draft: dynamics and structures in an … · draft: dynamics and structures in an unsteady pin fin...

Proceedings of GT2009Proceedings of ASME Turbo Expo 2009: Power for Land, Sea and Air

June 8-12, 2009, Orlando, Florida, USA

GT2009-60219

DRAFT: DYNAMICS AND STRUCTURES IN AN UNSTEADY PIN FIN CONVECTIONSIMULATION

Markus SchwanenFluids, Turbulence and Fundamental Transport Lab

Department of Mechanical EngineeringTexas A&M University

College Station, Texas 77843Email: [email protected]

Andrew T. DugglebyFluids, Turbulence and Fundamental Transport Lab

Department of Mechanical EngineeringTexas A&M University

College Station, Texas 77843

ABSTRACT

Internal cooling of the trailing edge region in a gas turbineblade is typically achieved with an array of pin fins. In orderto better understand and predict this flow and heat transfer, anunsteady Reynolds-averaged Navier Stokes computation is per-formed on a single row of cylindrical pin fins with a spanwisedistance to fin diameter ratio of 2 and height over fin diameterratio of one. With a locally adapted mesh, the boundary layeris resolved throughout the domain. For validation purposes, theflow Reynolds number of 12,800 was set to match experimentsavailable in the open literature.Numerical data were analyzed using Proper Orthogonal Decom-position (POD), where the correlation matrices were built usingthe internal energy in addition to the three velocity components.This enables a flow decompositon with respect to the flow struc-ture’s influence on surface heat transfer.The most energetic first three modes showed the same tempera-ture eigenfunction, which means that a considerable amount ofenergy is contained in flow structures that don’t contribute toincreasing endwall heat transfer. An eigenfunction linked to theshed vortices has the highest impact on heat transfer, followed bymore elongated structures originating from the leading edge vor-tex. It was also found that the vortex shedding frequency changesover time and both lift coefficient and Strouhal number increasecompared to experimental values.

NOMENCLATUREPOD Proper Orthogonal DecompositionS Spanwise pin spacingH Channel heightD Pin diameterReDH Reynolds number based on channel hydraulic diameterURANS Unsteady Reynolds Averaged Navier-Stokes Equa-

tionsTin Average inflow temperatureTout Average exit temperaturecv Specific heat capacity of airU0 Bulk flow velocityτ Size of snapshot sample in timeEc Eckert numbery+ Non-dimensional wall distance based on turbulence scalesMoS Method of Snapshots PODu,v,w Spanwise, wall-normal and streamwise velocity compo-

nent

INTRODUCTIONOptimization in gas turbines, demanded by the market or

increasingly strict environmental regulations, mostly concernsoverall engine efficiency. It is, amongst others factors, dependingon the combustor exit temperature, which in modern enginesranges well above the melting point of the blade and vanemetal. To maintain operability and durability (which decreaseswith higher temperature under oscillating load), cooling the

1 Copyright c© 2009 by ASME

surfaces exposed to the hot gas path is vital. Computationalfluid dynamics opens the possibility of closer examining flowphenomena and structures inside the pin fin array and its impacton surface heat transfer.At the first stage, turbine blade cooling is typically done bothexternally (film cooling) and internally. Internal cooling isrealized by casting channels into the vane or blade wherecooler compressor bleeding air is forced through the channelfor convective heat exchange. But in the trailing edge regionof a vane or blade, the channel height becomes rather small.Manufacturing ribbed cooling channels, which are typicallyfound in the main body of the airfoils, is therefor not possible.Instead, small solid cylinders, or pin fins, are put inside thechannel. This leads to an increased heat exchanging surface,higher turbulence levels and increased heat transfer.In case of pin fins, this turbulence level increase leads to ahigher pressure loss across the array. To avoid backflow of hotcombustion cases into the cooling region, the pressure loss hasto be compensated by the compressor. Since the pressured aircan not be used to drive the thermodynamic cycle, extractionof energy from the compressor via bleeding air decreases theoverall efficiency of the turbine.The engineering task is to find an optimal configuration, withlow pressure loss relative to high heat transfer. The turbulentflow inside pin fin passages is not well understood and not easilyaccessible by experiments.Computations can have a large impact on improving designs. Inindustry, this requires models as simple and as fast as possible.For this reason, the aim of this study is to evaluate a URANSmodel to solve the flow in a pin fin cooling passage. Due tothe unsteady nature of the flow and the relatively large amountof data created, effective anlytical tools are also required. Inthis paper, we present results of a numerical investigation of abase line pin fin geometry with a spacing over fin diameter ratioS/D = 2 and at an elevated Reynolds number of ReDH = 12,800.The data is analyzed with the help of POD. In extention tothe traditional formulation of the maximum value problemunderlying POD, we propose the inclusion of temperature interms of thermal energy.

