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19th Australasian Fluid Mechanics ConferenceMelbourne, Australia8-11 December 2014
Equivalent Direct Connect Free-Stream Shock Tunnel Conditions for Injection Locations in an
Axisymmetric Scramjet
K. Basore1, V. Wheatley1, R.R. Boyce2, and R. Starkey3
1Centre for Hypersonics
University of Queensland, Queensland, Australia
2Space Engineering
University of New South Wales at the Australian Defence Force Academy, ACT, Australia
3Busemann Advanced Concepts Laboratory
University of Colorado, Colorado, U.S.A
Abstract
Fuel injection optimisation into radical farming scramjets is one
of the key technologies being developed at the University of
Queensland [18, 6, 13]. To facilitate the experimental testing of
some these injection techniques, equivalent tunnel free-stream
conditions at the injection locations had to be determined. To
compare against previous work [19, 3] and intended flight test
data [20] the free-stream conditions were calculated relative to
the intended 48 kPa constant dynamic pressure trajectory of the
SCRAMSPACE vehicle [4].
Due to the variation of the equivalent standard atmosphere
through the test period (33-27 km [20]) and the discreet nature
of the nozzles available on the tunnel, the constraining variable
used to calculate the conditions was not the flight enthalpy but
the ramp injection Mach number; which is the reverse to the
standard procedure [10, 15]. By constraining the nozzle exit
Mach number, the equivalent flight enthalpy could be deter-
mined from the projected flight dynamic pressure and reverse
calculating the inviscid conical shocks of the truncated Buse-
mann inlet.
Using this equivalent flight enthalpy, a multi-dimensional Krig-
ing based surrogate model was developed to give a surface re-
sponse to the equivalent tunnel conditions of flows produced in
the T4 Hypersonic Shock Tunnel. The final calculated condi-
tions are estimated to be tuned but not necessarily tailored due
to the constant compression ratios used in the facility [9].
Introduction
The SCRAMSPACE flight experiment was intended to give ver-
ifiable flight data for an axisymmetric radical farming scramjet
engine [20]. This vehicle was intended to have a constant dy-
namic pressure flight trajectory, as is the current standard prac-
tice in flight testing [2]. One of the proposed uses of this flight
data was the comparison of the measured performance metrics
to predicted optimized fuel injection into an equivalent engine.
In order to test the optimised injectors, to validate the optimisa-
tion methods used, tunnel testing of the injectors at the equiv-
alent free-stream conditions that the injector would have expe-
rienced in flight was required. Due to the discrete nature of
the nozzles available in the facility, a window in the proposed
flight trajectory that matches the capability of the T4 free-piston
shock tunnel at the University of Queensland had to be found.
Due to the unfortunate failure of the SCRAMSPACE vehicle,
this condition algorithm was never fully implemented or vali-
dated but is still presented in an abridged version here.
Reverse Inviscid Taylor-MacColl Analytical Approximation
Calculation of the inviscid flow through the equivalent flight
conical shocks was done using the non-dimensionalized Taylor-
MacColl (equation 1) [1], with this formulation able to be used
due to the conical symmetry of the inlet [21].
γ−1
2
[
1−(
V ′r
)2−
(
dV ′r
dθ
)2]
[
2V ′r +
dV ′r
dθcot (θ)+
d2V ′r
dθ2
]
−dV ′
r
dθ
[
V ′r
dV ′r
dθ+
dV ′r
dθ
d2V ′r
dθ2
]
= 0
(1)
This equation was solved using a non-constant stepped 4th-
order Runge-Kutta ODE ray-tracing integrator [5] as described
in [21] to find the cone angle (θ) and the equivalent conditions
from an initial guess of the shock angle. Over the top of this
function was a trust-region Levenberg-Marquardt (LM) non-
linear least-squares algorithm [17] (equation 2) to solve for the
true shock angle based on the cone angle L1 error |θc −θdesg|.Where θc is the cone angle solved for from the root solving in-
put and θdesg is the constrained theta cone angle measured from
the SCRAMSPACE inlet turning angles.
φ(x) =1
2
m
∑i=1
f 2i (x) =
1
2||F(x)||2 (2)
Nested on top of the cone angle solver was another LM al-
gorithm that used the constant dynamic pressure to calculate
the approximate enthalpy altitude by varying the correspond-
ing free-stream conditions based on the L1 error |Mc −Mdesg|.Where Mc is the iteration Mach number at the inviscid surface
of the cone and Mdesg is the free-stream Mach number calcu-
lated from the altitude iteration and dynamic pressure.
