International Journal of Heat and Fluid Flow 44 (2013) 745–755

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International Journal of Heat and Fluid Flow

journal homepage: www.elsevier .com/ locate/ i jhf f

Computational modeling of vortex shedding in water cooling of 3Dintegrated electronics

0142-727X/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.10.004

⇑ Corresponding author. Tel.: +41 44 632 27 38; fax: +41 44 632 11 76.E-mail address: dpoulikakos@ethz.ch (D. Poulikakos).

Fabio Alfieri a, Manish K. Tiwari a, Adrian Renfer a, Thomas Brunschwiler b, Bruno Michel b,Dimos Poulikakos a,⇑a Laboratory of Thermodynamics in Emerging Technologies, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerlandb IBM Research GmbH, Zurich Research Laboratory, 8803 Rueschlikon, Switzerland

a r t i c l e i n f o

Article history:Received 16 July 2013Received in revised form 6 October 2013Accepted 10 October 2013Available online 30 October 2013

Keywords:3D integrated electronics coolingVortex sheddingLateral/longitudinal/vertical confinement

a b s t r a c t

This work aims at understanding the flow and heat transfer through a microcavity populated with micro-pins, representing a layer of a 3D integrated electronic chip stack with integrated cooling. The resultingvortex shedding behavior and its effect on the heat removal is analyzed in the Reynolds number (Re)range from 60 to 450. The lateral confinement, expressed as the ratio of diameter to lateral distancebetween two cylinders’ centers, is varied between 0.1 and 0.5; the longitudinal confinement (diameterto longitudinal distance between two cylinders’ centers) between 0.25 and 0.5; and vertical confinement(diameter to microcavity height ratio) between 0.1 and 0.5. For a single pin, as the lateral confinement isincreased, the Strouhal number (St) and the shedding frequency increase by up to 100%. The thermal per-formance represented by the spatiotemporal averaged Nusselt number (Nu), based on the average pinsurface and fluid temperatures, is also enhanced by over 30%. A direct relationship between Nu andthe shedding frequency was found. For a row of pins, Nu in the vortex shedding regime was found tobe up to 300% higher compared to the steady case. A decrease in the longitudinal confinement, testedwith rows of pins (either with 50 or 25 pins) in the streamwise direction, led to an upstream migrationof the vortex shedding location and in more homogeneous but higher wall temperatures. This coincidedwith a drastic reduction of pressure losses and a 30% Nu enhancement for the same pumping power.Finally, the vertical confinement is also investigated with 3D simulations around a single cylinder. Withincreasing Re and vertical confinement, the wake becomes strongly three-dimensional. For a given Re, theincrease of vertical confinement naturally shows a suppression or even a complete elimination of the vor-tex shedding due to a strong end-wall effect. Our results shed light on the effects of confinement on vor-tex shedding and related heat transfer in the integrated cooling of 3D chip stacks.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Given the constantly increasing demand for combined improve-ment in performance and energy efficiency in electronic devices,new technologies are emerging for the packaging and operationof microprocessor chip assemblies. A promising approach consistsof exploiting vertical integration of integrated circuits by means ofthrough-silicon-vias (TSVs). The TSV technology reduces wiringlength leading to higher bandwidth and less energy losses for com-munication. A major challenge for the three-dimensional (3D)stacking of chips is the cooling, which furnishes major designimpediments not only from the structural and packaging perspec-tive, but also with respect to the thermal reliability of the chips. Achip stack can dissipate up to 1000 W/cm2 heat (Brunschwileret al., 2010) which would lead to an immediate thermal failure

with conventional back-side air or water cooling techniques(Tuckerman and Pease, 1981). The integrated water cooling is anefficient strategy for heat removal. It implements water flowingthrough a microcavitiy – one for every die with heat enhancementstructures (that contain the TSVs) for improved heat transfer. Manystudies have shown advantages and disadvantages in hydrother-mal performances of various shapes such as round, square, ellipti-cal, droplet-shaped micropins or microchannels (Sahiti et al.,2007). The most obvious geometry is that where each TSV is acylindrical micropin in a cavity (Alfieri et al., 2012, 2010). Thecurrent work focuses on the hydrothermal investigation of flowsacross such micropins confined in a microcavity.

Many studies have investigated the transport phenomena in aflow around a single cylinder or an array of cylinders. For lowReynolds number (Re), a steady vortex pair is generated behindthe cylinder and the flow is characterized by a linear dependenceof both Nusselt number (Nu) and pressure losses on the Re. Withincreasing Re the vortices become unstable and they start shed-

Nomenclature

VariablesA surface area (m2)c specific heat capacity (J/kgK)cd drag coefficient (–)cdf friction drag coefficient (�)cdp pressure drag coefficient (�)cf friction coefficient (�)cl lift coefficient (�)cp pressure coefficient (�)d cylinder diameter (m)F force (N)f frequency (Hz)H cylinder height (m)h heat transfer coefficient (W/m2K)h� vertical confinement (�)k thermal conductivity (W/mK)L length of populated cavity (m)Nu Nusselt number (�)P pitch (m)Pr Prandtl number (�)P0d pumping power per unit height (W/m)p pressure (Pa)p� longitudinal confinement (�)_q00 heat flux (W/m2)Re Reynolds number (�)St Strouhal number (�)T temperature (K)

t time (s)u velocity (m/s)W width (m)w� lateral confinement (�)x, y, z coordinates (m)x�, y�, z� non-dimensional coordinates (�)

Indices1 first cell1 reference variablebulk bulkcell volume containing one pincs cross sectionf fluidi, o inlet, outlets separationw wall

GreekD small (e.g. numerical cell)q densityk oscillation periodl dynamic viscositym kinematic viscositys wall shear stressh anglex vorticityW arbitrary variable

