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I. Introduction
Nowadays, the most promising technology for drastical reduction of times to destination for long-distance (i.e.
Brussels to Sidney) civil flights is hypersonic air-breathing propulsion. At hypersonic flight speeds, ranging from Mach
5 to 8, turbofan engines need to be replaced by advanced propulsion concepts, like scramjet engines.
In the design process of Hypersonic Transport Vehicles, the extensive application of CFD tools is essential. The flow
paths envisaged for future scramjet engines will be highly three-dimensional and this requires the application of CFD
codes that are specifically developed for scramjet propulsion, including complex models to accurately describe relevant
physical and chemical effects1.
The aim of this work is to present the application of a 3D numerical solver for the simulation of chemically reactive
flows in typical scramjet engines operating conditions.
The code used in this work is the density-based module of the CIRA code C3NS2. It is able to solve RANS equations
for a generic mixture of gases for steady, subsonic-to-hypersonic flows around 2D, axis-symmetric or 3D geometries.
The correct modeling of the supersonic combustion processes is crucial in the simulation of a scramjet device;
therefore, a trade-off analysis and the selection of proper reduced combustion kinetic mechanisms for supersonic
combustion application is required. In fact, on one hand, due the short resident time of air and propellant inside the
combustion chamber, is necessary to correctly evaluate chemical non equilibrium effects. On the other hand, in order to
reduce CFD computational costs, the selection of a reduced kinetic schemes is highly preferable. Following these
considerations some reduced kinetic schemes for Air-H2 combustion have been analyzed (such as the 8-step model of
Evans and Schexnayder3, the 7-step mechanism of Jachimowski
4 and the single step mechanism of Marinov,
Westbrook and Pitz5) After this analysis, the code has been tested in typical scramjet configurations: some flow
problems have been selected from the relevant literature in order to obtain a verification of the implemented models and
a validation by comparing numerical and available experimental data.
In particular, 2D and 3D flow problems reproducing supersonic mixing and combustion processes have been
investigated:
� TC-1: Parallel Injection (mixing and combustion).
� TC-2: 2D and 3D supersonic combustion in rectangular channel (mixing and combustion).
� TC-3: 3D scramjet wind-tunnel model simulations.
For the last two cases some interesting aspects have been investigated, as the differences between 3D and 2D flow
features and the effects on the computed flowfield of the accuracy of the kinetic scheme in the prediction of the ignition
delay time.
II. C3NS Code Description
The CIRA code C3NS is a structured multiblock finite volume solver that allows the treatment of a wide range of
compressible fluid dynamics problems. The Chemkin II input interface allows to treat different mixture of reacting
gases specifying mixture composition and chemical kinetic scheme. The fluid can be treated as a mixture of perfect
gases in chemical non equilibrium. The thermodynamic model treats mixture of thermally perfect gases, whose specific
heats, enthalpy and entropy are computed by using Gordon McBride polynomial fits9. Species viscosity and thermal
conductivity, are calculated by means of the Eucken law10
, whereas the mixture viscosity and thermal conductivity are
calculated by using the semi-empirical Wilke mixing rules10
. The diffusion coefficients are computed through a sum
rule of the binary diffusivities for each couple of species. In the numerical formulation, the governing equations are
written in integral form, and discretized with a finite volume, cell centred technique. Eulerian fluxes are computed with
a second order Flux Difference Splitting method11
with a ENO reconstruction of the interface values. Viscous fluxes are
computed with a classical centred scheme. Time integration is performed by employing an explicit Euler forward
algorithm coupled with an implicit evaluation of the source terms in time marching approach.
III. Air-H2 Combustion: Kinetic Mechanisms Trade-Off Analysis.
In the study of supersonic combustion problems for advanced propulsion systems, a crucial role is played by
combustion models. The description of combustion processes in a scramjet engine requires finite-rate chemistry
models, because residence times are of the same order of magnitude (typically few milliseconds) as the chemical
characteristic times. Chemical kinetic schemes must be carefully selected, by taking into account the accuracy on one
hand and the computational costs on the other hand. A trade-off analysis of kinetic mechanisms for Air-H2 combustion
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has been carried out**
in order to select a reduced kinetic mechanism able to correctly reproduce ignition delay time and
adiabatic flame temperature. The Jachimowski ‘8813 detailed scheme is considered here as reference for the selected
reduced schemes. In Figure 1 a comparison between results obtained by the scheme of Jachimowski and experimental
data in term of ignition delay is reported.
