Validation of DSMC/Navier-Stokes Computations for
Laminar Shock Wave/Boundary Layer Interactions in
Hypersonic Flow
John K Harvey*, Michael S Holden† and Graham V Candler‡
*
†
Imperial College, London SW7 2BY, UK
Aerothermal and Aero-Optics Evaluation Center, Buffalo, New York 14225
‡
University of Minnesota, Minneapolis, Minnesota 55455
Abstract. This paper discusses the recent progress with a validation exercise to examine the effectiveness of DSMC and NavierStokes in predicting complex flows involving shock/shock and shock/boundary layer interactions under laminar hypersonic
conditions. The exercise is based on experiments conducted in the Aerothermal and Aero-Optics Evaluation Center shock
tunnels using nitrogen. Under moderately rarefied conditions Navier-Stokes methods have performed very well in capturing the
details of flow structure especially when non-equilibrium and slip effects are accounted for, whereas DSMC has not been as
successful. In particular, the size of the separation regions is underestimated. This paper presents redefined test conditions
accounting for vibrational non-equilibrium which have been substantiated by correlations obtained for pressure and heat transfer
on the hollow cylinder and cone forebodies. Details are announced of two new lower density test conditions for future validation
studies.
INTRODUCTION
The scope and versatility of CFD has advanced spectacularly over recent years and thus it is not surprising that
assumptions have been made that eventually numerical predictions will completely eliminate the need for wind
tunnel testing. Despite the success of CFD there appear to be certain flows - some of critical importance – for which
computations consistently appear unreliable and it is evident that at least in some instances this is due to a failure to
capture the governing physics correctly.
Writing and using effective CFD software are both demanding tasks, strewn with pitfalls. Difficulties arise
because of the complexity of the modelling and the numerical techniques employed. Developing the codes is a
protracted process and this increases the possibility of error. Implementing the codes is also a challenging task,
often requiring substantial computer resources and poor results may be the consequence of being tempted or forced
to use inappropriate spatial and temporal discretisation. Two procedures have been identified - termed validation
and verification - which are designed to assess the quality of codes and their solutions. In the former, results from
computations are compared with reliable external experimental and/or theoretical data. In the latter, results from
different implementations of the code(s) for identical boundary conditions are compared. Both procedures are
invaluable tools for assessing quality and effectiveness. It is self-evident that the validation process is critically
dependent on having high quality reference data against which the results can be compared.
Few would challenge that Bird’s Direct Simulation Monte Carlo (DSMC) method is the most effective CFD
method for predicting rarefied flows. Formulated in 1970 [1], it brought within grasp practical solutions to many
important low density problems. The method has significant numerical advantages over the alternatives and it can
readily be extended to multi-species reacting flows and geometrically complex shapes. As time passed solutions to
complicated high-temperature flows involving chemical reactions, ionisation and high degrees of molecular nonequilibrium have been produced but, because of the lack of suitable measured data, very little direct validation has
been possible to ascertain if these solutions are accurate. Bird developed his method from an intuitive standpoint
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
417
using physical reasoning. Until 1990 [2], even a limited formal proof was unavailable to show that the solutions
were equivalent to those obtained from the Boltzmann equation, and this led to emphasis having to be placed on
confirming its validity experimentally. Over time a series of closely matched experiments and computations have
been performed using a variety of flows selected specifically for this purpose. For practical reasons there is current
interest in validating DSMC for denser flows where it has been suggested that there are shortcomings in its
formulation.
In this paper we review the progress that is being made in validating the method for flows involving complex
shock-interaction phenomena critically relevant to those associated with manoeuvrable re-entry and air-breathing
vehicles. The results reported form part of an extensive code validation activity for both DSMC and NS codes that
has been conducted during the past decade in Europe and America in support of space vehicle design including
those developed in the NASP programme. As a component of this a combined American/European exercise has
been promoted by the NATO Research Technology Organization (RTO) under Working Group 10. Also, with the
support of the Air Force Office of Scientific Research (AFOSR), a number of experimental and numerical studies
have been undertaken to examine the complex viscous/inviscid interaction regions developed over simple model
configurations in laminar hypersonic flow. In both of these investigations, the flow associated with two
configurations – namely a hollow cylinder/flare and a bi-conic body – have been the objects of extensive
investigation and sharply focus on the shock/shock and shock/boundary layer interaction phenomena that are
specifically of great practical importance and they provide critical tests of the codes’ abilities to yield accurate
answers in the presence of viscous/inviscid interactions. These two simple configurations have proved to exhibit
suitable “well-posed” flow examples that include the required complex interactions. High quality experimental data
have become available for moderately rarefied conditions suitable for DSMC and NS code validation. These
examples provide demanding tests of the numerical schemes and especially of the meshing techniques used, and
very powerful computation resources are required for DSMC in particular. This exercise has been restricted to
laminar nitrogen hypersonic flows in which the only real gas effects are those involving the exchange of internal
energy thus avoiding the uncertainties of turbulence modelling. Important planned extensions to the programme
with these configurations will include high-enthalpy flows in which chemical reactions occur.