746 F. Alfieri et al. / International Journal of Heat and Fluid Flow 44 (2013) 745–755

ding, creating a Karman vortex street. The vortex shedding mech-anism is influenced not only by the physical factors of the fluidsuch as Re and the Prandtl number (Pr), but also from various geo-metrical factors. Given the obvious computational convenience,the majority of the literature on flow across cylinders has focusedon two-dimensional (2D) investigations. Several studies were per-formed on the hydrodynamics of the shedding from a single cylin-der; evaluating variables such as Strouhal number, pressure,friction, lift and drag coefficients, and vorticity as a function of Re(Son and Hanratty, 1969). The thermal characterization was alsoexhaustively carried out (Ahmad, 1996; Karniadakis, 1988;Sohankar, 2007). Hydrodynamic stability theory has been used todetermine the critical Re at which the transition to vortex sheddingregime occurs (Sahin and Owens, 2004). The effects of boundaryconditions (Kumar and Mittal, 2006) and cylinder geometry(Camarri and Giannetti, 2007) have also been analyzed. For inves-tigating multi-cylinder arrangements, placing two cylinders in tan-dem with unconfined surrounding is a common configurationemployed to study the longitudinal confinement. These studieshave revealed that the distance between the cylinders can controlthe shedding mechanisms and intensity (Sharman et al., 2005).Investigation of multi-cylinder rows show results similar to theones extrapolated from the two cylinders in tandem (Liang et al.,2009). On the other hand, placing two or more cylinders side byside (i.e. perpendicular to the streamwise direction) is essentialfor investigating the effect of lateral confinement (Meneghiniet al., 2001). A common practice to study the lateral confinementinvolves using a single cylinder and placing solid walls, repre-sented as lateral boundaries at various distances (Chakrabortyet al., 2004; Griffith et al., 2011). Studies have also been dedicatedto arrays, e.g., multiple side by side cylinders (Huang et al., 2006),four cylinders in square configuration (Farrant et al., 2000) andround array (Nicolle and Eames, 2011) etc., which show the com-bined effect of lateral and longitudinal confinement.

3D flow features arise for high enough Re (Mittal, 2001;Sohankar et al., 1999; Zhao et al., 2009). Therefore, although com-putationally expensive, 3D simulations are indispensable forresolving all physical characteristics. In the flows past a single cyl-inder, the main differences are noticed in the wake of the cylinder,where the 3D features are clearly recognizable (Kanaris et al.,2011; Williamson, 1996). Studies have also pointed out theinfluence of longitudinal and lateral confinements on flow pastfew cylinders using 3D models (Carmo and Meneghini, 2006;Lam and Zou, 2010) and elucidated changes in the wake structure.However, the overall influence on spatiotemporally averaged prop-erties such as drag coefficient, Nu etc. is not clearly understood. Onthe other hand, experiments have established a direct influence ofthe confinement on the hydrodynamics of flow past confinedmicroscale pin arrays (Renfer et al., 2011, 2013a,b), which areidentical to the geometric configurations in this work. The vortexinduced fluctuations reported in these experiments on confinedpins are likely to have positive influence on thermal transport aswell. If true, this will help explain the sharp drop in thermal resis-tance beyond a certain flow rate as reported by Brunschwiler et al.(2009) using a similar geometry. In addition, a key question thatremains to be addressed is that of the relative improvement inthermal transport with respect to the required pumping power.This question can only be addressed through a novel systematicinvestigation of vortex shedding across large arrays of heated, con-fined cylinders, in the parametric domain relevant to electronicscooling. With this motivation, the present work is focused on thefollowing three aspects:

(a) Analyzing the heat transfer enhancement due to thechange from steady to vortex shedding regime in TSVassemblies.

(b) Clarifying the effect of strong lateral, longitudinal and verti-cal confinements typical of TSV arrangements within anintegrated cooling cavity.

1 2 N

Li Lo

P

W

d

θ = 0°θ

cell

...

periodic B.C.

3 4

no-slip B.C.

y

x

T∞ p = 0

u∞

Fig. 1. Sketch of the computational two-dimensional model composed of a single ormultiple TSV pins in an inline arrangement. The indicated periodic boundaryconditions were used to capture the effect of lateral confinement typical of coolingstructures in a chip stack cooling cavity. In the three-dimensional model a solid wallwas used at bottom and symmetry was used at top to minimize computationaltime.

F. Alfieri et al. / International Journal of Heat and Fluid Flow 44 (2013) 745–755 747

(c) Demonstrating the effect of array periodicity by using lateralperiodic boundary conditions, which minimize the compu-tational effort. The approach is different from the oftenemployed solid wall boundary condition for investigatingthe lateral confinement (Chakraborty et al., 2004; Griffithet al., 2011). The periodic boundary condition is relevantfor microscale confined pin arrays such as those considered,especially for high confinement, where the growing bound-ary layers on the walls due to imposed no-slip conditionsinhibits the momentum exchange from adjacent rows.

In order to investigate the mentioned points, numerical simula-tions are conducted on representative portions of a 3D integratedchip stack with integrated water cooling using 2D and 3D conju-gate heat transfer models. Starting point is the 2D single cylinder,followed by the flow across a 2D array of inline cylinders, whichare compared with the 3D simulations with solid walls at the topand bottom; all configurations are characterized by lateral periodicboundary conditions.

2. Modeling

To understand the vortex shedding physics in a micro pin (TSV)array with lateral, longitudinal and vertical confinements, threedifferent configurations were considered. The first consists of aflow around a single, laterally confined cylinder. The second con-figuration is a flow across an inline row of pins with various longi-tudinal confinements. Lastly, a three-dimensional flow around apin with varying vertical confinements is considered. In every con-figuration the d ¼ 100 lm diameter cylindrical pins, were modeledin the presence of lateral neighbor pins in a periodic fashion. Theperiodic domain is fully characterized by the width W, the longitu-dinal distance P between the pins (in the case of a row), and theheight H (if three dimensional) and the length L of the cavity pop-ulated with pins (without taking into consideration inlet and outletregions). The L is thus always an integral multiple of the cell size.The geometrical dimensionless variables defining the computa-tional model are the lateral confinement w�, the longitudinal con-finement p�, and the vertical confinement h�, and the coordinatesx�, y�, z�:

w� ¼ dW;p� ¼ d

P;h� ¼ d

H; x� ¼ x

L; y� ¼ y

W; z� ¼ z

H: ð1Þ

The relevant dimensionless hydrodynamic parameters are theReynolds number Re, the Prandtl number Pr and the Strouhal num-ber St:

Re ¼ u1dm

; Pr ¼ clk; St ¼ fd

u1; ð2Þ

where f is the frequency obtained from the oscillating signal of thelift coefficient cl, u1 is the inlet velocity, m is the kinematic viscosity,l is the dynamic viscosity, c is the specific heat capacity and k is thethermal conductivity of water. The subscript1 is used to designatethe reference value at the inlet.

2.1. Two-dimensional model

The two-dimensional flow around a single cylinder as well asthrough a row of cylinders arranged in a periodic configuration issimulated for various Re and lateral and longitudinal confinementsw� and p�. The 2D model is used for reference and comparison withthe 3D model described next. A sketch of the computationaldomain is illustrated in Fig. 1.

