Home

Methods for Analysis of Preliminary Spacecraft Designs AAE450

1. P Po T T Purdue University AAE450 Spacecraft Design August 18 2003 33 This type of skin friction analysis is much simpler in concept and use than integral boundary layer methods Results obtained using the reference temperature method were within 10 of results obtained using a more complex integral boundary layer method Ref cited in paper even at high hypersonic Mach numbers Also the computation time required by the reference temperature method is very small when compared to an integral boundary layer method Boundary layer transition is predicted using a correlation of the local transition Reynolds number Re with the local edge Mach number Me as follows logio Rez 6 421 exp 1 209 x 10 4M 72 This correlation is based on the experimental data of DiCristina ref cited in paper for transition on sharp cones at zero angle of attack This correlation is used due to the lack of better methods of transition prediction in hypersonic flows It is also presented in Anderson HHTG p 280 eq 6 140 For 450 we will go ahead and use this even at angle of attack Gustafson suggests estimating Me for your angle of attack using Me M cosa This crude formula neglects the bow shock but it will have to do Note that for Me larger than about 35 Re is greater than about 10 For this case the flow will certainly be laminar so computation of this very large number is not needed 7 8 2 Low Speed Skin Friction C
2. Sin 21 Sin 22 cos 0 10 For the case of the Space Station and Iridium the results are plotted in Figure 3 Purdue University AAE450 Spacecraft Design August 18 2003 intersection Orbit 1 e g Space Station Orbit 2 e g Iridium equator Figure 2 Spherical Trigonometry for Heading Angle Differences 10 Purdue University AAE450 Spacecraft Design August 18 2003 XY 17 Aug 1999 sphetrig for i1 51 6000 deg i2 86 4000 deg 180 170 160 150 140 130 120 110 o 100 90 80 70 60 50 40 30 20 10 a de 0 G 0 51 6 deg inclination orbit intersecting a 86 4 deg orbit whose ascending node is eastward Angle is the difference in heading angles at the first intersection above the ascending node Angle 9 is the difference in longitude of the ascending nodes For 6 0 o 34 8 for 0 180 a 138 the only simple cases 20 40 60 80 100 120 140 160 180 0 deg Figure 3 Spherical Trigonometry for Heading Angle Differences Purdue University AAE450 Spacecraft Design August 18 2003 11 Point A latitude longitude line ground equator Figure 4 Spherical Trigonometry for Ground Track 5 3 Ground Track and Heading Angle for a Planar Orbit To keep track of the position of the various orbits it is often easiest to compute the ground track for them and plot the ground tracks This requires the use of spherical trigonometry see for example t
3. For a turbulent flat plate the constant N 0 8 and there are two different cases for the others For V lt 3962 m s M 3 37 and Cz 3 35 x 1078 cos 8 sin 6 8a 9 T 556 4 1 1 11 gy 55 For V gt 3962 m s M 3 7 and Ca 2 20 x 107 cos 8 sin 82 5 1 1 119 56 Laminar turbulent transition will be discussed later 7 5 2 Flat Plate Heat Transfer Rate for Small Angles of Attack Here we need a method of computing the approximate heat transfer to the windward surface when M sing lt 1 The flat plate at zero angle of attack will be used to provide a crude approximation primarily based on White Viscous Fluid Flow 2nd edition McGraw Hill 1991 For a laminar case follow section 7 3 3 The reference temperature T is computed using 7 42 T T 0 5 0 039M 0 5 57 z 05 0039M 057 57 Here Te is the temperature at the edge of the boundary layer T is the wall temperature and Me is the Mach number at the edge of the boundary layer The Chapman Rubesin parameter C is then evaluated at the reference temperature using 7 40 T 1 3 z l 58 Here we will apply these formulas by crudely taking edge conditions as equal to freestream condi tions which is not so bad for small angle of attack Then compute the adiabatic wall temperature using 7 41a Tou 1 SURVER m2 59 where we will take Pr as a constant equal to about 0 72 Then the
4. of 1 sec usually using 4th order Runge Kutta atmo76 This subroutine supplies the local atmospheric conditions on the earth given an altitude A variant is available for Mars trajheader inc This is a file which is included by reference in a number of the subroutines It defines the number of solution variables used and the length of the solution arrays These parameters are defined once in this file so that the parameters are automati cally consistent across all the subroutines For example when the number of surface temperature solution points is changed these parameters may need to be changed aerotest for This test program calls aerotrim for for a range of c g locations It can be used with aeroprop for to test aerotrim for to see if the results are ok altvmap for This test program also calls aerotrim for but for a fixed angle of attack and a range of altitudes and velocities It can be used to generate data for Tecplot contour plots of L D skin friction and so on aeroprop for This main program generates data for a plot of moment coefficient vs angle of attack for different flap angles This plot is used to determine stable trim ranges for a vehicle This is needed for design purposes It calls 6 Purdue University AAE450 Spacecraft Design August 18 2003 aerodat for See above The system is structured so this same subroutine can be called by both aeroprop and aerotrim Thus you need only generate one version of the aer
5. 7T is the thickness ratio of the wing and Se is the wetted or exposed area For the case where the leading edge is subsonic use 93 16 Cow a cot A 94 e Sref Purdue University AAE450 Spacecraft Design August 18 2003 37 To get the lift take dC da from Figure E 4 assume it s constant and use Cr dC da a Note that the symbols in this figure are the same as in Fig E 5 and that A seems to be the aspect ratio in both figures You may want to fit a curve to the data in the figure although this is a little tedious Assume that your vehicle remains below the stall angle To get the drag due to lift induced drag Cp take Cp C from Figure E 5 7 9 4 Other Notes We will ignore subsonic and transonic flow in 450 For supersonic flow equation 2 15 in Nicolai s design book gives the moment coefficient but no way to determine the angle of attack at which it becomes zero For subsonic and supersonic conditions we will just have to assume that the vehicle can be trimmed Although subsonic and transonic aerodynamics is often a critical design issue for reusable launch vehicles that are required to land it is just too difficult to try to cover this topic in 450 7 10 Viscous Interaction Effects At high altitudes the density becomes very low so although the speed is often very high the Reynolds number is low This causes the boundary layers to become very thick The thick boundary layers now act to displace
6. Design August 18 2003 35 correct the area normalization in Cp to the reference area Add in the skin friction due to the turbulent boundary layer which is assumed to have an origin at the leading edge but which is only switched on at x This is done by computing the turbulent boundary layer skin friction from the leading edge and then subtracting the turbulent boundary layer skin friction from the leading edge to x The total skin friction drag coefficient is thus Ax Cp E C D turb l CD lam z Eg CD turb xe A 85 where Aref bl and Ar bx Adding in the AOA correction and simplifying P pr 70125 Cp 0 074 Rel 0 074 Re 1 328 Reps i saa 86 Here Reet PooVootr Hoo Rely p Voorr m and Re p V u As when computing the local heating interpret x as the distance from the stagnation point when applying these flat plate formulas to your more complex geometries Be careful with the reference areas in the drag coefficient formulas The most convenient one is the entire surface area But in the end you will have to normalize the complete drag coefficient for the entire vehicle by one reference area which must be used consistently in all codes To analyze a wing like shape use strip theory Think of longitudinal strips along streamlines Divide the wing into several strips For a delta wing you might have transition near the root and not near the tip Put a few strips on the
7. are however given in Tauber Bowles and Yang Use of Atmo spheric Braking During Mars Missions J Spacecraft and Rockets v 27 no 5 pp 514 521 Sept Oct 1990 Away from the stagnation point Tauber et al use correlations developed for flat plates and swept leading edges in air These correlations for air are similar to those shown above Purdue University AAE450 Spacecraft Design August 18 2003 31 7 6 Leeward Surface Heating Any surfaces that are shadowed from the flow have much lower heat transfer which is difficult to analyze Designs should use thermal blankets in this region One simple method is to use blankets similar to those used on the lee side of the Shuttle The weight of the lee TPS should be a small fraction of the windward TPS 7 7 Material Properties for Thermal Protection System Most of these may be obtained from the public domain NASA Ames website http tpsx arc nasa gov Most of the properties of the Ultra High Temperature Ceramics can be found from public domain papers to be distributed by the instructor e g reports by Clougherty et al A detailed discussion of the various materials was recently given by Rasky et al Thermal Protection System Materials and Costs for Future Reusable Launch Vehicles J Spacecraft and Rockets v 38 no 2 pp 294 296 2001 Other data will be handed out in class A TPSX database is also available at the cited website however distribution is limited If you do use da
8. becomes very large for small angles of attack small a Since the hypersonic Newtonian pressure depends only on local incidence angle it would seem that the flat plate at small a has the best possible L D Of course a flat plate has zero internal volume for cargo however a wedge with a top face that is parallel to the flow has the same Newtonian aerodynamics as a flat plate Fig 9 shows the geometry Here is the included angle of the wedge Note that a is defined with respect to the lower surface of the 22 Purdue University AAE450 Spacecraft Design August 18 2003 wedge The Newtonian lift and drag coefficients for a flat plate are given by Anderson on p 51 of HHGD These were coded into wedgeaero for taking into account both upper and lower surfaces along with the shadowing effect The resulting L D is shown in Fig 10 Of course the flat plate XY 26 Jan 2000 Newtonian Aerodynamics for Wedge WEDGEAERO flat plate 6 2 0 deg y7 6 4 0 p 6 6 0 5 8 0 5 10 0 5 12 0 5 14 0 5 16 0 5 18 0 5 20 0 i A i D D The angle of attack is defined with respect to the lower surface of the wedge The included angle of the wedge is 5 The shadowing effect is accounted for Adapted from S Matsumura LEOTARD AAE451 Fall 1999 Fig 3 5 10 15 20 a deg aa amy ZN T Sz Figure 10 Newtonian Lift to Drag Ra
