Towards Robust CFD Based Design Optimisation of Virtual Engine

NATO/PFP UNCLASSIFIED

Towards Robust CFD Based Design Optimisation of Virtual Engine Shahrokh Shahpar4 Rolls-Royce plc PO Box 31 Derby DE24 8BJ, United Kingdom [email protected]

ABSTRACT TITLE Computational Fluid Dynamics (CFD) has become an indispensable tool in designing turbomachinery components. This paper reviews various sources of uncertainty in CFD based design methodology. There is a hierarchy of tools used at various stages of design, the level of accuracy needed and computational resources available dictate the choice of simulation codes. This paper focuses on turbomachinery blading; the importance of physical, numerical and geometrical fidelity is illustrated through a case study of a high-pressure turbine stage. A high fidelity design optimisation framework called SOPHY is used to provide parametric geometry, automatic meshing, advanced design-space search algorithms, accurate and robust CFD methodology and post-processing. Deterministic optimisation relies on CFD to capture the trends accurately. The significance of including the so-called real geometry, capturing secondary flow features and interaction of turbomachinery components in the optimisation cycle are discussed. In the last section of the paper, a robust optimisation methodology has been developed to mitigate the effects of uncertainties in simulation codes and external tolerances.

1.0 INTRODUCTION In order to reduce the costs of development of a new engine, increase the performance of a highly complex propulsive system and minimise the impact of aviation on the environment sophisticated simulation and design optimisation tools have been developed. The European union [1] has set the following stringent design goals for the aerospace community, to be achieved by 2020: 50% reduction in CO2 emissions from aircraft per passenger kilometre, halving of the perceived aircraft noise; 80% reduction in emissions of NOx; and, a five fold reduction in the average accident rate of global operators. Improvements in engine and aircraft aerodynamics reduce fuel-burn and lead to lower operating costs. Historically Rolls-Royce Trent engines have improved the fuel consumption per unit thrust at cruise at a rate of 1% per year [2], see figure 1. For a large passenger aircraft a 1% reduction in specific fuel consumption could save 560 tonnes of fuel per annum and reduce direct operating costs by 0.5% [3].

Figure 1: Historical Fuel Consumption Reduction. 4

RR Associate Fellow – Aeorthermal Design System – C.Eng., FRAeS., AFAIAA.

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Towards Robust CFD Based Design Optimisation of Virtual Engine Multi-disciplinary computer based simulation such as Aero-thermo-dynamic, aero-acoustic, aeromechanical simulation embedded in an automatic robust optimisation cycle are key technologies that can achieve the aforementioned EU objectives. Figure 2 illustrates the so-called “gas washed” surfaces and secondary air systems where CFD (Computational fluid Dynamics) is used to design and optimise components of a modern jet engine. Bypass Guide Vanes

Nacelle-Intake • Aerodynamics (Droop, Scarf angle, …) • Crosswind effects • Installed engine/pylon/wing interaction • Ground effect • Ice accretion and shedding

Combustion

• 3D Blade Design •circumferential OGV pattern design •Pylon - RDF - A Frame Interaction.

• NOx –CO2- Emission • Fuel injector and combustor cooling flows •Two-Phase Combusting flow • External aerodynamics

Noise Engineering

• Jet Noise

Fan

Exhaust

•3D aerodynamics • Fan flutter • Fan/OGV/pylon interaction • IGV forced response • Sand-hail indigestion

•Afterbody flow •Nozzle, mixer, jet flow

Turbine

• 3D Aerodynamic Design of

Noise Engineering

• Tone, Buzz saw, Broadband

Compressor

• 3D Aerodynamic Design • Multistage aerodynamics • Unsteady rotor/stator flow • Annulus leakage flow • Casing Treatment • ESS, LP stages ice accretion

Engine Systems

• Rotating disc cavity flows • Brush and labyrinth seals • Secondary air system losses • Thermal Matching • Pre-swirled cooling air system • Bearing Chambers

Blade, Endwall and Fillet • Multi-stage Aerodynamic • End wall and blade heat transfer • Film cooling • Internal Cooling • Rotor shroud leakage • Rim seals • Unsteady vane/rotor flow • Forced response

Figure 2: Aero-Thermo-Dynamic-Acoustic CFD based Optimisation Opportunities.

Indeed the performance improvements to date have only been possible through the use of advanced simulation techniques. This has given significant reductions in product design times and costs, with fewer options needing Rig testing. This is dramatically demonstrated in the table below derived from two of the Rolls-Royce engine design data.

Number of Compressor Tests

Number of Turbine Tests

RB211-524 (entry into service 1977) IPC 17 builds of rig test HP compressor 7 builds of rig test Trent 900 (entry service 2006) IP Compressor 1 rig test HP Compressor - no rig test

RB211-524 (entry into service 1977) HP model turbine 28 tests HP engine parts rig 63 tests Trent 900 (entry service 2006) HP Turbine - no rig test IP Turbine - no rig test

Table 1: Comparison of Number of Rig Tests Needed During Two RR Engine Programs [4].

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Towards Robust CFD based Design Optimisation of Virtual Engine If a rig test is still needed, CFD can help improve the productivity of the experimental methods, e.g. predicting the optimum locations of the strain or heat transfer gauges. A better understanding of flow phenomena will reduce development risk and allow rapid performance optimisation to be carried out. The ability of CFD to correctly predict the flow phenomena is hence fundamental in achieving a CFD based Virtual Engine. However, many uncertainties affect a CFD simulation, theses errors need quantifying, and the design process has to develop a methodology by which these uncertainties are taken into account in the optimisation process leading to a robust design. Failing to do so could lead to a high cost of recovery as illustrated in figure 3 [5]. Notice that the vertical axis is logarithmic. £M

Cost of Recovery

100 By Service Experience

10 By Engine Test By Rig Test

1 By Analysis

1

2

3

4

5

Years

Time to Recover Figure 3: Verification of Propulsion Design.

This paper reviews some of the sources of uncertainties that affect the reliability of a CFD based design process and recommends techniques that will lead to a robust deign optimisation process. Most robust design strategies proceed in three stages:

Identifying, qualifying and quantifying the sources of uncertainty related to CFD

Uncertainty propagation through the simulation system leading to the introduction of nondeterministic methodologies into the CFD simulation

Robust CFD based optimisation where not only objectives but also their variations are minimised.

The above steps are expanded in the following sections.

1.1 Uncertainty Parameters in CFD-based Simulation Although not exhaustive, the following categories should be considered: •

Physical Fidelity o Boundary conditions o Geometrical modelling, e.g. Fillets, Penny, Shroud,..etc

•

Numerical Fidelity o Code implementation, hence verification o Discretization error (2nd order, TVD, Compact method, ..) o Mesh dependencies o Turbulence & transition modelling, Wall functions

•

Modelling Fidelity o Potential ->Euler->NS->URANS->LES->DES-> DNS o Steady vs. Unsteady

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Towards Robust CFD Based Design Optimisation of Virtual Engine It would be outside the scope of this paper to discuss in detail all the sources of simulation modelling errors; however through a number of case studies attention is paid to the fidelity of geometrical modelling needed and mesh dependencies issues. All the above factors are determinate, i.e. one can change the mesh, the turbulence model or the boundary conditions and see the effect on the solution. However, one can never truly know the exact boundary condition, let’s say in a running engine or likewise know the true shape of the engine components. However, using a probabilistic modelling, one can arrive at the most likely running shape and the most likely boundary conditions, this topic will be further discussed in section 5 of this paper.

2.0 TURMOMACHINERY BLADE DESIGN

Complexity / Computational time

As was illustrated in figure 2, the aerothermal design of a modern gas Blading Design (3D) Single/Multi Row, Unsteady turbine comprises of many 3D Multistage including components, such as intake, Real Geometry Features turbomachinery (fan, compressor, 3D Multistage Steady and turbines), combustion flows, heat transfer, internal flows, exhaust Blading Design (3D) Single Row, Steady nozzle and nacelle. Arguably, one of Blading Design (2D) the most challenging aspects of S1/S2 design relates to the turbomachinery Throughflow Design blading. The aerodynamic design of (Performance Prediction, Viscous) which relies on a range of tools from Preliminary Design 1D Design/Prediction preliminary design analysis, to axisymmetric through-flow methods Accuracy to the full unsteady 3D Navier-Stokes Figure 4: Flow Models Used in the Turbomachinery Design [6]. based CFD solvers. As shown in figure 4, the predictive accuracy of the blade performance increases as the complexity of the flow modelling increases. However, this increase in complexity is usually associated with higher computational costs. It should be emphasised that the current design process makes use of all the listed tools at different, appropriate stages of design. It is not realistic to expect one code to cover all level of geometrical and physical fidelity, however key systems have been evolved that efficiently link various simulations codes over a range of applications. There are three main stages: Preliminary design, Through-flow design and Blading design. In preliminary design the basic outline of the turbomachine is established such as number of stages, the annulus shape and overall length. At this stage fast iterations are required to change the arrangements to fit in with whole engine requirements such as total thrust, weight and specific fuel ratio. The gas path is defined based on the velocity diagrams at meanline which would deliver a certain flow capacity, pressure ratio and surge margin for a range of engine operating points. The design methodology relies heavily on correlations and mature databases as well as one-dimensional flow solvers. The radial dimension is introduced in the throughflow design phase. Spanwise whirl distribution is selected and stage matching is established. The endwall boundary layers and their blockage as well as spanwise mixing are taken into account. At this stage the detailed velocity diagram can further be iterated upon to achieve the design goals set in the preliminary stage. The Rolls-Royce through-flow program is based on streamline-curvature methods and specified loss models. Currently the blading design i.e. generating aerofoil profiles starts in 2D using the data from the throughflow program along the stream sections. The through-flow data is then divided into a number of sections along the span. The design intent is to match the air angles and gas properties designed by the through-

