Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, July 1996A9636847, NCC3-413, AIAA Paper 96-2565

Prediction of 3-D unsteady flow in multi-stage turbomachinery using animplicit dual time-step approach

Roger L. DavisUnited Technologies Research Center, East Hartford, CT

Tonghuo ShangPratt & Whitney, East Hartford, CT

John ButeauPratt & Whitney, East Hartford, CT

Ron-Ho NiPratt & Whitney, East Hartford, CT

AIAA, ASME, SAE, and ASEE, Joint Propulsion Conference and Exhibit, 32nd, Lake

Buena Vista, FL, July 1-3, 1996

The application of an implicit dual time-step algorithm for the prediction of 3D unsteady transonic flow throughthe first stage of a high-pressure compressor is presented. The new dual-time step algorithm, in conjunction withan explicit Lax-Wendroff, multiple-grid procedure, greatly accelerates the convergence of the flow-field predictionto a time-periodic solution. The implicit dual time-step procedure has been used to predict 3D unsteady, multistagesolutions significantly faster than a Lax-Wendroff explicit algorithm alone. In addition, it is shown that the dualtime-step scheme is superior to an implicit residual smoothing approach in association with the present explicitsolution algorithm, providing a computational savings of at least a factor of 3. The new dual time-step procedure isused to simulate the unsteady transonic flow through one and a half stages of a gas turbine high compressor.Comparisons are made between the unsteady, time-averaged flow and a multistage steady flow predictionillustrating the unsteady effects on the overall flow. (Author)

Page 1

Prediction of 3-D Unsteady Flow in Multi-Stage TurbomachineryUsing an Implicit Dual Time-Step Approach

Roger L. Davis *United Technologies Research Center

E. Hartford, CT. USA, 06108and

Tonghuo Shang JJohn Buteau f and

Ron-Ho Ni§

Pratt & WhitneyE. Hartford, CT. USA, 06108

Abstract

The application of an implicit dual time-step al-gorithm for the prediction of three-dimensional un-steady transonic flow through the first stage of ahigh compressor is presented. The new dual time-step algorithm in conjunction with an explicit Lax-WendrofF, multiple-grid procedure is shown to greatlyaccelerate the convergence of the flow-field predictionto a time-periodic solution. In the present inves-tigation, the implicit dual time-step procedure hasbeen used to predict 3-D unsteady, multi-stage so-lutions significantly faster than a Lax-Wendroff ex-plicit algorithm alone. In addition, it is shown thatthe dual time-step scheme is superior to an implicitresidua] smoothing approach in association with thepresent explicit solution algorithm, providing a com-putational savings of at least a factor of 3. The newdual time-step procedure is used to simulate the un-steady transonic flow through one and a half stagesof a gas turbine high compressor. Comparisons are

* Senior Principal Engineer, Computational & DesignMethods

* Project Engineer, Computational Fluid Dynamics'Senior Programmer Analyst, Computational Fluid

Dynamics'Manager, Computational Fluid DynamicsCopyright ©United Technologies Corporation, 1996. Pub-

lished by the American Institute of Aeronautics and Astronau-tics, Inc. with permission.

made between the unsteady, time-averaged flow anda multi-stage steady flow prediction illustrating theunsteady effects on the overall flow.

Introduction

The prediction of the unsteady flow effects in axialflow multi-blade row turbomachinery has been the fo-cus of research over the last several years. The maindriving force behind the development of predictiontechniques for these applications has been the needto understand the unsteady flow mechanisms thatlead to aerodynamic performance degradation. Mostof the computational techniques developed to-datefor multi-stage axial flow turbomachinery solve theEuler/Navier-Stokes equations using a time-marchingscheme. Both implicit and explicit procedures havebeen used, each with their advantages and disadvan-tages in terms of computational efficiency.

Implicit techniques, such as those developed byRai [1] and Chen et al. [2], have the advantage of tak-ing large global time-steps in relation to those gov-erned by the local flow. Unfortunately, the advan-tages of alternate direction or direct implicit solutionalgorithms diminish somewhat for application withmulti-block grids typically used for multi-blade rowsimulations performed in a parallel computing envi-ronment due to decomposition of the imp licit solution

matrix and the non-efficient passing of informationbetween blocks.

