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Optimizing turbulent heat transferusing orthogonal decomposition-basedcontouringMarkus Schwänen a & Andrew Duggleby aa Department of Mechanical Engineering, Texas A&M University,College Station, TX, 77843, USA

Available online: 11 Aug 2011

To cite this article: Markus Schwänen & Andrew Duggleby (2011): Optimizing turbulent heattransfer using orthogonal decomposition-based contouring, Journal of Turbulence, 12, N31

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Journal of TurbulenceVol. 12, No. 31, 2011, 1–20

Optimizing turbulent heat transfer using orthogonaldecomposition-based contouring

Markus Schwanen∗ and Andrew Duggleby

Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA

(Received 31 December 2010; final version received 14 June 2011)

Large-scale simulations of cylindrical pin fins with a spanwise distance to fin diameterratio of 2 and height over fin diameter ratio of 1 are performed and optimized basedon orthogonal decomposition. The flow Reynolds number based on hydraulic channeldiameter and bulk velocity (Re = 12, 800) is chosen based on experimental resultsavailable in the open literature. The three-pin-wide domain was simulated using unsteadylarge eddy simulations.

The resulting flow field consisting of velocity and temperature data is analyzed us-ing an orthogonal decomposition technique based on temporal correlations of all flowvariables, including temperature. The first seven temperature basis functions shows anear-zero amplitude, indicating their irrelevance for turbulent heat transfer augmen-tation. A mode combination ranking methodology based on enthalpy confirmed thepotential for optimization. An isosurface of the most important mode’s temperaturebasis function is then used to contour the domain endwalls with a first-order approx-imation. Although some engineering thresholds are imposed to achieve a smooth andbounded wall deformation, the method as developed is still expected to be generallyapplicable to different flows and geometries. The mean heat transfer augmentation andpressure drop increased equivalently. Wall temperature fluctuations decreased on aver-age and locally. Even though lower order modes show a zero temperature amplitude afterwall contouring, the energy contained in those decreased. The increase in heat transferstems mainly from the effect of wall contouring on vortex shedding. This is achievedby manipulating high energy modes which contain vortex shedding motions.

Keywords: orthogonal decomposition; optimization; turbulence; heat transfer; pin fin;LES

Nomenclature

Greek Symbols

α Eigenvalue�2 Convective boundary layer enthalpy thicknessϕ Spatial basis function� Time basis function

Superscripts

ϑ Based on mean temperature and velocityς Based on fluctuating temperature and velocity

∗Corresponding author. Email: schwaenen@tamu.edu

ISSN: 1468-5248 online onlyC© 2011 Taylor & Francis

DOI: 10.1080/14685248.2011.602346http://www.informaworld.com

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2 M. Schwanen and A. Duggleby

− Time averagek,m, n Mode order′ Fluctuating part

Roman Symbols

Ai(x, t) Signal vector containing nondimensional velocity and temperatureCL,CD Lift and drag coefficientD Pin diameterEc Eckert numberH Channel heightLES Large eddy simulationL2 Function Space�n Unit vector normal on control surfaceNu0, f0 Nusselt number and friction factor of unobstructed channelOD Orthogonal DecompositionPOD Proper Orthogonal Decompositionq ′′

s Specific wall heat flux at boundaryRe = U02D

νReynolds number based on channel hydraulic diameter

S Spanwise pin spacingT Static fluid temperatureT Nondimensionalized and weighted temperatureU,U ∗, R, S Linear, bounded operatorsU0 Mean bulk velocityu, v,w Streamwise, spanwise, and wall-normal nondimensionalized velocity

componentx+, y+, z+ Nondimensional wall distance/grid point spacing based on turbulence scales

Subscripts

in At domain inletout At domain outletφ Available or potential

1. Introduction

Heat transfer in turbulent flows is not only an academically challenging problem, but alsoof great relevance to many real world applications. In a great number of thermal-fluid engi-neering systems, efficiency is directly related to the amount of heat a surface can withstandwithout melting. Unfortunately, turbulence enhancing methodologies to increase the rateof heat transfer often come at the cost of increasing pressure loss. In order to characterizeand understand such flows, well-resolved investigations are needed because steady simula-tions (Reynolds-averaged Navier–Stokes) do not compare well with experimental data andexperimental point or planar measurements cannot capture 3D turbulent structures. Recentadvances in supercomputing with large eddy simulation (LES) and structure identificationtechniques (proper orthogonal decomposition) now make an investigation into the flowdynamics and heat transfer accessible computationally.

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Journal of Turbulence 3

The advantages are significant as computational visualization and advanced analysissuch as proper orthogonal decomposition offer insight into the heat transfer dynamics. Thechallenge, as in any computational work, is to validate resolution and model accuracy bycomparing with experimental results. Once validated, the insight gained is applied to a morephysics-based methodology of optimizing turbulent heat transfer systems.

Many computational optimization strategies are based on user-defined design parame-ters and cost functions, for which global minima are searched. The number of simulationsto map out the solution space increases with the number parameters, which prohibits theuse of LES in many cases. Further, the solutions are constrained a priori by the designparameters – it is not known whether the iterated optimum is the best solution possible.

In this study, an analysis methodology that is based on orthogonal basis functions derivedfrom postprocessing data from a highly resolved simulation is developed and tested. Theorthogonal basis functions are ranked with respect to their contribution to surface heatflux. An isosurface of the most important mode is then used to optimize a heat exchangervia endwall contouring. As opposed to gradient-based optimizing strategies, this approachis led by the flow physics and does not require user constraints on the solution space.This method is expected to be generally applicable, although it might not always be mosteffective or efficient to contour a flow bounding wall for suppressing or stimulating certainmodes. To achieve a smooth and reasonably bound contouring, we apply certain thresholdvalues for transforming a mode isosurface to a wall contour. These engineering thresholdsdo not limit the method’s general applicability.

