Proceedings of the 1st Global Power and Propulsion Forum GPPF 2017

Jan 16-18, 2017, Zurich, Switzerland www.pps.global

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GPPF-2017-142

THE ROLE OF CURVATURE IN TURBOMACHINERY DESIGN

Mark G. Turner University of Cincinnati

mark.turner@uc.edu Cincinnati, Ohio, USA

ABSTRACT Streamline curvature has been used in axisymmetric

and blade-to-blade solvers ever since 1949. The physicalmeaning,numericalapproximations,andusesofstreamlinecurvature are presented here, particularly related toturbomachinerydesign.

The second derivative of a curve, which is directlyrelated to the curvature, is then described as the basis ofthe meanline creation for a geometry generator. Severalexamplesarepresentedwhichdemonstratethedistributionofcurvature in relation toblade loading. This technique isfurther extended for 3D optimization as elaborated withsubsequentexamples.

INTRODUCTION This paper discusses the overall role the use of curvature

has played in turbomachinery design. This is through its use in axisymmetric solvers, blade-to-blade solvers, and some recent work by the author on a 3D geometry generator that uses curvature concepts as the basis. This is not a treatise on the subject, which would require many more references and details. However, it does try to explain the overall role of curvature in design as well as understanding of flowfields that is not apparent in recent published work.

Streamline curvature has played a significant role in axisymmetric and blade-to-blade solvers since it was first used implicitly by Wu and Wolfenstein in 1949 [1]. Wu expanded on this in 1951 [2]. The Streamline Curvature (SLC) method then used curvature explicitly and has been used extensively in through-flow analysis to support turbomachinery design since [3-10]. Certain stream function formulations can have some of the qualities of the SLC methods [11, 12] and MISES, which uses an intrinsic streamline grid, also bears similarity to the SLC methods [13, 14]. An axisymmetric version of MISES is also available for turbomachinery design [15].

In addition to 2D analysis and simulation, the second derivative, which is directly related to curvature, can be used

to generate geometry. Korakianitis et al. [16] used curvature as a basis for geometry generation of the suction and pressure side of a blade separately. More recently, work has been done by the author and others on a geometry generator T-Bade3. T-Blade3 [17-20] is a parametric 3D turbomachinery geometry generator which uses the second derivative to define blade section meanlines along with a specified thickness distribution. These sections can be stacked smoothly in 3D along with several other critical geometry parameters to allow for novel shapes and integration with an optimization system [19-21]. The use of the second derivatives allows for a reduced number of high-level parameters to define geometry, which in turn reduces the number of design variables needed for optimization while allowing for a very large design space. These parameters are also related to the loading on the blade similar to angular momentum changes, another very important parameter for turbomachinery design.

HISTORICAL USES OF CURVATURE IN DESIGN Fig. 1 is taken from Wu’s [1] 1949 paper used to

describe the streamline curvature variation as the axisymmetric streamlines go through rotors and stators in an axial compressor (also applicable to turbines). This figure also shows the calculation stations between blade rows used to analyse flow properties upstream and downstream of each blade row. The radial equilibrium equation (a specific formulation of the radial momentum equation) is described that relates streamline curvature with the radial pressure gradient. The curvature is not explicit in the formulation of Wu, but is part of the finite difference term of the radial coordinates of a streamline which is derived from the derivatives of the radial velocity. The equations of motion are treated as uncoupled in an iterative fashion. Further, using the continuity equation and radial equilibrium equations, the streamline location and flow quantities are

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updated from hub to casing. Hand calculations were carried out for certain simplified cases.

Figure1.Stationsbetweenbladerowsandstreamlinedisplacement(fromWuin1949[1]).

Smith [2] and Novak [3] published distinct papers in 1966 on axisymmetric streamline curvature methods that used computers to solve the equations. Fig. 2 shows the coordinate system used by Novak [3] with the axial and radial (z,r) coordinates. Denton [5] used another approach for representing the coordinate system (Fig 3). The meridional angle φ is defined as

∅ = tan&' 𝑉) 𝑉* (1)

As Novak [3] describes, the radius of curvature has a

sign, and its convention is arbitrary. He defined positive such that the streamline is concave down and negative as concave up (Fig. 2).

