design of gas turbine engines using cfd.pdf

European Congress on Computational Methods in Applied Sciences and EngineeringECCOMAS 2004

P. Neittaanmaki, T. Rossi, S. Korotov, E. Onate, J. Periaux, and D. Knorzer (eds.)Jyvaskyla, 2428 July 2004

DESIGN OF GAS TURBINE ENGINES USING CFD

Leigh Lapworth? and Shahrokh Shahpar?

?Aerothermal Methods GroupRolls-Royce plc, P.O.Box 31, Derby, DE24 8BJ, England

e-mails: [email protected], [email protected] page: http://www.rolls-royce.com

Key words: Turbomachinery, Design, Optimisation, Adjoint.

Abstract. This paper describes a general purpose design system being developed at Rolls-Royce plc. The key elements of the system are a parametric design and rapid meshingcapability; a state-of-the-art CFD solver with an adjoint capability; and, an advanced op-timisation system consisting of a library of optimisers. A description is given of eachelement in the design system. To illustrate its use and flexibility, five different applica-tions of the system to a gas turbine are described. These are: optimisation of the guidevanes in the bypass duct to minimise excitation of the fan rotor; the same bypass guidevane optimisation using sensitivity gradients from the adjoint solver; optimisation of acompressor stage to improve efficiency whilst constraining flow rate, pressure ratio andoutlet flow angle; minimisation of the forced excitation of a turbine rotor by modifying thewake of the upstream nozzle guide vane; and, optimisation of a fan rotor to reduce tonenoise.

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Leigh Lapworth and Shahrokh Shahpar

1 INTRODUCTION

The design of individual gas turbine components using CFD is now commonplace[1]. Traditional design by analysis methods are increasingly being supplemented withautomated design systems [2] and the use of optimisation systems [3]. At the same time,fluid machinery can now be modelled and analysed to an unprecedented level using CFDon powerful multi-processor computers or clusters.

Although CFD can provide essential information to aid the physical understanding ofcomplicated flow fields, it is generally the requirement to design/modify geometry thatdrives the application of CFD. When applied to an individual component, the physi-cal understanding gained from CFD is often able to guide design improvements to thatcomponent. However, fluid machinery is generally characterised by the interaction of anumber of components, such as multistage turbomachinery; the intake and the fan rotor;the combustor and the upstream diffuser; and, so on. A truly optimal design can onlybe achieved by accounting for all the component interactions. It is here that the designby analysis approach becomes limited - a designer may know how he/she would like tochange the flow field, but changing the geometry to achieve that is much less intuitivethan in single component design. If the requirement to meet a number of constraints isadded, then design by analysis becomes a very crude tool.

This paper describes an automated design system that has been developed specificallywith multi-component fluid machinery in mind. Section 2 describes the main elements ofthe design system. Section 3 describes five novel applications of the system.

2 ELEMENTS OF THE DESIGN SYSTEM

The design systems consists of the following processes:

Parametric representation Geometry construction Mesh generation CFD solution Data extraction and functional evaluation OptimisationThe systems that implement these processes are described in the following sections.

An underlying theme of all the systems is that they have a batch execution mode thatallows them to be run automatically by the optimiser.

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2.1 Parametric representation

Although the parametric representation, geometry construction and mesh generationare logically separate processes, they are intimately linked with the objective of creating aCFD mesh in the shortest possible time. With this in mind, an integrated design systemhas been developed [4] called PADRAM (Parametric Design and Rapid Meshing System)which provides a very efficient and robust system for parametric geometry and meshcreation.

The parametrisation follows the strategy successfully adopted in previous work [2] ofusing parameters that are already familiar to the designer. In the context of turbomachin-ery blades, the parameters include: lean, sweep, stagger, inlet and outlet blade angles,camber distribution and thickness distribution. The parameters can be independentlyspecified at a number of radial heights to produce a complete 3D design space. Formulti-passage and/or multi-stage turbomachinery applications, the parameters can be in-dependently specified for every blade in the calculation domain. Figure 1 illustrates howa single blade within a ring of blades can be designed independently of the other bladesin the ring. For multipassage simulations, both the pitch and the relative axial positionbetween adjacent blades are also design parameters.

Figure 1: Independent design of a single blade within a ring.

The philosophy of using familiar engineering parameters has been extended into otherapplication areas. For example, the design system for nacelles uses parameters such asdroop angle, scarf angle, highlight radius and intake duct area ratio.

