Wind Energy in the Built Environment

Transcription

1 Wn Energy n he Bl Envronmen oncenraor Effec of Blng Saner Meren

2 Wn Energy n he Bl Envronmen oncenraor Effec of Blng Proefchrf er verkrjgng van e graa van ocor aan e Technche Unvere Delf, op gezag van e Recor Magnfc prof. r. r. J.T. Fokkema, voorzer van he ollege voor Promoe, n he openbaar e veregen op nag 5 epember 6 om 5. r oor Saner MERTENS werkgkng ngener en narkng ngener geboren e Haarlem

3 D proefchrf goegeker oor e promoor: Prof.r.r. G.A.M. van Kk Toegevoeg promoor: Dr. G.J.W. van Bel Samenellng promoecomme: Recor Magnfc, voorzer Prof.r.r. G.A.M. van Kk, Technche Unvere Delf, promoor Dr. G.J.W. van Bel, Technche Unvere Delf, oegevoeg promoor Prof.r.r. M.H. e W, Technche Unvere Enhoven Prof.r.r. Th. Van Holen, Technche Unvere Delf Prof.r.I. Parachvo, Ecole Polyechnqe e Monreal Prof. J. Twell, De Monfor Unvery Prof. r. M. Graham, Imperal ollege, Lonon Pblhe by: Ml-Scence 5 Wae Way, Brenwoo Eex, M5 9TB, Une Kngom Telephone +44 () ISBN Keywor: wn energy, bl envronmen opyrgh 6 by Saner Meren All rgh reerve. No par of h pblcaon may be reproce, ore n a rereval yem, or ranme, n any form or by any mean, elecronc, mechancal, phoocopyng, recorng, or oherwe, who he pror wren permon of he ahor. over pcre: ar mpreon of he hree man prncple of power agmenaon. From lef o rgh: cloe o a blng (valzaon: ahor), beween arfol-hape blng (valzaon: r.r. L. Aanen, DHV) an n a c hrogh a blng (valzaon: ahor).

4 Smmary Smmary Th he eal wh wn energy converon n he bl envronmen. I gve a ecrpon of he wn reorce n he bl envronmen ha can be convere no energy by a wn rbne. Wh a foc on maxmm energy yel of he wn rbne, epecally eal wh he negraon of wn rbne an blng n ch a way ha he blng concenrae he wn energy for he wn rbne. Three fferen bac prncple of ch blng ha concenrae he wn or concenraor are nghe: wn rbne a he roof or e of blng, wn rbne beween wo arfol-hape blng, wn rbne n c hrogh blng. The aeroynamc of hoe hree concenraor wh her poble wn rbne are nvegae wh a foc on negraon relng n maxmm energy yel of he wn rbne. The complcae concenraor effec of blng n he acal flow are mplfe o her bac aeroynamc qale n parallel flow. The propere of hee mplfe qale n parallel flow are explore hrogh he hree well-known cornerone of aeroynamc reearch: mahemacal moel, verfcaon wh mearemen an nmercal mlaon of he flow. The mahemacal moel are erve wh mplfe mahemacal flow ecrpon. The mearemen are carre o n he open je wn nnel of Delf Unvery of Technology, econ Wn Energy an he mlaon are performe wh a commercal ompaonal Fl Dynamc (FD) coe, whch olve he bac flow eqaon nmercally. The avanage of boh verfcaon ool: mearemen an FD calclaon are exploe by pre-elecng he ool wh he be propec for an accrae rel n a ere aon. Th he gve a broa ecrpon of he mo mporan e concernng he energy yel of a wn rbne n he bl envronmen. I prove ecrpon of he average/ global wn pee n he bl envronmen, he local wn pee, he wn pee near blng an verfcaon of he mahemacal moel of he hree poble concenraor prncple menone above. Frhermore, prove nformaon on able wn rbne for e n he bl envronmen. The pro an con of he hree concenraor prncple are mmarze, who mahemac, n he la chaper. Th la chaper how ha he a roof of blng confgraon an a varaon on he n c hrogh blng confgraon are promng. oncernng he a roof of blng confgraon, a phere-lke blng concenrae he energy n omnreconal free ream wn he mo: a facor of hree o for! Sch concenraor are able o overcome he problem of he low average wn pee n he bl envronmen an enable energy yel comparable o rral area. A varaon on he n c hrogh blng confgraon, wo ce ellpo n a cro wh he c a he cro cenre, able o concenrae he energy n omnreconal free ream wn wh a facor of approxmaely one an a half. The oher concenraor prove a maller energy concenraon. The cloe o a blng confgraon very effcen n ng he acceleraon by he blng an a relavely cheap olon compare o he oher poble concenraor prncple. Sll, h he how ha he energy yel for all concenraor confgraon lme becae he wn rbne can only prof from he concenraor effec when hey are relavely mall compare o he blng. Ye, hey elver he energy, where neee, he bl envronmen, an her energy yel fel a an energy avng n he blng. oneqenly, her energy yel reflece n energy avng on he comer bll from he ly company, whch a hgher rembremen compare o rral area. I h concle ha wn energy converon n he bl envronmen makng e of he concenraor effec of blng, a promng renewable energy orce. - -

5 Table of conen Table of conen SUMMARY...III LIST OF SYMBOLS...VII INTRODUTION... BASI THEORY DRAG-DRIVEN ROTOR VERSUS LIFT-DRIVEN ROTOR Drag-rven wn rbne Lf-rven wn rbne Hybr-rven wn rbne omparon of he wn rbne ONSTRAINTS Bl envronmen Wn rbne Blng....3 THE BUILDING-AUGMENTED WIND TURBINE THE TWISTED H-DARRIEUS WIND TURBINE WIND HARATERISTIS GLOBAL WIND IN THE BUILT ENVIRONMENT Log-law Sep n roghne hegh Trblence LOAL WIND IN THE BUILT ENVIRONMENT Blng characerzaon Flow characerzaon Wn pee probably rbon WIND AT BUILDINGS Sagnaon pon Separaon Wake an recrclaon zone moel Reaachmen ance The cavy moel Inflencng he eparaon velocy an backprere ANALYSIS TOOLS MATHEMATIAL Poenal flow Vorex hee Momenm heorem EXPERIMENTAL The open je wn nnel Tnnel correcon Error n power coeffcen from mearemen Scalng rle Wn rbne power OMPUTATIONAL FLUID DYNAMIS The rblence moel Near-wall regon The gr The amopherc bonary layer Some nal verfcaon v -

6 Table of conen 5 WIND TURBINES LOSE TO BUILDINGS THE WIND TURBINE S PERFORMANE LOSE TO A BUILDING Performance n parly accelerae flow The acceleraon a he roof THE LIFT-DRIVEN HAWT IN SKEWED FLOW THE H-DARRIEUS IN SKEWED FLOW Incon facor a mall loa Incon facor a hgh loa Power coeffcen Performance Verfcaon of he moel Dcon on he valy of he moel THE ENERGY YIELD AT THE ROOF THE ENERGY YIELD AT THE SIDES OF A BUILDING WIND TURBINES BETWEEN AIRFOIL-SHAPED BUILDINGS MOMENTUM THEORY Shroe wn rbne Dffer-agmene wn rbne D VORTEX MODEL FOR A SHROUDED ROTOR Se-p of he moel Ince veloce Relng veloce Power coeffcen Verfcaon of he moel Dcon on he valy of he moel APPLIATION OF VORTEX MODEL RESULTS TO MOMENTUM THEORY FINITE ASPET RATIO EFFETS THE DUTED SMALL WIND TURBINE Allowe ream be ze Velocy beween he ¼-chor pon THE SHROUDED H-DARRIEUS YAWED AND OPPOSITE FLOW Yawe flow Oppoe flow Zero power op THE ENERGY YIELD OF THE SHROUDED ROTOR... 7 WIND TURBINES IN DUTS THROUGH BUILDINGS THE PLATE ONENTRATOR MODEL Power coeffcen Je conracon VERIFIATION OF THE PLATE ONENTRATOR MODEL Verfcaon of he je velocy Verfcaon of he je conracon Verfcaon of he prere rop acro he acaor Verfcaon of he power coeffcen APPROXIMATED PLATE ONENTRATOR MODEL DISUSSION ON THE VALIDITY OF THE MODEL YAWED FLOW THE ENERGY YIELD OF A PLATE ONENTRATOR ENERGY YIELD OF A OMBINED PLATE ONENTRATOR RESULTS AND ONLUSIONS v -

7 Table of conen 8. SUMMARY OF RESULTS FOR GENERAL USE SUMMARY OF RESULTS FOR WIND ENERGY IN THE BUILT ENVIRONMENT Inegraon of wn rbne an blng Energy yel example of Blng-Agmene Wn Trbne Wn Trbne for he bl envronmen EXPETATIONS Expecaon for wn energy n he bl envronmen Reearch on aeroynamc of Blng-Agmene Wn Trbne REFERENES APPENDIES Appenx A Sream be lengh n vco flow 5 Appenx B Mearemen plae concenraor 54 Appenx Ue rblence moel FD calclaon 55 Appenx D Daa hgh-lf cambere arfol Da Appenx E Ince veloce by a wake 57 Appenx F Prony brake 59 Appenx G Tranglar chor-we vorex rbon 6 Appenx H Sable rblence moel for calclaon of flow aron arfol 66 Appenx I alclae lf- an rag coeffcen aa for a NAA 8 arfol 67 Appenx J Qck ng gelne 69 Appenx K rrclm vae ahor 7 - v -

