Aerodynamic performance effects of leading-edge geometry in gas-turbine blades

Applied Energy 87 (2010) 1591–1601

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Applied Energy

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Aerodynamic performance effects of leading-edge geometry in gas-turbine blades

I.A. Hamakhan, T. Korakianitis *

School of Engineering and Materials Science, Queen Mary, University of London, London, UK

a r t i c l e i n f o

Article history:Received 3 June 2009Received in revised form 8 September 2009Accepted 12 September 2009

Keywords:AirfoilBladeDesignTurbineGeometryAerodynamicsTurbomachineryLeadingTrailingEdgeCurvature

0306-2619/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.apenergy.2009.09.017

* Corresponding author. Tel.: +44 20 7882 5301.E-mail address: [email protected] (T. Kora

a b s t r a c t

The purpose of this paper is to illustrate the advantages of the direct surface-curvature distributionblade-design method, originally proposed by Korakianitis, for the leading-edge design of turbine blades,and by extension for other types of airfoil shapes. The leading edge shape is critical in the blade designprocess, and it is quite difficult to completely control with inverse, semi-inverse or other direct-designmethods. The blade-design method is briefly reviewed, and then the effort is concentrated on smoothlyblending the leading edge shape (circle or ellipse, etc.) with the main part of the blade surface, in a man-ner that avoids leading-edge flow-disturbance and flow-separation regions. Specifically in the leadingedge region we return to the second-order (parabolic) construction line coupled with a revised smooth-ing equation between the leading-edge shape and the main part of the blade. The Hodson–Dominy bladehas been used as an example to show the ability of this blade-design method to remove leading-edge sep-aration bubbles in gas turbine blades and other airfoil shapes that have very sharp changes in curvaturenear the leading edge. An additional gas turbine blade example has been used to illustrate the ability ofthis method to design leading edge shapes that avoid leading-edge separation bubbles at off-design con-ditions. This gas turbine blade example has inlet flow angle 0�, outlet flow angle �64.3�, and tangentiallift coefficient 1.045, in a region of parameters where the leading edge shape is critical for the overallblade performance. Computed results at incidences of �10�, �5�, +5�, +10� are used to illustratethe complete removal of leading edge flow-disturbance regions, thus minimizing the possibility of lead-ing-edge separation bubbles, while concurrently minimizing the stagnation pressure drop from inlet tooutlet. These results using two difficult example cases of leading edge geometries illustrate the superior-ity and utility of this blade-design method when compared with other direct or inverse blade-designmethods.

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1. Introduction

Design of high-efficiency airfoils and blades are essential foroptimum aerodynamic, thermoeconomic, and overall performanceof turbomachinery-based powerplants [1–3]. Gas turbine bladesare three dimensional objects operating in unsteady and complexflow fields. This complexity forces designers to decompose thethree dimensional blade-design problem to a series of two dimen-sional problems in the streamwise direction, and to assume thatthe blade operates in steady-flow conditions. The three dimen-sional variation of inlet and outlet flow angles is determined bystreamline curvature calculations [4–8], and they vary from hubto tip. Depending on design choices of blade aspect ratio ðh=bÞand the three-dimensional distribution of the centers of gravityof the two-dimensional sections, two-dimensional blade shapesare stacked to build the three dimensional geometry. The blade de-sign manufactures have different sequences to design their blades,

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kianitis).

but all of them follow essentially the above overall procedure. Thegoal of any blade-design method is to find a geometry that satisfiesflow requirements with minimum loss, tolerable mechanical stres-ses, minimum disturbances downstream and upstream, and in thecase of compressors adequate stall margin, among others. The de-sign of blade geometries is a very important step for the design ofefficient turbomachines, as the blade design process directly influ-ences the blade-row efficiency and thus the overall machineefficiency.

Early blade-design methods were based on specifying a thick-ness distribution around a camber line. In this method, for specify-ing turbine and compressor shapes, straight line, circular arc orparabolic camber lines were used, and then the thickness distribu-tion was added to these camber lines for both the suction and thepressure sides of the blade. Examples of this method [9,10] are de-signs of NACA compressor airfoils (NACA-65 series) and other ser-ies of NACA profiles for turbines by Dunavant and Erwin [11]. Thismethod did not provide enough flexibility to control both the suc-tion and pressure surfaces in order to obtain a desirable aerody-namic performance.