REVIEW OF RELEVANT LITERATUREPin fin arrays for heat transfer augmentation have been stud-

ied in a broad variety of experiments, see Armstrong and Win-stanley (1988), [1]. Pin fins have been examined by Lyall et al.(2007) [2], to which the domain is matched for experimental vali-dation. Lyall et al. (2007) [2] studied a single row of pin fins witha height-to-diameter ratio H/D = 1 in a high aspect ratio channelwith varying pin spacings at different Reynolds numbers. Resultsfor overall friction factor augmentation, drag on the fin and spa-tially resolved heat transfer augmentation are presented. A spac-

ing of S/D = 2 has been found to show the highest heat transferaugmentation compared to an unobstructed channel from all ge-ometries tested. It is highest for the lowest Reynolds numberused, ReDH = 5013, and remains on a rather constant level forReynolds numbers between 7500 < ReDH < 17500. The area ofhighest augmentation in the wake of the fin moves upstream, to-wards the the cylinder trailing edge, with increasing Reynoldsnumber.Ames and Dvorak (2005) [3] have computationally modeled aflow in a multi row pin fin array with the use of the k-ε turbulencemodel in conjuntion with a one-equation model in the boundarylayer, which was resolved down to the viscous sublayer. Onlyone quarter of the full domain was simulated, applying sym-metric boundary conditions at the cylinder mid span and alongeven and odd row pin centerlines. Steady calculations showed anunderprediction of heat transfer augmentation and pressure loss.The authors attribute this to the applied cylinder centerline sym-metry, which prevents transient vortex shedding in the trailingedge region of the pin.To examine if and why URANS predictions fail for this flowproblem, one can imagine decomposing this flow into a singlecylinder in crossflow and an unobstructed channel. The latter isa thouroughly investigated canonical flow problem, serving as avalidation source for turbulence model coefficients. More dif-ficulty arise from the cylinder case. Young and Ooi (2004) [4]examined different two-equation turbulence models for a cylin-der in cross flow and two dimensions in an infinte domain. Theresults showed a poor agreement with the experimental data forthe drag coefficient. The authors suggest improving turbulencemodeling. A known defficiency of eddy-viscosity models is theover prediction of turbulent kinetic energy in stagnating flow,which energizes the boundary layer on the cylinder surface andimpacts transition and separation. To improve this, Holloway etal. (2004) [5] propose a third transport equation for laminar ki-netic energy, representing the magnitude of non-turbulent streamwise fluctuations in the pre-transitional boundary layer. They re-port improved predictions for the drag coeffiecient and concludethat boundary layer transition is better captured with their modeladaptions. Harrison and Bogard (2008) [6] examined different2-equation models for a jet in cross flow and heat transfer. Theyshowed that surface heat transfer is reasonably predicted with theSST-k-ω model.Based on the previous research, we chose to use the SST-k-ω ina three dimensional, span wise periodic domain. The near wallmesh resolves the entire boundary layer and the Reynolds num-ber is set at an elevated level of ReDH = 12,800.

