Experimental and theoretical study on the internal pressure induced by the transient local failure of low-rise building roofs

Abstract

A case study on the internal pressure induced by a local failure on the vulnerable gable roof of a low-rise building was extensively conducted experimentally and numerically. Five roof opening configurations were tested in the wind tunnel under three different boundary layer conditions, based on 1:40 scaled models. The effects of opening shape, opening position, opening ratio, building internal volume, and wind speed on peak transient and steady-state internal pressures were studied. The study results indicate that the peak transient and steady-state internal pressures and the corresponding transient overshoot ratio all increase with an increasing opening ratio. The peak steady-state internal pressure is little affected by the approaching wind speed; while the peak transient internal pressure coefficient shows a significant linear relationship with the wind speed. The coupling effect of vortex shedding and Helmholtz resonance in double building volume compensation situation may cause larger fluctuating internal pressure. Both the vortex shedding and Helmholtz resonance reduce the internal pressure coherence to some extent. The agreement between the numerical and experimental results is much better for the mean internal pressure than that for fluctuating internal pressure or peak internal pressure.

Keywords

Helmholtz resonance internal pressure roof opening vortex shedding wind tunnel test

Introduction

The wind-induced internal pressure caused by the opening on a building surface is a concerning issue of wind-resistant design. The opening could be functional or destructive. The roof cladding of low-rise buildings often causes roof openings suddenly and locally fails during a severe wind event due to strong suction or the strike of flying objects. The peak fluctuating internal pressure of the building increases instantly due to the high-speed influx of airflow. Under the combined effect of internal and external pressures, the roof corner, eaves, and ridge area near the gable wall are all likely to suffer a secondary damage.

Extensive experimental and theoretical studies have been done on the response characteristics of internal pressures induced by dominant wall openings (Chen et al., 2012; Guha et al., 2012; Holmes and Ginger, 2012; Oh et al., 2007; Quan et al., 2012a, 2012b; Tecle et al., 2013). Besides, different numerical simulation methods were also employed to investigated the internal pressure induced by wall openings, with the flow regimes around the opening being presented (Xing et al., 2018a, 2018b). However, the wall openings usually were not caused by the direct action of wind load, resulting in the characteristics of internal pressure different from that induced by roof opening (Xu and Lou, 2018). It has been known that the coupling effect of external and internal pressures on damaged roof may cause further failure of the entire light roof. Sharma and Richards (2005), Sharma (2008), and Rajasekharan (2013) studied the net pressure characteristics of roof cladding in a building with dominant wall openings. The main factor determining the internal pressure characteristics is the external pressure characteristic along opening edge, as reported by Ginger et al. (1997) in their field measurement investigation on a full-scale building. Sharma and Richards (1997), Guha et al. (2011), and Sharma (2013) indicated that the wind-induced internal pressure around a building cavity also depended on other factors including opening area, building volume, envelope flexibility, and background leakage.

Previous studies mainly focused on internal pressure response induced by wall openings, with little addressing wind induced roof openings which are more likely, especially occur at roof corners, eaves, and ridge area. Field measurements and post-disaster damage investigations manifested that local roofs of low-rise buildings were the most vulnerable to be damaged in windstorms (Song and Ou, 2010; Wang et al., 2018), which might lead to various opening on the roofs. Therefore, investigations of internal pressure induced by roof openings need to be carried out to provide useful information for the wind-resistant design of a low-rise building. Xu and Lou (2018) investigated the internal pressure induced by the openings on a hemi-ellipsoidal roof and then compared the results with those of wall openings. Dai et al. (2020b) also studied the internal and external pressure characteristics of a large span roof caused by gaps on the roof. Wang and Li (2015) studied the characteristics of the internal and external pressures induced by the openings at the corner of a flat roof of a low-rise building, where only the geometry of opening was considered. Moreover, the transient internal pressure due to the suddenly created opening on roof was not considered in their studies. The Helmholtz model, first proposed by Holmes (1979), has been widely adapted in analyzing the internal pressure resonance in a building cavity. While the vortex shedding across an opening due to the external excitation of grazing flow may create another type of internal pressure resonance phenomenon, as discussed first by Sharma and Richards (2003). As verified in Wang and Li (2015), under the circumstances of diagonal wind angle and large turbulence, a large increase in fluctuating internal pressure may occur due to the coupling interaction of vortex shedding and Helmholtz resonance of internal pressures.

In this study, extensive wind tunnel tests were conducted to study the characteristics of internal pressures induced by the openings on vulnerable roof areas of a low-rise building. This paper describes and discusses the following important issues: the characteristics of peak transient internal pressure; the influencing factors including opening size, opening shape, opening location, and wind speed; the characteristics of steady-state fluctuating internal pressure in frequency domain; and the comparison between predicted and experimental internal pressures.