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Towards Robust CFD based Design Optimisation of Virtual Engine flow program. Although, the blade-to-blade design is 2D in essence the variation of the stream-tube height is taken into account for each stream section. The blade-to-blade programs are predominantly based on Euler plus an integral boundary-layer method [58] because of transitional boundary layers or 2D NavierStokes calculations with some turbulence model closure. These methods are relatively fast on modern computers requiring no more than a few minutes of computing time to simulate the flow through the blade’s passage. Inverse design routines are also available whereby the designer may specify the required surface Mach number distributions and the program calculates the aerofoil shape that delivers the specified flow. However, the accuracy of two-dimensional flow predictions is compromised where there are significant 3D flow features such as secondary flow effects especially near endwalls or strong shockboundary-layer interactions, e.g. as shown in Fig. 5. These flow features are three-dimensional in nature and can only be accurately predicted by solving the three-dimensional Navier-Stokes equations. The true turbomachinery flows are three-dimensional, unsteady because of the relative motion between successive blade rows and viscous effects play a dominant role due to boundary-layer separations, reverse flow and secondary flows. To simulate the flow as accurately as possible, there has been a great deal of effort for multi-stage and unsteady flow calculations. However, due to their high computing costs these methods are used towards the end of the design process to check the overall performance and matching rather than being used iteratively in the design phase. However, it is well known that almost 80% of design freedom is frozen in the first preliminary phase; hence as shown in figure 4 it is desirable to introduce high fidelity tools much earlier in the design process. It has been the aim of author’s research [78] on the use of optimisation and Response Surface methodology (RSM) to bring the more complex aerodynamic models earlier into the design cycle. With the relentless increase in computing power, it is now possible to include the steady three-dimensional Navier-Stokes CFD methods in the design loop. There are still many sources of uncertainty associated with 3D CFD methods such as transition modelling, turbulence modelling and minuscule region modelling for example as required for each film cooling hole on a HP turbine blade, detailed discussions of which are all outside the scope of this paper. Rather disappointingly, despite 30 years of research in the scientific community, there is still no universal turbulence model available and all CFD codes have shortcomings to model turbulent flows. Suffice to say that although designers cannot solely rely on the accuracy of CFD, it has become reliable enough to rank the relative performance of the designed geometries, as long as the simple model can adequately capture first order physical and geometrical phenomena. For simple blading applications where the flow is largely attached, the one-equation Spalart-Allmaras (SA) [9], even simple algebraic eddy viscosity models scaled with the correct length scale are good enough for design and for many optimisation methodology that mainly rely on accurate ranking. However, as more “real geometry” flows are introduced, scale becomes important, e.g. in the shear layers, hence a simple SA model which has been developed for non-separated flows may not be appropriate. In wall-bounded flows, small cells are required to resolve the flow; hence large computational resources are required to solve the NS equations directly, the Re3 computational requirement acts as a barrier to use DNS in the design loop, however, for non-bounded flows such as jets and combustion chambers, the LES plays an important simulation tool. For combustion flows, only LES (and DNS) is able to model the variation of Prandtl number; and the complex unsteady flow due to interacting jets. For realistic Re research is on going to make use of a hybrid LES-(U) RANS, e.g. in the propulsive jets [11]. The true flow though a gas turbine engine is unsteady. The interaction of multi-row turbomachinery blades can be taken into account by making use of a mixing plane (MP) analysis or in an unsteady mode running a sliding plane (SP) analysis. At start of each time iterations, the primitive variables are interpolated form one mesh to the next one. However, the unsteady SP approach requires significant larger CPU resources. For steady multi-stage analyses, the MP approach is often used, however, the implementation of the mixing plane boundary condition can significantly affect the results. The circumferential averaging is carried out so that the fluxes, e.g. mass, momentum, and energy is transferred in a conservative way. The assumption is that the flow quantities mixed out over one cell row. Characteristic boundary conditions are RTO-MP-AVT-147

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Towards Robust CFD Based Design Optimisation of Virtual Engine often used to avoid reflection of spurious waves into the domain, however; clearly the mixing plane approach does not represent the wakes going through the interface plane. This could affect the level of losses predicted [59] compared to time-averaged unsteady solutions due to wake loss augmentation and in some cases the level of CFD convergence due to trailing edge vortex shedding [60]. In some cases, e.g. high lift LP turbine blades [61], and OGV diffuser case [62] the inclusion of the wake is crucial to represent the true physics of the problem. On the former, the unsteady wake passing and interaction would lead to a higher lift that can otherwise be attained (due to the control of the size of a separation bubble on the aerofoil’s pressure-side), on the latter, the turbulent wake will interact with the diffuser boundary layer keeping it attached longer. The inadequacies of the MP approach had prompted some researchers to make use of an overlapping mesh to transfer body forces and deterministic stresses between the adjacent blade row [63]. However the overlapping grid approach is also CPU intensive compared to MP approach. The treatment of exchange of boundary conditions is another source of error in 3D multistage design, however, it will not be discussed in detail in this paper. Referring to figure 4, CFD design is about using imperfect tools at various stages of design due to their requirements of computational resources, robustness and accuracy.

2.1

Aerodynamic Design – A Complex Task

Figure 5 illustrates the three-dimensional viscous effects that occur in a turbine stator blade. The turbine endwall secondary flows are dominated by the inlet boundary layer which rolls up at the aerofoil leading edge to form the “horseshoe” vortex, this vortex interacts with the main passage vortex which sweeps to the suction side of the adjacent blade row, although not shown here, film cooling injection, shroud and other cavities bleeds further complicate the flow. Figure 5b shows the cooling air that flows from the internal passages that ejects to the surface to produce a protecting thin film-cooling sheet. The internal tubulators and impingement tubes designed to enhance mixing and hence heat transfer all need to be modelled to achieve at least the aforementioned “first-order effects” such as flow angle, total mass and capacity predictions.

Cooling (a)

air

(b)

Figure 5: Turbine Secondary Flow Model after Takeishi et.al. [12], Internal Cooling Passages of a Turbine Blade [13].

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Towards Robust CFD based Design Optimisation of Virtual Engine Depending on the design requirements, high fidelity geometry and physical modelling based on the RANS/URANS equations is required to adequately model such flows. Both of these complexities require a significant increase in computational resources. The trend is to move towards massively multi-core machines that need simulations that are easily parallelisable. As the order of complexity of modelling increases, the inherent non-linearity of the problem demands a more systematic approach to simulationbased design. More intelligent, automatic design search algorithms are needed in the design loop to help designers cope with a vast design space, dealing simultaneously with many conflicting objective functions and constraints in ever decreasing design time.

2.2 The SOPHY Design System SOPHY is the Rolls-Royce’s fully integrated, automatic aerothermal design system. It consists of three modules: SOFT, PADRAM, and HYDRA. It has been developed in-house and in conjunction with a number of its UTCs (University Technology Centres), hence it is completely license free for RR internal use. SOFT can be deployed on a range of operating system, e.g. UNIX, Linux, Interix and Windows (based on a QT interface). SOFT contains four advanced optimisation libraries as following: 1) Local and global optimisation library, e.g. SQP (sequential quadratic programming), ASA (Adaptive Simulated Annealing), evolutionary type methodologies such as MOGA (Multiobjective Genetic algorithm); 2) A DoE (a range of Design of Experiment) library, e.g. Latin Hypercube, Taguchi and Lptau; 3) A RSM (Response Surface Methodology) library, e.g. RBF (Radial basis function) and Krigging and Co-krigging (where adjoint gradients are used to produce a more accurate global RSM); and 4) ANOVA (statistically based ANaylsis Of VAriation), where the interaction between design parameters as well as ranking and screening of the design parameters can be carried out. For more details on the SOFT libraries readers are referred to references [14] and [15]. The flow chart of the SOPHY system is shown in figure 6, the SOFT system provides the user with a graphical interface for constructing the workflow, such as the one shown here, which defines the sequence of operations from the setting of design parameters to the evaluation of the cost and constraint functions. The GUI also provides a visualisation capability for plotting the progress of the optimisation. The designer can interactively change the design parameters, optimisation strategy and set multi-level optimisation runs. It is found that the automation of the design process is often as beneficial as the optimisation run itself. The SOFT GUI also allows for distributed runs, e.g. running the GA and DoE population on a heterogeneous PC cluster. As illustrated in figure 6 both Hydra and SOFT codes can both be run in parallel. Data Base Base design

Mesh Quality Checks – Mesh Adaptation

SOFT PADRAM/ UG

Preprocessing

Design parameters

Optimize HYDRA Cost constraints

New design

Yes Additional design parameters/ cost

Actran ,…

OK?

DoE Sensitivity

Sc03

(b)

No

Optimum design

Design Review

Figure 6: Flow Chart of the SOPHY Integrated Design System [14]. RTO-MP-AVT-147

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Towards Robust CFD Based Design Optimisation of Virtual Engine The second key element of the SOPHY system is PADRAM [16], which stands for Parametric Design and Rapid Meshing system. It is a unique system in RR where the geometry manipulation and parametric meshing are integrated into one system, hence avoiding the problems associated with import and export of geometry for meshing. Nevertheless, PADRAM can export IGES and STL file format onto CAD systems such as UG NX and ProEngineer for further analysis and/or manufacturing. The PADRAM design space consists of three libraries:

Engineering parameters,

Orthogonal bump functions

FFD (Free Form Deformation).