9. in Annual Reviews of Fluid Mechanics v 31 pp 459 494 1999 Surveys overall issues and computational methods M D Griffin and J R French Space Vehicle Design AIAA Education Series 1991 R W Humble G N Henry and W J Larsen Space Propulsion Analysis and Design McGraw Hill New York 1995 S J Isakowitz J P Hopkins Jr and J B Hopkins International Reference Guide to Space Launch Systems Third Edition AIAA Publications Virginia 1999 Kinney David J J V Bowles L H Yang and C D Roberts Conceptual Design of a SHARP CTV AIAA Paper 2001 2887 June 2001 Good introduction to preliminary or conceptual design discussion of methods for a high L D vehicle with a UHTC nose Penland J A J L Dillon and J L Pittman An aerodynamic analysis of several hypersonic research airplane concepts from Mach 0 2 to 6 0 AIAA Paper 78 150 Jan 1978 General consideration of aerodynamic design issues including static stability margins and minimum lift coefficient for landing Purdue University AAE450 Spacecraft Design August 18 2003 43 11 12 13 14 15 16 Rathbone Robert R Communicating technical information a new guide to current uses and abuses in scientific and engineering writing Addison Wesley 1985 On reserve in the Engineering Library Reuther James et al A reusable space vehicle design study exploring sharp leading edges AIAA Paper 2001 2884 June 2001 One o
10. skin friction coefficient can be computed as 7 41b _ 0 664VC Cre V Reze Here Rez is the Reynolds number based on edge conditions and the arc length from the leading edge Use freestream conditions as a crude approximation for edge conditions We won t use this skin friction coefficient for the drag but will use the Reynolds analogy to get the heat transfer coefficient 7 41c 60 Che 0 5C pe Pr y 61 This Stanton number is then used to compute the wall heat flux using 7 44 dw ChepeUeCpelTaw re Tw 62 30 Purdue University AAE450 Spacecraft Design August 18 2003 For the heat capacity of air Cpe a value of about 1100 J kg K seems reasonable based on Bolz and Tuve Handbook of Tables for Applied Engineering Science Table 1 2 For a turbulent case the formulas in White section 7 8 1 are rather complex requiring solution of a transcendental equation These seem to be rather more sophisticated than is warranted for 450 Continued use of the reference temperature concept seems warranted instead compare the present section 7 8 1 The incompressible formulas are used but with the properties evaluated at the reference temperature See Bertin Hypersonic Aerothermodynamics AIAA 1994 section 7 4 1 We then have that Cye 0 027 Rell 63 te following White eqn 6 70 The Stanton number Che is again computed using the Reynolds analogy eqn 61 as in White 7 122 The heat transfer rate is the
11. some average temperature from your TPS temperature solution even though this will be a crude average over the surface Here Re has no prime but is the freestream value computation of the flow behind the bow shock is too complex for this course 7 8 4 High Speed Turbulent Skin Friction Drag Coefficient Equation 70 gives the local skin friction coefficient Recall that p and u are evaluated at the reference temperature T The viscosity is found using equation 69 Assuming constant pressure across the boundary layer and across the oblique shock which is a poor approx the perfect gas law then gives p T pT Using a derivation similar to that used for the laminar case we will arrive at an equation analgous to equation 76 x Cori 20992 ee jae 82 Simplifying and skipping a number of steps and noting that 1 25 0 0592 0 074 Cp r 0 074 Re 83 For a plate at AOA a Cpr 0 074 cos a Rel 84 7 8 5 Summing Laminar and Turbulent Skin Friction For a complete plate one starts by computing the transition Reynolds number and the transition length x If x gt l the length of the plate then the entire plate is laminar and the laminar formulas should be used If x lt l then we have some laminar flow followed by some turbulent flow For 450 assume that the plate is fully laminar up to x and use the laminar skin friction to here making sure to Purdue University AAE450 Spacecraft
12. the UHTC sharp leading edge Thus it appears that this critical effect has in general been confirmed by detailed simulations and flight tests For example see NASA TM 110407 A thermostructural analysis of a diboride composite leading edge by Kowalski et al July 1996 For these solid nosetips one often finds that a decrease in nosetip radius decreases overall heating The minimum feasible nosetip radius is then a matter of interest In response to a query regarding this minimum radius Paul Kolodziej of NASA Ames stated that We ve been focusing on leading edges with a radius of 1 mm If you operate the sharp leading edge on its aerothermal performance constraint it will rapidly reach a steady state constant temperature email 04 27 00 Thus this is the minimum radius which one should probably look at 7 4 5 Temperatures of Ablating TPS For an ablating TPS there is no equivalent to the above simple lumped capacity method The simplest practical approach is to use a one dimensional heat conduction code that includes surface ablation effects One such method is the SODDIT code see for example A User s Manual for the Sandia One Dimensional Direct and Inverse Thermal Code by B F Blackwell R W Douglass and H Wolf Sandia Report SAND85 2478 May 1987 A iterative process is used the heating 28 Purdue University AAE450 Spacecraft Design August 18 2003 rate history is first computed for the TPS after which the TPS ablati
13. the shock in significant ways See Anderson Hypersonic and High Temperature Gas Dynamics Chapter 7 for a discussion of these effects At sufficiently high altitude the gas density becomes so low that the mean distance between molecular collisions the mean free path becomes significant or large compared to the vehicle dimensions This is the rarefied flow regime 7 10 1 Lockheed Viscous Interaction Correlation In the past AAE450 analysis has used a crude correlation provided by Lockheed Missiles and Space in order to correct the Newtonian inviscid L D for these low density viscous effects Figure 13 shows the correlation provided by them to AAE450 in 1990 The chart included the formula plotted y 1 0 30 851 V I 577 02 V I 4285 94 VI 95 where y L D L D iny If the formula gives y lt 0 275 then y is to be set to y 0 275 Here L D iny is the Newtonian value To use this Lockheed viscous interaction correlation compute VI M 4 Rex and use the correlation to correct L D Use this to correct the drag coefficient only COED Newtonian L D actual 96 Cp actual D Newtonian The drag coefficients at high altitude will be high but the forces will be small anyway At 75 km this will start to affect Cp noticeably Will have an effect for awhile and then go away at low altitude Assume C7 is totally unaffected in this approximation Do this for the Newtonian part of Cp only the inviscid part on
14. various simple shapes which your vehicle will be a composite of Be careful to note however that Clark s results are given in the frame of reference of the vehicle axial and normal force and need to be translated to the frame of reference of the velocity vector lift and drag This step was omitted by most of the Fall 1998 groups causing errors of unknown magnitude in their results You must also make appropriate allowances for shadowing effects Pitching moment coefficients should be computed based on the inviscid Newtonian aerody namics While the skin friction will also contribute to the pitching moment it has a smaller effect Therefore to save the non trivial effort of taking them into account skin friction effects on pitching moment should normally be neglected Rarefied flow viscous interaction will also have an effect on the pitching moments see Section 7 10 however this effect is difficult to predict so again it will have to be neglected 7 3 Wedge Aerodynamics Some Design Issues For an aeroassist orbital transfer vehicle a high lift to drag ratio is clearly critical cp Section 5 4 The following is a first look at the issues involved for a vehicle with small nose bluntness The material is adapted from the work of Shin Matsumura AAE451 Fall 1999 LEOTARD group Anderson works out the lift and drag of a flat plate as part of his discussion in section 3 2 of HHTG In Fig 3 6 he shows that the lift to drag ratio L D
15. velocity or perhaps high orbit The last term in equation 50 represents cooling of the surface via radiation from the surface see for example Reynolds Engineering Thermodynamics 1977 sec 14 8 Here is the emissivity of the surface g is the Stefan Boltzman constant o 5 669 x 1071 W cm K and T is the wall temperature in Kelvin Using these units the radiation heat transfer will again be in units of W cm If cp is taken in J kgK py in kg cm tw in cm and t in sec the units will be properly consistent Equations 46 and 50 are combined and solved at each time step in the reentry in order to obtain the stagnation point temperature for a reusable TPS Purdue University AAE450 Spacecraft Design August 18 2003 27 7 4 4 Lumped Heating at Solid Nosetip For a small bluntness nosetip that serves as a massive heatsink formulas 49 and 50 don t accurately reflect the geometry Consider a nosetip which is basically a blunted cone for example with a nose radius that is some small fraction of the base radius The entire nosetip is solid TPS yet the formula treats the nosetip as a plate whose thickness is small compared to surface distances over which the heat transfer varies significantly For such a solid nosetip it seems better to consider the whole nosetip as one solid heat sink by integrating 49 and 50 over the solid nose We assume that the internal heat conduction is large so that the nosetip is appr
16. with a known transfer function see for example the appendix of the AE421L notes by Prof Rotea The equations of motion for the vehicle are nonlinear Although a local linearization could be carried out for specific locations the effort involved is probably prohibitive for this course However PID control can still be used as suggested by Rotea and as discussed in Appendix A 3 of the 421L notes The controller should not be connected until the vehicle is reasonably close 18 Purdue University AAE450 Spacecraft Design August 18 2003 SPS 14 Aug 2001 AEROPROP Icg 5 70 ycg 0 00 Iplate 10 00 Iflap 2 00 rnose 0 0020 Trim and Static Stability betadeg 1 000 deg for Blunt Flat Plate with flap betadeg 0 500 deg 0 001 betadeg is flap angle deg betadeg 0 000 deg a is angle of attack of plate oe ane sie Cuce is the moment coeff betadeg 1 500 deg about the CG must be zero x betadeg 2 000 deg for trim and have neg slope zero reference for stability 0 Oo o O 0 001 0 5 10 15 a deg Figure 7 Sample Trim Plot for Blunt Flat Plate with Flap Purdue University AAE450 Spacecraft Design August 18 2003 19 to the target trajectory In the cruise altitude example the controller turn on altitud
