Research Article Design Optimization of a Transonic-Fan

Research ArticleDesign Optimization of a Transonic-Fan Rotor UsingNumerical Computations of the Full Compressible Navier-StokesEquations and Simplex Algorithm

M A Aziz1 Farouk M Owis2 and M M Abdelrahman2

1 Aircraft Engineering Department Institute of Aviation Engineering and Technology Giza Egypt2 Department of Aerospace Engineering Faculty of Engineering Cairo University PO Box 12613 Giza Egypt

Correspondence should be addressed to Farouk M Owis fowisengcuedueg

Received 30 June 2013 Revised 8 January 2014 Accepted 20 January 2014 Published 24 March 2014

Academic Editor S Acharya

Copyright copy 2014 M A Aziz et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The design of a transonic-fan rotor is optimized using numerical computations of the full three-dimensional Navier-Stokesequations The CFDRC-ACE multiphysics module which is a pressure-based solver is used for the numerical simulation Thecode is coupled with simplex optimization algorithm The optimization process is started from a suitable design point obtainedusing low fidelity analytical methods that is based on experimental correlations for the pressure losses and blade deviation angleThe fan blade shape is defined by its stacking line and airfoil shape which are considered the optimization parametersThe stackingline is defined by lean sweep and skews while blade airfoil shape is modified considering the thickness and camber distributionsThe optimization has been performed tomaximize the rotor total pressure ratio while keeping the rotor efficiency and surgemarginabove certain required values The results obtained are verified with the experimental data of Rotor 67 In addition the results ofthe optimized fan indicate that the optimum design is found to be leaned in the direction of rotation and has a forward sweep fromthe hub to mean section and backward sweep to the tip The pressure ratio increases from 1427 to 1627 at the design speed andmass flow rate

1 Introduction

Transonic fans are widely used in recent aircraft enginesto obtain maximum pressure ratios per stage High stagepressure ratios are important to reduce the engine weightsize and operational costs Performance of transonic fan hasreached a high level but further improvements are requiredby engine manufacturers

Recently axial flow fans have been developed to a pointwhere stage efficiency has exceeded 90 The goal of thecurrent study is to maximize the total pressure ratio and tosatisfy the required efficiency andmass flow rateHigh fidelitymethods are used for the design optimization of transonicfans of high pressure ratios Numerical methods offer afeasible approach to solve complex nonlinear optimizationproblems involving a multitude of design variables andconstraints in a systematic and efficient manner Applicationof these computational design optimization approaches for

fan blade designs can reduce design cost and design cycle andincrease efficiency of jet engines [1]

The problem of the design validation and optimizationof transonic compressors using numerical solution of theturbulent flow equations has been under investigation byseveral researchers [2ndash4] Hah and Reid [5] conducted anumerical study based on the three-dimensional Reynolds-averaged Navier-Stokes equation to investigate the detailedflow physics inside a transonic compressor Ning and Xu[6] performed a numerical investigation for the flow over atransonic compressor rotor using an implicit 3D flow solverwith one-equation Spalart-Allmaras turbulence model

The objective of the current study is the optimization ofthe rotor blade shape for a transonic fan to maximize therotor pressure ratio while satisfying certain geometrical andperformance constraints A parametric study for the fan bladegeometry is performed using a preliminary design methodfor transonic fans to find a suitable starting point for the highfidelity design method [7] Experimental correlations are

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2014 Article ID 743154 16 pageshttpdxdoiorg1011552014743154

2 International Journal of Rotating Machinery

used to predict the shock losses profile losses and deviationangle during the preliminary design and off-design stepsThemultiple circular arc (MCA) airfoil shape is found to be agood choice to design compressor blades during the prelimi-narily steps

The numerical simulations are done using the CFDRC-ACE multiphysics module which is a pressure-based solverThe code solves the time-dependent Reynolds-averagedNavier-Stokes equations for turbulent compressible flowsusing a finite volume time-marching approach onmultizonestructured grids Spatial accuracy is nominally second-orderupwind formulation Steady flows are simulated through aniterative process using local time stepping Turbulence iscomputed using the Standard 119896-120576model

For relative inlet Mach numbers in the order of 13and higher the most important design intent is to reducethe Mach number in front of the passage shock This isof primary importance due to the strongly rising pressurelosses with increasing preshock Mach number and becauseof the increasing pressure losses due to the shockboundarylayer interaction or shock-induced separationThe reductionof the preshock Mach number can be achieved by zeroor even negative curvature in the front part of the bladesuction side and by a resulting precompression shock systemreducing the Mach number upstream of the final strongpassage shock The thickness is also kept very low about2 of chord for the tip section of a transonic fan Besidesinducing energy losses the presence of shock waves makestransonic compressors particularly sensitive to variations inblade section design An investigation of cascade throat areainternal contraction and trailing edge effective camber oncompressor performance shows that small changes in meanline angles and consequently in the airfoil shape and passagearea ratios significantly affect the performance of transonicblade rows [8]

One of the most important airfoil design parametersaffecting the aerodynamics of transonic blading is the chord-wise location of maximum thickness Good performance isobtained for the lower shock front losseswith the finer sectionwhich results when the location of the maximum thicknessis moved aft An optimum maximum thickness location isassumed to exist in the range of 55 to 60 of the chordlength for transonic fan rotors [9] Not only the positionof maximum thickness but also the airfoil thickness has asignificant impact on the aerodynamic behavior of transoniccompressor rotors

The flow field in a compressor is not influenced bythe two-dimensional airfoil geometryThe three-dimensionalshape of the blade is also of great importance especiallyin transonic compressor rotors where an optimization ofshock structure and its interference with secondary flowsis required Many experimental and numerical studies havebeen done for the design and analysis of three-dimensionalshaped transonic blading (eg [10 11])

2 Mathematical Model

Anoptimization algorithm is used tomaximize the total pres-sure for the fan rotor blades using the CFD-RC package The

rotor blade geometry is parameterized in order to facilitateits handling through the design process The level of successin parameterizing the blade is dependent on two factors Thefirst factor is the flexibility and amount of coverage of allpossible solutions The second factor is the compactness ofthe parameters The more accurate the description of theblade the bigger the number of parametersThe optimizationprocess is used to determine the following dimensions

(i) thickness distribution of each section at differentradii

(ii) camber distribution of each section at different radii(iii) staking line of the different blade sections from hub

to tip(iv) blade twist distribution over the staking line

21 Classification of Design Optimization Parameters Theprementioned design parameters are the factors affectingblade shape optimization process They can be classified intotwo main groups The first group is the hub-to-shroud (119911-119903or meridional) plane group The second group is the blade-to-blade (119903-120579) plane group and the mixing between blade-to-blade and hub-to-shroud planes group In the followingsections the details of each group are described and theireffects on the rotor blade performance are investigated

22 Hub-to-Shroud Plane Group Thehub-to-shroud (merid-ional) plane group geometry is parameterized as follows

(1) blade inlet and exit hub radius(2) blade inlet and exit tip radius(3) equation describing the hub curve in the meridional

plane(4) equation describing the tip curve in the meridional

plane

This classification is based on the direct physical dimen-sions and their effect on the fan performance Inlet hub andtip radii together are affecting the inlet area and so the averageinlet Mach number The inlet tip radius affects the peripheralspeed and consequently the value of relative Mach numberat the tip which should be kept as low as possible tominimizelosses in this part The equations describing the hub andtip curves in the meridional plane can affect the pressuregradient on the hub and tip surfaces boundary layer growthand the associated velocity profile It is difficult to manipulatethese parameters efficiently using simple 1D or even 2Dmodels It should be manipulated using 3D CFD solvers toaccount for its different impacts on the flow The meridionalplane group is shown in Figure 1

23 Blade-to-Blade Plane Group The blade-to-blade planegeometry is parameterized as follows

(1) blades spacing(2) equation describing the mean blade camber line

International Journal of Rotating Machinery 3

Casing

Stream lineh1

Hub

rh1

rh2

rt2

rt1

Figure 1 Hub-to-shroud (meridionalplane) group

LE

TEBlade

camber line

Throatlocation

Incoming flowdirection

Rotation direction

Bladespacing

1205721

1205722

Blade thickness

Figure 2 Blade-to-blade plane group

(3) equation describing the blade thickness distributionalong meridional coordinates

(4) blade chord distribution(5) stagger angle

The first parameter (blades spacing) depends on thenumber of blades and the radius at this location Its combinedeffect with the second and third parameters and blade heightdetermines the chocking conditionsTheblade camber thick-ness distribution and the stagger angle determine the bladeinlet and exit angles The blade inlet angle affects directlythe velocity triangle at the inlet flow incidence angle andrelativeMach numberThe exit blade angle affects the work ofthe rotor through affecting the circumferential absolute exitvelocity Figure 2 shows the blade-to-blade plane group

24 Mixing between Blade-to-Blade Plane Group and Hub-to-Shroud Plane Group It is a set of geometry parameters thatmay appear in the two planes described above It could beparameterized as follows

(1) equation describing the axial position of the sectionstaking point over the radial direction

