AlAA 91 -0053 The Design of Hypersonic Waveriders for Aero-Assisted Interplanetary Trajectories

M. J. Lewis University of Maryland College Park, MD

A and A. D. McRonald Jet Propulsion Laboratory Pasadena, CA

29th Aerospace Sciences Meeting January 7-1 0, 1991 /Reno, Nevada ?

The Design of Hypersonic Waveriders for Aero-Assisted Interplanetary Trajectories

Mark J. Lewis' Department of Aerospace Engineering, University of Maryland a t College Park

Angus D. ~ c ~ o n a l d ' Jet Propulsion Laboratory

Abstract

The aerodynamic performance of a vehicle designed to execute an aero-gravity assisted maneuver, which combines a gravitational turn with a low-drag atmosphere pass, is examined. The advantage of the aero-gravity assisted ma- neuver, as opposed to a more traditional gravity-assist tra- jectory, is that, through the use of a controlled atmospheric flight, nearly any deflection angle around a gravitating body can be realized. This holds the promise of providing ex- tremely large values of AV. The success of such a maneuver depends on being able to design a vehicle which can execute sustained atmospheric flight a t Mach numbers in the range of 50 - 100 with minimal drag losses. Some simple modelling is used to demonstrate design rules for the design of such vehicles, and to estimate the deterioriation of their perfor- mance during the flight. Two sample aero-gravity-assisted maneuvers are detailed, including a close solar approach re- quiring modest AV, and a sprint mission to Pluto.

Nomenclature

. aerodynamic force coefficient drag

gravitational constant lift

waverider overall length Mach number

secondary body mass dynamic pressure

body radius orbital radius time variable

velocity magnitude normalized velocity

waverider base width linear dimension

change in velocity magnitude orbital angular deflection

viscosity gravitational constant times planetary mass

' h i s t a n t Professor, Member AIAA 'Member Technical Staff, Member AIAA

Copyright @American Institute of Aeronautics and Astronau- tics, Inc., 1991. All right. reserved.

Introduction

angle of arc through atmosphere density

waverider aspect ratio

subscripts

circular orbit value property above boundary layer

leading edge reference value

value a t periapsis wall value

initial or approach final or exit

value at infinite distance

Aero-gravity-assisted interplanetary trajectories are an application of hypersonic aerodynamics to orbital maneu- vering in which aerodynamic lifting forces are used to mod- ify a vehicle's trajectory. An aero-gravity-assisted flight is similar to aerobraking', although, whereas aerobraking uses aerodynamic drag forces to decelerate a vehicle, an aero- gravity-assisted flight would use lifting forces to redirect velocity with a minimum of losses2. Aerobraking would use high-drag hypersonic configurations to reduce orbital velocity in the atmosphere, but aero-gravity-assisted vehi- cles must be designed to provide minimum drag for a given desired lift.

The concept of using lifting forces for control during the atmospheric flight portion of spacecraft trajectories is not new. Reentry maneuvers typically use lifting forces to con- trol the trajectory, as will some proposed aerobraking con- cepts. For instance, the Apollo command module, with a LID .4, used lifting force to augment gravity in its high- speed entry from the moon. However, the dominant force in these maneuvers is still drag, and the vehicle designs reflect that philosophy.

In the aero-gravity-assist maneuver pictured in Figure 1, a high LID vehicle executes a cruising atmospheric pass through the atmosphere of a suitable planet and exits that

planet's atmosphere back out to space, in order to boost or reduce its velocity en route to a final destination. In this way, the aero-gravity-assisted maneuver is nearly identical to the more traditional gravity-assisted flight that has be- come common for interplanetary trajectories, with the ex- ception that the atmospheric flight permits far greater an- gular deflections around the planet, and thus substantially increased velocity increments. Randolph and McRonald have explored applications of this concept for close solar approaches and trajectories to the outer planets4. Figure 2 indicates how the motion of a spacecraft in either a grav- ity or aero-gravity trajectory around a planet, as measured relative to that planet, changes its velocity in the absolute frame. The more angular deflection that can be accom- plished, the greater the velocity increment in the absolute frame of reference, which is the primary advantage of the aero-gravity-assist.

The success of an aero-gravity maneuver will depend on whether configurations which supply the required lift can be built with correspondingly small drag. If drag is pro- hibitively high, energy losses during the atmosphere pass will negate the advantages of the maneuver. Indeed, anayl- sis has shown that the configurational ratio of lift-to-drag (LID) is the governing parameter in determining energy losses through the maneuver615, a fact which is not sur- prising given that the atmospheric flight is nearly constant speed cruise.

