Computational Investigation of Wind Loads on Tilted Roof-Mounted Solar Array

Abstract

A detailed computational investigation of the wind field around tilted solar modules mounted on a large building roof has been undertaken, utilizing the Reynolds-Averaged Navier-Stokesv (RANS) approach supplied with the SST k − ω turbulence model. The study investigated the flow field for various tilt angle of modules at normal wind directions relative to the wall. Then the shape factors and moment coefficients of modules were explored. The results show that the recirculation vortex generated by the building edge is disintegrated to smaller local vortices. With the increasing of the tilt angle, an increasing number of local vortices emerged at the leading rows, leading to a relatively large wind pressure and shape factor at the corner of the array. In most tilt angles at 0° and 180° wind direction the shape factors are negative. However, for the 40° and 55° tilt angles at 180° wind direction, the shape factors on the lower surfaces are positive, due to the dominating of approaching flow rather than the local vortices. The array is divided into six zones based on the distribution of shape factors. As the shape factors on upper and lower are similar, the shape factors in most zones for tilt angles from 5° to 55° are quite small. However, shape factors in the leading row for 30°, 40° and 55° are relatively large. This indicates that the shading effect of front rows can significantly reduce the shape factors of the rear rows. Compared to the values calculated by Chinese, American and Japanese standards, the shape factors by simulation are quite small. The moment induced by nonuniform wind pressure, which is often ignored in the literature and standards, is quite large at the leading zones, with a maximum of 0.28 for 55° tilt angle. Ignoring the wind induced moment on the leading zones may make the wind resistance design of the solar module support structure unsafe.

1. Introduction

Solar arrays on industrial building roofs are becoming increasingly popular for the possibility of reducing the effects of global warming and air pollution [1]. One of the most important issues related to engineering research on solar panels is analyzing the aerodynamic loads. As Doddipatla and Kopp point out, roof-mounted solar panels and their supporting structures are exposed to strong wind loads, dominating the load design [2].

The wind tunnel test, one of the most effective methods to investigate the aerodynamic loads on buildings and structures, has been used in wind load assessments on roof-mounted solar arrays [3,4,5,6,7]. As influenced by the building and modules, the flow field which develops over roof-mounted solar array is complex. The flow separates at the leading edge of building, resulting in a large scale building generated vortex. The modules themselves induce local separations and reattachments along the array [3]. The structure of the incident boundary layer was investigated by Pratt and Kopp [3]. As the module size is quite small compared to the building size, two particular issues on the surroundings and the model scale used in wind tunnel test are discussed by Kopp and Banks [4]. According to the additional requirements, the wind load features of a 12-row solar array were investigated in a wind tunnel [5]. Abiola-Ogedengbe et al. found that the inter-panel gap influences the module surface pressure field, and could not be ignored [6]. Dai et al. investigated the effects of building height on the module surface pressure, finding that buildings with lower heights tend to cause more significant fluctuations of wind pressure on solar panels [7].

Due to the rapid increase in demand for solar panel constructions, some building standards covering the wind loads experienced by solar arrays have been promoted [8,9,10]. In the Chinese standard NB/T 10115, the shape factors for the modules mounted on the ground and roof for various tilt angles at the normal wind directions are specified. When the solar array consists of more than seven rows, a reduction factor of 0.85 is suggested for the inner part [8]. In the Japanese standard JIS C8955, the pressure coefficients on modules mounted on the roof are divided into two zones, i.e., the inner and outer zones [9]. The positive and negative pressure coefficients of the two zones are functions of tilt angle, which are apparently different. In the American standard ASCE 7-22, the formulation of wind loads on roof-mounted modules is more complicated. The core part of the formulation, nominal net pressure coefficients, are different in the corner, end and inner zones. The height of the building, parapet, gap between module and roof surface, tilt angle, exposure and shading effect are considered in the formulation [10].

The Computational Fluid Dynamics (CFD) analysis is an important predictive tool for aerodynamic load investigation of a solar array. Aly and Bitsuamlak investigated the scale effects on wind loads of a standalone solar panel by CFD, finding that the peak loads are more sensitive to the model size than the mean loads [11]. Shademan et al. explored the influence of lateral spacing and longitudinal spacing on wind loads in a PV system by three-dimensional RANS simulations [12]. Jubayer and Hangan employed the unsteady RANS model to investigate the wind effects on a standalone panel as well as a solar array at variable wind directions [13,14]. Reina and Stefano evaluated the mean wind loads on sun-tracking ground-mounted solar arrays by employing the RANS model [15].