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Nomenclature

A coefficient (Eq. (9))a thickness-distribution coefficients (Eqs. (5) and (10)B coefficient (Eq. (9))b axial chord (nondimensionally b = 1)C ¼ 1=r curvature (Eqs. (1), (7), and Figs. 1, 3, and 12)C coefficient (Eq. (9))C0 slope of curvature (Eq. (2))CL tangential-loading coefficient (Eq. (4))k exponential function (Eq. (10))M Mach numbero throat diameter of the blade (Eq. (3))P points or nodes on the blade surfaces (Eqs. (5) and (10))r local radius of curvature (Eqs. (1) and (7))

S tangential pitch of the blades (Eqs. (3) and (4))t Bezier variable span (Eq. (6))ðx; yÞ Cartesian coordinatesðX;YÞ nondimensional (with b) distancesa flow angle

Subscripts1;2; k points (P, C), or with flow angle ac circle, or circle centeris isentropicle leading edgep pressure sides suction side

1592 I.A. Hamakhan, T. Korakianitis / Applied Energy 87 (2010) 1591–1601

After the 1950s, most turbine blades have been designed byspecifying the shape of the suction and pressure surfaces of theblades. Thus the passage was defined by the blades and the resul-tant blade thickness was obtained. Dunham [12] and Pritchard [13]used parametric methods to describe the blade surface geometry.In these methods segments of lines are joined in a variety of pointsalong the blade surfaces. The way in which the line segments arejoined generates shapes that look smooth to the eye, but theycause local flow accelerations and decelerations as will be ex-plained below.

Various categories for blade-design methods exist: direct; in-verse, semi-inverse, full-inverse or full optimization methods[14]; analysis and design modes [15]; optimization and designmethods; and others. The definitions of what is semi-inverse,full-inverse etc. methods vary in the literature. In our work, the di-rect design has been defined as a process in which the blade geom-etry is specified and the resultant aerodynamic performance onthat geometry is calculated; and the inverse method has been de-fined as a method in which the desired blade performance is spec-ified and the geometry that would accomplish such performance iscalculated.

Both methods have relative advantages and disadvantages. Inthe direct method, it is relatively easy to fulfill mechanical and geo-metric constraints, where the complete geometry is specified; butit is usually laborious to obtain the desired distribution of pressureor velocity along the profile with this method. On the other hand, itcan be difficult to get an acceptable geometry with an inversemethod, where the velocity is prescribed and the geometry calcu-lated [16]. The inverse design method [17] is based on the velocityor pressure distribution along the streamwise blade shape, and itrequires multiple variations of the velocity distribution until anacceptable profile geometry is obtained. The inverse design meth-od also has difficulties in both of the leading and trailing edges, dueto mathematical singularity at the stagnation points [16,18]. Theinverse method ends up with blades with zero thickness at thetrailing edge, which are impossible to manufacture. This last diffi-culty makes the inverse method acceptable for some compressorblade geometries of thin trailing edge, but unacceptable for turbinegeometries that have thicker trailing edges.

Korakianitis has worked on the design of airfoils by the directmethod. In his first attempt [19] he specified five points alongthe blade surface (excluding the trailing edge and the leading edgepoints) and two slopes on each surface. In the leading edge regionhe used two thickness distributions added perpendicularly to twoparabolic construction lines. He concluded that the blade aerody-namic performance, judged by the shape of the blade surface Machnumber distribution, is very sensitive to changes in the slope of thecurvature of the blades.

In follow on work Korakianitis [19–24,8,25] in a series ofimprovements to blade-design methods proposed a direct designmethod based on specifying blade surface-curvature distributions.He concluded that the aerodynamic and heat transfer performanceof the blades are determined by the surface-curvature distribution,and proposed methods to prescribe blade shapes with continuousslope of curvature which resulted in superior aerodynamic andheat transfer performance (avoiding local separation bubbles, andprescribing continuous slope of curvature joints among the linesegments, and of the line segments with the leading-edge andtrailing-edge shapes). He concluded that the changes in the localslope of curvature are more important than the (x,y) location ofthe blade shape, and that blade quality-assurance methods shouldnot be point-by-point comparisons of geometric accuracy, butshould be based on measurements of streamwise surface-curva-ture distributions instead.

The conclusion for this method is that blade surfaces with con-tinuous slope of curvature, or third derivative continuity in thespline knots, result in smooth Mach number and pressure distribu-tions, and thus can be used to avoid flow separation, and lead toimproved blade designs. Slope of curvature continuity means thatthe surface curvature C, as well as its slope along axial distancex;C0 ¼ dC=dx along the blade profile length, must both be continu-ous. This requires continuous third derivatives of the line segmentsyðxÞ, i.e. continuous y000 ¼ d3y=dx3, as illustrated by the followingtwo equations for curvature and its slope.