Theoretical background of internal pressure response

The theoretical calculation of internal pressure consists of a simple quasi-steady analysis on mean pressure and an estimation on fluctuating pressure. The internal pressure theory of Holmes (1979) was adopted to estimate the mean wind-induced internal pressure, that is, the airflow flowing through a building cavity being an ideal steady incompressible flow. The formula for calculating the mean internal pressure can be derived based on the Bernoulli theory and mass conservation principle, as follows:

{\bar{C}}_{pi} = \frac{A_{w}^{2}}{A_{w}^{2} + A_{L}^{2}} {\bar{C}}_{pw} + \frac{A_{L}^{2}}{A_{w}^{2} + A_{L}^{2}} {\bar{C}}_{pl}

(1)

where $A_{w}$ is the total opening area on the windward side, $A_{L}$ is the total background leakage area, ${\bar{C}}_{pi}$ is the mean pressure coefficient of building interior surface, and ${\bar{C}}_{pw}$ and ${\bar{C}}_{pl}$ are respectively the mean pressure coefficients of external opening and leakage area. A set of rigid non-porous models with only one opening on each respective roof were considered, that is, $A_{L} = 0$ leading to ${\bar{C}}_{pi} = {\bar{C}}_{pw}$ as per equation (1).

The internal pressure fluctuation shows obvious randomness, which increases the complexity of theoretical estimation on the fluctuating internal pressure induced by building openings. The internal pressure response can be described by a second-order differential equation with nonlinear damping terms as first proposed by Holmes (1979) based on Helmholtz resonator theory, appearing as

\frac{ρ_{a} l_{e} V_{0}}{γ A_{0} P_{a}} {\overset{\cdot\cdot}{C}}_{pi} + \frac{ρ_{a} {qV}_{0}^{2}}{2 {(γ k A_{0} P_{a})}^{2}} | {\overset{\cdot}{C}}_{pi} | {\overset{\cdot}{C}}_{pi} + C_{pi} = C_{pe}

(2)

The theoretical undamped Helmholtz frequency of internal pressure resonance is given by

f_{H} = \frac{1}{2 π} \sqrt{\frac{γ A_{0} P_{a}}{ρ_{a} l_{e} V_{0}}}

(3)

In equation (2), $C_{pi}$ and $C_{pe}$ are the internal and external pressure coefficients respectively, calculated by $C_{pi} = p_{i} / q$ and $C_{pe} = p_{e} / q$ with $q = 0.5 ρ_{a} U_{r}^{2}$ being the reference height dynamic pressure (Note: $U_{r}$ is the mean wind velocity at reference height); $ρ_{a}$ is the density of air; $γ$ is the specific heat ratio related to adiabatic process (= 1.4); $P_{a}$ is the standard atmospheric pressure (= 101.325kPa); $l_{e}$ is the effective length of the air slug at the opening; $A_{0}$ and $V_{0}$ are respectively the opening area and the effective internal volume of building model; and k is the discharge coefficient = 0.07∼0.4 as proposed by Ginger et al. (2010).

In this study, $C_{pe}$ is taken as the area-average wind pressure coefficient determined based on the time history of external pressure signals at different measurement points around the orifice being strongly correlated with internal pressures; and $l_{e} = l_{0} + C_{I} \sqrt{A_{0} / π}$ where $C_{I}$ is the inertial coefficient (Vickery and Bloxham, 1992) indicated $C_{I} = \sqrt{π / 4}$ based on potential flow theory) and $l_{0}$ is the physical length of an orifice. Different researchers proposed various values for discharge coefficient k based on different fluid diffusion hypothesis.

Experimental details

Model details

Wind tunnel tests were conducted at Hunan University of Science and Technology (China) on a 1:40 scale model of a low-rise building with gable roof, considering different opening configurations. The scaled model is made of perspex to satisfy the pressure measurement requirement (Figure 1(a)). The model has the outer dimensions of 300 mm (length) × 200 mm (width) × 233 mm (roof ridge height), the respective inner dimensions of 280 mm × 180 mm × 213 mm, and the roof inclination angle of 18.4°. The internal volume of the model is approximate 0.01 m³ which is limited due to the space between internal and external envelops of the model. Therefore, the internal volume of the model needs to be expanded somewhat using equation (4) to keep the Helmholtz frequency similarity in accordance with the theory of Holmes (1979). An adjustable cuboid cavity was designed and installed under a turntable of wind tunnel, as shown in Figure 1(b).

