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

Applied Energy 87 (2010) 1591–1601

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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 2009 Received in revised form 8 September 2009 Accepted 12 September 2009

Keywords: Airfoil Blade Design Turbine Geometry Aerodynamics Turbomachinery Leading Trailing Edge Curvature

a b s t r a c t The purpose of this paper is to illustrate the advantages of the direct surface-curvature distribution blade-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 design process, and it is quite difficult to completely control with inverse, semi-inverse or other direct-design methods. The blade-design method is briefly reviewed, and then the effort is concentrated on smoothly blending the leading edge shape (circle or ellipse, etc.) with the main part of the blade surface, in a manner that avoids leading-edge flow-disturbance and flow-separation regions. Specifically in the leading edge region we return to the second-order (parabolic) construction line coupled with a revised smoothing equation between the leading-edge shape and the main part of the blade. The Hodson–Dominy blade has been used as an example to show the ability of this blade-design method to remove leading-edge separation bubbles in gas turbine blades and other airfoil shapes that have very sharp changes in curvature near the leading edge. An additional gas turbine blade example has been used to illustrate the ability of this method to design leading edge shapes that avoid leading-edge separation bubbles at off-design conditions. This gas turbine blade example has inlet flow angle 0°, outlet flow angle 64.3°, and tangential lift coefficient 1.045, in a region of parameters where the leading edge shape is critical for the overall blade performance. Computed results at incidences of 10°, 5°, +5°, +10° are used to illustrate the complete removal of leading edge flow-disturbance regions, thus minimizing the possibility of leading-edge separation bubbles, while concurrently minimizing the stagnation pressure drop from inlet to outlet. These results using two difficult example cases of leading edge geometries illustrate the superiority and utility of this blade-design method when compared with other direct or inverse blade-design methods. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Design of high-efficiency airfoils and blades are essential for optimum aerodynamic, thermoeconomic, and overall performance of turbomachinery-based powerplants [1–3]. Gas turbine blades are three dimensional objects operating in unsteady and complex flow fields. This complexity forces designers to decompose the three dimensional blade-design problem to a series of two dimensional problems in the streamwise direction, and to assume that the blade operates in steady-flow conditions. The three dimensional variation of inlet and outlet flow angles is determined by streamline curvature calculations [4–8], and they vary from hub to tip. Depending on design choices of blade aspect ratio ðh=bÞ and the three-dimensional distribution of the centers of gravity of the two-dimensional sections, two-dimensional blade shapes are stacked to build the three dimensional geometry. The blade design manufactures have different sequences to design their blades, * Corresponding author. Tel.: +44 20 7882 5301. E-mail address: [email protected] (T. Korakianitis). 0306-2619/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2009.09.017

but all of them follow essentially the above overall procedure. The goal of any blade-design method is to find a geometry that satisfies flow requirements with minimum loss, tolerable mechanical stresses, minimum disturbances downstream and upstream, and in the case of compressors adequate stall margin, among others. The design of blade geometries is a very important step for the design of efficient turbomachines, as the blade design process directly influences the blade-row efficiency and thus the overall machine efficiency. Early blade-design methods were based on specifying a thickness distribution around a camber line. In this method, for specifying turbine and compressor shapes, straight line, circular arc or parabolic camber lines were used, and then the thickness distribution was added to these camber lines for both the suction and the pressure sides of the blade. Examples of this method [9,10] are designs of NACA compressor airfoils (NACA-65 series) and other series of NACA profiles for turbines by Dunavant and Erwin [11]. This method did not provide enough flexibility to control both the suction and pressure surfaces in order to obtain a desirable aerodynamic performance.

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Nomenclature A a B b C ¼ 1=r C C0 CL k M o P r

coefficient (Eq. (9)) thickness-distribution coefficients (Eqs. (5) and (10) coefficient (Eq. (9)) axial chord (nondimensionally b = 1) curvature (Eqs. (1), (7), and Figs. 1, 3, and 12) coefficient (Eq. (9)) slope of curvature (Eq. (2)) tangential-loading coefficient (Eq. (4)) exponential function (Eq. (10)) Mach number throat diameter of the blade (Eq. (3)) points or nodes on the blade surfaces (Eqs. (5) and (10)) local radius of curvature (Eqs. (1) and (7))

