Simulation of Hypersonic Laminar
Flow Validation Experiments
Michael A. Gallis, Christopher J. Roy,
Timothy J. Bartel and Jeffrey L. Payne
Sandia National Laboratories
Albuquerque NM 87185, USA
Abstract. Numerical simulations of the flowfield of a Mach 11, Kn 0.02 laminar flow of nitrogen over the forecone of
a spherically blunted 25/55 deg. biconic are presented. The numerical simulations are performed with a Direct
Simulation Monte Carlo (DSMC) and a Navier-Stokes (NS) code. The discrepancies between that measured and
calculated surface quantities and their sensitivity to the free stream conditions are examined.
INTRODUCTION
To promote the understanding of separated laminar flows, NATO, through the Research Technology
Organization (RTO), organized a combined experimental and computational study of the flow around a 25/55
degree axisymmetric biconic shape. The test conditions were chosen so that numerical reproduction of the
experimental results would be feasible. To reduce the complexity of the test case nitrogen was used as the gas of the
experiment. The flow was laminar and a recirculation zone formed at the intersection of the two cones. Surface
pressure and heat flux measurements were made at the LENS II CUBRC hypersonic wind tunnel. Experiments were
conducted with a sharp forecone and a number of spherically blunted ones [1].
The results of this exercise were presented at the 39th AIAA Aerospace Sciences Meeting as part of a blind
comparison validation exercise. Despite the general good qualitative agreement there were a number of
discrepancies between the calculated and measured results that were observed. The first problem was the failure of
DSMC codes to capture the size of the recirculation zone. This problem was studied by Roy et al [2] and Gimelshein
et al [3]. They independently concluded that the density of the flow in the area of the circulation zone made the
application of the DSMC method computationally impossible even on the most modern computational platforms. To
confirm this, Roy et al. [2] conducted a NS-DSMC comparison for a reduced density case, where the application of
the DSMC method was practical and all the requirements for its application were met. The two codes were found to
be in agreement, providing confidence in the ability of the DSMC method to deal successfully with hypersonic
recirculation flows.
A second problem witnessed was the lack of agreement between the measured and calculated surface pressure
and heat transfer. Figures 1 and 2, where the measured and calculated pressure and heat flux along the biconic are
respectively presented for run 31 (see table 1 for conditions), indicate that while the NS and DSMC solutions were
in agreement between themselves on the attached forecone, there was as much as 20% difference between the
measurements and the calculations for the pressure and 25%-35% for the surface heat flux.
In their original work Roy et al. [4] pointed out the sensitivity of the flow field and that of the surface properties
to the free stream conditions and in particular to the vibrational excitation of the flow. The minimum amount of
vibrational excitation in the flow is found by assuming thermal equilibrium in the test section, i.e., the vibrational
temperature is equal to the translational temperature. The maximum possible vibrational excitation level would
occur if the vibrational temperature was frozen at the stagnation temperature in the test section. While the true state
in the tunnel is somewhere in between these two limits (equilibrium and frozen flow), experience suggests that the
vibrational temperature could be closer to the stagnation temperature than to the translational temperature at the test
section. In a recent study of vibrational relaxation in nozzles Bird [5] verified this assumption. In his analysis Bird
[5] confirmed that vibrational non-equilibrium was inevitable in such expansions.
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
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To investigate the sensitivity of the flow to non-equilibrium freestream conditions and variations to the
freestream Roy et al [4] presented a sensitivity analysis where:
1) the total energy content of the flow was reduced by 5%. According to Harvey et al. [6] the maximum error in
the total enthalpy of free stream did not exceed 5%.
2) the vibrational temperature was assumed to be frozen at the reservoir temperature, thus leading to a reduction in
the translational energy of the flow.
10 6
8000
7000
6000
5000
Harvey et al.
DSMC
Navier-Stokes
4000
Harvey et al.
