Low Reynolds Number Effects on Hypersonic Blunt Body
Shock Standoff
George R. Inger
Department of Aerospace Engineering and Engineering Mechanics, Iowa State University,
2271 Howe Hall, Room 1200, Ames, Iowa, 50011-2271, USA
Abstract. An approximate analytical theory of the onset of viscous effects on hypersonic body nose shock standoff distance
is developed. Boundary layers displacement, shock layer vorticity and longitudinal curvature effects are all included, as are
the influence of both shock layer density ratio and arbitrary body surface temperature. Validating comparisons with both
CFD results and experimental data are also presented.
INTRODUCTION
The standoff distance ∆ of the bow shock on the nose region of a hypersonic blunt body (Fig. 1) is an important
observable in experimental studies of space vehicle aerodynamics [1]. It is therefore useful to have an accurate,
fundamentally-based theory for predicting and correlating data for shock standoff over a wide range of high altitude
flight conditions. When the combination of such conditions and body size Rb involve shock layer Reynolds
Numbers Res larger than 300, viscous effects have a negligible influence on ∆ and existing inviscid methods are
sufficient for the task. However, when simulating lower Reynolds Number flight conditions corresponding to flight
in rarefied atmospheric regimes, there arises the need to include the viscous effects on predictions/observations of
shock standoff and to estimate under what conditions these effects can be neglected.
Exact numerical solution codes are available to serve this need; however, such purely-CFD tools are expensive
to use for engineering parametric studies and do not readily yield the physical insight and similitude laws needed for
experimental design and data interpretation. The present work offers an alternative analytical treatment yielding a
closed form prediction method that accounts for the low Reynolds Number effects on the inviscid result and shows
how such effects depend on both the shock layer density ratio and body temperature. Such a theory is useful in the
design of simulation facilities for high altitude hypervelocity flight and the interpretation of their results.
OUTLINE OF THE THEORY
Our main focus is on the “threshold” of low Reynolds Number effects associated with the continuum end of the
rarefied flow spectrum, embracing the boundary layer/vorticity interaction shock layer regime wherein a distinct
discontinuous bow shock wave still occurs. To bring out the essential viscous aspects, we assume chemically frozen
shock layer conditions without vibrational or dissociative excitation; surface slip effects of velocity and temperature
are also neglected. A thin constant pressure shock layer is assumed in the stagnation region of either a 2-D or
axisymmetric hypersonic body. An arbitrary gas is treated, the body having an arbitrary temperature Tw relative to
the total T0 temperature of the flow.
A two-layered model of the shock layer is employed, consisting of an outer region of constant density vortical
inviscid flow behind the shock overlaying an inner viscous sublayer adjacent to the body surface (see Fig. 1).
Restricting attention to the immediate vicinity of the stagnation line where u ≈ u1 ( y) ⋅ x , ρ ≈ ρ 0 ( y) and v =
v0(y) (see Fig. 1), the continuity equation governing the flow along this line is
(1 + j )ρ 0U1 + (1 + j ) ρ 0V0
+
RB
d (ρ 0V0 )
=0
dy
(1)
where j = 0, 1 for 2-D and axisymmetric flows, respectively. Using the integrating factor exp[(1+j)y/RB], Eq. (1)
can be integrated from the solid non-ablating body surface (v0 =0) to any distance y in the shock layer to give the
mass flux there as
ρ 0V0 ( y ) = −(1 + j )e
− (1+ j )
y y
RB
∫e

y
 1+ j
RB




ρ 0U 1dy
0
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
514
(2)
Purely Inviscid Analysis
Suppose we were to neglect the inner viscous region and thus treat the entire shock layer as a purely inviscid
flow with a constant density ρ 0 = ρ F ≡ ρ ∞ ε F . Then applying Eq. (2) to the entire shock layer with
y = ∆ INV (the “inviscid” standoff distance) and using the mass conservation condition across the shock that
ρ 0 V0 (∆) = −ρ ∞ U ∞ , we obtain
ε FU ∞
1+ j
where
ζ ≡ y/∆.
1
= ∆ INV ∫ e
− (1+ j )
∆
(1−ζ
RB
)
ρ 0U 1, INV (ζ )dζ
(3)
0
Furthermore, for thin hypersonic stagnation shock layers the approximation
U1, INV ≈ βSH
(the stagnation point velocity gradient right behind the shock) is an accurate one. Using also the approximation that
∆ / R B in the integrand is necessarily small and hence can be evaluated from thin shock layer values, Eq. (3) yields
(1 + λ )ε F
∆ INV
≅
(1 + j ) )K (β w RB / U ∞ )
RB
(4)
where λ ≡ 1 / 2 ε F (β w R B / U ∞ ) is a longitudinal curvature parameter and K ≡ β SH / β w is the ratio of the
post-shock to body inviscid velocity gradients and hence reflects the inviscid shock layer vorticity. According to
Newtonian Theory [2], β W R B / U ∞ ≈ 2ε F (1 − ε F ) , while values of K obtained from Ref. 2 are given in
Table I as a function of the shock layer density ratio ε F ≡ ρ ∞ / ρ F . It should be noted that predictions from Eq.
(4) are in excellent agreement with both exact CFD-type calculations and experimental measurements on hypersonic
cylinders and spheres at high Reynolds Numbers [3].
Two-Layered Analysis Including Viscosity
We now include the presence of an inner region of thickness δ that contains viscous effects in the form of a
boundary layer-type of behavior. Returning to Eq. (2) and subdividing the integration into viscous and (outer)
inviscid region contributions, we have at the true shock location y=∆:
δ
ρ ∞U ∞ = (1 + j )∫ e
−(1+ j )
δ −y
RB
(ρ1U1 )BL dy + (1 + j )ρ F β SH
0
eff
δ − (1+ j )(1−ζ ) δ
 1 −(1+ j )(1−ζ ) R∆