2.2 Geometry construction

For constructing the geometry of turbomachinery blades, the approach taken is, typ-ically, one of perturbing an existing base geometry. The blade geometry may be defined

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using either a number of pre-defined blade-to-blade stream sections spanning the bladefrom hub to tip; or, as a single three-dimensional NURBS entity. If a NURBS entity isused then PADRAM creates its own set of pseudo stream sections on which to generatethe CFD mesh.

Since both the geometry and the meshes are constructed stream surfaces, an importantaspect of the design system is definition of these surfaces in three-dimensional space. InPADRAM, the stream surfaces are defined in polar coordinates as Si(r, , z), where i isthe section index. Using non-dimensional parametric coordinates, m and , a typicalsurface can be defined as:

rs = r(m, ) (1)

s = (m, )

zs = z(m, )

where, for axi-symmetric stream surfaces:

m =z

z0

dr2 + dz2

r(z)(2)

r = r(z)

z0 is an arbitrary reference and m is the non-dimensional distance along a stream

section which will be zero at z0.PADRAM starts by transforming the stream sections into parametric two-dimensional

planes, using the co-ordinates andm. As r is greater than zero for all z coordinates,m isa monotonic function of z, hence a unique inverse function exists to map the computationalcoordinates back to the physical, three-dimensional polar coordinates. The advantageof the above transformation is that the angles are preserved and the mesh-generationprocedure deals with plane sections only.

For non-turbomachinery geometries, PADRAM interfaces to CAD through IGES files.PADRAM uses a library of stylised geometries, so that it has a number of pre-definedrules for cleaning IGES files and constructing geometry.

2.3 Mesh generation

The PADRAM mesh generator can, very rapidly, produce good-quality viscous meshesfor multi-passage, multi-stage turbomachinery using 2D, quasi-3D or full 3D blade geome-try. PADRAM makes uses of both transfinite interpolation and elliptic grid generators togenerate hybrid C-O-H meshes [4]. An orthogonal body-fitted O-mesh is used to capturethe viscous region in the vicinity of the blade(s). A C-mesh is used for any bifurcationsin the domain, such as pylons and drive farings in bypass duct applications. ElsewhereH-meshes are used to patch adjacent blocks of O- or C- mesh together; and, to extend the

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mesh to meet the upstream, downstream and periodic boundaries. The mesh is indepen-dently generated for every stream section, hence three-dimensional meshes are producedeasily from stacked two-dimensional meshes with no mapping required to transfer themeshes radially. This avoids mesh morphing, and ensures that good quality meshes arecreated at every height even if the geometry varies considerably from hub to tip. Forblades with tip gaps, a separate H-mesh is generated in the gap. Mesh generation isextremely fast, requiring no more than a few seconds to generate a single passage 3Dmesh. A mesh of several million nodes can be generated in approximately one minuteon a 3.0GHz PC. A typical PADRAM mesh for a single blade passage is shown in figure2(a).

(a) Single blade mesh (b) Bypass duct mesh

Figure 2: PADRAM meshes for single and multi-passage applications.

As stated above, one of the key features of PADRAM is the fact that it is intimatelylinked with the parametric geometry definition. For example, within a ring of bypassguide vanes, the geometry of each vane can be specified independently of the otherseither by using a different geometry definition file or by using one of the several designparameters within PADRAM. Another important feature of PADRAM is that it is a truemulti-passage meshing system. The geometry of each blade in the annulus can be variedindependently and a good viscous mesh for the complete ring of blades can be generatedin one step. Figure 2(b) shows a PADRAM mesh for a bypass assembly consisting of 52guide vanes (each defined independently of the others) a pylon and a radial drive faring.

2.4 CFD solution

CFD solutions are generated using the Rolls-Royce plc. HYDRA-CFD code. HYDRA-CFD is a suite of non-linear, linear and adjoint solvers being developed collaboratively byRolls-Royce plc. and its University partners [5].

HYDRA-CFD is a general purpose code for hybrid unstructured meshes which uses an

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efficient edge-based data structure [6]. The multi-block PADRAM meshes are convertedinto this data structure using a pre-processor. The flow equations are integrated aroundmedian- dual control volumes using a MUSCL based flux-differencing algorithm. Thediscrete flow equations are predconditioned using a block Jacobi preconditioner [7] anditerated towards steady state using the 5-stage Runge-Kutta scheme [8]. Convergenceto steady state is further accelerated through the use of an element-collapsing multigridalgorithm [9]. The flow solver runs in parallel on both shared and distributed memorymachines using domain decomposition. The parallel multigrid capabilities are essentialfor generating CFD solutions in the elapsed time needed for effective use of optimisation.

The HYDRA-CFD framework is novel in the fact that it has linearised unsteady [10]and adjoint solvers [11] built on top of the non-linear solver. The linear and adjoint solversrepresent a full linearisation of the turbulent steady flow equations. Examples of the useof both the linear and adjoint solvers within an optimisation process are described in thispaper.