DSMC
Navier-Stokes
2
q (W/m )
2
p (N/m )
3000
2000
10 5
1000
Biconic
Case B55D , R un 31
Surface Heat F lux
Biconic
Case B55D, R un 31
Surface Pressure
0
0.05
0.1
10 4
0.15
0
0.05
x (m)
Figure 1. Biconic pressure distribution, run 31
7000
0.15
Figure 2. Biconic heat flux distribution, run 31
Base line
Minimum Ene rgy
Vib. Excite d
1.5E+06
Baseline
M inimum Energy
V ib. Excited
6000
0.1
x (m)
DSM C
1.25E+06
DSM C
2
q (W/m )
2
p (N/m )
5000
4000
3000
750000
500000
2000
250000
1000
0
1E+06
0.05
0.1
0
0.15
x (m)
0.05
0.1
0.15
x (m)
Figure 3. Biconic, DSMC pressure sensitivity study, run 31
Figure 4. Biconic, DSMC heat flux sensitivity study,
run 31
Figures 3 and 4 present part of the results of this sensitivity analysis using the DSMC method (for the full
analysis see Ref 2) for the case presented in Figures 1 and 2 (run 31). The effect of the reduced total energy was, as
expected, lower heating rates for the surface by about 5%. The effect of the increased vibrational energy was more
pronounced. The increase of the energy of the vibrational mode and the corresponding reduction of the directed
kinetic energy (i.e., velocity) of the gas resulted in a reduction of the pressure and heat flux to the surface by about
20-25% and an increase in the size of the recirculation zone. Similar results were found by the NS code. This
reduction could account for the difference between the measured and calculated surface properties. It must be noted
that vibrational temperature freezing at the plenum temperature is an extreme case that cannot be realized, but it
serves as a limiting case for the effect of the partial excitation of the vibrational mode on the flow.
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The suggestion that the vibrational mode of the flow was only partially excited was later confirmed by Candler
et al. [7] and Holden et al. [8] by simulating the nozzle flow. They also provided modified free stream conditions for
some of the validation workshop cases. Although these simulations helped shed more light onto the problem of the
freestream conditions the findings were not conclusive since a major parameter of the simulation was the point of relaminarization of the flow on the nozzle walls which was chosen to provide agreement with experimental pitot data.
The aim of this work is to present a combined DSMC-NS investigation of the problem of the freestream
conditions and their influence on the measured surface properties. The original and modified freestream conditions
are used and simulations of the forecone of the biconic shape are performed. Since modeling the whole domain with
DSMC turned out not to be feasible, simulations were performed on the forecone only to allow adequate resolution
in this region. For the NS simulations, the computational mesh used for the original simulations was simply
truncated at the cone-cone juncture. The DSMC mesh was truncated to include all the measurement locations on the
attached forecone. In the current study, finer grids and longer execution times for the DSMC code were afforded
since only a fraction of the body was simulated.
DSMC AND NAVIER-STOKES CODES USED
The DSMC code used was the Sandia DSMC code Icarus by Bartel et al [9]. The Icarus code was written for
massively parallel computing environments to accommodate the extreme computational requirements of the DSMC
method. The DSMC simulations in the present work were performed on the 4500 node (9000 processor) ASCI Red
platform (330 MHz Pentium II processors). The runs used 512 processors for 24 hrs. The Variable Hard Sphere
molecular model was used and energy exchange was modeled with the Borgnakke and Larsen model. The DSMC
code employs a diffuse reflection condition with 100% thermal accommodation. The effect of changing the thermal
accommodation coefficients has been shown to have a negligible influence on the results, at least for flows without
freestream vibrational excitation [10].
The SACCARA code was used to solve the Navier-Stokes equations for conservation of mass, momentum,
energy, and vibrational energy in axisymmetric form. The governing equations are discretized using a cell-centered
finite-volume approach. The convective fluxes at the interface are calculated using the Steger-Warming [11] flux
vector splitting scheme. Second-order reconstructions of the interface fluxes are obtained via MUSCL extrapolation
[12]. A flux limiter is employed which reduces to first order in regions of large second derivatives of pressure and
temperature. This limiting is used to prevent oscillations in the flow properties at shock discontinuities. The viscous
terms are discretized using central differences. At solid walls, the Navier-Stokes simulations use standard no-slip
wall boundary conditions with a fixed wall temperature of 297 K. For cases without freestream vibrational
excitation, the no-slip boundary condition used in the Navier-Stokes code is expected to be valid for blunted cones at
the experimental conditions [1]. However, the use of this boundary condition may lead to inaccuracies for the case
of the sharp cone due to Knudsen number effects.
