Rarified Flow Heat Transfer Model for Slender Bodies at Large
Angle of Attack
G. T. Chrusciel
Lockheed Martin Space Systems Company-MSO
Design Engineering, Org. L2-10, Bldg 157
Sunnyvale, CA 94088-3504 USA
Abstract. An approximate rarefied flow heating method has been developed for large angle of attack application,
based on extension of a rarefied transitional flow bridging technique obtained from direct simulation Monte Carlo
(DSMC) flowfield results at zero angle of attack. The method has used DSMC results for sphere cone shapes to
obtain a modified bridging function for angle of attack. Comparisons of detailed heat transfer coefficient
distributions using the improved method with large angle of attack DSMC flowfields have indicated satisfactory
agreement, considering the scatter in the reference DSMC heat transfer data.
BACKGROUND
Transitional rarefied flow phenomenology for reentering concepts has been primarily focused on the aerodynamic
force and moment aspects the vehicle experiences between the free molecular and continuum regimes. Limited
experimental or applicable theoretical results are available for first order assessment of the heating aspects these
vehicles may encounter, particularly at arbitrary angles of attack (α). An effort was undertaken to develop improved
methodology for high α heating predictions for slender configurations in the transitional rarefied flow regime.
IMPROVED RAREFIED FLOW HEATING MODEL
Existing Heating Model for Zero Angle of Attack
An existing heating method developed by Jain [1], for zero angle of attack application to blunted cones in the
rarefied transitional flow regime, was initially evaluated for extension to windward side heating at large α
conditions. A brief description of the methodology for α = 0 conditions will be provided to serve as a basis for
enhancements to angle of attack application.
Fitting functions (fhi) were used to bridge the flow regime from continuum values of the heat transfer coefficient
(Chi, con), incident to a surface of angle (θ), to the corresponding free molecular values (Chi, fm).
Ch = Cho + Ch1 sin θ
Chi = Chi, con + (Chi, fm – Chi, con) fhi
(1)
;
i = 0, 1
(2)
Cho, fm = [ 1.0 /(2 π γ)0.5 (M∞)] [ 1.0 – (γ + 1) Tw / γ ] exp(-Z2)
(3)
Ch1, fm = 0.5 (1.0 + erf (Z)) [ 1 + 2 / (γ (γ + 1) M∞2) – (γ + 1) Tw / 2 γ ]
(4)
Cho, con = Ch1, con = 0
(5)
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
fho = a2 (Reo)3/4 + exp (-b2 (Reo)3/4)
(6)
fh1 = 1 / [a3 (Reo)1/2 + exp (-b3 (Reo)1/2 ]
(7)
Z = molecular speed ratio normal to the surface = (γ / 2)0.5 M∞ sin θ
(8)
where:
a2 = function [ γ , Tw, Z, M∞, (Tr – Tw)]
(9)
b2 = function [ Tw ]
a3 = function [ γ , Tw, Z, Pr-, ( P’eo / P’∞), (T’∞ / T’eo) , (T’eo / T’w), (µ’w / µ’(T’eo) ), M∞, (Tw / Teo)]
b3 = function [ M∞ ]
(10)
(11)
(12)
(P’eo / P’∞) = stagnation to free stream pressure ratio = [ (γ + 1) / (2 γ M∞2 – (γ – 1))] (1 / (γ – 1))
x [ (γ + 1) M∞2 / ((γ – 1) M∞2 + 2) ] (γ / (γ – 1)) [ 1 + (γ – 1) M∞2 / 2 ] (γ / (γ – 1))
(14)
(T’eo / T’∞) = stagnation to free stream temperature ratio = [ 1 + (γ – 1) M∞2 / 2] = (T’o∞ / T’∞)
(15)
T’eo = T’o∞ = reference & free stream stagnation temperature = T’∞ (T’o∞ / T’∞)
(16)
Tr = recovery temperature to stagnation temperature ratio = T’r / T’o∞ = (Pr)0.5 ; for laminar flow
(17)
Ch = heat transfer coefficient = qW / 2 ρ'∞ V’3∞
(18)
Reo = ρ'∞ V’∞ s’ / µ’o∞
(19)
s’ = surface length from stagnation point = (S/Rn ) R’n
(20)
M∞ = freestream Mach number
Pr = Prandtl number = 0.76 ; for high temperature air
P∞ = freestream static pressure
qW = surface heat transfer rate
R’n = spherical nose radius
S/Rn = surface length normalized by nose radius
Tw = wall to stagnation temperature ratio = T’w / T’o∞
T∞ = freestream temperature
V’∞ = freestream velocity
γ = ratio of specific heats
ρ'∞ = freestream density
µ’w = viscosity based on wall temperature (T’w)
µ’(T’eo) = µ’o∞ = viscosity based on freestream stagnation temperature (T’o∞)
µ’ = viscosity, using Sutherland’s law; based on temperature T’
( )’ denotes a dimensional variable
Note that the parameters a2, b2, b3 of Jain’s analysis are similar in form to the parameters a, b1, b, respectively of
equation 6 given in the Kotov et al {3] bridging analysis for pressure and shear stress. Jain’s development for the a3
parameter was a significant advancement in predicting rarefied transitional flow heat transfer coefficient.
