AIAA-2001-0531Direct Calculation of Turbomachinery FlowsUsing the Space-Time Conservation Elementand Solution Element Method

Hsin-Hua Tsuei and Biing-Horng Liou

Concepts NREC

White River Junction, VT

and

S. T. John Yu

Dept. of Mechanical Engineering

Wayne State University

Detroit, MI

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AIAA-2000-0134

DIRECT CALCULATION OF TURBOMACHINERY FLOWS USING THE SPACE-TIME CONSERVATIONELEMENT AND SOLUTION ELEMENT METHOD

Hsin-Hua Tsuei1 and Biing-Horng Liou2

Concepts NRECWhite River Junction, VT

AndS. T. John Yu3

Dept. of Mechanical EngineeringWayne State University

Detroit, MI

1 Head of CFD Section/Corporate Fellow. AIAA Member.2 CFD Engineer. AIAA Member.3 Associate Professor. AIAA Member.

ABSTRACT

Computation of flows in turbomachinery bladerows is critical to aircraft engine design optimization.Traditional Computational Fluid Dynamics (CFD)techniques require an assumption of steady or unsteadyflow when these flows are known to include both steadyand unsteady effects. The space-time ConservationElement and Solution Element (CE/SE) method is abreakthrough CFD technique, allowing the treatment ofsteady and unsteady problems with the same calculation.Developed originally at NASA Glenn Research Center,this technique has never before been applied toturbomachinery. The current research applies andvalidates this new technique for turbomachinery for thefirst time.

A three-dimensional CE/SE code in cylindricalcoordinates has been developed. For the present study,the blade-to-blade flow field was first investigated toaddress the fundamental issues associated with rotation,and blade-to-blade loading. The algebraic Baldwin-Lomax turbulence model was also implemented to addressthe shock wave-boundary layer interaction phenomenon.This also marked the first time a traditional turbulencemodel has been user together with the CE/SE scheme.

A series of validation cases for the newlydeveloped CE/SE scheme for turbomachinery has beenconducted. Test cases include the blade-to-blade crosssections of a high pressure ratio, single-stage axialcompressor for a commercial small gas turbine engine,NASA Rotor 37, and a single-stage high pressure ratioturbine with a teardrop shaped choked stator. Futuredevelopments include extending the CE/SE scheme to fullthree-dimensional turbomachinery flow calculations,

especially in blade row interaction predictions in multi-stage turbomachines, and secondary flow such as leakagepath and seal calculations.

INTRODUCTION

Computational Fluid Dynamics (CFD) foraircraft engine and other turbomachinery design is nowcommonplace. To be effective for analyzing anddesigning turbomachinery, practical CFD calculationsrequire the capacity to predict the effects of variouscomplex steady and unsteady flow phenomena likeboundary layer formation and growth, flow separation andreattachment, shock wave propagation and reflection, flowoscillations, etc. It is desirable for these CFD models toinclude multiple blade rows and to model accurately theinteraction between the blade rows and the propagationforward and backward through the model of the effects ofmultiple blade rows, as occurs in the actual flow physics.Many CFD algorithms have been developed and are undercontinuous refinement, but the need for a new CFDparadigm applicable to wide classes of aerodynamic andaerothermodynamic flow problems has been wellestablished. Current techniques fall short in the grossapproximations they require to model the blade rowinteraction and in the need to separate spatial andtemporal effects to different parts of the solution.Traditional CFD methods postulate a steady or unsteadyflow field before the calculation is initiated, then based onthe steady or unsteady flow assumption, treat the spaceand time fluxes separately. As a result, a predefinedsteady-state calculation cannot perform for a physicallyunsteady problem, and a presumed unsteady calculationproved to be a waste of computational resource for aphysically stable flow field.

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Although there is the need to calculate moreaccurately all of the complex flow physics inturbomachinery, of particular interest in turbomachineryflow applications is the prediction of rotor-statorinteractions in multistage axial gas turbines. Theunderstanding that has been developed to date, regardingthe rotor-stator interaction phenomenon, has led to manynew design concepts, for both compressors and turbines,to take into account in the engine design the interactionsbetween blade rows. Nevertheless, this job has been mademore difficult because analytical techniques for predictingthese complex interactions have lagged the growing needsfor better understanding of these effects on theturbomachinery stability and performance. One of thereasons why development of sufficiently effectiveanalytical techniques has been difficult is that blade rowinteraction in turbomachinery includes both steady andunsteady effects and it has not been possible to treatsteady and unsteady effects together in the same solution.The proposed analytical method represents a breakthroughin CFD techniques in that it allows, for the first time, theability to treat steady and unsteady problems with thesame calculation. Developed originally at NASA JohnGlenn Research Center (GRC), this technique has neverbefore been applied to turbomachinery. This innovativeSTTR program proposes to apply this new technique toturbomachinery for the first time, offering a step increasein computational accuracy for analysis of turbomachinery.

The design of both compressor and turbine bladerows must account for blade row interaction. Forexample, in a compressor rotor stage, from the leadingedge of the rotor blade, low momentum gas tends tomigrate toward the tip section. The migration of the lowmomentum fluid is a steady blade row interactionphenomenon. If the gap between the rotor and stator istoo large, the low momentum gas quickly fills up the tipsection, causing degradation in the compressorperformance. From this point of view, the gap betweenthe rotor and stator should be kept as small as possible toreach optimum performance. However, because of theunsteady wake generated by the rotor trailing edge, if thegap is too small, the wake will not be well mixed outbefore it enters the stator blade passage. The unsteadywake-induced blade row interaction exhibits unsteadyflow physics. As a result, the pressure and velocitygradients in the wake flow impinging on the stator bladewill cause the stator structure to oscillate. From this pointof view, the gap between the rotor and stator should bedetermined on the rotor performance and blade geometry,instead of being based on the low momentum gasmigration alone. This is an example of where looking atsteady or unsteady physics separately would result in amisleading conclusion. Only when steady and unsteadyeffects are considered together can the correct designdecision be reached.

Wake-induced rotor-stator interaction oftenimpacts the design of a turbine stage as well. Acomprehensive review of the impact of unsteady bladerow interactions on turbine design concepts andprocedures was given by Sharma et al. [1]. The unsteadypressure fluctuation from the wake of the upstream statorcan cause the rotor cooling flow to be unable to exit thecooling holes on the rotor surface as expected, impactingthe effective cooling of the blade. There are severaloptions available to address this problem. First, the gapbetween the stator and rotor can be increased to allowenough mixing in the stator wakes to minimize thepressure oscillation effects on the rotor. However, over-compensating the unsteady fluctuation will again reducethe overall turbine stage performance. Second, thecooling hole location, size, and the amount of coolingflow can be redesigned to accommodate the pressureoscillations. The optimum design needs to seek abalancing point in the cooling effects and overall stageperformance. However, to achieve this delicate balance, adetailed understanding of these flow physics is required.Accurate computation of blade row interaction is one ofthe fundamental issues in turbomachinery performanceand flow field predictions and is clearly needed. Newtechniques for accurately calculating steady and unsteadyeffects in multiple blade rows is an important enablingtechnology for both aerospace and industrialturbomachinery.

