REDUCED-ORDER MODELING OF MULTISCALE TURBULENT CONVECTION: APPLICATION TO DATA CENTER THERMAL MANAGEMENT

Transcription

1 REDUCED-ORDER MODELING OF MULTISCALE TURBULENT CONVECTION: APPLICATION TO DATA CENTER THERMAL MANAGEMENT A Dssertaton Presented to The Academc Faculty by Jeffrey D. Rambo In Partal Fulfllment of the Requrements for the Degree Doctor of Phlosophy n the School of Mechancal Engneerng Georga Insttute of Technology May 2006 COPYRIGHT BY JEFFREY D. RAMBO 2006

2 REDUCED-ORDER MODELING OF MULTISCALE TURBULENT CONVECTION: APPLICATION TO DATA CENTER THERMAL MANAGEMENT Approved by: Dr. Yogendra Josh, Advsor School of Mechancal Engneerng Georga Insttute of Technology Dr. Shelton M. Jeter School of Mechancal Engneerng Georga Insttute of Technology Dr. P. K. Yeung School of Aerospace Engneerng Georga Insttute of Technology Dr. Benjamn Shapro School of Aerospace Engneerng Unversty of Maryland, College Park Dr. Marc K. Smth School of Mechancal Engneerng Georga Insttute of Technology Date Approved: February 23, 2006

3 ACKNOWLEDGEMENTS I would lke to thank my advsor, Professor Yogendra Josh, for hs support and gudance durng my doctoral studes. I am grateful for hs nsght and career advce, but most of all hs confdence n me. I would also lke to express my apprecaton to my commttee members for ther valuable suggestons and commtment to ths dssertaton. I am thankful for all my current and former colleagues for the valuable, or otherwse nterestng, dscussons we have had concernng ths and many other research topcs. I wsh to thank all my famly and frends who have supported me along the way and above all, I would lke to express my deepest grattude to my lovng wfe Allson for all her encouragement.

4 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS... LIST OF TABLES... v LIST OF FIGURES... v NOMENCLATURE... x SUMMARY... xv CHAPTER 1. Introducton to Data Center Thermal Management Power Densty Trends Arflow Confguratons Systems-Level Electroncs Thermal Management Objectves of Data Center Modelng and Characterzaton State of the Art and Future Trends n Data Center Thermal Management Numercal Modelng of Data Centers Flow Regmes and Scalng Revew of Data Center Numercal Modelng Lmtatons of Numercal Modelng Reduced-Order Modelng of Turbulent Flows Reduced-Order Model Taxonomy The Proper Orthogonal Decomposton: Lterature Revew Mathematcal Formulaton...25 v

5 The Galerkn Projecton Analyss of the RANS Equatons Flux Matchng Procedure Applcaton to Lamnar Flow Lamnar Flow Model Problem Lamnar Flow Results Applcaton to Turbulent Flow Turbulent Flow Model Problem Turbulent Flow Results Applcaton to Ar-Cooled Electroncs Rack Sngle Parameter RANS POD Mult-Parameter RANS POD Rack Optmzaton Reduced-Order Modelng of Forced Turbulent Convecton Model Parameters Low-Dmensonal Turbulent Flow Modelng Low-Dmensonal Turbulent Convecton Modelng Results Error Analyss Error Parttonng Error Estmates Turbulent Flow Error Turbulent Convecton Error...90 v

6 6. Interconnected Reduced-Order Models for Multscale Domans Indvdual Component Modelng Pressure Feld Approxmaton Systems-Level Modelng Full-Scale Results Addtonal Component Parametrzaton Concludng Remarks Summary Future Work REFERENCES v

7 LIST OF TABLES Table 1: Turbulent convecton observaton database...68 Table 2: Plenum flow observatons, P [Pa] and m& [kg/s]...95 Page v

8 LIST OF FIGURES Fgure 1.1: Data center photographs, courtesy of Lawrence Berkeley Natonal Laboratory...2 Fgure 1.2: Standard rased floor plenum and room return coolng scheme...4 Fgure 1.3: Systems-level electroncs thermal management heat generaton and length scale map...6 Fgure 3.1: State space model of a server...19 Fgure 3.2: Model descrpton and sze comparson, from [32]...21 Fgure 3.3: Lamnar flow model geometry and observaton database...43 Fgure 3.4: Lamnar flow ensemble mean and the frst three POD mode shapes...45 Fgure 3.5: Approxmate soluton results, (a) modal egenvalue spectra, (b) boundary condton error...45 Fgure 3.6: Lamnar flow approxmate soluton, (a) L 2 -norm error and (b) modal coeffcents...48 Fgure 3.7: Error reducton wth ncreasng system dmenson (a) on the vertcal md-plane and (b) over the entre doman...49 Fgure 3.8: Turbulent flow model geometry and observaton database...51 Fgure 3.9: Approxmate soluton results, (a) modal egenvalue spectra, (b) boundary condton error...52 Fgure 3.10: Turbulent flow system reference pont and the frst three POD mode shapes from the PODc...53 Fgure 3.11: Turbulent flow approxmate soluton results (a) L 2 error and (b) Galerkn projecton lmtng cases for constant vscosty assumpton...54 Fgure 3.12: Error reducton wth ncreasng system dmenson (a) on the vertcal md-plane and (b) over the entre doman...56 Fgure 3.13: (a) Rack and (b) server geometry and arflow patterns...57 Page v

9 Fgure 3.14: V n egenvalue spectrum and POD modes...59 Fgure 3.15: Sngle parameter spectrum and L 2 approxmaton error...60 Fgure 3.16: Mult-parameter POD modes...61 Fgure 3.17: Mult-parameter PODc error...62 Fgure 3.18: Local PODc and exact solutons near the entrance to servers 7 and Fgure 3.19: Local PODc approxmatons n regon of maxmum velocty error; exhaust of servers 6 and Fgure 3.20: Reduced-order model executon and tme requrements...65 Fgure 4.1: Model geometry from Yoo et al. [68]...67 Fgure 4.2: Comparson of numercal soluton and expermental measurements at Re = 13,690, data from Yoo et al. [68]...69 Fgure 4.3: Mean-centered velocty POD and orthogonal complement POD (PODc) modal energy content...70 Fgure 4.4: (a) Velocty absolute approxmaton error [m/s] and (b) detaled local velocty felds...71 Fgure 4.5: Temperature modal spectra for the POD and PODc procedures...78 Fgure 4.6: Local (a) mean-centered POD modes and (b) PODc modes for test case...78 Fgure 4.7: Exact and approxmate temperature felds [ C]...80 Fgure 5.1: a) POD subspace and b) n-plane (e ) and out of plane (e o ) error components...83 Fgure 5.2: Velocty weght coeffcents computed by the FMP and true values obtaned by projecton onto the POD subspace...89 Fgure 5.3: Temperature weght coeffcents computed by the FMP and true values obtaned by projecton onto the POD subspace...91 Fgure 5.4: Temperature flux matchng procedure error and error bounds...92 x

10 Fgure 6.1: a) Plenum b) server component model geometry and c) full-scale system...95 Fgure 6.2: a) Sub-component modal spectra and b) ntake plenum boundary condton error...96 Fgure 6.3: Intake plenum error feld...97 Fgure 6.4: Component ROM pressure feld approxmaton...99 Fgure 6.5: Response surface approxmaton for a) weght coeffcents and b) boundary condton approxmaton error Fgure 6.6: Comparson of response surface and true pressure weght coeffcents Fgure 6.7: Full-scale system nomenclature and flow resstance network Fgure 6.8: Exact and component-matched velocty and pressure felds Fgure 6.9: Local velocty msmatch at nterface a) Ω 1 Ω 4 and b) Ω 3 Ω Fgure 6.10: Ω 1 Ω 4 Interface for updated server ROM Fgure 7.1: Reduced-order model (ROM) methodology flowchart x

11 NOMENCLATURE Symbols A jk, C jk, D j, D jk...coeffcents n Naver-Stokes Galerkn projecton E... resolved modal spectrum energy F m, F h... mass and heat flux functon G, G h... mass flux and heat flux goal vector J...error functonal Nu... Nusselt number P k... orthogonal subspace projecton matrx Pr t... turbulent Prandtl number Q... heat flux R ( x, x')...correlaton functon Re... Reynolds number U, T, P...velocty and temperature feld observatons matrces W... optmal / robust soluton weght factor a,b... modal weght coeffcents d... POD subspace dstance e, e o... n-plane and out of plane error components k... turbulent knetc energy k eff... effectve thermal conductvty m...number of observatons n...number of system DOF x

12 nˆ... outward pontng surface normal p... number of retaned modes n approxmaton s...number of modes n orthogonal complement subspace u r, T, P...velocty, temperature and pressure felds Γ...control surface Φ r, Ψ, Π...velocty, temperature and pressure POD subspace bass vector matrx Ω, Ω... doman, doman boundary Ε... turbulent knetc energy dsspaton rate λ... egenvalue ν, ν eff... vscosty and effectve vscosty ρ...densty r ϕ, ψ, π... velocty, temperature and pressure POD modes...ensemble mean...nduced or L 2 norm u...area-averaged normal velocty Subscrpts err...relatve error 0...source functon Superscrpts () +...matrx pseudo-nverse x

13 () *... approxmate soluton or complex conjugate ()...orthogonal complement () obs () T...observaton quantty... matrx transpose Acronyms ESV...extended state vector DOF...degrees of freedom FMP... flux matchng procedure FNM... flow network modelng GP... Galerkn projecton POD... proper orthogonal decomposton PODc... orthogonal complement proper orthogonal decomposton RANS...Reynolds-averaged Naver-Stokes ROM... reduced-order model x

14 SUMMARY Data centers are computng nfrastructure facltes used by ndustres wth large data processng needs and the rapd ncrease n power densty of hgh performance computng equpment has caused many thermal ssues n these facltes. Systems-level thermal management requres modelng and analyss of complex flud flow and heat transfer processes across several decades of length scales. Conventonal computatonal flud dynamcs and heat transfer technques for such systems are severely lmted as a desgn tool because ther large model szes render parameter senstvty studes and optmzaton mpractcally slow. The tradtonal proper orthogonal decomposton (POD) methodology has been reformulated to construct physcs-based models of turbulent flows and forced convecton. Orthogonal complement POD subspaces were developed to parametrze nhomogeneous boundary condtons and greatly extend the use of the exstng POD methodology beyond prototypcal flows wth fxed parameters. A flux matchng procedure was devsed to overcome the lmtatons of Galerkn projecton methods for the Reynolds-averaged Naver-Stokes equatons and greatly mprove the computatonal effcency of the approxmate solutons. An mplct couplng procedure was developed to lnk the temperature and velocty felds and further extend the low-dmensonal modelng methodology to conjugate forced convecton heat transfer. The overall reduced-order modelng framework was able to reduce numercal models contanng 10 5 degrees of freedom (DOF) down to less than 20 DOF, whle stll retanng greater that 90% accuracy over the doman. xv

15 Rgorous a posteror error bounds were formulated by usng the POD subspace to partton the error contrbutons and dual resdual methods were used to show that the flux matchng procedure s a computatonally superor approach for low-dmensonal modelng of steady turbulent convecton. To effcently model large-scale systems, ndvdual reduced-order models were coupled usng flow network modelng as the component nterconnecton procedure. The development of handshakng procedures between low-dmensonal component models lays the foundaton to quckly analyze and optmze the modular systems encountered n electroncs thermal management. Ths modularzed approach can also serve as skeletal structure to allow the effcent ntegraton of hghly-specalzed models across dscplnes and sgnfcantly advance smulaton-based desgn. xv

16 1. INTRODUCTION TO DATA CENTER THERMAL MANAGEMENT Data centers are computng nfrastructure facltes used by ndustres wth large data processng needs, such as telecommuncatons swtchng, bankng, stock market transactons and supercomputng nodes. The trends of rapdly ncreasng heat fluxes and volumetrc heat generaton rates wthn these devces are causng many unque challenges for ther thermal desgn. Due to the hgh heatng rates, desgn gudelnes followed for human-occuped spaces such as audtora and theaters are napplcable for such computng spaces. The electroncs thermal management communty has, n the past, focused exclusvely on heat removal from the chps and sngle data processng unts or enclosures. The objectve has been to remove the heat generated by the chps and assocated devces out of the enclosure case and reject t to an extensve ambent. In a data center, hundreds or thousands of these data processng unts use the faclty as the ambent. Due to the vertcal stackng of these components and space constrants, the electronc equpment can nteract by the hot exhaust ar beng drawn from one data processng unt nto another unt. Fgure 1.1 shows 2 photographs of exstng data center facltes. Detaled expermental measurements and numercal computatons of heat transfer and flud flow are necessary for the proper placement of electronc equpment to ensure ts relable operaton. Due to the three-dmensonal, mult-mode, multscale nature of the transport nvolvng at least 10 decades of pertnent length scales from nanometer chp nterconnects to the faclty length scale of 10 s of meters, and prevalng turbulent flow 1

17 condtons, such characterzaton s hghly challengng. Only recently s electroncs thermal management at the systems level begnnng to receve systematc attenton. Improved characterzaton through expermental and computatonal thermal modelng has the potental of developng energy effcent desgns for these facltes, resultng n large fnancal savngs for the end users. Fgure 1.1. Data center photographs, courtesy of W. Tschud, Lawrence Berkeley Natonal Laboratory 1.1. Power Densty Trends The basc buldng block of a data center s a rack, whch s typcally ~2-m tall enclosure, n whch varous servers, data storage and networkng/swtchng equpment are stacked vertcally. In 1990, a typcal rack dsspated approxmately 1 kw of power [1], whle today s racks wth the same footprnt may dsspate up to 30 kw, based on current server heat loads. The decade from 1992 to 2002 has seen server power densty rse 300%, wth a projected annual ncrease of 5% over the next 4 years [2, 3]. The demand for data processng has drven data centers to grow as large as 7,500 m 2 (~80,000 ft 2 ) [3, 4], producng net power dsspatons on the order of several MW. Wth such an enormous amount of power dsspaton, provdng an envronment for the safe and 2

18 relable operaton of the data processng equpment s essental, as these facltes are ntended to operate contnuously. Practtoners and data center desgners measure data center heat fluxes as the rato of the net power dsspated to the total footprnt of the faclty. Recent energy benchmarkng studes have shown data centers are operatng n the W/m 2 (25 75 W/ft 2 ) range [3, 4] and growth to 100 W/ft 2 average over the faclty, wth local regons exceedng 200 W/ft 2, s expected n the near future. These power denstes are well beyond conventonal heatng, ventlaton and ar condtonng (HVAC) loads for the coolng of smlarly szed rooms such as audtora and offce spaces, whch are typcally 4 8 W/ft 2. A rack footprnt s approxmately 0.76 m by 0.91 m (2.5 x 3 ft), smlar to the area of an audtorum seat. However, a 30 kw rack has 300 tmes the power dsspaton of a sttng person. In other words, applyng standard HVAC gudelnes to data centers s equvalent to coolng an audtorum wth 300 people per seat. Wth facltes growng up to 5000 m 2 (~50,000 ft 2 ), the net power dsspated by the data processng equpment could be as large as several MW. The cost of just powerng these large computng facltes could be mllons of dollars a year, wth the cost of provdng adequate coolng not far behnd Arflow Confguratons A majorty of today s data centers use computer room ar condtonng (CRAC) unts to supply the faclty wth coolng ar desgned to provde an adequately low ambent temperature for relable server operaton. A major mprovement n arflow management has been to arrange the racks n rows wth alternatng drecton of arflow 3

19 [5] such that rack nlets are facng each other, to form a cold asle. The exhausts of the racks necessarly face each other and form a hot asle. Ths hot asle cold asle approach attempts to separate the supply from the exhaust ar and ncrease the overall effcency of the ar delvery and collecton from each rack n the data center. The predomnate coolng scheme n today s data centers s to use the CRAC unts to supply a rased floor plenum underneath the racks wth cold ar. Perforated tles are then located n the cold asles on the rased floor near the racks to delver the cool supply ar to the data processng equpment. The hot exhaust ar from the racks s then collected from the upper porton of the faclty by the CRAC unts, completng the arflow loop (see Fgure 1.2). Fgure 1.2. Standard rased floor plenum and room return coolng scheme Thermal performance and effcency of data centers could be mproved by consderng other possble supply and return schemes. To reduce the entranment of hot ar nto the cold asles, the exhaust could be collected above the hot asle and ducted back to the CRAC unts. Addtonally, the cold supply ar could be delvered to the racks from 4

20 above the cold asle, whch could also reduce the recrculaton effects. Fnally, both the supply and return could be overhead Systems-Level Electroncs Thermal Management Tradtonal electroncs thermal management studes have focused on the heat transfer characterstcs of ndvdual packages and multple devces n an enclosure such as an ndvdual server. Systems-level electroncs coolng s defned here as data processng systems comprsed of multple enclosures or modules, ncludng data processng racks contanng varous servers and data center facltes contanng numerous racks. Fgure 1.3 llustrates the current representatve volumetrc heat generaton rates at varous length scales n electroncs components and systems. It s the heat generaton at the nterconnect level that causes the temperature rse between the chp and the ambent and drves the need for a coolng system to mantan the chp operatng at an acceptable temperature. In systems-level thermal management, the coolng scheme s often developed at the largest length scales, such as the data center CRAC unt. Thus, thermal modelng and characterzaton s nherently multscale, as t s the heat generated at the smallest length scales that drves the need for a global coolng scheme at the largest length scales. 5

21 Fgure 1.3. Systems-level electroncs thermal management heat generaton and length scale map 1.4. Objectves of Data Center Modelng and Characterzaton Modelng of data centers ams to predct the domnant arflow and heat transfer characterstcs so a relable desgn methodology may be developed. An accurate modelng framework can be used to create more thermally effcent data center faclty desgns, elmnatng local hot spots and usng the cold supply ar from the CRAC unts n the most effcent way possble. Currently, overall energy balances on the faclty are used as a desgn gudelne. The expected net power dsspaton of the equpment s computed and an approprate number of CRAC unts s selected to provde the coolng. Consderng the rapd ncrease n power densty and the hgh relablty requrements, desgners often over-specfy the amount of coolng requred n a data center. The lfetme of a data center s approxmately 30 years, whereas the lfetme of the data processng equpment nsde the faclty s only about 2 years, requrng several cycles of equpment replacement, each wth hgher performance and power. Besdes the effcent desgn of new data centers, mproved modelng and characterzaton also ams to address questons 6