RB
B
dζ − δ ∫ e
dζ  (5)
∆ ∫ e
 0

0
where β SH ,
is the effective post-shock velocity gradient that now includes the viscous displacement effect on
eff
the inviscid tangential velocity field. Applying the above-mentioned treatment of the last two integrands for small
∆/RB and δ/RB, writing (ρ1U1 )BL = (ρu / ρ e U e )BL ⋅ β w ρ F and then dividing through by
(1 + j)(βSH, EFF )ρ F R B /(1 + λ ) Eq. (5) yields
(1 + λ )ε FU ∞ / RB
(1 + j )β SH ,eff
(1 + λ )β w e
∆  (1 + λ )  δ
≅
−
−

β SH ,eff ∫0
RB 1 + (δ / ∆ )λ  RB
δ
δ−y
− (1+ j )
RB
 ρu 

 dy
 ρ e ue  BL
(6)
At this point, we introduce several simplifications that reflect our goal of seeking a first order account of the
viscous effects on shock displacement: since the last two terms in Eq. (6) are already of order δ/RB and hence small,
we can neglect the second order contributions of (δ/∆)λ and (1+j)(δs-y)/RB to the coefficients therein. Moreover, to
515
[ (
*
the same order of approximation, we may consistently take βSH , EFF ≈ Kβ w 1 − δ / R B
δ
order boundary layer displacement thickness. Finally, we note that
∫ (ρu / ρ U )
e
e BL
)] where δ* is the first
dy ≡ δ − δ * . Implementing
0
all these steps and using Eq. (4), Eq. (6) simplifies to an equation indicating the explicit effects of viscosity on
standoff relative to the purely inviscid theory value, viz:
+
∆
∆
δ*
∆
 Κ − 1  δ − δ  
 +  λ + INV

≅ INV +
+ (1 + λ )

RB
RB
RB
RB
 Κ  RB  
 δ*

 RB
(7)
This relationship clearly brings out that the relative viscous effect on the inviscidly-predicted shock standoff is not
only due to the direct effect of the boundary layer displacement but is also composed of contributions from the
vorticity in the outer layer, the longitudinal curvature and the displacement influence on the inviscid flow speed.
Since δ* is negative for cooled bodies whereas the value of ∆-∆INV at lower Reynolds Numbers is known to be
greater than zero (see below), the naïve use of the intuitive estimate ∆≅∆INV+δ* is seen from Eq. (7) to give an
unsatisfactory result. As a final observation on Eq. (7), we note that consistent with the above-mentioned thin shock
layer approximations to evaluate ∆INV in the last term, the two terms in the square brackets can be shown to be equal
to the single combined term [K+2(1+λ)/(1+j)]λ/K.
The explicit dependence of the foregoing on Reynolds Number and wall temperature can now be revealed by
supplying values of δ and δ* as given by classical stagnation point boundary layer theory. Neglecting second order
wall velocity and temperature slip effects, we can obtain these values from Emmanuel’s [4] comprehensive
numerical study of self-similar boundary layer properties; for stagnation flow, they can be rendered in the following
very convenient format:
δ*
RB
δ*
RB
=
=
εF
(a + bg w )
(8A)
εF
(c + dg w )
(8B)
(1 + j )(β w RB / U ∞ ) Re s
(1 + j )(β w RB / U ∞ ) Re s
where Re s ≡ ρ F U F R B / µ F = ρ ∞ U ∞ R B / µ F is the shock layer Reynolds Number, gw≡Tw/T0 is the wall to
total temperature ratio, and the constants a, b, c and d depend only on the Prandtl Number and body dimensionality
as given in Table II. Incorporation of Eq. (8) into Eq. (7) thus produces the final working form of our closed-form
analytical solution:
 ∆ − ∆ INV

