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Improved turbine-blade design techniques using 4th-order parametric-spline segments
Improved turbine-blade design techniques using ,4th-order parametric-spline segments T Korakianitis and G I Pantazopoulos
The paper presents a method for generating blade shapes to be used as inputs to direct- or inverse-blade-design sequences. Isolated airfoils and gas-turbine-blade cascades are designed by a mixture of iterative and direct- and inverse-blade-design methods. In the direct-design method, the designer inputs the geometry of the airfoils, and flow solvers are used to study the resulting performance in terms of Mach numbers or pressure distributions along the airfoil surfaces. In the inverse-design method, the designer specifies as input the desirable performance along the airfoil surfaces, and flow solvers are used to compute the airfoil geometry that will provide the performance. The design iterations are enormously reduced if the initial blade shape has performance characteristics that are near the desirable ones. The method can be used to generate airfoils for compressors and turbines, or isolated airfoils, but the discussion in the paper is limited to subsonic-exit axial-turbine blade rows. The desirable performance characteristics are presented, and ways to specify the input aiming for such characteristics (minimizing design iterations) are discussed. It is shown that continuous curvature and continuous slope of curvature are necessary conditions for improved blade designs. A set of parametric curves that satisfy the above conditions is developed, and used to specify the geometry of turbine cascades. Thickness distributions satisfying the same conditions are used near the leading edge of the cascades. The performance characteristics of two turbine cascades generated by this method are included. The shape and distribution of curvature of the suction and pressure surfaces of the airfoil Department of Mechanical Engineering, Washington University, Campus Box 1185, One Brookings Drive, St Louis, MO 63130, USA Paper received: 16 April 1991. Revised:25 September 1992
volume 25 number 5 may 1 9 9 3
are particularly important for the aerodynamic and heat-transfer performance. The 4th-order (continuousslope-of-curvature) parametric splines should be used wherever separation in fluid streams at the walls would be detrimental to performance, and they have additional applications in smooth-surface-generation activities such as automobile-surface design. eontinuous curvatureslopes, 4th-ordersplines, airfoils, continuousthird derivatives
Axial gas-turbine blades are 3D objects operating in a complex flow field. Owing to its complexity, the problem is usually reduced to a series of radially 'stacked' 2D problems. Blade design is partly a science and partly an art. The performance of the blades is expressed in the form of some property distribution (such as pressure and velocity) on the blade surface and in the cascade passage. To minimize radial gradients in enthalpy, pressure and entropy, a condition which is called radial equilibrium, the inlet and outlet flow angles of the cascade vary along the span of the blades (from hub to tip). The centrifugal forces experienced at operating speeds impose constraints on the locus of the center of gravity of the 2D sections. The leading and trailing edges of the blade sections must blend smoothly in the radial direction. 3D turbine blades are generated by 'stacking' 2D designs, using empirical and analytic rules for the above considerations. Compromises in performance must be made to accommodate these 3D constraints. Blade cascades can be designed by direct or inverse methods. Various investigators use different definitions for the following terms: analysis and design modes t , optimization and design methods z, and direct, inverse, semiinverse, full-inverse and full-optimization methods 3.
0010-4485/93/050289-11 © 1993 Butterworth-Heinemann Ltd
289
T Korakianitis and G I Pantazopoulos
For the purposes of this paper, direct- and inverse-bladedesign methods are defined as follows. In the direct method, the designer inputs the geometry of the blade, and the output is the performance of the airfoil. The blade geometry is modified until a desirable performance is obtained. Examples of direct methods have been presented by Dunham 4, Prichard 5 and Korakianitis 6-8. In the inverse method, the designer usually starts from some initial blade shape and performance, and inputs the desired modifications to the performance. The output is a new shape and its performance, which is as close to the desired (input) performance as is permitted by the equations modelling the flow. Both methods are iterative, and are based on the assumption of steady-inflow and steady-outflow conditions (although the flow around the blades during engine operation is inherently unsteady). Both methods have relative advantages and disadvantages. The direct method is laborious, but the designer has direct control of the various geometric parameters, and infeasible blade shapes are excluded before they are analysed. Even if inverse-design programs are available, the inverse method should start from a blade shape that is close to an acceptable design, since this enormously reduces the number of inverse-design iterations. The inverse method is less laborious, but many (although not all) inverse-design programs must start from a blade geometry that is generated by a direct method. The designer has less control of the blade shape and the resulting performance. If some geometric parameters (such as stagger angle) need to be changed, the process must be started from the beginning (direct method) again. The final design is usually obtained by a judicious combination of both methods, although, if an inverse method is not available, the direct method can be used exclusively. The purpose of this paper is to provide a design tool for generating blade shapes for either method. In particular, the resulting blade shapes by 9eometric construction have inherently good performance, and, after the first design iteration, the required modifications for good performance are immediately obvious. The airfoil is specified by a series of geometric parameters. The blades presented in this paper have been analysed by direct methods exclusively. The geometry, performance and design iterations for two sample cases are included for illustration.
BACKGROUND Demands for improved performance require sophisticated methods for blade design, because the performance of the airfoils becomes critical. In turbine cascades, the flow expands: the pressure at the outlet is lower than the pressure at the inlet. In subsonic cases, from the cascade inlet to the outlet, the passage velocity is increased, and the area available to the flow should continuously decrease to the minimum area corresponding to the circle of diameter o at the throat (see Figure 1 ). Traditionally, turbine cascades have been designed by manipulating the shape of the passage between successive airfoils, and then
290
b
S
(
Z
Figure 1. Geometry of typical turbine cas(ade closing the shape at the leading edge by specifying a leading-edge circle, ellipse, or other appropriate shape. The flow accelerates around both sides of the cascade. The curvature of the streamlines is higher on the convex (suction) side, and lower on the concave (pressure) side, resulting in higher velocity and reduced pressure on the suction side, and lower velocity and increased pressure on the pressure side. These are shown in the performance figures (Mach numbers and pressure distributions ) of the sample turbine cascades discussed below. Although it is desired that the surface velocity and pressure distributions remain smooth, this is difficult to achieve, as shown in many of the examples from production turbines published by Sharma, Pickett and Ni 9, Stow! includes a turbine example with a sharp undesirable spike in the surface Mach number near the leading edge (this is due to a slope-of-curvature discontinuity). This spike was removed in a later design by modifying the blade surface with an inverse-design method. Korakianitis and Papagiannidis ~° have shown that slope-of-curvature discontinuities on the blade surfaces affect the local surface Math number and the pressure distributions, and the thickness of the boundary layer, and result in cascades with thicker boundary layers in the trailing edges, thus reducing efficiency. In compressor cascades, the flow diffuses" the pressure at the outlet is higher than the pressure at the inlet. In subsonic cases, from the cascade inlet to the outlet, the passage velocity is decreased, and the area available to the flow should continuously increase to a maximum area corresponding to the trailing edge of one blade and the pressure surface of the next. Traditionally, compressor cascades have been designed by designing individual airfoils in which thickness distributions are added to either side of a circular, parabolic, cubic or other-shaped camber line.
computer-aided design
Improved turbine-blade design techniques using 4th-order parametric-spline segments
In recent years, the performances of both types of cascade have been improved by the use of inverse-design techniques, which are otherwise known as prescri~dvelocity-distribution (PVD) techniques. Because the pressure is decreased along the turbine stages, any separation that may occur on the turbine cascades is likely to reattach, and the flow will continue its normal path from the turbine inlet to the outlet. In compressors, the pressure is increased along the stages, and any separation that may occur on compressor cascades may not reattach. This leads to compressor stall, and surge (a dangerous phenomenon characterized by the rapid periodic succession of flow from the compressor inlet to the outlet followed by flow from the high-pressure outlet to the low-pressure inlet). Bad turbine designs will still work, while bad compressor designs will not work. Compressors are harder to design than turbines. However, more attention has been given to compressor design, because of the deleterious effects of bad compressor performance, to the extent that many compressors are more efficient than the 'corresponding' turbines (for similar mass-flow rates and pressure ratios). This is because insufficient attention has been paid to turbine-blade design. The remainder of this paper concentrates on turbine cascades, although the method developed for turbine cascades is directly applicable to compressor cascades and isolated airfoils. Direct-blade-design methods in which the blade surfaces were designed with analytic polynomials of the form y = f(x) have been presented in previous work 6-s. These did not ensure slope-of-curvature discontinuity at the knots between the polynomials, and the throat diameter was not an input design parameter, but rather an output of the direct-design iterations. This paper presents a generalized blade-design method with parametric polynomials (y = f(s), x = f(s)) of order 4 which guarantee a continuous slope of curvature. In addition, the throat diameter is an input to the design process. One of the nodes between the splines of the suction side by construction specifies that the minimum distance between the pressure-side trailing-edge point and the suction surface of the previous blade is equal to the user-specified throat diameter o.