Due to the available nozzles on the facility and a secondary test-
ing requirement of a large core-flow for the work this method
was originally designed for, the Mach 7.6 nozzle was chosen
for the rest of the corresponding analysis. Due to this facility
constraint, only the first conical shock of the SCRAMSPACE
inlet was processed, otherwise, the corresponding free-stream
stagnation enthalpy, when examining the original injection lo-
cation on the second ramp of the inlet, was H0 = 8.15MJ
kgwith
M∞ = 12.74. Although the injection location could not be ex-
actly matched, the conditions processed for the first ramp would
Altitude M p T V P0 H0
31.88 7.6 2.11 294 2.61 14.8 3.70
km kPa K km/s MPa MJ/kg
Table 1. Inviscid Condition - SCRAMSPACE First Ramp.
have validated the optimization algorithms and thus, allowed
for an extrapolated correlation against the measured engine per-
formance metrics. Thus, from these parameters the following
test-section conditions were found from the nested root-finding
ray-traced algorithm shown in table 1.
Kriging Surrogate Model
In order to find an equivalent tunnel condition a Gaussian basis-
function Kriging surface response [8] was designed using the
database of previously achieved flow conditions in the T4 shock
tunnel(T4 database) and the DACE matlab package [12]. To de-
crease the complexity of the model, only shots using the Mach
7.6 nozzle were chosen to be analysed. This resulted in the
following independent variables being used to build the model
with: Reservoir Pressure; Compression Tube Pressure; Argon
%; and Shock Tube Fill Pressure. The independent variable re-
duction was done assuming that the diaphragm was a subset of
the compression tube fill pressure and the Helium percentage
was a function of the Argon percentage.
From the T4 database ≈100 shots were analysed using a cus-
tomized subset of PPM Basic [7] and ESTCj [10] to compute
the nozzle stagnation and exit conditions. It is recognized that
ESTCj does not account for the non-equilibrium chemistry that
is a substantial part of the nozzle exit conditions [14] but as this
is a first order large-sweep approximation it was considered an
acceptable incurred error.
For the shot analysis, a nozzle start-up time of 0.9 ms was as-
sumed with a test time of 1 ms. It was considered within the er-
ror range of the model to not use the steady pitot start-up time or
the 10% driver contamination for the end of the test time. From
this basic analysis, similar fill parameters were checked using
the variance of the independent parameters to average across
the response variables of: Stagnation Enthalpy; Velocity; Mach;
Stagnation Pressure; and Static Pressure. The reason this aver-
aging was performed is a Kriging model assumes deterministic
points in the random-field variable space. The averaging of the
response variables was considered to asymptotically approach
the ensemble average which is deterministic in the random field
and thus an acceptable data point for the surrogate model. The
Kriging model was chosen specifically due to the compact sup-
port and non-linearity necessary for the surface response as can
be seen in figure 1.
The surface shown in figure 1 was created through cubic splines
of 100 Monte-Carlo determined hypercube points plus the
deterministic data in the 4 dimensional independent-variable
space. The mesh points in each dimension was kept to the same
number of hypercube sampled points to keep the resulting di-
mensional projections of the variable space into 3 dimensions
as simplistic as possible.
The resulting optimized correlation vector θ∗ from the Kriging
model is shown in table 2. This table shows the influence of
each of the independent variables over the resulting response
surface and the most important parameters when trying to de-
termine conditions on the T4 tunnel.
To show the fitness of the model the percentage normalized
Root-Mean-Square Deviation(RMSD) (equation 3) is plotted as
23
45
6
x 106
1
2
3
x 105
2.5
3
3.5
4
4.5
5x 10
6
Reservoir Pressure, PaShock Tube Pressure, Pa
Sta
gnation E
nth
alp
y, J/k
g
Orig Data Points
Kriging Model Points
Figure 1. Surface Response-Full Model: Stagnation Enthalpy.
Variable θ∗i
Reservoir Pressure 17.82
Shock Tube Pressure 6.67
Compression Tube Pressure 9.36e-2
Argon % 9.90e-3
Table 2. Correlation Vector for Independent Variables.
a cubic spline interpolated mesh from the hypercube sampled
points as shown in figure 2. Here yi is all the response hyper-
cube sampled points for the ith dimension.
εimodel =
(
1−
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
yi −RMSDi
yi
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
)
(3)
12
34
56
x 1061
2
3
x 105
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Reservoir Pressure, PaShock Tube Pressure, Pa
||H
0||RMSD
Error
Conical Point
Figure 2. RMSD Error Full Condition Model.
Local Minimum Optimization for Output Conditions
In order to approximate the tunnel fill conditions necessary to
achieve the test section conditions found in table 1, the local un-
constrained direct search Nelder-Mead minimization algorithm
[11] was implemented with the constructed Kriging surrogate
model. To satisfy the scaler requirement for the simplex con-
struction of the Nelder-Mead algorithm, an unweighted objec-
tive function consisting of the norm of the error vector between
the desired conditions and the evaluated conditions was used
(equation 4). Due to the discrete nature of the diaphragms used
in the tunnel, an educated guess could be made as to the ini-
tial input vector to guarantee a convergence condition. Figure
3 shows the evaluated points in terms of the objective function
and the input variables of the shock tube pressure and reservoir
pressure which were shown in table 2 to have the most influence
on the final conditions.