RESULTS AND DISCUSSION: BLUNT CONE
The preliminary comparisons with experimental data for Run 31 shown in Figure 1 and 2 (see also Ref. 2)
indicated significant discrepancies for the attached flow over the forecone. These discrepancies are surprising
considering that the flow over the blunted forecone is laminar and undergoes negligible chemical reactions [2]. To
better resolve these discrepancies, additional DSMC and Navier-Stokes simulations were performed on the forecone
(omitting the aft cone) in order to allow higher resolution in this region. For the Navier-Stokes simulations, the
biconic mesh extends to the cone-cone juncture, while the DSMC mesh is truncated at the last experimental data
location on the forecone where the flow is still attached. For these truncated flow domains, very fine grids and
extended execution times are possible. The freestream conditions are those given in Table 1.
Two sets of calculations were performed: one with the freestream vibrational temperature assumed equal to the
translational and rotational temperatures (nominal conditions) and a second one where the free stream vibrational
temperature was assumed to be almost frozen to the reservoir temperature (non-equilibrium conditions). The free
stream conditions for run 35 are the ones proposed by Holden et al [7] and Candler et al [8] while for run 31 the free
stream conditions were calculated for this work in a similar manner.
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TABLE 1. Free stream conditions.
31
Conditions \ Run
Temperature (K)
Vibrational
Temperature (K)
Density (kg/m3 )
Velocity (m/s)
Cone Bluntness
Nominal
(equilibrium)
144.44
144.44
35
Nonequilibrium
107.22
2772.22
Nominal
(equilibrium)
138.9
138.9
Nonequilibrium
99.38
2768
0.5515×10-3
2713
0.6120×10-3
2571
0.5128×10-3
0.56735×10-3
2764.5
2621.3
0.25” spherically blunted
Sharp
To establish the appropriate grid resolution, time step and number of mo lecule simulators (for DSMC) for the
simulations a numerical accuracy analysis was performed. This analysis was performed for the equilibrium
conditions but the same simulation parameters are expected to be valid for the non-equilibrium ones.
Numerical Accuracy: Navier-Stokes
Iterative Convergence
Iterative convergence was determined for the forecone simulation by monitoring the L2 norms of the residuals
for the governing equations (mass, momentum, energy, and vibrational energy). All norms were reduced by ten
orders of magnitude except for the vibrational energy equation, which exhibited an eight order of magnitude drop.
These reductions in the L2 norms give confidence that the iterative convergence errors will be small relative to the
grid convergence errors.
Grid Convergence
Three meshes were employed in order to assess the numerical errors due to incomplete grid convergence for the
forecone simulations. The finest mesh employed was 512×512 cells. Medium and coarse meshes of 256×256 and
128×128 cells, respectively, were created by eliminating every other grid line in each direction. An estimate of the
exact solution is obtained via first-order Richardson extrapolation. The three meshes appear to be nearly asymptotic,
and the magnitude of the errors is generally small. The maximum errors in surface heat flux are approximately 0.5%
for the finest mesh. Employing a factor of safety of two, the grid convergence errors for the Navier-Stokes
simulations of the forecone on the fine mesh are estimated to be under 1% for both surface pressure and heat flux.
Numerical Accuracy: DSMC
A number of DSMC runs were used to judge the numerical accuracy of the forecone simulations. These runs are
summarized below in Table 2. A time step of 4×10-6 ms was used in each case.
Statistical Steady State
Two simulations were performed, one assuming steady state after 2 ms, and the other after 4 ms (runs 4 and 5,
respectively, in Table 2). The two simulations were in good agreement indicating that steady state was achieved
prior to 2 ms.
Grid Convergence
Four different grids were used for the simulations using 20,000 to 625,000 cells (runs 1, 2, 4, and 6,
respectively). In all cases the cells were clustered towards the surface of the body where the density is higher. The
last two simulations of 312,000 and 625,000 cells were in good agreement, indicating that grid resolved results were
achieved. In each refinement step from 20,000 to 312,000 cells, the number of cells was increased by two in each
direction. For the last refinement, only the number of cells in the direction normal to the surface was doubled.
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Particle Convergence
To investigate the influence of the number of molecule simulators on the solution, the average number of
particles per cell was doubled from 12.5/cell (Run 2) to 25/cell (Run 3) for the 80,000 cell case. The results showed
no significant difference between the two runs, thus supporting the assertion that an average of 10-12 simulators per
cell is enough to produce statistically meaningful results.
