New Thermal Conditions at the Wall in High Speed Flows
J.Gilbert Meolans∗ and Irina Graur†
∗
Université de Provence - Ecole Polytechnique Universitaire de Marseille, 5 rue E. Fermi, 13453 Marseille,
France
†
Institute of Mathematical Modeling Russian Academy of Science, 4a Miusskaya sq., 125047 Moscow, Russia
Abstract. This paper deals with new thermal slip boundary conditions at the wall in nonequilibrium polyatomic gas flows
when the dissociation effects may be neglected. The conditions usually employed are briefly described. Then the conditions
proposed in the paper are justified by referring to previous experimental results and by developing direct heat flux calculations
at the wall. Finally numerical calculations are presented in rotational nonequilibrium conditions, and lead to good agreements
with the experimental results.
INTRODUCTION
The continuum approaches using classical boundary conditions at the wall are generally unsatisfactory in high speed
flows [1], [2]. So various slip formulae have been used to describe kinetic and thermal conditions at the wall, especially
in spatial re-entry studies [1], [3] - [5]. But often the previous descriptions consider unstructured gas molecules; at best,
the internal degrees of freedom in polyatomic gases are only taken into account when a thermodynamic equilibrium
state is assumed. Then the majority of available expressions concerning the thermal jumps include a single temperature
characterizing the external degree of freedom as well as the internal ones. Moreover the reflection law employed is
often too approximate to describe the phenomena occurring at the wall realistically [1], [3], [5].
In this paper boundary conditions at the wall are established for nonequilibrium flows of polyatomic gases.
1. In the vibrational nonequilibrium case, the rotational and the translational degrees of freedom of the molecules
are characterized by the same temperature. Then new boundary conditions are derived, including an expression of the
external temperature jump and the vibrational (internal) energy flux at the wall equal to zero.
2. In lower temperature ranges, rotational nonequilibrium flows are analyzed neglecting the vibrational excitation.
A system of two equations giving the translational and the rotational temperature jumps are presented, including the
accommodation coefficients of translational and rotational energies.
These new conditions are first justified by analyzing previous experimental and theoretical results [4], [6] - [12].
Then direct calculations using the collisional transition probabilities allow us to obtain some expressions of the various
heat fluxes at the wall. So, in vibrational nonequilibrium conditions, it is once more shown that the vibrational flux
at the wall is very small compared to the translation energy flux, and explicit expressions of the accommodation
coefficients of energy could be easily obtained for some common gases. On the other hand, in rotational nonequilibrium
conditions it appears that the rotational accommodation coefficient and the translation one may be of the same order
of magnitude. Furthermore the rotational accommodation coefficient increases strongly when the departure from the
equilibrium value of the rotational macroscopic parameters increases. A basic assumption is used throughout the
present work: the influence of the gas/gas collisions on the internal macroscopic parameters (inter-mode exchange) is
neglected in the Knudsen layer appearing close to the wall.
Finally, in low temperature ranges, the flow field around a hollow cylinder has been calculated using the QGDR
model [13]: a good agreement is obtained with the parameters measured at the wall by implementing the present
nonequilibrium boundary conditions.
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
THEORETICAL EXPRESSIONS FOR THE VIBRATIONAL NONEQUILIBRIUM CASE
Low magnitude of the vibrational flux at the wall: consequences.
Various experimental results [6] - [9], previously obtained in vibrational nonequilibrium flows near a plane wall,
show a vibrational heat flux qV (normal to the wall) vanishing at the wall: that is to say a vibrational flux qVwall and,
of course, a vibrational energy accommodation coefficient αV which is very small compared to the corresponding
parameters of the external degree of freedom. Moreover, a recent theoretical study [10] concerning the Knudsen layer,
also evidenced this in strong vibrational nonequilibrium flows (for example behind reflected shock waves). Thus, it
seems clear that in such nonequilibrium flows the conditions qVwall ∼
= 0 and αV ∼
= 0 may be admitted.
Then, neglecting the gas/gas collision effects on the intermode exchange of the molecules in the Knudsen layer
as was usual up to now [5], [6], it can be admitted that qV is also negligible at the boundary separating the zones
governed by continuum fluid equations (for example the Navier-Stokes (NS) equations) and the Knudsen layer (see
Fig. 1a). Thus it can be written:
qVNS bound = 0,
αV = 0.