TURBINE-CASCADE GEOMETRY The blade-design sequence depends on the application and the global constraints of the engine. Examples of design sequences are described in various texts, such as Wilson's 1~. The designer has many choices. For the purposes of this paper, important geometric features are identified in the axial-turbine-cascade diagram shown in Figure 1. The flow is from left to right. One of the early design choices that must be made is the type of velocity diagram. It is assumed that the designer has chosen the blade-row velocity diagram and its parameters, such as the inlet flow angle ~ , the outlet flow angle ot2, and the exit Mach number M2 in the absolute frame for stators and in the relative frame for rotors. Four additional choices must be made: a specification for the
volume 25 number 5 may 1993
nondimensional tangential spacing S/b between the blades (where S is the tangential spacing of the blades, and b is the axial chord length (nondimensionally, b = 1 )), a specification for the stagger angle of the cascade 2, the inclination of the blades with respect to the axial direction x, the nondimensional throat diameter o/b, and the trailing-edge thickness. The stagger angle ). is defined as the angle between the axial direction and the line joining the leading edge of the cascade with the trailing-edge point on the concave (pressure) surface. Some guidelines for low-speed turbine blades have been published by Kacker and Okapuu ~2, although the designer has considerable flexibility in choosing 2. The value of ). also dictates whether the cascade is front- or aft-loaded, and different philosophies and guidelines are used by different manufacturers. The spacing between the blades (which is directly linked to the number of blades in the blade row) is a function of the tangential lift coefficient CL, defined by CL =
tangential aerodynamic force tangential blade area x outlet dynamic head
(1) This expression can be manipulated in a number of ways (for compressible flow, for incompressible flow, accounting for variations in axial-flow velocity etc. ). The incompressible-flow derivation ~ reduces CL to CL = 2 S cos2 at2 (tan =1 -- tan 0(2) b
(2)
The actual value of CL depends on the number of blades in the cascade, which dictates S. Typically, 0.8 < CL < 1.2; earlier designs had lower values of CL, and, in modern cascades, C L has been gradually increasing. It is difficult to design high-efficiency cascades that have both high values of CL and high values of the total flow deflection e in the cascade, where e = ~1 + ( 3600 - 0t2). The throat diameter is an extremely important design input, because it dictates the mass flow that can be passed through the cascade, and hence the work that can be delivered by the turbine. A good first approximation is o / S -~- cos-1 ~2. Experimental figures linking the throat diameter 0 to the blade spacing S, the throat Mach number M2, and the curvature of the convex (suction) blade surface near the trailing edge have been published by Ainley and Mathieson 13, and they are used extensively in the early stages of blade designs. The industry has for years relied on empirical relationships based on the Ainley-Mathieson data on improvements to it (e.g. by Dunham and Came 14) for the prediction of the outlet flow angle ~2 as a function of the exit Mach number M2, the spacing S and the throat diameter o. These empirical data have been simplified by others for design applications (for example by Wilson ~l and Dunham4). This is sufficient for preliminary design. Today's computer programs are more accurate, and they compute the outlet flow angle for a given geometry. Gostelow :5 has explained how the result depends not only on the geometry, but also to some extent on the exact location
291
T Korakianitis and G I Pantazopoulos
of the point at which the Kutta condition is applied at the trailing edge. The trailing-edge thickness should be as small as manufacturing considerations allow (to minimize the wake incident on the next blade row), but it is also affected by geometric constraints imposed by the cooling slots in cooled blades. In the sample cascade geometries shown below, and in Figure 1, the trailing edge is a point, because in this paper inviscid flow solvers have been used to analyse the blade performance. However, the method presented enables the designer to specify different trailing-edge points on the suction and pressure surfaces. In subsonic cascades, the area available to the flow from the inlet to the outlet should be continuously decreasing to ensure that the flow accelerates from the inlet to the throat. In successful designs, the velocity of the fluid adjacent to the concave (pressure) surface accelerates from the stagnation point near the leading edge to the trailing edge. The same is not possible for the velocity of the fluid adjacent to the convex (suction) surface, because the flow is 'unguided' on the suction surface downstream of the throat. In successful designs, the velocity of the fluid adjacent to the convex (suction) surface accelerates from the stagnation point near the leading edge to some point near the throat, and then it decelerates. This region on the suction surface from the throat to the trailing edge is called the region of'unguided diffusion'. If the flow decelerates too much, the boundary layer separates, the blade wake is thick, and the resulting cascade has a poor performance (low efficiency). Therefore, for high performance, the velocity should continuously and smoothly increase from the stagnation point to the trailing edge along the pressure surface. On the suction surface, the velocity should continuously and smoothly increase from the stagnation point to the point of maximum velocity that occurs near the throat. After the point of maximum velocity, designers strive for minimum and smooth diffusion from the point of maximum velocity to the trailing edge. The Reynolds number of typical production turbine blades is of the order of 5 x 105 based on the chord. The flow will probably become turbulent at some point along the surfaces. The smooth velocity distribution (and therefore the smooth pressure distribution) condition ensures that the boundary layer is not energized into turbulence unintentionally. (This can be done intentionally, by designing a curvature discontinuity at the point where the flow should become turbulent for performance considerations. )
THEORET~ CURVATURE
DEPENOENCE ON
Typically, the flow in turbomachinery cascades is analysed by considering Cartesian coordinates (x, y, z) or coordinate transformations in Cartesian systems of coordinates. However, the flow needs to curve around the blades. The dependence of flow on curvature is seen in the compressible-flow Navier-Stokes equations in the
292
limiting case of cylindrical coordinates (r, 0, z ). Since 2D solutions and geometries are 'stacked' radially, the equations of conservation of mass and momenta in polar coordinates (r, 0) are shown below (taken from the text by Bird et a/.16): dp ld 1 d -- + (prvr) + (pro) = 0 dt r~r r~
{ dVr u dVr VOOVr P ~ , ~ + "Or + r do _
dp
Or
V2I r/ + POt
(4)
r 2 Or (r:~,o) + -r d O ] + pod
(5)
d (r%,) +
Or
{dVo #vo vodvo p ~ - - ~ + v, d~ + - - dO lop
r dO
v,v.'~
(1 d
r dO
(3)
ldToo /
where p is the density, v is the velocity component, t is the time, # is the body force, p is the static pressure, and, for Newtonian fluids, the stress components z,, Too and T,0 are related to the dynamic viscosity/~ by the following equations :
r-~(v'*) 1 2d v'2 F [ 1 dvo
~rO = TOt
_ 2
= --# r ~
4- r dO3
( V - v ) = - I d ( r v , ) + 1 -dv- o
r~
r d0
(6)
where v is the velocity vector. A similar dependence of incompressible flow on curvature is also shown in most fluid-dynamics texts in cylindrical coordinates, and Schlichtingl 7 (p 112 ) attributes a set of similar equations for incompressible flow in curvilinear coordinates (x, r, y) to W Tollmien. The 1/r and 1/r 2 terms in all of these equations suggest a strong dependence of the local velocity on curvature. Once the flow passage has been manipulated until the turbine cascade looks like a turbine cascade, the performance depends on the behaviour of the boundary layer, where Equations 6 also indicate that the boundary layer is affected by the local radius. Since smooth velocity distributions along the blade surfaces are required (a continuous slope of velocity with respect to the surface length), the above equations require smooth curvature distributions along the blade surfaces (a continuous slope of curvature). A continuous slope of curvature requires continuous third derivatives, as shown in the following two equations for curvature and slope of curvature for y = f(x), y ' = d f ( x ) / d x ,
computer-aided design
Improved turbine-blade design techniques using 4th-order parametric-spline segments y" = d 2 f ( x ) / d x 2 and y" = d a f ( x ) / d x 3 :
C -
1 r
-
y"
[1
(7)
+ y,213/2
C' = y ' [ 1 + y,2] _ 3y,y,2 [1 + y , 2 ] s / 2
(8)
where C is the curvature and r is the local radius of curvature. In previous work 6-s, it was shown that turbine blades designed with polynomial functions with continuous first and second derivatives at the nodes resulted in surfaces of continuous curvature but discontinuous slope of curvature, which resulted in similar and corresponding discontinuities in the slopes of the velocity distributions. 3rd-derivative continuity is required to obtain smooth velocity distributions for high-efficiency cascades. Cascades designed with 3rd-derivative continuity result in thinner boundary-layer thicknesses at the trailing edge, and therefore have higher efficiencies ~°. Most parametric splines (e.g. cubic splines, B-splines, Brzier splines) have continuous first and second derivatives, and they result in surfaces with continuous curvatures but discontinuous slopes of curvature. In the following, a set of 4th-order parametric splines with continuous point, first, second and third derivatives at the nodes (and along the surfaces) are derived, and used to design a new set of turbine blades.
4TH-ORDER PARAMETRIC SPLINES The nodes of the parametric splines along the suction and pressure surfaces are illustrated in Figure 2. On the suction surface, the nodes are points {PI,, P2,, P3s, P4,}, and, on the pressure surface, the nodes are points {Pip, P2p, P3p, P4p}, although more nodes could be used for
each surface. For general applications, the analysis for 4th-order parametric splines for n number of nodes { P1, P2 . . . . . P, } is derived in the following. These nodes correspond to (n - 1 ) spline segments on either surface, where the parameter is s. Let (x, y~) be the Cartesian coordinates of any point P~ on the surface, and (Xj(s), Yj(s)) be the algebraic description of any spline segment j. The splines take the form x x S 2 + aj3s x 3 Xj(s) = aio + a]ls + a j2
x
+ aj,,s
4
Yj(s) = ago + a~ls + a~2 s2 + a~3 s3 + a~4s 4
(9)
where 0 ~ s ~< 1 in each of the (n - 1 ) segments j, and a are the spline coefficients for X(s) and Y(s). In the following, the derivation that is applicable to X(s) is shown; the derivation for Y(s) is exactly analogous. If fx~ and sx~ are the first and second derivatives of X(s) with respect to s at each of the n nodes i, Xi(O) = xi
Xi(1) =
Xi+ 1
dX~ -~s s=o = fxi dX_~i = fxi+ 1 I ds Is=l d2Xi -~$2
s=O = SX i
gives the solution for the spline coefficients of the spline segment i in terms of the first and second derivatives at the nodes : x
aio = X i
a';1 = fxi a~2 = sxi/2
a~'3 = 4(xi+1 - xi) - fxi+ l - 3fxi - sxi
(o,s)~/~
air = 3 ( x i -
(10)
The solution for the first and second derivatives at the nodes is found by requiring 2nd- and 3rd-derivative continuity :
P2v
d2Xi P2o
d2Xi+l
-fi~-s~,=,= daXi
(x.o,Y,o)
,~, (x,,,Y,o) (0, O)
x i + l ) + fxi+l + 2fxi + s x i / 2
P4, = P4 v
ds ~ daXi + 1
,=1=1 ds3 L=o which, for i = 2, 3 . . . . .
Plp
~=o
n -
1,
leads to
6fxi-1 + sxi-1 + 6fxi - sxi = 12(xi - x~-l) 5fxi-1 + sxi_ 1 + 6fxi + sxi + fxi + 1
P3~
= 4(xi+t + x i - 2xi-1)
p4 v Figure
P4o = P4 v
2. P a r a m e t e r s used to specify typical cascade
volume 25 number 5 may 1993
The last two equations can be manipulated into the matrix form shown in Equation 11, from which the values of the first and second derivatives at the nodes can be found. The only remaining unknowns are the boundary conditions shown in Equation 12, which are evaluated
293
T Korakianitis and G I Pantazopoulos below. -1
0
0
0
0
0
0
0
fx
0
1
0
0
0
0
0
0
SX 1
5
1
6
1
1
0
0
0
fx2
6
1
6
0
0
5
0
0
6
-1
0
0
0
0
SX 2
!