(2) equation describing the circumferential position ofthe section staking point over the radial direction

The first parameter is the staking point axial coordinateswhich should cause the blade to sweep back or forwardThis sweep has its effect on improving the blade adiabaticefficiency as mentioned by [13] A recent numerical andexperimental work shows that the axial blade curvature canhelp to influence the shock shape in the meridional planeinducing the shock to assume the meridional curvature ofthe blade leading edge In addition a considerable impacton the radial outward migration of fluid particles whichtakes place inside the blade suction side boundary layerafter the interaction with the shock has been confirmedNumerical and experimental analyses carried out to evaluatethe performance of a forward swept rotor and an aft sweptrotor show that the forward swept rotor has higher peakefficiency and a substantially larger stall margin than thebaseline unswept rotor The aft swept rotor has similar peakefficiency with a significantly smaller stall margin [14]

Detailed analyses of the measured and calculated flowfields indicate that twomechanisms are primarily responsiblefor the differences in aerodynamic performance among theserotors The first mechanism is the change in the radial shapeof the passage shock near the casing by the endwall effect andthe second is the radial migration of low momentum fluid tothe blade tip region Similar results are obtained in a parallelinvestigation which identified the reduced shockboundarylayer interaction resulting from reduced axial flow diffusionand less accumulation of centrifuged blade surface boundarylayer at the tip as the prime contributor to the enhancedperformance with forward sweep [15]

The second parameter is the staking point circumfer-ential coordinates which should cause the blade lean Asmentioned in [16 17] it is weakening the passagersquos shockand it is reducing loss core near the tip of the suctionsurface A recent numerical work gave a point of viewon the impact of blade curvature in transonic compressorrotors showing how the movement of blade sections in thetangential direction can influence the internal flow field [1819] Another research group investigated the aerodynamiceffects induced by several tangential blade curvatures on thesame rotor It is observed that when the curvature is appliedtowards the direction of rotor rotation the blade-to-bladeshock tends to move more downstream becoming moreoblique to the incoming flow This reduces the aerodynamicshock losses and entropy generation showing in some casesa peak efficiency increment of over 1 at design speed [20]Similar results were previously obtained using a numericaloptimization algorithm [21]

Higher performance can be achieved using a propercombination of two orthogonal blade curvatures that is theuse of a blade curved both axially and tangentially and sweptand leaned at the same time as applied in the current studyFigure 3 shows the blade-to-blade and hub-to-shroud planesmixing group

25 Blade Thickness Distribution and Camber Line CurveTreatment The blade camber line curve has a major role inthe design problem It describes the blade angle distributionalong the meridional path The inlet and exit blade angles


Compound leantip

Sweeptip

HubHub

1205793

1205792

1205791

12057521205751

Circumferential direction Axial direction

Figure 3 Definition of swept and leaned rotor blade geometry

affect directly thework transferred to the fluidDescribing thecamber line could be done using polynomial or Bezier curvesFive- or six-point Bezier curve is sufficient to describe acomplex curve where a polynomial of higher order is neededto do the same job In the present work three sections aredefined each section is defined using five-point Bezier curvefor description of the camber line curve That is to say ifevery point has two coordinates (119909 119910) a total number of (30)variables need to be defined Figure 4 shows the Bezier pointsused for the description of this curve In the present workand for the purpose of simplicity the number of sectionsconsidered is only three sections where themore sections arerecommended The meridional coordinates are taken as 025 50 75 and 100 of the chord length with the firstand last points being fixed

The blade thickness distribution along the meridionalcoordinates could be described using polynomial or Beziercurve In the present work this parameter is investigatedusing seven-point Bezier curve where the coordinates of thesecond and six points in the meridional directions representthe leading and trailing edges radiuses The remaining pointsare treated as described above in the camber line treatmentFigure 6 shows a typical blade thickness distribution usingBezier curve of Figures 4 and 5

26 Section Thickness and Camber Representation In orderto start the optimization for the section thickness and camberdistributions the Bezier control points should be determinedA Bezier curve is defined by a set of control points 119875

119899 where

119899 is the order A Bezier curve with 5 control points is a fourth-order curveThe parametric curves may be defined as follow

119910 (119909) =

119899

sum

119894=0

119887119894119899(119909) 119875119894 119909 isin [0 1] (1)

where the control points are 119875119894 119887119894119899(119909) are polynomials

defined as

119887119894119899(119909) = (119899

119894) 119909119894

(1 minus 119909)119899minus119894

(2)

and the ( 119899119894) is the binomial coefficient defined as

(119899

119894) =

119899

119894 (119899 minus 119894) (3)

Sect

ion

cam

ber (

m)

Section chord length (m)

Camber distributionControl points

3

25

2

15

1

05

00

001 002 003 004 005 006 007 008 009

times10minus3

Figure 4 Bezier curve describing blade camber line


25

2

15

1

05

00

001 002 003 004 005 006 007 008 009

times10minus3

Sect

ion

thic

knes

s (m

)

Thickness distributionControl points

Figure 5 Bezier curve describing blade thickness distribution

There are different methods to find the control pointscoordinates that accurately represent the section camberand thickness distributions One method that gives accurateresults is that using optimizations algorithm The algorithmsare used to locate the control points with the best fitting tothe original curve Figure 7 is a representation for applyingthe Particle Swarm Optimization (PSO) [22] on the camberline curve of a famous NASA Rotor 67

A fixed value for the tip clearance of 15 from bladetip chord is only considered in the current study The actualbehaviour of the rotor blade is affected by the combinationof the geometrical parameters together not by everyone

International Journal of Rotating Machinery 5y

(m)


003

002

001

0

minus001

minus002

minus003

0 001 002 003 004 005 006 007 008 009

Figure 6 Typical blade thickness and blade camber line in blade-to-blade plane using Bezier curve

y

OriginalPSO

Chord wise location

012

01

008

006

004

002

00 05 1 15 2 25 3 35 4

Figure 7 Bezier 5 control points representation for the camber lineobtained by PSO

alone That limits the ability of trusting empirical and simpleone-dimensional equations result This pushes the designertowards the obligatory 3D CFD simulations which deal withthe actual geometry as one unit combining all the previouslymentioned points The variation of the coordinates of anygeometric parameter will lead to a new geometry Thus it iseasy to manipulate the problem using an optimizer

3 The CFD Code

The CFD analysis or simulation is highly dependent on theboundary conditions because the flow is internal and theboundary conditions are applied in proximity to the complexflow features The first objective of this section is to describethe subsonic inflow and outflow boundary conditions that

Periodic boundary

Casing

Inlet Rotor

Outlet

Figure 8 The computational domain and boundary conditions

Tip clearance regionTangential AxialSparaice

Figure 9 Rotor mesh

have been implemented into theCFDcode and applied for theanalysis of flows through transonic fansThe second objectiveis to validate the utilized CFD code CFD-ACE through acomparison of the results with the previous computationaland the experimental studies

The CFDRC-ACE multiphysics module is a pressure-based solver It solves the time-dependent Reynolds-averagedNavier-Stokes equations for turbulent compressibleflows using a finite volume time-marching approach onmultizone structured grids Spatial accuracy is nominallysecond-order upwind formulation Steady flows are sim-ulated through an iterative process using local time steppingTurbulence is modeled using the Standard 119896-120576 model [23]CFD-ACE is capable of solving flows of speeds ranging fromlow subsonic flow to relatively high supersonic flow


Rela

tive M

ach

num

ber

Chord ()

15

14

13

12

11

1

09

08

minus100 minus50 0 50 100 150 200

20 span

LE TE

(a)

Rela

tive M

ach

num

ber

Chord ()

13

12

11

1

09

08

07

06minus100 minus50 0 50 100 150

50 span

LE TE

(b)

Figure 10 Comparisons between the experimental data [12] and the present CFD results for the relative Mach number at 20 and 50 spanmeasured from the tip section

Computed

Roto

r adi

abat

ic effi

cien

cy

Mass flow ratemass flow rate at choke

094

092

09

088

086

084

082092 093 094 095 096 097 098 099 1

Experimental reference

(a)

Computed

Mass flow ratemass flow rate at choke092 093 094 095 096 097 098 099 1

175

17

16

155

15

145

14

135

Roto

r tot

al p

ress

ure r

atio

Computed


175

17

16

155

15

145

14

135

Roto

r tot

al p

ress

ure r

atio 165


(b)

Figure 11 Comparison between the current computations of the rotor pressure ratio and the measured NASA Rotor 67

The computational domain for the rotor is constructedas a rotating domain The blade row is represented by asingle blade passage considering a 3D periodic sector alongthe whole rotor passage as indicated in Figure 8 Standardboundary conditions for subsonic flows are implementedAt the inlet the flow angles total pressure velocity andtotal temperature are specified At the outlet the averagevalue of the static pressure at the hub is prescribed whereascircumference pressure gradient is extrapolated to maintaina specified average static pressure The density and velocitycomponents are extrapolated from interior On the solid wall

the temperature is set constant as the total temperature at theinlet and the pressure is extrapolated from the interior Theno-slip boundary conditions and the temperature conditionare used together to compute the density and total energyPeriodic boundary conditions are applied from blade to bladepassage