𝐶 = 1/𝑟/ = 1/𝑟0 = −𝜕∅ ⁄ 𝜕𝑚 (2)

where m is the meridional direction, tangent to a streamline, as shown in Figs. 2 and 3. This form of a curvature definition that relates to an angle change allows for the sign to be used and becomes directly related then to a streamline angle. A simple form of curvature for a curve in 2D is

𝐶 = 𝑦77

1 + 𝑦79 :9 (3)

which shows how the curvature is related to the second derivative. This second derivative is used for geometry generation presented in a later part of this paper.

For a simple 2D flow, the pressure gradient normal to a streamline (the n direction) is represented as

𝜕𝑝 ⁄ 𝜕𝑛 = 𝐶𝑉9 (4)

For axisymmetric flows, Denton [5] shows a combined

equation in the q-direction (other grid coordinate often

aligned with a blade) which uses slightly different nomenclature here:

𝜕 𝑉/92

𝜕𝑞=𝜕𝐻ð𝑞

− 𝑇𝜕𝑆𝜕𝑞

−12𝑟9

𝜕 𝑟9𝑉C9

𝜕𝑞− 𝐹E

+𝐶𝑉/9 sin 𝛼 + 𝑉/IJKI/

cos 𝛼 (5)

This equation relates the total enthalpy H, the specific entropy s, a blade force in the q direction 𝐹E, the curvature C and the meridional and tangential velocities, 𝑉/ and 𝑉C. Except for the last term which is 0 if q is normal to the streamline, all derivatives are in the q direction (∝ is the angle between the m and q direction). It clearly shows the importance of curvature to the other flow quantities. Curvature is a better variable than radius of curvature since cylindrical endwalls would yield a zero curvature, but an infinite radius of curvature. Since a streamline is used, H and s can be readily tracked (with appropriate rotor work input and loss models) along streamlines without numerical dissipation.

Figure 2. Coordinate system showing meridional angle φ anddefinitionofsignconventionforradiusofcurvature(modifiedfromNovak[4)].

Additional tools were also created using the streamline curvature approach. Kieth et al. [10] describes a method using grid adaptation with a streamline method for inlets, as shown in Fig. 4. The method of iteration of moving the streamlines is shown in Fig. 5. This approach uses a type of cubic spline to define the streamlines. Details of curvature near a leading edge play a very large role in the resulting pressure distribution. It is interesting that splines were physical devices as shown in Fig. 6 where a flexible piece of wood was constrained by lofting ducks for the layout of boats, airplanes, compressors and turbines. These were still

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in limited use at GE, back in 1979 when the author started working there. The physical spline curvature can be readily related to simple beam theory with similar mathematical description. This connection relating curvature back to physical quantities keeps recurring.

Figure3.Coordinatesystemshowingmeridionalangleφ(modifiedfromDenton[5)].

Figure4.Axisymmetricinletshowingstreamlinesandnormalswith

refinement[10].

Figure5.Movementofstreamlines[10].

Fig. 7 shows the use of the axisymmetric solver using the streamline curvature (SLC) method for the GE design of the high pressure compressor and low pressure turbine as part of the NASA sponsored Energy Efficient Engine (EEE) program [23, 24]. It is interesting to note that redesigns of the compressor, designed in the late 1970s are still being used for new GE engines. In addition to axisymmetric analysis, streamline curvature methods were also used for 2D blade-to-blade flow solutions [25] as shown in Fig. 8. They allowed for extremely accurate solutions with no dissipation and a very coarse grid. Axisymmetric SLC methods with detailed loss and turning models continue to be used as the primary design approach for compressors. This is because

the dominant flow physics are represented by the streamline curvature, radial pressure forces, and blade turning in an appropriate fidelity. In addition, they are used to data-match component test data since empiricism can be readily applied.

Figure 6. Lofting Ducks used with wooden spline for boat layout[21].

Figure7.FlowpathofGE/EEEcomponentsusingtheaxisymmetricstreamlinecurvaturecode,CAFD[22,23].

Figure 8. Cascade analysis streamline grid and loading for aboosterstatorhubsection[25].