22 that arse from such equpment upgrades n an exstng data center. For nstance, f a new server that has twce the power dsspaton and arflow requrements s to be added to a rack, how wll ths affect neghborng servers? A robust modelng strategy can be used to predct f ths new server wll deprve neghborng servers of supply ar and how the hot exhaust ar wll affect nearby severs. Model effcency s a key concern as tradtonal numercal procedures for turbulent flows and heat transfer wll produce excessvely computatonally ntensve models, severely lmtng ther use as an analyss and desgn tool State of the Art and Future Trends n Data Center Thermal Management Wth contnually ncreasng power dsspaton n hgh performance computng equpment, the lmts of forced ar coolng wll be reached n the near future. Over the past several years, supplemental thermal solutons n the form or celng mounted ar coolng unts and addtonal rack level fans have been used to deal wth hgh power densty at the rack level. Currently, data center professonal have moved away from addng more ar coolng unts n favor of lqud coolng solutons. Lqud coolng has the ablty to reduce the power consumpton and nose levels n the data center although relablty s a major concern, especally f the coolng flud s not delectrc. Commercally avalable CRAC unts unversally use chlled water or glycol as the heat transfer medum between the data center ar to the envronment. The ppng nfrastructure to supply the CRAC unts wth coolng lqud exsts n the data center envronment. Most lqud coolng schemes tap nto ths lqud loop to provde addtonal coolng at the rack level. Such systems commonly use ar-to-lqud heat 7

23 exchangers mounted nsde the rack to cool the hot exhaust to form a self-contaned coolng loop. Ths approach reduces the dstance the hot exhaust ar must travel before reachng the heat exchanger, whch mnmzes the effects of recrculaton. Movng the heat exchanger to the rack stll reles on ar as the heat transfer medum and the performance of such systems are usually lmted by the ar sde heat transfer coeffcent. To date, there s ongong research and development of extendng the lqud coolng path to the chp level for maxmum effcency, but no commercally avalable products exst. Recently multple vendors have manufactured lqud coolng solutons to be used at the rack exhaust. The objectve s to cool the exhaust ar to mnmze the effect of a hgh power dsspaton rack on surroundng racks. Supplemental lqud coolng s not used at the rack nlet because of the possblty of condensaton near data processng equpment. 8

24 2. NUMERICAL MODELING OF DATA CENTERS Due to the complex nature of the flow nsde a data center, computatonal flud dynamcs and heat transfer (CFD/HT) are usually requred to nvestgate the thermal performance of data centers. The problem s nherently multscale, wth the smallest scale phenomena drvng the global temperature feld. It s the heat dsspaton at the chp level that causes the temperature rse across each electronc devce wthn each server. A rack contans a number of servers, each wth ther power dsspaton and arflow requrements. The chp may contan features as small as 10 nm, and the faclty length scale s on the order of 100 s of meters, mplyng a computatonal model resolvng all the features n a data center would contan 10 decades of length scales. It s computatonally mpossble to resolve all these features, but a numercal model should attempt to resolve as many length scales as necessary for the type of predctons desred Flow Regmes and Scalng The flow nsde a faclty typcally falls nto the turbulent mxed convecton regme, although there can be large spatal varatons n turbulence ntensty and buoyancy effects. The flow through the data processng equpment nsde a rack s manly forced convecton, but buoyancy effects may become mportant at the rack exhaust and as the hot ar returns to the CRAC unts. Reynolds numbers can be defned usng dfferent length scales and velocty scales and the range of both these scales n data center arflows makes quantfyng arflow characterstcs wth relatvely few dmensonless groups dffcult. Data center thermal performance s typcally consdered 9

25 usng a steady state approxmaton because a large factor n coolng and power equpment redundancy makes falures of such systems a rare event. As stated earler, electroncs coolng applcatons span a wde range of length scales. Wth current computng power, numercal smulatons of heat transfer and flud flow are lmted to approxmately three decades of length scales. Ths has lmted chplevel thermal analyss to resolvng some larger features contaned wthn the package (de and heat spreaders) but the smallest features are stll naccessble. Board-level smulatons typcally employ a smplfed representaton of the packages, treatng them ether as a block wth unform heat generaton or as a constant heat flux area. Numercal computatons of electronc enclosures, such as personal computers and servers, are able to resolve multple boards and devces, but only the components wth the largest power dsspaton are usually consdered. Modelng nfrastructures of electronc equpment must brdge the large dsparty n length scales between the board and the faclty. An Electroncs Industry Assocaton (EIA) standard rack measures 0.74 m wde and 0.94 m deep [1]. The vertcal drecton of a rack s standardzed n racks unts call U, where 1 U = 4.45 cm (1.75 nches) and a typcal rack can accommodate 40 U of equpment. Consder a rack contanng 10 servers, each measurng 4 U hgh and dsspatng 500 W. A flow rate of kg/s s requred to mantan a 10 C bulk ar temperature rse, resultng n a hydraulc dameter based Reynolds number for a sngle server of 5,730. The hydraulc dameter s calculated by assumng the server s 0.74 m wde by 0.18 m (4U) tall rectangular duct. The Raylegh number, based on the same 10 C temperature dfference and a characterstc length of (duct cross-sectonal area) / (duct permeter) = m s 4.46 x Usng Ra / Re 2 as an estmate of the rato of 10

26 buoyancy to nerta effects (whch s vald because Pr 1 for ar) gves , showng that buoyancy can be neglected based on such global consderaton. As the rack power dsspaton ncreases, the requred flow rate wll also ncrease to mantan constant component temperatures, causng the Reynolds number to ncrease, whle the Raylegh number remans the same, makng the rato of Ra / Re 2 decrease. These scalng arguments have also been expermentally justfed by demonstratng pont-wse measurements of rack nlet temperatures are lnearly proportonal to the CRAC exhaust temperature [6] Revew of Data Center Numercal Modelng Data center facltes are crtcal to the data processng needs they serve. They are desgned for contnuous operaton n order to provde a hghly relable envronment for the data processng equpment performng essental tasks and often contanng hghly senstve materal. Access to these envronments s lmted and performng detaled expermental measurements n operatonal data centers s very dffcult. Thus, most prevous nvestgatons n data center thermal management are computatonal n nature and rely on CFD/HT to predct arflow and heat transfer characterstcs. CFD/HT modelng of data centers was ntroduced n 2001 by Patel et al. [1]who performed computatons on a model faclty and by Schmdt et al. [7] who compared expermental measurements through rased floor data center perforated tles wth 2- dmensonal computatonal models. The varous numercal nvestgatons can be classfed nto the followng 4 man categores: 1) rased floor plenum arflow modelng to predct perforated tle flow rates, 2) thermal effects of rack layout and power 11

27 dstrbuton and 3) nvestgaton of alternatve supply and return schemes and 4) development of thermal performance metrcs. A revew of data center lterature s presented by Schmdt and Shaukatullah [8], whch serves to provde a hstorcal perspectve consderng the rapd growth n data center power densty. Schmdt et al. present a depth-averaged (2-dmensonal) numercal model for a m deep plenum [7] and for a m deep plenum [9]. Ther expermental valdaton shows far overall agreement wth select tle flow rates, wth large ndvdual predcton errors. In both papers, t s noted that a 0.1 m dameter ppe and a m tall cable tray are located n the bottom of the plenum, although there s no dscusson of how these obstructons are accounted for n the depth-averaged equatons. Schmdt et al. [9] state that for plenum depths less than 0.3 m deep, depth-averaged modelng s adequate as a tradeoff between computatonal cost and accuracy. Radmehr et al. [10] expermentally nvestgated the leakage flow, or porton of total CRAC arflow that seeps through the seams of the rased floor tles, n a 0.42 m deep faclty. Dstrbutng the leakage flow unformly throughout the perforated tle seams and modelng chlled water supply lnes, Radmehr et al. [10] were able to produce predctons wth an overall accuracy of 90%. Van Glder and Schmdt [11] present a parametrc study of plenum arflow for varous data center footprnts, tle arrangements, tle porosty and plenum depth. A vacant plenum was assumed, whch makes the perforated tle resstance much greater than any other flow resstance n the plenum and allows the resultng perforated tle flow rate to scale wth the net CRAC flow rate. Regardng data center rack layout, Patel et al. [1, 12] have focused on the asle spacng and the correspondng coolng load on the CRAC unts n conventonal rased 12

28 floor plenum data centers. These efforts dentfy the problem of recrculaton, or hot exhaust ar beng drawn nto a cold asle and through electronc equpment before returnng to the CRAC unts. The effects of recrculaton are also documented by Schmdt et al. [9, 13-15]. Geometrcal optmzaton of plenum depth, faclty celng heght and cold asle spacng for a sngle set of CRAC flow rates and unform rack flow and power dsspaton was performed by Bhopte et al. [16]. The results showed that ncreasng the plenum depth or the faclty celng heght, and decreasng the cold asle wdth produce an overall reducton n the average nlet temperature to all the racks n the faclty. Schmdt and Iyengar [15] have also documented thermal performance varablty n several operatonal data centers and attempted to provde some qualtatve explanatons for smlarly performng racks n dfferent data centers. A cluster of hgh-power dsspaton racks and ther effect on the remanng equpment of the faclty were consdered by Schmdt and Cruz [14]. The effect of varyng power throughout a data center s an mportant factor to consder, as operatonal facltes contan a wde varety of equpment, each wth ther own power dsspaton levels. Each data center has a unque geometrcal footprnt and rack layout and a common bass s needed to compare the thermal performance of varous coolng schemes. A unt cell archtecture of a rased floor plenum data center was formulated by consderng the asymptotc flow dstrbuton n the cold asles wth ncreasng number of racks n a row [17]. The results ndcated that for hgh flow rate racks, 4 rows of 7 racks adequately models the hot-asle cold-asle confguraton and s representatve of a long row of racks. 13

29 One of the earlest works n data center CFD/HT modelng analyzed a global flow confguraton other than that of the standard rased floor plenum wth return to the CRAC unt through the room. The model of Patel et al. [1] examned a coolng strategy where the CRAC unt supply and return are located over top of the cold asles. A novel concept of ntroducng cold supply ar nto the hot asles to mtgate recrculaton effects was proposed by Schmdt and Cruz [13] although computatonal results showed ths modfcaton could dmnsh the basc coolng scheme s effcency. Supplyng the cold ar from above the racks was nvestgated n [18] where the overhead supply and faclty heght were nvestgated parametrcally. Shrvastava et al. [19] parametrcally vary all possble locatons of the cold ar supply and hot exhaust return for fxed room geometry, unform rack power and fxed CRAC condtons. In the prevous analyss of data center arflows, researchers have ether modeled the rack n a black-box fashon wth prescrbed flow rate and temperature rse, or as a box wth fxed flow rate and unform heat generaton [1, 12, 14, 18]. Ths less computatonally expensve approach s useful only f faclty quanttes such as CRAC unt performance and rack layout are of nterest. However, ths level of descrpton cannot determne temperature varatons wthn the rack caused by recrculaton effects. To better characterze the performance varatons through a data center, a procedure to model ndvdual servers wthn each rack was developed n [20]. As computatonal lmtatons allow, the server sub-models can be ncreased n complexty to nclude dscrete heatng elements to mmc multple components on a prnted wrng board. A formal modelng approach to data centers consderng modelng of the CRAC unts, perforated tles and rack sub-models was presented n [21]. Ths nvestgaton 14

30 consders the dstrbuton of CRAC unts through the unt cell archtecture and the effect of rack orentaton relatve to the CRAC unts. To develop a mechanstc understandng of convectve processes n data centers, the global scheme was dvded nto the processes of 1) the CRAC exhaust to the cold asle, 2) cold asle dstrbuton of supply ar through the racks and 3) the hot exhaust ar return to the CRAC unts [22]. Numercal modelng of varous supply and return schemes, coupled wth varous orentatons between the racks and the CRAC unts, dentfed the causes of recrculaton and non-unformty n thermal performance throughout the data center. The models presented n [1, 12, 14, 15, 18, 23] have used a varety of orentatons between the CRAC unts and racks. The parametrc study presented n [21] s the frst attempt to generally quantfy these effects and followng work usng the same procedures was contrbuted n [19]. The performance of the assorted data center models are assessed n varous ways by dfferent authors and specfc comparsons can be dffcult. Sharma et al. [23] ntroduce dmensonless numbers to quantfy the effects of recrculaton. These dmensonless numbers are arrved at by consderng the ratos (cold supply ar enthalpy rse before t reaches the racks) / (enthalpy rse at the rack exhaust), and (heat extracton by the CRAC unts) / (enthalpy rse at the rack exhaust). These defntons requre the ar temperature evaluaton at arbtrary ponts near the rack nlet and exhaust. Sharma et al. [24] later computed these dmensonless performance metrcs n an operatonal data center by takng a sngle ar temperature measurement just below the top of each rack nlet and outlet. Norota et al. [25] used the statstcal defnton of the Mahalanobs generalzed dstance to descrbe the non-unformty n rack thermal performance. Shah et al. [26-28] have proposed an exergy-based analyss method that dvdes the data center 15

31 nto a seres of subcomponents and then CRAC unt operaton s optmzed gven nformaton regardng the rack power dsspaton. Thermal performance metrcs for systems level electroncs coolng based on the concept of thermal resstance (power dsspaton / temperature rse) were formulated and appled to data centers n [21]. The metrcs consder the spatal unformty of thermal performance to characterze poor desgns causng local hot spots. Entropy generaton mnmzaton was also proposed as a data center thermal performance metrc because poor thermal performance s often attrbuted to recrculaton effects. Snce the mxng of hot exhaust ar wth the supply ar n the cold asles generates entropy, cold asle entropy generaton mnmzaton was employed as a metrc. The results presented n [21] show that usng entropy generaton mnmzaton and thermal resstance wth spatal unformty consderatons predct the same desgn as beng the best. The same thermal performance metrcs were appled to a forced ar-cooled rack to optmze the server layout [29] because the metrcs were formulated generally for systems-level thermal management and should be applcable to a range of systems Lmtatons of Numercal Modelng In the absence of detaled expermental data on data center arflow and heat transfer, CFD/HT provdes a tool to estmate some of the domnant features of data center thermal management. The overall accuracy of models needs to be addressed due to many smplfcatons that are employed. RANS-based turbulence modelng procedures come nto queston for the strongly swrlng flow n data centers. All the numercal models employed the standard k-ε turbulence model [30], except for [21, 22, 29, 30] where the 16

32 full Reynolds stress model [30] was used to assess the sotropc eddy vscosty assumpton. Gven hghly accurate modelng procedures, the CFD/HT models are stll lmted due to the large amount of tme nvested n constructng a mesh, demonstratng mesh and teraton convergence and fnally post-processng the results. The models presented n [21, 22] exceed 1.5 mllon grd cells and complcated pressure-drven boundary condtons requre many teratons for the model to converge. For a 2-equaton turbulence model n 3 dmensons, the fnte volume method produces 7 degrees of freedom (DOF) per grd cell (P, u, v, w, k, ε, and T) or 13 DOF wth the addton of the 6 Reynolds stress components n the full Reynolds stress model. The total amount of tme consumed by a sngle soluton, ncludng mesh development, grd convergence and postprocessng was approxmately 100 hours, wth 80 hours dedcated computng tme on a dual Pentum Xeon 2.8 GHz processor workstaton wth 4 Gb of memory. All of the DOF solved for are necessary to model the arflow and heat transfer, but the key quanttes of nterest such as rack temperature rse and perforated tle flow rates, are evaluated usng only a small porton ( < 10% ) of the total model DOF. 17

33 3. REDUCED-ORDER MODELING OF TURBULENT FLOWS Desgn and analyss of complex engneerng systems nvolvng turbulent convecton often requre careful numercal smulatons usng computatonal flud dynamcs and heat transfer (CFD/HT) or detaled expermental data obtaned by full feld technques such as tomographc nterferometry and partcle mage velocmetry, to accurately descrbe the flud flow and heat transfer processes of the system [31]. Both methods of system characterzaton are tme ntensve, partcularly so for carryng out parametrc studes, and severely lmt the range of desgn varables that can be explored, renderng optmzaton algorthms mpractcally slow. In early stages of desgn, t may be desrable to trade ths effort for an expermentally-valdated reduced model that captures the domnant physcs, but s computatonally effcent. Such models may be used n conjuncton wth optmzaton routnes to quckly perform parameter senstvty studes. A low-dmensonal model of ths type can also be ntegrated wth multscale computatons to effcently brdge a range of length scales wthout requrng a sngle smulaton to resolve all length scales smultaneously Reduced-Order Model Taxonomy Methods of reduced-order modelng can generally be dvded nto state space and dstrbuted parameter system approaches. State space methods reduce a system to an nput/output map and many tools are avalable to analyze nterconnected state space components, see [32] for an applcaton-based overvew. Ths class of models s also synonymous wth lumped-parameter model and can utlze basc physcal prncples 18

34 such as mass and energy conservaton as well as correlatons to develop the model behavor. Fgure 3.1 below schematcally depcts a state space model of a server that returns the maxmum chp temperature and mean ar outlet temperature gven the nlet temperature, chp power dsspaton and flow rate. The maxmum chp temperature can be evaluated usng local Nusselt number relatons Nu x = f(re x,pr,gr x ) and Newton s law of coolng, Q = h x A(T x - T ). The model s based on a fxed geometry and materal propertes. Resstor network type models are often used to model electronc packages and are categorzed under state space methods as they typcally return a juncton temperature gven the power dsspaton, an external heat transfer and ambent condtons [33-37]. The fxed parameters are the chp geometry and materals, whch strongly affect the manner n whch the thermal resstance network s constructed. Fgure 3.1. State space model of a server Models of ths form may provde an adequate level of accuracy, but the descrpton of the underlyng physcal mechansms s very ncomplete. State space models cannot generally be examned to determne the root causes for specfc behavor and ther performance strongly depends on the modelng assumptons they are bult upon. Dstrbuted parameter systems am to approxmate the physcs over the entre doman, as opposed to returnng a vector of desred outputs. Ths approach s desrable n 19