The present work addresses this need through thedevelopment and validation of a unique numericalapproach that contrasts strongly with traditional CFDalgorithms. This new CFD technology, the so-called“Space-Time Conservation Element and Solution Element(CE/SE) Method,” or “CE/SE method” for short, wasdeveloped by Dr. Sin-Chung Chang [2,3], a SeniorResearch Scientist at NASA GRC (formally NASA LewisResearch Center). This technique promises a stepincrease in accuracy for blade row interactioncomputations in turbomachinery flows, especially in gasturbines. The proposed space-time CE/SE method is aninnovative breakthrough in CFD technology which treatsthe spatial and temporal fluxes as an integrated union,instead of separating the two by assuming steady orunsteady flows, as is done in traditional CFD schemes. Asa result, a steady or unsteady solution obtained using theSpace-Time CE/SE method depends entirely on the natureof the flow, not on an assumed steady or unsteadycondition. Therefore, time accurate solutions forcomplicated rotor-stator interactions often encountered inturbomachinery flows, including unsteady wake, dynamic(unsteady) shock reflection, and unsteady shock-tipleakage vortex flow interaction, can be obtained withoutadditional presumptive conditions. Further, the proposednumerical model is specifically intended to be general innature, so that little or no manual manipulation of themodeling parameters should be required to achieveaccurate solutions of flows within broad application areas,

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leading to substantial improvements as compared tocurrent CFD technology. For example, it is alreadyproven that greater accuracy can be achieved for “time-accurate” shock wave predictions with the proposedSpace-Time CE/SE technique, when compared to existingCFD methods, while no added artificial smoothing isnecessary [4,5].

The current study seeks not only to develop thismodel, but also to validate its effectiveness and accuracyspecifically for turbomachinery flows which are, without adoubt, some of the most complex of all flow conditions.When fully developed and validated for these complexturbomachinery flows, it is expected that this advancedCFD scheme will also be applicable to a wide range ofother flow conditions. These could include flows aroundairframes, engine ducting, control surfaces, weapons andseparable stores, plus general flows involving suchphenomena as rotation, separation, transition, convectiveheat transfer, skin friction, mixing, and combustion. Thebenefits of having such an improved CFD algorithm willbe more accurate CFD solutions and improvedunderstanding of steady and unsteady blade rowinteraction effects, leading to reduced engine design cycletime, improved design capabilities, and reduction in thenumber of physical model tests required to finalize adesign. The end results of better understanding of bladerow interactions, including how to account for them in thedesign process, could include greater engine stableoperating range, reduced specific fuel consumption,longer engine life, and improved safety. The proposedtechnology will have broad commercial use in CFDprograms for design optimization of aircraft turbineengines, plus a tremendous array of industrial equipment,including industrial compressors, turbochargers, refrigera-tion compressors, turbopumps, and both steam and gasturbines for power generation and marine applications.

Efforts and CFD Techniques on Rotor-StatorInteraction Flow Field Simulation in Turbomachinery

With the advancement of CFD technology andcomputer speed, three-dimensional (3D) CFD calculationshave become a routine part of the engineering designprocess, not only for turbomachinery, but for manyindustrial applications as well, ranging from chemicalmixing processes to automobile engines. Inturbomachinery flows, full-3D CFD analysis, for a singleblade row as well as for multiple coupled blade rows, hasgreatly enhanced our understanding of these complex flowfields and has enabled significant design improvements.Three-dimensional calculations serve as a numericallaboratory to provide visualization of the complicatedturbomachinery flow field. Details of gas turbine flowfield phenomena, such as shock-tip leakage vortexinteraction and unsteady shock-boundary layer structurethrough blade row interaction, have been discovered.With the understanding gained from the full-3D CFD

analysis, the turbomachinery design philosophy has againchanged over the past several years. Multiple blade rowsimulation has enabled the understanding of the influenceof blade row interaction on global machine performance,as well as local flow field characteristics. With this fact inmind, the impact of a better CFD predictive capability onthe overall improvement of the turbomachinery industry isprofound.

Turbomachinery flows, especially in gasturbines, include a variety of complex flow phenomena,including laminar-to-turbulent transition and relaminari-zation, adverse pressure gradients, shock wave and flowdiscontinuity, flow separation, secondary flow mixing,rotor-stator interaction, tip clearance flows, combustion,and heat transfer. Each of these flow characteristics mustbe taken into account in turbomachinery engineering. Incoupled rotor-stator analysis, all of the above mentioned,except combustion, are present in the flow analysis. Inthis proposal, we will limit the focus to non-reactingflows. Integrated chemically reactive combustor/turbineflow analysis using the Space-Time CE/SE scheme isdefinitely of value and might be addressed in the future asa separate technical effort, building on the foundationestablished by the present work.

For rotor-stator interaction computations, acommon thread is the blade row interface treatment.There have been a number of algorithms developed for theblade row interface modeling in rotor-stator interactioncalculations. The inter-row mixing plane method [6,7]assumed that the flow is steady, relative to each individualblade row. This model uses a circumferential averagingtechnique to provide communication between the bladerows. The gradients are smeared out in thecircumferential direction, yet preserved in the radialdirection. This inter-row mixing plane technique providesa fundamental mechanism to simulate rotor-statorinteraction. However, flow information can be lostthrough the averaging process. Obviously, the unsteadywake-induced rotor-stator interaction and its related issuesare not resolved by using this approach..

The average passage technique [8,9], like theproposed CE/SE method was developed at NASA GRC,provides row-by-row solution with sophisticatedaveraging to represent the “missing” rows in the multipleblade row environment. In this approach, steady 3Dequations are solved separately in each blade-to-bladepassage in each of the blade rows of a multistage machine.This approach allows a practical steady 3D methodologyfor multistage turbomachinery computations. Unsteadyeffects caused by neighboring blade rows are averaged outin time, and information about the neglected unsteadyeffects is transported from one row to another. Thecomputational domain of a given blade row is expanded toinclude only the space occupied by the downstream/upstream blade row, however, not the actual blade row

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itself. The effect of pressure difference across a bladesurface and the resultant flow turning is introducedthrough steady body-force for the missing blade row,while the unsteady effect is modeled by using‘deterministic’ stresses. Some of the modeling parametershave been modified specifically for multistage compressorcalculations [10,11].

Unsteady flow analysis using a combined“patched and overlaid” grid method for blade interfacehas been applied to 2D rotor-stator interaction simulationin an axial turbine [12]. This technique splits up theinterface region between blade rows into four zones, thenapplies appropriate interpolation procedures on thesepatched or overlaid grids. Computational results indicatedunsteady pressure existed in between the 2D turbinestator-rotor blades. The computed time-averaged resultscompared well with measured data, but the magnitude ofthe pressure fluctuation did not agree with measured data.This disagreement could be a result of the 2Dcomputational model, which is not appropriate for highly3D rotor-stator interactions.

A quasi-3D rotor-stator analysis was developedfor the first stage of the SSME high-pressure fuelturbopump [13]. In this study, the blade-to-blade planewas analyzed, though the stream surface thickness in thehub-to-tip direction was taken into consideration. Theblade row interface was modeled through a line of overlap“dummy cells” to impose periodic or overlap boundaryconditions. At this overlap gridline, flux variables in thestator are interpolated to obtain rotor quantities, while fluxvariables in the rotor are interpolated to acquire statorflow variables. Unsteady computations for a 1:1 and a 2:3stator-rotor blade ratio were made. The 2:3 blade ratiowas close to that of the real machine. The computationalresults showed that the blade count ratio has an influenceon the unsteady characteristics and fluctuation magnitude.

