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Fig. A.1.2.4.2.1 Normal Force Coefficient vs. Angle of Attack (Brian Budzinski) On the other hand, the axial force should be the greatest when flying directly into the relative wind, or at a zero degree angle of attack. As the angle of attack is increased, the axial force should decrease. This can be shown through Fig. A.1.2.4.2.2. Fig. A.1.2.4.2.2 Axial Force Coefficient vs. Angle of Attack (Brian Budzinski) Now we are ready to further discuss the performance of the drag coefficient versus angle of attack. Understandably, the drag coefficient increases with increasing angle of attack. This behavior can be seen through Fig. A.1.2.4.2.3 below. The addition of the wing will generate a drag coefficient of approximately 1.1 at a 90 degree angle of attack, as shown by Fig. A.1.2.4.2.3. Fig. A.1.2.4.2.3 Drag Coefficient vs. Angle of Attack (Brian Budzinski) A similar process can be used in order to determine the drag imparted through the addition of fins. Equation (A.1.2.4.1.1) and Eq. (A.1.2.4.1.2) still apply; however, the axial and normal force coefficients will be different. In order to determine the normal and axial force coefficients, we must look at Eq. (A.1.2.4.2.4) below. If we assume a pair of fins, (A.1.2.4.2.4a) (A.1.2.4.2.4b) where C_{N} is the normal force coefficient, R_{F} is the radius of the fin(s) leading edge, l_{F }is the length of the fin(s), k_{LE} is the correction factor for the leading edge, S is the reference area, Λ_{F} is the sweep of the fin(s), α is the angle of attack, C_{A} is the axial force coefficient, S_{F} is the fin area, and λ is the correction for the sweep angle.^{1} To help us better visualize this configuration, refer to Fig. A.1.2.4.2.4 on the following page. Fig. A.1.2.4.2.4 Launch Vehicle with a Pair of Fins (Kyle Donahue) Similar to the wing, once we know the axial and normal force coefficients for the fins, those values can be inserted into Eq. (A.1.2.4.1.2) in order to determine the generated drag. If a delta wing and a pair of fins are added to the launch vehicle, the individual axial and normal force coefficients are summed to determine the total axial and normal force coefficient, much like Eq. (A.1.2.4.2.3). For a pair of fins and a delta wing configuration the total axial and normal force coefficient is calculated as shown through Eq. (A.1.2.4.2.5) below.^{1} (A.1.2.4.2.5a) (A.1.2.4.2.5b) These values can then be inserted into Eq. (A.1.2.4.1.2) in order to determine the total induced drag generated by this configuration. A.1.2.4.3 Moment and Moment Coefficient Now that the drag and drag coefficient have been thoroughly covered, let us discuss in further detail the pitching moment that is incurred. As aforementioned, the addition of a wing will increase the nose up pitching moment, thus allowing the launch vehicle to pitch into a vertical configuration. Let us discuss this phenomenon in more detail. In order to determine the pitching moment by the addition of a wing, we once again must divide the wing up into two separate sections: the leading edge and the lower surface. The pitching moment coefficient for the leading edge is calculated by means of (A.1.2.4.3.1) where C_{m }is the moment coefficient, C_{N} is the normal force coefficient about the leading edge, x_{LE} is the axial distance from the leading edge to the center of mass, c is the chord, C_{A} is the axial force coefficient about the leading edge, and z_{LE }is the radial distance from the leading edge to the center of mass.^{1} Similarly we find the moment coefficient about the lower surface (A.1.2.4.3.2) where C_{m }is the moment coefficient, C_{N} is the normal force coefficient about the lower surface, x_{LS} is the axial distance from the lower surface to the center of mass, c is the chord, C_{A} is the axial force coefficient about the lower surface, and z_{LS }is the radial distance from the lower surface to the center of mass.^{1} Comparable to the total normal and axial force coefficients, the total moment coefficient is found by summing the leading edge term and the lower surface term. As one may assume, the moment coefficient will increase with increasing angle of attack. This is because the upward pitching exposes more of the lower wing surface to the relative wind, increasing the force applied. This increase in moment coefficient versus angle of attack can be seen through Fig. A.1.2.4.3.1 on the following page. Fig. A.1.2.4.3.1 Moment Coefficient vs. Angle of Attack (Brian Budzinski) In order to calculate the moment coefficient for the addition of a pair of fins, the mathematics become a little more involved. We now can calculate the moment coefficient for a pair of fins using Eq. (A.1.2.4.3.3). (A.1.2.4.3.3) where most of the variables were defined by Eq. (A.1.2.4.2.4) above, and x_{F} and z_{F} are the axial and radial distances from the fin leading edge to the launch vehicle center of mass respectively.