Large Eddy Simulation of a Harrier Aircraft at

Touch Down

Gary J. Page∗, Qinling Li†, James J. McGuirk‡

Loughborough University, Leicestershire, United Kingdom

Giles A. Richardson§

Engineering Department, Cambridge University, United Kingdom

The flow-field beneath a jet-borne vertical landing aircraft is highly complex and un-steady. Large Eddy Simulation is a suitable tool to predict both the mean flow and unsteadyfluctuations and could be a replacement for expensive rig testing. This work aims to demon-strate the viability of using LES to model a real aircraft geometry at touch down. Thenumerical method uses a compressible solver on a mixed element unstructured mesh. Thesmoothing terms in the spatial flux are kept small by the use of a monitor function sensitiveto vorticity and divergence. The WALE subgrid scale model is utilised. The method usesthe OPLUS library to allow parallel computation on large numbers of processors. The meshresolves the aircraft’s auxiliary intakes and under-fuselage strakes and dams. Hexahedralelements are used under the aircraft and in the intakes, whilst tetrahedral elements areused to resolve the less important geometrical details above the plane of the wing. Due tostability problems, the calculation was only able to be run for 100,000 time steps which istoo few to generate a statistically valid mean flow. Instantaneous results are presented. Thestrakes and dam are seen to be trapping the upwash fountain and directing the flow awayfrom the intake. Also, visualisation to show vortex cores indicates a structure entering theintake with a temperature of 20-30K higher than ambient. Instantaneous temperatures inthe intake are similar to typical experimental measurements. The relatively few numberof time steps means that the flow is unsteady, but symmetric. It is believed that a largernumber of time steps would result in this ‘breaking’ to produce an asymmetric flow. Whilstsignificant further work needs to be carried out, these calculations show that LES could bea practical tool to model Hot Gas Ingestion for the Joint Strike Fighter aircraft.

I. Introduction

The flow-field surrounding a vertical landing aircraft is a complex interaction of lift jet ground impinge-ment, upwash fountain flow and ground vortex flow. If hot gas enters the engine intake compressor stall,engine surge and performance loss may occur1–3 – this phenomenon is normally referred to as Hot GasIngestion (HGI). The multiple lift jets impinge upon the ground plane forming wall jets, these then collideand produce an upward flow, or ‘upwash fountain’. This fountain can potentially direct hot gas towardsthe engine intakes. A second mechanism for HGI is the forward motion of the aircraft, coupled with theoncoming wind, which rolls up the forward flowing wall jet to produce a ground vortex. Again, this can leadto hot gas entering the intake, although the path is relatively long and temperatures are likely to be lessthan the more direct fountain route.

Traditionally, model scale rig tests have been used to develop aircraft configurations which are lesssusceptible to HGI.3,4 However, experimental testing is expensive and provides only limited information onthe flow-field (such as ground plane flow visualisation and intake temperature histories). This testing haslead to the Harrier aircraft using ’strakes’ and ’dams’ under the fuselage to control the upwash fountainflow. CFD methods have the potential to complement or replace rig testing if they can be shown to model

∗Senior Lecturer, Department of Aeronautical and Automotive Engineering, AIAA Member. G.J.Page@lboro.ac.uk†Research Fellow, Department of Aeronautical and Automotive Engineering, AIAA Member‡Professor, Department of Aeronautical and Automotive Engineering§current address Rolls-Royce plc, Derby, United Kingdom

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accurately the HGI problem. In particular, with aircraft such as the STOVL variant of the Joint StrikeFighter, it is possible that issues will occur in service that will need to be rapidly studied and circumventedusing a validated CFD methodology.

CFD solutions using the Reynolds-averaged Navier-Stokes (RANS) equations with a turbulence modelcan provide reasonable predictions of the large ground vortex and turbulent mean flow,5–7 but are unableto provide information on important phenomena such as the unsteady fountain and instantaneous flowdistortion and swirl in the intake. The ability to predict both the mean flow and unsteady excursions isimportant for the design and development of future aircraft.

In a project linked to the current work, the dynamic descent towards the ground of a Harrier model hasbeen computed using Unsteady RANS.8–10 This showed that some of the important unsteady features dueto the dynamic behaviour of the ground vortex could be reproduced, and that it was possible to generatehigh quality meshes around a Harrier model that included important geometrical features such as strakesand auxiliary intakes.

