Arab J Sci Eng (2013) 38:351–364DOI 10.1007/s13369-012-0435-7

RESEARCH ARTICLE - SPECIAL ISSUE - MECHANICAL ENGINEERING

Multi-Level Multi-Objective Multi-Point Optimization Systemfor Axial Flow Compressor 2D Blade Design

A. Fathi · A. Shadaram

Received: 22 February 2012 / Accepted: 16 July 2012 / Published online: 17 January 2013© The Author(s) 2013. This article is published with open access at Springerlink.com

Abstract This paper introduces a multi-level frameworkto perform a multi-objective multi-point aerodynamic opti-mization of the axial compressor blade. This frameworkresults in a considerable speed-up of the design processby reducing both the design parameters and the computa-tional effort. To reduce the computational effort, optimiza-tion procedure is working on two levels of sophistication.Fast but approximate prediction methods has been used tofind a near-optimum geometry at the firs-level, which is thenfurther verified and refined by a more accurate but expen-sive Navier–Stokes solver. Surface curvature optimizationwas carried out in a first-level as a meta-function. Geneticalgorithm and gradient-based optimization were used to opti-mize the parameters of first-level and second-level, respec-tively. This procedure considers both the aerodynamic andmechanical constraints. An initial blade has been optimizedwith three different design targets to highlight the ability ofthe design method and to develop design know-how. Lead-ing-edge shape and curvature distributions of pressure andsuction surface had major effects on the design philosophiesof the blades. The result shows about −22.5 % reductionsin pressure-loss coefficient at design condition and 23.6 %improvement in the allowable incidence-angle range at off-design conditions compared to the initial blade.

Keywords Meta-function · Curvature distribution ·Genetic algorithm

A. Fathi (B) · A. ShadaramDepartment of Mechanical Engineering,K. N. Toosi University of Technology, Tehran, Irane-mail: alireza.fathy@gmail.com

A. Shadarame-mail: Shadaram@kntu.ac.ir

Nomenclatureb Bluntnessc Cord length (m)C CurvatureC p Curvature of pressure side, pressure coefficientCs Curvature of suction sideCt Curvature of thickness functioncl Lift coefficientH Weight distribution functionk Turbulence kinetic energyo Opening (m)P Point, local static pressure (pa)P1 Inlet static pressure (pa)q Primitive variable (P, u, v, T)

123

352 Arab J Sci Eng (2013) 38:351–364

r Radius (m)s Tangential spacing of blade (m)t Thickness of blade (m), time (s)T̂ Eigenvectors corresponding to the eigenvaluesUi Velocity componentsV Inlet flow velocity (m/s)Wi Weight of objective function termsx x Coordinates (m)xki Locations of maximum height of shape functionsx j Coordinatesy y Coordinates (m), distance to the nearest wallyi Weight of objective function terms

Greek Letters�α Incidence angle range (deg)∝ Incidence angle (deg)β Metal angle (deg)ω Loss coefficientγ Stagger angle (deg)ϕ Angle of opening diameter and y axis (deg)δ Leading edge thickness factor (deg)ρ Density (Kg/m3)

λ̂ Eigenvalues of the Jacobian matrixdetermined atRoe’s averaged condition

ω Turbulent frequencyδω Wave amplitudeξ, η Coordinate axis in computational domainμt Turbulent viscosityνt Turbulent kinematic viscosity

Subscripts

1, 2, 3 Derivative orderf First point of suction and pressure surfacef_, f_s First point in pressure and suction surfacesfo1, fo2 Point between leading edge and opening in the

suction surfacesft1, ft2 Point between leading edge and maximum

thickness in the pressure surfacesin Inletl Leading-edge parametermin Minimum lossmax Located at maximum thicknesso Locate at openingot1, ot2 Point between opening and trailing edge in the

suction surfacesout Outletp Pressure surfaceref Reference values Suction surfaces Parameters related to the thicknesstt1, tt2 Point between maximum thickness and trailing

edge in the pressure surfaces

t_p, t_s Last point at pressure and suction surfacest ThicknessW, E, N, S West, East, North, South

Superscript

L leftR right′ First order derivative

1 Introduction

The growing market of turbomachinery products causesa reduction in the times and costs of design by consid-ering alternative approaches, particularly in the aerody-namic design of blades. Many researchers have developedapproaches for optimal design of blades aided by numeri-cal algorithm. In this regard, one approach is to automatethe conventional design process by coupling an optimizationmethod, CAD-based blade generators, and a computationalfluid dynamics (CFD) code.

The earliest parametric design method was presented byDunham [1]. In this method, the thickness distribution ofblade was optimized around the camber line. However, thenature of flow phenomena in the pressure and suction sidesof the blades promoted the next activity to design blade sur-faces independently. Corral and Pastor [2] used the Beziercurve to generate turbomachinery blades’ surface. Keshinet al. [3] developed a tool for automatic multi-objective opti-mization of blade geometry with respect to loss and compres-sor operating range. Bonaiuti and Zangeneh [4] proposed anoptimization strategy, which enables the three-dimensionalmulti-point multi-objective aerodynamic optimization of tur-bomachinery blades. The design strategy was based onthe coupling of three-dimensional inverse design, responsesurface method, multi-objective evolutionary algorithms,and computational fluid dynamics analyzes. Korakiantis[5] presented the hierarchical development of three directblade-design methods of increasing utility for generatingtwo-dimensional blade shape. These methods can be usedfor generating inputs to the direct or inverse blade-designsequences in subsonic or supersonic airfoils in compressorsand turbines or isolated ones. These blade design methodscan be applied for improving the aerodynamic and heat-transfer performance of turbomachinery cascades, leadingto perform airfoils with in very little iteration. Many stud-ies in the turbomachinery field (Sonoda et al. [6], Shahparand Radford [7], Kammerer et al. [8], Buche et al. [9],Benini and Tourlidakis [10], Bonaiuti and Pediroda [11])have tried to exploit the time-saving potential of such a strat-egy and to integrate it into their design systems. Samad andKim [12] performed a multi-objective optimization of anaxial compressor rotor blade through genetic algorithm with