6
1
1
0
fx3
1
6
0
0
SX 3
-1
I
1
BCx,1( Py ) BCx.2(Pf) 4(x 3 + x 2
--
2xl )
I
12(x 2 - x 1 ) i
4(X4 4- X3
--
2x2)
12(x3 - x2) (11)
0
0
5
1
6
1
0
0
6
1
6
0
0
SX n _
0
0
0
0
0
0
1
0
fx,
0
0
0
0
6
1
6
--1
SX~
--1
4(x, + x , - i
fx n -
0
-
2Xn-2)
12(x,_ 1 - x, z)
BC~,a(P2)
I
J
1 2 ( x . - x._l)
appropriate blade-surface angle from Figure 2. On the suction surface, these are flsl and flsr- On the pressure surface, these are flvt and flpr. Similarly, the second derivatives can be expressed as :
dX 1
BClx'l = ~ - s s=o d2X1
BClx,2 = ~
1
~=o
d2y
dX._ 1 BC~'I = T ~=1
(12)
ds z
- A cos fl
dZx where BC is the boundary condition, and a subscript 1 indicates a left-side boundary condition, and a subscript r indicates a right-side boundary condition. The lst-derivative boundary conditions at the right and left sides of each spline set are related to the angles of the blades at the leading and trailing edge fist, fl~, flv~ and tips, as shown in Figure 2. These are in turn related to the flow angles ~1 and ~z, and the effect that the designer expects that the blade-surface angles will have on the flow angles. At the trailing edge, the difference between fist and flpr should be between 7 ° and 15 ° for manufacturing considerations, one being slightly larger and the other slightly smaller than the flow angle ~:. These angles are design inputs in this method (they are outputs in the inverse-design method). Near the leading edge fl,l and flpt are also design inputs, but the designer has a great deal of freedom in specifying these angles and the location of the leftmost points of the spline sets on each surface. For the lst-derivative boundary conditions tan fl - dy _ dy/ds _ sin fl dx dx/ds cos fl
A sin fl A cos fl
(13)
Thus an obvious choice for the first derivatives of the splines in X(s) and Y(s) at the first and last points of the spline sets of the suction or the pressure surfaces is
Asinfl
ds 2
(15)
Equations 14 and 15 give
dsZ ,] \ ds /
\ ds2 J \ ds ]
(16)
The curvature and the second derivatives are also related as follows :
c[1
~dx =
\~ds/
(d2y/ds 2 ) (dx/ds) - (d2x/ds 2 ) (dy/ds) (dx/ds) 3
(17)
Equations 16 and 17 are solved for the second derivatives with respect to s:
ds 2 -
d2y
\dsJL\dsJ
c(dX~[(dx']2
ds~ -
\~/L\ds/
+\ds,/
]
(dY~2] x/2
+ \~,/ J
(18)
which finally give the 2nd-derivative boundary conditions at points P1 : BClx'2
=
BCIy'2
=
d2~ ds2
el
= - A Z C ( P 1 ) sin fll
dy BCy'I
-
BC,,,x -
ds dx ds
-
A s i n fl
- A cos fl
(14)
where A is an arbitrary length chosen to be equal to the arc length of the appropriate spline segment, and fl is the
294
dZY
~2s2 el.
= A2C(P1)cosfll
(19)
The above procedure is applied for the splines in x and y for the suction and the pressure surfaces. Point P3s is special, and it is instrumental in the suction-surface design. The locus of P3 s is a circle, with the center being
computer-aided design
Improved turbine-blade design techniques using 4th-order parametric-spline segments
the trailing edge of the top blade (P4p), and the radius being o. Out of all the points that lie on the circle, only the ones that correspond to splines with slopes perpendicular to line P4pP3, at point P3, are chosen for further consideration.
LEADING-EDGE GEOMETRY The leading-edge geometry requires the specification of two lines, one for the suction and one for the pressure surface, joining the origin (0, 0 ) with points P 1s and P 1p, respectively. To maintain slope-of-curvature continuity at points P1, these two lines must join the splines of the blade surfaces with continuous first, second and third derivatives. First, two construction lines are specified. Parabolas have been used for simplicity (see Figure 2). The suction-surface construction parabola passes through points (0, 0) and (X,c, Y~¢) with a slope of fl~c at the origin. The pressure-surface construction parabola passes through points (0, 0) and (Xp¢, Yp¢)with a slope of flp¢ at the origin. Each parabolic construction line defines a parameter 09 and a corresponding baseline P(~o) as follows : x = a~
(20)
P(og) = eo + e,a~ + e2o~2 where o~ is the thickness-distribution parameter, and the constant coefficients of the construction lines, eo, el and e2, are evaluated from the boundary conditions of the parabolic construction lines. Next, the minimum distance of point P1, and the minimum distances of three additional points near P1, on the suction-surface spline P1,P2, from the suction-side parabola are evaluated. These are used to evaluate the distance and (numerically) the first, second and third derivatives of the distance of the spline at point PI, from the suction-side construction line. Similarly, the minimum distance of point Pip and the minimum distances of three additional points near Pip on the pressure-surface spline PlpP2p from the pressure-side parabola are evaluated. These are used to evaluate the distance and (numerically) the first, second and third derivatives of the distance of the spline at point Pip from the pressure-side construction line. Then, a thickness distribution is added perpendicularly above the suction-side construction line. A second thickness distribution is added perpendicularly below the pressure-side construction line. The thickness distributions are of the form Ts(~ ) =
~sgO 1/2 -1- Csl(D -~- Cs2(D 2 q- C,3(.D 3 -[- Cs4(D 4"
(21)
Tp(to) = 6pO~~/2 + cpltn + Cp2O~2 + Cp3tOa + Cp4~4 where the c are the thickness-distribution coefficients. The first, second and third derivatives of T(co) with respect to co are analytic expressions. The values of 5, and 6p, which are root-x coefficients for T, are specified by the designer. The only unknowns are the four coefficients cl, c2, c3 and c4. These are evaluated from
volume 25 number 5 may 1993
the four values of the distance and the first, second and third derivatives of the distance of the suction- and pressure-surface splines at points P1, and Plp from their respective construction lines. The above procedure ensures curvature and slope-ofcurvature continuity at points PI. The value of the parameters 5 can be used to specify thicker or thinner leading edges, which affect the design-point and off-design-point operating performance of the cascade, and are the subject of a future investigation. The origin (0, 0) is itself a singular point. To achieve lst-derivative continuity at this point, the angle of the parabolic construction lines should be equal for the suction and pressure surfaces (and approximately equal to the inflow angle ~1 ).
SEARCH ROUTINE The design choices of ct1, ~2, CL, 2, 0 and the trailing-edge thickness specify the location of the trailing edge and the circle of loci for point P3,. First, the suction surface is designed, and then the pressure surface. In the optimization routine for the suction surface, the designer specifies ranges of values for the y coordinate, the angle fl,i, and the curvature C(P1,) at point PI,, the x and y coordinates of points P2,, and the angle fl,,. For each combination of the above variables and location of P%, the parametric splines that pass through the nodes PI,, P2s, P3s and P4~ according to the above are computed, choosing the spline sets in which line P4pP% is perpendicular to the spline set at points P%, and eliminating all others. Each of the remaining spline sets results in a curvature distribution from Pls to P4,. From these curves, the ones that are chosen are those that pass two additional checks: (a) the curvature must have the same sign (positive) from PI~ to P4,, indicating that the suction surface must be continuously convex, and (b) the slope of the curvature must not change sign more than once from PIp to P4v. From the spline sets that pass all the above checks, the one that has the minimum value of ACx is chosen. ACx is numerically evaluated as follows: j=n-
ACx =
1(= 3) f P j + 1
~-' j=l
I
(C'(x))¢U)(nj(x))dx
(22)
d P.i
where C' is the slope of the curvature with respect to x, j is the number of the spline segment, ~ (j) is an exponent that is used to assign different degrees of importance to the slope of the curvature of different spline segments, and the Hi(x) are functions of x for each spline segment j used to assign different degrees of importance to the curvature distribution at different locations on the suction surface. For the first design iteration, ~ = 1 and H(x) = 1 are specified. After the performance of the resulting blade has been evaluated with a flow solver, the designer has a very good idea of where the curvature must be increased and decreased, and different values of ¢ and H(x) can be specified for the different spline segments.
295
T Korakianitis and G I Pantazopoulos 0.4
0.2
12
4th
Parametric
"~3rd
10
oo Y
\
8
-0.2
6
-0.4
4
-0.8 ! 0.0
1#
0 0.2
0.4
0.6
0.8
1.0
1.2
X
Figure 3. Comparison of 3rd-order and 4th-order splines and their curvature In the optimization routine for the pressure surface, the designer specifies ranges of values for the y coordinate, the angle tips, and the curvature C(Plv) at point Plp, the x and y coordinates of point P2p, the x and y coordinates of point P3p, and the angle tips. For each combination of the above variables, the parametric splines that pass through the nodes Plp, P2v, P3p and P4p according to the above are computed, and the corresponding curvature distributions from PI~ to P4~ are evaluated. From these curves the ones are chosen that pass three additional checks: (a) the flow area between the suction surface and the pressure surface should be monotonically decreasing for subsonic turbines (it should be monotonically increasing in subsonic compressors), (b) the curvature should either not change sign, change sign only once from PIp to P4v, indicating that the pressure surface must be continuously concave after one inflection point (usually near the leading edge), and (c) the slope of curvature should not change sign more than once from PI~ to P4~. From the spline sets that pass all the above checks, the one that has the minimum value of ACx is chosen. ACx is evaluated by an expression that is similar to Equation 22. A set of 4th-order splines that passes all the above checks, including the minimization of ACx, is different from a set of 3rd-order splines that passes through the same points. This is shown in Figure 3, in which the curvature of the 3rd-order splines indicates that the surface Math-number distribution of the corresponding 3rd-order cascade has a sharp 'dip' at x = 0.402, and a 'bulge' at x ~ 0.50.
