Parallel Computation of Turbine Blade Clocking

Parallel Computation of Turbine Blade Clocking Paul G. A. Cizmas Department of Aerospace Engineering Texas A&M University College Station, Texas 77843-34 Daniel J. Dorney Department of Mechanical Engineering Virginia Commonwealth University Richmond, Virginia 23284-305 ABSTRACT This paper presents a numerical study of airfoil clocking of a six-row test turbine configuration with equal pitches. Since the rotor-stator interaction flow is highly unsteady, the numerical simulation of airfoil clocking requires the use of time marching methods, which can be computationally expensive. The large turnaround time and the associated cost for such simulations makes it unacceptable for the turbomachinery design process. To reduce the turnaround time and cost/mflop, a parallel code based on Message-Passing Interface libraries was developed. The relative circumferential positions of the three stator and three rotor rows in an industrial steam turbine were varied to increase turbine efficiency. A grid density study was performed to verify the grid independence of the computed solutions. The clocking of the second-stage airfoils gave approximately a 50% greater efficiency variation than the clocking of the third-stage airfoils. This was true for clocking both rotor and stator airfoils. Rotor clocking produces an efficiency variation which is approximately twice the efficiency variation produced by stator clocking. For both stator and rotor clocking, the maximum efficiency is obtained when the wake impinges on the leading edge of the clocked airfoil. NOMENCLATURE p T η γ Pressure Temperature Efficiency Ratio of specific heats of a gas SUBSCRIPTS ca t t ta Circumferential-averaged Total-to-total Time-averaged

SUPERSCRIPTS Total (or stagnation) INTRODUCTION The requirement to further increase performance and improve reliability in turbomachinery has motivated designers to better understand unsteady effects. An important part of the unsteady effects in turbomachinery results from the rotor-stator interaction. The main sources of unsteadiness present in the rotor-stator interaction are potential flow interaction and wake interaction. Additional sources of unsteadiness include vortex shedding, hot streak interaction, shock/boundary layer interaction, and flutter. Potential flow interaction is a purely inviscid interaction due to the pressure variation caused by the relative movement of the blades and vanes. Potential flow interaction mainly affects adjacent airfoil rows. Wake interaction is the unsteadiness generated by the vortical and entropic wakes shed by one or more upstream rows. These wakes interact with the downstream airfoils and other wakes. Wake interaction is the primary contributor to unsteady forces on the blade for a large rotor-stator gap. The process of varying the circumferential relative position of consecutive stator airfoils is referred to as airfoil indexing or clocking. Consecutive rotor airfoils can be clocked as well. The effects of airfoil clocking on compressor performance have been investigated both experimentally and numerically. Capece [] was among the first to show the potential performance benefits of clocking. Saren et al. [2, 3, 4] have used theoretical, experimental and computational techniques to show that airfoil clocking can reduce unsteady forces on airfoil and increase compressor performance. Hsu and Wo [5], by clocking the downstream rotor, have experimentally shown a reduction of stator unsteady loading. Barankiewicz and Hathaway s experimental investigation of stator row clocking on a four-stage axial compressor showed a change in overall performance of about 0.2% [6]. The experimental investigation of Walker et al. studied the effects of inlet guide vanes on the boundary layer quantities and losses of a downstream stator row [7]. However, no firm conclusion could be drawn about the stator losses since the observed variation in losses was comparable in magnitude to the uncertainty in the data. The numerical results reported by Gundy-Burlet and Dorney [8, 9] for a 2-2 stage compressor predict efficiency variations between 0.5% and 0.8%, as a function of stator clocking position. For turbines, the effects of airfoil clocking have been experimentally investigated by Huber et al. [0]. The experimental results showed a 0.8% efficiency variation due to clocking. For the same turbine, a two-dimensional numerical analysis for the midspan geometry by Griffin et al. [] correctly predicted the maximum efficiency clocking positions. However, the predicted efficiency variation was only 0.5%. Clocking effects in a - 2 stage turbine have also been numerically simulated by Eulitz et al. [2] and Dorney and Sharma [3]. In all these analyses, the highest efficiencies occurred when the first-stage stator wake impinged on the leading edge of the second-stage stator, while the lowest efficiencies were observed when the first-stage stator wake was convected through the middle of the second-stage stator passage. The focus of the current investigation has been to study the effects of fully clocking a threestage industrial steam turbine. This paper presents for the first time the effects of clocking rotor rows, including the effects of clocking three rotor rows. This is also the first time that the effects of clocking three stator rows are presented. Finally, this paper shows the cumulative benefits of simultaneously clocking rotor and stator rows. NUMERICAL MODEL 2

