parallelcomputationofrotor-statorinteractiondayton.tamu.edu/pprs/1997cizmasravi.pdfthrough bilinear...

PARALLEL COMPUTATION OF ROTOR-STATOR INTERACTION

PAUL CIZMAS

Westinghouse Electric Corporation

Science & Technology Center

Pittsburgh, Pennsylvania 15235-5098

AND

RAVISHANKAR SUBRAMANYA

Pittsburgh Supercomputing Center

Pittsburgh, Pennsylvania 15213-2683

Abstract.

Numerical simulation of rotor-stator interaction is crucial for turboma-chinery improvement since it allows the optimization of blade path design.In addition, the numerical simulation of rotor-stator interaction makes itpossible to predict the location and variation of hot spots on blades, whichis extremely useful in the turbine thermal design. However, simulation ofrotor-stator interaction is notorious for being computationally expensive.To reduce the turnaround time and cost/MFLOP, a parallel code basedon the message-passing interface libraries was developed. This code runson symmetric multi-processors (Silicon Graphics Challenge) and massivelyparallel processors (Cray T3E). The parallel code was used to computethe rotor-stator interaction in two turbine configurations. A super-linearspeedup of 15.6 was obtained on a 12 CPU Silicon Graphics Challenge fora geometry distributed over ten processors.

1. Introduction

The need for increased performance and improved reliability in turboma-chinery has motivated designers to identify and evaluate unsteadiness ef-fects. An important part of the unsteady effects in turbomachinery is therotor-stator interaction. The main sources of unsteadiness present in therotor-stator interaction are potential flow interaction and wake interaction.

2 PAUL CIZMAS AND RAVISHANKAR SUBRAMANYA

Additional sources of unsteadiness are hot streak interaction, vortex shed-ding, shock/boundary layer interaction and flutter.

Potential flow interaction is a purely inviscid interaction due to thepressure variation caused by the relative movement of the blades and vanes.The effect of potential flow interaction increases if the rotor-stator gapdecreases and if the flow is supersonic. The unsteady forces developed onthe blades due to the potential flow interaction can be as large as 20% ofthe steady forces [14].

Wake interaction is the unsteadiness generated by the vortical and en-tropic wakes shed by one or more upstream rows. These wakes interactwith the downstream airfoils which are moving relative to the wakes. Wakeinteraction is the primary contributor to unsteady forces on the blade fora large rotor-stator gap. In this case, the unsteady forces on the blade canbe as large as 5% of the steady forces. Wake interaction induces intermit-tent boundary layer transition which causes a variation of the heat transferbetween a low, laminar value to a high, turbulent value [2]. Consequently,the mean heat transfer and the mean overall loss of the blade increase.

Experimental and numerical investigations of rotor-stator interactionwere developed to understand and control the governing physical mech-anisms. Dring and his colleagues have studied experimentally the aero-dynamic interaction between the blades and vanes of a large scale axialturbine. The turbine facility was assembled in both single stage [5] and 11

2-

stage [6, 7, 8, 9] configurations with various axial spaces between adjacentrows. These experimental results show that reducing the axial gap from65% of the blade axial chord to 15% has a strong effect on the unsteadypressure field.

The effects of axial spacing on the aerodynamic characteristics of aturbine were also experimentally investigated and numerically simulatedby Dunn and his colleagues [10]. The results of this investigation werein agreement with Dring’s conclusions, namely, increasing the axial gapbetween vane and blade reduces the unsteadiness in the pressure fields ofboth rows.

Huber et al. experimentally investigated vane indexing of the space shut-tle main engine turbine [20]. The experimental results showed a 0.8% ef-ficiency variation for the clocking positions investigated. The highest effi-ciency was obtained when the wake of the first vane was aligned with thesecond vane’s leading edge. A two-dimensional numerical analysis [15] ofthe flow at the turbine midspan was also performed using STATOR2 code[16]. The numerical simulation predicted correctly the maximum efficiencyindexing positions of the first and second stage vanes. However, the effi-ciency yielded by the numerical simulation was higher than the efficiencyobtained from experiment. In addition, the efficiency variation as a func-

Parallel Computation of Rotor-Stator Interaction 3

tion of clocking position was smaller in the case of the numerical simulationthan in the case of experiment.

