BLADE CAMBER SURFACE OPTIMIZATION FOR

006_06/119

BLADE CAMBER SURFACE OPTIMIZATION FOR TURBOMACHINERY DESIGN M. Ferlauto, H. Telib, A. Iollo and L. Zannetti DIASP, Politecnico di Torino, Torino, Italy [email protected], [email protected], [email protected], [email protected]

ABSTRACT The paper analyzes the 3D blade design problem from the fluid dynamics point of view. The design problem is solved imposing either flow variables, i.e. blade forces, distribution either a geometrical parametrization of lean and sweep along the blade span. The related blade geometry is then obtained by solving an inverse problem for the camber surface. Optimal distributions of blade lean and sweep are investigated by means of an adjoint optimization procedure. INTRODUCTION Three-dimensional shaping of the blades of modern gasturbines has become a very complex task. In the past the blade stacking was considered as the problem of extrapolating to three dimensions the results obtained by one or two-dimensional approaches. One of the first concepts introduced for this purpose was the radial equilibrium [1, 2]. The blade is obtained by assuming simple swirl distributions, e.g. the free-vortex law, along the blade span. Simple mechanical concepts, such as center of gravity or trailing edge stacking were also used. Innovative solutions were discovered by designers when advanced blade-to-blade analysis allowed for different stackings, setting blade sections according to optimal incidences. Modern design procedures adopt non-free vortex design methods. The benefits are the higher specific performances that allow the design of more compact gasturbines, with high flow rates and total pressure rises with moderate rotor diameter, blade number and rotor speed. Moreover, the prescription of moderate performance near the hub avoids unfavourably high solidity, stagger and camber angles at the blade root. The prescribed design performances are obtained at reduced rotor speeds, which is an effective strategy of noise reduction. As main drawback, the design results in a characteristic, fully three-dimensional interblade flow with spanwise change of outlet swirl and axial velocity. Fluid particles, traveling on the same entering stream surface, move radially inward and outward when passing the blade near the pressure side or suction side, respectively, and therefore they exit the rotor at different radii, where the axial velocity and swirl conditions are distinct. This effect increases the secondary flows and can be alleviated by introducing different blade sweep and lean design strategies. In literature, blade sweep and lean have been applied to improve stage performance and efficiency [3, 4] and to reduce noise [5, 6]. Motivations of the sweep and lean effectiveness are intuitive. The development and motion of secondary flows is governed by the balance of forces acting on the fluid. While the type of vortex law chosen

produces the radial equilibrium, the positioning in space of the blade elements generates additional axial and radial force components. The use of sweep and lean, within the mechanical constraints, is a straight way to exert a control on the secondary flow evolution [7]. Usual design procedures of blade stacking, leaning and sweeping act on geometric parameters that in literature are treated on an empirical basis and on designer esperience. In the present paper we suggest to use an inverse method of 3D blade designing [9, 10] and to reformulate the problem from a fluid dynamic standpoint. The inverse procedure find the geometry of the blade passing through a given spatial curve, here called the ”reference line” and realizing a given distribution of tangential forces, axial chords and blade thickness. As ”reference line” we adopt the line of the leading edges but other choices (e.g. the trailing edge line, the half-chord locus) are possible. Blade lean and sweep effects are included by an appropriate choice of the geometry of the reference line. For given blade loads, the procedure finds a different blade for each different choice of the reference line. Optimal choice of blade lean and sweep is then reduced in finding the most favorable flowfield that realizes the same main performances and it is obtained through an optimization procedure. The plan of the paper is as follows: in the first sections the mathematical model is explained, then the optimization procedure, based on the adjoint method, is discussed. Finally, some numerical experiments focusing on various aspect of the methodology are proposed. FLOW MODEL The flow deflection through rotors and stators of a turbomachine is the result of the forces that rotor and stator blades exert on the flow. The effect of solid blades can be modeled by volume forces orthogonal to stream surfaces. Let F = F x i +F r ξ +F ϑ η be the volume force, where i, ξ and η are the unit vectors pertinent to the axial, radial and tangential directions in a cylindrical frame of reference (x i, r ξ,ϑ η). The geometry of the blades, represented by 2D manifolds Θ(x, r, ϑ) = 0.