Figure 1 - Validation of Jachimowski ’88 detailed mechanism at p=1 bar13
.
The ranges of pressure (1÷2 bar) and temperature (1100-1300K) selected for the trade-off analysis are typical of
supersonic combustion in scramjet engines. The kinetic mechanisms considered here are the Marinov-Westbrook-Pitz5,
the Jachimowski4 (characterized by 7 steps), and the Evans-Schexnayder
3 schemes. The mixture has been considered
with the ER ranging from 0.5 to 2††
. In Figure 2 the test matrix including all the test conditions considered in the
analysis is graphically shown.
Figure 2 – Test matrix for Air-H2 combustion trade-off analysis.
**
The CIRA SPREAD12 1D code combustion module has been used in this analysis.
††
( )fuel ox
fuel ox st
m mER
m m= .
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a) b)
Figure 3 – a) Trade-off analysis results for T=1100K and p=1 bar; b) zoom in timescale.
a) b)
Figure 4 – a) Trade-off analysis results for T=1300K and p=2 bar; b) zoom in timescale.
The results of the analysis, carried out for Air-H2 combustion, are summarized in Figure 3 and Figure 4 . The
Jachimowski (7-steps) mechanism shows the best agreement with the reference scheme, in terms of adiabatic flame
temperature and ignition delay time, in the cases of stoichiometric and fuel-rich mixtures. For low ER cases the
Marinov-Westbrook-Pitz mechanism can be considered a valid alternative to be used in a CFD code, also due to its
simplicity. In the considered conditions the Evans-Schexnayder chemical kinetic scheme overestimates the ignition
delay time and underestimate the adiabatic flame temperature for high initial temperature and pressure, whereas the
combustion does not occur for low initial temperature and pressure.
In this work the effects of the modifications proposed by Craddock6 in order to improve Evans & Schexnayder
mechanism are investigated. To reduce the ignition delay time, the rate coefficients of some reactions has been
changed. In this way, the number of the reactions does not change and the computational efficiency does not decrease
(for figures labels this scheme is called Evans & Schexnayder trebled). The poor modeling of heat distribution with the
reaction model of Evans & Schexnayder is primarily due to the absence of the reactions involving the radical
hydroperoxyl (HO2).
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For this reason HO2 radical has been added to the reaction model to improve the heat release modeling capabilities and
the reaction related to this radical have been added to the model (the related scheme is Evans&Schexnayder_8sp). The
subsequent Figure 5 shows the effect of the chemical kinetic scheme modifications.
Figure 5 - Effects on ignition delay time (different time scales).
In the following Figure 6 is shown the comparison of the two scheme results with the other scheme results, where the
enhancement with respect to the original kinetic scheme is evident.
Figure 6 – Comparison with the other kinetic schemes (different time scales).
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IV. Supersonic Mixing and Combustion Simulations
In Tab. 1 are listed the simulations carried out with the C3NS code in order to test it in typical flow conditions of a
scramjet engine (other validation tests could be found in Ref.7).
Table 1. Test cases selected
TC Geometry Dim.
1 Parallel injection 2D
2 Supersonic combustion in a rectangular channel 2D – 3D
3 Scramjet tip to tail geometry 2D – 3D
A. TC-1: Parallel Injection The supersonic combustion experiment carried out by Burrows and Kurkov
15-17 has been selected to verify and
validate the code capabilities to simulate a typical Air-H2 parallel injection and combustion in a 2D planar flow.
Figure 7 – TC-1: Geometry and injection configuration
As described in Figure 7, a sonic jet of hydrogen is injected tangentially near the wall into a supersonic vitiated air-
stream flowing into a duct with diverging walls. The jet slot height is 0.4 cm and wall temperature is kept fixed at
298K. The operating conditions are reported in Table 2.
Table 2. TC-1: flow conditions
Air stream H2 Jet
Mach 2.44 1
Temperature (K) 1270 256
Pressure (MPa) 0.1 0.1
Velocity (m/s) 1764 1216
H2 mass fraction 0 1
O2 mass fraction 0.258 0
N2 mass fraction 0.486 0
H2O mass fraction 0.256 0
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The two-dimensional computational domain for the numerical simulation of turbulent mixing is divided in 3 blocks and
57267 cells (Figure 8).
Figure 8 – TC-2: computational domain.