After the 1950s, most turbine blades have been designed by specifying the shape of the suction and pressure surfaces of the blades. Thus the passage was defined by the blades and the resultant 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 points along the blade surfaces. The way in which the line segments are joined generates shapes that look smooth to the eye, but they cause local flow accelerations and decelerations as will be explained below. Various categories for blade-design methods exist: direct; inverse, semi-inverse, full-inverse or full optimization methods [14]; analysis and design modes [15]; optimization and design methods; and others. The definitions of what is semi-inverse, full-inverse etc. methods vary in the literature. In our work, the direct design has been defined as a process in which the blade geometry is specified and the resultant aerodynamic performance on that geometry is calculated; and the inverse method has been defined as a method in which the desired blade performance is specified and the geometry that would accomplish such performance is calculated. Both methods have relative advantages and disadvantages. In the direct method, it is relatively easy to fulfill mechanical and geometric constraints, where the complete geometry is specified; but it is usually laborious to obtain the desired distribution of pressure or velocity along the profile with this method. On the other hand, it can be difficult to get an acceptable geometry with an inverse method, where the velocity is prescribed and the geometry calculated [16]. The inverse design method [17] is based on the velocity or pressure distribution along the streamwise blade shape, and it requires multiple variations of the velocity distribution until an acceptable profile geometry is obtained. The inverse design method also has difficulties in both of the leading and trailing edges, due to mathematical singularity at the stagnation points [16,18]. The inverse method ends up with blades with zero thickness at the trailing edge, which are impossible to manufacture. This last difficulty makes the inverse method acceptable for some compressor blade geometries of thin trailing edge, but unacceptable for turbine geometries that have thicker trailing edges. Korakianitis has worked on the design of airfoils by the direct method. In his first attempt [19] he specified five points along the blade surface (excluding the trailing edge and the leading edge points) and two slopes on each surface. In the leading edge region he used two thickness distributions added perpendicularly to two parabolic construction lines. He concluded that the blade aerodynamic performance, judged by the shape of the blade surface Mach number distribution, is very sensitive to changes in the slope of the curvature of the blades.

S t ðx; yÞ ðX; YÞ

tangential pitch of the blades (Eqs. (3) and (4)) Bezier variable span (Eq. (6)) Cartesian coordinates nondimensional (with b) distances flow angle

a

Subscripts 1; 2; k points (P, C), or with flow angle a c circle, or circle center is isentropic le leading edge p pressure side s suction side

In follow on work Korakianitis [19–24,8,25] in a series of improvements to blade-design methods proposed a direct design method based on specifying blade surface-curvature distributions. He concluded that the aerodynamic and heat transfer performance of the blades are determined by the surface-curvature distribution, and proposed methods to prescribe blade shapes with continuous slope of curvature which resulted in superior aerodynamic and heat transfer performance (avoiding local separation bubbles, and prescribing continuous slope of curvature joints among the line segments, and of the line segments with the leading-edge and trailing-edge shapes). He concluded that the changes in the local slope of curvature are more important than the (x, y) location of the blade shape, and that blade quality-assurance methods should not be point-by-point comparisons of geometric accuracy, but should be based on measurements of streamwise surface-curvature distributions instead. The conclusion for this method is that blade surfaces with continuous slope of curvature, or third derivative continuity in the spline knots, result in smooth Mach number and pressure distributions, and thus can be used to avoid flow separation, and lead to improved blade designs. Slope of curvature continuity means that the surface curvature C, as well as its slope along axial distance x; C 0 ¼ dC=dx along the blade profile length, must both be continuous. This requires continuous third derivatives of the line segments 3 3 yðxÞ, i.e. continuous y000 ¼ d y=dx , as illustrated by the following two equations for curvature and its slope.

