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NASA TN D-1091

00

TECHNICAL NOTED-1091

lj-- FLIGHT-PATH CHARACTERISTICS FOR DECELERATING

S .FROM SUPERCIRCULAR SPEED

C4- By Roger W. Luidens

Lewis Research Center

Cleveland, Ohio

A NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

WASHINGTON December 1961

1PNATIONAL AERONAUTICS AND SPACE ADMINISTRATION

TECHNICAL NOTE D-1091

FLIGHT-PATH CHARACTERISTICS FOR DECELERATING

HFROM SUPERCIRCULAR SPEED

, By Roger W. Luidens

SUMMARY

Characteristics of the following six flight paths for deceleratingfrom a supercircular speed are developed in closed form: constant angleof attack, constant net acceleration, constant altitude, constant free-stream Reynolds number, and "modulated roll." The vehicles were requiredto remain in or near the atmosphere, and to stay within the aerodynamiccapabilities of a vehicle with a maximum lift-drag ratio of 1.0 andwithin a maximum net acceleration G of 10 g's. The local Reynoldsnumber for all the flight paths for a vehicle with a gross weight of10,000 pounds and a 600 swept wing was found to be about 0.7XIO6 .

With the assumption of a laminar boundary layer, the heating of thevehicle is studied as a function of type of flight path, initial G load,and initial velocity. The following heating parameters were considered:the distribution of the heating rate over the vehicle, the distributionof the heat per square foot over the vehicle, and the total heat inputto the vehicle. The constant G load path at limiting G was found togive the lowest total heat input for a given initial velocity. For avehicle with a maximum lift-drag ratio of 1.0 and a flight path with amaximum G of 10 g's, entry velocities of twice circular appear thermo-dynamically feasible, and entries at velocities of 2.8 times circularare aerodynamically possible. The predominant heating (about 85 percent)occurs at the leading edge of the vehicle. The total ablated weight fora 10,000-pound-gross-weight vehicle decelerating from an initial velocityof twice circular velocity is estimated to be 5 percent of gross weight.Modifying the constant G load flight path by a constant-angle-of-attacksegment through a flight- to circular-velocity ratio of 1.0 gives essen-tially a "point landing" capability but also results in an increasedtotal heat input to the vehicle.

INTRODUCTION

Studies of interplanetary missions indicate that atmospheric brakingcan be markedly more efficient (in terms of weight requirements) than

m2

propulsive braking (ref. 1). To apply atmospheric braking at the entryvelocities associated with interplanetary trips, several problems mustbe studied. One of these is the question of whether the achievable aero-dynamic entry corridors (e.g., refs. 2 and 3) are compatible with thecorridor capability of the planet approach guidance system. A secondproblem is protecting the vehicle from the heat generated in deceleratingfrom the entry velocity.

In terms of acceptable G loads, entry to Earth at velocitiesgreater than circular appears feasible. The advent of ablation heat-protection systems, in contrast to radiation- and heat-sink systems, also 0

0makes possible the consideration of supercircular entry speeds from thethermodynamic point of view. To minimize the weight of the heat-protection system requires the use of "deceleration flight paths" thathave inherently low total heat inputs. The deceleration flight path isdefined as that segment of the total flight path in the atmosphere whichis flown for the purpose of decelerating the vehicle from an initialvelocity and altitude to a desired terminal condition. It is subsequentto the maneuver which determines the entry corridor. The velocity changeduring the entry maneuver may be small (ref. 3), so that to a first ap-proximation the initial velocity for the deceleration flight is equal tothe entry velocity.

A number of studies of the heat input and flight mechanics of de-celeration flight paths have been made (refs. 4 to 7). The stagnation-point heat input per unit area nondimensionalized by functions of noseradius Rh, lift-drag ratio L/D, wing loading W/A, and drag coefficient

CD is the parameter usually presented. Items not considered in these

studies which are dealt with in the present analyses are: (1) the inter-relation among the parameters RN, L/D, W/A, and CD required by the

vehicle geometry, (L) the heat input to the windward side of the vehicle,(6) the integration of the heat input per unit area over the vehiclearea, and (4) the effect of differentiating between the vehicle maximumand operating lift-drag ratios. The parameter heat input per unit ve-hicle gross weighu is used as a measure of the heat input to the vehicle.This parameter is closely related to the ablated weight as a fraction ofthe vehicle gross weight.

The present analysis yields closed-form results. The flight mechan-ical approximations made are les3 restrictive than the assumptions usedto arrive at existng closed-form solutions. The characteristics ofthe following types of flight paths are derived: (1) constant angle ofattack, (2) const: nt G load, (7) constant heat rate, (4) constant alti-tude, (5) constant free-stream Reynolds number, and (6) "modulated roll."The fcllowing constraints are imposed: that the vehicle (1) remain inor near ;Jhe atmo-,sphure, (t) stay within its aerodynamic capabilities,and (,j) stay within Lhe G load capabilities of the payload and

3

structure. A Reynolds number constraint is also discussed. The totalheat input to the vehicle and the G load history are discussed as afunction of the type of flight path, initial velocity, and the initialG load. Finally, a method of controlling landing location is discussed.

ANALYSIS

HThe general objective of this analysis is to develop closed-formsolutions for the characteristics of the deceleration flight path.

Equations of Motion

The equations of motion including gravitation and aerodynamic forcesin the directions tangent and normal to the flight path may be written inthe nomenclature of figure 1 as

1 dV -CDpVAr -CDPV-2 Agr- = -2 - sin cp 2W -sin ()

V CIPV2Ar (2)2Agd S + (V2 -l)cos c. 2W + (V2 -l)cosq, (2)

The symbols are defined in appendix A.

The following assumptions are made: (1) that the flight-path anglerelative to the local horizontal is always small, so that cos p is ap-proximately unity and sin p is approximately p, and (2) that sin (is small compared with the nondimensional term representing the aerody-namic forces in equation (1).

Equations (1) and (2) then reduce to

1 dV _G CD (3)g d CR

V GCL v-2. l(4a)i dT CR

where G is defined as

CRpVA PVAgr2m 2W

4

G represents the net force on the vehicle in terms of the planet g's,or is the net acceleration vector felt by the pilot in planet g's.

For future reference, equation (4a), with the aid of equation (5)and the definition of CL (appendix A), may also be written as

L _L V2= =1 - +g dx(4b)mg g gdt

2 m 7V2 + V 2W . V2 + H

_ d L d (4c)PV- CAr CLAgr 4c

Also frcn figure 1, the component of lift in the vertical direction isrelated to the total aerodynamic lift CL,a and the roll angle e by

CL = CL,a cos e (6)

Vehicle Aerodynamics

The vehicle aerodynamics (fig. 2) are calculated using the equationsand assumptions given in appendix B. The vehicle design is specified by,or characterized by, its maximum lift-drag ratio (L/D)max. In general,

for a given lift coefficient these curves have double values of CD, OR,

and a. In the present study the low range of angle of attack is used.

Flight Paths

Many possible flight paths can be considered; the present reportconsiders six particular flight paths and a generalized flight path. Theflight paths are defined as follows:

(1) Constant a: e = o; CR, CD/CR, and CLCR are constant (see

fig. 2).

(2) Constant h: e = o; p is constant.

(3) Constant G: 0 = 0; CRpV2 is constant (see eq. (5)).

(4) Constant Re: e = o; pV is constant (see eq. (17)).

(5) Constant i: e = o; pl/2V 3 is constant (see eq. (10)).

5

(6) Modulated roll: e is varied from O0 to 900 (or 1800) in a man-ner to satisfy equations (4) and (6); the vehicle angle of attack is de-fined to be constant, and hence CR, CD/CR, and CLa/CR are constant.

Heat Transfer

The heat transfer for the present analysis is estimated assuming alaminar boundary layer. References 8 and 9 present a comprehensive re-

Hview of heat-transfer theories. These have been found adequate up to

Earth circular speeds and are assumed herein to be applicable also atsupercircular speeds.

Reference 10 gives the following form for stagnation-point heattransfer:

is 15 .5Xl09X2J/2p1/2V3 1 (7a)

RJ/ 2 hse

where j = 0 for the two-dimensional leading edge considered in thepresent analysis (j = 1 for a three-dimensional or spherical leading

edge). For entry from high velocities << 1.0 and with equationhse

(7a) in a form using V,

As = 15.5X10 9 V (gr)3/2 (7b)

This equation may be modified to include the effect of sweep and noseradius angle by the relation suggested in reference 8 (pp. 19 and 22):

= cosvAc cos = - sin 2A cos v/2 cos h (8)qs

Combining equations (7) and (8) gives

= 5.5x10-9(gr)3/2 1/2V3 cosvAc cos ? (9)1 /2

The theory and experiments of reference 11 indicate that, for the highvelocities of the present study, v has a value of 1.5 for sweep anglesfrom 00 to 600 and becomes less than 1.5 for sweeps greater than 600.

6

(The theory and experiment of ref. 12 indicate that the value of v maybe as low as unity.)

The symbols defining the vehicle geometry and various flow compo-nents are illustrated in figure 3. In calculating the heating rates andtotal heat inputs, a family of vehicles was considered which are delta-wing configurations having a wing sweepback angle A, a constant thick-ness d, and a constant nose radius RN. The configurations have noseparate fuselage, and the volume in the wing is considered the usablevolume.

The following analogously simple equation for the heat transfer tothe windward side of the vehicle is developed in appendix C:

Axlos VIAx~cos el 2 0-- 15.OlO9 kgr 1~sjV2 =10o.2 kc ~s1V (10

This result is independent of angle of attack between about 30 and 300for a constant roll angle 6.

General Form of Equations for Flight-Path Characteristics

In addition to those characteristics involved directly with theequations of motion there are other flight-path characteristics of in-terest and these are given in general form here.

Altitude. - With the assumption of an exponential atmosphere, theflight-path altitude may be calculated if the flight-path density isknown:

h = l n PL11P SL

For Earth the values 0-1 = 2.35xi04 feet and PSL = 0.0027 (slug)(ft-3 )are used as suggested in reference 4. Also for later use it is notedthat

dh 1 (12)

Radiation equilibrium temerature. - With the flight path and ve-hicle design known, equations (9) and (10) give the heating rate persquare foot 4. In sane applications the radiation equilibrium tempera-ture, determined by the heat rate, is of interest; it is given by

t = M (13)

7

Free-stream Reynolds number. - With the flight-path density pknown as a function of T and the vehicle characteristic length, thefree-stream Reynolds number may be calculated from

Re = pVx (14)

By use of equation (5), equation (14) may alternatively be written as

00 Re = (15)P p gr VC

The free-stream viscosity is assumed constant and independent of altitudefor the present analysis, p = 0.37x106 slugs per foot-second, and x istaken as f plus the root chord c (see fig. 3).

Local Reynolds number. - Local Reynolds number is related to thefree-stream Reynolds number by a function of the free-stream Mach number,the wing-sweep angle, and the angle of attack. The method of calculationof the ratio of local to free-stream Reynolds number Re,/Re is given

in appendix D and assumes r = 1.2. The local Reynolds number is then

= R 7e\ (16)R e , e R "

The value of Re,/Re depends on the value selected for T, and more

generally on the real gas properties. A more precise determination ofthe local Reynolds number than that of appendix D is of little value atpresent, because the transition Reynolds number is not accurately known.

Flight-path angle. - As mentioned previously, it is assumed thatthe flight-path angles are small, and hence

Vsin (p v - Z (17)

But

_ dh -h kh V (18)V dr dp dV dV

and by use of equations (3), (5), and (12)

CogV 2 (19)* 26W ~dV

A

8

The preceding characteristics are point functions along the flightpath. The succeeding characteristics result from an integration alongthe flight path.