2. Background

To develop and test the novel analysis methodology, a turbulent heat transfer applicationthat is both computationally tractable as well as practically relevant is needed. Such anapplication was found in pin fin heat exchangers, which have been studied in broad variety(see Armstrong and Winstanley (1988) [1]). A fairly low-Reynolds-number study with arelatively simple geometry that still contains all relevant features of such flows is presentedin Lyall et al. (2007) [2], which serves as an experimental validation for this work. A singlerow of pin fins with a height-to-diameter ratio H/D = 1 in a square duct with a very highaspect ratio was studied under varying Reynolds number and pin spacing. Results werereported in terms of friction and heat transfer augmentation compared to unobstructedchannel flow. The pressure measurements were performed on certain wall locations, wherethe temperature was measured with high spatial resolution. The heat transfer increase for apin spacing of S/D = 2 is highest for the lowest Reynolds number used, Re = 5013, andremains on a rather constant level for Reynolds numbers in the range 7500 < Re < 17, 500.

Computationally, this problem was examined by Ames and Dvorak (2005) [3] in amultirow pin fin array. They used the k–ε turbulence model and symmetric boundary con-ditions at the cylinder midspan and along even and odd row pin centerlines. Their steadycalculations showed an underprediction of heat transfer augmentation and pressure loss.One reason for the deviation might lie within the turbulence closure: for the cylinder incross-flow, Young and Ooi (2004) [4] examined different two-equation turbulence modelsand found poor agreement with the experimental data for the drag coefficient. They suggestimproving turbulence modeling. The overprediction of turbulent kinetic energy, a knowndeficiency of eddy viscosity models in stagnating flow, does not allow for correct determi-nation of the boundary layer separation point, which has an impact on the wake and thuswall surface heat transfer. Holloway et al. (2004) [5] introduce a third transport equation

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4 M. Schwanen and A. Duggleby

for laminar kinetic energy, which is expected to extend the turbulence model’s validity topretransitional, laminar boundary layers.

The underlying problem, that deriving turbulent viscosity from strain rate is not a validconcept in intermittent regions or laminarization zones, is addressed by Stolz (2005) [6],who developed a closure for large eddy simulations based on high-pass-filtered velocities.The model, which, in principle, is similar to an eddy viscosity ansatz for the subgrid stresses,is supposed to be inactive when no fluctuations (only low frequencies) are present in theflow. High shear in a laminar layer would thus not lead to excessive subgrid stress. A naturalway to implement such a model is the use of a spectral code, where the solved variables arerepresented as polynomials of different order. A highly parallel, open source and accuratespectral element code, NEK5000, has been developed by Fischer et al. (2008) [7]. Thesolver has shown 70% parallel efficiency for over 230,000 processors. As an alternativeto increasing viscosity in the momentum equation to account for unresolved scales, thestabilizing filter as proposed in Fischer and Mullen (2001) [8] is used as a fixed-coefficientturbulence model for the large eddy simulation.

The wealth of data obtained from spatial and temporal highly resolved simulations offersthe chance to gain insight from postprocessing beyond first- and second-order statistics.Ball et al. (1991) [9] apply proper orthogonal decomposition to data from direct numericalsimulation of channel flow in order to understand bursting and sweeping events in theturbulent boundary layer. Later, Duggleby et al. (2009) [10] applied the same methodologyto discover the energy transfer between turbulent structures in pipe flow. These studiesshow that orthogonal decomposition is an effective way to analyze turbulent flow data.Due to the canonical nature of their geometries, the direct computation of orthogonalspatial basis functions (proper orthogonal decomposition) could be done efficiently with apriori known results in the periodic directions. For the pin fin domain, this is not the case.Thus, an alternative method is needed which is shown in Aubry et al. (1991) [11]. Theirwork demonstrates the equivalence of orthogonal basis functions derived from time andspace autocorrelations. In addition to computing the time eigenfunctions from temporalcorrelations, as done in this study, or from projecting the spatial eigenfunctions onto theoriginal data set, as done, for example, in Duggleby et al. (2009) [10], it is also possible toapply a Galerkin projection of the Navier–Stokes equations onto the spatial basis functionsas originally done in Aubry et al. (1988) [12]. Orthogonal modes can also be used tobuild reduced order models. One application is flow control, for example, to reduce dragby affecting the vortex shedding in the wake of cylinders, by Bergmann and Cordier(2008) [13].

Some current computational fluid dynamics optimization strategies are presented inThevenin and Janiga (2008) [14]. One way to optimize turbulent heat transfer systems is bydeforming the wall surface. In one article, the wall of a wavy channel was optimized for lowpressure loss at high heat transfer. When using polynomial functions to parameterize thetwo-dimensional wall contour, some efficient design solutions could not be fabricated andhad to be filtered out. For more complex geometries, Lynch et al. (2011) [15] and Saha andAcharya (2008) [16], for example, use nonaxisymmetric endwall contouring to improveheat transfer in a turbine passage. The geometry can be determined computationally byparameterizing the surface, defining a cost function, and then trying to find optima byprobing the solution space. A high likelihood of finding global optima instead of local onesis provided by genetic algorithms as shown in Reising and Schiffer (2009) [17], who used acommercial computational fluid dynamics (CFD) code to optimize a compressor stage byhub and shroud contouring. Solutions are explored that are defined and constrained by the

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Journal of Turbulence 5

Figure 1. The computational domain consists of periodic boundaries in the lateral direction (brown),a pressure outlet (blue), no slip walls with a constant heat flux (gray) and a recycling inflow (twoblack planes), where velocity and temperature are copied from the downstream plane to the domaininlet after appropriate scaling. The top wall is omitted for better visualization.

user-defined design parameters and rely on a multitude of CFD runs, which were coarselyresolved (RANS turbulence model) to maintain computational feasibility.