Large Eddy Simulation (LES) resolves all of the large scale structures greater than the grid scale, andso predicts temporal variation of the flow. It has already been noted that the flow under the aircraft ishighly complex and unsteady and LES has the potential to provide, for example, time histories of intaketemperature rise. The potential usefulness of LES for the prediction of the upwash fountain was shown in the1980’s with pioneering work by Childs and Nixon,11 Childs et al.12 and Rizk and Menon,13 which, althoughlimited by available computing power, was extremely promising.

Previous work by the current authors used LES for simplified impinging jet flow problems that containedthe important fountain and ground vortex flow features, whilst avoiding the difficult mesh generation arounda realistic aircraft.14,15 Comparison with experiment showed improved accuracy over RANS predictions aswell as unsteady ingestion for a case with an intake.

Because of the complex geometry, grid generation is always a challenge, and the current work uses amixed element, unstructured solver to reduce the magnitude of this task. The unstructured grid approach isparticularly useful in this type of problem where high resolution is required underneath the aircraft, whilstalso handling features such as auxiliary intakes. Similarly, the combination of complex geometry and LESgives an extremely large computational problem and it is imperative that calculations are carried out onlarge scale parallel computation facilities.

The aim of this study is to demonstrate the capability of LES when applied to a geometrically detailedHarrier aircraft near touch down.

The following section gives an overview of the numerical methodology and the parallel implementation.The next section contains instantaneous solutions for a Harrier aircraft. Finally conclusions are drawn.

II. Methodology

A. Solution Algorithm

The starting point for this work is the Rolls-Royce CFD code Hydra16 which is an unstructured, mixedelement, compressible, density-based Reynolds Averaged Navier-Stokes solver. The discretisation was im-proved so as to avoid excessive dissipation of resolved eddies and subgrid scale models incorporated. Theimportant features are summarised below, and further details of the discretisation and testing on simplerLES flow problems can be found in Tristanto et al.17

B. Governing Equations

Employing Cartesian tensor notation and the conservative variables (ρ, ρui, E), the governing time dependentequations in terms of spatially filtered, Favre-averaged compressible N-S equations can be expressed as

∂

∂t

∫ ∫ ∫Γ

QdV +∫ ∫

∂Γ

F (Q).ndS +∫ ∫

∂Γ

G(Q).ndS = 0, (1)

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where,

Q =

ρ

ρu

ρv

ρw

E

, F (Q).n =

ρUn

ρUnu + nxp

ρUnv + nyp

ρUnw + nz p

Un(E + p)

,

and G(Q) contains viscous and conduction flux terms. The finite volume discretization provides animplicit filter for the large eddies. whilst ¯ denotes unweighted filtered variables and ˜ density weightedfiltered variables. The spatial filter size is computed at every node from the control volume surrounding thenode. The finite volumes are created from the median-dual of the original unstructured mesh which maycontain tetrahedra, hexahedra, pyramids and prisms.

C. Discretization

For an edge ij that connects nodes i and j, the flux is computed using a second-order accurate scheme ofMoinier18

Fij =12

[F (Qi) + F (Qj)− smoothing

], (2)

The smoothing term is defined as,16

smoothing = |Aij |ε1

(Llp

j (Q)− Llpi (Q)

), (3)

where L is the pseudo-Laplacian and

|Aij | = ∂F/∂Q (4)

For LES it is essential that the smoothing term should be kept as small as possible so as to avoidunphysical dissipation of the resolved eddies. This is achieved by the use the sensor function of Ducros etal.19 to control ε1, based upon the vorticity Ω and divergence ∇ • u

ε1 = max(ε2, ε3

(∇ • u)2

(∇ • u)2 + Ω2

), (5)

where ε2 and ε3 are user defined parameters. The sensor increases the level of smoothing for regions ofhigh divergence and reduces it to a base level of ε2 for regions of high vorticity. In some cases, paticularly atjet impingement, unphysical oscillations were observed in the near wall region and the smoothing was locallyincreased in the cells closest to the wall to damp the oscillations.

Temporal discretisation used a third order accurate, three-stage Runge-Kutta algorithm.20

D. Sub Grid Scale Model

The standard Smagorinsky SGS model defines the subgrid scale viscosity µt as

µt = C2s ρ∆2

√2SijSij , (6)

where the strain rate is

Sij =12

(∂ui

∂xj

+∂uj

∂xi

). (7)

and ∆ is the filter width.For the correct prediction of a laminar flow or the viscous sublayer of a turbulent flow, the SGS model

should tend to zero in these regions. This is not true for a fixed-coefficient Smagorinsky model, in particularin the near wall region where the Sij term becomes large. An improvement on the basic Smagorinsky modelis the wall-adapting local eddy-viscosity (WALE) model, proposed by Nicoud & Ducros21 for LES in complexgeometries. This model is based on the square of the velocity gradient tensor and accounts for the effectsof both the strain and the rotation rate of the smallest resolved turbulent fluctuations, and it also producesthe correct scaling at the wall (νt = o(y3)) without the explicit use of the local wall distance.