C ¼

1 y00 ¼ r ½1 þ ðy0 Þ2 ð3=2Þ

ð1Þ

C0 ¼

dC y000 ½1 þ ðy0 Þ2   3 y0 ðy00 Þ2 ¼ dx ½1 þ ðy0 Þ2 5=2

ð2Þ

where

y0 ¼

y ¼ yðxÞ;

df ðxÞ ; dx

2

y00 ¼

d f ðxÞ 2

dx

;

3

y000 ¼

d f ðxÞ 3

dx

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

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which is also used as an example later on in this paper). Corral and Pastor [28] report ‘‘... it is specially difficult to match the original geometry because of the strong i.e. discontinuity in the airfoil, which is not explicitly take into account in our method. The isentropic Mach number distributions computed with MISES ...” (MISES is a well known inverse design method). As will be shown later in this paper, the prescribed surface-curvature distribution method can be used in a direct design method to remove the leading edge separation regions from the HD and other blades. In consequence, the goal of this paper is to develop the proposed direct design method that can prescribe the leading edge geometry of the blade at design and off-design conditions for gas turbines in a manner that minimizes leading edge flow-disturbances and avoids separation bubbles. The original prescribed-curvature-distribution method is modified in the vicinity of the leading edge. A leading edge shape such as a circle or ellipse is prescribed at the leading edge. A polynomial function is used to join the leading-edge shape with the main part of the blade surface. The polynomial is prescribed as a thickness distribution in relation to a parabolic construction line. Furthermore, the polynomial matches point, first, second and third derivative at its two ends with the remaining blade shape, thus resulting in continuous curvature and slope of curvature at these crucial locations. This is a first publication of a technique that provides this advantage to leading-edge shapes. A similar technique is used to blend a trailing-edge circle to the blade surfaces while maintaining curvature and slope of curvature continuity at these locations. The HD blade has been used as an example to show the ability of this technique to remove the leading-edge separation bubbles around the leading edge. Additional blades have been used as examples at design and off-design conditions to further illustrate the ability of this method to remove the flow-disturbances and flow separation at off-design conditions, and minimize stagnation pressure losses from inlet to outlet. These improvements on the prescribed-curvature-distribution blade-design method provide direct control for the overall blade geometry, eliminating the problems of inverse method. 2. Geometric design of turbine blades There are a number of key issues that need to be addressed in the design of blade geometries that affect the choice of implementation methods. The first issue is the flexibility of any revised method to represent a wide range of different blade shapes, and ability to work on other similar shapes such as compressor, fan and wind turbine blades. The second issue is the accuracy or the optimizing objective functions, which must be compared with experimental data. The accuracy should be accounted for in both the geometric and aerodynamic sense. In the present paper the original method proposed by Korakianitis [19,22–24,8] has been further developed and optimized to enhance the ability of the method to provide optimum leading-edge shapes in geometrically difficult cases, for instance with thin leading edge shapes, small leading-edge wedge angles, and inflow angles around 0°. Based on the original method, the design of the main part of the suction and pressure sides of the blade shape (except the leading edge and trailing edge) is split into three major segments: the segment near the trailing edge; the main part of the blade; and the segment near the leading edge, as explained in the following. 3. The trailing-edge segments In the previous work [19,22], the trailing edge was represented by a point. Later on a trailing edge thickness (a cusp) was used to

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specify the trailing edge section [19]. Finally, Korakianitis and Wegge [8] used a circle at the leading and trailing edge, but the implementation still presented problems when the blade shapes were very near 0° and 90° in the vicinity of the trailing-edge circle. In the present modification to the basic method, the trailing edge thickness is based on a circle which is blended with the trailingedge segment of the main part of the blade with a polynomial. The suction-side trailing-edge segment is from the trailing-edge circle to the throat of the passage, where at the throat we use the location of the point and slope of the suction surface line at that location. For the pressure surface there is no corresponding guiding throat location, so a geometrically-sensible point and its slope are prescribed on the ðx; yÞ plane of the blade. The pressure-side trailing-edge segment is from the trailing-edge circle to the pressuresurface point described above. Essentially these two suction and pressure line segments, the blade-surface ‘‘departing” angles from the trailing-edge circles, and the throat and pressure points and their slopes chosen as inputs, prescribe the trailing-edge region ‘‘wedge” angle of the blade, and it’s thickness for manufacturing and stress considerations. The specification of the trailing-edge segment of the main part of the blade, from the throat point to the trailing-edge circle, is crucial in minimizing flow diffusion in this ‘‘unguided” (by a pressuresurface shape on the opposite side of the passage) flow region. Less deceleration (diffusion) means better designs. If the flow decelerates too much, then the blade wake will be thicker, the boundary layer may separate, and the resulting blade shape will have poorer performance (lower efficiency). The trailing-edge suction side is controlled by the throat diameter o. A first approximation [9] is

o ¼ cos1 a2 S

ð3Þ

but in practice this value is modified using experimental data such as the Ainley–Mathieson data [32]. Wilson and Korakianitis have modeled these data with equations using a main parameter cos1 a2 and a secondary parameter based on throat Mach number [9]. In the 2D design, the throat diameter o is calculated by an input parameter of throat/chord ðo=SÞ, the tangential lift coefficient ðC L Þ, inlet flow angle a1 , and outlet flow angle a2 :