3. Computational setup

The computational domain consists of three pins in a square channel, with height to pindiameter ratio H/D = 1 and spacing S/D = 2. Measured from the pin centers, the domainextends 8.5 pin diameters downstream and 9 pin diameters upstream. In the lateral direction(y in Figure 1), periodic boundaries simulate an infinite, single-row arrangement. Threepins instead of one were included to avoid locking the vortex shedding frequency. Top andbottom are modeled with no slip walls and a constant specific heat flux boundary condition.The heat flux applied on the no slip pin cylinder walls is equal to the heat conducted throughthe cylinder caps covered by the endwall. Starting 1.75D upstream of the outflow plane, asource term is added to the continuity equation. Its strength increases linearly. This resultsin an acceleration toward the outflow plane and straightens vortical flow structures thatmight create a backflow through the pressure outlet, which, in turn, might cause a divergingsimulation.

To generate a turbulent inflow, a recycling section is implemented (enclosed by the twoblack planes in Figure 1) where all three velocity components and the flow temperature areread during runtime from the downstream plane and then copied to the inflow. Within it, theflow is initialized with a Walsh vortex solution in the x/y-plane (see Walsh (1992) [18]).When recycling, all velocity components are scaled such that a domain-averaged meanstreamwise velocity of unity is maintained. The total added heat is subtracted from therecycling temperature. For the given diameter, the viscosity is set to yield the flow Reynoldsnumber based on hydraulic channel diameter and bulk velocity of Re = 12, 800. The fluidconductivity is fixed such that the Prandtl number is Pr = 0.71. The incompressible Navier–Stokes equations together with the energy equation, temperature being treated as a passivescalar, are solved using a spectral element method (Fischer et al. (2008) [7]). The domainis meshed with 3552 hexahedral elements (see Figure 2) and within each, the solution

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6 M. Schwanen and A. Duggleby

Figure 2. The mesh consists of hexahedral elements, which are smaller near the wall. The left figureshows a slice in the y–z plane through the pin row center. The right plot shows a slice in the x–y plane,where the pin fin bodies appear as unmeshed holes.

is approximated with 13th-order Lagrange polynomials. As seen in Figure 2, the gridresolution is increased near the walls to resolve the boundary layer. The maximum walldistance of the wall-nearest cell (in the region of highest wall shear) is z+ < 1. The gridpoint spacing increases away from the walls, but remains the same in the streamwise x andspanwise y direction. In the middle of the recycling inflow subsection (enclosed by the twoblack planes in Figure 1), the grid point spacing is �x+ ≈ 33,�y+ ≈ 22,�z+ ≈ 11. Theeffect of unresolved grid scales (subgrid-scale viscosity) is accounted for by filtering theequations, which is done by multiplying the Lagrange coefficients with a filter function.In this study, the interpolants of order 11, 12, and 13 are multiplied with a quadratic filterfunction with a maximum at order 11 such that the last mode is reduced by 5%.

The solution at each time step is considered converged when the residuals for pres-sure/continuity reach 10−6 and those for velocity and temperature 10−8. For a resolutionconvergence study, increasing the polynomial order would change the filter width becauseit is coupled to the grid spacing when the number of filtered modes is kept constant. This issimilar to eddy diffusivity LES on finite-volume meshes. Estimating convergence toward agrid-independent solution is thus prohibited as pointed out by Celik et al. (2009) [19]. Sinceno convergence study can be presented, validation against experimental values becomeseven more important.

4. Time orthogonal decomposition (TOD)

4.1. TOD with vectorial time eigenfunctions

The flow data generated by the LES are analyzed with an orthogonal decomposition methodthat is a variant of Aubry et al. (1991) [11]. Ultimately, this results in a scalar spatial fieldand a vector time signal which is more useful for extracting wall contouring information.Because the vector component of the orthogonal expansion is associated with time, theterm time-orthogonal decomposition will be used to describe this variant.

The following derivation is similar to [11], but uses a multidimensional signal Ai(x, t)from the start, where Ai(x, t) is a real, measurable, square-integrable, vectorial function ofspace and time

Ai(x, t) = [u, v,w,T], A ∈ L2(X × T), X ⊂ R1, T ⊂ R

4.