Figure 8 is a representation of the boundary conditionsspecified in the problem Initially the flow properties in thecomputational domain are assumed to be uniform and areset equal to the inlet free stream values The rotor passage isdiscretized using three blocks to represent the flow volume


XY

Z

Figure 12 Fan rotor structured grid

around the blade Two blocks are for the rotor pressure andsuction sides and the other block is for the tip regionThe firsttwo blocks represent a sector with the blade in the middle

The geometry and mesh of each block is generated usingPYTHON script file in the preprocessor of CFDRC packageThe mesh used for the model is mainly structured Figure 9shows the rotor and tip clearance region mesh

31 Validation of the CFD Code The geometry chosen tovalidate the code is the transonic high-speed axial fan rotorof NASA Rotor 67 This low aspect ratio rotor is the firststage rotor of a two-stage transonic fan designed and testedwith laser anemometer measurements at the NASA GlennResearch Center [12] The geometry and the grid are con-structed in the CFDRC geometry module and the boundaryconditions are applied in the solver module using PYTHONscript file The simulations are performed for different valuesof the back pressure to construct the rotor map at the designspeed

At the design mass flow rate the relative Mach numberdistribution along the blade-to-blade 50 passage chordlength at 50 and 20 span measured from tip section ispresented in Figure 10Thefigures show good agreementwiththe experimental data of Rotor 67The results indicate that theeffect of the shock system inside the rotor passage and at theboundaries is predicted accurately

Figure 11 compares the computed and themeasured rotormaps at the design speed and at different off-design massflow rates The numerical simulation reveals that at designspeed the computed pressure ratio for the rotor agrees verywell with the experimental data However the maximumefficiency obtained is less about 2 than that obtained fromthe experiment This difference has been observed by otherauthors NASA investigations revealed that this is due to thepresence of high axial gap in hub annulus line upstream of theblade leading edge which has detrimental effects on the rotorproprieties [24] Therefore we conclude that the numericalsimulations performed in the current study predict importantflow features and mechanisms

A grid sensitivity study is performed The objective is todetermine the level at which the solution is invariant withthe grid size The grids used in the simulations are generated

using characteristic grid spacing ℎ The finest grid spacingis denoted as ℎ

1 For each grid the simulation results in an

observed flow quantity 119891 such as the total pressure ratioThe change in the quantity 119891 between the grids is expressedin terms of the grid convergence index (GCI) GCI errorestimates can be used with minimum of two mesh solutionsIt provides less error estimate when used with three meshsolutions [25] The GCI between a finer grid with spacing ℎ

1

and coarser grid with spacing ℎ2is defined as

GCI =119865119878

1003816100381610038161003816(1198912 minus 1198911) 11989111003816100381610038161003816

119903119901 minus 1 (4)

where 119903 is the refinement ratio between the finer grid andcoarser grid and 119901 is the order of grid convergence observedin the simulations and they are given by the followingexpressions

119903 =ℎ2

ℎ1

119901 =ln ((119891

3minus 1198912) (1198912minus 1198911))

ln (119903)

(5)

A factor of safety of 119865119878= 125 is used based on the

experience of applying GCI in many situations as indicatedby Roache [25] A second-order solution would have (119901 =2) The GCI is a measure of the percentage difference ofthe computed quantity from the value of the asymptoticnumerical value It approximates an error band and itindicates how much the solution would change with furtherrefinement of the grid Verification assessment involvesperforming consistency checks One such check is that themass is conserved through the flow domain For inlets andducts mass conservation can be assessed spatially along thestreamwise coordinate of the duct Mass flow bookkeepingtracks the mass flow through the compressor with that of thecaptured mass flow The boundary conditions are indirectlyverified through comparison of the simulation results toavailable analytic results for the flow field The geometryand grid generation for rotor blade are constructed at thegeometry module

A grid sensitivity study is performed to ensure thatthe baseline grid has adequate sizes to resolve the solid


1435

1434

1433

1432

1431

143

1429

1428

1427

1426

Flow

par

amet

er (t

otal

pre

ssur

e rat

io)

First grid point distance normalized to blade height10minus29 10minus28 10minus27 10minus26 10minus25

Figure 13 Effect of grid spacing on the accuracy of the steady statesolution

11

1

09

08

07

06

05

04

03

02

010 500 1000 1500

Nor

mal

ized

mas

s flow

ratio

Iteration number

Figure 14 Nondimensional mass flow rate convergence history

wall boundary layers and the shock system [8] Simulationsare conducted on different grids with variable grid pointsTable 1 summarizes the sensitivity of the number of cellsfor structured grids shown in Figure 12 Figure 13 showsthe variation of the observed flow quantity (ie total pres-sure ratio) for different grids sizes while Figure 14 showsthe convergence history of the normalized mass flow rate(Design) through the rotor for the grid considered inthe design optimization The computations asymptoticallyconverge after 750 iterations to the same design mass flowrate

4 High Fidelity Optimization

The aim of the present study is to obtain an optimum bladegeometry for a given preliminary design of the transonicfan with some geometrical and performance constraints Thefinal task is to combine all the developed modules in associa-tion with the simplex optimization algorithm to complete the

optimization cycle Figure 15 illustrates the sequence of theoptimization flowchart with some modifications

41 Optimization Algorithm The simplex algorithm is usedfor the optimization process The algorithm is a direct (non-gradient) optimization method and requires only one objec-tive function evaluation per design iterationThe algorithm isrobust and is likely to converge The algorithm is easy to usebecause it has only three parameters to adjust (initial valuesof variables first step size and minimum and maximumvariables values) Some of the algorithm disadvantages arethat as with most algorithms the algorithm may find alocal minimum instead of the global minimum Differentminimum solutions can be found by starting the optimizerat different initial points Since the simplex algorithm doesnot use past information to accelerate movement through thedesign space convergence can be slow (especially with a largenumber of design variables)

A simplex is a polygon defined by (119899 + 1) verticesin 119899-dimensional space For example in 2D a simplexis a triangle (Figure 16) The simplex is termed ldquoregularrdquoif its vertices are equidistant Each vertex of the polygonrepresents a single design configuration with design variablevalues 119883(1) 119883(2) 119883(119873) each of which corresponds toan objective function value To progress towards an optimumsolution the simplex algorithm reflects the vertex associatedwith the worst design through the centroid of the polygonNew design variable values and the associated objectivefunction value define the new point

As the algorithm progresses through the design spacetwo setbacks can occurThe first setback occurs if the currentworst design is created in the previous iteration If this pointis again reflected the algorithmwould bounce back and forthbetween two configurations The algorithm instead reflectsthe second worst point The simplex moves in a differentdirection away from the stall point An objective functionthat has a steep valley leading to a local minimum will causethe simplex algorithm to cycle infinitely through the samedesign points at the rim of the valley The second setback isthat when simplex cycles through the same designs over aperiod of several iterations the algorithm is stalled Reducingthe physical size of the simplex allows it to fit into the valleyand get closer to the minimum solutionThe size reduction isdone at the first instance of a repeated design

An initial value for each design variablemust be specifiedThe optimizer uses initial variable values as a starting guessfor the optimization studies These values will be used tocreate the first design To start the optimization process onemust enter a value for the first step This value essentially setsthe geometric size of the simplex and affects the behavior ofthe algorithm A good rule of thumb for choosing a valuefor Delta is 20 of the size of the entire design space Theminimum and maximum values for each design variableare specified This will bound the optimizer preventing itfrom choosing designs that lie outside this range Constraintsare useful for preventing creation of unrealistic geometry orapplication of unrealistic boundary or volume conditionsThe maximum and minimum values were set as plusmn10 for


Preliminary designselected parameters

Gen

erat

es

cand

idat

e bla

de

Constraintsverifications

Meshgeneration Simulation Analysis

objective

Returnaerodynamicperformance

(cost function)

Performance analysis module

If co

nstr

aint

s vio

late

d

Simplexoptimization

algorithm

Figure 15 A flowchart for the design optimization process

X(1) X(1)X(2) X(2)

X(3)X(new)X(3)

Figure 16 New design point in simplex optimization algorithm

most of the variables such Bezier points coordinates Thiscycle is segmented into main steps that were previouslydeveloped and programmed They are as follows

(i) The lowfidelity (preliminary design) is started and thegeometry of the new design specified

(ii) The simplex optimization algorithm starts with thelow fidelity optimal as a baseline of optimization

(iii) The geometry construction variables are importedin CFD-GEOM and geometry is constructed Theconstructed geometry is passed through the con-straint verifications stage In this stage the geometry is

checked to make sure that it satisfies the constraintsThen the edge grid is generated and all steps arebeing carried out by executing the developed gridgeneration module

(iv) The boundary condition initial condition and solvercontrols are applied in the simulation module byexecuting the solver setting module

(v) Then the analysis objective module executes Outputof this module is a data file containing values whichis the average value of the cost function and thegeometric parameters