Time of flight. - The time of flight is obtained by integrating thetime increment from equation (3) as follows:

T2 g

dV dVSd = J(20) o

1 2V 2

Range. - Using equation (3) for d and approximating cos (P byunity give the range traversed during the deceleration:

R :VV cos dv d r V (21)

GCD

V2

Total heat input. - One of the important criteria of merit for aheat-protection system is its weight as a fraction of the vehicle grossweight. If, for example, the heat-protection system is of the ablationtype and has a characteristic heat-absorbing capacity C in Btu perpound, then the ablated weight may be expressed as

w 1 (22)W : W

The present analysis deals with the parameter Q/w which results froma double integration (along the flight path and over the vehicle). The

integration along the flight path yields the total heat input per squarefoot, which is given by

q dt = rE dV7 (23)

SCD2 G r

9

where 4 is given by equations (9) and (10) and dr is again given byequation (3).

The integration over the vehicle has the general form

a Add qjRNQdd (24)

00,1 This integration is discussed in detail in appendix E.

Sample Development of Flight-Path Characteristics

for a Constant Reynolds Number Path

The case of "constant free-stream Reynolds number" is consideredhere in detail. The velocity ratio V is taken as the independent vari-able. The free-stream Reynolds number is given by equation (14) or (15),'if the vehicle is specified (i.e., f+c and W/A are constant and given).Then for Re to be constant, pV must be constant. Since G (eq. (5))and all the terms collected on the left side of the equation

pVAgr (25)2W f V

are defined as constants, the right side must also be a constant. Thisis the relation given in column 2 of table I. The fir3t column identifiesthe type of flight path. The term in braces is a constant for the flightpath and is specified if the path Reynolds number is specified.

Equation (25) may then be solved for p as a function of V togive column 3 in table I:

p = 2(26)

Because {G/CRVj} is a constant along the flight path,

G = {}CRV (27)

10

which is the result given in column 4. To determine the flight-path

angle qp, the derivative dp/dV7 is required in equation (19). This may

be obtained from equation (26):

X = - 2 1 (28)dV A gr VV

Combining equations (19) and (28) leads to the result in column 5:

CDLrG 1

orV- (29)

With p having been determined, the term V can be discussed. Fromg dr

equation (29)

g L3 r _ 1 d (30)g dr 1r V g dV

where the term 1 dV is given by equation (3), and G in that equationg dr

is given by equation (27) (or column 4). With these substitutions equa-tion (30) becomes

g = -r - V12R CD -2 kRe (31)Or -)fi2%

The term V (column 6) is equal to -kRe (column 19). For CD/CRg d

and CR constant, an assumption required later, kRe is constant.

Equation (4a) may now be solved for the lift coefficient, using the

result of equation (31) and equation (27) for G, to arrive at the re-

sult in column 7:

CL = kRe (32)

The free-stresm Reynolds number (column 8) is given by equation (14)

and is a constant for this case. More generally it may be obtained from

the relations of p to V like equation (26) or column 3 of table I.

I

11

Obtaining the local Reynolds number (column 9) requires knowledgeof the wing sweep angle and the angle of attack. The wing sweep is partof the vehicle design specification. The angle of attack is determinedby figure 2 using CL from equation (32) and the vehicle (L/D)max .

The heating rate to the leading edge 4 (eq. (9)) now depends only

on the parameter p, which has been determined, and on the independentvariable V. This result is shown in column 10. The heating rate to

H the windward side jw (eq. (10)) depends on L/A, which may be determined

from equation (4b) and the result of equation (31) or column 6 as fol-Wlows:

L=WI 1 (1 k- e) (33)

Making this substitution in equation (10) gives the result listed incolumn 11 of table I.

The point functions have now all been determined, and it remainsto determine the integrated functions. The time of flight may be ob-tained from equation (20) and the equation for G from column 4 (oreq. (27)), where the term in braces is a constant, so that

V1

1 1 dv (34G CD V

C- 0

The variation of %/CR and CR (or Just CD) as a function of V re-

quired in equation (34) can be obtained from the curves of figure 2 andequation (32) for CL. Hence, equation (34) can be integrated graphi-

cally or by numerical methods for the most general cases. The presentreport, however, deals only with the more simple cases where CD/CR and

CR can be approximated by constant values and closed-form solutions can

be thus obtained. This assumption is different from the assumptions for

the case of constant a for which CL/CR in addition to CD/CR and

CR is constant. With the assumptions of CD/CR and CR constant,

12

which are noted in column 12, equation (34) becomes

S1 dv (35)

g HCR R CR 7

which integrates to the result in column 13.H

By similar substitutions and assumptions, equation (21) for thisrange becomes

R r dV (36)

C V2

which integrates to the result in column 14.

Again with the similar substitutions and approximations, and byuse of the result in column 10 or equation (9) for ql, equation (23)

for qj becomes

= 14.9 cosVA cos W. 1 1 f 1 V. 5 d7 (37)

R CR V 2

which integrates to the result in column 15.

When column 11 or equation (10) with e = 0 is used for 4 in

equation (23), the integral equation becomes

qw =10. 2 kc~ { } J K:~ -r V kReI dV (38)9GC D C 7

which integrates to the result in column 16.

The integration over the vehicle to obtain Q/W indicated by equa-tion (24) is carried out in detail in appendix E and gives the resultsshown in columns 17 and 18. Alternative forms for the denominator ofcolumns 17 to 18 are shown in the bottom row.

F

13

The results for all the other flight paths except constant a canbe derived in a similar manner. The case of constant angle of attack isdiscussed in appendix F.

RESULTS AND DISCUSSION

The results of the analysis for all the flight paths studied areHsummarized in tables I and II.00

Tables of Equations

The first column of table I gives the name of the flight path. Theterm in braces in the second and subsequent columns is constant over theflight path. The equations with the exception of those in columns 10,11, and 15 to 18 apply to any planet. The results in the aforementionedcolumns are given in table II in a form which is applicable to any planetthat has an atmosphere with heat-transfer characteristics similar toEarth. The equations for G are in terms of planet g's, and V is theflight velocity divided by the planet circular velocity.

Three heating parameters are presented:

(1) Heating rate q (in Btu/(sq ft)(sec)). This is the parameterof interest, for example, for radiation cooling.

(2) Heat input per square foot of vehicle q, which depends on thelocation with respect to the stagnation point of the area being con-sidered. (The stagnation point is not a fixed geometric point exceptfor the cases of constant a and modulated roll.) The parameter q re-sults from the integration over the flight path of the heat input to agiven location.

(3) Heat input Q = (Qj + Qw), which is related to the ablated weight

and involves an integration over the vehicle surface as well as over theflight path.

The heating parameters are presented for two regions on the airplane:(1) the region in the vicini ty of the leading edge or stagnation point,where the heating is a function of nose radius RN, nose radius coordi-

nate angle )\, and sweep angle A, and (2) the windward side of the ve-hicle, where the heat input is a function of the distance x from thestagnation point. The heat input to base and lee regions is not con-sidered.

14

The term k, (column 17) accounts for the reduction with angle of

attack of both the effective sweepback angle and the area of the leadingedge subject to heating. The value of kZ is approximately unity. The

term kc (columns 11, 16, and 18) accounts for the relative lift generated

by the windward side and the leading edge. The term k,, accounts for

the length of boundary layer ahead of the windward side. The term kwith a subscript to designate the flight path (e.g., kRe) accounts for

the amount that the flight-path curvature differs from the planet surfacecurvature. Both kc and kw approach unity for slender (or high H

(L/D) max) configurations and have a value of about 0.5 for theH

(L/D)max = 1.0 configuration of this report.

In table I the heading "Modulated angle of attack" indicates theflight paths for which the lift vector is constrained to the verticaldirection. For all but the constant case, the assumption of con-stant CD/CR restricts the angle of attack to the vicinity of 00 or 900,

as may be seen from figure 2. The present results are for m near 00.

"Modulated roll" in table I designates a specific case where angleof attack with respect to the free-stream direction is constant (hence,CR, CD/CR, and CLa/CR are constant) and the variation of the roll angle

e with V is given by the equation in column 1. This relation betweene and V gives a constant G and e = 0 at Vi.

Comparison with Other Analyses

To obtain closed-form solutions, a number of simplifying assumptionshave been made. As an indication of the accuracy of the present analysisin the stagnation region, a comparison is made for the constant-angle-of-attack path with the results from machine integrations of the equa-tions of motion and heating presented in reference 4. The procedure forreducing the results of the present report to the form of reference 4 isgiven in appendix G. Figure 4 presents the horizontal component of theG load GH and the heating-rate parameter * as a function of velocity

ratio V for an L/D of 0.1 and 1.0 and an initial velocity ratio Vof 0.99. The heat-input parameter q for decelerating from V = 0.99to V = 0 is given as a function of L/D. In the calculation of q,the function F of table I was evaluated at the maximum value of V.The present analysis generally gives the greater heat input. In allcases the agreement between the two analyses is considered satisfactory.

15

To indicate the accuracy of the present analysis in estimating theheat input to the windward side of the vehicle, a comparison is made withthe results of reference 13 for the constant-angle-of-attack path. Theresults of reference 13 have been extrapolated to the condition of zerowall temperature and apply for the case of F = 1 in table I. Figure 5compares the heating-rate parameter 4(x/(W/A))l/2 and the heat input

H per unit span parameter 1/2 as a function of velocity ratio

8 L f(c'W) 1_2 _

for an initial velocity V of 1.0. Also shown is the variation of heat-ing rate with angle of attack for constant lift. In general, the agree-ment between the two analyses is satisfactory. Compared with reference13, the present analysis indicates increasingly higher heat inats withincreasing angle of attack. The maximum angle of attack to which thepresent analysis is applicable depends on the degree of approximationconsidered acceptable. The present analysis is about 15 percent higherat m = 300.

In general, the accuracy of the present analysis is adequate forpreliminary studies.

Some General Considerations

For atmospheric braking it is desired to determine a combinationof vehicle geometry and flight path which will result in a low heat in-put to the vehicle. Generally there is a substantial increase in con-vective heat input to the vehicle if the boundary layer on the vehicleis turbulent rather than laminar. Thus it is desirable to restrict theflight path to conditions for which laminar boundary layer can be main-tained. An indication of the possible nature of the boundary layer isgiven by the magnitude of the Reynolds number. Two Reynolds numbers canbe considered, the free-stream value, and a local value at the edge ofthe boundary layer. The free-stream Reynolds number depends on thevehicle design parameters W/A and c and on the flight-path parameterG (see eq. (17) or (18)). The local Reynolds number is further influ-enced by the vehicle design factors of leading-edge bluntness and sweepand the operating parameter of angle of attack. Both local and free-stream Reynolds number are presented in the section Point Functions.

The local Reynolds number is interrelated with the problem of hot-gas radiation. At entry velocity ratios V greater than 1.0, the heatinput from hot-gas radiation can equal or become larger than the con-vective heat input to an unswept stagnation region. The lowest heat in-put to the vehicle will usually exist if hot-gas radiation is avoided

16

where possible. This requires that the vehicle surface angles be smallwith respect to the free-stream direction to avoid high air static tem-peratures. In general, unsweeping the wing leading edge decreases thelocal Reynolds number but tends to aggravate the problem of hot-gas radia-tion. The choice of wing sweep angle is, therefore, a compromise. Thecalculations for the present study were made for a sweep angle of 600.Also to avoid hot-gas radiation the low-angle-of-attack rather than high-angle-of-attack region of aerodynamic operation was chosen. The choiceof the low-angle-of-attack range and the 600 sweep is assumed to reducethe heat input from hot-gas radiation to a low value compared with theconvective heat input. Only the convective heat input is consideredsubsequently in this report.