SAMPLE BLADES The sample cases presented in this paper have been designed for a relative exit Mach number of 0.80, and they have been analysed with the use of the steady-flow-calculation option of Giles '~a UNSFLO computer program. (The blade-design method presented
296
in this paper can be used to design blades that will subsequently be analysed by any direct- or inversecalculation method.) A separate numerical routine has been developed to select points along the suction and pressure surfaces, clustering the points in regions where the curvature is large. This was done to ensure that the grid generators of the flow solvers had adequate points in regions where the computational grids might be clustered. Blade A has been designed for cq = 0 ~', ~2 = 300° ( = - 6 0 ° ) and CL = 1.00. This corresponds to a moderately loaded cascade for high-efficiency stages (in turbomachinery terms this blade could be used in a 50% reaction stage with a low work coefficient, resulting in a high-efficiency stage that is appropriate for industrial turbines). Blade B has been designed for ~ = 40 °, ~2 = 300° ( = -60°) and C L ~-- 1.00. This corresponds to a high-flow-deflection cascade (e = 100°) of higher loading that is suitable for aircraft applications. The high values of both CL and e classify this cascade as one in which good performance is harder to obtain. Table 1 shows the design parameters and inputs for the four cascades. Figure 4 shows the Mach number find pressure distribution of cascade A1. The surface Mach-number distribution is smooth, but it exhibits an undesirable increase and subsequent decrease in the suction-surface
Table 1. Parameters specifying blades A1, A2, BI and B2 Blade parameter
A1
A2
0.0000 300.0000 1.0000 0.5000 315.0000
0.0000 300.0000 1.(DO0 0.5000 315.0000
0.0000 0.2000 0.1500 0.1000
0.0000 0.2000 0.1500 0.I000
40.000 0.2000 0.0500 0.1000
40.0000 0.2000 0.0500 0.1000
C(PI~) fl~l fist x(Pls) y(Pls) x(P2s) y(P2s) x(P3s) y (P3~) x(P4s) y(P4~)
3.5994 4.8000 295.0000 0.I000 0.0700 0.2856 0.0200 0.5617 - 0.2211 1.0000 - 1.0000
3.5319 12.7050 295.0000 0.1000 0.0699 0.2856 0.0255 0.5772 - 0.2403 1.0000 -1.0000
-2.9980 31.8500 295.0000 0.1000 0.1417 0.4000 0.1832 0.6935 - 0.0389 1.0000 -0.5774
-2.8500 30.8000 294.0000 0.1000 0.1317 0.4000 0.1731 0.7036 - 0.0027 1.0000 -0.5774
ripe 6p
0.0000 0.2000 -0.0500 0.1000
0.0000 0.2000 -0.0500 0.1000
40.0000 0.2000 0.4100 0.I000
40.0000 0.2000 0.4100 0.1000
C(PIp) flpl fl~r x(Plp) y(Plp) x ( P2p ) y(P2p) x(P3p) y(P3p) x(P4p) y(P4p)
1.2900 343.8911 303.4967 0.1000 -0.0277 0.4500 -0.2470 0.7000 -0.5545 1.0000 --1.0000
1.4700 343.5109 303.4967 0.1000 -0.0279 0,4500 -0.2471 0.7000 -0.5546 1.0000 - 1.0000
0.8026 5.9300 304.9900 0.1000 0.0075 0.4000 -0.0018 0.7000 -0.2133 1.0000 -0.5774
0.8026 5.9300 304.9900 0.1000 0:0075 0.4000 --0.0018 0.7000 --0.2133 1.0000 --0.5774
~1 ~2 CL
o/S 2 fl~
XScs YS¢,~ 6s
XScp YScp
B1
B2
40.0000 40.0000 300.0000 300.0000 1.0000 1.0000 0.5000 p 0.5000 330.0000 330.0000
computer-aided design
Improved turbine-blade design techniques using 4th-order parametric-spline segments 1.2
1.1
1.0
1.0
0.8
0.9
0.6
0.8
0.4
0.7
Mach number
0.2
r~sure ~ (nondimensional)
v
0.0 -0.5
0.0
0.5 1.5
1.0
0.5
0.6
X a
2.0
contours indicates that the surface at x ~ 0.1 is at a location that is too high, and the curvature is too high at that location. The Mach-number contours of cascade A1 should be reshaped to be like the Mach-number contours of cascade A2, as shown in Figure 5. The Mach-number surface distribution and contours of cascade A2 are shown in Figure 6. The Mach-number contours also indicate that the stagnation point is too low on the pressure surface at x ~ 0.02. This leads to the next design iteration for cascade A2, in which the initial inputs to the optimization routine for point P1 s, BClsx.x and C ( P l s ) are lower, and the whole pressure surface is also lowered in the cascade passage. With this second design iteration, the performance of cascade A2 is satisfactory. For comparison purposes, the surfaces of a second cascade were derived, using 3rd-order parametric splines that pass through the same points and with the same boundary conditions (except for the 'extra' boundary condition for curvature at P1 ) as for cascade A2. The resulting surfaces were very different from those of cascade A2, with effects that were similar to those shown in Figure 3.
1.5
0.2
,
~
1.0 0.1 Y Y 0.5
0.0
0.0 ~.1
~.1
0.1 X
0.2
0.3
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a
-0.5
-1.0
0.0
--0.5
0.0
0.5
1.0
02t 01 /
1.5
X
b Figure 4. Cascade AI"
( a ) surface Mach-number distribution, ( b ) Mach-number contours of increment 0.05 oo
Mach-number distribution at x ~ 0.10. This is a typical problem in many production geometries, such as those shown in Reference 9. However, because, by construction, the blade surfaces have slope-of-curvature continuity, this is not a sharp spike with a corner (resembling an inverted V). It is different from the sharp spike in the turbine cascade in Reference 1, or the sharp dips and spikes at the nodes of the surface segments in past design efforts with lst- and 2nd-derivative continuity 6-s. Examination of the corresponding Mach number
volume 25 number 5 may 1993
-0.I 4
-0.1
0.0
X
b Figure 5. Cascade geometries and Mach-number contours
of increment 0.05; ( a ) cascade A1, ( b ) cascade A2 [i = o, o = -60, C = 1.0, I = -45, 4th-order.]
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T Korakianitis and G I Pantazopoulos 1.2
1.1
1.2
1.1
1.0
1.0
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0.8
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0.2
--
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Pressure (nondimensional)
1.0
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I
M~h number
0.8 0.7 0.6
nondiP:Tssi:::,)
0.0
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X
X
a
a 2.0
1.o
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~
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1.0
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Y 0.5
-0.5
-o.5
o.o
o.5
1.o
t.5
X
b 0.0
Figure 7. Cascade B1," ( a ) surface Mach-number distribution, ( b ) Mach-number contours of increment 0.05
-0.5
CONCLUSIONS -1.0
4---0.5
0.0
0.5
1.0
1.5
x
b Figure 6. Cascade A2; ( a ) surface Mach-number distribution, ( b ) Mach-number contours of increment 0.05
Similarly, Figures 7 and 8 show the performances of cascades B1 and B2. The (x, y) locations of their surfaces are very similar. Both cascades have acceptable performances. Cascade B1 has a maximum Mach number at x ~ 0.60. Cascade B2 has a slightly larger maximum Mach number at x ~ 0.75, resulting in a sharper deceleration region on the suction surface past the throat region. The more sudden deceleration in that region is more likely to separate.
298
A new blade-design method has been introduced. The method uses 4th-order splines which result in continuousslope-of-curvature airfoils. It can be used to design subsonic or supersonic blades for compressors or turbines, or isolated airfoils, but the discussion in this paper is limited to subsonic-exit turbine blades. The blade shapes are specified by a few points and other geometric parameters on the blade surfaces, and an optimization routine is used to minimize changes in the slope of curvature of the surfaces. This method permits the user to specify the leading edge by two thickness distributions around two independent construction lines, thus avoiding the overspeed regions near the leading edge (because it does not use the usual blending in of the curvatures near the 'leading-e~ge circle'). It is found that the blade performance, judged by the shape of the surface-Mach-number distribution, is very sensitive to changes in the curvature distribution and to small geometry changes of the blade surfaces. The design sequence for two representative blades is presented, and
computer-aided design
Improved turbine-blade design techniques using 4th-order parametric-spline segments 1.2
1.1
1.0
1.0
0.8
0.9 B2
0.6 0.4
0.8
0.7
Mach number
Pressure
0.2
~(nondimensionM) 0.6
0.0 -0.5
0.0
0.5 X
1.0
0.5 1.5
a
1.0
REFERENCES 1 Stow, P 'Blading design for multi-stage hp compressors' in Bladin9 Design for Axial Turbomachines (AGARD Lecture Series 167) (1989) 2 Bry, P F 'Blading design for cooled high-pressure turbines' in Blading Design for Axial Turbomachines (AGARD Lecture Series 167) (1989) 3 Meauze, G 'Overview on blading design methods' in Bladin9 Designfor Axial Turbomachines (AGARD Lecture Series 167) ( 1989 ) 4 Dunham, J 'A parametric method of turbine blade profile design' ASME Paper 74-GT-119 (1974) 5 Prichard, L d 'An eleven parameter axial turbine airfoil geometry model' ASME Paper 85-GT-219 (1985) 6 Korakianitis, T 'A design method for the prediction of unsteady forces on subsonic, axial gas-turbine blades' Doctoral Thesis Massachusetts Institute of Technology, USA (1987)
0.5
7 Korakianitis, T 'A parametric method for direct gas-turbine-blade design' AIAA Paper 87-2171 (1987)
Y
8 Korakianitis, T 'Design of airfoils and cascades of airfoils' AIAA J. Vol 27 No 4 (1989) pp 455-461
0.0
9 Sharma, O P, Pickett, G F and Ni, R H 'Assessment -0.5 -0.5
0.0
0.5 X
1.0
I.'5
b 8. Cascade B2; (a) surface Mach-number distribution, ( b ) Mach-number contours of increment 0.05 Figure
the gradual improvement in their performance through the design sequence is shown in the figures in the paper. This method ensures that the boundary layer, which is usually turbulent along the blade surfaces, is not excited unintentionally. The smooth curvature and Machnumber distributions also enable the designer to provide lower-loss cascades of higher loading for modern applications in airborne transportation. Continuousslope-of-curvature splines should be used wherever the aerodynamic, heat-transfer, hydrodynamic or aesthetic performances, appropriately defined in each case, are critical.
of unsteady flows in turbines' ASME Paper 90-GT-150 (1990) 10 Korakianitis, T and Papagiannidis, P 'Surfacecurvature-distribution effects on turbine-cascade performance' Trans. ASME J. Turbomachinery Vol 115 No 2 (1993)
11 Wilson, D G The Design of High-Efficiency Turbomachinery and Gas Turbines MIT Press (1984) 12 Kacker, S C and Okapuu, U 'A mean-line prediction method for axial-flow turbine efficiency' ASME Paper 81-GT-58 (1981) 13 Ainley, D G and Mathieson, G C R 'A method of performance estimation for axial-flow turbines' R & M 2974 British ARC (1951) 14 Dunham, J and Came, P M 'Improvements to the Ainley-Mathieson method of turbine performance prediction' Trans. ASME J. Eng. Pwr. Ser. A Vol 92 (Jul 1970) pp 252-256 15 Gostelow, J P 'Trailing edge flows over turbomachine
ACKNOWLEDGEMENTS The authors thank Professor Michael B Giles and Rolls Royce for their permission to use UNSFLOto compute the performance of the sample cascades, and the Department of Mechanical Engineering, Washington University, USA, for providing the computational resources for the program. They also thank P Papagiannidis for his assistance in designing cascade A2.