Here, u, v, and w are the velocity components in streamwise, spanwise, and wall-normaldirection nondimensionalized with the bulk inflow velocity U0. T represents a nondimen-sionalized and weighted temperature as shown in Equation (1), where the Eckert number

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Journal of Turbulence 7

Ec is defined in Equation (2)

T(x, t) = EcT (x, t) − Tin

Tout − Tin

, (1)

Ec = U 20

cp(Tout − Tin). (2)

Even though the problem is three-dimensional in space, the spatial coordinate x ∈ R1 is a

scalar and can be thought of as a pointer variable identifying a particular control volumeof a discretized domain. This does not hinder spatial integration because it is approximatedby a summation over grid cell volumes. For problems involving unstructured grids, as isthe case for the pin fin simulation under consideration, spatial correlations are hard to finddue to numerical reason. From the traditional POD methods, the “Method of Snapshot”overcomes this limitation by using correlations in time. By defining a linear operator

U : L2(X) → L2(T)

such that

∀ϕ ∈ L2(X), (Uϕ)i(t) =∫

X

Ai(x, t)ϕ(x)dx (3)

and the adjoint operator

U ∗ : L2(T) → L2(X)

such that

∀� ∈ L2(T), (U ∗�)(x) =∫

T

Ai(x, t)� i(t)dt, (4)

the analysis of the signal Ai(x, t) is reduced to the spectral analysis of U , which is compactfor A ∈ L2(X × T). This is motivated by the existence of orthonormal functions in X andT, as shown in [20] (Theorem VI.3.6).

We introduce another operator R = UU ∗ which is a nonnegative, integral (tensorial)operator whose kernel is the temporal correlation of the signal Ai(x, t). Consider a functionset � in L2(T), the operator kernel is shown to be

(R�)j (t) =∫

T

rij (t, t ′)�i(t′)dt ′, (5)

where rij (t, t ′) = ∫X

Aj (x, t)Ai(x, t ′)dx is the (tensorial) temporal correlation function ofthe signal. Similarly, considering a function ϕ(x) in L2(X), the kernel of operator S = U ∗Uwould be a spatial correlation. With spectral decomposition of the operators as shown inWerner (2007) [20] (Theorem VI.3.2), it can be derived that there exists a decomposition

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8 M. Schwanen and A. Duggleby

of the signal Ai(x, t) such that

Ai(x, t) =∞∑

k=0

αkϕk(x)�ki (t) (6)

with

α0 ≥ α1 ≥ ... > 0,

limN→∞

αN = 0,

(ϕk, ϕl) = (�k,� l) = δkl .

To compute the time eigenfunctions �ki , an eigenvalue problem involving the operator R

needs to be solved

U =∞∑

k=0

αk(·, ϕk)�ki (7)

UU ∗ =∞∑

k=0

αk((U ∗·), ϕk)�ki

=∞∑

k=0

αk(·, (Uϕk))�ki

=∞∑

k=0

(αk)2(·, �ki )�k

i , (8)

(R�k)i(t) = (αk)2�ki (t),∫

T

rij (t, t ′)�kj (t ′)dt ′ = (αk)2�k

i (t). (9)

The time integral on the left-hand side can be approximated numerically and written in ma-trix form so that the entire expression can be treated with standard numerical eigenproblemsolvers.

The spatial eigenfunctions are to be computed by projecting the time function set onthe signal

U ∗Uϕk = (αk)2ϕk,

U ∗αk�k = (αk)2ϕk,∫T

Ai(x, t)�ki (t)dt = αkϕk(x). (10)

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Journal of Turbulence 9

The appropriate basis function scaling for reduced order representations can be obtainedby combining Equations (10) and (6)

Ai(x, t) =∞∑

k=0

∫T

Ai(x, t)�ki (t)dt�k

i (t). (11)

The scalar eigenfunction ϕ(x) will not satisfy continuity or match boundary conditions,for example, no slip at walls. But because ϕ at the wall contains only the temperaturecontribution of the original data set (see Equation (10)), examining the decomposed surfacetemperature is still possible and is of most interest in this study. Another way of looking atthe resulting function sets is to see the spatial eigenfunctions as flow energy (originatingfrom kinetic and thermal sources lumped together) and the time eigenfunctions to resembleits componentwise time behavior. It can further be shown that the global energy of thesignal, given by the square norm in L2(X × T), equals the sum of eigenvalues as defined indecomposition (6). This highlights the optimality of the proposed orthogonal decompositionin a flow energy sense.

5. Heat transfer evaluation

5.1. Integral energy balance

For a three-dimensional, cubic control volume on a flat surface, an energy balance based onenthalpy can be written (see Kays et al. (2005) [21]). In addition to the standard assumptionspresented therein, we also assume that all turbulent fluxes on the boundary opposing thewall can be neglected compared to those at the wall; thus, only the streamwise (x) andpitchwise (y) variables, as well as the surface wall heat flux are significant. This upperboundary is the midplane and in the temporal mean, a symmetry plane. Then, the followingsimplified energy balance is obtained

q ′′s = ∂

∂x

∫ Z/2

0ρuhdz + ∂

∂y

∫ Z/2

0ρvhdz. (12)

We now define two enthalpy thicknesses

�2,u =∫ Z/2

0

u

u∞

(T − T∞Ts − T∞

)dz, (13)

�2,v =∫ Z/2

0

v

u∞

(T − T∞Ts − T∞

)dz, (14)

where u∞, Ts , and T∞ are constant reference values (the bulk velocity U0, mean sur-face temperature, and averaged inflow temperature, for example). Using the chain rule ofderivatives, Equation (12) can be rewritten in terms of the enthalpy thicknesses as

q ′′s = ∂

∂x[�2,uq

′′φ,s] + ∂

∂y[�2,vq

′′φ,s], (15)

q ′′φ,s = ρcpu∞(Ts − T∞). (16)

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10 M. Schwanen and A. Duggleby

The term ρcpu∞(Ts − T∞) has units of W/m2 and can be thought of as a potential heatflux available for surface heat transfer contained in the flow. Surface heat transfer is thusdetermined by the spatial change (gradient) of this “available heat flux” q ′′

φ,s multiplied withthe enthalpy thickness. Wherever the spatial enthalpy thickness change is high, the surfaceheat transfer will be elevated as well for given values of u∞, Ts , and T∞.