C ¼ 1r¼ y00

½1þ ðy0Þ2�ð3=2Þ ð1Þ

C0 ¼ dCdx¼ y000½1þ ðy0Þ2� � 3y0 ðy00Þ2

½1þ ðy0Þ2�5=2 ð2Þ

where

y ¼ yðxÞ; y0 ¼ df ðxÞdx

; y00 ¼ d2f ðxÞdx2 ;

y000 ¼ d3f ðxÞdx3

Tiow and Rooij [26,27] mentioned that the direct design meth-od does not provide any guidance to the designer to control theshape of the blade, and they say the direct design method alwaysrequires designer’s experience and is more laborious. However,this is only true if one is not considering the blade surface-curva-ture distributions. Corral and Pastor [28] used a parametric designmethod to remove the leading-edge separation bubbles in a tur-bine blade geometry for which extensive experimental data havebeen published (the Hodson–Dominy blade (HD blade) [29–31],

I.A. Hamakhan, T. Korakianitis / Applied Energy 87 (2010) 1591–1601 1593

which is also used as an example later on in this paper). Corral andPastor [28] report ‘‘... it is specially difficult to match the originalgeometry because of the strong i.e. discontinuity in the airfoil,which is not explicitly take into account in our method. The isen-tropic Mach number distributions computed with MISES ...” (MISESis a well known inverse design method). As will be shown later inthis paper, the prescribed surface-curvature distribution methodcan be used in a direct design method to remove the leading edgeseparation regions from the HD and other blades.

In consequence, the goal of this paper is to develop the pro-posed direct design method that can prescribe the leading edgegeometry of the blade at design and off-design conditions for gasturbines in a manner that minimizes leading edge flow-distur-bances and avoids separation bubbles. The original prescribed-cur-vature-distribution method is modified in the vicinity of theleading edge. A leading edge shape such as a circle or ellipse is pre-scribed at the leading edge. A polynomial function is used to jointhe leading-edge shape with the main part of the blade surface.The polynomial is prescribed as a thickness distribution in relationto a parabolic construction line. Furthermore, the polynomialmatches point, first, second and third derivative at its two endswith the remaining blade shape, thus resulting in continuous cur-vature and slope of curvature at these crucial locations. This is afirst publication of a technique that provides this advantage toleading-edge shapes. A similar technique is used to blend a trail-ing-edge circle to the blade surfaces while maintaining curvatureand slope of curvature continuity at these locations.

The HD blade has been used as an example to show the abilityof this technique to remove the leading-edge separation bubblesaround the leading edge. Additional blades have been used asexamples at design and off-design conditions to further illustratethe ability of this method to remove the flow-disturbances andflow separation at off-design conditions, and minimize stagnationpressure losses from inlet to outlet. These improvements on theprescribed-curvature-distribution blade-design method providedirect control for the overall blade geometry, eliminating the prob-lems of inverse method.

2. Geometric design of turbine blades

There are a number of key issues that need to be addressed inthe design of blade geometries that affect the choice of implemen-tation methods. The first issue is the flexibility of any revisedmethod to represent a wide range of different blade shapes, andability to work on other similar shapes such as compressor, fanand wind turbine blades. The second issue is the accuracy or theoptimizing objective functions, which must be compared withexperimental data. The accuracy should be accounted for in boththe geometric and aerodynamic sense.

In the present paper the original method proposed by Korakia-nitis [19,22–24,8] has been further developed and optimized to en-hance the ability of the method to provide optimum leading-edgeshapes in geometrically difficult cases, for instance with thin lead-ing edge shapes, small leading-edge wedge angles, and inflow an-gles around 0�.

Based on the original method, the design of the main part of thesuction and pressure sides of the blade shape (except the leadingedge and trailing edge) is split into three major segments: the seg-ment near the trailing edge; the main part of the blade; and thesegment near the leading edge, as explained in the following.

3. The trailing-edge segments

In the previous work [19,22], the trailing edge was representedby a point. Later on a trailing edge thickness (a cusp) was used to

specify the trailing edge section [19]. Finally, Korakianitis andWegge [8] used a circle at the leading and trailing edge, but theimplementation still presented problems when the blade shapeswere very near 0� and 90� in the vicinity of the trailing-edge circle.In the present modification to the basic method, the trailing edgethickness is based on a circle which is blended with the trailing-edge segment of the main part of the blade with a polynomial.

The suction-side trailing-edge segment is from the trailing-edgecircle to the throat of the passage, where at the throat we use thelocation of the point and slope of the suction surface line at thatlocation. For the pressure surface there is no corresponding guidingthroat location, so a geometrically-sensible point and its slope areprescribed on the ðx; yÞ plane of the blade. The pressure-side trail-ing-edge segment is from the trailing-edge circle to the pressure-surface point described above. Essentially these two suction andpressure line segments, the blade-surface ‘‘departing” angles fromthe trailing-edge circles, and the throat and pressure points andtheir slopes chosen as inputs, prescribe the trailing-edge region‘‘wedge” angle of the blade, and it’s thickness for manufacturingand stress considerations.

The specification of the trailing-edge segment of the main partof the blade, from the throat point to the trailing-edge circle, is cru-cial in minimizing flow diffusion in this ‘‘unguided” (by a pressure-surface shape on the opposite side of the passage) flow region. Lessdeceleration (diffusion) means better designs. If the flow deceler-ates too much, then the blade wake will be thicker, the boundarylayer may separate, and the resulting blade shape will have poorerperformance (lower efficiency). The trailing-edge suction side iscontrolled by the throat diameter o. A first approximation [9] is

oS¼ cos�1 a2 ð3Þ

but in practice this value is modified using experimental data suchas the Ainley–Mathieson data [32]. Wilson and Korakianitis havemodeled these data with equations using a main parametercos�1 a2 and a secondary parameter based on throat Mach number[9].