With the steady increase in availablecomputational power, numerous researchers are nowconducting CFD calculations for unsteady, 3D, fullNavier-Stokes equations for rotor-stator interactions. Thishas increased the understanding of rotor-stator interactionphysics one step further in order to provide a numericalwind tunnel for turbomachinery designs. The effects ofrotor-stator interactions were studied, with the help ofexperimental data, to conduct 3D unsteady CFDcalculations for stator and rotor separately for an axialturbine stage [14]. The measured inlet conditions wereused to calculate the stator flow field. Vortex sheddingfrom the stator blade trailing edge can be observed. Therotor inlet used measured time-variant data downstream inthe stator wake region to proceed the 3D unsteadyoscillation. It was found that the wake and secondaryflow structure in the rotor blade passage were largelyinfluenced by the unsteady inlet oscillation. The numericalsimulation suggested that the secondary flow/blade

interaction might have a dominant effect on unsteadyblade loading and noise production.

Investigation of the rotor-stator interactioncontinued to include the fully-coupled 3D unsteadycalculation for rotor-stator configurations. The unsteadyaerodynamics of a high through flow transoniccompressor stage were studied [15], and the time-averaged computational results were compared withexperimental data. The numerical simulation was limitedto a calculation of a specified blade count ratio for therotor and stator blades, with assumed periodicity on theoutermost tangential direction, instead of modeling theentire 360° compressor stage. Theoretically speaking,modeling the entire stage is possible, although thecomputational cost associated with such a calculation istoo high by today’s standards. To model the interface, thesliding mesh technique was used to providecommunication across the blade row interface. Thesliding mesh was established by using one overlaid gridline, then 2D interpolation for the flow variables andgradients was performed on this plane to transferinformation from rotor to stator, or vice versa. In thisstudy, higher order accurate schemes, both in space andtime, have to be utilized to prevent excessive numericaldissipation, and to provide enough resolution across theblade row interface. Computational results revealed aseparation region near the stator tip section, resulting froma high incidence angle caused by the rotor tip clearanceflow. Because of the unsteady wake, the separation wasnot found to be periodic for each stator blade passage. Asimilar study can be found in [16].

Because traditional CFD methods automaticallyseparate the space and time domains, the difficultiesencountered in modeling the blade row interface rotor-stator interaction are a subject of ongoing research.Research and development efforts have continued over thepast few years to seek better means within the traditionalCFD framework to improve the interface modeling inorder to enhance solution accuracy. A very comprehen-sive review of how these traditional CFD techniques, andtheir limitations when applied to turbomachinery flowpredictions, were given in [17]. Details on how eachindividual technique performed for turbomachinery flowswill not be repeated here.

In contrast to these traditional CFD schemes, theblade row interface boundary condition for the CE/SEscheme is surprisingly simple. As the solution marches inthe space-time domain, the space-time flux and flowvariables are extrapolated into the next time step for boththe rotor and stator. Since space and time fluxes areunited in the space-time domain, the rotor mesh simplyrotates to the next position in both space and time, and thestator mesh stays in the same spatial location, but has anew time frame. For example, for a two-dimensional (2D)rotor-stator mesh, the space-time mesh is 3D, with time

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frame being the third dimension. The 3D space-timemesh changes shape at each time step. Therefore, in thespace-time domain, the blade row interface simply turnsout to be one of the interior solution domains for theCE/SE scheme. As a result, the interface modeling issuesraised by conventional CFD schemes because of rotormotion, such as circumferential averaging or sliding meshand interpolation, do not apply for the CE/SE scheme.The mathematical formulation is given in the theoreticalsection of this proposal to describe this moving frameconcept in CE/SE scheme.

Another additional, and already proven,advantage of the CE/SE method over existing CFDmethods is CE/SE method’s ability to accurately captureshock waves. Shock capturing techniques are also asubject of ongoing research, not only for turbomachineryflows, but in all types of transonic or supersonic flow fieldpredictions. There have been many claims of improvedCFD algorithms which have later disappointed users. Thecommon trend of these CFD schemes is that one or acombination of several artificial damping techniques[18,19] are involved that can mask the real flow physicsby smearing out the discontinuity too much in order toprovide numerical stability. Past efforts at improvedshock wave capturing have historically exhibited poorresults because they improved accuracy only for asimplified flow condition, such as a one-dimensional (1D)shock tube. These techniques have often neglected morecomplicated issues, like shock reflections, shock wave-boundary layer interactions, and the unsteady shock wave-boundary layer structure that results from unsteady rotor-stator interactions in a turbomachinery environment.

The reason traditional shock capturingtechniques faltered was that they were developed underthe assumption that the 1D Riemann variable is valid forall three directions. Therefore, these techniques allappeared in a 1D form, even though they were used forpredicting 3D wave propagation characteristics. Directapplications of 1D based techniques to 3D problemsresulted in distorted 3D wave fronts [20]. In gas turbines,due to blade row interaction, the oblique shock near thetip section of a compressor rotor is often influenced by theunsteadiness resulting from the wake of the upstreamstator and the unsteady pressure oscillations caused bydownstream blade rows. The location of the shock, as aresult, will oscillate near the rotor tip. Existing techniquesare ill suited to predict this behavior, while the CE/SEtechnique has been shown to model this behavior well.

The CE/SE method treats space and timesimultaneously and has demonstrated accuracy inresolving 3D shock waves. The ability of the CE/SEmethod to capture 3D shocks while preserving the flowphysics is unprecedented. The proposed Space-TimeCE/SE method is based on integrated space-time flux(instead of the historically-separated spatial flux terms

plus the temporal flux, if unsteady) using a SolutionElement (SE) constructed on the space-time plane. Thedetails of constructing a unified space-time flux on the SE,based on a Conservation Element (CE) of the Space-Timeplane, will be given in the following section of thisproposal. The proposed CE/SE method was developed tosolve multi-dimensional problems and the importance ofsimplicity and generality weighed heavily in its design.Characteristics-based techniques were avoided toeliminate presumptive conditions for the Navier-Stokesequations.

NUMERICAL FORMULATION

The Space-Time Conservation Element andSolution Element Method, or the CE/SE Method,originally proposed by Chang [2-5], is a new numericalframework for conservation laws. The CE/SE method isnot an incremental improvement of a previously existingCFD method, and it differs substantially from other well-established methods. The CE/SE method has many non-traditional features, including a unified treatment of spaceand time, the introduction of conservation element (CE)and solution element (SE), and a novel shock capturingstrategy without using Rieman solvers. To date, numeroushighly accurate solutions have been obtained [2-5],including traveling and interacting shocks, acoustic waves,shedding vortices, detonation waves, shock/acousticwaves interaction, shock/vortex interaction, andcavitations. The design principles of the CE/SE methodhave been extensively illustrated in the cited references.In this paper, a brief description of the CE/SE method isprovided as the background of the present work.

Perhaps one of the most important features of theCE/SE method is the adoption of a space-time integralformulation as the cornerstone for the subsequentnumerical discretization. Note that one derives theconventional finite-volume methods based on Reynolds’transport theorem, in which space and time are not treatedequally. Thus, the space-time geometry of a finite volumeis restricted such that classical Riemann problems wereencountered in calculating flux conservation. In contrast,due to an equal footing treatment of space and time,Chang’s integral formula allows a better choice of space-time geometry of CEs in calculating flux conservation,and the Riemann problem is avoided.