The Marinov-Westbrook-Pitz chemical kinetic mechanism has been selected to simulate the hydrogen oxidation
process. This choice can be justified as follows: in Figure 9 are shown, in the mixing zone, just downstream the H2
injection, the equivalence ratio lines ranging from 0.01 to 2 overlapped to the temperature contour lines. According to
Ref.18 the ignition will take place in fuel lean zones (p=0.08 MPa, T>1000K, ER~0.2) where, as discussed in section
III, the Marinov-Westbrook-Pitz mechanism is the best choice among the analyzed ones.
800K 900K1000K
250K
250K
400K600K
800K 800K900K
1100K
1000K
900K
0.01
0.1
11.5
1.521
x[m]
y[m
]
0.11 0.115 0.12 0.125 0.13
0.08
0.085
0.09
0.095
ER: 0.01 0.1 0.5 1 1.5 2
TC-2 : Equivalence Ratio and Temperature contour lines
Figure 9 – TC-2: temperature (black) and ER contour lines in mixing zone.
The calculated mixture fraction profiles for H2 and H2O at 35.6 cm downstream of the injector (at the exit plane see
Figure 11) are compared with the experimental data, respectively in Figure 10 a) and Figure 10 b).
H2
Air
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a) b)
Figure 10 – TC-1: Predicted H2 (a) and H2O (b) mole fraction profiles vs. experimental data at x=35.6 cm.
The obtained results present a quite good agreement with experimental data in terms of H2O mole fraction prediction,
even if the flame position is not correctly predicted. In Ref. 15 it is remarked that the flame position is extremely
sensitive to the injection velocity profile used in the flow solver. In this simulation, before the injection in the
combustion chamber, hydrogen passes through a pre-injection duct that gives to the jet a realistic velocity profile before
mixing with air stream. Thus, the lack of accuracy in flame position prediction is probably due to the inlet values of
turbulence quantities, as also stated in Ref.16.
B. TC-2: Supersonic combustion in rectangular channel Differently from the previous test case, the TC-2 is a demonstrative flow case selected to show the code capabilities to
simulate the mixing and the combustion processes with flow features representative of a typical scramjet combustor.
In particular, 2D and 3D simulations of supersonic combustion in a rectangular channel with transverse hydrogen
injection have been carried out. The 2D and 3D geometries, the computational domains and flow conditions are
reported in Figure 11 and Figure 12. The flow conditions have been selected from Von Lavante et al.19
, and assure that
supersonic combustion takes place. Wall temperature is fixed to 293K.
Figure 11 – TC-2: 2D Geometry, grid and flow conditions
19.
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Figure 12 - TC-2: 3D Geometry, topology and flow conditions.
Besides the typical flow features of a transverse injection already studied in TC-1 (such as: Prandtl-Meyer expansion,
Mach disk and shock-wave/boundary-layer interaction), this test case is characterized by wall shock reflections, shock-
shock interactions and non-equilibrium combustion chemistry.
In order to demonstrate the flow-solver capabilities to describe such complex phenomena, the case has been initially
simulated by using a two-dimensional 77000-cell (648 in x and 120 in y direction) computational grid divided in 3
blocks. The 3D grid has been obtained by the extrusion of the 2D grid, and has 121 cells in z-direction for a total of
about 9300000 cells in the finest grid level. Its topology is characterized by 6 computational blocks (Figure 12).
The turbulent mixing process has been evaluated by using a classical two-equation k-ε compressible model20
, while
combustion chemical kinetics has been modeled with the Marinov-Westbrook-Pitz single-step mechanism in the 2D
case, and with both the Jachimowski seven steps mechanism4 and Marinov-Westbrook-Pitz single step mechanism in
the 3D case.
2D and 3D tests have been chosen in order to point out differences between 2D and 3D flowfield. Moreover, different
chemical schemes have been used in order to evaluate the effects, on the computed flowfield, of the accuracy in ignition
delay time prediction.
Our investigations begin with the analysis of the results concerning the 2D computations. Figure 13 and Figure 14
respectively provide mirrored views of the computed Mach number and temperature distributions in the combustor
respectively. The complex shock pattern that takes place downstream the injector holes is clearly visible.
Figure 13 – TC-2: Mach number flow field.
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A bow shock forms upstream the injector due to the interaction between the supersonic air stream and the hydrogen jet
and reflects on the symmetry axis. Both the bow shock and the reflected shock interact with the boundary layer
respectively upstream and downstream causing shock-induced separations.
