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References ^{1}Sutton, George P., and Oscar Biblarz. Rocket Propulsion Elements. New York: John Wiley & Sons, Inc., 2001. ^{2}The Martin Company, “The Vanguard Satellite Launching Vehicle”, Engineering Report No. 11022, April 1960. ^{3}Toft, Hans Olaf. “Aerolab” Software. [http://users.cybercity.dk/~dko7904/software.htm. accessed 1/30/08]. A.1.2.3.2 Design Considerations Aerodynamic forces quickly emerge as a major component of the Project Bellerophon design process. Center of pressure, normal and axial forces, pitching moments, bending moments and shear stresses all require analysis. In a more detailed design process (perhaps one with an operating budget), our analysis would include extensive wind tunnel testing. As a class however, our hands are tied to theoretical models and numerical solutions. If we were to build this launch vehicle (LV), wind tunnel testing would be absolutely necessary to ensure its operability. The following sections describe in detail the process by which we predict all of the aerodynamic forces on our launch vehicle. Our aerodynamic forces code was the main platform from which we offered solutions for the other members of the design team. D&C, Structures, and Trajectory were all affected by its results. The seed from which the code grew was the search for a valid center of pressure (CP). We surmised early on in the design process that the CP would be required by the dynamics and controls group. The CP is the point along the LV body where the various forces acting on the body act as one force (and by extension, one moment). Our initial research on this subject yielded several interesting processes by which we could calculate a reasonable CP location. The simplest method of determining the center of pressure is one very familiar to the model rocket builders of the world. For these cases, maintaining a CP behind the center of gravity (CG) is necessary for static stability. In subsonic conditions, a conservative estimate for the LV’s CP is located at the center of its lateral area.^{1} For an amateur rocketeer, a common way of using this information is to make a thin cardboard cutout in the shape of their rocket, and suspend the cutout across a sharp edge, like a ruler. Since cardboard is of uniform density, and is assumed to be of negligible thickness, the point about which the cutout is stable is the rocket’s CP. As a method of doing this computationally, we set up a computer code to determine the projected area of each launch vehicle section, using a triangle for the nose, a rectangle for the cylindrical stages, and a trapezoid for the “shoulder” or skirt sections. The code then summed these sections from the nose to determine the overall area. Halving this value, we designed the code to step from the nose until it reached the half area, and then determine the fraction of the current section that was on either side of the center point. This resultant point was the CP by the lateral area method. There were a few problems associated with this method of finding the Center of Pressure. First, the model essentially assumed an angle of attack of 90º, which is only present as a launch stand condition for most LVs. As a result of this initial assumption, the CP location is very conservative for the purposes of small LVs, and more importantly, not necessarily indicative of an actual value. Instead it gives a maximum limit for the CG if static stability is required. For the purposes of the design project, this stipulation was unnecessary, since with the use of gimbaling or LITVC, static stability is unnecessary, or even undesirable. With this in mind, our efforts turned toward the Barrowman Method. This is a method of analytically determining the CP by using the LV’s geometry. The advantages of this method are several: it gives a much less “conservative” location for the CP, it is relatively simple to calculate, and it is relatively accurate for the conditions it is designed for. The method involves dividing the LV into several portions (nose, cylinders, shoulders/boat tails, and fins), and determining the surface area and volume of each section. These geometric components are directly related to the coefficients of normal force and pitching moment (C_{N} and C_{m} respectively) as follows: where L is the length of the section, S is the crosssectional area of the section at the given location, d is a reference length equal to the diameter of the base of the nosecone, and V is the volume of the section. If we take X_{CP} to be the location of the center of pressure, then where C_{M }is computed from the tip of the nose cone. By computing X_{CP} for each section, it is possible to determine the overall location as so: For a more detailed treatment of the Barrowman method, please refer to “The Theoretical Prediction of Center of Pressure”.