\frac{V_{m}}{V_{f}} = \frac{{(l_{m} / l_{f})}^{3}}{{(U_{m} / U_{f})}^{2}}

(4)

Where, $V$ is the internal volume of the building, $l$ is the geometric length, $U$ is the mean wind speed, subscript m and f represent model and prototype, respectively.

Figure 1.

Wind tunnel experiments: (a) test model and (b) schematic of internal volume compensation.

Five roof opening configurations, four at the roof corner and one near the roof ridge being the potential damaged areas, were arranged according to the extreme suction distribution on the roof (Dai et al., 2017). The results of extreme suction partition under full wind directions were presented in a previous studies of the authors (Dai et al., 2020a). For the sake of brevity, repeated descriptions of the method for ensuring these five roof opening configurations were not presented herein. With different roof opening configurations, the layout of pressure measurement points on building roof differs partially. Nonetheless, the correspondence between internal and external pressure points is ensured. The ratio of opening area to the roof area containing the openings is defined as the opening ratio. For Configuration I, two opening shapes (square and rectangle) and six opening ratios (0.3%, 1.3%, 3%, 5.3%, 8.3%, and 12%) were considered in the experiment (Figure 2). For internal volume V = 4V₀ with V₀ being the model internal volume before compensation (≈0.011 m³), the internal and external pressures were sampled simultaneously for different wind directions (θ = 0∼360°) with an interval of 10°. In addition, wind angles of 45°, 135°, 225°, and 315° were also considered. While for V = 3V₀ and V = 2V₀, the wind direction range was narrowed to 0∼180°. The definition of wind azimuth and the layout of external pressure points on roof are shown in Figure 3. Tables 1 and 2 list all experimental cases considered in this study.

Figure 2.

(a) square opening and (b) rectangular opening.

Figure 3.

Definition of wind angles and layout of measurement points: (a) Configuration I, (b) Configuration II, (c) Configuration III, (d) Configuration IV, and (e) Configuration V.

Table 1.

Test conditions of roof openings.

Configurations	Opening shape and dimensions (mm)	Opening area ratios (%)	Test contents
I		12	See Table 2
II		12	See Table 2
III	Same as II	12	Wind velocity = 11 m/s; V = 4V₀
IV		6
V		9

Table 2.

Details of test conditions.

Configurations	Cases	Opening shape	Internal volume	Opening ratios/γ (%)	Steady-state or transient	Wind azimuth	Wind velocity/m/s
Configuration I	1	Square-opening	4V₀	0.3	Transient	0°, 180°	11
	2		4V₀	1.3	Transient		11
	3		4V₀	3	Transient		11
	4		4V₀	5.3	Transient		11
	5		4V₀	8.3	Transient		11
	6		2V₀	12	Both	0°, 30°, 45°, 90°, 120°, 150°, 180°	Case 6–7: 11 m/s
	7		3V₀	12	Both		Case 6–7: 11 m/s
	8		4V₀	12	Both		Case 8: 6~12m/s
	9	Rectangle-opening	4V₀	0.3	Transient	0°, 180°	11
	10		4V₀	1.3	Transient		11
	11		4V₀	3	Transient		11
	12		4V₀	5.3	Transient		11
	13		4V₀	8.3	Transient		11
Configuration II	14	“L shape” opening	2V₀	12	Both	0°, 30°, 45°, 90°, 120°, 150°, 180°	6–12
	15		3V₀	12	Both
	16		4V₀	12	Both

The volume distortion shape of the model cavity accords with the suggestions of Sharma et al. (2010) that the expansion box should be deep and narrow. However, this volume expansion may cause two cavity resonances due to the discontinuous cross section. A contrastive test has been carried out based on configuration I in terrain Category B. It can be concluded that the impact of discontinuous compensations of internal volume on characteristics of internal pressure is negligible.

Simulation of boundary layer

The boundary layer profile of the 1:40 scaled model having terrain Category B wind field condition specified in the Chinese load code (GB50009-2012, 2012) was simulated using turbulence generators including vertical spires, wooden roughness elements, and sawtooth barriers. The characteristics of simulated approaching flow was captured using a three-dimensional (3-D) fluctuating anemometer and a pitot tube located in the upstream of the model. Figure 4(a) shows the experimental mean wind speed and turbulence intensity profiles along with the code-specified values, where U_r is the mean wind speed at the reference height H_r = 233 mm and Z represents the real height from the wind tunnel floor. The mean wind speed and turbulence intensity at the reference height are 11 m/s and 14.9%, respectively. The corresponding Reynolds number, defined in terms of the roof length of the model (0.3 m), is approximately 2.75 × 10⁵. The friction velocity and the roughness length (z₀) estimated from the simulated boundary layer profile are 1.65 m/s and 0.26 m, respectively. The non-dimensional longitudinal wind speed spectrum along with three available theoretical spectra including Karman, Kaimal, and Davenport spectra are plotted against the reduced frequency, fL/U (f being the frequency and L being the integral length scale), as shown in Figure 4(b). A reasonable agreement is shown between the experimental and Karman spectra. The turbulence integral scale of the simulated profile is 0.732 m (equivalent to 29.3 m in full scale) at the ridge height. The Turbulence Flow Instrumentation (TFI) system was used to sample the wind pressure data which were acquired at a sampling frequency of 332 Hz over a period of 30.12 s.