In order to perform a practical aero-gravity-assisted ma- neuver, a vehicle with a lift-to-drag ratio of at least 5 must be employed. Figure 3 demonstrates this by presenting a plot of deflection angle that can be accomplished with 10% energy loss versus the incoming velocity normalized to cir- cular orbit velocity at periapsis, as a function of LID. Not surprisingly, a t higher LID, there are fewer losses for a given required lift. Figure 4 provides another indication of this ef- fect by displaying the velocity at exit from an aero-gravity- assisted flight which executes a 135' turn, versus the nor- malized incoming velocity, once again as a function of LID.

The requirement for L/D> 5 immediately suggests the class of configurations known as 'waveriders," first con- ceived by Nonweiler in 1959'. Waveriders are vehicles which are designed so that the bow shock is everywhere attached to the sharp leading edge. In this way, the vehicle is riding on a cushion of high-pressure fluid, which is contained bew- teen the bow shock and the undersurface. Because waverid- ers have shown both theoretical and experimental promise of providing the highest ~oss ib le values of LID, this paper will focus on that specific application.

If the aero-gravity-assisted flight is to be viable, several questions' must be explored in regard to waverider perfor- mance:

0 What values of realistic L/D ratios can be ex- pected for aero-gravity-assisted vehicles, and con these b sustained throughout the flight

What levels of heating can be expected on such a vehicle?

How much acceleration will be experienced by the vehicle, of particular importance if a crew is present?

What control issues must be identified, and are there reasonable hopes of controllable flight un- der required trajectory conditions?

Can the same aerodynamic configuration be used for more than one aeropass, possibly a t different planets?

It will be the goal of this present work to address some of the above questions, a t least in application to the specific trajectories cited, and to suggest possible limits for other aeregravity-aseisted flight operations.

Applicability of t h e Waver ider For High L/D

The above arguments have served to emphasize that these sorts of planetary maneuvers require LID ratios in excess of 5, and more likely, 10, in order to accomplish reasonable turning angles with acceptable losses in velocity. This will require the use of a waverider. Waveriders are a family of flight configurations that are derived by solving for a flow surface which intersects a known shock flowfield, such as that generated by a cone or power-law body. The process of designing a waverider is demonstrated in Figure @, in which a waverider shape is selected from a conical flowfield. The advantage of the waverider is that the shock is everywhere attached to the outer edge of the vehicle. This prevents spillage, and so permits the vehicle to derive maximum lift from the shock compression.

In fact, waverider designs have been analyzed which would provide reasonable values of LID. The work by Bow- cutt, Anderson, and Capriottielo, and Corda and Anderson8, resulted in a family of optimized waverider designs wherein the effects of skin friction drag were included within the optimization process. These waveriders demonstrate the potential for higher lift-to-drag ratios and/or lower mini- mum drag coefficients than other hypersonic configurations examined to date. To emphasize this point, Figure 7 (from Corda8) is a plot of maximum lift-to-drag ratio as a function of freestream Mach number, for a large number of hyper- sonic configurations under a variety of conditions - wind tunnel data, flight data, computational results - for a va- riety of Reynolds number. These are given by the open symbols. The solid curve in Figure 7 has been advanced by Kuchemannll as an upper limit to the maximum LID for hypersonic vehicles, and is approximately represented by LID = 4(M+3)/M. However, the waverider configurations generated in Referencess and9 break this "LID barrier," as shown by the solid symbols in Figure 7.

A typical Uviscous-optimizedn waverider from this work is shown in Figure 8, which is a top, front, side, and perapec- tive view of a waverider designed for Mach 20 operation in earth's atmosphere. Planetary waveriders have been shown to closely resemble terrestrial designs a t similar Mach num- ber and dynamic pressures13.

Aero-Gravity-Assisted Trajectories

Figure 5 presents a schematic of the trajectories followed by a waverider in an aeregravity-assisted flight to either the sun, or an outer planet such as Pluto. Following are two such representative trajectories:

Earth-Venus-Mars-Sun

The Earth-Venus-Mars-Sun aero-gravity-assisted trajec- tory of primary interest would be suitable for the proposed Jet Propulsion Laboratory Solar Probe Mission, the goal of which is to place an instrument package within four solar radii of the sun (0.0186 A.U.). To accomplish this mis- sion without benefit of an assist would require total AV of 24 km/s.

The EVMS mission begins on June 14, 2007, with an earth launch and AV of 6.204 km/s. The vehicle reaches Venus on August 10, 2007 with an approach velocity of 17.470 km/s, enters the Cytherean atmosphere a t Mach 80, and turns through 79.471°, of which 62.761' occurs in at- mosphere, which launches it on a trajectory towards Mars. Mars is reached on October 23, 2007, with an approach ve- locity of 24.808 km/s, leading to flight at Mach numbers in excess of 100. The vehicle turns through a total of 82.679' around Mars, including 80.430' in the atmosphere, reach- ing apehelion on November 16, 2007, and solar perihelion on March 20, 2008.