17. 41 8 3 Reaction Control System An RCS system is needed to control the attitude of the vehicle while it is out of the atmosphere or in the thin upper atmosphere The RCS system provides pitch roll and yaw In 450 we do a simplified design and select from available flight proven thrusters Data is available for various thrusters including the RS 45 43 25 42 and Peacekeeper engines Pairs of opposite thrusters should be placed in the fore and aft ends of the vehicle to give the biggest torque for a given thrust Save space for the engines and propellant Usually helium driven pressure fed thrusters have been used A feed pressure of 50 psi is sufficient The thrust of the RCS system is set by the need to be able to turn the vehicle around in preparation for reentry After the retroburn it is necessary to turn 180 deg for the reentry The vehicle has to be able to do the turn before the vehicle reaches the atmosphere The propulsion designer needs to look at the amount of RCS needed to do this turn and do some guessing to size the RCS system The time available for the turn should be estimated based on information from the trajectory people The turn can readily be computed using the torque angular momentum equation to get the angular acceleration A bang bang constant torque approximation will give a conservative estimate of the torque required Include a safety factor in your estimates Although the RCS system is used for translation as well a
18. 450 Spacecraft Design August 18 2003 Plane H oes J e E 3 a b Figure 1 Plane Section of Earth a with Vector Diagram in Plane H b fs Applying the law of cosines again allows one to find the angle as follows cos B V2 V2 V 2VV 5 Then Ye T 2 and thus the rotating values of velocity and heading angle are determined in terms of the corresponding inertial values The same figures can be used to work backwards from the rotating frame values to the inertial frame values Using Figure 1b 1 2 V V2 V 2V V cos 8 6 where 8 7 Ye Again using Figure 1b we then have cosp V V2 V2 QViV 7 Note that both of these relations were worked out based on vehicle velocities in quadrant I For retrograde orbits a modification of this analysis may be necessary 5 2 Spherical Trigonometry for Orbits This section gives a method for computing the differences in heading angle for two planar orbits that intersect away from the equator See Figure 2 where 7 and 72 are the inclination angles for the two orbits a is the difference between the two heading angles at the interaction point and 0 is the difference between the values of the RAAN for the two orbits Use the spherical law of cosines for angles cosa cos i cos m i2 sin i sin m 72 cos 9 8 or cos cos i1 cos iz sin sin iz cos 0 9 simplifying to COS Q cos 71 COS 22
19. 5 4 0 5 6 0 6 8 0 6 10 0 6 12 0 6 14 0 6 16 0 6 18 0 6 20 0 0 5 10 15 20 a deg Figure 11 Newtonian Lift Coefficient for 2D Wedge 24 Purdue University AAE450 Spacecraft Design August 18 2003 how small Cz is at a 5 deg where a 6 deg wedge has L D 11 Cr 0 0015 Although q 0 5pV is typically very large for a hypersonic vehicle these small lift coefficients may make lift to weight ratio a critical issue depending on altitude density and velocity Of course the drag coefficients are much smaller still for small a as shown in Fig 12 Here we see high drag until the upper wedge surface is shadowed after which the flat plate value is matched A slender wedge is required to shadow the upper surface at angles of attack where a high L D is possible This slender wedge may create problems with internal volume although these can be alleviated by increasing the vehicle length It is really the skin friction which will limit the slenderness used Increased slenderness means a longer vehicle and more surface area for the same internal volume at some point skin friction will dominate see design papers by W Hankey 7 4 Stagnation Point Heating Analysis The following contains the critical elements of the stagnation point heating analysis for Earth reentry as adapted from Prof Gustafson s notes Note that simplified methods for estimating heat transfer are also given in Hankey Reentry Aerodynamics AIAA Ed
20. 83 x 1078 2 Viis Tn 46 Purdue University AAE450 Spacecraft Design August 18 2003 XY 26 Jan 2000 Newtonian Aerodynamics for Wedge WEDGEAERO FOR The angle of attack is defined with respect to the lower surface of the wedge The included angle of the wedge is The shadowing effect is accounted for flat plate 2 0 deg y 4 0 s 6 0 i 6 8 0 5 10 0 o 6 12 0 as 14 0 16 0 v 6 18 0 6 20 0 20 Figure 12 Newtonian Drag Coefficient for 2D Wedge 25 26 Purdue University AAE450 Spacecraft Design August 18 2003 7 4 2 Stagnation Point Heat Transfer Rate Mars This can be taken from Tauber and Sutton Stagnation Point Radiative Heating Relations for Earth and Mars Entries J Spacecraft and Rockets v 28 no 1 pp 40 42 Jan Feb 1991 For entry velocities from 6 9 km s a similar correlation is given for stagnation point heating rates to a hemisphere If radiation can be neglected another relation is given in Tauber Bowles and Yang Use of Atmospheric Braking During Mars Missions J Spacecraft and Rockets v 27 no 5 pp 514 521 Sept Oct 1990 Eqn 6 in that reference can be used to obtain an equation similar to equation 46 es 8 P 173 04 Cowl w g 1 35 x 10 2l yey Be 47 See also p 3 of AIAA Paper 99 0348 New TPS Strategies for Planetary
21. Entry Vehicle Design by Olynick Loomis Chen Venkatapathy and Allen where this equation is discussed in a more general context 7 4 3 Stagnation Point Temperature Reusable TPS The temperature of the vehicle at the stagnation point depends on heat capacity and the balance of heat conduction and radiation In AAE450 for reusable TPS designs we ve evaluated the temperature of the surface at the stagnation point using a crude lumped heat capacity model This crude model is more refined than the even simpler approximation of radiative equilibrium to our knowledge it is the simplest approximation that allows determining a TPS weight From the first law AE X 4At 48 where E is the internal energy of an element of the surface at the stagnation point Per unit of surface area AE AT pata 49 where cp is the specific heat of the surface material per unit mass AT is the temperature change Pw is the density of the surface material and t is the thickness of the surface material This crude approximation assumes constant specific heats and neglects all heat transfer within the body but it is the kind of approximation used in preliminary design The heat transfer is composed of conduction from the fluid convective heat transfer and radiation dT ae Cpg Putu q dr coTh 50 where q is the radiation from the fluid to the surface We will neglect q for reentry from LEO it becomes significant mainly for reentry from escape
22. If the initial conditions for the differential equations are known in an inertial frame then they must be converted to the rotating frame before the solution begins In addition once the vehicle has left the atmosphere it is often convenient to perform analyses in the inertial frame for Hohmann transfers for example The following analysis was worked out by Prof Gustafson to handle this problem It will be assumed in the following analysis that the flight path angle is a small angle which would be the case for near circular orbit or during most of the reentry trajectory in a planetary atmosphere We are presently unaware of any reference book containing this analysis Consider an initial point at an altitude h and latitude Then the linear speed of rotation Vs is given by V rw 1 where w is the angular velocity of rotation of the planet and r is the radial distance from the axis to the point in question From Figure la it is seen that cos r re h 2 where re is the radius of the earth Thus we find that V w re h cos 3 In plane H which is perpendicular to re we must find the relation between the inertial flight velocity V and the rotating flight velocity V as well as the relation between the inertial heading angle w and the rotational heading angle pe This is done by applying the law of cosines to the triangle shown in Figure 1b giving Ve V2 V2 iV cosas 4 8 Purdue University AAE
23. Methods for Analysis of Preliminary Spacecraft Designs AAE450 Spacecraft Design Purdue University Written by Steven P Schneider Associate Professor Adapted from Methods Developed by W Gustafson Professor Emeritus School of Aeronautics and Astronautics Purdue University August 18 2003 Contents Introduction 2 List of Analysis Tools 3 Architecture of Framework Programs 4 Structural Analysis and Center of Gravity Analysis 6 Orbital Dynamics 6 5 1 Conversion Between Rotating and Inertial Coordinates 2 7 5 2 Spherical Trigonometry for Orbits 2 amp ve Bak eh ants bed a ete d ead amp 8 5 3 Ground Track and Heading Angle fora Planar Orbit 11 5 4 Single Point Cruise Design ge G hee kG Bah 6 A OUR AR 2606 4 AP a 14 pad Inclination Angle Changes lt 5 465 4 0 eae 6 ae Od eae 14 5424 Altitude Eff cts n e at Ochs 1 oh a es A we ae es doe es A eee es A We asso 15 Deo Reentry Trajectories na aie io oa ae cee nds he ia ENa ie d aena kar Be dal i a ana 16 Stability and Control 17 6 1 Vehicle Attitude Control oaoa a a 17 6 2 Design of Controllers for Trajectories o oo aa a eee eee 17 Aerothermodynamics 20 7 1 Atmospheric Conditions ooo aa a 20 T2 Pressure aa Carni yin Mies PR wd d So te da a de aa te ea ie a e a iia Pode 21 7 3 Wedge Aerodynamics Some Design Issues o0 o a a a pene 21 7 4 Stagnation Point Heating Analysis 600644 a a a 24 7 4 1 Stagnation
24. Point Heat Transfer Rate Earth 24 1 Purdue University AAE450 Spacecraft Design August 18 2003 7 4 2 Stagnation Point Heat Transfer Rate Mars 7 4 3 Stagnation Point Temperature Reusable TPS 7 4 4 Lumped Heating at Solid Nosetip o 4 44 Ate ae Ae ee ve 7 4 5 Temperatures of Ablating TPS 4 oka ak bee ae ee Ee eRe ES 7 5 Windward Surtace Heating Analysis 0 lt lt 625 decks Bi Rc Re Re 7 5 1 Flat Plate Heat Transfer Rate for Large Angles of Attack 7 5 2 Flat Plate Heat Transfer Rate for Small Angles of Attack 7 5 3 Heat Transfer to Wing Leading Edges 2 4 7 5 4 Heating Rates for Mare Entry 42 4 64 4 4 6 6 4 46 4 4 24 42 amp dk 7 6 Leeward Surface Heating 2 floes amp Sa Bk Ba a a ee a ee 7 7 Material Properties for Thermal Protection System 7 8 Hypersonic Skin Friction Analysis 2 Sa phew e eek es dee ak 7 8 1 Correlations for Local Compressible Skin Friction 2 7 8 2 Low Speed Skin Friction Coefficient and Drag 7 8 3 High Speed Laminar Skin Friction Drag Coefficient 7 8 4 High Speed Turbulent Skin Friction Drag Coefficient 7 8 5 Summing Laminar and Turbulent Skin Friction 7 9 Analysis of Supersonic Aerodynamics 2 0 0 a a Pool Tntrod ction lt a aor mait e ee Be ls eR ele ee el et a ee es CIZ Skin Friction Dra despre sake be B