Figure 4.

Simulated atmospheric boundary layer: (a) wind profile and (b) spectra of longitudinal wind-speed fluctuations.

Introduction of the experimental methods

In order to simulate the internal pressure overshoot effect in buildings caused by the sudden roof damage during a strong wind event, a transient opening device was designed (Figure 5). The device consists of a lightweight cover plate equipped with a permanent magnet, a roof opening equipped with electromagnet along its edge, an electromagnetic ejection device, and a string. The magnet on the cover plate edge and the electromagnet on the edge of the opening were used for orifice sealing before the transient opening. At the moment of opening, the two magnets changed from suction to repulsion, so as to achieve the purpose of opening the cover plate quickly. The tightness examination of the model was conducted before each transient opening case. The string was mainly used to connect with the model to prevent the cover plate from damaging the experimental equipment under the action of wind. It takes 0.1–0.2 s to open the opening completely.

Figure 5.

Schematic diagram of transient opening device (Configuration I).

Data processing

The pressure obtained from the wind tunnel test was converted into a dimensionless wind pressure coefficient, as follows:

C_{pi} (t) = \frac{P_{i} (t) - P_{ref}}{0.5 ρ U_{H}^{2}}

(5)

where $C_{pi} (t)$ is the time series of wind pressure coefficients at measuring point i, $P_{i} (t)$ refers to wind pressure time history obtained directly at measuring point i, $P_{ref}$ is the reference static pressure, ρ is the air density (= 1.25 kg/m³), and $U_{H}$ is the longitudinal mean wind speed at reference height $H_{r} = 233 mm$ .

Two kinds of tube length (60 and 80 cm) were adopted in the pressure measuring system. The pressure measuring tubes with the same aperture adopted in the wind tunnel test were selected to obtain the frequency response function which was used to correct the wind pressure signal.

Figure 6(a) shows the time series of the internal pressure coefficients in 1s roof opening process (before-during-after opening), where t₀ is the time of opening formation. It is found that peak transient internal pressures significantly exceed peak steady-state internal pressures. The overshoot ratio (equation (6)) adopted to describe transient internal pressure phenomenon.

R_{i} = {\hat{C}}_{pi, t}^{n} / {\hat{C}}_{pi, s}^{n}

(6)

where ${\hat{C}}_{pi, t}^{n}$ is the peak transient internal pressure obtained from the time series of internal pressures corresponding to the maximum internal pressure coefficient during the opening period; and ${\hat{C}}_{pi, s}^{n}$ is the peak internal pressure coefficient during the steady- state period, calculated by equation (7) using the data obtained during steady state period.

{\hat{C}}_{pi, s}^{n} = {\bar{C}}_{pi, s}^{n} + g {\tilde{C}}_{pi, s}^{n}

(7)

where ${\bar{C}}_{pi, s}^{n}$ and ${\tilde{C}}_{pi, s}^{n}$ are respectively the mean value and standard deviation of internal pressure coefficients during steady-state period; and g is taken as 3 as the probability density distribution of internal pressure coefficients during the steady-state period follows the Gaussian distribution (see in Figure 6(b)).

Figure 6.

Time history of the internal pressure coefficients during transient opening process (a) and the corresponding probability density of the internal pressure coefficients during steady-state stage (b).

Influencing factors of peak transient internal pressure

Opening ratio and geometry

The initial values of internal pressure were sampled before each experimental case to ensure the air-tightness of the test model. Without considering the background porosity, the peak transient internal pressure characteristics in Configuration I (Tables 1 and 2) were studied and the distorted internal volume of the model was found to be 0.042 m³ (i.e. 4V₀), meeting the requirement of internal pressure fluctuation similar to Holmes (1979). Figure 7 shows the effects of opening ratios (γ) on peak internal pressure coefficient, where ${\overset{̆}{C}}_{p, t}^{n}$ and ${\overset{̆}{C}}_{p, s}^{n}$ are respectively the area-average values of ${\hat{C}}_{pi, t}^{n}$ and ${\hat{C}}_{pi, s}^{n}$ and R is the overshoot ratio (equations (8)–(10)). $A_{i}$ is the auxiliary area at measuring point i. A is the total area of the roof.