8 L of Symbol L of Symbol All menon of qane e n h book are conen wh he Syem Inernaonal (SI). The menon of freqenly e qane an he efnon of he qane are gven hereafer. The menon of a qany menone a he nrocon of he qany. a ncon facor, cale parameer n probably rbon., m/ A area m A area of he ream be a nfne ance ownwn of he roor m e A percenage of earh rface occpe wh obacle. H A apec rao. A r roor wep area A area of he ream be a nfne ance pwn of he roor B nmber of blae of he roor. c chor lengh arfol m, probably on a ceran wn pee from ecor. c rag coeffcen. lf coeffcen of an arfol. l p prere coeffcen efne a p ( p p ) ( ρ ) 3 P power coeffcen efne a P P ( ρ A ) r =. =., acceleraon of free ream reference wn pee. T hr force coeffcen. placemen hegh m D characerc ze m D roor ameer m E energy yel kwh f vorex heng freqency, probably rbon of accelerae wn Hz f Webll rbon of reference or free ream free ream wn. F hr force of a wn rbne N F hr force calclae wh blae elemen heory N Bl F rag force N D F M hr force calclae wh momenm heory N h hegh m h knk hegh or hegh of he nernal bonary layer m k h hegh of he ngle-roor par m h hegh of he oble-roor par m H blng hegh m H agnaon pon hegh m H roor hegh (Vercal Ax Wn Trbne) m I rblence neny. k hape parameer n probably rbon, rblen knec energy. K reance facor of wre creen L lf force N L cavy lengh m L c P prere level a locaon P m m N m - v -

9 L of Symbol L W orce on prere level B n nmber of revolon Hz p ac prere N m p free ream or free ream ac prere N P power W Q orqe Nm r ra m R ra m R roor ra m Re Re nmber. ance along a free reamlne m S Srohal nmber. T Tme correponng wh he energy yel (z) horzonal wn pee m/ p poenal wn pee m/ r relan velocy m/ frcon velocy m/ * w velocy a he ¾-chor pon on he arfol m/ v vercal velocy, laeral velocy m/ W blng wh m x axal coornae m y laeral coornae m z vercal coornae m z roghne hegh m Sbcrp free ream, reference or free ream oble-roor par e en or nfne ownwn of acaor nce eparaon, ngle-roor par a roor or acaor max maxmm mn mnmm op opmal.e. relng n a maxmm ref reference + pwn - ownwn Spercrp flcaon n rblen velocy * non-menonal ance or velocy Greek α angle of aack of he flow on he arfol ra or eg β je conracon/ expanon. γ vorex rengh per mere m/ γ kew angle ra or eg Γ Gamma fncon, vorex rengh., m m - v -

10 L of Symbol δ bonary layer hegh m λ rao of p pee an free ream wn pee ρ eny of ar kg 3 m - x -

11 Inrocon Inrocon The Energy Informaon Amnraon [3] ae ha he nralze conre e more han half he worl oal energy conmpon n. They are he bg conmer an have he propery an knowlege o wch o renewable energy orce. Moreover, mo people n he nralze conre are aware of he vanhng fol energy reerve an he envronmenal mpac of he hge conmpon of fol energy nowaay. hange n energy orce an conmpon behavor wll probably be nae here. oneqenly, he nee for renewable energy n propero conre wll re. I no ceran ha he emperare re n he la ecae cae by he emon of carbon oxe from brnng fol energy reerve. Exper ffer n her opnon wheher or no clmae change a rel of or pollon (ee for nance [87]). Neverhele, here are ero ncaon ha he coneqence of he greenhoe effec may no allow he brnng of or fol energy reerve. The earh poplaon ncreae raply an energy eman follow. The Energy Informaon Amnraon [3] prec an average annal wo percen ncreae n energy conmpon over he comng weny year largely realze ng fol energy. Why hol we ake he rk of enconerng evere envronmenal problem? The rk by elf hol be enogh o wch o renewable energy. B here are more reaon o wch o renewable energy nea of fol energy. Fol reerve vanh. Exper [88] prec a peak n procon of fol energy n approxmaely 5 year. Agan, here mch conrovery on he meframe nce h largely epen on he efnon of or reerve an or energy conmpon n he fre. Neverhele, n he fre procon ecreae whle conmpon ncreae. Th wll have a hge mpac on energy prce an commny. No rnnng o b peak procon of fol energy eermne he rnng pon. Frhermore, he nable aon n he Mle Ea reponble for conerable varaon n he fol energy prce nce we largely epen on her fol fel procon. A wch o renewable energy can ecreae or epenence. However, here anoher mporan reaon for wchng o renewable energy from fol energy: he co of he energy. A far comparon of he prce of energy hol be bae on he oal co of energy. Th ncle exernal co, ch a he co of people geng ll becae of pollon. Tha ceranly ffcl o emae, b a ae n hanler [] o no ncorporae exernale n prce emonrably wrong. Among oher, hanler [] gve an example of Denmark where he oal co (nclng exernal co) of a kwh fol energy from coal (7- c /kwh) are hgher han ha of a kwh wn energy (4-4.5 c /kwh). Th, everal rong force ncae he ncreae e of renewable energy n he fre. Ye, we are e o fol energy avalably an energy eny an conme accorngly b renewable energy orce have a mch lower energy eny. We nee large area wh renewable energy procon n. I j no ha eay o flfl or crren energy nee, le alone or fre energy nee, wh renewable energy. We canno affor o rely on a few large renewable orce. Every poble renewable energy orce ha place an hol be lze. Moreover, alo we o verfy he orce (olar, wn, hyro, boma, geohermal) n orer o prove a conan energy orce. For nance, wn energy combne wh olar energy prove a more conan renewable energy pply nce here are ronger wn rng he wner when here le n. The energy pply yem wll change o a ecenralze one n he fre: anoher very rac change, whch nee a hge effor from he echncal commny. I oe no make ene o elver a hge effor n procng renewable energy, an hen o conme carelely. We hol combne renewable energy procon wh ecreae energy conmpon. Accorng o he Blng Energy Daa book [7] an he Mnry of Hong, Spaal Plannng an he Envronmen VROM [77], blng conme p o roghly 4% of oal energy conmpon, an a lo of work gong on o rece h. Several poneer have - -

12 Inrocon nroce Zero Energy Degn (ZED) or Lower Energy Degn (LED) blng o avo he hgh energy conmpon by blng. However, he enham for LED blng no alway volnary b omeme enforce by law. In he Neherlan for nance, an energy performance coeffcen EP (ee VROM [77]) e o enforce a recon of energy conmpon by blng. The ZED or LED blng e a local -wh he blng negrae- renewable energy orce an have a low energy eman. Afer all, why no generae he energy where neee an a he ame me avo ranpor co an horzon pollon n he rral area? Moreover, mall renewable energy orce negrae wh blng rel n le energy conmpon from he ly. oneqenly, hey are rembre wh he comer prce of he ly: a roghly hree me hgher prce han he one pa by he ly for large renewable energy orce n rral area (Meren [46]). On he oher han, he move for negrang renewable energy orce wh blng are no only rven by he foremenone envronmenal e. Archecre ha alway reflece ocey ren an one of hoe ren nowaay ceranly concern he nee o e renewable energy (Meren [46]). Wn energy one of he poble renewable energy orce for he bl envronmen. omparon of earh rface of he bl envronmen wh he earh rface n rral area how ha he laer mch mooher. In oher wor, he bl envronmen ha a hgh roghne compare o rral area. Th hgh roghne cae a mall wn pee n he bl envronmen compare o he wn pee n rral area. Some local acceleraon of he wn neee n orer o have a vable energy yel n he bl envronmen. The wn pee aron aller blng can be apprecably hgher han he average free ream wn pee n he bl envronmen. Maybe poble o lze h effec an egn blng wh ecae wn rbne n orer o e he ncreae of he average free ream wn pee. The blng h more han a ma for he wn rbne: e a a concenraor of he wn energy for he wn rbne. Normally concenraor have obfl economc. I no vable o bl an expenve concenraor f one col ealy make he roor omewha bgger o creae more power. However, f he concenraor avalable who large exra expene a wh blng ha are bl anyway, clear ha one hol e h opporny. The caegory of wn rbne h prove wh concenrae wn energy wll be calle he Blng- Agmene Wn Trbne (BAWT). Th he explan he aeroynamc of he BAWT, concenraor blng an wn n he bl envronmen. From h aeroynamcal backgron, he promng confgraon for wn energy converon n he bl envronmen are enfe. Th he eal wh wn energy converon n he bl envronmen. I h conan hree man opc: wn energy, wn energy converon an bl envronmen. The bl envronmen analogo o an envronmen wh a hgh roghne. Therefore, a large par of h he alo nrcve for hoe neree n wn or wn energy converon n an envronmen wh a hgh roghne. The he ve no 8 chaper. Apar from he nrocon n chaper, he chaper an 8 avo lenghy mahemacal calclaon. haper collec he bonary conon of wn energy converon n he bl envronmen an form he eparre of he reearch. The bonary conon are analye n more eal n chaper 3 o 7, whch prove mahemacal moel on wn energy converon n he bl envronmen. The eale analy ar wh a ecrpon of he wn n chaper 3. haper 4 gve he analy ool e n he reearch. haper 5 o 7 prove mahemacal moel for he bac concenraor avalable n he bl envronmen. haper 8 mmarze he man rel of he he an compare he man opon for wn energy converon n he bl envronmen. A qck ng gelne bae on h mmary can be fon n Appenx J. - -