Engineering parameters should be viewed as global super parameters acting on the aerofoil/blade as a whole and draw from the engineers’ knowledge about the geometry effects on the flow, e.g. SKEW, STAGGER for the blade, DROOP, and/or SCARF angle for a nacelle, etc. The bump function in PADRAM is based on the Hicks and Henne [19] shape functions, which have smooth properties, see figure 7. The idea of using shape perturbations stems from the fact that it may be difficult to define a Figure 7: PADRAM Bump Function Design typical blunt leading edge, high turning & high Space and a Typical Hybrid Mesh. curvature Turbine blade with low order splines, and the fact that there is already a good range of starting aerofoils in the blade bank to choose from. The PADRAM mesh generator can, very rapidly, produce good-quality viscous meshes for multi-passage turbomachnery using 2D, quasi-3D or full 3D blade geometry. PADRAM makes uses of both algebraic and elliptic grid generators to create hybrid C-O-H meshes [16]. An orthogonal body-fitted O-mesh is used to capture the viscous region in the vicinity of the blade(s), see Figs. 7 & 8. A C-mesh is used for any bifurcations in the domain, such as pylons and drive fairings in bypass duct applications. Any of the PADRAM multi-block structured meshes can be replaced with hybrid meshes, the internal block boundaries created for the structured mesh can also be removed and blocks merged to improve the hybrid mesh generation, see Fig. 7 for a typical PADRAM hybrid mesh. In section 3 the effects of mesh topology and quality on CFD solutions are further discussed and highlighted.

(a)

(b)

(c)

Figure 8: PADRAM and HYDRA Applications: a) Civil Aerospace, where the Bypass OGV, Splitter, Pylon and RDF are Shown, HYDRA Solution - contours of Static Pressure [17], b) Industrial Applications, where the OGVDiffuser and the Exhaust System are Meshed, HYDRA solution where the streamlines Indicate Two Main Vortices Being Convected Downstream [15] c) A Marine Water Jet Impeller & Guide vane - HYDRA Solution (Contours of Static Pressure) [15]. -8

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Towards Robust CFD based Design Optimisation of Virtual Engine Hydra is a suite of non-linear, linear and adjoint solvers being developed collaboratively by Rolls-Royce plc. and its University Partners [20, 21]. Hydra is a general-purpose CFD code developed for hybrid unstructured meshes which uses an efficient edge-based data structure [22]. The multiblock PADRAM meshes are converted into this structure using a pre-processor. The flow equations are integrated around median-dual control volumes using a MUSCL based flux-differencing algorithm. Turbulence is modelled using one-equation Spalarat-Allmaras [9], k-ε, or k-ω turbulence model [10]. The discrete equations are preconditioned using a block Jacobi preconditioner [23] and iterated towards the steady state using the 5stage Runge-Kutta scheme [24]. Convergence to steady state is further accelerated through the use of an element-collapsing multi-grid algorithm (refer to as JM56) [25]. The flow solver runs in parallel on both shared and distributed memory machines using domain decomposition. The parallel multigrid capabilities are essential for generating high-fidelity CFD solutions in acceptable time for effective use of optimisation strategy presented here. The SOPHY system has been used [15] on a range of Aerospace, Industrial and Marine applications, a typical example from each sector is shown in figure 8. Finally it should be noted that SOPHY system is developed in a modular form and although SOFTPADRAM-HYDRA represent a fully integrated design system, any one module can be replaced with other in-house or commercial codes, e.g. HYDRA can be replaced with FLUENT>, the SOFT library can be accessed by a different workflow manager e.g. from Model Centre> or the optimisation library can be replaced by another optimisation software, e.g. Mod Frontier> or iSIGHT>; and PADRAM can be replaced with ICEM>, e.g. as demonstrated in reference [18].

3.0 MESH GENERATION ISSUES As mentioned in section 2 there are many sources of uncertainties in CFD simulation of airflows in an engine. In this section some effects of grid shape, resolution and topology are discussed. Discretization of the flow domain often requires a mesh to be generated, which is structured or unstructured in nature. It has been found that body fitted grid produce better results as the boundary conditions are better reinforced and general dicretization errors are reduced. Experience has shown that a good grid has the following features:

Orthogonal or near orthogonal grid lines near walls (Skew angle is often used to monitor this)

Grid clustering where it is needed (e.g. to resolve high curvature regions, high gradient regions, and to capture shock waves), otherwise aim for isotropy

Not too excessive aspect ratio (very much code and flow dependant, ideally should be one, however values of O10 is OK, O1000 is not)

Not too excessive expansion ratio (code and flow dependant, keep < 30%)

Sufficient density (hence fit for the purpose, again very much code and flow dependent).

Sufficient Grid smoothness (no abrupt change in grid direction, or density, this minimises the discretization and truncation errors, note that most modern codes are 2nd order on a smooth mesh)

Orthogonal grids help to reinforce the CFD boundary conditions and minimises the discretization errors in the domain. Given a finite domain, there are different ways of generating a mesh, for example consider a >

For the sake of completeness names of some COTS products are given here, this is by no means an endorsement of these codes by the company, or an exclusive list. RTO-MP-AVT-147

Figure 9: Typical Letterbox Meshes Constant Axial Mesh and Curvy Type Mesh for a Turbine Blade. Plenary 2 - 9



Towards Robust CFD Based Design Optimisation of Virtual Engine Turbine blade section, shown in figure 9. A turbine blade has typically a blunt leading edge (LE) and considerable turning, hence large curvature variation along its surfaces. The easiest mesh to produce for such a passage is a single block mesh with constant axial lines (top picture of figure 9), however excessive shearing is evident near the LE. Although a single block curvier, letterbox type mesh chosen in the bottom picture of figure 9 alleviates some of the skewness (depending where the corner points on the aerofoil are chosen to be), it is far from perfect. A more suitable mesh is a body fitted, O mesh shown in the bottom of figure 10.

Corner points

Leading edge detail

Trailing edge detail

Figure 10: Top: Typical Letterbox Meshes (LE-TE regions) – Bottom: Typical Body fitted O mesh, Hydra Solution: Contours of Velocity – Purple Indicate a Low Value.

The HYDRA RANS solutions for a typical compressor blade on two types of mesh are shown in figure 10. The velocity contours on a relatively coarse, letterbox, mesh indicate that there are excessive thickening of the boundary layer, on the other hand the boundary-layer growth in the vicinity of the LE on the O-mesh, even for a slightly coarse mesh is significantly smaller and although not shown here matches the fine mesh resolution solution better. It should be pointed out that a different in-house NS solver based on a pressure-correction algorithm performs quite adequately on the letterbox meshes presented here. Hence, it is concluded that different solvers perform better on their own preferred meshes. It is a common mistake, and source of errors, where designers switch a solver to run on a mesh which has been calibrated for a different solver. For example, a code based on an edge based scheme may require a much smaller near wall spacing than a code based on cell centred finite volume scheme, although a similar wall function may have been implemented in both codes. In some COT CFD codes, in order to improve the robustness of a code, it may also be the case that the boundary conditions and numerical scheme automatically switch depending on the mesh used, e.g. from a no slip boundary condition formulation to a slip boundary condition where a mesh is too coarse without the user knowing about it. Hence will not be surprising if another solver which rigoursly enforces the no-slip boundary conditions will suffer from reaching an acceptable accuracy or even convergence when running on a mesh that is not suitably generated. Similarly, solvers could switch from a 2nd order scheme to a first order one if the residuals are too high or difficult to reduce. Hence, it is concluded that a CFD code should not be used as a black box and designers must be aware of all the switches external or internal.

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Towards Robust CFD based Design Optimisation of Virtual Engine

(a)

(b)

(c)

Figure 11: Different style of PADRAM meshes for a turbine blade. a) and b) multi-block structured, c) Hybrid mesh with wake clustering.

The TE flow for the O-mesh solution shown in figure 10 does not show significant variations between the two types of meshes shown. Although the mesh corner points put in a critical region of the flow where the flow is accelerating around the leading edge may not seem sensible, a similar conclusion may not be drawn for the TE solution presented here, where the mesh corner points may actually help the flow to separate at a critical point. In order to improve the TE solution, three different mesh topologies are considered as shown in figure 11. Figure 11a represent a mesh that has zero inlet and exit angle, hence avoids excessive shearing in the passage H mesh, especially on the suction side near the TE, figure 11b illustrates a mesh that the righthand-side H-grid is aligned with the blade metal angle, hence producing a better mesh for the wake at the expense of higher shearing angles in the blade passage, and finally in figure 11c, a hybrid mesh is shown consisting of an isotropic looking mesh in the blade passage, but has a specified clustering line source downstream of the TE in order to capture the wake better. Numerical investigations have revealed that the mesh in figure 11b although has a higher number of skew cells in the blade passage produces marginally better results (nearer to the fine mesh solution), mainly because of aligning the downstream H mesh with the wake. During this investigation, however, it was revealed that special care must also be given to the Fish Tailing phenomenon, which is discussed next.

3.1 Fish Tailing The imbalance of lift on the trailing edge of the aerofoil is referred to as fish tailing. As shown by the surface pressure plot in figure 12. Investigations with a number of modern COTS and in-house RANS solvers on a similar mesh reveal that they all suffer from the same phenomena, however, to a slightly different degree depending on which turbulence modelling is used. Note that due to mechanical and thermal life requirements turbomachinery blades have blunt TE. For a blunt TE, the true physics should dictate an unsteady Karman type vortex shedding, however, various experimental evidence suggests that for a range of aerofoil considered here the mean pressure distribution should not exhibit the significant pressure imbalance seen in the simulation. A systematic investigation that has been presented in reference [26] reveals that a better mesh resolution would alleviate this problem leading to exit flow angles matching the data. Higher mesh resolution helps to resolve the TE double vortices better as well as leading to a smaller y+ in the TE region. Predicting the flow angle is extremely important for turbmachienry, as the overall turning dictates the level of pressure rise across a row as well as setting the angles for stage matching. From tests on a range of aerofoils it has been found that fish tailing could lead to 1o to 1.5o change in the deviation angle. A significant value considering engineers try to design the blade to 0.1o.