2.5 Data extraction and function evaluation

In order to drive the optimisation process, the values of both cost and constraint func-tions must be extracted from the CFD solution. The key element here is that the designsystem must be able to extract these values as a batch process. Typically, design func-tionals involve integrations over boundary planes. These integrals may be over the entireboundary plane, as in the case of inlet or outlet flow rate; or, in the circumferentialdirection at a series of radial positions, as in the case of prescribed inlet or outlet profiles.

Two approaches to data extraction have been adopted. The first is to export theHYDRA-CFD solution into a multi-block PLOT3D flow file - this is only possible formultiblock meshes and relies on a stored mapping generated by the HYDRA-CFD pre-processor. Once the flow solution is in PLOT3D format, data can be extracted usinga range of structured mesh post-processors. The second approach is to process the un-structured mesh directly - this relies on constructing a series of interpolating lines in thecircumferential direction which overlay the unstructured mesh.

2.6 Optimisation

Optimisation is performed using the SOFT (Smart Optimisation For Turbomachin-ery) system [3] being developed by Rolls-Royce plc. and its university partners. SOFTprovides a library of different optimisers; design of experiments techniques; statisticalanalysis of variations; and, advanced response surface models. It is well known that nosingle optimisation technique performs better than others across a range of engineeringapplications. Hence, SOFT provides a library of optimisers which fall into four broadcategories:

Explorative methods, such as Simplex based methods.

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Gradient based methods, such as the Modified Feasible Descent and SequentialQuadratic Programming techniques which are used in this paper.

Evolutionary type methods, such as the Simulated Annealing technique used in thispaper.

Hybrid methods, such as Tabu search which combine elements of the previous meth-ods.

An issue that must often be addressed with CFD based optimisation is the run-timeassociated with the performing a simulation. With access to a compute cluster, thiscan be addressed through the parallel capabilities in the flow solver. In addition, SOFTprovides a number of approximation techniques for reducing the number of expensiveCFD simulations which must be performed [3].

The SOFT system provides the user with a graphical interface for constructing the workflow, such as the one shown in figure 5, which defines the sequence of operations from thesetting of design parameters to the evaluation of the cost and constraint functions. TheGUI also provides a visualisation capability for plotting the progress of the optimisation.

2.7 Adjoint CFD

Consider an objective function I(U, ) which is a function of the flow variables U anda set of design parameters . The sensitivity of the cost function to the design variablesis:

dI

dj=

I

Uk

Ukj

+I

j(3)

Using the fact that the flow variables satisfy the Navier-Stokes euqations R(U, ) = 0and introducing the adjoint variables, v,

vTi =I

Uk

(RiUk

)1(4)

equation 3 can be rewritten as:

dI

dj= vTi

Rij

+I

j(5)

Equation 4 can be re-written to give the adjoint equation(R

U

)Tv =

(I

U

)T(6)

The advantage of the adjoint equation is that it depends only on the cost function I andnot on the design parameters . Hence, for applications where there are many designvariables but only one, or a small number, of cost and constraint functions the adjointapproach is a computationally efficient approach for computing design sensitivities.

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3 DESIGN APPLICATIONS

In order to demonstrate the flexibility of the design system, the following applicationsare presented:

Optimisation of bypass guide vanes using steady state CFD, Optimisation of bypass guide vanes using adjoint CFD, Optimisation of a compressor stage using multi-stage CFD, Reduction of turbine forced response using linear unsteady CFD, Reduction of fan rotor noise using steady state CFD.These applications make use of a range of capabilities from both the CFD solver and

the optmisation library. The use of both approximation techniques and PC clusters tomake the CFD simulation times tractable for the optimiser is also demonstrated.

3.1 Optimisation of bypass duct guide vanes

The layout of a modern high bypass ratio gas turbine engine is shown in figure 3(a).Within the bypass duct there are large scale blockages due to the pylon (which attachesthe engine to the wing) and the faring around the radial drive shaft (RDS). Upstream ofthese blockages is a ring of outlet guide vanes (OGVs) which remove the swirl from theflow leaving the fan rotor, see figure 3(b). It is has been shown [12] that the pylon andRDS generate a circumferential distortion in the static pressure field which can propagateupstream, through the bypass OGVs, and excite the fan rotor. It has also been shown in[12] that by circumferentially varying the OGV geometry the level of pressure distortionreaching the fan, and hence its excitation, can be drastically reduced. Theoretically, thedistortion reaching the fan can be completely eliminated if every OGV is allowed to bedifferent. However, this is not a cost-effective solution and design options include: a singleOGV with a variable stagger pattern; and, a small number, typically 3 or 5, of differentOGVs arranged in blocks around the annulus.