13 Bac heory Bac heory The bonary conon of he hree man opc: wn energy, wn energy converon an bl envronmen nroce he poble an mpoble for wn energy converon n he bl envronmen. From hee bonary conon, h chaper formlae he pon of eparre n earch of feable opon for wn energy converon n he bl envronmen.. Drag-rven roor ver lf-rven roor The wn rbne conere n h he are nene for he procon of elecrcy. The work h rece o he energy yel of he wn rbne n relaon o he co of he wn rbne. Moern wn rbne are e n open rral area an look alke: hey have a horzonal ax wh a hree-blae roor. Inee, for open rral area, h roor ha ome avanage compare o oher roor ype. Wn rbne for he bl envronmen however are e fferenly. oneqenly, hey o no necearly nee he ame roor. We h nee o y he roor n omewha more eal. There are wo way o conver wn energy no mechancal power n he roor ax. The converon can be realze wh a rag-rven roor, a lf-rven roor or a combnaon of boh concep: he hybr roor. The converon mechanm of wn power no mechancal power of he lf-rven an rag-rven roor fferen. A lf-rven wn rbne can be a horzonal ax wn rbne (Fgre, lef) or a vercal ax wn rbne (Fgre 4, rgh). The rag-rven wn rbne have a vercal ax (Fgre 4, lef). They how no clear wake. I wll be hown ha he roor concep mporan for he effcency of he converon. Th effcency or power coeffcen of a roor efne a P P = () A P 3 ρ In h eqaon, P he mechancal power a he roor ax, A he wep area of he roor, 3 ρ he power of he ncomng free ream wn per qare mere roor area, ρ he eny of he ar an he free ream wn pee... Drag-rven wn rbne The rvng force of he rag-rven roor orgnae from he fference n rag of (roang) blff boe. Fgre how an example of a rag-rven roor. I con of cp. The cp wh he phercal pwn e ha he lowe rag of boh cp. Hence, h cp pae power a move n he recon of he wn, whle he oher cp ha move ownwn proce power. I h characerc for he rag-rven roor ha he power proce wh a boy ha move ownwn., l ω, h R Fgre Example of a rag-rven wn rbne (lef) an chemac op vew of a mlar roor (rgh)

14 Bac heory. Drag-rven roor ver lf-rven roor The orqe Q of he roor ax follow from he fference n rag of he rnng blff boe. I clear from he chemac confgraon n Fgre ha he relave veloce on he rnng boe hol be e o calclae he rag. A he epce orenaon n Fgre (rgh), he orqe of he roor ax per qare mere roor area fon from ( ωr ) R, l ρ( R ) R Q = ρ + (), h ω where he conan, h an, l are he rag coeffcen of he roang blff boe or cp. A fr orer approxmaon for he power of he roor fon by amng a pre ranlaon of he cp wh ame conan relave veloce ωr an + ωr. The power P of he ax hen follow from P = Qω an he power coeffcen rea [ ( λ) ( λ) ] P =, h, l λ + (3) where λ = ωr he rao of he p pee of he roor ω R an he free ream velocy. Acally, he boe on he roor carry o a roaonal movemen. They elver a lower average orqe han calclae by (). Eqaon (3) h prove an pper vale for P of a rag-rven roor. If he rag of he boy ha move pwn neglece ( ), he maxmm power coeffcen P, max an fon o be, l = fon by fferenaon of (3) wh repec o λ, acheve a λ = 3 (4) 7 P,max, h 3 Hoerner [6] gve, h. 5, whch gve P, max 7.. In oher wor, he rag-rven roor conver a maxmm of % of he power avalable n he free ream wn no mechancal power a he roor ax... Lf-rven wn rbne The P, max of a lf-rven roor can be calclae wh e of he acaor concep nroce by Froe []. Th acaor efne a an energy-exracng k or plane of nfne mall axal ze wh a normal force on he rface ha ecelerae he normal velocy hrogh he k or plane. Th a moel for he roor of he lf-rven wn rbne a nfne hgh λ. The acaor locae n a ream be n parallel flow wh he enrance an ex of he ream be a nfne ance from he acaor (Fgre, rgh). op In h he he wor large an mall are e when relae o menon whle hgh an low are e n al oher cae

15 Bac heory. Drag-rven roor ver lf-rven roor Sreamlne on conrol area A e A A e Acaor Fgre Example of a lf-rven wn rbne an a chemac op vew of an acaor n parallel flow. We efne a conrol area bone by he ream be ha form he bonary of he flow hrogh he acaor. Thoma [7] an Glaer [] howe ha he prere force on he conrol area gve no conrbon o he momenm balance. Hence, conervaon of axal momenm whn he conrol area n he ame rroaonal an vergence-free flow how ha he hr force of he acaor fon wh F M = ρ A ρ A (5) e e Sbon of he rel from conervaon of ma whn he ream be: ρ A = ρ A = ρe Ae hen gve F M = A ( ) ρ (6) The hr force of a bare acaor eqal o he ma flow rae hrogh he acaor me he rae of change of velocy n he ream be. Th mporan rel wll be e hrogho he he. The hr force of he acaor can alo be fon wh Bernoll heorem on a reamlne pwn of he acaor an ownwn of he acaor (no acro he acaor becae energy exrace a he acaor). Th gve e ( ) F = ρ A (7) e Eqang (6) an (7) gve he velocy hrogh he acaor ( ) + (8) = e I hol be noe ha momenm heory brake own a e. In ha cae, he ame ream be efnon no longer val. In mo exbook, an ncon facor a e o enoe he eceleraon of he ar by he roor. In ha cae, he velocy hrogh he roor efne a = ( a). Wh (8), h e = a. Hence, he nce velocy nfnely far ownwn of he acaor: rel n ( ) a, wce he nce velocy a he locaon of he acaor: a. Th mporan rel, whch can alo be obane wh mplfe vorex heory (econ 4..), wll be calle bare acaor wake expanon. The power aborbe by he acaor can be fon by bracng he ole power from he nle power n he ream be

16 Bac heory. Drag-rven roor ver lf-rven roor P 3 3 A = ρ ρ e A e (9) Wh ma conervaon n he ream be ( ρ A = ρ A = ρe Ae ) h rel n a power coeffcen of Wh bon of (7), h rel n P ( ) ρ A e = () ρ A P 3 F = () ρ A 3 By eqang wh (), h how ha he power aborbe by an acaor can be fon by mlplyng he hr force of he acaor wh he velocy hrogh he acaor. Th anoher mporan rel ha we wll e hrogho he he. Wh e of (8) he power coeffcen fon n () can be wren a = e + e P () P,max or he Lanceer-Bez lm, afer Lanceer [37] an Bez [4], fon by eng he ervave of () wh repec o e eqal o zero. Th prove e, op =, whch gve 6 =.59 (3) P, max 7 The acaor aborb a maxmm of approxmaely 59% of he power avalable n he free ream wn. The ervaon of h follow from he e of he acaor concep, whch P, max a moel for he ealze lf-rven roor. We h fon. 59 for a lf-rven roor. P, max..3 Hybr-rven wn rbne The Savon wn rbne wh a gap beween he rnng blff boe hown a he rgh of Fgre 3. 3 D g Fgre 3 Example of a Twe Savon (lef) an op vew of a Savon roor wh gap beween he rnng boe (rgh)

17 Bac heory. Drag-rven roor ver lf-rven roor Th evce manly rag-rven. However, e o he gap n h parclar Savon confgraon, he blff boe are rven by rag an con, whch make h parclar confgraon hybr-rven. The aonal rvng force cae by con reponble for a hgher P, max han fon wh a pre rag-rven roor. The hghe P, max fon for a wo blae Savon wh a gap of g D =.. 5. Mearemen a Sana laboraore howe ha h parclar confgraon ha. 4 a λ. 9 (ee Blackwell e al. [5] or Parachvo [53]). P, max op..4 omparon of he wn rbne The projece blae area of he rag-rven roor approxmaely eqal o he roor area. The projece blae area of he lf-rven roor a fracon of ha area. The fference n maeral e clear from he followng fgre. Fgre 4 Example of wo Vercal Ax Wn Trbne. Lef, a rag-rven an rgh a lf-rven wn rbne. From he prevo econ we know ha he lf-rven roor ha a mch hgher he rag-rven roor. The rag-rven roor h combne a low e whle he lf-rven wn rbne combne a hgh P, max P, max P, max han wh a hgh maeral wh a low maeral e. Perhap, he maeral of a rag-rven roor can be cheaper b n orer o proce energy a 6 3 he ame co, he rag-rven wn rbne nee o be a facor cheaper. Th only poble for rare aon nce he man fference beween he rag-rven an lfrven wn rbne he roor, whch co approxmaely % of he oal wn rbne (Ha [4]). Hence, he lf-rven wn rbne ha he be propec o elver energy a low co. The rag-rven wn rbne herefore abanone for he generaon of elecrcy n h he. P,max of he Savon roor wh gap apprecably hgher han ha of he rag-rven roor. ompare o he lf-rven roor, he Savon wh gap ll cople a hgh maeral e wh a moerae P, max. oneqenly, he Savon wh gap hol be he fr o be examne a an alernave for he lf-rven roor. Th he foce on lf-rven wn rbne.. onran Th econ eal wh a fr analy of he bonary conon of wn energy converon n he bl envronmen ha are nroce by bl envronmen, wn rbne an blng. Table Man bonary conon for wn energy converon n he bl envronmen. Bl envronmen Wn rbne Blng Wn Roor ze Acceleraon Noe emon Yawe flow Reonance Vbraon Re nmber effec - 7 -