The computer code used to simulate the flow in the turbine is presented in detail in [4]. The numerical approach used in the code is based on the work done by Rai [5]. The code was developed as a parallel version of the STAGE-2 analysis, which was originally developed at NASA Ames Research Center. The numerical approach is briefly described here. The quasi three-dimensional, unsteady, compressible flow through a multistage axial turbomachine with arbitrary blade counts is modeled by using the Navier-Stokes and Euler equations. The computational domain associated with each airfoil is divided into an inner region, near the airfoil, and an outer region, away from the airfoil. The thin-layer Navier-Stokes equations are solved in the regions near the airfoil, where viscous effects are strong. Euler equations are solved in the outer region, where the viscous effects are weak. The flow is assumed to be fully turbulent. The eddy viscosity is computed using the Baldwin-Lomax model and the kinematic viscosity is computed using Sutherland s law. The Navier-Stokes and Euler equations are written in the strong conservation form. The fully implicit, finite-difference approximation is solved iteratively at each time level, using an approximate factorization method. Two Newton-Raphson sub-iterations are used to reduce the linearization and factorization errors at each time step. The convective terms are evaluated using a third-order accurate upwind-biased Roe scheme. The viscous terms are evaluated using second-order accurate central differences and the scheme is second-order accurate in time. Grid Generation Two types of grids are used to discretize the flow field surrounding the rotating and stationary grids. An O-grid is used to resolve the Navier-Stokes equations near the airfoil, where the viscous effects are important. An H-grid is used to discretize the Euler equations away from the airfoil. The O-grid is generated using an elliptical method. The H-grid is algebraically generated. The O- and H-grids are overlaid. The flow variables are communicated between the O- and H-grids through bilinear interpolation. The H-grids corresponding to consecutive rotors and stators are allowed to slip past each other to simulate the relative motion. Boundary Conditions Since multiple grids are used to discretize the Navier-Stokes and Euler equations, two classes of boundary conditions must be enforced on the grid boundaries: natural boundary conditions and zonal boundary conditions. The natural boundaries include inlet, outlet, periodic and the airfoil surfaces. The zonal boundaries include the patched and overlaid boundaries. The inlet boundary conditions include the specification of flow angle, average total pressure and downstream propagating Riemann invariant. The upstream propagating Riemann invariant is extrapolated from the interior of the domain. At the outlet, the average static pressure is specified, while the downstream propagating Riemann invariant, circumferential velocity, and entropy are extrapolated from the interior of the domain. Periodicity is enforced by matching flow conditions between the lower surface of the lowest H-grid of a row and the upper surface of the top most H-grid of the same row. At the airfoil surface, the following boundary conditions are enforced: the no slip condition, the adiabatic wall condition, and the zero normal pressure gradient condition. For the zonal boundary conditions of the overlaid boundaries, data is transferred from the H- grid to the O-grid along the O-grid s outermost grid line. Data is then transferred back to the H-grid along its inner boundary. At the end of each iteration, an explicit, corrective, interpolation procedure is performed. The patch boundaries are treated similarly, using linear interpolation to update data between adjoining grids [6]. 3