This study has highlighted the requirement to define the properties of the test flow just as accurately as the
measurements of the model flow. Published results rarely include information about shortcomings in wind tunnel
flow quality and this can lead to difficulties in interpreting the results from a validation exercise. In this paper issues
will be discussed relating to the flow characterisation, and in particular the determination of the vibrational
temperature in the nitrogen flow.
CONFIGURATIONS TESTED
The two configurations selected for this exercise – namely a hollow cylinder/flare body and a bi-conic body
(sometimes referred to a cone/cone body) are shown in Figure 1. The latter has three variant geometries with either
an aerodynamically sharp tip or one of two spherical blunted noses being fitted (not shown). The tests were
conducted at Mach numbers in the range of 9.5 to 12.5 and for unit Reynolds numbers going from 1.19E+05/m to
1.46E+06/m. Full details of the model geometries and revised test conditions are given in Holden et al [3] and in an
FIGURE 1. The two configurations tested: a) The hollow cylinder/flare model; b) the bi-conic model.
418
extensive database called CUBDAT [4]. Both configurations are axisymmetric and so avoid the experimental
uncertainties that occur due to the finite span of the wind tunnel models with supposedly “two-dimensional” tests
on, for example, flat plates and wedges. The non-existence of three-dimensional disturbances and unsteadiness was
verified by using very fast responding gauging, some distributed azimuthally around the models, and by using
thermal and high speed Schlieren imaging.
In every case tested, the flow separates from the fore-body and reattaches on the aft cone or flare producing a recirculating region roughly centred on the corner. For one set of conditions the flow is quite close to the incipient
separation case with only a very small separated region. The extent of the recirculation region varies considerably
with changes in the Reynolds number, Mach number and in the case of the bi-conic model, nose bluntness. For both
configurations the shock wave generated by the separation interacts with the leading edge shock to form a complex
shock/shock interaction close to the reattachment point on the aft cone. This is more intense and complex with the
bi-conic model and hence this geometry posed a more challenging problem for the CFD codes. Each model was
fitted with a large number of heat transfer and pressure gauges from which the pressure coefficient, Cp, and Stanton
numbers, St, that form the basis of the CUBDAT dataset were inferred. Schlieren pictures revealed the flow
structure and these were compared with the density plots that were requested from the CFD participants. The wind
tunnel tests for this study were conducted very carefully and the pressure and heat transfer profile are generally very
smooth.
The cylinder/flare body is the axisymmetric equivalent of a flat plate/ramp configuration and the flows are
similar except for the effects of body curvature. The leading edge is sharp and the flow entering the hollow body,
being supersonic, does not interact with the exterior flow to any measurable degree except for minimal forward
molecular scattering from the under surface near the tip. The angle of the flare is 30º and the model was tested with
two flare lengths. The first shorter one matched a model tested in the ONERA R5Ch wind tunnel. With this model,
under most conditions the reattaching flow impinged very close to the lip of the flare giving rise to a strong
possibility of distortion of the reattachment process due to downstream interference from the model base flow and
support. This is not seen to be a well defined test flow and for this reason, a longer flare model was constructed,
which gave an opportunity for the flow to return nearly to the undisturbed cone conditions before leaving the flare.
The pressure and heat transfer could be predicted with reasonable certainty here and this provided a valuable check
that the flow was in no case transitional or turbulent.