In spite of the differences which still exist between experimental dataand numerical results, the numerical simulation of rotor-stator interactionis a useful tool for turbomachinery improvement. The numerical simulationof unsteady flows in turbomachinery started with the work of Erdos et al.[11] who solved the inviscid, compressible, two-dimensional, unsteady flowon a blade-to-blade stream surface through a compressor stage. The firstthree-dimensional model of the rotor-stator interaction was solved by Koyaand Kotake [24]. Similarly to Erdos code, Koya’s code used phase-laggedperiodic boundaries. The first spatially periodic boundary conditions, aswell as the first model using Navier-Stokes equations to model the rotor-stator interaction, were introduced by Rai [28] in 1985. Since then, nu-merous computer codes for rotor-stator interaction simulation have beendeveloped, especially in the last two years [1, 12, 22, 26, 27, 31].

Some of the unsteady multi-row codes solve the two-dimensional flow[12, 13, 21, 25, 27, 28] and some solve the three-dimensional flow [1, 22,26, 30, 31]. The computation time for solving the three-dimensional rotor-stator interaction problem is very large, especially if airfoil clocking is thefinal goal. In the case of a 11

2-stage turbine, with the simplest vane/blade

count of 1:1:1, the computation time reported by Gundy-Burlet and Dorney[19] is about 21 days on the Cray YMP C90. Obviously, the computationtime becomes prohibitive for a realistic vane/blade count.

To reduce the high turnaround time and the associated cost of such asimulation, a parallel code was developed for the more cost effective mas-sively parallel processing platforms. This code is based on the STATOR2code developed at NASA Ames [16]. Similarly to Eulitz’s code [12], the codepresented in this paper uses message-passing interface (MPI) libraries.

The next section briefly presents the numerical method inherited by theparallel code from the sequential code. Details about the parallel compu-tation are given in section three. Section four presents some rotor-statorinteraction results for two turbine configurations. Conclusions and futurework suggestions are covered in the last section.

2. Numerical Model

The two-dimensional, unsteady, compressible flow through a multistageaxial turbomachine with arbitrary blade counts is modeled by using theNavier-Stokes/Euler equations. The computational domain associated witheach airfoil is divided into an inner region, near the airfoil, and an outerregion, away from the airfoil. The thin-layer Navier-Stokes equations aresolved 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.Some viscous effects are present in the outer region due to the wakes comingfrom the upstream rows. However, the numerical dissipation in the outerregion is much larger than the viscous effects, even if the Euler equationsare replaced by fully viscous Navier-Stokes equations [15].

The integration method of the Euler/Navier-Stokes equations writtenin strong conservation form is a third-order-accurate, iterative, implicit,upwind scheme. The method is presented in detail in the paper by Rai andChakravarthy [29]. The nonlinear finite-difference approximation is solvediteratively at each time level using an approximate factorization method.Three Newton iterations are used at each time step to reduce the associatedlinearization and factorization error.

2.1. GRID GENERATION

Two types of patched and overlaid grids are used to discretize the flow fieldsurrounding the rotating and stationary grids. An inner “O” grid is used toresolve the viscous effects near the airfoil. The “O” grid which discretizesthe Navier-Stokes equations is generated using an elliptical method. Anouter “H” grid is used to resolve the Euler equations away from the airfoil.The “H” grid is algebraically generated. The “O” and “H” grids are over-laid. The flow variables are communicated between the “O” and “H” gridsthrough bilinear interpolation. The outer grids corresponding to consecu-tive blades and vanes slip past each other to model the relative motion.