(1)

is found by solving (q −ωr η) · ∇Θ = 0,

(2)

as the blades are to be stream surfaces of the absolute or relative motion for stators and rotors respectively. In the equation above q = u i +w ξ +v η is the flow velocity vector, ω is the angular velocity of rotors. The components of the volume force F x and F r are determined enforcing the blade manifolds to be orthogonal to the volume forces F ×∇Θ = 0.

(3)

which implies Fx = r

Θx ϑ F , Θϑ

Fr = r

Θr ϑ F Θϑ

(4)

We now detail the solution technique of the inverse problem taking as known F ϑ (x, r, ϑ). In a cylindric frame of reference, the compressible Euler equations with volume forces acting on the fluid are ∂A ∂B ∂U + + +Q=0 ∂t ∂x ∂r

(x, r) ∈ Ω

(5)

Ωb1

Ωb 2

Ω b3

Ωb 4

r

x

Figure 1: Schematic representation of a multistage turbomachine in the meridional plane. The areas Ωb are the projection of the blade surfaces. where

U = {ρ, ρu, ρv, ρw, e}T

 ρu      p + ρu2 ρuv A=   ρuw    u(p + e)

          

      , B=

    

and

ρw ρuw ρvw p + ρw2 w(p + e)

       

          

. Q=

      

ρw/r + ρuα ρuw/r − F x + ρu2 α 2ρvw/r − F θ ρ(w2 − v 2 )/r − F r w(p + e)/r − F · q +u(p + e)α

              

(6)

as usual ρ is density, p pressure, e total internal energy per unit volume. The volume force F is defined on the region Ωb ⊂ Ω obtained as the projection of the blade surface onto the meridional plane. The blade blockage is taken into account by the source term containing α=

∂(log Ξ) , ∂x

Ξ = 2πr − T

(7)

Ξ being the free passage per unit radius and T = T (x, r) the sum of the estimated blades thickness, including the boundary layers. System (5) is approximated by a finite volume discretization. The numerical procedure is solved according to the technique described in [8]. In our approach, instead of modifying the blade shape, as classical shape optimization does, we give the force which the blades exert on the flow, and let the geometry accommodate such distribution of forces by solving an inverse problem. In [9, 10, 11] the volume force distribution itself is varied according to an adjoint optimization process, so that the objective functional is maximized. A blade surface changes its shape during the transient to obey the condition of impermeability. Let us express eq. 1 as Θ(x, r, ϑ, t) = ϑ − g(x, r, t) = 0

(8)

so that eqs. 4 become F x = −rgx F ϑ ,

F r = −rgr F ϑ

(9)

Flow particles on Θ(x, r, ϑ, t) = 0 must remain on the manifold for the impermeability condition. It follows that during the transient the Lagrangian derivative of the function Θ(x, r, ϑ, t) has to be null dΘ = Θt + (q − ωrη) · ∇Θ = 0 dt

(10)

that can be written as gt = −ugx − wgr +

v − ωr r

(11)

The above equation is solved coupled to the Euler equations, and it is integrated in time upwinding the spatial derivatives of g, while enforcing the boundary condition g(xref (r), ϑref (r), t) = g(xref (r), ϑref (r), 0)

(12)

where the spatial curve Γref = {xref (r), ϑref (r)} is the geometric parametrization of the reference line. OPTIMIZATION BY ADJOINT METHOD In this section we derive briefly the adjoint technique to minimize a predefined objective functional T . More details on the technique adopted can be found in [9]. A recent review of adjoint methods and other optimization techniques for turbomachinery design can be found in [13]. For instance, we take Z ³ Z ´2 w H(U ) dΩ (13) T = − σx dΩ = u Ωb Ωb where σx is the slope of the grid lines. Intuitively, this functional aims to reduce the flow path distortion. For sake of simplicity we assume that the control is the optimal blade lean, but the blade loads should also be included in the optimization process, as already done in [9]. β(r) = g(xl , r) at the leading edge, while the blade force distribution F θ is kept constant. In general, one could select a generic stacking line as control and simultaneously optimize blade forces, lean and sweep. The minimum of T is constrained by the steady state Euler equations E(F ϑ ) = Ax + B r + Q = 0