Table 1 Rotor pressure ratio for different grid sizes

Number of cells 21198645 51198645 81198645 91198645 101198645

Normalized 1st grid spacing 323119864 minus 3 20119864 minus 3 14119864 minus 3 12119864 minus 3 11119864 minus 3

(stage total pressure ratio) 1426 14317 14327 1433 14334

(vi) The simplex optimizer continues to run on the othergeometry and the path of the optimization is storedin a data file to monitor the history of results duringthe optimization run

All the above steps are arranged and programmed usingthe PYTHON language and conducted in the simulationmanager module which is one of the modules in the CFDRCpackage

5 Results

51 Low Fidelity Design Results The current transonic rotorunder consideration is a first-stage rotor of a three stage fanrecently designed [7]The baseline data of the fan preliminarydesign is presented in Table 2 The results of the preliminarydesign using low fidelitiy modules are presented in Table 3The initial estimation of the number of stages indicates that3 stages are required to obtain an overall pressure ratioof 26 A parametric study is performed to investigate theeffect of different design parameters on the fan performanceand to choose the design parameters such as the rotationalspeed blade geometry and the stagger angle of the rotorand stator The fan performance is computed in terms ofthe surge margin fan efficiency and pressure ratio at thedesign and off-design conditions as shown in Figure 17 Theselections that are based on minimum number of stageswith maximum isentropic efficiency allow producing therequired fan pressure ratioThe three-stage fan is the result ofcompromise between the isentropic efficiency and tip speedconstraintThe fan stages have pressure ratios of 155 136 and125 for the 1st 2nd and 3rd stages respectively [7] Thenthe optimization process using the CFD is conducted forthe first-stage rotor only in order to reduce the optimizationparameters

52 High Fidelity Design Results Thehistory of the optimiza-tion process for the high fidelity design is shown in Figure 18CFD simulations conducted with and without optimizationare presented in Table 4 The difference in pressure ratioand efficiency for the low fidelity rotor in Table 3 and CFDsimulation in Table 4 is due to the inaccuracy of the lowlevel models considered in the preliminary design phaseThe comparison presented in Table 4 shows an increase inthe total pressure ratio by 138 The isentropic efficiencyincreases as well The number of blades computed for theoptimal design is less than the original low fidelity design by3 blades

The high fidelity design is found to lean toward thedirection of rotation The rotor blade is swept forward fromthe hub to mean portion of the blade and have a backwardswept for the rest of the blade as presented in Figure 19

4

35

3

25

2

15

190 95 100 105 110 115 120 125 130

Design pointDPSpeed linesSurge limit Efficiency contours

Tota

l pre

ssur

e rat

ioMass flow rate (kgs)

078

078

072

05

08

08

064

086

084

078

072

088

086

08

084

084

086

088

088086

DP084

6070

80

9095

100

110

115

Figure 17 Low fidelity fan performance map

17

165

16

155

15

145

14

1350 100 200 300 400 500 600

Iteration number

Roto

r pre

ssur

e rat

io

Optimization history

Figure 18 Variation of the rotor pressure ratio during the optimiza-tion process

Previous study of [9] concludes that more leaned rotor bladeincreases the rotor isentropic efficiency and the operatingrange The forward swept rotor is found to have higher peakefficiency and a substantially larger stall margin than thebaseline of nonsweep rotor

It is clear fromFigure 20 that the blade chord length tendsto increases at the hub section In addition the stagger angleslightly increases At the mean section the chord increaseswhile the stagger angle remains almost constant The tip


Table 2 Baseline data of the selected case

Requirements Inlet conditions Selected parametersMass flow rate 120 kgsec Total inlet temperature 300K 119873 (rpm) lt10500Fan pressure ratio 26 Total inlet pressure 101325 kPa Hub-to-tip ratio 02ndash07

Diffusion factor lt055Inlet axial Mach number 07

Table 3 Low fidelity design parameters

Parameter First stage Second stage Third stageRotor Stator Rotor Stator Rotor Stator

Blades number 25 27 34 35 34 35Mean radius (m) 032 032 032 032 032 032Aspect ratio 345 315 4 38 35 355Blade height 028 021 02 019 017 016Hubtip ratio 039 049 052 053 057 06(rpm) 9800 mdash 9800 mdash 9800 mdashTip speed (ms) 450 mdash 4113 mdash 398 mdash119872rel at tip 163 mdash 115 mdash 108 mdashPressure ratio 155 136 125Isentropic efficiency 0944 09337 09052

Design withoutoptimization

(a)

High fidelity optimaldesign

(b)

Low fidelity designHigh fidelity design

(c)

Figure 19 Comparison between the blade shape of the high fidelity optimal design and the design without optimization


Airfoil shapeat tip section

(a)


Airfoil shapeat mean section

(b)


Airfoil shapeat hub section

(c)

Figure 20 Comparison between airfoil shapes with and without optimization at 0 50 and 100 span from the hub


24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot

(a) Design without optimization

24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot

(b) High fidelity optimal design

Figure 21 Comparison between total pressure contours for high fidelity and low fidelity designs

Incomingflow

Mach18

16

14

12

1

08

06

04

02

00

(a) Without optimization

Mach18

16

14

12

1

08

06

04

02

00

(b) High optimal fidelity design

Figure 22 Comparison between high fidelity design and low fidelity design Mach contour at mean section

section has a greater stagger angle with a shorter chord lengththan the low fidelity design

Similar transonic stages with inlet Mach number of 07 to11 limited by a pressure ratio from 115 to 16 and an isentropicefficiency from80 to 85are obtained as indicated byBoyce[26] The isentropic efficiencies decrease with the increase ofthe inlet relative Mach number The current transonic rotorproduces a pressure ratio of 162

The total pressure contours at the inlet and exit planes ofthe rotors are presented in Figure 21The high fidelity optimaldesign has a high pressure distribution near the hub regionthan the low fidelity one

The Mach number contours of the high fidelity optimaldesign at the mean section are compared to those of thelow fidelity and the results are presented in Figure 22 Thehigh fidelity design has a lower Mach number in front of the


Mach14

12

1

08

06

04

02

00

Incomingflow

(a) Near hub section

Mach2

2

18

16

14

12

1

08

06

04

02

00

Incomingflow

(b) Near tip section

Figure 23 High fidelity design Mach contour near the hub (a) and tip (b) sections

Table 4 CFD simulations with and without optimization

Withoutoptimization High fidelity optimum design

Pressure ratio 143 1627Isentropicefficiency 082 0842

Leaned Non In direction of rotation

Swept Non Forward swept (hub to mean)Backward swept (mean to tip)

Numbers ofblades 25 22

passage shock Thus the losses across the shock are reducedand the passage shock is moved toward the blade leadingedge The shock system inside the passage is reduced to onestrong shock Researchers explain the shape of the shock atthe peak efficiency operation as an oblique shock followed bynormal shock [27 28] The location of the normal shock iscontrolled by the back pressure applied to the rotor

The same observations are shown in Figure 23(a) near thehub section Near the tip section of Figure 23(b) the flowenters the passage with relative Mach number of about 14The Mach number in front of the shock reaches 155 whichreduces to 095 after the shock

Close to the tip section the shock structure is affectedby the tip clearance flow Figure 24 shows the relative Machnumber contours at the tip clearance and how the shockstructure at this zone is affected by the tip flow Figure 25shows the total pressure contours for four meridional planesat 5 50 75 and 100 of the tip chord from leadingedge where the tip clearance effect appears strongly at themidchord The flow path over the blade tip leading edge at90 95 and 100 span from hub is presented in Figure 26

Incomingflow

Mach1629

16

14

12

1

08

06

04

02

00

Figure 24 High fidelity design Mach contour at tip section

Because of the very high inflow angle of attack the main flowcannot follow the direction given by the blade geometry asclear from Figure 26(b) This makes the flow slightly deflectfrom the suction side Flow through the tip clearance shownin Figure 26(c) interacts with the deflected flow and decreasessignificantly the main flow velocity at the tip region

Figure 27 shows the total pressure contour and velocityvector at the tip clearance region where the effect of the flowthrough the clearance on the main flow is noticeable

Performance of the high fidelity optimum design duringthe off-design operation is presented in Figures 28 and 29


25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow

Suction sidePressure side

307E + 005

(Nm2)Ptot

Figure 25 High fidelity design total pressure contours at four meridional planes

LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100

Figure 26 Flow path at different sections near the blade tip leading edge

Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)

Figure 27 Total pressure contour and flow path at the blade tip clearance

The results show that the total pressure ratio tends to increasewith the back pressure to a certain value As the back pressureis increased the rotor starts to stall Decreasing the backpressure increases the isentropic efficiency to certain beakpoint then decreases rapidly near the choke point as shownin Figure 29

The same trend is observed for the different operatingspeeds but the operation range decreases with the increasein rotor speed This result sets a limit on the range of theoperating speed The operation range measures the stabilityof the rotor performance One of the definitions for thesurge margin is that defined by Gostelow et al [29] Surge

International Journal of Rotating Machinery 15Ro

tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point

Normalized mass flow rate

107 ND100 ND

82 ND

mmchoke

Figure 28 Variation of the total pressure ratio with the mass flowrate at different rotational speeds for high fidelity design

Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1

DP design point107 ND100 ND

82 ND

085

084

083

082

081

08

079


DP

mmchoke

Figure 29 Variation of the isentropic efficiency with the mass flowrate at different rotational speeds

margin depends on the mass flow and the pressure ratio atthe operating point This margin is used to measure the rotorstability The current design has a 12 surge margin whichis a reasonable value compared to high loaded fans For highloaded rotors the surge margin varies from 10 to 20 [28]

6 Conclusion

In the current study the design of a transonic fan isoptimized using numerical simulation of the compressible-viscous flow equations and simplex optimization algorithmThe results obtained using the CFDRC code are verified withthe experimental data of Rotor 67 A grid sensitivity analysis is

performed for the numerical simulations The cost functionof the optimization process is the rotor total pressure ratioThe blade geometry is defined in terms of set of optimizationgroups describing the section chord stagger angle stakingposition the section thickness and camber distributionsTheoriginal total number of variables for the three sections is 84The optimum design is found to be leaned in the directionof rotation and has a forward sweep from the hub-to-meansection and backward sweep to the tip The pressure ratioincreased by 14 at the design speed and mass flow rateThe peak efficiency increments were numerically observedusing a blade prevalently curved towards the direction ofrotation and slightly backward inclined near the tip A fewernumber of blades is achieved to reduce the rotor weightThe performance of the new design shows a stable operationduring a wide range in the off design

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Oyama M-S Liou and S Obayashi ldquoTransonic axial-flow blade optimization evolutionary algorithmsthree-dimen-sional Navier-Stokes solverrdquo Journal of Propulsion and Powervol 20 no 4 pp 612ndash619 2004

[2] Y Lian and M-S Liou ldquoMulti-objective optimization of tran-sonic compressor blade using evolutionary algorithmrdquo Journalof Propulsion and Power vol 21 no 6 pp 979ndash987 2005

[3] Y Lian and N-H Kim ldquoReliability-based design optimizationof a transonic compressorrdquoAIAA Journal vol 44 no 2 pp 368ndash375 2006

[4] A Oyama LM Liou and S Obayashi ldquoHigh fidelity swept andleaned rotor blade design optimization using evolutionary algo-rithmrdquo in Proceedings of the 16th AIAA Computational FluidDynamics Conference Orlando Fla USA 2003

[5] C Hah and L Reid ldquoA viscous flow study of shock-boundarylayer interaction radial transport and wake development in atransonic compressorrdquo Journal of Turbomachinery vol 114 no3 pp 538ndash547 1992

[6] F Ning and L Xu ldquoNumerical investigation of transonic com-pressor rotor flow using an implicit 3D flow solver with one-equation Spalart-Allmaras turbulence modelrdquo in Proceedingsof the ASME Turbo Expo Power for Land Sea and Air NewOrleans La USA 2001

[7] M A Aziz F M Owis and M M Abdelrahman ldquoPreliminarydesign of a transonic fan for low by-pass turbofan enginerdquoInternational Review of Aerospace Engineering vol 6 no 2 pp114ndash127 2013

[8] A R Wadia and W W Copenhaver ldquoAn investigation of theeffect of cascade area ratios on transonic compressor perform-ancerdquo Journal of Turbomachinery vol 118 no 4 pp 760ndash7701996

[9] R Biollo and E Benini ldquoRecent advances in transonic axialcompressor aerodynamicsrdquo Progress in Aerospace Sciences vol56 pp 1ndash18 2013


[10] C Hah D C Rabe and A R Wadia ldquoRole of tip-leakagevortices and passage shock in stall inception in a swept tran-sonic compressor rotorrdquo inProceedings of theASMETurboExpoPower for Land Sea and Air pp 545ndash555 Vienna Austria June2004

[11] S L PuterbaughWW Copenhaver C Hah and A J Wenner-strom ldquoA three-dimensional shock loss model applied to an aft-swept transonic compressor rotorrdquo Journal of Turbomachineryvol 119 no 3 pp 452ndash459 1997

[12] A J Strazisar J R Wood M D Hathaway and K L SuderldquoLaser anemometer measurements in a transonic axial-flow fanrotorrdquo NASA Technical Paper 2879 NASA 1989

[13] C-M Jang P Li and K-Y Kim ldquoOptimization of blade sweepin a transonic axial compressor rotorrdquo Journal of ThermalScience and Technology International B vol 48 no 4 pp 793ndash801 2006

[14] C Hah S L Puterbaugh and A R Wadia ldquoControl of shockstructure and secondary flow field inside transonic compressorrotors through aerodynamic sweeprdquo in Proceedings of theInternational Gas Turbine amp Aeroengine Congress amp Exhibitionpp 1ndash15 Stockholm Sweden June 1998

[15] A R Wadia P N Szucs and D W Crall ldquoInner workings ofaerodynamic sweeprdquo Journal of Turbomachinery vol 120 no 4pp 671ndash682 1998

[16] J Bergner S Kablitz D K Hennecke H Passrucker and ESteinhardt ldquoInfluence of sweep on the 3D shock structure in anaxial transonic compressorrdquo in Proceedings of the ASME TurboExpo Power for Land Sea and Air pp 343ndash352 Reno NevUSA June 2005

[17] S Kablitz H Passrucker D K Hennecke and M EngberldquoExperimental analysis of the influence of sweep on tip leakagevortex structure of an axial transonic compressor stagerdquo inProceedings of 16th International Symposium on Air-BreathingEngines (ISABE rsquo03) Cleveland Ohio USA 2003

[18] E Benini and R Biollo ldquoAerodynamics of swept and leanedtransonic compressor-rotorsrdquoApplied Energy vol 84 no 10 pp1012ndash1027 2007

[19] R Biollo and E Benini ldquoImpact of sweep and lean on theaerodynamic behavior of transonic compressorrotorsrdquo in Pro-ceedings of the 4th International Conference on Future of GasTurbine Technology Brussels Belgium 2008

[20] E Benini and R Biollo ldquoEffect of forward and aft lean onthe performance of a transonic compressor rotorrdquo InternationalJournal of Turbo and Jet Engines vol 25 no 1 pp 13ndash26 2008

[21] C-S Ahn and K-Y Kim ldquoAerodynamic design optimizationof an axial flow compressor rotorrdquo in Proceedings of the ASMETurbo Expo Power for Land Sea and Air pp 813ndash819 Amster-dam The Netherlands June 2002

[22] Q Bai ldquoAnalysis of particle swarm optimization algorithmrdquoComputer and Information Science vol 3 no 1 pp 180ndash1842010

[23] B E Launder and D B Splading Lectures in MathematicalModels of Turbulence Academic Press London UK 1972

[24] J Dunham ldquoCFD validation for propulsion system compo-nentsrdquo AGARD Advisory Report 355 1998

[25] P J Roache ldquoPerspective a method for uniform reporting ofgrid refinement studiesrdquo Journal of Fluids Engineering vol 116no 3 pp 405ndash413 1994

[26] M P Boyce Gas Turbine Engineering Handbook Butterworth-Hienemann 2nd edition 2003

[27] K M Boyer An improved streamline curvature approach for off-design analysis of transonic compression systems [PhD thesis]Virginia Polytechnic Institute and State University BlacksburgVa USA 2001

[28] G S Bloch Flow losses in supersonic compressor cascades [PhDthesis] Virginia Polytechnic Institute and State UniversityBlacksburg Va USA 1996

[29] J P Gostelow KW krabacber and L H Smith Jr PerformanceComparisons of High Mach Number Compressor Rotor BladingNational Aerodynamics and Space Administration Washing-ton DC USA 1968

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



used to predict the shock losses profile losses and deviationangle during the preliminary design and off-design stepsThemultiple circular arc (MCA) airfoil shape is found to be agood choice to design compressor blades during the prelimi-narily steps

The numerical simulations are done using the CFDRC-ACE multiphysics module which is a pressure-based solverThe code solves the time-dependent Reynolds-averagedNavier-Stokes equations for turbulent compressible flowsusing a finite volume time-marching approach onmultizonestructured grids Spatial accuracy is nominally second-orderupwind formulation Steady flows are simulated through aniterative process using local time stepping Turbulence iscomputed using the Standard 119896-120576model

For relative inlet Mach numbers in the order of 13and higher the most important design intent is to reducethe Mach number in front of the passage shock This isof primary importance due to the strongly rising pressurelosses with increasing preshock Mach number and becauseof the increasing pressure losses due to the shockboundarylayer interaction or shock-induced separationThe reductionof the preshock Mach number can be achieved by zeroor even negative curvature in the front part of the bladesuction side and by a resulting precompression shock systemreducing the Mach number upstream of the final strongpassage shock The thickness is also kept very low about2 of chord for the tip section of a transonic fan Besidesinducing energy losses the presence of shock waves makestransonic compressors particularly sensitive to variations inblade section design An investigation of cascade throat areainternal contraction and trailing edge effective camber oncompressor performance shows that small changes in meanline angles and consequently in the airfoil shape and passagearea ratios significantly affect the performance of transonicblade rows [8]