2.2. TURBULENCE MODEL

The flow is assumed to be fully turbulent. The eddy viscosity is computedusing Baldwin-Lomax model and the kinematic viscosity is computed usingSutherland’s law. The Baldwin-Lomax turbulence model is written in astationary reference frame for the vanes and in a rotating reference framefor the blades [16].

2.3. BOUNDARY CONDITIONS

Since multiple grids are used to discretize the Euler/Navier-Stokes equa-tions, two classes of boundary conditions must be enforced on the gridboundaries: natural boundary conditions and zonal boundary conditions.The natural boundaries include inlet, outlet, periodic and the airfoil sur-faces. The zonal boundaries include the patched and overlaid boundaries.

Boundary conditions enforced at the airfoil surface are: the “no slip”condition, the adiabatic wall condition, and the zero normal pressure gra-dient condition. The adiabatic wall condition and the pressure gradient


condition are implemented in an implicit manner. For the blade, the “noslip” boundary condition enforces that the fluid velocity at the blade surfaceis equal to the rotor speed [28].

The inlet boundary conditions specify flow angle, average total pressureand upstream Riemann invariant. The downstream Riemann invariant isextrapolated from the interior of the domain. At outlet, the average staticpressure is specified, while the upstream Riemann invariant, circumferentialvelocity, and entropy are extrapolated from the interior of the domain.Periodicity is enforced by matching boundary conditions between the lowersurface of the lowest “H” grid of a row and the upper surface of the topmost “H” grid of the same row.

For the zonal boundary conditions of the overlaid boundaries, data istransferred from the “H” grid to the “O” grid along the “O” grid’s outer-most grid line. Data is then transferred back to the “H” grid along its innerboundary. At the end of each iteration an explicit, corrective, interpolationprocedure is performed. The outer boundaries of the “O” grids are interpo-lated from the interior grid points of the “H” grids. The inner boundariesof the “H” grids are interpolated from the interior grid points of the “O”grids. This information update is performed using a nonconservative, lin-ear interpolation technique. Stability improves by increasing the overlaparea of the “O” and “H” grids [16]. The patch boundaries are treated in anonconservative manner as well, using linear interpolation to update databetween the adjoint grids [28].

3. Parallel Computation

The high turnaround time and the associated cost of running a sequentialcode to simulate the rotor-stator interaction is unacceptable for the turbo-machinery design process. To reduce the turnaround time and cost/MFLOP,a parallel code was developed based on the sequential code STATOR2 [16].The parallel code uses message-passing interface (MPI) libraries and runson symmetric multi-processors (Silicon Graphics Challenge) and massivelyparallel processors (Cray T3E). The parallel code should run with minormodifications on all platforms that support MPI.

The development of the quasi three-dimensional parallel code was donesuch that a three-dimensional parallel version can be an easy extension. As aconsequence of this requirement, one processor was allocated for each airfoilin the two-dimensional simulation. Consequently, the number of processorsnecessary for a typical three-dimensional turbomachinery configuration willnot exceed the number of processors available on today’s computers.

The processors allocation is presented in Figures 1 and 5. One processoris allocated for each inlet and outlet “H” grid. One processor is allocated


Figure 1. Processors allocation for sin-gle stage geometry.

Figure 2. Grid of single stage geome-try (every other point in each directionshown).

for the “O” and “H” grids corresponding to each airfoil. In the case ofthe single stage turbine, the airfoil count allows that the computationaldomain be reduced to two passages for the vane row and three passages forthe blade row, such that the number of allocated processors is ten.

Interprocessor communication is used to match boundary conditionsbetween grids. Periodic boundary conditions are imposed by cyclic com-munication patterns within rows. Slip boundary conditions are imposedby gather-send receive-broadcast communication patterns between adjacentrows. Load imbalance issues need to be considered at grid generation timeto reduce synchronization overhead.

4. Results

In this section, results of using the parallel code are presented for twoturbine configurations. In the first case, single stage turbine results arepresented and the CPU time compared to that of the sequential code. Thesecond geometry investigated is a 11

2-stage turbine.

