwind locations by Lo, Kim, and Li (2016) and (2020).Unfortunately, so far, there is no reference given to examine how different these interference mechanisms are.This study aims to distinguish the mechanisms of different downstream interference effects from previous works, including the elliptical resonant vibration and the amplified across-wind motion, and validate the rare occurrence of the former motion through a series of welldesigned experiments.Five specific interference locations are selected for interfering buildings in four different shapes.By changing the reduced velocity, windinduced responses are estimated from the measured overturning moments of the principal building based on the high-frequency-force balance test.Besides the downstream interference effects, the upstream interference effects are also investigated for the interfering buildings in different shapes.This study discusses the optimum oblique-downwind locations for interfering buildings in different shapes in terms of response spectra and resonant-component interference factors.The conclusion is then given for the occurring mechanisms of different downstream interference effects.

Experimental setting
The approaching flow is simulated in an atmospheric turbulent boundary layer wind tunnel at Tamkang University in Taiwan.The wind tunnel has a testing section of 2.2 m in width, 1.8 m in height, and 12.0 m in length, enabling the passive development of an ideal turbulent boundary layer flow with a length scale of 1/400.Wooden spires and roughness blocks are adequately equipped to simulate eddies in different sizes and good vertical wind profiles specified by the power law. Figure 2 shows the photo of the experimental setting of the principal and interfering buildings in square cross-sections inside the wind tunnel.In the photo, the interfering building is located at one of the selected oblique-upwind locations to generate the upstream interferences.
Figure 3 shows the vertical profiles of the simulated turbulent boundary layer flow, including the normalized mean wind speeds, turbulent intensities, and the estimated turbulent length scales.The simulated terrain is open country terrain with a power law index of 0.13.The models adopted in this study are all manufactured to have the same height of 60 cm, which is normalized to 0.46 with the simulated boundary layer height of 130 cm, Z g , inside the wind tunnel.Figure 4 shows the reduced wind speed spectrum at the model height, showing the consistency between the generated turbulences inside the wind tunnel and the Karman spectrum in the field.This study adopts different cross-section shapes for the interfering buildings and remains the principal building in the square cross-section.To ensure the subsequent comparisons meet this study's purpose, all interfering models are manufactured to have the same volume of 6,000 cm 3 and model height of 60 cm.The geometric information of all the models is listed in Table 1.
The mean wind speed at the model height is 9.2 m/s for all the measurements.The turbulent intensity ranges from 5% to 15% over the model height.The turbulent length scale is about 60 cm at the model 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 principal building model are measured through the   installation of the JR3 Universal Force-Moment Sensor System from Nitta Co. Figure 5 shows the coordinate system of the six base reactions, including three base shears and three overturning moments.The sampling rate is 1,000 Hz for each measurement at any interference location.The sampling length is assumed to be 180 seconds, long enough to meet the stable ensemble averaging requirement.To apply the highfrequency-force-balance (HFFB) tests, all models are rigid and light enough to avoid the high-frequency noise signals and not disturb the estimation of structural responses based on the spectral analysis approaches.Table 2 ensures that the experimental setting meets the similarity rules of the wind tunnel test and avoids the experimental limitation due to the model manufacture.
According to the Taiwan Wind Code(2015), the 50year-return-period design wind speeds for a 240m-high building range from 32.5 m/s to 65 m/s, which can be ideally transferred to reduced velocities from 4 to 8 in a 0.5 interval with the assumption of fundamental frequency equals 0.2 Hz based on Tamura (2006Tamura ( , 2012)).The target building's fundamental frequency of 0.2 Hz is then converted based on the estimated time-scale factors to be 11.5 Hz for U r = 8 and 22.9 Hz for U r = 4.The converted frequency range is relatively lower than the identified frequency of 62 Hz of the principal building model, which allows the correct calculation of resonant-component responses under all reduced velocities.The estimated time-scale factors also divide the continuous 180second measurements into at least 17 records of 10min samples in the field scale, ensuring ensemble stability in calculating force spectra and aerodynamic coefficients.The Reynolds number of the model in the wind tunnel is estimated to be 6.3 × 10 4 , fulfilling the essential requirement in AWES(2019) for a standard square building model.The Pitot tube is set up at the elevation of the model height, providing the reference velocity pressure for force coefficient normalization.It is worth noting that when the interfering building is in a circular cross-section, the flow field around it shall be recognized as in a sub-critical Reynolds number condition.The Reynolds number estimated by actual sizes should be in the 10 7 -10 9 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 simulated 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 selected to install interfering buildings in four different  shapes, while the principal building remains in the square cross-section.The red intersectional dots in Figure 6 are the chosen interference locations.This study ignores the direction effects caused by the different incidental winds.Both the windward faces of the buildings are perpendicular to the wind direction.In some references, for example, Kim, Tamura, and Yoshida (2011), the directional effect could be crucial for estimating local peak loading designs.