volume 25 number 5 may 1993
blades and the Kutta-Joukowsky condition' ASME Paper 75-GT-94 (1975) 16 Bird, R B, Stewart, W E and Lightfoot, E N Transport Phenomena John Wiley (1960) 17 Sehlichting, H Boundary-Layer Theory (6th Ed.) McGraw-Hill ( 1968 ) 18 Giles, M B 'Calculation of unsteady wake/rotor
interactions' AIAA J. Propulsion & Pwr. Vol 4 No 4 (1988)
299
The paper presents a method for generating blade shapes to be used as inputs to direct- or inverse-blade-design sequences. Isolated airfoils and gas-turbine-blade cascades are designed by a mixture of iterative and direct- and inverse-blade-design methods. In the direct-design method, the designer inputs the geometry of the airfoils, and flow solvers are used to study the resulting performance in terms of Mach numbers or pressure distributions along the airfoil surfaces. In the inverse-design method, the designer specifies as input the desirable performance along the airfoil surfaces, and flow solvers are used to compute the airfoil geometry that will provide the performance. The design iterations are enormously reduced if the initial blade shape has performance characteristics that are near the desirable ones. The method can be used to generate airfoils for compressors and turbines, or isolated airfoils, but the discussion in the paper is limited to subsonic-exit axial-turbine blade rows. The desirable performance characteristics are presented, and ways to specify the input aiming for such characteristics (minimizing design iterations) are discussed. It is shown that continuous curvature and continuous slope of curvature are necessary conditions for improved blade designs. A set of parametric curves that satisfy the above conditions is developed, and used to specify the geometry of turbine cascades. Thickness distributions satisfying the same conditions are used near the leading edge of the cascades. The performance characteristics of two turbine cascades generated by this method are included. The shape and distribution of curvature of the suction and pressure surfaces of the airfoil Department of Mechanical Engineering, Washington University, Campus Box 1185, One Brookings Drive, St Louis, MO 63130, USA Paper received: 16 April 1991. Revised:25 September 1992
volume 25 number 5 may 1 9 9 3
are particularly important for the aerodynamic and heat-transfer performance. The 4th-order (continuousslope-of-curvature) parametric splines should be used wherever separation in fluid streams at the walls would be detrimental to performance, and they have additional applications in smooth-surface-generation activities such as automobile-surface design. eontinuous curvatureslopes, 4th-ordersplines, airfoils, continuousthird derivatives
Axial gas-turbine blades are 3D objects operating in a complex flow field. Owing to its complexity, the problem is usually reduced to a series of radially 'stacked' 2D problems. Blade design is partly a science and partly an art. The performance of the blades is expressed in the form of some property distribution (such as pressure and velocity) on the blade surface and in the cascade passage. To minimize radial gradients in enthalpy, pressure and entropy, a condition which is called radial equilibrium, the inlet and outlet flow angles of the cascade vary along the span of the blades (from hub to tip). The centrifugal forces experienced at operating speeds impose constraints on the locus of the center of gravity of the 2D sections. The leading and trailing edges of the blade sections must blend smoothly in the radial direction. 3D turbine blades are generated by 'stacking' 2D designs, using empirical and analytic rules for the above considerations. Compromises in performance must be made to accommodate these 3D constraints. Blade cascades can be designed by direct or inverse methods. Various investigators use different definitions for the following terms: analysis and design modes t , optimization and design methods z, and direct, inverse, semiinverse, full-inverse and full-optimization methods 3.
0010-4485/93/050289-11 © 1993 Butterworth-Heinemann Ltd
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T Korakianitis and G I Pantazopoulos
For the purposes of this paper, direct- and inverse-bladedesign methods are defined as follows. In the direct method, the designer inputs the geometry of the blade, and the output is the performance of the airfoil. The blade geometry is modified until a desirable performance is obtained. Examples of direct methods have been presented by Dunham 4, Prichard 5 and Korakianitis 6-8. In the inverse method, the designer usually starts from some initial blade shape and performance, and inputs the desired modifications to the performance. The output is a new shape and its performance, which is as close to the desired (input) performance as is permitted by the equations modelling the flow. Both methods are iterative, and are based on the assumption of steady-inflow and steady-outflow conditions (although the flow around the blades during engine operation is inherently unsteady). Both methods have relative advantages and disadvantages. The direct method is laborious, but the designer has direct control of the various geometric parameters, and infeasible blade shapes are excluded before they are analysed. Even if inverse-design programs are available, the inverse method should start from a blade shape that is close to an acceptable design, since this enormously reduces the number of inverse-design iterations. The inverse method is less laborious, but many (although not all) inverse-design programs must start from a blade geometry that is generated by a direct method. The designer has less control of the blade shape and the resulting performance. If some geometric parameters (such as stagger angle) need to be changed, the process must be started from the beginning (direct method) again. The final design is usually obtained by a judicious combination of both methods, although, if an inverse method is not available, the direct method can be used exclusively. The purpose of this paper is to provide a design tool for generating blade shapes for either method. In particular, the resulting blade shapes by 9eometric construction have inherently good performance, and, after the first design iteration, the required modifications for good performance are immediately obvious. The airfoil is specified by a series of geometric parameters. The blades presented in this paper have been analysed by direct methods exclusively. The geometry, performance and design iterations for two sample cases are included for illustration.
BACKGROUND Demands for improved performance require sophisticated methods for blade design, because the performance of the airfoils becomes critical. In turbine cascades, the flow expands: the pressure at the outlet is lower than the pressure at the inlet. In subsonic cases, from the cascade inlet to the outlet, the passage velocity is increased, and the area available to the flow should continuously decrease to the minimum area corresponding to the circle of diameter o at the throat (see Figure 1 ). Traditionally, turbine cascades have been designed by manipulating the shape of the passage between successive airfoils, and then
290
b
S
(
Z
Figure 1. Geometry of typical turbine cas(ade closing the shape at the leading edge by specifying a leading-edge circle, ellipse, or other appropriate shape. The flow accelerates around both sides of the cascade. The curvature of the streamlines is higher on the convex (suction) side, and lower on the concave (pressure) side, resulting in higher velocity and reduced pressure on the suction side, and lower velocity and increased pressure on the pressure side. These are shown in the performance figures (Mach numbers and pressure distributions ) of the sample turbine cascades discussed below. Although it is desired that the surface velocity and pressure distributions remain smooth, this is difficult to achieve, as shown in many of the examples from production turbines published by Sharma, Pickett and Ni 9, Stow! includes a turbine example with a sharp undesirable spike in the surface Mach number near the leading edge (this is due to a slope-of-curvature discontinuity). This spike was removed in a later design by modifying the blade surface with an inverse-design method. Korakianitis and Papagiannidis ~° have shown that slope-of-curvature discontinuities on the blade surfaces affect the local surface Math number and the pressure distributions, and the thickness of the boundary layer, and result in cascades with thicker boundary layers in the trailing edges, thus reducing efficiency. In compressor cascades, the flow diffuses" the pressure at the outlet is higher than the pressure at the inlet. In subsonic cases, from the cascade inlet to the outlet, the passage velocity is decreased, and the area available to the flow should continuously increase to a maximum area corresponding to the trailing edge of one blade and the pressure surface of the next. Traditionally, compressor cascades have been designed by designing individual airfoils in which thickness distributions are added to either side of a circular, parabolic, cubic or other-shaped camber line.
computer-aided design
Improved turbine-blade design techniques using 4th-order parametric-spline segments
In recent years, the performances of both types of cascade have been improved by the use of inverse-design techniques, which are otherwise known as prescri~dvelocity-distribution (PVD) techniques. Because the pressure is decreased along the turbine stages, any separation that may occur on the turbine cascades is likely to reattach, and the flow will continue its normal path from the turbine inlet to the outlet. In compressors, the pressure is increased along the stages, and any separation that may occur on compressor cascades may not reattach. This leads to compressor stall, and surge (a dangerous phenomenon characterized by the rapid periodic succession of flow from the compressor inlet to the outlet followed by flow from the high-pressure outlet to the low-pressure inlet). Bad turbine designs will still work, while bad compressor designs will not work. Compressors are harder to design than turbines. However, more attention has been given to compressor design, because of the deleterious effects of bad compressor performance, to the extent that many compressors are more efficient than the 'corresponding' turbines (for similar mass-flow rates and pressure ratios). This is because insufficient attention has been paid to turbine-blade design. The remainder of this paper concentrates on turbine cascades, although the method developed for turbine cascades is directly applicable to compressor cascades and isolated airfoils. Direct-blade-design methods in which the blade surfaces were designed with analytic polynomials of the form y = f(x) have been presented in previous work 6-s. These did not ensure slope-of-curvature discontinuity at the knots between the polynomials, and the throat diameter was not an input design parameter, but rather an output of the direct-design iterations. This paper presents a generalized blade-design method with parametric polynomials (y = f(s), x = f(s)) of order 4 which guarantee a continuous slope of curvature. In addition, the throat diameter is an input to the design process. One of the nodes between the splines of the suction side by construction specifies that the minimum distance between the pressure-side trailing-edge point and the suction surface of the previous blade is equal to the user-specified throat diameter o.