Periodic boundary conditions were imposed in lateral directionin the 2D model (see Fig. 1). The choice of periodic boundary con-dition is based on the lateral periodicity of the vortex shedding ob-served experimentally by Renfer et al. (Renfer et al., 2011). Thecylinder in single cylinder case, or the first cylinder in the case ofcylinder row, were placed at inlet and outlet region lengths ofLi ¼ 12d and Lo ¼ 35d, respectively. These parameters yield solu-tions independent from the boundary conditions (Camarri andGiannetti, 2010). The lateral confinement w� is varied in regularsteps from 0.1 to 0.5.

At the inlet, constant Re and constant static temperature (corre-sponding to Pr values of 3, 5 and 7) were specified over the entirecross section. At the outlet, zero averaged pressure and an averageoutlet temperature were set due to the opening-type boundarycondition. At the interface between solid and fluid no slip bound-ary conditions were imposed, symmetry conditions were used forthe bottom and top surfaces (infinite long cylinder, no endwall ef-fects) and periodic boundary conditions are specified on the lateralinterfaces. A spatially constant heat flux _q00 is imposed on the cyl-inder surface.

2.2. Three-dimensional model

In the three dimensional model, the main distinction comesfrom the introduction of the vertical confinement h� ranging be-tween 0.5 and 2. Since the hydrothermal flow field was symmetricwith respect to the center plane, only half of the height is modeled.Thus in the reduced geometry the top surface had symmetry con-dition imposed, whereas on the bottom surfaces no slip boundarycondition was used.

Upon introducing the third dimension, important change had tobe made to the inlet boundary condition. A fully developed velocityprofile, derived from the integration of the momentum equation,was used for the velocity component in the x-direction

uðzÞ ¼ 4u1zðH � zÞ

H2 : ð3Þ

The other velocity components were set to 0. Given Pr > 1 forwater, a uniform inlet liquid temperature was imposed. To betterapproximate reality, in the 3D model the heat flux is imposed onthe bottom surface (and not on the lateral surface of the cylinder).

2.3. Mesh and time step independence study

The sensitivity of the grid and the time step on the computedflow characteristics was rigorously tested in order to optimizethe balance between accuracy and computational costs. First, themesh and time step independence study for the 2D case atRe = 100 and 450 with w� = 0.1 and 0.5 was carried out by compar-ing three grids with various refinements (G1: coarse, G2: medium,

748 F. Alfieri et al. / International Journal of Heat and Fluid Flow 44 (2013) 745–755

G3: fine) as well as three different dimensionless time stepsDt� = 0.02, 0.01 and 0.005, where

Dt� ¼ Dt � u1d

: ð4Þ

The meshes for different lateral confinements with equivalentrefinement were generated by keeping the same first cell size aswell as the same growth ratio and by increasing the number ofnodes for decreasing w�. The first normalized cell sizes were Dy1/d = 0.2, 0.5 and 2, the growth ratios were 1.04, 1.029 and 1.014,and a geometric distribution of nodes was used. Based on the com-parison of average hydrothermal values, the mesh G2 and thedimensionless time step Dt� = 0.02 were found to be accurate en-ough (<2% difference from G3) and were therefore used for the3D case as well. Additionally, in the 3D case three more nodes dis-tributions were tested in the third dimension (G2C: coarse, G2M:medium, G2F: fine). The first cell was chosen to be Dz1/d = 0.02,0.01 and 0.005 with respective growth ratio of 1.05, 1.07 and 1.1and geometrical law distribution. As for the 2D case, the meshG2M was chosen after testing its accuracy based on the predictedaverage hydrodynamic parameters such as lift coefficient and pres-sure drop. A difference of less than 3% was observed between G2Mand G2F meshes. The node distribution was scaled proportional tothe cavity height. The meshing tool used for all the configurationswas ANSYS ICEM 13�. The CFD solver ANSYS CFX 13�, fed withtemperature dependent material properties, was used to solvethe governing equations of conjugate heat transfer and unsteadylaminar flow. The solver employs finite volume approach for dis-cretization of the governing equations. The differencing schemefor the advective term was second order upwind and the time inte-gration method used was second order backward Euler. The con-vergence criteria were set to 10�7 for the normalized residualsfrom the momentum and energy equations.

2.4. Hydrodynamic and thermal variables

The variable W can be expressed either through its local instan-taneous value (W); or its local time average, obtained by averagingall the local instantaneous values over one shedding period k(W ¼ 1=k

R k0 Wdt); or its spatiotemporal average obtained by inte-

grating the local time average over the circumference of the cylin-ders (W ¼ 1=2p

R 2p0 Wdh).

2.4.1. HydrodynamicsFrom the hydrodynamic point of view a number of variables can

be employed to describe the mechanism of vortex shedding anddevelopment.

The pressure coefficient defined as

cp ¼p� pi;cell12 q1u2

1ð5Þ

where p is the static pressure acting on the cylinder surface, pi;cell isthe static reference pressure at the inlet of the cell (see Fig. 1) andq1 is the water density at the inlet. The friction coefficient

cf ¼s

12 q1u2

1; ð6Þ

where s is the wall shear stress acting tangentially to the cylindersurface.

The total drag coefficient

cd ¼dFd

12 q1u2

1dh¼ cdp þ cdf ¼

�pxAcell12 q1u2

1dhþ sxAcell

12 q1u2

1dhð7Þ

is calculated by means of the total resistance force in the x directiondFd, obtained by summing up the x components of the pressure and

of the shear stress forces acting on the cylinder. The frontal area ofthe cylinder dh was employed as a reference. The symbols cdp andcdf denote the pressure, respectively the friction component of thedrag. The components used are the pressure px (with negative sign)and the shear stress sx in x direction acting on the cell surface areaAcell. From the total drag, the power needed to drive the flow pastthe pin resistance can be calculated as

Pd ¼ Fdu1 ¼12

cdq1u31dh: ð8Þ

This value can be understood as the pumping power needed topush the fluid through the micro cylinder array and corresponds to_VDp, where Dp is the total pressure loss from inlet to outlet. Resultsare presented as pumping power per unit height P0d in W/m. Final-ly, the separation angle hs is the angle at which the flow detachesfrom the surface causing separation (see Fig. 1).