Response estimation based on the HFFB test
From the measured base forces and overturning moments of the principal building model in the wind tunnel tests, aerodynamic coefficients can be defined in a non-dimensional format for further investigations on wind loadings or wind-induced responses of buildings in the field scale.Equations (1) define the force and moment coefficients.
In Equations (1), F i t ð Þ and M i t ð Þ are measured instantaneous base forces and moments as indicated in Figure 4.For a high-rise building, F z is usually ignored, and M z is the twisting loading along the vertical z-axes of a high-rise building.The overturning moment M y is linearly related to the base force F x , as well as M x to F y .The force coefficient of C F i is calculated by normalizing the measured base force by the reference force, q H BH, where the velocity pressure q H ¼ 0:5ρ � U 2 H (pa) with ρ the air density in kg/m 3 and � U H the mean wind speed at the model height in m/s.BH is the projected area in m 2 of the windward face.Same as the force coefficient, the moment coefficient C M i is calculated by normalizing the measured overturning moment by the reference moment, q H BH 2 .
For a high-rise building under the assumption of a continuously mass-distributed linear system, the governing equation of the motion can be described by Equation (2) in terms of the space variable z and the time variable t.
where m, c, k the systematic mass, damping, and stiffness; u z; t ð Þ the lateral displacement against height z at time instant t, and f z; t ð Þ the external wind loading.Based on the orthogonal vibration mode shapes estimated from the eigenvalue analysis, Equation ( 3) can be further transformed to Equation ( 4), the generalized governing equation of the jth mode: where In Equation ( 3), ϕ j is the jth vibration mode and U j t ð Þ is the jth generalized coordinate.Normally for a high-rise building, the first model dominates its oscillation behavior when excited by earthquakes or winds.The highfrequency-force-balance test provides a convenient calculation of Equation ( 3) for response estimations.Since the building model is made in a rigid format, the first mode shape can be assumed to be linear, which is ϕ 1 ¼ z=H.The generalized force F j in Equation ( 3) is then written in the format of the overturning moment in Equation ( 4).In this study, M t ð Þ is the measured overturning moment M y for the estimation of the along-wind response x and is M x for the estimation of the across-wind response y.H is the model height.As a result, the governing equation of the motion for a high-rise building based on the HFFB test can be provided as Equation ( 5) in conjunction with Equation (2).
where � 1 the damping ratio is assumed for the principal building, � 1 ¼ 1% in this study, ω 1 the circular frequency of the first mode, ω 1 ¼ 2πn 0 in this study, M 1 the generalized mass of the first mode.This study assumes that the mass density is 150 kgf/m 3 .Equation ( 5) can be derived by the spectral analysis method through Equation ( 6) for the along-wind and acrosswind responses.
where The mechanical function H n ð Þ j j 2 is a function to indicate how amplified or reduced the dynamic response is when a structural system is excited by a dynamic loading.It is assumed to be identical in the along-wind and the acrosswind directions in this study because the principal building has a square cross-section.Same as Equation ( 5), n 0 represents the fundamental frequency of the first mode in both directions.� 1 represents the damping ratio of the first mode in both directions.In Equation ( 6), S x n ð Þ and S y n ð Þ are response spectra of the along-wind and the across-wind responses.
Based on the Davenport Chain, the response variance obtained by integrating the response spectra can be considered to consist of two components -the background component and the resonant component, as Equation ( 7).The background component can be derived from the approaching fluctuating wind based on the quasi-static assumption; the resonant component is estimated by the narrow-band assumption of a Gaussian distributed variable.
where C 0 M y and C 0 M x the fluctuating force coefficients of the along-wind and the across-wind directions; S C My n 0 ð Þ and S C Mx n 0 ð Þ the spectrum values at the fundamental frequency n 0 from force coefficient spectra.Table 3 is given to verify the acceptable precision of the response estimation method by Equation ( 7).In this table, the results from Equation ( 7) and the direct integration method of the first generalized mode under different chosen reduced velocities are compared to confirm a good agreement between the two methods.The error percentage between the two methods decreases as the 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 considered acceptable for the subsequent response estimation in this study.

