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Project Bellerophon Appendix A.1.0 Aerothermal A.1.1 Introduction We discuss in detail all of the required aerodynamic and aerothermal information that is needed in order to launch a 200g, 1kg, and 5kg payload into low earth orbit. We cover everything from the research and development costs to detailed numeric values for all of the aerodynamic coefficients, aerodynamic loads, and aerothermal heating. Aerodynamic terms, such as coefficients of drag, lift, pressure, and moment (among others) are needed in order to predict the trajectory, devise the external structure, and confirm the control system. Generally aerodynamic characteristics are determined via wind tunnel testing; however, due to the nature of the coursework, making a model and performing wind tunnel tests is not viable. Therefore, all of our coefficients are devised numerically through extensive codes and analysis. The aerodynamic coefficients are predicted to the best of our ability through a multitude of engineering methods. One such method, Linear Perturbation Theory, is implemented in order to determine a majority of the aerodynamic coefficients. Programs such as Gambit, FLUENT, MATLAB, and EXCEL are also employed to aid in determining the aerodynamic coefficients. We will discuss our methods in detail, in the sections to follow. A.1.2 Design Methods A.1.2.1 Research and Development The key component for research and development from an aerodynamic standpoint is wind tunnel testing. Wind tunnel testing is essential in many aeronautical design processes. Wind tunnel testing gives us data on a scaled down model, which we can then relate to our current design. This data includes: drag, lift, moment, dynamic and static stability, surface pressure distributions, flow visualization, wind effects, and heat transfer properties. Due to the nature of the coursework, we are not using a wind tunnel for our design process. However, if a full design and build process were to be done, a wind tunnel would be necessary. The wind tunnel needs to be applicable for subsonic, transonic, supersonic, and possibly hypersonic regimes. We also need the wind tunnel to allow for changes in temperature, pressure, and density. This is because the launch vehicle will be traveling through the atmosphere where these values will vary, thus affecting the design parameters. We also need to take into account the scaling effects, flow blockage, presence of the model in the test section, and wall boundary layers. To simulate the real conditions, we must keep the dimensionless parameters constant when building our scaled down model (Reynolds, Mach, and Prandtl). Flow blockage occurs in wind tunnels of limited size when testing relatively large models. The blockage is defined as the ratio of the frontal area of the model to the area of the test section. Ideally blockage ratios of less than 5% are necessary for aeronautical testing. The presence of the model in the test section blocks the incoming flow and has the effect of increasing the pressure on the tunnel walls. The size of our scaled down model depends on the wind tunnel we use, and there are a variety of candidates available for us to use. The tunnel needs to take into account the blockage we mentioned earlier. Therefore, we investigated three different locations, which we chose based on the upper limits of the free stream velocity they could achieve. However, each location also has limits on the size of the model that can be used. The three locations are the NASA Glenn Research Center (GRC), located in Cleveland, Ohio, NASA Langley Research Center in Hampton, Virginia, and Purdue University in West Lafayette, Indiana. Both Purdue and GRC are able to reach a maximum test section Mach number of 6.0. Glenn Research Center also provides ten discrete flow velocities between Mach 1.3 and 6.0 for their 1’x1’ Supersonic Wind Tunnel (SWT). Glenn also has four additional and distinct wind tunnels located at the same facility. Table A.1.2.1.1 gives a comparison of the three different tunnels available, showing parameters which would be important to future testing.
Purdue’s “Quiet” Mach 6 wind tunnel is the most feasible for testing a scaled down model of our particular launch vehicle. It offers the cheapest running rate, the largest allowable model size, and also proximity since it is located near the main campus in West Lafayette. A.1.2.2 Sizing Function The purpose of this part of the project is to come up with a method for determining the shape of the launch vehicle. The first method we use is to linearly scale the vehicle by payload mass. To accomplish this, we use the dimensions of two rockets for data points to make the sizing functions; the Vanguard rocket and the Purdue Hybrid Launch Vehicle. This method, however, is ineffective at sizing the vehicle because it yields unrealistic overall lengths for small payload masses. We choose to abandon the linear scaling in favor of sizing the vehicle based on the volume of propellant in each stage. The method of sizing the vehicle based on fuel volumes yielded realistic lengths for every vehicle. The size was more realistic because it was based off of how much propellant each stage needs instead of a scaling factor based off of the payload mass. However, we had to manually optimize the length and diameter of the vehicle to obtain the final vehicle dimensions. Since this proved time consuming, we discontinued use of the Excel version due to a similar method employed in a large optimization code (MAT code). To begin the initial sizing of the vehicle, a sizing function was needed. We decided to size the vehicle by linearly scaling the Vanguard rocket based on payload mass. The linear relationship was calculated using Vanguard payload mass data along with stage length and diameter data found from an online source for historical rockets.^{1} For a second set of data points, the payload mass, stage length, and stage diameter data from the Purdue Hybrid Launch Vehicle were used.^{2} This data was then entered into Excel and a linear relation between length and diameter was found with respect to payload mass for each stage. An example of how the sizing functions were calculated is shown in Fig. A.1.2.2.1 below. Fig. A.1.2.2.1: Sizing function regression plot for vehicle second stage. (Chris Strauss) Figure A.1.2.2.1 shows the regression plot for the length of the second stage of the launch vehicle along with the sizing function associated with the stage. This was created by entering the data for second stage length of Vanguard and the Purdue Hybrid Launch Vehicle versus the payload mass of each. A linear regression line between the points was then plotted and the equation of the line was used as the sizing function for the stage length where x is the payload mass. We used a similar method on each stage length and diameter until a complete set of dimensions was calculated for each launch vehicle, and each different payload mass. The results of this scaling are shown in Table A.1.2.2.1 below for the overall length of the rocket. The results by stage are shown in Tables A.1.2.2.2 through A.1.2.2.4.