2.4.2. Heat transferThe Nusselt number Nu needed to quantify the cooling perfor-

mance is defined as

Nu ¼ hdk1¼

_q00d=k1Tw � Tf

; ð9Þ

where Tw is the average cylinder wall temperature, _q00 is the heatflux imposed on the cylinder, k1 is the water thermal conductivityat inlet temperature T1 and Tf is the bulk temperature defined asthe average bulk temperature on the planes (with cross sectionalarea Acs) at the inlet and outlet of the cell

Tf ¼12ðTbulkj@cell;in þ Tbulkj@cell;outÞ with Tbulk ¼

RAcs qucTdAcsRAcs qucdAcs

: ð10Þ

3. Results and discussion

3.1. Two-dimensional flow around a single TSV pin: lateralconfinement

The flow around a single two-dimensional TSV pin with variouslateral confinements is the first step towards understanding thephysics of vortex shedding and propagation. We studied both theadiabatic and heated cylinder cases in order to show the interac-tion between the flow field and thermal transport, the latter beingparticularly relevant for electronic cooling application. The z-vor-ticity xz and temperature contours for a heated pin with 50 W/cm2 heat flux, are illustrated in Fig. 2a. The figure clarifies the cou-pling between temperature and velocity fields, and shows someclear effects of increase in the lateral confinement. The first effectis the increase in the shedding frequency, which occurs due to in-crease in the local value of Re close to the cylinder. The second con-sequence of increasing w� lies in the interaction of vortices fromneighboring rows in the wake of the cylinder. In fact, when w� isvery low (minimal confinement), the vortices from adjacent rowsare not interacting with each other, instead they individually movedownstream in a straight line. With increasing w� vortex clustersare diverging from the center line until they meet another vortexwith the same magnitude, merging into a ‘‘vortex column’’. Thetwo typical alternating vortices are therefore substituted by alter-nating ‘‘vortex columns’’. The merging of two vortex clusters of thesame type (positive or negative) appeared to happen close to thecylinder for higher w�. Clearly, the lateral confinement does not al-low the vortices to move freely and they are forced to merge andmove together.

0 100 200 300 400 500

0.18

0.22

0.26

0.3

Re

St

0.14

Rajani et al.

(b)

w*=0.5

w*=0.4

w*=0.3

w*=0.2

w*=0.1

(a)

w*=0.1

w*=0.3

w*=0.5

Vorticity

Temperature

Fig. 2. (a) Vorticity (range between �40,000 1/s < xz < 40,000 1/s) and temperature (range between 293 K < T < 293.6 K) contour plots of periodic flow around a heatedcylinder (50 W/cm2), for Re = 200, w� = 0.1–0.5 and Pr = 7; values are represented by colors in the order of the visible spectrum, where red is the positive (maximum) and bluethe negative (minimum). The heat dissipated from the pin is transported downstream trapped in the vortices, which detach with higher frequency as the confinement isincreased. (b) St vs. Re for flow around single adiabatic pin, w� = 0.1–0.5 and Pr = 7. Increase in confinement and/or Re leads to higher St; computed results for w� = 0.1 are alsocompared with other 2D simulations of an unconfined cylinder (Rajani et al., 2009) (circles) showing good agreement. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

-0.005

0

0.005

0.01

0.015

0.02

0.025

0

0.2

0.4

0.6

0.8

1

1.2

-6

-5

-4

-3

-2

-1

0

1

0 100 200 300 400 50010-1

100

101

Re

c d

cd

cdp

cdf

0.1

0.2

0.40.3

0.5

Rajani et al.

{}

w*

(b)(a)

(d)(c)

c pc fc d

Homann - Re=114.5Rajani et al. - Re=100

Dimopoulos et al. - Re=104Rajani et al. - Re=100w*=0.1, Re=60w*=0.1, Re=100w*=0.1, Re=200w*=0.1, Re=300w*=0.1, Re=450w*=0.2, ”w*=0.3, ”w*=0.4, ”w*=0.5, ”

0Angle θ (deg)

18030 90 15060 120

0Angle θ (deg)

18030 90 15060 120

0Angle θ (deg)

18030 90 15060 120

Fig. 3. Local time averaged (a) friction coefficient (b) pressure coefficient and (c) drag coefficient. (d) Spatiotemporal averaged drag coefficient vs. Re for w� = 0.1–0.5. Thesymbols denote data from literature for validation. The total drag is plotted as solid lines, the friction drag as dotted line and the pressure drag as dashed line. The former issum of the latter two.

F. Alfieri et al. / International Journal of Heat and Fluid Flow 44 (2013) 745–755 749

3.1.1. Hydrodynamics and validationAn important global parameter used to characterize the shed-

ding phenomena is the Strouhal number (St) plotted in Fig. 2b. Stis monotonically increasing with increasing inlet Re as well as withincrease in lateral confinement. This is because the increase in localvelocities near the cylinder leads to higher shedding frequencies.The effect is more pronounced at low lateral confinements. InFig. 2b results for 2D flow around a single unconfined cylinder from(Rajani et al., 2009) are also shown and agree well with our resultsin the lowest confinement case with w� value of 0.1. Note that dueto small diameter of our pins, the reported St values correspond tooscillation frequencies that can exceed 10 kHz. This highlights thefine time scale involved in our transient model.

To better understand the initiation of vortex shedding, localprofiles of selected variables – friction, pressure and drag coeffi-cients – over the pin surface are extracted and plotted Fig. 3a–c.Due to symmetry, the data are presented only from 0� to 180�.The friction coefficient attains a value of zero at three differentlocations. This corresponds to locations of vanishing velocity gradi-ent, namely the front and rear stagnation points as well as theboundary layer separation point. From the front stagnation pointthe friction coefficient increases over about the first quarter ofthe pin as a consequence of the flow acceleration. Then it dropsagain to zero or slightly negative values and maintains at valuesclose to zero in the region between the boundary layer separationpoint (at 55–70�) and the rear stagnation point. The pressure coef-

750 F. Alfieri et al. / International Journal of Heat and Fluid Flow 44 (2013) 745–755

ficient is inversely proportional to the fluid velocity and thus de-creases as the flow accelerates around the pin. It slowly increasesin the wake and is almost constant in the rear stagnation region.The drag coefficient results from the sum of the friction and pres-sure drag contributions and assumes a ‘‘V-shaped’’ profile (Fig. 3c).