COMPUTATIONAL PROCEDUREIn this section, the settings, mesh setup and boundary condi-

tions for the numerical solver are outlined. A grid independencestudy was performed to obtain an estimate on the uncertainty of


Figure 1. THE WALL MESH ON ENDWALL AND CYLINDER FOR APART OF THE STAGATION REGION ARE SHOWN. GRID REFINE-MENT IN REGIONS OF HIGH VELOCITY GRADIENTS AS WELL ASTHE BOUNDARY LAYER MESH AROUND THE CYLINDER ARE VISI-BLE.

the computational results. The second subsection outlines themodified POD procedure for data analysis.

Solver settings and gridThe commercial Navier-Stokes solver Fluent 6.2.16 was

used in conjunction with the SST-k-ω model. This model blendsa k-ω closure near the wall with a modified k-ε formulation inthe free stream. Furthermore, the transitional option was en-abled, which activates a Low-Reynolds number modification ofthe model and damps the eddy viscosity in the near wall region[7]. The discretization schemes for time, pressure and all trans-port scalars are of second order, where an upwinding schemehas been used for the convective terms, enenergy and turbulencescalars. Time is advanced with an implicit algorithm. Pressureand velocity are coupled with the PISO algorithm since the meshhad some skewed cells at the interface between the channel gridand the cylinder grid (Fig. 1). The PISO scheme also yields fasterconvergence and allows all under-relaxation paramaters to be setto unity. The Courant number is close to unity for the employedtime step of ∆t = 0.001sec. This is roughly equal to 0.034% ofone cylinder lift cycle. One time step was considered convergedwhen all residuals fell on the order of 10−5. The entire simula-tion was run for a little over 100 cycles of cylinder lift coefficient.The domain was modeled with periodic boundaries in the span-

wise direction, 2D apart, yielding an infinte row of pin fins. Apressure outlet boundary condition was used 14D downstreamof pin fin. All required boundary values at the inflow, 4D up-

Figure 2. ERROR IN TERMS OF DIFFERENCE TO EXPERIMENTALRESULTS AS FUNCTION OF DOMAIN SIZE IS PLOTTED. A LOGA-RITHMIC DECREASE, AS EXPECTED FOR SECOND ORDER ACCU-RACY, IS SHOWN.

stream of the pin, were obtained from a steady state simulationof an unobstructed channel beforehand and then linearily inter-polated to fit the pin fin domain mesh. A specific wall heat fluxof qBottom = 20W/m2 was imposed on the no-slip bottom wall.Since the fin is only heated by conduction, its surface heat fluxwas calculated from the cylinder cap area covered by the heatedchannel walls and also imposed as a constant heat flux boundarycondition. Only one half of the domain in wall normal directionis modeled, using a symmetry boundary condition on top.The structured grid, Fig. 1, was created with Gambit and containsaround 6.2 million hexaedral cells. Since wall functions wouldnot be able to correctly model the complex flow phenomena, theboundary layer is resolved down to the viscous sublayer with asolution based adaptation. Before data for the POD analysis wasrecorded, cells with high gradients in velocity (near the cylinderwall), high skewness and distance too far from the wall, wererefined at multiple instants in time to retain symmetry withinthe mesh. Around 98% of the wall nearest cells range between0 < y+ < 2 and no cell is farther from the wall then y+ = 5.4.For the grid independence study as shown in Fig. 2, all wall adap-tations were removed yielding a domain with 1 million cells. Forthe second domain with 2.2 million cells, only the first adapta-tion level was retained. To stay consistent with the literature, theNusselt number averaged over an area spanning 0.5D upstreamand downstream of the fin was calculated and related to an openchannel correlation found in [8]. The local reference temperatureis calculated from an energy balance as shown in [2].