TURBINE-CASCADE GEOMETRY The blade-design sequence depends on the application and the global constraints of the engine. Examples of design sequences are described in various texts, such as Wilson's 1~. The designer has many choices. For the purposes of this paper, important geometric features are identified in the axial-turbine-cascade diagram shown in Figure 1. The flow is from left to right. One of the early design choices that must be made is the type of velocity diagram. It is assumed that the designer has chosen the blade-row velocity diagram and its parameters, such as the inlet flow angle ~ , the outlet flow angle ot2, and the exit Mach number M2 in the absolute frame for stators and in the relative frame for rotors. Four additional choices must be made: a specification for the
volume 25 number 5 may 1993
nondimensional tangential spacing S/b between the blades (where S is the tangential spacing of the blades, and b is the axial chord length (nondimensionally, b = 1 )), a specification for the stagger angle of the cascade 2, the inclination of the blades with respect to the axial direction x, the nondimensional throat diameter o/b, and the trailing-edge thickness. The stagger angle ). is defined as the angle between the axial direction and the line joining the leading edge of the cascade with the trailing-edge point on the concave (pressure) surface. Some guidelines for low-speed turbine blades have been published by Kacker and Okapuu ~2, although the designer has considerable flexibility in choosing 2. The value of ). also dictates whether the cascade is front- or aft-loaded, and different philosophies and guidelines are used by different manufacturers. The spacing between the blades (which is directly linked to the number of blades in the blade row) is a function of the tangential lift coefficient CL, defined by CL =
tangential aerodynamic force tangential blade area x outlet dynamic head
(1) This expression can be manipulated in a number of ways (for compressible flow, for incompressible flow, accounting for variations in axial-flow velocity etc. ). The incompressible-flow derivation ~ reduces CL to CL = 2 S cos2 at2 (tan =1 -- tan 0(2) b
(2)
The actual value of CL depends on the number of blades in the cascade, which dictates S. Typically, 0.8 < CL < 1.2; earlier designs had lower values of CL, and, in modern cascades, C L has been gradually increasing. It is difficult to design high-efficiency cascades that have both high values of CL and high values of the total flow deflection e in the cascade, where e = ~1 + ( 3600 - 0t2). The throat diameter is an extremely important design input, because it dictates the mass flow that can be passed through the cascade, and hence the work that can be delivered by the turbine. A good first approximation is o / S -~- cos-1 ~2. Experimental figures linking the throat diameter 0 to the blade spacing S, the throat Mach number M2, and the curvature of the convex (suction) blade surface near the trailing edge have been published by Ainley and Mathieson 13, and they are used extensively in the early stages of blade designs. The industry has for years relied on empirical relationships based on the Ainley-Mathieson data on improvements to it (e.g. by Dunham and Came 14) for the prediction of the outlet flow angle ~2 as a function of the exit Mach number M2, the spacing S and the throat diameter o. These empirical data have been simplified by others for design applications (for example by Wilson ~l and Dunham4). This is sufficient for preliminary design. Today's computer programs are more accurate, and they compute the outlet flow angle for a given geometry. Gostelow :5 has explained how the result depends not only on the geometry, but also to some extent on the exact location
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of the point at which the Kutta condition is applied at the trailing edge. The trailing-edge thickness should be as small as manufacturing considerations allow (to minimize the wake incident on the next blade row), but it is also affected by geometric constraints imposed by the cooling slots in cooled blades. In the sample cascade geometries shown below, and in Figure 1, the trailing edge is a point, because in this paper inviscid flow solvers have been used to analyse the blade performance. However, the method presented enables the designer to specify different trailing-edge points on the suction and pressure surfaces. In subsonic cascades, the area available to the flow from the inlet to the outlet should be continuously decreasing to ensure that the flow accelerates from the inlet to the throat. In successful designs, the velocity of the fluid adjacent to the concave (pressure) surface accelerates from the stagnation point near the leading edge to the trailing edge. The same is not possible for the velocity of the fluid adjacent to the convex (suction) surface, because the flow is 'unguided' on the suction surface downstream of the throat. In successful designs, the velocity of the fluid adjacent to the convex (suction) surface accelerates from the stagnation point near the leading edge to some point near the throat, and then it decelerates. This region on the suction surface from the throat to the trailing edge is called the region of'unguided diffusion'. If the flow decelerates too much, the boundary layer separates, the blade wake is thick, and the resulting cascade has a poor performance (low efficiency). Therefore, for high performance, the velocity should continuously and smoothly increase from the stagnation point to the trailing edge along the pressure surface. On the suction surface, the velocity should continuously and smoothly increase from the stagnation point to the point of maximum velocity that occurs near the throat. After the point of maximum velocity, designers strive for minimum and smooth diffusion from the point of maximum velocity to the trailing edge. The Reynolds number of typical production turbine blades is of the order of 5 x 105 based on the chord. The flow will probably become turbulent at some point along the surfaces. The smooth velocity distribution (and therefore the smooth pressure distribution) condition ensures that the boundary layer is not energized into turbulence unintentionally. (This can be done intentionally, by designing a curvature discontinuity at the point where the flow should become turbulent for performance considerations. )
THEORET~ CURVATURE
DEPENOENCE ON
Typically, the flow in turbomachinery cascades is analysed by considering Cartesian coordinates (x, y, z) or coordinate transformations in Cartesian systems of coordinates. However, the flow needs to curve around the blades. The dependence of flow on curvature is seen in the compressible-flow Navier-Stokes equations in the
292
limiting case of cylindrical coordinates (r, 0, z ). Since 2D solutions and geometries are 'stacked' radially, the equations of conservation of mass and momenta in polar coordinates (r, 0) are shown below (taken from the text by Bird et a/.16): dp ld 1 d -- + (prvr) + (pro) = 0 dt r~r r~
{ dVr u dVr VOOVr P ~ , ~ + "Or + r do _
dp
Or
V2I r/ + POt
(4)
r 2 Or (r:~,o) + -r d O ] + pod
(5)
d (r%,) +
Or
{dVo #vo vodvo p ~ - - ~ + v, d~ + - - dO lop
r dO
v,v.'~
(1 d
r dO
(3)
ldToo /
where p is the density, v is the velocity component, t is the time, # is the body force, p is the static pressure, and, for Newtonian fluids, the stress components z,, Too and T,0 are related to the dynamic viscosity/~ by the following equations :
r-~(v'*) 1 2d v'2 F [ 1 dvo
~rO = TOt
_ 2
= --# r ~
4- r dO3
( V - v ) = - I d ( r v , ) + 1 -dv- o
r~
r d0
(6)
where v is the velocity vector. A similar dependence of incompressible flow on curvature is also shown in most fluid-dynamics texts in cylindrical coordinates, and Schlichtingl 7 (p 112 ) attributes a set of similar equations for incompressible flow in curvilinear coordinates (x, r, y) to W Tollmien. The 1/r and 1/r 2 terms in all of these equations suggest a strong dependence of the local velocity on curvature. Once the flow passage has been manipulated until the turbine cascade looks like a turbine cascade, the performance depends on the behaviour of the boundary layer, where Equations 6 also indicate that the boundary layer is affected by the local radius. Since smooth velocity distributions along the blade surfaces are required (a continuous slope of velocity with respect to the surface length), the above equations require smooth curvature distributions along the blade surfaces (a continuous slope of curvature). A continuous slope of curvature requires continuous third derivatives, as shown in the following two equations for curvature and slope of curvature for y = f(x), y ' = d f ( x ) / d x ,
computer-aided design
Improved turbine-blade design techniques using 4th-order parametric-spline segments y" = d 2 f ( x ) / d x 2 and y" = d a f ( x ) / d x 3 :
C -
1 r
-
y"
[1
(7)
+ y,213/2
C' = y ' [ 1 + y,2] _ 3y,y,2 [1 + y , 2 ] s / 2
(8)
where C is the curvature and r is the local radius of curvature. In previous work 6-s, it was shown that turbine blades designed with polynomial functions with continuous first and second derivatives at the nodes resulted in surfaces of continuous curvature but discontinuous slope of curvature, which resulted in similar and corresponding discontinuities in the slopes of the velocity distributions. 3rd-derivative continuity is required to obtain smooth velocity distributions for high-efficiency cascades. Cascades designed with 3rd-derivative continuity result in thinner boundary-layer thicknesses at the trailing edge, and therefore have higher efficiencies ~°. Most parametric splines (e.g. cubic splines, B-splines, Brzier splines) have continuous first and second derivatives, and they result in surfaces with continuous curvatures but discontinuous slopes of curvature. In the following, a set of 4th-order parametric splines with continuous point, first, second and third derivatives at the nodes (and along the surfaces) are derived, and used to design a new set of turbine blades.
4TH-ORDER PARAMETRIC SPLINES The nodes of the parametric splines along the suction and pressure surfaces are illustrated in Figure 2. On the suction surface, the nodes are points {PI,, P2,, P3s, P4,}, and, on the pressure surface, the nodes are points {Pip, P2p, P3p, P4p}, although more nodes could be used for
each surface. For general applications, the analysis for 4th-order parametric splines for n number of nodes { P1, P2 . . . . . P, } is derived in the following. These nodes correspond to (n - 1 ) spline segments on either surface, where the parameter is s. Let (x, y~) be the Cartesian coordinates of any point P~ on the surface, and (Xj(s), Yj(s)) be the algebraic description of any spline segment j. The splines take the form x x S 2 + aj3s x 3 Xj(s) = aio + a]ls + a j2
x
+ aj,,s
4
Yj(s) = ago + a~ls + a~2 s2 + a~3 s3 + a~4s 4
(9)
where 0 ~ s ~< 1 in each of the (n - 1 ) segments j, and a are the spline coefficients for X(s) and Y(s). In the following, the derivation that is applicable to X(s) is shown; the derivation for Y(s) is exactly analogous. If fx~ and sx~ are the first and second derivatives of X(s) with respect to s at each of the n nodes i, Xi(O) = xi
Xi(1) =
Xi+ 1
dX~ -~s s=o = fxi dX_~i = fxi+ 1 I ds Is=l d2Xi -~$2
s=O = SX i
gives the solution for the spline coefficients of the spline segment i in terms of the first and second derivatives at the nodes : x
aio = X i
a';1 = fxi a~2 = sxi/2
a~'3 = 4(xi+1 - xi) - fxi+ l - 3fxi - sxi
(o,s)~/~
air = 3 ( x i -
(10)
The solution for the first and second derivatives at the nodes is found by requiring 2nd- and 3rd-derivative continuity :
P2v
d2Xi P2o
d2Xi+l
-fi~-s~,=,= daXi
(x.o,Y,o)
,~, (x,,,Y,o) (0, O)
x i + l ) + fxi+l + 2fxi + s x i / 2
P4, = P4 v
ds ~ daXi + 1
,=1=1 ds3 L=o which, for i = 2, 3 . . . . .
Plp
~=o
n -
1,
leads to
6fxi-1 + sxi-1 + 6fxi - sxi = 12(xi - x~-l) 5fxi-1 + sxi_ 1 + 6fxi + sxi + fxi + 1
P3~
= 4(xi+t + x i - 2xi-1)
p4 v Figure
P4o = P4 v
2. P a r a m e t e r s used to specify typical cascade
volume 25 number 5 may 1993
The last two equations can be manipulated into the matrix form shown in Equation 11, from which the values of the first and second derivatives at the nodes can be found. The only remaining unknowns are the boundary conditions shown in Equation 12, which are evaluated
293
T Korakianitis and G I Pantazopoulos below. -1
0
0
0
0
0
0
0
fx
0
1
0
0
0
0
0
0
SX 1
5
1
6
1
1
0
0
0
fx2
6
1
6
0
0
5
0
0
6
-1
0
0
0
0
SX 2
!