The computed results from the present work are validated withdata from other studies, as illustrated in Fig. 3a, b and d. There isexcellent agreement for all the considered coefficients with other2D simulations at Re = 100 (Rajani et al., 2009) and slight localdeviations from experimental results at Re = 104 and 114.5 for fric-tion and pressure coefficients (Dimopoulos and Hanratty, 1968;Homann, 1936).

Fig. 3a–c also capture the effect of increasing confinement andRe. An increase in Re or lateral confinement leads to a friction coef-ficient increase in the front region (before separation) and to a de-crease in the rear region. The pressure coefficient is decreasing forboth increasing Re and lateral confinement. Their variations getcombined in the drag coefficient.

The lift coefficient is the variable used to characterize the vortexshedding through its oscillation frequency, but the drag coefficientand its components can also give important quantitative informa-tion about the shedding physics. Fig. 3d depicts the spatiotemporalaverage drag coefficient cd, along with friction and pressure com-ponents cdf and cdp, as a function of Re for adiabatic conditions.The data for w� = 0.1 are validated against simulations for flowaround an unconfined cylinder (Rajani et al., 2009), which differfrom our result by less than 10%. Since the trend is the same forw� > 0.1, the values of friction and pressure drag are not reportedfor the sake of clarity. As a general trend cd decreases at low Re val-ues and then plateaus with increase in Re. That is due to thedecreasing trend of the friction drag cdf coefficient and the almostconstant trend of the pressure drag cdp. It is important to noticethat for our investigated configurations and operating conditions,cdp is much larger than cdf. Moreover, cd is increasing with increaseconfinements due to local acceleration of flow around the pin andincrease in the pin resistance to the flow.

One variable which may explain the influence of the lateral con-finement on the trends of all physical quantities investigated is theseparation angle. An early detachment of the boundary layer, corre-sponding to a large separation angle, is not desired since it deterio-rates both the hydrodynamic as well as the thermal performance.For increasing lateral confinements the separation angle decreasesdue to the compression of the fluid against the pin walls, delayingthe detachment from the surface.

For completeness, the effect of the global viscosity changes forvarious lateral confinements was also investigated by changingthe Prandtl number (Pr = 3, 5 and 7), i.e., by increasing the inletwater temperature. The results, not illustrated in Fig. 3 for brevity,

0

10

20

30

40

50

60w*=0.1

w*=0.2

w*=0.3

w*=0.4

w*=0.5

(a)

Re=450

Re=200

Re=100

Re=60

Re=300

0Angle θ (deg)

18030 90 15060 120

Nu

Fig. 4. (a) Local time average Nusselt number (Nu) profiles. Nu is enhanced by increasingaverage Nu vs. Re for various lateral confinements and Pr = 3, 5, 7. Both increasing w� andlow Pr.

showed increase in the shedding frequency and the water velocitydue to the decrease in viscosity, leading to a nearly constant St. Onthe other hand, as expected, the change in Pr will have consider-able effects on the thermal development of the vortex shedding(see Fig. 4 below).

3.1.2. Heat transferAn important aim of the present work is to understand and

quantify the effect of high confinement on the heat transfer frommicropins in the vortex shedding regime. We start the discussionby mentioning that heat is removed by the shed vortices and istransported downstream (see Fig. 2a). This leads to heat transferenhancement and therefore to higher Nu with shedding comparedto the steady vortex pair situation. The heat flux applied to the pinfor the cases presented hereafter is 50 W/cm2 and unless otherwisespecified, the Prandtl number is 7. A comparison with a higher heatflux of 125 W/cm2 was also made (not shown). It only altered theliquid velocity locally near the pin and the effect on hydrothermalvariables except temperature was less than 2%.

The local time averaged Nusselt number Nu, directly related tochip cooling efficiency, is first evaluated and shown in Fig. 4a. Forthe sake of clarity, the profiles are for w� = 0.1 at Re = 60–450 andfor w� = 0.2–0.5 only at Re = 450. The trends of the omitted profileswere similar. At the front stagnation point, where Nu attains itsmaximum, heat transfer was most efficient and the wall tempera-ture was lowest (not shown for brevity). The opposite is true at theseparation point due to the boundary layer detachment. Withincreasing Re the heat transfer is visibly enhanced. Similarly, theincrease of w� for a constant Re, leads to higher local Re close tothe pin as well as smaller separation angle. By combining thesetwo effects an increase of Nu at any angular position is noticeable.

The spatiotemporally averaged Nusselt number Nu, used tocompare the thermal transport at various lateral confinements, isshown in Fig. 4b. As expected from other studies (Acrivos et al.,1965; Ahmad, 1996), Nu is a function of Re and Pr. The figure alsoshows it to be clearly a function of w� as well. Although Pr did notplay a major role in the hydrodynamics of vortex shedding, it isvery important for the heat transfer because it controls the growthof the boundary layer over the pin. With increase in Pr the Nu in-creases non-linearly; Fig. 4b shows the influence of Pr only forw� = 0.1 since all other confinement ratios exhibited similar trend.To quantify this phenomenon, the data were fitted using the fol-lowing correlation:

Nu ¼ 1:0308Re0:4822Pr0:3828w�0:1354: ð11Þ

The exponents were obtained using a non-linear least squarestechnique, and the fitted data were within ±10% of the original

0 100 200 300 400 5005

10

15

20

25

30

35

40

Re

Nu

}Pr=7

Pr=5

Pr=3

(b)

w*=0.1

w*=0.2

w*=0.3

w*=0.4

w*=0.5

Re and w�; the effect is more pronounced at low Re and large w�. (b) SpatiotemporalPr lead to Nu enhancement; higher relative enhancement is observed at large w� and

10-3 10-2 10-1 100 101

15

25

35

45

Pd (W/m)

Nu

Nufit = 24.Pd

1/6

w*=0.1 w*=0.2w*=0.4

w*=0.3w*=0.5

0

15

10

5

20

5

f (kH

z)

ffit = 5189.Pd0.3691

`

`

`

Fig. 5. Nusselt number (Nu) and shedding frequency (f) plotted as a function ofpumping power P0d . The unit W/m emerges from the 2D geometry modeled.