In the 2D design, the throat diameter o is calculated by an inputparameter of throat/chord ðo=SÞ, the tangential lift coefficient ðCLÞ,inlet flow angle a1, and outlet flow angle a2:

Sb¼ CL

2 cos2 a2 tan a1 � tan a2ð Þ ð4Þ

where CL is the tangential lift coefficient and its value is determinedby the number of the blades in the annulus. Korakianitis [19]pointed out that tangential lift coefficient typically is between 0.8and 1.2. In blade designs of the 1940–1960 the value of tangentiallift coefficient was low, but in modern cascades the value of CL

has been gradually increasing. For instance for the 3D blade shapeof the 2D blade B3 shown later in this paper CL is 0.86, 1.03 and1.17 at the hub, mean and tip regions (the mean geometry withCL ¼ 1:03 is shown in this paper). In the 3D design process the num-ber of blades, and therefore the resultant S at each radius, is an in-put. Therefore o is calculated from cos�1 a2 as modified by thethroat Mach number using the Ainley–Mathieson data or theirmathematical expressions in [9].

Similarly to the previous approaches as in [33], the blade-surface segment from the trailing-edge circle to the throat isdescribed with an equation of the form

y ¼ a0 þ a1xþ a2x2 þ a3x3 þ a4x4k1½x� xðPs2Þ� þ a5x5k2ðx� xðPs2ÞÞð5Þ

where in the above expression k1 and k2 are exponential functions,resulting in terms of increasing importance as we approach pointPs2. This type of expression requires derivation of six parameters,


a0; . . . ; a5. These are derived from the conditions of point, first, sec-ond and third derivative continuity (four conditions) of the blade-surface line at the trailing-edge circle; and prescribing the pointand slope of the blade-surface line at the tangent to the throatdiameter (two additional conditions). This approach enables easycontrol of the shape of the blade surface in the vicinity of the trail-ing edge, and control of its curvature at the blending point to thetrailing-edge circle (or other trailing-edge shape).

4. Main curvature-mapped part of the blade surface

The main part of the blade is the mid portion of the surface thatconnects the trailing and leading edge segments of the blade. As inthe previous approaches in [20,22,24,8], Bezier splines have beenused to prescribe the surface curvature of this part of the blade. Be-zier curves are optimal from the point of view of geometric model-ing aiming to control the inner location of the points and thedegree of its differentiability. The surface-curvature distributionof this part of the blade is prescribed as shown in the example ofFig. 1, where the example shown the curvature of the main partof the blade is controlled by a five-segment Bezier curve controlledby the six control points C1s; . . . ;C6s. The blade surface points onthe ðx; yÞ plane are derived from the prescribed curvature pointsin the ðC; yÞ plane, where the latter is illustrated in Fig. 1.

The main part of the blade begins at the upstream point of thetrailing-edge segment (the throat on the suction side), which ispoint Psm on the blade surface; and ends at a point Psk on the bladesurface (which will be later connected to the the leading edge seg-ment). This point Psk will be derived as the last point in the curva-ture mapping sequence from the ðC; yÞ to the ðx; yÞ plane.

Referring to Fig. 1, the trailing-edge segment is a prescribedpolynomial given by Eq. (5), which results in the curvature distri-bution shown from x ¼ 1:0 to x ¼ xðC6sÞ in the figure. The slopeof the curvature line C0 at x ¼ xðC6sÞ is prescribed by Eq. (6). Akey input at this part of the design sequence is the x location ofpoint C5s, which also prescribes the length of the line segmentC5sC6s. The conditions describing line C5sC6s are derived so that thisline results in continuous slope of curvature ðC0Þ conditions atpoint Psm on the blade surface, and point C6c on the ðC; yÞ planein Fig.1. This approach ensures slope of curvature continuity atblade surface point Psm, which joins the trailing-edge segment tothe main part of the blade.

Fig. 1. Sample of curvature distribution of the whole suction side of the blade.

The surface curvature of the main part of the blade can be spec-ified by a Bezier curve of two or more controlling segments. Themore the controlling segments the higher the degree of flexibilityin designing the blade surface, and the higher the complexity. Inthis work we have used six control points C1s; . . . ;C6s for the Beziersegments as shown in Fig. 1.