The Reynolds Transport Theorem

The conventional finite-volume methods forsimulating the conservation laws were formulatedaccording to flux balance over a fixed spatial domain.The conservation laws state that the rate of change of thetotal amount of a substance contained in a fixed spatialdomain V is equal to the flux of that substance across theboundary of V, i.e., S(V). Let the density of the substance

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be u and its spatial flux be f, the convection equation canbe written as

0=⋅∇+∂∂

ft

u rr (1)

According to the Reynolds transport theorem, the integralform of the above equation can be expressed as:

∫∫∫ ⋅+∂∂

==)()(

0VSVtV

sdfudVt

udVdt

d rr(2)

The transport equation provides the bridge between theLagrangian frame the Eulerian frame. The conventionalfinite-volume methods concentrated on calculating thesurface flux, i.e., the last term of Equation (2). The timederivative term of Equation (2) is usually calculated by afinite difference method, e.g., the Runge-Kutta method, ora temporal integration:

∫ ∫∫ ⋅−=2

1

2

1

)()(

t

t VS

t

tVdtsdfudV

rr (3)

Due to the fixed spatial domain, the shape of theSpace-Time Conservation Elements (CEs) in one spatialdimension for Equation (3) must be rectangular. Refer toFigure 1(a). These elements must stack up exactly on thetop of each other in the temporal direction, i.e., nostaggering of these elements in time is allowed. Forequations in two space dimensions, as depicted in Figure1(b), a conservation element is a uniform-cross-sectioncylinder in the space-time domain, and again nostaggering in time is allowed.

t

x

x

y

t

( a )

( b )

Figure 1. Space-time integration forconventional finite-volume methods.

This arrangement results in vertical interfacesextended in the direction of time evolution betweenadjacent space-time conservation elements. Across theseinterfaces, flow information travels in both directions.Therefore, an upwind bias method (or a Riemann solver)must be employed to calculate the interfacial fluxes.The Space-Time Integral Formula

Equation (1) is a divergence free condition, i.e.,

0=⋅∇ hr

, (4)

where the current density vector hr

= ( fx, fy, fz, u).According to the divergence theorem, we have

0)(

=⋅∫ VSsdhvv

(5)

Figure 2 is a schematic for Equation (5) in one spatialdimension.

r + d r

r

d r

d s V

S ( V )

t

x

Figure 2. A schematic of the space-timeintegration [1].

We remark that space and time are treated in an equalfooting manner, and there is no restriction on the space-time geometry of the CEs.

The CE/SE Method

Based on the above integral equation for space-time flux balance, Chang proposed a new numericalframework for conservation laws. The CE/SE method is afamily of schemes, i.e., thea scheme, the a-ε scheme, andthe a-α scheme. The a determines the space-timegeometry of the numerical mesh employed. The a-ε andthe a-α schemes are extensions of the a scheme fornonlinear equations and for shock capturing.

First, the space-time domain of interest is dividedinto many Solution Elements (SEs). In each SE, flowvariables are assumed continuous. A first-order Taylorseries is used by Chang to discretize the flow variables.Thus the scheme is second-order accurate. Across theboundaries of neighboring SEs, flow discontinuities areallowed. Flow variables are calculated through a local

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space-time flux balance, which is enforced by integratingover the surfaces of a Conservation Element (CE). UnlikeSEs, various CEs could be imposed for local and globalspace-time flux balance.

In Chang’s a scheme, the number of the CEsemployed matches the number of unknowns designated bythe scheme. In addition to the flow variables, the spatialgradients of flow variables are also treated as unknowns.As a result, two CEs are used to solve a one-dimensionalconvection equation, Equation (2.1.4), because thevariable u and its spatial derivative ux are the unknowns.Similarly, three CEs are used for two-dimensionalequations, because u, ux, and uy are the unknowns, andfour CEs are used for the three-dimensional convectionequation. As shown by Chang, triangles and tetrahedronsare used as the basic mesh stencil to construct thenecessary CEs for two- and three-dimensional equations.We remark that unlike the modern upwind schemes, flowvariable distribution inside a SE is not calculated througha reconstruction procedure using its neighboring values atthe same time level. Instead, they are calculated as a partof local space-time flux conservation.

In matching the flow solution, the flow variablesat neighboring locations leapfrog each other in time in azigzagging manner. Refer to Figure 2.1.3 for equations inone spatial dimension. This is achieved by imposing thespace-time integration over rhombus-shape CEs. In thiscase, the CEs coalesce with SEs. Through each obliqueinterface between adjacent CEs, flow informationpropagates only in one direction, i.e., toward the future.Therefore, no Riemann problem is encountered and theuse of a Riemann solver to catch shocks is avoided. Theresultant a scheme is non-dissipative (or neutrally stable)for linear waves. One can match the solution from aspecific space-time point first forward in time and thenbackward to recover the initial condition of the flowvariables the a-scheme is space-time inversion [2-5].As pointed out by Chang, conventional schemes lackspace-time inversion due to the inherent artificialdamping.

x

t

Figure 3. A schematic of the CE/SE method in one spatialdimension.

It is noteworthy that the treatment of the non-reflective boundary condition in the CE/SE method is verysimple [2-5]. Usually, one simply extrapolates the flowvariables to the mesh point in the staggered position at afuture time step. In general, nearly perfect non-reflectiveboundary condition can be achieved.

When applied to nonlinear equations, e.g., theEuler equations, the above a scheme is modified byadding artificial damping for numerical stability. Theresult is the a-ε scheme. In particular, within one timematching step, only the calculations of umx, umy, and umz areimpacted, while the calculation of um is kept intact as thatin the original a scheme. Note that m = 1, 2, 3, 4, and 5for the five equations in the Euler equation set in threespatial dimensions. When contact discontinuities appearin the flow solution, the calculation of umx, umy, and umz isfurther modified by a re-weighting procedure to filter outspurious oscillations. The result is the a-α scheme with αas the weighting parameter. Since the original a scheme isneutrally stable, locally, the calculation of u based on thea scheme is not contaminated by artificial damping and/orthe re-weighting procedure. The careful management ofthe numerical damping by Chang contributes to the highaccuracy of the CE/SE method, when the method is onlysecond-order accurate.

However, by adding the artificial damping, theflow variables gradients, calculated by the a-α and a-εschemes, do not satisfy the space-time flux conservationover the original CEs. Thus, the property of space-timeinversion and non-dissipation is lost. Therefore, inapplying the CE/SE method to nonlinear equations, thecalculations of (umx, umy, umz) are distinctly different fromthe calculation of um. This deviation in calculating (umx,umy, umz) from that in the original a scheme provides awindow of opportunity to modify the CE/SE method suchthat the CEs are specified only for um. The algorithm ofcalculating (umx, umy, umz) could be modified based on ausual central differencing scheme. As it will be illustratedin the following sections, this modification allows theflexibility of using general polygons and polyhedrons fortwo- and three-dimensional flow equations.

The original CE/SE method is modified in thepresent project such that only one CE at each grid point isemployed for equations in one, two and three spatialdimensions. As a contract to the original CE/SE method,the CE in the present method is used only to calculate theflow variables, while the spatial gradients of the flowvariables are calculated by a central differencing method.For equations in two spatial dimensions, the presentmethod allows the use of quadrilaterals and/or polygons ineither structured or unstructured meshes. For equations inthree spatial dimensions, general polyhedrons can be usedas CEs. Thus, the present modified CE/SE method isapplicable to general unstructured meshes with mixedelements of various shapes. As such, it can serve as an

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alternative solver for time-accurate solutions in well-established CFD codes.

In the present method, the flow variable gradients(umx, umy, umz) are calculated based on a central differencetype reconstruction method, which is similar to Chang’sa-ε scheme for one-dimensional equations. Because ofthe finite-difference reconstruction, the present methodcan be straightforwardly applied to structured mesh forflows in two and three spatial dimensions.