The PADRAM-HYDRA-SOFT system has been applied to the design of a ring of 52bypass OGVs to reduce fan forcing. In this example, all 52 OGVs are identical but theirstagger, or setting, angle is allowed to vary around the annulus up to a user specifiedmaximum value.

Notionally, each OGV stagger angle is independent of the others and the design spacehas 52 degrees of freedom. In practice, the stagger angle should vary more smoothlyaround the annulus its circumferential distribution can be represented by a Fourier seriesof the form:

i = A0 +Mj=1

[Aj sin

(2pii

Nj)+Bj cos

(2pii

Nj)]

(i=1,N) (7)

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(a) Layout of a high bypass ratio gas turbine engine. (b) CFD solution for bypass duct.

Figure 3: Engine cut-away and bypass duct components: OGVs, pylon and radial drive faring.

where i is the stagger angle of the ith OGV; N is the number of OGVs; and, M is

the number of harmonics used in the Fourier expansion. In the current calculations,seven Fourier harmonics have been used, leading to a design space with 15 parameters(A0, A1, B1, . . . , A7, B7). In order to ensure that the all the stagger angles are less thanthe user specified constraint, const 0, the stagger angles from equation 7 are modifiedby:

i =i

maxmin (max, const) (8)

where max = maxi=1,N (|i|) is the maximum, unmodified, stagger angle.The objective function, I, for the design is the root mean square of the deviation in

static pressure from a uniform value at the inlet to the bypass duct:

I =

Ni=1 (pi p)2N

(9)

where p = 1N

Ni=1

pi is the mean inlet static pressure, with N the number of mesh

nodes on the inlet boundary. Hence, I = 0 corresponds to a completely uniform pressuredownstream of the fan rotor and, therefore, a zero excitation.

Using SOFT, the OGV stagger pattern has been optimised using the dynamic hillclimbing (DHC) approach. Two constraint values, const were used: 3

and 6.43.Figure 4(a) shows the convergence of the optimiser in the case when the stagger vari-

ation is limited to 3. Figure 4(b) shows the variation in static pressure around thecircumference at the inlet to the bypass duct. With uniform OGVs there is a peak to

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! "

#$%"" &"

#$%"" &"

(a) Convergence of the optimiser. (b) Pressure variation at OGV inlet.

Figure 4: Optimisation of bypass OGV stagger variation.

trough variation in static pressure of approximately 6kPa. With the stagger variationlimited to 3, the optimiser achieves a 57% reduction in the objective function. Withthe stagger variation limited to 6.43, a reduction of 85% is achieved.

3.2 Bypass OGV optimisation using adjoint CFD

The bypass OGV optimisation described in 3.1 is an example where the number ofdesign parameters is significantly larger than the number of cost and constraint functions.In fact, the constraints apply directly to the design parameters and do not require theevaluation of an auxiliary equation. This is precisely the situation where the use ofadjoint CFD, as described in section 2.7, should perform well. This section demonstratesthe adjoint capabilities of the PADRAM-HYDRA-SOFT system using the same bypassOGV case as used in 3.1.

In order to utilise adjoint CFD, the chosen optimiser must be one that is able to usegradient information effectively. In this example, the Sequential Quadratic Programming(SQP) technique is used. This is well suited to the use of adjoint CFD since it uses apre-defined sequence of perturbations to each design parameter in turn, in order to buildan approximation to the Hessian matrix of second derivatives. The Hessian matrix is thenused to compute a new, more optimal, point in the design space.

When using adjoint CFD, one factor that must be addressed is that only first ordersensitivity gradients are computed. In a design space that is highly non-linear, the adjointgradients must be coupled with a procedure for re-evaluating the base and adjoint flowfields at regular intervals in the optimisation cycle. The decision of whether to use theexisting adjoint gradients or recompute the steady and adjoint flow fields is based on asimple Euclidean distance test:

d =

Ni=1

(i oldi

)2(10)

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where, in this instance, N is the number of design parameters, i.When the Euclidean distance is smaller than a user specified threshold, e.g. 0.01, the

adjoint sensitivity gradients are used to compute the new value of the objective function;otherwise, the objective function is computed from a new non-linear steady flow solutionalong with a new adjoint solution.

Figure 5 shows the work flow for the optimisation of bypass OGVs using adjoint CFD.The modules setadj and rhslin are part of the HYDRA-CFD suite which compute theadjoint boundary conditions and flow residuals on a perturbed mesh respectively.