Interference factor definition
The discussions are divided into two parts.In the first part, the interference effects on aerodynamic forces and estimated responses at five location series are investigated in Sections 3.2 and 3.3.In the second part, the downstream interference effect focuses on those chosen locations to examine the optimum areas to cause resonant vibration for different building shapes Section 3.4.In both parts, the principal building is in a square cross-section, while the interfering building is in four cross-sections, as indicated in Table 1 and Figure 6.The interference effects on aerodynamic forces are based on the normalized interference factors of measured force/moment coefficients; meanwhile, the interference effects on the estimated responses are based on reduced velocities ranging from 4 to 8 and Equation ( 7).
Interference factors for the aerodynamic forces and the estimated responses are determined by Equations ( 8) ~ (10).In Equation ( 10), an enveloped interference factor is defined to identify each location's most significant interfered case under all reduced velocities (Yu, Xie, and Gu 2018).

Interference effects on aerodynamic forces
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 close locations, say (x/B, y/B) = (1.5, 0) or (0, 1.5), interfering buildings in rectangular (R2.0 and R0.5) shapes are not installable since their model plate sizes will collide with the installation of the high-frequencyforce-balance sensor.

Static aerodynamic force
Figure 7 shows the interference factors of the mean along-wind forces.Among five location series, the interference effect on the mean along-wind force has a noticeable reduction effect at upwind locations, while in other series, nearly no effect is identified.The interfering building in the rectangular shape with a wider windward face and narrower side faces, the R2.0 case, essentially blocks the approaching wind force in the along-wind direction.At very close locations, for example, x/B < 4.0, the positive drag force turns negative since the principal building is submerged in the wake of the R2.0-interfering building.
The negative force acting on the windward face due to the wake of the R2.0-interfering building is even more significant than the negative force acting on the leeward face of the principal building itself.The same reduction and the sign switch can also be indicated in the interfering building in the square shape; however, it is not as significant as in the R2.0 case.The CIRand the R0.5-interfering buildings can reduce the mean along-wind force to nearly zero.

Along-wind fluctuating aerodynamic forces
Figure 8 shows the interference factors of the fluctuating along-wind forces coefficients.Like Figure 7, the upwind location series has the most significant effect on the force coefficients; however, it is not just the reduction effect.
When the interfering building is square and is located upstream of the principal building, the amplification effect is observed at (x/B, y/B) = (3.5 ~ 5, 0).To explain this amplification effect, Figure 9 shows the power spectra of the wind force coefficient of the along-wind fluctuating forces at some selected locations under the open country and suburban terrains.According to the authors' previous conclusion (Chen, Li, and Lo 2022), the windinduced response due to the open terrain is more apparent than the suburban and urban ones, especially in explaining the interference effect mechanisms.To concentrate on the subsequent discussion of the resonant buffeting, the open country terrain is the main simulated flow condition.The results of the suburban and urban terrains can be referred to Chen, Li, and Lo (2022).It is interesting to indicate that when the principal building is submerged in the wake of the interfering building, the windward face of the principal building is directly affected by the vortices in the wake.In the case of isolated square building case, the approaching turbulence will generally lower the Strouhal number.When the terrain changes from the open country, the suburban, to the urban terrain, the isolated square principal building has a Strouhal number changing from 0.096, 0.089, to 0.086 gradually (Chen, Li, and Lo 2022).The power spectra of the wind force coefficient at (x/B, y/B) = (3.5 ~ 5, 0) show small spectral peaks near the reduced vortexinduced frequency of 0.082, which is a little biased from 0.096 and is supposed to be the increased turbulence in between the two buildings.However, when it changes to the suburban terrain, with a higher approaching turbulent intensity, such a spectral peak is no longer seen, which means the wake structure behind the interfering building is also changed.For the interfering buildings in a circular or rectangular shape, such a vortex-induced amplification in the along-wind fluctuating forces is hardly seen, perhaps due to weak wake structures or different spaces for development.
The oblique-upwind location series also show an apparent amplification effect at (x/B, y/B) = (2.5 ~ 4, 0) for the SQ-interfering building.The vortex shed from the upstream SQ-interfering building hit or partially hit the windward face of the principal building to slightly amplify the along-wind fluctuating force.However, other interfering buildings do not have the same effect at the oblique-upwind location series.As for the side, oblique-downwind, and downwind location series, nearly no interference effect is found from the interfering buildings in all four shapes.