(a)

(c)

(b)

Fig. 6. Comparison of the investigated configurations at Re = 100, by means of thez-vorticity (depicted values of xz are, in order of appearance, between ±75,000 1/s,±25,000 1/s and ±10,000 1/s); values are represented by colors in the order of thevisible spectrum, where red is the positive (maximum) and blue the negative(minimum). The sketches on the right, also used in Fig. 7, show the confinementlevels. (a) p� = 0.5, w� = 0.5, in transition regime (from steady oscillations to fullydeveloped shedding); the flow is periodic every two pins. (b) p� = 0.25, w� = 0.5,shedding regime; the flow around every pin is periodic. (c) p� = 0.5, w� = 0.1,shedding regime, the flow is periodic every five pins with large counter rotatingvortices moving on the right and left side between adjacent rows. (For interpre-tation of the references to color in this figure legend, the reader is referred to theweb version of this article.)

F. Alfieri et al. / International Journal of Heat and Fluid Flow 44 (2013) 745–755 751

data. Eq. (11) summarizes the thermal characteristics of 2D flowpast a single micropin factoring in all the tested parameters.

In order to assess a relative improvement in heat transfer at agiven pumping power, the Nusselt number is plotted as a functionof the pumping power in Fig. 5. The data for different lateral con-finements collapse on a single line. This indicates that due to lat-eral confinement, the increase in local fluid momentum aroundthe pin and the changes in the vortex structure monotonicallyinfluence the vortex shedding frequency and thermal transport.This result is of significant practical value since it implies that fora given confinement, determining the vortex shedding frequencyis enough to get an accurate sense Nu for the heat transfer fromthe pin. Since both functions shown in Fig. 5 are dependent onthe pumping power, by solving one of them for P0d and insertingthe result in the other one, a direct relationship between the vortexshedding frequency and Nu can be determined as Nu ¼ 132:81f�0:2.This relation can be used for thermal characterization withoutrequiring knowledge of the detailed temperature distributionwithin the chip itself.

3.2. Two-dimensional flow across a row of TSV pins: longitudinal andlateral confinement

This section focuses on investigating single rows of inline pinsand analyzing the role of lateral and longitudinal confinement.Rows with 50 or 25 pins are used for changing the longitudinalconfinement p� between 0.5 and 0.25. The large number of pinscorrespond to typical TSV arrays in 3D chip cavities and are chosenfor the purpose of distinguishing the three main flow regimes:laminar stable, transition and eventually vortex shedding (Renferet al., 2011). The 25 pin row is referred as half populated case forsimplicity. Both adiabatic and diabatic conditions are considered.Fig. 6 shows a qualitative picture of the flow characteristic in threedifferent pin arrangements (with different levels of confinement)for Re = 100. The figure clearly shows the major effect of confine-ment on the flow field, which is what modifies the trajectory ofthe vortices (characterized by waves with periodicity of one ormore cells) through the micropin arrays, consequently causingchanges in the hydrothermal parameters investigated hereafter.

3.2.1. Flow regimes identification and characterizationThe current section concentrates on characterizing the flow

transition for the configurations in Fig. 6. The cRMSl ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N

PNi¼1c2

l;i

q

profiles shown in Fig. 7a, which are calculated using the data forN time steps, for Re values of 100, 150 and 200, provide an easymeans to detect changes in flow regime. N was kept sufficient to

include at least 10 oscillations. In particular, the instabilities orig-inating at the flow transition (i.e. at the onset of vortex shedding)lead to a steep rise in cRMS

l . Following the transition zone, cRMSl val-

ues plateau to nearly constant value, characterizing the regime ofstable vortex shedding. These profiles can thus be used to developa flow regime map shown in Fig. 7b, which demarcates the regionsof transitional flow and its evolution as a function of Re. The Fig. 7bonly considers the configuration in Fig. 6a for brevity. Some simpleinferences can be readily drawn by close observation. Clearly, for agiven high enough Re, vortex shedding initiates close to the outletand the transition point moves upstream with increase in Re. Bothof these facts are in clear agreement with experimental results ofRenfer et al. (2011). The Fig. 7a and b also capture the effect ofimposing a heat flux on the pin surface. The flow transition isclearly influenced by heat flux, which is to be expected due to astrong decrease in liquid viscosity with temperature increase. Infact, the change in the flow regime from steady to transition seemsto occur earlier for the heated cases, while the passage to fullyestablished vortex shedding depends on the Re. In particular, atlow Re the transition to fully developed shedding regime is re-tarded, leading to a relatively larger transition region comparedto the adiabatic case.

The three flow regions identified above are also expected tostrongly modify the thermal performance. For the analyzed pinconfigurations in Figs. 6 and 7c plots the spatiotemporally aver-aged Nusselt number for every pin at an inlet Re = 100. The plotsclearly confirm that the hydrodynamic changes during the transi-tion result in a remarkable modification of the thermal transport.For the configuration with p� = 0.5 and w� = 0.5, it can be observedthat in the steady portion of flow the Nu profile reflects a typicaldeveloping flow trend. As instabilities start to develop due to initi-ation of flow transition and vortex shedding, the convective heattransfer is subjected to an abrupt increase. Upon full establishmentof vortex shedding, Nu plateaus to a constant high value. Neverthe-less, the most important outcome of Fig. 7c is that the Nu in thevortex shedding region is up to 300% higher compared to the valueexpected with thermally developed flow without shedding (shownwith red dashed line in Fig. 7c). Fig. 7 also includes the results onthe effect of confinement. For convenience of presentation theyare described separately in the next two subsections.

3.2.2. Effect of longitudinal confinementThe effect of longitudinal confinement (pitch variation) on ther-

mal transport in a row of pins is analyzed in this section, while

0 0.2 0.4 0.6 0.8 10

0.4

0.8

1.2

1.6

x*

c l

adiabatic

heated

Re=200(adiabatic)

Re=150(adiabatic)

RM

S

80 120 160 200

0.7

Re

x*

0.6

0.3

0.4

0.5

0.2

30

50

70

90

Tw

all(°

C)

2

6

10

14

18

Nu

expected steady flow curve

+300%

22

0 0.2 0.4 0.6 0.8 1x*

0 0.2 0.4 0.6 0.8 1x*

vortex shedding

transition

steady

(c) (d)

(b)(a)0.8

heated

adiabatic

Fig. 7. Characterization of flow regimes and their effect on hydrothermal parameters: (a) RMS lift coefficient variation in the streamwise direction. If unspecified Re = 100 andsimulation is diabatic, i.e., pins are heated. (b) Identification of steady (pre-transition), transition and stable vortex shedding regions for the adiabatic and heated (125 W/cm2)fully populated configurations. (c) Nusselt number variation, highlighting an increase of up to 300% in the vortex shedding region compared to expected results from steadyflow. (d) Streamwise variation of wall temperature. Note that in (a), (b) and (d) the lines are used to connect the discrete data for each pin in a particular row of pins.