RESULTS

Single-Stage Compressor for a Small Commercial GasTurbine Engine

To evaluate the CE/SE turbomachinery resultscompared with traditional CFD schemes, we started byrunning the solver through a series of test cases. Thecomputed CE/SE results are compared to results obtainedfrom the traditional Runge-Kutta (RK) scheme developedin [6,7], by using identical geometry, grid size anddistribution, and machine operating conditions to providea fair assessment.

The first test case was a one stage transoniccompressor. This compressor was developed for a smallgas turbine engine for business jets (Yoshinaka, T. andLeBlanc, A. D., 1981).

Figure 4. Single-stage transonic compressor geometry.

In the single-stage compressor, the rotor contains16 blades, and the stator has 40 blades. The design massflow rate was 4.0 kg/s at 37,500 rpm. The total pressureand temperature at the inlet were 101 KPa and 288 K,respectively. The rotor hub section operated at a relativeMach number about 0.9, while the tip section was at about1.4. The geometry of this single-stage compressor wasshown in Figure 4, which is a screen shot from Concepts

NREC’s axial blading design software AXCAD4. Inthis figure, the 3D graphics showed the machinegeometry, while the 2D window showed the gas path andthe rotor blade shapes. Both the rotor and stator bladesare designed by using DCA blades.

To begin with, we calculated the rotor tipsection. The rotor tip section was made of a DCA blade,with an inlet blade angle of 60°, exit angle of 53°. Theaxial chord length was about 2.5 cm, while the maximumthickness to chord ratio was 2%.

To model the blade-to-blade flow field correctly,we need to include streamtube thickness effects in thecode. In other words, it is a quasi-three-dimensionalproblem. The computational domain in the hub to tipdirection is of one cell thin, and with radius and curvatureeffects included in the cell volume calculations. By usingthe two-dimensional CE/SE solver, the streamtubethickness effects is missing, hence, the rotation sourceterms can not be balanced out through pressure rise alonein the control volume. The only way to overcome thisproblem is to develop a three-dimensional CE/SE code forturbomachinery. The blade-to-blade flow then becomes aspecial case of the general 3D CE/SE solver, with thedistance in the hub to tip direction being trimmed down toone cell thin.

In order to address the turbulent shock wave-boundary layer interactions, we also implemented thestandard algebraic Baldwin-Lomax turbulence model(with inner-outer layer matching to calculate eddyviscosity) to better assess the loss mechanism associatedwith the blade row and to compare with the solutionobtained from the traditional RK code.

To evaluate that the Euler part of the CE/SEscheme is functioning correctly, we performed calculationusing the tip section of the single-stage compressor rotor.To begin with, we chose a relatively coarse grid size of 71x 41 in the streamwise direction and the blade-to-bladedirection, respectively.

Figures 5a and 5b showed the relative Machnumber contours obtained from the RK code and the EulerCE/SE code. The freestream Mach number is close to1.5, which was very close to the test data at 1.45. As canbe seen in these two figures, the leading shock from theRK results indicated a more spread-out resolution, whilethe CE/SE results showed a crisp and tight resolutionacross the shock. The CE/SE results converged to anearly steady-state solution, with very minor oscillationsin the downstream blade wake region. In contrast to theshock oscillation generated by 2D CE/SE code shown inthe previous paragraphs, the location and the strength ofthe shock remained very steady from the 3D CE/SE code 4 AXCAD is a trademark of Concepts ETI, Inc.

9

prediction – an anticipated result with the inclusion of thestreamtube thickness effects. The comparisons betweenthe traditional CFD method used in RK code and theCE/SE code also showed the high fidelity solutioncapability of CE/SE algorithm.

For designing highly loaded transoniccompressors, the high accuracy prediction capability forshock wave is highly desirable. Upon completion, theCE/SE code can provide a step increase in predictingturbomachinery flow field as well as performances. Toexamine how the shock was resolved by CE/SE, Figure 5cshows a close-up view with the computational gridsuperimposed on top of the relative Mach numbercontours obtained from CE/SE. We can see the width ofshock wave was being resolved to about one cell thin, atremendous improvements over traditional CFDtechniques, which at least requires several cells to resolvethe shock wave.

There is another interesting phenomenonassociated with the computed results, from both the RKand CE/SE. We can clearly see that the shock wavestarted to smear out a little bit after it entered the bladerow. The shock strength after blade LE is somehowweakened, which was predicted by both codes. Thismight be a result of the curvature effects on the suctionsurface of the blade, which caused the flow to accelerateand to lose a lot of energy prior to the shock. Afterconsulting with Mr. Tsukasa Yoshinaka, who designedthis particular compressor during his tenure at Pratt &Whitney Canada, we are still not sure about the reason,that caused the shock smearing effect inside the bladepassage.

Figures 6a and 6b show the predicted staticpressure contours from the RK code and the CE/SE code.Similar to the relative Mach number contours, the RKresults show a more smeared LE shock wave, especiallynear the suction side of the blade surface, while the CE/SEscheme resolves shock wave to one cell thin. The peakpressure appeared near the LE of the blade, which waspredicted at about 160 KPa by the RK code, 170 KPa bythe CE/SE code. The lowest static pressure, whichoccurred along the first 1/3 chord of the suction surface,showed the flow was being accelerated near this region,which was predicted by both methods. The blade-to-bladeloading pattern, with high pressure at the pressure surfaceof the blade, than the suction surface at the same axiallocation, was predicted by both codes. The exit staticpressure was about 130 Kpa, although predicted by bothcodes, however, the CE/SE results showed a more smoothsolution near the blade trailing edge. Keep in mind thatthe RK code was a well-established turbomachinery CFDcode, written specifically for turbomachinery flows. It hasbeen calibrated over the years for all types ofturbomachinery flows. For the CE/SE scheme to bedeveloped to work on rotating machinery, then to predict

blade-to-blade flow results at a level which werecomparable or better than the RK results, was a trueaccomplishment.

1.4

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M1.51.41.31.21.110.90.80.70.60.50.40.30.20.10

Figure 5a. Relative Mach number contours predicted bythe RK code.

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M1.51.41.31.21.110.90.80.70.60.50.40.30.20.10

Figure 5b. Relative Mach number contourspredicted by the CE/SE code.

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1.51.41.31.21.110.90.80.70.60.50.40.30.20.10

Figure 5c. Close-up view of relative Mach numbercontours near the leading edge, CE/SE results.

10

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Figure 6a. Static pressure contours predicted by theRK code.

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P17000016000015000014000013000012000011000010000090000800007000060000

Figure 6b. Static pressure contours predicted by theCE/SE code.

After evaluating the Euler solution, we thenproceeded to evaluate the Navier-Stokes solution, with theeffects of turbulence included in the computations. Forthe rest of the test cases calculated, we will only includethe Navier-Stokes results, since it represents the flowphysics. We use the same blade tip section to begin theevaluation. The grid size has been increased to 201 x 101in the axial and blade-to-blade directions, respectively, to

provide resolution near the blade surfaces for possibleshock wave-turbulent boundary interaction computations.The grid size remained constant throughout thecomputation, there was no grid-adaptation method beingemployed to favor one solver over another in thisinvestigation. The LE blade angle is 60° measured frommeridional, while the TE blade angle is 53°.

