identifying inefficiencies in unsteady pin fin heat transfer using … · 2018-05-29 · markus...

Markus Schwanen1

e-mail: [email protected]

Andrew DugglebyDepartment of Mechanical Engineering,

Texas A&M University,

College Station, TX 77843

Identifying Inefficiencies inUnsteady Pin Fin Heat TransferUsing Orthogonal DecompositionInternal cooling of the trailing edge region in a gas turbine blade is typically achievedwith an array of pin fins. In order to better understand the effectiveness of this configura-tion, high performance computations are performed on cylindrical pin fins with a span-wise distance to fin diameter ratio of 2 and height over fin diameter ratio of one. Forvalidation purposes, the flow Reynolds number based on hydraulic channel diameter andbulk velocity (Re¼ 12,800) was set to match experiments available in the open literature.Simulations included a URANS and LES on a single row of pin fins where the URANS do-main was 1 pin wide versus the LES with 3 pins. The resulting time-dependent flow fieldwas analyzed using a variation of bi-orthogonal decomposition (BOD), where the corre-lation matrices were built using the internal energy in addition to the three velocity com-ponents. This enables a detailed comparison of URANS and LES to assess the URANSmodeling assumptions as well as a flow decomposition with respect to the flow structure’sinfluence on surface heat transfer. This analysis shows low order modes which do notcontribute to turbulent heat flux, but instead increase the heat exchanger’s global ineffi-ciency. In the URANS study, the forth mode showed the first nonzero temperature basisfunction, which means that a considerable amount of energy is contained in flow struc-tures that do not contribute to increasing endwall heat transfer. In the LES, the first nonzero temperature basis function was the seventh mode. Both orthogonal basis functionsets were evaluated with respect to each mode’s contribution to turbulent heat exchangewith the surface. This analysis showed that there exists one distinct, high energy modethat contributes to wall heat flux, whereas all others do not. Modifying this mode couldpotentially be used to improve the heat exchanger’s efficiency with respect to pressureloss. [DOI: 10.1115/1.4004873]

Keywords: URANS, LES, pin fin, turbulence, orthogonal decomposition, heat transfer

1 Introduction

Optimization in gas turbines, demanded by the market andincreasingly strict environmental regulations, mostly concernsoverall engine efficiency. It is, among other factors, depending onthe combustor exit temperature which in modern engines rangeswell above the melting point of the blade and vane metal(Schobeiri [1]). To maintain operability and durability whichdecreases with higher temperature under oscillating load, coolingthe surfaces exposed to the hot gas path is vital. Computationalfluid dynamics opens the possibility to study the cooling effective-ness of a pin fin array by examining the details of the flow in bothspace and time.

At the first stage, turbine vane and blade cooling is typicallydone both externally (film cooling) and internally. Internal coolingis realized by casting channels into the vane or blade where coolercompressor bleeding air is forced through for convective heatexchange (Fig. 1). However, in the trailing edge region of a vaneor blade, the channel height becomes rather small. Manufacturingribbed cooling channels, which are typically found in the mainbody of the airfoils, is therefore not possible. Instead, small solidcylinders, or pin fins, are put inside the channel. This leads to anincreased heat exchanging surface, higher turbulence levels, andincreased heat transfer (Han et al. [3]). For the same reason, pinfins are also used for endwall cooling and can also be found on in-ternal airfoil surfaces.

In case of pin fins, this turbulence level increase leads to ahigher pressure loss across the array compared to unobstructedchannel flow. The cooling air is delivered from the compressorand bypassing the combustion chamber. A higher pressure lossacross any turbine cooling passage is thus an overall engine effi-ciency penalty. The engineering task is to find an optimal configu-ration, with low pressure loss relative to high heat transfer.

To understand how to improve the efficiency, the turbulent flowdynamics inside pin fin passages need to be characterized as theyare currently not well understood and difficult to access experi-mentally. With the latest computing systems and massively paral-lel algorithms, computational fluid dynamics can have a largeimpact on exploring the underlying physics. Due to the unsteadynature of the flow and the relatively large amount of data created,effective analytical tools are also required to mine the data toextract the relevant physics and phenomena.

With a proper characterization of the flow physics in the pin-finpassage, the flow structures responsible for heat transfer can beidentified. Likewise, the flow structures which are high in dragand have a low effect on heat transfer can also be found, yieldinga goal for future designs (contouring, pin alignment, etc.) todecrease their effect.

In this paper, we present results of a numerical investigation ofa base line pin fin geometry with a spacing over fin diameter ratioS/D¼ 2 and at an elevated Reynolds number of Re¼ 12,800,where the Reynolds number is based on bulk velocity and hydrau-lic channel diameter. The data are analyzed with orthogonaldecomposition that is based on time correlations (a variation ofbi-orthogonal decomposition by Aubry et al. [4]). As an extensionto traditional decomposition methods, we propose the inclusion of

1Corresponding author.Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF

HEAT TRANSFER. Manuscript received May 11, 2010; final manuscript received February8, 2011; published online December 13, 2011. Assoc. Editor: Phillip M. Ligrani.

Journal of Heat Transfer FEBRUARY 2012, Vol. 134 / 020904-1Copyright VC 2012 by ASME

Downloaded 09 Jan 2012 to 165.91.74.118. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

temperature in terms of thermal energy. To improve the modelingof turbulence in the simulation, a LES study of the same geometrywas done instead of a URANS and evaluated in the same manner.The basis functions were then ranked with regard to their impacton turbulent surface heat flux. It is hypothesized that low order ba-sis functions with no temperature dynamics constitute the heatexchanger’s inefficiency. If those flow structures can be identifiedand eliminated, the overall efficiency will likely increase.