752 F. Alfieri et al. / International Journal of Heat and Fluid Flow 44 (2013) 745–755

keeping the lateral confinement at w� = 0.5 (see Fig. 6a and b). Todo so, first a configuration with 25 inline pins with p� = 0.25, i.e.a half-populated (HP) configuration where every second pin wasremoved and imposed with a heat flux of 250 W/cm2, was consid-ered. The results obtained were compared with those on 50 pinrow, fully populated (FP) configuration, with each pin having a heatflux of 125 W/cm2. For the FP configuration p� = 0.5. The heat fluxeswere chosen to keep the total amount of heat input to the cavityconstant.

A major effect of doubling the longitudinal pitch between thepins is the enlargement of the vortex shedding region, whichestablishes itself very close to the inlet, as shown in Fig. 7a (orangedash-dotted line). The cRMS

l value on the first pin of the HP config-uration is nonzero, which suggests that a transition from steady totransitional flow regime has already begun at the inlet. The averageNu plotted in Fig. 7c is approximately constant and high for the HPconfiguration compared to the FP configuration, which becomeshigh only after the transition region. In the HP configuration, theonset of vortex shedding or strong transition is observed fromthe very first lateral row of pins. The corresponding pin wall tem-peratures are plotted in Fig. 7d for both HP and FP configurations.The maximal temperature reached in the FP row configuration is�11% lower. The large change in Nu in the transitional regime isalso clearly reflected by the abrupt decrease in the pin temperaturein the FP case. However, this affects the temperature homogeneity,which is important for minimizing thermal stresses in a chip pack-age (Sharma et al., 2012). In the HP row case, where flow instabil-ities are stronger and vortices are shed from nearly all the pins, thewall temperature increases linearly in the streamwise directionand the pin temperatures are more homogeneous.

3.2.3. Effect of lateral confinementThe effect of lateral confinement of a pin row is analyzed by

employing a small lateral confinement of w� = 0.1. This lateral

confinement should interfere with the formation, growth andpropagation of the vortices. The z-component of the vorticity plot-ted in Fig. 6c shows the presence of large, alternating vorticesspread over half of the channel width. The instabilities resultingfrom these large vortices cause the flow to oscillate with a wavelength of 5 cells. The effect of the reduced lateral confinement isreadily apparent with instabilities occurring earlier as the pinsare less confined. This observation confirms the results obtainedby Rehimi et al. (2008). The two plateaus in the right half of thew� = 0.1 curve in Fig. 7a are intriguing. After being stable at a valueof �0.4 for a while, the RMS lift coefficient seems to decrease by�25% and flatten again. In fact, in the last dozen of pins, the vor-tices were slightly weaker than those shed upstream. This indi-cates that the influence of the channel unpopulated outletportion, where vortices spread and dissipate due to the absenceof confining pins (see Fig. 1), propagates upstream and alter theflow vorticity. The lower vorticity is expected to translate in lowerflow coefficients and possible worsening of the convective heattransfer. The average Nu in Fig. 7c is also influenced by changesin lateral confinement. In the vortex shedding regime, the profilefor w� = 0.1 case exhibits values �58% lower than the w� = 0.5 case.This is due to the weakening of the vortices over the last 10–15pins for the w� = 0.1 case and to the low local Re. The non-unifor-mity of Nu also results in less homogeneous wall temperatures, asindicated in Fig. 7d. Similarly to the w� = 0.5 configuration, thewall temperature rises due to the boundary layer development,before dropping abruptly due to the creation of periodically oscil-lating eddies. Once these instabilities reach a plateau, the walltemperature stabilizes itself and starts to increase linearly. The re-duced heat transfer over the last pins also leads to a correspondingincrease in wall temperature. The maximal wall temperature of�91 �C in the w� = 0.1 case is unacceptable in cooling of micropro-cessor and points to the beneficial effect of confinement on reduc-ing wall temperature.

F. Alfieri et al. / International Journal of Heat and Fluid Flow 44 (2013) 745–755 753

3.2.4. Overall effect of lateral and longitudinal confinementsFor overall comparison of the different configurations with mul-

ti-pin rows, the average Nusselt number for a row (represented byNurow) and the the pumping power per unit height P0d were deter-mined. The Nurow was calculated by averaging the profiles in Fig. 7cand P0d by averaging the pumping power across different pins alongthe streamwise co-ordinate. The increase in the Reynolds numberin the fully populated (FP) and w� = 0.5 case is found to enhancethe Nurow. Reducing the lateral confinement resulted in a 65% low-er pumping power due to the negligible contribution of flowimpingement on the pin walls, but also produces poor heat transferperformance, with a �35% reduction in Nurow. On the other hand,the decrease in longitudinal confinement (HP case compared tothe FP) was beneficial, resulting in a gain of �23% in Nurow and a15% reduction in pressure drop at Re = 100.

3.3. Three-dimensional flow around a single TSV pin: verticalconfinement

The two-dimensional flow around a single pin and through aninline array of pins investigated in the previous sections can bethought as representing the case where the pins are tall-enoughand therefore the effect of upper and lower cavity walls can be dis-regarded. In the present section, results obtained from the simula-tions of a 3D pin with diameter-to-height ratios h� from 0.1 (2D-similar) to 0.5 are meant to enrich the understanding of vortexshedding physics and elucidate the effect of vertical confinement.This will also give an insight in similarities and differences withthe 2D models. The reference case is the one with w� = 0.5, but lat-eral confinement 0.1 and 0.3 are also investigated in order to con-firm the trends from the vertically unconfined 2D cases presentedabove. To keep the boundary conditions comparable the fullydeveloped velocity profile from Eq. (3) is used. For 3D simulations,a heat flux of 125 W/cm2 was imposed on the wall. The 3D model-ing is limited to a single pin since considering that an array wouldbe computationally prohibitively expensive and out of the mainscope of the present work.