Figures 7a and 7b showed the computed relativeMach number contours obtained from the RK code and theCE/SE code. In Figure 7a, the RK results showed adetached shock wave near the LE, and the shock wavebeginning to spread out close to the suction surface of theblade. This is the location where the shock wave impingingand interacting with the boundary layer near the bladesuction surface. The RK results showed a minor growth ofthe boundary layer caused by the shock wave and there wasno shock wave induced separation observed. The boundarylayer by the suction surface remained thin throughout theblade surface. The size of the turbulent wake is small andbeing mixed out quickly by the core flow. In Figure 7b, theCE/SE solution shows a detached LE shock wave, whichwas computed at the same level of resolution before it splitup into a classical λ wave, due to the interaction with theboundary layer. The two branches of the λ wave wereclearly observed. This also explains the very widely spread-out solution predicted by the RK code, because it failed toresolve the λ wave on the present grid system. The shockwave induced separation, which was clearly observed at thefoot of the λ wave, impacts the downstream wake to extendfurther into the mixing region. In a highly loaded axialcompressor, this phenomenon is usually observed because itcauses the subsequent rotor-stator interaction due the non-uniformity of the rotor TE wake in a multiple blade rowaxial turbomachinery environment. The CE/SE solutionshows a very detailed local flow field surrounding the LE,while the RK results suggest the presence of the LE was notbeing resolved well.

If we put Figure 7a and 7b together, we can seethe location of the oblique shock wave predicted by thetwo solvers is different. The oblique shock wave ispushed further away from the LE as predicted by the RKcode, while the oblique shock wave is much closer to theLE as indicated by the CE/SE results. Both computationswere based on the same exit static pressure at 125 KPa.The reason for the farther shock location from LE in theRK result is because of the under-resolved shock waveinduced separation to lead to a smaller loss associatedwith the computation. Both solvers predicted, veryaccurately, the maximum relative Mach number at about1.4, at a location upstream of the oblique shock wave,prior to the LE. In addition, the passage shock wave wasalso clearly predicted by both solvers.

11

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0.06Mr1.41.31.21.110.90.80.70.60.50.40.30.20.10

Figure 7a. Relative Mach number contours predicted bythe RK code, NS solution.

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Figure 7b. Relative Mach number contours predicted bythe CE/SE code, NS solution.

NASA Rotor 37

The second test case that we selected was thelegendary NASA rotor 37. This was a research programfor axial flow compressors being conducted at NASAGlenn Research Center about 20 years ago. There werefour rotors being studied in this research, namely rotor 35,36, 37, and 38. Rotors 37 and 38 represented a pressureratio about 2.05, while rotors 35 and 36 generated apressure ratio close to 1.85. Each compressor stagecontained a rotor and a stator. Measurement data weretaken downstream of the stator to address the pressureratio, efficiency, and flow angle at different design speeds.There were no data taken at the rotor exit, prior to thestator LE because the spacing between the blade row wastoo narrow to allow proper instrumentations and datasamplings. All four stages were designed for a mass flowof 20.0 kg/s and a rotor tip speed of 455 m/s. The radialdistributions of rotor inlet meridional velocity are almost

the same for all four rotor geometry, with a peak value ofabout 215 m/s at the mean section, and a minimum valueof 185 m/s at both hub and tip locations.

The test data for Rotor 37 was later used as ablind test case for CFD code validations. Turbomachineryinvestigators worldwide were invited to submit theircomputational results based on the published geometryand test conditions before the test results were revealed.Once the test data were compared to all the CFD resultssubmitted and released to the public, CFD researcherswere surprised by the discrepancies between the test datathe computational results. The CFD results spread over awide range in terms of pressure ratio, stage efficiency, andpitched averaged radial flow profiles. The comparisonswith the test data turned out to be rather disappointing.The advancements in CFD technology over the pasttwenty years have been significant enough to make CFD apractical tool for turbomachinery designs. However, rotor37 test data still presents a challenge for most of the CFDcode to predict close matches for the radial flow profiles.Based on the interesting story and its rich history in themaking, we selected rotor 37 as one of the test cases toverify the prediction capability of CE/SE. Although theblade-to-blade flow field in the present Phase I projectwas not appropriate to be compared with the full three-dimensional test data, the intriguing history and thechallenging supersonic rotor flow field has providedenough reason to look into the heart of rotor 37. Inpossible Phase II project, full three-dimensionalcalculations of Rotor 37 using the traditional CFDtechnique and the CE/SE scheme definitely presents thebest opportunity to address the advantages and/orshortcomings of the CE/SE scheme when compared to thetraditional CFD methods.

The geometry of the Rotor 37 was shown inFigure 8. In this figure, the 3D graphics showed themachine geometry, while the 2D window showed the gaspath and the rotor blade shapes. The rotor was composedof 36 blades, while the stator had 46 blades. At the designpoint, the rotational speed was 17,200 rpm. The inlet totalpressure and total temperature were 101.3 KPa and 288 K.The design relative Mach number was about 1.3 at thehub, then increased to about 1.5 level for the meansection, then to 1.8 near the blade tip section.

We first looked at the rotor hub blade cross-section. The hub blade shape was designed using theMCA blade family. The inlet blade angle was 50°, whilethe exit blade angle was 23° to provide a turning of 27°.The axial chord length was 4.3 cm, and the tangentialchord length was 3.5 cm. The maximum thick to chordratio was 8%.

12

Figure 8. The full stage geometry, meridional flow path,and the rotor blade hub, mean, and tip cross-sections ofthe legendary NASA Rotor 37.

We used a computational grid size of 201 x 101in the axial and the blade-to-blade directions, respectively,to conduct full Navier-Stokes calculation for both the RKcode and the CE/SE code. Once again, the geometry andboundary condition inputs were identical for both solvers.The computational mesh remained the same throughoutthe numerical simulation, there was no grid adaptationtechnique employed in either of the two solvers to bias thecomputational results.

The relative Mach number contour for the hubblade cross-section was plotted in Figures 9a and 9b byusing the RK code and CE/SE code, respectively. Themaximum relative Mach number about 1.2 was clearlycaptured in front of the shock wave in both cases. TheCE/SE code, however, indicated a sharper shock waveand thicker boundary layer. The strength and location ofthe shock were a bit different, too. Recall that both theRK and the CE/SE codes predicted nearly the sameinviscid flow physics, as shown in Figure 5. Therefore,the discrepancies in shock strength and location weremainly due to the presence of a boundary layer. Thepresence of a boundary layer on the blade suction surfaceresulted in an altered boundary condition for the shockwave. These interactions are nonlinear phenomenon and,thus, were not likely to be predicted with a simplifiedboundary layer theory. The question now is whichmethod predicted a more realistic boundary layerthickness. Unfortunately, we did not have test data tojustify this issue. Nevertheless, we know that the shockwave will likely induce flow separations and suddenboundary layer growth due to the pressure jump across ashock wave. The CE/SE results seemed to better justifythe flow physics. The passage shock, which strength isweak in nature, was merely captured by the RK code, asseen in Figure 9a, and the results suggested the RK wouldnot be able to resolve the passage shock if the grid density

is lower than the current one. On the other hand, theCE/SE code showed a distinctive resolution of the passageshock and more detail flow gradients across the passageshock wave. This was shown in Figure 9b. The wakeregion was captured well by both codes, and the CE/SEmethod showed an unsteady wake pattern coming off theblade TE.

The other interesting observation can be seenfrom the loading predictions of these two codes. The RKcode showed the flow accelerated for most of the bladepassage. For about 80% of the chord length, the flow wasexperiencing acceleration. Because of this acceleration,the flow speed, once again, stayed at or above sonic speedfor approximately 80% of the blade passage. Thisindicated the RK code predicted the hub section wasoperating at its choke condition. If this happened, therotor would not be able to add work efficiently to the fluidto provide the necessary pressure rise. The test data,however, showed the rotor was operating at a very highperformance level with a measured stage efficiency atnearly 88%. This represented a condition, which for achoked rotor, could not possibly to achieved. The CE/SEresults also showed slight flow acceleration downstreamof the shock wave, then the flow started to diffuse, to slowdown to a relative Mach number of 0.8 at the blade TE.