18 Bac heory. onran Shaow flcker Safey They prove he conor of wn energy converon n he bl envronmen an are analye n more eal laer on n h he... Bl envronmen Wn One of he mo omnan effec of he hgh roghne n he bl envronmen ha cae a low average velocy. Wn energy converon become feable for hgher wn pee. The combnaon: wn energy converon an bl envronmen h no o obvo from an energy yel pon of vew. Se well above he average blng hegh are mo promng. A hoe e, he wn pee hgh enogh o harve he wn energy. Noe emon Bron e al. [8] ae ha he aeroynamc noe emon of a lf-rven HAWT approxmaely proporonal o he ffh power of he p pee. A low p pee h ere for a wn rbne n he bl envronmen. Frhermore, all orce of vorex heng from he roor hol be avoe nce he velocy fference an coneqenly he prere fference n he flow are reponble for he noe emon. Sall of he roor blae hol h be avoe. The oal noe level a home ha o ay below ome maxmm allowable noe level enforce by law. The allowable noe level a home rng he ngh he lowe noe level. I approxmaely 4 B(A) n he Neherlan. Bron e al. [8] how ha h eqal o he noe level n a rral area wh a wn pee of approxmaely 9 m/. Accorng o Verkerk [73] he 4 B(A) alo eqal o he noe level of a refrgeraor a m ance. Bron e al. [8] gve ome eqaon ha can be e for he calclaon of he noe level. If he wn rbne moelle a a pon orce wh phercal preang, he noe a a ceran ance r from he orce L P a locaon P how a recon n noe level fon wh L P ( π ) = L log 4 r (4) W where L W he orce on prere level n B. Th how a 6 B ecreae per oblng of ance o he orce. A an allowe on prere level of 4 B, (4) gve he followng allowe L W a a fncon of r. Table Allowable L W of a wn rbne n he bl envronmen a a fncon of he ance r o a on prere level of 4 B. The oal on prere level r [m] Allowe L W [B] L P, n from n orce a a ceran pon can be calclae wh - 8 -

19 Bac heory. onran L P, n j = n j=.l P ( j ) = log (5) Sppoe we have a backgron on prere level of 4 B an wn rbne wh a relng on prere level a locaon P of 4 B each. Then he oal on prere level e o backgron noe level an wn rbne 5 B. Normally he on prere level A-weghe -enoe by B(A)- o accon for he elecve envene of he hman ear. Th A-weghng rece on prere wh freqence oher han Hz becae Hz he mo enve regon of he hman ear. The roor generae a broaban aeroynamc noe wh omnang freqence of everal khz perceve a a whng noe. oneqenly, for he omnang freqence of a wn rbne, he A-weghe fler oe no have mch nflence on he noe level. The on prere level n B aron Hz fon wh (4) an (5) are roghly eqal o he on prere level n B(A). There are wo frher mporan e for noe emon. Fr, n conra o low freqence, he hgher freqence of everal kloherz have a rong recon epenency. They can only be hear whn he ragh pah of he orce or a reflecon from ha ragh pah. Seconly, alernang noe level are a nance. Th kn of noe emon hol h be avoe. Vbraon A HAWT nce vbraon n he blng wh freqence eqal o he roaonal freqency of he roor n H an freqence Bn H where an neger an B he nmber of blae of he roor. The lowe nce vbraon n H a coneqence of ma nbalance an fference n aeroynamc loa of he blae. The ower haow an rblen rcre cae he hgher freqence. More pecfcally, he ower haow cae an nce freqency Bn H becae -a each revolon-, B blae move hrogh he low velocy regon pwn or ownwn of he ower. Frhermore, he blae ha move hrogh a rblen rcre wh a velocy ha ffer from he average velocy nce a freqency Bn H. Th phenomenon calle roaonal amplng. If he blae move hrogh wo regon wh a velocy fferen from he average velocy, hey nce a freqency Bn an o on. We h fn an nce freqency cae by roaonal amplng eqal o Bn. A VAWT wll nce freqence n he blng a he roaonal freqency n V becae of ma nbalance an fference n aeroynamc loa of he blae an freqence BnV cae by roaonal amplng. Hence, compare o he HAWT, he roaonal amplng freqency of a VAWT wce a hgh, becae he blae of he VAWT pa he rblen rcre wce: once a he pwn e of he VAWT an once a he ownwn e of he VAWT. are hol be aken o avo freqence of he HAWT or VAWT cloe o he egenfreqence of he ppor rcre (blng roof, blng wall, ma, ec.) on whch hey are mone (ee econ..3 below reonance). Shaow flcker Saon where he wn rbne blae are whn he rec pah of he nray o he eye or reflecon of he n ray on he wn rbne blae hol be avoe. The laer mple o olve wh ll pan. Saon where he blae are whn he rec pah of he n ray are a nance f he oberver cloe o he wn rbne an a vble freqence H H - 9 -

20 Bac heory. onran below ome Hz (ee Bron e al. [8] for more eal). ompare o he Darre, HAWT have more problem n avong hoe low freqence becae of he ngle-blae paage where he Darre ha a oble-blae paage beween he nray an he oberver nea. The HAWT h more lkely o cae hnrance becae of haow flckerng below Hz... Wn rbne Roor ze The bl envronmen aeroynamcally ffer from rral area by he hgh average hegh of he (roghne) obacle.e. he average blng hegh H. The large rblen rcre n he wn, cale wh h average characerc ze of he bl envronmen. Thee bg rblen rcre wll change he local wn recon n he bl envronmen wh a me cale τ rb eqal o he characerc ze of he rblen rcre H ve by he average velocy. H τ (6) rb = For H = 5 m an = 5 m/ (a lower no o nereng from an energy yel pon of vew) we fn τ rb = 5 for he large me cale of mechancal rblence n he bl envronmen. In he momenm heory ha lea o he Lanceer-Bez lm, he ream be where he energy exrace of nfne lengh. In mlae vco flow, a mlar ream be expan p o a ceran ameer an hen ecreae n ameer agan. Th a coneqence of he reenergzng/ peeng p of he flow n he wake by he flow aron he wake. The area where he reamlne of he ream be are roghly parallel aken a he vral begnnng or en of he ream be where he energy exrace. Wh h efnon for vco flow he ream be lengh where he energy exrace fne. A FD calclaon wh an acaor (Appenx A) how ha h efnon gve a vral ream be lengh of approxmaely x me he ameer of he roor ( 6 D ) n vco flow. A me cale τ fon by 6D τ = (7) h characerze he me neee o creae a qa-aonary ream be for an acaor n whch he Lanceer-Bez lm applcable. For τ rb < 6D, neay effec become mporan an he Lanceer-Bez lm bae on eay flow no longer applcable. In orer o avo he fa change n loa ha are cople wh he neay effec we nee o nre ha τ rb > 6D. Wh (6) an (7) h rel n τ rb > τ D 6 < H (8) For H = 5 m we fn D < 4 m. The ream be ame o be efne by he average pah of parcle n he flow wh a very mall ma. - -

21 Bac heory. onran Of core, h calclaon can only gve a very cre emae of he able ameer of a wn rbne for he bl envronmen b he pon clear: he wn rbne for he bl envronmen lme n roor ze. Yawe flow Unl now, he qeon wheher he lf-rven wn rbne a Vercal Ax Wn Trbne (VAWT) or a Horzonal Ax Wn Trbne (HAWT) ha no been menone. Th nee rrelevan for he rercon n ze a a ngle acaor can moel boh roor ype. The neceary yawng of he HAWT gve aonal eman. Sppoe ha we have a roor ha flfl (8). Then, f boh roor ype are moelle a a ngle acaor n a ngle ream be, he lf-rven VAWT a well a he lf-rven HAWT can acheve he Lanceer-Bez lm. However, e o he change n wn recon, he HAWT nee o yaw a lea every τ rb econ. For an average blng hegh H = 5 m an = 5 m/, (6) how ha h approxmaely every 5 econ. Th very emanng becae of he nera of he HAWT aron he yaw ax ha preven a fa yaw. rb rb Ofen, mall HAWT have a vane bae yaw yem. The combnaon of nera aron he yaw ax an vane force rel n an egen-freqency of he yaw yem. Reonance a h egen-freqency poble f he freqency cople wh he large rblen rcre f = τ eqal he egen-freqency of he yaw yem. Obervaon of mall vanege HAWT operae n he bl envronmen confrm h by howng large noal yaw movemen ha ofen rel n pnnng aron of he roor (Vre [75], Plomp [58]). The roor of a HAWT n he bl envronmen h freqenly malgne wh he flow. In oher wor, he roor perform n an average yaw ha rel n an average power rop compare o ragh flow (econ 5.). Frhermore, he freqen yawng rel n an accompanyng freqen loa change, whch rel n an ncreae fage loa. The lfrven VAWT or Darre roor ha oe no nee o yaw h preferre for he general wn conon 3 n he bl envronmen. Safey Wn rbne for he bl envronmen are mean o operae n a ene poplae area. A a coneqence he probably of njre n cae of malfnconng of he wn rbne hgher han ha for a wn rbne n rral area. In orer o arrve a he ame afey level for repaer a wh rral wn rbne, a wn rbne for he bl envronmen rerce o a maller malfnconng probably. B he rblence level an a a rel, he fage loa on he blae are hgher n he bl envronmen ha n rral area. A a coneqence of he rercon o a maller malfnconng probably an he hgher fage loa of he blae, he blae egn a very mporan e. In orer o ecreae he rk of blae rp-off, ome manfacre mon eel cable whn he roor blae (Trby [89]) whle oher hro he roor wh a eel cage (Wnwall [9]). Reynol nmber effec Reynol effec play a major role for wn rbne n he bl envronmen. A we have een n h econ, he roor of wn rbne for he bl envronmen nee o be mall. The coneqence of he combne low nmber of revolon (noe) an he mall wn rbne ze ha he Re nmber of he flow aron he roor blae mall. ompare o hgh Re nmber, a low Re nmber cae he rag of blae o ncreae an he lf o ecreae. Th combnaon of lf recon an rag ncreae cae a large ecreae n power op. Mearemen n he wn nnel on he prooype of he Darre Trby n 3 Parclar wn conon ha occr n concenraor may reqre fferen roor concep. - -