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The WALE model defines the subgrid scale viscosity as

µt = C2wρ∆2 OP1

OP2, (8)

where

OP1 = (SdijS

dij)

3/2, (9)

OP2 = (SijSij)5/2 + (SdijS

dij)

5/4, (10)

Sdij =

12

( ∂ui

∂xk

∂uk

∂xj+

∂uj

∂xk

∂uk

∂xi

)− 1

3δij

∂uk

∂xk

∂uk

∂xk. (11)

This model has been demonstrated to be independent of near-wall distance with natural recovery ofsublayer flow conditions; zero subgrid scale viscosity is recovered at the wall without an explicit dampingfunction. Cw is a model constant and for the calculations presented here, Cw has been set to a value of 0.5as recommended by Nicoud & Ducros.21 The filter width, ∆, is determined from the cube root of the controlvolume,

∆ = Γ1/3. (12)

E. Parallel Implementation

The complex geometry and the requirement to resolve small scales, necessarily leads to a large number ofgrid points; coupled with the need to run for a large number of time steps because of the disparity in timescales between the smallest and largest eddies means that these calculations are unfeasible on current singleprocessor machines and must be run on large scale parallel facilities to achieve a reasonable turnaround time.

Whilst structured multiblock CFD codes are relatively straightforward to implement in parallel usingdomain decomposition by block, the unstructured solver requires an efficient partitioning strategy and carefulhandling of the message passing to achieve good efficiency on large numbers of processors. The unstructuredsolver uses the OPLUS library22 with message passing subsequently implemented in MPI. The partitioningis carried out in parallel using the ParMetis library. More information is provided by Hills23 on how theparallel implementation has been tuned for large scale problems and near linear speed-up is demonstratedup to 1024 processors on an IBM Power5 system.

III. Results

A. Test Conditions

Figure 1 shows the 1/15 scale model Harrier which is based on the mould lines of the AV-8B/GR-7 version.The lower half of the model is well detailed, but above the plane of the nozzles, simplifications have beenmade in order to allow the correct feed of air to the nozzles and extraction of air from the intake. The modelincludes auxiliary intake blow-in doors, longitudinal strakes and transverse dams. Whilst the model shown inthe figure includes under-carriage and gun-pods, these were not included in the CFD geometry. The aircraftfuselage datum is at a 7.5 nose up angle to the ground plane. The centre of the intake is 0.591m from theground plane and is representative of an aircraft whose wheels are about to touch the ground.

Model testing uses scaling laws and the conditions tested here are not necessarily representative of thefull scale aircraft. The rear jets are hot at a total temperature greater than 700K and a nozzle pressureratio of 2.0. The front jet temperature is above ambient at 50% of the rear jet temperature and at a slightlyhigher nozzle pressure ratio of 2.5. The nozzle pressure ratios are sufficiently high that they are choked andare mildly under-expanded. The approximate dimensions of the non-circular nozzles are 21x28mm (front)and 38x42mm (rear). The jet Reynolds numbers are 8 million (front) and 7 million (rear). In both cases theReynolds number is based on jet exit conditions and the largest dimension of the non-circular nozzles. Thecrossflow is at 6m/s and the intake static pressure is set to 3kPa below ambient. Even at this model scale,the jet Reynolds numbers are high for large eddy simulation and it will not be possible to resolve accuratelythe small scale turbulence in the initial jet shear layers.

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Figure 1. 1/15 scale model Harrier

Figure 2. Harrier mesh: side view

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Z

X

-0.1 -0.05 0 0.05 0.10.3

0.35

0.4

0.45

0.5

(a) top view

Z

Y

-0.3 -0.2 -0.1 0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(b) front view

Figure 3. Harrier mesh: top and front view

B. Mesh

The mesh is of a mixed element type and is generated using the ICEM Hexa package. The majority ofthe elements underneath the aircraft are hexahedral, with tetrahedral elements used to handle the complexrig geometry at the top of the aircraft. Hexahedral elements were also used in the engine main intake andauxiliary intakes. O-mesh features were used to capture the geometry of the jet nozzles. The nozzle internalflow was not meshed - only flow downstream of the nozzle exit planes was predicted.