(14)

and by the kinematic constraint on the blades G(U (F ϑ )) = ugx + wgr −

v − ωr =0 r

(15)

In order to solve such constrained minimization problem we introduce the Lagrangian function Z Z H(U ) dΩ + tΛ E(U , F ϑ , g) dΩ+ L(U , g, Λ, µ, χ, β) = Ω Ωb Z Z (16) χ[g(xl , r) − β(r)] dΓ + µ G(U , g) dΩ + Ω

Γbi

where Λ = t(λ1 (x, r), λ2 (x, r), λ3 (x, r), λ4 (x, r), λ5 (x, r)), µ = µ(x, r) and χ = χ(r) are Lagrange multipliers. Γbi is the blade inlet contour. The Lagrangian will allow us to treat the maximization problem as an unconstrained problem, so that a stationary point is found when the variation of L with respect to all its arguments, that are now considered independent functions, is 0. Let us compute δL. We have δL = δLU + δLg + δLΛ + δLµ + δLχ + δLβ

(17)

where Z

Z

∂H δ U dΩ + ∂U

δLU = Ωb

Z t

Λ δ E U dΩ +

Ω

µ Ωb

∂G δ U dΩ ∂U

(18)

Z δLΛ = Z

Ω

Z

Ω

δLµ =

t

δ Λ E(U , F ϑ , g) dΩ

(19)

G(U , g)δ µ dΩ

(20)

Z

Z

t

δLg = Ω

Λ δ Qg dΩ +

Z

δLβ = −

µ δGg dΩ + Ω

χ δg dΓ

(21)

Γbi

χ δβ dΓ

(22)

Γbi

Z δLχ =

[g(xl , r) − β(r)] dΓ

(23)

Γbi

In order to have δL = 0, all the single contributions to δL must be 0 at the maximum / minimum, so that to find a stationary point we enforce δLU = 0

δLΛ = 0

δLµ = 0

δLg = 0

δLχ = 0

In general this results in δLβ 6= 0. Z δL = δLβ = − χδβ dΓ

(24)

(25)

Γbi

To reach the minimum we take δβ such that δL = δLβ < 0, for example using a conjugate gradient method, β is updated as £ ¤ (δβ)k = % χk − τ k−1 χk−1 (26) where

Z τ k−1

[χk − χk−1 ] χk dΓ Γ = biZ [χk−1 ]2 dΓ

(27)

Γbi

and % > 0 . From δLU = 0 we obtain the so called adjoint of the Euler equations, that is t

Λx AU + tΛr B U − tΛ

∂G ∂Q −µ =0 ∂U ∂U

on the domain Ω, and also the related conditions · ¸ ∂I t + Λ (AU nx + B U nr ) δ U = 0 ∂U on its boundary Σ with unit normal vector n = (nx , nr ).

(28)

(29)

1.2 Z

1.1 Y

X

Y

1

0.9

0.8

0.7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X

Figure 2: Aft-swept fan rotor. Pressure field (left) and blade geometry (right). The gray regions represent the blade projection onto the meridional plane, i.e. the blade planform. The condition δLg = 0 yields to the adjoint of the kinematic constraint as (µu)x + (µu)r + ∇ · (tΛ K) = 0

(30)

where  0 0      1 0 0 0 K = rF ϑ   0 1    u w

          

together with the boundary condition [µ (q · n) + (tΛ K) · n +χ] δg = 0

(31)

From (31) also the value of χ is deduced at each control point. The adjoint equation of the kinematic constraint is coupled to eq. (28) in the same way as the kinematic constraint is coupled to the flow equations. Note that the variations of L with respect to the Lagrange multipliers Λ and µ simply yield the flow equations and the kinematic constraint respectively. The system of eqs. (29,30) is solved numerically as proposed in [9]. NUMERICAL RESULTS In this section some numerical experiments are proposed. At this stage, these should be considered as numerical experiments that illustrate the possible range of application of the proposed methodology.