6
1
1
0
fx3
1
6
0
0
SX 3
-1
I
1
BCx,1( Py ) BCx.2(Pf) 4(x 3 + x 2
--
2xl )
I
12(x 2 - x 1 ) i
4(X4 4- X3
--
2x2)
12(x3 - x2) (11)
0
0
5
1
6
1
0
0
6
1
6
0
0
SX n _
0
0
0
0
0
0
1
0
fx,
0
0
0
0
6
1
6
--1
SX~
--1
4(x, + x , - i
fx n -
0
-
2Xn-2)
12(x,_ 1 - x, z)
BC~,a(P2)
I
J
1 2 ( x . - x._l)
appropriate blade-surface angle from Figure 2. On the suction surface, these are flsl and flsr- On the pressure surface, these are flvt and flpr. Similarly, the second derivatives can be expressed as :
dX 1
BClx'l = ~ - s s=o d2X1
BClx,2 = ~
1
~=o
d2y
dX._ 1 BC~'I = T ~=1
(12)
ds z
- A cos fl
dZx where BC is the boundary condition, and a subscript 1 indicates a left-side boundary condition, and a subscript r indicates a right-side boundary condition. The lst-derivative boundary conditions at the right and left sides of each spline set are related to the angles of the blades at the leading and trailing edge fist, fl~, flv~ and tips, as shown in Figure 2. These are in turn related to the flow angles ~1 and ~z, and the effect that the designer expects that the blade-surface angles will have on the flow angles. At the trailing edge, the difference between fist and flpr should be between 7 ° and 15 ° for manufacturing considerations, one being slightly larger and the other slightly smaller than the flow angle ~:. These angles are design inputs in this method (they are outputs in the inverse-design method). Near the leading edge fl,l and flpt are also design inputs, but the designer has a great deal of freedom in specifying these angles and the location of the leftmost points of the spline sets on each surface. For the lst-derivative boundary conditions tan fl - dy _ dy/ds _ sin fl dx dx/ds cos fl
A sin fl A cos fl
(13)
Thus an obvious choice for the first derivatives of the splines in X(s) and Y(s) at the first and last points of the spline sets of the suction or the pressure surfaces is
Asinfl
ds 2
(15)
Equations 14 and 15 give
dsZ ,] \ ds /
\ ds2 J \ ds ]
(16)
The curvature and the second derivatives are also related as follows :
c[1
~dx =
\~ds/
(d2y/ds 2 ) (dx/ds) - (d2x/ds 2 ) (dy/ds) (dx/ds) 3
(17)
Equations 16 and 17 are solved for the second derivatives with respect to s:
ds 2 -
d2y
\dsJL\dsJ
c(dX~[(dx']2
ds~ -
\~/L\ds/
+\ds,/
]
(dY~2] x/2
+ \~,/ J
(18)
which finally give the 2nd-derivative boundary conditions at points P1 : BClx'2
=
BCIy'2
=
d2~ ds2
el
= - A Z C ( P 1 ) sin fll
dy BCy'I
-
BC,,,x -
ds dx ds
-
A s i n fl
- A cos fl
(14)
where A is an arbitrary length chosen to be equal to the arc length of the appropriate spline segment, and fl is the
294
dZY
~2s2 el.
= A2C(P1)cosfll
(19)
The above procedure is applied for the splines in x and y for the suction and the pressure surfaces. Point P3s is special, and it is instrumental in the suction-surface design. The locus of P3 s is a circle, with the center being
computer-aided design
Improved turbine-blade design techniques using 4th-order parametric-spline segments
the trailing edge of the top blade (P4p), and the radius being o. Out of all the points that lie on the circle, only the ones that correspond to splines with slopes perpendicular to line P4pP3, at point P3, are chosen for further consideration.
LEADING-EDGE GEOMETRY The leading-edge geometry requires the specification of two lines, one for the suction and one for the pressure surface, joining the origin (0, 0 ) with points P 1s and P 1p, respectively. To maintain slope-of-curvature continuity at points P1, these two lines must join the splines of the blade surfaces with continuous first, second and third derivatives. First, two construction lines are specified. Parabolas have been used for simplicity (see Figure 2). The suction-surface construction parabola passes through points (0, 0) and (X,c, Y~¢) with a slope of fl~c at the origin. The pressure-surface construction parabola passes through points (0, 0) and (Xp¢, Yp¢)with a slope of flp¢ at the origin. Each parabolic construction line defines a parameter 09 and a corresponding baseline P(~o) as follows : x = a~
(20)
P(og) = eo + e,a~ + e2o~2 where o~ is the thickness-distribution parameter, and the constant coefficients of the construction lines, eo, el and e2, are evaluated from the boundary conditions of the parabolic construction lines. Next, the minimum distance of point P1, and the minimum distances of three additional points near P1, on the suction-surface spline P1,P2, from the suction-side parabola are evaluated. These are used to evaluate the distance and (numerically) the first, second and third derivatives of the distance of the spline at point PI, from the suction-side construction line. Similarly, the minimum distance of point Pip and the minimum distances of three additional points near Pip on the pressure-surface spline PlpP2p from the pressure-side parabola are evaluated. These are used to evaluate the distance and (numerically) the first, second and third derivatives of the distance of the spline at point Pip from the pressure-side construction line. Then, a thickness distribution is added perpendicularly above the suction-side construction line. A second thickness distribution is added perpendicularly below the pressure-side construction line. The thickness distributions are of the form Ts(~ ) =
~sgO 1/2 -1- Csl(D -~- Cs2(D 2 q- C,3(.D 3 -[- Cs4(D 4"
(21)
Tp(to) = 6pO~~/2 + cpltn + Cp2O~2 + Cp3tOa + Cp4~4 where the c are the thickness-distribution coefficients. The first, second and third derivatives of T(co) with respect to co are analytic expressions. The values of 5, and 6p, which are root-x coefficients for T, are specified by the designer. The only unknowns are the four coefficients cl, c2, c3 and c4. These are evaluated from
volume 25 number 5 may 1993
the four values of the distance and the first, second and third derivatives of the distance of the suction- and pressure-surface splines at points P1, and Plp from their respective construction lines. The above procedure ensures curvature and slope-ofcurvature continuity at points PI. The value of the parameters 5 can be used to specify thicker or thinner leading edges, which affect the design-point and off-design-point operating performance of the cascade, and are the subject of a future investigation. The origin (0, 0) is itself a singular point. To achieve lst-derivative continuity at this point, the angle of the parabolic construction lines should be equal for the suction and pressure surfaces (and approximately equal to the inflow angle ~1 ).
SEARCH ROUTINE The design choices of ct1, ~2, CL, 2, 0 and the trailing-edge thickness specify the location of the trailing edge and the circle of loci for point P3,. First, the suction surface is designed, and then the pressure surface. In the optimization routine for the suction surface, the designer specifies ranges of values for the y coordinate, the angle fl,i, and the curvature C(P1,) at point PI,, the x and y coordinates of points P2,, and the angle fl,,. For each combination of the above variables and location of P%, the parametric splines that pass through the nodes PI,, P2s, P3s and P4~ according to the above are computed, choosing the spline sets in which line P4pP% is perpendicular to the spline set at points P%, and eliminating all others. Each of the remaining spline sets results in a curvature distribution from Pls to P4,. From these curves, the ones that are chosen are those that pass two additional checks: (a) the curvature must have the same sign (positive) from PI~ to P4,, indicating that the suction surface must be continuously convex, and (b) the slope of the curvature must not change sign more than once from PIp to P4v. From the spline sets that pass all the above checks, the one that has the minimum value of ACx is chosen. ACx is numerically evaluated as follows: j=n-
ACx =
1(= 3) f P j + 1
~-' j=l
I
(C'(x))¢U)(nj(x))dx
(22)
d P.i
where C' is the slope of the curvature with respect to x, j is the number of the spline segment, ~ (j) is an exponent that is used to assign different degrees of importance to the slope of the curvature of different spline segments, and the Hi(x) are functions of x for each spline segment j used to assign different degrees of importance to the curvature distribution at different locations on the suction surface. For the first design iteration, ~ = 1 and H(x) = 1 are specified. After the performance of the resulting blade has been evaluated with a flow solver, the designer has a very good idea of where the curvature must be increased and decreased, and different values of ¢ and H(x) can be specified for the different spline segments.
295
T Korakianitis and G I Pantazopoulos 0.4
0.2
12
4th
Parametric
"~3rd
10
oo Y
\
8
-0.2
6
-0.4
4
-0.8 ! 0.0
1#
0 0.2
0.4
0.6
0.8
1.0
1.2
X
Figure 3. Comparison of 3rd-order and 4th-order splines and their curvature In the optimization routine for the pressure surface, the designer specifies ranges of values for the y coordinate, the angle tips, and the curvature C(Plv) at point Plp, the x and y coordinates of point P2p, the x and y coordinates of point P3p, and the angle tips. For each combination of the above variables, the parametric splines that pass through the nodes Plp, P2v, P3p and P4p according to the above are computed, and the corresponding curvature distributions from PI~ to P4~ are evaluated. From these curves the ones are chosen that pass three additional checks: (a) the flow area between the suction surface and the pressure surface should be monotonically decreasing for subsonic turbines (it should be monotonically increasing in subsonic compressors), (b) the curvature should either not change sign, change sign only once from PIp to P4v, indicating that the pressure surface must be continuously concave after one inflection point (usually near the leading edge), and (c) the slope of curvature should not change sign more than once from PI~ to P4~. From the spline sets that pass all the above checks, the one that has the minimum value of ACx is chosen. ACx is evaluated by an expression that is similar to Equation 22. A set of 4th-order splines that passes all the above checks, including the minimization of ACx, is different from a set of 3rd-order splines that passes through the same points. This is shown in Figure 3, in which the curvature of the 3rd-order splines indicates that the surface Math-number distribution of the corresponding 3rd-order cascade has a sharp 'dip' at x = 0.402, and a 'bulge' at x ~ 0.50.