^{1} Once the moment coefficients have been calculated, we determine the pitching moment using Eq. (A.1.2.4.3.4). (A.1.2.4.3.4) where M is the moment, C_{m} is the moment coefficient, q is the dynamic pressure, S is the reference area, and c is the chord length. Though it may be difficult to tell from the previous equations, through the addition of a wing, the nose up pitching moment is increased. Seeing as the wing is mounted on the first stage of the rocket, it is aft of the aerodynamic center. Since the moment caused through the addition of the wing is aft of the aerodynamic center, the launch vehicle pitches upward. A.1.2.4.4 Lift Coefficient Lift is yet another important aerodynamic characteristic that should be reviewed. Any structure or body can generate lift once an angle of attack is encountered. Moreover, the addition of a wing, referred to previously as a lifting body, will create lift due to reaction forces. The lift force is the equal and opposite force created from an object, such as an airfoil, turning the relative fluid flow perpendicular to its original direction. Therefore, the lift coefficient, much like the drag coefficient, is calculated using the axial and normal forces as shown in Eq. (A.1.2.4.4.1). (A.1.2.4.4.1) where C_{L }is the lift coefficient, C_{N} is the normal force coefficient, α is the angle of attack, and C_{A} is the axial force coefficient.^{1} As expected, the lift coefficient increases with increasing angle of attack. We can see this through Fig. A.1.2.4.4.1. Fig. A.1.2.4.4.1 Lift Coefficient vs. Angle of Attack (Brian Budzinski) At approximately 53 degrees angle of attack, the wing reaches the maximum lift. Once the angle of attack is pushed beyond the maximum, the lift begins to decrease dramatically. Though an angle of attack of 53 degrees may seem excessive for a traditional configuration, for a hypersonic vehicle with a delta wing design, this is commonplace. To help better understand this phenomenon, let us briefly discuss how a delta wing generates lift. A delta wing uses vortices to generate lift rather than straight air flow. Since straight flow is disrupted by high angles of attack, a traditional wing becomes dysfunctional at high angles. However, with a delta wing configuration, high angles of attack increase vortices, thus increasing the lift.^{2} Additionally, the relationship between lift and drag is shown in Fig. A.1.2.4.4.2. Fig. A.1.2.4.4.2 Drag Coefficient vs. Lift Coefficient (Brian Budzinski) A.1.2.4.5 Shear Coefficient The final aerodynamic force that we discuss is shear force. A shear force occurs when shear stress is encountered. Shear stress is defined as the stress that acts parallel or tangential to the face of a material as opposed to normal stress which acts in a perpendicular manner. Though the details of shear stress are not thoroughly covered in this section, particularly because shear is a structural problem, the results from the addition of a wing and/or fins are covered. For the simplicity of an aerodynamic viewpoint, the shear force imparted on the launch vehicle through the addition of a wing is considered equal to the normal force acting on the wing itself. This concept is more easily seen through Fig. A.1.2.4.5.1 below. Fig. A.1.2.4.5.1 Shear Imparted on the Launch Vehicle by the Wing (Brian Budzinski) Therefore, as the angle of attack of the wing increases, the normal force also increases. This increase in normal force thus increases the shear induced on the launch vehicle. The maximum shear coefficient is found to be approximately 1.1 which can be shown through Fig. A.1.2.4.5.2 below. Fig. A.1.2.4.5.2 Shear Coefficient vs. Angle of Attack (Brian Budzinski) The analysis of the shear induced on the launch vehicle from the addition of fins follows suit. We find the shear force imparted on the launch vehicle through the addition of fins by assuming that it is equal to the normal force acting on the fin itself. Once again, this can be more easily shown through Fig. A.1.2.4.5.3 below. Fig. A.1.2.4.5.3 Shear Imparted on the Launch Vehicle by the Fins (Brian Budzinski) A more in depth analysis is required in order to determine the cost effectiveness of fins. We neglect to go into great detail of this matter. The addition of fins would require less stabilization control from D&C. However, the method for stabilization control that we implement does not require the addition of fins. In summary, the use of a wing and/or fins is very beneficial if an aircraft launch configuration is to be considered. The additional nose up pitching moment is advantageous if the launch vehicle is launched from a horizontal configuration. Furthermore, fins are a favorable method for stabilizing the rocket as they eliminate