The resulting Bezier spline equation is

xC ¼ t3xC1s þ 3t2ð1� tÞxC2s þ 3tð1� tÞ2xC3s þ ð1� tÞ3xC4s

þ ð1� tÞ4xC5s þ ð1� tÞ5xC6s

yC ¼ t3yC1s þ 3t2ð1� tÞyC2s þ 3tð1� tÞ2yC3s þ ð1� tÞ3yC4s

þ ð1� tÞ4yC5s þ ð1� tÞ5yC6s

ð6Þ

The evaluating variable t is still 0 and 1 at each end, so t ¼ 0 atstarting point C6s and t ¼ 1 at ending point C1s. The remaining ap-proach is similar to [22]. The curvature, given by

C ¼ 1r¼

d2y=dx2� �

1þ ðdy=dxÞ2h ið3=2Þ ð7Þ

is written with central differences as

Ci ¼CF1=CF2

CF3ð8Þ

CF1 � 2yiþ1 � yi

xiþ1 � xi� yi � yi�1

xi � xi�1

� �CF2 � ðxiþ1 � xi�1Þ

CF3 � 12

yiþ1 � yi

xiþ1 � xiþ yi � yi�1

xi � xi�1

� �� 2

þ 1

( )ð3=2Þ

The above equation can be solved numerically or by manipulat-ing into a sixth-order algebraic equation to find out the value ofsuccessive yiþ1 from inputs xiþ1; Ci and xi; yi; xi�1; yi�1. Korakiani-tis [22] used the regula-falsi solution for yiþ1. In this paper thesame procedure has been used to solve numerically this part ofthe blade. The the control points C1s; . . . ;C6s of the Bezier splinein Fig. 1 are iteratively manipulated until the slope of the curvatureand the resultant output blade shape on the ðx; yÞ plane are bothacceptable.

The procedure described above is identical for the pressure side,but with different values of variables.

5. The leading edge region

This part of the blade has significant effects on the aerodynamicand heat transfer performance of the blade, and the result is signif-icantly affected by small changes in the blade shape. Joining theleading edge circle or other shape to the main blade results in flowseparation bubbles, and local accelerations and decelerations in thepressure and Mach number distributions. Both direct and inversedesign methods have difficulties around this area (e.g. [34,16,35]).

In this work we implement a hybrid method based on modifica-tions of the earlier methods [19,20,22,8]. First we introduce theleading edge shape, such as a circle or ellipse. Then a parabolic con-struction line and thickness distribution is added (as in [20,22]),where the construction line now starts from a key geometric pointsuch as the center of the leading edge circle. Finally the thicknessdistribution is added about this parabolic construction line in amanner that the thickness distribution (and therefore also theblade surface) have continuous point, first, second and third deriv-ative (continuous y; y0; y00; y000 and therefore continuous C0) at thepoints where it joins the leading edge shape (circle) and the mainpart of the blade (point Psk and C1s in Fig. 1). This is analogous tothe circle-joining work of [8] with exponentials in the polynomials,

Fig. 2. Original and redesigned leading edge for HD, I1, I4, and I9 blades.


and the above section on joining the trailing-edge circle to thetrailing-edge segment of the blade.

Thus the suction-side parabolic construction line passingthough the center of the leading edge circle is of the form:

yðxÞ ¼ Ax2 þ Bxþ C ð9Þ

and the thickness distribution added about the parabolic construc-tion line is of the form

y¼a0þa1xþa2x2þa3x3þa4x4k1½x�xðPs1Þ�þa5x5k2½x�xðPskÞ�þa6x6k3½x�xðPs1Þ�þa7x7k4½x�xðPskÞ� ð10Þ

where functions k1; k2; k3 and k4 are exponential polynomialswhich acquire increasing importance as we approach Ps1 and Psk

on the blade surface. Point Psk is the joint point with the main cur-vature-mapped part of the blade; and point Ps1 is the joint pointwith the leading edge circle. The eight parameters of the thicknessfunction a0; . . . ; a7 enable us to evaluate eight conditions of thisthickness distribution. Four parameters are evaluated to matchy; y0; y00 and y000 (and thus C0) at point Ps1; and four parametersare evaluated to match y; y0; y00 and y000 (and thus C 0) at point Psk.This approach ensures continuity of curvature from the main partof the blade surface through the leading-edge thickness distributionand into the leading edge circle.

The procedure is similar for the pressure side of the blade, ex-cept with different geometric parameters.

This overall approach enables us to remove any flow disconti-nuities due to surface-curvature discontinuities from the leading-edge stagnation point to the trailing edge stagnation point. Ofcourse flow discontinuities will still occur where we demandtoo-high adverse pressure gradients for the flow conditions, butthe method ensures that such flow discontinuities are not due tosurface-curvature discontinuities.

6. Aerodynamic effects of leading edge geometry

Experimental data published in [36–40] clearly indicate localkinks in the surface pressure or Mach number distribution ofblades, which, as will be illustrated later, is the result of surface-curvature discontinuities. These ‘‘kinks” affect boundary-layer per-formance and blade efficiency, and they could be smoothed by theprescribed surface-curvature blade-design method.

Hodson [29] experimentally detected the presence of the lead-ing edge spike as a result of a curvature discontinuity in the blend-ing point between the circle and the rest of the blade. Hodson andDominy experimentally tested this blade (which we designate theHD blade) extensively [29–31]. Several authors have computation-ally confirmed this spike [19,15,8,28] around the leading edge ofthe same blade. Stow [15] illustrated the removal of the leadingedge spike locally using an inverse design method.