Parallel Computation The parallel code uses Message-Passing Interface (MPI) libraries and runs on symmetric multiprocessors (e.g., Silicon Graphics Challenge) and massively parallel processors (e.g., Cray T3E). The quasi three-dimensional parallel code was developed such that it could be easily extended to a three-dimensional parallel version. As a result, one processor was allocated for each airfoil in the two-dimensional simulation. Consequently, the number of processors necessary for a typical threedimensional turbomachinery configuration does not exceed the number of processors available on today s computers. The processor allocation is presented in Figure. One processor is allocated for each inlet and outlet H-grid. One processor is allocated for the O- and H-grids corresponding to each airfoil. Interprocessor communication is used to match boundary conditions between grids. Periodic boundary conditions are imposed by cyclic communication patterns within rows. Inter-blade-row boundary conditions are imposed by gather-send receive-broadcast communication routines between adjacent rows. Load imbalance issues need to be considered at grid generation time to reduce synchronization overhead. NUMERICAL RESULTS The results reported in this section are for a three-stage test turbine. This test turbine has 58 first-stage stators, 46 first-stage rotors, 52 second-stage stators, 40 second-stage rotors, 56 thirdstage stators and 44 third-stage rotors. A dimensionally accurate simulation of this geometry would require the modeling of 29 first-stage stators, 23 first-stage rotors, 26 second-stage stators, 20 second-stage rotors, 28 third-stage stators and 22 third-stage rotors. To reduce the computational effort, it was assumed that there were an equal number of blades (58) in each turbine row. As a result, the airfoils were rescaled by the factors shown in Table. Note that modifying the blade count represents a form of airfoil clocking. The effects of clocking a turbine with an equal number of airfoils per row are larger than the effects of clocking a turbine with a different number of airfoils per row. Consequently, the efficiency variation obtained for the airfoil count ::::: will be larger than the efficiency variation for the airfoil count 29:23:26:20:28:22. However, the focus of this paper is to estimate the relative contribution of clocking different stator and rotor rows. The inlet temperature in the test turbine is 293 degrees Kelvin and the inlet Mach number is 0.073. The inlet flow angle is 0 degrees and the inlet Reynolds number is 53494 per inch, based on the axial chord of the first-stage stator. The rotational speed of the test turbine is 2400 RPM. The results presented in this paper were computed using two Newton sub-iterations per timestep and 3200 time-steps per cycle. Here, a cycle is defined as the time required for a rotor to travel a distance equal to the pitch length at midspan. Since the airfoil count is :::::, the flow repeats after each cycle, i.e. the flow period is equal to the duration of a cycle. To ensure time-periodicity, each simulation was run in excess of 80 cycles. All time-averaged quantities were calculated over 20 cycles. Three computational grids were used to verify that the numerical solutions were grid independent. The number of grid points per row for the coarse, medium and fine grids are shown in Table 2. For a given grid size, the same number of grid points per airfoil were used for all the rows. Details of the grid around the first-stage stator are shown in Fig. 2 for the three different grid densities. Before verifying that the solution is grid independent, one has to verify that the solution is periodic. Since the flow in the last row of airfoils is more likely to be the last to become periodic, 4

the flow periodicity will be monitored in this last row. To assess periodicity, the pressure variation on the row-six airfoil is compared for three consecutive cycles, as shown in Fig. 3. Maximum, minimum and time-averaged pressures denote maximum, minimum and averaged pressures over a blade-passing cycle. The close agreement of the pressure values indicates that the solution is not a function of the cycle number, i.e., the solution is periodic. To verify that the solution is grid independent, the pressure variation obtained using the three different size grids is compared for the last row airfoil. The results presented in Fig. 4 indicate very good agreement for the time-averaged pressure and good agreement for the results corresponding to the maximum and minimum pressure variation. The overall good agreement among the results corresponding to the three grids gives confidence that the solution is grid independent. To reduce the computational effort, the coarse grid shown in Fig. 5 is used in the remainder of the paper. The average value of y +, the non-dimensional distance of the first grid point above the surface, is less than for all the rows. Approximately 25 grid points are used to discretize the boundary layers. In this analysis, the effects of airfoil clocking are estimated by performing simulations with the clocked stator or rotor located at five different locations equidistantly spaced over one pitch. Figure 6 shows the clocking locations for the second-stage stator. The computations were performed on a twelve-processor SGI Challenge computer. Eight processors were used for this analysis, as suggested by Fig.. The computation time for this simulation was 6.24x0 6 secs/grid point/iteration. Instantaneous Mach contours are presented in Fig. 7 in order to visualize the velocity distribution. For the given flow conditions, the maximum Mach number is approximately 0.5 and is located on the suction side of the third-stage rotors. In this investigation, the total-to-total efficiency is defined as [7]: η t t = ( T exit,ca,ta ) Tinlet,ca,ta ( p exit,ca,ta p inlet,ca,ta where subscript ca denotes circumferential-averaged, ta denotes time-averaged, and the superscript * denotes total (or stagnation). The efficiency variation as a function of the second-stage stator clocking position is shown in Fig. 8. The time-averaged (over a cycle) entropy contours on the second-stage stators are shown in Figures 9-3. The red contours correspond to the high entropy value and the blue contours correspond to the low entropy value. The maximum efficiency corresponds to the case in which the wakes impact the stator at the leading edge, slightly shifted to the pressure side. The minimum efficiency corresponds to the case in which the wake is located in the passage between the stator airfoils. The correlation between the wake impact and the efficiency value agrees well with the results of previous experimental and numerical studies [8, 9]. A slight increase in entropy can be observed on the suction side of the airfoil, at about 80% of the chord. This slight increase in entropy corresponds to the inner part of the H-grid. The magnitude of the entropy rise is exaggerated because the entropy levels were restricted, in order to increase the wake contrast. This local entropy rise does not affect the conclusions of this investigation. The efficiency variation due to clocking the third-stage stator is shown in Fig. 4. To allow for easy comparison between the efficiencies corresponding to clocking different rows, the same scale is used for all the efficiency plots. The efficiency variation obtained by clocking the thirdstage stator is only 48% of the efficiency variation obtained by clocking the second-stage stator. The clocking of the third-stage stator is done with the second-stage stator clocked for maximum efficiency. Consequently, the maximum efficiency in Fig. 8 is equal to the efficiency at clocking ) γ γ 5