εsearch =∣
∣
∣
∣yi − ydesgi
∣
∣
∣
∣ (4)
4.4
4.6
4.8
x 106
1.5
1.6
1.7
1.8
1.9
x 105
0
0.5
1
1.5
2
2.5x 10
6
||U
nw
eig
hte
d O
bje
ctive F
unction||
Reservoir Pressure, PaShocktube Pressure, Pa
Figure 3. Norm Error Vector for Condition Search.
In equation 4: εsearch is the scalar error value; yi is the evalu-
ated response metrics from the surrogate model; and ydesgiis
the designated test conditions as defined in table 1.
The tunnel fill conditions and the percentage absolute error as
determined from this minimization direct search algorithm are
shown in table 3. The error values seen are within the exper-
imentally determined tunnel uncertainty of approximately ±5-
10% [16] except for the static pressure. This large percentage
error is estimated to be due to the resolution of the model at this
level as the absolute difference between the two values is under
500 Pa.
Tunnel Conditions Test Conditions
Units Error
Res 4.60 MPa H0 3.63 MJ/kg 1.9 %
Driver 67.0 kPa V 2.67 km/s 2.3 %
Argon 72 % M 7.44 Non 2.1 %
Shock 166 kPa P0 15.1 MPa 2.0 %
Ps 2.53 kPa 19.8 %
Table 3. Tunnel Conditions from Minimization of Error to Desired.
The final error % from the objective function was 20.2 % with
a RMSD model uncertainty of ±15.8% as can be seen in figure
2.
Reduced-Order Kriging Model
To examine the recognized “global” model’s resolution issue, a
3 dimensional variable subset consisting of constant compres-
sion tube pressure was examined. Due to the limited number
of condition data points available, the 3 mm diaphragm (40.2
kPa compression tube pressure) subset was chosen. Figure 4
shows the surface response for the 3 variable subset for H0 with
100 hypercube Monte-Carlo determined points again used in
the evaluation. The RMSD error vector associated with this re-
sponse surface is shown in figure 5 which shows the increased
overall uncertainty confidence in the reduced dimension model.
2.5
2.6
2.7
x 106
1
1.5
2x 10
5
3
3.5
4
x 106
Shock Tube Pressure, PaReservior Pressure, Pa
S
tagnation E
nth
alp
y, J/k
g
Kriging Model Points
Orig Data Points
Design Point
Figure 4. Reduced Order Surface Response.
2.5
2.55
2.6
2.65
2.7
x 106
1
1.5
2
x 105
0
0.05
0.1
0.15
Shock Tube Pressure, PaReservior Pressure, Pa
||H
0||RMSD
Error
Conical Point
Figure 5. Reduced Order RMSD.
The correlation vector for the reduced model can be seen below
in table 4 which shows at the single compression tube pressure,
that the shock tube fill pressure has the greatest influence.
Variable θ∗i
Shock Tube Pressure 19.0
Reservoir Pressure 0.0680
Argon % 0.0224
Table 4. Correlation Vector for 3 Dim.
The 3 mm diaphragm condition is not able to produce a 3.7
MJ/kg enthalpy but the closest approximation found by the
Nelder-Mead algorithm is shown in table 5. These values show
that even at the lower diaphragm condition, the subset higher
resolution model was able to achieve a much lower condition
and modelling error than the full global model.
Tunnel Conditions Test Conditions
Units Error
Res 2.60 MPa H0 3.24 MJ/kg 12.4 %
Driver 40.2 kPa V 2.55 km/s 2.3 %
Argon 100 % M 7.58 Non 0.3 %
Shock 143 kPa P0 13.5 MPa 9.1 %
Ps 1.97 kPa 6.9 %
Table 5. 3mm Diaphragm Closest Approximation.
The 3mm objective function error was 9.57 % with a RMSD
model uncertainty of ±5.9% as seen in figure 5.
Conclusions
The inviscid flight conditions of an axi-symmetric scramjet in-
let were able to be evaluated using the Taylor-MacColl equa-
tions given a designated Mach number and constant dynamic
pressure. Using these equivalent flight conditions, a Kriging
surrogate model was able to be built, using the T4 database, to
provide equivalent flight tunnel-testing conditions that matched
specific locations on the axi-symmetric scramjet inlet. The abil-
ity to determine these conditions will allow for quick evaluation
of the necessary testing conditions for optimized injectors that
need to be experimentally validated.
This work was originally planned to verified against the
SCRAMSPACE, Mach 8 48kPa dynamic pressure condition but
was never fully implemented. This tool can be easily converted
to any other axi-symmetric engine and if used in the future
should incorporate a further analysis of the T4 database as well
as subset surrogate models for each diaphragm thickness to im-
prove the model uncertainty error as was shown with the Mach
7.6 3 mm diaphragm condition.
Acknowledgements
The author would like to acknowledge ARC Grant:
DP120101009 for funding as well as Luke Doherty for
his help with the T4 PPM Basic code.
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