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Towards Robust CFD Based Design Optimisation of Virtual Engine

Fish tailing

Figure12: Mid-height Pressure Variation and Y+ for the NASA Rotor 37 Obtained using Two PADRAM Meshes with Different Wall Spacing.

Figure 12 shows the pressure variation at mid-height of a public domain, transonic fan, namely NASA rotor 37 where experimental data for code validation is available [27]. It is observed that as y+ falls towards one, fish tailing disappears. The reason SA model can produce, arguably, a better result with low y+ is due to its wall function implementation. The Spalding approximation is used to bridge the log-law region to the laminar sublayer; hence as y+ approaches unity, effectively SA turbulent production terms are naturally reduced. It has been found that improving the prediction of the flow angle would lead to a better overall fan characteristic, e.g. its pressure ratio versus mass flow rate. Figure 13 shows that a better agreement with experimental data is obtained when a fine mesh (that eliminates fish tailing) is used (denoted as PADRAM fine in the legend), note that a similar fine letterbox mesh does not predict the flow near the choke region correctly.

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Pressure ratio / mass flow 2.2

Fine Mesh (470K nodes)

2.15 Pressure ratio

The fine mesh spacing at the TE has meant that the radial mesh has a higher mesh resolution too (to void large aspect ratio in the third direction), hence leading to an overall larger mesh than may be needed for design. As commented earlier most commonly used turbulence models, e.g. one-equation SA and two-equation k-ε struggle to predict all the flow features correctly, e.g. accurately predicting the rate of mixing and wake decay, i.e. its width and depth especially at off design conditions, where separations are larger. This could lead to a wrong level of losses for the blade row, however, it is found that CFD gets the trend in loss generation right.

2.1

Exp AGARD 1998

2.05 2

LBox Fine Mesh

1.95 1.9 1.85 19.2 19.4 19.6 19.8 20 20.2 20.4 20.6 20.8 21

PADRAM Coarse mesh

mass flow (kg/s)

Figure 13: NASA rotor 37 Characteristic - Effect of Mesh Spacing and Topology.

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3.2 Hexahedra versus Tetrahedral Meshing It is generally accepted that for a complex geometry, unstructured grids provide a more automatic and robust meshing. Furthermore, mesh adaptation that can minimize the CFD discretization errors lend itself to unstructured meshes better. Although the multi-block structured mesh approach can also be used to mesh a complex geometry, as it is shown in section 4.2 of this paper, automatic blocking has not yet been achieved. The research on medial axis [28] is promising but has not yet achieved an automatic blocking needed for any arbitrary defined complex geometry. In PADRAM a library of templated features has been coded to ease the automatic mesh generation for a range of turboamchienry components [7]. Based on a number of aerodynamic and more recently aeroacustic simulations, it has been found that hexahedral meshes offer significant advantages compared to tetrahedral cells, in terms of accuracy and memory requirements. For tetrahedral grids, the ratio of cells to the number of vertices is close to 6 (in 3D) while this ratio remains close to one for hexahedral cells. Hence, the polygon meshes [29] often obtained by an agglomeration process of the simple background mesh or the dual volume presented recently in some COT tools is designed to provide a compromise between the two aforementioned approaches. The Hydra code within the SOPHY system is an edge-based unstructured FV code which accepts a variety of cell types, such as hexahedra, tetrahedral, prism and pyramid elements. However, most of the work presented here is based on multi-block structured meshes. To illustrate the strong grid influence, tests are recently carried out for Hydra LES simulation using the unstructured grids shown in Figures 14c&d, intended to model a subcritical Tollmien-Schlichting wave in a two-dimensional channel [30]. Snapshot of grids for the lower channel half are shown. Essentially, for these more triangulated grids, where possible, the nodal locations are the same as used for the hexahedral grid. However, due to the inclusion of triangulation the number of edges typically increases by around 30%. To permit the high aspect ratio cells, most suited to modelling the near wall region, clustering in the radial direction is used.

(c)

(a)

(d)

(b) (b) Figure 14: Hydra LES Simulation of an Open Jet – Curtsey of Tucker et. al . [30].

Figure 14b shows the decay of (vertical velocity fluctuation) for the subcritical Tollmien-Schlichting wave. The analytic solution to the Orr-Sommerfeld equation is represented by symbols. From Figure 14b it is clear that Grid 14d gives very poor accuracy. However, as shown, in Figure 14c, the modest hexahedral grid shows reasonable agreement with the analytic solution. Further grid topologies have been studied in reference [30] and has been found some grids are not suitable

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Towards Robust CFD Based Design Optimisation of Virtual Engine for the channel flow when tested for the vortical and cut-on acoustic waves gave encouraging results. Hence it is concluded that when using a temporal and spatial second order accurate code attention must be given to the grid topology and quality that matches the flow requirements.

4.0 CASE STUDY – SOPHY SYTEM APPLIED TO A HP TURBINE STAGE Having introduced readers to different aspects of the RR SOPHY system, the importance of geometry fidelity representations is highlighted in this section. The test case presented here is a high pressure (HP) turbine blade of a modern jet engine. This component is arguably the most challenging to design in a turbine assembly because of the highly loaded components that could produce considerable secondary flows and the surface cooling arrangements that keep the metal temperature within an acceptable range. The focus of this study is to minimise the endwall losses by the non-axisymmetric design of endwall surfaces. The methodology of non-axisymetric endwall contouring represent an additional extension to the PADRAM design space previously introduced in section 2.2. The details of RR methodology for endwall design contouring and application to a turbine blade is presented in references [31, 32], and novel application to a compressor blade in reference [8]. The methodology is briefly described below. The hub’s radius is incremented circumferentially by zero and the first three terms in the Fourier series as follows,

δ (θ ) = AA +

1 3 ⎛ 2πiθ 2πiθ ⎞ ⎜⎜ ai ⋅ sin( ) + bi ⋅ cos( )⎟ ∑ coe i =1 ⎝ p p ⎟⎠

where AA is the zero harmonic (i.e. an axisymmetric change), p is the blade pitch and coe is a normalizing coefficient. A BSpline curve is used to distribute the amplitude of the perturbations in the axial direction. The tensor product of the Bspline and Fourier harmonics produces a 3D surface by the which the endwall geometry is generated. Higher Fourier harmonics lead to more complicated shapes, whilst sinusoidal curves ensure pitchwise periodicity which in turn help to preserve the throat area. In the present study six axial stations are used and only the first Fourier harmonic is considered, hence a total of 18 design parameters needs optimising for each endwall. A typical endwall perturbation is shown in figure 15 below:

Baseline geometry

Apply Fourier Harmonics (tangential)

Profiled geometry

• Zero harmonic • First harmonic (sin) • First harmonic (cos)

Apply B splines (axial)

Figure 15: Turbine Nozzle Guide Vane with Axisymmetric and Profiled Endwalls.

The design objective consists of identifying the endwall geometry that can minimise the total pressure losses and the secondary kinetic energy multiplied by helicity (SKEH, a variable closely linked to the secondary flow loss) with a minimal deviation in absolute whirl exit angle compared to the baseline geometry. Secondary kinetic energy (SKE) is defined by using the velocity components: UC,m and Utotal,m, which refer to the local cross-flow velocity and the averaged cross-flow velocity respectively. The helicity

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Towards Robust CFD based Design Optimisation of Virtual Engine h is computed by the scalar product of the velocity V and the vorticity ω.

SKE =

U C2 ,m 2 U total ,m

h = V •ω SKEH =

SKE h huser

The datum flow capacity and specific work (for the stage) are also used as tight side constraints.

4.1 Mesh Sensitivity

Capacity/C apacity_ref

1.05 It is recommended that upstream of every 1.04 design study a mesh sensitivity study to be 1.03 carried out in order to identify the best suitable 1.02 mesh during the design cycle. A mesh 1.01 independent solution needs to be obtained for 1.00 0.99 the final design, this means that the mesh Struct-1 Struct-2 0.98 needs to be refined until an asymptotic Hybrid 0.97 behaviour is obtained for all the objective 0.96 functions and constraints under study. The 0.95 three types of mesh topologies shown in figure 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 mesh size [in M of nodes] 11 are examined for the HP turbine blade considered here. Figure 16 illustrates that as Figure 16: Variation of Blade Capacity with Mesh Size. the mesh is refined, an asymptotic behaviour is obtained for the inlet flow capacity. The data is normalised with the coarsest structured mesh values as unity. Note that the mesh independent solution does not mean the most accurate solution is obtained. Attention needs to be paid to the type of mesh needed to resolve various flow features, e.g. adequate mesh in the tip gap and high curvature regions, e.g. around the leading edge and fillet shoulders, etc. It should be noted that during the design optimisation iterations, as long as the optimiser is not mislead, it is not necessary to have a grid independent solution. Based on the sensitivity studies carried out, 0.5M mesh point is deemed adequate for a single row clean annulus calculations carried out here.