One of the most important airfoil design parametersaffecting the aerodynamics of transonic blading is the chord-wise location of maximum thickness Good performance isobtained for the lower shock front losseswith the finer sectionwhich results when the location of the maximum thicknessis moved aft An optimum maximum thickness location isassumed to exist in the range of 55 to 60 of the chordlength for transonic fan rotors [9] Not only the positionof maximum thickness but also the airfoil thickness has asignificant impact on the aerodynamic behavior of transoniccompressor rotors

The flow field in a compressor is not influenced bythe two-dimensional airfoil geometryThe three-dimensionalshape of the blade is also of great importance especiallyin transonic compressor rotors where an optimization ofshock structure and its interference with secondary flowsis required Many experimental and numerical studies havebeen done for the design and analysis of three-dimensionalshaped transonic blading (eg [10 11])

2 Mathematical Model

Anoptimization algorithm is used tomaximize the total pres-sure for the fan rotor blades using the CFD-RC package The

rotor blade geometry is parameterized in order to facilitateits handling through the design process The level of successin parameterizing the blade is dependent on two factors Thefirst factor is the flexibility and amount of coverage of allpossible solutions The second factor is the compactness ofthe parameters The more accurate the description of theblade the bigger the number of parametersThe optimizationprocess is used to determine the following dimensions

(i) thickness distribution of each section at differentradii

(ii) camber distribution of each section at different radii(iii) staking line of the different blade sections from hub

to tip(iv) blade twist distribution over the staking line

21 Classification of Design Optimization Parameters Theprementioned design parameters are the factors affectingblade shape optimization process They can be classified intotwo main groups The first group is the hub-to-shroud (119911-119903or meridional) plane group The second group is the blade-to-blade (119903-120579) plane group and the mixing between blade-to-blade and hub-to-shroud planes group In the followingsections the details of each group are described and theireffects on the rotor blade performance are investigated

22 Hub-to-Shroud Plane Group Thehub-to-shroud (merid-ional) plane group geometry is parameterized as follows

(1) blade inlet and exit hub radius(2) blade inlet and exit tip radius(3) equation describing the hub curve in the meridional

plane(4) equation describing the tip curve in the meridional

plane

This classification is based on the direct physical dimen-sions and their effect on the fan performance Inlet hub andtip radii together are affecting the inlet area and so the averageinlet Mach number The inlet tip radius affects the peripheralspeed and consequently the value of relative Mach numberat the tip which should be kept as low as possible tominimizelosses in this part The equations describing the hub andtip curves in the meridional plane can affect the pressuregradient on the hub and tip surfaces boundary layer growthand the associated velocity profile It is difficult to manipulatethese parameters efficiently using simple 1D or even 2Dmodels It should be manipulated using 3D CFD solvers toaccount for its different impacts on the flow The meridionalplane group is shown in Figure 1

23 Blade-to-Blade Plane Group The blade-to-blade planegeometry is parameterized as follows

(1) blades spacing(2) equation describing the mean blade camber line


Casing

Stream lineh1

Hub

rh1

rh2

rt2

rt1


LE

TEBlade

camber line

Throatlocation


Rotation direction

Bladespacing

1205721

1205722

Blade thickness














Compound leantip

Sweeptip

HubHub

1205793

1205792

1205791

12057521205751






119899 where


119910 (119909) =

119899

sum

119894=0

119887119894119899(119909) 119875119894 119909 isin [0 1] (1)


defined as

119887119894119899(119909) = (119899

119894) 119909119894

(1 minus 119909)119899minus119894

(2)


(119899

119894) =

119899

119894 (119899 minus 119894) (3)

Sect

ion

cam

ber (

m)



3

25

2

15

1

05

00

001 002 003 004 005 006 007 008 009

times10minus3



25

2

15

1

05

00

001 002 003 004 005 006 007 008 009

times10minus3

Sect

ion

thic

knes

s (m

)






(m)


003

002

001

0

minus001

minus002

minus003

0 001 002 003 004 005 006 007 008 009


y

OriginalPSO

Chord wise location

012

01

008

006

004

002

00 05 1 15 2 25 3 35 4



3 The CFD Code


Periodic boundary

Casing

Inlet Rotor

Outlet



Figure 9 Rotor mesh




Rela

tive M

ach

num

ber

Chord ()

15

14

13

12

11

1

09

08

minus100 minus50 0 50 100 150 200

20 span

LE TE

(a)

Rela

tive M

ach

num

ber

Chord ()

13

12

11

1

09

08

07

06minus100 minus50 0 50 100 150

50 span

LE TE

(b)


Computed

Roto

r adi

abat

ic effi

cien

cy


094

092

09

088

086

084

082092 093 094 095 096 097 098 099 1


(a)

Computed


175

17

16

155

15

145

14

135

Roto

r tot

al p

ress

ure r

atio

Computed


175

17

16

155

15

145

14

135

Roto

r tot

al p

ress

ure r

atio 165


(b)






XY

Z











1


GCI =119865119878

1003816100381610038161003816(1198912 minus 1198911) 11989111003816100381610038161003816

119903119901 minus 1 (4)


119903 =ℎ2

ℎ1

119901 =ln ((119891

3minus 1198912) (1198912minus 1198911))

ln (119903)

(5)





1435

1434

1433

1432

1431

143

1429

1428

1427

1426

Flow

par

amet

er (t

otal

pre

ssur

e rat

io)



11

1

09

08

07

06

05

04

03

02

010 500 1000 1500

Nor

mal

ized

mas

s flow

ratio

Iteration number












Gen

erat

es

cand

idat

e bla

de



objective


(cost function)


If co

nstr

aint

s vio

late

d

Simplexoptimization

algorithm


X(1) X(1)X(2) X(2)

X(3)X(new)X(3)











Number of cells 21198645 51198645 81198645 91198645 101198645





5 Results




4

35

3

25

2

15

190 95 100 105 110 115 120 125 130


Tota

l pre

ssur

e rat


078

078

072

05

08

08

064

086

084

078

072

088

086

08

084

084

086

088

088086

DP084

6070

80

9095

100

110

115


17

165

16

155

15

145

14

1350 100 200 300 400 500 600

Iteration number

Roto

r pre

ssur

e rat

io













(a)


(b)


(c)




(a)



(b)



(c)



24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot


24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot



Incomingflow

Mach18

16

14

12

1

08

06

04

02

00


Mach18

16

14

12

1

08

06

04

02

00








Mach14

12

1

08

06

04

02

00

Incomingflow


Mach2

2

18

16

14

12

1

08

06

04

02

00

Incomingflow












Incomingflow

Mach1629

16

14

12

1

08

06

04

02

00






25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow


307E + 005

(Nm2)Ptot


LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100


Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)





tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Casing

Stream lineh1

Hub

rh1

rh2

rt2

rt1


LE

TEBlade

camber line

Throatlocation


Rotation direction

Bladespacing

1205721

1205722

Blade thickness














Compound leantip

Sweeptip

HubHub

1205793

1205792

1205791

12057521205751






119899 where


119910 (119909) =

119899

sum

119894=0

119887119894119899(119909) 119875119894 119909 isin [0 1] (1)


defined as

119887119894119899(119909) = (119899

119894) 119909119894

(1 minus 119909)119899minus119894

(2)


(119899

119894) =

119899

119894 (119899 minus 119894) (3)

Sect

ion

cam

ber (

m)



3

25

2

15

1

05

00

001 002 003 004 005 006 007 008 009

times10minus3



25

2

15

1

05

00

001 002 003 004 005 006 007 008 009

times10minus3

Sect

ion

thic

knes

s (m

)






(m)


003

002

001

0

minus001

minus002

minus003

0 001 002 003 004 005 006 007 008 009


y

OriginalPSO

Chord wise location

012

01

008

006

004

002

00 05 1 15 2 25 3 35 4



3 The CFD Code


Periodic boundary

Casing

Inlet Rotor

Outlet



Figure 9 Rotor mesh




Rela

tive M

ach

num

ber

Chord ()

15

14

13

12

11

1

09

08

minus100 minus50 0 50 100 150 200

20 span

LE TE

(a)

Rela

tive M

ach

num

ber

Chord ()

13

12

11

1

09

08

07

06minus100 minus50 0 50 100 150

50 span

LE TE

(b)


Computed

Roto

r adi

abat

ic effi

cien

cy


094

092

09

088

086

084

082092 093 094 095 096 097 098 099 1


(a)

Computed


175

17

16

155

15

145

14

135

Roto

r tot

al p

ress

ure r

atio

Computed


175

17

16

155

15

145

14

135

Roto

r tot

al p

ress

ure r

atio 165


(b)






XY

Z











1


GCI =119865119878

1003816100381610038161003816(1198912 minus 1198911) 11989111003816100381610038161003816

119903119901 minus 1 (4)


119903 =ℎ2

ℎ1

119901 =ln ((119891

3minus 1198912) (1198912minus 1198911))

ln (119903)

(5)





1435

1434

1433

1432

1431

143

1429

1428

1427

1426

Flow

par

amet

er (t

otal

pre

ssur

e rat

io)