S CL ¼ b 2 cos2 a2 ðtan a1  tan a2 Þ

ð4Þ

where C L is the tangential lift coefficient and its value is determined by the number of the blades in the annulus. Korakianitis [19] pointed out that tangential lift coefficient typically is between 0.8 and 1.2. In blade designs of the 1940–1960 the value of tangential lift coefficient was low, but in modern cascades the value of C L has been gradually increasing. For instance for the 3D blade shape of the 2D blade B3 shown later in this paper C L is 0.86, 1.03 and 1.17 at the hub, mean and tip regions (the mean geometry with C L ¼ 1:03 is shown in this paper). In the 3D design process the number of blades, and therefore the resultant S at each radius, is an input. Therefore o is calculated from cos1 a2 as modified by the throat Mach number using the Ainley–Mathieson data or their mathematical expressions in [9]. Similarly to the previous approaches as in [33], the bladesurface segment from the trailing-edge circle to the throat is described with an equation of the form

y ¼ a0 þ a1 x þ a2 x2 þ a3 x3 þ a4 x4 k1 ½x  xðPs2 Þ þ a5 x5 k2 ðx  xðPs2 ÞÞ ð5Þ where in the above expression k1 and k2 are exponential functions, resulting in terms of increasing importance as we approach point Ps2 . This type of expression requires derivation of six parameters,

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a0 ; . . . ; a5 . These are derived from the conditions of point, first, second and third derivative continuity (four conditions) of the bladesurface line at the trailing-edge circle; and prescribing the point and slope of the blade-surface line at the tangent to the throat diameter (two additional conditions). This approach enables easy control of the shape of the blade surface in the vicinity of the trailing edge, and control of its curvature at the blending point to the trailing-edge circle (or other trailing-edge shape).

The surface curvature of the main part of the blade can be specified by a Bezier curve of two or more controlling segments. The more the controlling segments the higher the degree of flexibility in designing the blade surface, and the higher the complexity. In this work we have used six control points C 1s ; . . . ; C 6s for the Bezier segments as shown in Fig. 1. The resulting Bezier spline equation is

xC ¼ t 3 xC 1s þ 3t2 ð1  tÞxC 2s þ 3tð1  tÞ2 xC 3s þ ð1  tÞ3 xC 4s þ ð1  tÞ4 xC 5s þ ð1  tÞ5 xC 6s

4. Main curvature-mapped part of the blade surface The main part of the blade is the mid portion of the surface that connects the trailing and leading edge segments of the blade. As in the previous approaches in [20,22,24,8], Bezier splines have been used to prescribe the surface curvature of this part of the blade. Bezier curves are optimal from the point of view of geometric modeling aiming to control the inner location of the points and the degree of its differentiability. The surface-curvature distribution of this part of the blade is prescribed as shown in the example of Fig. 1, where the example shown the curvature of the main part of the blade is controlled by a five-segment Bezier curve controlled by the six control points C 1s ; . . . ; C 6s . The blade surface points on the ðx; yÞ plane are derived from the prescribed curvature points in the ðC; yÞ plane, where the latter is illustrated in Fig. 1. The main part of the blade begins at the upstream point of the trailing-edge segment (the throat on the suction side), which is point P sm on the blade surface; and ends at a point Psk on the blade surface (which will be later connected to the the leading edge segment). This point P sk will be derived as the last point in the curvature mapping sequence from the ðC; yÞ to the ðx; yÞ plane. Referring to Fig. 1, the trailing-edge segment is a prescribed polynomial given by Eq. (5), which results in the curvature distribution shown from x ¼ 1:0 to x ¼ xðC 6s Þ in the figure. The slope of the curvature line C 0 at x ¼ xðC 6s Þ is prescribed by Eq. (6). A key input at this part of the design sequence is the x location of point C 5s , which also prescribes the length of the line segment C 5s C 6s . The conditions describing line C 5s C 6s are derived so that this line results in continuous slope of curvature ðC 0 Þ conditions at point P sm on the blade surface, and point C 6c on the ðC; yÞ plane in Fig.1. This approach ensures slope of curvature continuity at blade surface point Psm , which joins the trailing-edge segment to the main part of the blade.