35 modelng convectve flows, as the model s not lmted to returnng a prescrbed quantty, such as a set of velocty, temperature or heat flux nformaton, rather the complete velocty and temperature felds are avalable for further analyss of the transport processes nvolved. The fundamental prncple of dstrbuted parameter modelng s to fnd a sutable set of modes to project the governng equatons onto, reducng the soluton procedure to fndng the approprate weght coeffcents that combne the modes nto the desred approxmate soluton. Tradtonal modal expansons usng Fourer seres or orthogonal polynomals (Legendre, Chebyshev, Laguerre) form the bass for spectral methods. Complex boundary condtons can be problematc n spectral methods and the types of boundary condtons often dctate the functons employed n the expanson. For example, Fourer seres are the natural choce for perodc domans whle the propertes of Chebyshev polynomals are often exploted for nhomogeneous boundary condtons. Hgher order terms are requred to resolve sharp gradents n the soluton and many terms n the seres are needed for accurate predctons. The process of takng a model from a large number of DOF, ether from detaled numercal smulatons or full-feld expermental measurements, to a model nvolvng sgnfcantly fewer DOF s termed model reducton. A number of tools exst for reducng the number of nternal states of large lnear systems, such as those resultng from descretzng dfferental equatons [32]. System dentfcaton s the process of dentfyng model structure or estmatng unknown parameters through expermentng wth an unknown system. Consderng the black box type system depcted n Fgure 3.1, system dentfcaton would am to estmate T chp,max n the most accurate and effcent 20

36 manner by varyng the system nputs. Fgure 3.2 llustrates ths taxonomy of effcent modelng procedures [32]. Fgure 3.2. Model descrpton and sze comparson, from [32] Wth objectve of constructng a reduced-order modelng framework for systemslevel electroncs thermal management, a key model development would nvolve the constructon of models for parametrc systems. A parametrc system s defned here as a system contanng a source term or boundary condton that can be vared over a specfed range n order to produce dfferent system responses. For a convectve flow, such parameters nclude a geometrcal length, mass flow rate, boundary heat flux or temperature, or volumetrc heat generaton, to produce dfferent flow patterns, and/or transport characterstcs. Ths can be quantfed wth one or more relevant dmensonless groups, such as Reynolds or Strouhal number, non-dmensonal heat generaton rates, or a set of aspect ratos to defne geometry. Changes n thermophyscal property varatons 21

37 are excluded from ths defnton because they may arse naturally even n the absence of the parametrc varabltes defned here. In summary, the parametrc nature may result from a prescrbed boundary condton, for example nlet velocty or wall heat flux, nteror condton, such as volumetrc heat generaton, or geometrc parameter such as an aspect rato The Proper Orthogonal Decomposton: Lterature Revew The proper orthogonal decomposton (POD) s a stochastc tool used to assemble the model-specfc optmal lnear subspace from an ensemble of system observatons. Owng to the stochastc nature of the subspace calculatons, the POD s deally suted for nonlnear phenomena and has been used extensvely n low-dmensonal modelng of turbulent flows, see [38] for a more complete descrpton and revew of ts use n prototypcal turbulent flows. A major shortcomng of the exstng POD methodology to date s ts neffcent treatment of a range of model parameters. Prevous reduced-order flow and heat transfer modelng studes have nvestgated the dynamcs of a prototypcal system under a sngle Reynolds or Raylegh number or lmted range of varaton. Lamnar flows were nvestgated by Deane et al. [39] and ad hoc mode scalng showed mxed results (± 15% n perod predctons) for approxmatng flows from 52 observatons over a small range of Reynolds numbers. For lamnar flows wth varous nlet profles, both Park and Km [40] and Ravndran [41, 42] suggest homogenzng the POD modes by subtractng a reference feld that satsfes the governng equatons. Park and Km [40] constructed a low-dmensonal controller for flow over a backward facng step based on 1,000 observatons of 2 nlet velocty profles. Ravndran 22

38 homogenzed 100 observatons for developng a flow controller usng blowng at Re = 200 [41, 42]. In the nvestgaton of transtonal behavor, Ma and Karnadaks used 40 modes to study the lmt cycle of 3-dmensonal flow around a cylnder at a crtcal Re = 188 [43] and at Re = 610 [31]. No specal boundary treatment was requred for ether flow because perodc boundary condtons were employed. In low-order modelng of heat transfer, Park and Cho parttoned the lnear governng equaton nto homogeneous and nhomogeneous components to account for boundary condtons n order to model conducton [44] and temperature and speces transport for a fxed velocty feld [45], usng 200 and 400 observatons respectvely. Srovch [46-48] analyzed the dynamcs of natural convecton by workng wth homogeneous devatons from the mean flow, allowng the mean to take the fxed nhomogeneous boundary condtons and typcally usng around 200 observatons. Park and L [49] modeled natural convecton wth 30 snusodal boundary heat flux profles for a total of 3,000 observatons. Recent developments n low-dmensonal flow modelng have been made by Srsup and Karnadaks [50] who have proposed usng a penalty functon Galerkn method to treat tme varyng boundary condtons. Geometrcal scalng has also been nvestgated by Taylor and Glauser [51] who constructed a low-dmensonal model of a varable angle dffuser at the expense of 30,720 observatons. Uttakar et al. [52] used POD for reduced turbulent smulatons of flows wth movng boundares; however they do not descrbe any reduced-order model development, only the accuracy and data compresson assocated wth POD representaton of the observatons. Gallett et al. [53] modeled lamnar flow over a confned square block by nterpolatng the modal weght 23

39 coeffcents at dfferent Reynolds number to correct the pressure drop across the duct from 160 observatons. To summarze the work to date n developng POD-based models of flows over a range of varyng Reynolds and Raylegh numbers, nhomogeneous boundary condtons are ether treated through expensve homogenzaton procedures or through a very large, and often mpractcal, number of system observatons. A key concern n the exstng POD methodology s determnng the mnmum number of observatons requred to construct a POD subspace that fathfully represents the physcs of the system. In dynamc systems, each smulaton tme step s avalable to be ncluded n the ensemble. Expermentally-based POD models of turbulent flows also beneft from large data ensembles, as many repeated measurements are requred to generate statstcally sgnfcant turbulence data. For parametrc reduced-order modelng of statonary turbulent flows, each observaton s from an ndependent system snapshot. To the best of the author s knowledge, the only other publshed attempt of usng PODbased modelng for statonary analyss of thermo-flud systems was by Ly and Tran [54]; however ther soluton method was based on nterpolatng splnes between weght coeffcents to match a desred parameter value. Ths method would requre hgher order mult-dmensonal nterpolaton to model a complex system wth multple parameters and also does not guarantee that the desred parameter level wll be acheved. There are two major defcences wth the standard POD procedure as descrbed above. In the context of dynamcal systems, the Galerkn projecton has been demonstrated to produce false lmt cycles [55] and deemphasze mportant modal contrbutons under varyng bfurcaton parameters n parameter dependent flows [56, 57], ultmately leadng to unphyscal results. Secondly n prevous reduced-order flow 24

40 modelng studes, homogeneous boundary condtons n the form of ether closed [48, 58] or perodc domans [39, 43] are employed. Inhomogeneous boundary condtons have also been treated by subtractng reference velocty felds from each member of the ensemble to homogenze the boundary condtons before the POD modal bass s computed [42]. The reference feld must satsfy the governng equatons, mplyng that f the boundary condtons are to be altered a new reference feld for each set of boundary condtons must be obtaned Mathematcal Formulaton The POD uses prncpal component analyss to decompose a large DOF system nto a seres of fundamental modes and an approxmaton to the governng equatons s sought usng the expanson theorem: m u( x, t) = a ( t) ϕ ( x) (3.1) = 1 Soluton methods based on (3.1) requre the specfcaton of a famly of functons formng the modal bass Φ = { ϕ1, ϕ 2,..., ϕ m } that span the doman Ω. The bass functons usually satsfy homogeneous boundary condtons ndvdually and nhomogeneous boundary condtons are treated by addng a source functon: m 0 ( x, t) + a ( t) ( x) = 1 u r ( x, t) = u r r ϕ (3.2) Note n the most general case the source functon, u r 0, may account for tme dependent boundary condtons. In the context of flud flow, the soluton and modal bass wll be consdered vector-valued functons. In tradtonal POD analyss of turbulent flows, the 25

41 r r source functon n (3.2) often assumes the form of the ensemble mean, u0 = u, whch renders the POD modes akn to the Reynolds stresses. The POD s a stochastc tool that computes the optmal lnear bass for the modal decomposton n (3.2). Gven an ensemble of system observatons, r { u k H k = 1, 2,...,m}, Φ r can be computed by mnmzng the projecton error onto the ensemble, mn{ u r r k Pk u k } where P s the orthogonal projector, s the nduced k norm on the Hlbert space H and denotes the ensemble average. Ths s equvalent to maxmzng the energy (n the sense of the nduced norm) of the projecton of the observatons onto the bass functons [38, 59]. Standard varatonal calculus methods to r r 2 r 2 extremze the functonal ( u, ϕ) λ( ϕ 1), where the ( r 2 ϕ 1) term s ncluded to produce a normalzed bass and λ s a Lagrange multpler, show the POD bass vectors are the egenvectors of the vector-valued egenvalue problem: r r r R( x, x' ) ϕ( x' ) dx' = λkϕ( x' ) (3.3) Ω r r r where R( x, x' ) u( x) u * ( x' ) s the cross correlaton functon and * ( ) denotes the complex conjugate. The assocated egenvalue wth each emprcal bass vector r r 2 λ = (, Φ) s a measure of the average energy (n the sense of the nduced norm) of k u k the projecton of the ensemble onto Φ r because R r s a self-adjont lnear operator R : H H. r For dscrete data, R (x,x' ) s computed by takng m observatons of the system contanng n DOF and assemblng them nto the observaton matrx 26

42 r r r r U = u, u,..., } R { 1 2 u m n x m are then the egenvectors of the covarance matrx. The emprcal bass functons (referred to as POD modes ) r r T n x n ( 1 m) UU R, whch may be computatonally nfeasble for large DOF problems snce egenvalue algorthms can typcally only handle matrces on the order of 10 5, renderng the POD mpractcal for large data ensembles. Ths problem can be crcumvented by realzng r r r r r r span{ u1, u 2,..., u m } = span{ ϕ1, ϕ 2,..., ϕ m} and the POD bass can be expressed as a lnear combnaton of the lnearly ndependent observatons, known as the method of snapshots, see [60], vz: m r r ϕ ( x) = b u ( x) (3.4) r r r The weght coeffcents n (3.4) are egenvectors of the soluton to R ' ϕ = λϕ where r R r r T m x m ' = ( 1 m) U U R, allowng the number of observatons and not the number of system DOF to dctate the sze of the bass functon computaton. Usng the method of snapshots to assemble the bass functons as admxtures of the system observatons = 1 mples that Φ r ntrnscally contans any lnearly nvarant propertes of u r k from the fact that Φ r has been computed only through lnear operatons on U r. Thus, the POD modes ndvdually satsfy the ncompressblty condton, r ϕ = 0 k k and homogeneous boundary condtons, r ϕ( Ω) = 0 where u r ( Ω) = 0. For the ncompressble flow consdered here, H s the space of smooth, vector-valued, solenodal functons on Ω r r equpped wth the nner product ( u,v ) u vdx r r Ω u r r and (,v) u v for dscrete vectors. The egenvalues of the kernel n (3.4) are postve sem-defnte because the kernel s postve sem-defnte, however, egenvectors belongng to the null space λ = 0 are 27

43 gnored because they contan no nformaton about the system n the sense r r 2 2 λ = (, Φ) = 0. Ths leads to an ncomplete L ( Ω ) k u k space but poses no problem because the subspace only needs to descrbe the physcs contaned wthn the observaton set u r k and ncludng all admssble bass functons as would drectly oppose the reducedorder modelng framework. In the computaton of the POD modes, a numercal cut-off crteron on the order of the machne precson s used to elmnate egenvectors assocated wth λ < O( ), as these modes are laden wth numercal error [56]. From an mplementaton standpont, the POD mode may be computed by assemblng the observaton matrx U r = { u r,u r,..., u r } R m n x m 1 2 and then decomposng U r usng the sngular value decomposton (SVD). Gven a matrx A R n x m, the SVD produces the decomposton A T = UΣV. U n x n R s a matrx whose columns form the left sngular vectors of A, V m x m R s a matrx whose columns form the rght sngular vectors of A and Σ s a pseudo-dagonal matrx of the sngular values. It can be shown that A T A 2 T = VΣ V and that the egenvalues n (3.4) are equal to k Σ kk λ =. The method of snapshots can be mplemented as B SVD( U r T = U r ) wth B m x m R and then assemblng Φ r = U r B. The energy captured by each POD mode s then computed as E k λ k = m λ j = 1 j and the total energy resolved usng the frst p modes p λ k = 1 k 1 p = m. E j= 1 λ j The Galerkn Projecton The standard method of evaluatng the weght coeffcents n (3.2) s to project the governng equatons onto the modal subspace, known as the Galerkn method or Galerkn 28

44 projecton. Consder the 3-dmensonal ncompressble flow governed by the Naver- Stokes wthout external forcng and nhomogeneous boundary condtons over a porton of the doman boundary: u r = 0 (3.5a) r r 2 r 1 u u ν u + P = 0 (3.5b) ρ r r u (3.5c) Ω = u b where ρ s the flud densty and ν s the knematc vscosty. The Galerkn projecton of the modal bass onto the governng equatons s computed by takng the L 2 nner product between the two, recallng the ntegral defnton, r r r r u, v u vdx : Ω Ω r r r 2 r 1 ϕ ( u u u ) + P ) dx = 0 ρ (3.6) Ths procedure orthogonalzes the resdual to each bass functon of the modal subspace. Substtutng the modal expanson n (3.2) onto (3.6) yelds the m-dmensonal system of equatons (3.7a): C a a D a + A a a + S + B = 0 (3.7a) jk j k C D A S jk j jk j j Ω ν jk j r r r ϕ ( ϕ ϕ )dx Ω Ω Ω j k r r ϕ ϕ dx k k (3.7b) (3.7c) r r r r r ϕ (u ϕ + ϕ u )dx (3.7d) o k k o r r r r ϕ (u u ν u )dx (3.7e) o o o B 1 ρ Ω r ϕ Pdx (3.7f) 29

45 where summaton over repeated ndces s mpled. The C jk term results from the convectve operator ( u r u r ), Dj from the dffusve term ( u r 2 ν ), A jk s the cross term between the source functon and the POD modes, S comes the source term only and B s the projecton of the POD modes onto the pressure term, ( 1 P ). ρ The pressure term B can be ntegrated by parts to obtan B P r ϕ nˆ dx = Ω wth the ad of the dvergence theorem and the ncompressblty property of r ϕ. In (3.7b), the pressure term s a Lagrange multpler to satsfy the contnuty constrant (3.5a) and therefore should be unnecessary over the doman because r ϕ = 0 k k by constructon. The pressure on the boundary has physcal sgnfcance because t provdes the drvng force for the flow. The man obstacle for nhomogeneous boundary condtons s the treatment of the boundary pressure and specfcally couplng the pressure to the velocty a least squares manner as EMBED Equaton.3 where EMBED Equaton.3 r sons, B = 0 because ϕ nˆ Ω = 0 and for perodc boundary condtons, Ω P r ϕ nˆ dx = 0 f there s no mean pressure gradent, such that the boundary term n (3.7a) s elmnated. For flows wth nhomogeneous boundary condtons, some authors r [40-42, 49] homogenze the boundary condtons by a reference feld ( u r ) such that r (u k r - u r ) Ω = 0 before computng the POD modes to elmnate the need for boundary pressure-velocty couplng. As noted earler, u r r must satsfy (3.5a-c) and needs to be redefned for each new nhomogeneous boundary condton, whch could possble requre another full numercal computaton for each new set of boundary condtons. 30

46 The varatonal formulaton of the POD bass functons produces the Sobolev space r 1 ϕ H wth frst dervatves belongng to L 2 ( Ω ) and the use of homogeneous boundary condtons further restrcts the approxmaton to the Soboloev space of 1 homogenous functons r ϕ H 0. However, ths need not be the case n Galerkn methods, as the tau method was frst ntroduced by Lanczos n 1938 and presented n a more modern context by Gottleb and Orszag [61]. The tau method s a spectral method that does not requre the modes to satsfy the boundary condtons ndvdually, but the sum of the weghted modes together must satsfy the boundary condtons. As noted by Rempfer [55], the Galerkn system n (3.7a-e) could potentally be solved subject to constrants n an attempt satsfy the nhomogeneous boundary condtons although these constrants lmt the soluton space to smaller subspace of Φ r where only admssble combnatons of the POD modes satsfyng the nhomogeneous boundary condtons resde Analyss of the RANS Equatons In modelng and computaton of ndustral turbulent flows, the Reynolds-averaged Naver-Stokes equatons (RANS) are commonly used because the more detaled descrptons of large eddy smulaton and drect numercal smulaton (DNS) are computatonally mpractcal for complex geometres and boundary condtons. The RANS equatons model the effect of turbulence on the mean flow as a spatally dependent effectve vscosty and the steady momentum equaton n the absence of body forces s: r r u u ( ν eff r 1 u ) + P = 0 ρ (3.8) 31