Equation (15) can be further simplified by introducing an enthalpy thickness vector��2 = �2,u�ex + �2,v�ey with which one arrives at an equation or heat flux similar to thedivergence of a flow field

q ′′s = �∇ · ( ��2q

′′φ,s). (17)

As the goal is to evaluate modes of a turbulent flow, Reynolds decomposition (u = u + u′,v = v + v′, and T = T + T ′) is now applied on every fluctuating part and the entireexpression is averaged. Note that all reference variables (Ts , T∞, and u∞) as well as thespecific wall heat flux (q ′′

s ), which is a computational boundary condition, are constantand thus have no fluctuating part. After introducing the definition of turbulent enthalpythickness (compare with Equations (13) and (14))

�ς

2,u =∫ Z/2

0

u′T ′

u∞(Ts − T∞)dz, (18)

�ς

2,v =∫ Z/2

0

v′T ′

u∞(Ts − T∞)dz. (19)

Equation (17) is rewritten to yield

q′′s = �∇ · ( ��ϑ

2 q ′′φ,s

) + �∇ · ( ��ς

2 q ′′φ,s

). (20)

5.2. Evaluation of OD modes

The assessment of modes averaged over an area of interest (the stagnation and wake regionof the pin row) is of greater interest than the local OD mode ranking. Thus, Equation (20)is integrated over the surface and simplified using Gauss’ divergence theorem

∫S

q′′s dS =

∫S

�∇ · ( ��ϑ2 q

′′φ,s

)dS +

∫S

�∇ · ( ��ς

2 q′′φ,s

)dS

qs =∮

L

[∫ Z/2

0ρucp(T − T∞) dz

]�ex · �n dL + . . .

. . . +∮

L

[∫ Z/2

0ρvcp(T − T∞) dz

]�ey · �n dL + . . .

. . . +∮

L

[∫ Z/2

0ρcpu′T ′ dz

]�ex · �n dL +

∮L

[∫ Z/2

0ρcpv′T ′ dz

]�ey · �n dL.

(21)

The unnormalized enthalpy thickness terms can be expressed with OD modes. This proce-dure is computationally tractable since no derivatives have to be computed and the integrals

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Journal of Turbulence 11

only need to be evaluated along the lines enclosing the area of interest. For a square area de-fined on the x, y-plane with four edges (West, North, East, South), inserting OD modes andneglecting the averages (mode 0), the right-hand side can then be written as (exemplarilyshown for the west side, where �ex · �n = −1)

−∫

W

⎡⎢⎣

∫ Z/2

0

⎛⎝ρ

∑m =0

αmϕm(�x)�mu (t)

⎞⎠

⎛⎝cp

∑n =0

αnϕn(�x)�nT (t)

⎞⎠ dy

⎤⎥⎦ dL + . . . (22)

6. Results

6.1. Baseline

To validate the LES study against experimental results, friction factor and Nusselt numberaugmentation (in comparison to the unobstructed channel) are evaluated (see Figure 3).The friction factor correlation that gives higher augmentation values is used in Lyall et al.(2007) [2]. It is expected to be valid for square channels even though it was developed forcircular cross-sections assuming the Reynolds number is based on the hydraulic diameter.Patel and Head (1969) [22] show that this concept breaks down for channels with highaspect ratios. They developed another correlation (valid for 3000 < ReC < 105, where ReC

is based on the channel height). Normalizing the CFD results with this correlation leads toa lower augmentation, as shown in Figure 3, left. The desired result (f/f0 = 1) is within a6–7% range between the two correlations.

Figure 3, right, shows a comparison of the pitchwise (y)-averaged Nusselt numberaugmentation on the channel wall to experiments. Downstream of the pin and whenusing the same Nusselt correlation as in the experiments (from Kays and Crawford(1980) [23], Nu0 = 0.022Re0.8Pr0.5), the agreement is quite good. Upstream, the desiredvalue of Nu/Nu0 = 1 is achieved when using the Dittus–Boelter correlation from Incroperaand DeWitt (2002) [24]. For completeness, another baseline heat transfer correlation isshown when using the friction factor for high aspect ratios defined by Patel and Head(1969) [22]. The differences can be explained by geometry, uncertainty, and different

Figure 3. Left: The local, pitchwise-averaged friction factor augmentation is plotted as a functionof streamwise distance. The pressure data are normalized with different baseline correlations. Right:The local, pitchwise-averaged Nusselt number augmentation compared to experiments is plottedas a function of streamwise distance. The temperature data are normalized with different baselinecorrelations.

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12 M. Schwanen and A. Duggleby

Figure 4. The plot shows time eigenfunctions �ki (t) of eight modes for velocity (three left) and

temperature (right), scaled for better visualization. The orthogonal expansion was computed within abox enclosing all three pins, covering the entire domain in the spanwise y and wall-normal z directionand extending 3D upstream of the pins and 5.75D downstream. Mode 7 is the first mode with anonzero temperature function. The streamwise and pitchwise velocity functions of modes one, two,four, and five closely resemble the time history of lift and drag.

turbulence conditions (for example, Lyall et al. (2007) [2] used sand paper to trip theboundary layer far upstream of the measurement location, where the computation usedsmooth walls).