Whilst RANS predictions for an un-yawed aircraft can exploit symmetry and only model half the aircraft,LES is instantaneously asymmetric and the complete aircraft must be modelled. Consequently, the meshcontains 17.3 million nodes with 22.6 million elements, of which 15.7 million are hexahedral. The meshspacing in the near wall region and jet shear layers was set to be similar to that found to be acceptable inthe earlier simulations of simplified impinging jet problems.15

The coordinate system is set such that the nose of the aircraft is set to x = 0 with the domain extendingforward by 3m and aft by 3m. The spanwise (z) extent is set to 1.956m from the aircraft centreline and theheight (y) of the domain is 2.184m. These locations are chosen to be representative of the experimental rigat Rolls-Royce, Bristol.

The mesh is illustrated in Fig. 2 and Fig. 3 and more details of similar meshes used for Unsteady RANSpredictions may be found in Richardson et al.10

C. Touch down condition

Some difficulties were encountered in achieving a successful Large Eddy Simulation. In particular the largetemperature gradients created by the interaction of the hot and cold jets seemed to be prone to provokingnumerical instabilities that ultimately caused a floating point error. The large problem size, and hence longrun time makes it difficult to pin down the exact cause of the problem. The solutions presented here haverun for approximately 100, 000 time steps at a time step value of 5 × 10−8s. The small time step is due tothe high speed (supersonic) jet and the fine mesh resolution.

The calculations were run on the HPCx IBM Power5 system with 128 processors. Throughput wasestimated at 300 time steps per hour of wall time, and the results presented here are the equivalent of 43,000CPU hours. The calculation could also be run on a local 64-bit AMD Opteron cluster using smaller numbersof processors (typically 16-24).

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Figure 4. Isosurface of velocity magnitude, coloured by temperature: front view

Figure 5. Isosurface of velocity magnitude, coloured by temperature: rear view

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Figure 6. Isosurface of velocity magnitude, coloured by temperature: top view

Instantaneous solutions are shown to give an overall understanding of the flow-field. Figures 4 to 6 showan isosurface of velocity magnitude at a value of 175m/s; the isosurface is then coloured by the local statictemperature (red indicating hot and blue indicating cold). In this type of multiple impinging jet flow, theturbulence in the jet shear layers is generated by two mechanisms: the instability of a high Reynolds numbershear layer; and the fountain turbulence feeding into the initial shear layer of the jet. For the front jets(Fig. 4), the strakes shield the jet shear layer from the fountain and, in comparison to the rear jets (Fig. 5),have a much lower level of turbulent fluctuation. At this high Reynolds number it would be expected thatincreased mesh resolution in the jets would lead to a more realistic fine scale turbulent structure in the jetshear layers. Nevertheless, significant turbulence is generated in the impingement process and collision ofthe wall jets creating the fountain. Increased resolution of the fine scale jet turbulent structures is unlikelyto have a significant influence on the overall upwash fountain and under aircraft flow-field. It should also benoted that the nose up configuration means that, in terms of jet diameters, the rear jets are closer to theground and so create a stronger fountain.

The fountain can be seen on the centreline of the aircraft, and the green colour indicates that this ishotter than the front jets. The longitudinal strakes and transverse dams are partially successful in divertingthe fountain flow away from the intake - at the left and right intersection of the dam with the strakes(Fig. 4) a region of high velocity (175m/s and above) can be seen streaming off at the angle of the strakes.Nevertheless, there are still patches of high velocity on the fountain centreline forward of the dam whichmay contribute to hot gas ingestion. The influence of the strakes on the upwash fountain can also be seenin Fig. 7.

Figure 6 shows a top view of this isosurface, with the aircraft set to be transparent to avoid obscuringthe flow visualisation. There is a significant temperature gradient across the convoluted interface betweenthe hot and cold wall jets. The interface sweeps back outboard and this is similar to that observed inexperimental surface flow visualisation. The symmetry apparent in this isosurface structure is not realisticof a true turbulent simulation; experience with the the simpler impinging problems indicate that running fora larger number of timesteps usually breaks the asymmetry. This calculation has not been run for sufficienttime steps to achieve a true asymmetric large eddy simulation.