yC ¼ t3 yC 1s þ 3t2 ð1  tÞyC 2s þ 3tð1  tÞ2 yC 3s þ ð1  tÞ3 yC 4s

ð6Þ

þ ð1  tÞ4 yC 5s þ ð1  tÞ5 yC 6s The evaluating variable t is still 0 and 1 at each end, so t ¼ 0 at starting point C 6s and t ¼ 1 at ending point C 1s . The remaining approach is similar to [22]. The curvature, given by

 1 C ¼ ¼ h r

2

d y=dx

2



1 þ ðdy=dxÞ2

ið3=2Þ

ð7Þ

is written with central differences as

CF1=CF2 CF3   y  yi yi  yi1  CF1  2 iþ1 xiþ1  xi xi  xi1 Ci ¼

ð8Þ

CF2  ðxiþ1  xi1 Þ (  )ð3=2Þ 2 1 yiþ1  yi yi  yi1 CF3  þ þ1 2 xiþ1  xi xi  xi1 The above equation can be solved numerically or by manipulating into a sixth-order algebraic equation to find out the value of successive yiþ1 from inputs xiþ1 ; C i and xi ; yi ; xi1 ; yi1 . Korakianitis [22] used the regula-falsi solution for yiþ1 . In this paper the same procedure has been used to solve numerically this part of the blade. The the control points C 1s ; . . . ; C 6s of the Bezier spline in Fig. 1 are iteratively manipulated until the slope of the curvature and the resultant output blade shape on the ðx; yÞ plane are both acceptable. The procedure described above is identical for the pressure side, but with different values of variables. 5. The leading edge region

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

This part of the blade has significant effects on the aerodynamic and heat transfer performance of the blade, and the result is significantly affected by small changes in the blade shape. Joining the leading edge circle or other shape to the main blade results in flow separation bubbles, and local accelerations and decelerations in the pressure and Mach number distributions. Both direct and inverse design methods have difficulties around this area (e.g. [34,16,35]). In this work we implement a hybrid method based on modifications of the earlier methods [19,20,22,8]. First we introduce the leading edge shape, such as a circle or ellipse. Then a parabolic construction line and thickness distribution is added (as in [20,22]), where the construction line now starts from a key geometric point such as the center of the leading edge circle. Finally the thickness distribution is added about this parabolic construction line in a manner that the thickness distribution (and therefore also the blade surface) have continuous point, first, second and third derivative (continuous y; y0 ; y00 ; y000 and therefore continuous C 0 ) at the points where it joins the leading edge shape (circle) and the main part of the blade (point Psk and C 1s in Fig. 1). This is analogous to the circle-joining work of [8] with exponentials in the polynomials,

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and the above section on joining the trailing-edge circle to the trailing-edge segment of the blade. Thus the suction-side parabolic construction line passing though the center of the leading edge circle is of the form:

yðxÞ ¼ Ax2 þ Bx þ C

ð9Þ

and the thickness distribution added about the parabolic construction line is of the form

y ¼ a0 þ a1 x þ a2 x2 þ a3 x3 þ a4 x4 k1 ½x  xðP s1 Þ þ a5 x5 k2 ½x  xðPsk Þ þ a6 x6 k3 ½x  xðPs1 Þ þ a7 x7 k4 ½x  xðPsk Þ