22 Bac heory. onran Meren [44] how ha a egn λ = 3 wh D =. 5 m gve an accepable. 3 a P, max = m/. Fgre 5 epc he mearemen rel, whch how a rong epenence of on he average Re nmber 4 5. No rprng becae he Re nmber orer an a P,max h parclar Re nmber he lf an rag of he e Naca 8 arfol change racally 5 wh Re nmber. A egn Re nmber above Re herefore erable. However, he noe emon rerc he egn λ o low vale. A goo gelne fon by lmng he egn o λ ~ 4. Th gve an accepable maeral age, power coeffcen an noe emon P [.] mearemen f Re/ 5 [.] Fgre 5 P, max from mearemen a a fncon of he Re nmber on he blae of he Trby prooype 5 ee Meren [44]...3 Blng Acceleraon All boe cae an acceleraon of he free ream wn pee a ceran locaon cloe o he boy. A hgher ance o he boy he velocy approache he free ream wn pee. In orer o prof from he acceleraon, he wn rbne hol h be cloe o he boy an ze hol be lme compare o he blng ze. Larger BAWT ha are able o acheve apprecable are coneqenly cople wh aller blng. P, max A he velocy aron a boy approache he nrbe velocy a larger ance from he boy, a roor cloe o a boy operae n a paal non-nform flow. Th cae changng blae loa an coneqenly fage of he blae. Reonance Ell [] gve mearemen of he egenfreqency f e of blng a a fncon of he f e = 46 H. Small wn rbne have blng hegh H. The large amon of aa f wh roaonal freqence of everal Hz. Reonance of he oal blng h only lkely for low blng hegh. Fornaely, h eem no bg problem nce low blng are no o 4 The average Re nmber for he arfol bae on he velocy λ. 5 The prooype ha he followng roor characerc: hegh=.[m], ameer=.5[m], chor lengh=.57[m], arfol=naca

23 Bac heory. onran nereng for negraon of wn rbne an blng a he average wn pee mall cloe o he earh rface. The egenfreqence of par of he blng are more lkely o gve problem. The egenfreqence of floor, wall, wnow, ec., are conerable hgher an mall wn rbne nce freqence ha are able o cae reonance n par of he blng. Sch reonance hol be avoe by choong very ff ppor rcre for he mall wn rbne o ha he egenfreqency of he ppor rcre well above he nce freqence by he mall wn rbne..3 The Blng-Agmene Wn Trbne In rral area, he average wn pee conerably hgher han he wn pee n he bl envronmen. Ye everyone know ha he wn pee n he bl envronmen cloe o hgh blng omeme aonhngly hgh. Sch locaon wh concenrae wn energy are perhap nereng for wn rbne. Th ea rel n he man reearch opc of he he: he blng ha concenrae he wn energy for he wn rbne an he wn rbne ha operae n he concenrae wn. The reearch opc h concern: he performance of a wn rbne a a locaon near a blng where he blng concenrae he wn energy. The name Blng-Agmene Wn Trbne or BAWT frhermore refer o he wn rbne ha operae n he wn energy concenrae by he blng. Accorng o., he ze of he BAWT lme by he ze of he blng n rec envronmen. Among oher, qalavely hown ha he characerc menon D of he blng ha concenrae he wn energy for he BAWT rerc he BAWT ze. Laer on, hown ha he rercon of he BAWT ze o O(.D ) are an effcen e of he concenrae wn energy. There are hree bac BAWT confgraon ha can be nghe by her aeroynamc operang prncple: cloe o (on op or bee) a blng, beween arfol-hape blng or n a c hrogh a blng. Fgre 6 Ar mpreon of BAWT. From lef o rgh: cloe o a blng, beween arfol-hape blng (valaon Loren Aanen, DHV) an n a c hrogh a blng. All oher poble BAWT confgraon have operang prncple bae on hee hree man aeroynamc prncple (Meren [4]). ombnaon of he aeroynamc operang prncple are alo poble. The aeroynamc operang prncple of hee BAWT are ce n he econ 5, 6 an

24 Bac heory.4 The we H-Darre wn rbne In he econ.,. an.3, hown ha a promng wn rbne for he general wn conon n he bl envronmen : a mall (< 4m), lf-rven, vercal ax wn rbne, wh a low λ (approxmaely ~4). Sch a wn rbne known a he Darre wn rbne afer he French nvenor G.J.M. Darre (U.S. paen offce, 93). We wll refer o h wn rbne a Darre. Parclar wn conon ha occr n concenraor a hown n Fgre 6 may reqre oher roor concep. The Darre roor ha an eggbeaer hape (Fgre 7), whch rel n pre enon force n he blae. Fgre 7 A Darre. The Darre roor no elf-arng becae, a low λ, he average abole angle of aack on he blae hgh. Hence, a low λ he blae are alle an he lf force of he blae mall compare o he hgh rag force of he alle blae. A hgher λ, he blae are below all o ha he lf force ncreae whle he rag force of he blae ecreae. A mple egn gelne for he Darre can be fon n Wlon & Laman [84]. They moelle a Darre wh a ngle acaor an mlple ream be. The mlple ream be approach enable a local performance o be calclae. Th neceary for an accrae moel nce he par of he Darre where he blae move parallel wh he wn ffer n operaon from he par where he blae move perpenclar o he wn. They eermne ha, n orer o acheve P, max, any lce of he roor ha o flfl Bcλ =.4 D (9) In h eqaon, B he nmber of blae of he roor wh chor lengh c an local roor ameer D. Th mple geomery conon fon wh an ame lf coeffcen of he blae eqal o l = π nα where α enoe he angle of aack on he blae. In oher wor, ame ha he flow aache o he blae. Th a val ampon for hgh λ. Accorng o (9), for P, max an ceran λ an B, c hol be proporonal o D. In oher wor, c nee o be maller a he op an boom of he roor nce D goe o zero here

25 Bac heory.4 The we H-Darre wn rbne Th of core no allowe nce he hghe enon force occr a he blae roo. Moreover, he blae are commonly manfacre wh he exron proce, whch rel n a conan c. A Darre wh a conan D eem o be favorable: he H-Darre 6 nroce. Fgre 8 An H-Darre. Alhogh h egn eem o work well a he enre blae pan, ffer from fne apec rao effec of he blae. The blae creae a prere fference acro rface an h nce a flow from he hgh-prere o he low-prere e of he blae. The relng vorce a he p of he blae rece he lf of he oal blae an wh h ecreae he of he H-Darre. Frhermore, ro are neee o keep he blae fxe a a ceran P,max D. Thee ro o alo rece P, max a hey ncreae he rag of he roor. Fornaely, he loe cae by he jon of he ro an blae can be rece by changng he confgraon hown n Fgre 8 o he confgraon hown n Fgre 9 (Meren [44]). In h confgraon, he blae ben o he ro wh a ceran ronng ra an are no longer exenng oe he ro a hown n Fgre 8. Fgre 9 Recng he loe cae by he jon of ro an blae. Th confgraon choen a mo promng roor for general e n he bl envronmen. The H-Darre blae move from a poon wh zero angle of aack a he e of he H- Darre o a poon of maxmm angle of aack a he pwn an ownwn e of he H- Darre. Accorngly, he lf force on he blae vare perocally. Th cae a peroc loa on he H-Darre b moreover h can rel n peroc noe emon. An o nmber of blae ha he coneqence ha he loa of he blae no n phae. Th cae a maller orqe rpple n he roor ax, whch nce maller force on he blng compare o he confgraon where he blae are n phae wh her orqe. The H-Darre h be fe wh an o nmber of blae. Frhermore, a few blae a poble are ere becae each blae ncreae he roor co. Only one blae rel n nrealc large 6 The name of he H-Darre wh conan roor ameer bae on he H-hape ha nroce by he confgraon of wo oppoe arfol wh a ro o fx he arfol a conan D

26 Bac heory.4 The we H-Darre wn rbne wng of he blae n orer o acheve a conan orqe. We h chooe hree blae for he bl envronmen H-Darre. Peroc loang of he blae a well a peroc noe emon hol be avoe. Peroc loang of he blae rel n hgh fage loa an vbraon an peroc noe emon a nance. Whn he ampon an lmaon of he moel ha preene hereafer, can be hown ha a w of he H-Darre roor wh angle θ able o change he peroc loa a he roor ax an peroc noe emon o a conan loa an noe emon (Meren [44]). The confgraon of he H-Darre wh we roor epce n Fgre. D z H θ Fgre onfgraon wh a we H-Darre. The blae hae. Agan, we e he mple moel of Wlon & Laman [84] for a Darre wh hgh λ an lf coeffcen of he blae l = π nα. Bae on hee ampon an wh e of a mlple ream be ngle acaor approach, hey eermne ha he orqe Q of a lce of a Darre per blae per mere Darre hegh can be fon wh ( a) ( θ ) wh local ncon facor of he Darre gven by Q = ρπcd n () θ Bcλ a = nθ () D Accorng o he confgraon, he angle θ of a ceran blae a a ceran hegh z can be efne by π z θ θ = θ + ( j ) + () B where j enoe a ceran blae ( j = B ). The oal orqe of he roor wh B we blae can be fon wh H - 6 -