{\overset{̆}{C}}_{p, t}^{n} = \sum_{i} {\hat{C}}_{pi, t}^{n} \times A_{i} / A

(8)

{\overset{̆}{C}}_{p, s}^{n} = \sum_{i} {\hat{C}}_{pi, s}^{n} \times A_{i} / A

(9)

\overset{̆}{R} = {\overset{̆}{C}}_{p, t}^{n} / {\overset{̆}{C}}_{p, s}^{n}

(10)

Figure 7.

Influences of opening ratios on peak internal pressure coefficient: (a) square opening (i: θ = 0°; ii: θ = 180°) and(b) rectangular opening (i: θ = 0°; ii: θ = 180°).

${\overset{̆}{C}}_{p, t}^{n}$ and ${\overset{̆}{C}}_{p, s}^{n}$ at windward roof openings (θ = 0°), i.e., Cases 1–5 and 8–13 are larger in magnitude than those at leeward roof openings (θ = 180°). For the cases with square opening, ${\overset{̆}{C}}_{p, t}^{n}$ and ${\overset{̆}{C}}_{p, s}^{n}$ increase with the increasing open area ratio, with the former more significant. The peak steady-state pressure coefficient for windward corner openings and leeward corner openings approach to −0.9 and −0.5, respectively, when $γ$ (opening ratio) exceeds 3%. For the cases with rectangular opening, the peak transient internal pressure coefficient increases with γ, while the peak steady-state internal pressure coefficient varies slightly (−0.8 to −1.0 for windward cases and −0.4 to −0.6 for leeward cases). ${\overset{̆}{C}}_{p, t}^{n}$ and ${\overset{̆}{C}}_{p, s}^{n}$ for the cases with square opening are larger in magnitude than those with rectangular opening when $γ$ is less than 3%, while the peak internal pressures of the two opening shapes are relatively close to each other when γ > 3%. This indicates that the increase of opening length will induce peak internal pressure larger in magnitude, but such increasing effect gradually diminishes with the increase of opening width. It is also found that there are obvious transient internal pressure overshoot phenomena at either windward or leeward openings and there is an apparent linear relationship between opening ratio and overshoot ratio.

Opening position and wind direction

In order to study failure mechanism of building due to roof opening, Figure 8 depicts the contours of transient internal pressure overshoot ratio (R) for Case 8 (corner opening) and Case 16 (ridge opening) with the wind speed of 11 m/s. As shown, the distribution of R values is significantly affected by the opening positions and wind direction (θ ). The largest overshoot ratio for Cases 8 and 16 are about 1.66 and 1.87, occurring at θ = 30º. In Case 8, the R values are distributed evenly, with the maximum value ≈1.64–1.66 occurring in the downwind area of windward roof and in the middle area of leeward roof. In Case 16, the R value in the windward roof corner area and the entire leeward roof is relatively small at ≈1.48–1.65, with the maximum value occurring in the downwind area of windward roof. At θ = 0°, the extreme R value happens in the windward eave area in both cases. The R values at leeward openings are relatively large. However, the peak transient and steady-state pressures induced by windward openings are larger than those caused by leeward openings. At θ = 180°, R values for the entire roof in Case 16 are generally larger than those in Case 8, while the magnitudes of transient and steady-state internal pressures in both cases are nearly equal. This indicates that the instantaneous opening around roof ridge increase the internal pressure more significantly when the incoming flow is perpendicular to an eave. At θ = 90°, in Case 8, the larger R values are mainly distributed in the middle and downwind roof opening areas, while in Case 16, the larger R only appears at the corner of downwind roof opening area. The opening of the two cases has the same windward width but with different opening length. The results indicate that the overshoot ratio (R) is significantly affected by the opening length. The increase of the opening length in ridge areas can weaken the overshoot effect on the no-opening side.

Figure 8.

Contours of transient internal overshoot ratio at different wind directions for Case 8 (a) and Case 16 (b).