123

Arab J Sci Eng (2013) 38:351–364 353

total pressure and adiabatic efficiency as objective functions.By the optimization, maximum efficiency and total pressureare increased by 1.76 and 0.41 %, respectively. Ju and Zhang[13] established a multi-point and multi-objective optimi-zation design method, particularly, aiming at widening theoperating range while maintaining good performance at theacceptable expense of computational load. They used the arti-ficial neural network and the genetic algorithm technique andback-propagation algorithm along with the computationalfluid dynamics. Compared with their original design, opti-mized cascade decreases the total pressure loss coefficientby 1.54, 23.4, and 7.87 % at the incidences of 5◦,−9◦, and13◦, respectively.

Pierret and Van den Braembussche [14] presented a multi-level method for the automatic design of more efficient tur-bine blades. An artificial neural network (ANN) has beenused to construct an approximate model using a databasecontaining Navier–Stokes solutions for all previous designs.This approximate model has been used for the optimization ofthe blade geometry by means of simulated annealing, whichis then analyzed by a Navier–Stokes solver. This procedureresulted in a considerable speed-up of the design process byreducing the interventions of the operator and the computa-tional effort.

The parametric description of the blade is related to thedifficulties, which are conventionally performed by meansof purely geometrical parameters [15]. A possible solutionto this problem is to use a blade parameterization techniquebased on the parameters related to the fluid flow. Utilizing anoptimized curvature distribution on blade surface preventsthe occurrence of most of these drawbacks.

This paper presents a framework to perform multi-objec-tive multi-point aerodynamic optimization of axial compres-sor blade which considers all aerodynamic and mechanicalconstraints. A multi-level optimization is used to speed-updesign time; therefore, the blade geometry parameters wereclassified into two categories and optimized in two loopswith the distanced objective function. A combination of thegradient-based method and genetic algorithm is used to opti-mize the profiles. The main advantage of this approach isto use optimized surface curvature distribution to build upblade profile which was done in the first-level optimization.The aerodynamic performance of blade section in the designand off-design conditions is evaluated by a two-dimensionalNavier–Stokes blade-to-blade flow solver. An initial blade isoptimized with three different design philosophies to high-light the ability of design method in developing designknow-how.

2 Problem Description

High aero-performance is not the only objective of an opti-mization. A good design should perform well at off-design

Fig. 1 Loss variation versus incidence angle

conditions and respect the mechanical and manufacturingconstraints. Figure 1 schematically shows the loss variationsversus incidence angles for a sample blade. The value andlocation of minimum loss and incidence angle range betweenblade’s stall and choke can be observed in this figure. Ideally,design is done with the aim of increasing �α and reducingωmin at specific ∝min. However, boosting the blade loadingwill demand a decrease in �α and an increase in ωmin. Whenincidence variation at the off-design is not high, a blade canbe designed with the aim of reducing the minimum loss, andwhen incidence variations at the off-design is high, a bladeshould be designed with the aim of increasing �α.

Performance of a blade is only one of the many consider-ations among the design objectives. In fact, the mechanicalconstraints and the imposed flow angle must also be satisfied.The general approach to this problem is to build an objectivefunction, which is the summation of penalty terms, to limitthe violations of the constraints. Equation (1) shows the pro-posed objective function in this paper for blade design whereωmin.ref , αmin.ref and �αref are the reference values calcu-lated in the through-flow design step. w1, w2, and w3 are theweights of each term at the objective function. These weightswere used to dictate design philosophies to the design system[16].

OF = w1 × ωmin

ωmin .ref+ w2 × |αmin .ref − αmin|

αmin

+ w3 × �αref

�α+ PT (1)

where PT is the penalty term which is used to consideroptimization constraints. Computational cost of mentionedobjective function is high, because it needs a calculation ofcascade flow filed at several operational points (multi-pointoptimization). Increasing the number of parameters improvesblade geometry generation flexibility, but time and cost ofblade design would increase. If blade parameters have acloser relationship with the parameters influencing the flowfield, the speed of design process may be increased. Bound-ary layers on the blade surfaces and blade losses are affectedby both curvature value and curvature gradient of the blade

123

354 Arab J Sci Eng (2013) 38:351–364

Fig. 2 Parameterization of atypical cascade on the presentblade-design process

surfaces. The terms 1/r and 1/r2 in compressible Navier–Stokes equations in polar coordinate show strong dependenceof velocity and pressure on the curvature. Small slope-of-curvature discontinuities in the blade surfaces result in Machnumber or pressure coefficient surface-distribution spikes orhumps, which may introduce regions of local accelerationand deceleration, separation, or undesirable loading distri-butions along the length of the blades [15].

In this paper, velocity diagram and its parameters werepredefined from axisymmetric through-flow design. The fol-lowing non-dimensional parameters were also specified: thetangential spacing (s/c) between blades, opening diame-ter o/c, the trailing-edge thickness and the blade maximumthickness (the minimum allowable value of blade’s maximumthickness was specified from mechanical constraints).

3 Airfoil-Design System

Typically, the blade-design system developed in this workincludes a geometry generation, a cascade flow analysis tool,

and an optimization algorithm. Because of high computa-tional cost of the objective function, the optimization andgeometry tools are modified to optimize the airfoil with aminimum iteration. However, fast and reliable flow solver isalso necessary.