Without the Cytherean aero-assist, a simple gravity assist could accomplish a t most 16.53', assuming closest passage at 160 km altitude, the lower extent of the Cytherean ex- osphere. This means that an extra 62.94' is accomplished by flying through the atmosphere of the planet. Note that this is slightly higher than the actual flight through the at- mosphere, because the aero-gravity assist makes possible hyperbolic trajectories with smaller radius at periapsis, and thus more gravity-only turning.

This aero-assisted trajectory looks like a normal grav- ity assisted maneuver with an "effective radiusn of 600.7 km closest approach, about 5,568 km beneath the planet's surface. At Mars, a simple gravity assist would provide 2.198' deflection with 130 km altitude closest approach, so the aero-assisted maneuver adds 80.482' to the deflection angle. In this way, i t resembles a simple gravity assist with an effective periapsis radius of 35.77 km from the center of the planet.

The Earth-Venus-Mars-Pluto aero-gravity-assisted tr* jectory has been proposed to provide a rapid flight to the outermost planet. The proposed trajectory, which would launch on October 26, 2013, reaches Pluto almost exactly five years later. This is in contrast to the 12 years that would be required for a more conventional trajectory.

Initial AV is 5.089 km/sec, with an apehelion two days later, followed by an aero-gravity-assisted turn a t Venus on January 5, 2014, with an approach velocity of 13.849 krn/s, and a turning angle of 116.051°, of which 91.195" is accom- pished in the atmosphere while flying at Mach 70. This puts the vehicle on course to Mars, approaching on March 27, 2014 with velocity 24.426 km/s, and a resulting 76.696' turn, of which 74.352' occurs in atmosphere, with subse- quent approach to Pluto on October 26, 2018.

The atmospheric maneuver a t Venus adds 91.707' to the 24.344' that could be accomplished with a traditional grav- ity assist. The aero-gravity assist looks like a traditional gravity assist with periapsis radius of 302.95 km from the center of the planet. At Mars, only 2.314' could have been accomplished with a gravity assisted flight; the remaining 74.383' is realized in the aero-assist. This would correspond to a gravity turn at 43.95 km periapsis radial distance from the planet's center.

Aero-Assisted Hyperbolic Tkajectory

The dynamics of an aero-gravity-assisted vehicle will be reviewed briefly. A trajectory with an atmospheric passage provides unlimited capability to modify the flight direction, with the constraint that velocity magnitude is lost to drag during the course of the flight. A vehicle entering the at- mosphere will experience a deceleration given by:

where D/m is the drag force divided by the vehicle mass. Since it is most convenient to characterize the vehicle in terms of the lift-to-drag ratio, L/D, this is written as:

The time of flight can be related to the arc length 8 travelled at radius r in the atmosphere since Vdt = rd@. If i t is assumed that approximately constant-altitude flight is to be achieved, lift must be used to add to gravitational forces in counteracting the excess centrifugal effects. This means that the waverider is flying "upside-downn with respect to the planet's surface, using its lift to stay in the atmosphere:

so that equation (2) can be written as: In order to treat the entire maneuver in ublack boxn man- ner, with an input velocity and output velocity, i t is conve-

(4) nient to relate final velocity a t infinity in terms of the initial relative approach velocity:

This can be integrated, with the goal of expressing the velocity as a function of 8, provided that the variation of the L I D ratio with flight velocity is known. Off-design per- formance of a hypersonic waverider is not well known, and is in fact a current topic of research. There is a traditional suspicion that waveriders will perform poorly a t off-design conditions, although the rule of hypersonic Mach number independence suggests that in fact, L I D ratio should be relatively insensitive to changes in Mach number alone. In- deed, the governing geometry becomes insensitive a t high Mach numbers14 .

The dominant variation in vehicle performance will be a result of the reduced lift requirements as flight velocity is lost to drag. Consider for instance a vehicle which enters an atmosphere a t a speed in excess of orbital velocity, and at some assumed high value of L I D . At the point where drag reduces the velocity to orbital speed, no aerodynamic lift is required, so L I D will have dropped to zero.

In the above example, lift could be reduced by either changing angle of attack, increasing flight altitude, or both. Changes in angle of attack will result in variations in the coefficient of lift, CL, and the coefficient of drag, CD, and therefore the lift-to-drag ratio, L I D . On the other hand, in- creasing flight altitude will have minimal effect on the lift and drag coefficients, while changing the magnitude of lift by reducing the ambient density since lift is proportional to p. Increasing altitude will decrease the Reynolds num- ber, and thereby increase viscous effects such that drag is increased.