Figure 14 – TC-2: Temperature flow field.
As shown in Figure 14, combustion takes place in the region upstream the injector holes, where the mixing occurs at
high temperature. The predicted flame temperature is about 2000 K and the levels of H2O are fairly comparable with
the ones obtained by Von Lavante19
.
A zoom of the separation region is given in Figure 15 where the two recirculation zones, upstream and downstream the
injector hole, are highlighted.
Figure 15 – TC-2: zoom of the separation region (2D case).
In the separation region ahead of the hydrogen transversal jet (see Figure 16), non-zero water concentration can be
found upstream of the jet. In fact, as also observed in Ref. 19, the H2O is carried upstream of the injection opening by
recirculating fluid in the boundary layer. Downstream of the region close to the hydrogen jet, the flow is essentially
chemically frozen, and the H2O is only convected downstream. Moreover, some of the hydrogen still reacts with the
surrounding air in the free shear layer.
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Figure 16 – TC-2: 2D H2O mass fraction distribution.
After the analysis of the 2D results, the 3D results will be presented hereinafter. As highlighted in Figure 12 the 3D
geometry is a thin region with a supersonic stream of air in x-direction and with an orthogonal injection of hydrogen.
Although, the geometry is very close to a 2D one, some significant differences must be remarked.
In Figure 17 and Figure 18, respectively, Mach number and temperature contours are shown; respectively the same
flow structures, present in 2D computations, can be recognized. However, in this case, is clearly noticeable the 3D
feature of the flow field. All the flow characteristics, typical of this kind of problem, can be analyzed (by looking also
to Figure 19). The barrel shock, the oblique bow shock, the free shear layer behind the H2 jet and the separation regions
downstream and upstream of the injector are clearly visible. In particular, Figure 18 and Figure 19 show that the flow
forms a horseshoe vortex around the H2 jet, thus relieving the pressure ahead the injection. This feature explains the fact
that the separation regions near the injector hole are smaller in 3D than in 2D (Figure 15). Finally, near the exit plane,
the shock induced separation is not present, because in the 3D configuration the shock reflection is weaker and moved
downward (compare Figure 13 and Figure 17) and the turbulence levels are stronger with respect to the 2D case (as
observed also in Ref.19).
Figure 17 – TC-2: Mach number contours: Section Z=0.02 and X varying sections.
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Figure 18 - TC-2: Temperature contours: Section Z=0.02 and X varying sections.
Figure 19 – TC-2 : Zoom of the injector region: Mach contour and streamlines colored with H2 mass fraction.
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In this case the predicted flame temperature is around the 2200 K and the levels of H2O production are comparable with
those in Ref. 19.
Figure 20 – TC-2: 3D H2O mass fraction distribution on Z=0.02 m plane.
In particular the levels of H2O in the z-plane symmetry section (Figure 20) are much lower (about 45 %) with respect
to those predicted the 2D case. Figure 21 explains that the H2O distribution is very different with respect to the 2D
case, because of the 3D nature of the flow structures; the H2O is, in fact, in the 3D case, convected downstream of the
injector by the presence of the horseshoe vortex around the H2 jet. This causes the combustion products to round on the
hydrogen jet, thus determining the distribution shown in Figure 21 characterized by the maximum H2O concentration
across the injection.
Figure 21 – TC-3: Zoom of the injector region: H2O mass fraction‡‡
contours and streamlines colored with
Temperature.
A further aspect has been analyzed for TC-3: the effect of kinetic mechanism on the flow field feature. A correct
prediction of ignition delay time is very important in supersonic combustion, because of the small residence time of the
mixture in the combustor.
‡‡
The value of H2O mass fraction is cut off below 0.006, for better visualization purposes, so some zones are blanked.
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As stated in the introduction section of this chapter, both Marinov-Westbrook-Pitz and Jachimowski reduced schemes
have been used to perform this 3D test. In chapter III it has been noticed how Marinov-Westbrook-Pitz kinetic
mechanism underpredicts the ignition delay time and the adiabatic flame temperature (Figure 3-Figure 4) with respect
to the detailed one (Jachimowski ‘88). On the contrary, better results have been obtained with the 7-steps Jachimowski
kinetic scheme. In the following figures is shown how the accuracy in ignition delay time prediction could affect the
temperature distribution, shock patterns and H2O production.