ð10Þ

where functions k1 ; k2 ; k3 and k4 are exponential polynomials which acquire increasing importance as we approach Ps1 and P sk on the blade surface. Point Psk is the joint point with the main curvature-mapped part of the blade; and point Ps1 is the joint point with the leading edge circle. The eight parameters of the thickness function a0 ; . . . ; a7 enable us to evaluate eight conditions of this thickness distribution. Four parameters are evaluated to match y; y0 ; y00 and y000 (and thus C 0 ) at point Ps1 ; and four parameters are evaluated to match y; y0 ; y00 and y000 (and thus C 0 ) at point P sk . This approach ensures continuity of curvature from the main part of the blade surface through the leading-edge thickness distribution and into the leading edge circle. The procedure is similar for the pressure side of the blade, except with different geometric parameters. This overall approach enables us to remove any flow discontinuities due to surface-curvature discontinuities from the leadingedge stagnation point to the trailing edge stagnation point. Of course flow discontinuities will still occur where we demand too-high adverse pressure gradients for the flow conditions, but the method ensures that such flow discontinuities are not due to surface-curvature discontinuities. 6. Aerodynamic effects of leading edge geometry Experimental data published in [36–40] clearly indicate local kinks in the surface pressure or Mach number distribution of blades, which, as will be illustrated later, is the result of surfacecurvature discontinuities. These ‘‘kinks” affect boundary-layer performance and blade efficiency, and they could be smoothed by the prescribed surface-curvature blade-design method. Hodson [29] experimentally detected the presence of the leading edge spike as a result of a curvature discontinuity in the blending point between the circle and the rest of the blade. Hodson and Dominy experimentally tested this blade (which we designate the HD blade) extensively [29–31]. Several authors have computationally confirmed this spike [19,15,8,28] around the leading edge of the same blade. Stow [15] illustrated the removal of the leading edge spike locally using an inverse design method. In this paper we illustrate the use of the prescribed-curvature blade design technique: first to reproduce a blade (designated I1) designed with the prescribed-curvature-distribution direct-design method (not locally, but from trailing edge to leading edge) that has geometry and leading edge spike similar to those of the HD blade; then to redesign the blade shape with the prescribed-curvature blade-design method (again from trailing edge to leading edge) in a series of successive improvements (blades I4 and I9 are shown) that remove the leading edge separation spikes. The details of the computations are included below. The design parameters and representative data points for blades I1 and I4 are included 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. We have restricted the geometry to use the same leading-edge circle diameter, and in order to maintain the same blade chord from

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x ¼ ½0; 1, as the blade became progressively thinner near the leading edge we limited the reduction in the leading edge wedge angle so that the foremost point on the blade did not move forward from x ¼ 0 to values x > 0. Thus the foremost point on blade I9 is at ðx; yÞ ¼ ð0:0000; 0:006723Þ as shown in Fig. 2. The HD blade profile is a thin, hollow, castable root section from the rotor of a low-pressure turbine. It was designed to operate at air inlet flow angle 38.8° relative to the axial direction and to provide approximately 93° of flow turning. The design velocity ratio across the cascade is equal to 1.41. The nominal aspect ratio was 1.8. The test Reynolds number is 2:3  105 and the test inlet and outlet Mach numbers are 0.496 and 0.702 respectively. Further details can be found in references [15,29–31]. The left side of Fig. 3 shows the surface-curvature distribution for the HD blade (jagged line, evaluated numerically from the original data points) and the curvature distributions of blades I1, I4 and I9. The surface-curvature distributions of blades I1, I4 and I9 are smoother lines, as these blades have been reproduced with the prescribed-curvature-distribution blade-design method. The figure also shows the curvature of blade I1 trying to follow the curvature of the HD blade in the vicinity of the leading edge ‘‘spike” on the 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, in order 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 computed isentropic Mach number distributions are shown on the right side of Fig. 3. The sharp local acceleration–deceleration region in the leading edge of blade I9 has also been smoothed. Mesh generator GAMBIT and flow solver FLUENT have been used in the computations. A reasonably high number of computational mesh elements is required for reasonably accurate calculations. The exact numbers depend on the geometry of the blade and its model. The mesh elements used for the HD and I1, I4 and I9 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-unstructured mesh have been used in the viscous calculations. The pave mesh consisted of a combination of structured and unstructured regions. The mesh around the airfoil consisted of twelve structured clustered O-grid layers with yþ less than 6 (quadrilaterals), and the remaining majority of the flow field was discretized with quadrilateral and a small numbers of triangular cells. The k  x and

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

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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.

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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°, increment 0.05.

Fig. 7. Left, B1 isentropic surface Mach number distribution at off-design condition, inflow at +5°. Right, B1 isentropic surface Mach number distribution at off-design condition, inflow at 5°.

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-design condition, inflow at 10°.

k  e turbulence models have been used. The results for both turbulence models are approximately the same, and the results shown are those for the k  x turbulence model.

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

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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°, increment 0.05.