With the assumption of constant L I D , equation (4) can be solved directly. The vehicle will travel through an angle O a t altitude r , where

in which it is assumed that L I D is approximately constant, or a t least, can be characterized by some constant value during the flight. Note that this assumption will fail as the vehicle's velocity aproaches the circular orbit velocity, for under those circumstances, lift goes to zero, and with fixed L I D , so does drag.

The atmospheric exit velocity can be written in terms of the atmospheric entrance velocity:

Note that, as expected, drag in the planetary atmosphere has decreased the incoming velocity by an amount governed by the exponential scale factor 2 8 / [ L / D ] . The total turning angle, including gravity-only and atmospheric components, is5 :

Aerodynamic Modeling and Performance Losses

Viscous Scaling

A viscous-optimized waverider is designed with a single goal, usually of providing maximum L I D under suitable geo- metric constraints; however, since waveriders are configured for a single design point (both Mach number and Reynolds number) attention must be paid to the off-design perfor- mance of the vehicle in an aero-maneuver.

It is likely that, during flight, the L I D will decrease from the optimimum design value. Furthermore, i t is likely that, under the extreme heating conditions characteristic of an aero-gravity-assist atmospheric cruise, the waverider will be losing leading edge material, and the resulting bluntness may also decrease L I D . In this section, we will attempt to model the off-design performance of a waverider, and determine some simple strategies for adjusting the generted lift so as to maximize L I D throughout the flight.

Although waveriders have associated with them flowfields which are fundamentally three-dimensional, the very na- ture of the waverider permits some important simplifica- tions. Streamlines on a waverider tend to be highly two- dimensional, because the attached bow shock prevents sec- ondary around the leading edge. Thus, it is relatively accu- rate to model waverider flow as moving straight back along the surface of the vehicle. The waveriders of most interest are designed from axisymmetric flowfields, such as that as- sociated with a cone; however, experience has shown that optimization processes tend to result in shapes which re- side in regions of the design flowfield with relatively small curvature. In other words, even though the waverider may be carved from a three-dimensional flowfield, it is still gen- erally accurate to model i t as a two-dimensional surface, with nearly planar shocks. In this respect, waveriders have

a gross resemblence to delta wings, though the promised LID is higher. Finally, i t is reasonable to assume that the requirement for a sharp leading edge will result in insignifi- cant bluntness effects.

In hypersonic aerodynamics, a fundamental nondimen- sional number is the viscous interaction parameter, x , which indicates the strength of interactions between a shockwave and the boundary layer on a surface:

where C = (pp)wdl/(pp)invis&j. At values of > 3, the shock above a surface interacts strongly with the boundary layer on that surfacel2. From this, a minimum length can be derived beneath which strong interactions are important:

Assuming constant pressure normnal to the surface, and using a power-law approximation for viscosity, C can be written in terms of temperature only:

With the assumption of a weak shock and adiabatic wall, this can further be approximated as:

Using Chapman's result that relates viscosity to mean free path, p 1! .67pAa, the interaction length becomes Xstrong = . 74M2t3 (Y)"-'A. In air, with y = 1.4 and n = .67,

t r t rong = 1.26MkS4A, while in a COz atmosphere such as that of Mars or Venus, with y = 1.33 and n = .79, xstrong = ~ . o ~ M & ' ~ A .

For conditions associated with all of the portions of the Venus and Mars aero-assisted trajectories, The strong in- teraction length will be on the order of 100's to 1000's of meters. For instance, in the Venus entry portions, t s t rong x 5,000 m, primarily as a result of the extreme Mach numbers. Thus, ezcept in cases of highly cooled walls, these planetary waveriders must be designed with strung in- teractions between boundary layer and shock included.

Lift and Required Al t i tude Approximation

Using the principles stated above, flow over a waverider may be modelled with straight streamlines over a delta- shaped flat plate at some apparent angle of incidence. If the

waverider is a t apparent incident angle B and has area A, Lee's modified Newtonian theory predicts for small angles and high Mach numbers, lift

where C,,,, = 1.84 for terrestrial air, and 1.86 for the COz of the Martian and Cytherean atmospheres.