0.04 0.035 0.03

H 1 (U)

0.025 0.02 0.015 0.01 0.005 0 0

50

100

k

150

200

Figure 3: Blade Lean optimization. Optimization history of functional H1 (U ) vs. optimization steps (right).

Z

Y

Z

X

Y

X

Figure 4: Blade Lean optimization. Blade geometry at K = 50 (left) and k = 200 (right), when minimizing the functional H1 (U ) . Aft-swept and Leaned Fan Rotor The example explains how the concept of sweep and lean are introduced into the inverse problem formulation. We observe a major difference between the proposed approach and the classical design. In latter case, the introduction of the sweep and of straight or compound lean modifies the blade forces, hopefully in a manner that some benefits on performances arise. In our approach the blade forces are a design parameter, so that lean and sweep are introduced here without modifications to the blade force distribution, with the aim of obtaining more advantageous flowfields. Conversely the resulting blade geometry does not maintain any similarity with the unleaned or unswept case. As example a fan stage with pressure ratio of 1.5 is designed. The inlet Mach number is 0.35. The rotor has 10o sweep at hub and tip and a parabolic leaned leading edge. Sweep is introduced also at the stator leading edge, as visible in figure 2. A three-dimensional sketch of the resulting blade geometry is also shown in the same figure. The computation has been

performed on a 50 × 25 stretched grid with at least 12 points per blade in the axial direction. Blade Lean Optimization In this section some numerical experiments of blade optimization are presented. The control is the blade lean, introduced by imposing a prescribed leading edge geometry. Once a blade force distribution is selected, the inverse problem is solved to obtain the blade shape. The adjoint procedure described in previous sections is then implemented to obtain a new update of blade lean at each computational node on the trailing edge. At a desired level of convergence, the final results are a blade geometry and the related flowfield, “optimal” with respect to a predefined a cost function. R In a first case we attempt to minimize the functional H1 (U ) = ρw2 dΩ on the blade. This functional has been chosen as an example on how the flowfield can be manipulated by the radial forces exerted by a properly designed blade. Results of the optimization are shown in Figs. 3-4. Figure 3 shows the functional H1 (U ) vs optimization steps The number of control variables to be optimized are 2 × 20 = 40, since the computational grid has 50 × 20 nodes. At the end of the optimization process, H1 (U ) is found to be about one order smaller than the initial value. The blade geometry obtained after k = 50 (blade A) and k = 200 (blade B) optimization steps, respectively, are shown in Figure 4. Although the two solutions have comparable merits in term of H1 (U ) the blade geometries are quite different and blade B k = 200 could be unrealistic from the structural point of view. In fact, since no mechanical constraints have been included in the design procedure, the final blade geometry may not satisfy some requirements of the mechanical design. For a correct treatment of the structural constraints those should be included in the optimization procedure. In our case, even for the uncostrained procedure, one can still find a compromise by selecting a unconverged solution that still satisfies the missing constraint. Using the same design parameters and distribution of tangential blade-forces, a second R optimization experiment is performed adopting a different objective functional. H2 (U ) = M 5 dΩ on the blade. The functional H2 (U ) expresses a rough dependence of trailing edge contribution to fan noise [14, 15] and the Mach number M . The optimization history and the final blade geometry are presented in Figure 5 . As visible, the cost function H2 (U ) has been reduced again of one order of magnitude. Turbine Blade Leaning In general, the simultaneous imposition of blade forces and geometrical constraints needs special care to avoid inverse problem overposedness[12]. Nevertheless, the selection of a reference line other that the leading edge locus does not require any special treatment in the present procedure. Since the equation of motion (11) is hyperbolic, the required boundary conditions can be used to impose a generic 3D curve that will lay on the blade camber surface. Intuitively the inverse problem solves for a blade geometry fastened to the reference line. Figure 6 shows the solution of the blade inverse design of a turbine stage. The stator reference line has been selected as the half-chord locus, while the trailing edge has been adopted as stacking line for the rotor. In both cases the stacking curve is a quadratic function of radius, so that compound lean appears. The inverse problem has been solved on a computational grid of 50 × 20 nodes for two different sets of blade-forces F θ (r), so that, for instance, two rotor blades are shown in Fig.6. The reference line is the locus of points shared by both the camber surfaces (e.g. the trailing edge line).

