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J. Leith Potter 

(NASA-CB-181395) AERODYNAMICS OF N87-29429 

(Vanderbilt Ouiv. ) 14 p Avail: NTIS BC 

A02/MF A 01 CS CL 01A Unclas 

G3/02 0102688 

NASA Research Grant NAG-1 -549 

11 October 1987 


School of Engineering 

Department of Mechanical & Materials Engineering 

Nashville TN 37235 



J. Leith Potter 

NASA Research Grant NAG-1 -549 

11 October 1987 

Aerodynamics of 3-Dimensional Bodies 
in Transitional Flow 

J. Leith Potter* 

Vanderbilt University, Nashville, Tennessee 


Based on considerations of fluid dynamic simulation appropriate to hypersonic, vis- 
cous flow over blunt-nosed lifting bodies, a method was presented earlier by the author for 
estimating drag coefficients in the transitional-flow regime. The extension of the same 
method to prediction of lift coefficients is presented in this paper. Correlation of available 
experimental data by a simulation parameter appropriate for this purpose is the basis of the 
procedure described. The ease of application of the method makes it useful for preliminary 
studies which involve a wide variety of 3-dimensional vehicle configurations or a range of 
angles of attack of a given vehicle. 


Cd = drag coefficient 

Ce = (|ie/|-too)(T<x/re) 

Cl = lift coefficient 

Cp = pressure coefficient (p- poo)/qoo 

D = drag 

H = enthalpy 

L = lift 

M = Mach number 

PFA = projected frontal area 

PPA = projected planform area 

Pn = similarity parameter (Eqs. 2 and 5) 

p = static pressure 

q = dynamic pressure 

Re 0 = Reoc(|W|i 0 ) 

s = streamwise wetted length 

s* = characteristic length (Eq. 3) 

T = temperature 

U =velocity 

Ve = Me(Ce/Re e ) 1/2 

WA = wetted area 

a = angle of attack 

♦Consultant and Research Professor, Department of Mechanical and Materials Engineer- 

Y = ratio of specific heats 

T| = angle between surface unit normal and free- stream velocity vectors 
p = absolute viscosity 

V = kinematic viscosity (|i/p) 

p = density 

0) = exponent in viscosity-temperature relation 


e = edge of boundary layer 

fm = free molecular flow 

i = inviscid flow 

o = stagnation region 

w = wall 

<» = free stream 


Many experimental and theoretical investigations of the drag of simple shapes in hy- 
personic, rarefied flow have been reported. Very few reports on lift and static stability have 
appeared. During this time it has been recognized that even the best laboratory facilities 
available for this work did not meet all of the desired simulation requirements, and com- 
putational techniques had not been developed to the level necessary for dealing with com- 
plex shapes having unknown gas/surface interaction, slip-flow, and nonequilibrium gas 
processes. It is remarkable that adequate re-entry vehicles have been designed on the basis 
of this earlier research. That continues the history of aeronautics which also is a record of 
successful but not generally optimal aircraft produced despite obvious deficiencies in the 
theoretical and laboratory base for the work. 

The low air densities that characterize the transitional flow flight regime lead to low 
aerodynamic forces and heating. For many vehicles, that makes precision in aerodynamic 
predictions of lesser importance. Large errors in coefficients lead to only small errors in ab- 
solute magnitudes of forces or heating rates. This had made many aerospace designers in- 
different to rarefied flow phenomena despite the rather spectacular changes in aerodynamic 
coefficients that often occur in the transitional regime. However, aerospace vehicles of in- 
creased aerodynamic and structural sophistication, capable of maneuvering through use of 
aerodynamic controls, and dependent upon much more accurate predictions of fluid 
dynamics for safe and efficient operation are now of greater concern. Lift, lift-to-drag ratio, 
and stability have assumed increased importance. 

The prediction of aerodynamic, or more generally gasdynamic forces under hyper- 
velocity, rarefied flow conditions is based upon approximate theoretical methods or correla- 
tions of the relatively small collection of relevant experimental data. At present, and probably 
for several more years, the two avenues to follow are (1) the use of experimental data in 
conjunction with carefully chosen simulation and scaling parameters and (2) the direct 


simulation Monte Carlo (DSMC) computational technique. The latter is a powerful tool, but 
it requires certain arbitrary inputs; the necessity to assume a gas/surface interaction model 
perhaps is the main point in this case. On the other hand, the shortcomings of wind tunnels 
and other laboratory devices in regard to duplication of in-flight real gas and gas/surface ef- 
fects are well known and do not need repeating here. Complex configurations, with inter- 
fering flows associated with different components of the vehicle usually are no problem in 
wind tunnels. However, inability to fully duplicate the full-scale vehicle’s flight environ- 
ment, possibly including the gas/surface interaction, raises various questions. Thus, the data 
must be scaled or extrapolated with care, to say the least. 