Point Functions

The evaluation of the various flight paths for decelerating in theEarth atmosphere is discussed with the aid of figures 6 to 11 in whichis plotted the variation of the point functions of table I with V. Asecond abscissa shows elapsed time in minutes with time equal to zero atV = 2.0. To arrive at numerical results, a vehicle design defined byPf = 20 pounds per cubic foot, W = 10,000 pounds, A = 600, and

(L/D)max = 1.0 was assumed. (Higher values of (L/D)max result in

larger heat inputs.) For a vehicle like that of figure 3 and the abovenumerical values, the following vehicle characteristics may be determinedby the equations given in appendix E: W/A = 83 pounds per square foot,RN = 2.06 feet, and c =14.5 feet. A length of 18.8 feet was used inthe calculation of the Reynolds numbers. Vehicle operation is in thelow-angle-of-attack range, and the assumptions are made that CD =.CD,o

and CD/CR = 1.0, although for some cases more precise approaches to the

calculation of the point functions are possible. All the flight pathshave a comnmon point, at V = 2.0, G = 10 g's, and e = 0. This resultsin approximately the same values of a and h, about 5.30 and 162,000feet, respectively. When appropriate, the following physical limits arenoted on the curves: ICL,maxl = 0.9, Gma x = 10 g's, hmin = 0, and

= 190 Btu/(ft-2 )(sec), or tV = 44600 R for an emissivity of unity.

Flight-path angle. - The fact that for all cases the flight-pathangle cp is small except in the vicinity of V = 0 justifies the assump-tion of small angles. For all cases except constant a (fig. 6), cp isalso continuous. For the constant a case, cp changes discontinuouslyfrom plus to minus near V = 1.0 and the altitude becomes indefinite.This situation is physically approached by a shallow skip out of theatmosphere at V = 1.0. In spite of this indefiniteness in the altitude,the heat input to the vehicle remains finite as is shown later (see alsocolumns 17 and 18 of table I).

ZP17

Altitude and lift coefficients. - For all cases except constant h(fig. 7) the altitude becomes less than zero for V approaching zero.On the constant h path excessive lift-coefficient requirements are en-countered at V's approaching zero. The constant Re path (fig. 8)also requires excessive lift coefficients at low V's. In a practicalflight these and other low-speed anomalies are usually precluded byspecifying some terminal condition of velocity, lift coefficient, oraltitude. The actual flight path may deviate from the specified path inthe low-speed range with little effect on the total heat input calculated

ofrom an initially supercircular velocity, as is shown later.0

The lift coefficient is negative; that is, lift force is directed

toward the Earth for flight velocities greater than circular. Thisoccurs fundamentally because the centrifugal force tending to "throw"the vehicle out of the atmosphere is greater than the gravity force andthe difference in these forces must be counteracted by the lift. Forall cases except constant G (fig. 9), constant 4 (fig. 10), and modu-lated roll (fi . 11), the lift coefficient is positive (directed awayfrom the Earth) for subcircular velocities, because at subcircular speedthe centrifugal force is less than the gravity force. However, to satisfythe assigned values of 4 and G for the cases of constant 4, constantG, and modulated roll, negative lift coefficients are required at sub-circular speeds as well as at supercircular speeds (because of the term(V/g)(d#/dt) in eq. (4)).

G load. - For the constant G path (G = 10 g's), the abscissa offigure 9 shows that the time of deceleration in the region of strongheating, that is, from V = 2.0 to 7- 0.5, is about 2 minutes. Thiscombination of g's and time is about the presently accepted limit ofhuman tolerance if useful functions are to be performed by the man. Forthis reason Gmax = 10 g's was used in the present examples.

For all cases except constant A (fig. 10) the G load is at orbelow the initial G value at V = 2.0. For the case of constant qthe G increases with decreasing V to maintain q constant. At aV higher than 2 the CL becomes excessive. These same characteristics

exist if other values of constant 4 are specified. Thus it is con-

cluded that the constant q path does not have application over a widerange of V.

Angle of attack. - The angle of attack is shown as positive. Forthe flight paths where the lift coefficient varies from minus to plusalong the flight path, one has the choice of permitting the sign of theangle of attack to change or of rolling the vehicle 1800. In the latterinstance the same side of the vehicle remains the windward side. Thevehicle angle of attack is for all cases equal to or less than the angleof attack at V = 2.0, where a is about 5.30 (except for the very lowV's). From figure 2 it can be seen that CD, CD/CR, and CR vary less

18

than ±5 percent from a mean value over the angle of attack range 00 to5.3 ° . More generally, for Gi = 10 g's and (CL/CR)i = (1 - V2 )/Gi

(eq. (4a) with dt +0), CD for example, varies less than ±7 percent

from a mean value up to Vi = 2.25 for (L/D)max = 1 and 2, the values

of (L/D)max examined. An increase in Gi will increase Vi, for which

the variation of the coefficients is small. Hence, for many problems ofinterest it is a good approximation to use constant values for CD, tF

II-CD/CR, and CR as is required to arrive at some of the integrated func- C

tions.

Free-stream Reynolds number. - The free-stream Reynolds number Reis also shown in figures 6 to 11. The case of constant h shows alinearly decreasing Reynolds number with decreasing V. The case of con-stant G for G = 10 g's shows increasing Reynolds number with decreasingV and values much larger than for the other cases in the lower V range.The various flight paths illustrated thus offer a choice of increasing,constant, or decreasing Reynolds number with decreasing V.

Local Reynolds number. - The transition from laminar to turbulentflow in the boundary layer is more significantly related to the localReynolds number than to the free-stream R_"nolds number. The variationof the local Reynolds number Re, with V is shown for the condition

that the wing leading edge is blunt and swept 600. All the flight pathsexcept modulated roll (fig. 11) show a dip in Re, in the vicinity of

V= 1 followed by a rise in Re, for V < 1.0. For the flight paths of

constant G and constant Re, this dip is associated with a lower angleof attack in the vicinity of V = 1.0. For the constant a case it isassociated with a low air density (see fig. 6). In the vicinity ofV= 1.0, the local Reynolds number for both the constant m (m - 5.30)and constant G (G = 10) cases is substantially lower than for themodulated roll case (G = 10 g's and a. = 5.30). This illustrates P dis-advantage of using modulated roll. The high local Reynolds number formodulated roll is associated with the combination of the high air den-sity and the 5.30 angle of attack at V = 1.0. The effect of angle ofattack on the ratio Re,/Re is discussed in appendix D. In general,

for the flight paths considered, it appears less difficult to achievelow local Reynolds number at supercircular than at subcircular speeds.

The constant a, constant h, constant Re, and constant G flightpaths show rather small variation in Re, with V and little variation

among the several flight paths over the speed range from V = 3.0 to0.8. In this speed range the general level of the local Reynolds number,0.7X10 6, suggests an extensive run of laminar boundary layer on the ve-hicle is possible.

19

Heating rate. - The values of the heating rate q shown are for avehicle density pf = 20 (lb)(ft-3 ) and apply to the swept leading edge

(A = 600) (note that, in column 10 of table I, W/RNA = 2pf). For all

cases except constant 4, values of 4 increase with increasing V.The temperature limits of presently known materials (e.g., ceramics) areabout 44600 R. By equation (16) this corresponds toj = 190 (Btu)(ft-2)(sec'l). For this value the best case, constant a,shows that entry velocity ratios V up to 1.3 may be possible withradiation cooling, for the present vehicle and flight path. (The flight

-4 path and vehicle design shown do not yield the lowest possible radiationtemperatures.) For entry velocity ratios greater than approximately 1.4radiation cooling is probably not possible, and hence there is interestin the total heat input to the vehicle.

Maximum entry velocity. - From equation 4(a) for Gi = 10 g's andthe maximum values of CL/CR shown in figure 2, it can be determined

that entry velocities of 2.8 times circular are aerodynamically possiblewith a vehicle with an (L/D)max of 1.0. Still higher entry velocities

are possible with a vehicle with a higher (L/D)max.

Summary of point functions. - The more important conclusions fromthe discussion of the point functions may be summarized as follows:

(1) All the flight paths considered except constant j are or ap-proximate physically possible flight paths.

(2) For Gi = 10 g's and (L/D)max = 1.0 or 2.0 the assumption of

CR, CD, and CD/CR constant is a valid approximation up to an entry

velocity ratio V of about 2.25.

(3) With the vehicle designs and flight paths assumed, radiationcooling appears impossible for entries at supercircular speeds corre-sponding to V > 1.4. Hence, the total heat input into the vehicle isimportant.

(4) The local Reynolds number for many of the flight paths of in-terest is about 0.7X106 . This value suggests the possibility of alaminar boundary layer.

Total Heat Input

For the assumptions of laminar boundary layer and hot-gas radiationnegligible because of low vehicle surface angles and with heat input tobase and lee regions neglected, the total heat input to the vehicle is

20

given by the sum of columns 17 and 18 of table I or II. The total heatinput depends on many factors, and these may be categorized as follows:

(1) Factors that depend on the flight plan: terms in braces or incolumn 2 that define the variation of G with V and the terms inbrackets in columns 15 and 16 that depend on the initial and final veloc-ity of the deceleration flight path.

(2) Factors that depend on the vehicle design: vehicle density,pf; gross weight, W; wing loading, W/A; and wing sweep angle, A.

0

(3) Factors that depend on both vehicle design and flight plan: CL

and L/D or CD/CR and CR. These depend on design through the con-

figuration maximum lift-drag ratio and on the flight plan through theoperating lift-drag ratio or angle of attack.

The heat input along families of flight plans related in two waysto the curves of figures 6 to 11 will be considered. For the first familythe flight plans start at various points along the curves of figures 6 to11 and the flight path then follows these curves. The second family isdefined by the condition that, at the initial velocity Vi, the initial

G load Gi is 10 g's.

Figure 12 considers the first family of flight paths, which have incommon a net acceleration of 10 g's at V = 2.0, and presents the heatto the windward side of the vehicle Qw and the total heat input Q

(the sum of the heat inputs to the windward side and to the leading-edgeregion) as a function of Vi . Though not shown by the curves, from the

equation of tables I and II, the heat input Q equals zero for Vi = 0.

Each point in figure 12 can be interpreted as giving the heat inputfor a complete "deceleration flight path" having a terminal velocity ofzero and the initial velocity and initial G given by figures 6 to 11.The history of the flight is also given by figures 6 to 11. For thesupercircular speed rang2 the flight paths in order of increasing totalheat input for a given Vi rank as follows: constant G, modulatedroll, constant Re, constant h, and constant L. At Vi = 2, the con-

stant m case has about twice the heat input of the constant G case.

The curves of figure 12 can also be interpreted to show the distri-bution of heat input as a function of V. For example, for a flightstarting at any initial velocity, the contribution to the heat input indecelerating from V = 0.5 to 0 is given by the Q at V = 0.5. Withthis interpretation for all the flight plans the heat, input for decelerat-ing from V = 0.5 to 0 is generally an order of magnitude less than the

F21

total heat input for decelerating from any supercircular speed. A precisedetermination of the total heat input at the low speeds is not essentialto an accurate estimate of the total heat input in decelerating fromsupercircular speeds. This implies that neglecting the low-speed anom-alies previously discussed in regard to the point functions has littleeffect on the accuracy of calculating the total heat input for the de-celeration from supercircular speeds, and therefore these anomalies havebeen neglected in the present calculations. For most flight paths the

Htotal heat input to the windward side of the vehicle is about 15 percento of the total heat input, for a vehicle with an (L/D)max of 1.0.0

The second family of deceleration flight paths considered is shownin figure 13. Each point in the figure represents a complete decelera-tion flight path having the Vi given by the abscissa and Gi = 10 g's.

The values for Vi = 2.0 for all the flight plans and for the complete

curve for constant G are the same for figures 12 and 13. In the super-circular speed range the flight paths generally rank in the same orderas for the preceding family of curves. The equat'ons for the constant apath are of questionable accuracy for the very low L/D's (L/D < 0.03)tbat 2esult for Vi close to unity, and hence results are not shown at

V less than 1.2.