Across-wind fluctuating aerodynamic forces
Figure 10 shows the interference factors of the acrosswind fluctuating force coefficients.Different from Figures 7 and 8, the interfering buildings in four shapes produce various interference mechanisms.A reduction effect is generally seen at the upwind locations when the four interfering buildings are close to the principal building, say at x/B < 3.0.As the interfering building moves farther from the principal building, the SQ-and R2.0-interfering buildings show amplified across-wind fluctuating forces.In contrast, the CIR-and R0.5-interfering buildings maintain almost the same reduction effect as when they are close to the principal building.The distance of x/B = 12 seems insufficient to recover the zero-interference effect.At the obliqueupwind locations, the SQ-and R2.0-interfering buildings again produce apparent amplification effects when x/B = y/B ≥ 3.0 but do not last longer than x/ B = y/B = 6.0.For the CIR-and R0.5-interfering buildings, when the interfering building moves farther than x/B = y/B = 3.0, the across-wind fluctuating wind force is almost no different from the situation of the isolated principal building.Comparing the upwind and the oblique-upwind locations, the amplified interference effects at the oblique-upwind locations show a narrower area than at the upwind locations, indicating that the wake from the upstream interfering building strongly depends on the interference location as well as the building shapes.
The side locations have shown a reduction effect in the across-wind fluctuating wind force when the interfering building, no matter which shape, is close to the principal building, for example when x/B < 3.0.The R2.0-interfering has a longer distance for reduction effect than the other three because of its longer geometric shape in the across-wind direction.The oblique-downwind locations have a similar tendency as the side locations.The CIR-interfering building shows a less reduced effect in a shorter distance between two buildings among four interfering buildings.
The downwind locations have noticeable variations when the location of the interfering building is close or far from the principal building for four cases.For the SQinterfering building, a general reduction effect is found except for the location of (x/B, y/B) = (−2, 0), which shows the exact downstream interference mechanism mentioned in Lo, Kim, andLi (2016) and(2020).For the CIR-interfering building, the location of (x/B, y/ B) = (−1.5,0) has an amplified effect, which may be the same phenomenon as the SQ-interfering building at (−2, 0).However, a further examination based on a more detailed experiment is necessary.This amplification effect gradually reduces when the CIR-interfering building moves farther.It reaches the maximum reduction effect at (x/B, y/B) = (−3, 0).After this location, the interference effect again turns to an amplification effect at (x/B, y/B) = (−4, 0).Interestingly, the R2.0-interfering building has the same variation as the CIR-interfering building; however, all the interference factors shift upward and show only the amplification effect.The location of (−2, 0) has the largest amplification effect.As for the R0.5-interfering building, no regularity is found to vary with the downwind locations.

Interference effects on estimated responses
Wind-induced responses are estimated based on Equation ( 7) and the structural information mentioned in Section 2.2.In Table 2, those design wind speeds in the first row are referred to Taiwan Code for a 240 m high building.The magnitude range is similar to other international codes.Different velocity and time-scale factors are obtained by choosing different design wind speeds.Therefore, the fundamental structural frequency of the building can vary to show various reduced velocity conditions.The converted frequency, n_0, in the ninth row of Table 2, is then substituted to Equation ( 7) for response calculations.By doing so, interference factors are calculated, and the maximum is picked up for discussion.Figures 11 and 12 show the enveloped interference factor of estimated responses at five location series in the along-wind and acrosswind directions, respectively.Figure 13 shows results from other references to validate the observed interference effects in this study; however, only the two commonly compared locations are shown.