position 0 in Fig. 4. As in the case of clocking the second-stage stator, the maximum efficiency is obtained when the wake impinges on the leading edge of the stator, as shown in Fig. 5. The minimum efficiency is obtained when the wake is located in the passage between the stators, as shown in Fig. 6. The maximum efficiency clocking position for the second-stage stator is modified when the indexing of the third-stage stator is modified. As a result, the maximum efficiency obtained by first clocking the second-stage stator, and then the third-stage stator, may not be the absolute global maximum. To obtain the absolute global maximum efficiency, the clocking of the second-stage should be repeated with the third-stage stators clocked in clocking position. This iterative process is very computationally intensive and the analysis will be limited here to one iteration only. During the clocking of the second-stage rotor, the second- and third-stage stators were indexed for maximum efficiency. The efficiency variation due to clocking the second-stage rotor is presented in Fig. 7. The efficiency variation in this case is.83 times larger than the efficiency variation obtained by clocking the second-stage stator. Similar to clocking stators, the maximum efficiency is obtained when the wake impinges on the leading edge of the rotor airfoil, as shown in Fig. 8. The minimum efficiency corresponds to the case when the wake is located in the passage between the rotor airfoils, as shown in Fig. 9. The clocking of the third-stage rotor is accomplished using the second-stage rotor indexed in position 2. The efficiency variation due to clocking the third-stage rotor is shown in Fig. 20. In this case, the efficiency variation is 28% of the efficiency variation obtained by clocking the second-stage stator. However, no absolute increase in efficiency is obtained by clocking the thirdstage rotor, since the third-stage rotor was already in the optimal position. As in the previous clocking cases, the minimum efficiency corresponds to the case where the wake is located in the passage between the rotor airfoils, as shown in Fig. 2. The summary of efficiency variation and efficiency increase is presented in Table 3. The total increase in efficiency obtained by clocking the rotors and the stators is 2.45 times larger than clocking the second-stage stator only. This additional increase in efficiency represents a significant reward for clocking multiple stator and rotor rows. CONCLUSIONS A quasi three-dimensional unsteady Euler/Navier-Stokes analysis, based on a parallel code, has been used to investigate the effects of fully clocking a three-stage turbine. The effects of simultaneously clocking rotor rows, including the effects of clocking three rotor rows, are presented for the first time. This is also the first time that the effects of clocking three stator rows have been presented. Previous experimental and numerical investigations have shown that in the case of stator clocking, maximum efficiency is obtained when the wake impinges on the leading edge of the clocked stator. The present numerical simulation reconfirmed this observation for stator clocking and extended it for rotor clocking. The fact that the wake impinging on the leading edge produces the highest efficiency holds true for clocking multiple stator or rotor rows as well. For the turbine investigated, the clocking of the second-stage gives larger efficiency variations than the clocking of the third-stage. This conclusion is true for both rotor and stator clocking. The predicted results also showed that rotor clocking produces an efficiency variation which is approximately twice the efficiency variation produced by stator clocking. ACKNOWLEDGMENTS 6