SAMPLE BLADES The sample cases presented in this paper have been designed for a relative exit Mach number of 0.80, and they have been analysed with the use of the steady-flow-calculation option of Giles '~a UNSFLO computer program. (The blade-design method presented
296
in this paper can be used to design blades that will subsequently be analysed by any direct- or inversecalculation method.) A separate numerical routine has been developed to select points along the suction and pressure surfaces, clustering the points in regions where the curvature is large. This was done to ensure that the grid generators of the flow solvers had adequate points in regions where the computational grids might be clustered. Blade A has been designed for cq = 0 ~', ~2 = 300° ( = - 6 0 ° ) and CL = 1.00. This corresponds to a moderately loaded cascade for high-efficiency stages (in turbomachinery terms this blade could be used in a 50% reaction stage with a low work coefficient, resulting in a high-efficiency stage that is appropriate for industrial turbines). Blade B has been designed for ~ = 40 °, ~2 = 300° ( = -60°) and C L ~-- 1.00. This corresponds to a high-flow-deflection cascade (e = 100°) of higher loading that is suitable for aircraft applications. The high values of both CL and e classify this cascade as one in which good performance is harder to obtain. Table 1 shows the design parameters and inputs for the four cascades. Figure 4 shows the Mach number find pressure distribution of cascade A1. The surface Mach-number distribution is smooth, but it exhibits an undesirable increase and subsequent decrease in the suction-surface
Table 1. Parameters specifying blades A1, A2, BI and B2 Blade parameter
A1
A2
0.0000 300.0000 1.0000 0.5000 315.0000
0.0000 300.0000 1.(DO0 0.5000 315.0000
0.0000 0.2000 0.1500 0.1000
0.0000 0.2000 0.1500 0.I000
40.000 0.2000 0.0500 0.1000
40.0000 0.2000 0.0500 0.1000
C(PI~) fl~l fist x(Pls) y(Pls) x(P2s) y(P2s) x(P3s) y (P3~) x(P4s) y(P4~)
3.5994 4.8000 295.0000 0.I000 0.0700 0.2856 0.0200 0.5617 - 0.2211 1.0000 - 1.0000
3.5319 12.7050 295.0000 0.1000 0.0699 0.2856 0.0255 0.5772 - 0.2403 1.0000 -1.0000
-2.9980 31.8500 295.0000 0.1000 0.1417 0.4000 0.1832 0.6935 - 0.0389 1.0000 -0.5774
-2.8500 30.8000 294.0000 0.1000 0.1317 0.4000 0.1731 0.7036 - 0.0027 1.0000 -0.5774
ripe 6p
0.0000 0.2000 -0.0500 0.1000
0.0000 0.2000 -0.0500 0.1000
40.0000 0.2000 0.4100 0.I000
40.0000 0.2000 0.4100 0.1000
C(PIp) flpl fl~r x(Plp) y(Plp) x ( P2p ) y(P2p) x(P3p) y(P3p) x(P4p) y(P4p)
1.2900 343.8911 303.4967 0.1000 -0.0277 0.4500 -0.2470 0.7000 -0.5545 1.0000 --1.0000
1.4700 343.5109 303.4967 0.1000 -0.0279 0,4500 -0.2471 0.7000 -0.5546 1.0000 - 1.0000
0.8026 5.9300 304.9900 0.1000 0.0075 0.4000 -0.0018 0.7000 -0.2133 1.0000 -0.5774
0.8026 5.9300 304.9900 0.1000 0:0075 0.4000 --0.0018 0.7000 --0.2133 1.0000 --0.5774
~1 ~2 CL
o/S 2 fl~
XScs YS¢,~ 6s
XScp YScp
B1
B2
40.0000 40.0000 300.0000 300.0000 1.0000 1.0000 0.5000 p 0.5000 330.0000 330.0000
computer-aided design
Improved turbine-blade design techniques using 4th-order parametric-spline segments 1.2
1.1
1.0
1.0
0.8
0.9
0.6
0.8
0.4
0.7
Mach number
0.2
r~sure ~ (nondimensional)
v
0.0 -0.5
0.0
0.5 1.5
1.0
0.5
0.6
X a
2.0
contours indicates that the surface at x ~ 0.1 is at a location that is too high, and the curvature is too high at that location. The Mach-number contours of cascade A1 should be reshaped to be like the Mach-number contours of cascade A2, as shown in Figure 5. The Mach-number surface distribution and contours of cascade A2 are shown in Figure 6. The Mach-number contours also indicate that the stagnation point is too low on the pressure surface at x ~ 0.02. This leads to the next design iteration for cascade A2, in which the initial inputs to the optimization routine for point P1 s, BClsx.x and C ( P l s ) are lower, and the whole pressure surface is also lowered in the cascade passage. With this second design iteration, the performance of cascade A2 is satisfactory. For comparison purposes, the surfaces of a second cascade were derived, using 3rd-order parametric splines that pass through the same points and with the same boundary conditions (except for the 'extra' boundary condition for curvature at P1 ) as for cascade A2. The resulting surfaces were very different from those of cascade A2, with effects that were similar to those shown in Figure 3.
1.5
0.2
,
~
1.0 0.1 Y Y 0.5
0.0
0.0 ~.1
~.1
0.1 X
0.2
0.3
0.1
0.2
0.3
a
-0.5
-1.0
0.0
--0.5
0.0
0.5
1.0
02t 01 /
1.5
X
b Figure 4. Cascade AI"
( a ) surface Mach-number distribution, ( b ) Mach-number contours of increment 0.05 oo
Mach-number distribution at x ~ 0.10. This is a typical problem in many production geometries, such as those shown in Reference 9. However, because, by construction, the blade surfaces have slope-of-curvature continuity, this is not a sharp spike with a corner (resembling an inverted V). It is different from the sharp spike in the turbine cascade in Reference 1, or the sharp dips and spikes at the nodes of the surface segments in past design efforts with lst- and 2nd-derivative continuity 6-s. Examination of the corresponding Mach number
volume 25 number 5 may 1993
-0.I 4
-0.1
0.0
X
b Figure 5. Cascade geometries and Mach-number contours
of increment 0.05; ( a ) cascade A1, ( b ) cascade A2 [i = o, o = -60, C = 1.0, I = -45, 4th-order.]
297
T Korakianitis and G I Pantazopoulos 1.2
1.1
1.2
1.1
1.0
1.0
1.0
1.0
0.8
0.9
0.8
0.9
0.6
0.8
0.6
0.7
0.4
0.6
0.2
0.5
0.0 -0.5
0.4
Mach number
0.2
--
0.0 -0.5
0.0
0.5
Pressure (nondimensional)
1.0
1.5
I
M~h number
0.8 0.7 0.6
nondiP:Tssi:::,)
0.0
0.5
1.0
1.5
0.5
X
X
a
a 2.0
1.o
1.5
~
0.5 Y
1.0
0.0
Y 0.5
-0.5
-o.5
o.o
o.5
1.o
t.5
X
b 0.0
Figure 7. Cascade B1," ( a ) surface Mach-number distribution, ( b ) Mach-number contours of increment 0.05
-0.5
CONCLUSIONS -1.0
4---0.5
0.0
0.5
1.0
1.5
x
b Figure 6. Cascade A2; ( a ) surface Mach-number distribution, ( b ) Mach-number contours of increment 0.05
Similarly, Figures 7 and 8 show the performances of cascades B1 and B2. The (x, y) locations of their surfaces are very similar. Both cascades have acceptable performances. Cascade B1 has a maximum Mach number at x ~ 0.60. Cascade B2 has a slightly larger maximum Mach number at x ~ 0.75, resulting in a sharper deceleration region on the suction surface past the throat region. The more sudden deceleration in that region is more likely to separate.
298
A new blade-design method has been introduced. The method uses 4th-order splines which result in continuousslope-of-curvature airfoils. It can be used to design subsonic or supersonic blades for compressors or turbines, or isolated airfoils, but the discussion in this paper is limited to subsonic-exit turbine blades. The blade shapes are specified by a few points and other geometric parameters on the blade surfaces, and an optimization routine is used to minimize changes in the slope of curvature of the surfaces. This method permits the user to specify the leading edge by two thickness distributions around two independent construction lines, thus avoiding the overspeed regions near the leading edge (because it does not use the usual blending in of the curvatures near the 'leading-e~ge circle'). It is found that the blade performance, judged by the shape of the surface-Mach-number distribution, is very sensitive to changes in the curvature distribution and to small geometry changes of the blade surfaces. The design sequence for two representative blades is presented, and
computer-aided design
Improved turbine-blade design techniques using 4th-order parametric-spline segments 1.2
1.1
1.0
1.0
0.8
0.9 B2
0.6 0.4
0.8
0.7
Mach number
Pressure
0.2
~(nondimensionM) 0.6
0.0 -0.5
0.0
0.5 X
1.0
0.5 1.5
a
1.0
REFERENCES 1 Stow, P 'Blading design for multi-stage hp compressors' in Bladin9 Design for Axial Turbomachines (AGARD Lecture Series 167) (1989) 2 Bry, P F 'Blading design for cooled high-pressure turbines' in Blading Design for Axial Turbomachines (AGARD Lecture Series 167) (1989) 3 Meauze, G 'Overview on blading design methods' in Bladin9 Designfor Axial Turbomachines (AGARD Lecture Series 167) ( 1989 ) 4 Dunham, J 'A parametric method of turbine blade profile design' ASME Paper 74-GT-119 (1974) 5 Prichard, L d 'An eleven parameter axial turbine airfoil geometry model' ASME Paper 85-GT-219 (1985) 6 Korakianitis, T 'A design method for the prediction of unsteady forces on subsonic, axial gas-turbine blades' Doctoral Thesis Massachusetts Institute of Technology, USA (1987)
0.5
7 Korakianitis, T 'A parametric method for direct gas-turbine-blade design' AIAA Paper 87-2171 (1987)
Y
8 Korakianitis, T 'Design of airfoils and cascades of airfoils' AIAA J. Vol 27 No 4 (1989) pp 455-461
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9 Sharma, O P, Pickett, G F and Ni, R H 'Assessment -0.5 -0.5
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b 8. Cascade B2; (a) surface Mach-number distribution, ( b ) Mach-number contours of increment 0.05 Figure
the gradual improvement in their performance through the design sequence is shown in the figures in the paper. This method ensures that the boundary layer, which is usually turbulent along the blade surfaces, is not excited unintentionally. The smooth curvature and Machnumber distributions also enable the designer to provide lower-loss cascades of higher loading for modern applications in airborne transportation. Continuousslope-of-curvature splines should be used wherever the aerodynamic, heat-transfer, hydrodynamic or aesthetic performances, appropriately defined in each case, are critical.
of unsteady flows in turbines' ASME Paper 90-GT-150 (1990) 10 Korakianitis, T and Papagiannidis, P 'Surfacecurvature-distribution effects on turbine-cascade performance' Trans. ASME J. Turbomachinery Vol 115 No 2 (1993)
11 Wilson, D G The Design of High-Efficiency Turbomachinery and Gas Turbines MIT Press (1984) 12 Kacker, S C and Okapuu, U 'A mean-line prediction method for axial-flow turbine efficiency' ASME Paper 81-GT-58 (1981) 13 Ainley, D G and Mathieson, G C R 'A method of performance estimation for axial-flow turbines' R & M 2974 British ARC (1951) 14 Dunham, J and Came, P M 'Improvements to the Ainley-Mathieson method of turbine performance prediction' Trans. ASME J. Eng. Pwr. Ser. A Vol 92 (Jul 1970) pp 252-256 15 Gostelow, J P 'Trailing edge flows over turbomachine
ACKNOWLEDGEMENTS The authors thank Professor Michael B Giles and Rolls Royce for their permission to use UNSFLOto compute the performance of the sample cascades, and the Department of Mechanical Engineering, Washington University, USA, for providing the computational resources for the program. They also thank P Papagiannidis for his assistance in designing cascade A2.
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blades and the Kutta-Joukowsky condition' ASME Paper 75-GT-94 (1975) 16 Bird, R B, Stewart, W E and Lightfoot, E N Transport Phenomena John Wiley (1960) 17 Sehlichting, H Boundary-Layer Theory (6th Ed.) McGraw-Hill ( 1968 ) 18 Giles, M B 'Calculation of unsteady wake/rotor
interactions' AIAA J. Propulsion & Pwr. Vol 4 No 4 (1988)
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