In practical use, the Dome will be subjected to daily wind loading. On occasion, it will be exposed to severe windstorms. Wind loads will produce a non-uniform loading over the surface of the Dome, as well as an overturning moment. In order to ensure that the Dome will be able to withstand such wind loads, the team has modeled severe wind loading using computational fluid dynamics (CFD).
Fluid mechanics analysis predicts that the velocity will be lowest at the upwind side of the Dome, where the Dome meets the ground, and on the downwind side where the Dome meets the ground. Also, the flow velocity should be highest at the top of the Dome, where the flow accelerates due to the Bernoulli effect. Essentially, the Dome acts as a large wing. The accelerated flow at the top causes a lifting force on the Dome, while the low pressure on the downwind side and high pressure on the upwind side cause drag. Moreover, flow should be symmetric about an axis through the diameter of the Dome parallel to flow.
The fundamental equations of fluid motion applied by the computational method are those of continuity, momentum, and energy, the Navier-Stokes Equations. The continuity equation makes use of the law of conservation of mass. The mass flow into a control volume minus the mass flow out of the control volume is equal to the accumulation of mass within the control volume.
The momentum equation makes use of the law of conservation of momentum. The momentum entering a control volume is equal to the momentum leaving the control volume plus the diffusion terms.
The energy equation performs an energy balance on a control volume. The rate of change of energy within the control volume is equal to the rate at which energy flows into the volume minus the rate of outflow. In this equation, The Dome has a 20 foot in diameter. It has three openings at the base for entry and exit as well as ventilation. It also has a small hole at the top for additional ventilation. In order to simplify the modeling process, the team approximated the Dome as a perfect hemisphere. This is an acceptable model, since the holes in the structure are insignificant compared to the overall size of the Dome.
The surface of the Dome using a program in the Fortran computing language. The coordinates used are standard spherical coordinates converted to x, y, and z. After generating the surface in Fortran, the author applied surface grids to it using the Grid-ed tool within Overgrid. The grids are most closely spaced at the base of the Dome and at the top of the Dome, since these are the points where the most severe discontinuities in geometry occur. The near body grids are splayed outwards at the edges of the grid, with a singular axis point at the top of the Dome. The grids are periodic in sweeping around the Dome, since the Dome is symmetric and the grids are evenly spaced. After generating the surface and near body grids in Overgrid, the team utilized Overflow to generate off-body volume grids (Reference 14.9.). These are used to analyze flow in areas far from the Dome itself. Overflow generated a total of 37 off body volume grids. The distance from the center of the grid system to the far-field boundaries is 100 feet. Since the Dome has a radius of 10 feet, this means that the grids extend 90 feet beyond the Dome. The overall combined number of grid points is 249,520.
A freestream velocity of Mach 0.2 was run through Overrun, a component of Overflow. For the assumed conditions of freestream air temperature = 288.16 K (15oC) and an elevation of zero meters. The surface of the Dome is viscous, therefore, the velocity at the surface is zero. The minimum CFL number, a stability condition, was set to 5.0. The ratio of specific heats is 1.4. The angle of attack is zero – this means that airflow is assumed to be horizontal. The second and fourth order dissipation factors were set to 2.0 and 0.04, respectively. The dissipation factors are used to minimize numerical error in the solving of the Navier-Stokes equations and to ensure stability. The model ran for 3000 time steps. The analysis ran on two appbots, a total of 4 processors. The appbots are part of a computer cluster, located in the CFD lab at Northern Arizona University. The job took 30 hours and 22 minutes to run. The overall processor efficiency was 74.9%.
The results of the CFD model were intended to be put into an FEA model of the Dome, in order to determine the stresses, deflections, and overturning moments acting on the structure. However, the FEA model of the Dome is made of sections of rebar, with wide gaps in between. The CFD model gives results acting upon a solid shell. The CFD model must be a solid shell, because the wind loads act upon a continuous surface.
The results obtained by the CFD model and apply the average values along the length of the rebar. However, that would lose the resolution inherent in a CFD model. Using industry standards to approximate the wind load on the structure and comparing the CFD model will be useful to validate the application of the industry standards to the Dome.
The Uniform Building Code (UBC) provides guidelines for applying wind loads to structures. According to the UBC, the wind load is a function of the average highest velocity of the wind, the type of terrain surrounding the structure, the height of the structure, and what the structure is used for. The UBC recognizes two components of pressure acting on a structure. One component tends to push the structure downstream (drag), and the other component tends to lift the structure off the ground.
In the event that the height to width ratio is 0.5 or less, the loads acting on the structure can be reduced by 1/3. Since the height to width ratio for the Dome is 10/20 = 0.5, the team can take advantage of this in its analysis. Based on these parameters, the total force pushing the Dome downstream is 10,895-pound force, or 0.722-pound per square inch on the surface of the Dome. The lifting force acting on the Dome is 11,734-pound force, which is equivalent to 0.389-pound per square inch on the surface of the Dome.
In applying these wind loads to the FEA model, it was assumed to act uniformly on the Dome. That is, the 10,895-pound force load is evenly distributed as a horizontal distributed load over the projected vertical area of the Dome (a semicircle of a diameter 20 foot). Similarly, the lifting force is applied as uniformly distributed vertical load acting on the projected horizontal area of the Dome (a circle of a diameter 20-foot).
In order to validate the applicability of the UBC on the Dome, the pressure was sampled at approximately six hundred points on the Dome within the CFD model. The team used Fieldview, a powerful post-processing CFD package, to work with the data generated by the analysis (Reference 14.10.). By averaging these values, the pressure allowed calculates the lifting force acting on the Dome. The total lift on the Dome, according to the averaged CFD model, is 10,675-pound force.
The values obtained by the CFD model and from the UBC are quite close to each other. The percent difference is 100 – (10,675/11,734)*100 = 9.03%. These values are within 10% of each other, which is considered good for a numerical method such as CFD. Therefore, the team is confident in the accuracy of the CFD results, and can conclude that the CFD model as valid.