Generally, in the current literature the wind load on roof-mounted solar array has been studied, but the effect of module angles has rarely been systematically studied. Also, the consistency of the wind load parameter using the wind tunnel test, as well as the effectiveness of practical standards such as the Chinese standard (NB/T 10115), Japanese standard (JIS C8955) and American standard (ASCE 7-22) have not been investigated.

The objective of the current study is to investigate the wind field around a roof-mounted solar array with various tilt angles, and examine the consistency of the wind load parameter (shape factor and moment coefficient) using the wind tunnel test and practical standards. To achieve this goal, a numerical simulation model of a roof-mounted solar array using the RANS approach was developed in Section 2. Then the flow field around the array and building was analyzed through the streamlines (Section 3.1 and 3.2). Furthermore, the shape factors and moment coefficients of the modules calculated by numerical simulation and practical standards were investigated and compared (Section 3.3 and 3.4). Finally, some useful conclusions are expressed in the Section 4.

2. Computational Algorithm and Settings

2.1. Turbulence Model

The Shear–Stress Transport (SST) $k-\omega$ turbulence model proposed by Menter, which is commonly exploited in computational wind engineering [13,14,15], was adopted to simulate the wind field around the roof-mounted solar arrays. The effectiveness of the turbulence model has been validated. Jubayer and Hangan validated the numerical modeling approach for a stand-alone photovoltaic system, finding that the surface pressure on the panel as well as wind flow field around the panel are consistent with the experiments performed in the wind tunnel test [13,14]. Reina and Stefano compared the mean surface pressure from their simulation against the wind tunnel measurements in Abiola-Ogedengbe et al. [6], showing a good agreement with the experiments for both surfaces of the panel. The SST $k-\omega$ turbulence model is suitable in the wind flow simulation around bluff bodies in the atmospheric boundary layer as it retains the robust and accurate formulation of the Wilcox $k-\omega$ model in the near wall region, and takes advantage of the freestream independence of the $k-\epsilon$ model in the outer part of the boundary layer [16]. The transport equations of SST $k-\omega$ turbulence model are Equations (1) and (2):

$$\frac{\mathrm{D}\rho k}{\mathrm{D}t}={\tau }_{ij}\frac{\partial u_i}{\partial x_j}-{\beta }^{*}\rho \omega k+\frac{\partial }{\partial x_j}\left[(\mu +{\sigma }_k{\mu }_t)\frac{\partial k}{\partial x_j}\right]$$

(1)

$$\frac{\mathrm{D}\rho \omega }{\mathrm{D}t}=\frac{\gamma }{{\nu }_t}{\tau }_{ij}\frac{\partial u_i}{\partial x_j}-\beta \rho {\omega }^2+\frac{\partial }{\partial x_j}\left[(\mu +{\sigma }_{\omega }{\mu }_t)\frac{\partial \omega }{\partial x_j}\right]+2\rho (1-F_1){\sigma }_{\omega 2}\frac{1}{\omega }\frac{\partial k}{\partial x_j}\frac{\partial \omega }{\partial x_j}$$

(2)

where $\frac{\mathrm{D}\cdot }{\mathrm{D}t}=\frac{\partial \cdot }{\partial t}+u_i\frac{\partial \cdot }{\partial x_i}$, t is time, ρ is air density, μ is kinematical viscosity, ${\mu }_t$ is turbulent viscosity, k is turbulent kinetic energy, $\omega$ is specific dissipation rate, $u_i(i=1,2,3)$ is the mean wind speed in the direction of $x_i$, ${\tau }_{ij}$ is the Reynolds stress tensor and ${\sigma }_k$, ${\sigma }_{\omega }$, $\beta$, ${\beta }^{*}$, $\gamma$ are the constants in the SST $k-\omega$ turbulence model. Expressing the constants as $\varphi$, it can be calculated from the constants ${\varphi }_1$ and ${\varphi }_2$ by Equation (3).