4.2 Optimisation Strategy 60

Delta% (SKEH)

The main optimisation strategy used here is based St1-Lptau 40 on one of SOFT’s multi objective genetic St2-MOGA algorithm called ARMOGA [33] and a RBF St3-DRAM 20 response surface method. Three strategies referred DATUM 0 to as ST1, ST2 and ST3 has been followed, and all the results are compared after 300 non-linear CFD -20 runs. In ST1 a DoE based on SOFT’s LPtau -40 Opt1 algorithm is used to generate an initial population with 100 members, an RBF RSM is then formed -60 -4 -3 -2 -1 0 1 and searched with ARMOGA, the Pareto front is Delta% f1 (Ploss) updated ten times with 20 points added each time. Figure 17: Pareto Front for Hub Endwall In ST2 approach, ARMOGA is directly used to Optimisation. search the design space, using eight individuals per generation and 37 generations to generate the Pareto front. ST3 is similar to ST1, except that the DoE is

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Towards Robust CFD Based Design Optimisation of Virtual Engine replaced with the initial ARMOGA population Figure 17 shows the Pareto front obtained indicating a significant reduction in loss of the order of 1-2% and a reduction in SKEH of the order of 30-40%. The benefits of SKEH reduction is two folds, on the one hand, secondary flow formation is reduced whilst on the other hand a more uniform flow field benefits the downstream blade row. For this particular test case, where there are 18 design parameters for the hub non-axi endwall, the hybrid strategies ST1 and ST2 which makes use of the RSM techniques performs better than a strategy based solely on the direct ARMOGA search methodology.

4.2 Real Geometry Validation

The optimised geometry shown in figure 17 as opt1 represents a compromise between the two extreme of the Pareto front where total pressure loss and SKEH are minimised. The performance of this geometry is then assessed in the presence of the high fidelity stage (vane-rotor) geometry shown in figure 18. The socalled real geometry features for the rotor compromise of the shroud including non-axisymmetric fences, upstream and downstream cavities, surface cooling, whilst trailing edge cooling slot is added to the upstream vane assembly.

(c) (b)

(a) (d)

(f)

(e)

Figure 18: Meridian view of PADRAM Multi-block Mesh for a HP Turbine Stage [34]: a) Trailing Edge Cooling Slot b) Surface Film Cooling c) Shroud Model d) Non-axisymmetric Fences e) Full Cavity Bleed, f) Upstream Cavity and Shingling Model.

The PADRAM multi-block structured mesh was chosen to provide a fine mesh for the real geometry,

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Towards Robust CFD based Design Optimisation of Virtual Engine stage configuration. The mesh consists of 4M nodes and is the result of a mesh independence study carried out to ensure all complex flow features such as shear layers between cavities and main gas path is captured adequately. Further detail of this geometry and mesh independent studies can be found in reference [34]. The optimum non-axisymmetric endwall design was tested in the presence of the real geometry features and mesh shown in figure 18. The high-fidelity fine-mesh configuration requires a much higher CPU than the model used for the clean annuls optimisation. A stable convergence of the CFD solver is obtained in about 600 multigrid iterations, which nearly corresponds to 10h using 16 CPUs of a modern computer cluster. However, for a single row simulation, an acceptable convergence (3-4 level drop in residual) is obtained in 150 multi grid iterations (staring from a fully converged datum solution) requiring no more than 2 hours CPU time on a single processor machine (for the 0.3 million mesh size). Hence, the high fidelity simulation requires a 80 times increase in CPU resources.

[Radius, 0-25% span]

Unfortunately the pressure loss reductions obtained in the single row computation did not lead to any stage efficiency gain. Using part converged CFD solution on a relatively coarse mesh has previously been successfully demonstrated in reference [6]. The low fidelity geometry i.e. where the secondary flow features have not been taken into account is the most common simplifications when optimising the blade or non-axi endwalls, e.g. see the recent publications [35 and 36]. This is mainly due to significant reduction in computational cost and complexity in an iterative design loop where a fast turn around is required. However, it was initially thought that the single row simplification where the component interaction is not adequately taken into account is responsible for this shortcoming. Hence, the next optimisation was carried out using a stage model of the HP turbine, however still without any secondary flow features. A relatively coarse mesh size of the order of (0.5 Million mesh nodes) was used for the stage; the computation takes about 4 hours still on a single CPU. This represents a 40 times reduction in CPU requirements compared to the full fidelity model. The design space was also increased to 36 parameters (18 hub non-axi endwall per blade row). The optimisation was conducted with 16 individuals per generation and for 50 generations, corresponding Figure 19: Second Optimisation Results: Radial to 800 CFD runs. This optimisation lead to an overall Distribution of the Total Pressure Vane Outlet. increase of 0.4% increase in stage efficiency. As before the high fidelity model was used to check the performance of the optimised geometry. It was found that although the non-axisymmetric hub contouring produces a small reduction of the total pressure loss at the vane outlet, the stage efficiency benefit is entirely lost at the rotor outlet. As shown in figures 18 and 20, the upstream and downstream cavities have significant impact on the main flow path near the endwall. Figure 20 shows that the pressure difference across the passage is responsible for indigestion of the main flow near the stagnation region (leading edge) leading to a non-axisymmetric bleed pattern. As the streamlines in figure 20 show, the leakage flow interacts with the secondary flow formation in the main passage. Hence it is expected to affect its formation and its development.

Figure 20: Leakage flow streamlinesShingling of an HP Turbine Vane – Radial Velocity Contours.

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Towards Robust CFD Based Design Optimisation of Virtual Engine Next, a third optimisation run was performed using a geometry configuration that includes the trailingedge cooling slot and the hub upstream and downstream bleed cavities. Film cooling flows and the rotor shroud were not taken into consideration in the CFD model. This corresponds to a compromise between higher-fidelity geometry modelling and an affordable computational cost. Similar to the previous optimisation runs, a CFD strategy with relatively coarse mesh size and partially-converged solution has been applied. The total grid size for this model is around 1.2M nodes. This leads to a higher computational cost but still an order of magnitude cheaper to run than the full fidelity model.



The optimisation run was stopped after 800 individuals, which corresponds to 50 generations of the genetic algorithm. Similar to the former runs, a response surface model (namely a Radial Basis Function algorithm) was then formed to continue the MOGA’s design space exploration. An overall 0.30% increase on stage efficiency was obtained for the optimised geometry.

Figure 21: Relative Whirl Angle at the Stage Outlet.

Figure 22: Relative Total Pressure at the Stage Outlet.

Figure 21 shows the radial distribution of the relative flow whirl angle at the rotor outlet. A significant decrease of secondary flows is obtained, as shown by the large reduction of flow under/overturning near hub endwall. A corresponding reduction of total pressure losses between ~ 20 to 70% of blade span is also evident in figure 22. Similar to the first optimisation runs, the main performance improvement at the rotor outlet is attained between ~ 20 to 40% of blade span and not very close to the hub endwall region where the profiled endwall is applied.

Based on an axisymtric hub endwall configuration, figure 23 shows the radial distribution of the relative whirl angle at the stage outlet for four differentfidelity models of the stage under investigation. The blue line refers to the fine-mesh fully featured model of the stage, whilst the green and red lines respectively to the geometry used in the second optimisation run with and without the film cooling. The grey line corresponds to a fully featured model of stage with a relatively coarse mesh size (~1.5M nodes). It is shown that, in terms of hub-endwall


As before, the optimised geometry was assessed by means of the fine-mesh real geometry model. This shows that although higher than the first optimisation run, the fidelity of the current setting is still not accurate enough to reproduce exactly the stage improvement expected. Some of the benefits achieved for the low fidelity geometry are lost when accounting for the actual stage model. Nevertheless, a global performance gain is confirmed to be of the order of 0.15% improvement in the stage efficiency with respect to the base geometry, against the 0.30% computed using the lower fidelity model.

Figure 23: Outlet Whirl Angle for different-Fidelity Geometry Configuration of the Stage. - 18

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Towards Robust CFD based Design Optimisation of Virtual Engine optimisation, film-cooling flows have the largest influence in the differences between simplified models and the fully featured geometry configuration. Shroud flows have strong impact only between 60-100% blade span. Furthermore, it is interesting to note that the mesh size has lower influence than real geometry and flow features. Based on these findings, it was decided to add the film-cooling model to the previous stage configuration which included the hub bleeds but still without including the shroud geometry. The outcome of this higher fidelity optimisation run was an increase in stage efficiency of 0.32% with respect to the base axisymmetric hub geometry. The efficiency gain is confirmed by running the high-fidelity fine-mesh model of the stage. A significant reduction of secondary flows is observed near the hub endwall with a corresponding increase in stage performance. It is confirmed that leakage flows directly impact the boundary layers in the endwall regions, which contribute to a large extent in increasing the overall stage loss. The optimised profiled hub endwall reduces the penetration of these low-pressure flows in the span wise direction and allows a more uniform flow distribution in the tangential direction. Hence it is concluded that the secondary flow geometrical features must be included in the simulation model when optimising the endwall profiles.