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CURVATURE USED TO DEFINE GEOMETRY As shown in Eq. (3) the curvature and second derivative

of a function are directly related. The use of the second derivative allows for direct integration. Based on this, a method of airfoil construction has been developed that uses the second derivative of the meanline. Fig. 9 shows the cubic B-splines used to define the second derivative of an airfoil meanline as shown by Nemnem [17]. The blades are defined in a u-v plane with u along the normalized chord. Using only a few control points a smooth curve can be created which can be integrated analytically. It is desired to specify both the leading edge and trailing edge angle for blade design. In addition v(1)=0. Therefore, the magnitude of the second derivative (scaling) is also a variable. In other words, the curvature shape is prescribed, but not the magnitude since the overall camber is known. The slope of the meanline is shown in Fig. 10; integrating that gives the meanline relative to the chord of the blade rotated by the stagger angle. After a smooth thickness is added to each side of the meanline, a blade shape is defined as shown in Fig. 11 along with its loading (from Mises [13, 14]) in Fig.12. Mises is an accurate quasi-3D solver that uses a streamline oriented grid with a coupled integral boundary layer.

Figure9.SecondderivativeofthecamberlineasaB-spline[17]

Figure10.Slopeofthecamberline[17].Theslopeofthemeanlineisthefirstderivative.Theleadingandtrailingedgeslopeisbasedontheprescribedleadingandtrailingedgeanglesandthecalculatedstaggerangle.

Figure11.CamberlinefromsecondderivativeinFig.9[17].

Figure12.BladeandloadingfromthesecondderivativedefinedinFig9[17].

A study was recently conducted on a swirler-deswirler vane with an inlet angle of -45 deg and exit angle of -38.6 deg i.e., only 6.4 deg of camber or turning. A typical second derivative distribution for a compressor is shown in Fig. 13 as the baseline case. The corresponding blade is shown in blue in Fig. 14, and the loading is shown in Fig. 15. Because of the low amount of turning, the loading is small. Several cases were then explored with the second derivative changing sign. This creates an s-shaped blade. For the 3 cases, the second derivative at the front part of the blade was kept the same while the back varied. Several conclusions are apparent from this study: an unconventional blade can be readily designed with only few control points to define a complex shape, since higher-order information is used. The blade shape and loading are shown in Fig. 16 for case 2 using MISES. The loading (difference in pressure) as a function of chord is related to the second derivative specification. This demonstrates that the use of the second derivative is not only useful for geometry construction, but also has physical significance to the function of an airfoil.

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Figure 13. Second derivative for a baseline blade and 3 otherblades(showninFig.14).

Figure 14. Blades produced using the second derivativedistributionsshowninFig.13.

Figure15.LoadingcalculatedwithMisesforbaselineblade.

Figure 16. Loading from Mises for case 2. There is a reverse-loading,andtheloadingisrelatedtothesecondderivativeshowninFig.13.

The geometry generator was applied to the optimization of a subsonic rotor [20]. Fig. 17 shows the baseline meanline and second derivative for which the optimization started. The figure uses the term curvature, but in fact the plots show the second derivative. In addition, the second derivative and meanline curves for the hub, pitch and tip sections are shown. In Fig. 17, the B-spline and second derivative are different because of the scaling and because for the mid-span optimized plot the section second derivative uses spanwise interpolated control points. With only a few control points, a novel blade shape is produced as shown in Fig. 18. It has a pronounced feature of a dimple on upper half of blade due to the s-shape produced by two sign changes of the second derivative. The unique blade shape has the effect of reducing the tip loading and thereby the strength of the tip vortex. In addition to the second derivative, the leading and trailing edge angles were perturbed spanwise as shown in Fig. 19. The spanwise variation is itself defined using a cubic B-spline where only 3 control points are used. To allow for a smooth curve spanwise, the second derivative control points are specified as cubic B-splines spanwise as shown in Fig. 20. This allows any number of sections to be defined spanwise with smooth variation. The number of sections is also not dependent on the number of spanwise control points.

To demonstrate the general capability, a transonic rotor was optimized and a vortex dynamics method was used to analyse the results of the optimization [21]. An s-shape at the tip was the result of that optimization process too. The vorticity contours and skin friction vectors on the suction side are shown in Fig. 21 for the baseline and optimized case. From the plot, the shock strength and weak flow behind it is shown to be reduced.