$$\varphi =F_1{\varphi }_1+(1-F_1){\varphi }_2$$

(3)

where $F_1$ is the multiplied function. The constants of set 1 (${\varphi }_1$) taken from the Wilcox $k-\omega$ model are: ${\sigma }_{k1}=0.85$, ${\sigma }_{\omega 1}=0.5$, ${\beta }_1=0.0750$, ${\gamma }_1={\beta }_1/{\beta }^{*}-{\sigma }_{\omega 1}{\kappa }^2\sqrt{{\beta }^{*}}$, ${\beta }^{*}=0.09$, $\kappa =0.41$. The constants of set 2 (${\varphi }_2$) taken from the $k-\epsilon$ model are: ${\sigma }_{k2}=1.0$, ${\sigma }_{\omega 2}=0.856$, ${\beta }_2=0.0828$, ${\gamma }_2={\beta }_2/{\beta }^{*}-{\sigma }_{\omega 2}{\kappa }^2\sqrt{{\beta }^{*}}$ [16].

2.2. Roof-Mounted Solar Arrays

The roof-mounted solar arrays are shown in Figure 1. The three-dimensional size of the building was 51 m in length, 35 m in width and 25 m in height. The solar array consisted of 12 rows, with each row made up of 44 modules, located symmetrically on the roof. The plan dimension of each module was 1.038 m (B) × 2.094 m (W), as shown in Figure 2. The height from the roof to lower edge of module was 0.5 m. The distance between rows was 2.7 m. The setbacks from the longer and shorter edges of the roof were 1.5 m and 3.5 m, respectively. As the setbacks in two directions were less than 0.4 $H$ = 0.4 × 25 m= 10 m, the two outer rows were in the zone of vortices shedding caused by the building [17]. According to Lin and Surry [18], the wind pressures on the roof and its auxiliary structures are more sensitive to building height than building length and width. Therefore, although the building length and width were similar to Kopp’s study [5], the building height in this study was 3.4 times of that in Kopp’s study, making the wind field and loads different.

To investigate the effects of tilt angle on wind field around roof-mounted solar arrays, and the wind loads on the modules, the modules were modeled with 6 different tilt angles: 5°, 10°, 20°, 30°, 40° and 55°, consistent with the Chinese standard NB/T 10115 [8]. The relative position relationship of tilt angle α and wind angle θ is shown in Figure 2. When θ = 0°, the wind flow approaches the upper surface of the module. The wind field simulations were taken at wind direction of 0° and 180°.

2.3. Computational Domain Discretization

The wind field was simulated by the commercial software ANSYS Fluent v19.2. To reach a compromise between wind field development and computational efficiency, the computational domain is usually set as 10~20 times the building’s width or depth and 2~5 times the building’s height. Thus, the computational domain was 800 m in x direction, 600 m in y direction and 200 m in z direction, as shown in Figure 3a. A cylindrical computational domain with a diameter of 70 m and a height of 100 m around roof-mounted solar array was set. An unstructured mesh was generated in the cylindrical domain, while a structured mesh was generated in the rest computational domain. As a high Reynolds number turbulence model, the $y^{+}$ in SST $k-\omega$ model should be less than 5. To meet the accuracy requirements of wall function, 5 layers of meshes were generated near the surfaces of the modules. The thickness gradually increased, with the first layer’s thickness of 0.05 m, and the thickness growth rate of 1.1. The mesh generated around the building and solar modules are shown in Figure 3b,c. About 8 × 10⁷ cells were generated in the computational domain. The blockage ratio in this computational domain was 1.12%, less than the 5% requirement specified in the Chinese order GB 50009-2012 [19].

2.4. Boundary and Turbulence Setting

The oncoming wind was simulated by imposing the following power law boundary layer velocity profile (Equation (4)) at the inlet of the computational domain:

$$U_z=U_{10}{(\frac{z}{z_{10}})}^{\alpha }$$

(4)

where $U_z$ is the mean wind speed at the height of z, $U_{10}$ is the reference wind speed of 36 m/s, $z_{10}$ is the reference height of 10 m above the ground and α is the ground Roughness Exponent of 0.16, according to the B-category terrain in GB 50009-2012 [19]. The turbulence intensity $I_{\mathrm{u}}$ also followed the power law as seen in Equation (5).

