Wind tunnel

The primary device used in this research was the open-circuit wind tunnel located at the Autonomous University of Yucatan, Mexico (Figure 2). The wind tunnel has a 1.5 m - long test area with a cross-section of 1 m². In the current research, a uniform wind flow was considered without generating the Atmospheric Boundary Layer (ABL) profile. This can be associated with a structure in a Terrain Category 0 or I according to EN 1991-1-4 (2005) and ^{CFE (2020)}, respectively.

Figure 2 Wind Tunnel at the Autonomous University of Yucatan 

The analyzed model

The research aimed to obtain the pressure coefficients of an air-inflated membrane structure composed of six circular arches. The prototype structure dimensions (Figure 3) are 13.0 m height and length, and 26.0 m width. The inflated arches have a diameter of 3 m.

Figure 3 Dimensions of the prototype structure: a) front view, b) top view, and c) cross-section of the inflated walls 

The equilibrium shape of the inflated membrane was determined by the Dynamic Relaxation Method (DRM), which is a step-by-step method developed by ^{Day, 1965}. DRM is often used for the form-finding and static analysis of different tensile structures (^{Barnes, 1988}; ^{Hegyi et al., 2006}; ^{Lengyel & Hincz, 2019}). The synclastic surface of the inflated membrane was approximated by a network of planar, triangular surface elements during the form-finding. A linear and orthotropic material was considered with the same modulus of elasticity in mutually perpendicular fiber directions (warp and fill). The warp fiber direction was assumed to be parallel with the centerline of the inflated arches. The applied material properties were: E _w =E _f =400 kN/m, G=10 kN/m, v_wf=v_fw=0.3. The pressure in the tubes was 25 mbar during the form-finding procedure. Figure 4 shows the inflated equilibrium shape of the numerical model.

Figure 4 Equilibrium shape of the inflated model 

After the form-finding, the equilibrium shape of the inflated but unloaded membrane was 3D printed. The material used by the 3D printer was Polylactic Acid (PLA), with a layer thickness of 0.25 mm and a hatch filling ratio of 15 %. The two side walls were 3D printed with similar characteristics but as independent parts. Figure 5 shows photos of the model with and without side walls.

Figure 5 Model of the structure: a) with side walls, b) without side walls, and c) top view 

Considering that the blockage ratio required for wind tunnel experiments should not be higher than 10 % of the cross-section testing area (^{Meseguer, 2001}), the selected geometric scale was 1:72.5, which allowed a proper study in the considered wind directions with a maximum blockage ratio of 5 %. Hence, the height and length of the model were 17.93 cm, and its width was 35.86 cm (Table 1).

The next step in the modeling was to determine the distribution of the pressure taps. There were nine measurement points on every arch; the angle between the adjacent measurement points on the same arch was 10°. The pressure taps were composed of two parts, the sensor allocated at the measurement holes on the model surface and the tube which connects the sensor to the pressure transducer. Since the pressure taps were conducted through the arches wall, one hole with two diameters was drilled at every measurement point; the smaller diameter was used for the pressure tap itself (sensor) and the larger one, for the tube on the opposite side of the current arch. According to the model symmetry, measurement holes were drilled on two opposite quarters of the model (Figure 5c). It means that three arches were modeled to measure the pressure on the external surface, and the other three were used to measure the internal pressure. This arrangement was considered to measure both external and internal pressures on the structure.

By proper rotation of the model in the wind tunnel, the 27 external and 27 internal measurement points provided pressure values in 102 points on each, the external and the internal sides of the model for every wind direction. On the one hand, this density of the measurement points enables a smooth interpolation between the measured pressure values, as well as there is a convenient distance between the measurement points to fit the pressure taps. Besides the measurement points on the external and internal surfaces of the inflated arches, there were 15 measurement points on each side-wall, which resulted in altogether 54 pressure values on the two side-walls.

The pressure taps were installed into the model in groups during the tests. The groups of active taps were selected to minimize the disturbance of the flow by the tubes around the active taps. Therefore, the measurements of external and internal pressures were not recorded simultaneously.

Both cases of the model (with and without end-walls) were analyzed in three wind directions 0°, 45° and 90°. The wind directions are measured from the structure’s central axis (Figure 6).

Figure 6 Applied wind directions 

The pressures were recorded using a pressure scanner Esterline 9116, which can take up to 16 simultaneous measurements. As a control technique, all the tests were repeated at least three times, improving the validity of the results. The sampling time for each test was 120 seconds, and the sampling frequency was 10 Hz.