For the bi-conic body the fore and aft cone angles were 25º and 55º respectively. Earlier tests on a 65° aft cone
proved to be unsatisfactory as the flow was not steady. However, with the lower cone angles of 55° and 60° steady
flows were realised as witnessed by the time history from the gauges on these models and from high-speed Schlieren
videos. No unsteadiness was evident in any of the flows with the 55° flare. Again the recovery of the pressure and
heat transfer towards the simple attached flow laminar values at the downstream end of the second cone was
convincing evidence that the flows were not transitional or turbulent even after the separation.
RESULTS
Figure 2 shows the pressure and heat transfer coefficient results produced by Moss [3] using DSMC for the
hollow
1.50
0.080
St
Cp
0.020
0.80
St
Cp
0.060
0.60
0.040
0.40
0.020
0.20
0.000
2.5
0.00
1.00
0.010
0.50
0.00
0
0.5
1
1.5
2
X/L
0.000
0.3
0.8
X/L
1.3
FIGURE 2, a & b. Pressure and heat transfer results for the hollow cylinder/flare. Run 9; M=12.49, Re#/m =
3.58E+05; Moss DSMC results: Pressure: ■ measured, ▬DSMC; Heat transfer: ○ measured, - - - DSMC.
419
cylinder/flare body for test case 9 (see Table I). In this flow the weak-interaction bow shock coalesces with the
shock generated by the separation (which occurs at about x/L = 0.4) to form a strengthened shock wave. Towards
reattachment a third shock wave or a sequence of compression waves is produced by the outward deflection of the
shear layer that defines the edge of the separation bubble being forced outwards as it approaches the surface of the
flare. This takes place beneath the other shock wave with which it ultimately interacts. This compression process
leads to a sharp rise in density and a sudden overshoot in pressure on the flare surface. The consequential thinning
of the boundary layer on the flare gives rise to a corresponding peak in heat transfer at about the same place.
Downstream of this both Cp and St recover rapidly towards the attached laminar flow values on the rearward end of
the flare. A similar process of shock interaction occurs on the bi-conic body but the process is more intense as
illustrated in the composite computed density contours and Schlieren image shown in Figure 3. Full details of the
results of the experimental studies have been presented at the 2001[5] and 2002 [3] AIAA Reno meetings together
with a number of computed solutions. In general, the comparisons between Navier Stokes computations and
experiment have been excellent - possibly better than expected - with the size of the separated region and the
magnitude of the heat transfer and pressure distributions through the separated and even in the reattachment region
being in good agreement with the experimental measurements. Comparisons between the computed density contours
and the pattern of complex shock waves revealed by Schlieren photographs are in excellent agreement as illustrated
by the example shown in Figure 3. Heat transfer for the same example, illustrated in Figure 4, shows that the
measured distributions and Candler’s computed solutions are very similar except for the region ahead of separation
(x/L about 0.6) if slip and non-equilibrium effects are included. Of particular note is the precision with which the
point of separation and the details of the complex reattaching flow are modelled. The success of this computed
solution is typical of several others’ NS results for both bodies covering different test conditions.
In contrast, except for the forebody flows, the DSMC method has not yielded as good agreement as the NavierStokes method. This is typified by results already presented from Moss (Figure 2) which is generally representative
of the DSMC solutions. The primary shortcomings appear to be linked to under-predicting the size of the separation
region generally by a factor of around 1.5 to 2. The separation point can be identified in Figure 2b by the sharp rise
in pressure that occurs at about x/L = 0.5 in the experiment whereas a value of 0.8 is computed. L is the length of
0.14
Experiment
Nominal
Noneq with slip
0.12
0.10
0.08
St
0.06
0.04
0.02
0.00
0.0
FIGURE 3. Composite picture of
Bi-conic model showing Candler’s
NS density contours superimposed on
Schlieren picture. Run 35
0.2
0.4
0.6
0.8
1.0
x/L
1.2
1.4
1.6
1.8
FIGURE 4. Heat transfer results for the bi-conic model
including Candler’s NS results with and without vibrational nonequilibrium and slip included.; Run 35, M=11.3, Re#/m=
1.56E+5; Pressure: ■ Measured, ▬ NS; Heat transfer: ○
Measured, - - - N-S, uncorrected, ▬ ▬ N-S with nonequilibrium and surface slip.