(1)
But here it is convenient to make precise that (1) does not mean that the vibrational temperature TVNS bound is equal
to Tw . In fact the vibrational temperature jump ∆TV should be deduced from the calculation process based on the NS
equations (or on other continuum fluid equations) supplemented by the slip conditions. This point may be cleared if one
reminds that it is easy to prove that "far from the wall": qVNS = A (∂ TV /∂ x)NS , using for example a distribution function
of Chapman-Ensgok type and taking into account the Ensgokian constraint on the momentum [14] (furthermore, a
similar result also has been obtained inside the Knudsen layer [10]). So, using the previous remarks, it can be further
written:
qVNS bound → 0,
and
(∂ TV /∂ x)NS bound → 0
simultaneously.
(2)
The consequences of the last relations on the calculation of the temperature jumps appear in the following section.
Practically useful thermal slip conditions
Now, it must be remembered that almost all the easy-to-use jump temperature formulae, assume a negligible
influence of the gas-gas collisions on the inter-mode exchanges within the Knudsen layer [4], [5], which means that,
the vibrational macroscopic parameters are actually considered to be frozen from the Knudsen layer boundary up to
the wall. For such admitted conditions, some explicit expressions of vibrational temperature jumps are theoretically
available [4], [8], [9]. When a single temperature T characterizes both the rotation and translation degrees while
another temperature TV refers to the vibrational degree it can easily be deduced, for example, from the reference [4]:
√
5/4 +Cr /2k π 2 − k1 αT R 1
2kT0 ∂ T
∆T =
,
(3)
1 +Cr /2k 2
αT R θT R
m
∂ x NS bound
√
2kT0 ∂ TV
π 2 − k2 αV 1
.
(4)
∆TV =
2
αV
θT R
m
∂ x NS bound
Where T0 is the gas temperature at the wall, αT R is the accommodation coefficient for the translation-rotation energy,
θT R is the collision frequency including elastic and pure rotational processes, Cr is the heat capacity per molecule for
the rotational degree of freedom; k is the Boltzmann constant. Subsequently k1 is equal to 0.82 whatever the gas and
k2 increases from 0.82 to 0.85 with increasing Cr . But, the equation (4) does not give a practical result, as according to
(1) and (2) ∆TV becomes here indeterminate. So, instead of equation (4), equation (1) must be associated to equation
(3) giving the external temperature jump.
Theoretical energy flux calculation at the wall and basic comments
Some basic calculations and comments may be added showing the low magnitude of (qV /qT )w and αV /αT ratios
the transition
(where the subscript T refers to the translation degree of freedom) compared to one. Let us call Pi,iN,N
probability of a colliding wall atom/gas molecule pair. Here the subscript i = (ir , iν ) denotes the rotation-vibration
energy state of the incoming molecule, and i = (ir , iν ) the corresponding state of the reflected molecule, while N and
N mean the vibrational levels of the solid atom, before and after the collision respectively. It is possible to obtain an
approximated expression of the energy fluxes at the wall involving this probability.
Energy flux calculations at the wall. For each degree of freedom of the molecules, the energy flux at the wall
includes a term of exchange with the wall and exchange terms with the other molecule modes. For a simpler
description, the translation and rotation fluxes may be grouped together in a single term qT R (of the same order as
qT ). Then:
(qT R )w = ϕT R⇔w + ϕT R⇔V ,
(qV )w = ϕV ⇔w + ϕV ⇔T R ,
(5)
where ϕ represents the various partial fluxes, the subscript w refers to the wall and T R, V characterize the translationrotation and the vibration modes of the gas respectively. Moreover the condition ϕT R⇔V + ϕV ⇔T R = 0, is obviously
verified. Let us now consider the effective molecule collisions per time unit occurring on a surface unit of the wall:
when multiplying the number of collisions acting in each exchange process by the energy quantum transferred per
collision, the corresponding flux term is obtained. For example, for the translation energy transfer from the gas to the
wall (direct and reverse processes)
N,N ,
(6)
ϕT R⇔w = ∑ n−
C
σ
P
A
h
ν
N
0
el
N,N
i
i,i
i,i ,N,N th
where n−
i is the numerical density of the incoming molecules on the i rotational-vibrational level, AN is the atomic
population on the N th energy level per wall surface unit, σel is the elastic cross-section characterizing the collisions of
the gas molecules with the solid atoms, and hνN,N is the energy quantum transferred to (or from) the wall in the atomic
transition, counted algebraically. Finally C0 is an average thermal velocity close to the quadratic thermal velocity.