Figure 22 –TC-2 : Z=0.02 section. Temperature contour comparison.
Figure 23 –TC-2 : Z=0.02 section constant X planes section. Mach number profiles comparison.
In Figure 22, the temperature contours obtained by 7-steps Jachimowski and Marinov-Westbrook-Pitz kinetic schemes
are compared. It is evident that the results obtained with the simplified mono-step mechanism over-predict the flame
temperature. In fact, in this case, the very underestimated ignition delay time (see section III) causes a stronger
temperature gradient than the one predicted in the computation with the Jachimowski scheme. In Figure 23 Mach
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number distributions along y for the sections x1=0.104 m and x2=0.122 m (highlighted in yellow in Figure 22) are
shown. The aim is to show how the shock patterns vary in dependence on the kinetic schemes. In particular the
computation carried out with Marinov-Westbrook-Pitz predicts a steeper shock wave (see Figure 23), which is mainly
due to the different accuracy prediction of the ignition delay time. Moreover, a comparison of the H2O molar fraction
levels is showed in Figure 24, also in this case the simulation with Marinov-Westbrook-Pitz scheme over predicts the
H2O concentration, coherently with the results of section III.
To complete the analysis of TC-3, some comparisons with the available experimental data have been made. In Figure
25 the comparison of the exit velocity of both the 2D and 3D computations with the experimental data is shown.
Differently from 2D, 3D simulations are in good agreement with the experimental data. These results are coherent with
the ones reported in Ref. 19 obtained by Von Lavante using LES methods.
Figure 25 – TC-2: exit velocity comparison with experimental data.
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Furthermore, a qualitative comparison of the numerical density gradient compared with the Schlieren experimental
picture is presented. It can be noticed (Figure 26) that the predicted flow structures are similar to those visible in the
experimental flow visualization§§
. This is verified mainly near the injection region, where grid resolution is high, but is
worse in normal direction because the grid rapidly becomes coarser. It is remarkable that the positions of the reflected
shock waves are quite well predicted. It should be actually noticed that the RANS methodology does not allow to
obtain an accurate vortex structures description.
Figure 26 – TC-2: Density gradient comparison (CFD results VS Experimental Schlieren).
C. TC-3: Tip to Tail Scramjet simulations This section reports the 2D and 3D numerical simulations of the shock-tunnel test, carried out in the free piston T4
facility of the Queensland University, of the scramjet-like configuration shown in Figure 27. The experimental
configuration21
involves injection of hydrogen fuel into the scramjet inlet, followed by mixing, shock-induced ignition,
and combustion. This configuration has been subsequently investigated by CFD analysis in the work of Star22
.
Figure 27 - Photograph of the scramjet model mounted in the test section of the T4 shock tunnel21
.
§§
In the figure all the shock patterns are remarked for both experimental and numerical picture in order to clarify the analysis.
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The 3D geometry is reported in Figure 28 where one of the sidewall has been removed for illustration proposal. The
scramjet model has a rectangular cross-section consisting of an inlet section with an integrated fuel delivery system, a
combustion chamber and a simply ramped nozzle section. The total length of the model is 0.625 m, the width is 0.075
m, and the combustor section considered here is of 0.024 m. The two inlet ramps are respectively inclined by 7 deg and
9 deg with respect to the horizontal. The fuel delivery subsystem injects choked gaseous hydrogen at a 45 deg angle
(relative to the surface) through four circular discrete holes located on both upper and lower surfaces of the first inlet
ramp section upstream of the combustor section. The four fuel holes are evenly spaced in the spanwise direction, with
the intent to provide the fuel and air with enough time to mix and properly ignite, of course only when the shock
induced temperature rises sufficiently.
Table 3. TC-4: model geometric specifications.
Figure 28 – Three-Dimensional Scramjet Model and Centerline Cross-Section.
The experimental campaign carried out by Odam e Paul21
considered both fuel-off and fuel-on runs. In this work two
representative points of both these categories of experiment have been considered; fuel-off simulation in order to
analyze the air flow feature inside the scramjet engine model and assesses grid topology and refinement, fuel-on
simulation in order to predict all the processes typical of a scramjet thruster such as compression, induced shock
ignition, supersonic combustion and acceleration. In the following tables are reported the conditions that have been
identified with the ID number of the related experimental run21
.