11

1

09

08

07

06

05

04

03

02

010 500 1000 1500

Nor

mal

ized

mas

s flow

ratio

Iteration number












Gen

erat

es

cand

idat

e bla

de



objective


(cost function)


If co

nstr

aint

s vio

late

d

Simplexoptimization

algorithm


X(1) X(1)X(2) X(2)

X(3)X(new)X(3)











Number of cells 21198645 51198645 81198645 91198645 101198645





5 Results




4

35

3

25

2

15

190 95 100 105 110 115 120 125 130


Tota

l pre

ssur

e rat


078

078

072

05

08

08

064

086

084

078

072

088

086

08

084

084

086

088

088086

DP084

6070

80

9095

100

110

115


17

165

16

155

15

145

14

1350 100 200 300 400 500 600

Iteration number

Roto

r pre

ssur

e rat

io













(a)


(b)


(c)




(a)



(b)



(c)



24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot


24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot



Incomingflow

Mach18

16

14

12

1

08

06

04

02

00


Mach18

16

14

12

1

08

06

04

02

00








Mach14

12

1

08

06

04

02

00

Incomingflow


Mach2

2

18

16

14

12

1

08

06

04

02

00

Incomingflow












Incomingflow

Mach1629

16

14

12

1

08

06

04

02

00






25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow


307E + 005

(Nm2)Ptot


LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100


Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)





tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Compound leantip

Sweeptip

HubHub

1205793

1205792

1205791

12057521205751






119899 where


119910 (119909) =

119899

sum

119894=0

119887119894119899(119909) 119875119894 119909 isin [0 1] (1)


defined as

119887119894119899(119909) = (119899

119894) 119909119894

(1 minus 119909)119899minus119894

(2)


(119899

119894) =

119899

119894 (119899 minus 119894) (3)

Sect

ion

cam

ber (

m)



3

25

2

15

1

05

00

001 002 003 004 005 006 007 008 009

times10minus3



25

2

15

1

05

00

001 002 003 004 005 006 007 008 009

times10minus3

Sect

ion

thic

knes

s (m

)






(m)


003

002

001

0

minus001

minus002

minus003

0 001 002 003 004 005 006 007 008 009


y

OriginalPSO

Chord wise location

012

01

008

006

004

002

00 05 1 15 2 25 3 35 4



3 The CFD Code


Periodic boundary

Casing

Inlet Rotor

Outlet



Figure 9 Rotor mesh




Rela

tive M

ach

num

ber

Chord ()

15

14

13

12

11

1

09

08

minus100 minus50 0 50 100 150 200

20 span

LE TE

(a)

Rela

tive M

ach

num

ber

Chord ()

13

12

11

1

09

08

07

06minus100 minus50 0 50 100 150

50 span

LE TE

(b)


Computed

Roto

r adi

abat

ic effi

cien

cy


094

092

09

088

086

084

082092 093 094 095 096 097 098 099 1


(a)

Computed


175

17

16

155

15

145

14

135

Roto

r tot

al p

ress

ure r

atio

Computed


175

17

16

155

15

145

14

135

Roto

r tot

al p

ress

ure r

atio 165


(b)






XY

Z











1


GCI =119865119878

1003816100381610038161003816(1198912 minus 1198911) 11989111003816100381610038161003816

119903119901 minus 1 (4)


119903 =ℎ2

ℎ1

119901 =ln ((119891

3minus 1198912) (1198912minus 1198911))

ln (119903)

(5)





1435

1434

1433

1432

1431

143

1429

1428

1427

1426

Flow

par

amet

er (t

otal

pre

ssur

e rat

io)



11

1

09

08

07

06

05

04

03

02

010 500 1000 1500

Nor

mal

ized

mas

s flow

ratio

Iteration number












Gen

erat

es

cand

idat

e bla

de



objective


(cost function)


If co

nstr

aint

s vio

late

d

Simplexoptimization

algorithm


X(1) X(1)X(2) X(2)

X(3)X(new)X(3)











Number of cells 21198645 51198645 81198645 91198645 101198645





5 Results




4

35

3

25

2

15

190 95 100 105 110 115 120 125 130


Tota

l pre

ssur

e rat


078

078

072

05

08

08

064

086

084

078

072

088

086

08

084

084

086

088

088086

DP084

6070

80

9095

100

110

115


17

165

16

155

15

145

14

1350 100 200 300 400 500 600

Iteration number

Roto

r pre

ssur

e rat

io













(a)


(b)


(c)




(a)



(b)



(c)



24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot


24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot



Incomingflow

Mach18

16

14

12

1

08

06

04

02

00


Mach18

16

14

12

1

08

06

04

02

00








Mach14

12

1

08

06

04

02

00

Incomingflow


Mach2

2

18

16

14

12

1

08

06

04

02

00

Incomingflow












Incomingflow

Mach1629

16

14

12

1

08

06

04

02

00






25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow


307E + 005

(Nm2)Ptot


LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100


Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)





tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(m)


003

002

001

0

minus001

minus002

minus003

0 001 002 003 004 005 006 007 008 009


y

OriginalPSO

Chord wise location

012

01

008

006

004

002

00 05 1 15 2 25 3 35 4



3 The CFD Code


Periodic boundary

Casing

Inlet Rotor

Outlet



Figure 9 Rotor mesh




Rela

tive M

ach

num

ber

Chord ()

15

14

13

12

11

1

09

08

minus100 minus50 0 50 100 150 200

20 span

LE TE

(a)

Rela

tive M

ach

num

ber

Chord ()

13

12

11

1

09

08

07

06minus100 minus50 0 50 100 150

50 span

LE TE

(b)


Computed

Roto

r adi

abat

ic effi

cien

cy


094

092

09

088

086

084

082092 093 094 095 096 097 098 099 1


(a)

Computed


175

17

16

155

15

145

14

135

Roto

r tot

al p

ress

ure r

atio

Computed


175

17

16

155

15

145

14

135

Roto

r tot

al p

ress

ure r

atio 165


(b)






XY

Z











1


GCI =119865119878

1003816100381610038161003816(1198912 minus 1198911) 11989111003816100381610038161003816

119903119901 minus 1 (4)


119903 =ℎ2

ℎ1

119901 =ln ((119891

3minus 1198912) (1198912minus 1198911))

ln (119903)

(5)





1435

1434

1433

1432

1431

143

1429

1428

1427

1426

Flow

par

amet

er (t

otal

pre

ssur

e rat

io)



11

1

09

08

07

06

05

04

03

02

010 500 1000 1500

Nor

mal

ized

mas

s flow

ratio

Iteration number












Gen

erat

es

cand

idat

e bla

de



objective


(cost function)


If co

nstr

aint

s vio

late

d

Simplexoptimization

algorithm


X(1) X(1)X(2) X(2)

X(3)X(new)X(3)











Number of cells 21198645 51198645 81198645 91198645 101198645





5 Results




4

35

3

25

2

15

190 95 100 105 110 115 120 125 130


Tota

l pre

ssur

e rat


078

078

072

05

08

08

064

086

084

078

072

088

086

08

084

084

086

088

088086

DP084

6070

80

9095

100

110

115


17

165

16

155

15

145

14

1350 100 200 300 400 500 600

Iteration number

Roto

r pre

ssur

e rat

io













(a)


(b)


(c)




(a)



(b)



(c)



24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot


24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot



Incomingflow

Mach18

16

14

12

1

08

06

04

02

00


Mach18

16

14

12

1

08

06

04

02

00








Mach14

12

1

08

06

04

02

00

Incomingflow


Mach2

2

18

16

14

12

1

08

06

04

02

00

Incomingflow












Incomingflow

Mach1629

16

14

12

1

08

06

04

02

00






25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow


307E + 005

(Nm2)Ptot


LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100


Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)





tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Rela

tive M

ach

num

ber

Chord ()

15

14

13

12

11

1

09

08

minus100 minus50 0 50 100 150 200

20 span

LE TE

(a)

Rela

tive M

ach

num

ber

Chord ()

13

12

11

1

09

08

07

06minus100 minus50 0 50 100 150

50 span

LE TE

(b)


Computed

Roto

r adi

abat

ic effi

cien

cy


094

092

09

088

086

084

082092 093 094 095 096 097 098 099 1


(a)

Computed


175

17

16

155

15

145

14

135

Roto

r tot

al p

ress

ure r

atio

Computed


175

17

16

155

15

145

14

135

Roto

r tot

al p

ress

ure r

atio 165


(b)






XY

Z











1


GCI =119865119878

1003816100381610038161003816(1198912 minus 1198911) 11989111003816100381610038161003816

119903119901 minus 1 (4)


119903 =ℎ2

ℎ1

119901 =ln ((119891

3minus 1198912) (1198912minus 1198911))

ln (119903)

(5)





1435

1434

1433

1432

1431

143

1429

1428

1427

1426

Flow

par

amet

er (t

otal

pre

ssur

e rat

io)



11

1

09

08

07

06

05

04

03

02

010 500 1000 1500

Nor

mal

ized

mas

s flow

ratio

Iteration number












Gen

erat

es

cand

idat

e bla

de



objective


(cost function)