4.3 Virtual Engine – A Grand Challenge: In the previous section, it is demonstrated that capturing the interaction of components is important to predict the performance of the overall stage. The geometry and physical fidelity needed should be built parametrically to systematically investigate the importance of different features. The SOPHY system presented here provides all the components needed to build a virtual engine. Advanced optimisation search engines in SOFT, parametric geometry manipulation and automatic mesh generation in PADRAM, highly efficient and accurate RANS/LES solver provided by Hydra are key enablers. The solver must be highly scalable as recently demonstrated in reference [37] to thousands of processors. However, massively multi-core machines are needed to produce a practical deign tool. The Stanford researchers [38] had recently simulated a 20o sector of the entire flow path of a Pratt & Whitney jet engine. Two levels of mesh consisting of 15M and 75M cells had been considered. The coarser grid has run on 700 processors US department of energy Xeon Linux cluster requiring no more than 24 hours wall-clock time. It was estimated that 4000 processors are needed for the finer grid computation and an order of magnitude longer time steps needed for a full wheel revolution of the LP components. Automatic optimisation requires at least a further one, possibly two orders of magnitude more resources, hence, to optimise the whole virtual engine it is argued that 4,000,000 processors will be needed. Arguably, robust optimisation methodology presented in the next section would still require an order of magnitude larger computing resources than the single point optimisation that most researchers seem to concentrate on. Although in a recent article, researchers in [39] argue that the engineering community should make better use of HPC at the national level. It does not seem sensible to simply throw many components requiring a very large mesh in the design loop requiring immense computing resources. As demonstrated here, judicious choice of components to capture first order effects is required. It is also argued that further research is needed to produce smarter algorithms requiring less mesh points and computing resources to achieve the same and even higher level of accuracy. It seems, despite 20 years of research on high resolution and compact schemes, almost all the current industrial CFD codes make use of second-order accurate schemes. Why is that? Higher order schemes inevitably require high order boundary conditions and robust shock capturing schemes, which need further developments. Undoubtedly, the virtual engine capability will provide many benefits to the design by simulation community, e.g. automatic, rapid meshing of a complicated geometry, automation of the whole simulation process from pre processing to post processing, however, zooming is still required to focus on a particular region or engine sector. The zooming techniques provide a multi-fidelity simulation environment as

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Towards Robust CFD Based Design Optimisation of Virtual Engine demonstrated by the NASA’s NPSS (Numerical Propulsion System Simulation) for a GE90 engine [40]. This will make the task of design optimisation tractable. It is interesting to note that neither of the above cited references who have simulated flow for the whole engine have considered the real engine features described in the previous section. As mentioned earlier, after nearly 50 year of intense research efforts, CFD simulation still cannot replicate exactly all the physical phenomena such as turbulent flow losses, hence the question that arises is how can engineers confidently use CFD in design? One further mitigation strategy is discussed in the next section.

5.0 ROBUST DESIGN CONSIDERATIONS Robust design methodologies are developed in order to obtain insensitive design in respect to CFD uncertainties and manufacturing tolerances. Designing for robust design requires the following 3 stages: •

Identification and quantification of uncertainties sources associated with all the CFD simulation parameters, e.g. geometrical parameter, physical boundary conditions, modelling parameters such a turbulence model constants, etc.

•

Prorogation of the probability distribution identified in step one into the simulation codes, leading to probabilistic description of the objective functions and constraints,

•

Minimisation of the mean cost functions and their variances.

Although there are many sources of uncertainty in the CFD simulation, some which were discussed in the previous sections, the focus of this section is on robust optimisation with respect to geometrical variations, for example due to uncertainty in manufacturing process and/or in service deteriorations. As shown in figure 24 the leading edge of a compressor blade could become blunted during its service leading to degradation in performance. The author believes that the geometrical fidelity has a higher, first order effect compared to other sources of uncertainty, such as unsteady flow features or even the turbulence.

Figure 24: Leading edge Modifications – PADRAM Design Perturbations Applied to a Compressor Blade.

It is not practical to consider every minute change in the blade shape due to manufacturing process or in service deterioration, Hence the most important, influential changes in geometry needs identifying. Principle Component Analysis (PCA) minimises the number of random variables used to describe the geometrical variations by using uncorrelated shape functions. Garzon and Darmofal [41] have demonstrated design principles that lead to compressor blades with improved robustness subject to manufactured geometrical variability. The PCA technique is similar to Fourier decomposition where the overall behaviour can be represented by a relatively a smaller number of mutually orthogonal basis functions, m

X = x o + x + ∑ σ i Z iυ i

5.1

i =1

where xo is nominal blade, x is the average variation from all measurements, σ is the eignvalue from PCA

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Towards Robust CFD based Design Optimisation of Virtual Engine analysis, ν is the eigenvector from PCA modes, Z is a random variable, i is the manufactured mode number, m is the total number of manufactured mode number. Figure 25 shows the first PCA mode obtained for a relatively small (25) sample of manufactured rotor blades. The leading edge at midspan and trailing edge near the tip show the largest variations. Minimising the variability of a manufactured product is the objective of the Taguchi methods [42 and 43]. A more common measure of quality is sigma level. Performance can be characterised as a number of standard deviations from the mean performance. Initially from a manufacturing perspective, Motorola has introduced a 6-sigma quality measure into engineering design [44]. Designing for 6σ means designing for reliability of 99.999999% (or Figure 25: Contours of Geometrical Variations probability of failure of 0.000001%). However, tightening Obtained from the First PCA mode for the Suction of the manufacturing tolerances may prove prohibitively and Pressure Side of a Rotor Blade. expensive or impractical to achieve. Hence mitigation actions are needed to minimise the risk of significant performance and/or life deterioration. Robust design formulation requires the following: Minimize: F ( μ y ( x ), σ y ( x )), Subject to g i ( μ y ( x ), σ y ( x )) ≤ 0

i = 1,....r ,

5.2

x L + nσ x ≤ μ x ≤ xU − nσ x . Here, x represent the design vector including the random parameters, both the input and output parameters are formulated to include a mean (μ) and desired variation (σ) level which are defined as following, +∞

μ y ( x ) = ∫ f ( x ) p x dx , −∞

+∞

σ y2 ( x ) = ∫ { f ( x ) − μ y }2 p x dx.

5.3

−∞

where px is the joint probability function, however since a closed form of these integrals cannot be obtained for general CFD applications, hence, uncertainty propagation is performed in a discrete approach. There are a number of uncertainty propagation techniques that vary in CPU resources and accuracy as following, •

Direct Simulation Methods, e.g. Monte Carlo (MC) [45, 46]

•

Stochastic probability PDE approach, e.g. Polynomial Chaos (PC) [47, 48]

•

Weighted sum approach (WS) [49], e.g. Sigma-Point (SP) [50]

•

The Moment Method (MM) [51-53]

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Reduction in CPU requirements and Complexity, but also accuracy.

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Towards Robust CFD Based Design Optimisation of Virtual Engine The Monte Carlo method randomly simulates a design process by sampling the random variables making use of their probabilistic distributions, usually expressed as pdf (probability density function). It is recognised that MC is the most accurate probabilistic method. However, the standard Monte Carlo technique is intractable for large CFD problems. Other sampling techniques such as Latin hypercube (LHS) [54] and Lptau based on semi-random Sobol Sequence [55] can be used to reduce the number of simulations, however the number of simulations remains relatively large. An alternative sampling technique is based on stochastic PDE whereby one or more stochastic variables is added to the deterministic CFD equations and solved using the same disretization scheme. The PC expansion provides a means for expanding second-order random processes in terms of orthogonal polynomials such as Hermite polynomials. For turbulent flows the stochastic part of the Polynomial Chaos method scales as (Npc+1)7 for a 1D nozzle flow [48], where NPC is a function of the order of the expansion and number of random variables, hence the computational cost grows very rapidly. The PC methods is also intrusive and requires compatibly with the underlying CFD methodology, hence a new PC expansion is required every time the industrial RANS code is modified. There are various approaches to obtain robust design based on a Weighted Sum approach as follows, l

F = ∑ ± w1,i μ y ,i + w2,iσ y2,i

5.4

i =1

where w1,y and w2,y are weights for mean performance and variation components of objective i, l is the number of performance responses under considerations. Padulo et.al. [50] present an alternative of weighted sum propagation derived from the signal filtering theory of Controls called Sigma-Point (SP) method. SP considers the two limits of deviation, hence required 2n+1 function analysis, as following, n

μ y ( x ) = w0 f ( x0 ) + ∑ wp [ f ( x p + ) + f ( x p − )], p =1

σ = 2 y

∑ {w [ f ( x n

1 2

p =1

p

]

p+

[

) − f ( x p − ) + ( wp − 2 w ) f ( x p + ) + f ( x p − ) − 2 f ( x0 ) 2

2 p

]}

5.5

2

The weights can be chosen as follows:

h2 − n , h2 1 wp = 2 , for 1 ≤ p ≤ n. 2h w0 =

The sampling points are:

5.6

x0 = μ x , x p ± = μ x ± hσ xp e p ,

5.7

Where n is the dimension of the input space, ep is the pth column of the identity matrix of size n and h is equal to the square root of the kurtosis of the input distribution. It is argued that SP is more accurate than first order moment methods because the accuracy of the mean estimation is higher, and further more it does not require one or high level of differentiations [50]. However, it should be noted that weighted sum objective function are limited when generating a Pareto set

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Towards Robust CFD based Design Optimisation of Virtual Engine which will arguably be better generated by a MOGA type optimisation approach. This will be further discussed in the next section. A similar approach to SP is goal programming which makes use of decision support problem to formulate a robust design [49]. The p+ and p- in equation 5.5 represent deviation variables between the target level and the actual attainment of the goal. The Moment Methods are based on expanding the problem variables around their mean value using a truncated Taylor series. First or second order methods are often used [51-53]. The mean and variance of the first order method is obtained as following,

μ y = f ( μ x ), 2

⎡ ∂f ⎤ 2 σ = ∑⎢ ⎥ σ xp . x ∂ p =1 ⎢ ⎥⎦ p ⎣ n

2 y

5.8

This approach is limited to small perturbations, which are also prone to inaccuracies due to CFD noise, and does not readily provide information on high-order statistics of the response. Nevertheless, in both the SP and the Moment methods, the gradients of the robust objective can be efficiently obtained from an Adjoint code. However, similar to the PC expansion method the cost of maintaining an industrial adjoint code is relatively high. Although AD (Automatic Differentiation) techniques can help to maintain codes, some restructuring of the source may still be required. Hence attention is diverted to techniques that are less intrusive and easier to maintain. In order to overcome the prohibitive computational resources required to introduce probabilistic design into the optimisation cycle, surrogate or RSM technique introduced in the previous sections are linked to MCS methods. When using an approximate model to replace high fidelity model, it is necessary to conduct many updates to make sure the surrogate model is a good representative of the more expensive to run simulation code. Failing to do so will introduce another level of uncertainty into the model. In reference [57] a Gaussian field is used to evaluate the integrals shown in equation 5.3. A Gaussian process is fully characterised by its mean and covariance function. The covariance of the observed data is very similar to the Krigging RSM method [8], hence the maximum likelihood estimation is used to tune the hyper parameters, this methodology is referred to as Bayesian MCS [56]. For more information on derivation of BMCS refer to [57].