$$I_u=I_{10}{(\frac{z}{10})}^{-\alpha }$$

(5)

where $I_{10}$ is the turbulence intensity at the height of 10 m, equaling 0.12. To satisfy the requirement of modeling an equilibrium ABL flow at the inlet boundary, the wind velocity profile must be coupled with suitable profiles of resolved turbulence variables [20], which are the turbulence kinetic energy and the specific turbulence dissipation rate for the SST $k-\omega$ model. Therefore, the following inlet profiles (Equations (6) and (7)) of these two variables were considered:

$$k=1.5{\left(U_z{I}_u\right)}^2$$

(6)

$$\omega =\frac{k^{0.5}}{C_{\mu }^{0.25}l}$$

(7)

where $C_{\mu }$ equals 0.09, $l$ is the turbulence integral scale. The inlet of the computational domain was set as velocity-inlet. The inlet of wind velocity, turbulence kinetic energy and specific turbulence dissipation rate were imported in to Fluent by compiling. The outlet of the computational domain was set as pressure-outlet. Ground, building surfaces and module surfaces were set as wall, while left and right boundaries of the computational domain were set as symmetry boundaries. The SIMPLEC algorithm was used for pressure and velocity coupling. The Second Order Upwind format was used for the discretization of momentum equation, turbulent kinetic energy and specific dissipation rate transport equation. The convergence precision of each variable was set as 1 × 10⁻⁵.

2.5. Data Processing

The wind pressure coefficients on the upper and lower surfaces of modules, denoted as $C_{pu}$ and $C_{pl}$, respectively, are defined as Equations (8) and (9):

$$C_{pu}(i)=\frac{P_u(i)-P_{\infty }}{0.5\rho U_H^2}$$

(8)

$$C_{pl}(i)=\frac{P_l(i)-P_{\infty }}{0.5\rho U_H^2}$$

(9)

where $P_u(i)$ and $P_l(i)$ are the wind pressures on the upper and lower surfaces, respectively, $P_{\infty }$ is the ambient atmospheric pressure, $U_H$ is the mean wind speed at the roof height and i is the number of the pressure measuring point. The drag coefficients on the upper and lower surfaces of modules, denoted as $C_{Du}$ and $C_{Dl}$, respectively, can be calculated by Equations (10) and (11):

$$C_{Du}=\frac{{\displaystyle \sum C_{pu}(i)}A(i)}{A}$$

(10)

$$C_{Dl}=\frac{{\displaystyle \sum C_{pl}(i)}A(i)}{A}$$

(11)

where $A(i)$ is the tributary area of measuring point i. Then the shape factors of upper and lower surfaces of each module, denoted as ${\mu }_{su}$ and ${\mu }_{sl}$, respectively, can be obtained by Equations (12) and (13):

$${\mu }_{su}=\frac{C_{Du}}{\cos \alpha }{\left(\frac{U_H}{U_z}\right)}^{2\alpha }$$

(12)

$${\mu }_{sl}=\frac{C_{Dl}}{\cos \alpha }{\left(\frac{U_H}{U_z}\right)}^{2\alpha }$$

(13)

As the wind pressure $P_u(i)$ and $P_l(i)$ are positive if the surface is under compression, the total shape factor on each module can be determined by ${\mu }_s$ = ${\mu }_{su}$ − ${\mu }_{sl}$. If the wind pressures are nonuniform in their distribution on the module surfaces, a moment ($M_y$) will be generated, as shown in Figure 2, and can be calculated by Equation (14):

$$M_y={\displaystyle \sum (P_i-P_{\infty })d_i}$$

(14)

Then the moment coefficients ($C_{My}$) can be acquired by Equation (15):

$$C_{My}=\frac{M_y}{0.5\rho U_{\infty }^{2}BW}$$

(15)

where $U_{\infty }$ is the ambient mean wind speed, and B and W are the size of module, as defined in Figure 2.

3. Results and Discussion

3.1. Preliminary Validation

Before showing the results of the current study for the tilted roof-mounted solar array, the preliminary validation of the overall computational modeling approach is presented for a small-scale roof-mounted solar array.