27 Bac heory.4 The we H-Darre wn rbne Q o z= H B ( = ) cd ( ) ρπ a n θ z (3) H j= z= Th eqaon ogeher wh he eqaon () an () may be e o calclae he orqe rpple. I reveal ha he oal orqe of an H-Darre wh we blae conan ( Q no a fncon of θ ) f π θ = (4) B The calclaon ha lea o (4) are carre o nmercally. Whn he moel ampon, (4) efne a kn of helx hape of he H-Darre ha cae a conan loa an conan noe emon. The followng fgre how he rel of he egn of a able wn rbne for he bl envronmen: a hree-blae helx hape H-Darre calle Trby. o Fgre Trby on a roof n Tlbrg, The Neherlan (rgh: overvew, lef: zoom n). The aeroynamc egn of he prooype of Trby ogeher wh everal wn nnel e for verfcaon of he egn are gven n Meren [44]. The flow on he roof of harp-ege blng no parallel o he roof. The flow approache he H-Darre on he roof of mo common qare blng from below. One of he e n Meren [44] howe a power ncreae for flow from below. Th make he H-Darre very able for operaon on a roof of harp-ege blng. Secon 5. goe no eal on he behavor of an H-Darre n ch flow. The egn gelne n h econ can be e for a fr menonng of he H-Darre. However, hey are oo mple for egn of a prooype. I h eem we o accompany a rogh egn of he H-Darre for he bl envronmen wh wn nnel e n orer o verfy he emane performance

28 3 Wn characerc 3 Wn characerc In h chaper, he propere of he wn ha concern wn energy converon n he bl envronmen wll be analye. The chaper ve no hree econ. Secon 3. eal wh he global propere of he wn n he bl envronmen. Secon 3. zoom n a he local propere of he wn aron blng. Fnally, econ 3.3 eal wh he flow a he blng. 3. Global wn n he bl envronmen Th econ analye he wn n he bl envronmen ha nflence by he average roghne of he bl envronmen. Becae of he average characer, he analye are no applcable cloe o he nval roghne elemen ch a blng. Nmero propere of he bl envronmen an amophere nflence he wn pee. The wn pee n he bl envronmen h how a very complex behavor ha ffcl o moel. Only few parclar aon allow moellng of he amophere an wn pee n he bl envronmen. Th ofen forgoen an rel of calclaon are e who he rercon of he moel. 3.. Log-law In orer o flfl he no-lp conon a he earh rface, he wn pee ecreae o zero a he gron. Th rel n he o-calle amopherc bonary layer (Fgre ). z z Fgre Skech of an average horzonal velocy profle a a fncon of he hegh z n he amopherc bonary layer evelope over gralan wh roghne hegh z. Log-law a low roghne Mechancal rblence he man rvng force for he rcre of he amopherc bonary layer above an average wn pee of 6 m/ a m hegh (Wernga & Rjkoor [79]). Above h wn pee, he flly evelope rblen amopherc bonary layer moly neral an emperare effec can be neglece. Frher obervaon how ha he flow n he neral bonary layer can be ve no wo regon wh eqal hear re b fferen calng: an nner an an oer layer (Pranl [59]). Machng of he velocy graen n he oer an nner regon rel n a logarhmc bonary layer profle or log-law (ee for nance Newa [5]). ( z) = * ln κ z z (5) In (5) (z) enoe he wn pee a hegh z, * he frcon velocy, κ he Von Kármán conan aken a κ =. 4, z he hegh above he earh rface an z calle he earh rface roghne hegh. The ervaon of he log-law how ha z a meare - 8 -

29 3 Wn characerc 3. Global wn n he bl envronmen for he mall rcre n he bonary layer. The log-law val p o a lea 5~ m (Panofky & Don [5], Sm & Scanlan [67]) n a neral amophere wh rong wn. Wn map, ch a for nance hoe preene n he wn ala [8], gve he average reference wn pee a z ref = m an z, ref =. 3m. Th reference wn pee calle he poenal wn pee p. The velocy for a fferen z an z can be fon by machng of he velocy from reference aon an new aon a 6 m (Wernga & Rjkoor [79]). The velocy a z an z fon from he reference aon h rea z ln z ( z) =.3 6 ln z p (6) Log-law a hgh roghne For he bl envronmen, he log-law nee a mofcaon o accon for he hgh roghne. Fng of he log-law wh mearemen of (z) well above he average hegh of he roghne elemen how a new vral rface level a + z above he earh rface (ee Fgre 3), where calle he placemen hegh. z z mn + z H Fgre 3 Skech of an average local horzonal velocy profle n he amopherc bonary layer evelope over he bl envronmen (ol lne). The average local horzonal velocy profle for gralan roghne hown n Fgre alo hown n orer o valze he profle fference (oe lne). The log-law for he bl envronmen h rea = * z ( z) ln κ z (7) loe o he roghne elemen, he wn pee nflence by he local roghne an he log-law ceae o be val. The hgh roghne n he bl envronmen nflence he wn pee well above he average blng hegh. I herefore naccrae o ame a hegh of 6 m above he cy rface where he average velocy approxmaely conan a wa one for he ervaon of (6). The calclaon of he velocy n he bl envronmen reqre a fferen procere ha howe n econ 3... Reference on np for he log-law Several reference (Table 3) prove recommene vale for z mn, an z. Ther nformaon refer o he average hegh of he roghne elemen H an he percenage of he oal area occpe by he roghne elemen A H. Only he eqaon for z gven by - 9 -

30 3 Wn characerc 3. Global wn n he bl envronmen Wernga & Rjkoor [79] refer o he hegh H efne a he hegh of he mo mporan reglarly occrrng roghne elemen. Table 3 Recommene vale for, z, z mn an H fon n lerare. Reference [m] z [m] Pannofky & Don [5].7H ~. 8H Wernga & Rjkoor [79].5H ~. 75H.5A H H z + Sm & Scanlan [67] z H κ ESDU 86 [4] H 4.3z( AH ).5 f..8 ESDU 86 [4], Typcal for cy A H z mn [m] H [m] z H.5z.8 5 Dplacemen hegh ESDU 86 [4] prove he mo ol ba for a calclaon of becae her rel bae on nmero wn nnel mearemen. = H.3z ( A ) f..8 (8) 4 H Eqang of (8) an he ypcal = H.5z accorng o ESDU 86 [4] gve A =.4. Typcally, 4% of he oal area of a cy occpe wh blng. H Roghne hegh If goo mearemen of ( z) a fferen hegh are avalable, z hol be calclae from a f of he log-law wh he mearemen. Ye, calclaon of z for fferen ce how a large varaon an are bjec of con (Wernga [78], e W [85]). Apparenly z vale above ce are ffcl o emae. Who mearemen, z can be emae by comparon of appearance of he roghne wh reference pcre. Sch reference are nfornaely no avalable for he bl envronmen. The eqaon for z gven by Wernga & Rjkoor [79] eem o be efl. Ye, he e of z =.5A H H omewha ffcl becae of he bjecve efnon of H (ee efnon j before Table 3). Some mall change make more efl. Sppoe he rbon of he roghne elemen how a anar evaon n blng hegh σ. Then, f A whle σ = (enely packe H blng wh he ame hegh) he change moel hol gve z. I wll h be ame ha A H H z ( ) = cz σ H AH H (9) where c ( ) z σ H ha o be emae. The bonare of he area A H hol be approxmaely wo klomere pwn from he pon where z emae n orer o prove an average vale of he roghne acro whch he bonary layer evelope. Wh z =. 8, A =.4 an H = 5, (9) gve H c ( σ ).8 (3) z H = H - -

31 3 Wn characerc 3. Global wn n he bl envronmen whch ypcal for a cy accorng o he np. e wh oher σ H gve a fferen c ( ) z σ H. In cae of lack of aa, c ( ) z σ H can be aken eqal o he ypcal.8. Sbon of (9) n (8) how a parabolc characer of a a fncon of A H wh a mnmm a A H =.5. The orgn of he parabolc characer perhap fon n he growh of a recrclaon zone pwn an ownwn of he roghne elemen a A H =.5. Sch recrclaon zone enable a mooher flow or kmmng flow (Han [8]) acro he roghne elemen wh le rblen mxng an coneqenly a lower. Mnmm hegh for log-law Sbon of he ypcal vale of ESDU 86 [4] (Table 3) n he eqaon for gve he followng recommene vale for z mn n a cy. Table 4 Reference vale for z mn n a cy. Reference z mn [m] Pannofky & Don [5] 8 Wernga & Rjkoor [79] 39 ESDU 86 [4] 34.5 The z mn = 8 [m] of Pannofky & Don [5] far below he ypcal vale for n a cy accorng o ESDU 86 [4]. Hence, her eqaon rejece. The z mn vale accorng o boh oher reference are comparable b he eqaon of ESDU 86 [4] wll be e a bae on he nmero mearemen ha lea o (8). z mn hol accorngly be calclae by z = (3) mn.5 z mn Wn energy converon below z mn nee a (me-conmng) y on he ably of each pecfc e whle wn energy converon above z mn can raghforwarly be bae on (7). Smmary np for he log-law The followng eqaon an reference vale wll h be e for calclaon wh he loglaw. Table 5 Recommene vale for he log-law n a cy. Qany Eqaon Eqaon No. Typcal for a cy A.4 H H 5 c ( ) z σ H.8 = H 4.3z( AH ) (8) 3 z z ( ) = c σ A H (9).8 mn z H H z ( H, ypcal ) = wh c σ. 8 z zmn =.5 (3) 35 The eqaon of he log-law, he np for he log-law an he reference vale hol be e wh grea care. Thee eqaon are all em-emprcal an oo mple o moel he complex - -