47 The velocty, u r, appearng n (3.8) represents the ensemble-averaged velocty feld n the r r r Reynolds decomposton U = u + u where U r r s the total velocty and u tme varyng devaton from the ensemble mean. Equaton (3.5a) serves as the contnuty equaton n the RANS system for the ncompressble flows consdered here. There are many dfferent eddy vscosty closure schemes, wth the standard k-ε model and full Reynolds Stress model beng dscussed earler [30]. Examnaton of the turbulent knetc energy (k) and turbulent energy dsspaton rate (ε) transport equatons as well as the Reynolds stress transport equatons shows that these equatons model the effect of turbulence as a functon of local velocty gradents r only,.e. ν = f ( u ). Equaton (3.8) can be vewed as a lamnar flow model of a flud eff wth a stran-rate dependent vscosty. Ths modelng elmnates all of the small-scale turbulent dynamcs, makng t reasonable to expect that the requred number of observatons and modes to construct a low-dmensonal model s smlar to that of a lamnar flow rather than a hghly detaled set of DNS data, whch often ncorporate thousands of observatons. In the coupled phenomena of natural convecton, Srovch and Tarman [48] demonstrate that the velocty and temperature feld can be decomposed separately n leu r r of workng wth an extended state vector of the form y = { u, ν }. Assumng the a eff smlar decomposton for the velocty and eddy vscosty felds, u r = m a ϕ r and ν eff m = j b ψ j j, the Galerkn projecton of the POD modes onto the RANS momentum equaton yelds: 32

48 Ω r r r ϕ ( u u ( ν eff r 1 u ) + P ) dx = 0 ρ (3.9a) C a a D b a + A a a + S = 0 (3.9b) jk j k jk j k jk j k S Ω D jk r r ϕ ( u Ω o r r ϕ ( ψ ϕ )dx r u o j k r ( ψ u o o )) dx (3.9c) (3.9d) where the C jk, A jk, and B terms are the same as n (3.7a-f) and the boundary pressure term can be gnored at ths pont wthout loss of generalty. In the analyss of (3.5a-c), the governng equatons are second order spatally. As was prevously gnored, the POD modes, and therefore the resultng approxmatons, belong to the space 1 H. In lamnar flow approxmatons, ths poses no problems 1 because the smoothng acton of the Laplacan mples that solutons startng n H wll 2 rapdly enter H, [38]. Ths s a potental problem for the RANS equatons, as no such smoothng s guaranteed because the behavor of ν eff s unknown a pror. The dffusve term D jk could possbly be ntegrated by parts to acheve a term that contans only frst dervatves, however ths operaton wll then requre ν eff Ω to be specfed. Specfyng the correct boundary condtons for the turbulence transport equatons on nhomogeneous regons such as nflows and outflows can be dffcult for any flow computaton and must often be justfed for complex flows. Another large ssue n usng the Galerkn projecton for the RANS equatons appears when the turbulence transport equatons are used to couple the eddy vscosty modes to the momentum equaton. The standard k-ε model wll be used here, but the arguments equally apply to other turbulence models. Addtonal modes descrbng k and 33

49 ε would need to be generated from the flow database, ncreasng the computatonal cost. The fundamental problem can be seen n the defnton of the effectve vscosty: 2 k eff = ν + c (3.10) ε ν µ Assumng modal decompostons for k and ε of the form k = g ζ and = ε hη, substtutng these terms nto (3.10) would result n the term g ζ g jζ j( 1 hkη k ) j k But η k s a vector and ts nverse s undefned. Terms of the form ε/k also arse n the ε transport equaton. To avod specal treatment of the nverse terms that arse from the need to close the effectve vscosty weght coeffcents, refer back to comments regardng the nature of the RANS equatons as a model for the lamnar flow of a flud wth a stran-rate dependent vscosty Flux Matchng Procedure To mprove the computatonal effcency of POD analyss by avodng the Galerkn projecton and specal treatment of nhomogeneous boundary condtons, a flux matchng procedure s ntroduced. Snce the POD modes are themselves solutons to the governng equatons (3.5a-c and 3.8), new solutons can be generated by usng nhomogeneous modes to satsfy the boundary condtons, loosely based on the tau method. The structure of the POD modes lets them resolve the flow feld over the remander of the doman. Snce the POD bass s the optmal lnear subspace for a set of u r k, the approxmate analyss s treated as a lnearzed problem rather than usng the Galerkn projecton to construct a possbly unstable nonlnear model. The mplct 34

50 assumpton s the modal contrbuton can be unquely determned by satsfyng the boundary condtons. Ensurng that only the boundary condtons are satsfed and gnorng the resdual over the rest of the doman s a famlar concept. Usng nformaton from only a small subset of the entre doman to estmate the velocty feld s the bass for flow control because t s mpractcal to dstrbute sensors through the entre flow feld. One example used lnear stochastc estmaton to correlate pressure measurements at the wall and POD modal coeffcents n order to develop a low-dmensonal flow controller [51]. In the analyss of complex flows, the exact velocty profle on the boundary s often unknown. However, the desgn objectve s often based on ntegral condtons, such as the approprate mass flux through a porton of the boundary, promptng the ntroducton of the flux functon: r r F( u, β ) = ρβu nˆ dx, Γ Ω Γ (3.11) Ths functon generally returns a vector because Γ may be a fnte set of dscontnuous surfaces. Dependng on the transport problem at hand, the flux functon could descrbe the flow of any scalar, such as mass ( β = 1), energy ( β = E ), or speces concentraton ( β = c ), but mass flux s consdered here to demonstrate the methodology and s denoted F m. Note momentum s not consdered a vable term n the flux functon of the lnearzed analyss because s depends on the profle. Knowng the mass flux only s not suffcent to evaluate the momentum flux because 2 2 u u. In the absence of any knowledge about the specfc velocty profles on Γ, the mass flux s an deal choce to construct velocty feld approxmatons, as wll be detaled n further sectons. 35

51 To construct a new approxmate soluton, the fluxes can be expressed as the vector of goals q G R correspondng to the desred mass flux through the set of control surfaces = Γ, Γ,... Γ } whch defne the desred flow feld Γ { 1 2 q * r * u r such that G = F( u ). The soluton procedure s then to fnd the set of weght coeffcents that mnmzes the error on the set Γ: mn{ G' p r r a Fm( ϕ ) } where G' = G Fm( u ) (3.12) = 1 The modal summaton s carred to p m because the optmal approxmate soluton may requre less than the total number of modes avalable. The weght coeffcents can * generally be computed as a u r r ϕ, whch n the flux matchng procedure can be =, + r r* solved as a = F ( ϕ )G' ( u ), where m m F( r ϕ ) R x q s the matrx obtaned by operatng (3.11) on the q control surfaces of the m POD modes and () + s a sutably defned generalzed matrx nverse. The soluton procedure s carred out as a seres expanson wth the ordered POD modes formng the expanson sequence and terms are successvely added to the seres untl the boundary condtons are satsfed. Algorthmcally, ths can be expressed as: G = G F r ( u 1) m, m, 1 m (3.13a) = ( r + ϕ ) (3.13b) a Fm Gm, u r = u r + r 0 ϕ (3.13c) a j j= 1 j where F + T 1 T ( F F ) F s the Moore-Penrose matrx pseudo-nverse producng the least r squares approxmaton [62]. To ntate the calculaton, G = G F ) and the m, 1 m m ( u 0 36

52 frst modal weght coeffcent s computed as a1 = F r + m ϕ1) G, 1. The process s repeated ( m untl the desred set of mass fluxes s satsfed. The soluton process s akn to a perturbaton expanson where the source functon, u r 0 n (3.2), acts as the leadng order soluton and each modal contrbuton serves as the next order correcton. The dfference from tradtonal perturbaton methods s that the successve correctons occur n state space to satsfy the desred mass fluxes, wthout consderng the remander of the doman. Ths s a hghly desrable property as the exact velocty profle on the boundary of a complex flow may be unknown. Snce POD modes are solutons to the governng equatons (to wthn a multplcatve constant), they contan physcally correct velocty profles and satsfy the dvergence free condton. There s no need to account for the pressure term n (3.8) as t can be vewed as a Lagrange multpler to enforce the dvergence-free condton of (3.5a), whch s already satsfed by Φ r because r ϕ = 0 k k. Ths also mples that the approxmatons n (3.12) satsfes an overall mass balance on the doman because f r r ϕ = 0 k, then ϕ dx = k Ω k ϕr k nˆdx = 0 Ω k by the dvergence theorem. Ths formulaton s a sgnfcant mprovement over the work of Gallett et al. [53] who used POD to model flow over a block n a channel. The pressure drop through the channel vared wth Reynolds number and a lnear correcton of the form (, ) p r ϕ was ncorporated to correct a Galerkn system of the form n (3.7a-f). For parameter-dependent flows (as defned n 3.1), takng the system reference pont to be the ensemble mean may not produce the best results over a range of parameters because dfferent modes may become more or less mportant n k 37

53 approxmatng the flow feld as parameter values change. Usng (3.12) wth the ensemble mean as the source functon for the approxmaton would only produce accurate r results for solutons near the ensemble mean, G Fm ( u ), but the soluton method must be robust n that t constructs accurate solutons over the entre range of parameters present n POD subspace. To allevate the poor approxmatons for solutons beng far from the system reference pont n parameter dependent flows, Graham and Kevrekds [57] proposed takng averages over arc lengths n phase space and Chrstensen et al. [56] proposed preweghtng certan modes to ncrease ther contrbuton on the superposton. Followng [56], observatons could be repeatedly added nto the ensemble to shft the system reference pont n order to satsfy the nhomogeneous boundary condtons wth the ensemble mean actng as the source functon: * F ( u r ) G (3.14) m The mmedate problem s that weghtng s not unque as there are multple sets of w r r r that solve F( w1 u1 + w2u wm um ) = G and addtonal nformaton about whch modes to weght s requred. A more sgnfcant problem les n the fact that as the weghtng factor on any one member ncreases, the modal spectrum asymptotcally approaches λ = { 1, 0, 0,...}, collapsng the POD subspace to a pont near that sngle observaton. Hgher order modes are prematurely excluded n (3.12) because G' 0 makng the weght coeffcents n 0 for > 1, resultng n the loss of nformaton from all other modes. Approxmatons can be mproved n lght of (3.14), but all the features present n Φ r must be retaned and accessble to (3.12) n order to develop a robust soluton a 38

54 methodology. To accomplsh ths, decompose the POD subspace nto orthogonal complement subspaces: r Φ r r n x s = ϕ + ϕ', where ϕ R and ϕ' r r R n x m s (3.15) Ths decomposton s referred to as the PODc from here on. In a parametrc flow, dfferent modes should become more or less mportant under varous parameter values [57]. The source term should then be a functon of the mass flux goals, u r ( G ), whch can be accomplshed by choosng members of the ensemble as the source functon and constructng the POD subspace as an orthogonal complement. Ths method s superor to the standard mean-centered POD method (referred to as smply the POD from here on), where the source functon s taken to be the ensemble mean because solutons far from the mean tend to ncur larger errors and ths dstance from the mean has prevously been used as an error measure [56]. Also note, usng the mean-centered POD modes n the above flux matchng procedure to compute the modal weght coeffcents may not satsfy the boundary condtons to the requred accuracy for all requred approxmate solutons wthn the parameter wll be satsfed. The orthogonal complement r ϕ boundary condtons ( forcng modes) and r ϕ' s chosen to best satsfy the nhomogeneous represents the flow features over the rest 0 m of the POD doman ( responsve modes). Lettng u r denote the observatons used to construct r ϕ, the modal expanson and mnmzaton problem are modfed to: mn{ G' p = 1 m + * u r = u r a r ϕ (3.16) r r a F( ϕ ) } where G' = G F( u ) (3.17) 39

55 The observatons n u r are be selected by the algorthm: r* r mn{ F( u ) F( uk ) }, k = 1, 2,..., m 1 (3.18) * Ths method s based on determnng observatons that are geometrcally close to u r n r r the parameter space of F. Once the set u u s selected, the POD s performed on the r r r orthogonal complement mean-centered observatons u = u u. The remanng observatons r r u u are orthogonalzed to u r and the POD s performed wthout meancenterng. The full POD subspace s assembled as n (3.15) and reordered based on descendng magntudes of the combned egenvalue spectrum. Thus, the approxmaton procedure conssts of selectng the closest members of the ensemble to the desred soluton to serve as the source functon and the nformaton about the flow physcs contaned n the remanng observatons s converted nto an orthogonal seres expanson about the source functon to make hgher order correctons to the approxmate soluton. Essentally, the orthogonal complement subspace converts the POD bass nto a purely spatal functon to a parameterzaton of approxmate soluton, vz: r r ϕ( x) ϕ( x, G) (3.19) The sze of the orthogonal complement subspace (s) generally depends on the densty of observatons. If enough modes are present to accurately represent the model behavor, s = 1 wll usually be suffcent. Increasng s to 2 wll result n margnal approxmaton mprovements and s > 2 wll generally degrade the approxmaton. The purpose of the orthogonal complement s to shft the source functon n the POD subspace and usng s = 1 causes the source functon to assume the form of the nearest observaton. 40

56 3.5. Applcaton to Lamnar Flow The methodology s demonstrated on two complex flow geometres, one lamnar flow and one turbulent flow. The lamnar flow stuaton s used because the proposed flux matchng procedure can be drectly evaluated aganst the Galerkn method. The turbulent flow stuaton s used to llustrate how the flux matchng procedure can be used to develop low-dmensonal models when there are modelng uncertantes, or even n stuatons where the governng equatons are unknown. At ths pont, t s reterated that just as wth the standard POD, the method s applcable to both expermental and numercal data. Numercally generated data are used here for convenence and the numercal solutons are consdered as exact solutons, see [63] for a dscusson on the effect nherent errors assocated numercal and expermental data collecton has on POD analyss. Currently, the POD modelng methodology conssts of 2 components; the POD mode calculaton and the determnaton of weght coeffcents, whch are generally coupled through the boundary condtons. In most prevous POD analyses, the components are decoupled through homogenous or perodc boundary condtons. The proposed reduced-order modelng methodology presented here sgnfcantly mproves both of these components and develops a robust framework that satsfes parametrc boundary condtons by couplng the two soluton components. The frst enhancement to the POD procedure s usng orthogonal complement subspaces to satsfy the boundary condtons, denoted the PODc. The second development concerns the computaton of the weght coeffcents n (3.2). The flux matchng procedure (FMP) was developed on physcal arguments and ts superor computatonal effcency to the Galerkn projecton 41

57 for steady flows wll be demonstrated. Accordngly, there exst the followng 4 combnatons of possble soluton procedures: the standard mean-centered POD wth Gakerln projecton (denoted POD-GP), the orthogonal complement POD wth Galerkn projecton (PODc-GP), the POD wth the flux matchng procedure (POD-FMP) and combnng the orthogonal complement POD wth the flux matchng procedure (PODc- FMP) Lamnar Flow Model Problem Consder the flow n a manfold wth a sngle nlet that dstrbutes ar at standard densty to 5 outlet ports. The doman measures 5L x 9L x 3L wth L = 0.05 m. The nlet measures 3L x 3L and the pressure constrant P = 0 was appled. The outlet ports measured L x L and were able to vary mass flow rates from 3.06 x 10-5 to 1.68 x 10-3 kg/s for a correspondng Reynolds number range of ( Re = m & / µ L ) 34 to These mass flow rates are based on an area-averaged velocty between 0.01 and 0.55 m/s. The fnal numercal model conssted of 135,000 grd cells and the 3 velocty components plus pressure solved for at each grd cell gves a total of 540,000 DOF. The mesh was shown to be convergent to less than 2% n terms of each velocty component and was demonstrated to be ndependent of teraton convergence crtera. The governng equatons were solved usng second order upwndng, the SIMPLEC pressure-velocty couplng algorthm and the PRESTO pressure nterpolaton scheme for rectlnear staggered grds [64]. Fgure 3.3 schematcally depcts the model geometry and defnes the m = 10 randomly generated observatons used to generate the POD modes. The observatons are lsted n terms of mass flux through the c outlet control surfaces accordng to (3.11) 42

58 because the objectve of the flux matchng procedure s to satsfy the net mass flux and let the structure of the POD modes determne the correct velocty profles requred by contnuty. Fgure 3.3. Lamnar flow model geometry and observaton database The number of system observatons was arbtrarly chosen to be 10 as no rgorous theores exst for the mnmum number of observatons for fxed parameter flows or the dstrbuton of observatons for parameter-dependent flows needed a pror to construct the optmal data ensemble. Ths s an nherent lmt n POD analyss and ths nvestgaton does not am to extend those lmts, rather t provdes a computatonally effcent methodology to treat a predefned range of nhomogeneous boundary condtons. To valdate that enough system observatons were made to construct a meanngful POD subspace, an addtonal 5 randomly generated observatons were added to the data and the egenvalue spectrum was recomputed. The frst 9 egenvalues of the mean-centered spectrum computed from the addtonal 5 observatons dd not change sgnfcantly, suggestng that the orgnal 10 observatons were adequate to construct the POD subspace. 43

59 The ensemble mean and the frst 3 mode shapes generated by the basc meancentered POD procedure are llustrated on the vertcal md-plane of the doman (z = m) n Fgure 3.4. The mode shapes demonstrate the decreasng energy of each POD mode and how lower order modes resolve domnant flow features whle the hgher order modes represent more localzed effects. Also note that operatng the mass flux functon defned n (3.11) on the ensemble average produces a nearly unform mass flow rate for each outlet port of 8.48 x 10-4 kg/s for a correspondng area-averaged velocty of 0.28 m/s. Thus, F m ( u r ) produces a system reference pont of unform mass flow rates equal to the average value the parameter range spans, further ndcatng that a suffcent number of system observatons have been made Lamnar Flow Results r A test case of G( u * ) = { , , , , } x 10 4 kg/s was randomly chosen and many other test cases were analyzed to ensure that the presented results are representatve. Fgure 3.5a-b plots the egenvalue spectrum for both the POD and PODc procedure and the error assocated wth the satsfyng the boundary condtons descrbed by G and L 2 norm of the error of the approxmate velocty feld. The error n r* matchng the boundary condtons s defned relatvely as G = F( u ) G 2 / G 2 r* r r and lkewse for the error over the doman, u = u u 2 / u 2. err exact err exact 44