5.1 Robust Design of a Compressor Blade The SOPHY system discussed in earlier sections is used to build an efficient method for robust design optimisation. The first step is that a series of suitable geometry which represent the manufactured variations is generated. The test case is the mid section of a typical low-pressure compressor blade. The blade is parameterised using linear combinations of ten Hicks-Henne shape perturbations presented in section 2.2; possible resultant aerofoils is shown in figure 24. PADRAM generates the geometry and the mesh for Hydra automatically based on its template file. Next an initial database is created consisting of 100 members using the LHS technique. Hydra is run on this data set and the Gaussian stochastic process emulator is trained on this data set. Figure 26 shows the histogram of the pressure loss obtained using 100000 samples using the Gaussian process emulator. It takes only a few seconds on a single processor machine to generate the uncertainty data which is negligible compared to the time needed to run the highfidelity CFD. It can be seen that the manufacturing variations can lead to significant deterioration in the aerodynamic performance of the blade. In the worst case, there can be up to 14% degradation in pressure

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Towards Robust CFD Based Design Optimisation of Virtual Engine loss coefficient. It also suggests almost 4% shift in the mean performance. Next a robust optimisation is carried out using a MOGA using the Gaussian process emulator; Bayesian Monte Carlo Method is used for all points in the population at each generation to evaluate the performance statistics [57]. The objectives are then ranked to obtain the Parito-optimal solutions. Figure 27 shows the initial data set, subsequent update points, and the final Pareto front after 10 updates. In reference [57] a standard deterministic optimisation is also carried out to illustrate the differences between the robust design and a design where the nominal performance is improved. Next, probabilistic performance analysis for the robust and the deterministically optimal blades is conducted by using the emulators trained for each data set. Figure 26 also compares the histogram of the pressure loss coefficient of the robust blade with that of the deterministically optimal blade. This figure was generated by sampling the respective Gaussian process emulator (100000 samples) in the noise space around the robust and the deterministically optimal blades. As expected, the histogram of the robust geometry shows less variability in pressure loss coefficient when compared with the design obtained using a deterministic approach.

Robust Solution

t

Figure 26: Histogram for Robust Design and Deterministic Optimal Design of a Compressor Blade [57].

Figure 27: Pareto Front for the Robust Design Optimisation [57].

A comparison between the nominal, mean, and worst-case performance of the baseline, deterministic, and robust design blades is presented in Reference [57]. It is found that the robust design to have better performance than the deterministic optimal design in all respects (mean, standard deviation, mean shift (2.5% better), and worst-case performance (3% better)), except at the nominal performance (4% reduction in loss compared to 6% obtained for deterministic optimum design). Significant computational savings for the robust design studies is demonstrated, for this test case 350 non-linear CFD was needed. However, further work is needed to develop a more rigorous statistical update of the surrogate (RSM) model. This is essential to ensure errors are not crept in the regions of interest and the subsequent statistical analysis.

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6.0 CONCLUSIONS The aerospace industry is faced with stringent targets to minimise the impact of aviation on environment yet stay competitive. The latter goal requires a reduction in cost and design cycle time whilst achieving multiple often-conflicting engineering objectives, e.g. minimising specific fuel consumption and weight. It has been discussed that different fidelity simulation codes are used at different stages of tubomachinery design; this necessitates the use of different codes which will have different uncertainty associate with it. It has been demonstrated that high fidelity geometrical representation where primary and secondary gas path are taken into account is the key to achieve a design environment which is the representative of the actual engine. It is demonstrated that high fidelity RANS based SOPHY system provides a flexible framework for building a virtual turbojet engine for Aerospace, Industrial and Marine applications. Based on the results presented here, it may be concluded that optimisation approach would seem to take intuition out of the design loop. However, intuition is still needed in setting mesh resolution, choosing the level of geometry fidelity, design parameters and define objective functions and constraints. All these depend on the experience of the designer and the calibrations carried out. Reducing geometrical complexity and the mesh size are necessary not just because of the CPU requirements but also because of difficulties in both mesh generation and the solver convergence. However, care must be taken when simplifying a model, as CFD still need to capture the first order effects and rank good designs. As the complexity of a product is increased the greater is the chance of the uncertainties in the simulation codes and or the manufacturing process could lead to potentially undesirable effects. It is therefore of paramount importance to develop techniques to mitigate the uncertainty risks in design optimisation process. The uncertainty analysis has been presented here in the context of automatic optimisation where not a more optimal design but a more insensitive design is sought. Robust design should be insensitive to uncertainties in design, modelling simulation as well as external noises and manufacturing tolerances and in-service deteriorations. Although, direct stochastic model is the most accurate technique, its use with high fidelity CFD is impartial. Hence techniques based on a Bayesian Monte Carlo are developed which provide an efficient robust design methodology for computationally expensive problems. The proposed method makes use of a multi-objective GA to ensure explicit trade offs between the mean and standard deviation are obtained. The application of the robust design to a compressor blade has lead to a design which is less sensitive in manufacturing variations when compared with the deterministic optimal design. Further work is required to extend the methodology to 3D multi-stage CFD taking into account the real geometry features that were also discussed in this paper.

7.0 ACKNOWLEGMENTS The author gratefully acknowledges the permission of Rolls-Royce plc to publish this paper. The author would like to acknowledge Dr Andrea Milli for the application of the SOPHY optimisation system to the HP Turbine stage presented in section 4 and Professor Paul Tucker of Cambridge whittle Laboratory for providing the Hydra LES solutions for the jet. The author would also like to thank his colleague and friend, Dr Leigh Lapworth for useful discussions and reviewing this paper.

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8.0 REFERENCES [1]

Senior figures from European industry, “European Aeronautics – A vision for 2020”, January 2001, available at http://europa.eu.int/comm/research/growth/aeronatics2020/en/index.html

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Shahpar, S., "Automatic Aerodynamic design Optimisation of Turbomchinery Components - an Industrial Prospective", Invited Lecture at VKI, Belgium, 2005.

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Shahpar, S., "Design of Experiments, Screening and Response Surface Modelling to Minimise the Design Cycle Time", invited lecture at VKI, Belgium, 2005.

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Towards Robust CFD based Design Optimisation of Virtual Engine Nozzle Design”, ASME GT 2006-90249, 2006, also published in ASME J. of Engineering Gas Turbines and Power, vol. 129, pp 387-393, 2007. [19] Hicks, R.M., and Henne, P.A., “Wing Design by Numerical Optimization”, Journal of Aircraft, vol. 15, no. 7., pp. 407-412, 1978. [20] Lapworth, L., Shahpar, S., " Design of Gas turbine Engines using CFD", ECCOMAS 2004, Eds., Neittannmaki, et. al., Finland, July 2004. [21] Lapworth, L., "Hydra-CFD: A Framework for Collaborative CFD development", International Conference on Scientific and Engineering Computation (IC-SEC), Singapore, July 2004. [22] Moninier, P., Muller, J.D., and Giles, M.B., “Edge-based Multigrid and Preconditioning for Hybrid Grids”, AIAA Journal, Vol 40, No, 2001. [23] Moinier, P., and Giles, M.B., “Preconditioned Euler and Navier-Stokes Calculations on Unstructured Grids”, 6th ICFD Conference on Numerical Methods for Fluid Dynamics, Oxford, UK, 1998. [24] Martinelli, L., “Calculations of Viscous Flows with a Multi-grid Method”, Ph.D. Thesis, Dept. of Mech. And Aerospace Eng., Princeton University, USA, 1987. [25] Muller, J.-D., and Giles, M.B., “Edge-Based Multigrid Schemes for Hybrid Grids”, 6th ICFD Conference on Numerical Methods for Fluid Dynamics, Oxford, UK., 1998. [26] Diego, B. and Shahpar, S., “Automatic Multi-Stage Design-Optimisation of an Axial Compressor”, ISABE –2005-1181, Munich 2005. [27] Dunham, J., “CFD Validation for Propulsion System Components”, AGARD Advisory Report 355, ISBN 92-836-1075-X, 1998. [28] http://www.fegs.co.uk/medial.html , Feb 1992. [29] http://www.cd-adapco.com/products/STAR-ccm_plus/common/meshing.html [30] Tucker P.G., Eastwood S. and Xia H., “Exploring the LES Model’s Role With Jet Noise in Mind”, 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 7 – 10th Jan 2008, Paper No. AIAA-2008-0527, 2008. [31] Harvey, N. W., Rose, M.G., Taylor, M. D., Shahpar, S., Hartland, J., and Gregory-Smith, D.G., 1999, “Non-Axisymmetric Turbine Endwall Design: Part I – Three Dimensional Linear Design System”, Paper No. 99-GT-337, ASME Turbo Expo, also transcription of J. of Turbomachinery, April 2000, Vol. 122, Issue 2, pp 278-285. [32] Harvey, N. W., Brennan, G., Newman, D. A. and Rose, M. G., 2002, “Improving Turbine Efficiency Using Non-Axisymmetric Endwalls: Validation in the Multi-Row Environment and with Low Aspect Ratio Blading”, Paper No. GT-2002-30337, ASME Turbo Expo, Amsterdam, The Netherlands [33] Sasaki, D., and Obayashi, S., “Adaptive Range Multi-Objective Genetic Algorithms and SelfOrganizing Map for Multi-Objective Optimization Problem”, von Karman Lecture Series 2004-7, 2004.