420
FIGURE 5. Photograph showing the hollow cylinder for body mounted in
the wind tunnel with an array of calibration probes.
the forebody for both configurations. The weaker performance of DSMC may, in part, be attributed to the
relatively high density of these test cases and to the difficulties in defining suitably fine and well proportioned
meshes for flowfield in the reattachment region. But for these complex recirculating flows, there are questions on
the level of convergence of the solutions and whether sufficient time is allowed in the calculations for the flow to
circulate around the separation bubble. Premature apparent convergence may be a consequence of the settling of the
initial pressure disturbances within this region.
Test Flow Characterisation
When reporting at the 2001 AIAA meeting, it was noted that in the Navier-Stokes solutions the heating rates
over the forebody ahead of separation were consistently overpredicted. Similar observations were be made for the
hollow cylinder flare model where the flow is more rarefied. Questions were raised concerning the evaluation of the
vibrational temperature in the freestream, and this has led to a second experimental program and computations to
select the most effective techniques to determine the properties of the freestream in the presence of vibrational
nonequilibrium. This has led to a redefinition of the incident flows for this series of test cases.
An uncertainty that arises in the validation of codes is the precision to which the test conditions are specified.
This is of particular note in hypersonic experiments where a hot high-pressure gas is rapidly expanded to produce
the test flow. Non-ideal gas behaviour arising from intermolecular force effects, chemical reactions, and vibrational
excitation affect the free-stream conditions and exacerbate the difficulty in determining the flow characteristics. In
the present experiments flow characterization is done from a knowledge of the nozzle reservoir conditions (derived
from precise measurements of the shock speed before reflection in the shock tube) and from data from an array of
pressure and heat transfer probes mounted in the test section (see Figure 5). These data are used to determine the
free-stream conditions using a real gas code. Because the effective area ratio of the nozzle is not known due to
nozzle wall boundary layer displacement, the code is run to the point where the computed pitot pressure matches the
measured values. This approach has been used successfully in the past, but usually for air for which the vibrational
relaxations processes are faster than in nitrogen due to the role played by oxygen. The tests are also generally
performed at higher pressures than in the present set of experiments.
To understand better the effect of vibrational nonequilibrium, we performed a series of CFD simulations of the
nozzle flows, solving the axisymmetric compressible Navier-Stokes equations, along with a vibrational energy
conservation equation, see Candler’s et al [6]. We allow finite-rate vibrational energy relaxation using Millikan and
White rates. Nitrogen dissociation and recombination is included with standard rate constants, but under the present
conditions the dissociation levels are so low that they can be neglected. The Blottner curve-fits for viscosity are
used, and because the nozzle wall boundary layer is turbulent, we use the Baldwin-Lomax turbulence model. A
second-order accurate modified Steger-Warming flux vector splitting approaches is used, along with the DataParallel Line-Relaxation method. The nozzle grids are constructed directly from the CAD files of the nozzle
contours. Typical grids use 1786 points in the axial direction and 128 points in the surface normal direction, with
stretching at the surface. Because the nozzle throats are elongated, making the specification of the sonic line
421
0.7
Pitot Pressure (psi)
0.6
0.5
0.4
0.3
0.2
Experiment
Computed
0.1
0
-20
-10
0
y (inch)
10
20
FIGURE 6. Pitot pressure profile computed and
measured at nozzle exit.
FIGURE 7. Flush instrumentation on hollow
cylinder model.
impractical, the calculations were started from the subsonic flow settling chamber adjusting the initial conditions so
that the total enthalpy is set correctly in the computed flow. Initial simulations were discouraging with the
centreline pitot pressure significantly over-predicted due to the turbulence model exaggerating the wall boundary
layer displacement thickness in the nozzle exit plane. The rapid expansion in the nozzle is highly stabilizing and can
lead to relaminarisation of the wall boundary layers. We therefore performed parametric studies with
“relaminarisation” assumed to occur at different locations to simulate the observed boundary layer growth. In this
way a close match with the measured pitot pressure profiles and displacement thickness at the nozzle exit was
achieved. For example, consider Figure 6 which plots the measured pitot pressures for the Run 35 calibration run
with the results of the simulation. It is argued that the conditions in essentially inviscid central core of the flow will
be reliably predicted. Table 1 summarizes the redefined test conditions obtained from the non-equilibrium
calculations. The previously published values [5] are also shown. Note that in contrast to air flows there is
substantial vibrational freezing, resulting in a lower velocity and higher density in the test section.