2 Background

Pin fin arrays for heat transfer augmentation have been studiedin a broad variety of experiments (see Armstrong and Winstanley[5]). Lyall et al. [6], to which the domain is matched for experi-mental validation, studied a single row of pin fins with a height-to-diameter ratio H/D¼ 1 in a high aspect ratio channel with vary-ing pin spacings at different Reynolds numbers. They reportedresults for overall friction factor augmentation, drag on the fin andspatially resolved heat transfer augmentation. A spacing of S/D¼ 2 has been found to show the highest heat transfer augmenta-tion compared to an unobstructed channel from all geometriestested. It is highest for the lowest Reynolds number used,Re¼ 5013, and remains on a rather constant level for Reynoldsnumbers between 7500<Re< 17,500. The area of highest aug-mentation in the wake of the fin moves upstream, towards the cyl-inder trailing edge, with increasing Reynolds number.

Ames and Dvorak [7] have computationally modeled the flowin a multirow pin-fin array with the use of the k-e turbulencemodel in conjunction with a one-equation model in the boundarylayer, which was resolved down to the viscous sublayer. Only onequarter of the full domain was simulated, applying symmetricboundary conditions at the cylinder mid span and along even andodd row pin centerlines. Steady calculations showed an under-prediction of heat transfer augmentation and pressure loss. Theauthors attribute this to the applied cylinder centerline symmetry,which prevents transient vortex shedding in the trailing edgeregion of the pin.

To examine the complexity of this flow problem, one can imag-ine decomposing the flow into a single cylinder in crossflow andan unobstructed channel. The latter is a thoroughly investigatedcanonical flow problem, serving as a validation source for turbu-lence model coefficients. More difficulty arise from the cylindercase. Young and Ooi [8] examined different two-equation turbu-lence models for a cylinder in cross flow and two dimensions inan infinite domain. The results showed a poor agreement with the

experimental data for the drag coefficient. The authors suggestimproving turbulence modeling. A known deficiency of eddy-viscosity models is the over prediction of turbulent kinetic energyin stagnating flow, which energizes the boundary layer on the cyl-inder surface and impacts transition and separation. To improvethis, Holloway et al. [9] propose a third transport equation forlaminar kinetic energy, representing the magnitude of nonturbu-lent streamwise fluctuations in the pretransitional boundary layer.They report improved predictions for the drag coefficient and con-clude that boundary layer transition is better captured with theirmodel adaptions. Harrison and Bogard [10] examined differenttwo-equation models for a jet in cross flow and heat transfer. Theyshowed that surface heat transfer is reasonably predicted with theSST-k-x model. Thus we chose to use the SST-k-x in a threedimensional, spanwise periodic domain for the URANS. The nearwall mesh resolves the entire boundary layer and the Reynoldsnumber is set at an elevated level of Re¼ 12,800.

Once time-dependent data is generated (by URANS and LES), adynamical analysis technique known as Karhunen-Loeve or properorthogonal decomposition (POD) can decompose a long-time dataset into its most optimal basis functions in L2 [11]. This methodwas first applied to turbulent flows by Lumley [12] for homogene-ous turbulence, and was extended to channel flow by [13–15]. Suc-cesses with this method include Ref. [15] who explained burstingand sweeping events, and Ref. [16] who explained energy transferbetween the structures. POD was recently extended to turbulentpipe flow by Duggleby et al. [17] and was used to explain dragreduction by spanwise wall oscillation in Ref. [18]. In order tofocus on heat transfer, a modified orthogonal decomposition foruse in heat transfer analysis is developed. It is based on spatialeigenfunctions and vector time coefficients instead of vector eigen-functions and scalar time coefficients.

3 Computational Procedure

For all simulations presented, the incompressible Navier-Stokesequations

@iUi ¼ 0 (1)

@tUi þ Uj@jUi ¼ �1

q@iPþ @j �@jUi þ

1

qsSGS

ij

� �(2)

@tT þ Uj@jT ¼ @jða@jT þ qSGSj Þ (3)

were solved, with velocity vector Ui, pressure P, viscosity �, ther-mal diffusivity a, density q, and models for the unresolved turbu-lence effects of subgrid stress sSGS

ij and subgrid thermal flux qSGSj .

For the URANS, the term sSGSij is approximated with mean veloc-

ity gradients and turbulent viscosity (diffusivity for the energyequation). For the LES, the sub grid scales are not computed butaccounted for by filtering (removing energy from) velocity andtemperature solutions at each time step.

In the following section, the settings, mesh setup and boundaryconditions for the URANS part are outlined. A grid independencestudy was performed to obtain an estimate on the uncertainty of thecomputational results. The LES domain and boundary conditionsare described in Sec. 3.2. The Sec. 3.3 closes with a description ofthe modified signal decomposition procedure for data analysis.

3.1 Solver Settings and Grid for URANS Study. The com-mercial Navier-Stokes solver Fluent 6.2.16 was used in conjunc-tion with the SST-k-x model. This model blends a k-x closurenear the wall with a modified k-e formulation in the free stream.Furthermore, the transitional option was enabled, which activates aLow-Reynolds number modification of the model and damps theeddy viscosity in the near wall region (Durbin [19]). The discreti-zation schemes for time, pressure, and all transport scalars are ofsecond order, where an upwinding scheme has been used for theconvective terms, energy and turbulence scalars. Time is advanced

Fig. 1 Schematic of a turbine blade with a pin fin cooling arrayin the trailing edge region from Kindlmann [2]

020904-2 / Vol. 134, FEBRUARY 2012 Transactions of the ASME


with an implicit algorithm. Pressure and momentum are coupledwith the pressure-implicit with splitting of operators (PISO) algo-rithm since the mesh had some skewed cells at the interfacebetween the channel grid and the cylinder grid (Fig. 2). The PISOscheme also yields faster convergence and allows pressure and mo-mentum under-relaxation parameters to be set to unity. The Cou-rant number is close to unity for the employed time step ofDt¼ 0.001 s. This is roughly equal to 0.034% of one cylinder liftcycle. One time step was considered converged when all residualsfell on the order of 10� 5. The entire simulation was run for a littleover 100 cycles of cylinder lift coefficient. On a 40 processor clus-ter, computing 5 lift cycles for this study with 15,000 time stepsand 6 iterations per time step took about 7 days wall clock time.