Along-wind estimated responses
At the upwind locations, the SQ-interfering building and the R2.0-interfering building produce amplification effects to the principal building's along-wind responses at all observed upwind locations.The maximum amplification is observed at (x/B, y/B) = (2, 0) for the SQ-interfering building under the reduced velocity of U r = 4.5.Compared to the same SQ-interfering building in Figure 8, the maximum effects occur at different locations, indicating that the resonant component significantly increases the along-wind vibration of the principal building when the SQ-interfering building is getting closer to it.In Figure 13(a), Bailey and Kwok (1985) and Huang and Gu (2005) show a similar tendency for the SQ-interfering building at the upwind locations under the reduced velocity of U r = 6, even their aspect ratios of testing building models are different.Unlike the SQ-interfering building, the maximum amplification for the R2.0-interfering building is observed at a more upstream location, (x/B, y/ B) = (8, 0), under the reduced velocity of U r = 7.This location does not show a relatively larger value.Instead, the amplification effects at almost all upwind locations for the R2.0-interfering building are pretty consistent.The same observation point as the SQinterfering building is that the resonant component contribution is significantly large when the interfering building is close to the principal building.The CIRinterfering building only has a least limited amplification effect at (x/B, y/B) = (1.5 ~ 2, 0), while the R0.5-interfering building is challenging to find any amplified or reduced effects.Examining the interference factors under all reduced velocities for four interfering buildings indicates that the estimated responses at upwind locations are not quite dependent on the reduced velocity, confirming the applicability of the enveloped interference factor in this study.
For the interfering buildings located at the obliqueupwind locations, the estimated responses of the principal building have a consistent enveloped interference factor variation with the interference factors in Figure 8.The SQ-interfering building and the R2.0-interfering building show apparent amplified along-wind fluctuating responses in the range of (x/B, y/B) = (2, 2) ~ (4,4).The location of (x/B, y/B) = (3, 3) indicates the maximum amplified effects for both interfering buildings under a reduced velocity of 5.5.The amplified effects caused by the CIR-interfering building and the R0.5-interfering building are not as apparent as the SQ-and the R2.0-interfering buildings.
Their maximum amplified effects occur at the location of (x/B, y/B) = (2.5, 2.5) under the reduced velocity of 6 and 4, respectively.The reduced velocity dependency is less significant for the estimated responses at all the oblique-upwind locations.In Figure 13, the comparisons at the oblique-upwind locations show similar tendencies to those in Bailey and Kwok (1985).
All the interfering buildings at the side locations do not have essential contributions from the resonant component of the along-wind estimated responses under all reduced velocities.Therefore, the variations of the enveloped interference factors are all around unity and look almost the same as the variations of the interference factors by simple aerodynamic forces in Figure 8.
In the figure of the oblique-downwind location, the SQ-interfering building and the R2.0-interfering building produce the maximum amplified responses of the principal building when they are at the locations of (x/B, y/B) = (−1.5,1.5) and (−2, 2), respectively.The former case occurs under a reduced velocity of U r = 5.5, while the latter occurs under U r = 8.The along-wind estimated responses shall be discussed later with the across-wind estimated responses for a general understanding of the elliptical resonant vibration induced by the obliquedownwind interference effects.
Nearly no apparent amplified estimated response of the principal building is found at the downwind locations by all the four interfering buildings under the investigated range of reduced velocity in this study.

Across-wind estimated responses
The across-wind responses of the principal building due to the presence of the interfering building in four different shapes are normalized with that of the isolated principal building and then plotted in Figure 12 by the enveloped interference factors.
When the interfering building is located at the upwind locations, the interfered across-wind responses of the principal building show different patterns for four shapes.The apparent amplification effects are indicated when the CIR-interfering building is located at (x/B, y/B) > 2.5.The most significant one is (x/B, y/B) = (6, 0) under a reduced velocity of U r = 7.5.The R0.5-interfering building has a consistent trend at similar locations but with smaller amplification effects.The location of (x/B, y/B) = (6, 0) is still the most prominent; however, the reduced velocity is decreased to U r = 6.5.When the interfering building is in a square shape, the principal building has a general amplification effect of EIF y = 1.4 in the across-wind responses at all observed upwind locations, which is slightly higher than the general amplification effect of EIF x = 1.3 in the along-wind responses.When the shape changes to the R2.0-interfering building, the enveloped interference factor varies more or less around unity, with no apparent interference effect.Compared to the interfered along-wind responses in Figure 11, the interference mechanisms for four interfering buildings are very different.When the SQ-interfering building is at upwind locations, it enlarges the principal building's responses in both the along-and the across-wind directions.
When the interfering building turns to the R2.0 shape, only the along-wind responses are increased at all the upwind locations.When the interfering building is in the CIR or the R0.5 shape, the amplification effect is identified in the across-wind responses, just the other way around.
For the interfering building at the oblique-upwind locations, the SQ-and the R2.0-interfering buildings produce amplified across-wind response of the principal building by 50%.The maximum amplification of the former occurs at (x/B, y/B) = (2.5, 2.5) and (x/B, y/ B) = (3, 3) for the latter.Combining the interfered along-wind and the across-wind responses from Figures 11 and 12, the two interfering buildings generate the commonly mentioned upstream interference effect due to the wake shed from the upstream building.However, when the wake structure of the upstream is weaker or narrower, for example, the wakes from the R0.5-and the CIR-interfering buildings, such an amplified effect only occurs in the along-wind responses with a much smaller amount.
Compared to the reference shown in Figure 13, the observations in this study at the upwind and obliqueupwind locations under the reduced velocity of U r = 6 are validated with a good agreement.
Unlike the almost un-interfered along-wind responses, the across-wind responses of the principal building are significantly amplified by the SQ-and the R2.0-interfering building at the side locations.The reduction effect is first identified for both interfering buildings closer to the principal building.As the interfering building moves farther from the principal building, the interference effect dramatically changes from reduction to amplification and soon reaches the maximum amplification effect at (x/B, y/B) = (0, 2.5) for the SQ-interfering building and at (x/B, y/B) = (0, 4) for the R2.0-interfering building, respectively.The CIR-and R0.5-interfering buildings have a slight amplification effect near (x/B, y/B) = (0, 2 ~ 2.5); however, the amplification effect soon decreases to zero when (x/B, y/B) > (0, 4).Generally speaking, the observed interference effects when the interfering building is located at the side locations are mostly due to the channel effect between two side-by-side buildings.
The along-wind response of the principal building is amplified when the interfering building is at very close oblique-downwind locations, no matter which shape the interfering building is.The across-wind response is amplified differently and significantly depends on the interfering building's shape.The maximum amplification occurs at (x/B, y/B) = (−2, 2) for the SQ-and the CIR-interfering buildings.The former has a larger amplification effect than the latter.The R2.0-interfering building produces a similar but smaller amplification effect than the SQ-interfering building.The maximum amplification occurs at (x/B, y/B) = (−2.5,2.5), a bit farther from the SQ-interfering building.The R0.5-interfering building produces its maximum amplification effect at (x/B, y/B) = (−2, 2); however, from its tendency and compared with the other three shapes, it seems the R0.5-interfering building has not yet reached its maximum amplification effect.The observations at the oblique-downwind locations imply that it may be interesting to dig further into the interference effect at those regions in the distance smaller than two times the building breadth.
As for the downwind locations, no interference effect is identified in the across-wind responses for all four interfering buildings under the chosen reduced velocities.It is worth mentioning that, in Lo, Kim, and Li (2016) and ( 2020), the downstream interference effects due to the presence of the interfering building at the downwind locations are indicated under higher reduced velocities and the assumption that the Scruton number of the principal building is low.In this study, the estimated responses of the principal building are not expected to be significantly interfered with by the downwind interfering building.
It is interesting to point out that, except for the oblique-downwind locations, the interference factors of fluctuating responses in Figures 11 and 12 generally follow the interference factors of fluctuating forces in Figures 8 and 10.The interference factors of fluctuating responses are those picked-up enveloped maximum values under various reduced velocities, which means that except for the oblique-downwind locations, all other locations show a less velocity-dependent interference mechanism.As the reduced velocity changes, the response estimation based on the resonant components varies with the fundamental frequency.The response might be amplified even with a reduced spectral area for fluctuating wind force in Figure 10.The resonant motion of the principal building contributes significantly to the amplified responses as long as the vortices from the two buildings merge to form a more prominent spectral peak.The following section details the downstream interference effects under different reduced velocities.