!

"

"#

#$"#

#%

! #$

"#

&&'&&()

!

"

"

Figure 5: Flow chart for adjoint optimisation

In this example, the maximum stagger variation is limited to 3. Figure 6(a) showsthe convergence of the SQP optimiser, and figure 6(a) shows the solution after 280 designcycles which achieves a reduction of 38% in the objective function. Comparing the con-vergence paths of the DHC (figure 4) and the SQP (figure 6) optimisers, it is seen thatthe SQP optimiser consists of a series of plateaux. Each plateau consists of the seriesof perturbations to each design parameter - it is these sensitivities that are computedusing the adjoint solution. Once all the perturbations have been evaluated, the Hessienmatrix is assembled and the optimiser moves to a new point in the design space. At thispoint, a new steady state solution is evaluated and if the Euclidean distance thresholdhas been exceeded a new adjoint solution is also computed. Hence, in the 280 iterationsof the optimiser shown in figure 6 there are only 16 steady and adjoint CFD simulations.Whereas, the 206 iterations shown in figure 4 each require a steady CFD solution.

The convergence of the adjoint SQP optimisation (figure 6) shows a number of over-shoots in the latter stages of the convergence. These are due to the fact that part convergedsteady and adjoint solutions were being used. This practice has been used successfullyin non-linear optimisation [13] - as the optimiser converges toward the optimum, the

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!"#

!$

(a) Convergence of the optimiser. (b) Pressure variation at OGV inlet.

Figure 6: Optimisation of bypass OGV stagger variation using adjoint CFD.

convergence of the steady CFD becomes tighter because the solution from the previousoptimisation step is used as an initial guess. This practice appears to be less success-ful when using adjoint CFD because the part convergence can lead to one or more poorsensitivity gradients which are propagated forward to the next adjoint solution. Thisalso explains why the adjoint optimisation does not achieve as large a reduction in thecost function as the non-linear approach. Further research is ongoing to understand howbest to use adjoint sensitivity gradients within a computationally efficient optimisationscheme.

3.3 Optimisation of a compressor stage

Modern axial flow compressors consist of many stages of closely coupled blade rowswhich must be modelled using multistage CFD techniques to obtain accurate performancepredictions. Similarly, design/optimisation of each blade row in turn is generally over-constrained by the need to maintain the matching with the adjacent rows. More optimumdesigns are, clearly, possible if the compressor is designed in its entirety as a multi-stagemachine. The capability of the PADRAM-HYDRA-SOFT system to design multistagecompressors is demonstrated in this section.

The example chosen is that of designing the 3rd stage of a 4-stage low speed compressorrig. The datum stage is shown in figure 7. The rotor 3 inlet Mach number is approximately0.2. The compressor is cantilevered with tip gaps of 1.18% span on the rotor and 1.12%span on the stator. There are 101 rotors and 134 stators and the calculation is run witha mixing plane between the rotor and stator.

The design space consists of lean, sweep, stagger, and camber angle at the trailing edge.The same set of design parameters are used for both the rotor and stator but, otherwise,the rotor and stator designs are independent. The design parameters are specified via

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Figure 7: Datum stage 3 of a 4 stage low-speed compressor.

control points at 0, 25, 50, 75 and 100% span on both the rotor and stator. A cubic splineis fitted through the control points to produce design perturbations at the interveningradii. In this example, the sweep, stagger and camber angles are not allowed to vary ateither the hub or casing - this would allow existing fittings on the rig to be re-used. Thelean angle is also fixed at the casing since this is invariant to any bulk circumferentialoffset. This reduces the design space to 26 independent parameters for the stage.

The objective function is

I =1 stage1 base (11)

with the constraint that

4mnew mbasembase

+ Rnew RbaseRbase+

new basebase (12)

where m is the inlet flow rate; R is the stage pressure ratio; is the stator exit flowangle; and, is a user defined threshold. By constraining the mass flow, pressure ratioand exit whirl angle the matching between the stage being designed and the other 3 stagesshould be maintained.

The optimisation is performed using the Simulated Annealing optimiser within SOFT.Figure 8 shows the initial convergence of the optimiser together with snapshots of thedesign at three points in the convergence history. After running the optimiser for 400iterations a 0.72% improvement in stage efficiency was achieved.

3.4 Minimisation of turbine forced response using linear unsteady CFD

Forced response is the excitation of one blade row by another due to a periodic forcingfrom either the wake from the upstream blade; or, the potential field from the downstreamblade. With trends towards increased blade loading and reduced axial gaps between bladerows, the higher unsteady forces and hence vibratory stresses levels can lead to High CycleFatigue (HCF).