 RB
where F( j, ε F ) ≡ Κ
−1
   2λ 

 1/ 2
 Re s = (1 + λ )F ( j, ε F ) 1 + 
(a + bg w ) + (k − 1)(c + dg w )
   1 + j 


(9)
ε F /(1 + j)(β w R B / U ∞ ) is tabulated for convenience in Table I along with values for
βwRB/U∞ and λ.
COMPARISONS WITH NUMERICAL SOLUTIONS AND EXPERIMENT
We first note that the breakdown into various contributions emerging from the present analysis, Eq. (7), is
consistent with the classification of low Reynolds Number effects in hypersonic blunt-nose regions given some time
ago by Van Dyke [5]. Secondly, the present analytical predictions that ∆ - ∆INV increase with reducing Res in a
manner scaling with Res-1/2, and that this increase be considerable larger in the two-dimensional case (j=0) than in the
axisymmetric case (j=1), are both qualitative agreement with the results given by Probstein and Kemp [6] based on
numerical solutions of the complete viscous shock layer equations. This is shown in Fig. 2. The present theory is,
in fact, in excellent quantitative agreement as well for the case of the sphere.
516
TABLE 1. Vortical Shock Layer Properties vs. Density Ratio
j=0
εF
λ
j=1
F
Κ
F
Κ
.250
.204
1.053
.607
1.112
.406
.200
.177
1.095
.543
1.185
.355
.167
.158
1.135
.496
1.250
.318
.143
.144
1.170
.459
1.309
.290
.125
.134
1.204
.429
1.367
.267
.111
.125
1.234
.405
1.419
.249
.100
.118
1.263
.384
1.467
.234
.083
.106
1.314
.351
1.560
.209
.067
.095
1.384
.315
1.685
.183
.050
.081
1.486
.271
1.870
.152
.010
.036
2.333
.114
3.568
.053
TABLE 2. Parameters in the Shock Standoff Theory (Eq. 9)
j=0
PR
A
b
c
j=1
d
a
b
c
d
.725
.304
.902
2.220
.480
.256
1.024
2.319
.804
1.000
.158
.806
1.956
.423
.109
.914
2.043
.708
517
518
519
520
Turning to a comparison with available experimental data, Fig. 3 shows a comparison of the present theoretical
predictions of the low Reynolds Number effect with data obtained at AEDC [7] for the case of a cool (gw ≈ .30)
sphere in Argon at several different high freestream Mach Numbers. Allowing for the fact that our theory is a
“strongly-hypersonic” one, its predictions of an increase in standoff distance with decreasing Reynolds Number are
seen in good agreement with this data.
PARAMETRIC STUDIES
The aforementioned validating comparisons and the analytical nature of our results make the present theory
attractive for several parametric studies. In particular, it permits direct examination of the role of both wall
temperature and shock layer density ratio effects on the low Reynolds Number behavior of shock standoff. The
former effect, for example, is evident by direct inspection of Eq. (9): cooling the body significantly reduces ∆ .
REFERENCES
1.
2.
3.
4.
5.
6.
7.
Morgan, R. G., “A Review of the Use of Expansion Tubes for Creating Superorbital Flows,” AIAA Paper 97-0279, 1997.
Hayes, W. D., and Probstein, R. F., Hypersonic Flow Theory, Academic Press, New York, 1959, pp. 150-162.
Inger, G. R., Higgins, C., and Morgan, R., “Shock Standoff on Hypersonic Blunt Bodies in Nonequilibrium Gas Flows,”
Journal of Thermophysics and Heat Transfer 16, No. 2, pp. 245-250 (2002).
Emmanuel, G., Analytical Fluid Dynamics, Chem Rubber Press, Boca Raton, 1994, pp. 265-272.
Van Dyke, M., “Second Order Compressible Boundary Layer Theory with Application to Blunt Bodies in Hypersonic
Flow” in Hypersonic Flow Research (F. R. Riddell, Ed.), Academic Press, New York, 1962, pp. 37-76.
Probstein, R. F., and Kemp, N. M., “Viscous Aerodynamic Characteristics in Hypersonic Rarefied Flow,” Journal of the
Aeronautical Sciences 27, No. 3, pp. 174-192 (1960).
Bailey, A. B., and Sims, W. H., “Shock Detachment Distance for Blunt Bodies in Argon at Low Reynolds Number,” AIAA
Journal 1, No. 12, pp. 2867-68 (1963).
521