25. acecraft Design August 18 2003 a For the TPS the specific heat cp is temperature dependent Fit a curve to this data and use the local value of cp T 2 26 p 3 b The internal temperature of the aluminum Shuttle has to be held below 350 F Will see a peak temperature during reentry perhaps at 70 80km then temperature will drop For some trajectories the total heat load can be an issue but usually not 3 26 Orbital Intercept The period of an inner circular orbit is less than that of an outer orbit To get in the right relative position to intercept can circularize in a larger or smaller orbit and then wait a bunch of orbits 2 26 p 4 Four AV s drop orbit circularize wait raise orbit circularize Instead can go to an elliptic orbit but have to hit target Choose period of elliptic orbit by major axis wait to line up Then one AV to start another to circularize when arrive at target 3 5 p 3 Some References J D Anderson Hypersonic and High Temperature Gas Dynamics HHTG for short McGraw Hill New York 1989 Reprinted by AIAA Publications Fall 2000 H Ashley Engineering Analysis of Flight Vehicles Dover New York 1992 R D Bate D D Mueller and J E White Fundamentals of Astrodynamics Dover New York 1971 G F Franklin J D Powell and A Emani Naeini Feedback Control of Dynamic Systems 2nd edition Addison Wesley 1991 Gnoffo P A Planetary Entry Gasdynamics
26. altitude change One then determines y by solving Try y h 43 simultaneously with the other equations Begin with a small amount of filtering by using a very small value for the parameter Ty Note that if Tp 0 the original system is recovered Since excessive filtering can make the system unstable proceed cautiously Consult Prof Rotea for further suggestions 20 Purdue University AAE450 Spacecraft Design August 18 2003 S P Schneider Purdue AAE 28 Aug 2001 atmo76 and Mars atmospheres in SI units 1 0 Pearth Pmars 350 10 j ere 325 10 i eee 14300 3 Pe 10 0 4275 gt 210 Z A 42505 e i 225 10 7 200 p 175 10 150 0 50000 100000 altitude m Figure 8 Properties of Atmospheres of Earth and Mars 7 Aerothermodynamics Simplified aerodynamics are also used to allow multidisciplinary design iterations The vehicle shape is generated by adding together simple shapes like cones and cone frustra flat plates spherical sections and so on A more sophisticated surface panel method of defining the geometry wo
27. ample Rotea suggests u t ky ho h t kah t 42 where k gt 0 and kg gt 0 Here we assume increasing u gt 0 increases the force that tends to increase h otherwise change the sign of kg Be wary of including feedback on an ho t term in the controller of changing the target altitude within the controller since this can cause instability In the target cruise altitude example a number of trajectories could be simulated with increas ing values of the gains until a suitable trajectory is obtained The properties of the PID controller will be discovered through this process In particular it will be found that it is impossible to reach and remain at the target altitude with kg 0 using only information about the current altitude the vehicle will simply oscillate about the target altitude To achieve the target altitude without large oscillations it is essential to include information about the rate of change of the altitude This information provides the damping term in the mass spring damper analogy In some cases small high frequency oscillations may remain particularly in the bank angle or other control parameter These can be removed by low pass filtering the bank angle or other parameter such as the rate of change of altitude This is done by adding a differential equation to the system For example if is the rate of change of altitude we can create a new parameter y hy where h y is the filtered rate of
28. an be a small portion of the maneuver If V is maintained near orbital speed can be near 90 degrees and the efficiency can approach L D making an aeroassist vehicle useful and practical 5 4 2 Altitude Effects To maintain the cruise altitude we require thrust equal to drag This requires 1 T z V 4Cp 34 16 Purdue University AAE450 Spacecraft Design August 18 2003 where T is thrust and A is the reference area for the Cp computation We can also write equation 29 in terms of the lift coefficient Z V AC cosa m g V R 35 The lower the altitude the higher p is This increases both L and D in the same proportion Note that T L D coso m g V R 36 For a given L D ratio the allowable bank angle increases with maximum thrust and with cruise velocity If we normalize this equation we get T mg L D coso 1 V gR 37 7 T mg L D coso 1 RV t1y 38 j T mg L D coso 1 V ug R 39 7 T mg L D cos 1 V Vore 40 If V V then a 90 deg bank angle can be maintained independent of T mg and L D For smaller V these two parameters must increase to allow the constant condition cruise At lower altitudes heating is greater which will require a heavier thermal protection system TPS see Section 7 Clearly the altitude should be low enough to minimize viscous interaction drag Section 7 10 in order to obtain maximum L D Heating is then the domin
29. ant concern can V be maintained near V r without excessive heating Otherwise it may be desirable to decrease V in order to reduce heating while increasing T mg in order to be able to maintain the altitude and high bank angle Note that the viscous interaction parameter VI M Re Approximate that the sound speed is fairly independent of altitude Then VI is proportional to V p From equation 46 the heat transfer seems approximately proportional to p V 5 5 Reentry Trajectories See Vinh for a discussion of these Chapter 5 covers methods of computing the retroburn AV to achieve a given flight path angle at atmospheric interface ye Using these techniques or trial and error one would normally try to obtain 1 5 gt ye gt 3 degrees An angle of 5 to 6 degrees may be too steep and will often cause excessive heating A lifting vehicle lets the trajectory person control the heating Typically the altitude oscillates during reentry if the angle of attack and bank angle are not modulated If the wing area is too small the vehicle will sink too fast The wing area or lift to mass ratio is a critical parameter Gus s sample vehicle has too small of a wing area Typically one runs some trajectories and tries out some modulation algorithms and evolves the design using trial and error For a shuttle like vehicle one usually holds a bank angle Purdue University AAE450 Spacecraft Design August 18 2003 17 of rou
30. bridge from the very low density free molecular region to the continuuum results This concept is discussed in various papers as are methods for computing free molecular effects If this issue is important to the present design these papers may be handed out and discussed in class 7 11 Sample Case Blunt Flat Plate with Body Flap The full set of aerothermodynamic methods was coded up by the instructor for a simple sample case The objective of this is twofold 1 to provide a good framework of FORTRAN programs with a reasonable architecture from which students can develop their own programs and 2 to show by example how to use the methods for a particular design The sample case was purposely selected to be an oversimplified geometry so that all groups will have to develop their own codes for their own vehicles It is nevertheless representative of an idealized hypersonic vehicle with best case performance so it can be used by the trajectory analysts in their early design stages before a complete vehicle design has been generated by their group The sample case is shown in Fig 14 It is a blunt flap plate with a body flap The nose is a hemicylinder with radius rnose shown as R Since the top part of the hemicylinder is shadowed when at angle of attack the force coefficients for the hemicylinder are different from those for the full cylinder The coefficients that were coded into the aerodat f subroutine were taken from Clark and Trimmer s
31. cated 2 of the fuselage length behind or downstream of the moment reference or c g of the airplane and the vehicle is said to possess a static margin of 2 For Penland s vehicle where this is 4 of the wing chord the static margin is said to be minimal but adequate Penland 1978 p 6 These simple requirements cause critical interactions between the various design components and make the design truly multidisciplinary For example a change in the location of the propellant tanks for the propulsion system will cause a change in the center of gravity which will thus cause a change in Cm Due to fuel usage Cm and the associated aerodynamics and stability may change during your trajectories and this needs to be monitored An example of a static stability diagram is shown in Fig 7 for the blunt flat plate that is discussed in Section 7 11 6 2 Design of Controllers for Trajectories In several cases it is desirable to control the vehicle so as to reach a target altitude or trajectory For example it may be desirable to reach a target cruise altitude during an aeroassist maneuver without excessive overshoot oscillations by appropriate control of the bank angle as a function of time How should the bank angle be set so as to achieve this goal Control theory can be used to help with this part of the design see for example the reference by Franklin et al Much of introductory control theory is limited to the case of a linear system
32. e Ee he Be Be AE 4 7 9 3 Wave Drag due to Thickness 0 2 0 00 00 eee eee TIA t Other Notes Sr 8 and 88a dae ds 1 Rea EH hie OAS ace tA es A pase a 7 10 Viscous Interaction Effects ark os geal edhe aaa ee ee al a a gal an og 7 10 1 Lockheed Viscous Interaction Correlation 04 7 10 2 Viscous Interaction Effects Bridging Formulas 7 11 Sample Case Blunt Flat Plate with Body Flap Propulsion Sul Boost Propulsion Ss 3h yates Wok ee Mak See Wa EES heehee 8 2 Vehicle PropulsiOMy 24k os hi wien te ee ee ath A OD eee oO hoes OES 8 3 Reaction Control System secs Lg fen a pee ea ee ee ee a ee ees 9 Other Design Information from Gus 10 Some References 1 Introduction 39 40 40 41 41 42 The following is a collection of notes describing the methods used for analysis of preliminary designs in the senior capstone Spacecraft Design course at Purdue University These notes were first collected in Fall 1999 They are preliminary and subject to continual revision Please inform the author of any errors you may detect It is very important to realize that for preliminary design you just try to get the big picture The analysis methods used here need only be accurate and reliable enough to give a correct idea Purdue University AAE450 Spacecraft Design August 18 2003 3 of the general tradeoffs between vehicle size shape trajectory and so on During the prelim