The orthogonal expansion was computed within a box enclosing all three pins, cov-ering the entire domain in the spanwise y and wall-normal z direction and extending 3Dupstream of the pins and 5.75D downstream. Two data sets were used: a small one spanningapproximately one lift cycle (19 samples) and one for the entire flow time of approximately13.3 seconds (a total of 443 samples). The resulting time basis functions from cases arequalitatively similar and are shown in Figure 4 for the larger sample. Mode 0 shows themean flow in x and an increase in mean temperature due to heating the endwalls (thenonzero temperature time function of mode 0 is not visible in Figure 4 due to the scalingof the plot). The velocity functions of the higher nonzero modes show the time trace offlow components responsible for lift and drag. Mode 7 is the first mode of all modes withαk = 0 that has a nonzero temperature function. Modes 1 and 2 are paired and form thespanwise wave pattern of shed vortices downstream of the pin. There is a shift in each of thetemporal and spatial eigenfunctions which is a characteristic of a modulated traveling wave(see Aubry et al. (2003) [25]). If the domain had spanwise periodic boundary conditionsenclosing only one pin (total domain width of 2D), the vortex shedding is locked and thereis no possibility of irregular shedding cycles. In that case, the time eigenfunctions of modes

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Journal of Turbulence 13

Figure 5. The plot shows a ranking of mode combinations with respect to their heat transfer contri-bution (see Equation (21)). Mode combination 7/7 has the highest positive result.

1 and 2 appear as sine waves with no change in amplitude or frequency over time. Modes1–6, which are potentially wasteful in terms of turbulent heat transfer, contain 32.14% ofthe total energy in all nonzero modes. To further distinguish between useful and nonusefulmodes, the integrals from the enthalpy-based analysis (Equation (21)) are evaluated formodes 1–10 and are shown in Figure 5. Globally, mode combination 7/7 has the highestpositive heat transfer contribution. Higher heat transfer is desired in this application sincethe internal passage is intended to cool the blade walls. Had the ranking shown equallyimportant modes, the highest energy, lowest order one among them would have been usedfor deriving a wall contour.

6.2. Contouring the endwall

As seen from the evaluation of orthogonal basis functions, the scalar mode 7 has thehighest positive heat flux contribution (as desired for cooling the blade wall). In order totest whether this information can be used to improve heat transfer while at the same timeincreasing pressure drop by as little as possible, it might be beneficial to stimulate that modesuch that it contains more energy, climbs up the OD spectrum, and replaces lower ordermodes that are identified as wasteful in terms of heat transfer. One possible approach is tocontour the domain endwall to resemble an isosurface of a given mode value. The scalarmode threshold which determines the surface elevation in this case needs to be high enoughto capture important spatial features, but should not be too high to avoid contacting thechannel. The latter would result in increasing heat transfer merely by globally acceleratingthe flow. Figure 6, left, shows the chosen isosurface at a certain fraction of the absolutemaximum of scalar mode 7. The elevation of this surface over the bottom endwall wasused as a basis for the wall contour: by postprocessing the spatial eigenfunction mode 7,the wall-normal coordinate (Z) was saved of every grid point on which the scalar valueof mode 7 fell within a given tolerance of the desired isosurface value (13% ± 1.7% of

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Figure 6. The left figure shows an isosurface at ≈ 13% of the maximum value of scalar mode 7. Theright plot shows the resulting contoured endwall of the computational domain, which is a first-orderapproximation in the sense that only element corners were deformed and grid points in betweenlinearly interpolated.

the absolute maximum of scalar mode 7). This threshold was chosen because it provides agood compromise between channel contraction (if the isosurface values are too high andthe distance to the wall too big) and capturing surface features (if the values are too low,the isoshape tends to flatten toward the wall). In the absence of a precise mathematicalformulation at present, this step requires some engineering judgment. From the pointswithin the desired range, the lowest Z-values were chosen to lift the element corners offthe smooth bottom wall. The Gauss–Lobatto grid points between element corners are theninterpolated with a first-order function. This deformation was applied symmetrically to thebottom and top walls of the domain. As seen in Figure 6, right, which shows the deformedendwall of the computational domain, this leads to sharp edges. Due to the coarseness ofthe element mesh, it is only an approximation of the smooth surface in Figure 6, left.

Another representation of the contoured endwall is given in Figure 7. The lines markthe new wall contour sliced normal to the streamwise direction. Upstream of the pins, thechannel is only contracted, using a fairly flat, horizontal profile. This decreases the flowarea by about 11.3%. Approaching the pin in the streamwise direction, the profile thendrops toward the circular pin bottom and forms a trough around it where the stagnationhorse shoe vortex is located. Between the pins (at X/D = 0), the trough sides form a ridgewith a flat top, located in the middle of the pins. Those ridges become sharper downstreamand stay at the same height for about one pin diameter. Then, two ridges decrease towardthe undeformed profile and the remaining ridge widens at the top to form a bump. Thespanwise-averaged elevation of the wall stays on a constant level in the wake of the pinsand yields a flow area (cross-sectional) reduction of 5.7%. The deformation then decreasesto a horizontal profile, resulting in 2.5% area reduction downstream of X/D = 2.6 (notshown in Figure 7). Overall, the contoured endwall increases the top and bottom wall areaavailable for heat exchange by ≈ 0.31% and can be neglected which allows keeping thespecific heat flux boundary condition the same for the smooth and contoured case.