Newtonian theory is relatively inaccurate a t small angles of incidence, and is best used as a basis of comparison for other more accurate solutions. Lift can also be estimated with the tangent wedge theory, which predicts tha the pres- sure rise through a shock on a shallow-angled surface is given approximately by

In the limit of large 8, such that M2B2 > 1, this apprxo- mation yields a lifting force of

whereas in the limit of very small B such that MZB2 < 1,

Equation (17) approaches the limit of

Waveriders designed for aero-gravity-assisted maneuvers may in fact fall into any of these three categories, depending on flight conditions and design constaints. In general, flight Mach numbers for these craft , especialy on interplanetary trajectories, will be in the range of 30-100, and effective deflection angles will be a t most 5' - 10'. This will lead to values of M2B2 x 7 - 300, so lift will be L = 1 . 2 p ~ ~ B ~ ~ in air, and L = 1.17pU2B2A in the Cytherean and Martian atmospheres, which is only 20% higher than the Newtonian prediction.

Newtonian theory also predicts drag of D = $,,,,pU2~B3; the tangent-wedge concepts once again pre- dict slightly higher drag, D = (y + 1)/2pUZ~B3. If wave drag is the dominant source of drag (as opposed to leading edge bluntness and skin friction), L 1! 1 . 1 7 p ~ ~ ( $ ) ~ . The required altitude is then that at which density satisfies:

so that Leading Edge Bluntness

Not surprisingly, increased mass, m, or decreased area A forces the vehicle to fly at higher densities, and thus lower altitudes. Increased aerodynamic performance, as measured by LID, also requires lower altitudes, because higher LID generally means lower lift. Correspondingly, the higher den- sities a t lower altitudes dictate higher Reynolds numbers, and resulting in reduced viscous effects and correspondingly higher LID. Planetyary waveriders will generally operate best a t lower altitudes, although heating problems are o b viously exacerbated under thoee conditions.

Drag Prediction

Using the above arguements, waverider aerodynamic per- formance will be analyzed by dividing the drag into three basic components:

0 wave drag

skin friction

leading edge bluntness

In order to do so, it is of interest to parameterize the wa- verider configuration, as shown in Figure 9. The waverider will be assumed to have a delta planform, which ignores the fact that waveriders in fact tend to have curved lead- ing edges, and often blunted forntal regions. With length L, width w, and aspect ratio A= w/L, with a leading edge radius R1.,., planform area can be expressed in terms of overall length and aspect ratio as A = L2.4t/2, and leading edge length is ! d m .

Wave Drag

Wave drag can be estimated with the shock pressure ratio approach taken above to estimate lift: D = 1.17pU2~83 in a COz atmosphere. Substituting for planform area in terms of overall length and aspect ratio,

Once again, as with lift, the angle 19 is an effective angle of incidence ot the flow, which is best defined in terms of the aerodynamic performance; however, since additional drag terms will also be included in the following formulation, i t would be inconsistent to define LID entirely with Newtonian drag, so i t is not possible to relate 8 to LID directly without the addtion of viscous and leading edge terms.

Newtonian theory is quite accurate in predicting forces on blunted objects. Force on a cylinder with radius RI.,. is &... = ~Cp,,pU~or,d&.e.s, where s is the length of the cylinder and is the velocity normal to the cylinder. Note that the leading edge only resembles a half cylinder, but the downstream portion of a cylinder in Newtonian flow does not contribute to the drag. For a COz atmosphere, the two waverider leading edges will therfore experience a normal force 4.,. = l.24pU~0,,a1 RI... LJ-.

The normal velocity, Unor,d which appears in the drag equation, is the component of the freestream velocity per- pendicular to the leading edge; in terms of delta plan- form dimensions. If the leading edge makes an angle 4 with the freestream, the component of force on the cylin- der opposing the motion of the vehicle will be the net drag, &... = 1.24pu2 sin 43&.e.f Jm. Note that the sine of the relative angle 4 is cubed in this equation, since it appears not only in the velocity term, but also the direction in which net drag is transmitted to the vehicle.

The leading edge radius is most readily considered in terms of the overall length of the vehicle, so it will be more convenient to define R.,. = Rl.,./L. The angle 4 can be defined in terms of the dimensions defined above: sin q5 = A L / ~ - . Thus, the leading edge bluntness drag is :

The above development has assumed that all of the wa- verider leading edge bluntness contributes to drag. This is true if there is neither anhedral or dihedral. In fact, wa- verider designs tend to exhibit anhedral, so some of the force generated on a blunt leading edge may contribute to the lift of the vehicle if the leading edge has an effective in- cident angle with the freestream. If the apparent anhedral angle of the leading edge goes to 90' downward, leading edge bluntness makes no contribution to drag if the leading edge incidence angle is zero relative to the flight direction. For small leading edge incidence, bluntess produces more lift than drag, so large anhedral can be used to increase LID with bluntness. Thus, the above development is a worst case drag model.