The relation of the ablated weight to the total heat input is givenby equation (26). An ablation system based on the extrapolation of thedata of references 14 and 15 to the conditions at supercircular speedsand a system utilizing the boiling of lithium (ref. 16) both indicatethe possibility of an effective heat-absorbing capacity of about 10,000Btu per pound of ablated weight. Using this value for C in equation(26) for Vi = 2.0 and Gi = 10 g's gives values of the ablated weight

as a fraction of the vehicle gross weight of 0.03- and 0.085 for theflight paths of constant G and constant a, respectively. Halving thevalue of C would double the ablated weight. From the consideration ofthe ablated weight it is concluded that entries at V = 2.0 and greaterare feasible.

The distribution of heat between the windward side and the leadingedge of the vehicle depends on the G load, as illustrated in figure 14for a family of constant a flight paths with Vi = 2.0 and for a ve-

hicle with an (L/D)max of 1.0. As the initial G load decreases, the

vehicle angle of attack and initial flight altitude must increase tosatisfy equations (4a) and (5). Because of the assumption of constant

CR and CD/CR the other flight plans cannot be considered over this

wide a range of G or a. The curves show that the heat input to both

22

the leading-edge region and the windward side of the vehicle increaseswith decreasing G load. Reducing the initial Gi from 10 to 4.25 g's

about doubles the total heat input. The value Gi = 4.25 g's is the

minimum at which the constant . path can be initiated. Also the pro-portion of the heat input to the windward side of the vehicle increasesfrom about 19 percent at Gi = 10 g's to 47 percent at Gi = 4.25 g's.

The increased angle of attack accounts for the increased proportion ofheat to the windward side of the vehicle.

CWhile the heat input for the constant m, Gi = 4.25 g path is 2.2 C

times that for the constant m, Gi = 10 g path, it is 4.9 times larger

than that for the constant G = 10 g path. This indicates the importanceof both the G level and the type of deceleration flight path to thevehicle heat input.

Figure 14 also shows the Reynolds numbers at Vi = 2.0. Decreasing

the G load decreases the free-stream Reynolds number almost directly,as would be expected from equation (18) (for constant CR). Increasing

angle of attack, however, increases the ratio of local to free-streamReynolds number (see appendix D), so that decreasing Gi has essentially

no effect on the local Reynolds number.

Control of Landing Location

One of the practical requirements that may be placed on an entryvehicle is that it have the capability of landing at a predesignatedlocation. The problem of achieving a low total heat input on a flightpath that will also yield control of the landing location can be studiedto a first-order approximation with the present solutions.

To land at a predesignated location requires control of motion bothdown and cross range with respect to the Earth's surface. By column 14of table I the down range for the constant a. case is indefinitely large.For this case the analysis implies that at V = 1.0 the vehicle glidesup out of the atmosphere into an orbit of indefinite duration. (The de-tails of this maneuver are not defined by the equation. If the vehicleexits from the atmosphere at a small angle as indicated by fig. 6, asmall rocket may be required to "circularize" the orbit, and then a secondsmall rocket may be required to initiate reentry.) Cross range movementmay be obtained by waiting for the Earth to rotate through the plane ofthe orbit If, for example, the vehicle had glided into a polar circularorbit, all of Earth's surface passes through the plane of the vehicle ini/2 revolution of Earth or in 12 hours. This time corresponds to about8 Levolutions of the vehicle about Earth. A down-range capability of 8

23

revolutions thus yields a control of cross range. With the aerodynamiccross-range capability such that half the distance between the traces onthe Earth's surface of two successive orbits can be traversed during thedeceleration from the circular orbit, the vehicle can potentially landat any predesignated location.

The present discussion deals with the down range and total heat in-put for some possible flight plans that include control of landing loca-tion by control of the flight through V = 1.0. The flight plans con-

Hsidered are illustrated in figure 15 in terms of the G history against0o V. The flight paths are composites of the constant G and constant a

flight paths, and all begin at V = 2.0 with an initial deceleration ofG = 10 g's. Figure 15(a) shows the constant m case modified by a con-stant G segment through V = 1.0. The magnitude of the decelerationduring this constant G segment is designated Gmin, because it is the

minimum deceleration that occurs during the deceleration. Then Gmincan be considered an independent variable. For Gmin = 0 the general

path reduces to the constant a case and for Gmin = 10 g's to the con-

stant G case.

The variation of Q and range with Gmin is presented in figure

16. The minimum Q and range occur for Gmin = 10, and the values are

'90 Btu per pound of gross weight and 0.032 of the Earth circumference,respectively. The maximum Q and range occur at Gmin = 0. At

Gmin = 0 the value of Q is a little more than twice the minimum, and

the range is indefinitely large. These data show that increased rangeresults in an increase in total heat input and that the maximum totalheat input is finite and occurs for an indefinitely large range.

Figure 15(b) illustrates a family of flight paths, all of which,according to the present-analysis, have indefinitely large range. Thedeceleration begins at V = 2.0 along a constant G path. At a speci-fied velocity designated V. the flight path is changed to a constant a

path, which yields an indefinitely large range. With V considered an

independent variable, for Va = 2.0 this case reduces to the constant a

case previously considered. Figure 17 presents the total heat inputplotted against Vc. Decreasing V. reduces the total heat input. At

Va = 1.3 the total heat input is 32 percent more than the value for a

constant G of 10 g's, compared with 120 percent more for V.= 2.0,

which is the constant a case.

To obtain control of the range the flight paths of figure 15(b),like those of figure 15(a), can also be modified by a constant G segmentthrough V = 1 as shown in figure 15(c). From the discussion of the

24

flight paths of figure 15(a), it can be concluded that the heat inputsfor the flight paths of figure 15(c) will fall between the two curves offigure 17.

Several comments about the limitations of the present range controlcalculations may be made. The composite flight paths considered in thissection have mathematical discontinuities in flight-path angle at thejunctions of the various types of flight paths, so that some transitionmaneuver is required. The details of the transitions, however, are notexpected to have a significant effect on the total heat input to the 1'

vehicle. c

The details of the flight path through V = 1.0 for the constantflight plan and the determination of a minimum permissible value of

V both require a more detailed study than can be made with the present

assumptions.

Generalizing the Results with Respect to Vehicle

Gross Weight and Vehicle Density

The numerical results for the heat input have been calculated fora specific vehicle with a gross weight of 10,000 pounds and a vehicledensity of 20 pounds per cubic foot. With the assumption of a laminarboundary layer for all vehicles, these results can be applied to vehiclesof different gross weight and average density by the following reasoning.Consider the results of columns 17 and 18 with the substitutions given atthe bottom of the table. In the present application the product(CD/CR)CR is approximately CD,o* For configurations like that of fig-

ure 3, the drag coefficient at a. = O, CDo, depends only on the vehicleproportions, that is, on the thickness ratio d/c and sweep angle A,if the contribution of friction is neglected. The contribution of fric-tion is small for the low L/D configuration considered in this report.The /W for a configuration with (LD)max = 1 but with any given

weight and fuselage density may hence be estimated from the curves ofthis report by the relation:

1 11 l/3(0l/6W ) Wcurves X W /

CONCLUSIONS

An approximate analysis of flight paths for the descent from cir-cular and supercircular speeds has y:ielded closed-form solutions for the

25

following flight-path characteristics: altitude, lift coefficient, Gload, path angle, free-stream and local Reynolds numbers, heating rate,heat input per square foot, total heat input to the vehicle, range, andtime of flight. The heating analysis considers separately the heatingof the leading-edge region and the windward side of the vehicle. Thesix deceleration flight paths considered are constant values of altitude,angle of attack, G load, heating rate, and free-stream Reynolds number,and a case of modulated roll. Limitations on altitude, G load, liftcoefficient, and Reynolds number have been considered. The followingconclusions may be drawn:

00-, 1. Of the flight paths considered, all except the constant-heating-

rate path stayed within the following physical limitations: Altitudegreater than 0, net acceleration G equal to or less than 10 g's, andlift coefficient less than 0.9 for flight- to circular-velocity ratiosequal to or less than 2.0 and a vehicle maximum lift-drag ratio of 1.0.Entry velocity ratios up to 2.8 are aerodynamically possible.

2. Changing the type of flight path or the G level of the flightpath has little effect on the value of the local Reynolds number, al-though the value of the free-stream Reynolds number may vary a largeamount. For the present examples the value of the local Reynolds numberis about O.7XlO6 , a value for which one can be hopeful of achieving alaminar boundary layer.

3. The flight path with the minimum total heat input is the one withthe highest allowable G load.

4. With a laminar boundary layer and a maximum deceleration of 10g' s, entry velocities of twice circular and higher appear aerodynamicallyand thermodynamically feasible for a vehicle with a maximum lift-dragratio of 1.0. About 85 percent of the total heat input to the vehicle isto the leading-edge region. The ablated weight is estimated to be about5 percent of the vehicle gross weight to decelerate from twice circularto zero velocity along a constant 10 G path. A constant-angle-of-attackpath with an initial G load of 4.25 g's has a heat input 4.9 timeshigher than that of the constant 10 G path.

5. For an initial velocity of twice circular, incorporating landingpoint control into a constant 10 G path by controlling the flight pathin the vicinity of circular velocity increases the heat input about 32percent.

6. For a delta wing vehicle for which the useful volume is thevolume in the wing, the heat input per square foot varies directly withthe one-half power of the average fuselage density and inversely withthe one-half power of the G load and resultant force coefficient. Forvehicles of this type, with the same sweep back, the total heat input

26

varies inversely with the one-half power of the G load, the one-thirdpower of the gross weight, and the one-sixth power of the vehicle averagedensity and drag coefficient.

Lewis Research CenterNational Aeronautics and Space Administration

Cleveland, Ohio, August 7, 1961

Li

H

27

APPENDIX A

SYMBOLS

A reference area for aerodynamic coefficients, plan area ofvehicle (see fig. 3), sq ft

H Af projected frontal area of vehicle (see fig. 3), sq ft

b span of vehicle (see fig. 3), ft

C effective heat-absorbing capacity of ablation heat protectionsystem, Btu/lb

CD aerodynamic drag coefficient, 1 D

2 pV2A

CD,o aerodynamic drag coefficient at zero lift

Cf friction coefficient, dimensionless

CL component of aerodynamic lift coefficient in vertical direc-

tion, 1 LV2

2 p 2

CL,a total aerodynamic lift coefficient

C P pressure coefficient, 2(Pe - p)/pV 2

C p specific heat at constant pressure, Btu/(slug)(°R)

CR aerodynamic resultant-force coefficient

c P2W

CRAgr -

C C~gA

c root chord of vehicle, ft

cA leading-edge axial force coefficient,

leading-edge axial force/A. PV2 Af

28

cD leading-edge drag coefficient,

leading-edge drag/i pV2 Af

cL leading-edge lift coefficient,

leading-edge lift/.2 pV2Af

cN leading-edge normal force coefficient,

leading-edge normal force i 2

c1 general wing chord, ft

D aerodynamic drag, lb

d thickness of vehicle (see fig. 3), ft

F see column 19 of table I

f linear distance along streamline from leading-edge stagna-tioln point to end of leading-edge region (beginning ofwindward side) (see fig. 3), ft

G net acceleration, g's

g acceleration due to gravity, essentially the planet surfacevalue, ft/sec

2

H see column 19 of table I

h altitude, ft

hse total enthalpy at edge of boundary layer, Btu/lb

h: wall enthalpy, Bcu/lb

h heat-transfer coefficient, Btu/(sq ft)(sec)(0 R)