In this paper we illustrate the use of the prescribed-curvatureblade design technique: first to reproduce a blade (designated I1)designed with the prescribed-curvature-distribution direct-designmethod (not locally, but from trailing edge to leading edge) thathas geometry and leading edge spike similar to those of the HDblade; then to redesign the blade shape with the prescribed-curva-ture blade-design method (again from trailing edge to leadingedge) in a series of successive improvements (blades I4 and I9are shown) that remove the leading edge separation spikes. Thedetails of the computations are included below. The design param-eters and representative data points for blades I1 and I4 are in-cluded in Appendices A and B.

Fig. 2 shows the leading-edge geometry of the original HD blade(which coincides with the I1 blade) and of blades I1 and I9. Wehave restricted the geometry to use the same leading-edge circlediameter, and in order to maintain the same blade chord from

x ¼ ½0;1�, as the blade became progressively thinner near the lead-ing edge we limited the reduction in the leading edge wedge angleso that the foremost point on the blade did not move forward fromx ¼ 0 to values x > 0. Thus the foremost point on blade I9 is atðx; yÞ ¼ ð0:0000;0:006723Þ as shown in Fig. 2.

The HD blade profile is a thin, hollow, castable root section fromthe rotor of a low-pressure turbine. It was designed to operate atair inlet flow angle 38.8� relative to the axial direction and to pro-vide approximately 93� of flow turning. The design velocity ratioacross the cascade is equal to 1.41. The nominal aspect ratio was1.8. The test Reynolds number is 2:3� 105 and the test inlet andoutlet Mach numbers are 0.496 and 0.702 respectively. Further de-tails can be found in references [15,29–31].

The left side of Fig. 3 shows the surface-curvature distributionfor the HD blade (jagged line, evaluated numerically from the ori-ginal data points) and the curvature distributions of blades I1, I4and I9. The surface-curvature distributions of blades I1, I4 and I9are smoother lines, as these blades have been reproduced withthe prescribed-curvature-distribution blade-design method. Thefigure also shows the curvature of blade I1 trying to follow the cur-vature of the HD blade in the vicinity of the leading edge ‘‘spike” onthe suction side. This ‘‘spike” in the surface curvature of blade I1(which we would not normally use in this region of a blade design)is now required in order to reproduce the flow spike for blade I1, inorder to reproduce the spike in the HD blade. The ‘‘spike” is not‘‘prescribed” in the curvature distributions of blades I4 and I9,which are by specification and design smooth. The resultant com-puted isentropic Mach number distributions are shown on theright side of Fig. 3. The sharp local acceleration–deceleration regionin the leading edge of blade I9 has also been smoothed.

Mesh generator GAMBIT and flow solver FLUENT have beenused in the computations. A reasonably high number of computa-tional mesh elements is required for reasonably accurate calcula-tions. The exact numbers depend on the geometry of the bladeand its model. The mesh elements used for the HD and I1, I4 andI9 blades are: 19,705 quadrilateral cells; 38,967 2D interior faces;and 20,148 nodes for all zones. A 2D O-mesh and a pave-unstruc-tured mesh have been used in the viscous calculations. The pavemesh consisted of a combination of structured and unstructuredregions. The mesh around the airfoil consisted of twelve structuredclustered O-grid layers with yþ less than 6 (quadrilaterals), and theremaining majority of the flow field was discretized with quadri-lateral and a small numbers of triangular cells. The k�x and

Fig. 3. Left, isentropic surface Mach number distributions of the reproduced HD blade (I1) and of redesigned I4 and I9 blades. Right, curvature distributions for all blades (HD,I1, I4, I9).

Fig. 4. Left, Mach contours for the original HD blade. Right, Mach contours for the redesigned I9 blade.

Fig. 5. Left, isentropic Mach number distributions for the HD and I1 blades. Right, isentropic Mach number distributions for the HD and I9 blades.


Fig. 6. Blade B1: Left, isentropic surface Mach number distribution at design condition, inflow at 0�. Right, Mach number contours at design condition, inflow at 0�, increment0.05.

Fig. 8. Left, B1 isentropic surface Mach number distribution at off-design condition, inflow at +10�. Right, B1 isentropic surface Mach number distribution at off-designcondition, inflow at �10�.



k� e turbulence models have been used. The results for both tur-bulence models are approximately the same, and the results shownare those for the k�x turbulence model.

Fig. 4 shows the Mach number contours around the leadingedge of the HD and I9 blades, and shows the success of theprescribed-curvature-distribution method to remove the local

Fig. 9. Blade B3: Left, isentropic surface Mach number distribution at design condition, inflow at 0�. Right, Mach number contours at design condition, inflow at 0�, increment0.05.





deceleration regions (and the resultant separation bubbles) at theedges of the leading-edge circle region.