TABLE 2. DSMC run summary for the forecone.
Run
1
2
3
4
5
6
7
Particles
per cell
30
12.5
25
10
10
10
10
# of
cells
20,000
80,000
80,000
312,000
312,000
625,000
625,000
Moves to
Steady state
50,000
50,000
50,000
50,000
100,000
50,000
50,000
Moves to
Final state
120,000
120,000
120,000
120,000
120,000
120,000
100,000
Conditions
Nominal
Nominal
Nominal
Nominal
Nominal
Nominal
Non-equilibrium
Results
Presented in figure 5 is a comparison of the temperature contours for the Navier-Stokes simulation using
512×512 cells and the DSMC prediction of Run 6. The temperature reaches a maximum of 4000 K near the
stagnation point and then gradually falls off along the body. The simulations are in good qualitative agreement, with
the DSMC approach predicting a more diffuse temperature shock as expected. In the area of the shock precursor, the
molecular velocity distribution is a bimodal combination of the freestream Maxwellian and the distribution
emanating from the shock layer. Under such conditions, the definition of temperature in the traditional sense is no
longer valid.
0.06
0.05
0.04
T (K)
NavierStokes
0.03
4000
3750
3500
3250
3000
2750
2500
2250
2000
1750
1500
1250
1000
750
500
0.02
y (m)
0.01
0
-0.01
-0.02
-0.03
D SMC
-0.04
-0.05
F orecone Only
-0.06
0
0.05
0.1
x (m)
Figure 5. Temperature contour lines, run 31
Figure 6 presents the surface pressure comparison between the experimental data and the DSMC and NavierStokes predictions for both equilibrium and non-equilibrium conditions. For run 6 (equilibrium conditions), the
pressure distribution for both simulations overpredict the experimental data by between 5% and 15%, although,
qualitatively they are in good agreement. The agreement between the two numerical methods is quite good. Using
the non-equilibrium free stream conditions (Run 7) results in a slight reduction in the surface pressure by
approximately 2-3%, improving the agreement with the experimental data. It is interesting to note that the agreement
between the experimental and the numerical data improves as we move downstream along the body.
437
The surface heat flux distributions are presented in Fig. 7. The comparison between the two simulation
approaches is good (for both nominal and non-equilibrium conditions), while there are significant differences
between the predictions and the experimental results. Although not shown, the heat transfer achieves a maximum at
the stagnation point of roughly 1.2×106 W / m2 in both simulations, drops rapidly after the expansion around the
spherical nose region, and then drops gradually along the cone to about 1.6×105 W/m2 . Figure 7 indicates that for the
nominal conditions both simulation methodologies overpredict the heating by 30% at the first data station. The
predictions become slightly better downstream until the last experimental data station in the attached flow region,
where the predictions are 20% high. This improvement in the agreement could be attributed to the fact that as the
flow moves downstream the stagnation point non-equilibrium effects are absorbed by the effect of collisions, i.e.,
the flow becomes less sensitive to the stagnation point flow conditions.
1500
500000
H arvey et al.
D SM C (Nom inal)
N S (nominal)
D SM C (non-equilibrium)
N S (non-equilibrium)
1000
H arvey et al.
D SM C (nominal)
N S (nominal)
D SM C (non-equilibrium)
N S (non-equilibrium)
400000
2
2
p (N /m )
q (W/m )
300000
200000
500
100000
0
0
0.01
0.02
0.03
0.04
0.05
0
0.06
0
0.01
0.02
x (m)
0.03
0.04
0.05
0.06
x (m)
Figure 6. Pressure distribution, run 31
Figure 7. Heat flux distribution, run 31
Applying non-equilibrium conditions (Run 7) results in a marked improvement to the calculated heat flux to the
surface. The difference between the calculated and measured heat flux drops to almost a third of the previous value.
Unlike the equilibrium case there is a small difference between the DSMC and CFD predictions. This difference can
be attributed to the lack of slip modeling in the NS code. The effect of slip becomes more pronounced in the case
where vibrational non-equilibrium exists in the free stream.
From the above comparisons we conclude that the inclusion of non-equilibrium effects in the free stream
changed significantly the surface predictions. However, it must be pointed out that the non-equilibrium free stream
conditions of Run 7 are calculated and not measured. The vibrational excitation of the gas at the test section was not
addressed in the original experimental description Holden et al [1].