The authors wish to thank the Westinghouse Power Generation and Westinghouse Science & Technology Center for supporting this work. The authors are thankful to Pittsburgh Supercomputing Center for making the computing resources available. The authors are especially grateful to Dr. Karen Gundy-Burlet of NASA-Ames Research Center for her assistance with code related issues and interpretation of the physics. The authors would also like to thank Mr. Harry Martin and Dr. Shun Chen for the helpful discussions and suggestions during this project. References [] Capece, V. R., Forced Response Unsteady Aerodynamics in a Multistage Compressor, Ph.D. Thesis, Purdue University, West Lafayette, IN, 987. [2] Saren, V. E., Some Ways of Reducing Unsteady Loads Due to Blade Row Hydrodynamic Interaction in Axial Flow Turbomachines, Second International Conference EAHE, Pilsen, Czech Republic, pp. 60-65, 994. [3] Saren, V. E., Relative Position of Two Rows of an Axial Turbomachine and Effects on the Aerodynamics in a Row Placed Between Them, Unsteady Aerodynamics and Aeroelasticity of Turbomachines, Elsevier, pp. 42-425, 995. [4] Saren, V. E., Savin, N. M., Dorney, D. J., and Zacharias, R. M., Experimental and Numerical Investigation of Unsteady Rotor-Stator Interaction on Axial Compressor Stage (with IGV) Performance, 8th International Symposium on Unsteady Aerodynamics and Aeroelasticity of Turbomachines, Stockholm, Sweden, September 997. [5] Hso, S. T., and Wo, A. M., Reduction of Unsteady Blade Loading by Beneficial Use of Vortical and Potential Disturbances in an Axial Compressor with Rotor Clocking, ASME Paper 97-GT-86, Orlando, FL, June 997. [6] Barankiewicz, W. S., and Hathaway, M. D., Effects of Stator Indexing on Performance in a Low Speed Multistage Axial Compressor, ASME Paper 97-GT-496, Orlando, FL, June 997. [7] Walker, G. J., Hughes, J. D., Kohler, I., and Solomon, W. J., The Influence of Wake-Wake Interactions on Loss Fluctuations of a Downstream Axial Compressor Blade Row, ASME Paper 97-GT-469, Orlando, FL, June 997. [8] Gundy-Burlet, K. L., and Dorney, D. J., Physics of Airfoil Clocking in Axial Compressors, ASME Paper 97-GT-444, Orlando, FL, June 997. [9] Gundy-Burlet, K. L., and Dorney, D. J., Investigation of Airfoil Clocking and Inter-Blade Row Gaps in Axial Compressors, AIAA Paper 97-3008, Seattle, WA, July 997. [0] Huber, F. W., Johnson, P. D., Sharma, O. P., Staubach, J. B., and Gaddis, S. W., Performance Improvement Through Indexing of Turbine Airfoils: Part - Experimental Investigation, ASME Journal of Turbomachinery 8, 996, pp. 630-635. [] Griffin, L. W., Huber, F. W., Sharma, O. P., Performance Improvement Through Indexing of Turbine Airfoils: Part2 - Numerical Simulation, ASME Journal of Turbomachinery 8, 996, pp. 636-642. [2] Eulitz, F., Engel, K., and Gebbing, H., Numerical Investigation of the Clocking Effects in a Multistage Turbine, ASME Paper 96-GT-26, Birmingham, UK, 996. 7