After recording data from the smooth wall case in a statistically stationary state, theendwall contour is applied at the top and bottom and the computation is restarted from someearlier point in time. This is done by mapping the existing smooth case field (includingtemperature) to the deformed domain grid point by grid point, without interpolation. Thesimulation is run for the same flow time so that a comparison can be made. Figure 8 showslift and drag coefficients on the middle pin of the smooth wall case (left) and the contouredwall case (right). There is a strong correlation between lift amplitude and drag in both cases,but the overall drag coefficient is slightly higher in the contoured case. As indicated by a

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Figure 7. The plot shows the endwall contour in spanwise slices at different streamwise locationsX. The pin bodies at X /D = 0 are shown in gray. The dotted lines represent a deformation involvingall grid points as opposed to only the element corners with linear interpolation (solid line). For thisstudy, the linear interpolation is used.

significant decrease in lift on the middle pin of the smooth case, the vortex shedding breaksdown for some time and a relatively wide and long wake develops. This behavior cannotbe seen in the contoured case, where the pins are somewhat isolated from each other bythe ridges in the endwall. In this flow, the heat transfer downstream of the pins is mainlydetermined by large-scale motions from pin vortex shedding. Thus, the most importantmode with respect to turbulent heat transfer does not coincidentally contain the vortexshedding motions near the endwall. Since the orthogonal decomposition is energy-optimal,low-order modes contain relatively more high-energy flow features – in this case, the large-scale structures in the pin wake from vortex shedding. In fact, the dominant frequency ofmodes 0–7 is the vortex shedding frequency. It is thus not a coincidence that manipulating

Figure 8. The left plot shows the lift and drag coefficient of the middle pin from the smooth wallcase as a function of flow time. The vortex shedding appears to break down, generating a relativelywide wake and lower lift, but it recovers thereafter. In comparison, the lift on the middle pin from thecontoured case, right, shows that the time periods of lower lift are shorter, indicating a faster recovery.

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16 M. Schwanen and A. Duggleby

Figure 9. The plot shows the total pressure (wall-static pressure and dynamic pressure derivedfrom the mean streamwise velocity) normalized with total inlet pressure as a function of streamwisedistance. The total pressure loss of the contoured case is higher than for the smooth wall, with lossesmainly occurring in the mixing zone downstream of the pin row. Upstream (in the region of theleading edge horse shoe vortex), the pressure loss is slightly lower in the contoured case.

those high-energy modes will change the effects of shed vortices on the endwall heattransfer. It is further hypothesized that the horse shoe vortex oscillation is dampened dueto the trough around the pin base, which could also stabilize the flow in the wake.

A key goal of this heat exchanger optimization is to increase surface heat transferwithout increasing the pressure loss of the passage. In order to judge the pressure penaltythat is paid for changing heat transfer, the total pressure drop as a function of streamwisedistance is computed from the wall static pressure, averaged in the spanwise (y) direction,and the dynamic pressure. For the latter, only the mean flow velocity in the streamwisedirection is used since both spanwise and pitchwise velocity are zero in the mean. Thestreamwise velocity is derived from the volumetric flow rate in the recycling section (whichis the same for the smooth and contoured case) divided by the cross-sectional area at anygiven streamwise distance. Figure 9 shows the total pressure for both the cases. In thesmooth wall case, there is an almost constant total pressure in the recycling inflow section.The pressure then drops approaching the region where the stagnation horse shoe vortex isactive. At X/D = 0, where in the given geometry the cross-sectional area of the channel isreduced by half due to the pin bodies, the total pressure recovers, but a significant decreasefollows due to mixing and losses in the pin wake. The pressure slightly recovers furtherdownstream. For the contoured case (line with circles in Figure 9), one can see a disturbancein the pressure curve at X/D = −2. This is where the first deformed wall grid points arelocated. Between X/D = −2.5 and X/D = −2, the wall is formed as a sharp ramp whichleads to losses possibly due to flow separation. Compared to the smooth case, the pressuredrop in the pin wake is larger. Toward the channel end, the total pressure difference remainsconstant and the pressure recovery phase appears to be completed.

The local effectiveness of the wall contouring can be judged by comparing the frictionfactor change and heat transfer augmentation. Figure 10 allows this comparison by plottingthe local Nusselt number augmentation based on spanwise averaged wall temperature of thecontoured case over the smooth case as well as the friction factor augmentation. The highestlocal heat transfer increase takes place at the start of the wall contour, at X/D = −2.5.The friction factor, which is based on wall static pressure, drops in this region due to theacceleration of the flow, but increases above the smooth wall case after the ramp, indicatinglosses. Another drop can be seen approaching the pin centers (X/D = 0), where the cross-section is reduced due to the pin bodies. The friction factor remains almost constant

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Figure 10. The plot shows the Nusselt number based on spanwise averaged wall temperature of thecontoured case over that of the smooth case, solid line. The circled line shows the friction coefficientas derived from the pressure drop between a certain streamwise location and the inlet plane, again forthe contoured case over the smooth case.

thereafter. The heat transfer augmentation is highest in the ramped wall section, but this isonly due to accelerating the flow which has the effect of increasing the Reynolds number.The highest heat transfer increase due to changing the pin flow by wall contouring is seenbetween X/D = 0 and X/D = 1. Care needs to be taken when comparing the friction andheat transfer augmentation at a specific streamwise location: there can be delays in pressurerecovery and the wall static pressure does not always equal the mean planar pressure.Further, both values are integral quantities in a sense that the friction factor is based onpressure differences between a varying streamwise location and the (fixed) inlet plane andthe Nusselt number is based on a reference temperature obtained from an integral energybalance.