4.1. SINGLE STAGE TURBINE

The turbine airfoil count allows that the computation domain be reduced totwo vane and three blade passages, as shown in Figure 2. The number of gridpoints is presented in Table 1. For the given “O” grid point distributionsand flow conditions, the average value of the y+ number is 1.

The sequential code STATOR2, which was used to develop the paral-


Grid points

H inlet 2x(30x65)

H vane/ O vane 2x(61x95) / 2x(151x30)

H blade / O blade 3x(53x95) / 3x(151x45)

H outlet 3x(30x65)

Total grid points 65890

TABLE 1. Grid points for single stage turbine.

Serial SGI Octane 397.2x10-6 second/(iteration * grid point)

Serial Cray YMP C90 33.5x10-6 second/(iteration * grid point)

Parallel SGI Challenge 25.5x10-6 second/(iteration * grid point)

Parallel Cray T3E 25.0x10-6 second/(iteration * grid point)

TABLE 2. Wall-clock running time.

lel code presented in this paper, had been extensively calibrated againstexperimental data [15, 16, 17, 18]. As a result, the focus will be here oncomparing the performance of the parallel code to that of the sequentialcode. The parallel code was implemented on a Silicon Graphics Challengecomputer and on a Cray T3E. The sequential code was implemented on aSilicon Graphics Octane and a Cray YMP C90. A comparison of the wall-clock time is presented in Table 2. Parallelization yields better turnaroundtimes on low end symmetric multi-processor (SMP) machines such as Sili-con Graphics Challenge as compared to a serial vectorized code on a CrayYMP C90. The economic benefits of this are readily apparent. Not only isthe turnaround time shorter for the parallel code, but also the computingcost is smaller, since the ratio of the aggregated CPU time on the parallelcode by the CPU time on the sequential code is exceeded by the ratio ofthe sequential CPU time cost by the parallel CPU time cost.

Parallelization of the code resulted in super-linear speedups on the Sili-con Graphics Challenge of 15.57 as opposed to the expected speedup of 10,due to the 10 processors used. The reason for this greater than expectedspeedup is the availability of 10 times the cache of a single processor on theSMP. Increased cache availability leads to faster memory accesses for theCPU and consequent higher MFLOPS.

The Cray T3E is a distributed memory machine and should have greateroverhead for communication than does the Silicon Graphics Challenge whichis a shared memory machine. The faster turnaround time is due to theslight edge in performance of the 21164 Alpha Microprocessor (450 MHz)as compared to the R10000 MIPS processor (200 MHz). Clock speeds do not


Figure 3. Instantaneous Mach contours. Figure 4. Instantaneous entropy contours.

directly translate into sustained MFLOPS due to chip architecture issues,such as functional unit design and on-chip cache sizes.

Instantaneous Mach contours are presented in Figure 3 in order to vi-sualize the velocity distribution. Instantaneous entropy contours, shown inFigure 4, were generated to visualize the airfoil wakes. The wakes of thefirst vanes are chopped by the blades and then interact with the wakes ofthe blades, increasing blades loses.

4.2. 1 1

2-STAGE TURBINE

In this section the flow characteristics of a 11

2-stage turbine are presented.

The specifications of the test turbine are similar to those of the single stagecase except for the pressure ratio. The first two rows of this turbine areidentical to those presented in the previous section. The number of gridpoints of the third row are 97x33 for the “O” grid and 61x65 for the “H”grid. The total number of grid points is 78272. The number of allocatedprocessors is 11, as shown in Figure 5.

Pressure coefficient, Cp, is defined as Cp = (p−p∗inlet)/1

2ρinletU

2m, where

Um is the wheel speed at mid-span. Pressure coefficient variation on theairfoils of each row is presented in Figures 7, 8 and 9. The largest variationbetween the maximum and minimum values of pressure coefficient occurson the suction side of the blade, where the variation is ∆Cp/Cp = 24.5%.