3.1 Geometry Tools

As shown in Fig. 2, the cascade geometry is specified fromthree curves: leading edge, suction surface and pressure sur-face. Each part of the blade surface has an analytical polyno-mial, which is evaluated by mapping a described curvaturedistribution. System of differential equations with boundaryvalue [Eq. (2)] is solved to calculate surface coordinates usingthe curvature distribution.

y1 = y′

y2 = C(x) · 3√

1 + y21 (2)

y(a) = ya y(b) = yb

123

Arab J Sci Eng (2013) 38:351–364 355

where a and b are boundary points and C(x) is the curvaturedistribution on surfaces.

The suction surface is made of two polynomials: one frompf (leading edge) to po (indicating the minimum distancebetween suction surface and stagnation point, evaluated byopening and the ϕ); another, from po to the trailing edge.Each part of curvature distribution is made of a curve withfour points. The first part of suction surface is evaluated bytwo conditions: locations of pf and po, and curvature dis-tribution specified by the curvature of these points plus twointermediate points. The second part is evaluated by four con-ditions: location of po, slope of suction surface at po, locationof pt (trailing edge point), and slope of suction surface at thetrailing edge (βs−out). The second part of the suction surfaceis defined by four boundary conditions and by Bezier curvewith four points. This is because the curvature value of twopoints of Bezier is a parameter.

Pressure surface consists of two curves: one from leadingedge to maximum thickness location and another from max-imum thickness to trailing edge. Pressure surface should bedefined to satisfy the maximum thickness location and maxi-mum thickness value. First and second derivative of thicknessdistribution should be zero and negative, respectively. There-fore, suction surface coordinate is calculated by Eq. (3).

yp = ys − t. (3)

There are three conditions for calculating the thicknessdistribution of the first part: thickness value at the leadingedge, maximum thickness value and thickness slope at xmax.Curvature distribution of pressure surface is also determinedby the four-point Bezier curve. Equation (4) is solved to cal-culate the thickness distribution of each part of the suctionsurface.

t1 = t ′

t2 = Ct (x) · 3√

1 + t21 (4)

Ct (x) =⎡⎣Cs − Cp · 3

√1 + (s′ − t1)2

1 + s2

⎤⎦ · 3

√1 + s′2

1 + t21

t(xf_p

) = tf_p t (xmax) = tmax t1 (xmax) = 0

There are three and four boundary conditions for the firstand second sections, respectively [value and slope of thethickness curve at xmax, and the trailing edge where (t ′ =y′

p − y′s)]. Pressure surface curvature distribution in each

section is defined by the four-point Bezier curve. Therefore,curvature value of one point at the first section and two pointsat the second section should be a parameter at each Beziercurve.

A leading-edge bluntness mode is implemented by fittinga profile to the nose of the airfoil section.

Fig. 3 Leading-edge bluntness

The profile is defined by Eq. (5).

f (x) = (4x × (1 − x))1/b (5)

1 ≤ b.

For b = 1, f (x) is a concave-down parabola centered atx = 1/2 with maximum value of 1. As b increases, so doesthe bluntness of the leading edge. Figure 3 shows the nose ofan airfoil to which the five different profiles with b values of1–5 have been fitted.

Trailing edge is defined by a semicircle. The semicirclediameter is a manufacturing limit which should not be lessthan the allowed minimum in the optimization process.

3.2 Flow Solver

Accurate calculations of the cascade loss in different condi-tions are necessary to determine an objective function. Thus,a flow analysis tool including mesh-generation tool and com-putational fluid dynamics solver are used to calculate cascadelosses. The flow field is automatically divided into severalzones in a way that one structured mesh is used for eachzone which is able to make a fine mesh near the walls. Atwo-dimensional compressible viscous flow code is used tosimulate the cascade fluid flow. This code is developed basedon Roe scheme, so that it could be used to design transonicblades also. A method has been presented by Kermani andPlett [17] is used to solve the fluid equations by the Roescheme in the computational domain and to give a formulafor the Roe’s numerical fluxes in generalized coordinates.The fluid governing equations for the viscous, unsteady andcompressible flow in generalized coordinates with no bodyforce could be shown as follows:

123

356 Arab J Sci Eng (2013) 38:351–364

∂ Q1

∂t+ ∂ F1

∂ξ+ ∂G1

∂η= ∂G1VT

∂η(6)

where Q1 is the conservative vector, F1 and G1 are the invis-cid flux vectors, and G1VT is the viscous flux vector. Becauseof high-speed flow in cascade, all the viscous derivativesalong the mainstream of the flow are neglected (thin-layerapproximation). Therefore, the governing equation can bediscretized as follows.

∂ Q1

∂t+ F1E − F1w

�ξ+ G1N − G1S

�η= G1VTN − G1VTS

�η(7)

For the time discretization, the two-step explicit scheme,from the Lax–Wendroff family of predictor–correctors, isused. The inviscid numerical flux F1E based on the Roescheme is written in generalized coordinates, according toRef. [17]. Other flux term can be written the same as F1E .

F1E = 1

2

[FL

1E+ FR

1E

]

−1

2

4∑

k=1

∣∣∣λ̂(k)E

∣∣∣ δω(K)E T̂ (k)

E

⎡⎣

√ξ2

x + ξ2y

J

⎤⎦

E

(8)

To obtain the left (L) and right (R) flow conditions, a third-order upwind-based algorithm with the MUSCL extrapo-lation strategy [18] is applied to the primitive variables(pressure, velocity components and temperature). For exam-ple, at the east cell face of the control volume, E, the L andR flow conditions are determined as follows (k = 1/3 in thisstudy).

qLE = q j,k + 1

4

[(1 − k)�wq + (1 + k)�Eq

]

qRE = q j+1,k − 1

4

[(1 − k)�EEq + (1 + k)�Eq

] (9)

For successful modeling of the turbulent effects on theflow, the K −ω (SST) method has been utilized which incor-porates modifications for low-Reynolds number effect, com-pressibility and shear flow spreading. k − ω (SST) modelconsists of a transformation of the k − e model in the outerregion to a k − ω formulation near the surface by a blendingfunction F1. Equations (10, 11) are k and ω equations of themodel, respectively.