Fig. 5 shows the comparison of the experimental data for theHD blade with the results of viscous calculations blades HD, I1,I4 and I9.

Table 1Stagnation pressure losses at design and off-design conditions.

Blade names Design and off-design conditions

�10 �5 0 +5 +10

B1 0.002430 0.002450 0.002480 0.002518 0.002560B2 0.002424 0.002446 0.002474 0.002508 0.002552B3 0.002357 0.002404 0.002433 0.002467 0.002512

Fig. 12. Surface-curvature distributions of blades B1 and B3.

Table 2Suction-side diffusion ratio at design and off-design conditions.

Blade names Design and off-design conditions

�10 �5 0 +5 +10

B1 0.87110 0.869975 0.86985 0.86772 0.86780B2 0.849743 0.84831 0.84685 0.84538 0.843875B3 0.838995 0.836926 0.834855 0.832745 0.83042

7. Design and off-design performance

Curvature discontinuities have a more severe impact around theleading edge at design and off-design conditions because at leastone of the discontinuities takes place in a region of a high velocity.Benner [41] mentioned that the discontinuity in the (flow) curva-ture is increased at off-design conditions, and he argued that theoff-design behavior of the blade is influenced by the magnitudeof the discontinuity in the surface curvature between the circleand the rest of the leading edge part of the blade.

The prescribed-curvature blade design technique has been usedto design three example blades, designated B1, B2 and B3, to illus-trate the effect of the surface-curvature distribution at design andoff-design conditions. Three dimensional blade designs B1, B2 andB3 have been obtained, but this paper reports the representativemean-line results. The design parameters and representative datapoints for blade B3 are included in Appendices A and B. BladesB1, B2 and B3 have been designed for the following conditions:

Inlet total temperature 1000 K.Inlet total pressure 532 kPa.Inlet flow angle (design point) 0.00�.Outlet flow angle (design point) 64.30�.Pitch to axial chord ratio S=b ¼ 1:3371.Throat diameter to pitch ratio o=S ¼ 0:441.Tangential-loading coefficient (at mean-line) CL ¼ 1:04500.Stagger angle k ¼ �47�.Leading edge circle radius to axial chord ratio rle=b ¼ 0:0300.Trailing-edge circle radius to axial chord ratio rte=b ¼ 0:0150.Inlet Mach number 0.2052.Outlet Mach number 0.5346.Inlet Reynolds number of 1:7� 105.Outlet Reynolds number of 3:7� 105.

These specifications have been specifically chosen around inletflow angles of zero degrees (to illustrate the effect of positive andnegative inflow angles from this value). The flow solutions havebeen obtained with FLUENT specifying free-stream turbulenceintensity of 10% and using the k�x turbulence model. The solu-tions have been obtained for design point conditions as well asfor incident angles of �5�, +5�, �10�, and +10�.

The computational meshes are: 25,076 quadrilateral cells;49,604 2D interior faces; and 25,624 nodes for all zones. A 2D O-mesh and a pave-unstructured mesh have been used in the viscouscalculations. The same procedure for the HD blade have been usedto generate mesh around the B1, B2 and B3 blades.

Fig. 6 illustrates the isentropic surface Mach number distribu-tions and the Mach contours of blade B1 at design conditions. Figs.7 and 8 illustrate the off-design conditions. Fig. 9 illustrates thesurface Mach number distributions and the Mach contours of bladeB3 at design conditions. Figs. 10 and 11 illustrate the off-designconditions. The surface-curvature distribution of blades B1 andB3 is shown in Fig. 12.

At design point conditions blade B1 exhibits perfectly accept-able behavior, without any leading-edge flow-disturbances at thejoining points of the blade surface with the leading edge circle.At +5� a small disturbance is shown on the suction surface at thelocation of joining the leading edge circle with the blade surface.At �5� a small acceleration–deceleration region is shown on the

pressure surface at the location of joining the leading edge circlewith the blade surface. These flow-disturbances are made worseat +10�, and �10�.

These disturbances at off-design conditions are progressivelyimproved in blades B2 and B3 by small changes in the blade sur-face-curvature distribution, as illustrated in Fig. 12. Furthermorethese flow effects have consequences in the flow deceleration re-gime in the region of unguided diffusion, and the resultant com-puted total pressure losses on the blade. For the purposes ofdiscussion we will define as the suction-side diffusion ratio themaximum isentropic Mach number on the surface of the suctionside divided by the Mach number at the trailing edge.

The results show that with the blade modification made usingthe prescribed-curvature blade-design method the effect of joiningthe leading edge circle to the blade surfaces has been minimizednot only at design conditions, but also at relatively large anglesof incidence. The computed stagnation pressure losses from inletto outlet at design and off-design conditions are shown in Table1. Table 2 shows the resultant suction-side diffusion ratios at de-sign and off-design conditions. There is a direct correlation be-tween increases in suction-side diffusion ratio and stagnationpressure losses.