An isosurface of the positive invariant of velocity divergence (Qc)24 identifies vortex cores and allows a

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(a) front nozzles (b) aft nozzles

Figure 7. Fountain flow interactions with dam and strakes

Figure 8. Turbulent structures, Qc: front view

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Figure 9. Turbulent structures, Qc: top view

better visualisation of the turbulent structures. The individual features shown in the isosurfaces of Fig. 8and 9 each represent the core of a vortical structure. Again, these isosurfaces are coloured by temperature.Whereas the isosurface of velocity magnitude shows a ‘wrinkly’ front jet, this technique highlights thestructures formed early near the jet nozzle. Turbulent structures are also seen in the wall jet and fountain.Longitudinal structures are visible at the edge of the strakes - these are due to the fountain being deflectedby the strakes and the generation of vorticity at the sharp edge. This structure would be visible in themean flow and is not due to turbulence. Similarly, around the lip of the intake a local acceleration creates astructure that would be visible in the mean. Of more significance is the structure identified in the centre ofthe intake which is an unsteady feature emanating from the fountain that has been sucked into the intake.The turquoise colour indicates that this is 20-30K above ambient and would be a significant hot gas ingestionevent.

Figures 10 and 11 show the static temperature at a longitudinal cut displaced outboard from the aircraftsymmetry plane, a transverse cut in the intake and on the groundplane. As noted earlier, the physicalsimulation time for this simulation is very short and the solution is still symmetrical. The colour scale inFig. 11(a) differs to that in Figures 10 and in this case the peak temperature rise in the lower half of theintake corresponds to a temperature rise of 30K above ambient. The main route for ingestion is directlyunderneath the intake corresponding to flow penetrating past the dam. A comparison of a typical exper-imental instantaneous temperature distribution from Richardson et al.10 shows a similar, but asymmetricdistribution.

IV. Conclusions

A successful Large Eddy Simulation of a model Harrier at touch down has been carried out using aparallel unstructured CFD algorithm. The large problem size coupled with a requirement for small timesteps means that insufficient samples have been generated for a statistically valid mean. Nevertheless,important phenomena have been observed in the instantaneous flow, in particular the turbulent structuresin the fountain entering the intake and the influence of the strakes and dam on controlling the upwashfountain.

Whilst there is still an issue regarding the small time step needed for stability, the results show that LEScould be a practical tool to model HGI for the Joint Strike Fighter aircraft and, more generally, indicatethat LES can be applied to practical engineering problems that require detailed geometrical resolution.

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(a) side view (b) ground plane

Figure 10. Instantaneous total temperature

(a) prediction (b) typical experiment10

Figure 11. Instantaneous total temperature at intake face

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V. Acknowledgements

Funding for this project was from the UK Engineering and Physical Sciences Research Council (EPSRC)grant GR/S21021/01. The computer time was provided through the UK Applied Aerodynamics Consortium(UKAAC) under EPSRC grant GR/S91130/01.

The authors would like to thank Dr. Nick Nills at the University of Surrey for guidance on carrying outthese calculations in parallel.

References

1Curtis, P., “A review of the Status of Ground Effect/Environment Technologies,” No. 2002–5985, Nov. 2002, 2002 BiennialInternational Powered Lift Conference and Exhibit, Nov. 2002, Williamsburg, Virginia.

2Seddon, J. and Goldsmith, E., Intake Aerodynamics, Blackwell Science Ltd, 2nd ed., 1999.3Lewis, W. and Hurd, R., “Augmented Vectored Thrust Engines and the Problem of Avoiding Hot Gas Recirculation,”

1979, ASME Gas Turbine Conference, March 12-15, 1979, San Diego, California.4Hill, W. and Jenkins, R., “Effect of Nozzle Spacing on Ground Interference Forces for a Two-Jet V/STOL Aircraft,”

Journal of Aircraft , Vol. 17, No. 9, Sept. 1980, pp. 684–689.5Page, G. J., McGuirk, J. J., Harper, L. R., and Penrose, C. J., “Application of Computational Fluid Dynamics to Hot

Gas Ingestion Modelling,” Aug. 1998, Proceedings of the Royal Aeronautical Society International Powered Lift Conference,London, UK.

6Chaderjian, N. M., Pandya, S. A., Ahmed, J. U., and Murman, S. M., “Progress Toward Generation of a Navier-StokesDatabase for a Harrier in Ground Effect,” No. 2002–5966, Nov. 2002, 2002 Biennial International Powered Lift Conference andExhibit, Nov. 2002, Williamsburg, Virginia.