Fig. 10. Left, B3 isentropic surface Mach number distribution at off-design condition, inflow at +5°. Right, B3 isentropic surface Mach number distribution at off-design condition, inflow at 5°.

Fig. 11. Left, B3 isentropic surface Mach number distribution at off-design condition, inflow at +10°. Right, B3 isentropic surface Mach number distribution at off-design condition, inflow at 10°.

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deceleration regions (and the resultant separation bubbles) at the edges of the leading-edge circle region. Fig. 5 shows the comparison of the experimental data for the HD blade with the results of viscous calculations blades HD, I1, I4 and I9.

7. Design and off-design performance Curvature discontinuities have a more severe impact around the leading edge at design and off-design conditions because at least one of the discontinuities takes place in a region of a high velocity. Benner [41] mentioned that the discontinuity in the (flow) curvature is increased at off-design conditions, and he argued that the off-design behavior of the blade is influenced by the magnitude of the discontinuity in the surface curvature between the circle and the rest of the leading edge part of the blade. The prescribed-curvature blade design technique has been used to design three example blades, designated B1, B2 and B3, to illustrate the effect of the surface-curvature distribution at design and off-design conditions. Three dimensional blade designs B1, B2 and B3 have been obtained, but this paper reports the representative mean-line results. The design parameters and representative data points for blade B3 are included in Appendices A and B. Blades B1, 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) C L ¼ 1:04500. Stagger angle k ¼ 47 . Leading edge circle radius to axial chord ratio r le =b ¼ 0:0300. Trailing-edge circle radius to axial chord ratio r te =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 inlet flow angles of zero degrees (to illustrate the effect of positive and negative inflow angles from this value). The flow solutions have been obtained with FLUENT specifying free-stream turbulence intensity of 10% and using the k  x turbulence model. The solutions have been obtained for design point conditions as well as for 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 Omesh and a pave-unstructured mesh have been used in the viscous calculations. The same procedure for the HD blade have been used to generate mesh around the B1, B2 and B3 blades. Fig. 6 illustrates the isentropic surface Mach number distributions and the Mach contours of blade B1 at design conditions. Figs. 7 and 8 illustrate the off-design conditions. Fig. 9 illustrates the surface Mach number distributions and the Mach contours of blade B3 at design conditions. Figs. 10 and 11 illustrate the off-design conditions. The surface-curvature distribution of blades B1 and B3 is shown in Fig. 12. At design point conditions blade B1 exhibits perfectly acceptable behavior, without any leading-edge flow-disturbances at the joining points of the blade surface with the leading edge circle. At +5° a small disturbance is shown on the suction surface at the location of joining the leading edge circle with the blade surface. At 5° a small acceleration–deceleration region is shown on the

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

pressure surface at the location of joining the leading edge circle with the blade surface. These flow-disturbances are made worse at +10°, and 10°. These disturbances at off-design conditions are progressively improved in blades B2 and B3 by small changes in the blade surface-curvature distribution, as illustrated in Fig. 12. Furthermore these flow effects have consequences in the flow deceleration regime in the region of unguided diffusion, and the resultant computed total pressure losses on the blade. For the purposes of discussion we will define as the suction-side diffusion ratio the maximum isentropic Mach number on the surface of the suction side divided by the Mach number at the trailing edge. The results show that with the blade modification made using the prescribed-curvature blade-design method the effect of joining the leading edge circle to the blade surfaces has been minimized not only at design conditions, but also at relatively large angles of incidence. The computed stagnation pressure losses from inlet to outlet at design and off-design conditions are shown in Table 1. Table 2 shows the resultant suction-side diffusion ratios at design and off-design conditions. There is a direct correlation between increases in suction-side diffusion ratio and stagnation pressure losses.