J mechanical equivalent of heat, 778 ft-lb/Btu

J see eq. (7a)

k see columns 11, 17, 18, and 19 of table I

L component of aerodynamic lift in vertical direction

La total aerodynamic lift, lb force

29

L/D lift-drag ratio

(L/D)max (LJD)ma x

I distance along leading edge of wing (see fig. 3), ft

M Mach number, dimensionless

m entry vehicle gross mass, slugs00

n exponent defining variation of altitude density with V for

general flight path of table I

p static pressure, lb force/sq ft

Pr Prandtl number, dimensionless

Q total heat input, Q, + Qw, Btu

Qj total heat input to leading edge, Btu

Qw total heat input to windward side, Btu

q heat input per square foot, Btu/sq ft

dimensionless function Q of ref. 4, proportional to totalheat input per unit area for laminar flow

q dimensionless function q of ref. 4, proportional to heatingrate for laminar flow

cheating rate, Btu/(sq ft)(sec)

at ?\=O, maximum heating rate on swept leading edge (see

fig. 3 for 'A), Btu/(sq ft)(sec)

R range, ft

RN leading-edge radius, ft

R gas constant, 1716 ft-lb/(slug)(°R)

Re free-stream Reynolds number, dimensionless

Re Z Reynolds number at outer edge of boundary layer, dimension-less

30

r radius of planet, ft

St Stanton number, dimensionless

T total temperature, OR

t static temperature, OR

V flight velocity, ft/secH

Vc component of velocity normal to leading edge (see fig. 3) 0

VN component of velocity normal to any surface (see fig. 3)

V ratio of flight to circular velocity, V/-VJ, dimensionless

V dimensionless velocity at which change from constant G toconstant m type of flight path occurs

W entry vehicle gross weight, Earth lb

W/A wing loading, Earth lb/sq ft

Wp Pablated weight, Earth lb

x distance from leading edge in stream direction, ft

xt , coordinate systems on vehicle (see fig. 3)

M angle of attack with respect to free stream

CL cross-flow angle of attack (see fig. 3)

exponent describing exponential variation of density with

altitude in equation -P- = e-Oh, ft (0-1 = 2.35x10 4 ftPSL

for Earth)

T ratio of specific heats, dimensionless

5 angle of surface with respect to free-stream direction

E emissivity, dimensionless

e roll angle (see fig. 1) (zero for lift in vertical plane di-rected upward)

K conductivity, Btu/(ft) (sec) (°R)

31

A gecmetric leading-edge sweep angle (see fig. 3)

A C effective sweep angle (see fig. 3)

leading-edge radius coordinate angle (see fig. 3)

pviscosity, slugs/(ft)(sec) or (lb force)(sec)/sq ft

H Vexponent defining variation of laminar heating rate with wing

8 sweepback angle

p atmospheric density0 slugs/cu ft

Pf average entry vehicle density, Earth lb/cu ft

pi average entry vehicle density, slugs/cu ft

a Stefan-Boltzmann constant, 0.48xi0 " 1 2 Btu/(sq ft) (sec) (0R4 )

Ttime, sec

pflight-path angle with respect to local horizontal(see fig. 1)

Wleading-edge surface angle (see fig. 3)

Subscripts:1

aw adiabatic wall

e at edge of boundary layer

i beginning of deceleration flight path

I leading edge

max maximum

min minimum

n general subscript for summation in appendix B

SL sea level

s stagnation point

iFlight quantities of M, 1), i, Re, t, t/T, V, yy p., and p withno subscript or superscript are free-stream values.

32

w windward side of vehicle

w wall

v vertical component of flight velocity

1.,2 stations along deceleration flight path

Superscript :1

H* reference condition in "reference temperature" or "reference 0

0enthalpy" method of calculating heat transfer H

1FlighL quantities of M, p, , Re, L, t/T, V, r, [i, and p withno subscript or superscript are free-stream values.

3P

33

APPENDIX B

VEHICLE AERODYNAMICS

The vehicle aerodynamics are estimated by assuming hypersonic flowand using the approximation to the leading edge of the vehicle shown infigure 3.

H The pressure coefficient on windward surfaces is in general

!2

Cp = (T + l)sin2b = (T + i) (v N (BI)

and for a surface on the leading edge

= (r + I)(.#) Icos(W - aC%)Icos(W - Me) (B2)

In equation (B2) if Cp is positive, the surface is windward, if

negative, the surface is leeward. The pressure coefficient on leewardsurfaces is assumed to be zero (rather than the value given by eq. (B2)).From figure 3

2(PC sin2cL + cos 2 m cos2A = 1 _ cos2cL sin2A (B3)

and

c = arc tan tan. (B4)cos A

Equation (B2) hence becomes

Cp =(,+ 1)(l- cos 2 m sin2A) cos(w arc t tan c--joA -arc tan t-nm

cos A, 1 cos A)

(B5)

For a leading edge composed of multiple flat surfaces of equal width, theaxial force coefficient of the leading edge using the projected frontalarea of the vehicle as a reference area is

34

F CP,n cos cn

CA (B6)Cos%

Similarly the normal force coefficient for the nose isH

n

R C,n sin (CN= n(37)

cos A cos on1

The contribution of the leading edge to the drag and lift, frm the re-sults of equations (6) and (B7), is

cD a cA Cos M + cN sin m (B8)

cL = c N cos m - cA sin m (B9)

The total lift and drag coefficients of the vehicle based on the refer-ence area of the vehicle (see fig. 3) are then

2f (BlO)CL,a = (r + l)sin cL cos M + CL (0

CD = ( + l)sin3c + CD A(Bf)

where the first terms on the right side of the equations are the contri-bution of the reference area A to lift and drag coefficients, and thesecond terms are the contribution of the leading edge.

If one considers a family of vehicles having the general plan formshown in figure 3, that is, having the same sweepback angle but variousvalu.s of c/d, or what is equivalent, various values of (L/D)max, then

the following relation may be derived for low angles of attack:

Af

CD - CD,o = CD o A

35

where cD,o is a constant for a given sweepback angle. From equation* (El)

A- =bc2

and

H Af = bd

so that

CD 2 cD,o= Constant (B12)

36

APPENDIX C

WINDWARD-SIDE HEATING ANALYSIS

A simple analytic expression for the laminar heat transfer to thewindward side of the vehicle at high velocities is developed herein.

Off the stagnation point for moderate angles of attack the heattransfer to the windward side is analyzed using an approximation to the"reference temperature" (ref. 9). The relations required are: 0

q = (taw - )(l)

St = ' (C2)P*VeCp

St = f (Pr*) "2/3 (C3)2

Cf = 0.664.Je (C4)P~ex

v2 -P- 2

taw =t+ 2CpJ 2CpJ(C5)

*C*

Pr*= " (C6)

t* = te + 0-5(t7 - te) + 0.22(taw - te) (C7)

Assuming tv << taw and combining equations (Cl) to (C5) yield

= 0.166 ._____ V2 (C)jpr.O.167V x8'

The three terms in the numerator of the square root term of equation(CB) may be evaluated as follows, as an approximation of Southerland's

37

viscosity equation for high temperatures (ref. 17, eq. (A3)):

8 t*3/2 2.27x10 8 t*1/2 (C9)227xlO'8 t* + 198

The reference density is given by

SWPe t (CO)

p t*

At moderate angles of attack, by Newtonian aerodynamics,

Ve = V cos M (Cil)

The terms t* and pe/p in equations (C9) and (ClO) must now be

evaluated. In equation (C7) for low and moderate angles of attack, te

and (t-w - te ) are small compared with (taw - te) and taw) so that equa-

tion (C7), combined with equation (05), becomes

0.11P1' V2

t* - 0.22 taw = CpJ (C7a)

From hypersonic aerodynamics

Cp,e = -_= (y + l)sni 2(013)

where at supercircular speed (M > 25) and for moderate angles of attack

Pe/P > > 1.0. At the low angles of attack where the assumption produces

maximum error, the heat input to the windward side of the vehicle is gen-erally small compared with the heat input to the leading edge (see fig.14). The error in estimating the total heat input to the vehicle canthus still be small at small angles of attack. Then, since

M2 = V

2

equation (C13) becomesPe V2

2e + y + 1 sin2 A T + 1Y- sin 2a (C14)p 2 Rt 2 Rt

38

Making the appropriate substitution in equation (C8) gives

0.166 12.27Xo-8 _ 2 v COe i V 2 gr (C15)J" .,r*O.29 2

The following values are assumed to be constants:

Pr* - 0.71

1 -1.2

- 1716 ft-lb/(slug)(OR)

C p =- 13.2 Btu/(slug)(OR)

J = 778 ft-lb/Btu

From the hypersonic aerodynamics of the configuration, equation(Bli), and equation (6)

L La j~ 22= A 2k2 pV sin cos (C16)A cos 0 2

where

kc + AfCL 2-/ C7c + A(T + 1)sin2 cos M)(c)

so that equation (C15) becomes, with the assigned constants,

=w l.5xlo'8k gr I a , -V2 (c18)c TIco lOS

The absolute value signs are used to indicate that the heating is realand positive irrespective of the sign of L and cos e. By use of the

value of V k given in column 6 of table I, one may solve for L ing dequation (4b) and substitute in equation (C18) to arrive at the resultsgiven in column 10 of table I. For example, for the constant Re case

39

with e = 0

=, l.5o>lo-kg3/2r i4 .- - 1.e 2 10.2, c 4 -o V- - 1 e

2The term k2 is the ratio of the lift produced by the vehicle wind-

S ward side to the sum of the windward-side and leading-edge lift. Approx-imately, for angles of attack from 00 to 150

Af dc L 1 -/

kc = + -A d(sin m) (T + 1)sin

where dced(sin m) is approximately constant and depends on the sweep

angle. For the present configuration with A = 600 anddcjd(sin m) = 0.88, Af/A = 0.569, m - 5.30, and kc = 0.535.

The term kc, in general, varies over the flight path. For super-

circular entry velocities a conservative estimate of the heat input isobtained by evaluating kc at the maximum velocity. This is done in

the present calculations. The value of kc approaches unity and con-stancy for small values of Af/A or high values of (L/D)max .

40

APPENDIX D

METHOD USED FOR CALCULATING RATIO OF LOCAL

TO FREE-STREAM REYNOLDS NUMBER

The procedure for calculating the local Reynolds number at super-sonic speeds for cones and wedges with unswept edges is presented inreference 18. A relation for calculating the local Reynolds number athypersonic speeds for the case of a flat plate at angle of attack with aswept blunt leading edge is developed herein. The nomenclature is illus-trated in figures 3 and 18.

At a station sufficiently far downstream from the leading,edge,where the surface static pressure for a blunt body is approximately thatfor a nonblunt body, the ratio of local to free-stream Reynolds numberper foot or for the same length may be written

Re z = 0e Ve j_ (D )Re p V

In equation (Dl) the velocity ratio is taken as the Newtonian value,

V-cos a.(D2)

The viscosity approximation is the same as that used in equation(C9),

2.27X10 8 tl/2 (D3)

where for this case

te = t ( ) (D%)

where

V2 cos 2Ac V2 cos2Acts = t + 2J J (D5)s 2JCp 2JCp

41

Hence equation (D3) becomes

Y-1

Pe = 2.27X10 8 V cosA e) 2 (D6)

The term pe/p in equation (Dl) may be written

Pe Ps Pe8, =eP~ (D7)

r- P P Ps

and equation (Dl) then becomes

Re P cos 2/2JCp ----

Re 2.27... 8 V-V/r coB (De)

The ratio Pe/Ps may be written by use of the perfect gas law as

Pe- _j - (Le _rP l/T~~e ~ l/ U_ lI P/ (D9)

Hypersonic aerodynamics gives (see also eq. (C14))

mLe + I + l1 V2 sin2cL (DIO)

p 2 R

and

Ps + T + 1V 2 cos2^ c + i1V 2 c s A D l

p 2 Rt 2 Rt

where the approximation is valid for sweep angles less than about 870.From figure 3 (see also eq. (B3))

cos2Ac = (l - sin 2A cos2aL) (D12)

The density ratio across the shock at the cross-flow stagnation point is

SE = + 1(D13)

P T-i

42

With the appropriate substitution, equation (D8) gives finally

2rRel ( " CO rt 'fi + sinRe 2.27X10-8 7 ./r( - sin 2A Cos2,)1/2 1 - sin 2A cos2 a ( 4)

The following values were assumed in the present calculations: -

S=1.2

= 1716 ft-lb/(slug)(0 R)

C p 13.2 Btu/(slug)(OR)

t = 5000 R

0 0.37X10"6 (lb force)(sec)/sq ft

J = 778 ft-lb/Btu

The term V is the independent variable, and . is determined by thelift coefficient (column 7 of table I) and the vehicle aerodynamics.