The computational domain models a single row of pin finsinside a rectangular duct, where the ratio of duct height to cylinderdiameter is unity, H/D¼ 1 (Fig. 3). The influence of walls in thespanwise direction is neglected by assuming an infinite row of finsand applying periodic boundary conditions on the lateral domain

faces, which are positioned such that one full pin fin is includedwithin the domain. Experimentally, this can be achieved by exam-ining a long row of fins with the spanwise test section dimensionmuch larger than the duct height and assuming that the middlepins will not see the influence of the spanwise walls.

The spanwise periodic boundaries are 2D apart, yielding an in-finite row of pin fins with a spacing of S/D¼ 2. Only one half ofthe domain in the wall normal direction is meshed and computedto save computational resources. A symmetry boundary conditionis applied on the top face. A pressure outlet boundary condition isused 14D downstream of the pin fin center. All required boundaryvalues at the inflow, which is modeled as a velocity inlet 4Dupstream of the pin, were obtained from a steady state simulationof an unobstructed channel beforehand and then linearly interpo-lated to fit the pin fin domain mesh. At the upstream side of thepost processed inflow channel, a turbulence intensity of 0.5% wasdefined and boundary conditions for k and x derived from correla-tions of intensity and channel hydraulic diameter. The streamwisevelocity is constant and the magnitude chosen to yield the desiredReynolds number. The inflow into the domain is turbulent andfully developed (both in terms of momentum and heat transfer). Aspecific wall heat flux _qWwas 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 ð _qPinFin ¼ _qW=2Þ. The values for material propertieswere evaluated at room temperature.

Initially, a structured grid was created with Gambit. Since wallfunctions cannot correctly model the complex flow phenomena,the boundary layer is resolved down to the viscous sublayer with asolution based grid adaptation. Figure 2 shows some adaptedmesh regions on the pin fin surface and the endwall. The finalmesh contains around 6.2� 106 hexahedral cells. Before, data forthe Orthogonal Decomposition (OD) analysis were recorded, cellswith high gradients in velocity (near the cylinder wall), highskewness, and distance too far from the wall, were refined at mul-tiple instants in time to retain symmetry within the mesh. Around98% of the wall nearest cells range between 0< yþ< 2 and nocell is farther from the wall then yþ¼ 5.4. For the grid independ-ence study as shown in Fig. 4, first all cell adaptations wereremoved yielding a domain with 1� 106 cells. For the second do-main with 2.2� 106 cells, only the first adaptation level wasretained. The third domain consists of 5.1� 106 cells and was cre-ated by removing refinements in the upper boundary layer parts.The Nusselt number averaged over the entire bottom surface (Fig.3) was then calculated and related to an open channel correlationfound in Kays and Crawford [20]. The local reference temperatureis calculated from an energy balance, in detail shown in Lyall et

Fig. 2 The URANS domain mesh close to the pin fin is dis-played (top). A top view of the bottom wall mesh is shown in thelower left figure. Grid refinement in regions of high velocity gra-dients on the pin fin surface viewed from the front are visible inthe lower right figure.

Fig. 3 The computational domain of the URANS study withapplied boundary conditions is shown schematically

Fig. 4 The deviation of Nusselt number augmentation for threecoarser grids compared to the finest grid solution is plotted forthe URANS simulation

Journal of Heat Transfer FEBRUARY 2012, Vol. 134 / 020904-3


al. [6]. This Nusselt number augmentation for the three coarserdomains is then related to the finest grid solution in terms of per-centage deviation and plotted in Fig. 4.

3.2 LES Study. For the large eddy simulation, the domainwas extended to include three cylindrical pins as opposed to onein order to avoid locking the shedding frequency in the wake ofthe pin by imposing periodic boundary conditions enclosing onlyone single pin. The spanwise periodic boundaries are therefore 6Dapart, the outflow with a pressure outlet was placed 9.5D down-stream of the pin row middle and the inflow is 9D upstream. Thedomain was extended to contain the full pin height and is thusbounded by two walls in z (Fig. 3).

To generate a fully developed, turbulent inflow, the inflowplane is a recycling periodic boundary that picks up velocity andtemperature values from a recycling plane located 3D upstream ofthe pin middle. When copying the different variables from therecycling plane to the inflow plane, the velocity components arescaled to maintain a constant flow rate. The heat added due to theconstant heat flux at the walls within this quasi periodic section iscomputed from a first law energy balance and substracted fromthe recycling temperature. All material properties were set to unityexcept for viscosity (used to set the desired Reynolds number fora bulk velocity of unity) and conductivity (used to set a Prandtlnumber of 0.74).

The Navier-Stokes equations are solved using a spectral elementmethod (for details see Fischer et al. [21]). The mesh consists of3552 elements and within each, the solution is approximated by13th order Legendre polynomials. To account for the unresolvedgrid scales, the last 3 modes of the solution polynomials are multi-plied with a second order filter function such that the last mode isreduced by 5%. This grid resolution provides a maximum wall dis-tance of the wall nearest cell (in the region of highest wall shear)of zþ< 1. Away from the walls, the grid point spacing increases inthe wall normal direction z (but stays the same in the streamwise xand spanwise y direction), such that the grid point spacing is Dxþ

� 33, Dyþ � 22, Dzþ � 11 in the middle of the recycling inflowsubsection. In the channel middle upstream of the pin row, thestreamwise turbulence intensity is around 4.4%.

For the spectral element LES, a convergence study could bedone by increasing the polynomial order, which would couple thefilter width with grid spacing, similar to eddy diffusivity LES on fi-nite volume meshes. As pointed out by Celik et al. [22], this pro-hibits estimating convergence towards a grid independent solution.For this reason, no convergence study is presented herein and vali-dation against experimental values becomes even more important.

