1
A.1.2.4Lifting Bodies
Though lifting bodies are not implemented on the final design, they were still researched in order to determine a cost effective means of launch. Lifting bodies, such as a wing are beneficial for an aircraft launch. Wediscuss in detail the aerodynamic coefficients which include lift, drag, and moment that are created with the addition of lifting bodies. Lifting bodies create additional nose up pitching moments that would allow for the launch vehicle to pitch from an initial horizontal configuration, which is assumed to be angle of attack zero degrees, to a final vertical configuration, which is assumed to be an angle of attack of 90 degrees. This extra nose up pitching moment is needed if an aircraft launch configuration is considered.
To help us better visualize this configuration, refer to Fig. (A.1.2.4.0) below.
Figure A.1.2.4.0Launch Vehicle with a Delta Wing Configuration
(Kyle Donohue)
Though the pitching moment is a known benefit of the wing, induced drag is disadvantageous. Induced drag is defined as a dragforce which occurs whenever a lifting body or a finite wing generates lift. If all other parameters are held constant, the induced drag will increase with increasing angle of attack.Let us look deeper into this subject.
The induced drag is calculated using
(A.1.2.4.1)
where D is the induced drag, ρ is the air density, V is the true airspeed, S is the reference area, and CD is the coefficient of drag.1
It was previously noted that induced drag increases with increasing angle of attack. But this is not apparent from Eq.(A.1.2.4.1). Therefore, in order to see this relation we must further dissect Eq.(A.1.2.4.1). The variable that changes with angle of attack is the coefficient of drag. This is shown using
(A.1.2.4.2)
where CD is the coefficient of drag, CN is the normal force coefficient, α is the angle of attack, and CA is the axial force coefficient.1
The normal force and the axial force coefficients can then be computed for a lifting body. The derivation of the coefficients follow three basic steps: first we must determine the geometric shape of the body, next we must integrate the theoretical pressure coefficients over the body and evaluate the basic force coefficients, and finally we must determine the appropriate moment coefficients from the vehicle center of mass. All of the extensive integrations necessary to derive the aerodynamic force coefficients are omitted and only the results are presented.
For this analysis, we assumean aircraft launch, being that an aircraft launch is the only launch configuration that would require a wing. In order to determine the normal and axial force coefficients we make several assumptions. We implement the Newtonian Model; this assumption is made because the launch vehicleis traveling at supersonic and hypersonic speeds throughout most of the trajectory. We assume turbulent flow; once again this is a valid assumption due to the high speeds. Finally a delta wing configuration isemployed.
With the assumptions stated, we can now determine the axial and normal force coefficients. In order to determine the total axial and normal force coefficients we must divide the wing surface up into two separate parts, the leading edge and the lower surface. The leading edge and the lower surface are chosenbecause they are the two portions of the wing that are exposed to the relative wind given an angle of attack. The normal and axial force coefficients from the leading edge are found using
(A.1.2.4.3a)
(A.1.2.4.3b)
where CN is the normal force coefficient, RLE is the radius of the leading edge, lLE is the length of the leading edge, S is the reference area, kLE is the correction factor for the leading edge, Λ is the wing sweep, Λe is the effective wing sweep, α is the angle of attack, and CA is the axial force coefficient.1
Next we must look at the lower surface of the wing. The normal and axial force coefficients from the lower surface can be found using
(A.1.2.4.4a)
(A.1.2.4.4b)
(Laminar Flow)
(A.1.2.4.4c)
(Turbulent Flow)
where kLS is the lower surface correction factor, SLSis the lower surface area, S is the reference area, α is the angle of attack, Sw is the wing area, V∞ is the relative velocity, c is the chord length, μ∞is the relative air viscosity, , n = 0.5 laminar, n = 0.8 turbulent, and m is the planform taper ratio.1
Once we find the normal and axial force coefficients for the leading edge and the lower surface, the total normal and axial force coefficients are determined by summing the two.1
(A.1.2.4.5a)
(A.1.2.4.5b)
Now that the axial and normal coefficients are known, they can be substituted back into Eq.(A.1.2.4.2)in order to solve for the coefficient of drag. Prior to doing that though, let us first look at the behavior of the normal and axial force coefficients against angle of attack. Logically the normal force should be the greatest when the launch vehicle is at a high angle of attack. Therefore, as the angle of attack is increased, the normal force should also increase. This can be shown through Fig.(A.1.2.4.1).