This paper addresses the problem of predicting lift and lift-to- drag ratio of aerospace 
craft in transitional flow. The approach used earlier by this author 1 in scaling drag coeffi- 
cients is extended for the present purpose. That may be described as the correlation of nor- 
malized aerodynamic coefficients with a simulation parameter which is designed to account 
for the principal flow phenomena that cause the coefficients to vary. The derivation of this 
simulation parameter is fully described in Ref. 1, and only its extension to include lift coef- 
ficients is covered here. 

Lift in Hypersonic Transitional Flow 

In marked contrast to the amount of data now available on drag of bodies in transition- 
al flow, very few measurements of lift have been published. Several reports from the Ar- 
nold Engineering Development Center in the 1960’s (e.g., Refs. 2-3) contain the only wind 
tunnel data on lift, drag, and pitching moment in the midst of the transitional regime that the 
author has knowledge of. Space Shuttle re-entry data 4, 5 are now available and are, of 
course, extremely valuable because no wind tunnels can fully match the flow conditions of 
hypersonic re-entry. No theoretical/numerical computations of these coefficients for lifting 
bodies throughout the transitional regime are known to the writer, although it would seem 
likely that further development of the DSMC method will soon lead to some results on this 
problem. Therefore, it has been necessary to work with only the experimental data of Refs. 

The Parameters _ 

The simulation parameter described in Ref. 1 was devised as a hybrid of Ve and Reo 
with the purpose of obtaining a parameter suitable for scaling viscous, hypersonic flow ef- 
fects on typical hypersonic re-entry configurations, either lifting or non- lifting. These were 
visualized as blunted, cold-wall bodies at varying angles of attack. The characteristic length 
in the parameter was taken to be a modified wetted length in the streamwise direction. The 
modification was made to obtain a length more representative of the overall body. For ex- 
ample, a simple wetted length or body diameter tells nothing about fineness ratio or angle 
of attack of the body. Yet it is well known that both projected frontal area and wetted area 


are significant in regard to forces on bodies in transitional flow. Therefore, a "shape factor" 
was applied to the wetted length to yield the characteristic length. 

To briefly summarize, in Ref. 1 , it is shown that the normalized form of drag coeffi- 


Cd = (Cd - CDi)/(CDfm - CDi) (1) 

of a variety of practical vehicle shapes is correlated with a simulation parameter defined as 

Pn = V (U/v)°° s* (Hw/H*)® (2) 

s* = s V PFA/WA (3) 

H«/H* = [IW(0.2Ho + 0.5Hw] (4) 

co * 0.63 for high-stagnation-temperature conditions in air or N 2 

Note that PFA was denoted as CS A in Ref. 1 , The chan ge to PFA makes the terminol- 
ogy more precise. Also note that the shape factor, V PFA/WA, is empirically based and there- 
fore provisional. 

The alteration proposed here for the accommodation of both drag and lift by the same 
basic parameter is to now define the pair, 

PnD = drag parameter in Eq. (2) (5 a) 

PnL = PnD(PPA/PFA) 1/4 (5b) 

Thus, for correlating lift coefficients, the shape factor is based upon projected planform 
area (PPA), while projected frontal area (PFA) is retained for correlating drag coefficients. 
The counterpart of Eq. (1) in the case of lift is 

a=(a-ai)/(CLf m -cu) (6) 

It will be noticed that the form of Eqs. (1) and (6) encourages scatter in the correlation 
of experimental data when Cd or Cl is close to the corresponding value for near-inviscid 
flow, i.e., at large PnD or PnL. That deficiency is not as serious as it may seem at first, be- 
cause the designer often will have both experimental and theoretical results available for the 
higher Reynolds number, continuum, viscous-flow part of the transitional regime for which 
the scatter may be troublesome. 