POD procedurePOD provides a decomposition of the tensor A(xi, t), defined

by Eqn. (6), according to Eqn. (3),

λncn(t ′) = limτ→∞

1τ

Z τ

0

Z

ΩA(xi, t)A(xi, t ′)dΩcn(t)dt, (1)

Φn(xi) =1τ

1λn

Z τ

0cn(t)A(xi, t)dt, (2)

cn(t) =Z τ

0A(xi, t)Φn(xi)dt, (3)

where the eigenfunctions Φn(xi) with corresponding eigenvaluesλn are accompanied by their respective time-dependent coeffi-cients cn(t) and τ is the time spanned by the used snapshots. Thisprojection is optimal in the sense that it maximizes the energycontained in each eigenfunction or mode. Thus the lowest pos-sible number of modes is needed to reconstruct a given fractionof the original data. An estimate for the dimension of a system,D90, is typically derived from the number of modes needed toressemble 90% of the original data set’s energy content.To include the effect of heat transfer into the POD basis, the orig-inal formulation was extended using a non-dimensionalized fluidtemperature, Eqn. (5), weighted with the Eckert number Ec ac-cording to Eqn. (4).

Ec =U2

0

cv(Tout −Tin)(4)

T ∗(xi, t) = EcT (xi, t)−Tin

Tout −Tin(5)

A(xi, t) = [u(xi, t),v(xi, t),w(xi, t),T ∗(xi, t)] (6)

Due to the lack of two periodic spatial directions, the direct(Fourier) POD is not feasible. Instead, the Method of Snapshots(MoS) as developed by Sirovich (1987) [9] was used. The timecoefficients are obtained from numercially integrating Eqn. (1)and solving the resulting matrix system as an eigenvalue prob-lem. The time independent eigenfunctions can then be obtainedby projecting the time coefficients onto the data field, Eqn. (2).The volume integration in Eqn. (1) is numerically approximatedby a summation over all control volumes, which are known fromthe meshed geometry. Since the solver is a Finite Volume code,all calculated variables are already available at the cell centersand do not have to be interpolated from grid points.

RESULTSTo ensure a fully developed flow, the lift coefficient on the

cylinder surface, Eqn. (7),

CL(t) =F

ρU20 Anormal

, (7)

Figure 3. THE PLOT OF LIFT COEFFICIENT OF THE PIN FIN SHOWSTHAT THOUGH THE SOLUTION IS STEADY IN THE MEAN, A CHANGEIN PEAK CAN STILL BE OBSERVED. THE VORTEX SHEDDING FRE-QUENCY IS NOT CONSTANT.

Table 1. TIME SAMPLE SIZES AND DIMENSIONS, EXCLUDING THEFIRST MODE. D90, D95 AND D99 IS THE NUMBER OF MODES RE-QUIRED TO RETRIEVE 90, 95 AND 99 PERCENT OF THE FLOW, RE-SPECTIVELY.

Sample size D90 D95 D99

89 3 6 19

268 4 7 35

446 4 9 51

was monitored over the simulation time and is shown in Fig. 3.One data sample was taken every 30 time steps so that roughly90 snapshots are collected per cycle of lift coefficient. The fre-quency of vortex shedding, which causes the oscillations of thelift coefficient, is not constant but ranges between 0.333Hz and0.342Hz. On average for the plotted time range, the Strouhalnumber St = 0.326, which is considerably higher than experi-mantally obtained results for a single cylinder in crossflow. Theamplitude of the lift coefficient is also higher than what is ex-pected. POD analysis was performed on three data sets of dif-ferent size in time, namely for one, three and five lift cycles.The snapshot indices can be found in Table 1. As Duggleby etal. (2008) [10] pointed out, the MoS is relatively slow to learnabout the dynamics of a system. To ensure convergence of themethod, three different dimensions of the system are also givenin Table 1. When plotted over the number of snapshots used toacquire POD modes, the dimension should converge towards aconstant value. That is not the case for the three samples underconsideration. But since the system is periodic in time and turbu-lent fluctuations are filterd out by the chosen turbulence model,


Figure 4. THE TEMPERATURE EIGENFUNCTION OF MODE 0SHOWS THE AVERAGE TEMPERATURE PROFILE ON THE END-WALL. THE COOLING EFFECT OF THE LEADING EDGE HORSESHOE VORTEX AS WELL AS THE COOLING DUE TO TURBULENTMIXING IN THE CYLINDER WAKE CAN BE SEEN.