Wind speed

The relationships of peak internal pressure coefficient and overshoot ratio versus wind speed (U_r) are shown in Figure 9 for Cases 8 and 16, with the most unfavorable wind direction of θ = 30°. The peak steady-state internal pressures of both cases are sparsely affected by the upstream U_r, while the peak values of Case 8 (≈−0.69 to −0.83) are much larger in magnitude than those of Case16 (≈−0.39 to −0.44). The peak transient internal pressure is sensitive to the change of U_r. In Case 8, the peak transient internal pressure increases in magnitude with the increasing U_r, while the counterpart in Case 16 decreases in magnitude with the increase of U_r. This is because when the wind speed increasing, the airflow separation around the opening at the windward roof corner is intensified, while the opening region at the leeward roof is affected by the roof ridge, leading to the airflow going upwards. The peak transient pressure increasing in magnitude is mainly induced by the intensified shear flow along the opening edge due to the increasing approaching flow speed. It was demonstrated by Wang and Li (2015) in their investigation of internal pressure induced by opening at a flat roof corner. When the opening at leeward roof ridge area where is the high negative pressure zone because of flow separation occurs there, the separated airflow rises with the increasing U_r, which leads to the decreasing speed of the airflow instantaneously entering into the orifice, thus reducing the peak transient internal pressure in magnitude. Figure 9 also depicts the fitted results of overshoot ratio (R) versus wind speed (U_r) at θ = 30°. As seen, R increases linearly with U_r for Case 8, while it decreases linearly with U_r for Case 16.

Figure 9.

Effects of wind speed on peak internal pressure coefficients: (a) Case 8 and (b) Case 16.

Steady-state internal pressure characteristics

Spectral density analysis

The fluctuating internal pressure is affected by the fluctuating characteristics of incoming flow, and the vortex shedding caused by the airflow separation at opening edge and the Helmholtz resonance of internal pressure. Figure 10 display the power spectral density (PSD) curves of internal pressures for Cases 1–5 and 8–13 at θ = 0°. As shown, except for Cases 1 and 2 (respectively $γ$ = 0.3% and 1.3%) and Case 9 ( $γ$ = 0.3%), there are two obvious peaks in each spectrum. Comparing with the external pressure power spectra (Figure 10(a)(iii) and (b)(iii)) at the orifice and the theoretical Helmholtz resonance frequency, it is found that the peak in the high frequency range is dominated by the vortex shedding and the other peak is induced by the Helmholtz resonance of fluctuating internal pressure due to the constraint of finite building volume (Wang and Li, 2015). By comparing these figures, the second peak of the internal pressure spectrum is not completely matched with the peak of the external pressure spectrum, which indicates that the orifice parameters have a great influence on the vortex shedding frequency in addition to the action of the airflow separation at the eaves. Both vortex shedding frequency and Helmholtz frequency under the two different opening shapes (square and rectangle) show an increasing trend with the increase of opening area ratios. With the same opening ratio (γ), the opening shape has little effect on Helmholtz frequency, while the vortex shedding frequency for a rectangular opening is higher than that for square openings. This phenomenon is more obvious when γ is small. Under the condition of same opening ratio, the orifice friction of passing-through flow is increased by the slender opening due to the increasing opening width, resulting in the increase of the vortex shedding frequency of orifice airflow eventually. The PSD value of internal pressure increase with the increasing γ and more so with the square opening condition. However, the PSD values obtained for large γ values (=5.3%–12%) are approximately equal.

Figure 10.

The PSD curves of square opening (a) and rectangular opening (b) in Configuration I, (i) internal pressure PSD ( $γ = 0.3 % ~ 3 %$ ); (ii) internal pressure PSD ( $γ = 5.3 % ~ 12 %$ ); (iii) external pressure PSD (Tap E3).

Figure 11 shows the influences of building volume expansion on internal pressure power spectrum at the wind directions (θ) of 0° and 45° (Cases 6–8). In Case 6 (i.e. $V_{e} = 2 V_{0}$ ), there is only one peak on the internal pressure spectrum curve and the corresponding frequency is 104.33 Hz (θ = 0°) or 104.43 Hz (θ = 45°). When $V_{e} = 3 V_{0}$ , there is also only one peak on the spectrum curve with the corresponding frequency of 69.6Hz (θ = 0°) or 60.9 Hz (θ = 45°), which is significantly larger than the other volume cases. This is because of the coupling effect of vortex shedding on the orifice edge and the Helmholtz resonance of internal pressure. When $V_{e} = 4 V_{0}$ , additional peaks were observed on the internal pressure spectra, with the Helmholtz frequencies f_H = 34.8 Hz (θ = 0°) and 35.3 Hz (θ = 45°) and the respective vortex shedding frequencies $f_{V} = 99.4$ Hz and 104.7 Hz. These indicate that the vortex shedding is somewhat strengthened under oblique wind, while the effect of wind direction on Helmholtz frequency is negligible.

Figure 11.

Effects of building volume expansion on the fluctuating internal pressure spectrum: (a) θ = 0° and (b) θ = 45°.