3.3 Signal Decomposition. Due to the lack of two periodicspatial directions, the direct POD cannot be computed efficiently.In variation of a decomposition published by Aubry et al. [4]termed bi-orthogonal decomposition, the following approach istherefore based on time correlations. First, the signal vectorAið~x; tÞin index notation is defined including the fluid temperature

Aið~x; tÞ ¼ ½u; v;w;T� (4)

Here, u, v, and w are the nondimensionalized velocity componentsin streamwise, spanwise, and wall normal direction, respectively.T represents a nondimensional, weighted temperature to take theeffect of heat transfer into account. The temperature is nondimen-sionalized and weighted as shown in Eq. (5), where the Eckertnumber Ec is defined in Eq. (6).

Tð~x; tÞ ¼ EcTð~x; tÞ � Tin

Tout � Tin

(5)

Ec ¼ U20

cvðTout � TinÞ(6)

The signal is decomposed into scalar spatial functions ukð~xÞandvector temporal function sets Wi

k(t) for the kth eigenfunction

Aið~x; tÞ ¼X1k¼0

akukð~xÞWki ðtÞ (7)

These are computed by first solving an eigenvalue problem

ðT0

rijðt; t0ÞWkj ðt0Þdt0 ¼ ðakÞ2Wk

i ðtÞ (8)

rijðt; t0Þ ¼ð

X

Ajð~x; tÞAið~x; t0Þdx (9)

whereÐ

X stands for a spatial integral andÐ T

0is an integration over

time. Then, projecting the time function set Wki ðtÞ onto the signal

ðT0

Aið~x; tÞWki ðtÞdt ¼ akukð~xÞ (10)

yields the spatial eigenfunctions ukð~xÞ. For later evaluation ofindividual mode’s contribution to heat transfer, the basis functionsmust be scaled appropriately and redimensionalized to correctlyreproduce the original variables. This is done by combining Eqs.(7) and (10) to yield

Aið~x; tÞ ¼X1k¼0

ðT0

Ajð~x; tÞWkj ðtÞdtWk

i ðtÞ (11)

The spatial integration in Eq. (9) is numerically approximated bya summation over all control volumes, which are known from themeshed geometry. All time integrations are approximated using atrapezoidal quadrature, where the integration weight in Eq. (8) isapplied such that the matrix symmetry is preserved (Ball et al.[13]). Similar to the traditional POD as for example described inDuggleby et al. [23], the eigenvalues in Eq. (8) represent a frac-tion of the global system energy (kinetic and thermal). The projec-tion in Eq. (7) is optimal in the sense that it maximizes the energycontained in each eigenfunction or mode. Thus, the lowest possi-ble number of modes is needed to reconstruct a given fraction ofthe original data. An estimate for the dimension of a system, D90,is typically derived from the number of modes needed to resemble90% of the original data set’s energy content.

To estimate the contribution of individual modes to overall heattransfer, a control volume similar to external flow analysis (Kaysand Crawford [20]) is defined from the bottom wall to channelhalf height (a symmetry plane in the URANS study). On the uppersurface, heat transfer, shear and mass flux are neglected, whichestablishes a relation between the streamwise and spanwise flowvariables (enthalpy) and surface heat transfer. By using Reynoldsaveraging and Gauss’ divergence theorem, an expression for timeaveraged wall heat flux into a (control) volume of interest isderived (see Kays and Crawford [20])

�_qW ¼þ

L

ð10

q �Ucpð �T � �T1Þdz

� �~ex �~ndL

þþ

L

ð10

q �Vcpð �T � �T1Þdz

� �~ey �~ndL

þþ

L

ð10

qcpU0T0dz

� �~ex �~ndL

þþ

L

ð10

qcpV0T0dz

� �~ey �~ndL (12)

The modes of order k¼ 0 and k= 0 are used to express the aver-age and time-varying parts, respectively.



4 Results

To ensure a fully developed and statistically stationary flow,the lift coefficient on the cylinder surface

CLðtÞ ¼FLðtÞ

12qU2

0ANormal

(13)

was monitored over the simulation time and is shown in Fig. 5,top, for the URANS simulation. One data sample was taken every30 computational time steps so that roughly 90 snapshots are col-lected per one lift coefficient period (that is from one zero cross-ing of lift to the second closest one). The sampling frequency forthe orthogonal expansion is thus ninety times the vortex sheddingfrequency for the URANS. In comparison, Fig. 5, bottom, showslift and drag coefficient (the respective forces are normalized as inEq. (13)) for the LES on the middle pin. Here, roughly 34 datasamples are collected for one lift coefficient period. The periodicboundary enclosing only one pin does indeed lock the sheddingfrequency, whereas the 3-pin arrangement allows for the lift onthe middle pin to vary more freely.

The frequency of vortex shedding, which causes the oscillationsof the lift coefficient, is nearly constant and ranges between 0.333Hz and 0.342 Hz in the URANS study. On average for the plottedtime range, the Strouhal number St¼ 0.326, which is considerablyhigher than experimentally obtained results for a single cylinderin crossflow. The amplitude of the lift coefficient is also higherthan what is expected. The Strouhal number obtained from theLES and the middle pin is slightly higher.

The friction factor augmentation upstream of the pins from theLES study is shown in Fig. 6. Two different experimental correla-tions for the baseline friction factor f0 were used: the one that

gives higher augmentation values is used in Lyall et al. [6] andwas developed for geometries of circular cross section, but isexpected to be valid for rectangular channels when basing theReynolds number on the hydraulic diameter. As pointed out byPatel and Head [24], this concept breaks down for high aspect ra-tio channels, and another expression is shown therein (valid to3000<ReC< 105, where ReC is based on the channel height)which leads to the lower augmentation in Fig. 6. The computa-tional results lie roughly in the middle between the two correla-tions within a 6-7% range. From the plot, it is noted that the flowwithin the recycling section is statistically fully developed as thefriction factor is a straight horizontal line. Lyall et al. [6] alsopresent an array friction augmentation (f� f0)/f0¼ 3.8 for thegiven geometry and Reynolds number.