∂(ρk)

∂t+ ∂

∂x j(ρU j k)

= ∂

∂x j

[(μ + μt

σk3

)∂k

∂x j

]+ Pk + β ′ρkω (10)

∂(ρω)

∂t+ ∂

∂x j(ρU jω)

= ∂

∂x j

[(μ + μt

σ ω2

)∂ω

∂x j

]

+(1 − F1)2ρ1

σω2ω

∂k

∂x j

∂ω

∂x j+ a3

w

kPk + β3ρω2 (11)

Table 1 k-ω (SST) model constants

β ′′ α1 β1 σk1 σω1 α2 β2 σκ2 σω2

.09 0.5556 0.075 2 2 0.44 0.0828 1 1.168

Table 2 Blade-to-blade domain mesh

Boundary Grid number Grid property

Inlet periodic 101 Grid ratio is 1.02

Inlet 51 Grid ratio is 1

Blade walls 101 Grid ratio is 1.02 double side

Outlet periodic 91 Grid ratio is 1.02

Leading andtrailing edge

5 Grid ratio is 1

Boundary layer 10 First row height is 0.01(mm)and growth factor is 1.25

In this model, eddy-viscosity formulation by a limiter isused to consider the transport of the turbulent shear stress.

μt = ρa1k

max(a1ω, SF2)(12)

F2 is a blending function which restricts the limiter to thewall boundary layer. S is an invariant measure of the strainrate. The blending functions are defined based on the distanceto the nearest wall and on the flow variables by Eqs. (13, 14).

F1 = tanh

⎛⎝

(min

(max

( √k

β ′ωy,

500ν

y2ω

),

4ρk

CDkwσω2y2

))4⎞⎠

(13)

where:

CDkω = max

(2ρ

1

σω2ω

∂k

∂x j

∂ω

∂x j, 1.0 × 10−10

)

F2 = tanh

⎛⎝

(max

(2√

k

β ′ωy,

500ν

y2ω

))2⎞⎠

(14)

All coefficients of model are listed in Table 1.To achieve accurate and reliable results, the tools could

make the mesh coarser or finer considering the input condi-tion and the turbulence model. Details of domain mesh havebeen presented in Table 2. Boundary layer mesh is used tosatisfy y+criteria. Domain mesh has been shown in Fig. 4.

The surface pressure coefficient [Eq. (15)] distributions atdesign point conditions resulting from the experimental cas-cade tests [19] and Navier–Stokes solver calculations for acontrolled diffusion blade are shown in Fig. 5. A good agree-ment between experimental data and simulated by presentedflow solver was observed. This agreement was pronouncedalong the whole blade surfaces. Therefore, the comparisonindicated that the present flow solver has enough accuracy tobe used in this work.

123

Arab J Sci Eng (2013) 38:351–364 357

Fig. 4 Cascade domain mesh

X/C

CP

0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

Experiment Data[19]

Presented flow solver

Fig. 5 Surface pressure coefficient distributions

Cp = p − p1

0.5ρV 2 (15)

3.3 Algorithm and Optimization Tool for Blade Design

High computational cost of objective function and largenumber of parameters lead to high design computational

Fig. 6 Flowchart of presented blade design process

cost. To reduce optimization time, design variable param-eters are divided into two categories. For each category,an optimization level is considered. The first category isrelated to the curvature and the curvature slope of blade sur-faces which directly related to the both Mach and the pres-sure distribution. The second category is used to define theoverall shape of blades (maximum thickness location, thick-ness and shape of the leading edge). These parameters areoptimized using the objective function in Eq. (1). Figure 6depicts blade-design system components. The input bladehas been re-parameterized to create initial guess of optimi-zation process. The parameters are calculated with least devi-ations between input blade and resulting geometry of theseparameters. In addition to coordinates of points, slope andcurvature of the surfaces are compared to calculate thesedeviations.

The whole shape of blade including the leading-edgeand trailing-edge thickness to the maximum thickness,the location of maximum thickness, and the leading-edgeshape factor is optimized at an outer loop with Eq. (1)as objective function. Table 3 define optimization prob-lem for both optimization levels. As shown in Fig. 6,in each iteration of outer loop, first-level variables need

123

358 Arab J Sci Eng (2013) 38:351–364

Table 3 Definition of optimization problem for the outer loop on thepresent blade-design process

Min OF(ωmin, αmin,�α)

Design variable (tt/tmax, tl/tmax, tmax, xtmax, y1, y2, y3)

Subject to tmax > tmax,ref

tt/tmax = (tt/tmax)ref

βin = Specified by designer

cl = Specified by designer

� = Specified by designer

o = Specified by designer

First-level variables = must be optimized in the inner loop

to be optimized. Constraints of design problem includingmechanical considerations (maximum thickness), manufac-turing limitations (trailing-edge thickness) and through flowdesign parameters (inlet metal angle, outlet metal angleand lift coefficients) must be satisfied through optimizationprocess.

In the first-level optimization, the surfaces curvature dis-tribution optimization is carried out at an inner loop usingthe integration of normalized curvature slope as its objectivefunction which summarized in Table 4. Considering that thecost of optimization based on the first-level objective func-tion is low, therefore, optimization of these parameters haslittle impact on the total time of the blade-design process.