Model Extension to Large Angle of Attack
Initial approaches to extend the method to large α considered using a simple tangent cone approximation for the
effective surface angle (θeff), retaining the α = 0 bridging relations. The effective cone angle for the windward side
(θeff > 0), replacing the cone angle θ in the above relations, was determined from geometric relations for a body of
revolution
(θeff) = sin-1 [ sin θ cos α + cos θ sin α cos φ ]; θeff > 0
(21)
where
φ = meridional angle; φ = 0 on windward ray
θ = (π/2) – S/Rn , on spherical nose
(22)
θ = θc = cone half angle, on conical surface
(23)
Ch predictions for the windward side using this approach demonstrated poor agreement with sphere cone DSMC
data at α = 750 for values of λ∞ / Rn = 0.23 and 1.28. λ∞ represents the free stream mean free path, and λ∞/Rn is
defined as the Knudsen number based on nose radius. In addition, the tangent cone approach could not be utilized on
the leeward side for combinations of angle of attack and cone angles which produce negative values of the effective
cone angle.
Analysis of the parameters affecting the heat transfer predictions for the windward side was performed to develop
improved modeling. Evaluating the relative contributions of the terms affecting the fitting functions, it was
determined that the Cho contribution was negligible for hypersonic speeds. Focus was directed at modifying the
most significant portion of the fh1 function, the [a3 (Reo)0.5)] contribution. Parametric variations of the Reo exponent
were evaluated, including terms containing first and second order angle of attack functions. The following was
selected as providing the best overall fit of the windward heating distributions for the two 7 deg sphere cone cases α
= 750 and a 10 deg sphere cone flowfield at α = 200, using the effective cone angle θ eff replacing θ in the heating
expressions.
fh1 = 1 / {a3 (Reo) N + exp (-b3 (Reo)1/2 }
(24)
where
N = [ 0.45 – 0.1072 sin2 α]
(25)
Leeside heating predictions (θ eff < 0) are based on semi- empirical methods using the DSMC data base for the
three rarefied flowfields (α / θ ) > 1.5, and 3DVSL results for lower values of the ratio α / θ.
for (α / θ ) < 1.5
π
Ch 
π/2
= {[ (Ch) φ =0 – (Ch) φ =90 ] [cos φ ] / [1+ (α / θ )1.5 ] } + (Ch) φ =90
(26)
= [1 + cos φ ] (Ch) φ =90
(27)
for (α / θ ) > 1.5
π
Ch 
π/2
where
θ = surface angle, previously defined
(Ch) φ =0 , (Ch) φ =90 are the Ch predictions for the windward side at φ = 0 and 90 deg, respectively.
Evaluation of Modified Model for Zero Angle of Attack
The exponent N (equation 25) for α = 00 was adjusted slightly from the value of 0.5 obtained from the original
Jain model, to provide overall improved agreement with available DSMC data. Jain used DSMC data for spheres,
wedges, and 50 sphere-cones (S/Rn < 8) at various altitudes in determining the constants employed in the fitting
functions. Reasonable comparisons with the DSMC data of Moss were demonstrated for the bridging technique in
[1]. Comparisons of DSMC and 3DVSL continuum heat transfer data for 70 spherically blunted cones with the Jain
original and modified bridging parameter are provided in Fig. 1; small differences in results for the two models are
indicated. Note that DSMC data at the various S/Rn locations were interpolated from results for a large number of
axial stations, typically providing increments in ∆S/Rn ≈ 1. Reasonable agreement is demonstrated for the bridging
similarity parameter (M∞ / Reo)0.5 > 0.2. For values of (M∞ / (Reo)0.5) < 0.2, significant overprediction of the
3DVSL data occurs. These comparisons serve to indicate that a criteria for the lower limit of applicability must be
imposed on the bridging technique; this will be established as Jain’s similarity parameter greater than 0.8.