Downstream interference effects
Comparing the oblique-downwind figure and the downwind figure in Figure 10 implies two different downstream interference effects.For the obliquedownwind locations, the across-wind aerodynamic forces are reduced due to the close existence of the interfering building, which effectively disturbs or collapses the vortex shed from the upstream building or the wake structure formation.For the downwind locations, the across-wind aerodynamic forces are potentially amplified at certain distances for different buildings, indicating that the disturbance from the straight downstream building does not necessarily reduce the wake behind the principal building.With the contribution from the resonance induced by the fundamental structural characteristics under different reduced velocities, the across-wind responses at the oblique-downwind locations are potentially amplified.This section focuses on the interfered responses at the oblique-downwind locations.Section 1 explains the difference between the two downstream mechanisms from the results by Bailey and Kwok (1985) and Lo, Kim, andLi (2016) and(2020).This section continues to discuss how to identify the mechanism of the former, the elliptical motion induced from the rhythmic (a) SQ-interfering building at (-1.5, 1.22) under U r = 6.2 from Bailey and Kwok (1985) (b) SQ-interfering building at (-1.5, 1.5) under U r = 5.5 (from this study) vibration of two buildings, in terms of force spectra and resonance-derived interference factor.As indicated in Bailey and Kwok (1985), when the force spectra in the along-wind and the across-wind directions both show spectral humps or peaks at the same vortex-induced frequency, the resonant responses in the two directions will be simultaneously amplified under the same reduced velocity.If drawn in terms of the along-wind and the across-wind displacements, the interfered responses will behave like an inclined hollow elliptical trajectory motion due to the high correlation between the responses in two directions.Figure 14 shows two examples of force spectra, one is the extracted case from Bailey and Kwok (1985), and the other one is from the location of (x/B, y/ B) = (−1.5,1.5) under the reduced velocity U r = 5.5.Although these two cases are slightly different regarding the aspect ratio of buildings, the observed location, and the reduced velocity, it is supposed to indicate the same interference effect mechanism.The vertical red line in Figure 14(b) represents the reduced velocity of U r = 5.5.From the interfered along-wind and acrosswind force spectra, it is reasonable to conclude a more significant portion of resonant responses compared to the background responses if multiplied by the mechanical functions.The elliptical motion shall behave like an inclined hollow elliptical trajectorya combination of two harmonic movements with two different amplitudes but in the same frequency.
Two other interesting examples occur at the locations of (x/B, y/B) = (−2, 2) under the reduced velocity of U r = 6.5 and (x/B, y/B) = (−2.5,2.5) under the reduced velocity of U r = 8 for the SQ-interfering building.Although the spectral peak is still indicatable in the former example, the one in the along-wind force spectrum occupies a relatively smaller portion than the background.On the other hand, the spectral peak in the across-wind spectrum is the dominant feature.When the reduced velocity approaches 6.5 at this location, a resonant movement is expected in the acrosswind response, while the along-wind movement is expected to be a buffeting response due to the background fluctuating force.Consequently, the overall motion of the principal building is supposed to be a standing elliptical trajectory without a hollow core part.The same motion was shown at the location (−2, 2) under the reduced velocity of U r = 6.8 in Lo, Kim, and Li (2016).The two elliptical motions in Bailey and Kwok (1985) and Lo, Kim, and Li (2016) are different because of their different portion ratios of the background and the resonant responses.When the SQ-interfering building moves farther to (x/B, y/ B) = (−2.5,2.5), the resonant response is still dominant in the across-wind direction; however, there is no interference-induced spectral peak in the along-wind force spectrum for the possibility of any resonant response to occur.Figure 16 shows the hand-drawn schemes of an inclined hollow elliptical motion and a standing elliptical motion.
Examining the other three interfering buildings shows that all the along-wind force spectra do not indicate any resonance-induced spectral peak at the oblique-downwind locations, which means no inclined hollow elliptical motion is expected.However, in some cases, as long as the spectral peak caused by the interfering building is inspired, a standing elliptical motion in the across-wind response is still possible.To simplify the identification of the inclined hollow elliptical motion, Equation ( 11) is given to show the estimated resonant-component response.Equation ( 12) is then given as the resonant-component interference factor based on Equation (11).
Figure 17 shows the resonant-component interference factor varying with the reduced velocity at all the oblique-downwind locations for the SQ-interfering building cases.Figure 17 correctly identifies the amplified resonant responses in the along-wind and the across-wind directions for the case with the SQinterfering building located at (−1.5, 1.5) under Ur = 5.5.The other two examples of the standing elliptical motion are also identified in Figure 16.Unfortunately, at the oblique-downwind locations in this study, the CIR-, R2.0-, or R0.5-interfering building cannot interfere with the principal building to generate the elliptical resonant motion.Moreover, the identified location and reduced velocity for the case with the SQ-interfering building to occur in such a motion is not the exact location and reduced velocity, as mentioned in Bailey and Kwok (1985).The optimum location of (−1.5, 1.22) under U r = 6.2 in Bailey and Kwok (1985) suggests that the optimum locations for the other three interfering buildings of different shapes may not be at the oblique-downwind locations.Instead, a larger oblique-downwind area between the side locations and the downwind locations is required for future searching for their optimum locations.Table 4 lists the augmented experiments to examine the elliptical resonant vibration by switching the buildings' principal and interfering roles.The first label of the combination name means the interfering building's cross-section shape, and the second label means the principal building's cross-section shape.Among all these combinations, only the SQ-R2.0combination, the principal building in an R2.0 shape with the interfering building in a square shape shows a standing elliptical motion at the (x/B, y/B) = (−1.5,1.5) location under the reduced velocity of U r = 6, similar to that in Figure 15 (a).Other combinations at the oblique-downwind locations do not show any features in their force spectra that fulfill the conditions required to inspire an inclined hollow elliptical or standing elliptical motion.
From the above observations, the standing elliptical motion is found to occur a bit easier than the inclined hollow elliptical motion since the latter requires the same shedding vortex frequencies from both buildings and proper space between the two buildings for motion development.The shedding vortex in the along wind force spectra is the crucial element to validate the existence of the inclined hollow elliptical motion.Therefore, it may not be surprising that, with very different flow separation phenomena, the combinations of the SQ and CIR buildings have little possibility of inspiring such a motion.While for the varieties of the SQ and the two rectangular buildings, it might be worth examining further the relative distance and the inclinational angle of the align line of the two buildings.In this study, the geometric centers assign the relative position of two buildings.However, due to the different cross-sections, the actual space for the flow between the two buildings may be slightly different.Unfortunately, this study cannot cover all crosssections to conclude an empirical formula for the abovementioned variables.The novelty of this study lies in providing a suggestive explanation for distinguishing two different kinds of downstream interference effects.