Figure 5. MODE 3 IS THE FIRST TEMPERATURE EIGENFUNCTIONDIFFERENT FROM THE AVERAGE CONTOUR. COLUMN LIKE VORTI-CAL STRUCTURES WITH A DIAMETER ON THE ORDER OF THE FINARE REPRESENTED.

the first 5 modes under examination are expected not to changewhen building the POD modes with more snapshots. The timeaveraged values for temperatue and velocity were not substractedfrom the data before hand, thus the eigenfunction of the firstmode is a representation of the average. It is depicted in Fig. 4.The cooling effect of the leading edge region horse shoe vortex,winding around the cylinder at its junction with the endwall, canbe seen as an area of relatively low temperature where cool coreflow is pushed towards the heated endwall. Downstream of thepin fin, vortices spinning off from the cylinder create a rather ho-mogeneous, relatively low temperature profile. All temperatureeigenfunction plots are linearily interpolated from the cell cen-tered data onto an evenly spaced grid at the wall. This does notlead to an excessive error since the wall nearest cells are withinthe viscous sublayer, where the non-dimensionalized flow tem-perature varies linearily with wall distance. The fist temperatureeigenfunction that does not show a similar, average temperaturecontour is plotted in Fig. 5. The biggest temperature fluctuationsoccur in the separation region of the fin boundary layer and fur-

Figure 6. THE TEMPERATURE EIGENFUNCTION OF MODE 4SHOWS THE HIGHEST MAGNITUDE FURTHER DOWNSTREAM OFTHE FIN COMPARED TO MODE 0. ELONGATED STRUCTURES AREFOUND AROUND THE FIN, INDICATING THAT THIS MODE REPRE-SENTS DYNAMICS FROM THE LEADING EDGE VORTEX.

ther downstream, were the vortex shed from the fin interacts withthe leading edge horse shoe vortex. An isosurface of the absolutevalue of swirl (defined as the dot product of vorticity and velocitydevided by dynamic pressure) at 0.1% of its highest value, showsrich dynamics especially downstream of the fin. The column likestructures have a diameter on the order of the fin and ressemblethe shed vortices. All swirl isosurfaces are interpolated onto anevenly spaced grid that starts about 0.013D above the channelwall in order to filter out the near wall swirl contours that orig-inate from high shear in the boundary layer rather than vorticalstructures. With higher mode order, the region of high temper-ature change shifts further downstram, as shown in Fig. 7 formode 4. The vortical structures are rather elongated, especiallyaround the fin. In the wake of the fin, the vortices have been ro-tated with respect to the bulk flow direction due to the interactionwith the vortices shed from the cylinder. The region of highesttemperature fluctuaction and, in turn, high heat transfer, appearsto follow this rotation and has a preferred direction at 45deg.with respect to the mean flow. As the regions of high tempera-ture fluctuations shift downstream, they also move towards thefin centerline, where they will ultimatley merge to a region withrather homogeneous wall heat transfer far downstream of the fin,as was seen in Fig. 4 for the average temperature contour. Acorrelation between visualized reduced order flow structures andphenomena in the full simulation cannot be established readilyfrom the plots. Turbulent mixing of momentum, diffusing vorti-cal structures downstream of the pin fin, is the reason for higheroverall heat transfer levels without specific flow structures caus-ing it.The POD analysis showed that, because the first three modeslook essentially the same with respect to wall temperature, theflow field represented by modes 1 and 2 does not contribute to


Figure 7. THE TEMPERATURE EIGENFUNCTION OF MODE 5SHOWS A FURTHER SHIFT OF HIGH TEMPERATURE CHANGEDOWNSTREAM OF THE FIN AND TOWARDS THE CENTERLINE.PARTS OF THE LEADING EDGE VORTEX APPEAR STRONEGERTHAN IN THE LOWER ORDER MODES, BUT DO NOT HAVE AN IM-PACT ON WALL TEMPERATURE.

a change in wall temperature and heat transfer. The energy con-tained in these modes can therefor be considered a loss and acloser examination might yield some potential for either opti-mizing the fin geometry or arrangement.