The building was 51.0 m in length, 19.0 m in width and 25.0 m in height. Five rows of solar modules were located symmetrically on the roof, with each row consisting of 44 modules. The size of each module was the same with that in numerical study. The tilt angle was 10°. The distance between rows was 1 m, and the setbacks from the two edges were 2.16 m.

A geometric scale of 1:50 was used to make the test model. The aerodynamic wind pressures on solar array were measured in the wind tunnel test. The pressure taps were drilled on half of the array because of symmetry, as shown in Figure 4. The wind pressures on the upper and lower module surfaces were measured. There were 48 taps on Row 1#, 3# and 5#, respectively, and 16 taps on Row 2#, 4#, making a total of 496 taps on the five rows of modules. The experiments were carried out at the ZD-1 Wind Tunnel Laboratory of Zhejiang University, a return-flow boundary layer wind tunnel of 18 m in length, 4.0 m in width and 3.0 m in height. The blockage ratio of the mode in the wind tunnel was about 1%, meeting the requirements in GB 50009-2012. The test terrain belonged to category-A, with a power exponent of 0.12. The wind velocity profile and turbulence intensity can also be expressed by Equations (4) and (5). The reference wind speed at the reference height (roof height of 50 cm) was 10.65 m/s. Pressure signals were acquired at a sampling frequency of 312.5 Hz over a period of 90 s. Thus, over 28,000 data points were recorded for each measured point. Measurements were taken for 24 incident wind angles at 15° interval for the full 360° azimuth, where 0° was perpendicular to the wide face acting in the shorter direction of the building, as defined in Figure 4. Figure 5 shows the model arrays mounted on the roof of low-rise building.

The computational settings of the validation case are the same as in Section 2.1 and 2.3, except that the scales of the building and solar array are smaller. To check the adequacy of the grid refinement, a grid sensitivity analysis was also conducted. Three grids with different cell numbers are shown in

Table 1. Cell numbers of Grid 1–3.

Grid Program	Inclination (°)	Cell Number
1	5	6,197,292
2	5	8,842,786
3	5	11,989,451

. The wind loads acting on the solar arrays were simulated using Grid Programs 1–3. Mesh around building in Grid Program 2 are displayed in Figure 6. Figure 7 shows the mean velocity profiles of approaching flow extracted from numerical simulation, experimental measurements and the Chinese standard GB 50009-2012. The mean wind speed profiles were normalized at the roof height for comparison, using a model scale of 1:50. The simulated and experimental mean profiles are in excellent agreement with the target profile from GB 50009-2012. Figure 8 shows the shape factors of Row 1#–5# calculated by wind tunnel data and CFD simulation data with Grid Programs 1–3. It could be seen that the simulated values are comparable with the tested values. The shape factors first decrease and then increase from Row 1# to 5#, and the smallest values in magnitude were taken at Row 3#. The shape factors by Grid Programs 1–3 were quite close, indicating that the simulation results are insensitive to the grid number. Therefore, the overall computational modeling approach can be considered fully validated. In the following sections, the computational results of the target roof-mounted solar array with 12 rows will be discussed.

3.2. Mean Flow Field

The streamlines of the mean velocity over the configurations and wind directions are presented in Figure 9 and Figure 10. The figures show that a recirculation vortex forms at the leading edge of the building for all tilt angles, making the flow accelerate vertically to pass over the leading row of modules.

However, when the tilt angle and wind direction change, the flow within the recirculation vortex is significantly altered. At the wind direction of 0°, the flow approaches the module on the upper surface. With the increasing of tilt angle, the recirculation vortex is gradually divided by the modules into smaller local vortices. When the title angle α = 5°, the centers of local vortices are at the trailing edge of the building, while when the tilt angle increases to 40°, the recirculation vortex is dragged and elongated, and the local vortices are generated almost on each row of the module.

At the wind direction of 180°, the flow approaches the module on the lower surface. When α = 5°, the center of the recirculation vortex has left the tailing edge of the building. However, according to the study by Pratt and Kopp [3], when wind flow approaches the upper surface of a module at a 2° tilt angle, the center of recirculation vortex is located above the roof. The reason for the difference may be that the wind flow approaches the longer edge (51 m) of the building, generating a larger size of recirculation vortex, which is far larger than the building width (35 m). The local vortices for 20° tilt angle are also larger than those in the study by Pratt and Kopp [3], because the module size is larger than the latter, with the length of 1.65 m and the width of 1.0 m. The streamlines at 180° are also different from that at 0°, especially for the 40° and 55° tilt angles. At 180°, the local vortices underneath the first two rows rotate faster than at 0°, because the local vortices cannot move but rotate with the upper streamline of the circulation vortex due to the shade of module.