32 3 Wn characerc 3. Global wn n he bl envronmen flow phenomena n he bl envronmen. They are herefore rerce n applcaon. Ye, hey are ofen fon o be e who her rercon, mply becae of lack of oher aa or n gnorance. Ther rel are of core qeonable n ha cae. 3.. Sep n roghne hegh Le ame ha he qane ha eermne he log-law oe a cy, where z = z mall, are eermne va -for nance- he wn ala [8]. We wol lke o fn he wn pee n a cy where z = z hgh. The flow ha ener a cy experence a ep n roghne from z o z. A new bonary layer, calle nernal bonary layer, wll evelop a he new z. The effec of he ep n roghne are no nananeoly preen n he whole amophere b are lme o he hegh h of he nernal bonary layer. Oe he nernal bonary layer, he amophere k behave accorng o he pwn roghne z (Sm & Scanlan [67]). De o rblen mxng, h wll grow ownwn of he roghne change. The aon epce n Fgre k 4. Exernal bonary layer z z + z z mn z h k Inernal bonary layer x Roghne Roghne Fgre 4 Skech of a bonary layer profle change e o a ep n roghne hegh. Woo [86] gve an emprcal moel for he growh of he nernal bonary layer bae on a menonal analy an a conerable amon of aa x hk ( x) =.8z,max z,max.8 (3) where z,max he hgher of z an z. Eqaon (3) val for mooh-o-rogh an rogh-o-mooh change n he wall regon h k ( x) <.δ, where δ enoe he bonary layer hegh, whch approxmaely m for neral flow an rong wn acro a cy. Afer ome ranen effec cloely ownwn of he ep n roghne, he nernal bonary layer wll be logarhmc n a neral amophere. Accorng o Sm & Scanlan he log-law - -

33 3 Wn characerc 3. Global wn n he bl envronmen for he nernal bonary layer can be apple f 5 m x 5 km. Whn he approxmaon of he moel of he nernal an exernal bonary layer a epce n Fgre 4, machng of he velocy above an below h ( x ) h gve he velocy n he nernal bonary layer. k h k z ln ln z z ( z) =.3 6 hk ln ln z z p (33) For freqenly changng roghne hegh, Wernga & Rjkoor [79] arge ha h k = 6 m hol be aken nce roghne change occr from hgh o low roghne an va vera o ha her effec cancel. Sbon of h k = 6 m n (33) rel n (6) erve n econ Trblence The rblence neny I efne a I σ = = (34) wh = + an where σ enoe he anar evaon of he wn pee. The anar evaon moly wren n erm of *. For a neral amophere, Pannofky & Don [5] gve Sbon of (7) an (34) n (35) gve I z ln z σ =.4 (35) f * z > z Th how an ncreang rblence level wh ncreang roghne an ecreang hegh o he earh rface. Sbon of he ypcal vale for he bl envronmen from Table 3 an e of (9) an (3) wh z = z gve Imax.77. mn 3. Local wn n he bl envronmen Th econ ecrbe he flow propere ha are oberve aron blng. Frly, he blng hape allow a characerzaon n aeroynamc, blff an bln blng. Seconly, he flow aron he blng characerze by he e of he Reynol an Srohal nmber. Fnally, he probably rbon of he velocy a a ceran locaon aron a blng erve. 3.. Blng characerzaon A fr characerzaon of blng hape nghe aeroynamc blng from blff blng. Aeroynamc blng have a hn bonary layer aache o he rface of vrally he whole blng. They are characerze by a mall wake. Blff blng are characerze by early eparaon of he bonary layer from her rface an a large wake (ee Fgre 5). Bln blng how a combnaon of he flow phenomena of blff an aeroynamc blng. mn (36) - 3 -

34 3 Wn characerc 3.Local wn n he bl envronmen Fgre 5 FD calclaon of he reamlne aron an aeroynamc boy (lef) an a blff boy (rgh) n parallel flow. Flow recon from lef o rgh. I hol be noe ha he characerzaon of blng a aeroynamc or blff epen on he flow recon. An aeroynamc blng for flow from recon φ (Fgre 5, lef) wll change o a blff blng for flow from a recon perpenclar o φ (Fgre 5, rgh). A a blng wh harp pwn ege, he bonary layer eparae a he pwn ege (Schlchng [63]) an eparaon bbble are forme on he e an on op of he blng. The man ream eflece aron he blng an a large wake ownwn of he blng forme (ee Fgre 6). Sch a blng h calle a blff blng. Fgre 6 FD calclaon of he reamlne aron a harp-ege blng. The rag force of a blng mae p of wo conrbon. The fference n local velocy aron a blng rel n prere fference a he rface of he blng. Th gve a conrbon n he oal rag calle prere rag. The no lp conon a he rface of he blng cae he conrbon n oal rag calle vco rag. The prere rag of an aeroynamc blng fon o be zero by amng poenal flow aron he blng (Paraox of D Alember, ee for nance Bachelor []). oneqenly, he vco rag omnae he prere rag a an aeroynamc blng. A a blff blng, he vco rag of he ame orer a he vco rag a he aeroynamc blng b he prere rag hgh a a rel of he eparae bonary layer. oneqenly, a a blff blng, he prere rag omnae he vco rag

35 3 Wn characerc 3.Local wn n he bl envronmen 3.. Flow characerzaon Th econ gve bac flow propere ha are fon from he menon of he flow, he Re nmber an he Srohal nmber. 3D ver D flow Boe wh her hegh, wh an lengh approxmaely eqal are characerze a 3D boe. Boe wh one ze very large compare o he oher ze can be characerze a D boe. The flow propere aron D or 3D boe are fferen a wll be hown laer n h econ. Accorng o mearemen mmarze n Hoerner [6], a D flow approxmae aron boe ha have one ze more han en me hgher han he oher ze. The flow aron all boe wh one ze maller han en me he oher ze can coneqenly be characerze a 3D flow. The fference beween he flow aron 3D an D boe have mporan coneqence for BAWT. Thee fference can be llrae by comparon of flow aron a phere an a cylner. The (D) cylner fon from he (3D) phere by nfne elongaon of he phere n one recon. oneqenly, boh boe have he ame bac hape an only her menon vary. The fference n flow aron he phere an cylner h prove an llraon of he fference n flow beween 3D an D boe. The velocy a he rface of a phere an cylner can be fon wh poenal heory. If aache flow ame, he velocy on he rface of boh bln boe θ rea (ee for nance Bachelor []) 3 θ R n( θ ) = + r phere (37) a a phere wherea a he cylner θ rea θ R θ = n( ) + r cylner (38) The confgraon for θ = 9 o hown n Fgre 7. Fgre 7 Poenal flow veloce a he e of a phere (3D) an a cylner (D). The fference n θ beween 3D an D boe can be neroo wh he followng argmenaon. In D flow, he flow ha j one plane o move aron he boy: he plane wh normal vecor parallel wh he large ze of he boy. A a 3D blff boy, he flow can - 5 -

36 3 Wn characerc 3.Local wn n he bl envronmen alo move perpenclar o ha plane. Therefore, he veloce a he e of D boe are hgher. A Re, he bonary layer wll eparae a he along-wn e of he bln boe an a wake wll be forme. Th rel n a fference beween he poenal flow rel for aache flow (37) an (38), an he acal velocy a Re. The velocy a Re can be calclae from mearemen of he prere coeffcen n Hoerner [6]. The prere coeffcen efne a p p p p = (39) ρ where p enoe he prere a he locaon where p efne an where he nex refer o he free ream qane. Now, f we calclae p wh Bernoll heorem, he angenal velocy θ a a ceran pon wh prere coeffcen p fon from θ = p (4) 5 5 omparng (37), (38) an (4) wh p a. < Re < 4. fon from mearemen hown n Hoerner [6] gve he followng rel..5 hea / [.].5 from p cylner po. cylner from p phere po. phere hea [egr.] Fgre 8 θ a he rface of a cylner an phere, fon from poenal heory wh aache flow, compare wh θ fon from meare wh 5 5. < Re < 4.. p an (39) a eparae flow Fgre 8 how ha he phere an cylner wh eparae flow can be ffcenly accrae (error <%) moelle wh (37) repecvely (38) p o a crcmferenal angle θ 9 o. o Wh r R. an θ = 9, (37) an (38) how ha θ ( r).9θ ( R). In oher wor, he velocy a.r from he rface rece o 9% of he rface velocy. For mporan conclon are exrapolae from he prevo rel an (37) an (38): θ ( θ = 9, R) a a D bln boy hgher han θ ( θ = 9, R) a a 3D bln boy, - 6 -