Vibrational relaxation rates can be affected by the presence of water vapour which could be introduced by outgassing from the wind tunnel walls in the period prior to a test for the low Reynolds number, low pressure tests.
Experiments have been performed, measuring the forebody heat transfer with different levels of water vapour
saturation. These have been conducted under conditions ranging from holding the facility under vacuum for several
days prior to filling with high purity nitrogen, to deliberately adding water to the test gas. It is concluded from these
sensitivity tests that there is no measurable effect due to presence of water vapour in any of the reported results.
Hollow Cylinder/Flare and Bi-Conic Model Tests
As part of this study to validate the freestream properties, measurements have been made over the two fore
bodies used in the code validation study, namely the hollow cylinder detached from its flare and the 25° cone
without the second conical section of the model. Measurements were made over a large range of stagnation
temperature and Reynolds numbers conditions to obtain conditions that varied from rarefied to continuum over the
forward part of the body, and where the degree of vibrational nonequilibrium varied from being excited (nominally
T0 = 2,800°K) as in the earlier experiments to unexcited (nominally T0 = 1,100°K) in colder flows.
Pressure measurements for these tests for some stations have been performed with flush mounted transducers
(see Figure 7) to avoid the difficult-to-quantify orifice effect correction that is required for rarefied flows.
Correlations with these measurements have been used to correct those values made with conventional tappings
elsewhere on the model where necessary. The heat transfer and pressure measurements made in these studies were
compared with detailed Navier-Stokes calculations and correlated in terms of the parameters used in simple
predictive techniques. Sample results from these correlations for the less dense flows are illustrated in figures 8 and
9 for the cylinder and cone respectively. The heat transfer coefficient has been multiplied by M3 to remove Mach
number dependence and plotted against χ . It is evident that the data for the hollow cylinder generally correlates
well and agrees with Candler’s NS computations when they include vibrational relaxation and slip and Cheng’s
corrected theory. The cone data also correlate well, falling a little below the theoretical lines with increasing rare-
422
Run 26 (Re/ft = 0.8E5, To = 5000)
Run 43 (Re/ft = 0.7E5, To = 5000)
Run 25 (Re/ft = 0.6E5, To = 5000)
20
Run 12 (Re/ft = 3.0E5, To = 5000)
1
Run 11 (Re/ft = 1.6E5, To = 5000)
Run 26 (Re/ft = 0.8E5, To = 5000)
Run 38 (Re/ft = 3.5E5, To = 2600)
M^3Ch ((M^3*q)/(Rhoe*Ue*(Ho-Hw)))
M^3Ch ((M^3*q)/(Rhoe*Ue*(Ho-Hw)))
Run 40 (Re/ft = 4.4E5, To = 3300)
Run 35 (Re/ft = 3.5E5, To = 2400)
15
Cheng (Weak)
Cheng (Weak) (Prandtl Number
Correction)
Candler (Run 11)
10
5
Run 25 (Re/ft = 0.6E5, To = 5000)
0.8
Run 40 (Re/ft = 4.4E5, To = 3300)
Run 38 (Re/ft = 3.5E5, To = 2600)
Run 35 (Re/ft = 3.5E5, To = 2400)
0.6
Cheng (Weak)
Candler (Run 35)
0.4
0.2
0
0
0
10
20
0
30
M^3(sqrtC(star))/(sqrt(Re))
FIGURE 8 Correlation of non-dimensional heat
transfer for cylinder
0.5
1
M^3(sqrtC(star))/(sqrt(Re))
1.5
FIGURE 9 Correlation of non-dimensional heat transfer
for 25° fore-cone.
action. Good correlations are also obtained for the pressures.
New computations for the bi-conic body have been produced by Candler for the redefined test conditions for
test 35, including vibrational non-equilibrium in the approaching stream and around the body and surface slip. The
results are added to Figure 6 which illustrates a very close agreement with experiment for the entire flow. This
appears to confirm that the overprediction seen before in the NS solutions submitted for the validation exercise
appears to be attributable to neglecting these two effects.
During the course of this study techniques have been developed to make precise measurements under more
rarefied flow conditions than those originally defined in CUBDAT and used for the 2001 AIAA paper. These new
standard test conditions are presented in Table II, and it is evident that they are at about one half and one third of the
lowest previous density. It is planned to repeat the experiments on both configurations for these conditions before
January 2003.
