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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