The method is also applicable to radial machines, and has been applied to a novel multistage centrifugal design [25].

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Figure17. BaselineandOptimal secondderivative,andmeanlineforsubsonicrotor[20].

Figure18.Optimizedsubsonicrotorshowinginflection[20].

Figure19. Spanwisevariationofangle tocreate theblade inFig.18[20].

Figure 20. Normalized second derivative control point valuesdistributionspanwisearecubicB-Splines[20].

Figure 21. Baseline (left) and optimum (right) transonic rotorshowingradialvorticitycontoursandskinfrictionvectors[21].

CONCLUSIONS & FUTURE WORK The historical role of curvature in turbomachinery

design has been presented along with some physical insight to its importance. In addition, the use of curvature (specifically the second derivative) as the basis of geometry definition has also been established. The relationship of the meanline curvature to the blade loading has been described showing its physical importance in design and analysis of airfoil characteristics. Geometry created with this approach has been demonstrated for two applications where it has been integrated with optimization systems effectively.

Future work efforts will be to further explore the effectiveness of the blade geometry generation in additional optimization applications. Also the approach will get applied for creating or modifying axisymmetric flowpaths.

NOMENCLATURE C Curvature H Total Enthalpy q direction of a grid line m meridional coordinate in direction of streamline r radial coordinate

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𝑟/ radius of Curvature 𝑟0 radius of Curvature (alternate definition) s entropy u coordinate in chord direction of blade v coordinate normal to u V velocity z axial coordinate ∝ angle between the m and q direction φ meridional angle

ACKNOWLEDGMENTS I learned of streamline curvature (among many other

things) first from Jim Keith, my first manager at GE. I also had the privilege to work for Dick Novak and indirectly Roy Smith, two individuals who were pioneers in streamline curvature methods.

The work on using curvature for meanlines came about through the following former and current students who I’ve had the pleasure to work with: Ahmed Nemnem, Kiran Siddappaji, Syed Moez Hussain Mahmood, and KarthikBalasubramanian (Karthik also provided some of theimages in thispaperon theswirler-deswirlerstudy). Inaddition, I acknowledge the work of a visiting scholarHuanlong Chen who also worked with the curvature-basedgeometrygenerator.

REFERENCES [1] Wu, Chung-Hua and Lincoln Wolfenstein, “Application of Radial-Equilibrium Condition to Axial-flow Compressor and Turbine Design,” NACA Tech Note 1796, Lewis Flight Propulsion Laboratory, Cleveland, Ohio, January, 1949. [2] Wu, Chung-Hua, “A General Through-Flow Theory of Fluid Flow with Subsonic or Supersonic Velocity or Supersonic Velocity in Turbomachinery of Arbitrary Hub and Casing Shapes,” NACA Tech Note 2302, Lewis Flight Propulsion Laboratory, Cleveland, Ohio, March, 1951. [3] Smith, L. H. Jr., “The Radial-Equilibrium Equation of Turbomachinery,” J. of Engineering for Power, Jan. 1966, pp 1-11. [4] Novak, R. A., “Streamline Curvature Computing Procedures for Fluid-Flow Problems,” J. of Engineering for Power, Jan. 1966, pp 1-13. [5] Denton JD. “Throughflow Calculations for Transonic Axial Flow Turbines.” ASME. J. Eng. Power. 1978;100(2):212-218. doi:10.1115/1.3446336. [6] Adkins, G. G., Jr., and L.H. Smith, Jr., “Spanwise Mixing in Axial-Flow Turbomachines,” J. of Engineering for Power, Jan. 1982, Vol. 104, pp.97-110, also ASME paper 81-GT-57. [7] Jennions, I.K., and P. Stow, “A Quasi-Three-Dimensional Turbomachinery Blade Design System: Part I - Throughflow Analysis,” J of Engineering for Gas Turbines and Power, Vol 107, April, 1985, pp 301-307. [8] Abdallah, S., Henderson, R.E., "Improved Approach to the Streamline Curvature Method in Turbomachinery," Trans. ASME Journal of Fluids Engineering, Vol. 109, pp. 213-217, 1987.