Explicit techniques, such as those developed byChima [3], Rao and Delaney [4], and Ni et al. [14, 15,16, 17], are easy to implement and require no specialinter-block discretization treatment but are limiteddue to stability considerations by the extremely smalltime-steps set by the smallest grid cell in the compu-tational domain. This stability limit can be some-what overcome through the use of implicit residualsmoothing techniques as demonstrated by Jorgensonand Chima [5]. Other "hybrid" techniques have beendeveloped, such as that by Giles [11], that incorpo-rate a combination of explicit solution techniques incoarse-grid regions and implicit techniques in the fine-grid regions in an attempt to overcome the explicitstability limitation.

Recently, a new approach has been developed byJameson [6] in which explicit techniques with their as-sociated steady-flow multiple-grid convergence accel-eration schemes could be applied to unsteady flows.This procedure, called the implicit dual time-step ap-proach, is a two-level iterative algorithm in whichthe error associated with an approximation of thetime-rate change of the primary variables is drivento zero in an inner iteration for each global time-step. Venkateswaran and Merkle [8] have shown thatthe implicit dual time-step approach is very similarto an implicit approximate factorization technique inwhich a Newton iteration is used to reduce lineariza-tion and factorization errors for each global time-step.Arnone [9, 10] has successfully used the dual time-step approach for the prediction of 2-D unsteady flowsin axial rotor/stator configurations and has demon-strated a savings in CPU of a factor of 30 over apurely explicit scheme.

In this paper, the implicit dual time-step approachis used for the first time in conjunction with aLax-Wendroff/multiple-grid procedure. The new ap-proach is demonstrated using a distributed memoryparallel computing environment for the efficient pre-diction of 3-D unsteady flows in the first stages ofan advanced high compressor including an upstreaminlet guide vane. A comparison between the implicitdual time-step approach and the conventional explicitscheme with and without implicit residual smoothing

will be given. Numerical issues including inner iter-ation convergence, global time-step size, and compu-tational efficiency will be addressed along with a dis-cussion of the unsteady flow physics associated withthe high compressor multi-stage configuration.

Governing Equations

The prediction of turbulent, viscous flows in tur-bomachinery can be performed through the time in-tegration of the time-dependent Navier-Stokes equa-tions. These equations for three-dimensional flow canbe written in conservative form as:6U _dt ~

where

g.)dx dy dz + S

(1)

u =p

pupvpwE

pupu2 +1

puvpuwpuH

pvpuv

pv<pvwpvl

pwpuwpvw

pw2' +1pwH

0TxyTyy

yxThy

(2)

(3)

'y*TZZThz

(4)

where p is the density, u, v, and w are the velocitycomponents in the x—, y—, and z—directions respec-tively, E is the total energy, P is the static pressure,H is the stagnation enthalpy, and S is a vector con-taining any source term components such as thosecontributing to body forces. The vectors, Fu, Gv,and Hv contain the shear stress terms associated with

the viscous flow assumption. In the current approach,Sutherland's law is used to relate the laminar coeffi-cient of viscosity to the static temperature for air andthe turbulent eddy viscosity is obtained from eitherthe Baldwin-Lomax [12] algebraic turbulence modelor Menter's [13] two-equation k-u turbulence model.

In addition, the perfect gas assumption is usedalong with the relations for the total energy, totalenthalpy, and pressure:

Z71 .J2/ —

H = h+±(v? + v2 + w2)

P = pRT

Numerical Solution Procedure

(5)

(6)

(7)

(8)

called the dual time-step technique, that allows ex-plicit, multiple-grid time-marching procedures to beused as the basic integration scheme with very largetime-steps. Arnone [9, 10] has shown the advantageof using this approach in conjunction with an explicitRunge-Kutta, multiple-grid code. The dual time-steptechnique in conjunction with the previously usedLax-Wendroff, multiple-grid scheme developed by Niet al. [14, 15,16, 17] will now be described along withits application to 3-D transonic flow in a multi-stagehigh compressor.