There are other features of the normalized coefficients that must be pointed out. These 
concern the specific bases of the inviscid- and free-molecular-flow coefficients used in this 
paper. The use of other approaches toobtaining these "baseline" coefficients and then com- 
bining those results with the Cd or Cl from the correlations of Ref. 1 or this paper could 
easily lead to considerable error in the final result. It should be noted that, for all cases in 
this paper either the modified Newtonian theory or an experimental high Reynolds number 


CLi has been used. In the first instance, Cy has been calculated on the basis of a windward 
surface pressure coefficient, 

Cp = K cos 2 T] (7) 

where, following Lees^ 

K = (y + 3)/(y + 1) (8) 

is chosen for blunt bodies with detached shock waves at large Mach numbers. The conven- 
tional assumption, Cp = 0 is made for leeward surfaces. The user of the correlations must 
not substitute an experimental value for the inviscid term unless it is essentially free of vis- 
cous-flow influence, i.e., a very high Reynolds number result. An admonition also is neces- 
sary concerning use of a more sophisticated and perhaps more accurate calculation for 
obtaining CDi or CLi- If any of these improvements are attempted, the correlation curve 
should also be adjusted by recalculation of the normalized coefficient on the basis of the 
same inviscid coefficient. A similar warning applies to any changed basis for obtaining the 
coefficient for free molecular flow, as discussed below. 

The values of CDfm or CLfm are based upon the gas/surface interaction model of fully 
accommodated, completely diffuse reemission. This is the most common model assumed, 
even through it is well known that some degree of specular remission and less- than-com- 
plete accommodation are more realistic assumptions. Blanchard 5 has deduced from NASA 
Space Shuttle flight data that approximately 90% energy accommodation and on the order 
of 10% specular reflection would lead to the best agreement of conventional free-molecular 
theory 7 and flight results for lift- to-drag ratio. It is worth noting that these small changes 
in gas/surface interaction parameters correspond to very large percentage changes in L/D. 
In any event, the basis for Cd and Cl must be kept in mind and consistency in the use of in- 
viscid and free molecular coefficients is essential. 

Correl at i on of Da ta 

Not only is the amount of data on lift in transitional flow small, but accuracy is also a 
concern because of the low forces acting in rarefied flows. Nevertheless, several sets of data 
are available. 2 ' 5 These have been normalized according to the procedure described and the 
results are plotted in Fig. 1. The solid curve in Fig. 1 is the same curve shown in Ref. 1 where 
only drag coefficients were correlated. It is not clear if the points for the experimental Cl 
indicate that a different curve for lift coefficients is necessary. If a separate curve for Cl 
were drawn, it would consist of a fairing through the points for the STS orbiter. 

There is evidence that drag coefficients of slender, sharp-nosed bodies at low angles of 
attack may exceed the free-molecular level in the near-free-molecular or first-collision 
regime. Therefore, the part of the curve arvery low Pn’s is uncertain in regard to both level 
and suitability of a single curve for both Cd and Cl of bodies fitting that description. 

It was stated earlier that blunt-nosed bodies of the type that may be considered for most 
lifting re-entry vehicles are the primary focus of this paper. However the experimental Cl 
for a sharp, 18-deg apex angle cone at 20-deg angle of attack was compared with the 


predicted Cl based on Fig. 1 as a test case. These experimental data had not been included 
in the preparation of Fig. 1. When the predicted Cl is based on the solid-line curve in Fig. 
L_the error in Cl at PnL = 19.2 is +5% and at PnL = 9.5 it is +10%. If a revised curve for 
Cl, passing through the STS points in Fig. 1 is used, these errors in Cl become 0% and 
+7%, respectively. In both cases, the experimental uncertainty is at least as great as the dis- 
crepancies cited. 

Considering the limited data, uncertainty in those data, and the inherent limitations on 
a correlation of the type presented, an anticipated uncertainty in Cd or Cl of approximate- 
ly ±10% seems justified at this time. Thus, it is suggested that Fig. 1 represents a useful tool 
for preliminary design studies wherein numerous variants may be screened, and for other 
problems where wind tunnel experiments or CFD solutions are not feasible. 

Application to AFE Spacecraft 

There is current interest in spacecraft which will utilize aerodynamic lift to transfer 
payloads between different orbits. To obtain flight data on one vehicle design, an Aeroas- 
sist Right Experiment (AFE) is being considered by NASA. A brief description of the con- 
figuration and its trajectory are given in Ref. 8, where DSMC calculations of drag coefficient 
of an axisymmetric approximation to the AFE are also presented. Figure 2, from Ref. 8, 
shows the elevation or side view of the AFE and the trajectory. A more detailed description 
of this configuration and discussion of its advantages are in Ref. 9. 