60 Fgure 3.4. Lamnar flow ensemble mean and the frst three POD mode shapes, (n. b. the mode shapes have been nterpolated to a grd 3 tmes less dense then the computatonal grd for llustratve purposes) Fgure 3.5a shows that the PODc procedure produces a steeper spectrum, ndcatng that fewer PODc modes are requred to obtan a smlar order of approxmaton accuracy as the POD modes. The PODc spectrum was computed usng s = 1 and s = 2 observatons to construct r ϕ, whch were found to be observatons k = 9 and k = 2, through the ad of (3.18). Fgure 3.5. Approxmate soluton results, (a) modal egenvalue spectra, (b) boundary condton error 45

61 Note the mean-centerng procedure reduces the rank of the covarance matrx by one, therefore only m-1 modes have nonzero λ and are avalable for the reconstructon. In the PODc procedure, the mean-centerng reduces the number of modes n r ϕ by one, therefore, s must be greater than or equal to 2 produce forcng and responsve modes. Usng s = 1 n the PODc causes the system reference pont (source functon) to correspond to a sngle observaton and n the lmt s m, the mean-centered POD procedure s recovered. Usng s = 1 produces slghtly more accurate approxmatons than s = 2 and ncreasng s > 2 does not sgnfcantly mprove the approxmaton and ncreases the computaton tme, thus s = 1 s the best dmenson for r ϕ. The POD generates fxed modes ndependent of G. Fgure 3.5a demonstrates there may exst a G that cannot be satsfactorly usng the mean-centered POD. Ths s remeded by the PODc procedure, whch essentally extends F( r ϕ ) to F( r ϕ,g ). For all test cases consdered,, the boundary condtons can be satsfed to O(10-2 ). The soluton method for the Galerkn system of equatons deserves some comments. In prevous POD flud flow analyss, the low-dmensonal models were created to nvestgate flow dynamcs or to develop a control scheme. The resultng Galerkn system s a m-dmensonal system of ODEs n tme for the evoluton of the modal coeffcents, whch are then numercally ntegrated. A smlar approach could be taken for parametrc steady flows where (3.7a-e) are ntegrated untl a steady soluton s obtaned. The boundary pressure term n (3.7f) s set to zero because of the specalzed treatment of nhomogeneous boundary condtons presented here. In the scope of reduced-order modelng and fast approxmatons, (3.7a-e) are solved drectly because the 46

62 numercal ntegraton would be computatonally neffcent. The soluton s posed as the lnearly-constraned mnmzaton problem: mn { C a a D a + A a a + S } subject to Fa G' = 0 a jk j k j j jk j k (3.20) whch s solved usng standard sequental quadratc programmng methods. The mnmzaton problem s treated as a sngle objectve optmzaton based on a. The mnmum s defned as the locaton n the m-dmensonal vector space of weght coeffcents that when substtuted nto (3.7a) returns the mnmum of a. The soluton to + a = F( r ϕ ) G' s used as the ntal guess, and the convergence to a mnmum from that pont was verfed by usng other ntal guesses. The termnaton crtera were also vared to ensure the soluton was ndependent of the convergence tolerance. Fgure 3.6a plots u err for the POD-GP, PODc-GP and PODc-FMP soluton procedures whle the results from the POD-FMP procedure are not shown because the boundary condtons are never satsfed. The results show usng the Galerkn projecton wth the standard mean-centered POD produces large errors over the doman. The L 2 error norm (u err ) error s greatly reduced wth the PODc modes and the flux matchng procedure results n a decayng error whle the Galerkn projecton produces an error that ncreases wth the addton of modes. Fgure 3.6b lsts the modal weght coeffcents and t can be seen that the Galerkn projecton and flux matchng procedures produces a smlar set of coeffcents usng the POD modes. 47

63 Fgure 3.6. Lamnar flow approxmate soluton, (a) L 2 error norm and (b) modal coeffcents Each result presented n Fgure 3.6a for the Galerkn soluton to the weght coeffcent problem s based on a dfferent system because C jk, D j, A jk and S change wth m. Equaton (3.20) cannot be solved for the full m modes and then only use the frst p < m to approxmate the soluton because the boundary condtons are only satsfed usng the full number of modes from whch the Galerkn system was computed. Ths may mpose sgnfcant computatonal effort when decdng the number of modes to retan n the reduced-order model, as the numercal evaluaton of C jk, D j, A jk and S s tme consumng and needs to be performed for each p. The mode shapes n the PODc procedure dffer from those n the POD, but the Galerkn projecton and flux matchng soluton procedure produces smlar modal weght coeffcents for a gven set of modes. These weght coeffcents are lsted n Fgure 3.6b. The soluton to the Galerkn system requres p q because there s no feasble soluton to (3.20) f there are more constrants than the dmenson of system. 48

64 Fgure 3.7a llustrates contours of the absolute error supermposed on the vertcal md-plane of the approxmate solutons for varous numbers of modes computed from the PODc-FMP. The absolute error s plotted because the relatve error becomes unrepresentatvely large where the true velocty feld goes to zero. The results show that the reduced-order model captures the domnant flow physcs very well and the largest errors are ncurred n regons of large velocty gradents near the outflow ports of the doman. Fgure 3.7b plots sosurfaces of the absolute velocty error over the entre doman. Fgure 3.7. Error reducton wth ncreasng system dmenson (a) on the vertcal mdplane and (b) over the entre doman 49

65 3.6. Applcaton to Turbulent Flow As a turbulent flow example, consder a smlar 3-dmensonal turbulent flow of ar at standard densty and temperature n a sngle nlet manfold wth fve equal szed outlet ports. The outlet ports are specfed to have mass fluxes rangng from kg/s, for correspondng hydraulc dameter based Reynolds numbers rangng between 10,269 and 27,383. The doman s 6L x 9L x 3L wth the outlet ports measurng L x L and the nlet measurng 3L x 3L wth L = 0.05 m. The full numercal model contaned 162,000 grd cells wth 10 DOF at each grd cell (3 velocty components, pressure and 6 Reynolds stress components) for a total 1,620,000 flow DOF. The mesh was shown to be convergent to less than 1% n terms of each velocty component and was demonstrated to be ndependent of teraton convergence crtera Turbulent Flow Model Problem Fgure 3.8 schematcally depcts the model geometry and defnes the m = 10 observatons used to generate the POD modes. The frst two observatons were chosen to have the unform mnmum and maxmum velocty of the parameter range and the remanng observatons were generated randomly based on the correspondng areaaveraged velocty rounded to the nearest nteger. 50

66 Fgure 3.8. Turbulent flow model geometry and observaton database Turbulent Flow Results A number of test cases were randomly chosen and the results shown here are for the set 2 G( u*) r = { 1. 74, 1. 08, 1. 16, 1. 64, 1. 58} 10 kg/s, whch s typcal of other test cases. Fgure 3.9a plots the egenvalue spectra of the standard POD and PODc procedures and Fgure 3.9b plots the boundary mass flux error. Fgure 3.10 llustrates the PODc mode shapes on vertcal md plane (z = m). The u r subspace s assembled usng s = 2 observatons whch were determned to be k = 3 and k = 4 from (3.18). Increasng s > 2 does not ncrease the accuracy of the approxmate solutons and slghtly ncreases the computatonal tme, and agan s =1 s optmal. 51

67 Fgure 3.9. Approxmate soluton results, (a) modal egenvalue spectra, (b) boundary condton error The PODc produces two modes wth relatvely large egenvalues, correspondng to the domnant drvng and responsve modes whle the remander of the PODc spectrum exhbts a sharper decay relatve to the POD spectrum, ndcatng an mprovement n accuracy for approxmatons employng the same number of POD modes. Fgure 3.10 llustrates that the domnant forcng mode contans sharper gradents and more secondary effects, but better reproduces the boundary mass fluxes. Snce the mode shape s a soluton to (3.8) up to a multplcatve constant, the resolved secondary effects are a drect cause of the boundary condtons. 52

68 Fg Turbulent flow system reference pont and the frst three POD mode shapes from the PODc The error norm results n Fgure 3.11a show that the standard POD-GP method produces poor results wth ncreasng error as addtonal modes are added. The data show the FMP generally produces better approxmatons than the Galerkn projecton and the orthogonal complement subspace PODc s a sgnfcant mprovement over the meancentered POD for treatng parametrc boundary condtons. The flux matchng procedure reduces the L 2 error norm proportonally wth the goal resdual, ndcatng convergence. The PODc-FM outperforms the POD-FMP due to the segregaton of the POD subspace nto drvng and responsve modes. The data n Fgure 3.11a ndcate the frst m q modes wll not reduce the error over the doman because the boundary condtons are not satsfed, but for m > q, the error s rapdly reduced, although no guarantee exsts that the error wll tend to zero as more modes are added to ensemble and used n the approxmate soluton. 53

69 Fgure Turbulent flow approxmate soluton results (a) L 2 error and (b) Galerkn projecton lmtng cases for constant vscosty assumpton. Recall the RANS equatons appear as a model for the lamnar flow of a stran rate dependent vscosty flud and that the results n Fgure 3.11a demonstrate that usng a constant vscosty n the RANS Galerkn projecton produces a reasonably accurate approxmaton consderng the level of smplfcaton nvolved. Thus, t would be natural to consder some lmtng cases of the Galerkn projecton of the RANS equatons, (3.9ac). Snce the flow s boundary drven and the source functon S s used to satsfy the boundary condtons, one could argue that settng C jk = D jk = A jk = 0 would result n a good approxmaton to the full PODc-GP soluton. Other lmtng cases would be droppng only the dffusve term (D jk ) snce t s domnated by the convectve term n hgh Re turbulent flows and to drop the convectve and dffusve terms such that only the source term and the cross term (A jk ) reman. The results presented n Fgure 3.11b show that ndeed the source functon domnates the soluton to Galerkn system, further justfyng the use of the flux matchng procedure, where only the boundary condtons are satsfed, over solvng the full Galerkn system. 54

70 the exact To further llustrate the error feld of the approxmate soluton, Fgure 3.12a plots * u r soluton and approxmate solutons usng ncreasng number of modes wth supermposed u e contours on the vertcal md-plane. Fgure 3.12b plots sosurfaces of u e over the entre doman. Fgures 3.12a-b demonstrate that approxmaton errors occur n regons of large velocty gradents because the POD procedure orders the modes n descendng energy and the small scale flow features are lost n the fnte truncaton of the modal expanson. Determnng the number of observatons to generate the POD modes s a major concern of all POD analyses. An addtonal 5 randomly generated observatons were added to ensemble and u e showed approxmately the same rate of convergence as n Fgure 3.11 for p > 5 modes, mplyng the flux matchng procedure converges as p. In dynamc smulatons each tme step serves as an observaton, however n steady parameter dependent flows, each observaton s ndependent and may take sgnfcantly more tme to generate. Thus, t s desrable to compute as few observatons as necessary and t s noted that the 5 addtonal observatons do not sgnfcantly change the egenvalue spectrum, suggestng an adequate representaton of the POD subspace. 55

71 Fgure Error reducton wth ncreasng system dmenson (a) on the vertcal mdplane and (b) over the entre doman 3.7. Applcaton to Ar-Cooled Electroncs Rack To llustrate the PODc-FMP methodology for a representatve problem encountered n systems-level electroncs coolng, consder a 2-dmensonal representaton of an ar-cooled data processng cabnet contanng 10 servers. Cold supply ar s delvered to the rack through a 0.39 m cutout n the bottom and s drawn nto each ndvdual server to mantan a safe operatng envronment for the data processng equpment. Each server contans an nduced draft fan model to produce the necessary flow and lumped resstance at the nlet to account for the pressure drop across the server. Two 0.30 m tall by 0.50 m long blocks n each server are gven a constant heat flux to mmc the power dsspaton of hgh performance central processng unts (CPUs). The rack dmensons are based on commercally avalable unts and all walls are modeled as 56

72 adabatc. Fgure 3.13 below descrbes the rack and server geometry and schematcally depcts the arflow patterns. In the sngle parameter case, the server fan model s fxed to produce a nomnal kg/s (310 CFM) flow rate and each CPU dsspates 20 W for a total rack power of 400 W. The nlet velocty (V n ) s vared between 0.0 and 2.0 m/s and mantaned at a constant 288 K. For the mult-parameter case, the nlet velocty s fxed at 0.5 m/s and the followng 3 dfferent types of servers are used: a low power wth nomnal 0.08 kg/s flow rate and 20 W per CPU, a medum power wth nomnal 0.13 kg/s flow rate wth 30 W per CPU and a hgh power wth 0.17 kg/s flow rate and 40 W dsspated per CPU. The rack s assumed to contan 3 low-powered, 4 medum-powered and 3 hgh-powered servers to lmt the desgn space. Both of these cases llustrate real world type desgn problems. The sngle parameter objectve s to fnd the optmal nlet flow rate to ensure relable operaton of all the servers n the rack. The mult-parameter case s an example of fndng the thermally optmal arrangement of servers to mnmze hot exhaust ar from servers pollutng the nlet supply of others [29]. Fgure (a) Rack and (b) server geometry and arflow patterns 57

73 The full CFD model neglects buoyancy effects and solves the steady ncompressble RANS momentum and energy equatons wth no body forces, usng second order upwndng and SIMPLEC pressure-velocty couplng wth PRESTO pressure nterpolaton [64]. The fnal converged model contaned grd cells for a total of DOF. Note that the sum of the server flow rates s always greater than the rack net flow rate for all cases consdered n the sngle and mult-parameter studes. Ths flow rate mbalance requres a large degree of recrculaton, or a server s hot exhaust beng drawn through another server before extng the rack, and a correspondngly complex flow feld Sngle Parameter RANS POD The technques to create reduced-order models of RANS based CFD/HT computatons for facltatng desgn studes such as optmzaton are presented frst for the sngle parameter flow case n order to clarfy the development of the methodology. The results wll then be extended to the mult-parameter case, whch represents a pressure-drven flow. The observatons for the sngle parameter case where created by varyng the nlet velocty between 0.0 and 2.0 m/s on 0.25 m/s ncrements, V obs = { 0. 0, 0. 25, 0. 50, 0. 75, 10., 125., 150., 175., 2. 0 } m/s, for a total of 9 observatons,. Based on nlet hydraulc dameter and the mnmum nonzero and maxmum values of V o, the Reynolds number of the observatons ranges from 8,829 to 70,629. The normalzed egenvalue spectrum (λ) correspondng to the mean-centered POD modes s shown n Fgure 3.14 below. Note velocty magntude contours are shown to better llustrate the overall flow patterns. 58

74 Fgure V n egenvalue spectrum and POD modes The rapd decay of λ ndcates that the frst 2 POD modes capture the domnant modes of the system and the frst 4 modes are able to reconstruct any observaton wth less than L 2 error. Note that λ 0 wth ncreasng mode number, ndcatng that the hgher order POD modes do not contrbute to the mechancs of the system and ther computaton may be laden wth numercal error [56]. A numercal cutoff crteron for the mnmum value of λ that produces meanngful modes s defned. 59

75 The results are demonstrated for 4 test cases. To approxmate the soluton for the test case not n the range of the observatons ( V t n = m/s), the largest two values of o V n were used to construct r φ. Fgure 3.15 below compares the PODc egenvalue spectrum wth the orgnal and shows the resultng u err error measure. Fgure Sngle parameter spectrum and L 2 approxmaton error The data show the error decreases monotoncally as more modes are added and at about p = 5 modes, u err has converged. For each test case, the desred nlet mass flux s matched accurately, G e = O(10 2 ) p. The RMS error n velocty magntude s less o than 0.05 for all cases n the range of V n, and only slghtly larger for the case outsde the range of o V n. The results also ndcate that a majorty of the error s ncurred near the nlet and exhaust of the rack, whle the flow nsde the ndvdual servers s accurately approxmated. It s these regons of the flow feld that have the most practcal sgnfcance n the case of data processng cabnet arflow management Mult-Parameter RANS POD Recall for the mult-parameter case, the nlet velocty s fxed at 0.5 m/s and 3 dfferent server flow rates (0.08, 0.13 and 0.17 kg/s) are specfed. The rack s assumed 60

76 to contan 3 low, 4 medum and 3 hgh flow rate servers to lmt the desgn space. Varous arrangements of the servers were solved wth the full CFD/HT model for a total of m = 21 observatons and addtonal confguratons were solved to serve as test cases. Fgure 3.16 shows the mean-centered POD egenvalue spectrum and the frst 3 POD mode shapes. Fgure Mult-parameter POD modes 61

77 Each ndvdual server s flow rate s unknown as t wll devate from the nomnal value dependng on ts poston due to the pressure feld nsde the rack. Thus, each observaton s characterzed by a vector contanng 10 values of ether 1, 2 or 3 for low, medum and hgh flow rate servers. Gven an nput vector, the mass of flow rates for a new approxmate soluton can be computed by takng a weghted average of the two nearest observatons, as determned by (3.18). Fgure 3.17 plots the error measures for the ones of the test cases for varous orthogonal complement subspace dmenson, s. The data n Fgure 3.17 show that usng more observatons to construct r φ wll reduce the error for small p, but the errors become the equal as p m 1 ndependent of s. The flux matchng procedure s able to satsfy the ndvdual server mass flux goals wth a relatve error on the order of O (10 2 ). Fgure Mult-parameter PODc error A fnal pont to be made s that snce the POD modes are themselves solutons to the governng equatons, they predct the correct velocty profle nto each ndvdual server. Fgure 3.18 plots the exact and approxmate solutons bear the entrance to servers 7 and 8, where strong changes n mean stran rate result from recrculaton effects. Ths s the prmary reason why an ntegral term (mass flux) s used as the matchng condton. 62

78 If a specfc profle was to be matched, the desred profle mght not correspond to an actual soluton. By matchng the flux, the correct mass flow rate s acheved and the detaled structure contaned wthn the POD modes produce the local velocty feld that corresponds to a physcally realzable soluton. Ths also mples that ths method wll produce the closest physcal soluton n the event that an unphyscal goal s specfed. Fgure Local PODc and exact solutons near the entrance to servers 7 and 8 Fgure 3.19 shows the exhaust regon of servers 6 and 7 whch s the regon where the maxmum error of all the test cases occurred. The PODc procedure s not able to completely resolve the sharp gradents that occur n secondary flows. But snce the objectve was to retan the domnant flow patterns for desgn type analyses, ths tradeoff s more than acceptable. 63