Implicit Dual Time-Step Procedure

The basis of the dual time-step approach is the ad-dition of a pseudo-time-rate derivative of the primaryvariables to the left hand side of Eq. 1 in the followingform:

For steady flow solutions in multi-blade-row tur-bomachinery, Eq. 1 has previously been solved withan explicit, multiple-grid [14, 15, 18] algorithm as-suming a constant CFL stability number usually setto 0.8. For the simulation of unsteady flows, themultiple-grid convergence acceleration scheme is re-placed with a time-accurate implicit residual smooth-ing scheme, similar to that implemented by Jorgensonand Chima [5] assuming a constant time-step over theentire domain. The global fixed time-step typicallyused with this implicit residual smoothing approachis 50 times the minimum time-step in the entire do-main (or a CFL number of 50). Unfortunately, evenwith a CFL number of 50, the computational timesrequired to predict time-periodic solutions in multi-stage turbomachinery can be unacceptably large.

The current strategy to increase the time-step sizefor multi-stage simulations is to move to a point-implicit algorithm. This type of algorithm avoids anyissues related to discretization of the governing equa-tions across block boundaries and greatly expeditesthe execution of multi-stage simulations with dis-tributed memory parallel computer systems. Jame-son [6] has developed a new point-implicit approach,

dU_dr

Fv)dt dxd(Gj+Gv)

dy dz ^ + 5 (9)

where r is the pseudo-time. The true time-ratederivative of the primary variables, which has beenmoved to the right hand side of Eq. 9, is discretizedwith a second-order accurate backward finite differ-ence.

dU~dt (10)

This discretization requires that the primary vari-ables at two additional previous levels of time bestored in addition to the current values.

Inner Iteration Integration Algorithm

For each global time step, an explicit, Lax-Wendroff (Taylor-Galerkin), time-marching proce-dure [14, 18] is used to iteratively solve Eq. 9 ac-cording to the Taylor series approximation:

(11)

where T is again, pseudo-time and Ar is the pseudo-time-step. Local pseudo-time-steps and the multiple-grid convergence acceleration scheme [14] are usedin this inner iteration of the current dual time-stepprocedure to accelerate convergence of the inner iter-ation. This approach was used for all of the solutionsin the current investigation. A second inner iterationapproach in which a fixed pseudo-time-step can beused with an implicit residual smoothing scheme [5]to solve Eq. 9 has also been tried but has not beenfound to provide any advantage over the multiple-gridapproach.

The second term on the right hand side of Eq. 11,the first-order pseudo-time rate change, is obtainedthrough an integration of the flux (spatial derivative)terms of Eq. 9 over primary control volumes con-structed from a grid meshing of the computationaldomain. The primary variables, U, are located atthe intersection of the grid mesh lines (nodes). Thefirst-order pseudo-time rate change at the grid nodesis obtained from an averaging of the adjacent pri-mary control volume values. In addition, the truetime-derivative of the primary variables is obtaineddirectly at the nodes from the discretization given byEq. 10.

The third term on the right hand side of Eq. 11,the second-order pseudo-time rate change, could beobtained directly at the grid nodes from an integra-tion of the pseudo-time-derivative of the flux terms ofEq. 9 over a secondary control volume which is con-structed by connecting the centroids of the adjacentprimary control volumes. Instead of using this ap-proach, however, the second-order pseudo-tune ratechange is calculated from each primary control vol-ume in a piece-wise manner. Viscous effects are in-cluded by a similar piece-wise integration of the shearstress and conduction terms of the Navier-Stokesequations around the secondary control volumes. Inthe current approach, the turbulent viscosity appliedto the viscous terms of the Navier-Stokes equationsis computed during the first inner iteration of thedual time-step procedure and is frozen for all furtherinner iterations in order to enhance the convergenceof the inner iteration. As will be shown in the cur-rent results, this treatment has only a minor effect onthe solution accuracy. The second-order pseudo-timerate change of the true primary variable time deriva-

tive associated with the dual time-step algorithm isneglected.

Finally, in order to capture shocks and eliminateoscillations in the flow field in regions of coarse gridspacing, a numerical smoothing model which consistsof a combination of second and fourth difference op-erators [7] are added to the right-hand side of thenumerical approximation of the Navier-Stokes equa-tions.

Thus, the overall integration of the flux terms ofEq. 9 is performed completely as a cell-based opera-tion in which the first-order pseudo-time-rate changesare computed and then transported along with thepiecewise contribution of the second-order pseudo-time-rate changes and the viscous effects to the gridnodes. This feature enables domain decomposition tobe applied as necessary for distributed memory par-allel computing with a minimal amount of informa-tion exchanged (only at the interface itself) betweenprocessors. The inner iteration of the dual time-stepscheme is then used to balance these flux terms withthe discretized true time-derivatives of the primaryvariables at each node.