It is usual to refer all pressure values measured on a structural surface to the mean dynamic pressure (12 ρU∞2) of the free stream wind at a certain distance from the structure (^{Simiu & Yeo, 2019}). The non-dimensional Pressure Coefficients, Cp, are defined as:

where p is the local pressure in a specific point on the structure surface detected by the pressure tap, p _∞ is the nominal undisturbed free stream static pressure, p is the density of air, and U_∞ is the velocity of the undisturbed flow at a reference point distant from the model. These non-dimensional values make it possible to transfer the experimental model results to the full-scale structure (prototype) to present the aerodynamic properties of the given geometric shape. During the experiments, the value p - p _∞ for each point was measured by the pressure scanner in Pa and the mean wind velocity (U_∞) was obtained in m/s using a Pitot tube. For the calculation of the pressure coefficients, standard air conditions with p = 1.23 kg/m³ density was considered. The setup of the experiment with the model and the Pitot tube can be seen in Figure 7.

Figure 7 Setup of the experiment 

A sensitivity analysis was carried out before all the experiments; this included the evaluation of the velocities, sampling frequency, and time length, which contributed to register the stabilization of the flow turbulence around the model. In the sensitivity analysis, speeds between 15-20 m/s were considered, but the conclusions were that velocities above 19 m/s would give steady pressures. Then, the applied velocities in all tests were 17, 19, 21, and 23 m/s to verify that in all directions 19 m/s was the lower boundary velocity to register stabilized pressures. This criterion of ±2 m/s was determined just to evaluate if there could be any significant difference when the wind speed changes. This way it was verified that pressure coefficients recorded with wind velocities equal or higher than the lower boundary velocity were above the transition gap of the Reynolds number and thus independent of its value (stabilized turbulence region). The Reynolds number defined in Equation (2) is a non-dimensional parameter which indicates the relation between the inertial and the viscous forces (^{Simiu & Yeo, 2019}).

where p and U are the density and velocity of the air, respectively and their values were defined in previous paragraphs. The air viscosity (μ) was set up to 1.79e-5 kg/(m-s). Meanwhile, the characteristic length (L) was adopted according to each wind direction. The corresponding Reynolds number in each direction is presented in Table 2.

Figure 8 shows the pressure coefficients for different wind speeds and the average values for the current wind directions and side-wall conditions. In these diagrams, the Cp values are presented in two sets of points, A) on the external and internal surface of the arches in the symmetry plane parallel with the main axis, B) on the external and internal surface of the two central arches; either set A or B are also visualized in Figure 8. Table 3 presents the mean percentage difference with respect to the average Cp obtained for each side condition case and every wind direction. The mean percentage differences reported in Table 3 were computed based on Equations (3) and (4):

where Cp was the resulting average pressure coefficient, Cp ₁₉, Cp ₂₁, and Cp ₂₃ were the pressure coefficients obtained when the wind speed was 19, 21, or 23 m/s. The percentage difference for each point at a certain velocity was obtained by:

Figure 8 Mean Cp for the enclosed structure: a) 0° wind direction, point set A, b) 90° wind direction, point set B. Also, for the case of the open sides, 45° wind direction, point set B, c) mean external, and d) mean internal Cp 

Then the mean percentage difference was the average resulting from applying Equation (4) to all the points included in each configuration set (A or B). In most of the measurement points, the results showed that the wind speed had a minor effect on the Cp, which meant that the turbulence wake around the model was stabilized.

Pressure coefficients

In this section, the mean pressure coefficient (Cp) distribution on the external and internal surfaces of the inflated membrane and the end-walls is discussed. The following nomenclature is referred to in the figures: CPE and CPI, which mean External Pressure Coefficient and Internal Pressure Coefficient, respectively. Implicitly CPI values were presented only for open sides case (model without side walls). Furthermore, CPN refers to Net Pressure Coefficient. Evidently, for the enclosed structure, CPE is the same as CPN, but for the structure without end-walls, CPN refers to the sum of the mean external and internal pressures. The mean Cp‘s presented in this section was calculated as the average of the reported data from the three wind speeds mentioned above. All the graphics in this section have been plotted with the same color scale to compare the different cases easily. In the case of symmetric cases (wind directions 0° and 90°), the pressures were measured only on half of the surface; the color Cp maps were prepared by taking into account symmetry.