Comparison of predictions using the original and modified exponents with DSMC heating results for spheres and
sphere cones are provided. Fig. 2 shows results on the conical afterbody for 70 sphere-cones; Fig. 3 evaluates
stagnation heating results from Ref. 2; and Fig. 4 illustrates heating distributions for 50 sphere-cones, taken from
Ref. 2. The present application stresses the afterbody heating methodology; consequently, somewhat improved
comparisons far aft of the nose region were found with predictions using the modified exponent at α = 0.
Predicted, Kn=3.031;Jain
Predicted, Kn=3.031;Mod. Jain
1.0E+00
1.0E-01
3DVSL, DSMC
data; S/Rn=2.7
1.0E-02
Predicted, Mod.
Jain; S/Rn=2.7
1.0E-03
3DVSL, DSMC
data; S/Rn=10.4
Predicted, Mod.
Jain; S/Rn=10.4
1.0E-04
0.01
0.50
M ∞ / (Reo)
0.1
1
10
100
, Viscous Similarity Parameter
FIGURE 1. Heat transfer correlation with rarefied
flow parameter; θc =7 deg; α= 0.
DSMC, Kn=3.031
Predicted, Kn=0.238;Jain
Ch, Heat Transfer Coefficient
Ch, Heat Transfer Coefficient
1.0E+00
Predicted, Kn=0.238;Mod. Jain
DSMC, Kn=0.238
1.0E-01
1.0E-02
0
2
4
6
8
10
S / Rn, Normalized Surface Distance
FIGURE 2. Sphere cone heat transfer distribution
comparison; θc = 7 deg; α = 0.
DSM C,70km
; M oss
1
0.8
0.6
DSMC, Moss
0.4
Predicted, Jain
0.2
Predicted, Jain,
mod.
0
0.01
Ch, Heat Transfer Coefficient
1.2
Ch, Heat Transfer Coefficient
Predicted,
70km; Jain
1
Predicted,
70km; M od.
Jain
DSM C,80km
; M oss
0.1
Predicted,
80km; Jain
Predicted,
80km; M od.
Jain
DSM C,100k
m; M oss
0.01
Predicted,
100km; Jain
Predicted,
100km; M od.
Jain
0.001
0.1
1
10
100
λ ∞ / Rn, Knudsen Number
FIGURE 3. Comparison of stagnation point heat
transfer coefficient; α = 0.
0
2
4
6
8
10
S / Rn, Normalized Surface Distance
FIGURE 4. Heat transfer coefficient distribution
comparison; θc= 5 deg; α = 0.
EVALUATION OF IMPROVED HEATING MODEL AT ANGLE OF ATTACK
7 Degree Sphere Cone
Comparisons of the improved rarefied heating model with DSMC results are provided for the flowfields
investigated. Windward side results for a 70 sphere cone at α = 750 for λ∞ / Rn = 0.229 are shown in Fig. 5a; and
leeside results are provided in Fig. 5b. λ∞ / Rn represents the free stream Knudsen number, the ratio of the
molecular mean free path to a reference length, using nose radius for the length scale. The same DSMC cell
structure was used for the higher altitude simulation at λ∞ / Rn = 1.283. Ch results are shown in Fig. 6a for the
windward side and Fig. 6b for the leeward side.
Scatter in DSMC heat transfer results, particularly for the λ∞/ Rn = 0.229 case, render an accurate assessment of
the improved modeling difficult for the windward side of the body for the α = 750 cases. In general, the improved
predictions over the conical surface are in reasonable agreement with DSMC trends and magnitudes. Good
agreement is indicated for λ∞ / Rn = 0.229 near the most windward side (φ = 4.7 and 14.2 deg) and for φ = 83.50 for
S/Rn >2. Considering the sparse number of cells devoted to the nosetip region, accurate DSMC results are not
expected along the nose and near the nose afterbody juncture; accuracy degradation in heat transfer predictions has
been demonstrated when flow gradients are not adequately resolved. The most windward side heating is slightly
overpredicted with the improved model at λ∞ / Rn = 1.283; comparisons similar to the lower altitude case are noted
for 14.2 < φ < 83.50 deg.