The demonstrated numerical accuracy of, and the agreement between, the DSMC and Navier-Stokes simu lation
approaches provide strong evidence for the presence of a bias error in the experimental data of reference 14 or in the
calculated free stream conditions. The same observation seems to be true for the free stream conditions that have
included the non-equilibrium. Another major parameter that has not been addressed adequately is the point of relaminarization on the nozzle. The location of this point was chosen in references 7 and 8 to provide a good match
with the experimental data for the free stream and can play an important role in the specification of the freestream.
RESULTS AND DISCUSSION: SHARP CONE
An abbreviated summary of calculations for run 35 are also presented here to indicate that the differences
observed here are not unique to this case, Run 31, but are repeated for all cases of the validation workshop. The
conditions for Run 35 appear in Table 1. A numerical analysis similar to the one for run 31 was performed using
grids of 20,000, 40,0000 and 200,000 cells. In agreement with Moss [12], it was concluded that a 40,000 cell grid
would be sufficient. That said, all simulations presented here were performed with a 200,000 cell grid and a time
step of 2×10-9 s and an average of 30 molecule simulators per cell. Much of this analysis was based on the work of
438
Moss [10] and it was concluded that the chosen configuration meets all criteria for the application of the DSMC
method.
Figure 8 presents the temperature contour lines for both the DSMC and the Navier-Stokes code. As in the
previous studies in case 31 the two runs are in qualitative agreement. The DSMC simulation presents a more diffuse
shock layer especially in the area of the sharp nose. This is to be expected since the sharp nose is an area where the
applicability of the continuum approach is questionable.
Figure 9 presents the measured and calculated surface pressure. DSMC simulations of the nominal and nonequilibrium conditions are presented as well as Navier-Stokes predictions of the non-equilibrium conditions in
comparison with the experimental data. The pressure drops from a value of 1,100 N/m2 for the DSMC and 1,200
N/m2 for the NS code on the nose tip to an almost constant pressure of about 800 N/m2 . We note that, similar to the
previous case, the non-equilibrium free stream conditions lead to a reduced surface pressure, which is in better
agreement with the experimental data.
0.03
NS
2300
2150
2000
1850
1700
1550
1400
1250
1100
950
800
650
500
350
200
0.02
y (m)
0.01
0
-0.01
-0.02
-0.03
-0.04
DSMC
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
x (m)
Figure 8. Temperature contours, run 35
Figure 10 presents the heat flux to the surface for the same conditions. Again the maximum heat flux appears at
the nose tip and then the heat flux falls off gradually for all three simulations to about 150,000 W/m2 . We note that
in this case the non-equilibrium NS prediction is higher than the DSMC one. This difference is attributed to the noslip boundary condition of the NS code.
1000000
1200
1100
DSMC (non-equilibrium)
Harvey et al.
DSMC (nominal)
NS (non-equilibrium)
1000
750000
900
q (W/m )
700
2
2
p (N/m )
800
600
500
DSMC (non-equilibrium)
NS (non-equilibrium)
Harvey et al.
DSMC (nominal)
400
300
500000
250000
200
100
0
0
0.01
0.02
0.03
0.04
0
0.05
x (m)
0
0.01
0.02
0.03
x (m)
Figure 9. Pressure distribution, run 35
Figure 10. Heat flux distribution run 35
439
0.04
0.05
Unlike the spherically blunted case (run 31), slip seems to play a more important role in this case. The simulations
are again in very good overall qualitative agreement with the experimental data. The inclusion of the nonequilibrium effects in the free stream resulted in a reduction of the difference between the experimental and
computational values from 30% to an average difference of about 5%.
For both the pressure and heat transfer profiles we note that the agreement between the simulations and the
experimental data improves as we move along the body towards the back of the cone in a similar fashion as in Run
31.
CONCLUSIONS
In this work the flow around the forecone of a 25/55 deg. biconic configuration is examined as part of an
attempt to validate DSMC and CFD codes. A spherically blunted and a sharp cone case are examined and compared
to experimental data. Numerical investigation indicates that the flow is sensitive to the free stream conditions and in
particular to the vibrational energy of the flow. Slip effects seem to have a significant effect on the surface
properties of the sharp cone, and less of an effect on the spherically blunted case. The issue of the free stream
conditions has not been resolved since no direct measurements are available for comparison; the simulations of the
nozzle flow are subject to parameters that could influence the reliability of the predictions.