This calculation takes no detailed account of the boundary-layerhistory on the leading edge of the vehicle.

Typical curves of the ratio of local to free-stream Reynolds numberare shown in figure 18 as a function of angle of attack for severalvalues of V. Both 7 and angle of attack have a strong effect on theReynolds number ratio.

43

APPENDIX E

INTEGRATION OF HEAT INPUT OVER VEHICLE

The vehicle gemetry over which the integration of the local heatper square foot q is carried out is shown in figure 3 together withthe symbols and coordinate systems used. The constant Re case is usedas an example.

Vehicle Gecetry

The following approximate relations can be derived frm the vehiclegeametry:

1 c2 W m 1

A = bc = t- = m2 an AW7 VA 7

/W 1/2 (M 1/2Af bd - -tn p:t A

W1 =pfd

Af 2d 2 (W/A) / 2

A c (W tan A)I 2 (El)

RN = d

b - 21 cos A

c - tan < 12c = orA

R-W 2pf or m = 2p'

44

Integration Over Leading-Edge Region

The integration over the vehicle is divided into two parts, thatover leading-edge regions, and that over the windward side of the ve-hicle. To illustrate the integration over the leading-edge regions, qz

may be taken from column 15 of table I (hence the subscript 15 on k) orthe integration of equation (37) and written as

q= k -5[1 sin 2A cos2cL]v/2 cos (E2)

where for constant Re

k15 = 12.1xlO3 W 1

~CR VCVThe general form of the integration desired is (see fig. 3)

-c)2 q1RN d? dl (E3)

Carrying out the interior integration and using equations (El) and (E2)yield

q1 = klsd/ [ - sin 2A cos2m] v/2 (1 + cos c)dz (E4)

0

where from figure 3

c= arc tan cos A (E5)

Equation (E4) may be written

Qj = 2k1 5d k, cosVA dl (E6)

0

45

where

{ - sin2A cos2 /l + cos arc tan/tan A,

IL= _ 1cosA, (E7)2 cos A

For a = 0, k1 = 1.0; for a. = 120, v - 1.5, and A = 600, k2 = 1.05;

thus, for the integrations over the flight paths for which the angle of

attack is small, k1 is assumed to be unity. For the constant m case,

which is used at higher a,, k, is a constant and may be evaluated. With

kj a constant and by use of equation (El), equation (E6) can be written

b/2q, = kl5kjd cos(v'I)A db (E8)

so that finally integration gives

= kskbd cs('-l)A klskZAf cos(v-l)A (Eq)

Writing this result in terms of the gross weight and wing loading, by

using equation (El), yields

Qj kl 5 k, cos(v-l)A A /2

W Pf tan (ElO)

or

Q, k15kI cos(v-l)A A )1/2m pj. I~tanA,

and substituting for k1 5 (eq. (E2)) and for W/RNA frum equation (El)

results in

-( W/A l/2 L -2- i (

12. l0l3Wk, cOs k ' fW tan A C D Rl12~~~~ ~~ .IICk o~'~

46

or

21.9X10-9mgr /2k z COs(v-i), kOf t C

Camparing equations (E2) and (Ell) yields M8

Q1 ____________I ____ -/Tij cos(v1')Aw-= --- (i- sin 2A cos2m)v/2 cos , (PfWtan A1/2 (El2)

Equation (El2) results from an integration over the vehicle and henceis true for all flight paths. It is listed in column 17 of table I anda similar relation is given in table II.

Integration Over Windward Side of Airplane

For this case q. may be taken fram column 16 of table I (hence

the subscript 16 on k) or the integration of equation (38) and may bewritten as

k1= (E13)

where, for example, for the constant Re case for V > 1,

~VR1~Vk,6=~ ~ ~ 8.X0]kcC tG 2

The integration desired is ther. (see fig. 3)

a/2 c

Qw = 2k161 1t d/ p - E14n

Of f+ tan A / a

47

The integration results in

8w . (f + c ) b/ 2 f 3 / 2 ] (El5)% 3 k16 tan^ "A ta

Equation (E15) may be written

=8 c5/2

H- kl6kw 3 (El6)t

where, using equation (El),

kw 3)/2 - (f)3/2 - 3(f 1/2 (E17)

In terms of gross weight and wing loading and by use of relations fromequation (El),

V , 8/ 1 (E18)_ _k M /4 (W tan A)1/4

Substituting for k16 (eq. (El3)) gives

= (V - 1 + e)3/2JlVI

3

8.Z'(103W 3 CD ~ ~ tn / E9

Comparing equations (E13) and (E19) gives

S qw 8 (E2)( wW tan A)

Equation (E20) holds for all flight paths and is given in column 18 of

table I. A similar result is given in table II.

48

From section AA of figure 3 the ratio f/c may be written approxi-mately for small angles of attack as

E 4 c cos A (E21)

and by use of equation (E5)

1 a tan (E22)cf4c + ns a tan c A J

For the configuration used as an example in this report to generatenumerical values, RN - 2.06, c = 14.5, and A - 600. For m - 00 and5.30, f/c is 0.34 and 0.31, respectively. The corresponding values of

kw (eq. (E17)) are 0.48 and 0.49, respectively. Note that kw does notvary significantly for small changes in m, and hence it is assumed tobe constant. A value of 0.49 was used for the calculations presented.

The most desirable form of some of the terms in equations (Ell) and(E19) depends on what information is known. The following useful alter-nate forms may be derived if the parameter W/A is eliminated by use ofone of the equations (El):

PW tan 1/2 ( tan A)1/3 wl/3pl/6 (E23)

w tan A)=/4 tanI/3A wl/3 P/6 (E24)(c/d)l16 f

The term c/d depends on the vehicle sweep angle and maximum lift-dragratio. For the present examples with A = 600, c/d = 3.51 and 21.7 formaximum lift-drag ratios of 1 and 2, respectively. Still another inter-esting form of the results of columns 17 and 18 may be written using theresults of equations (E23), (E24), and (B13). For vehicle operation atlow angles of attack where CR CD - CD,o, which is approximately the

same assumption made to arrive at the results of columns 17 and 18,

(pf W tan A 1/21

\ W/A = c /8w /3p /(2cD tan A)/ (E25)

4P

49

W ta A) 1/4= c 1/61/6,1/6 tanl/3A (E26)

where cD is constant for a given sweepback angle.

An example of the significance of this result is the following. Fora family of vehicles having the same sweepback angle, gross weight, and

Hfuselage density, the total heat input to the vehicle leading edge variesas

rz~ ___(E27)

(by the results of column 17 and eq. (E25) for ki = Constant). On the

other hand (by column 15) the heat input per square foot to the leadingedge varies as

1 (E28)

Thus the change in the total heat input to the vehicle is much lesssensitive to changes in CD (or (L/D)ma x ) than is the heat input per

square foot.

50

APPENDIX F

CASE OF CONSTANT ANGLE OF ATTACK

The determination of the term V df is more difficult for the caseg d

of constant angle of attack than for the other flight paths. However,the solution of this case permits a check on the accuracy of the resultsof the present analysis by comparison with the numerical integrations ofreferences 4 and 13. The equations to be manipulated are repeated here

1dV G C(

G -= 1 -V2 +v (4a)CR g dr

G CPV2 Agr ()2W(5

(19)=20W dV

Using the mathematical identity

Y -q 2 !LV(Fl)g d g dV dT

and equation (3) permits equation (4a) to be written

Soln1 g V2 (bfG

Solving equation (S) for p, by use of (F2) for G, yields

p = C (F3)- -

51

where

C_ 2W CL (F4)

Differentiating equation (19) gives

a 2]00 C ~+ V j (F5)

dV qdV dV2

where

CgAC = ~A(F6)

The term di/dV can be eliminated between equations (F3) and (F5) toyield a single differential equation for which a solution must be found.Herein, however, a solution to equations (F3) and (F5) was found bysuccessive approximations. For a first approximation the flight-path

angle is assumed to be a constant ( = 0). Then from equation (F3)

dv 2C P (F7)dV V3

and

d 2 ..ac (F8)dV2 V4

and equation (F5) becomes

-4 Cpp 4 (F'9)dV V3 OrV3 L

The value for di/dV given by equation (F9) may now be introduced intoequation (F3) to obtain a second approximation. Thus,

p C )( + (FlO)

52

where

k4 (FII)k or(L/D)'

Repeating the steps of equations (F7) to (F9) results in

,V1~V 1 k2 V 2JV 2 1)J-122 V22

(F13)

di qi V42 k 2k /J (F14)d rV-$DL(1 + )q k

Higher order approximations may be obtained. If this degree of approxi-

mation is accepted, equation (F2) for G becomes

G = 1 _ V2 (FI5)FCL

where

F =k1 +ik + (1i- 72)k/ 2k i ] (F16)

With G known, p can be determined from equation (5), and this is the

result given in table I.

From equations (19) and (F12)

2H (F17)

prV D

where

1+ L k(l-V) (F18)

;72

53

Fromn equation (Fl), by use of equations (3) and (F14),

va 1-1~~(- 2 (F19)g dr F

which is the result given in column 6 of table I.

54

APPENDIX G

CCMPARISON WITH OTHER ANALYSES

The results of reference 4 are essentially for a three-dimensionalstagnation point, or j = 1 in equation (7a). The result of column 10of table I for constant angle of attack for a three-dimensional stagna-tion point can be written in the modified form

q -2 14.9.- %RNALT 1IF V (01)

From equation (36) of reference 4 for Earth, and in the nomenclature ofthis report,

-q (G2)590 T

VRNASubstituting equation (Gl) in (G2) gives

.0.202 , V (G3)

The equation from column 15 of table I for a three-dimensional nose is,in modified form,

q - -2 12.05X103 .,rg ~ arc sin v v I - V21 (G4)2(1

From equation (39a) of reference 4, and in the nomenclature of thisreport,

q ,. (05)15 ,9 0J

55

Evaluating (G4) for V1 - 0.99 and V2 - 0, the values used in reference

4, and substituting in equation (G5) yield

-3 9 1 FjU (0G)

As a point of interest, the total heat input per square foot q or qis quite sensitive to the initial velocity for this case. For example,

Hevaluating equation (G4) from V1 = 1.0 to V2 = 0 gives for equation

(G6)

q = 4.76,F L (G7)

which is an increase of 22 percent over the case for V, = 0.99. In the

present calculation the term F is evaluated at the highest value ofV. If the flight path is divided into several parts, F is evaluated atthe highest value of V in each part.

Reference 4 also presents the horizontal component of the G load,GH . For small flight-path angles, an assumption already made in the

present analysis,

m CD (08)

CR

With the result given for G in column 4 of table I for the constant acase,

GH =FL V (G9)

D

Part of the difference in heating rate between the present analysisand reference 4 is due to the use of a slightly different value of theconstant in equation (7a).

REFERENCES

1. Himmel, S. C., Dugan, J. F., Jr., Luidens, R. W., and Weber, R. J.:A Study of Manned Nuclear-Rocket Missions to Mars. Paper 61-49,Inst. Aerospace Sci., Inc., 1961.