In DSMC computations, almost universally one of the two popular extensions to the basic hard sphere collision
model – namely Bird’s [7] Variable Hard Sphere (VHS) model and the Variable Soft Sphere (VSS) model devised
more recently by Koura and Matsumoto [8] - are used. In both, the viscosity of the simulated gas is matched to that
of its real counterpart. The addition of anisotropic scattering in the VSS model permits the diffusion coefficient to
be correctly reproduced but neither model addresses the issue of matching explicitly the thermal conductivity. It is
generally accepted that this is not a problem, but the availability of high quality heat transfer data in the present
experiment and the new accurate free stream characterisation, provides a good opportunity to assess the
effectiveness of these models in reproducing the thermal processes with fresh DSMC computations.
CONCLUSIONS
In this paper the latest developments our discussed in the CFD code validation exercise to examine the
effectiveness of DSMC and Navier-Stokes in predicting complex flows involving shock/shock and shock/boundary
layer interactions under laminar hypersonic conditions. This exercise has been an important component of the
exercise which has been promoted by RTO Working Group 10. It has already been demonstrated in this exercise
that Navier-Stokes codes are capable of capturing with considerable accuracy the complex viscous/inviscid
interaction regions developed over the hollow cylinder/flare and bi-conic configurations under laminar hypersonic
non-reacting circumstances.
In contrast it has been evident that DSMC codes have not been successful in predicting the scale of the
separation region which has affected the peak heating and pressure levels in the reattachment region have not been
calculated very accurately. The test cases are at the upper end of viability for DSMC and the indifferent
performance may well be attributable to difficulties in resolving the flow adequately with the computational mesh,
the results may not be fully converged and hence they may not be truly representative of DSMC. The present paper
423
addresses the question of flow characterisation and in particular the level of vibrational excitation in the flows.
From new non-equilibrium computations for the nozzle flow, revised test conditions have been defined which
include differing values for the static and vibrational temperatures for each test case. Further test have also been
performed on the fore-bodies of both configurations and the data for these is included in the updated CUBDAT data
base. It has been shown that with the new test conditions the heat transfer and pressure data from these tests
correlates well with the rarefaction parameter χ . It is hoped that fresh computations will be made with both DSMC
and NS codes as a final stage of this validation exercise, which will be concluded in 2003. The inclusion of new
computations for the two new low density test conditions will ease the difficulty of performing these and will be of
particular value to the DSMC community in assessing the methods available to predict complex viscous/inviscid
interaction flows.
Table I. Revised Test Conditions. Cy/F – Cylinder Flare model; C/C – Bi-conic model
Test #
Cy/F 8
Cy/F 9
Cy/F 11
Cy/F 14
C/C 24
C/C 28
C/C 31
C/C 35
M
11.36
11.40
11.10
9.50
9.48
9.59
11.30
11.30
Re#/m
Ref 4
3.51E+5
2.59E+5
1.56E+5
1.84E+5
2.26E+5
1.39E+5
1.42E+5
1.56E+5
M
12.46
12.49
12.23
10.30
10.38
10.50
12.43
12.49
ρ kg/m3
1.314E-3
9.326E-4
5.848E-4
7.471E-4
1.371E-3
1.430E-3
5.668E-4
6.100E-4
U m/s
Re#/m
T°K
REDEFINED CONDITIONS
2.548E+3
4.79E+5
1.006E+2
2.441E+3
3.58E+5
9.222E+1
2.485E+3
2.05E+5
9.870E+1
2.327E+3
2.42E+5
1.141E+2
2.612E+3
3.41E+5
1.522E+2
2.538E+3
2.55E+3
1.400E+2
2.621E+3
1.99E+5
1.072E+2
2.571E+3
2.21E+5
9.940E+1
Tvib °K
2.480E+3
2.490E+3
2.497E+3
2.269E+3
2.640E+3
2.589E+3
2.770E+3
2.768E+3
Table II. New Low Density Test Conditions for both configurations.
Test #
1319
1318
U∞ m/s
2.13E+07
2.10E+03
P∞ Pa
4.86E+00
2.74E+00
Ho m2/s2
2.58E+06
2.52E+06
ρ∞ kg/m
3.60E-04
1.92E-04
Re/m
2.67E+05
1.30E+05
T °K
48.18
45.23
Tvib °K
2.17E+03
2.16E+03
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