Figure A.1.2.4.1Normal 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).
Figure A.1.2.4.2Axial 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.1.2.4.3) 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.1.2.4.3).
Figure A.1.2.4.3 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. Eq.(A.1.2.4.1)and Eq.(A.1.2.4.2)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.1.2.4.6)below. If we assume a pair of fins,
(A.1.2.4.6a)
(A.1.2.4.6b)
where CN is the normal force coefficient, RF is the radius of the fin(s) leading edge, lF is the length of the fin(s), kLE is the correction factor for the leading edge, S is the reference area, ΛF is the sweep of the fin(s), α is the angle of attack, CA is the axial force coefficient, SF 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.1.2.4.0a) below.
Figure A.1.2.4.0aLaunch Vehicle with a Pair of Fins
(Kyle Donohue)
Similar to the wing, once we know the axial and normal force coefficients for the fins, those values can be inserted into Eq.(A.1.2.4.2)in order to determine the generated drag. If a delta wing and a pair of fins are to be added to the launch vehicle, all of the individual axial and normal force coefficients would be summed in order to determine the total axial and normal force coefficient, much like Eq.(A.1.2.4.5). For a pair of fins and a delta wing configuration the total axial and normal force coefficient is calculated as shown through Eq.(A.1.2.4.7)below.1
(A.1.2.4.7a)
(A.1.2.4.7b)
These values can then be inserted into Eq.(A.1.2.4.2)in order to determine the total induced drag generated by this configuration.
Now that the drag and drag coefficient have been thoroughly covered, let us discuss in further detail the pitching moment that will be 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 more easily. 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.1.2.4.8)
where Cm is the moment coefficient, CN is the normal force coefficient about the leading edge, xLE is the axial distance from the leading edge to the center of mass, c is the chord, CA is the axial force coefficient about the leading edge, and zLE 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.1.2.4.9)
where Cm is the moment coefficient, CN is the normal force coefficient about the lower surface, xLS is the axial distance from the lower surface to the center of mass, c is the chord, CA is the axial force coefficient about the lower surface, and zLSis 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, thus increasing the force applied. This increase in moment coefficient versus angle of attack can be seen through Fig.(A.1.2.4.4) below.
Figure A.1.2.4.4 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
(A.1.2.4.10)
where most of the variables were defined by Eq.(A.1.2.4.6)above, and xF and zF 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
(A.1.2.4.11)
where M is the moment, Cm 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.
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 toreaction forces. The lift force is the equal and opposite force that is 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
(A.1.2.4.12)
where CL is the lift coefficient, CN is the normal force coefficient, α is the angle of attack, and CA is the axial force coefficient.1
As expected, the lift coefficient increases with increasing angle of attack. We can see this through Fig.(A.1.2.4.5).
Figure A.1.2.4.5Lift 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 point of maximum lift, the lift starts 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. Additionally, the relationship between lift and drag is shown.
Figure A.1.2.4.6Lift Coefficient vs. Angle of Attack
(Brian Budzinski)
The final aerodynamic force that we discuss is the 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 stress 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.1.2.4.7)below.
Figure A.1.2.4.7Shear 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.1.2.4.8)below.
Figure A.1.2.4.8Shear 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 stress 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.1.2.4.9 below.
Figure A.1.2.4.9Shear 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 would be 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., Re-Entry Aerodynamics, AIAA, WashingtonD.C., 1988,
pp. 70-73
2. Rhode, M.N., Engelund, W.C., and Mendenhall, M.R., “Experimental Aerodynamic Characteristics of the Pegasus Air-Launched Booster and Comparisons with Predicted and Flight Results”, AIAA Paper 95-1830, June 1995.
Author: Brian Budzinski