79 Fgure Local PODc approxmatons n regon of maxmum velocty error; exhaust of servers 6 and Optmzaton The reduced-order models produced by the PODc-FMP procedure allow desgners of thermal systems to quckly assess canddate desgns and perform optmzaton studes wthout cumbersome full-scale numercal models. To demonstrate ths utlty, robust desgn prncples were used n conjuncton wth the reduced-order modelng methodology to determne the optmal nlet flow rate to the rack usng the model and boundary condtons of Robust desgn ams to not only mnmze the value of the optmzaton functon but also to mnmze the curvature of the functon such that small changes n the both controllable and uncontrollable desgn varables do not cause sgnfcant devatons away from the optmzaton pont and result n system operaton outsde the feasble desgn space. Further detals are provded n [65, 66] especally those concernng the applcaton of robust optmzaton to desgn of electroncs thermal management. Usng robust desgn prncples, the ar-cooled rack was optmzed to dsspate 50% more power (2400 kw ) and reduce the chp temperature varablty 20% to 60% 64

80 dependng on whether an optmal or robust confguraton s employed [65, 66]. These robust desgn prncples are accessble wth reduced-order models where full-scale CFD/HT may be too tme consumng to perform the analyss. The reduced-order flow model for the ar-cooled rack and temperature solver have also be used n other optmzaton studes, ncludng genetc algorthm based optmzaton routnes [67]. The reduced-order model averaged 13 seconds per functon executon. Fgure 3.20 detals the program flow and tme requred for each step. A majorty of the tme s spent assemblng the lnear system for the energy equaton. Fgure Reduced-order model executon and tme requrements Valdaton was performed by solvng the full CFD model, whch yelded chp temperatures wthn an average of 5 o C of the reduced-order computed soluton. On a hgher level of valdaton, the power dstrbuton of the servers found to be most effcent yelds an approxmate hyperbolc tangent, demonstrated to be a hghly effcent confguraton n [29]. Even f very precse optmzaton ponts are requred, the reducedorder model can be used wth desgn optmzaton prncples to compute a very good ntal guess and then the full-scale model can be used to refne the optmzaton. 65

81 4. REDUCED-ORDER MODELING OF FORCED TURBULENT CONVECTION The POD procedure developed n 3.4 provdes a reasonably accurate and hghly effcent method for constructng full feld approxmaton of parametrc turbulent flows and overcomes the dffcultes assocated wth takng the Galerkn projecton of the RANS equatons wthout detaled knowledge of the effectve vscosty. As seen n Fgure 3.20, solvng the temperature feld gven the reduced-order flow approxmaton creates a bottleneck n the rapd soluton methodology. To overcome ths, the flux matchng procedure s extended to ncorporate the RANS energy equaton and specal treatment s ntroduce to ensure the proper couplng of the velocty and temperature feld n forced convecton. The methodology s developed through an example of a prototypcal forced convecton stuaton n electroncs thermal management Model Parameters The methodology s llustrated for a RANS smulaton of two-dmensonal duct flow of ar over two alumnum heated blocks n tandem (see Fgure 4.1). The geometry s dentcal to the expermental measurements of Yoo et al. [68], whose data were used to valdate the turbulence modelng n the CFD/HT code. The steady, ncompressble, constant propertes RANS contnuty, momentum and energy equatons wthout external forcng or buoyancy effects used to model the flow and heat transfer are: u r = 0 (4.1a) r r r 1 u u ( ν eff u) + P = 0 (4.1b) ρ r ρ c u T ( k T ) = 0 (4.1c) p eff 66

82 where 2 k eff = ν + C and ε ν µ k eff c pν t = k + wth Pr t = 0.85 and can be computed ρ Pr t through any RANS-based turbulence model. The standard k-ε model wth nonequlbrum wall functons [69] was used to model the effect of turbulence on the mean flow and the nlet velocty, turbulent knetc energy and turbulent dsspaton rate boundary profles were calculated assumng a fully developed flow usng Prandtl s 1 / 7 power law. The fully converged CFD/HT model conssted of 10,000 grd cells and solvng for two velocty components, pressure, k and ε at each grd cell resulted n approxmately 50,000 DOF to model the flow. The soluton was demonstrated to be ndependent of grd sze and convergence crtera, and each soluton requred approxmately 500 teratons to converge. Fgure 4.1. Model geometry from Yoo et al. [68] The flow parameter range of the model was chosen to be 13,690 Re 41,070, for Re = Hu/ν whch corresponds to an average velocty of 5.0 u 15.0 m/s n ar, and the block power was assumed to range from 25 to 200 W. The nlet temperature was fxed at 288 K and all flud propertes were evaluated at ths temperature. All results wll be reported as the temperature rse above ths nomnal value. Table 1 summarzes the set of observatons used to construct the reduced-order model subspace. 67

83 Table 1. Turbulent convecton observaton database The heat nput to each block was appled as a unform heat flux on the bottom surface (y = 0). The local Nusselt number ( Nu = hb/k ), usng a runnng coordnate over the surface of blocks, s plotted aganst the expermental data of Yoo et al. [68] n Fgure 4.2. The numercal smulaton agrees farly well wth the expermental data wth some error n magntude over the surface of the frst block. Chen et al. [70] have expermentally nvestgated a smlar geometry for smlar Reynolds numbers and have suggested the standard low Reynolds number turbulence model of Jones and Launder [71] provdes accurate local heat transfer coeffcent predctons. The code s based on wall functons and the pressure-gradent senstve wall functons employed provde the most accurate results wthout sgnfcant code modfcatons. It should also be noted that the mesh employed here s of smlar sze to that of Chen et al., even though the wall functons are used n ths nvestgaton whle the Jones-Launder low Reynolds number model reles on a dampng functon to model the near wall effects. 68

84 Fgure 4.2. Comparson of numercal soluton and expermental measurements at Re = 13,690, data from Yoo et al. [68] The basc motvaton of low-dmensonal modelng s to create a more computatonally effcent way of reproducng the physcs descrbed by a hgh fdelty numercal smulaton or detaled expermental dataset. It s mportant to note that both numercal and expermental observatons wll depart somewhat from the true system behavor. The purpose of ths nvestgaton s to present a reduced-order modelng framework for turbulent forced convecton, not to model a partcular system. Thus, the numercal data wll be treated as the exact system response n the descrpton of the methodology below, and some dscusson of varous errors and ther contrbuton to total error wll be subsequently provded. 69

85 4.2. Low-Dmensonal Turbulent Flow Modelng The orthogonal complement POD (PODc) wth flux matchng wll frst be demonstrated to approxmate the velocty feld for a randomly selected test case * correspondng to Re = 36,320 ( u = m/s n ar). The set of mass flux control surfaces reduces to a sngle surface concdent wth the doman nlet (conversely, the doman outlet could be used to produce the same results by contnuty). Ths smple flow has only a sngle parameter to be used as a matchng condton, ndcatng the 2-term expanson u r = u r + a r ϕ s all that s avalable for the soluton approxmaton. It has been demonstrated that the cumulatve energy resolved by the frst k modes produces an error bound on the approxmaton [72]. Fgure 4.3 shows E , ndcatng the 2-term approxmaton should produce errors on the order of 2% n the sense of the L 2 -norm. Equaton (3.18) selects the observaton k = 9 n Table 2, correspondng to u = 13.0 m/s, as the source functon. Fgure 4.3. Mean-centered velocty POD and orthogonal complement POD (PODc) modal energy content 70

86 Fgure 4.4a llustrates the decayng error n the velocty magntude for the 1- and 2-term approxmate solutons. Fgure 4.4b plots the exact soluton and the approxmate velocty feld n the vcnty of the leadng edge of the frst block, whch s where the maxmum error occurs. Both Fgures 4.4a and 4.4b show that approxmate soluton s very accurate, especally consderng the full CFD model requres 40,000 DOF to solve the flow and the reduced-order model contans only 2 DOF. The 2-term approxmaton captures the nlet velocty profle exactly, produces a maxmum absolute pont-wse error of m/s and an L 2 error norm over the doman of Even though the observaton space s dense wth varous mass flow rates, weghted averagng and scalng observatons wll generally result n poor approxmatons. For example, rescalng the observaton wth the dfference n mass flow rate from the nearest observaton (u = 13.0 m/s) produces errors 3 tmes as large as the PODc based approxmaton. Fgure 4.4. (a) Velocty absolute approxmaton error [m/s] and (b) detaled local velocty felds 71

87 At ths pont, t s reterated that ths reduced-order modelng procedure has been demonstrated on sgnfcantly more complex flows, comprsed of multple control surfaces, wth very successful results [73] and the objectve of the present study s to couple the energy equaton nto the methodology to extend the low-dmensonal modelng framework to convectve flows. As the system grows n complexty and more parameters are ncorporated, more matchng condtons are generated, whch requres more modes to be retaned n the approxmaton. Thus, the level of approxmaton keeps pace wth growng system complexty Low-Dmensonal Turbulent Convecton Modelng An effcent and accurate reduced-order modelng methodology for turbulent flows has been demonstrated n the prevous secton, and the objectve s now to extend the procedure to nclude a low-dmensonal soluton to the energy equaton. The orthogonal complement POD and flux matchng procedure from the prevous secton wll be employed because of ther smplcty, and the man challenge wll be n couplng the temperature and velocty felds. To begn, ndependent velocty and temperature decompostons wll be assumed: r r u = u 0 + aϕ and T = T0 + r j b ψ j j (4.2) The temperature POD modes are computed wth the same procedure, gven the temperature observaton matrx T obs n x m = {T1,T2,...,Tm } R. A natural way to couple the velocty and temperature felds s the Galerkn projecton, but as wth the ν eff term n RANS momentum equaton, the RANS energy equaton would requre k eff to be specfed. Thus, substtutng the approxmate velocty 72

88 feld * u r nto (4.1c) and projectng onto the subspace spanned by Ψ {ψ1,ψ 2,...,ψ m } = s consdered neffectve here. The flux matchng procedure (FMP) wll be extended to nclude the energy equaton, accordngly the heat flux functon can be defned analogous to (3.11) as: F h dt ( T ) = k dx (4.3) Γ h dn ˆ The heat flux control surfaces correspond to the 3 surfaces of each block exposed to the arflow. Alternatvely, the bottom surface of each block where the heat flux s appled could also be used as the control surface because the system s steady. POD modes are solutons to the governng equatons (4.1a-c) and nhomogeneous modes can be vewed as a soluton wth arbtrary boundary condtons. The flux functons (3.11 and 4.3) defne an nverse problem of fndng the correspondng boundary condtons. When the flux functon nvolves a gradent, approxmaton wth dscrete data can produce large errors, especally f the gradent s sharp relatve to the measurement pont spacng. Ths can be especally dffcult f the observatons were generated through CFD/HT data, where wall functons were used to allevate near-wall grd resoluton requrements when ntegratng the turbulence transport equatons. Temperature wall functons based on T + + '' ( y ) ( Tw Tp ) ρc put / qw are used to lnk the wall boundary condton to the temperature n the frst grd cell, T p. When generatng the observatons ether T w or q '' w s specfed, but n the temperature POD modes, only T p s known, renderng the evaluaton of wall heat flux or temperature an under-determned problem. 73

89 The wall fluxes can be evaluated by recallng that the method of snapshots expresses the POD modes as a lnear combnaton of the observatons. Ths can generally be wrtten as: Φ r = LU r (4.4) where the lnear transform L nvolves a SVD operaton. If the POD procedure s thought of as fndng the prncple axes of the data contaned n U r, then L can be thought of as the m x m rotaton matrx between U r and Φ r. The transformaton matrx L can be computed as the projecton of the observatons onto the modes utlzng the pseudonverse agan and the modal fluxes can be drectly computed, vz: r r L = U + Φ (4.5a) F ) T T m = ( v L (4.5b) obs The vector v defnes the observaton mass fluxes, F (U ), and the transpose operaton n (4.5b) s to mantan the same dmenson between F m and v. Defnng the temperature obs observaton matrx and the assocated matrx of block heat nputs, Q = F (T ), the modal heat flux can be computed as: m h T T F h = ( Q T + Ψ) (4.6) Ths procedure can be used to evaluate any flux functon that defnes the same quantty contaned n the goal vector G regardless of where Γ s located n the doman. A common method of treatng coupled phenomena s to work wth an extended state vector, ESV, see [48]. Defnng Y [ u T ] T to: r * =, the modal expanson s modfed 74

90 p u r u r r * ϕ = + a (4.7) * T T 0 = 1 ψ and the goals of boundary condtons are concatenated to G = [ G ] T m G h. The mass and heat flux functons are also concatenated to form the extended modal flux functon r F = [ F m ( ϕ) F ( ψ )], leadng to the ESV optmzaton problem to fnd the weght coeffcents, h mn{ F(u r p * *, T ) a F( r ϕ, ψ ) }, k = 1,2,... m (4.8) = 1 whch can be solved by the goal resdual technque of (3.13a-c). Snce the weght coeffcents must smultaneously account for the velocty and temperature goals, a slower rate of converge s expected, mplyng more modes wll generally be requred to satsfy the boundary condtons. Ths could be a major shortcomng n parametrc system model reducton where observatons are expensve to generate. Note that the reduced-order r r r =, flow model uses the ESV method to couple the velocty components, U [ u v] T otherwse the dvergence free condton would be volated. Solutons based on the ESV method wll not be demonstrated because poor results should be expected as satsfyng both flow and thermal condtons smultaneously mposes conflctng restrctons on the weght coeffcents. Ths can be seen by consderng the separate decompostons (4.2) and operatng the mass and heat flux functons on the velocty and temperature modes. The vector of modal mass fluxes has an order of magntude of F ~ 10 3 m and the modal heat fluxes are ~ 10, usng F h ether the POD or PODc procedure. The mass flux goal s G m ~ 10 1 kg/s and the heat 75

91 transfer rate goal s G h ~ 10 2 W. Notng the order of magntude of a matrx and ts pseudo-nverse are related by F + ~ F 1, the modal weght coeffcents are respectvely a 2 = Fm Gm ~ 10 and b ~ Fh Gh ~ 10. Thus, tryng to combne mass flow and temperature goals wll result n poor approxmatons and rescalng ether velocty or temperature modal fluxes (.e workng n unts of kw nstead of W) wll greatly ncrease the error n other. A new method for couplng the temperature feld to the velocty feld needs to be devsed. Ideally, the temperature source functon would be a soluton to (4.1c) wth * u r as the velocty feld and correspondng k eff. Then, the lnearty of (4.1c) for a known k eff r could be used to rescale the soluton as T = ct( * ). An approxmaton to ths would be 0 u to borrow the temperature feld assocated wth u r 0 and use the POD modal expanson to perturb the soluton untl the boundary condtons are satsfed. The source functon can also be scaled to mprove the approxmaton snce t s treated as the domnant mode of a lnear system: T p r = ct ( u o + b ψ 0 ) = 1 (4.9) Reasonably accurate temperature feld solutons may possbly be obtaned wthout mplctly couplng the velocty feld, however ths would dsregard (4.1c), possbly producng unphyscal results and lack the rgor needed for a robust methodology ntended for more complex flows. The mplct couplng and source functon scalng ntroduce no addtonal complextes mplementng (4.9) algorthmcally. The same PODc procedure of ( ) s employed on the temperature observatons wth: 76

92 Ψ = ψ 0 + ψ ', where ψ 0 = T ( u0 ( Gm )) and ψ ' r R n x m 1 (4.10) To scale the source functon properly, concatenate ψ = [ ψ 0 ψ '] and apply the sequental flux matchng procedure of (3.13a-c) as: G = G F ( T 1) (4.11a) h, h, 1 h = + ( ψ ) (4.11b) b Fh Gh, 4.4. Results T = b j ψ j (4.11c) j= 1 To demonstrate the reduced-order temperature soluton, the prevous test case of Re = 36,320 ( r u * = m/s ) wll be used wth an arbtrary power dsspaton G h [ Q ] [ 96 66]W = Q 1 2 =. Fgure 4.5 plots the POD and PODc temperature modal spectra and t can be seen that the PODc produces a slghtly steeper spectrum n the lower order modes. Ths s a favorable property, as resolvng more of the domnant physcs wth fewer modes allows one to truncate the expanson (4.9) earler for a gven accuracy requrement. It s generally the case that the temperature spectrum decays less sharply than the velocty spectrum, as noted by other researchers [31], mplyng more temperature modes are requred for the same order of accuracy as the velocty approxmaton. Fgure 4.6a llustrates a few basc mean-centered POD modes and Fgure 4.6b shows the PODc modes, both for a secton of the doman near the heated blocks surface where the largest temperature gradents occur. The PODc procedure produces dfferent temperature modes than the POD procedure because the PODc formulates the modes as a functon of parameter values contaned n G, convertng ψ x) ψ ( x, G ). ( h 77

93 Fgure 4.5. Temperature modal spectra for the POD and PODc procedures Fgure 4.6. Local (a) mean-centered POD modes and (b) PODc modes for test case 78

94 Usng all p = 10 modes, the mplct couplng has a maxmum pont-wse error of C. The relatve error over the doman and boundary condton satsfacton s defned as T err,k * Tk Texact 2 = (4.12a) Texact 2 Q err,k * * Fh (Tk ) Fh (Texact ) Fh (Tk Texact ) 2 2 = = (4.12b) Fh (Texact ) F (T ) 2 h exact 2 and the mplct couplng procedure wth the PODc subspace and flux matchng method of evaluatng the weght coeffcents produced T err = and Q err = 4.17 x Fgure 4.7 llustrates the exact and approxmate solutons. The largest errors occur near the surface of the blocks where the largest temperature gradents occur. A large truncaton n system DOF can allow the domnant physcs to be captured, but at the expense of some small-scale features beng dscarded, usually n the form of sharp gradents. However, these errors are of the order of 3%, makng the temperature approxmaton qute accurate consderng the system was reduced to 10 DOF from the orgnal 60,000 DOF requred to compute the turbulent flow and heat transfer. The approxmate soluton shows a slghtly overly dffusve temperature feld near the tralng edge of both blocks, whch may be partally attrbuted to a fnte error between the desred and approxmate soluton boundary condtons. 79