To assess the efficiency, the overall pressure drop across the array can be compared tothe mean Nusselt number augmentation. Table 1 shows the area averaged Nusselt numberaugmentation (relating the smooth wall case to the contoured case) for the endwall only andcombined from pin surface and wall. The contouring has a higher effect when evaluating theendwall heat transfer only and when using a larger area for the evaluation. The experimentalvalues found in Lyall et al. (2007) [2] further enable a comparison to the Nusselt numberaugmentation that could be achieved by just changing the Reynolds number (bulk velocity).The published data indicate that the averaged, combined Nusselt number augmentation forpin and wall remains constant over the Reynolds number range and the area in question.For an increased Reynolds number that would yield a 5.02% increase in heat transfer, thefriction factor would be raised by 13.22%, where this value was found from a second-order

Table 1. Mean Nusselt number change of the pin finarray comparing the smooth to the contoured case.

Area for average Pin and endwall Only endwall

–D to D 2.63% 4.21%–2.5 D to 2.5 D 5.02% 6.26%

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18 M. Schwanen and A. Duggleby

Figure 11. The contour plots show temporal wall temperature RMS from the smooth case (left) andthe contoured case (right).

interpolation of the data plotted in Lyall et al. (2007) [2]. The friction factor augmentation(based on total pressure and measured 8.25D upstream and 5.75D downstream of the pinrow) due to the wall contouring in this study is 14.76%. The orthogonal decompositionanalysis shows that the energy fraction contained in the highest modes with zero temperaturebasis function compared to all modes with αk = 0 decreases from 32.14% (smooth case)to 31.35%. The highest nonzero temperature mode is 8 in the contoured case instead of 7.

Another important aspect of part durability besides overall cooling effectiveness is thetemperature fluctuation in the material. Figure 11 shows the temporal root mean square(RMS) of wall temperature comparing the smooth to the contoured case. Particularly, inthe recirculation zone close to the trailing edge of the pin (most notably downstream ofthe middle pin), the RMS appears to have decreased. The region of intense fluctuationsin the intersection of the pin wakes is shortened, possibly due to the stabilizing effect thewall contouring has on the vortex shedding (see Figure 8). In the horse shoe vortex region,there is an increase in temperature RMS compared to the smooth case, but the fluctuationlevels are overall more uniform. Similarly, comparing the smooth to the contoured case,the spatially resolved heat transfer augmentation is shown in Figure 12. The increase inheat transfer augmentation is most notable immediately downstream of the pins, where thespanwise uniformity is also greater.

Figure 12. The heat transfer augmentation of the undeformed geometry (left) and the wall-contouredcase (right) is shown. The same definition for unobstructed channel Nusselt number Nu0 as in [2] wasused.

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Journal of Turbulence 19

7. Conclusion

Unsteady heat transfer in a single-row pin fin heat exchanger is examined with LES.Comparison of the integral quantities friction and Nusselt number show good agreement.The data are then analyzed with an orthogonal decomposition technique based on temporalcorrelations and including internal energy. Since the low-order (high-energy) modes showa zero amplitude of temperature basis functions, a mode combination ranking methodologybased on enthalpy is developed. This confirmed that the mode with the highest basisfunction amplitude is most relevant to turbulent heat transfer. An isosurface of this mode’stemperature basis function is then used to deform the domain endwall, leading to a maximumdecrease in cross-sectional area of about 11%. The wall contour forms a trough aroundthe pin body upstream of the separation zone and develops sharp ridges at the interfaceof the wakes. This arrangement seems to stabilize the vortex shedding as can be seenfrom a more uniform lift coefficient history on the middle pin. Due to the orthogonaldecomposition’s energy optimality, the targeted high-energy mode contains mainly vortexshedding motions. Manipulating this mode will consequently change the effects of shedvortices on the endwall heat transfer. Depending on the averaging area, the mean heattransfer augmentation increased between 2.6% and 6.2%. If a heat transfer augmentationincrease of 5% was to be achieved by merely increasing the Reynolds number, the frictionfactor augmentation would have been the same as due to endwall contouring. This isremarkable given that the contoured surface is only a first-order approximation of themode’s isosurface, which is not smooth. Even though the number of low-order modes withnear-zero temperature amplitude has increased, the amount of energy contained in themfell from 32.14% to 31.35% of all nonzero modes.

The wall contour was derived by analyzing flow data and utilizing a generalizableranking method for orthogonal modes. Since this methodology is based on the physics ofthe flow, it is expected to yield usable results for other flow scenarios as well. It might,however, not always be most effective or efficient to contour a flow bounding wall forsuppressing or stimulating certain modes. Instead of a static, deformed wall, the timeinformation of targeted basis functions could, for example, be used to pulse the base flowor move wall parts in a time-varying manner. Also, the wall contouring as done in thisstudy requires the ad-hoc application of certain threshold values for transforming a modeisosurface to a wall contour. These engineering thresholds do not limit the method’s generalapplicability, but are necessary to achieve a smooth and reasonably bound contouring inthe absence of a precise mathematical description at present.

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cooling applications, J. Turbomach. 110 (1988), pp. 94–103.[2] M. Lyall, A. Thrift, and K.A. Thole, Heat transfer from low aspect ratio pin fins, in Proceedings

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[4] M. Young and A. Ooi, Turbulence models and boundary conditions for bluff body flow, inProceedings of the Fifteenth Australasian Fluid Mechanics Conference, Sydney, Australia,2004.

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[20] D. Werner, Funktionalanalysis, 6th ed., Springer-Verlag, Heidelberg, Germany.[21] W. Kays, M. Crawford, and B. Weigand, Convective Heat and Mass Transfer, 4th ed., McGraw-

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