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

B1 B2 B3

Design and off-design conditions 10

5

0

+5

+10

0.002430 0.002424 0.002357

0.002450 0.002446 0.002404

0.002480 0.002474 0.002433

0.002518 0.002508 0.002467

0.002560 0.002552 0.002512

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

B1 B2 B3

Design and off-design conditions 10

5

0

+5

+10

0.87110 0.849743 0.838995

0.869975 0.84831 0.836926

0.86985 0.84685 0.834855

0.86772 0.84538 0.832745

0.86780 0.843875 0.83042

1600

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

Appendix A (continued)

8. Conclusions Redesign of the HD blade to remove the leading edge problems had been attempted before with inverse design methods [15], but this is the first reported attempted to remove the leading edge spikes with a surface-curvature direct blade-design method. It is noted that suction-surface spike and the pressure-surface region of diffusion near the leading edge have been completely removed. The results show that the prescribed surface-curvature distribution blade-design method is a robust tool for blade design, providing a manageable and accurate way to control the blade surface and the aerodynamic properties around the blade. The prescribed surface curvature blade design technique results in smoother boundary layer flows, affecting aerodynamic as well as heat transfer performance. Sample blades B1, B2 and B3 have been designed to show the capability of the method to eliminate the spike and dips in the surface Mach number distribution at design and off-design conditions in the vicinity of the leading edge circle at both design and off-design conditions. Stagnation pressure losses decrease with smoothing the blending of the circle with the blade surfaces, as predicted by Benner [41]. We conclude that the prescribed surface curvature blade design technique can be used to provide accurate guidance and control for the design of blade shapes. Furthermore the method can be used to eliminate the flow problems resulting from blending a leading edge circle or other shape with the blade surfaces at design and off-design conditions. The prescribed-curvature blade design technique has been used to design stacked 3D turbine blades, compressor and fan blades, isolated airfoils, and wind turbines. These will be reported in other papers.

Appendix A. Parameters specifying the blades Some symbols defined in Ref. [22]. I1

I9

B3

S=b k o=S [22] / [22] r le =b X c;le Y c;le rte =b X c;te Y c;te bs2 [22]

38.80 53.90 0.6000 19.60 0.5850 43.10 0.01650 0.01650 0.01164 0.00485 0.99515 0.34921 64.00

38.30 38.80 0.6000 19.60 0.5850 43.50 0.01650 0.01650 0.01164 0.00485 0.99515 0.34921 64.00

0.00 64.30 1.3371 47.00 0.4410 41.40 0.03000 0.03000 0.01182 0.01500 0.98500 1.03430 76.501

XC1s YC1s XC2s YC2s YC3s YC4s XC5s bs1 [22] X sc [22] Y sc [22] bsc [22]

0.0900 2.3000 0.2011 2.2000 2.40700 2.4997 0.5500 74.00 0.1800 0.0900 25.00

0.0900 2.1000 0.2011 2.1000 2.1870 2.2200 0.5500 119.00 0.1400 0.0900 40.00

0.0800 4.1820 0.2220 4.2200 2.4150 2.2200 0.400 69.00 0.1400 0.0100 7.00

bp2 [22] bpm [22] X pm [22] Y pm [22]

33.50 39.27 0.7100 0.0570

32.50 40.15 0.7100 0.0520

56.00 48.00 0.5600 0.3900

a1 a2

XC1p YC1p XC2p YC2p YC3p YC4p XC5p bp1 [22] X pc [22] Y pc [22] bpc [22]

I1

I9

B3

0.1100 1.5700 0.2080 1.4520 2.1620 2.2300 0.5850 24.00 0.1800 0.0900 27.00

0.1100 1.6800 0.1880 1.7800 1.8000 1.8000 0.5800 15.00 0.1800 0.0900 27.00

0.0800 0.6300 0.1700 0.6800 0.7600 0.7950 0.5000 40.50 0.1300 0.0008 0.00

Appendix B. Representative blade points Arbitrary numbers of blade data points, with accuracy up to any number of desired decimal points, can be derived using the blade parameters in Appendix A. Blade X

I1 Y

I9 Y

B3 Y

1.00 0.95 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.05 0.00 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95

0.349 0.275 0.203 0.064 0.054 0.139 0.188 0.210 0.207 0.179 0.121 0.076 0.012 0.011 0.035 0.070 0.086 0.084 0.062 0.018 0.049 0.139 0.245 0.303

0.349 0.272 0.199 0.061 0.054 0.137 0.189 0.212 0.210 0.182 0.125 0.080 0.007 0.010 0.033 0.068 0.083 0.082 0.061 0.021 0.044 0.136 0.246 0.304

1.034 0.887 0.754 0.523 0.334 0.182 0.063 0.027 0.085 0.110 0.096 0.070 0.012 0.030 0.052 0.103 0.164 0.237 0.327 0.436 0.567 0.720 0.898 0.997

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