95 Fgure 4.7. Exact and approxmate temperature felds [ C] Both velocty and temperature low-dmensonal models are constructed by usng a lnear subspace to descrbe the physcs for a range of parameters. Poor approxmatons wll result for the nonlnear RANS momentum equaton (4.1b) f the POD or PODc procedures are used outsde the parameter range, however the lnearty of (4.1c) wth known * u r and keff allows one to predct a temperature feld from any parameter value as long as Φ r and Ψ subspaces adequately descrbe the physcs. If the boundary heat fluxes were large enough to nduce sgnfcant buoyancy or even phase change n the case of a lqud medum, the Ψ subspace would not descrbe the thermal physcs, for nstance. For an nlet Reynolds number of Re = 23,221 (u = 8.48 m/s) and block power dsspaton of [ Q ] [ ]W G h = Q, the maxmum pont-wse temperature error was 3.07 C 1 2 = (out of a maxmum of 127 C) and T err = The ntegral boundary condton formulaton was satsfed to Q err ~ The approxmate velocty feld had a 0.48 m/s 80

96 maxmum error and a relatve L 2 error of Note the observaton data n Table 4.1 ranges from 1/8 Q1/Q2 8, so t s reasonable to expect that any forced convecton flow wthn the parameter range of Re and Q 1 /Q 2 would perform wth comparable accuracy. 81

97 5. ERROR ANALYSIS In the POD methodology, the queston of the mnmum number of modes to be retaned n the reduced-order model often arses. The Galerkn projecton produces m- coupled ODE s n tme for the weght coeffcent evoluton and reducng the number of equatons to be ntegrated n tme can result n sgnfcant economes for long term dynamcs nvestgatons. The objectve of ths work was to produce accurate steady models usng the mnmum number of system observatons. In ether case, t must be demonstrated that the POD subspaces ( Φ r,ψ, Π,...,) suffcently capture the system physcs. Some authors use projecton energy of the un-retaned POD modes as a total error estmate e m m λ = + 1 k = 1 m = / k p k 1 k = 1 E λ, although ths assumes there s no nplane error [72]. Note denotes the 2-norm throughout ths secton unless otherwse noted. Chrstensen et al. [56] have suggested usng the dstance n the POD subspace between the approxmaton and the ensemble mean as an error measure. The PODc was developed to specfcally correct ths lmtaton n the classcal POD methodology and new error estmates need to formulated. The parametrc modelng methodology s based on a low number of system observatons, leadng to a relatvely few number of bass functons and generally requrng that most, f not all, modes wll be retaned. It must be noted that the number and dstrbuton of observatons cannot be determned a pror for a gven nonlnear system and s currently the largest concern and lmtaton n the POD methodology [38]. 82

98 5.1. Error Parttonng Rathnam and Petzold [63] dvde the error nto the subspace projecton error (e o ) and n-plane error assocated wth evaluatng the modal weght coeffcents (e ). Fgure 5.1 sketches the POD subspace as the optmal lnear ft to an ensemble of data and schematcally depcts ths error parttonng for a general POD subspace Φ. The term u obs s the system observaton to be approxmated, u * s the approxmate POD soluton and u p s the affne orthogonal projecton of u obs onto Φ and represents the best POD approxmaton of u obs. To show Φ contans suffcent nformaton, the a posteror error estmate of e o 0, or at least e o << e, can be used to show the error s domnated by the n-plane contrbuton. Fgure 5.1. a) POD subspace and b) n-plane (e ) and out-of-plane (e ) error components The error between the observaton and the true soluton wll not be consdered as t s the user s task to ensure that numercal or expermental data fathfully represent the true system. The POD approxmate soluton (u * ) s an effcent soluton to the full model (u obs ) and only descrbes the physcs contaned wthn Φ. Thus, the low- 83

99 dmensonal model wll generally not be more accurate that the full model n the sense of beng closer to the true soluton, but t can produce nearly as accurate results as the full model n an exceedngly more effcent manner. To examne the convergence of the sequental soluton procedure of (3.13a-c) and (4.19a-c), a dual weghted resdual technque [74] wll be used. Consder the canoncal non-square optmzaton problem: mn{ G a F } Fa G (5.1) = Ths could be solved drectly n a least squares manner as + a = F G where F + = T 1 T ( F F) F s the matrx pseudo-nverse. The POD modes are normalzed and ordered n descendng projecton energy so the modal weght coeffcent magntude should generally decay. Computng the a s sequentally wll mmc ths spectral decay because the goal resdual wll decrease wth each successve mode whle computng the vector of a s all at once as magntude. F + G does not guarantee ths decrease n coeffcent Defne d as the vector of dstances between the approxmate weght coeffcents (a * ) and the projected weght coeffcents ( a ~ ) n the modal subspace. The true projected and approxmate solutons to (4.13) are then: F a ~ = G ( true projected soluton ) (5.2a) * Fa = G ( approxmate soluton) (5.2b) The error functonal s defned as T J ( a) = d a, resultng n an error of ( ~ * J a) J ( a ) = J ( e) = ( e, d) and a resdual of r * = G Fa. The boundary condton error n state space (e) s analogous to the n-plane error (e ) n the POD subspace. Note a 84

100 small resdual does not mply a small error. The dual problem can then be formulated as a lnear problem drven by the error functonal, see (5.3), and the error functonal can be expressed: F T a = d (5.3) T J ( e) = ( e, d) = ( e, Fa ) = ( F e, a ) = ( r, a ) (5.4) The fourth term n (5.4) was derved from the thrd term usng Lagrange s dentty ( u, Kv) = ( Ku ˆ, v) where Kˆ s the adjont of K, whch reduces to K T for K R. From (5.4) t can be deduced that the error for the th term n the sequental soluton s bounded by [74]: J ( e) ( r a ) (5.5) k = 1 Ths estmate provdes an a posteror error bound because knowledge of the vector of weght coeffcent perturbatons from the true projected soluton s requred. k k 5.2. Error Estmates The POD subspace representaton of the physcs has been demonstrated to be adequate and the sequental flux matchng procedure has shown to be bounded and weakly converges to the desred boundary condtons. However, the overall objectve s to demonstrate how well the reduced-order model approxmates the full model. Consder a full nonlnear steady model n the canoncal form N(u) = 0, u R n. The reduced-order canoncal soluton of the form m a = 1 u = u0 + ϕ can be expressed n the arbtrary POD subspace Φ = { ϕ1, ϕ 2,..., ϕ m } through the acton of the orthogonal projector P from

101 as u ~ = P T ( u u0 ) + u0, P n x m R. The full (u) and reduced-order approxmate (u * ) solutons can be wrtten as [63]: T T T u = P x + ( P' ) z + u = P x e + u (5.6a) 0 o 0 u * T T T = P x + P y + u = P x + e + u (5.6b) 0 0 where P s the orthogonal complement to P. As above, the approxmatons wll employ all m POD modes such that e 0 renderng u u~. Solvng for the n-plane error T e = P y and substtutng n the modal expanson gves: o y = = T + * T T + * ( P ) ( u P x ) = ( P ) ( u u ~ ) T + * T + ( P ) ( a φ a ~ φ ) = ( P ) φ ( a * a ~ ) (5.7) It can be seen that the projecton matrx s smply the concatenaton of the POD modes, P Φ = { ϕ 1, ϕ 2,..., ϕ }, such that Φ T U obs = a. Snce Φ s an orthogonal matrx, = m m Φ =1 ϕ = I R m, and (5.7) reduces to: * y a = = a ~ d (5.8) Comparson of the reduced-order approxmaton to an affne projecton of the full model on the POD subspace produces the same error estmate as the adjont problem to the boundary condton error. Therefore, the boundary condton error bound can be vewed as an n-plane error estmate e, suggestng the mplct assumpton of matchng only the relevant control surface ntegral condtons n the flux matchng procedure (FMP) s a vald approxmaton. Agan, ths assumes that the out plane error s neglgbly small, e 0 relatve to the n-plane error. As a fnal note, the trangle nequalty can be used o 86

102 to bound the total error e + o + e eo e even though o e e << for parameter values wthn the predefned range. Error bounds are generally dffcult to establsh for POD-based models because the bass functons are problem dependent. The ntroducton of a flux functon to evaluate the weght coeffcents n leu of standard Galerkn methods poses addtonal hardshp n establshng error bounds because of the defnton of the flux functon of (3.11) depends on the parameters of the specfc model. However; a heurstc a pror error estmate s avalable because of the rapdly decayng contrbuton from each successve POD mode. The prncple of the orthogonal complement subspace decomposton s to allow the source functon to satsfy a domnant porton of the parametrc nhomogenety and use the remanng POD modes to perturb the approxmaton to satsfy the resdual on the control surface. Assumng that the contrbuton of the POD modes to the approxmaton s less than the source functon, the followng a pror error estmate s avalable: u * 0 uexact > uk uexact where u0 = u (5.9) 2 2 The asserton s that u domnates the approxmaton n the sense u 2 > u * k u 2 k, whch s generally satsfed by the applcaton of (3.18) to construct the orthogonal complement subspaces. The POD methodology has a tendency to localze errors, whch can be attrbuted to the POD subspace not completely resolvng all the physcs of the full model. The 2-norm s used to quantfy the error over the entre doman and the subscrpt wll be dropped throughout the remander of ths secton for brevty. The k th approxmate soluton s gven by u * k = u + k = 1 a ϕ, where the flux matchng procedure determnes k k 87

103 the k th weght coeffcent as a k + = F ( ϕ ) G, see (3.13). The weght coeffcent s thus k k on the order of a k ~ F + ( ϕ k ) G k = F ( ϕ ) G and the contrbuton of k 1 k each successve mode s then u k ~ a k ϕ k = F( ϕ ) G. The term Gk k 1 k exhbts decay by the applcaton of the flux matchng procedure and the magntude of s fxed because the POD modes are normalzed. Ths causes a decay n F( ϕ ) k 1 a k so the contrbuton of each successve to the total approxmaton, and hence the ncurred error, decays. These arguments are heurstc n nature because of the defnton of the flux functon, but holds for all applcatons of the methodology demonstrated here Turbulent Flow Error The rack model ntroduced n wll be used to demonstrate the error assocated wth the PODc and FMP reduced-order modelng methodology usng the a posteror error analyss of the prevous secton. The frst step s to valdate the POD subspace by determnng the out of plane error by consderng the relatve error between norm of the observaton (test case) and the norm of the test case projected onto the POD e r r r = u u u O p obs obs 2 subspace. Ths s computed as ( )/ = = (10 ) o showng the subspace constructed by the PODc procedure adequately represents the full model flow characterstcs. Ths s consstent wth the boundary condton error from 3.7.2, whch was also shown to be O (10 2 ). Fgure 5.2 compares the weght coeffcents obtaned by projectng the test case onto the POD subspace defne the true projected soluton, denoted wth the subscrpt true wth the weght coeffcents obtaned by the FMP. The modal weght coeffcents 88

104 obtaned by the FMP are very close for the frst 10 modes and the varatons for the remanng modes do contrbute a sgnfcant amount of error because ther magntude s small relatve to the frst few domnant modes of the system. The FMP values for the weght coeffcents for p > 10 all have the correct sgn and the magntude error s due to the mode shape not producng a sgnfcant mass flux through some of the control surfaces. The modal flux s relatvely small for these modes and the weght coeffcent s artfcally enlarged as the FMP attempts to satsfy the mass flux resdual. The classes of flows consdered for ths methodology nvolve closed domans wth multple nlets and outlets and the resultng POD mode shapes do not generally exhbt ths recrculaton behavor near the control surfaces. Small recrculaton regons may exst n the POD modes, but ths behavor s relegated to much hgher order modes where the mass flux resdual s already very small. The error n weght coeffcent evaluaton may also be partally attrbuted to a nonzero out of plane error component. Fgure 5.2. Velocty weght coeffcents computed by the FMP and true values obtaned by projecton onto the POD subspace 89

105 The error bound estmate for the test case was O(10 2 ) for each successve mode. Ths may not appear to be a partcularly sharp error bound, but t does mply the FMP coeffcents are of the correct order of magntude, mplyng the overall velocty approxmaton s bounded. In relaton to the a pror error bounds of 5.2, the flux functon s on the order of F( ϕ ) ~ 10 3 and goal resdual decays from ~10-1 to ~10-2. The weght coeffcents are k then ~10 2, renderng at most u k 2 ~ 10. The source functon s u 3 ~ 10 and thus represents an error bound on the ROM approxmate soluton Turbulent Convecton Error For the approxmate temperature soluton to the test case presented n secton p obs 4.4, the out of plane error was e = T T = O( 10 4 ), consstent wth the heat o flux boundary error, mplyng the temperature POD modes adequately capture the system physcs. Fgure 5.3 plots the true projected and FMP-computed weght coeffcents for the temperature feld approxmaton. The temperature modal weght coeffcents demonstrate the same behavor of closely matchng the frst few domnant modes and then show small devatons for the hgher order modes. The resultng approxmatons beneft form the strong decay n weght coeffcent magntude from O(10 3 ) to O (10 3 ). 90

106 Fgure 5.3. Temperature weght coeffcents computed by the FMP and true values obtaned by projecton onto the POD subspace Fgure 5.4 plots the boundary condton matchng error and the error bound estmate of (5.8) for the temperature feld. The error bound s sharp and decays wth ncreasng modes, ndcatng that the PODc-FMP approxmate temperature soluton s weakly convergent for forced turbulent convecton. 91

107 Fgure 5.4. Temperature flux matchng procedure error and error bounds 92

108 6. INTERCONNECTED REDUCED-ORDER MODELS FOR MULTISCALE DOMAINS Flud and thermal transport processes occur across a range of length scales n systems-level electroncs thermal management. The prmary scales of nterest span the ndvdual server to the largest length scale of coolng scheme when modelng racks and data centers. Such domans are consdered multscale as they contan a seres physcally separated domans, such as the ndvdual servers nsde a rack and the ndvdual racks nsde a data center. Characterzaton and modelng of theses systems are challengng because each rack nsde a data center faclty may contan a varety of dfferent servers and each server may contan dfferent electronc components. An effcent strategy to brdge length scales s to develop separate models for varous system components and assemble them together to model the full system. From an analyss and desgn vewpont, modularty s a key beneft as varous subsystem models can be ntegrated to form a full system n order to nvestgate ther nteractons. Decomposng a system nto subsystems can greatly mprove model effcency by usng varous levels of descrpton for dfferent components. Complex engneerng systems often nvolve transport processes across several length scales and a sngle computatonal model would requre a large number of grd cells to stretch across these length scales. Modularty affords the ablty to quckly ntegrate new components nto an exstng model wthout developng a new computatonal grd for alterng the subcomponent models. It should also be noted that system decomposton can be appled to any system and there s not nherent error assocated wth treatng a system as an nterconnected set of subcomponents, [32]. 93

109 6.1. Indvdual Component Modelng For a representatve example of systems-level thermal management of electroncs, consder a seres of channels connectng 2 plena, where each channel represents a server and the plena are representatve of front and rear plena of a data processng cabnet. These systems are modular n nature, consstng of a seres nested sub-domans (see Fgure 6.1). The ablty to mantan effcent models of the ndvdual components and connect them together to evaluate varous desgns would result n a very effcent analyss tool. To demonstrate the methodology, consder a hghly smplfed 2-dmesonal model of an ar-cooled rack contanng a few servers. The model wll be constructed from reduced-order model (ROMs) of an ntake plenum and the repeated use of a sngle server model, as s shown n Fgure 6.1. The plenum measures 2L x 5L and the server measures 4L x L, wth L = 0.1 m, and both models were developed for the range 6,900 Re L 31,300. Table 2 lsts the observatons used to construct the component ROMs. In leu of specfyng the boundary velocty, the observatons may be constructed wth pressure boundary condtons and the flux functon of (3.11) can be used to compute the boundary mass flux and locate the observaton n parameter space of G. Specfyng pressure boundary condtons may be less ntutve and more dffcult to generate observatons wth a specfed Re range, but s an equally applcable method of generatng observatons. To the best of the author s knowledge, the only works developng POD-based flow models wth pressure effects are that of [53] whch redefned the nduced norm to model compressble flow, [59] whch ntroduced a lnear correcton for the pressure drop 94

110 through a channel and flows drven by dstrbuted pressure-velocty momentum sources to model fans was ntroduced by [73]. Fgure 6.1. a) Plenum b) server component model geometry and c) full-scale system Table 2. Plenum flow observatons, P [Pa] and m& [kg/s] 95

111 Results for the ntake plenum sub-model wll brefly be presented to llustrate the domnant flow features. The velocty and pressure modal spectra for the plenum and server models for representatve test cases are plotted n Fgure 6.2. The use of the pressure modal spectrum wll be explaned shortly. Fgure 6.2. a) Server and plenum sub-component modal spectra and b) ntake plenum boundary condton error The relatve L 2 * error, defned agan as uerr, k = u r k uexact / uexact for the velocty feld 2 and lkewse for the boundary condtons, showed that the boundary condtons were satsfed to wth 3% and velocty approxmaton error was 6.2%. Error contours for a test case correspondng to a mass flux of G = {0.2070, , } kg/s are plotted n Fgure 6.3. The plenum model contaned 4000 grd cells, or total DOF to model the flow consderng u, v, P, k, ε are solve for n each grd cell, whle the reduced-order model contans only 12 DOF, for an O(10 2 ) reducton n DOF. 2 96

112 Fgure 6.3. Intake plenum error feld 6.2. Pressure Feld Approxmaton The ROMs do not need to be autonomous because the nput mass flux can be computed wth a full-scale CFD smulaton or be derved from another ROM. To couple the ROMs, t must be noted that specfyng the component nterface mass fluxes a pror decouples the ROMs because there s no drvng force between the two components and they behave essentally ndependent from one another. Many flows n electroncs coolng are pressure-drven, such as fans movng ar through a seres of vents and channels, and t s the pressure dfference across these components that drve the flow. A muchcelebrated property of POD analyss s the elmnaton of pressure for ncompressble 97