As it is well known, for the significantly populated low levels, the main vibrational transition processes in the gas
are the single-quantum exchanges up to very high temperatures [15]; and in such single-quantum exchanges, θV k and
θA k (where θ = hν /k) measure the quantum of energy lost (or gained) by the vibrational mode of the molecule and
the energy amount gained (or lost) in the bounded states of the solid atom respectively when an efficient collision
occurs (see Fig. 1 b). Thus, the vibrating molecules and atoms may be described as harmonic oscillators; moreover,
a Maxwell-Boltzmann distribution (at temperature Tw ) of the atom population and a nonequilibrium Boltzmann
distribution (at T and TV temperatures) for the numerical density of the molecular energy levels may also be admitted.
This last assumption is confirmed keeping in mind the low magnitude of the corrective term of the Maxwell-Boltzmann
distribution function in the Knudsen layer [10], [16]. Such a simplified model requires N = N ± 1 in (6). Then
considering only the terms corresponding to iν = iν , the result given below in equation (7) is obtained. Let us note
that, to be totally exact, these terms should be supplemented with two terms from the coupled transitions (N → N + 1,
iν → iν + 1) and (N → N − 1, iν → iν − 1): actually a translation energy change is involved in these transitions to
equilibrate the energy balance, but these terms include two non diagonal matrix elements of atomic and molecular
internal degrees and so are completely negligible. On the other hand, the transitions involving a rotational number
change ir → ir practically exist only when ir → ir ± 2 for common diatomic gases and are rarer than those occurring
with unchanged ir [17]. So these rotational transitions are disregarded. Moreover the probability dependence on ir is
also neglected because of its slight influence on the calculation [18].
0,1
Using these approximations which imply PiN,N+1
= (N + 1)P0,0
, the sum in (6) is then easily calculated and leads to
ν ,iν
0,1
ϕT R⇔w = σel C0 An− kθA P0,0
[exp(−θA /T0 ) − exp(−θA /Tw )] / [exp(θA /Tw ) − 1] .
With similar considerations the vibrational transfer to and from the wall may be written successively
N,N+1
ϕV ⇔w = C0 σel k ∑ PiN,N−1
−
P
θA AN n−
iν ,
,i
+1
,i
−1
i
ν ν
ν ν
(7)
(8)
iν ,N
0,1 exp((θA − θV )/T0 ) exp(θV /TV0 ) − exp(θA /Tw ) / [exp(θA /Tw ) − 1] exp(θV /TV0 ) − 1 ,
ϕV ⇔w = σel C0 An− kθA P1,0
(9)
where TV0 is the vibrational temperature of the gas on the wall. And finally, the inter-mode exchange term:
N,N
N,N+1
N,N−1
N,N+1
N,N−1
(
ϕT R⇔V = C0 σel k ∑ PiN,N
−
P
θ
+
P
−
P
θ
−
θ
)
+
P
−
P
θV AN n−
V
V
A
iν ,
,i
−1
,i
+1
,i
−1
,i
+1
,i
−1
,i
+1
i
i
i
i
i
ν ν
ν ν
ν ν
ν ν
ν ν
ν ν
iν ,N
(10)
TABLE 1. Flux ratios at the wall at T = 2200 K, TV = 1500 K, Tw = 600 K. PiN,N
=
ν ,iν
√
2
128S/π 3πσel Wiν ,iν VN ,N FT , S is the steric factor, σel is the dimensionless elastic cross
section, Wiν ,iν is the vibrational matrix element, VN ,N is the atom matrix element, FT =
13/6
1/3 2
T /θµ
exp (∆EC /2kT ), θµ = α h2 /2π 4 µ k,
(∆EC /kT )7/3 exp −3 (∆EC /kT )2/3 T /θµ
α is interaction
potential
range, µ is the reduced mass of the pair of colliding partners, ∆EC /kT =
εiν + ηN − εiν + ηN , εiν is the dimensionless energy of the ith
ν vibrational level, ηN is the atomic
level dimensionless energy (W atoms). The fluxes are calculated using transition probabilities given
in [17], the molecular parameters are taken in [19] and [21].