Leeside DSMC heating distributions, shown in Fig. 5b at λ∞/Rn = 0.229 and Fig. 6b for λ∞/Rn = 1.283, indicate
relatively smooth behavior over the afterbody. Considering the significantly reduced heating levels on the leeside,
reasonable agreement with the improved modeling using the expression for (α / θ ) > 1.5 is demonstrated for both
altitudes.
DSMC,
phi=4.7
Predicted
, phi=4.7
DSMC,
phi=14.2
Predicted
, phi=14.2
DSMC,
phi=23.7
Predicted
, phi=23.7
DSMC,
phi=33.8
Predicted
, phi=33.8
DSMC,
phi=45
Predicted
, phi=45
DSMC,
phi=57.4
Predicted
, phi=57.4
DSMC,
phi=70.5
Predicted
, phi=70.5
DSMC,
phi=83.5
Predicted
, phi=83.5
Ch, Heat Transfer Coefficient
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
DSM C, phi=97.5
DSM C, phi=114.4
DSM C, phi=135
DSM C, phi=162.8
1.0E+00
Ch, Heat Transfer Coefficient
0.8
Predicted, phi=97.5
Predicted, phi=114.4
Predicted, phi=135
1.0E-01
Predicted, phi=163
1.0E-02
1.0E-03
1.0E-04
0
0 1 2 3 4 5 6 7 8 9
S / Rn, Normalized Surface Distance
1
2 3
4
5
6 7
8
9
S / Rn, Normalized Surface Distance
FIGURE 5. Comparison of 7 deg sphere cone heating distributions; α = 75 deg; λ∞ / Rn = 0.23.
a) windward side
b) lee side.
Ch, Heat Transfer Coefficient
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0 1 2 3 4 5 6 7 8 9
S / Rn, Normalized Surface Distance
DSMC,
phi=4.7
Predicted,
phi=4.7
DSMC,
phi=14.2
Predicted,
phi=14.2
DSMC,
phi=23.7
Predicted,
phi=23.7
DSMC,
phi=33.8
Predicted,
phi=33.8
DSMC,
phi=45
Predicted,
phi=45
DSMC,
phi=57.4
Predicted,
phi=57.4
DSMC,
phi=70.5
Predicted,
phi=70.5
DSMC,
phi=83.5
Predicted,
phi=83.5
DSMC, phi=97.5
DSMC, phi=114.4
DSMC, phi=135
DSMC, phi=162.8
Predicted, phi=97.5
Ch, Heat Transfer Coefficient
1.0
Predicted, phi=114.4
1.0E+00
Predicted, phi=135
Predicted, phi=163
1.0E-01
1.0E-02
1.0E-03
1.0E-04
0 1 2 3 4 5 6 7 8 9
S / Rn, Normalized Surface Distance
FIGURE 6. Comparison of 7 deg sphere cone heating distributions; α = 75 deg; λ ∞ / Rn = 1.28.
a) windward side
b) lee side.
10 Degree Sphere Cone
Improved predictions are compared to DSMC heating results for the 100 sphere cone configuration at α = 200 for
λ∞ / Rn = 0.467 in Fig. 7a for the windward side, and Fig. 7b for the leeside. Scatter in the DSMC results is
significantly lower than for the α = 750 cases. Windward results are shown for S/Rn > 1 to emphasize the heating to
the conical surface; the sparse cell distribution in the nose region prevented accurate assessment of the method for
S/Rn < 1. Considering the accuracy degradation in DSMC heat transfer results when flow gradients are not
adequately resolved, realistic comparisons for 1 < S/Rn < 2 are not expected. However, relatively good agreement is
indicated for the windward side for S/Rn > 2. Leeside heating predictions, again using the expression for (α / θ ) >
1.5, are in good agreement except for the most lee ray (φ = 1650).
DSMC results for the 100 sphere cone shape at λ∞ / Rn = 1.809 at α = 200 are compared with predictions in Fig.