Several other effects of leading edge bluntness have not been included in this analysis, and require a development which is beyond the scope of this treatment. As the leading edge blunts, the pressure gradient at the leading edge will become increasingly more favorable to the surface boundary layer. Thus, the increase in drag due to bluntness my be offset by diminished viscous losses on the rest of the vehicle. Furthermore, increased bluntness may mean reduced tem- peratures a t the leading edge, which will also reduce visous losses there. This once again suggests that the above model

will overpredid the effect of bluntness in increasing drag. A negative aspect of leading edge bluntness that has not been included will be that the bow shock detachment die tance will increase, thereby permitting flow leakage from the lower surface and reducing the flowfield containment effect of the waverider shape.

Viscous Surface Drag

The very high Mach numbers associated with aero- assisted flight suggest that boundary layers will be entirely laminar over all but the largest of waverider bodies. Once again, the waverider permits the assumption that stream- lines run straight back along the vehicle surface, so the drag may be estimated by integrating over the entire vehicle area as a series of two-dimensional streamlines.

Reynolds analogy can be modified for a hypersonic flat plate, and the result is that the skin friction is given by:

C, t 0 . 6 6 4 J E pao Retao (23)

where p,/p, is the ratio of inviscid shock layer pressure to the freestream pressure, and C is the Chapman-Rubesin pa- rameter defined above. If the pressure ratio across the shock is nearly constant (i.e. no strong interaction effect, so that

> 3, the integrated coefficient of drag over a pathlength 1 is:

The drag on a delta planform with length f and width w will therefore be:

Once again substituting for the Chapman-Rubesin pa- rameter with a power law viscosity relation, and using the approximation pe/p, t 1 + +Me + )+(+ + l)(M@)', the viscous drag becomes:

In a CO2 atmosphere,

In practice, planetary waveriders will have values of MB be- tween 5 and 300, although skin temperatures may be much

cooler than the adiabatic wall temperature; as a result, it is difficult to accurately model skin friction, but it will be in the range of as:

where i t is assumed that on the top surface, 0 s 0. Note that in the range of flight of interest here, viscous drag is very insensitive to flight Mach number, but is relatively sensitive to incident angle on the lower surface.

Lift/Drag Sensitivities

From the above development, the overall lift-tedrag ratio of a waverider can be expressed in terms of the components of drag, and modified Newtonian lift:

We can return to the question of whether changes in lift should be accomplished by varying angle of attack or al- titude. If changes in required lift are satisfied by varying altitude, and thus p, the derivative of drag with respect to a change in lift is:

. = 0 + ( 1 . 0 5 ~ ~ ...[A]+ a L constant e

In contrast, the derivative of drag with respect to changes in lift as a result of changing angle of attack is:

Note that the viscous term is sensitive to changes of in- cidence angle only a t relatively large angles. As such, de- creasing lift will tend to decrease the viscous effects on the lower surface, while having little effect on the upper surface of the waverider.

Thus, if the effective incidence angle is such that

(31) in the Mach number range of interest, changes in lift should be accomplished by changing angle of attack. For the case of very small leading edge radius, this requirement is gener- ally satisfied if O(CD,,, dras) > O(2CDskin friction). In most

of 10" to lo-'. The resulting near independence of aspect

~ ~ b l ~ M - ~ ~ ~ LID and ~ ~ ~ ~ ~ ~ ~ ~ d i ~ ~ 8 for venus ratio has been confirmed by detailed computational codes,

Aero-Assist Waveriders which have occasionally produced multiple optimized wa- veriders for certain flight conditions which vary dramatically in width for a given length, but have nearly identical LID. Waverider length, L 8,,,. LIDrnax. This also implies that ablation will have negligible effect on

1 m 10.2O 2.8 the waverider performance, unless leading edge bluntness is so severe that the pressure containment provided by the

10 5.8' 5.0 leading edge-bow shock attachment is lost.

The LID ratio will decrease from maximum with changes in altitude. Taking the derivative of LID with respect to changes in L for fixed 8 ,

waveriders, wave drag dominates over skin friction drag, so f l L / D ] I = (L)' . 6 9 ~ . " it will usually be best to adjust altitude to modify required aL constant B D U28312At& lift. This suggests that a waverider control law will primar- ily maintain the waverider a t the same attitude, but adjust altitude as needed. If wave drag dominates other drag sources, 8 [LID]- ' ;

in fact, for waveriders studied for the Venus flight, 812 z With the above formulation for lift and drag, the lift-to- [LID]-' :

drag ratio is of the order:

Several trends are apparent in order to maximize LID: The LID ratio is more sensitive at higher values of LID: this is understandable, since high LID typically translates into