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Table 4. Fuel-off run conditions (RUN#767521
)
Air stream H2 Jet
Mach 6.25 NP
Temperature (K) 487 NP
Pressure (Pa) 10230 NP
Velocity (m/s) 2766 NP
H2 mass fraction 0 NP
O2 mass fraction 0.233 NP
N2 mass fraction 0.766 NP
TWall (K) 300
Table 5. Fuel-on run conditions (RUN#767821
)
Air stream H2 Jet
Mach 6.42 1
Temperature (K) 412 250
Pressure (Pa) 8958 647058
Velocity (m/s) 2612 1201
H2 mass fraction 0 1
O2 mass fraction 0.233 0
N2 mass fraction 0.766 0
TWall (K) 300
Computational grids are referred to the quarter-symmetry scramjet geometry showed in Figure 29, where in both the 2D
and 3D computational grids meshed geometries are presented.
Figure 29 – TC-3 : Computational domain.
The 3D grid used is characterized by 38 blocks (Figure 30) and has 2000134 cells on the finest grid level. The
minimum normal spacing for both sidewall and combustor wall is about 1.0e-6 m.
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a) b***
)
Figure 30 – TC-3: 3D blocks decomposition (a) and grid (b).
The two dimensional grid was extracted from the mid-plane of the quarter-symmetry three dimensional grid with an
irrelevant rearrangement in the fuel injector zone. The cells number of the finest grid level is 10786 and the stretching
at the wall is the same of the 3D grid, of course.
Figure 31 – TC-3: 2D grid.
Fuel-off computations, as previously said, are carried out in order to preliminary identify main air flow features in the
scramjet geometry and, furthermore, to asses grid refinement effects. In the following, 3D simulations results, in terms
of upper wall pressure distributions, are presented and compared with the 2D ones and with experimental data†††
(see
Figure 32) .
***
For visualization purposes wall fitting has been relaxed. †††
The experimental pressure measurements are obtained from pressure taps located on the centreline of the model21,22
.
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Figure 32 – TC-3 : Comparison between numerical and experimental static pressure distribution along the wall
(fuel-off case).
The fuel-off solution on the finest grid level has been analyzed in order to have an accurate description of air flow
features inside the scramjet model. The plot of the magnitude pressure gradient is also showed in Figure 32.
In the Figure 32 the key flow features can be distinguished and described as follows.
A. First inlet ramp shock.
B. Second inlet ramp shock.
C. Corner expansion fan arising from the flow near the wall turning to enter the combustor section.
D. Wall interaction of the first ramp shock with the transmitted shock from the opposite wall.
E. Wall interaction of the second inlet ramp shock with the transmitted shock from the opposite wall.
F. Wall interaction of the expansion fan from C transmitted from the opposite wall.
G. Wall interaction with the transmitted combined shock structure of D and E.
H. Wall interaction of the expansion fan from F transmitted from the opposite wall.
I. Wall interaction with the transmitted shock structure of G.
J. Corner expansion fan arising from the flow near the wall turning to enter the nozzle section.
K. Final transmitted shock system.
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Moreover, it can be noticed from Figure 32 the good agreement of the 3D simulation results with the experimental
data. It is remarkable that the 3D computation better predicts the position of the corner expansion fan and the final
shock system with respect to the 2D computation‡‡‡
, as expected.
In conclusion, a detailed view of the 3D flow field is shown in the following figure. In particular, Figure 33 shows the
3D temperature distribution. It can be noticed that the temperature peaks are in proximity of the wall shocks
impingements, moreover the angularity of the flow in the corner of the combustion chamber is visible on the
temperature maps. These recirculation zones will be significant for the air-fuel mixing in the fuel-on conditions.
Figure 33 – TC-4: Fuel-off case 3D. Temperature contour slices.
The main focus of the fuel-on test-case is to asses code capability in the simulation of all the structures typical of a
scramjet engine flow, such as: compression, shock induced ignition, supersonic combustion and expansion. In addition
particular performance parameters such as thrust and wall heat flux have been derived.
In particular, Figure 34 and Figure 35 show that the fuel jet initially penetrates through the thin boundary layer, then the
fuel is turned toward the wall downstream of the initial shock impingement point and resides within the boundary layer
before partially combusting midway through the combustion chamber, where is visible the conversion of the O2 in H2O
(Figure 36).
‡‡‡
Even if there is a good agreement between the computed and the experimental data a further grid refinement could improve the shock structures
resolution.
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