^{1} Using Vanguard geometry, we designed a computer program to calculate X_{CP} using the Barrowman method. This gave us a result of 25.9% of the body length from the tip of the nosecone. This was where we ran into Barrowman’s limitations. The report states a series of rather strict assumptions, including that the angle of attack is approximately zero, and that the flow is both steady state and subsonic. The Vanguard report published a wind tunnel center of pressure graph that starts at the beginning of the transonic regime, but it suggests that the subsonic CP was around 40%. Furthermore, the Vanguard report includes data for the CP vs. angle of attack, which deviates quite significantly from the initial zero case. The Barrowman Method is unable to account for this change and thus we renewed our search for a serviceable aerodynamic model. We then considered a third model, Newtonian Theory. The advantages of this model are apparent simplicity, and a high degree of accuracy even at very high angles of attack. The main disadvantage is that the method is only valid at very high Mach numbers; starting at about Mach 5 – hypersonic speeds. It was certainly a valuable tool if we found our LV would reach those speeds, but lacked the range of Mach numbers we would need for the overall design. For the purposes of Project Bellerophon, our aerodynamic model required several characteristics. First, we needed it to function over a large range of Mach numbers; i.e. we needed subsonic, supersonic, and perhaps even hypersonic values. Since transonic flows are poorly understood even by state of the art computational models, we were forced to “fudge” these values in the final design, keeping some eye to historic launch vehicles – realizing that wind tunnel testing would be required for proper analysis here. We also needed our model to accurately predict CP’s across a range of angles of attack. A study of the Vanguard report revealed that our angle of attack range would probably not progress past six degrees, but we wanted to be prepared for as much as fifteen. ^{4}_{ } We searched through aerodynamic texts and consulted knowledgeable professors, and finally came upon the most serviceable method for our needs. Linear perturbation theory allowed us to compute all of the major aerodynamic forces in a relatively simple fashion. By computing coefficients of pressure over the surfaces of the launch vehicle, perturbation theory gave us the building blocks of normal forces, axial forces, shear stresses, bending moments, and the ever elusive center of pressure. It is important when using linear perturbation to study the theory’s underlying assumptions. The first is that the flow is steady and isentropic. The second is that the airflow is irrotational. Third, that the flow is inviscid. The theory goes on to assume that the changes in the vehicle geometry (the perturbations) are small, and that the vehicle is at a small angle of attack. The resulting equation is as follows where is the velocity potential, x and y are Cartesian axis directions, and M is mach number.^{2} The above equation may be written as well in cylindrical coordinates as follows where φ is the perturbation velocity potential, r is the radial direction and x is the axial direction.^{3} Using Eq. (A1.2.3.2.6), the term that governs C_{p} for slender, axially symmetric bodies is where θ (please note that theta is used for more than one quantity in this report) is the angular direction in the cylindrical coordinate system and u_{0} is the free stream velocity.^{3} The problem that the aerothermodynamics group ran into was the difficulty in applying the velocity potential φ – itself a differential equation – to a “real world” problem. As a result, the decision was made to use a method more correct for airfoils than for launch vehicles, for which a simpler equation was available. The reason the equations are not the same is that the shape of an airfoil allows certain cross flow velocities to be neglected.