2. Luidens, Roger W.: Approximate Analysis of Atmospheric Entry Cor-ridors and Angles. NASA TN D-590, 1961.

a,

56

6. Chapman, Dean R.: An Analysis of the Corridor and Guidance Require-ments for Supercircular Entry into Planetary Atmospheres. NASATR R-55, 1960.

4. Chapman, Dean R.: An Approximate Analytical Method for Studying Entryinto Planetary Atmospheres. NASA TR R-11, 1959. (Supersedes NACATN 4276.)

5. Eggers, Alfred J., Jr., Allen, H. Julian, and Neice, Stanford E.: AComparative Analysis of the Performance of Long-Range HypervelocityVehicles. NACA TN 4046, 1957. (Supersedes NACA RM A54L1O.)

6. Loh, W. H. T.: Dynamics and Thermodynamics of Re-Entry. Paper pre-sented at 4th U. S. Symposium on Ballistic Missiles &nd Space Tech.,Univ. Calif., Aug. 24-27, 1959.

7. Marshall, Francis J.: Optimum Re-Entry into the Earth's Atmosphereby Use of a Variable Control Force. TR 59-515, WADC, Oct. 1959.

8. Hankey, Wilber L., Jr., Neumann, Richard D., and Flinn, Evard H.:Design Procedures for Computing Aerodynamic Heating at HypersonicSpeeds. TR 59-610, WADC, June 1960.

9. Wisniewski, Richard J.: Methods of Predicting Laminar Heat Rates onHypersonic Vehicles. NASA TN D-201, 1959.

10. Lees, Lester: Space Technology. Lecture 6A, Eng. Extension, Univ.Calif., 1958.

11. Goodwin, Glen, Creager, Marcus 0., and Winkler, Ernest L.: Inves-tigation of Local Heat-Transfer and Pressure Drag Characteristicsof a Yawed Circular Cylinder at Supersonic Speeds. NACA RM A55H31,1956.

12. Feller, William V.: Investigation of Equilibrium Temperatures andAverage Laminar Heat-Transfer Coefficients for the Front Half ofSwept Circular Cylinders at a Mach Number of 6.9. NACA RM L55FO8a,1955.

13. Becker, John V., and Korycinski, Peter F.: The Surface Coolant Re-quirements of Hypersonic Gliders. NASA MEMO 1-29-59L, 1959.

14. Georgiev, Steven, Hidalgo, Henry, and Adams, Mac C.: On AblatingHeat Shields for Satellite Recovery. Res. Rep. 65, Avco-EverettRes. Lab., Avco Corp., July 1959.

15. Steg, Leo: Materials for Re-Entry Heat Protection of Satellites.Paper 836-59, Am. Rocket Soc., Inc., 1959.

57

16. Esgar, Jack B., Hickel, Robert 0., and Stepka, Francis S.: Prelim-inary Survey of Possible Cooling Methods for Hypersonic Aircraft.NACA RM E57L19, 1958.

17. Ames Research Staff: Equations, Tables, and Charts for CompressibleFlow. NACA Rep. 1135, 1953. (Supersedes NACA TN 1428.)

18 Moeckel, W. E.: Some Effects of Bluntness on Boundary-Layer Transitionand Heat Transfer at Supersonic Speeds. NACA Rep. 1312, 1957.(Supersedes NACA TN 3653.)

rz

TABLE 1. - FLIGHT-

[a, CR1 C/CRI CL~a

Point r ... tio..441 P140.

fgh h 1It :.144 11 Flgtp-t44 .,1

1lod

04o141.4 MI4. or 0tak 0

00.10441)14

'0-I-

ICLI, (Col. Ica) iTrz ' I 1 4 L4n {} iv5

(a)4.0 va

0047.141

C}q - 2 jp tLr)

C,.sai pr44as

ft047.4 11 0

Col*)i

404. la.'w "l" 0 -AW. _____

___________ __________A44lj l.tja0o- or 4.?*flnlttms_______ ___

.. 'r, .- I..lI0* .Ocrd 704 0000 0.,apl 0.. 1q~ o . 1h. I00 10 .0 .I elfuO

59

PATH CHARACTERISTICS

are always positive.]

10 _ ~ 3 h13 14

My p1.at bf00.1My0..

30n.- I I.tI 0 .- mw~ II.0 rt. 1. I.fd 046101.0. 1 I.. Or f11500. 0.160,.m. 4.I 4. .r .." 30p, -. uta.0

IT I*

ftG ~~44 -- ; Vt c:c:I&~J~ CA-C/-I.M v

4 W .00.1. (2-

RP, 120 20 .3(2n

*At,00 0 R'

__ ___ LW~ i±~ ~__ ____

60

TABLE I. - Concluded.

(0, CR, CD/CR, CL,a

Punction. Integrated o-er the flight path

Earth ,orly

Filght path Hest input per square root to lsmdlng-dge0

Heat inp.t per square foot to elnjward .!do ror

region for , - 00 to 30, 0 to 30ot, q5 /A 2X0O

I q 1/12. Iol ONi - Sin2 A co2ql

v/2 Co. 0, htu/sq ft

Bt/Sq rt

Mcdulatd sn i of nte. w ; a * 0

Constant alitude 7 2 1

Constant I2 [I0 R 1 74 v

(,V V-i + . VF) / ] V,

V2

V22

Constant 0

0 . . . ' .. . R -0 ) ' D][ "( / ) - " n 2 )l - - . 6 1 -

eo C le t4flLl

Constantfretrn

Reynolds no.r . I/

"o C 7 7

General name

d1 rd rol I \onrnn' n

I = tV~I i.~ WI~L/')2 t '1

-/y-v^

~Ii V0' 1_ /r~ f

Aoiifny rei Ir I I Ii-n

O W113p2/31 .11

/3

'ro.,-'Iio,", ibId r- Icoio c-,c

'ihcc n r~n .c-'mlI,'i"i~~~c c.23110fI 'W ii 0 nVc'i'Inyiinm lne

61

FLIGHT-PATH CHARACTERISTICS

are always positive.]

17 18 19

Functtons lntegrated over flight path and over vehtcle

Earth only

Heat input to l.&dlng-. .0

reglon for Meet inpt to Itng..rd aid. for Definitions. - 0. to S0. Qz/W. Q - 00 to 30, qw/W,

Btu/lb of vehicle &mo. weight Btj/1b or vehicle gross eI ght(e)(f)

Modulated angl* of attlCk ' 0

_______________-/2k,___ co.'________ ~ 6 X,

I/I

5 A q Kw k,

- 2 . )/2 co. - - - -oA r o))2 oosh -

)'2(W tan 1)- -

0~~- - ci2 o2, 1 f2 0 2 oD 0 S2

+ ,.n .. o''o.( ) y tao o)~

kn r

Modulated roll

q,' -/k c..,' KW £ 0 12/(c3(1- sln

2A -. 1.),/2 .(Pr

0 tan0A 112 W ----- T7 ")2

AXoll ry rey tl,,os ,r jeriniti-a.,2 . ..... /2.. (1 + 1)/2 3"Al -

2 r tA r rT .'A I 5110

1/2 -. ) W XwtnA1,4 thhl/5A W13 1

) ,1, . - ) ... a 1).0

rq;w F5Ri , , r , ,

62

H

tn

*)- o . N

= oI

H U

-.o '

,~. x

.

0.

C.) a ______

63

(U

:0

P4.

He

d)ON. 'T

4.4,

P4p

4

U

0

64

1.0lift -drag

.8

.6

0

4w-)

0

-. 210 20 40 60 80 100

Angle of attack, m., deg

(a) Lift coefficient.

Figure 2. - Aerodynamic characteristics.

I 5

65

2.8 -- Imaximum

_____ _____lift -drag __

- - ___ ____ - rat io,- -

(L/D)mjk

H 2.4 -

2.2

.' 1.6

.4*4

0000

0 20 40 60 80 100Angle of attack, mx, deg

(b) Drag coefficient.

Figure 2. -Continued. Aerodynamic characteristics.

66

2.8 -- - -

Maximumlift-drag

rat io,2.4 - (L/D)ma -

CC/0

2.0 -A-

*43

42 1

C) 1.6

0

0

4- 1.2

.8

.4 -- _ _ _ _ - _ _ -

0 20 40 60 80 100Angle of attack, a., deg

(c) Resultant-force coefficient.

Figure 2. - Continued. Aerodynamic characteristics.

67

1.0

8 .H2

0

.44

0

4-)

0

0

0

Angle of attack, m., deg

(d) Ratio of lift to resultant force.

Figure 2. -Continued. Aerodynamic characteristics.

68

1.01

C.9 Maximum

- [lift-dragrat io,(L/D) /

C)

0

4-)

H .7cn

.60

0

4-,

.40 20 40 60 80 100

Angle of attack, aL, deg

(e) Ratio of drag to resultant force.

Figure 2. - Continued. Aerodynamic characteristics.

69

2.0___

Maximumlift -drag

rz 1.6 ratio,(L/D)max

1.2

.8

.41 ,., ,

0 20 40 60 80 100Angle of attack, m, deg

(f) Lift-drag ratio.

Figure 2. - Concluded. Aerodynamic characteristics.

70

Vcos a cos A

V Cosa M C

A Bv

A AH

Plane of wing Plane of V, Vc

Stagnation

point

V CVcaV V sin a

B-B

CIC VC

Figure 3. - Vehicle peometry. coordinate systems, and flow angles.

71

Reference area for aerodynamiccoefficients and wing loading;also used for volume estimation

'-4IC

Approximation to leading edgeused in aerodynamic calculations

~450-Ib

C-C

V cos n cos A etes

% VN W- L

• Figure 3. - Concluded. Vehicle geometry, coordinate systems, and flow angles.

72

6 - -- Present analysis

Ref. 4, fig. 10

0

4

00

.24

Present analysis-- Ref. 4, fig. 14 -L/D

0.1

.1 -- - _ _ /110_

0

14

0 .2 .4 .3.8 1.0Flight- to circular-velocity ratio,V

4-4

o 0

'0~ PI

S010

Present analysisw~n- Ref. 4, fig. 22(b)

00 .2 .4 .8.8 1.0

Vehicle operating lift-drag ratio, L/D

Figure 4. -Comparison of results of present analysis forstagnation-point heating with results of reference 4. Veloc-Ity ratio, 0. 99 to 0; constant-angle-of-attack fli(,.ht path.

73

I I I I I -

Present analysis -

Ref. 13, fig. 4 /

__ _/_

3/

43

0 .2 .4 .6 .8 1.0

12 0 - Pr esent Ianalysis is

~Ref. 13, f ig. 7

8/

4--

.2 .4 .6 .8 1.0

Flig~ht- to circular-velocity ratio,---- 5 Present analys'is _ I _1 I

44 0Ref. 13, fi1g.7 -__ - _ _

08P M- I/

0 8 16 24 32 40

Vehicle angle of attack, a, deg

Figure 5.-Comparison of present analysis for windward-side heatingwith results of reference 13. Initial velocity ratio, 1.0; wallstatic temperature, 00 RI constant-angle-of-attack flight path.

74

'C4)H _8

-1.6 - --

I "'

fH

.2

4;

-°

4)

-21

°. Olimit

z

0.b 1.0 1.5 2.0 2!.5 3.0Ratio of flight to circular velocity, V

1._ L __ _ j _ I _ J- 1.04 0 -.51 -. 81

Time, r , min

Figure 6. - Constant-angle-of-attack flight path for Earth atmos-phere. Angle of attack, 5.301 initial net acceleration, 10 g's;Initial velocity ratio, 2.0; maximum lift-drag ratio, 1.0; entryvehicle gross weight, 10,000 pounds; average entry vehicle den-sity, 20 sounds per cubic foot; geometric leading-edge sweepangle, 60 .