A gradient-based optimization method is used to optimizethe second-category parameters on an outer loop. Becauseof the design is established aiming at minor changes on theoverall shape of the blade. Objectivity, computational cost ofgradient-based methods is low. A genetic algorithm is alsoused to optimize the parameters of first category on an inte-rior loop. The genetic algorithm is applied to prevent theoptimization process from falling into local optimal points.Since the computational cost of an inner-loop objective func-tion is low, the use of genetic algorithms for an inner loop isjustifiable. Finally, the output geometry is determined basedon the criteria of objective function in Eq. (1). Interior opti-mization loop are used to reduce the number of influencingparameters and optimization time.

4 Results and Discussion

Ability of each level of multi-level optimization system isinvested in this section. An initial profile is optimized withnine different objects by first-level optimization process toevaluate effect of curvature distribution. Three blades withdifferent objective function have been optimized with two-level optimization. In the end, performance of optimizedprofiles is compared with initial profile and a well-knownNACA-65 profile.

4.1 Firs-Level Optimization by Curvature DistributionAssessment

The objective of airfoil design is to curve flow to create liftforce with minimal stagnation pressure drop which is usu-ally due to: (1) shear forces between fluid and airfoil, and (2)an imbalance input and output pressures; both of which areaffected by the side-wall boundary layer. The blade designaffects boundary-layer development in three ways: first, thepositive pressure gradient on the blade suction surface thatcauses flow separation; second, drastic changes in surfacecurvature of the leading edges that make over-speed zonesand, consequently, the local separation, and, third, diffusingregions created on the pressure surface. The designer shoulddefine the surface curvature avoiding drastic changes on thetrailing-edge curvature and optimizing surface curvature dis-tribution which controls the boundary layer at pressure andsuction surfaces.

Blade curvature distribution can be changed using theweight function, H(x), which has been defined as a linearcombination of three sinusoidal-shape function illustrates byEq. (16). The weighing coefficients, yi , are the outer-loopvariables which are determined through multi-level optimi-zation process. Actually, the optimization loops are relatedto each other by these weighting coefficients.

H(x) =3∑

i=1

yi ×[Sin

(π

( x

2

)+ 0.25

)ei]0.3

(16)

Table 4 Definition ofoptimization problem forfirst-level on the presentblade-design process

Min obj = ∫tl H(x)

∂c(x)∂x dx

variables (xfo1·s, xfo1·s, cfo2·s, cfo·s, xot1·s, cft1·p, xft1·p, cft2·p, xft2·p, ctt1·p, xtt1·p, cfp, cmax·p)

Subject to:

y1, y2, y3 = specified from outer optimization loop

Other geometrical parameters = specified from outer optimization loop

CS(x) > 0

Max(

Numb(

∂c(x)∂x = 0

))=

{1 suction side

2 pressure side

123

Arab J Sci Eng (2013) 38:351–364 359

Fig. 7 Effect of yi variation on H(x)(y1 + y2 + y3 = 1, y2 = 0.1 :0.5, y3 = 0 : 0.3)

where

ei = log(0.5)

log(

xki2 + 0.25

)

xki = 0, 0.5, 1.0.

Here, xki values are the locations of maximum height ofcorresponding shape functions. Figure 7 shows the effect ofyi variation on H(x) where y1 + y2 + y3 = 1, y2 = 0.1 :0.5, y3 = 0 : 0.3. For example, if y1 become larger, locationof the maximum height move toward the front of the blade,consequently front curvature slope of blade became larger(front loaded). This blade had relatively long surface lengths,with relatively less adverse pressure gradients toward thepressure and suction surface.

The surfaces curvature optimization is carried out at thefirst level. Three groups of curvature distribution are inves-tigated. In the first one, weight coefficients of H(x) are ofsame order (y1 ∼ y2 ∼ y3), the second group is mid-loaded(y3 > y2 > y1), and third group is also front-loaded (y1 >

y2 > y3). An initial blade with inlet metal angles of 50◦,outlet metal angle of 30◦ and tmax/c = 0.2 has been used asinitial guess. Design variables which are presented in Table 2are summarized in the three mentioned groups and the othergeometric parameters are kept fixed. Figure 8 shows losscoefficient distribution and curvature distribution compari-son between the reference and optimized blades. In the firstgroup, the maximum curvature is located near leading edgewhich is reduced with almost uniform slope. Allowable inci-dence rang has been increased and minimum loss has beendecreased in this case. In the second group, the maximum cur-vature is located in the middle of the blade surfaces and onlythe minimum loss is reduced. In the third one, only allow-able incidence angles range has increased. In the best cases,about 16 % decreasing in minimum loss and 20 % incrementin allowable incidence angle rang has been achieved. Similar

changes on loss curve of optimized blade in each group resultin a very close relationship between design philosophy andcurvature distributions of surfaces. A blade with an expectedperformance can be designed by setting the proper weightsof H(x) which are design variables of the outer optimizationloop.

4.2 Multi-Level Optimization of Blade Profile

To investigate design tool abilities, three blades with differentdesign philosophies were optimized. First case is aimed toreduce the value of loss at the design point (w1 = 0.8, w2 =0.2, w3 = 0.0). In the second case, the goal is to widen theoperating range for off-design conditions (w1 = 0.0, w2 =0.2, w3 = 0.8). The third case represents an intermediatecase (w1 = 0.4, w2 = 0.2, w3 = 0.4). Figures 9, 10,and 11 compare the geometrical and aerodynamic charac-teristic of the mentioned cases with the initial geometry anda well-known NACA-65 profile with same inlet and outletmetal angle and maximum thickness. All figures consist offollowing features:

(a) Geometric comparison between the blades.(b) Comparing the curvature distribution of the suction sur-

face, pressure surface and camber line.(c) Comparing loss coefficient versus angle of attack

curves.(d) Comparison of Mach number distribution on pressure

and suction surfaces on α = +5◦, α = −5◦ and α =−15◦.