Gases
H2
O2
N2
CO
NO
θV (K)
6333
2270
3390
3123
2742
θµ (K)
11.68
15.48
16.42
16.65
16.08
6.62 × 10−2
0.88 × 10−2
0.67 × 10−2
0.98 × 10−2
0.88 × 10−2
ϕV ⇔w /qT R
−0.31 × 10−13
−0.19 × 10−6
−0.36 × 10−8
0.14 × 10−7
0.42 × 10−7
qV /qT R
−0.77 × 10−11
−0.19 × 10−4
−0.50 × 10−6
0.20 × 10−5
0.51 × 10−5
Wi2ν ,iν +1
−1 0,0 P1,0 exp(−θV /T0 ) exp(θV /TV0 ) − 1 +
ϕT R⇔w = −ϕV ⇔T R = σel C0 An− kθV exp(θV /TV0 ) − 1
1,0
+P1,0
[exp(θA /Tw ) − 1]−1 exp(−(θV + θA )/T0 ) exp(θA /Tw ) exp(θV /TV0 ) − 1 + (1 − (θV /θA )) ϕV ⇔w .(11)
0,1
0,0
1,0
0,1
Comments. In the compact expressions given above P0,0
, P1,0
, P1,0
and P1,0
are calculated using the Tanczos formulae
[17] shown in Table 1. As is well known, these probabilities include the vibration matrix elements characterizing the
changes in internal energy levels; they also contain a function FT of the molecular kinetic energy departure ∆EC
occurring in the corresponding colliding processes (see the detailed expression in Table 1). FT strongly decreasing
with increasing ∆EC becomes very small for the large non resonant transitions (∆EC /kT0 ≥ 1).
Now let us note that θV the vibrational characteristic temperature of molecules is much higher than θA the
characteristic temperature of the solid atom motion. Typically θV is close to 3390K for Nitrogen molecules [19]
compared to 205K for θA of solid Tungsten atoms [20]. Let us then analyze the following two cases.
1. For the transitions where no vibrational change occurs in the gas, the molecule kinetic energy change is only
provided by atom transition N → N = N + 1 (for transfers heating the wall); then as previously seen ∆EC /k may be
evaluated as statistically close to θA = 205K (see Fig. 1 b); then according to Tanczos expressions [17], data in Table 1
lead to FT = 1.64 × 102 for N2 . On the other hand, iν = iν ; consequently Wiν ,iν = Wiν ,iν ∼
= 1 (as a diagonal matrix
element of a normalized operator) [21].
2. For the transitions involving a single-quantum change in the vibrational molecular energy, the total energy
balance of the collision gives a quantum defect transferred to the kinetic energy equal to ∆EC /k = θV − θA , and
θV − θA = 3185K for the collisions between Nitrogen molecules and Tungsten atoms (see Fig. 1 b). Then FT is very
much smaller than in the previous case, FT = 2.80 × 10−4 . On the other hand, iν = iν − 1; then Wiν ,iν = Wiν ,iν −1 , and
for the significantly populated low levels Wiν ,iν −1 ∼
= 10−2 [21].
Then the probability that vibrational transfers occur will be very much slighter than the probability of translation
energy transfers to (or from) the wall: this last process is much easier because mach more resonant.
By means of the formulae (7), (9) and (11), the ratios of various fluxes can be calculated precisely enongh and
finally (qV /qT R )w is obtained. The results for the most common diatomic gases are given in Table 1. It is found
that, whatever the nonequilibrium state, qV remains negligible compared to qT R , especially for the heavy molecules
posessing a large moment of inertia (N2 , O2 , CO, NO). It should be noted, that generally the inter-mode exchange term
ϕV ⇔T R in (5) represents the largest part of the vibrational flux. Consequently, when all the degrees of freedom reach
equilibrium together, qV becomes smaller and smaller since, as a result of equilibrium, the detailed balancing causes
the predominant term ϕV ⇔T R (the first term in the right member) to vanish.
In any case it might be said that the wall is almost "non catalytic for the vibration" [10].
ROTATIONAL NONEQUILIBRIUM FLOWS
Temperature jumps at the wall
By neglecting the intermode exchanges which results from collisions in the Knudsen layer, as in the previous
sections, it is possible to refer once more to the case described in [4] i.e. that of a gas with a single internal degree
whose energy is frozen in the Knudsen layer. In a low temperature range it is now pertinent to use the formulae [4]
adapted to describe the rotational nonequilibrium state. The following is obtained:
√
√
5 π 2 − k1 αT 1
2kT0 ∂ T
π 2 − k2 αR 1
2kT0 ∂ TR
∆T =
,
∆TR =
.
(12)
8
αT
θel
m
∂ x NS bound
2
αR θel
m
∂ x NS bound
In (12) TR is the rotational temperature, αR is the accommodation coefficient of the rotational energy, and θel is
the frequency of the elastic collisions. The ratio of coefficients αR /αT is noticeable on account of its very different
behavior from those of the αV /αT R ratio: for example in [11], by comparing the numerical and experimental results,
the authors conclude that the correct values of αR are close to those of αT for experimental conditions characterized
by significant rotational nonequilibrium effects [11].