Overall length, L, should be as large as possible very small drag, so and change in vehicle performance which produces noticably more drag will have a strong effect on

When significant leading edge bluntess is LID. It is further apparent that the waverider aerodynamic aspect be (i'e' performance will be more constant through an aero-gravity-

wider waveriders are desirable) assist if

0 In the absence of leading edge bluntness, aspect ratio has little effect on LID for a given overall

0 The waverider is as long and wide as possible length.

0 Altitude should be as low as possible. Flight is at lowest possible altitude

Velocities are as high as possible, since required Since L I D is insensitive to either Mach number or flight lift scales approximately with the square of

velocity, a waverider designed for one maneuver should per- flight velocity form about as well in a second aeropass. Thus, in the Earth- venus-Mars trajectories, a single waverider shape should provide good performance a t both Mars and Venus. Unfortunately, the trend to flight at lower altitudes will

also increase the heating problems on the vehicle. Table 1 presents maximum LID values for the Venus

cruise portion of the Earth-Venus-Mars-Sun trajectory, as a function of vehicle length. Note that the larger the vehi- ~ ~ ~ d i ~ ~ ~ d ~ ~ ~ ~ ~ ~ ~ ~ ~ t ~ ~ ~ ~ ~ t i ~ ~ t ~ ~ cle, the higher the value of L I D that can be obtained. Also shown is the effective incident angle, 8 , associated with max- imum LID. Smaller vehicles tend towards larger 8 because wave drag must offset viscous effects. These simple analyt- ical results match computational results extremely well; for instance, a 80 m waverider has been designed for the Venus flight with L/D=6.9, as shown in Figure 8 .

The primary hinderance to the performance of a wa- verider will be the extremes of heating rates experienced at the leading edges. This not only because velocities will be extremely high and densities relatively high as well, but also because the leading edge radius of these vehicles is gen- erally very small. Indeed, the design trends that improve

Generally, leading edge bluntess will have a negligible ef- waverider aerodynamic performance tend also to increase fect on L I D , as typical values of z... will be on the order the severity of the heating problem.

8

Heat transfer to a stagnation point is most simply a p proximated with the dimensional correlation:

for a CO2 atmosphere and wall enthalpies much lower than freestream, , we can estimate K x 1.64 x lo-' when q, has units of w/m2, p has units of kg/m3 and velocity is in m/s.

At the flight conditions associated with aero-gravity- assisted trajectories, nearly all of the gas surrounding the vehicle leading edge will be dissociated. The complete die sociation of COz liberates 382 kcal/g-mole, or 3.63 x 10' J/kg. Thus, in radiative equilibrium, assuming black body conditions in the shock layer (which may not be a good a s sumption!), the stagnation point equilibrium temperature will be approximately given by the energy balance:

with a typical emissivity of about .8, the equilibrium stag- nation point temperature will be:

Heating conditions at the flatter surfaces of the waverider will obviously be less severe than those at the leading edge; indeed, the upper surface of the waverider will be in rela- tively benign thermal conditions, since it is in the lee of the flowfield.

Sample Atmospheric Flight

In the Earth-Venus-Mars-Sun mission, the above con- cepts can be demonstrated for a sample waverider vehicle. Consider a satellite with mass 5000 kg, on a waverider that is 10 m long and 2 m wide.

With an approach velocity of 17.470 km/s, the spacecraft will enter the Venus atmosphere a t 20.25 km/sec, at about Mach 80 (compared to Mach 65 for Galilee's Jovian entry) at 80 km altitude. With a 20 cm leading edge, stagnation temperatures will reach 10,000 K; 14,000 K with a 10 cm radius, neglecting ionization effects. The spacecraft flies through approximately 62' for 5 minutes 34 seconds; with constant L/D=7, the spacecraft loses 2.53 km/sec to drag; with a more likely L/D=5, the loss is 3.37 km/s. Although the initial velocity could be increased to compensate for this loss, i t is best made up for with an engine firing a t the end of the maneuver, as found in Reference's.

Acceleration will be 5.75 terrestrial g's a t the start of the maneuver, droppping down to 4.2 g's a t atmospheric

exit. These values are higher than t h e typically specified than sustained human flight (3 g's), but within the range commonly experienced by pilots.

The above modelling estimates i t will emerge with an ini- tial L/D=7 find L/D=5.8; with initial L/D=5, final LID= 4.8. Higher LID vehicles will suffer a noticable loss in per- formance, reducing their overall advantages versus lower LID shapes. At exit, stagnation heating will have abated, so the 10 cm radius leading edge will have equilibrium temper- ature of 13,500 K, and the 20 cm radius will be a t 8,350 K. The waverider leaves Venus at 14.459 km/sec relative speed if initial L/D=7, having lost 3.011 km/sec to drag losses; with initial L/D=5, the loss is 4.05 km/sec. The total AV of the maneuver is 21.64 km/sec a t L/D=7.