8a for the windward side and Fig. 8b for the leeside. These results were not included in the development of the
method to include angle of attack effects on the heat transfer coefficient. Trends similar to results at the lower λ∞ /
Rn case are noted with the DSMC data; increased heat transfer coefficient levels are indicated at the higher altitude
conditions. Windward side heating predictions show somewhat larger errors than at the lower altitude case;
however, leeside predictions are in reasonable agreement on the conical surfaces.
DSM C, phi=98.3
0.3
0.2
0.1
0.0
DSMC,
phi=5.6
Predicted,
phi=5.6
DSMC,
phi=16.9
Predicted,
phi=16.9
DSMC,
phi=28.1
Predicted,
phi=28.1
DSMC,
phi=40.8
Predicted,
phi=40.8
DSMC,
phi=54.8
Predicted,
phi=54.8
DSMC,
phi=68.9
Predicted,
phi=68.9
DSMC
0
2
4
6
8
10 12
S / Rn, Normalized Surface Distance
DSM C, phi=116.7
DSM C, phi=138.3
1.0E+00
Ch, Heat Transfer Coefficient
Ch, Heat Transfer Coefficient
0.4
DSM C, phi=165
P redicted, phi=98.3
P redicted, phi=116.7
P redicted, phi=138.3
1.0E-01
P redicted, phi=165
1.0E-02
1.0E-03
0
2
4
6
8
10
S / Rn, Normalized Surface Distance
FIGURE 7. Comparison of 10 deg sphere cone heating distribution; α = 20 deg; λ ∞ / Rn = 0.47.
a) windward side
b) lee side.
12
0.6
DSMC, phi=116.7
DSMC, phi=138.3
DSM C,
phi=16.9
Predicted,
phi=16.9
DSM C,
phi=28.1
0.4
Predicted,
phi=28.1
DSM C,
phi=40.8
0.3
Predicted,
phi=40.8
DSM C,
phi=54.8
Predicted,
phi=54.8
0.2
DSM C,
phi=68.9
Predicted,
phi=68.9
DSM C,
phi=83.0
0.1
Predicted,
phi=83.0
0.0
0
2
4
6
8
10
DSMC, phi=165
1
Ch, Heat Transfer Coefficient
0.5
Ch, Heat Transfer Coefficient
DSMC, phi=98.3
DSM C,
phi=5.6
Predicted,
phi=5.6
Predicted, phi=98.3
Predicted, phi=116.7
Predicted, phi=138.3
Predicted, phi=165
0.1
0.01
0.001
0
12
S / Rn, Normalized Surface Distance
2
4
6
8
10
S / Rn, Normalized Surface Distance
FIGURE 8. Comparison of 10 deg sphere cone heating distributions; α = 20 deg; λ ∞/ Rn = 1.81.
a) windward side
b) lee side.
CONCLUSIONS
An improved rarefied flow heating method has been developed for large angle of attack application, based on
extension of a rarefied transitional flow bridging technique obtained from DSMC flowfield results at zero angle of
attack. The method has used DSMC results for sphere cones to obtain a modified bridging function for angle of
attack. Comparisons of detailed heat transfer coefficient distributions over conical surfaces using the improved
method with large angle of attack DSMC flowfields have indicated satisfactory agreement.
A criteria for the lower limit of applicability was imposed on the bridging technique, based on comparisons of
DSMC and 3DVSL heating distributions at α = 0; these results corresponding to Jain’s similarity parameter = 0.8.
ACKNOWLEDGEMENT
Appreciation is expressed for Dr. Amock C. Jain’s consultations and advanced copy of his aerothermal analysis
methodology, which served as the basis for this effort. Gratitude is expressed to Dr. Daniel McGregor of TRW for
the DSMC results used in this study.
REFERENCES
1. Jain, A. C., “Hypersonic Aerothermal Analysis Program for a Maneuvering Missile/Fighter Aircraft”, STC
Technical Report 3163, January 1998.
2. Moss, J. N., Cuda, V., Simmonds, A. L., “Nonequilibrium Effects for Hypersonic Transitional Flow”, AIAA
Paper- 87-0404.
3. Kotov, V. M., Lychkin, E. N., Reshetin, A. G., Schelkonogov, A. N., “An Approximate Method of Aerodynamic
Calculation of Complex Shape Bodies in a Transitional Regime”, Thirteenth International Symposium of Rarefied
Gas Dynamics, Rarefied Gas Dynamics, Vol. 1, Plenum Press, New York, 1985, pp 453-460.