3.3. Shape Factors

In the Chinese [8], American [10] and Japanese [9] order, the solar array is divided into exposed and non-exposed zones. Based on the expose feature, the array is divided into six zones in this study, as shown in Figure 11. The dimensions of each zone are based on the sizes of modules and solar array. The dimensions of the corner zones (zones 1 and 5) are B × W, consistent with the module size. The remaining zones of Row 1#, 12#, and the ends of Row 2#–11# with width of W, are edge zones (zones 2, 3 and 6). Then the non-exposed part of solar array is the inner zone (zone 4). In the following study, the shape factors in each zone will also be investigated.

Figure 12 and Figure 13 show the contours of shape factors on the modules’ upper (${\mu }_{su}$) and lower (${\mu }_{sl}$) surfaces at the wind directions of 0° and 180°, which are calculated by Equations (12) and (13). Because of symmetry, only contours on one half modules are shown. At 0° wind direction, the shape factors on the upper surfaces are relatively comparable with that on the lower surfaces, and both are negative. When tilt angle $\alpha \le 30°$, the shape factor on each row changes gradiently along with the incoming flow. The shape factors reach the maximums in magnitude of Row 12#, and decrease first due to the shading effect, then increase due to the reattachment of local vortices shown in Figure 9, from Row 11# to Row 1#. The shape factors reach the minimums at Row 6# and 7#. When the tilt angle increases from 5° to 30°, the shape factors on upper and lower surfaces increase because the windward surfaces of rows increase and the flow accelerates at the leading edge of building. When $\alpha =30°$, the maximum shape factor is −1.4 at the end of Row 12# (leading row). After this tilt angle, the shape factor decreases. When $\alpha =55°$, the shape factor at the end of Row 12# decreases to −0.84. The happens because when $\alpha =30°$, the velocity of flow at the leading row is the largest.

The shape factors at θ = 180° exhibit similar features with those at θ = 0° except that when $\alpha \ge 40°$, the shape factors on the lower surface of Row 1# (leading row) are positive. This indicates that the wind pressure on leading row is significantly influenced by the incoming flow as the windward area increases, and less influenced by local vortices, while the rear rows are sheltered by the front rows, and their wind pressures are mainly affected by the local vortices.

It can be seen in Figure 11, Figure 12 and Figure 13 that there is a clear partition of shape factors for the outer and inner modules. At θ = 0°, when $\alpha \le 40°$, the shape factors in inner modules are less than the outer module because of the shading effect. However, when $\alpha =55°$, the local vortices almost attach on all rows, making the wind pressure comparable on the whole module. At θ = 180°, the shape factors on the upper and lower surfaces of leading row (Row 1#) are larger than that of the trailing row (Row 12#). By dividing Row 1# and 12# in the same zone, the wind pressure on the leading row may be underestimated, while the wind pressure on the trailing row may be overestimated. However, in ASCE 7-22 2022 [10], the leading and trailing rows are in the same zone.

The shape factors on modules in zones 1–6 were calculated according to the zone division (see in Figure 11). The results are presented in Figure 14 and Figure 15. As the shape factors on the upper and lower surfaces are close, the shape factors in most zones for tilt angle from 5° to 55° are quite small. However, shape factors in the leading row (zones 5 and 6 at θ = 0°, and zones 1 and 2 at θ = 180°) for 30°, 40° and 55° range from −0.2 to −0.4. This indicates that the shading effect of the leading row significantly reduces the shape factors of the rear row, and local vortices generate comparable wind pressure on the upper and lower surfaces of the rear module. When wind approaches the upper surface at θ = 0°, α = 30° is the most unfavorable tilt angle, with the maximum shape factor of −0.28 at the corner zone (zone 5). When wind approaches the lower surface at θ = 180°, α = 40° is the most unfavorable tilt angle, with the maximum shape factor of −0.38 at the corner zone (zone 1). The shape factors in the leading zone (zone 6 at θ = 0°, and zone 2 at θ = 180°) are slightly smaller than the corner zone.