Boundary conditions are applied at the end of eachexplicit and multiple-grid cycle of the inner iteration.No-slip and adiabatic wall conditions are used on allof the solid boundaries. Giles [19] two-dimensional,steady non-reflecting freestream boundary conditiontreatment is used at each spanwise grid location tohold the upstream stagnation pressure, stagnationtemperature, and flow angles as well as the down-stream static pressure. At the inter-blade-row bound-aries where the computational grid of each rotor slides(rotates) relative to the adjacent stator grid(s), thepseudo-time-rate change of the primary variables areinterpolated from the adjacent blade row and addedto the time-rate changes computed from the Lax-Wendroff treatment.

In order to accelerate the convergence of the inneriteration, the primary variables are first initializedusing a quadratic extrapolation based upon the threeprevious time-steps using the formula:

UT =-AT _4r/T-2Ar

(12)

Approximately 50% of the computational time sav-ings resulting from the dual time-step approach canbe attributed to this extrapolation. The dual time-step scheme could be viewed as a predictor/correctoralgorithm with the extrapolation serving as the pre-dictor and the inner iteration as the corrector..

In the current approach, the inner iteration is con-sidered converged when the average residual (righthand side) of Eq. 9 becomes less than 1 x 10~6. Nu-merical investigations have shown that for some con-figurations, a savings in computational time can berealized without a loss in accuracy by relaxing the in-ner iteration convergence tolerance somewhat. Otherconfigurations, including the high compressor geome-try shown below, yield little savings with an increasein the convergence criteria. Further work is neces-sary to determine an optimal inner iteration conver-gence criteria that is universal to all configurationsand flows.

Time-Step Restriction for Stability

As shown by Jameson [6] from a stability analy-sis, there is a limit to the pseudo-time-step size thatcan be used in Eq. 11 for the explicit, multiple-gridinner iteration procedure. The physical rational be-hind this time-step limitation is that the distance awave can propagate in the local, inner iteration pro-cedure must be less than that allowed by the global,or true time step. This relationship can. be given bythe stability equation:

GridRegion

Grid NumberDimensions of Points

2CFL At (13)

As shown in Eq. 13, the pseudo-time-step is signif-icantly limited by the multiple-grid level used by theinner, iteration due to the ability of the convergenceacceleration procedure to propagate waves much fur-ther through the domain. As a result, the use ofmultiple-grid in the inner iteration of the dual time-step scheme has been found to be only advantageousfor two-dimensional flows or when very large globaltime-steps are used.

ResultsModel problem calculations have been performed

Inlet Guide VaneRotor

Rotor Tip ClearanceInlet Guide Vane

113 x 33 x 49 365,442113 x 41 x 49 227,017

73 x 9 x 9 5,913113 x 33 x 49 182,721

Total 781,093

Table 1: Computational Grid Densities for EachBlade-Row of High Compressor Simulation

with the new dual time-step algorithm in an inves-tigation of computational efficiency and solution ac-curacy. In addition, several unsteady, multi-blade-row simulations have been successfully executed withthe implicit, dual time-step approach and the solu-tions have been compared with the previously usedexplicit procedure (with implicit residual smoothing)to ensure solution accuracy. One of these cases, forviscous, turbulent flow through the first stage of ahigh compressor, including the inlet guide vane, isshown below. In the inviscjd and viscous simulationsexecuted thus far, the dual time-step approach hasprovided both superior computational efficiency andnumerical stability in terms of robustness comparedto the previously used explicit algorithm with implicitresidual smoothing.

High Compressor 1-| Stage Simulation

Figure 1 shows the geometry and computationalgrid used for an unsteady simulation for the first stageof a high compressor, including an inlet guide vane.The solution of the unsteady flow in the true geome-try would require the inclusion of 76 inlet guide vanes,40 rotors, and 62 stators. Since an unsteady solutionfor all of these passages would require excessive so-lution time and computer resources, the current sim-ulations have been reduced in scope to include twoinlet guide vanes, one rotor, and one stator. The in-let guide vanes and downstream stator were scaledin the axial and circumferential directions in orderto achieve the same blade-to-blade pitch of the ro-tor. Table 1 lists the computational grid densities foreach blade-row. The average inlet Mach number andReynolds number to the inlet guide vanes is 0.44 and4.5 x 105, respectively. At this flow condition, the

17.0

11.0-1.0 1.0 3.0 5.0

Blade Surface Grid

-2.0-2.0

Midspan Section Blade-to-Blade Grid

Figure 1: Computational Grid of High Compressor

relative inlet Mach number to the rotor becomes su-personic from midspan to the tip and a rotor leadingedge bow shock propagates upstream and interactswith the inlet guide vanes. The need to efficientlypredict the unsteady aerodynamics associated -withinteractions such as this has led to the current inves-tigation into the acceleration of the existing unsteadyflow solution procedure through the use of the dualtime-step approach.