33. e would be another design parameter The control law in the time domain is u t kye t kael ki ere 41 where u t is the control signal e t is the error signal and kp ka and k are the gains associated with each component of the controller proportional derivative and integral In the example of the target cruise altitude e t ho h would be the difference between the current altitude h and the target altitude ho and u t would be the bank angle Rotea suggests setting k 0 at least initially If steady state errors occur when k 0 then increasing k should solve this problem although large k can destabilize a system Of the two remaining gains ky controls the extent and amplitude of the oscillations and k controls their frequency These parameters can be used in a trial and error study to obtain sufficient control Rotea suggests starting with kp kg 0 and slowly increasing the two one at a time starting with kp It is critical to keep close track of the signs of the feedback parameters which must be such as to provide damping and a restoring force when examined using a mass spring damper analogy In this analogy one notices that u t provides a restoring force so that u mch where m is the vehicle mass and c is some constant which is actually a constant only in a linear system It is immediately apparent that the system is analogous to the mass spring damper system In the cruise altitude ex
34. ection 2 1 3 equations 38 and 42 as discussed in the comments in the code The equations given by Clark and Trimmer are for a pair of swept cylinders so the results must be divided by a factor 2 See the check derivation given in the file hemicylinder pdf in the class directory The flap length is 1flap the plate length is lplate the angle of attack is defined with respect to the plate as a and the body flap deflection with respect to the plate is defined as 3 positive downwards The width of the vehicle is width 8 Propulsion The propulsion system is to be designed using an existing flight proven engine with a known weight ISP thrust and so on Your instructor is open to hearing of methods by which new propulsion systems can be reliably designed within the constraints of this course Prof Gustafson 40 Purdue University AAE450 Spacecraft Design August 18 2003 R Iplate Piped t yeg gt leg Iflap Figure 14 Schematic of Blunt Flat Plate with Body Flap was not able to find a set of analysis tools which could result in reasonable designs with realistic performance The properties of existing propulsion systems can be found in many places The best current source is Isakowitz 1999 Other sources include the Jane s handbooks Jane s Space Directory is one that we have and Marc Wade s Encyclopedia Astronautica on the web Data is also provided on p 234 and pp 302 304 of Humble and on p 282 and p 286 of Sutto
35. elocity and altitude are assumed constant and heating should be considered separately Figure 6 shows the geometry We want Ai Figure 6 Inclination Angle Change for Single Point Cruise Design to change from a velocity V at inclination 7 to a velocity V2 at inclination i Az For small At the same law of cosines used earlier gives Ai AV V 26 Consider a vehicle at bank angle 0 the component of the lift in the vertical plane is then L cosa while the component in the horizontal plane is Lsin For a small time At AV aAt where a is the acceleration in the horizontal plane L sina m where m is the vehicle mass Thus AV LsinoAt m 27 or Aisy LsinaAt mV 28 This same equation can be obtained from the third part of equation 2 31 in Vinh s book by looking at the equator where heading angle equals inclination angle latitude is zero and flight path angle is zero Purdue University AAE450 Spacecraft Design August 18 2003 15 Constant horizontal velocity is maintained by holding thrust T equal to drag D The single point design condition also assumes we have enough vertical force at this bank angle to maintain constant altitude which requires Lcoso m g V R 29 where R is the height above the center of the earth g ug R is the local gravitational acceler ation and this last term is the centripetal acceleration Outside the atmosphere L 0 and the two right hand terms balance to make a ci
36. es given such an initial state Another common issue is the magnitude of the retroburn that should be Purdue University AAE450 Spacecraft Design August 18 2003 7 carried out to transfer from a circular orbit to a reentry orbit with a particular flight path angle at atmospheric entry Vinh section 5 4 works out this particular problem While the earth will be assumed spherical nodal regression is substantial for Earth polar orbits This effect should be accounted for using the empirical data presented in Bate Mueller and White Fundamentals of Astrodynamics Dover 1971 pp 156 158 You will need to use various elements from spherical trigonometry in order to determine the ground track of various orbits and how to get them to rendezvous Information for this will be provided in a handout taken from the CRC Standard Math Tables When two orbits with different inclinations intersect away from the equator the difference in heading angle must be known in order to compute the pure propulsive impulse needed to change from one to the other These relations are given in section 5 2 There are several references describing trajectory designs for aeroassist Among these are Optimal plane change of an elliptic orbit during aerocruise by Medepalli and Vinh AAS Paper 91 417 Aug 1992 5 1 Conversion Between Rotating and Inertial Coordinates The differential equations given in Vinh et al are written in an Earth fixed rotating coordinate system
37. es precisely over a pole so latitudes within about 0 1 degrees of the North or South pole must be avoided The orbital mechanics in this course will be described using Vinh s coordinates The equations described simulate only the 3 spatial degrees of freedom in which the vehicle moves They do not simulate the 3 degrees of freedom associated with vehicle attitude It will be assumed that the bank angle of the vehicle can be somehow set using the reaction control system RCS Section 6 1 describes the requirement for pitch plane static stability While this requirement must be met there is no dynamic simulation of pitch plane attitude either it is simply assumed that the vehicle angle of attack can be set as an input The user thus has arbitrary control over the vehicle angle of attack within static stability limits bank angle thrust within engine specifications and thrusting angle within reasonable limits on nozzle gimballing and the aerodynamics of the vehicle attitude A more realistic simulation of the vehicle attitude must be left to a more refined design stage Vinh s book is an excellent reference for general orbital trajectory information For example once a vehicle has left the atmosphere it is automatically in an elliptic orbit assuming the thrust is turned off One may wish to know the elliptic orbit properties from the initial state upon atmospheric exit Section 3 5 of Vinh shows how to compute these elliptic orbit properti
38. f several papers from the 35th Thermophysics meeting reporting design studies for a crew return vehicle and exploring use of UHTC TPS and sharp leading edges Papadopoulos P D Prabhu D Olynick Y K Chen and F M Cheatwood CFD code comparisons for Mars entry simulations AIAA Paper 98 0272 Jan 1998 Recent analysis of Mars aeroheating shows typical approach G P Sutton Rocket Propulsion Elements John Wiley and Sons New York 1992 Special Section Planetary Entry Systems Aeroassist Systems Journal of Spacecraft and Rockets v 36 n 3 May June 1999 Special Section HL 20 Personnel Launch System Journal of Spacecraft and Rockets v 30 n 5 Sept Oct 1993 114 pages of discussion of preliminary design of a crew return vehicle for the Space Station Good reference for design methods subsystem weights etc
39. find the value of flap angle 8 at which the moment is zero trim It brackets a region where the moment coefficient passes through zero zbrent for This Numerical Recipes subroutine is used to find the trimmed value of the flap angle It iterates on aerodat for to find the accurate zero in the moment coefficient aerodat for This function computes the aerodynamic parameters for a vehicle at a given angle of attack and flap angle You will need to modify this function to represent the hypersonic and supersonic properties of your design Right now it is programmed to generate data for a flat plate An earlier version aerodat Gus is also available this earlier version computes aerodynamics for Gus s sample vehicle The aerodat function returns the moment coefficient about the center of gravity c g which needs to be zeroed for trim The major part of the semester s aero dynamics work involves generating aerodat and heatflux subroutines for your vehicle design and using them to iterate your designs trajsub This subroutine contains the equations for the spacecraft dynamics It calls heatflux for This subroutine computes the heatflux at the stagnation point and at selected points along the windward surface of the body This is presently set up for a blunt flat plate It will also have to be modified to reflect your design rk4 This is a Numerical Recipes subroutine to advance the solution of the differential equa tions by one time step
40. ft is nearly zero 3 Pitch plane static stability analysis with sample embodied in aeroprop for and aerotrim for At flight points Cm 0 trim and the slope of the Cm vs a curve must be negative static stability 10 11 12 13 14 15 16 17 3 Purdue University AAE450 Spacecraft Design August 18 2003 Stagnation point heating equations given in section 7 4 and in Tauber paper Includes method of approximating TPS temperature Embodied in heatflux for Heating analysis for other sample points on vehicle Equations given in section 7 5 The TPS analysis must be added to this and coded into heatflux for For an ablating TPS the SODDIT code should be used Skin friction correlations see section 7 8 Includes method for estimating boundary layer transition Supersonic aerodynamics analysis Section 7 9 Brief discussion of RCS and life support systems to be presented later TPS data from TPSX and other papers to be handed out Rocket data handout Detailed data for booster selected Elementary orbital mechanics e g for pure propulsive baseline case Orbital precession data from Bate Spherical trig data for orbital crossing angles Website and structures handouts for structural analysis Rocket boost analysis using trajectory equations sample code in boost for This needs additional development Cg analysis using Matlab tools from Jason Bowman see website Visco