[3] Dorney, D., and Sharma, O. P., A Study of Turbine Performance Increases Through Airfoil Clocking, AIAA Paper 96-286, Lake Buena Vista, FL, 996. [4] Cizmas, P., and Subramanya, R., Parallel Computation of Rotor-Stator Interaction, 8th International Symposium on Unsteady Aerodynamics and Aeroelasticity of Turbomachines, Stockholm, Sweden, 997. [5] Rai, M. M., and Chakravarthy, S., An Implicit Form for the Osher Upwind Scheme, AIAA Journal 24, pp. 735-743, 986. [6] Rai, M. M., Navier-Stokes Simulation of Rotor-Stator Interaction Using Patched and Overlaid Grids, AIAA Paper 85-59, Cincinnati, Ohio, 985. [7] Lakshminarayana, B., Fluid Dynamics and Heat Transfer of Turbomachinery, Wiley, New York, 996, p. 58. 8

Figure : Processor allocation. Airfoil Rescaling Factor first-stage stator first-stage rotor 46/58 second-stage stator 52/58 second-stage rotor 40/58 third-stage stator 56/58 third-stage rotor 44/58 Table : Airfoil rescaling factors. Coarse Medium Fine Grid Grid Grid H-grid inlet 75x45 00x60 00x60 H-grid airfoil 67x45 90x60 08x72 O-grid airfoil 2x37 50x50 80x60 H-grid outlet 75x45 00x60 00x60 Total 49704 89400 23456 Table 2: Number of grid points 9

Figure 2: Detail of the coarse, medium and fine grids around the row-one airfoil 0

6.0 Pressure coefficient, averaged 6.5 7.0 7.5 8.0 8.5 9.0 6.0 Coarse grid, Cycle 9 Coarse grid, Cycle 20 Coarse grid, Cycle 2 Pressure coefficient, maximum 6.5 7.0 7.5 8.0 8.5 9.0 6.0 Coarse grid, Cycle 9 Coarse grid, Cycle 20 Coarse grid, Cycle 2 Pressure coefficient, minimum 6.5 7.0 7.5 8.0 8.5 Coarse grid, Cycle 9 Coarse grid, Cycle 20 Coarse grid, Cycle 2 9.0 7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4 Axial distance, X/Chord Figure 3: Pressure variation on the row-six airfoil during three consecutive cycles

6.0 Pressure coefficient, averaged 6.5 7.0 7.5 8.0 8.5 Coarse grid Medium grid Fine grid 9.0 6.0 Pressure coefficient, maximum 6.5 7.0 7.5 8.0 8.5 Coarse grid Medium grid Fine grid 9.0 6.0 Pressure coefficient, minimum 6.5 7.0 7.5 8.0 8.5 Coarse grid Medium grid Fine grid 9.0 7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4 Axial distance, X/Chord Figure 4: Pressure variation on the coarse, medium and fine grids 2

Figure 5: Computational grid (every other grid point in each direction shown) 4 3 2 0 Figure 6: Clocking position of second-stage stator airfoils Figure 7: Instantaneous Mach contour distribution 3

Steam turbine Clocking stator2 Efficiency 0 2 3 4 5 Clocking Position Figure 8: Efficiency variation for clocking the second-stage stator. 4

Figure 9: Entropy variation on second-stage stator, clocking 0. Figure 0: Entropy variation on second-stage stator, clocking. Figure : Entropy variation on second-stage stator, clocking 2. 5

Figure 2: Entropy variation on second-stage stator, clocking 3. 6

Figure 3: Entropy variation on second-stage stator, clocking 4. Steam turbine Clocking stator3 Efficiency 0 2 3 4 5 Clocking position Figure 4: Efficiency variation for clocking the third-stage stator. Figure 5: Entropy variation on third-stage stator, clocking. 7

Figure 6: Entropy variation on third-stage stator, clocking 3. Steam Turbine Clocking rotor2 Efficiency 0 2 3 4 5 Clocking Figure 7: Efficiency variation for clocking of second-stage rotor. Figure 8: Entropy variation on second-stage rotor, clocking 2. 8

Figure 9: Entropy variation on second-stage rotor, clocking 4. Steam turbine Clocking rotor3 Efficiency 0 2 3 4 5 Clocking position Figure 20: Efficiency variation for clocking the third-stage rotor. 9

Figure 2: Entropy variation on third-stage rotor, clocking 2. Efficiency Efficiency Clocking Variation Increase second-stage stator 00% 00% third-stage stator 48% 4% second-stage rotor 83% 245% third-stage rotor 28% 245% Table 3: Efficiency variation and efficiency increase. 20