The maximum shape factors were extracted from Figure 14 and Figure 15 for six tilt angles, and compared to those estimated by Chinese [8], American [10] and Japanese [9] standards. Figure 16 shows a comparison of shape factors predicted by simulations and recommended by codes. It should be noted that the method proposed in ASCE 7-22 [9] is not suitable for a tilt angle greater than 30°. The shape factors increase with the tilt angle, as expected. In the American standard, the shape factors in the end and corner zones, which are defined as exposed zones, are much larger than those in the center zone, and larger than those determined by Chinese and Japanese standards. This may be because the height of the building and solar module above the roof are considered in the American standards, while they are not in the Chinese and Japanese standards. However, as the wind field is closely related to the building height, the specifications in the Chinese and Japanese standards with few parameters about building and module size cannot estimate the shape factor convincingly. The simulated shape factors are much smaller than those determined by the standards, supporting the conservative nature of the standards.

3.4. Moment Coefficients

The nonuniform wind pressures on a module generate a moment around the center axis of module, as shown in Figure 2. Figure 17 and Figure 18 display the moment coefficients in the six zones for tilt angles from 5° to 55°. At the 0° wind angle, the moment coefficients in most zones and tilt angles are close to 0, except for the leading zone (zones 5 and 6) when tilt angle α = 40° and 55°. This is because in most zones and tilt angles, the wind pressures (shape coefficients) on the modules are uniform (see in Figure 10). However, when α = 40° and 55°, the leading row is subject to the combined action of incoming flow and recirculation vortex, making a certain gradient of wind pressure distribution, then the moment is generated. Similar to 0° wind angle, the moment coefficients are close to 0 in most zones and tilt angles at 180° wind angle. However, when α = 55°, there is a relatively large moment in the leading row (zones 1 and 2), and the maximum moment coefficient is 0.28, which is much larger than that at 0° wind angle (0.06). A local vortex attaches on the lower surface of leading row, while no vortex attaches on the upper surface. As a result, the wind pressures on the upper and lower surface of leading row are negative and positive, respectively, leading to a relatively large moment coefficient. Unfortunately, the wind induced moment on solar modules are not considered in the Chinese, American or Japanese standards, which may make the wind resistance design of the solar module support structure unsafe.

4. Conclusions

A detailed computational investigation of wind field around tilted solar modules mounted on a large building roof was undertaken. The incoming flow was simulated for various module tilt angles at normal wind directions relative to the wall by using the RANS approach, supplied with the SST $k-\omega$ turbulence model. Then the shape factors and moment coefficients of modules were explored. The main findings are as follows.

The mean flow is significantly accelerated at the leading row of solar array, and a recirculation vortex is generated above the array. Because of the modules, the recirculation vortex is disintegrated to smaller local vortices. With the increasing of tilt angle, more local vortices are generated and attached on the front rows, leading to a relatively large wind pressure and shape factor at the corner of array. Although in most tilt angles the shape factors are negative, for the 40° and 55° tilt angle at 180° wind direction, the shape factors on the lower surfaces are positive, due to the dominating of approaching flow rather than the local vortices.

For a solar array with 12 rows, the wind pressures on the modules are nonuniform in their distribution, and can be divided to the corner, end and center zone. As the shape factors on upper and lower are close, the shape factors in most zones for tilt angle from 5° to 55° are quite small. However, shape factors in the leading row for 30°, 40° and 55° are relatively large. This indicates that the shading effect of front rows can significantly reduce the shape factors of the rear rows. Comparing the shape factors estimated by computational simulation and the Chinese, American and Japanese standards, the values obtained by simulation were much smaller than the standards. From a safety perspective, the standards for wind resistant design of solar modules are quite conservative.

The moment induced by nonuniform wind pressure is often ignored in the literature and standards. However, for the large tilt angles (30°, 40° and 55°) at 180° wind direction, the moment coefficients are quite large at the leading zones, with a maximum of 0.28 for 55° tilt angle. This is caused by positive wind pressure on the lower surface and negative wind pressure on the upper surface. Ignoring wind induced moment on the leading zones may make the wind resistance design of a solar module support structure unsafe.