If co

nstr

aint

s vio

late

d

Simplexoptimization

algorithm


X(1) X(1)X(2) X(2)

X(3)X(new)X(3)











Number of cells 21198645 51198645 81198645 91198645 101198645





5 Results




4

35

3

25

2

15

190 95 100 105 110 115 120 125 130


Tota

l pre

ssur

e rat


078

078

072

05

08

08

064

086

084

078

072

088

086

08

084

084

086

088

088086

DP084

6070

80

9095

100

110

115


17

165

16

155

15

145

14

1350 100 200 300 400 500 600

Iteration number

Roto

r pre

ssur

e rat

io













(a)


(b)


(c)




(a)



(b)



(c)



24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot


24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot



Incomingflow

Mach18

16

14

12

1

08

06

04

02

00


Mach18

16

14

12

1

08

06

04

02

00








Mach14

12

1

08

06

04

02

00

Incomingflow


Mach2

2

18

16

14

12

1

08

06

04

02

00

Incomingflow












Incomingflow

Mach1629

16

14

12

1

08

06

04

02

00






25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow


307E + 005

(Nm2)Ptot


LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100


Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)





tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










XY

Z











1


GCI =119865119878

1003816100381610038161003816(1198912 minus 1198911) 11989111003816100381610038161003816

119903119901 minus 1 (4)


119903 =ℎ2

ℎ1

119901 =ln ((119891

3minus 1198912) (1198912minus 1198911))

ln (119903)

(5)





1435

1434

1433

1432

1431

143

1429

1428

1427

1426

Flow

par

amet

er (t

otal

pre

ssur

e rat

io)



11

1

09

08

07

06

05

04

03

02

010 500 1000 1500

Nor

mal

ized

mas

s flow

ratio

Iteration number












Gen

erat

es

cand

idat

e bla

de



objective


(cost function)


If co

nstr

aint

s vio

late

d

Simplexoptimization

algorithm


X(1) X(1)X(2) X(2)

X(3)X(new)X(3)











Number of cells 21198645 51198645 81198645 91198645 101198645





5 Results




4

35

3

25

2

15

190 95 100 105 110 115 120 125 130


Tota

l pre

ssur

e rat


078

078

072

05

08

08

064

086

084

078

072

088

086

08

084

084

086

088

088086

DP084

6070

80

9095

100

110

115


17

165

16

155

15

145

14

1350 100 200 300 400 500 600

Iteration number

Roto

r pre

ssur

e rat

io













(a)


(b)


(c)




(a)



(b)



(c)



24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot


24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot



Incomingflow

Mach18

16

14

12

1

08

06

04

02

00


Mach18

16

14

12

1

08

06

04

02

00








Mach14

12

1

08

06

04

02

00

Incomingflow


Mach2

2

18

16

14

12

1

08

06

04

02

00

Incomingflow












Incomingflow

Mach1629

16

14

12

1

08

06

04

02

00






25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow


307E + 005

(Nm2)Ptot


LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100


Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)





tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1435

1434

1433

1432

1431

143

1429

1428

1427

1426

Flow

par

amet

er (t

otal

pre

ssur

e rat

io)



11

1

09

08

07

06

05

04

03

02

010 500 1000 1500

Nor

mal

ized

mas

s flow

ratio

Iteration number












Gen

erat

es

cand

idat

e bla

de



objective


(cost function)


If co

nstr

aint

s vio

late

d

Simplexoptimization

algorithm


X(1) X(1)X(2) X(2)

X(3)X(new)X(3)











Number of cells 21198645 51198645 81198645 91198645 101198645





5 Results




4

35

3

25

2

15

190 95 100 105 110 115 120 125 130


Tota

l pre

ssur

e rat


078

078

072

05

08

08

064

086

084

078

072

088

086

08

084

084

086

088

088086

DP084

6070

80

9095

100

110

115


17

165

16

155

15

145

14

1350 100 200 300 400 500 600

Iteration number

Roto

r pre

ssur

e rat

io













(a)


(b)


(c)




(a)



(b)



(c)



24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot


24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot



Incomingflow

Mach18

16

14

12

1

08

06

04

02

00


Mach18

16

14

12

1

08

06

04

02

00








Mach14

12

1

08

06

04

02

00

Incomingflow


Mach2

2

18

16

14

12

1

08

06

04

02

00

Incomingflow












Incomingflow

Mach1629

16

14

12

1

08

06

04

02

00






25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow


307E + 005

(Nm2)Ptot


LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100


Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)





tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Gen

erat

es

cand

idat

e bla

de



objective


(cost function)


If co

nstr

aint

s vio

late

d

Simplexoptimization

algorithm


X(1) X(1)X(2) X(2)

X(3)X(new)X(3)











Number of cells 21198645 51198645 81198645 91198645 101198645





5 Results




4

35

3

25

2

15

190 95 100 105 110 115 120 125 130


Tota

l pre

ssur

e rat


078

078

072

05

08

08

064

086

084

078

072

088

086

08

084

084

086

088

088086

DP084

6070

80

9095

100

110

115


17

165

16

155

15

145

14

1350 100 200 300 400 500 600

Iteration number

Roto

r pre

ssur

e rat

io













(a)


(b)


(c)




(a)



(b)



(c)



24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot


24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot



Incomingflow

Mach18

16

14

12

1

08

06

04

02

00


Mach18

16

14

12

1

08

06

04

02

00








Mach14

12

1

08

06

04

02

00

Incomingflow


Mach2

2

18

16

14

12

1

08

06

04

02

00

Incomingflow












Incomingflow

Mach1629

16

14

12

1

08

06

04

02

00






25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow


307E + 005

(Nm2)Ptot


LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100


Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)





tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Number of cells 21198645 51198645 81198645 91198645 101198645





5 Results




4

35

3

25

2

15

190 95 100 105 110 115 120 125 130


Tota

l pre

ssur

e rat


078

078

072

05

08

08

064

086

084

078

072

088

086

08

084

084

086

088

088086

DP084

6070

80

9095

100

110

115


17

165

16

155

15

145

14

1350 100 200 300 400 500 600

Iteration number

Roto

r pre

ssur

e rat

io













(a)


(b)


(c)




(a)



(b)



(c)



24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot


24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot



Incomingflow

Mach18

16

14

12

1

08

06

04

02

00


Mach18

16

14

12

1

08

06

04

02

00








Mach14

12

1

08

06

04

02

00

Incomingflow


Mach2

2

18

16

14

12

1

08

06

04

02

00

Incomingflow












Incomingflow

Mach1629

16

14

12

1

08

06

04

02

00






25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow


307E + 005

(Nm2)Ptot


LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100


Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)





tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















(a)


(b)


(c)




(a)



(b)



(c)



24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot


24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot



Incomingflow

Mach18

16

14

12

1

08

06

04

02

00


Mach18

16

14

12

1

08

06

04

02

00








Mach14

12

1

08

06

04

02

00

Incomingflow


Mach2

2

18

16

14

12

1

08

06

04

02

00

Incomingflow












Incomingflow

Mach1629

16

14

12

1

08

06

04

02

00






25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow


307E + 005

(Nm2)Ptot


LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100


Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)





tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot


24E + 005

22E + 005

2E + 005

18E + 005

16E + 005

14E + 005

12E + 005

1E + 005

8E + 004

6E + 0045759E + 004

26E + 005

(Nm2)Ptot



Incomingflow

Mach18

16

14

12

1

08

06

04

02

00


Mach18

16

14

12

1

08

06

04

02

00








Mach14

12

1

08

06

04

02

00

Incomingflow


Mach2

2

18

16

14

12

1

08

06

04

02

00

Incomingflow












Incomingflow

Mach1629

16

14

12

1

08

06

04

02

00






25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow


307E + 005

(Nm2)Ptot


LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100


Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)





tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Mach14

12

1

08

06

04

02

00

Incomingflow


Mach2

2

18

16

14

12

1

08

06

04

02

00

Incomingflow












Incomingflow

Mach1629

16

14

12

1

08

06

04

02

00






25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow


307E + 005

(Nm2)Ptot


LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100


Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)





tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










25E + 005

2E + 005

3E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

TE

LE

100

75

50

5

Incoming flow


307E + 005

(Nm2)Ptot


LE

(a) 90 Span

Incoming flow

(b) 95 (c) 100


Casing

LERotor tip

TE

(a)

Casing

LE Rotor tip

TE

(Nm2)

3E + 005

25E + 005

2E + 005

15E + 005

1E + 005

5E + 004

3987E + 004

307E + 005

Ptot

(b)





tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










tor p

ress

ure r

atio

120587R

2

19

18

17

16

15

14

13

086 088 09 092 094 096 098 1

DP

DP design point


107 ND100 ND

82 ND

mmchoke


Roto

r ise

ntro

pic e

ffici

ency

120578

086 088 09 092 094 096 098 1


82 ND

085

084

083

082

081

08

079


DP

mmchoke



6 Conclusion





References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Design Optimization of a Transonic-Fan