CONCLUSIONIn this paper, we examined an unsteady RANS calculation

of a single, infinite row of pin fins which can be found in theinternal cooling passage at the trailing edge region of a gas tur-bine blade. Compared with experiments, the simulations showedreasonable agreement. The computational demand, originatingfrom resolving the wall boundary layers even though a simpleturbulence model was used allowing for the coarsest mesh pos-sible, makes computational studies of pin fin arrays more suit-able for academic research rather than application in industrialdesign processes. Since modern jet engines use far more com-plex fin arrangements than the one presented, experiments mightnot be able to provide enough insight for future optimization.This study shows that inspite of the deficiencies stemming fromthe Reynolds stress closure in RANS, reasonable predictions forsurface heat transfer can be achieved. The presence of a wall in-creases Strouhal number and lift coeffiecient compared to a sin-gle cylinder in crossflow. It was also observed that the Strouhalnumber is not constant. The leading edge horse shoe vortexmight accelerate or break vortex shedding as soon as the pin finvortices reach a certain size.Higher POD modes showed a shift of temperature fluctuationsaway from the fin and merging towards the centerline. The high-est impact on unsteady heat transfer is attributed to the columnlike vortical structures from the pin fin itself. The elongated and

rotated leading edge horse shoe vortex parts have the next high-est impact on transient temperature change.With POD, flow features that do not contribute to heat transferaugmentation but contain relatively high amounts of energy areidentified within the second and third mode. This means thatthere is plenty of room for optimization towards more efficientinternal cooling.

REFERENCES[1] Armstrong, J., and Winstanley, D., 1988. “A Review of

Staggered Array Pin Fin Heat Transfer for Turbine CoolingApplications”. Journal of Turbomachinery, 110, pp. 94–103.

[2] Lyall, M., Thrift, A., and Thole, K., 2007. “Heat Transferfrom Low Aspect Ratio Pin Fins”. ASME Paper GT-2007-27431.

[3] Ames, F., and Dvorak, L., 2005. “Turbulent Transportin Pin Fin Arrays - Experimental Data and Predictions”.ASME Turbo Expo, GT2005-68180.

[4] Young, M., and Ooi, A., 2004. “Turbulence Models andBoundary Conditions for Bluff Body Flow”. AustralasianFluid Mechanics Conference.

[5] Holloway, D., Walters, D., and Leylek, J., 2004. “Predic-tion of Unsteady, Separated Boundary Layer over a BluntBody for Laminar, Turbulent, and Transitional Flow”. In-ternational Journal for Numerical Methods in Fluids, 45,pp. 1291–1315.

[6] Harrison, K., and Bogard, D., 2008. “Comparison of RansTurbulence Models for Prediction of Film Cooling Perfor-mance”. ASME Paper GT-2008-51423.

[7] Durbin, P., 1996. “On the k-3 stagnation point anomaly”.International Journal of Heat and Fluid Flow, 17(1),pp. 89–90.

[8] Kays, W., and Crawford, M., 1980. Convective Heat andMass Transfer. McGraw-Hill Book Company, Inc., NewYork, NY.

[9] Sirovich, L., 1987. “Turbulence and the Dynamics of Co-herent Structures. Part 1: Coherent Structures”. Q. Appl.Math. XLV, 561-571.

[10] Duggleby, A., Ball, K., and Schwaenen, M., 2008. “Struc-ture and Dynamics of low Reynolds number Turbulent PipeFlow”. Phil. Trans. Roy. Soc. A, to appear.


draft: dynamics and structures in an … · draft: dynamics and structures in an unsteady pin fin...

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