All of the simulations in this investigation wereexecuted with four SUN SPARC 20 workstations inparallel with one workstation per blade passage. Asteady, multi-blade-row simulation of the inlet guidevane/rotor/stator simulation was executed prior tothe unsteady flow simulation for comparison with theunsteady flow predictions and to initialize the un-steady flow simulations. The Bald win-Lomax [12]turbulence model was used to compute the turbulentviscosity for all of the simulations shown.

A comparison of the computational efficiency of thedual time-step scheme with the Lax-Wendroff explicitalgorithm with and without the use of implicit resid-ual smoothing is given in Table 2. As shown in thisTable, the average number of iterations to reach con-vergence of the inner iteration increases as the globaltime-step size increases. Fortunately, the increase inthe average inner iteration count is more than offsetby the reduction in the number of global time-stepsused during a global cycle, thus leading to an over-all savings of CPU time as the global time-step sizeis increased. Table 2 shows that the current dualtime-step approach is capable of achieving very largetime-steps. Reduction in the number of time-stepsper global cycle below 100 for this high compressorconfiguration resulted in non-convergence of the in-ner iteration and eventual divergence of the overallprocedure due to the dispersion error. Other geome-tries have been executed with as few as 50 time-stepsper global cycle. However, as will be discussed below,care should be taken to use a small enough time-stepto properly resolve the various wave frequencies in thedomain in order to insure temporal accuracy of thesolution. Thus with the dual time-step approach, thetime-step size for unsteady flow simulations such asthis are no longer limited by stability considerations,but instead are now limited by the temporal resolu-tion of the pressure and vorticity wave propagationnecessary to achieve a desired level of unsteady flowsolution accuracy.

Figure 2 shows the history of the lift of each air-foil as a function of the global (outer) iteration num-ber for the explicit/residual smoothing algorithm andthe dual time-step approach. A total of 12,000 time-steps per global cycle were used in the explicit schemewhereas 100 time-steps per global cycle were usedin the dual time-step simulation. This figure shows

6

Algorithm

ExplicitExplicit w/ Residual Smoothing

Implicit Dual Time-Step3)

)J

n

n

Iterations/Global Cycle

600,00012,0001,000500400200100

CFLNumber

0.850600

1,2001,5003,0006,000

Average Number ofInner Iterations

N/AN/A

1625283552

CPU/Passage/Cycle (Hrs)

47009411974655232

Table 2: Comparison of Computational Efficiency between Dual Time-Step and Explicit Algorithms

that the pressure history of the dual time-step sim-ulation agrees well with that of the explicit/residualsmoothing prediction. The initial transients in theflow are predicted by the dual time-step scheme tobe the same as the explicit/residual smoothing ap-proach. The amplitude of the lift is predicted by thedual time-step scheme using 100 time-steps per globalcycle to be slightly less than the explicit solution forall of the airfoils. Figure 2 shows that the stator liftis the last to reach some level of convergence to atime periodic solution, as expected, since it incursall of the wakes from the upstream blade rows. Theinlet guide vane and rotor lift, however, converge toa time-periodic solution within 10 cycles. Furthercycles are probably necessary, however, for the aero-dynamic losses to become time-periodic.

A detailed comparison of the time-history of the fi-nal 4 cycles of the inlet guide vane, rotor, and statorlift for the explicit as well as the dual time-step simu-lations using 100, 400, and 1000 time-steps per globalcycle are shown in Fig. 3. The lift time-histories ofall of the blade-rows for the dual time-step solutionsagree well with each other demonstrating that 100 it-erations per global cycle corresponds to a time-stepsize small enough to properly resolve the unsteadysurface pressures. In addition, the dual time-step so-lutions match well with the explicit solution for theinlet guide vane and the rotor except for a small dif-ference in lift amplitude. Larger differences in liftamplitude between the explicit and dual time-stepsolutions exist for the stator. These lift amplitudediscrepancies could possibly be due to the strategyof freezing the turbulent eddy viscosity in the inner

iteration of the dual time-step procedure. The largeramplitude differences for the stator could also be dueto a lack of convergence. As more blade-rows areincluded in the unsteady simulations, increasing lev-els of unsteadiness become apparent in the flow ofthe later rows, necessitating more global iterations toreach a time-periodic solution.