The first-case airfoil Comparison of the airfoils geometryis presented in Fig. 9a, the optimized profile shows almostsame stagger angles and thinner leading edge than the initialairfoils. Similar to initial profile, the new airfoils show morecamber in the mid-part and less camber in the front and rear-part (Fig. 9b). In as a starting profile, the design losses werereduced by more than 22 %; the incidence range is reducedby more than 3◦. As shown in Fig. 9c, design incidence anglehas not changed more.

The corresponding design and off-design Mach numberdistributions are shown in Fig. 9d. At design condition, firstcase is characterized by a smaller Mach number peak on lead-ing edge than initial profile, flow acceleration at beginning ofthe suction surface, boundary-layer transition at about 20 %chord, and smaller deceleration gradients in the mid-part andrear-part of the airfoils which result in lower pressure lossesthan initial profile. At positive incidence angle (α = +5◦),Mach number peak on the leading edge is smaller than ini-tial profile but boundary-layer separation and re-attachmentoccur at pressure surface; therefore, pressure losses is notsignificantly improved. Compared to initial blade, optimized

123

360 Arab J Sci Eng (2013) 38:351–364

Axi

al-p

ositi

on

Curvature

00.

51

-.01

5

-.01

-.00

50

.005.0

1

.015.0

2

.025

4

Axi

al-p

ositi

on

Curvature

00.

51

0

.005.0

1

.015.0

2

.025

1

Axi

al-p

ositi

on

Curvature

00.

51

-.010

.01

.02

.03

5

Axi

al-p

ositi

onCurvature

00.

51

-.010

.01

.02

.03

6

Axi

al-p

ositi

on

Curvature

00.

51

0

.005.0

1

.015.0

2

.025.0

33

Axi

al-p

ositi

on

Curvature

00.

51

0

.005.0

1

.015.0

2

.025

2

Axi

al-p

ositi

on

Curvature

00.

51

0

.02

.04

.06

.08

7

Axi

al-p

ositi

on

Curvature

00.

51

0

.005.0

1

.015.0

2

.025

9

Axi

al-p

ositi

on

Curvature

00.

51

-.050

.05.1

8

Inci

denc

e

Loss

-15

-10

-50

502468

32 1

Inci

denc

e

Loss

-15

-10

-50

502468

456

Inci

denc

e

Loss-1

5-1

0-5

05

02468

978

y 1~y

2~y 3

y 3>

y 2>

y 1

y 1>

y 2>

y 3

First group Second group Third group

Fig

.8

Eff

ecto

fcu

rvat

ure

dist

ribu

tion

onth

epr

ofile

loss

(sol

idli

ne:o

ptim

ized

blad

e,da

shli

ne:i

nitia

lbla

de).

Incu

rvat

ure

char

ts:u

pper

,mid

dle

and

low

ercu

rves

are

rela

ted

tosu

ctio

nsu

rfac

e,ca

mbe

rlin

ean

dpr

essu

resu

rfac

e,re

spec

tivel

y

123

Arab J Sci Eng (2013) 38:351–364 361

Cord fraction

Cur

vatu

re

0 0.2 0.4 0.6 0.8 1-0.01

0

0.01

0.02

0.03Suction

Camber

Pressure

Incidence(Deg)

Loss

-15 -10 -5 0 50

2

4

6

8

10

12

14

X

Y

0 5 10 15 20 25

0

5

10

15

20

Initial-geometryFirst caseNACA

b=1.45

b=1.05

(a) (b) (c)

Cord fraction0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7Incidence=-5°

Cord fraction0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7Incidence=+5°

Cord fraction

Mac

hN

um

ber

Mac

hN

um

ber

Mac

hN

um

ber

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7Incidence=-15°

(d)

Fig. 9 Comparison of geometry and aerodynamic performance of ini-tial profile and optimized profile of the first case. a Geometry of designedblade and initial profile, b curvature distribution for designed blade and

starting profile, c loss curve for designed blade and initial profile, ddesign and off-design Mach number distributions of initial profile anddesigned profile

blade pressure loss has been slightly improved at nega-tive off-design incidence angle (α = −15◦) because ofthe following flow characteristics: Mach number declara-tion gradient is less on the suction surface, flow separa-tion on the pressure surface is occurred at x/c = 0.7 thanx/c = 0.3, and a smaller Mach number peak at the leadingedge.

The second-case airfoil Figure 10a, b compares geometry andsurface curvature distributions of the second case and initialprofile. In the second surface case, curvature peak is locatednear the leading edge same as second airfoil group in lastsection. As shown in Fig. 10c, the considerable improvementin the off-design behavior of the optimized profile was car-ried out where the incidence range increased and the designlosses decreased. Comparison of Mach number distributionsis shown in Fig. 10d; the most significant difference is vis-ible in the upstream propagation of the peak suction sideMach number for the optimized airfoil. Suction side diffusionstarts shortly after the Mach number peak location, when the

turbulent boundary layer is still thin. On the other hand, thisleads to a smaller deceleration gradient for the optimizedairfoil in the downstream of the peak Mach number. On theone hand, this means that pressure loss at design incidenceslightly improved by optimization.

The most significant improvement is observed at off-design negative incidence angle. Compared to the initial pro-file, the second-case airfoil is characterized by leading-edgeshape treatment (the lower suction side peak Mach numberat the optimized airfoil), a flattened mid-part, a front-loadedMach number distribution, and removing the flow separa-tion at pressure side result in lower total pressure losses andconsiderably higher operating ranges at negative incidenceangle.