At a rotational speed of 3600 rpm and the given blade count, the timewhile a composite pitch (two vane passages or three blade passages) isswept is T=1.04 ms. During one period T , three blades perturb the flow ina passage of the first row of vanes. Figure 10 shows the pressure variationat 50% axial chord on the suction side of the first vane. As expected, three


Figure 5. Processors allocation for1 1

2-stage geometry.

Figure 6. Grid of 1 1

2-stage geome-

try (every other point in each directionshown).

0.0 0.2 0.4 0.6 0.8 1.0

Chord

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

Pre

ssu

re c

oe

ffic

ien

t, C

p

Time−averaged

Maximum

Minimum

Figure 7. Pressure coefficient variationon the first vane.

0.0 0.2 0.4 0.6 0.8 1.0

Chord

−3.5

−3.0

−2.5

−2.0

−1.5

−1.0

Pre

ssu

re c

oe

ffic

ien

t, C

p

Time−averaged

Maximum

Minimum

Figure 8. Pressure coefficient variationon the first blade.

perturbations can be observed during one period T , due to the potentialflow interaction with the blades.

The pressure variation at 25% axial chord on the suction side of the firstblade is presented in Figure 11. During one period T one observes two spikesdue to the interaction with two vanes from the adjacent row. The pressurevariation on the blade is more jagged than the pressure variation on the firstvane. This can be caused by the fact that while the first vane is exposedonly to potential flow interaction, the blade is perturbed by additionalsources of unsteadiness such as wakes and vortices shed from the first rowof vanes. Also, the potential flow interaction of the blade is more complexthan the potential flow interaction of the first row of vanes, since the blade


0.0 0.2 0.4 0.6 0.8 1.0

Chord

−6

−5

−4

−3

−2

Pre

ssu

re c

oe

ffic

ien

t, C

p

Time−averaged

Maximum

Minimum

Figure 9. Pressure coefficient variationon the second vane.

41.0 42.0 43.0

Time (ms)

0.65

0.66

0.67

0.68

0.69

0.70

0.71

Pre

ssu

re r

atio

, p

/pin

let

Figure 10. Pressure variation on thesuction side of the first vane, at 50% ax-ial chord.

41.0 42.0 43.0 44.0 45.0

Time (ms)

0.64

0.65

0.66

0.67

0.68

0.69

0.70

Pre

ssu

re r

atio

, p

/pin

let

Figure 11. Pressure variation on thesuction side of the first blade, at 25%axial chord.

42.0 43.0 44.0 45.0

Time (ms)

1.22

1.23

1.24

1.25

1.26

Tem

per

atu

re r

atio

, γT

/Tin

let

Figure 12. Temperature variation onthe suction side of the first blade, at 25%axial chord.

interacts with two adjacent rows of vanes. In Figure 11 it is shown that thepressure pattern repeats after 1.5 ms. Maximum pressure variation at 25%axial chord on the suction side is 7.4%. The temperature variation at thesame point on the blade is shown in Figure 12. The maximum temperaturevariation at this point is 2.3%.

5. Conclusions

This paper presents the development of a two-dimensional, time-accurate,Navier-Stokes equations parallel code from a sequential multiple-grid code.The primary objective of this investigation was to compare the performanceof the parallel code to that of the sequential code. The secondary objective


was to use the parallel code for the numerical simulation of flow in a singlestage and 11

2-stage turbine.

Parallelization of the code resulted in super-linear speedups of morethan 50% for a simulation running on 10 processors. The reason for thisgreater than expected speedup is the availability of more cache on the SMP.Increased cache availability leads to faster memory accesses for the CPUand consequent faster MFLOPS.

6. Acknowledgments

The authors wish to thank the Westinghouse Power Generation and West-inghouse Science & Technology Center for supporting this work. The au-thors are thankful to Pittsburgh Supercomputing Center for making thecomputing resources available. The authors are especially grateful to Dr.Karen Gundy-Burlet of NASA/Ames Research Center for her help in usingthe STATOR2 code and to Ms. Anjana Kar of Pittsburgh SupercomputingCenter for her help in visualizing the numerical results.

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