The pressure loss was measured over 15 hydraulic diameters.Since the computational LES domain was not that long, pressureloss from plane averaged data and wall pressure values were eval-uated at locations 7 hydraulic diameters apart. The downstreampressure measurement location was 5.75D away from the pin cen-ter. At this point, the pressure has not yet fully recovered and thefriction factor is not quite constant. But due to the accelerated out-flow which started at 5.75D, this location is as far away from thepins as possible. This resulted in (f� f0)/f0¼ 3.75 when using thestatic wall pressure line-averaged in the spanwise direction, and aslightly higher value of 4 when using y/z-planar averaged pressures.

A comparison of Nusselt number augmentation on the channelwall is shown in Fig. 7. The experiments, described in Lyall et al.[6], were conducted using the same geometry and Reynolds num-ber as in the present study. The surface Nusselt number augmenta-tion is defined as the ratio of actual Nusselt number (with areference temperature depending on the streamwise position) overthe Nusselt number for an unobstructed channel (found in Kaysand Crawford [20], Nu0¼ 0.022Re0.8Pr0.5). For Fig. 7, bottom,two additional correlations for Nu0 are used: the Dittus-Boeltercorrelation from Incropera and DeWitt [25] and a baseline Nusseltnumber computed based on the friction factor from Patel andHead [24], where Nu0¼ cf/2 Re Pr/(0.88þ 13.39 (Pr(2/3)� 0.78)(cf/2)0.5) is computed from Kays and Crawford [20]. The equa-tions are also given in the figure legend. Results are plotted interms of a spanwise line average.

Since the computational domain did not include the solid pinfin body, no data are presented in the region where the pin ismounted onto the channel surface (gray area in Fig. 7). Close tothe pin fin, both URANS AND LES show significant peaks thatdeviate from the experiments. This can be explained by the heattransfer augmentation on the channel surface due to conductionthrough the fin. This effect, as well as conduction within the wallitself, is not included in the simulations. A similar trend is shownby Su [26], who used a RANS approach for an array of pin fins (atslightly different Reynolds number and H/D¼ 2). Compared to

Fig. 5 The plot shows normalized lift force on the pin for the URANS study (top) and lift and drag for theLES on the middle pin (bottom). The vertical lines indicate the time span of the smaller sample used for or-thogonal decomposition. The lift and drag coefficients vary in magnitude and slightly in frequency. Eventhough the shedding breaks down for some time in the LES, it recovers to its former extent.

Fig. 6 The local, span-wise averaged friction factor augmenta-tion from the LES study is plotted as a function of streamwisedistance. The pressure data is normalized with different base-line correlations.



experimental values, the computed endwall heat transfer showssimilar peaks near the pin fin row independent of row number.

Upstream of the pin fin, heat transfer augmentation is predictedaround unity for the URANS study, which is expected but belowexperimental values. This might be due to different turbulence inletconditions compared to the experiment or an inappropriate correla-tion for Nu0. Lyall et al. [6] used a sand paper strip upstream of thefin row to trip the boundary layer, whereas the URANS simulationinlet conditions were computed from a smooth turbulent channelflow. The URANS computation shows a reasonable agreement inthe pin fin wake, but those results are improved with the LES.

Using the same correlation as Lyall et al. [6] for baseline heattransfer, the augmentation in the wake is matched closely. Thepeak locations of this curve are identical to the element corners ofthe spectral element grid. Upstream of the pin, the LES data showgood agreement with the Dittus-Boelter correlation from Incropera

Fig. 7 The local, span-wise averaged Nusselt number augmentation from the URANS (top, blacksolid line labeled “computations”) and LES (bottom) study compared to experiments is plottedas a function of streamwise distance. For the LES (one temperature field for NuLES), the tempera-ture data from the simulation is normalized with three different baseline correlations for Nu0.

Table 1 Time sample sizes and dimensions from URANS,excluding the first mode. D90, D95, and D99 is the number ofmodes required to retrieve 90, 95, and 99% of the flow,respectively.

Sample size D90 D95 D99

89 3 6 19268 4 7 35446 4 9 51

Fig. 8 The contour plots of the average bottom wall tempera-ture (left) and the scalar spatial eigenfunction uk ð~xÞ of mode 0at the bottom surface (right) from the URANS computation areidentical. The data are scaled to a common color range.

Fig. 9 Time eigenfunctions Wki ðtÞ from the URANS computation of the first 6 modes for ve-

locity (3 left) and temperature (right). The constant average streamwise velocity component(top left), two lift modes oscillating in the spanwise velocity m and three shearing modes 3-5can be identified.



and DeWitt [25]. When using the same correlation as Lyall et al.[6] for the LES upstream of the pin (Fig. 7, bottom, solid blackline, X/D<�2), the LES results are closer to the experiment thanURANS. For X/D> 2, the LES also provides better results thanURANS. In the region right downstream of the pin, URANS seemsto come closer to the experiments but that might also be due to thelack of turbulent interaction with neighboring pins as a result of theperiodic boundaries. The heat transfer augmentation right upstreamif the pin, in the region of the horse shoe vortex, is much higher inboth computations compared to experimental because the wallswere not conducting. An overprediction of heat transfer in the stag-nating region of a circular shaped body for RANS models was alsoseen in Carmine et al. [27] and attributed to the underlying modelURANS deficiencies. The difference in inlet turbulence of URANSand LES might also play a role, but with the given data cannot bedistinguished from modeling effects.

4.1 Orthogonal Decomposition URANS The decompositionanalysis was performed on three URANS data sets of differentsizes in time, namely for one, three, and five lift coefficient peri-ods. The time sample sizes can be found in Table 1. To ensureconvergence of the decomposition method, three different dimen-sions of the system are also given in Table 1. When plotted overthe number of time samples used, the dimension should convergetowards a constant value. That is not the case for the three samplesunder consideration. But since the system is periodic in time andturbulent fluctuations are filtered out by the chosen turbulencemodel, the first 6 modes under examination are expected not tochange when extracting the modes from data taken over a longerflow time.