After finishing the investigation for the hubsection of Rotor 37, we proceeded to study the blade-to-blade flow field of the rotor mean blade section. Themean section was designed by using the MCA bladefamily. The axial chord length was 3.4 cm and thetangential chord length was 4.4 cm. The inlet blade anglewas 54°, while the exit blade angle was 47°. Themaximum thickness to chord ratio was about 5.2%. Thegeometry of the mean blade section was shown in Figure3.18.

The computational grid size was 201 x 102 in theaxial and blade-to-blade directions, respectively, and thegrid size and number remained constant throughout thecomputations. At the mean section, the computed relativeMach number contours were shown in Figures 10a and10b, obtained from the RK and the CE/SE methods. Firstof all, similar to other supersonic blade sections studied,the CE/SE code showed excellent resolution of thepassage shock wave, as shown in Figure 10b. Although inFigure 10a, the RK code also predicted the presence of thepassage shock, the RK results showed a very washed outcompression wave with minor flow gradient across it. Thedesigned maximum relative Mach number was around 1.5.This was computed accurately by both codes. Other thanthe maximum relative Mach number, the rest of the blade-to-blade flow results showed significant differencebetween the two codes.

13

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Mr1.31.21.110.90.80.70.60.50.40.30.20.10

Figure 9a. Relative Mach number contours predicted bythe RK code, NS solution.

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Figure 9b. Relative Mach number contours predicted bythe CE/SE code, NS solution.

First, the RK code, similar to the hub section,predicted the mean section was also operating at chokecondition. The flow accelerated throughout the entireblade-to-blade passage, from LE to TE, then formed ashock wave at the TE. The flow speed reduced tosubsonic and the pressure rose outside the blade passage.This was similar to what was observed in RK results forthe hub blade section. As stated earlier, Rotor 37 had ahigh performance curve, with peak stage efficiency at88%. For the mean section to be operating at chokecondition was not likely to happen based on the test data.On the contrary, the CE/SE predicted a very reasonableblade-to-blade loading patter, with an oblique shock wavein front of the LE, then the flow speed was reduced tosubsonic, gradually, to a relative Mach number of 0.8 atthe blade exit. This was consistent with the test result. Inaddition to the loading pattern prediction, the CE/ SEcode also predicted a shock induced separation at theblade suction surface, and a nicely resolved wake region.Because of the shock induced separation and the thinker

boundary layer calculated by the CE/SE code, the solutionpredicted a larger loss system compared to the RK results.

To investigate why the RK code would predict achoked condition for both the hub and mean section, weperformed an additional test to better understand themechanism. We started to increase the exit staticpressure, which was fixed as the downstream boundarycondition at the end of the computational domain, to 5%,10%, even to 20% higher than its original value, to see ifthe shock wave structure changed or not. To our surprise,the RK code still predicted a choked flow condition forthe mean section, under a circumstance which the exitpressure was raised 20% higher than the original value.Under such a high-pressure situation, the shock wave wasnot pushed out of the LE, but instead, it only movedslightly upstream of the original position, as shown inFigure 10a. The reason remained unclear as to why theRK prediction was unrealistic.

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Mr1.51.41.31.21.110.90.80.70.60.50.40.30.20.10

Figure 10a. Relative Mach number contours predicted bythe RK code, NS solution.

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Figure 10b. Relative Mach number contours predicted bythe CE/SE code, NS solution.

14

Finally, we moved to examine the tip section ofRotor 37. Once again, the tip section was designed byusing the MCA blade family. The axial chord length was2.6 cm and the tangential chord length was 4.9 cm. Theinlet blade angle was 64°, while the exit blade angle was53°, providing 11° of turning. The maximum thickness tochord ratio was about 3.0%.

For Navier-Stokes calculations of the tip section,we again chose a grid size of 201 x 101 in the axial andthe blade-to-blade directions. Figures 11a and 11bshowed the computed relative Mach number contoursfrom the RK and the CE/SE code. The designedmaximum relative Mach number at the tip section wasclose to 1.7, a relative high Mach number for supersonicblade sections. This was a result of the increasing radiustoward the tip section while the machine is moving at aconstant speed. The RK results, shown in Figure 11a,indicated the LE oblique shock wave impinging almost atthe TE of the blade suction surface, then bounced off thewall, to form a nearly normal shock and terminated at alocation at about 50% on the pressure surface. There wasa small shock induced separation region near the TE ofthe blade suction surface observed. The computationalmesh was not sufficient to resolve the further reflection orthe ‘shock chain’ pattern for the RK code. The wakeregion was clearly larger compared to the ones predictedby the RK code in the hub and mean sections, probablybecause of the induced separation in front of the wake.The passage shock wave was almost not being resolved,there only existed a very small gradient to indicate acompression wave, as predicted by the RK code.

In Figure 11b, the CE/SE code showed anextremely interesting result. The shock wave pattern, ingeneral, was similar to what was computed by the RKcode. However, the CE/SE code solution showed finerresolution of the complex shock structure and flowpattern. In this figure, the attached LE oblique wasformed and started to interact with the suction surfaceboundary layer at a location close to the TE. Theboundary layer immediately grew thicker and an obviousinduced separation was observed. This agreed with whatwas calculated by the RK code, only the resolution of theshock wave was better. Then the shock wave wasreflected off the suction surface boundary layer, across theblade passage, to impinge upon at about half way on thepressure surface. This reflected shock wave theninteracted with the pressure surface boundary layer tocause the boundary layer to grow thicker in theneighborhood of the interaction point. Then the shockwave, already being reflected once from the suctionsurface, being bounced off the pressure surface again, toout of the blade passage to interact with the wake comingoff the suction surface. The second reflection wascompletely unseen in the RK results. This shock sequenceformed a shock chain throughout the blade passage and inthe wake region. The blade wake, hence, grew into a

highly unsteady state, formed a wavy wake patterndownstream of the blade passage. The CE/SEcomputation has been carried on long enough in the timedomain to confirm the unsteady nature of the turbulentwake, due to its interactions with the shock chain. In theCE/SE results, the shock chain pattern was clearlyresolved. This indicated that in the CE/SE code, thenumerical dissipation effect was at a minimum level topreserve the reflected shock strength. On the contrary, thenumerical dissipation in the traditional central-differencing scheme used in the RK code was not able topreserve the reflected strength to resolve the shock chain.This also explained why the very weak passage shock wasstill captured beautifully by the CE/SE method, yet not bythe RK code.

Throughout the study of Rotor 37’s hub, meanand tip blade sections, we can conclude that the CE/SEproduced much detail and accurate computational resultscompared to the traditional RK turbomachinery CFDcode, by using the same computational grid and machineoperating conditions.

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Figure 11a. Relative Mach number contours predicted bythe RK code, NS solution.

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Figure 11b. Relative Mach number contours predicted bythe CE/SE code, NS solution.

15

High Pressure Ratio Turbine

Two previous test cases were axial compressors.To demonstrate the validity of the developed CE/SEmethod is suitable for all turbomachines, we chose asingle-stage high pressure ratio turbine as a test case. Thegeometry and aerodynamic data was taken from the paperentitled “The Design, Performance and Analysis of a HighWork Capacity Transonic Turbine,” co-authored byBryce, J. D., Litchfield, M. R., and Leversuch, N. P.,published in the October, 1985 issue of Journal ofEngineering of Gas Turbine and Power, ASME.