Energy fluxes at the wall
Furthermore, the relationships corresponding to equation (5) in the present physical conditions are:
(qT )w = ϕT ⇔w + ϕT ⇔R ,
(qR )w = ϕR⇔w + ϕR⇔T ,
(13)
with ϕT ⇔R + ϕR⇔T = 0. Structurally ϕT ⇔w is written as the (6) right member, but here, (i, i ) means only (ir , ir )
because iν = iν = 0. In the same way ϕR⇔w and ϕR⇔T involve the same structures as the terms ϕV ⇔w and ϕV ⇔T R
given previously in (8) and (10) respectively. On the other hand the transition probability of the colliding partners
differs here from that of T R − V process [17]. Even so, it presents similar properties: the transfer described are rarer
and rarer for increasing amounts of transferred energy to (or from) the molecular kinetic energy. So, noting that the
rotational characteristic temperature θR is much smaller than θA and using similar arguments to those of previous
section it is easily shown that:
– far from equilibrium, rotational and translational energy fluxes are of the same order of magnitude,
– far from equilibrium the flux ϕR⇔T deriving from the inter-mode exchange is much greater than ϕR⇔w ,
– reaching the equilibrium state, ϕR⇔T decreases strongly because of the detailed balancing, and so (qR )w becomes
much smaller than (qT )w .
The obvious consequences are the following: far from equilibrium the rotational accommodation coefficient is close
to the translational one; moreover the rotational coefficient decreases when decrease the nonequilibrium effects and
finally becomes very small when reaching thermodynamic equilibrium.
Calculation along the hollow cylinder flare
The axisymmetric flow with strong shock wave boundary layer interaction is calculated around a hollow cylinder
(see Fig. 2 a). The flow conditions are obtained under nominal stagnation conditions of p0 = 2.5 · 105 Pa and T0 = 1050
K, which yield an upstream flow characterized by the following parameters: p∞ = 6.3 Pa, T∞ = 51 K, M∞ = 9.91 and
Tw = 293 K. The experiment is carried out in ONERA [22], the surface pressure coefficient and Stanton number
distributions are measured.
The translational-rotational nonequilibrium in a gas with two rotational degrees of freedom is accounted for by
means of the QGDR model (continuum equations model [13]). As matter of fact, in these conditions (temperature
range less than 1000 K), the vibrational energy is negligible compared to the total thermal energy per molecule.
So the thermal jumps at the wall have been described according to (12) characterizing a translational and rotational
nonequilibrium thermodynamic state; in addition the velocity slip from [1] is used.
In fact, a significant difference between the translation and rotation temperature (about ∼ 30%) is found from
a previous DSMC calculation [23]. Then, according to [11] and the previous comments on the rotational flux, the
following values of the accommodation coefficients are considered: αT = 0.8, αR = 0.5. Figure 2b demonstrates
the deviation of the translational and rotational temperatures from the wall temperature along the cylinder surface.
The greatest difference is observed near the leading edge and it decreases downstream, up to the separation point
(X/L ∼ 0.7).
The calculated values of the Stanton number and of the pressure coefficient are in good agreement with the
measurements [22] (see Figs. 3 a and b). Otherwise, when comparing with a mean temperature approach (QGD
model) associated to a single temperature jump [13], the improvement brought by the new boundary conditions is
not clearly demonstrated everywhere. More complete calculations describing more significant nonequilibrium effects
(from a rotational or a vibrational point of view) seem necessary to appreciate the improvements clearly when using
the present boundary conditions, especially because of the probable experimental uncertainties.
CONCLUSION
The previous arguments drawn from experimental and theoretical fields allow us to describe the thermal slip conditions
in high speed and high temperature flows as defended here above.
In vibrational nonequilibrium gas flows, it is pertinent to use two conditions including a flux equation (1) instead of
available equations similar to (4), which are not of practical interest, and, in addition to use, another equation giving
the jump of the translation temperature at the wall. Moreover, it is of some interest to note that explicit expressions of
the energy accommodation coefficients might be easily derived from (5), (7), (9), (11), also using the definition of the
general accommodation coefficients [4], [11].
On the other hand, in the rotational nonequilibrium flow field, the implementation of a system of two equations,
giving the temperature jumps, is possible under the assumption previously admitted (negligible effects of the intermode exchange in the Knudsen layer). Furthermore it has been shown that the accommodation coefficients defined
respectively for the thermal kinetic energy and for the rotational energy may be of the same order of magnitude. When
the departure (T − TR )/T increases to one, the two coefficients are close to each other, while the ratio αR /αT becomes
very small when reaching equilibrium conditions.