With the use of the trajectory information in Ref. 8 and supplemental information from 
Robert C. Blanchard of NASA Langley ResearchCenter, the values of PnD and PnL have 
been computed. Corresponding values for Cd and Cl were taken from Fig. 1 , using the curve 
shown. Experimental data for CDi and CLi at high Reynolds number and computed CDfm 
and CLfm for fully accommodated, diffuse reemission were taken from unpublished NASA 
sources. The results for Cd, Cl, and L/D appear in Figs. 3-5. 

On Fig. 3, for qualitative comparison, the results from Ref. 8 are also shown. It must 
be emphasized that the DSMC calculation was carried out for an axisymmetric shape which 
approximates the actual AFE design. That was done to simplify and shorten the computa- 
tion, according to Ref. 8, and it is not known how much that may have affected the results. 
It is noteworthy that the major real gas effects were modeled in the calculation, but the im- 
portance of those effects on overall aerodynamic forces in rarefied flow is not yet fully 
evaluated. Simply based on Eqs. (7-8), it is inferred that Cp on windward surfaces is in- 
creased by real gas processes which effectively lower the ratio of specific heats. However, 
when densities are very low, thermo-chemical-kinetic processes may be nearly frozen. 
Therefore, no conclusions regarding accuracy can be justified on the basis of Fig. 3, but it 
may be appropriate to view Fig. 3 as indicative of the current state of the art of predicting 
transitional, hypersonic drag coefficients by different approaches. At this time the writer is 
not aware of other published Cl or L/D estimates for the AFE vehicle although work toward 
that end undoubtedly is in progress. 



This work was performed as part of a program supported by NASA Research Grant 
NAG- 1-549, with Dr. J. N. Moss as Technical Officer. The assistance of NASA staff, par- 
ticularly Dr. Moss, D. C. Freeman, Jr., and R. C. Blanchard, is gratefully acknowledged. 


^Potter, J. L. "Transitional, Hypervelocity Aerodynamic Simulation and Scaling," Ther- 
mophysical Aspects of Re-entry Flows, J. N.. Moss and C. D. Scott, eds., Vol. 103 of Progress 
in Astronautics and Aeronautics series, AIAA, New York, 1986, pp. 79-96. 


Boylan, D. E. and Potter, J.L., "Aerodynamics of Typical Lifting Bodies Under Conditions 
Simulating Very High Altitudes," AIAA Journal, Vol. 5, Feb. 1967, pp. 226-230. 

3 Boylan, D. E., "Aerodynamics of Blunt Heat Shields at Simulated High Altitudes," AEDC- 
TR-67-126, July 1967. 

4 Blanchard, R. C. and Buck, G. M., "Rarefied-Flow Aerodynamics and Thermosphere 
Structure from Shuttle Flight Measurements," Jour, of Spacecraft and Rockets, Vol. 23, Jan. 
-Feb. 1986, pp. 18-24. 

5 Blanchard, R. C., "Rarefied Flow Lift-to-Drag Measurements of the Shuttle Orbiter," ICAS 
Paper 86-2.10.2, 15th Congress of International Council of Aeronautical Sciences, London, 
Sept. 7- 12, 1986. 

6 Lees, L., "Hypersonic Flow," Proc. of the Fifth International Aeronautical Conf., Los An- 
geles, Inst, of the Aero. Sci., June 1955, pp. 241-276. 


Schaaf, S. A. and Chambre, P. L., "Flow of Rarefied Gases," Fundamentals of Gas 
Dynamics, H. W. Emmons, ed., Princeton University Press, 1958, pp. 687-738. 


Dogra, V. K., Moss, J.N., and Simmonds, A. L., "Direct Simulation of Aerothermal Loads 
for an Aeroassist Flight Experiment Vehicle," AIAA Paper 87-1546, 22nd Thermophysics 
Conf., June 8-10, 1987. 

^Scott, C. D., Ried, R. C., Maraia, R. J., Li, C-P, and Derry, S. M., "An AOTV Aeroheat- 
ing and Thermal Protection Study," Thermal Design of Aeroassisted Orbital Transfer 
Vehicles , H. F. Nelson, ed., Vol. 96 of Progress in Astronautics and Aeronautics series, 
AIAA, New York, 1985, pp. 309-337. 


7 0 g io 

Velocity, km/s 

Figure 2. Aeroassist Flight Experiment (AFE) vehicle shape and trajectory' 
(from Reference 8). 


Figure 4. Lift coefficients predicted for the AFE during re-entry. 

Figure 5. Lift-to-d nig ratios predicted for the AFE during re-entry.