The time averaged values for temperature and velocity were notsubtracted from the data before hand, thus the eigenfunctions ofthe first mode 0 are a representation of the average. Its spatial rep-resentation is depicted in Fig. 8 and is very similar to the averagewall temperature contour in the same figure. The cooling effect ofthe leading edge region horse shoe vortex, winding around thecylinder at its junction with the endwall, can be seen as an area ofrelatively low temperature where cool core flow is pushed towardsthe heated endwall. Downstream of the pin fin, vortices spinningoff from the cylinder create a rather homogeneous, relatively lowtemperature profile.

The time eigenfunctions Wki ðtÞ, Fig. 9, show that the only non-

zero average component is the streamwise velocity and the tem-perature in mode 0. Both modes 1 and 2 contain the flowfieldresponsible for spanwise flow fluctuations, causing a lift force onthe pin fin. The wall temperature fluctuations start deviating fromthe mean for modes higher than 2, as seen by the respective timeeigenfunctions which increase by an order of magnitude. The timeeigenfunctions of temperature have an overall much lower valuedue to the weighting when assembling the correlation matrix. Thisdoes not mean that they are less significant. The time basis func-tions from the medium and small sample size have similar qualita-tive features as the time functions shown in Fig. 9.

The decomposition analysis (Fig. 9) shows that the first threemodes look essentially the same with respect to wall temperature,therefore the flow field represented by modes 1 and 2 does not con-tribute to a change in wall temperature and thus heat transfer. If theheat exchanger efficiency is defined as the amount of heat trans-ferred into the flow via turbulent convection over the pressure loss

Fig. 10 Different modes k= 0 have been used to evaluate theintegrals in Eq. (12) for the URANS and show that the dominantmode combination in terms of surface heat flux is 3/3. Thismode combination is the most useful in meeting the device’sdesign goal of augmenting heat transfer. Modes 1 and 2 can beconsidered parasitic since they contain drag-producing flowstructures but have no impact on turbulent heat flux.

Fig. 11 The figure shows contours of the scalar spatial eigenfunction uk ð~xÞat 10%D. The plots from left to right are: Mode 2URANS, mode 2 LES, mode 3 URANS, mode 3 LES, mode 4 URANS, mode 7 LES.



in the passage, then the modes containing large scale unsteady flowfeatures but no wall temperature dynamics can be considered inef-ficient. The energy contained in these two modes (83.22% of theenergy in all nonzero modes) can therefore be considered a lossand a closer examination might yield some potential for either opti-mizing the geometry or fin arrangement. The evaluation of the inte-gral from Eq. (12) is shown in Fig. 10, where different modecombinations have been inserted for the fluctuating parts and theaverage (k¼ 0) was not considered. Mode combination 3/3 has thehighest contribution to turbulent heat flux for a control volumearound the pin. Due to the optimality of the orthogonal expansion,the higher order modes show a decreasing importance. This analy-sis framework provides the physical basis for categorizing modesin “useful” and “parasitic” in terms of heat transfer. The latter isthe case for the low order modes 1 and 2 which do not significantlycontribute to the turbulent heat exchange, but contain flow struc-

tures responsible for drag. A useful mode in this regard is mode 3,which has a distinctly higher peak than the other combinationsdepicted in Fig. 10 and thus enhances heat transfer, which is whatthe pin fin array has been designed for.

In Fig. 11, the near wall spatial eigenfunctions (which includeall four components of the flow field) are plotted on a plane at10%D above the bottom wall. The spatial eigenfunctions of theparasitic mode 2 is shown left in Fig. 11. Compared to the usefulspatial eigenfunctions of modes 4 and 7 (Fig. 11, right), whichcontribute to heat transfer increase, they appear as a one-dimensional wave function of streamwise distance downstream ofthe pin. Mode 3, in turn, has a higher spatial complexity and cov-ers the region of the stagnating horse shoe vortex.

4.2 Orthogonal Decomposition LES. For the LES, an or-thogonal expansion was computed within a box around all 3 pins,where the box covered the domain between 3D upstream of thepins and 5.75D downstream, for two data sets—one spanning theentire lift history as seen in Fig. 5 and another one for approxi-mately one lift coefficient period as indicated by the vertical linesin the same figure. Every second data sample (for a total of 443samples) was used to compute the orthogonal expansion. The plotof time basis functions for the larger sample size is shown in Fig.12, which is qualitatively similar to the basis functions obtainedfrom the smaller sample. Again, since the time averages wherenot subtracted before hand, mode 0 shows the mean flow rate in xand an increase in mean temperature. As seen from the basis func-tions from URANS, the high order v-modes are very similar to thelift oscillation, but as there are now also fluctuations in the stream-wise direction, the u-functions exhibit the same dynamics as thedrag curve in Fig. 5, bottom. The LES proves useful in that it cap-tures more flow dynamics (namely in the streamwise direction asseen from the low order modes) and avoids filtering out largescale motions via a turbulence model assumption. The first non-zero mode in the temperature functions is mode 7, where it wasmode 3 in URANS. This is due to the fact that LES data containmore large energy fluctuations (large eddies are now resolved)which spread the mode spectrum over a higher energy range com-pared to URANS. But it can still be hypothesized that there aremode combinations which do not contribute to heat transfer aug-mentation. Modes one to six contain 32.14% of the total energy inall nonzero modes. Unlike in the URANS, the temperature func-tion of mode 7 does not strongly correlate with velocity and does

Fig. 12 Time eigenfunctions Wki ðtÞ of 8 modes for velocity (3

left) and temperature (right) from the LES. The constant averagestreamwise velocity component (top left), two lift modes oscil-lating in the spanwise velocity v and two mainly shearingmodes 4-5 can be identified. The first mode with nonzero tem-perature function is mode 7. The functions are scaled for bettervisualization.