Coherence analysis

The measured data of Cases 8 and 16 at θ = 30° were used to closely examine the coherence characteristics of fluctuating internal pressures. Figure 12 depicts the coherence function between external and internal pressures at EN5 and FN1 (the pressure measuring points at the orifice). As shown, the coherence function curve of Case 8 is generally smooth in the low frequency range (0–35 Hz). However, the coherence function curve of Case 16 in this frequency range is not so smooth and its coherence value is as low as 0.75 which is significantly lower than Case 8. This is mainly due to the influence of characteristic turbulence formed at the leeward ridge and gable. The coherence function curves of both cases descend sharply at a frequency of 35 Hz, as affected by the Helmholtz resonance of fluctuating internal pressures. The functions reach their lowest value at different frequencies. The internal pressure coherence of a leeward roof opening starts at the frequency of about 3 Hz more than a windward roof opening, reflecting the influence of Helmholtz resonance on coherence. The Helmholtz resonance lowers the coherence value to be below 0.7 in the frequency range of 40–55 Hz and weakens the coherence between the internal pressure on opening edge and inner roof area in different degrees, especially in Case 16. The coherent functions in Cases 8 and 16 have a second steep drop at frequencies of 93.8 Hz and 99.5 Hz respectively, caused by the vortex shedding on opening edge.

Figure 12.

Internal pressure coherence functions between opening edge and inner roof area in Case 8 (a) and Case 16 (b).

Figure 13 shows the coherence function curves of the area-average external pressure on opening edge and internal pressure at typical wind directions in Cases 8 and 16. As shown in the figure, the coherence in the low frequency range is significant but gradually drops in high frequency range (f > 120 Hz) to a value below 0.3. This indicates that the large-scale vortex with low frequency in the incident flow is an important factor affecting the fluctuating characteristics of internal pressure. The attenuation slope of the coherence function in the low frequency range of Case 18 is 0.01, while it is doubled in Case 8. It is also found that the oblique wind direction (e.g. θ = 30°) has a great influence on the coherence function in the low-frequency range more so at 35.5 Hz (Case 8) and 36.3 Hz (Case 16). This manifests that the Helmholtz resonance has a greater influence on the coherence of internal and external pressures at oblique wind directions. In Case 8, vortex shedding increases the coherence only at θ = 30°, while in Case 16, the increase occurs at more wind directions with θ values of 0, 30, and 90°. This is mainly due to the difference in opening positions and the discrepancy in the characteristics of body-induced flow caused by the airflow on the roof.

Figure 13.

Coherent function of external pressure at opening edge and internal pressure in Cases 8 (a) and Case 16 (b).

Internal pressure admittance function

Based on the spectra of area-averaged external and internal pressure fluctuations for different opening configurations, the admittance functions were obtained to further investigate the effect of external pressure on the dynamics of internal pressure. The admittance functions were calculated using the equation proposed by Guha et al. (2014), as follows:

{| χ_{pi - pe} |}^{2} = S_{pi} (f) / S_{pe} (f)

(11)

where $S_{pi} (f)$ is the internal pressure spectrum and $S_{pe} (f)$ is the spectrum of external area-averaged pressure. Considering the frequency domain characteristics of the external pressure at the orifice, only the measurement points on the orifice side edge were selected for the area averaging, that is, measurement points E3–E4 for configuration I, E12, and E13 for configuration II, E3–E5 for configuration IV, and E10 and E11 for configuration V.

As shown in Figure 14, when the model volume conforms to the similarity relationship (i.e. $V_{e} = 4 V_{0}$ ), the contribution of external pressure excitation to internal pressure response is mainly concentrated around the Helmholtz resonance frequency. The peak of admittance function for these opening configurations appears at near the Helmholtz frequency, which shows that the Helmholtz resonance is mainly caused by the internal effect of the airflow and less affected by the opening ratio and opening shape. The peak frequency of the admittance function related to Helmholtz resonance is little affected by the opening shape, but the peak value increases with the increase of opening areas. The peak related to vortex shedding disappears in high frequency range, which manifests the second peak is mainly induced by the external pressure at the opening edge.

Figure 14.

Influence of opening configuration on admittance function ( $V_{e} = 4 V_{0}$ , U_r = 11 m/s): (a)opening shapes and (b) opening ratios.