(a)

(c)

0.6

0.4

0

0.2

0

4

2

050

150100

Angle θ (deg)

0.5

0

z* 0.25

050

150100

Angle θ (deg)

x10-3

0.6

0.4

0

0.2

0.5

0

z* 0.25

2

1

3x10-3

2.5

1.5

0.5

c fc d

Fig. 8. Hydrothermal coefficients on pin surface extracted from 3D simulations (w� = 0.dimensional z coordinate. (a) Friction, (b) pressure (c) drag coefficients and (d) Nusselt nwith higher values in the middle of the channel. Near the wall, the boundary effects red

The supplementary video captures the qualitative features ofvortex shedding mechanisms and the wake of the pin producedby the 3D simulation. The video demonstrates the complexity ofthe flow field, which comprises of small vortical structures thatare symmetric with respect to the longitudinal mid-plane. Anotherfeature observable in the video is related to the origin of vortices;contrary to the 2D case, the generation of the vortices is not occur-ring on the pin surface only. The study of the 3D vortical structuresin the pin wake, which has already been extensively investigated(Saha et al., 2003; Thompson et al., 1996) and is not the focus ofthe present work. Rather, we concentrate on the region very closeto the pin, which determines the hydrothermal characteristics suchas the Nusselt number, pressure losses and the related flow coeffi-cients. These parameters are discussed below.

The hydrothermal characteristics produced from a 3D simula-tion of the flow around a pin are illustrated in Fig. 8, where the lo-cal time averaged friction, pressure and drag coefficients as well asNusselt number are plotted as a function of the angular positionand the dimensionless z-coordinate z�. At any z�, the profiles ofcf, cp, cd and Nu are very similar to the vertically unconfined(2D) cases shown in Fig. 3a–c and Fig. 4a. The only difference ob-servable is due to the effect of the boundary layers, which changesthe parameters along the height of the pin. In fact, all the param-eters in Fig. 8 decrease from the middle plane towards the wall,mostly because of the local Re reduction due to the boundarylayer.

From a practical point of view, the quantification of the changesoccurring from vertical confinement is important in order tounderstand whether the changes in the wake of the pin are influ-encing the heat transfer mechanism from the pin to the fluid. Forthis purpose, Fig. 9 summarizes the deviations of the relevantparameters between vertically confined (3D) and unconfined(2D) cases. The percentage deviation in a parameter W is expressedas

nVUC�VC ¼WVC �WVUC

WVUC� 100; ð12Þ

(b)

(d) 050

150100

80

60

20

40

Nu

0

0.5

0

z* 0.25

-20

60

40

20

0

-20

Angle θ (deg)

050

150100

Angle θ (deg)

2

0

-4

-2

-60.5

0

z* 0.25

1

-1

-3

-5

0

-2

-4

c p

5, h� = 0.5 and Re = 200). Data are presented as function of the angle and the non-umber. The angular variations show trends that are very similar to the 2D profiles,uce the coefficients.

St Nu cd-20

-10

0

10

20

30

40ξ V

UC

-VC

w*=0.1, h*=0.1, Re=200w*=0.3, h*=0.5, Re=200w*=0.5, h*=0.5, Re=200w*=0.5, h*=0.5, Re=100

Fig. 9. Relative deviation of vertically confined (3D) results from verticallyunconfined (2D) ones.

754 F. Alfieri et al. / International Journal of Heat and Fluid Flow 44 (2013) 745–755

where the subscript VC denoted the vertically confined and VUC thevertically unconfined quantity. The results in Fig. 9 help describeand quantify the effect of the vertical confinement (additionally tothe lateral confinement). Clearly, the deviations for Nu and cd arefar below ±20% for the cases analyzed at Re = 100 and 200. Thismeans that the difference in the wake structure, as captured bythe 3D simulations of a vertically confined pin, influences the rele-vant hydrothermal parameters at pin surface, but not in a dramaticfashion. On the other hand, the St is shown to reach larger devia-tions of about 40%.

The above discussed trends and differences between verticallyconfined and unconfined cases are likely valid for any vertical con-finement, but vortex shedding is not triggered for any arbitrary h�.In fact, diagnostic simulations at Re = 200 on the configurationswith w� = 0.5 and h� = 1 and 2 as well as on w� = 0.1 and h� = 0.5,surprisingly showed no vortex shedding, but only steady recircula-tion zones behind the pin. This fact has also been observed exper-imentally in measurements with micropin arrays (Renfer et al.,2011). Thus a vertical confinement seems to act as natural stabi-lizer of the steady and stationary vortex pair in the wake of thepin and avoids their shedding.

The above discussed 3D simulation using a single pin clearly re-veals some important effects of vertical confinement on flowacross a pin. Future investigations focusing on 3D simulations ona row of pins are necessary to gain a more complete understandingof hydrothermal in flows across confined pins.

4. Conclusions

In conclusion, the effect of lateral, longitudinal and vertical con-finement on the vortex shedding in the flows across a single line ofpins placed in a periodic fashion side-by-side or through an arrayof in-line pins was studied. The focus was on analyzing the hydro-dynamic and thermal parameters such as friction, pressure, anddrag coefficients, Strouhal number, pumping power, chip tempera-ture and Nusselt number.

The increase in lateral confinement on the flow around a singlepin caused all the hydrothermal parameters to increase consider-ably: the drag coefficient up to 230% and Nu up to 30%. Interest-ingly, at any given pumping power, Nu and the sheddingfrequency were not influenced by the lateral confinement. More-over, it was possible to characterize Nu by means of a correlationincluding Re, Pr and the level of lateral confinement as independentparameters.

Next, by considering the flow across a periodic array of inlinepins, it was shown that Nu in the shedding regime can be enhancedup to 300% compared to the expected steady conditions. It was also

shown that despite the simultaneous increase in pumping power, acooling system operating in this regime is still beneficial. Further-more, the increase in the longitudinal pitch facilitated the sheddingin the entire chip, thereby leading to more homogeneous but high-er chip temperature. On the other hand, the reduction of the lateralconfinement drastically modified the shedding behavior.

Finally, the increase in vertical confinement on the flow arounda single three-dimensional (including end wall effects) pin wasalso investigated. The results showed differences compared tothe two-dimensional solution; up to �20% Nu and 15% drag coeffi-cient variations were observed with increase in vertical confine-ment. Vortex shedding suppression was observed in the wake forincreased vertical confinement.

The presented results can be used to infer guidelines such asthat for best performance, vertically confined micropins in inte-grated electronic cooling device should be organized to have smalllongitudinal confinement, whereas the lateral confinement can beas high as 0.5 (pin diameter to pin gap ratio) without affecting thecooling performance appreciably.

Acknowledgments

This work was partially supported by the Swiss Confederationthrough the SNSF evaluated RTD Project No. 618_67-CMOSAIC-funded by Nano-Tera.ch. The support is gratefully acknowledged.

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