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Towards Robust CFD Based Design Optimisation of Virtual Engine [34] Milli, A., Shahpar, S., “Full-Parametric Design System to Improve the Stage Efficiency of a HighFidelity HP Turbine Configuration”, ASME GT2008-51348, Berlin, 2008. [35] Praisner, T.J., Allen-Bradley, E., and Grover, E.A., Knezevici, D.C., and Sjolander, S.A., “Application of Non-axisymmetric Endwall Contouring to Conventional and High-lift Turbine Airfoils”, GT2007-27579, ASME Turbo Expo 2007, Montreal, Canada, 2007. [36] Dofner, C., Nicke, E., Voss, C., “Axis-Asymmtric Profiled Endwall Design using multi-objective Optimization Linked with 3D RANS-Flow-Simulations”, GT2007-27268, ASME Turbo Expo 2007, Montreal, Canada, 2007 [37] Hills, N.J., “Whole Turbine CFD Modelling”, GT2007-27918, proceedings of ASME Turbo Expo 2007, Montreal, Canada, 2007. [38] Medic, G., Kalitzin, G., You, D., Weide, E.V.D., Alonso, J.J, and Pitsch, H., “Integrated RANS/LES Computations of an Entire Gas Turbine Jet Engine”, AIAA 2007-1117, 45th AIAA Aerospace Sciences Meeting and Exhibit, Reno, January 2007. [39] Mavriplis, D.J., Darmofal, D., Keyes, D., and Turner, M., “Petaflops Opportunities for the NASA Fundamental Aeronautics Program”, AIAA-2007-4084, 18th AIAA Computational Fluid Dynamics Conference, Miami, Florida, June 2007. [40] Turner, M.G., Reed, J.A., Ryder, R., Veres, J.P., “Multi-Fidelity Simulation of a Turbofan Engine With Results Zoomed Into Mini-Maps for a Zero-D Cycle Simulation”, NASA-TM-213076, 2004. [41] Garzon, V.E. and Darmofal, D.L., “Impact of Geometric Variability on Axial Compressor Performance”, ASME Turbo Expo GT2003-38130, 2003. [42] Phadake, M.S., “Quality Engineering Using Robust Design”, Prentice-Hall, Englewood Cliffs, N.J., 1989. [43] Fowlkes, W., and Creveling, C., “Engineering Methods for Robust Design using Taguchi Methods in Technology and Product Development”, Addison Wesley Longman”, Reading, MA, 1995. [44] Pande, P.S., Neuman, R.P., and Cavanagh, R.R., “The Six Sigma Way: How GE, Motorola, and other Top Companies are Honing Their Performance”, McGraw-Hill, New York, 2000. [45] Hammersley, J.M., and Handscomb, D.C., “Monte Carlo Methods”, Chapman and Hall, London, 1964. [46] Halton, J.H., “A Retrospective and Prospective Survey of the Monte Carlo Method”, SIAM Review, 1970, vol 12, no 1, pp 1-63. 1970. [47] Xiu, D., Karniandakis, G.E., “Modelling Uncertainty in Flow Simulations via Generalization Polynomial Chaos”, J. Computational Physics, Vol. 187, No. 1, pp. 137-167, 2003. [48] Mathelin, L., Yusuff Hussaini, M., Zang, T.A., Bataille, F., “Uncertainty Propagation for Turbulent, Compressible Flow in a Quasi-1D Nozzle Using Stochastic Methods”, AIAA 2003-4240, 16th AIAA Computational Fluid Dynamics Conference, Orlando, Florida, June 2003. [49] Park, G.-J.,Lee, T.-H., Lee, K.-H., Hwang, K.-H., “Robust Design: An Overview”, AIAA Journal,

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Towards Robust CFD based Design Optimisation of Virtual Engine vol. 44, No. 1., pp181, 191, Jan 2006. [50] Padulo, M., Campobasso, M.S., Guenov, M.D., “Derivative-free Uncertainty Propagation Methods for Airfoil Design”, Euromech Colloquium on Efficient Methods for Robust Design and Opitmization, Queen Mary London University, London, U.K., Set. 2007. [51] Taylor, A.C., Green, L.L., Newman, P.A., Putko, M.M., “Some Advanced Concepts in Discrete Aerodynamic Sensitivity Analysis”, AIAA 2001-2529, 2001. [52] Green, L.L., Lin, H. and Khalessi, M.R., “Probabilistic Methods for Uncertainty Propagation applied to Aircraft Design”, AIAA 2002-3140, 2002. [53] Du, X., and Chen, W., “Efficient Uncertainty Analysis Methods for Multidisciplinary Robust Design”, AIAA Journal, vol. 40, no. 3, pp 545-552, 2002. [54] McKay, M.D., Beckman, R.J., Conover, W.J., “A Comparison of three methods for selecting values of input variables in the analysis of output from a Computer”, Technometrics, vol 21, no 2, pp 239245, 1979 [55] Sobol, I.M., “On the Systematic Search in a Hypercube”, SIAM Journal of Numerical Analysis, vol. 16, pp. 790-793, 1979. [56] Rasmussen, C., “Gaussian Processes to speed up hybrid Monte Carlo for Expensive Bayesian Integrals”, Bayesian Statistics, Bernardo MBJM, West, M. (eds.), vol. 7, Oxford University Press: pp 651-659, 2003. [57] Kumar, A., Nair, P.B., Keane, P.B., Shahpar, S., “Robust Design using Bayesian Monte Carlo”, International Journal For Numerical Methods in Engineering (in press), Wiley, 2007. [58] Youngren, H., Drela, M., “Viscous/Inviscid Method for Preliminary Design of Transonic Cascades”, 27th Joint Propulsion Conference, AIAA/SAE/ASME conference, AIAA-91-2364, 1991. [59] Praisner, T.J., Clark, J.P., Nash, T.C., Rice, M.J., and Grover, E.A., “Performance Impacts Due To Wake Mixing In Axial-Flow Turbomachinery”, ASME Turbo Expo 2006, ASME GT2006-90666, Barcelona, 2006. [60] Yang, H., Roeber, T., Kozulovic, D., “Hybrid-grid Simulation of Unsteady Wake-Boundary Layer Interaction on a High Lift Low Pressure Turbine Airfoil”, ASME Turbo Expo paper 28111, Montreal 2007. [61] Hazelbach, F., Schiffer, H.,P., Hosman, M., Dressen, S., Harvey, N., and Read, S., “The application of ultra-High Lift Blading in the BR715 LP Turbine”, ASME Paper No., 2001-GT-0436, 2001. [62] Walker, A.D., Carrotte, J.F., McGuirk, “Enhanced External Aerodynamic of A Generic Combustor using An Integrated OGV/Pre-Diffuser Design Technique”, ASME Turbo Expo 2006, GT200690184, Spain, May 2006. [63] Adamcyzk, J.J., Celestina, M., Beach, T.A., Barnett, M., “Simulation of Viscous Flow within a Multistage Turbine”, ASME Journal of Turbomachinery, vol. 112, 1990.

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7.0 APPENDIX A: GLOSSARY OF TERMS

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AD

Automatic Differentiation

ANOVA

Analysis of Variation (to predict interaction of parameters)

ARMOGA

Adaptive Range Multi Objective Genetic Algorithm

ASA

Adaptive Simulated Annealing (global optimisation algorithm)

BMCS

Bayesian MCS (Monte Carlo Simulation)

CFD

Computational Fluid Dynamics

COTS

Commercial Off The Shelf

CPU

Central Processing Unit (measure of computational resource)

DES

Detached Eddy Simulation

DNS

Direct Navier Stokes Simulation

DOE

Design of Experiment (or design space filling algorithm)

EU

European Union

FFD

Freed Form Deformation

FV

Finite Volume

GA

Genetic Algorithm

HP

High Pressure (Compressor or Turbine)

HYDRA

RR suite of CFD codes (Non-linear, Linear and Adjoint RANS FV code)

IGES

Initial Graphics Exchange Specification

LE

Leading Edge (of an aerofoil)

LES

Large Eddy Simulation

LHS

Latin Hyper-Cube Sampling (one of SOFT’s orthogonal DOE)

MC

Monte Carlo Simulation

MOGA

Multi Objective Genetic Algorithm

MP

Mixing Plane (Boundary condition used for the adjacent the blade rows)

MSC

Monte Carlo Simulation

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Monotone Upstream Centred Schemes for Conservation Laws

NASA

National Aeronautics and Space Administration

NPSS

Numerical Propulsion System Simulation

NGV

Nozzle Guide Vane

NS

Navier-Stokes (Equations)

OGV

Outlet Guide Vane (stator blade)

PADRAM

Parametric Design and Rapid Meshing (RR geometry and meshing system)

PC

Personal Computer – also used as Polynomial Chaos

PCA

Principal Component Analysis

PDE

Partial Differential Equations

PDF

Probability Density Function (e.g. as used in Monte Carlo Sampling techniques)

RANS

Reynolds Averaged Navier-Stokes (Equations)

RBF

Radial Basis Function (kind of RSM)

Re

Reynolds number

RSM

Response Surface Modelling

SKE

Secondary Kinetic Energy

SKEH

Secondary Kinetic Energy times Helicity

SOFT

Smart Optimisation For Turbomachinery (RR Optimisation Library)

SOPHY

SOFT-PADRAM-HYDRA

SQP

Sequential Quadratic Programming (a gradient based 2nd order optimisation algorithm)

SP

Sliding Plane (boundary condition used for unsteady simulation of blade rows)

STL

File format (of triangles) used in Stereolithography CAD (Computer Aided Design)

TE

Trailing Edge (of an aerofoil)

TVD

Total Variation Diminishing (which would not allow spurious entropy generation)

UTC

University Technology Centre

URANS

Unsteady Reynolds Averaged Navier Stokes (equations)

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