the need for a costly thrust vectoring method. References ^{1} Hankey, Wilbur L., ReEntry Aerodynamics, AIAA, Washington D.C., 1988, pp. 7073 ^{2 }Rhode, M.N., Engelund, W.C., and Mendenhall, M.R., “Experimental Aerodynamic Characteristics of the Pegasus AirLaunched Booster and Comparisons with Predicted and Flight Results”, AIAA Paper 951830, June 1995. A.1.2.5 Computational Fluid Dynamics As computer technology has greatly advanced, it has become an industry standard to use Computational Fluid Dynamics, CFD, as a preliminary form of aerothermodynamic analysis. A cheaper alternative to wind tunnel testing, CFD allows engineers to obtain accurate solutions to a variety of aerothermodynamic concerns. Because most aerodynamic theory falls apart in the transonic regime, it is hard to get accurate results using basic equations and analytical solutions. It is much more accurate to create a mock up of the launch vehicle and place it in a wind tunnel to retrieve physical results. Creating a mock up of the launch vehicle becomes a very time consuming and costly task however, when the design begins to advance. As the design progresses, the launch vehicle geometry begins to change; since most aerothermodynamic loads are based on geometry, they are constantly changing as well. Every time the geometry of the launch vehicle changes, a new launch vehicle mock up needs to be built, and more wind tunnel tests need to occur. The alternative to these costly wind tunnel tests is CFD. A CFD analysis can output the same type of information as a wind tunnel test in a timelier, more cost effective manner. Instead of paying for new launch vehicle mock ups to be created with each change in geometry, changes can simply be made in a computer aided design, CAD, software program such as CATIA, ProEngineer, or SolidWorks. CFD can then be completed for each phase of the design, and costs associated with wind tunnel testing become obsolete. Completing a CFD analysis on a launch vehicle can be broken down into a four step process:
The results can then be used to determine whether or not the aerothermodynmic loads exceed tolerable values. If they do, a new design will need to account for these loads, and if not, more analysis can be done on other components of the launch vehicle design. What makes CFD nearly as accurate as wind tunnel testing are the numerical methods imbedded internally within the CFD program. By meshing the CAD model first, the launch vehicle is broken down into small pieces. When placed into the CFD program, solutions to NavierStokes equations are integrated across each of these small pieces, and summed in order to solve for a multitude of aerodynamic loads. Outputs can range from pressure, temperature, and velocity distributions to coefficient of drag, coefficient of pressure, and moment coefficient acting on the launch vehicle. CFD is an incredibly advantageous tool because it allows for geometry changes as well as environmental changes to be taken into consideration. By specifying the appropriate boundary conditions one can change the speed and angle of attack of the launch vehicle, account for changes in temperature, density, and pressure of the surrounding atmosphere, and even include viscous effects and shock waves. Due to the cost, time, and inaccessibility of a wind tunnel, we decided to use CFD as a means of determining aerothermodynamic loads at designated intervals throughout the launch. In order to exploit Fluent’s symmetry capability we created a model of half of the 1 kg launch vehicle using CATIA. Splitting the launch vehicle in half reduces the complexity along with the amount of time to needed to solve the problem. We then saved this model as a “.igs” file, and imported into GAMBIT. Once in GAMBIT, the model was nearly ready to be meshed. In order to account for the fact that air flows around the launch vehicle and not through it, the area surrounding the launch vehicle model needed to be meshed, rather than the launch vehicle itself. To do this, we created a large rectangular prism surrounding the launch vehicle. The launch vehicle geometry was then subtracted from this rectangular prism leaving only the area surrounding the launch vehicle to be meshed. To begin, we meshed the edges of the rectangular prism with a spacing of 0.8. Next, the longest symmetry plane edges of the launch vehicle were meshed with a spacing of 0.13, and the smallest symmetry plane edges of the launch vehicle were meshed with a spacing of 0.05. Using the edge mesh sizes as guides, we meshed the faces of the launch vehicle and the faces of the rectangular prism next. We created both of these face meshes using a triangular mapping pattern. Finally, the volume surrounding the launch vehicle was meshed using a tetrahedral hexcore pattern. The results of the mesh can be seen in Fig. A.4.1.2.5.1 below. 