The higher local deceleration gradient results in the higherlocal momentum thickness growth rate. At positive inci-dence angle (α = +5◦), between 5 and 80 % of chord(Fig. 10d), the diffusion on the optimized geometry is signif-icantly higher and the boundary-layer momentum thickness

123

362 Arab J Sci Eng (2013) 38:351–364

Cord fractionC

urva

ture

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03Suction

Camber

Pressure

Incidence(Deg)

Loss

-15 -10 -5 0 50

2

4

6

8

10

12

14 b=1.45

b=3.2

X

Y

0 5 10 15 20 25

0

5

10

15

20

Initial-geometrySecond caseNACA

(a) (b) (c)

Cord fraction

Mac

hN

um

ber

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7Incidence=-5°

Cord fraction

Mac

hN

um

ber

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7Incidence=-15°

Cord fraction

Mac

hN

umb

er0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7Incidence=+5°

(d)

Fig. 10 Comparison of geometry and aerodynamic performance ofinitial profile and optimized profile of the second case. a Geometry ofdesigned blade and initial profile, b curvature distribution for designed

blade and initial profile, c loss curve for designed blade and initial pro-file, d design and off-design Mach number distributions of initial profileand designed profile

grows with a higher gradient; therefore, the total pressurelosses for optimized airfoil has been deteriorated at positiveincidence angle.

The third-case airfoil An equal weight was used for thefirst and third terms of the objective function. Geometry andsurface curvature distributions of third case and initial onehave been compared in Fig. 11a and b, respectively. In thesecond surface case, curvature peak is located near leadingedge as same as the second airfoil group in last section. Itwas observed that the minimum loss reduced about 6.6 %and the range of incidence angle improved by 9.6 % thanthe initial profile (Fig. 11c). To explain the reasons for theconsiderable increase in operating range, the Mach numberdistributions at +5◦ and −15◦ incidence for the third cas-cade are compared in Fig. 11d. The corresponding incidenceflow angles are marked in the total pressure-loss diagram inFig. 11c. The importance of leading-edge treatment is dem-onstrated by the off-design behavior of optimized profile,

while the peak Mach numbers of initial blade is about 0.8,the new airfoil avoids these peaks at −15◦ incidence andstays at a lower Mach number level. This peak reductionleads to a significant boundary layer unloading in the vicin-ity of the leading edge and finally results in an improve-ment of the separation behavior. The more camber in thefront, the almost flattened mid-part and the lower curvatureof rear-part of the profile lead to the reduced design pointlosses.

The NACA-65 profile minimum loss is greater than allother cases because of great much number at leading edgeand great curvature slope near leading edge. But off-designbehavior of this blade is better than that of initial profile andthe first case. To explain the reasons, there is no boundary-layer separation at negative off-design condition (Fig. 11d).In positive incidence angles, separation is occurred near lead-ing edge rather than initial profile and first case, therefore,its loss at α = +5◦ is somewhat greater.

123

Arab J Sci Eng (2013) 38:351–364 363

Cord fractionC

urva

ture

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03Suction

Camber

Pressure

Incidence(Deg)

Loss

-10 -5 0 50

2

4

6

8

10

12

14

X

Y

0 5 10 15 20 25

0

5

10

15

20

Initial-geometryThird caseNACA

b=1.45

b=1.41

(a) (b) (c)

Cord fraction

Mac

hN

um

ber

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7Incidence=-15°

Cord fraction

Mac

hN

um

ber

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7Incidence=-5°

Cord fraction

Mac

hN

um

ber

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7Incidence=+5°

(d)

Fig. 11 Comparison of geometry and aerodynamic performance of ini-tial profile and optimized profile of third case. a Geometry of designedblade and initial profile, b curvature distribution for designed blade and

initial profile, c loss curve for designed blade and initial profile, d designand off-design Mach number distributions of initial profile and designedprofile

Table 5 Comparison of runtime and aerodynamicperformance of optimized blade

ωmin �α OF CPU Time

Value Change % Value Change %

Initial 3.95 16.4

NACA-65 4.5 13.5 18 9.7

First case 3.06 −22.5 13.6 −8.6 0.62 5 h and 22 min

Second case 3.58 −2.5 19.7 23.6 0.75 7 h and 44 min

Third case 3.69 −6.6 18 9.6 0.74 6 h and 50 min

Table 5 summarized run time and aerodynamic perfor-mance of optimized blade. Optimized profile results are com-pared with initial one and a well-known NACA-65 profilewhich has the same inlet and outlet metal angle as initialprofile. As expected, Maximum reduction of design pointloss coefficient has been occurred in case 1 by −22.5 % andthe greatest increase in allowed angles of attack ranges werebeing entailed to the second case by 23.6 %. Calculations

have been done by a computer with Intel core(TM) 2 quadsCPU 2.8 GHz. For typical problem, presented multi-leveloptimization is about 2.5 times faster than single-level opti-mization via genetic algorithm because of numbers of param-eters that are optimized using Navier–Stokes analysis inmulti-level procedure (7 parameters) are lesser than that insingle-level optimization (17 parameters). The main defi-ciency of using multi-level optimization is the risk that the

123

364 Arab J Sci Eng (2013) 38:351–364

discrepancies between the predictions by the meta-functionand the Navier–Stokes results drive the optimizer to a falseoptimum. Presented meta-function for first-level optimiza-tion is fast but inaccurate and using it may drive optimizationprocess to a non-optimum combination of design parameters.Any further control by optimization of yi using an accurateNavier–Stokes solver will diminish the inherent inaccuracyof the fast calculation method but there is no mechanism toeliminate it.