41. ghly 45 deg and then reduces it at some point to increase lift The shuttle has used bank angles to 60 65 deg at times It is necessary to compute the footprint of where you can land from a given deorbit location what is the down and cross range Where can you land To get crossrange it is usual to run a zero bank trajectory to get the nominal conditions and then run the banked trajectory to get the crossrange The crossrange is the distance along the surface of the earth in the direction normal to the zero bank trajectory Note that the engine nozzles should not be exposed to the reentry flow The shuttle has fairings to preclude this 6 Stability and Control 6 1 Vehicle Attitude Control The stability and control methods used in the course are limited to requiring static stability in the pitch plane during hypersonic flight The static stability issues are discussed in H Ashley Engineering Analysis of Flight Vehicles Dover 1992 pp 247 248 You must fly your vehicle in a trimmed state with Cm 0 where Cm is the pitching moment coefficient To accomplish this some kind of flap control is needed In addition you must demonstrate static stability that is dCm da lt 0 where a is the angle of attack The amount of margin that is required does not seem to be a clearcut issue Penland 1978 calls for dC dC lt X where X is at least 0 02 Penland notes that a value of dCm dCr 0 02 indicates that the lift vector is lo
42. have that cos 17 2 Y cosicos 7 2 sini sin 7 2 cos A9 23 so sinw sini cos A9 24 and cos A0 ne 25 sini This gives the angle A0 between the local longitude and the RAAN given 7 and w Figure 5 shows the computed results for the space station orbit plane Note the difference from the intuitive formula which one might incorrectly guess Purdue University AAE450 Spacecraft Design August 18 2003 XY 16 Aug 1999 ground track for i 45 0 RAAN 90 0 deg Ground Track for Planar Orbit with inclination angle 51 4 deg and 300 deg for the RAAN right ascension of the ascending 50 node sps 9 30 98 40 4 A 30 20 10 F o degrees oO 10 20 30 _ latitude 0 Intuitive i sin A 8 40 WRONG X e 300 400 500 600 0 degrees Figure 5 Ground Track for Space Station Orbit 14 Purdue University AAE450 Spacecraft Design August 18 2003 5 4 Single Point Cruise Design Here consider a single point cruise design for an aeroassist mission in order to have a relatively simple basis to trade altitude wing loading and so on 5 4 1 Inclination Angle Changes Consider a single point cruise design carried out at the equator for a small time At The vehicle is at point B in Figure 4 Look at a tangent plane near B We have velocity V at inclination 7 and want to change to velocity V2 at inclination 7 Ai The v
43. he CRC Standard Math Tables Figure 4 shows the geometry Note that there is a spherical right angle at point C Here however we will use the general formulas for oblique spherical triangles From the law of sines sng 11 sinA sinB where A is the angle at point A a is the length of the opposite side and so on This implies that sinA sing 12 sin A sini 12 where 2 is the inclination angle and in AO sini mAs aeee 13 sin 12 Purdue University AAE450 Spacecraft Design August 18 2003 From the law of cosines cos B cos cos A sin sin A cos b 14 so cos i cos T 2 cos A sin 7 2 sin A cos 15 where cos 7 2 0 and sin 7 2 1 Therefore cost sin A cos 16 Using equation 13 we then have cosi ae 9 sin A sin i 17 sin or cot i cot dsin Ad 18 Note that A9 is the longitude measured from the Right Ascension of the Ascending Node RAAN point B This can be rewritten as tan tan isin Ad 19 where tant is a nonzero constant Simplifying arctan tani sin A0 20 Note that this does NOT account for the rotation of the earth as it is derived in inertial space Using Bate p 142 we have that cosi sin 1 2 y cos 21 or cosi cosy cos 22 This gives the inclination angle 7 in terms of the heading angle w and the latitude Also using Fig 2 15 2 in Bate and the fact that the angle B 7 2 and the law of cosines we
44. inary design you should accumulate a list of details that need to be verified later The main effort is to determine the validity of the overall concept and attempt to be sure that you aren t missing any show stoppers Your vehicle design and the supporting analysis will end up embodied in a computer program This program will both define your vehicle and serve to implement the analysis of its performance A number of such programs are available from previous semesters beginning in Fall 1998 to serve as a basis for your designs see the public access course directory The instructor s sam ple programs are also available in the subdirectory sps_prgm See also the AAE450 website Prof Schneider s section possibly still located under the Spacecraft part of the 451 website at http roger ecn purdue edu aae450s Should you choose to begin with an existing code you can take advantage of previous efforts as one normally does However you are then responsible for the accuracy of the codes so check them carefully for bugs Since your code is used to demonstrate the performance of your design any errors in the coded analysis make your performance results dubious at best Your codes should thus be well documented and well checked Any past codes that you use should be well understood and improved Put your name and the last date of modification at the top of the code Use generous commenting to describe the variables and the logic All e
45. lodziej does indicate that SiC is used with the diboride several papers For Zr B this suggests that compound V in Clougherty et al is a reasonable approximation suggesting a density of about 6 gm cc Table 13 or 6000 kg m For emissivity Kolodziej reports a representative value of 0 8 in NASA TM 112204 Aerothermal performance constraints for hypervelocity small radius unswept leading edges and nosetips July 1997 p 5 He indicates that a 1 mm nose radius is reasonable for an unswept hemicylinder leading edge and gives a single use temperature limit of 2760 C While Clougherty gives emissivity ranging from 0 6 to 0 4 for Zr B in Fig 45 depending on temperature the more recent value of 0 8 from Kolodziej is suggested 32 Purdue University AAE450 Spacecraft Design August 18 2003 7 8 Hypersonic Skin Friction Analysis The following contains the critical elements of the simplified skin friction analysis as adapted from Prof Gustafson s notes The method is primarily taken from AIAA Paper 90 0538 Hypersonic waveriders for planetary atmospheres by John Anderson et al For some additional detail see Gustafson s notes as saved in the file skinfric backup pdf in the course directory This method should be used above Mach 5 for supersonic skin friction see Sec 7 9 2 7 8 1 Correlations for Local Compressible Skin Friction This section is nearly a direct quote from the Anderson paper The skin friction distribution a
46. long the streamlines is calculated using the reference temper ature method of Eckert In the reference temperature method approximate formulas are used to predict the skin friction with the physical properties evaluated at an appropriate reference temperature For a flat plate in laminar flow the local skin friction coefficient is given by Cy 0 664 Reg T T 07 66 Re is the local Reynolds number defined as so Vog Re tee oe 67 Hes where p is the freestream density V is the freestream velocity x is the local distance from the leading edge of the plate and uo is the freestream value of the viscosity Also T is the reference temperature defined as T T 1 0 032M 0 58 1 68 T ork T 68 where M is the freestream Mach number and T is the wall temperature In 450 put in some average wall temperature from your trajectory differential equation solution even though this results in some ad hoc averaging over the surface Finally w is the exponent of an assumed exponential variation of u namely u T w Me ees ES 69 7 69 A value of w 0 75 is used in the present study The flat plate skin friction coefficient for turbulent flow is given by 0 0592 r Re J02 70 where y Re T 71 Here p and p are evaluated at the reference temperature T Note that this change to the reference temperature is done at constant pressure Thus p p RT Po P RT and p po SO
47. ly After doing the correction add in the skin friction drag 38 Purdue University AAE450 Spacecraft Design August 18 2003 XY 16 Aug 1999 lockheed viscous interaction curve fit gt 2 N 0 6 L D L D w Viscous Interaction Effects on Maximum L D Lockheed Missiles and Space Co 1990 as transmitted to Prof Gustafson for AAE451 05 0 025 0 VI M_ sqrt Re_ Figure 13 Lockheed Correlation for Viscous Interaction Purdue University AAE450 Spacecraft Design August 18 2003 39 Unfortunately the source of this correlation is unknown Therefore the accuracy and reliability are also unknown and suspect However the correlation is very simple to use Another relatively simple correlation is given by Maslen in Synergetic Turns with Variable Aerodynamics J Spacecraft vol 4 pp 1475 1482 Nov 1967 on p 1480 e g eqn B4 This should be investigated if improved estimates are needed 7 10 2 Viscous Interaction Effects Bridging Formulas A good review of the experimental data for this effect is contained in Boylan and Potter Aerody namics of typical lifting bodies under conditions simulating very high altitudes AIAA J v 5 n 2 pp 226 232 Feb 1967 Boylan and Potter also suggest a more accurate and reliable method of computing the viscous interaction The lift and drag should be computed using free molec ular methods and then a bridging formula should be used to
48. n Prof Heister has noted that non cryogenic liquid rockets are OK for launch on warning missions as in the early ICBM s although solid rockets require less maintenance for these missions 8 1 Boost Propulsion An existing booster is normally to be selected Data for the thrust ISP weight empty weight nozzle diameter and so on need to be found in the literature The boost trajectory is then analyzed using the same equations of motion used for the reentry analysis An existing code can be modified for this purpose However as usual when using an existing code be sure to check the code Be warned that the BOOST codes derived from that written by Dan Harris in Fall 1998 have some major bugs The best existing code is probably that written by T J Hoverman for Columbiad in Fall 1999 Wind loads should be taken into account A control algorithm for the gimballing of the rocket nozzle is also to be designed and used The vacuum thrust of the engines is to be corrected for altitude 8 2 Vehicle Propulsion Here also an existing flight proven engine is normally required The propellant tanks and the propellant feed system should be designed following Chapter 5 of Humble et al 1995 Another authoritative source may be substituted upon consultation with the instructor Try to balance the tank positions about the center of gravity so that the c g doesn t move as the fuel is used Purdue University AAE450 Spacecraft Design August 18 2003