Conclusions
This study discussed the effects of four different crosssectional interfering buildings at typical locations.The first part gives the variations of the interference factors for fluctuating forces and estimated responses in the along-wind and the across-wind directions.By examining these variations, the discrepancies in the obliquedownwind locations were found to be different from other locations.In the second part, the downstream interferences induced by the interfering buildings at the oblique-downwind locations were examined in  detail.A schematic diagram and a necessary condition for the rhythmic resonant motion, i.e., the inclined hollow elliptical motion, were given, along with potential reasons for those interfering buildings not being able to inspire such a motion.Here are some conclusions given as follows.
1. Cooper and Wardlaw (1971) first mentioned the downstream interference effect.Bailey and Kwok (1985) then pointed out a specific rhythmic vibration happening at one relative distance and under a specific low reduced velocity.Recently Lo, Kim, and Li (2016) and ( 2020) pointed out the existence of another downstream interference effect.This study conducted a series of wind tunnel experiments to point out the velocity-dependent features of the downstream interferences.In contrast, the upstream interference effects show a more location-dependent feature.
2. When both the principal building's along-wind and across-wind force spectra show a similar magnitude of vortex-induced spectral peaks at the same frequency, the first downstream interference mechanism, an inclined hollow elliptical motion, is expected to occur as long as the reduced velocity happens to inspire the resonance with a larger contribution than the background.The crucial condition to this motion is the spectrum shape of the along-wind forcea necessary and relatively large vortex-induced spectral peak as the one in the across-wind force spectrum.The resonant-component interference factor was provided to identify the occurrence of the inclined hollow elliptical motion.
3. When the vortex-induced spectral peak is only dominant in the across-wind force spectrum, the buffeting response in the along-wind direction and the harmonic motion in the across-wind direction together produce a non-hollow standing elliptical motion, recognized as the second mechanism at the obliquedownwind locations.The third mechanism is the amplification effect at close downwind locations, which Lo, Kim, and Li (2016) and ( 2020) investigated and ignored in this study.Comparing the response magnitude caused by this motion with the upstream interference effect suggests that the inclined hollow elliptical motion may not be crucial in practical design.The second or third mechanisms of the downstream interference effects may play a more critical role.
4. The shedding vortex in the along wind force spectra was found to be the crucial factor for the existence of the incline hollow elliptical motion.Since the SQ and the CIR buildings have very different wake structures after the flow separation, it is not surprising to conclude that there is only little possibility for the incline hollow elliptical motion happening in such combinations.Although this study could not cover all cross-sectional combinations to validate the motion, it is worth exploring in the future that not just the reduced velocity but the relative distance and the inclinational angle of the aligned line of the two buildings may play essential roles in inspiring the incline hollow elliptical motion.