Fig. 13 The figure shows isosurfaces at 24 (red) and 4 (blue) of the scalar spatial eigenfunction uk ð~xÞ of the LES. Upper left:Mode 0, lower left: Mode 2, not contributing to heat transfer. Upper right: Mode 3, a shearing mode. Lower right: Mode 7, usefulmode in terms of heat transfer.



not vary in time. Similarities between URANS and LES also existfor the spatial functions, which are shown in Fig. 13. The moderepresenting the average flow field (whose only nonzero time ba-sis function is in the streamwise component), top left, extents far-ther downstream from the middle pin compared to the side pins.This is due to the break up of vortex shedding, which makes thewake of the middle pin longer for a certain part of the simulatedtime. A lift mode, lower left, whose time basis function has thehighest amplitude in the v-component, is very similar to what wasseen from the URANS results. The shearing mode 3, which in theURANS study used to be the most important mode for turbulentheat flux, also has a very similar spatial basis function structure asit is active in the region of the horse shoe vortex. In comparingLES and URANS derived basis functions with regard to potentialdevice optimization, it becomes apparent that the modes onewould likely pick (3 for URANS and 7 for LES) show a differentbehavior in time and space. Most notably, the first nonzero LESbasis function for temperature is not a function of time. Howeverone would derive an optimization from those modes, the improvedgeometry would look different depending on what turbulencemodel was used to compute the basis. Employing the computa-tionally more intense LES seems appropriate because the model iscloser to the physics of the “real” heat exchanger. The orthogonaldecomposition presented herein only guarantees optimality for thedata base it was generated from. Modes identified as useful inURANS thus do not have to be the same in LES, considering thatLES does not only allow a frequency spectrum but also trueunsteadiness in the upstream channel part. The quantitative modeassessment is shown in Fig. 14, proving that the mode combina-tion 7/7 has the highest positive heat transfer contribution. In thisapplication, higher heat transfer is desired since the internal pas-sage is intended to cool the blade walls. Combinations of tempera-ture mode 7 with other velocity modes, shown in Fig. 14 right,have a negative impact on wall heat flux.

5 Conclusion

Using a URANS calculation and a LES, the heat transfer effec-tiveness of a single, infinite row of pin fins was examined. The in-finite row was simulated with one pin (URANS) and 3 pins (LES)enclosed by periodic boundaries. Compared to experiments, thecomputationally intense simulations showed good agreement,depending on the correlations for unobstructed flow from the liter-ature. The level of resolution in space and time allows for a highorder analysis of the pin fin heat transfer effectiveness that wouldbe impossible to obtain experimentally.

A new orthogonal decomposition and analysis technique is cre-ated to analyze heat transfer effectiveness. Inefficiencies are iden-tified as flow features that do not contribute to heat transferaugmentation but contain relatively high amounts of energy. Themost inefficient modes are those responsible for the oscillating lifton the pin, and the mode with the highest heat transfer (per LESsimulation) is the seventh most energetic. This means that there isplenty of room for optimization towards more efficient internalcooling. Potential improvements would include geometric adjust-ments that cause higher streamwise flow fluctuations relative tothe spanwise fluctuations. The most efficient way to increase heattransfer might thus not be to just increase turbulence levels in anisotropic fashion, but to stimulate or dampen certain directions orfrequencies. If the turbulence generation is caused by objects onthe scale of the geometry, such as pin fins in channels with H/D¼ 1, it is the large scale structures that have the highest contri-bution to convective heat transfer. Due to the orthogonal decom-position, these can be analyzed in a decoupled fashion fromsmaller scale eddies which they feed energy into, even thoughthey are nonlinearly related. While the large scale structures arenecessary for the existence of more heat transfer efficient smallerscales, a redistribution of the energy towards the more usefulmodes will enable more efficient heat transfer.

Nomenclatureak¼ Eigenvalue of mode k_qW ¼ specific wall heat fluxT¼ nondimensionalized and weighted temperature

Ec¼ Eckert numberNu0, f0¼ Nusselt number and friction factor of unobstructed

channelRe ¼ U02D

�¼ Reynolds number based on channel hydraulic

diameterSt¼ Strouhal number

Tin¼ average inflow temperatureTout¼ average exit temperature

Wik(t)¼ time basis function of mode k and index i

q¼ densityukð~xÞ¼ scalar spatial basis function of mode k

Aið~x; tÞ¼ signal vector containing nondimensional velocityand temperature

BOD¼ bi-orthogonal decompositionCL,CD¼ lift and drag coefficient

D¼ pin diameterH¼ channel height

LES¼ large Eddy simulationPISO¼ pressure-implicit with splitting of operators

pressure-velocity coupling schemePOD¼ proper orthogonal decomposition

S¼ spanwise pin spacingT¼ static temperature

u, v, w¼ streamwise, spanwise and wall-normal nondimen-sionalized velocity component

U0¼ bulk flow velocityURANS¼ unsteady Reynolds averaged Navier-Stokes

equationsxþ, yþ, zþ¼ nondimensional wall distance/grid point spacing

based on turbulence scales

References[1] Schobeiri, M., Turbomachinery Flow Physics and Dynamic Performance

(Springer-Verlag, Berlin, Heidelberg, 2005).[2] Kindlmann, G. L., 1999, “Semi-Automatic Generation of Transfer Functions

for Direct Volume Rendering,” MS Thesis, Cornell University, Ithaca, NY, Jan-uary, http://www.cs.utah.edu/�gk/MS/html/thesis.html, accessed February 10th2009.

[3] Han, J.-C., Dutta, S., and Ekkad, S. V., 2000, Gas Turbine Heat Transfer andCooling Technology, Taylor and Francis, New York.

[4] Aubry, N., Guyonnet, R., and Lima, R., 1991, “Spatiotemporal Analysis ofComplex Signals: Theory and Applications,” J. Stat. Phys., 64, pp. 683–739.