The method presented in this paper can consider differentdesign philosophies using different weights in the objectivefunction (multi-objective ability). Weights in objective func-tion can be determined based on the expected application ofthe blades and the designer considerations. For instance, theblade designed in the first case is suitable for the middle stageof compressors, because the range variation of their angle ofattack is not very high at the off-design condition; wherethe compressor performance can be optimized by reducingthe loss at the design condition. The third case is appropri-ate for the rear and front stages of the compressors whereblade incidence variations are high. In optimum conditions,the spectrum of different weights may be considered by thedesigner for different sections of compressor blades.

5 Conclusions

In this paper, a multi-level airfoil design tool was developedaiming to decrease loss in design and off-design conditions.To verify the ability of the first-level optimization (meta-function) in the blade design process, an initial blade wasoptimized with nine different design objectives. In best cases,optimization for off-design condition resulted in an increaseof 20 % in allowed incidence angle range. Design incidenceangle loss has been reduced by 16 % when optimization fordesign condition was desired.

Three different blade design philosophies are consideredto optimize an initial blade with multi-level optimization;the first one is to reduce blade loss at the design condition;the second one aims to increase the range of incidence angleat the off-design, and the third one is to reduce the mini-mum loss and increase the incidence angle range. The high-est performance improvement at design condition in the firstcase is −22.5 % reductions in loss coefficient at design inci-dence angle. In the second case, 23.6 % increase in allow-able incidence angle range, improves blade’s performance atoff-design conditions. It is also found that the design phi-losophy affects the leading-edge shape, maximum thicknesslocation and maximum curvature location and value. Pre-sented multi-level optimization is about 2.5 times faster thansingle-level optimization via genetic algorithm with samegeometry parameters and objective function. Therefore, thepresented system is successful in both time-saving and opti-mization with various targets.

Open Access This article is distributed under the terms of the CreativeCommons Attribution License which permits any use, distribution, andreproduction in any medium, provided the original author(s) and thesource are credited.

References

1. Dunham, J.: A parametric method of turbine blade profile design.American Society of Mechanical Engineers, No 74-GT-119 (1974)

2. Corral, R.; Pastor, G.: Parametric design of turbomachinery airfoilsusing highly differentiable splines. J. Propuls. Power 20(2) (2004)

3. Keshin, A.; Dutta, A.K.; Bestle, D.: Modern compressor aerody-namic blading process using multi-objective optimization. ASMETurbo Expo, No.GT2006-90206 (2006)

4. Bonnaiuti, D.; Zangeneh, M.: On the coupling of inverse design andoptimization techniques for the multiobjective, multipoint designof turbomachinery blades. J. Turbomach. 131(021014) (2009)

5. Korakiantis, T.: Hierarchical development of three direct-designmethods for two-dimensional axial-turbomachinery cascades.J. Turbomach. 115(021014), 314–324 (1993)

6. Sonoda, T.; Yamaghuci, Y.; Arima, T.; Sendhoff, B.; Schreiber, H.:Advanced High Turning Compressor Airfoils for Low ReynoldsNumber Condition. ASME Paper, No.GT2003-38458 (2003)

7. Shahpar, S.; Radford, D.: Application of the FAITH Linear DesignSystem to a Compressor Blade. Proceedings of ISAABE Confer-ence, No. 99–7045 (1999)

8. Kammerer, S.; Mayer, J.F.; Paffrath, M.; Wever, U.; Jung, A.R.:Three-Dimensional Optimization of Turbomachinery BladingUsing Sensitivity Analysis. ASME Paper, No. GT2003-38037(2003)

9. Buche, D.; Guidati, G.; Stoll, P.: Automated Design Optimizationof Compressor Blades for Stationary, Large-Scale Turbomachin-ery. ASME Paper No. 2003-38421 (2003)

10. Benini, E.; Tourlidakis, A.: Design Optimization of Vanned Diffus-ers of Centrifugal Compressors Using Genetic Algorithms. AIAAPaper No. 2001-2583 (2001)

11. Bonaiuti, D.; Pediroda, V.: Aerodynamic optimization of an indus-trial centrifugal compressor impeller using genetic algorithms.Proc. Eurogen (2001)

12. Samad, A.; Kim, K.Y.: Shape optimization of an axial compressorblade by multi-objective genetic algorithm. Proc. Inst. Mech. Eng.A J. Power Energy 222(6599-61) (2008)

13. Ju, Y.P.; Zhang, C.H.: Multi-point and multi-objective optimizationdesign method for industrial axial compressor cascades. Proc. Inst.Mech. Eng. C J. Mech. Eng. Sci. 225(1481–1496) (2011)

14. Pierret, S.; Van den Braembussche, R.A.: Turbomachinery bladingdesign using Navier Stokes solver and artificial neural network.ASME J. Turbomach. 121, 326–332 (1999)

15. Korakiantis, T.: Prescribed-curvature-distribution airfoils for thepreliminary geometric design of axial- turbomachinery cascades.J. Turbomach. 115(JTM000325), 325–333 (1993)

16. Koller, U.; Mönig, R.; Küsters, B.; Schreiber, H.A.: Development ofadvanced compressor airfoils for heavy-duty gas turbines—part i:design and optimization. J. Turbomach. 122(JTM000397), 397–405 (2000)

17. Kermani, M.J.; Plett, E.G.: Roe Scheme in Generalized Coordi-nates; Part I- Formulations. American Institute of Aeronautics andAstronautics Paper 2001-0086 (2001)

18. Van Leer, B.: Towards the ultimate conservation difference scheme,V, a second order sequel to Godunov’s method. J. Comput. Phys.32, 110–136 (1979)

19. Schreiber, H.A.; Starken, H.: Experimental Cascade Analysis of aTransonic Compressor Rotor Blade Section. West Germany ASME288 (1984)

123