The stator blade row is composed of 40 vanes.Each blade is made of large leading edge radius for thepurpose of circulating cooling fluid inside the blade toaccommodate the incoming high temperature combustiongas. Only the hub cross-section contour of the blade wasgiven in the paper while leading edge and trailing edgeblade angles were missing. Because of the very largeleading edge radius, the stator blade cross-sections lookslike a “Tear Drop.” The rotor blade row is composed of89 blades. The designed pressure ratio through the single-stage turbine was 4.48. The geometry, flow path, statorblade cross-sections, and the rotor blade cross-section ofthe single-stage high pressure ration turbine was shown inFigure 12.

The inflow conditions were given in terms ofperformance parameters (pressure, work capacity,aerodynamic loading, and mean blade speed function,etc.). Therefore, we have to transform these parametersinto standard testing conditions (absolute pressure andtemperature, mass flow rate, flow swirl angle, androtational speed). The gas path was deduced from a plotshown in the reference. Measurement data of the statormean section was presented in terms of Mach number inthe paper. The measured absolute Mach number at the exitof the stator mean section was 1.295. In this particulardesign, in which the design pressure ratio was 4.48 (total-to-total), the measured flow in the rotor blade was nearchoke, and the exit relative flow is close to sonic. Thiswas a very rare design to have both the stator and the rotorchoked.

We began by examining the stator flow field.The stator had a relatively two-dimensional geometry,therefore we chose to analyze the mean section of the‘tear-drop’ nozzle. The mean section was designed usingPritchard turbine blade family. The inlet blade angle was50°, and the exit blade angle was 74°. The axial chordwas 3.6 cm, while the tangential chord was 7.1 cm. Thelarge LE radius was 7 mm, and TE radius was 0.6 mm.The throat was about 1.1 cm.

The computation mesh used for this calculationwas 71 x 41. The computed Mach number contours wereshown in Figure 13a and 13b for the RK and the CE/SE

code results. The inlet Mach number was about 0.1, asshown by both codes. Both codes predicted the exit Machnumber to be 1.3, which was in excellent agreement withthe test data of 1.295. This turbine stator provided a largeamount of acceleration, and the was the reason why theLE was shaped like a tear-drop, in order to provide morecontraction effects for flow acceleration. The RK resultsonly captured a minor wake region generated by the TE,as shown in Figure 13a. The CE/SE code, showed anexcellent wake formation and mixing with the core flow,can be seen in Figure 13b. Nevertheless, the CE/SE codepredicted the Mach number contours were almost normalto the flow path while the RK code showed the contourlines tilted toward one surface after the flow turningsupersonic. The CE/SE code also predicted a thickerboundary layer for this particular blade section.

Figure 12. The geometry, gas path, stator blade cross-sections, and the rotor blade cross-sections of the single-stage high pressure ratio turbine.

Next, we studied the rotor mean section of thissingle-stage high pressure ratio turbine. The rotor blademean section was designed by using the Pritchard turbineblade family. The inlet blade angle was 60°, while theexit angle was –58°, providing a turning of 118°. Theaxial chord length was 3.0 cm, while the tangential chordlength was 0.15 cm. Both the LE and TE radius were onthe order of 0.5 mm. The throat was 1.0 cm.

16

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1.31.21.110.90.80.70.60.50.40.30.20.10

Figure 13a. Mach number contours predicted by the RKcode, NS solution.

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M1.31.21.110.90.80.70.60.50.40.30.20.10

Figure 13b. Mach number contours predicted by theCE/SE code, NS solution.

We used a computational grid of 71 x 41 in theaxial and blade-to-blade directions for this mean bladesection calculations. Figure. 14a and 14b showed thecomputed relative Mach number contours obtained fromthe RK code and the CE/SE code. As stated earlier in thedescription of this high pressure ratio turbine, themeasured exit flow of the rotor was near sonic, close tochoke condition. In Figure 14a, the RK code predicted anexit relative Mach number of 0.6 to 0.7, a level much toolow compared to the test data. There was also a weakmomentum layer near the suction surface of the blade, dueto the incidence at the inlet caused by the thicker turbineblade near the LE. In Figure 14b, the CE/SE codepredicted an exit number about 0.95, at a level muchcloser to the test data compared to the RK results. Inaddition, the majority of the blade passage flow at the exitplane was uniformly composed of Mach 0.95 flow. Onthe other hand, the RK results showed a more distributedexit Mach number.

For this single-stage high pressure ratio turbinecase, the CE/SE results compared very well with the testdata for both the stator and rotor calculations. The RKresults showed a reasonably good agreement with data forthe stator blade mean section, however, the rotor resultswere off by at least 30% compared to data. This study hasagain indicated the CE/SE code not only can resolve detaillocal flow field better than the traditional RKturbomachinery CFD code, but also can provide highlyaccurate computational results when compared with testdata. For this reason, the future development for theCE/SE code applied to turbomachinery flow field analysisto improve turbomachinery design capability is definitelyneeded and highly valuable.

x

y

0.07 0.08 0.09 0.1 0.11 0.12

-0.01

0

0.01

0.02

0.03

0.04

Mr0.90.80.70.60.50.40.30.20.10

Figure 14a. Relative Mach number contours predicted bythe RK code, NS solution.

x

y

0.07 0.08 0.09 0.1 0.11 0.12

-0.01

0

0.01

0.02

0.03

0.04

M r0.90.80 .70 .60 .50 .40 .30 .20 .10

Figure 14b. Relative Mach number contours predicted bythe CE/SE code, NS solution.

17

SUMMARY

The present research emphasized thedevelopment and validation for the space-time CE/SEmethod for rotating machinery to complement an existingturbomachinery CFD code, in order to provide betterpredictions for turbomachinery flows. The CE/SE methodhas demonstrated the potential to replace the widely used,explicit Runge-Kutta algorithm in the turbomchinery CFDindustry to provide more accurate and reliable results. Itis expected that the CE/SE method will be applied torotor-stator interaction and a wide range ofturbomachinery related flow fields when the developmentand validation processes are completed under researchefforts.

In summary, the accomplishments of the presentresearch are as follows.

• The three-dimensional space-time CE/SE code forrotating machinery has been developed for the firsttime, with emphasis on the effects of adverse pressuregradient, rotational flow predictions forturbomachinery flows.

• Algebraic Baldwin-Lomax turbulence model has beenimplemented into the developed CE/SE code toprovide loss prediction mechanism forturbomachinery flow calculations. This is the firsttime for a traditional turbulence model to beintegrated with the CE/SE method.

• Fundamental validation studies have been performedto provide understanding of the prediction capabilityof the CE/SE method for rotating machinery for bothEuler and Navier-Stokes solutions.

• The CE/SE code was used to calculate a single-stagecompressor rotor blade sections. The computationalresults have been compared with the RK code results.

• The CE/SE code was validated to calculate the NASARotor 37 blade sections. Results have been comparedwith the RK results and summarized.

• The CE/SE code was tested for a single-stage highpressure ratio turbine for both the stator and rotorblade sections. The CE/SE results showed excellentagreement with test data and superceded thecomputed results by the RK code.

ACKNOWLEDGMENT

The research was funded by the NASA GlennResearch Center under Contract No. NAS3-00005. Thefinancial support is highly acknowledged.

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