7Pandya, S. A., Murman, S. M., and Sankaran, V., “Unsteady Computations of a Jet in Crossflow with Ground Effect,”No. 2003–3890, June 2003, 33rd AIAA Fluid Dynamics Conference and Exhibit, June 2003, Orlando, Florida.

8Richardson, G. A., Dawes, W. N., and Savill, A. M., “Hot Gas Ingestion Modelling for the Vertical Descent Phase of aHarrier,” No. 2005–5338, 2005, 17th AIAA Computational Flow Dynamics Conference; Toronto, Ontario; Canada; 6-9 June2005.

9Richardson, G. A., Dawes, W. N., and Savill, A. M., “CFD Analysis of Hot Gas Ingestion Mechanisms for the VerticalDescent Phase of a Harrier Aircraft,” No. GT2006–90766, 2006, ASME Turbo Expo 2006, May 8-11, 2006, Barcelona, Spain.

10Richardson, G., Dawes, W., and Savill, M., “Unsteady, Moving Mesh CFD Simulation Applied to Harrier VSTOL Aircraftfor Hot-Gas Ingestion Control Analysis,” The Aeronautical Journal , Vol. 111, March 2007, pp. 133–144.

11Childs, R. and Nixon, D., “Unsteady Three-Dimensional Simulations of a VTOL Upwash Fountain,” No. 86–0202, Jan.1986, AIAA 24th Aerospace Sciences Meeting, January 6-9, 1986, Reno, Nevada.

12Childs, R., Nixon, D., Kuhn, G., and Perkins, S., “Further Studies of Impinging Jet Phenomena,” No. 87–0017, Jan.1987, AIAA 25th Aerospace Sciences Meeting, January 12-15, 1987, Reno, Nevada.

13Rizk, M. and Menon, S., “An Investigation of Excitation Effects on a Row of Impinging Jets Using Large-Eddy Simula-tions,” No. 88–0043, Jan. 1988, AIAA 26th Aerospace Sciences Meeting, January 11-14, 1988, Reno, Nevada.

14Page, G. J., Li, Q., and McGuirk, J. J., “LES of Impinging Jet Flows Relevant to Vertical Landing Aircraft,” No.2005–5226, 2005, 23rd AIAA Computational Flow Dynamics Conference; Toronto, Ontario; Canada; 6-9 June 2005.

15Li, Q., Page, G. J., and McGuirk, J. J., “Large Eddy Simulation of Twin Impinging Jets in Cross-Flow,” The AeronauticalJournal , Vol. 111, March 2007, pp. 195–206.

16Crumpton, P. I., Moinier, P., and Giles, M. G., “An Unstructured Algorithm for High Reynolds Number Flows on HighlyStretched Grids,” 1997, Tenth International Conference on Numerical Methods for Laminar and Turbulent Flow.

17Tristanto, I. H., Li, Q., Page, G. J., and McGuirk, J. J., “On the Effect of Convective Flux Formation for LES ofCompressible Flows using Hybrid Unstructured Meshes,” No. 2006–54940, 2006, 36TH AIAA Fluid Dynamics Conference, SanFrancisco, California, June 5-8, 2006.

18Moinier, P., Algorithm Developments for an Unstructured Viscous Flow Solver , Ph.D. thesis, University of Oxford, 1998.19Ducros, F., Ferrand, V., Nicoud, F., Weber, C., Darracq, D., Gacheriu, C., and Poinsot, T., “Large-Eddy Simulation of

the Shock/Turbulence Interaction,” Journal of Computational Physics, Vol. 152, 1999, pp. 517–549.20Gottlieb, S. and Shu, C. W., “Total Variation Diminishing Runge-Kutta Schmes,” Tech. Rep. 96-50, ICASE, 1996.21Nicoud, F. and Ducros, F., “Subgrid-Scale Stress Modelling Based on the Square of the Velocity Gradient Tensor,” Flow,

Turbulence and Combustion, Vol. 62, 1999, pp. 183–200.22Burgess, D., Crumpton, P., and Giles, M., “A Parallel Framework for Unstructured Grid Solvers,” Programming Envi-

ronments for Massively Parallel Distributed Systems, 1994, pp. 97–106, eds K.M. Decker and R.M. Rehmann, Birkhauser.23Hills, N., “Achieving High Parallel Performance for an Unstructured Unsteady Turbomachinery Code,” The Aeronautical

Journal , Vol. 111, March 2007, pp. 185–193.24Jeong, J. and Hussein, F., “On the Identification of a Vortex,” Journal of Fluid Mechanics, Vol. 285, 1995, pp. 69–94.

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