^{3} Due to the small size of our LV, we decided that the simplification was reasonable. Also, because we continue to integrate around the longitudinal axis of the LV, our theory retains some of the accuracy that would otherwise be lost. Discussion with knowledgeable professors and accuracy of our final numbers has served as justification of our choice. ^{4, 5, 6} We make use of an equation from Anderson to calculate our data: where θ here is the geometric angle of the launch vehicle geometry with respect to the free stream velocity, and C_{p},_{0} is the incompressible pressure coefficient, for which we also used 2θ. We implement Eqs. (A.1.2.3.2.8) and (A.1.2.3.2.9) respectively, using the code CP_Linear_two.m, which runs off the master call_aerodynamics.m. We use the lengths and diameters of each vehicle section to find the angle θ of the geometry. We then take angle of attack (α) into account by adding α to θ for the lower surface of the LV and subtracting α from θ for the upper surface geometry. We compute C_{p} at many different points along the launch vehicle surface at regular intervals, creating a pressure distribution vector. The pressure coefficients form a distribution along the launch vehicle body as shown below. Please note that the geometry used for the figures in this section is not final. Please refer to the detailed design of the report for numbers related to our final designs: Fig. Section A.1.2.3.2.1: Pressure distribution over the length of a 3 stage launch vehicle at Mach 4.5 and 0° angle of attack (Alex Woods) We can see here that geometry, Mach number, and angle of attack are the primary variables that affect pressure distribution. As geometry changes, the shape of the C_{p} spikes change, with higher, thinner spikes coming with shorter, higher angle changes in geometry. The overall magnitude of the distribution changes with Mach number and the difference between the upper and lower surfaces grows with angle of attack. A.1.2.3.3 Normal Force Coefficient Once CP_Linear_two.m forms pressure distributions, we can integrate those distributions to derive the aerodynamic forces acting along the LV body. The first of these is the normal force coefficient, C_{N}. We can integrate using the equation: where S is a reference area in square meters, r is the radius of the LV at a given point in meters, L is the overall vehicle length in meters, and θ is the angle of the LV, in radians, with respect to the windward point. For the purposes of Project Bellerophon, we use the base of the first stage as the reference area. Within CP_linear_two.m, we make this equation work by first integrating numerically around the LV body for the lower and upper surface, resulting in an “average” C_{p} for each. This is then integrated along the axial direction by subtracting the upper surface C_{p} from that of the lower surface, giving a resultant pressure difference, and multiplying by the radius and the step size (taken to be 0.1 meters in the analysis). Summing and dividing by the reference area, we compute C_{N} for the launch vehicle. The behavior of the resulting coefficient may be seen in the figures to follow. Fig. A.1.2.3.3.1, Normal coefficient vs. angle of attack for a 3 stage launch vehicle (Alex Woods) Fig. A.1.2.3.3.2, Normal force coefficient vs. vehicle length at 6º aoa and M = 3.5 for a 3 stage launch vehicle (Alex Woods) Fig. A.1.2.3.3.3, Normal Force Coefficient vs. Mach number for a 3 stage launch vehicle at 0º angle of attack (Alex Woods) We can see from Fig. A.1.2.3.3.1 that normal forces increase with angle of attack. Also we see from Fig. A.1.2.3.3.2 that C_{N} is distributed over the length of the launch vehicle in a fashion similar to that of the C_{p} distribution. Of note is that for zero angle of attack, the output of C_{N }from CP_Linear_two.m is nonzero, when theoretically it should be zero. This is caused by a flaw in the code that could not be resolved before the conclusion of this project. The slope of the C_{N} vs α curve should be steeper than is represented as well. Furthermore, theory predicts a linear relationship between C_{N} and α, but in the real world this relationship is nonlinear, with C_{N} increasing at a greater rate than predicted. This nonlinearity begins around 6º angle of attack, and becomes too great to neglect at least as early as 14º. Finally, bear in mind that the values within the transonic region of the graph are of placeholder value only; they are not based on any valid theory. A.1.2.3.4 Moment Coefficient Directly related to the coefficient of normal force is the pitching moment coefficient, C_{M}. We chose the pitching moment to be the moment about the nose, caused by the normal force acting at the center of pressure. This quantity is determined theoretically as such: A.1.2.3.4.1 where z is the distance of the current point from the tip of the nose cone. By this method, each point along the distribution vector generates a separate moment, and the magnitude of that moment tends to increase as we move along the body of the launch vehicle. By summing the vector (integrating in theory) we calculate the overall scalar value of C_{M}. Since C_{M} is directly related to C_{N}, the changes of C_{M} with angle of attack and Mach number are very similar in behavior, as can be seen in the figures to follow. 