8. Conclusions

Redesign of the HD blade to remove the leading edge problemshad been attempted before with inverse design methods [15], butthis is the first reported attempted to remove the leading edgespikes with a surface-curvature direct blade-design method. It isnoted that suction-surface spike and the pressure-surface regionof diffusion near the leading edge have been completely removed.The results show that the prescribed surface-curvature distributionblade-design method is a robust tool for blade design, providing amanageable and accurate way to control the blade surface and theaerodynamic properties around the blade.

The prescribed surface curvature blade design technique resultsin smoother boundary layer flows, affecting aerodynamic as well asheat transfer performance. Sample blades B1, B2 and B3 have beendesigned to show the capability of the method to eliminate thespike and dips in the surface Mach number distribution at designand off-design conditions in the vicinity of the leading edge circleat both design and off-design conditions. Stagnation pressurelosses decrease with smoothing the blending of the circle withthe blade surfaces, as predicted by Benner [41].

We conclude that the prescribed surface curvature blade designtechnique can be used to provide accurate guidance and control forthe design of blade shapes. Furthermore the method can be used toeliminate the flow problems resulting from blending a leadingedge circle or other shape with the blade surfaces at design andoff-design conditions.

The prescribed-curvature blade design technique has been usedto design stacked 3D turbine blades, compressor and fan blades,isolated airfoils, and wind turbines. These will be reported in otherpapers.

Appendix A. Parameters specifying the blades

Some symbols defined in Ref. [22].

I1
I9 B3
a1
38.80 38.30 0.00 a2 53.90 38.80 64.30 S=b 0.6000 0.6000 1.3371 k �19.60 �19.60 �47.00 o=S [22] 0.5850 0.5850 0.4410 / [22] 43.10 43.50 41.40 rle=b 0.01650 0.01650 0.03000 Xc;le 0.01650 0.01650 0.03000 Yc;le 0.01164 0.01164 0.01182 rte=b 0.00485 0.00485 0.01500 Xc;te 0.99515 0.99515 0.98500 Yc;te �0.34921 �0.34921 �1.03430 bs2 [22] 64.00 64.00 76.501
XC1s 0.0900 0.0900 0.0800
YC1s 2.3000 2.1000 4.1820 XC2s 0.2011 0.2011 0.2220 YC2s 2.2000 2.1000 4.2200 YC3s 2.40700 2.1870 2.4150 YC4s 2.4997 2.2200 2.2200 XC5s 0.5500 0.5500 0.400 bs1 [22] 74.00 119.00 69.00 Xsc [22] 0.1800 0.1400 0.1400 Ysc [22] 0.0900 0.0900 0.0100 bsc [22] 25.00 40.00 7.00
b [22] �33.50 �32.50 �56.00
p2
bpm [22]
�39.27 �40.15 �48.00 Xpm [22] 0.7100 0.7100 0.5600 Ypm [22] �0.0570 �0.0520 �0.3900
Appendix A (continued)

I1
I9 B3
XC1p 0.1100 0.1100 0.0800
YC1p 1.5700 1.6800 0.6300 XC2p 0.2080 0.1880 0.1700 YC2p 1.4520 1.7800 0.6800 YC3p 2.1620 1.8000 0.7600 YC4p 2.2300 1.8000 0.7950 XC5p 0.5850 0.5800 0.5000 bp1 [22] 24.00 �15.00 �40.50 Xpc [22] 0.1800 0.1800 0.1300 Ypc [22] 0.0900 0.0900 0.0008 bpc [22] 27.00 27.00 0.00
Appendix B. Representative blade points

Arbitrary numbers of blade data points, with accuracy up to anynumber of desired decimal points, can be derived using the bladeparameters in Appendix A.

Blade
I1 I9 B3 X Y Y Y
1.00
�0.349 �0.349 �1.034 0.95 �0.275 �0.272 �0.887 0.90 �0.203 �0.199 �0.754 0.80 �0.064 �0.061 �0.523 0.70 0.054 0.054 �0.334 0.60 0.139 0.137 �0.182 0.50 0.188 0.189 �0.063 0.40 0.210 0.212 0.027 0.30 0.207 0.210 0.085 0.20 0.179 0.182 0.110 0.10 0.121 0.125 0.096 0.05 0.076 0.080 0.070 0.00 0.012 0.007 0.012 0.05 0.011 0.010 �0.030 0.10 0.035 0.033 �0.052 0.20 0.070 0.068 �0.103 0.30 0.086 0.083 �0.164 0.40 0.084 0.082 �0.237 0.50 0.062 0.061 �0.327 0.60 0.018 0.021 �0.436 0.70 �0.049 �0.044 �0.567 0.80 �0.139 �0.136 �0.720 0.90 �0.245 �0.246 �0.898 0.95 �0.303 �0.304 �0.997
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Aerodynamic performance effects of leading-edge geometry in gas-turbine blades