A common approach to reducing forced response levels is the so called wake-shapingtechnique. The rationale is that if the vane wake is completely in phase when it reachesthe rotor, the latter feels a large impulsive force at discrete instants in time. Whereas, if

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Figure 8: Optimisation for stage efficiency

the vane wake is out of phase (i.e. leant circumferentially relative to the rotor), the forceexperienced by the rotor is distributed over a longer time interval and the peak forcinglevel, and hence likelihood of HCF, is lower. Leaning the vane wake can be achieved byrestacking the stream sections defining the vane geometry. Since the vane, typically, hasa high exit flow angle this is a very powerful design mechanism and the vane wake can beleant significantly with relatively small changes in stacking.

The design objective is usually the HCF endurance limit, the evaluation of which isa 2 step process. In the first step, the unsteady forces, i.e. pressures, on the rotorare determined from a CFD simulation. In the second, a mechanical analysis of therotor is performed to determine the forcing levels. The mechanical analysis uses a modalrepresentation of the structure and solves an equation of the form:

Mx+ Cx+Kx = F1 sin(t+ ) + F2 (13)

where x is the displacement; M , C and K are the mass, damping and stiffness matricesrespectively; F1 is the unsteady force from the CFD calculation at a frequency andan inter-blade phase angle ; and, F2 contains any additional forces such as non-lineardamping.

The result of the mechanical analysis is the amplitude of the maximum displacement,usually at the rotor tip. This is fed into a database of material properties along with thesteady and alternating stress levels on the rotor and the metal temperature to producean estimate of the HCF endurance level.

The evaluation of the unsteady rotor pressures requires an unsteady CFD simulation,which is expensive if a time-accurate calculation is used. Fortunately, time linearisedunsteady CFD methods have proved a successful means of capturing the primary forcingmechanisms [14] and are computationally very efficient. Here, the unsteady flow in the

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rotor is solved as a perturbation to its steady state flow field at a given frequency andinterblade phase angle (defined by the numbers of rotors and vanes). The unsteadiness inthe rotor is driven by the incoming wake. This is extracted from a steady CFD solutionof the upstream vane at the axial position corresponding to the rotor inlet. At each radialheight, the vane wake is decomposed into Fourier harmonics. The amplitude and phaseof the harmonic of interest provide the inlet boundary conditions for the linear unsteadyrotor calculation.

The capability of the PADRAM-HYDRA-SOFT system to combine aerodynamic andmechanical analysis codes; and, to utilise advanced techniques such as linearised unsteadyCFD is demonstrated in this section. The example chosen is that of designing a HighPressure (HP) turbine vane which minimises the HCF endurance level of the downstreamHP rotor. More exactly, it is the HCF limit of the 2nd edge (2E) mode which is to beminimised; with the constraints that the HCF limits in the 1st torsion (1T) and 2nd flap(2F) modes should not increase.

The design is broken into 2 phases:

Phase 1: Taking the wake amplitudes from the base vane, use the linear unsteadyCFD solver to find the phasing, i.e. lean, of the wake which minimise the 2E HCFlimit.

Phase 2: Taking the optimal phasing of the wake, use inverse design techniques tofind the stack of the vane that delivers the required wake.

The design space consists of perturbations to the phase of the wake from the base vanespecified via control points at 0, 25, 50, 75 and 100% height. A cubic B-spline is fittedthrough the values at the control points to produce perturbations at the intervening radii.Since, the wake phase is invariant up to an additive constant, the phase perturbation at0% height is set to zero. The phase perturbations are also normalised to lie within 2pi/Nwhere N is the number of rotor blades. Hence, the wake can be leant by no more thanone rotor pitch in either circumferential direction.

3.4.1 Phase 1

The evaluation of the cost and constraint functions consists of: perturbing the incomingwake according to the design parameters; generating the corresponding linear unsteadysolution; performing the forced response analysis; and, computing the HCF endurancelevels for the 2E, 1T and 2F modes.

This example is also used to demonstrate the approximation techniques available inSOFT. The Response Surface Method (RSM) is used to construct an approximate modelof the design space which can be used to provide function evaluations in lieu of the simu-lation codes. The RSM requires an initial population of simulations is order to construct

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a first approximation to the response surface. The design parameters for the initial popu-lation are determined using the Taguchi Design of Experiments (DoE) technique. In thisexample, a three level Taguchi DoE was used which, for 4 design parameters, consists of 9experiments. A further 10, manually prescribed, experiments were also performed. Thisgave a population of 20 simulations (including the initial design) from which the initialresponse surface was constructed. The RSM method proceeds by performing an optimi-sation using the response surface to evaluate the HCF levels rather than the simulationcodes. Once an optimum has been obtained, a verification run of the simulation codes isperformed using the optimum wake shape parameters. If the results of verification runare sufficiently close to the RSM value then the process has converged; otherwise, the newresults are added to the RSM population and the response surface is updated.