[9] Casey M, Robinson C, “A New Streamline Curvature Throughflow Method for Radial Turbomachinery,” ASME. J. Turbomach. 2010;132(3):031021-031021-10. doi:10.1115/1.3151601. [10] Keith, J.S., D.R. Ferguson, C.L. Merkle, P.H. Heck, and D. J. Lahti, “Analytical Method for Predicting the Pressure Distribution About a Nacelle at Transonic Speeds,” NASA CR-2217, July 1973. [11] M. G. Turner, and J. S. Keith, “An Implicit Algorithm for Solving 2D Rotational Flow in an Aircraft Engine Fan Frame,” AIAA-85-1534, 1985. [12] Mark G. Turner, “Design and Analysis of Internal Flowfields using a Two Stream Function Formulation,” MIT Sc.D. thesis and MIT GTL Report #201, 1990. [13] M. Drela and M.B. Giles, “Viscous-inviscid analysis of transonic and low reynolds number airfoils,” AIAA Journal, pages 1347–1355, Oct,1987. [14] H.H. Youngren and M. Drela, “Viscous/inviscid method for preliminary design of transonic cascades," AIAA Journal-91-2364, 1991. [15] Mark G. Turner, Ali Merchant, and Dario Bruna, “A Turbomachinery Design Tool for Teaching Design Concepts for Axial-Flow Fans, Compressors, and Turbines,” Journal of Turbomachinery, Vol. 133, Issue 3, July 2011. [16] Korakianitis, T., “Prescribed-Curvature-Distribution Airfoils for the Preliminary Geometric Design of Axial Turbomachinery Cascades,” Journal of Turbomachinery, Vol. 115, April 1993, pp. 325–333. [17] Nemnem, Ahmed, Mark G. Turner, Kiran Siddappaji, and Marshall Galbraith, “A Smooth Curvature-Defined Meanline Section Option for a General Turbomachinery Geometry Generator,” ASME Paper GT2014-26363, Dusseldorf, Germany, June 16-20, 2014. [18] http://gtsl.ase.uc.edu/t-blade3/ for open source software or http://gtsl.ase.uc.edu/3DBGB/ for executables. [19] Nemnem, Ahmed F., Mark G. Turner, Kiran Siddappaji, and Anthony J. Gannon, “An Automated 3D Turbomachinery Design and Optimization System,” Journal of Multidisciplinary Engineering Science and Technology, Vol. 2, Issue 11, Nov 2015, ISSN: 3159-0040 [20] Mahmood, Syed Moez Hussain*, Mark G. Turner, and Kiran Siddappaji*, “Flow Characteristics Of An Optimized Axial Compressor Rotor Using Smooth Design Parameters,” ASME Paper GT2016-57028, Seoul, South Korea, June 13-17, 2016. [21] Chen, Huanlong, Mark G. Turner, Kiran Siddappaji*, and Syed Moez Hussain Mahmood*, “Vorticity Dynamics Based Flow Diagnosis for a 1.5-Stage High Pressure Compressor with an Optimized Transonic Rotor,” ASME Paper GT2016-56682, Seoul, South Korea, June 13-17, 2016. [22] http://digiitalarchfab.com/portal/wp-content/uploads/2011/03/LOFTING-DUCKS-Spline-Weights.pdf [23] Holloway, P. R.; Koch, C. C.; Knight, G. L.; Shaffer, S. L., “Energy efficient engine. High pressure compressor detail design report,” NASA-CR-165558, May 1, 1982.

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[24] Cherry, D. G.; Gay, C. H.; Lenahan, D. T., “Energy efficient engine. Low pressure turbine test hardware detailed design report,” NASA-CR-167956, Aug 1, 1982. [25] Sullivan, T. J., “Energy efficient engine: Fan test hardware detailed design report,” NASA-CR-165148; Oct 1, 1980.[26] Mishra, Shashank, Shaaban Abdallah, and Mark G. Turner, “Flow Characteristics of a Novel Centrifugal Compressor Design,” ASME Paper GT2016-58103, Seoul, South Korea, June 13-17, 2016.