The applied calculation, presented here in a flow field around a hollow cylinder, gives a good agreement with
the experimental results. But the rotational nonequilibrium, even if significant, is not great enough to measure the
improvements induced by the new formulation exactly. More complete calculations describing greater nonequilibrium
effects (from rotational or vibrational point of view) seem necessary to appreciate the improvements clearly when
using the boundary conditions presented above.
Of cause, some arguments used in this comment (and the exactness of (3)) fail partially when the usual assumption
of the negligible effects of the inter-mode exchanges in the Knudsen layer is no longer admitted. Of course various
attempts exist to account for these effects. They generally require more accurate and refined processing. Two groups
of approaches to this topic can be especially singled out. The first is based on detailed balances of energy exchanges
within the Knudsen layer and leads to slip expressions including formal undetermined coefficients. Their results are
very difficult to use in practice [4], [9], [11]. The second group mainly concerns vibrational nonequilibrium gas flows.
It includes the search for approximated solutions of the Boltzmann equation in the Knudsen layer [10], [16] or various
derivations of gas-wall interaction laws [24] - [26]. However, up to now, sufficient results have not been achieved in
this way to give slip formulae directly usable in practical flow field calculations.
ACKNOWLEDGMENTS
This work has been partially supported by the Russian Foundation for Basic Research, grant No 00-01-00061. The
authors thank Prof. M.S. Ivanov and Dr. G.N. Markelov for the results of the DSMC calculations put at our disposal
and also Dr. P.Perrier for useful discussions.
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22. Gorchakova, N.G., Chanetz, B., Kuznetsov, L.I., Pigache, D., Pot, T., Taran, J.P., Yarygin, V.N.: "Electron - Beam - Excited Xray Method for Density Measurements of Rarefied Gas flow Near Models". In: Rarefied Gas Dynamics, 21th Int. Sym. on RGD,
ed. by R. Brun et al. (Cepadues-Edition 1999), pp. 591 - 598.
Vibration molecular levels
wall
Knudsen layer
flow field
q
=0
δT
X
wall atom
translation
"quantum defect" transferred from
the gas translation mode
}
Gas translation energy
( δ x v )0 =0
V, wall
{
V, wall
equations
(for example NS equations)
{q
i =iv
Atomic levels
N+1
205 K
N
described with
continuum fliud
q v NS bound
’
v
gaseous molecule
unvarying level
Wall energy
Vibration molecular levels
Atomic levels
iv
3185 K
gaseous molecule
de-excitation
3390 K
}
N+1
iv -1
0
205 K
N
"quantum defect" transferred to
the gas translation mode
{
Gas vibration energy
FIGURE 1. a) Knudsen layer, schematic view; b) Transfers to the wall.
}
Wall energy
wall atom
transition
23. Markelov, G.N., Kudryavtsev, A.N., and Ivanov, M.S. " Continuum and kinetic simulation of laminar separated flow at
hypersonic speeds", Journal of spacecraft and rockets, v. 37, N 4, July-August 2000, pp. 499-506.
24. Billing, G.D., "Semi-Classical Treatment of the Dynamics of Molecule-Surface Interaction", Molecular Physics and
Hypersonic Flows, M. Capitelli (ed.), 1996, pp. 231-257.
25. Lord, R.G., "Some Extensions to the Cercignani-Lampis Gas Surface Scattering Kernel", Physics of Fluids, Vol. A3, N 4,
1991, pp. 706-710.
26. Meolans, J.G. and Aufrere, L., "Gas-Wall Interaction : Scattering Kernel for Polyatomic Molecules", Comptes Rendus,
Academie des Sciences, Paris, v. 329, Serie IIb, 2001, pp. 201-205.
1.3
T/Tw
L = 0.1017 m
0.0433
0.025
1.2
o
30
15
1.1
o
0.0575
0.0325
TT
TR
1
0
0.5
1
1.5
X/L
FIGURE 2. a) Hollow cylinder, schematic view; b) Temperature distribution along the cylinder surface.
0.05
St 0.04
Cp
0.03
Experiment
DSMC
QGDR
10
-1
10
-2
0.02
0.01
Experiment
DSMC
QGDR
0
0
0.5
1
1.5
0.5
X/L
FIGURE 3. a) Pressure coefficient; b) Stanton number.
1
1.5
X/L