It is worth noting that interaction between experimentalists and modelers after the initial blind comparison led
to the improvement in the quality of the data. Further stronger interactions are necessary to improve our
understanding of these flows and our ability to simulate and measure them. Simulation technology (numerical codes
and computational power) have matured to a point where reliable predictive simulations with a high level of
confidence are now possible.
REFERENCES
1. M. Holden, “Experimental Studies of Laminar Separated Flows Induced by Shock Wave/Boundary Layer and Shock/Shock Interaction in
Hypersonic Flows for CFD Validation,” AIAA Paper 2000-0930, Jan. 2000.
2. Roy C.J. Gallis, M.A., Bartel, T.J. Payne, J.L., “Navier-Stokes and DSMC simulations for Hypersonic Laminar Shock-Shock Interaction
Flows,” AIAA 2002-0737, 40th Aerospace Sciences Meeting and Exhibit, Jan. 2002.
3. Gimelshein S.F., Levin, D.A., Markelov, G.N., Kudryavtsev, A.N. and Ivanov, M.S., “Statistical Simulation of Laminar Separation in
Hypersonic Flows: Numerical Challenges,” AIAA 2002-0736, 40 th Aerospace Sciences Meeting and Exhibit, Jan. 2002.
4. Roy, C.J., Bartel, T. J., Gallis, M.A., and Payne, J.L., “DSMC and Navier-Stokes Predictions for Hypersonic Laminar Interacting Flows,”
AIAA Paper 2001- 1030, 39 th Aerospace Sciences and Exhibit Meeting, Jan. 2001.
5. Bird, G.A., “A Criterion for the Breakdown of Vibrational Equilibrium in Expansions”, Physics of Fluids, 14, 5, pp1732-1735, May 2002.
6. Harvey, J.K., Holden, M.S., and Wadhams, T.P., “Code Validation Study of Laminar Shock/Boundary Layer and Shock/Shock Interactions in
Hypersonic Flow, Part B: Comparison with Navier-Stokes and DSMC Solutions,” AIAA Paper 2001-1031, Jan. 2001.
7. Holden M.S., Wadhams T.P. Harvey, J.K., Candler, G.V., “Comparisons between DSMC and Navier-Stokes Solutions and Measurements of
Laminar Shock Wave Boundary Layer Interactions in Hypersonic Flows,” AIAA 2002-0435, 40th Aerospace Sciences Meeting and Exhibit, Jan.
2002.
8. Candler G.V., Nombelis I., Druguet M-C, Boyd I.D., Wang W-L., “CFD Validation for Hypersonic Flight: Hypersonic Double-Cone Flow
Simulations,” AIAA paper 2002-0581, 40 th Aerospace Sciences Meeting and Exhibit, Jan. 2002.
9. Bartel, T. J., Plimpton, S.J., and Gallis, M.A., “Icarus: A 2-D Direct Simulation Monte Carlo (DSMC) Code for Multi-Processor Computers:
User’s Manual Version 10.0,” Sandia National Laboratories, Report. SAND 2001-2901, Albuquerque, NM, October 2001.
10. Moss, J. N., “Hypersonic Flows around a 25 o Sharp Cone,” NASA TM-2001-211253, Dec. 2001.
11. Steger, J.L., and Warming, R.F., “Flux Vector Splitting of the Inviscid Gasdynamic Equations with Applications to Finite Difference
Methods,” Journal of Computational Physics, 40, pp. 263-293, 1981.
12. Van Leer, B., “Towards the Ultimate Conservative Difference Scheme. V. A Second Order Sequel to Godunov’s Method,” Journal of
Computational Physics, 32, 1, pp. 101-136, 1979.
ACKNOWLEDGEMENTS
The authors would like to thank Michael Holden of Calspan-University at Buffalo Research Center and John Harvey
of Imperial College in London for providing the experimental data and for organizing this CFD validation study. We
would also like to thank Jim Keenan and Ed Piekos of Sandia National Laboratories who provided thoughtful
technical reviews of this paper. Sandia is a multiprogram laboratory operated by the Sandia Corporation, a Lockheed
Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000.
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