Enclosed structure, 90° wind direction

Figure 9 depicts the CPE on the whole membrane surface for the enclosed building with the flow perpendicular to the main central axis (90°). Both side-walls are parallel with the incoming flow; that is why there is wind suction only with CPE between -0.37 and -0.17. The Cp absolute values decrease from the windward to the leeward side. On the surface of the inflated arches, both positive and negative Cp’s were obtained between -0.76 and 0.75. As it was expected, the largest compression was at the windward side, close to the ground level, while on the top and leeward side, only suction was detected, with the minimum CPE values close to the top. It is important to recall that the fluid detachment on circular shapes has a strong relationship with the Reynolds number (^{Simiu & Yeo, 2019}). So, because of the turbulence developed around the model, it was the mid-central part where the flow separation took place. In this research the Reynolds number was not calculated but with the applied velocities it could be seen that the flow detachment for velocities over 19 m/s was approximately at the same location.

Figure 9 CPE distribution on the model, enclosed structure, 90° wind direction 

To validate the CPE results in this direction, Figure 10 depicts the standard CPE given in the EN 1991-1-4 (2005), ^{CFE (2020)}, and ^{ASCE (2010)} for circular cylindrical roofs with a rectangular plan. The presented line refers to points set B. The experimental curve follows a similar tendency as the values established in the codes. In the middle of the model, the experimental results underestimated the values given by the codes. The European and American standards provide conservative parameters in the whole suction area and on the positive pressure zones. However, the ^{CFE (2020)} only does it on the separation zone, meanwhile it underestimates the wind effects on the pressure zones.

Figure 10 Experimental CPE values compared with European, Mexican, and American standards 

Enclosed building, 0° wind direction

For this condition, the wind was parallel with the central axis of the structure. Figure 11 shows the CPE on the whole membrane surface. The windward end-wall blocked the incoming flow, which resulted in relatively high positive Cp between 0.28 and 0.86. Simultaneously, wind suction appeared on the leeward end-wall with almost constant Cp between -0.30 and -0.27. On the inflated arches, only negative pressures were detected; the largest wind suction appeared along the first arch, and the absolute Cp magnitudes decreased along the length. The reported CPE varied between -1.81 and -0.37. Unfortunately, the EN 1991-1-4 (2005) does not include any CPE for curved roofs in this direction, but there are some researchers who have been developing wind tunnel tests on curved roofs subjected to different wind directions and considering various geometric ratios (^{Natalini et al., 2013}; ^{Blackmore & Tsokri, 2006}). Both ^{CFE (2020)} and ^{ASCE (2010)} provide constant CPE for the whole surface, which depends on the geometric ratios of the structure. In this case, ^{CFE (2020)} included two cases with constant CPE equal to -0.90 and -0.40. At the same time, ^{ASCE (2010)} also provides two cases with values of -0.90 and -0.18. The worst case should be applied in the wind analysis.

Figure 11 CPE distribution on the model, enclosed structure, 0° wind direction 

Enclosed building, 45° wind direction

Figure 12 shows the CPE on the membrane surface obtained in the 45° wind direction. Compression was detected on most of the windward end-wall; the Cp varied from -0.06 to 0.62. Meanwhile, there was wind suction on the whole leeward wall, and the Cp range was from -0.72 to -0.50. On the surface of the inflated arches, there were wind suction and compression also. The largest positive pressure coefficient (0.41) was detected at the windward corner, close to the ground level. At the same time, the most significant wind suction appeared close to the top of the first arch at the windward side, caused by strong flow separations. The lowest pressure coefficient was -1.17.

Figure 12 CPE distribution on the model, enclosed structure, 45° wind direction 

Open sides case, 90° wind direction

In the case of open sides, the pressure was also measured on the internal surface of the inflated arches. Figure 13 shows both CPI and CPE for 90° wind direction. The pressure distribution on the external surface was similar to the enclosed case. The maximum CPE (0.73) was detected at the windward side, close to the ground level, while the minimum CPE (-0.80) was detected close to the top of the structure. As abovementioned, for the given geometry, the flow separation point depends on fluid regimes. In the case of fully developed turbulence around the model, the fluid separation took place at the mid-central part of the model. On the internal surface, all CPI values were negative; they varied between -0.64 and -0.04. The lower CPI values were detected close to the top of the model.

Figure 13 CPI and CPE distribution on the model, open sides structure, 90° wind direction 

For 90° wind direction, the CPN (Figure 14) varied between -0.27 to 1.11. The positive pressures on the lower part of the windward side were induced by the addition of the mean external pressure and internal suction. Furthermore, on the leeward side, the magnitude of the mean internal suction was larger than the magnitude of the mean external suction, resulting in positive CPN, except for small areas at the corners. The mean external suction in the model central zone was larger than the mean internal suction, resulting in negative CPN.