113 flows; however, couplng of ROMs for the class of flows presented here requres knowledge of the pressure feld. The ROM pressure felds can be approxmated by usng the PODc procedure to construct a pressure modal subspace, Π {π1,π 2,...,π }. The egenvalue spectra of Π = m strongly follow the velocty modal spectra, as s expected due to the role of pressure n ncompressble flows. Usng the velocty modal weght coeffcents from the PODc-FMP produces accurate sobars shapes, however the magntude of the feld s poorly scaled. The velocty coeffcents based pressure approxmaton s reasonably accurate up to a constant,.e. P * r = c P( u ) + bπ, the unknown constant cannot be determned. To approxmate the pressure feld accurately, the pressure POD modes are obs projected back onto the observatons ensemble P P, P,..., P } to obtan the set of observaton weght coeffcents, = { 1 2 m b * = Π + P obs R m x m (6.1) A G-dmensonal quadratc response surface of the form b * = f ( G obs ) s computed for the k th pressure mode as a functon of the observatonal mass fluxes, G obs. The k th weght obs coeffcent s then evaluated as b = f ( G ) and the approxmate pressure feld s assembled as k k P * obs π f ( G ) π (6.2) = b = The response surface methodology can be appled one mode at a tme because the pressure modes are lnearly ndependent and b k are uncorrelated. Regresson methods are the only avalable tool to compute b because the ROM are parametrzed by the boundary 98

114 mass flux and no pressure nformaton about the new approxmate soluton s avalable to use n a matchng procedure. Fgure 6.4 below plots the velocty weght coeffcent and response surface approxmatons for a test case usng the ntake plenum component ROM. Fgure 6.4. Component ROM pressure feld approxmaton Wth the response surface methodology producng satsfactory results for the pressure feld, t would be natural to consder the same soluton process to determne the velocty weght coeffcents. The mmedate ssue s that the boundary condtons are not enforced n the weght coeffcent computaton and one reles on the assumpton of the response surface accurately approxmatng the parametrc condtons as well as the soluton over the entre doman. Fgure 6.5a compares the frst couple velocty weght coeffcents usng the FMP and response surface (denoted rs ) approxmatons and Fgure 6b plots the boundary condton error usng the response surface method. 99

115 Fgure 6.5. Response surface approxmaton for a) weght coeffcents and b) boundary condton approxmaton error Fgure 6.6 plots the pressure weght coeffcents and compares the values aganst the true weght coeffcents obtaned by projectng the pressure test observaton onto the POD pressure modal subspace, b true P + obs = Π. The reason the response surface method produces acceptable results for the pressure coeffcents and not the velocty coeffcents s that pressure coeffcents tend to be larger n magntude and exhbt a strong spectral decay so that error n the response surface predctons are less amplfed n the fnal feld approxmaton relatve to the velocty approxmaton. Note ths strong spectral decay was also exhbted by the scalar temperature feld n

116 Fgure 6.6. Comparson of response surface and true pressure weght coeffcents 6.3. Systems-Level Modelng The system-level model s constructed by jonng the 3 server ROMs to the nlet plenum and the exhaust plenum, whch s a mrror mage of the ntake plenum both n the x- and y-drectons, wll collect the outlet ar from the servers and exhaust t to the ambent. To drve the flow, fan models wll be placed at the outlet of the server models to mmc an nduced fan and lumped resstance models are ncluded between the ntake plenum and sever nlet to model the pressure drop across an nlet vent. The fan s modeled wth the cubc pressure-velocty model (S f ) and the lumped resstance model (S s ) s representatve of nerta losses n hgh Re flows. The nlet and outlet pressures to the full system can be assumed to be zero wthout loss of generalty. The source functons for the lumped parameter models are evaluated usng area-averaged velocty u : p( u 2 3 ) = u + 20u 4u S ( u ) (6.3a) f 101

117 p( u 2 ) = u S ( u ) (6.3b) A new POD bass needs to be computed for the exhaust plenum, even though the exhaust plenum has the same geometry because the flow observatons are hghly nonlnear and reversng the flow through the plenum produces sgnfcantly dfferent flow patterns. The exhaust plenum ROM was developed from the same observatons as the ntake plenum wth the mass fluxes specfed n the opposte drecton from the nlet to the outlet. It wll be noted that the ROM geometry s fxed because geometrc scalng of POD modes s a source of ongong research and beyond the scope of ths nvestgaton. Each component sub-doman wll be dentfed wth a superscrpt as s j Ω, j = 1,.., 5, and the k th control surface (nterface) for the j th sub-doman n (3.11) s termed j Γ k, provdng the j G k mass flux. The mnmzaton of (3.12) s altered to: mn{ G j a j F j r ( ϕ ) } j where F j r j ( ϕ ) = j Γ j ρϕ nˆ dx (6.4) Contnuty can be used to reduce the number of unknown ROM fluxes to j Gk 1 and the ROMs do not need to be solved concurrently to ensure satsfacton of (6.4). Concepts from flow network modelng (FNM) [75] are used to generate the matchng condtons between the component models and assemble the full system. FNM calculates the pressure at specfc nodal locatons and the flow between two nodes s drven by momentum equatons of the form P P = S( ). Ths technque produces acceptably 1 2 u accurate results when the momentum equaton models condut-type components, such as ppes, channels or even the servers n ths example because the flow resstances can be expressed as 2 P 1 2 = Au + Bu. For manfold-type flows, such as the ntake plenum of Fgure 6.1, P = f P, P ) and no smple relatons are readly avalable from 1 (

118 flow resstance handbooks. Accurate pressure feld approxmatons are avalable from the ROMs and pressure drop between nodes can be computed by ntegratng the pressure over the control surfaces. The system nomenclature and FNM pressure locatons are llustrated n Fgure 6.7. Fg 6.7. Full-scale system nomenclature and flow resstance network The standard SIMPLE algorthm [76] s used to solve for the nodal pressure and momentum lnk flow rates wth the only alteraton beng that the momentum source s evaluated usng the ROM pressure feld. The procedure s completely analogous to pressure-velocty couplng methods n ncompressble CFD and wll be outlned here wth more detals avalable n [75]: FNM SIMPLE Algorthm 1. Guess a nodal pressure dstrbuton P 2. Use the momentum lnk equatons P = S(u ) to calculate the momentum lnk flow rates gven the nodal pressures 103

119 = 3. Use the nodal contnuty equaton ρ u A = G 0 to combne the momentum lnk equatons nto a lnear system and solve for a new pressure dstrbuton 4. Return to Step 2 and repeat untl convergence In general, the momentum equaton wll be nonlnear n u and an approprate lnearzaton must be ntroduced to perform step 3 above. Ths source term lnearzaton, also equvalent to standard CFD methods, can be obtaned by expressng the momentum source term as constant plus the lnear varaton n u: P = C + Lu (6.5) The constant and lnear varatons are obtaned by takng a Taylor seres about the prevous teraton: S S ( u) S 1 + ( u u 1) (6.6a) u 1 S S C = S 1 + u 1 and L = (6.6b) u u Full-Scale Results Usng the SIMPLE procedure to assemble the above system showed that the relatve L 2 error norm for the approxmate soluton mass flux over all nterfaces was and the relatve L 2 error norm over pressure nodes P 1 to P 12 was Fgure 6.8 plots the true and approxmate velocty and pressure felds. The pressure felds n the varous sub-domans can be properly scaled after the FNM soluton calculates the nodal 2 pressure dstrbuton n Fgure 6.7. For example, P 4 s known and P = P S( ) 7 4 Ω 104

120 2 where S ( Ω ) s obtaned by ntegratng the pressure over the nlet and outlet of Ω 2. The 2 * 2 lnearty of pressure can than be used to shft the pressure feld P(Ω ) = P (Ω ) + P1 + P4 relatve to the full-scale model reference pressure P = 0. Fgure 6.8. Exact and component-matched velocty and pressure felds [Pa] The regons of greatest error occur at the nterfaces where there s severe msalgnment between the component velocty felds. The flow at the entrance to the uppermost server 1 ( G 4 3 G1 ), plotted n Fgure 6.9a, s almost entrely n the vertcal drecton whch leads to a dscontnuty n the approxmate soluton because the server ROM has been parametrzed n terms of nlet mass flux only and has assumed a unform nlet velocty wth no vertcal component. The lumped resstance only acts to attenuate the x-velocty 105

121 component and does not affect the y-velocty component. The recrculaton regon near the nlet to domans Ω 3 and Ω 4 s not resolved by the approxmate soluton only because the server ROM dd not contan ths behavor. Ths s not an nherent lmt of the methodology presented here because, as wth any methodology that couples component models, the nterconnected models must contan the behavor passed to them from other components. Fgure 6.9b compares the flow at the exhaust of a server (the Ω 4 Ω 5 nterface). The predctons are much more accurate and resolve the domnant flow features because the velocty from Ω 4 has a sgnfcant component normal to 2 Γ 3 and the ROM for Ω 5 was constructed assumng a unform nflow normal to the surface 5 Γ 2. Fgure 6.9. Local velocty msmatch at nterface a) Ω 1 Ω 4 and b) Ω 3 Ω 5, n.b. the vectors have been nterpolated to a coarse grd for llustratve purposes 106

122 The error of the assembled system can be attrbuted to the ROM mass flux and pressure feld approxmatons. The ROMs presented here satsfy the target mass flow rate to ~3% and the pressure feld approxmaton s accurate pont-wse on the control surfaces to ~5%. These error are coupled n the FNM procedure durng the pressure correcton step when the nodal pressures are updated as a functon of the nodal mass mbalance. Another concern n connectng component models together s the propagaton of error and the compoundng of errors as more components are added to the system. POD-based ROMs manage ths error because the ndvdual models satsfy overall mass and energy balances, makng the O(10-2 ) varatons between desred and ROM mass and energy fluxes the prncpal error contrbuton Addtonal Component Parametrzaton The dscontnuty n the velocty feld at the nterface regon s not an nherent lmtaton n assemblng POD-based ROMs, rather s an ssue pertanng to all nterconnected systems modelng approaches. One such example occurs n block dagram modelng of dynamc systems. Suppose one component model has ts behavor tuned to the frequency range 5 f 15 Hz and ts output s connected to a model that has a resonance at 25 Hz. The full-scale system wll never see the effects of that resonance because t cannot be exted by the upstream model. The recrculaton regon not resolved by the server ROM n Fgure 6a s caused by the same effect; the upstream plenum model ntroduces a non-normal velocty component whch the server ROM cannot account for. Ths poses no problem n assemblng the components snce the matchng condtons of 107

123 mass flux and pressure are stll avalable, t just results n an unphyscal dscontnuty n the fully-assembled velocty feld. The server ROM was parameterzed n terms of nlet mass flux only, but had velocty nlet drecton been accounted for the parameterzaton, the server model would produce much more accurate velocty feld predctons near the nterface. An updated server ROM s developed from modes parametrzed by the nlet mass flux and average y- velocty on nlet,.e. F = f ( G, v). Incorporatng ths model nto the full-scale system shows that the recrculaton regon can be resolved n the server ROM, see Fgure The pressure feld s nearly dentcal to that of Fgure 6.8 except for small regons n the upper porton of the nterface between the ntake plenum and servers 2 and 3 where the nlet velocty mpnges on the upper wall. Fgure Ω 1 Ω 4 nterface for updated server ROM The full-scale model presented here llustrates a number of ssues concernng the nterconnecton of full-feld models. The man ssue concerns the severe msalgnment of the velocty feld from ether sde of the nterface between two component models, whch can be resolved by addtonal parametrzaton of the component ROM. Ths example s 108

124 especally dffcult because the lumped parameter model does not act as a flow straghtener by attenuatng the velocty component parallel to the nterface. Even wth such lmted ROM model behavor, the result systems-level model produces reasonably accurate representaton of the flow and pressure felds. 109

125 7. CONCLUDING REMARKS The reduced-order modelng framework developed here presents an accurate methodology for full-feld approxmatons of steady turbulent flows and turbulent convecton. The methodology presented here s equally applcable to expermental data and could be used to create low-dmensonal models of stochastcally characterzed systems and ntegrated nto large-scale smulatons, creatng an expermentally valdated, computatonally effcent modelng methodology for complex systems. Slghtly less accurate models may be far superor to large expensve models durng early system desgn and optmzaton, where many dfferent parameter values and component nteractons may need to be evaluated. The low-dmensonal framework developed here also has the advantage of characterzng dstrbuted parameter systems n state space usng ntegral condtons, allevatng the need to specfy detaled flow and heat transfer profles that are often unknown. It also does not requre the evaluaton of the governng equatons, makng t well suted for nverse problems and parameter dentfcaton studes. Reduced-order models can be used to effcently brdge length scales n modelng and analyss of complex thermal fluds systems. Meshes constructed for CFD/HT smulatons of systems-level electroncs thermal management must adequately resolve sharp gradents at the smallest modeled length scale, often producng excessvely large models. Conventonal CFD/HT technques requre a contnuous computatonal grd that causes the smallest mesh features to dctate the overall grd sze. Computatonal lmtatons often lmt the smallest features than can be accurately modeled. Reduced- 110

126 order models (ROMs) based on the framework presented here can be ntegrated nto large-scale CFD/HT smulatons to construct effcent models across varous length scales. These models accurately couple the data present on the CFD/HT grd and produce full-feld approxmatons at length scales below the computatonal grd sze wthout requrng the smallest features n the smulaton to determne the full model sze. The true effcency n the reduced-order modelng framework s the ablty to assemble the models as varous components n a full-scale model Summary The tradtonal proper orthogonal decomposton (POD) methodology has been reformulated to treat steady parametrc turbulent flows and forced convectve flows for a predefned range of boundary condtons. Orthogonal complement POD subspaces were ntroduced to treat nhomogeneous boundary condtons, elmnatng the addtonal effort requred by homogenzaton procedures and extendng the reduced-order methodology to a wde range of flow parameters. A flux matchng procedure (FMP) was formulated to evaluate the modal weght coeffcents after the standard methods of Galerkn projecton were shown to be neffectve for developng parametrc reduced-order models. The ntegral condtons of the FMP are used n a state space resdual expanson to mmc the POD subspace egenvalue spectra and converge toward the desred parametrc condtons. An mplct couplng procedure was developed to lnk the temperature and velocty felds, greatly mprovng the accuracy of low-dmensonal temperature predctons. The overall reduced-order modelng framework presented here was able to reduce numercal models 111

127 contanng DOF down to less than 20 DOF for an order reducton, whle stll retanng greater that 90% accuracy over the doman. Rgorous a posteror error bounds were developed by parttonng the error nto n-plane and out of plane components wth specal attenton to the error assocated wth usng the FMP to evaluate the modal weght coeffcents. Dual resdual methods were used to show that the flux matchng procedure converges and s computatonally superor approach for low-dmensonal modelng of steady turbulent flows and convecton. Parametrc reduced-order models were then used as component-level models and assembled together to model a full-scale system. Dstrbuted parameter ROMs were combned wth lumped parameter models of flud movng devces and flow resstances to model a pressure-drven system. Accurate approxmatons of the component pressure felds were constructed usng response surface technques, wth the pressure becomng the drvng force between the dfferent sub-domans. Flow network modelng was used as the component handshakng procedure to couple the component ROMs through the nterfacal mass flux. The methodology to construct POD-based ROMs for parametrc flows s schematcally summarzed n Fgure 7.1 below, whch segregates the methodology nto model development, sngle component ROM soluton and the nterconnecton of ROMs for pressure-drven flows. The model development conssts of constructng a numercal or expermental representaton of the physcal system to be modeled and then observng the system under varous parameter values to form the observaton database. To use the ROM to construct a new approxmate soluton, the desred approxmaton s expressed n terms of the model parameters, snce ths s the only known porton of the soluton. The 112

128 reduced-order soluton begns wth the selecton of an observaton to serve as the source functon. Orthogonal complement subspaces POD subspaces are then be constructed and the flux matchng procedure s used to evaluate the modal weght coeffcents. The nterconnecton of ROMs to model systems-level flows requres the computaton of a pressure response surface n the model development porton of the process. Addtonal lumped parameter models are specfed and the component nterfacal mass fluxes are computed usng the standard SIMPLE algorthm. The soluton procedure to solve for the ndvdual component velocty and pressure felds s repeatedly used n connectng ROMs because the SIMPLE procedure requres the soluton for each POD-based ROM durng each teraton. Fgure 7.1. Reduced-order model (ROM) methodology flowchart 113

129 7.2. Future Work The low-dmensonal modelng methodology has focused on transport processes for electroncs coolng, especally nternal turbulent convecton wth conjugate conducton. However, the framework s general enough to accommodate a wde range of other thermal flud mechansms ncludng radaton, electrohydrodynamcs and chemcally reactng flows. Ths procedure can also be extended to non-contnuum smulatons such as molecular dynamcs wth varyng parameters or boundary condtons to ntegrate mcro- and nanoscale mass, momentum and energy transfers nto full-scale engneerng computatons. Ths would allow one to effcently lnk non-contnuum models nto complex engneerng systems, wthout requrng mathematcal homogenzaton procedures. The development of reduced-order models represents a paradgm shft n computatonal thermal scences. Much research n ths area s currently focused on hgher order approxmaton schemes and advanced mesh generaton, ntrnscally enlargng the computatonal effort. Gven the lmtatons n modelng complex geometres and nonlnear phenomena, model reducton provdes a tool to rapdly characterze the domnant behavor of the system, whch can be used n conjuncton wth robust desgn technques to perform optmzaton studes and desgn control schemes for systems wth modelng and operatonal uncertanty. The reduced-order modelng framework was developed for use n conjuncton wth desgn and optmzaton tools. Ths research brdges the gap between the large data sets generated by expermental and numercal solutons of thermal flud transport n complex geometres, and forms a lnk to research n desgn. Low-dmensonal modelng can also be ncorporated nto the 114

130 emergng feld of smulaton-based desgn, where expermentally valdated, reducedorder models can be used to effcently analyze and desgn thermal-flud systems nvolvng varous transport mechansms across several decades of length and tme scales. 115

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