Structural mode Initial HCF level Minimum HCF level Redesigned Nozzle1st Torsion 0.39467 0.0518 0.070012nd Flap 0.14444 0.0636 0.048152nd Edge 10.91056 1.7974 4.68130

Table 1: Initial and optimum HCF endurance levels.

For this case, the optimisation is performed using the Simulated Annealing (SA) tech-nique and a total of 12 verification simulations are needed for the RSM to converge. The2nd and 3rd columns of table 1 show the initial and optimised HCF endurance levels forthe three modes of interest The optimisation has not only achieved a reduction of 87%in the objective function it has also bettered the constraints with reductions of 83% and56% reductions in the 1T and 2F modes.

Figure 9: Inverse designed nozzle giving required wake shape.

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3.4.2 Phase 2

The inverse design of the nozzle to deliver the required wake shape follows the processdescribed in [2]. The resulting nozzle is shown in figure 9. The HCF levels were recom-puted using the wake from the redesigned nozzle and are shown in the 4th column of table1. The rise in the 2E HCF level is due to the fact that the inverse design procedure isnot able to deliver a new nozzle which gives exactly the desired wake shape. Also, it canbe seen from figure 9 that the changes in the stack of the nozzle are somewhat extreme,particularly from the point of view of internal cooling passages. This was not imposedas a constraint in the design which is why such large reductions in the objective functionwere achieved.

3.5 Low noise fan design

In a modern high bypass ratio gas turbine engine, a significant amount of noise isgenerated by the fan rotor. The noise falls into 2 categories: tone noise which is generatedat discrete frequencies and generally associated with blade row interactions and shocks;and, broadband noise which is generated over a broad spectrum and generally associatedwith boundary layers and turbulence phenomena. Tone noise is much more amenable toanalysis by CFD and the steady and linearised unsteady solvers in HYDRA-CFD havebeen successfully applied to a number of tone noise test cases [15]. Indeed, there issufficient confidence in the calculation of tone noise to consider using the optimisation todesign components specifically to reduce noise levels. To demonstrate this, the PADRAM-HYDRA-SOFT design system has been applied to the reduction of tone noise generatedby a fan rotor.

Figure 10: High bypass ratio fan rotor with static pressure field and PADRAM generated mesh.

Figure 10 shows a typical high bypass ratio fan rotor along with its static pressure field

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on the hub, casing and rotor surfaces. It is the strong shocks in outer part of the span,as seen in the casing surface pressures, which are the source of the noise. Figure 10 alsoshows a blade to blade section from the PADRAMmesh. This mesh has approximately 1.7million nodes. This is much finer than would be needed for an aerodynamic optimisationand is needed to resolve the acoustics with a sufficient number of points per wavelength.

The design space consists of sweep and lean of the rotor blade sections at 90% and 100%height. Below 80% the blade sections are unmodified. Between 80-90% and 90-100% thedesign parameters are linearly interpolated to the other sections. The objective functionis the amplitude of the 1st radial harmonic of the tone at one blade passing frequency (1BPF). The objective functions is evaluated one chord upstream of the fan leading edge andat aerodynamic conditions corresponding the fan working line. Multi-point simulationsare used to ensure the optimisation is performed at a point on the working line.

Figure 11: Optimisation of fan rotor tone noise.

Figure 11 shows the convergence of SOFT using the dynamic hill climbing optimiser.The reduction in the amplitude of the first radial harmonic corresponds to a potentialreduction in noise of approximately 9dB. In this case, the HYDRA-CFD calculations wererun on a PC cluster. Each calculation used 60 processors and took approximately 2 hourselapsed time. Hence, the majority of the predicted reduction in the 1 BPF tone noise wasachieved after 2 days of running the optimiser.

4 CONCLUSIONS

A new design system based around parametric design and mesh generation; and, ad-vanced CFD and optimisation techniques has been presented. Five applications of thesystem have been presented which demonstrate the following, novel, attributes of thesystem:

A true multi-passage, multi-row design and meshing capability.

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The use of gradient sensitivities from adjoint CFD. The simultaneous design of multiple components. The coupling of aerodynamic and mechanical simulation codes. The use of advanced CFD techniques, such as linearised unsteady methods. The use of approximation techniques to supplement expensive CFD simulations The effective use of parallel CFD and PC clusters to reduce CFD simulation timesto an acceptable level.