Z

0.14 Y

X

H 2 (U)

0.12 0.1 0.08 0.06 0.04 0.02 0 0

10

20

30

40

50

60

70

k

Figure 5: Blade Lean optimization. Final blade geometry (left) that minimize the functional H2 (U ) and optimization history of the functional vs. the optimization steps (right). CONCLUSIONS An automated procedure of 3D blade designing, limited by the through-flow model approximation adopted for the flow solver, has been derived. Blade design has been approached as solution of an inverse problem. The numerical technique, well assessed in finding optimal blade force configurations, has been extended to include the use of lean and sweep as additional geometrical controls. Some preliminary numerical experiments have been proposed in explaining the possible range of application of the method. Among the approximation introduced into the mathematical model, currently accepted in the industrial practice, a limitation is due to the absence of mechanical constraints that need to be included in future developments of the procedure. Viscous effects and secondary flows cannot be treated explicitly but through empirical correlations, as common practice for this class of models. Nevertheless a pure 3D extention of the methodology overcome this restriction. REFERENCES [1] C.H. Wu and L. Wolfestein. Application of radial equilibrium conditions to axial flow compressor and turbine design. NACA Report n. 955, 1955. [2] M.H. Vavra. Aero-Thermodynamics and Flow in Turbomachinery. John Wiley and Sons Inc., New York and London, 1960. [3] H.D. Weingold, R.J. Neubert, R.F. Behlke, and G.E. Potter. Bowed stators: An example of CFD applied to improve multistage compressor efficiency. ASME Journal of Turbomachinery, 114:161–168, 1997. [4] T. Wright and W.E. Simmons. Blade sweep for low-speed axial fans. ASME Journal of Turbomachinery, 112, 151-158, 1960. [5] N. Yamaguchi, T. Tominaga, S. Hattori, and T. Mitsuhashi. Secondary-loss reduction by forward-skewing of axial compressor rotor blading, Proc. Yokohama International Gas Turbine Congress, Japan, 1993. [6] Carolus T.H. and Beiler M.G. Computation and measurement of the flow in axial flow fans with skewed blades, ASME Journal of Turbomachinery, 121,59–66, 2001

Z Y

X

Figure 6: Turbine blade leaning. Blade geometry obtained for two different blade forces distributions but the same reference lines. The stator reference line is the half-chord locus, while trailing edge locus is prescribed for the rotor. [7] Breugelmanns F.A.E., Experimental investigation of sweep and dihedral in compressors, in ” Turbomachinery Blade Design Systems” VKI Lecture Series 1999-02, 1999. [8] C. Bena, F. Larocca, and L. Zannetti. Design of multistage axial flow turbines and compressors. IMech-E 3rd European Conference on Turbomachinery, London, 1999. [9] A. Iollo, M. Ferlauto, and L. Zannetti. An aerodynamic Optimization Method Based on the Inverse Problem Adjoint Equations. J. Comp. Phys., 173:87–115, 2001. [10] M. Ferlauto, A. Iollo, and L. Zannetti. Propfan Optimization Based on Inverse Problem Adjoint Equations. 4th Eur. Conf. on Turbomachinery, Florence, 2001. [11] M. Ferlauto, A. Iollo and L. Zannetti. Set of Boundary Conditions for Aerodynamic Design. AIAA J., 42(8), 1582–92, 2004. [12] L. Zannetti, M. Pandolfi. Inverse Design Technique for Cascades. NASA CR 3836, 1984. [13] AA.VV., OPtimization Methods and tools for multicriteria/multidisciplinary Design, VKI LS 2004-03, Bruxelles, Belgium,2004. [14] H. Hubbard, K. Shepherd, Aeroacoustics of large wind turbines, J. Acoust. Soc. Am. 89:2495–2508, 1991. [15] W. Blake, Mechanics of Flow-induced Sound and Vibration, Vol.1, Orlando, Academic Press, 1986.