Figure 14 CPN distribution on the model, open sides structure, 90° wind direction 

Open sides case, 0° wind direction

In the 0° wind direction, all the CPE and CPI were negative (Figure 15). The magnitude of the wind suction on the external surface decreased from the windward to the leeward zone; CPE varied between -1.23 and -0.30. The CPI on the internal surface varied from -0.79 to -0.07. The minimum value was detected on the last arch at the leeward side of the model.

Figure 15 CPI and CPE distribution on the model, open sides structure, 0° wind direction 

The CPN map for 0° wind direction is presented in Figure 16. The maximum CPN was 0.18, while the minimum was -0.86. The positive CPN values were obtained in the leeward area; this represented that the mean suction was higher on the internal than on the external surface. Meanwhile, the CPN values were negative in the windward zone because the mean external suction was higher than the internal one.

Figure 16 CPN distribution on the model, open sides structure, 0° wind direction 

Open sides case, 45° wind direction

The Cp maps on the external and internal surfaces of the model are presented in Figure 17. On the inner surface, the obtained CPI varied between -1.20 and 0.60; it is important to recall that this was the only case with positive CPI. The maximum and minimum CPI were found on the first and last arch, respectively. The maximum was located close to the ground level. On the external surface, both the minimum and maximum CPE were found on the first arch; the CPE range was from -1.39 to 0.77.

Figure 17 CPI and CPE distribution on the model, open sides structure, 45° wind direction 

The CPN values for the 45° wind direction are presented in Figure 18. Those CPN varied between -1.28 and 1.36. The maximum wind pressure was detected at the windward corner close to the ground level because of mean external pressure and internal suction. At the same time, the largest suction was detected close to the opposite support of the same arch due to the mean internal pressure and external suction.

Figure 18 CPN distribution on the model, open sides structure, 45° wind direction 

External Pressure Coefficients (CPE) and Net Pressure Coefficients (CPN)

In the 90° wind direction, the wind was parallel with the side walls so, they did not mean a significant blockage between the upcoming wind and the external surface of the inflated arches. Figure 19a presents the CPE and CPN measured in point set B for the current flow direction. The distributions and the magnitudes of the CPE are similar to the values obtained in the open sides case; however, on a relatively small area at the central part of the model, the CPE values were higher for the open sides structure. The largest differences were detected at the top of the structure where the CPE obtained on the open structure was 1.5 times the CPE measured on the enclosed structure (-0.80 versus -0.53, respectively). The positive CPN values at the windward side are larger in the case of the open structure. Also, the absolute magnitudes of the negative CPN values are larger for the enclosed structure. Thus, the minimum CPN on the enclosed building was -0.65 while on the structure without end-walls was -0.27.

Figure 19 CPE and CPN, a) 90° wind direction, point set B; b) 0° wind direction, point set A; and 45° wind direction, c) point set B, and d) point set A 

When the upcoming wind direction was 0°, the windward end-wall of the closed structure meant a significant obstruction, and it had a substantial effect on the flow separation. Figure 19b presents the CPE and CPN obtained in point set A for this direction. Nevertheless, the diagram also shows that the maximum difference could be detected at the top of the first arch, where the magnitude of the CPE is more than two times larger for the enclosed condition than in the case without end-walls: -1.63 versus -0.62, respectively. So, the CPE on the open sides condition only represents 40 % of the CPE on the enclosed structure. The CPN values are significantly larger in the case of the enclosed model, especially on the windward side of the surface. For example, the CPN on the first arch for the enclosed and open model were -1.63 and -0.17, respectively; this means that the CPN obtained on the open structure only represents approximately 10 % of the one measured on the enclosed structure.

Figures 19c and d present the CPE and CPN for 45° wind direction in point sets B and A, respectively. Similarly to the 0° wind direction, on the surface of the first arch, the negative pressure coefficients are significantly smaller if the windward end-wall blocks the flow. In most of the points presented in Figure 19c and d, the CPE followed the same flow pattern in both end-walls conditions (closed and open). However, Figure 19d shows that the CPE measured on the first arch was larger for the closed structure case than the one obtained on the structure without end-walls (-1.07 versus -0.62, respectively). It can be stated that the negative CPN values are more significant on the enclosed structure; however, the positive CPN values are larger on the open structure. Figure 19c depicts that the maximum CPN on the enclosed model only represents the 24 % of the CPN obtained on the open sides building: 0.23 versus 0.97, respectively. This supports the idea about considering the analysis of openings in the structures and how they could affect the global wind forces.