Figure 4 shows the time-averaged pressure distri-bution at the midspan of the rotor for the standardexplicit (with residual smoothing) and dual time-step(100 iterations per cycle) solutions. The two pressuredistributions agree well with only minor differencesexisting primarily on the suction surface. Again,these small differences could be due to the differentstrategies used to compute the turbulent viscosity be-tween the two numerical approaches.

The steady and time-averaged pressure distribu-tions as well as the unsteady pressure envelope atmidspan of each airfoil for the dual time-step solu-tion using 100 time-steps per global cycle is shownin Fig. 5. The minimum (maximum) pressure distri-butions represent the minimum (maximum) pressureoccurring at each point on the airfoil surface duringthe coarse of one global cycle. The difference betweenthe minimum and maximum pressure distributions inthe figure are an indication of the level of unsteadinessin each blade row due to the interaction with adja-cent blade rows. In addition, the difference betweenthe time-averaged and steady pressure distributionsare also an indication of the non-linear effects of theunsteady interaction.

The instantaneous pressure and entropy contours

16.0

8.0-

IGVLIFT

.0

-8.0

EXPLICITDTS (100 ITS/CYCLE)

4.0 8.0CYCLE

12.0

92.0

84.0

ROTORLIFT

76.0

68.0

—— EXPLICIT- - - DTS (100 ITS/CYCLE)

4.0 8.0CYCLE

12.0

-63.Q.

-68.0-

STATORLIFT

-73.0-

-78.0'

- EXPLICIT• - DTS (100 ITS/CYCLE)

4.0 8.0CYCLE

12.0

16.0

8.0

IGVLIFT

.0

16.0-8.0

10.0

92.0

84.0-

ROTORLIFT

76.0

16.068.0

10.0

-63.0-

-68.0-

STATORLIFT

-73.0'

16.0 10.0

EXPLICIT- - - DTS (100 ITS/CYCLE)- - - - DTS (400 ITS/CYCLE)- - - DTS (1000 ITS/CYCLE)

11.0 12.0CYCLE

13.0

• EXPLICIT- - - DTS (100 ITS/CYCLE).... DTS (400 ITS/CYCLE)- - - DTS (1000 ITS/CYCLE)

11.0 12.0CYCLE

13.0

• EXPLICIT- - - DTS (100 ITS/CYCLE).... DTS (400 ITS/CYCLE)- - - DTS (1000 ITS/CYCLE)

11.0 12.0CYCLE

13.0

14.0

14.0

14.0

Figure 2: Time-History of Lift for IGV, Rotor, and Figure 3: Detailed Time-History of Lift for ExplicitSta.tor and Dual Time-Step Simulations with 100, 400, and

1000 Time-Steps per Global Cycle

1.10-1.60'

1.20-

P/Pr

.80-

.40

DTS (100 ITS/CYCLE)EXPLICIT

1.00 2.00 3.00

.90-

P/Pr

.70-

- STEADY--- TIME-AVERAGE- - - - MINIMUM UNSTEADY- - - MAXIMUM UNSTEADY

.50-.50

1.60-1

Inlet Guide Vane

.00 .soX

1.00

4.00

Figure 4: Comparison of Rotor Midspan Time-Averaged Pressure Distribution between Explicit andDual Time-Step Solutions

1.20'

P/Pr

at the midspan of the flow path resulting from thedual time-step solution (100 iterations per cycle) isshown in Fig. 6 to further illustrate the various inter-action mechanisms occurring in the high compressorconfiguration. The pressure contours show the bowwave extending upstream of the rotor leading edgeinto the inlet guide vane passages. It is this waveand the potential interaction with the rotor that isresponsible for the unsteadiness in the inlet guidevanes shown in Fig. 5. The migration of wakes tothe downstream blade-rows, illustrated by the en-tropy contours in Fig. 6, is responsible for much ofthe unsteadiness in the rotor and stator flow-fields.The potential interaction between the rotor and sta-tor is not as strong as between the rotor and inletguide vane as illustrated in the pressure contours ofFig. 6 and in the pressure distributions of Fig. 5.