The incorporation aerodynamic, aeromechanic and aeroacoustic design objectives.

5 ACKNOWLEDGEMENTS

The authors gratefully acknowledge the permission of Rolls-Royce plc to publish thispaper. We would also like to acknowledge the contribution of our colleague John Coup-land to section 3.5; and, to Davide Giacche, Diego Benito and Piero Distefano for theircontributions to sections 3.1, 3.2, 3.3 and 3.4.

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REFERENCES

[1] B.L. Lapworth. Challenges and Methodologies in the Design of Axial Flow Fans forHigh Bypass Ratio Gas Turbine Engines using Steady and Unsteady CFD. Advancesof CFD in Fluid Machinery Design, Ed. Elder, Tourlidakis and Yates, ProfessionalEngineering Publishing, 2002.

[2] S.Shahpar and B.L. Lapworth. A Forward and Inverse Three-Dimensional Linear De-sign System for Turbomachinery Applications. 4th ECCOMAS Computational FluidDynamics Conference, Athens, 1998.

[3] S.Shahpar. SOFT: A New Design and Optimisation Tool for Turbomachinery. Evolu-tionary Methods for Design, Optimisation and Control, Ed. Giannakoglou, Tsahalis,Periaux, Papailiou, Fogarty, CIMNE, Barcelona, 2002.

[4] S.Shahpar and B.L. Lapworth. PADRAM: Paramtric Design and Rapid MeshingSystem for Turbomachinery Optimisation. Paper GT-2003-38698, ASME Turbo Expo,Atlanta Georgia, June 16-19, 2003.

[5] B.L. Lapworth. Hydra-CFD: A Framework for Collaborative CFD Development In-ternational Conference on Scientific and Engineering Computation (IC-SEC), Sin-gapore, June 30 - July 02, 2004.

[6] P. Moinier, J.-D. Muller and M.B. Giles. Edge-based multigrid and preconditioningfor hybrid grids. AIAA Journal, Vol. 40, No 10, 2001.

[7] P. Moinier and M.B. Giles. Preconditioned Euler and Navier-Stokes Calculations onUnstructured Grids. 6th ICFD Conference on Numerical Methods for Fluid Dynam-ics, Oxford, UK, 1998.

[8] L. Martinelli. Calculations of Viscous Flows with a Multigrid Method. Ph.D Thesis,Dept. of Mech. And Aerospace Eng., Princeton University, USA, 1987.

[9] J.-D. Muller and M.B. Giles. Edge-Based Multigrid Schemes for Hybrid Grids. 6thICFD Conference on Numerical Methods for Fluid Dynamics, Oxford, UK, 1998.

[10] M.C. Duta, M.B. Giles and M.S. Campobasso. The harmonic adjoint approach tounsteady turbomachinery design. International Journal for Numerical Methods inFluids, 40(3-4), p.323-332, 2002.

[11] M.S. Campobasso, M.C. Duta, M.B. Giles. Adjoint Methods for TurbomachineryDesign. Paper No. 1055, ISABE Conference, Bangalore, India, 2001.

[12] A.B. Parry. Optimisation of bypass fan outlet guide vanes, Paper No. 96-GT-433,ASME Gas Turbine Conference, 1996.

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[13] S. Shahpar, D. Giacche and B.L. Lapworth. Multi-objective Design and Optimisationof Bypass Outlet-guide Vanes. Paper GT2003-38700, ASME Turbo Expo, AtlantaGeorgia, June 16-19, 2003.

[14] J.G. Marshall and M.B. Giles. Some Applications of a Time Linearized Euler Methodto Flutter and Forced Response in Turbomachinery. Unsteady Aerodynamics andAeroelasticity of Turbomachines, edited by T.H.Fransson, Kluwer Academic, Dor-drecht, NL, 1998.

[15] A.G. Wilson. Application of CFD to Wake/Aerofoil Interaction Noise - A Flat PlateValidation Case. Paper AIAA-2001-2135, 7th AIAA/CEAS Aeroacoustics Confer-ence, Maastricht, 28-30 May, 2001.

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INTRODUCTIONELEMENTS OF THE DESIGN SYSTEMParametric representationGeometry constructionMesh generationCFD solutionData extraction and function evaluationOptimisationAdjoint CFD

DESIGN APPLICATIONSOptimisation of bypass duct guide vanesBypass OGV optimisation using adjoint CFDOptimisation of a compressor stageMinimisation of turbine forced response using linear unsteady CFDPhase 1Phase 2

Low noise fan design

CONCLUSIONSACKNOWLEDGEMENTS

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