49. n computed again using equation 62 but now using Taw Te with the change in the power of the Pr based on Bertin p 340 and y 1 4 This is crude with uncertain accuracy but it is better than using the Tauber formula at zero angle of attack where the flat plate heat transfer rate is taken as zero 1 Pr 1M 64 7 5 3 Heat Transfer to Wing Leading Edges This is taken from equation 6 in Tauber which is qie 0 5 qo cos A Grp sin A 65 where A is the sweep angle of the wing leading edge gj comes from using the stagnation point heat transfer formula for the local conditions and r p comes from using the flat plate heat transfer formula for the local conditions Presumably all lengths are to be given in meters These correlations are validated against Shuttle data in the paper The leading edge thickness of any wings is clearly a critical issue For 450 this heat transfer analysis can only be done for representative points in your design Pick a few points along the leading edges and a few points on the body surfaces and compute the heating results only there This limited analysis can then be used to determine a TPS distribution and weight that is accurate enough for preliminary design Clearly the windward surface will be the critical region 7 5 4 Heating Rates for Mars Entry Most recent papers analyzing the heating rates for Mars entry use CFD methods for design pur poses Some simple correlations
50. odynamics 4 Structural Analysis and Center of Gravity Analysis These are carried out using methods developed by Jason Bowman AAE451 TA for several years ca 1996 1998 These methods are documented on the AAE450 website in considerable detail Microsoft Word versions of this documentation are also available A method for determining the loading and required structural weight is given A simplified finite element method should be used to compute structural weight in place of the approximate method used prior to Fall 2001 Also Matlab software CGMOI for computing the combined vehicle center of gravity and moments is given and documented on the website These codes may be written in a language of your choice however get the language approved by Prof Schneider or the TA prior to use since one of them will need to review your code upon completion 5 Orbital Dynamics The orbital dynamics is analyzed using the equations for a point mass vehicle flying above a spherical planet usually the Earth These equations are taken from N Vinh A Busemann and R Culp Hypersonic and Planetary Entry Flight Mechanics Univ of Michigan Press 1980 Chapter 2 This book is out of print but a copy of this chapter will be provided to you In addition the book is on reserve in the Engineering Library These equations will be integrated numerically over your trajectory using a Runge Kutta or other subroutine The equations are singular for an orbit that pass
51. oefficient and Drag The following information for low speed computations is given for background only to help you in understanding and applying the high speed methods At low speed the local flat plate laminar skin friction coefficient is T C x Here 7 is the local shear stress The drag on one surface of the flat plate is l l D of T x dxz 0 332bp 0V2 Re dz 74 0 0 where b is the width of the flat plate and l is the length We can make a drag coefficient for a length l of this plate using bl for the reference area as follows D _ 0 664 G i Re de 75 G50 te 18 Simplifying TAn Hes f a dy 76 Po V Note that fede 20 77 0 so 0 664 55 Cou fe oe l Poo Voo 34 Purdue University AAE450 Spacecraft Design August 18 2003 or Cp 1 328 Re 7 79 Equation 79 is the drag coefficient for laminar flow on one side of a low speed flat plate of length l and span b Compare White Viscous Fluid Flow 2nd ed 1991 eqn 4 53 Let the angle of attack AOA be a Assume that at AOA Cp cos a C p 1 flatplate 80 This appears to be no more than a guess 7 8 3 High Speed Laminar Skin Friction Drag Coefficient Similarly for laminar flow on a flat plate at high speed the local skin friction on one surface is pr 70 128 Cp cos a 1 328 Re 81 OO The reference area is again bl T is again the reference temperature Prof Gustafson suggests putting in
52. on history is then computed separately This is the same method used by Sandia personnel for various NASA Ames designs and sample input files were provided by Sandia personnel for the Mars entry problem 7 5 Windward Surface Heating Analysis The following contains the critical elements of the windward surface heating analysis for hypersonic flow as adapted from Prof Gustafson s notes These methods are for the earth s atmosphere unless otherwise noted 7 5 1 Flat Plate Heat Transfer Rate for Large Angles of Attack This is also taken from Tauber Menees and Adelman Aerodynamics of Transatmospheric Vehicles J Aircraft v 24 n 9 Sept 1987 pp 594 602 The formulas are also available in Anderson Hypersonic and High Temperature Gasdynamics McGraw Hill 1989 p 291 Equation 4 in Tauber et al gives the heating rate per unit area as CoV 52 Here q is the heat transfer rate into the body per unit area p is the freestream density and V is the flight velocity The appendix of Tauber et al gives the constants for a fully catalytic surface a conservative approximation The constants give the heating rate in W cm if the velocity is given in m s and the density in kg m3 For a laminar flat plate the constants are M 3 2 N 0 5 and Cy 2 53 x 107 cos sin o 712 1 gw 53 where is the local body angle with respect to the freestream x is the distance from the stagnation point measu
53. oximately at a single temperature In this case we have dT Cog PwV 4 dr coTa dA 51 where V is the volume of the solid nosetip and dA is an element of the surface area The left hand side of equation 51 is easily evaluated but the right hand side requires further simplification Depending on geometry it is probably best to evaluate the integrand on the right hand side at roughly 2 4 points on the nosetip and use some approximation to the actual integral Discuss this analysis with your instructor and document it in your report For blocks with sharp noses this solid nosetip exhibits some interesting phenomena The heat transfer to the stagnation point is commonly roughly 10 times that observed at 90 deg to the stagnation point This rough analytical result is confirmed by experiment compare the cylinder heat transfer measurements shown in Fig 6 18 of Anderson s text 1989 So overall heat transfer to the block is not that high with a sharp nose Furthermore at some temperature a steady state condition is reached where radiative cooling from the rear sides of the block balances the heat transfer at the stagnation point Paul Kolodziej from NASA Ames has commented that Bingo This last phenomena contributes to the remarkable performance of the UHTC leading edges We ve done extensive finite element analysis both thermal and structural to improve our understanding of this coupled interaction between the flow field and
54. quations coded in the program should have comments referring to the text handout or section of your report in which the equations are written out or described FORTRAN will continue to be used as the main language for this course allowing continued use of the existing code base and of the instructor s expertise in that language You may need to refresh your knowledge by picking up a reference book and studying it All of your analysis should be traceable back to standard engineering methods either through references to standard works or detailed presentations of your own Don t reinvent the wheel but do refer equations back to their source so they can be checked Reports from previous semesters are on file in the Design Room in Grissom 100 It may be profitable to consult these when performing designs similar to those from past years You may wish to cite them and their results when performing trade studies However these design reports are not authoritative so analysis methods should not be referenced back to them 2 List of Analysis Tools 1 Trajectory equations embodied in 451traj for Assumes spherical planet small flight path angle etc Normally set up for earth 2 Newtonian hypersonic aerodynamics Sample in aerodat for Use handouts or your own analysis to obtain integrals for your vehicle configuration Define the vehicle axes carefully Angle of attack is normally zero when the vehicle is nearly symmetric to the flow when li
55. rcular orbit with the usual velocity Vorb Mg R Here ug is the gravitational constant for the planet and should not be confused with the viscosity Equation 29 can be obtained from the second part of equation 2 31 in Vinh s book again looking at the equator with zero latitude and flight path angle However in this case we need to add some terms that come about from the rotating earth angular velocity w 0 L m coso g V R QV cosi w R 30 The 2wV term is the Coriolis acceleration according to Vinh and may be important for long range flight The w R term can usually be neglected according to Vinh Here we will neglect both of these terms for an initial cruise point analysis For a maneuver done with pure propulsion outside the atmosphere the same AV is needed For a small time At AV aAt still where a is again the acceleration in the horizontal plane which is now the thrust of the rocket If we use the same thrust for the same length of time we will use the same amount of fuel and achieve AVpp D m At 31 Thus Mop DAt mV 32 Taking the ratio we have Ai n L D sino 33 This is the best case for efficiency Efficiency will be reduced due to viscous interaction during entry and exit due to operation away from the equator where bank affects not only i but also the RAAN and other effects Ideally the maneuver can be carried out in a short distance near the equator and entry and exit c
56. red along the body surface and gw is the ratio of wall enthalpy hw to total enthalpy ho The body angle is the angle between the tangent and the velocity vector for a flat plate it is the angle of attack For zero angle of attack as for a flat plate this formula goes to zero which is incorrect A query to Micheal Tauber at Ames produced the following response The angle phi is the local body angle with respect to the free stream i e for a flat plate it s the angle of attack The expression was derived assuming that Newtonian theory was valid for the product of pu the surface pressure x boundary layer edge velocity that appears in the laminar and turbulent convective heat transfer equations The Newtonian approximation is only valid when the free stream Mach number component that is normal to the surface is supersonic Therefore the simple expressions given in the 1987 paper fail at zero or small angles of attack The limit is M sing gt 1 For smaller angles of attack it seems better to estimate using the flat plate formulas given in White Viscous Fluid Flow 2nd edition Sec 7 3 3 and 7 8 1 see section 7 5 2 From thermodynamics hy Cpwlw The total enthalpy is ho ha 0 5V where ha is the local enthalpy of the atmosphere However for reentry ha is usually much smaller than 0 5V and will be neglected Thus as before E Chay eV Gu 54 Purdue University AAE450 Spacecraft Design August 18 2003 29