The results from this investigation demonstrate thecapability of the new dual time-step algorithm to ef-ficiently predict 3-D unsteady flows in multi-stageturbomachinery consistent with the previously usedexplicit/residual smoothing approach. The point im-plicit nature of the dual time-step scheme and its abil-ity to use very large time-steps make it an attractive

.80

.40

- STEADY--- TIME-AVERAGE- - - - MINIMUM UNSTEADY- - - MAXIMUM UNSTEADY

Rotor

1.001.40

2.00 3.00X

4.00

1.00-

.60-

.20

STEADYTIME-AVERAGEMINIMUM UNSTEADYMAXIMUM UNSTEADY

Stator

4.00 7.00S.OO 6.00X

Figure 5: Steady and Time-Averaged Pressure Distri-bution and Unsteady Pressure Envelope at Midspanof Airfoils

7.0

-1.0-2.0 2.0 6.0

X

Entropy Contours

Figure 6: Instantaneous Pressure and Entropy Con-tours at Midspan of Airfoils for Dual Time-Step So-lution

and robust approach for these simulations.

Summary

A point implicit, dual time-step algorithm hasbeen implemented for the simulation of unsteadyflow in multi-blade-row gas turbine turbomacbinery.For each global time-step, a Lax-Wendroff explicit,

multiple-grid solution procedure is used to reach con-vergence of an inner iteration. The results of the cur-rent investigation show that a significant savings incomputational time can be achieved with the dualtime-step technique as compared with the traditionalexplicit, Lax-Wendroff algorithm with and withoutthe use of implicit residual smoothing. This studyalso has shown the effect of varying global time-stepsize on the solution times and temporal solution accu-racy in conjunction with the dual time-step approach.As a demonstration of the prediction capability of thenew approach, the results of an unsteady flow simu-lation for the first one and a half stages of a highcompressor have been shown.

AcknowledgementsThis effort was supported by NASA Lewis Research

Center as part of the NASA Cooperative AgreementNCC3-413 on Scalable Distributed Parallel SystemsSoftware for High-Performance Defense Applications.The authors would like to thank Dr. Andrea Arnonefor his helpful discussions concerning the dual time-step approach.

References

[1] Rai, M. M., "Three-Dimensional Navier-StokesSimulations of Turbine Rotor-Stator Interac-tion," AIAA Journal of Propulsion and Power,Vol. 5, May-June, 1989.

[2] Chen, J. P., Celestina, M. L., and Adamczyk, J.J., "A New Procedure for Simulating UnsteadyFlows Through Turbomachmery Blade Rows,"ASME Paper 94-GT-151, June, 1994.

[3] Chima, R. V., "Viscous Three-DimensionalCalculations of Transonic Fan Performance,"AG ARD 77th Symposium of the Propulsion andEnergetics Panel, CFD Techniques for Propul-sion Applications, (also NASA Technical Mem-orandum 103800), May, 1991.

[4] Rao, K. V. and Delaney, R. A., "Investigationof Unsteady Flow Through a Transonic TurbineStage, Part I, Analysis," AIAA Paper 90-2408.

[5] Jorgenson, P. C. E. and Chima, R., "An Ex-plicit Runge-Kutta Method for Unsteady Ro-tor/Stator Interaction," AIAA Paper 88-0049.

10

[6] Jameson, A., "Time Dependent Calculations Us-ing Multigrid with Applications to UnsteadyFlows Past Airfoils and Wings," AIAA Paper91-1596.

[7] Jameson, A., Schmidt, W., and Turkel, E., "Nu-merical Solutions of the Euler Equations by Fi-nite Volume Methods Using Runge-Kutta Time-Stepping Schemes," AIAA Paper 81-1259.

[8] Venkateswaran, S. and Merkle, G. L., "DualTime Stepping and Preconditioning for Un-steady Computations," AIAA Paper 95-0078,January, 1995.

[9] Arnone, A. and Pacciani, R., "Rotor-StatorInteraction Analysis Using The Navier-StokesEquations and A Multigrid Method," ASME Pa-per 95-GT-117, June, 1995.

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