Figure 1 .
Figure 1.Two closely located high-rise buildings in Taipei City.

Figure 3 .
Figure 3. Vertical profiles of the simulated turbulent boundary layer flow.

Figure 4 .
Figure 4. Reduced wind speed spectrum at the model height.Figure 2. Photo of the experimental setting.

Figure 2 .
Figure 4. Reduced wind speed spectrum at the model height.Figure 2. Photo of the experimental setting.

Figure 5 .
Figure 5. Coordinate system of base forces and overturning moments.

Figure 6 .
Figure 6.Interference locations of interest in this study.

Figure 7 .
Figure 7. Interference factors of the along-wind mean force coefficients.

Figure 8 .
Figure 8. Interference factors of the along-wind fluctuating force coefficients.

Figure 9 .
Figure 9. Power spectra of force coefficient of along-wind fluctuating forces for the SQ-interfering building cases.

Figure 10 .
Figure 10.Interference factors of the across-wind fluctuating force coefficients.

Figure 11 .
Figure 11.Interference factors of the along-wind fluctuating displacement coefficients.

Figure 12 .
Figure 12.Interference factors of the across-wind fluctuating displacement coefficients.

Figure 13 .
Figure 13.Comparison of interference factors of the estimated responses for the SQ-SQ arrangement.

Figure 14 .
Figure 14.Examples of an inclined hollow elliptical motion due to dominant two-directional resonant responses.

Figure 16 .
Figure 16.Hand-drawn schemes of elliptical motions induced by oblique-downwind interference effects.

Figure 17 .
Figure 17.Resonant-component interference factor for the SQ-interfering building cases.

Table 2 .
Scale factors under different assumed reduced velocities.

Table 1 .
Geometric information of all building models in this study.

Table 3 .
Comparison of response estimations based on Equation (7) and the direct integration method.

Table 4 .
Augmented experiments for elliptical resonant motion searching.