57. s rotation we will not design trans lation motions in 450 The vehicle requires 4 thrusters in the pitch plane to achieve both signs of rotation In addition redundant thrusters should be provided at each position so the vehicle does not become catastrophically unmaneuverable In each of the 3 planes you thus need 8 thrusters for a total of 24 It is very difficult to estimate the amount of propellant one might need so the RFP gives the AV requirement from which the needed propellant can be determined 9 Other Design Information from Gus Following is some miscellaneous design information from my notes from the last time Prof Gustafson taught 451 1 Yaw Stability could use a vertical tail but will mostly be shadowed Could use outboard fins but these create a heating issue RCS system is mostly used for yaw control on hyper sonic vehicles 1 29 98 2 The electronics package must be in the pressurized cabin for thermal control 1 29 98 3 Life Support Need oxygen and nitrogen with pressure regulated into the cabin Find out on the internet how much a person consumes If the vehicle is opened then the cabin will have to be repressurized Use lithium canisters to remove carbon dioxide as on the shuttle Get a basic idea of the amount needed and allow for the mass and volume 4 Aerodynamics a For a flat surface the force acts at the centroid 2 5 98 p 5 5 Structures TPS 42 10 10 Purdue University AAE450 Sp
58. ta obtained from TPSX be careful not to release this data contrary to regulations In particular be careful what you put in your reports which must be available for public distribution The thermal properties of the materials are temperature dependent If the temperature de pendence is significant greater than 10 20 say then this temperature dependence should be included in your analysis Clougherty et al give the properties of the UHTC compounds in Research and development of refractory oxidation resistant diborides Part II Volume V Thermal Physical Electrical and Optical Properties AFML TR 68 190 1969 DTIC citation AD 865321 This is the most re cent publicly available source so it will be taken as the best available data for the hafnium and zirconium diborides A variety of different material types were tested as listed in Table 1 in Clougherty et al For zirconium diboride Table 10 lists values of specific heat ranging from 0 17 to 0 21 cal gm C depending on temperature 1000 to 2000 C and composition A value of 0 19 cal gm C is therefore recommended as sufficiently accurate for the present preliminary design In MKS units this is C 795 J kg K Table 13 shows density data which ranges from 4 44 to 10 95 gm cc depending on material composition This is the largest source of uncertainty in the spec ification of these compounds Unfortunately accurate specifications for current compounds are not publicly available Ko
59. tio for 2D Wedge has the best L D Anderson shows that the Newtonian L D goes like cot a and becomes infinite as a 0 Of course this neglects skin friction etc For a lt 6 2 L D is negative since L lt 0 As angle of attack increases for a particular L D rises until the upper surface is shadowed at a after which L D is the same as for a flat plate For smaller 6 the upper surface becomes shadowed for smaller a where L D is larger For example for an included angle of 6 6 deg L D gt 10 is possible This is a best case limit since skin friction nose bluntness trim drag and viscous interaction will reduce L D However it points to the classically critical importance of a slender vehicle Like the Carnot efficiency in thermodynamics this simple analysis shows a best possible case These high values of L D at low a come at a price Fig 11 shows the lift coefficients for these same shapes Again the flat plate is the best case The wedges have lesser or negative lift coefficient until the upper surface is shadowed after which they match the flat plate Note Purdue University AAE450 Spacecraft Design August 18 2003 XY 26 Jan 2000 Newtonian Aerodynamics for Wedge WEDGEAERO FOR 0 25 The angle of attack o is defined with respect to the lower surface of the wedge The included angle of the wedge is 5 The shadowing effect is accounted for 0 2 flat plate 6 2 0 deg
60. ucation Series 1988 The primary purpose of the heating analysis is to enable designing the thermal protection system TPS which must protect the vehicle from the high temperatures of planetary entry 7 4 1 Stagnation Point Heat Transfer Rate Earth This is taken from Tauber Menees and Adelman Aerodynamics of Transatmospheric Vehicles J Aircraft v 24 n 9 Sept 1987 pp 594 602 see also Anderson HHTG p 291 eq 6 169 Equation 4 gives the heating rate per unit area as g Co Vv 44 Here q is the heat transfer rate into the body per unit area p is the freestream density and V is the flight velocity Note that despite the use of for heat transfer following Tauber et al no time derivatives are involved and the plain symbol q is used for the same quantity in most references The appendix of this paper gives the constants for a fully catalytic surface a conservative approximation The constants give the heating rate in W cm if the velocity is given in m s and the density in kg m The constants are M 3 N 0 5 and C 1 83 x 1078 r5 7 1 gy 45 where r is the body nose radius in meters and gw is the ratio of wall enthalpy h to total enthalpy ho From thermodynamics hy CpwTw The total enthalpy is ho ha 0 5V where ha is the local enthalpy of the atmosphere However for reentry ha is usually much smaller than 0 5V and will be neglected This results in Cra Lao 0 5V2 0 5 1
61. uld be too complex to allow multidisciplinary iteration during the time available for the course 7 1 Atmospheric Conditions For earth these are taken from the 1976 Standard Atmosphere as coded by Prof Gustafson and checked against tables e g Handbook of Tables for Applied Engineering Sciences Bolz and Tuve CRC Press For Mars and Venus atmospheric data can be taken from the references in Seiff Atmospheres of Earth Mars and Venus as Defined by Entry Probe Experiments J Spacecraft and Rockets v 28 no 3 pp 265 275 May June 1991 A FORTRAN program has been written to generate the properties of the Mars atmosphere Figure 8 shows the exponential decrease of density with altitude Purdue University AAE450 Spacecraft Design August 18 2003 21 u AAL N Figure 9 2D Wedge for Newtonian Flow Analysis 7 2 Pressure The aerodynamic pressure on the vehicle surfaces is evaluated using Newtonian methods These are described for example in J Anderson Hypersonic and High Temperature Gasdynamics AIAA Publications 1989 Chapter 3 The Newtonian pressures have been integrated over simple shapes by E L Clark and L L Trimmer Equations and Charts for the Evaluation of the Hypersonic Aerodynamic Characteristics of Lifting Configurations by the Newtonian Theory AEDC TDR 64 25 March 1964 A copy of this report will be provided to each group Clark s results allow you to proceed directly to coding the analytical results for the
62. us interaction analysis using Lockheed correlation and Boylan paper If needed this may be improved upon using free molecular flow analysis methods similar to the Newtonian flow analysis methods and using a bridging formula Architecture of Framework Programs Several Fortran 77 programs are provided to you at the AAE450 course directory These can serve as baseline or framework programs from which you can develop the code you need to analyze your design The FORTRAN codes developed by previous students beginning in Fall 1998 are also provided to you again on the course directory These are part of the full design history which is available The architecture of the baseline programs is as follows Most of them were set up to work for Earth and will need to be modified if used for other planets traj for This is the main program which computes trajectories around a rotating spherical planet It contains one call to a Microsoft time and date routine which you may have to eliminate or modify to run on the Unix system It calls Purdue University AAE450 Spacecraft Design August 18 2003 5 aerotrim for This subroutine provides the trimmed aerodynamic parameters for the vehicle that is the aerodynamic parameters for flap angles which provide a zero moment that is stable If no stable flap condition exists for the requested angle of attack an error flag is returned It calls zbrak for This Numerical Recipes subroutine is used to help
63. w we do not neglect the shadowed side at supersonic speeds The entire vehicle is assumed to have a turbulent boundary layer The incompressible skin friction is Cri 0 074 Re 88 as discussed in section 7 8 Correct this for compressibility using Cri Cr 89 EO 0 144M2 ee Correct for conical geometries using 2 Cirone z Fiflatrate 90 Add up the various elements of the vehicle as follows O fasela e Daas Siini Stail C C usela ce C nose C win A C ailn 91 D F F fuselag Do TOR Gt F wing Seep OF tail Sa 91 7 9 3 Wave Drag due to Thickness The wave drag is modeled with crude approximations from Fundamentals of Aircraft Design by Leland Nicolai University of Dayton 1975 self published See the figures from this book which will be handed out See also ss subsaero pdf in the class account directory On a conical forebody Cp from Fig 11 18 is the nose wave drag coefficient based on the body cross sectional area md 4 where d is the base diameter of the cone Note that 6 M2 1 and fy ly d where ly is the axial length of the cone For example sin 0 d 2ly where is the half angle of the cone The wing wave drag is approximated differently depending on whether the leading edge is supersonic For a supersonic leading edge 4 M2 1cot A gt 1 92 where A is the sweep angle 0 degrees for an unswept wing In this case 167 Se Cow me 3 M2 1 Sref where
64. wing and get an approximate result Neglect the shadowed regions of your vehicle since p 0 here Get p M etc from the main program Note that Cp total C Dnewtoman CD skinfriction 87 in the hypersonic case For a cone or cylinder just try to unroll the cone or cylinder and apply the formulas as best you can 7 9 Analysis of Supersonic Aerodynamics 7 9 1 Introduction According to Prof Gustafson the supersonic portion of the mission is still important enough that it cannot be neglected The Newtonian result will not give enough lift coefficient at lower altitudes and speeds to give a reasonable first approximation In real supersonic flow L D will increase and occur at lower angles of attack In the aerodynamics subroutine Gus suggests making the change between analysis methods for a range between Mach 5 and 4 In between use a weighted average relative to Mach number The change must be carried out smoothly and the angle of attack must be reduced smoothly so oscillations do not develop in the trajectory and solution Remember that the trajectory person selects angle of attack 36 Purdue University AAE450 Spacecraft Design August 18 2003 7 9 2 Skin Friction Drag Prof Gustafson stated that the following were taken from A Shapiro The Dynamics and Ther modynamics of Compressible Fluid Flow but the exact reference remains to be found They do not use the reference temperature Both sides of the vehicle are analyzed no

Methods for Analysis of Preliminary Spacecraft Designs AAE450

Contents

Download Pdf Manuals

Related Search

Related Contents