Comparison with the theory

Figure 15 shows various comparisons of mean, root mean square (RMS), and peak internal pressure coefficients between the predicted and measured results, considering the impact factors of opening ratio, opening configuration, internal volume, turbulence intensity, and wind speed, in which the horizontal and vertical axes respectively represent the experimental and predicted results of internal pressure. The solid red lines with unity slope in the middle of a figure indicate that the predicted values are identical to the test ones. The location of the colored points relative to the red line indicates the effect of theoretical estimation. The mean internal pressures are calculated using equation (1) and the RMS and peak internal pressures are obtained by solving the second-order nonlinear differential equation (i.e., equation (2)) using the measured external pressure signals around the opening edge. The corresponding experimental values for comparison in Figure 15 were calculated by using the data collected at θ = 0°. The peak resonance at Helmholtz frequency under different roof opening conditions is calculated and the discharge coefficient k is taken as the calculated value when the difference between the test and theoretical values is the least. The k in different cases considered in this study ranges from 0.06 to 0.3.

Figure 15.

Comparison between measured and predicted mean, RMS and peak internal pressure coefficients for (a) different opening ratios in Configuration I, (b) different opening configurations, (c) different building internal volumes and different turbulence intensities, and (d) different wind speeds.

In each influencing factor considered, the predicted mean internal pressures are generally in agreement with the experimental values, with a maximum error of 15%. While the predicted RMS and peak internal pressures are poorer, with a higher error of 35%. This suggests that the estimation of mean internal pressures based on the Bernoulli theory is efficient, while the fluctuating internal pressure response characteristics are difficult to predict due to the randomness of fluctuation and the complex mathematical expression in fluid diffusion characteristics. As shown in Figure 15(a), it is difficult to estimate the fluctuating internal pressure induced by the roof openings with a small area, caused mainly by the ill-defined parameters k and l_e. In Figure 15(b), there is a large deviation between the predicted and experimental fluctuating internal pressures in Configurations I and III, suggesting that the estimation method of fluctuating internal pressure needs to be further improved for leeward roof openings and more complex opening shapes. It can be seen from Figure 15(c) and (d) that under the conditions of different building internal volumes, turbulence intensities, and wind speeds, the internal pressure transfer equation (i.e. equation (2)) estimates fluctuating internal pressures reasonably well.

Conclusions

Extensive experimental and numerical investigations on the internal pressures induced by roof openings are presented. The characteristics of internal pressure induced by the transient failure in the vulnerable roof area of a low-rise building was also studied. Five opening configurations were considered in the wind tunnel tests with three-boundary layer wind fields in different terrain categories. Sixteen cases with Configurations I and II were further investigated, considering the influencing factors of opening shape, opening ratio, and building internal volume. This paper mainly focuses on the peak transient internal pressure characteristics and its influencing factors, steady-state internal pressure characteristics, and comparison between the predicted and experimental internal pressures. Based on this study, the following major findings are offered:

For roof corner openings with different shapes, both peak transient and steady- state internal pressures increase in magnitude with the increase of opening ratios and the overshoot ratio increases linearly with the opening ratio.

The transient overshoot ratio for roof corner opening is larger in magnitude than that for roof ridge opening. Moreover, the increasing opening length weakens the transient internal pressure overshoot effect on the roof without opening.

The peak steady-state internal pressure coefficient changes little with wind speeds, while the peak internal pressure coefficient caused by a windward roof opening is much larger in magnitude than that induced by a leeward roof opening. The peak transient internal pressure coefficient increases linearly with wind speeds, which is contrary for the situation of a leeward roof ridge opening.

The vortex shedding frequency and Helmholtz frequency for square and rectangular openings increase with increasing opening ratios. The vortex shedding and Helmholtz resonance produce a coupling action when the internal building volume is doubly expanded, which results in a sharp increase in fluctuating internal pressure energy. The wind direction has little influence on Helmholtz frequency, while the vortex shedding frequency increases under the action of oblique wind.

The coherence between external and internal pressures induced by windward roof openings is stronger than that by leeward ridge roof openings. Both vortex shedding and Helmholtz resonance reduce the internal pressure coherence function. The attenuation slope of the internal-external pressure coherence function of windward roof corner opening in the low frequency range is twice that of a leeward roof ridge opening.

The estimation accuracy of mean internal pressure is higher than that of fluctuating and peak internal pressure, using equation (2). The estimation of fluctuating internal pressure becomes more difficult for leeward roof openings or complex opening shapes.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is funded by the National Nature Science Foundation of China (51578237) and Degree & Postgraduate Education Reform Project of Hunan Province (2019JGZD063), Hunan Taught Information (2018, NO. 436, 1017), and Hunan Education Department of Science Research Project (19A168), to which the authors are very grateful. Mr. Kui Guo and Mr. Qinghang Gu are acknowledged for their help in the wind tunnel tests and relevant studies.

ORCID iDs

Yangjin Yuan

Shu Jiang

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