Conclusions

The use of aerodynamic lifting force for the manipulation of spacecraft trajectories depends ultimately on the feasi- bility of designing hypersonic configurations with LID val- ues of approximately 7-10, and retaining that perfromance for the duration of the flight. Regardless of the details of the particular atmospheric trajectory chosen, higher LID always translates into a more effective aero-assisted flight, because i t enables larger angular deflections for a given en- ergy loss.

This work has used simple modeling to address some basic performance questions. For the trajectories featured here, acceleration levels and heating rates are beyond the limit of current technology. Although unmanned spacecraft may be capable of 6 g sustained flight, 14,000 K leading edge temperatures for 5 minute intervals goes beyond even the extremes of temperatures predicted for the Galileo atmo- sphere probe (11,000 K). However, they are not so large that i t is impossible to envision future technology capable of accepting such a thermal environment.

Aerodynamic performance will certainly suffer during an aero-assisted flight. In the example presented, a drop in LID from 7 to 5.3 is not particularly dramatic, but it does translate directly into losses that must be compensated for with onbaord propulsion. This also suggests that the ad- vantages of high L/D are not as great as the constant LID predicts. Also, the dependence of LID on viscous effects, and the strong influence of ambient density through the Reynolds number, casts doubt on the viability of using one waverider shape for more than one aero-pass a t different flight conditions.

If they are viable, some general design rules should be adhered to in designing planetary waveriders. The dimen- sional arguements suggest that, for maximum aerodynamic efficiency, they should be as long as possible, though width has little impact on performance. Additionally, low alti- tude flight, which results from high required wing loading,

provides better performance, albeit a t the cost of higher heating rates.

This work suggests that more concern should be devoted to the off-design characteristics of waveriders, not only for changes in Mach number and angle of attack, but changes in ambient density as well. A study of the effect of leading edge bluntness on the leading edge attachment is sorely lacking. In addition, more detailed studies of heating patterns, and an address of control issues will be paramount if the aero- gravity-assisted flight concept is to be made a viable option for space mission planners.

Acknowledgements

The authors gratefully acknowledge the assistance of Prof. John D. Anderson, Jr., of the Department of Aerospace Engineering at the University of Maryland in preparing this document and discussing many of the ideas contained within. Dr. James Randolph, David Bender, and Stacy S. Weinstein of the Jet Propulsion Laboratory s u p plied much information on the aero-assisted trajectory anal- ysis. A portion of this work was supported by a grant from NASA and the University Space Research Association, to whom appreciation is also expressed. Additional computer support has been provided by the Jet Propulsion Labora- tory.

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Figure 1: Generic aero-gravity-assisted maneuver, in planet-relative frame.

Figure 2: Transformation of spacecraft velocity from planet-relative frame to absolute coordinates. Planet's ve- locity could be in any direction, though a representative value is shown.

Figure 3: Total deflection angle that can be achieved if final velocity is 90% of initial velocity, for various values of LID.

0 1 2 3 4 5 Normalized V

Figure 4: Velocity a t completion of an aero-gravity-assisted maneuver, as a function of L/D ratio, assuming 135' total turning angle

@Initial position of Position of Venus planets during aero-pass

0 Position of Mars during aero-pass Figure 5: Schematic of a representative flights performing aero-gravity-assists in the atmospheres of Venus and Mars.

Wentry) 17 km/seC M a c h : 76 ~ l t . 80 km

Base H e i g h t : 1 Om Base W l d t h : 28m

Figure 6: Generation process for a conical flowfield wa- verider. The conical shock is defined, and the waverider lower Surface is Selected by identifying a Specific stream Figure 8: Typical waverider shape, in this case optimized mrface which intersects the shock- Other generating shapes for flight at Mach 76 in the Venusian atmosphere. Reference may be used, and selection of the stream surface is best o p timized for minimum drag, maximum LID, or appropriate

[151

volume constraints.

0 Non-waveriders Conid flow waveriders Powerlaww1vaidem

A Planetary waveriders - K u c h e m p ~ L/D " burier"

Figure 7: A comparison of LID values for various hypersonic configurations, including waveriders. Planetary waveriders have been designed for flight in Mars and Venus atmosphere. Open circles represent various non-waverider flight vehicles, both ground tests and flight data. The viscous optimized waveriders break the traditional "LID barrier" of Kuche-

Figure 9: Nomeclature for simple model of waverider flow- field.

mann [ll]. From References [8,9]