[5] Armstrong, J., and Winstanley, D., 1988, “A Review of Staggered Array Pin FinHeat Transfer for Turbine Cooling Applications,” J. Turbomach., 110, pp. 94–103.

Fig. 14 Different modes k= 0 have been used to evaluate theintegrals in Eq. (12) for the LES and show that the dominantmode combination in terms of positive surface heat flux is 7/7.There are 6 lower order modes (energetically greater) that donot contribute to the heat transfer.



www.cs.utah.edu/&hx223C;gk/MS/html/thesis.html



http://dx.doi.org/10.1007/BF01048312

http://dx.doi.org/10.1115/1.3262173

[6] Lyall, M., Thrift, A., and Thole, K., 2007, “Heat Transfer from Low Aspect Ra-tio Pin Fins,” ASME Paper GT-2007-27431.

[7] Ames, F., and Dvorak, L., 2005, “Turbulent Transport in Pin Fin Arrays—Ex-perimental Data and Predictions,” ASME Paper GT-2005-68180.

[8] Young, M., and Ooi, A., 2004, “Turbulence Models and Boundary Conditionsfor Bluff Body Flow,” Proceedings of the 15th Australasian Fluid MechanicsConference, Sydney, Australia.

[9] Holloway, D., Walters, D., and Leylek, J., 2004, “Prediction of Unsteady, Sepa-rated Boundary Layer Over a Blunt Body for Laminar, Turbulent, and Transi-tional Flow,” Int. J. Numer. Methods Fluids, 45, pp. 1291–1315.

[10] Harrison, K., and Bogard, D., 2008. “Comparison of Rans Turbulence Modelsfor Prediction of Film Cooling Performance,” ASME Paper GT-2008-51423.

[11] Holmes, P., Lumley, J. L., and Berkooz, G., 1996, Turbulence, CoherentStructures, Dynamical Systems, and Symmetry, Cambridge University Press,Cambridge, UK.

[12] Lumley, J. L., 1967, “The Structure of Inhomogeneous Turbulent Flows,” Atmos-pheric Turbulence and Radio Wave Propagation, Nauka, Moscow, pp. 166–178.

[13] Ball, K. S., Sirovich, L., and Keefe, L. R., 1991, “Dynamical EigenfunctionDecomposition of Turbulent Channel Flow,” Int. J. Numer. Methods Fluids, 12,pp. 585–604.

[14] Sirovich, L., Ball, K. S., and Keefe, L. R., 1990. “Plane Waves and Structuresin Turbulent Channel Flow,” Phys. Fluids A, 12, pp. 2217–2226.

[15] Sirovich, L., Ball, K. S., and Handler, R. A., 1991, “Propagating Structuresin Wall-Bounded Turbulent Flows,” Theor. Comput. Fluid Dyn., 2, pp.307–317.

[16] Webber, G. A., Handler, R. A., and Sirovich, L., 2002, “Energy Dynamics in aTurbulent Channel Flow Using the Karhunen-Loeve Approach,” Int. J. Numer.Methods Fluids, 40, pp. 1381–1400.

[17] Duggleby, A., Ball, K. S., Paul, M. R., and Fischer, P. F., 2007, “DynamicalEigenfunction Decomposition of Turbulent Pipe Flow,” J. Turbulence 8(43),pp. 1–24.

[18] Duggleby, A., Ball, K. S., and Paul, M. R., 2007, “The Effect of Spanwise WallOscillation on Turbulent Pipe Flow Structures Resulting in Drag Reduction,”Phys. Fluids, 19, pp. 125107-1–125107-12.

[19] Durbin, P., 1996, “On the k-3 Stagnation Point Anomaly,” Int. J. Heat FluidFlow, 17(1), pp. 89–90.

[20] Kays, W., and Crawford, M., 1980, Convective Heat and Mass Transfer,McGraw-Hill, New York.

[21] Fischer, P. F., Lottes, J. W., and Kerkemeier, S. G., 2008, “Nek5000 Webpage,” http://nek5000.mcs.anl.gov

[22] Celik, I., Klein, M., and Janicka, J., 2009, “Assessment Measures for Engineer-ing les Applications,” J. Fluids Eng., 131(3), 031102.

[23] Duggleby, A., Ball, K., and Schwanen, M., 2009, “Structure and Dynamics ofLow Reynolds Number Turbulent Pipe Flow,” Phil. Trans. R. Soc. A, 367,pp. 473–488.

[24] Patel, V., and Head, M., 1969, “Some Observations on Skin Friction and Veloc-ity Profiles in Fully Developed Pipe and Channel Flows,” J. Fluid Mech., 38,pp. 181–201.

[25] Incropera, F., and DeWitt, D., 2002, Fundamentals of Heat and Mass Transfer,Wiley, New York.

[26] Su, G., 2005, “Computation of Flow and Heat Transfer in Rotating RectangularChannels (ar¼ 4:1) With Pin-Fins by a Reynolds Stress Turbulence Model,”ASME Paper GT-2005-68390.

[27] Carmine, E. D., Facchini, B., and Mangani, L., 2008, “Investigation of InnovativeTrailing Edge Cooling Configurations With Enlarged Pedestals and Square orSemicircular Ribs. Part II—Numerical Results,” ASME Paper GT-2008-51048.



http://dx.doi.org/10.1002/fld.v45:12


http://dx.doi.org/10.1063/1.857808

http://dx.doi.org/10.1007/BF00271470



http://dx.doi.org/10.1080/14685240701376316

http://dx.doi.org/10.1063/1.2825428

http://dx.doi.org/10.1016/0142-727X(95)00073-Y

http://dx.doi.org/10.1016/0142-727X(95)00073-Y

http://nek5000.mcs.anl.gov

http://dx.doi.org/10.1115/1.3059703

http://dx.doi.org/10.1017/S0022112069000115

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