75

8

0

41 1

I- II

20)Q06

-

H0

IDI

00 - 1

temperature,

500 /

.51..152. 5 .20-

100-

50---

Ratio of flight to circular velocity,

Figure 6. - Concluded. Constant-angle-of-attack flight path for Earthatmosphere. Angle of attack, 5.30j initial net acceleration, 10 g's;

initial velocity ratio, 2.01 maximum lift-drag ratio, 1.0; entry vehiclegross weight, 00,000 pounds; average entry vehicle density, 20 pounds per

cubic foot; geometric leading-edge sweep angle, 600.

r6

.8

4,

H

-H

HH

6. 0

.2x106 0

0

.4

.2

4..

"4

4)

C- ----741

)

c°

L -- --- - - I--

10 0 .. _0.

Al td,16,0 eL nta no iceclet lo 101,3

dnt 0 { ..

ag,-.i

T0 0, ' lmitlC) Ie / (u t u -l l te i l;Lp t i r E r l t op e e

o r l .sa 1.0 1. r5e, 2.0; ra llI. t-r; 2.5io .0

anglee,, i0°.-

77

4) - -_ -

4) bO

00

Id0

0 0210 6

_ ____

0 )

0 C

4xO-6 - _

OE

0 Z > 0 0 0 , T ..Limit on wall/

1 "m 000 temperature, -r. t W, 44600 R

;4 200 'H i0

00

0 .5 1.0 1.5 2.0 2.5 3.0Ratio of flight to circular velocity, V

Figure 7. - Concluded. Constant-altitude flight path for Earthatmosphere. Altitude, 162,000 feet; initial net acceleration,10 gs; Initial velocity ratio, 2.0; maximum lift-drag ratio,1.0; entry vehicle gross weight, 10,000 pounds; average entryvehicle density, 20 pounds per c~ibie foot; geometric leading-edge sweep angle, 600 .

j

78

0

4> I -.4

4-

U

00

20,

-41

)to

10

4)

0 .5 1.0 1.5 2.0 2.5 3.0

Ratio of flight to circular velocity,V

3.74 1.87 .78 0 -. 60 -1.09Time, r, min

Figure A. -Constant free-stream Reynolds number flight path for Earth atmos-pher. Free-stream Reynolds number, 7.5XI61 initial net acceleration,3,0 g's; initial velocity ratio, 2.0; maximum lift-drag ratio, 1.0; entryvehicle 1ross weight, 10,000 pounds_ average entry vehicle density, 0

pounds per cubic foot; geometric leading-edge sweep angle, 600 .

79

H

0 4-_

4 r-4j )

4 i ' 0 61_

_ _ +

0:4)

0

1000 t- , 4 4 60 0 R _ _ _ _

41__ _ _ _

ertin 10 1.0; inta eoct .a, 2.0 maxmu litda rto .0;

entry vehicle gross weight, 10,000 pounds; average entry vehicle density,20 pounds per cubic foot; geometric leading-edre sweep angle, 7.00

80

- -4 -

4

. 2X1O6

434

o ,I-,,.4

/.,

0

4,

'44

0 _. 2

'-4

4.,

200

10

0 .5 1.0 1.5 2.0 2.5 3.0Ratio of flight to circular velocity, V

I I I I I I2.70 2. 02 1.35 .(C7 0 -. 67 -1.35

Time, r, min

Fi:ure t. - Constant-net-acceleration fli.ht, path for Earth atmosphere. Netacoeleration, .10 g's; initlal velocity ratio, 2.0; maximum lift-drag ratio,.1.0; entry vehicle gross weight, 10,000 pounds; averag~e entry vehicle den-atty. 20 pounds per cubic foot; geometric leadinfg-edge sweep angle. 600.

ep81

12 - -

414

20x106 --- __ - - _ -

4100

.- 4 106-

041

0- -

4, 3000- _ _ - - -

S2000__ __

w1000 temperaturep

10 4) 4.1

20 _

* 0 -- 1__ -5_ 2. 2. -.

Ratio of flight to circular velocity, 7

* Figure 9. -Concluded. Constant-net-acceleration flight path forEarth atmosphere. Net acceleration, 10 g's; initial velocityratio, 2.0; maximum lift-drag ratio, 1.0; entry vehicle groissweight, 10,000 pounds; average entry vehicle densBity, 80 poundsaper cubic foot; geometric leading-edge sweep angle, 60

82

O00o-

-4

-8

.3x10

5

-. 4'

4

;4 0

0

-12

.5 1.0 1.A 2.-- -I.0.

10

Fiur-0.- o -tn -hetn-aefihtpt o at atmhrt aica

4)

0 .5 1. 1.5 20 li 5 3.03t

10ue 0 Cnt-hetngrt flgh -at fo Eat tmshre ai

mum leading-edge heating rate, 900 Btu per square root per second; tni-tial net accelerati~on, 10 g's; initial velocity ratio, 2.0; maximum lift-drag ratio, 1.0; entry vehicle gross weight, 30,000 pounds; average

entry vehicle density, 20 pounds per cubic foot; geometric leadlng-edgesweep angle, 600.

83

24

0

0

lO6

- - \0 --

09

40

200

200-----5000

oc

100

50 6

'-4

, 3e 000- - - - - - -

M 200 1 00-

100 .5 1.0 1.5 2.0 2.5 3.0Ratio of flight to circular velocity, V

Figure 10. - Concluded. Constant-heating-rate flight path forEarth atmosphere. Maximum leading-edge heating rate, 900 Btuper square foot per second; initial net acceleration, 10 g's;Initial velocity ratio, 2.0; maximum lift-drag ratio, 1.0;entry vehicle gross weight, 10,000 pounds; average entryvehicle density, 20 pounds per cubic foot; geometric leading-edge sweep angle, 600.

84

d -4

4,

0

2

0 i4-)

0 .5 1.0 1.5 2.0 2.5 3.0Ratio of flight to circular velocity,

2.70 2.02 1.35 67 0 -.67 -1.35

Time, T, min

Figure ii1. - Modulated roll flight path for Earth atmosphere. Initial roll

angle, 0; initial net acceleration, 10 g's; initial velocity ratio. 2.0;maximum lift-dag ratio, 1.0; entry vehicle gross weiglht, 20.000 pounds;average entry vehicle density, 20 pounds per cubic foot; f(cometric leading-edge sweep angle, 600.

85

160

01 1)-4

to r

10

0

'0-4

10

H!

EOE

000 --

2000 --Limit on wall.-

10-temperature,00

41 W

'.4

C----.-- -

F0Nur . - onuded M

0d4)

X-404 10-- -

0 .5 1. -. -2.-- 2. - .

Rati of flgtt-icla eoiy

at'mosphere. Initial roll angle, 0; inIti~l net aeeeleratior,10 P's; -nitial velocity rao, 2.0; maximum lift-drag ratio,

1.0; entry vehicle gross weight, 10,000 pounds; average entryvehicle density, 20 pounds per cubic foot; eometrc leading-ed2e sweep ane, 600.

86

000 800 - Fight path-

600 _____ Constent angle

00,Constant altitude400 '_Constant Reynolds

1100 / /- -Modulated roll

f/-Constant net00 ___. acceleration~ 200 ____H 200

100 , /,0<0, 10' '0 80 0S! _/

____ "_\-Constant angle19 i/*J "o f a t t a c k

40 -Constant altitude

& Modulated roll!//! eConstant Reynolds

20 Constant net04 acceleration

0

4-)

4-)

03 1

0"4

- Total heat input,Q

Heat input towindward side, ,

1 = I I I

.5 1.0 1.5 2.0 _ 2.5Ratio of flight to circular velocity, V

Figure 12. - Total heat input at Earth for flight paths withnet acceleration of 10 C's at velocity ratio of 2.0. Maxi-mum lift-drag ratio, 1.0; entry vehicle gross weight, 10,000pounds; average entry vehicle density, 20 pounds per cubicfoot; geometric leading-edge sweep angle, 600.

V87

800- Flight path - -- 1

Constant angle -

600- of attack -/-

Constant altitude --.

Constant Reynolds /__ ___

400 number- Modulated roll-,,/_

Constant net /,-Constant angleacceleration--- of attack

W 200 F

8O /ll It/ // I100 ___ z_ _

60- Constant altitudeo / /l - -Modulated roll

:a l7 1 Constant ReynoldsI40, number

* .... Constant net

-acceleration

20

0

0

4.

0

0

Total heat input,Heat input to

windward side, Qv.8o I II

.5 1.0 1.5 2.0 2.5Initial ratio of flight to circular velocity, V

Figure 13. - Total heat input at Earth for flight path with initialnet acceleration of 10 gs. Maximum lift-drag ratio, 1.0; entry

vehicle gross weight, 10,000 pounds; average entry vehicle den-sity, 20 P ounds per cubic foot; geometric leading-edge sweepangle, 60

88

0

80 7 8 ............... 0...

FiueA e- Totalo heat input fo osatageo-tac wigtw a th o

1.0; entry vehiclee t gross wit 1.0 ons;aerigeh nryvhil

de0 t.~0pud e ui ot cmtl edn-desepage

89

Constant anglelfo attack a 7

"'--.-.--Constant 0, Ga

0-

(a) Modified constant-angle-of-attack flight path.

12

kDConstant 0,

Constant ca

'-4,o 4\ - -

0

(b) Modified constant-net-acceleration-flight path.

12 -

8Contant 0,

Constant ,

4 -

0.4 .8 1.2 1. 6 2.0

Ratio of flight to circular velocity,

(c) Composite flight path.

Fifure 1.5. - Flirht pnths with control of landing location.

90

Constant angle- -of-attack flight

path

850 - __ L _-0" 01

750Gin

1.0

650

0

5

'4

450450Flight path for

0 constant net0o4 acceleration

350 - __1 ~_ 1_ 1_ 1-_0 .08 .16 .24 .32 .40

Range/circumference, R/2xr

Figure 16. - Total heat input at Earth for modified constant-angle-of-attack flight path (fig. 15(a)). Initial net acceleration,10 g's; initial velocity ratio, 2.0; maximum lift-drag ratio, 1.0;entry vehicle gross weight, 10,000 pounds; average entry vehicledensity, 20; geometric leading-edge sweep angle, 600.

91

Constant-angle-of-attack flight path

850

750

= 650

0

bo

4.)

-A 5---------------

4.)

A

450 Con- -et-

0o accelerat ion0 flight pah

3501.2 1.4 ..6 1.8 2.0

Ratio of flight to c! - cu~lar velocity at beginningof constant-angle-of-attack flight, VM

Figure 17. - Total heat input at Earth for modifiedconstant-net-accelerat-on flight path (fig. 15(b))and infinite range. Initial net acceleration, 10 g'sjinitial velocity ratio, 2.0; maximum lift-drag ratio,

1.0; entry vehicle gross weight, 10,000 pounds; aver-age entry vehicle density, 20; geometric leading-edgesweep angle, 60.

92

Ratio oiflight tocircularvelocity,

Iv

.4

00

A .2 2.0 000-

I

0 .1. ! .040,00

00

.06Fe sra

I .o,.I!I I I I152

.Anleof.t - Ag le of attac

. u6 18.ree stream l t ,es a Ryosu rShock wave / -

.04Local conditions - /

Edge of boundary layer ed

Wing section A-A of fig. 3

• 002r I I I I I I I I I0 5 10 15 20 25

Angle of attack) m, deg

Figure 18. - Ratio of local to free-stream Reynolds number forswept blunt leading edge for Earth atmosphere. Leading-edge

sweep angle, 60; specific-heat ratio, 1.2.

NASA-LOu., INI 3-1001

vii

& I i Ii !j '!ll,!J! , ! ll

II 1, 1 !.

______________i ,I! I°. thl~