37 3 Wn characerc 3.Local wn n he bl envronmen o a θ = 9, for he phere a well a he cylner, he velocy a.r from he rface rece o 9% of he rface velocy, for r R = conan, he acceleraon nepenen from he abole ze of he boy, for BAWT, blng ha cae D flow are preferre above blng ha cae 3D flow becae he wn pee cloe o a D blng conerable hgher. Reynol an Srohal nmber The Reynol nmber Re of a fl efne a ρ D Re = (4) η where ρ he eny of he fl, a characerc velocy of he fl, D a characerc ze of a boy whn he fl an η he ynamc vcoy of he fl. The Re nmber a meare for he rao of neral an vco force n he flow (ee for nance Bachelor []). A hgh Re nmber how ha neral force omnae he vco force whle a low Re nmber how ha he vco force omnae he neral force. Wh he Re nmber we are h able o jge wheher vco effec n he flow can be neglece. A ceran Re nmber, a reglar flow paern can be oberve n he wake of a blff boy. Vorce of oppoe gn are he from he pwn ege of he blff boy n an alernaely an reglar way a epce n Fgre 9. x x / D Fgre 9 Vorex ree ownwn of a blff boy. Lef: velocy conor fon wh a FD calclaon (econ 4.3.5). Rgh: efnon Th flow paern moly referre o a vorex ree an ofen calle afer Von Kármán. Sheng of vorce mporan for blng an BAWT. By mean of her hgh local velocy, hee he vorce nce a reglar con a he e of he blff boy. Th can cae reonance a he egenfreqency of he boy. I ha o be ceran ha he freqency of he vorex heng f no n he neghborhoo of he egenfreqency of he boy ( f e = 46 H for a blng ee econ..3). oncernng he operaon of he BAWT, he reglar hgh veloce a he e of blff boe cae an alernang power procon of he BAWT

38 3 Wn characerc 3.Local wn n he bl envronmen In non-menonal form, he Srohal nmber S repreen he freqency of he vorce leavng one e of he boy. Hoerner [6] how ha a goo f of S wh mearemen a D bln a well a blff boy hape acheve wh S 3 4 = (4). For mo blff boe, he rag coeffcen (approxmaely) known. Th allow o calclae an (approxmae) S. By efnon, S fon from fd S = (43) where D he characerc ze of he boy an he free ream velocy. The heng freqency of he vorce can h be fon from f. D 3 4 = (44) Sppoe ha he me cople wh (44) enogh for allowng a qa-eay approach of he BAWT n he heng regon. Wh (7) we hen fn ha a eay approach of he operaon of he BAWT wh ameer D allowe f f > 6 D or D D < (45) For bln o blff boe,.5... (Hoerner [6]) o ha a rercon of he BAWT ze o D D <.5 avo neay operaon of he BAWT. Th no exra rercon nce he BAWT ameer lme o D D <. n econ.3 n orer o prof effcenly from he acceleraon effec cloe o blng. The vorex ree orgnae from he neracon of he flow on he wo e of he boy an h more freqenly oberve behn D boe compare o 3D boe. Ajmen on he boy ha rece he neracon beween he vorce can preven he vorex ree. Sch ajmen are for nance errae pwn ege of he boy. An example of h fon a he TU Delf blng n he Neherlan epce n Fgre 3. A o calle pler plae, a plae a he ymmery ax n he wake of he boy, can alo preven he vorex ree Wn pee probably rbon Th econ fr how he ervaon of he probably rbon of he wn for a BAWT.e. he probably rbon a he locaon where he wn pee change by a concenraor blng. Seconly, h econ how he ervaon of he energy yel of a BAWT from h probably rbon. Probably rbon The probably rbon of wn n he amopherc bonary layer can be gven by a Webll rbon. Wernga [79] how ha he ecrpon wh a Webll rbon work well for wn pee 4~6 m/, he range where a wn rbne operae mo of he me. A locaon: cloe o a harp-ege blng, - 8 -

39 3 Wn characerc 3.Local wn n he bl envronmen n a c hrogh a harp-ege blng an beween arfol-hape blng, he change of he free ream wn pee n he local wn pee near he blng Reynol-nepenen. Hence, he change n free ream wn pee cae by he majory of blng e a a concenraor Reynol-nepenen an wll be ame ha he change of he free ream wn pee by he concenraor blng Reynol-nepenen. Then, he wn pee a he concenraor, for free ream wn from wnroe ecor, can be fon from he free ream wn pee, a a ceran reference hegh (for nance roof hegh) wh = (46) r,, where r, gve he Re-nepenen change of, o. The Webll rbon f, of he free ream wn pee, from wn roe ecor rea k k k,, f, = exp a a a (47) where he cale parameer a of he free ream wn pee, can be fon from a, = Γ + k (48) where Γ he gamma fncon k Γ + = exp( x) x x k (49) an k he hape parameer. Wernga an Rjkoor [79] how ha k vare wh hegh an roghne. Ther rel for he epenence of k on hegh a abaw, The Neherlan, gven n Fgre

40 3 Wn characerc 3.Local wn n he bl envronmen 5 z [m] k [.] Fgre Shape parameer k a a fncon of he hegh above he earh rface z a abaw, The Neherlan (Wernga an Rjkoor [79]). For a hegh of approxmaely m Fgre how ha k = a goo approxmaon. Wernga an Rjkoor [79] gve a varaon of k wh roghne of 5% for a roghne change from z =. 5 o z =. 3. We hall ame ha k =. For k = he Webll rbon calle he Raylegh rbon an he vale of he gamma fncon fon o be π Γ + =. (5) 4 The probably of a wn pee nerval a he concenraor f where f he probably rbon of he wn a he concenraor, eqal o he probably of a free ream wn pee from ecor : c, f,,, where c, he probably of wn from wn roe ecor. We h have f = f (5) c,,, wh N c =, = (5) where N he nmber of ecor of he wn roe. Sbon of (46) an (47) n (5) gve k k k f = exp c, a r, a r, a r, (53) The probably rbon of he wn pee a he concenraor from wn roe ecor h rea f for free ream wn - 3 -

41 3 Wn characerc 3.Local wn n he bl envronmen k k k f = c, exp a r, a r, a r, (54) Hence, he probably rbon of he wn a he concenraor for all wn recon f gven by k k N k f = c, exp a = r, a r, a (55) r, omparng (47) an (55) how ha he probably rbon of he wn pee a he concenraor he m of a weghe (facor, ) Webll rbon wh fferen cale parameer a r, compare o he free ream cale parameer a. oneqenly, n general f no Webll rbon. The banwh of f can be mch hgher (ee econ 5..). Wh (54) he average wn pee a he concenraor fon by c f N = (56) = A meare for he energy eny a he concenraor N 3 3 f = =. (57) Energy yel Generally, he power op of he wn rbne wll be a fncon of he wn recon. For nance n yawe flow, he power op ecreae f he wn rbne oe no have a yaw mechanm. In ch cae, he energy yel E for a ceran wn pee an wn recon rea E = TP f, (58) where P = ρ A enoe he power of he wn rbne a wn recon an T he 3 P me he wn rbne rn (a year o fn he yearly energy procon). The oal energy yel for a ceran wn recon h fon wh = co E = T P f (59) = c where c he c-n wn pee of he wn rbne, co he c-o wn pee of he wn rbne an P a fncon of. The oal energy yel for all wn recon an wn pee hen fon from - 3 -

42 3 Wn characerc 3.Local wn n he bl envronmen = N = co E T P f =. (6) = = c 6 All concenraor moel n h he have = for he bare acaor a hey are bae on P 3 6 he acaor concep. Wh efnon P = P ρ A n (6), P = an 7 = r, o,, he power of ch acaor n a concenraor rea 7 ( ) 3,, P = ρ A = ρ A. (6) r I no alway r, ha known for a parclar concenraor. Someme, he performance coeffcen of he concenraor P, conc known nea of r,. The power of he acaor n 3 ch concenraor efne a P = P, conc ρ, A where P, conc known va expermen, FD calclaon or mahemacal moel. Eqang wh (6) lea o 3 P, conc, r, = 6 7. (6) 3.3 Wn a blng Th econ eal wh he flow phenomena own o he rface of he blng ch a: he agnaon pon a he pwn rface of he blng, he velocy n he eparae bonary layer for harp-ege blng an he hape of he eparaon bbble a he e of harp-ege blng Sagnaon pon The hegh of he agnaon pon H an mporan aeroynamc qany. I characerze he flow aron he blng an gve he pon on he pwn blng façae wh he hghe prere. In a cy, he aon look a epce n Fgre. H H H Fgre Locaon of he agnaon pon. The hegh of he agnaon pon on a harp-ege blng n a bonary layer wh neglgble wll be enoe by H,. A approxmaely qare blng, he - 3 -

43 3 Wn characerc 3.3 Wn a blng mearemen of Sharan [65] how a % change n H, for a change n z wh a facor 7. Jenen [3] gve H,.8 for relave roghne H z = 3 ~ 44. The nflence of z on H, eem mall an wll frhermore be neglece. H, accorng o everal reference gven n Table 6. Table 6 Rao of agnaon pon hegh an blng hegh a neglgble placemen hegh accorng o varo reference. Reference Sharan [65] Bane [3] Blng ze eph : wh : hegh H, H ::.85 ::3.85 ::.8 ::8.9 : :.85 Jenen [3] ::..8 From Table 6 clear ha he nflence of he blng ze on can be neglece. I wll accorngly be ame ha H, H alo mall an H,.85 (63) H nepenen of blng ze an z. The agnaon pon hegh a a fncon of, enoe by H. For experence flow a he pper par of he blng an accorngly H H, he blng H. We alreay know ha for, H H,. A a fr approxmaon, le ame a lnear epenence of H on for H. Th gve H H, H, = + H H H H (64) Wh (8), (9), (3) an (63) h rea H H H = (.5 AH ( AH )) (65) H Th relaon wll be e a a rle of hmb for emaon of roghne propere. H a a fncon of he 3.3. Separaon A a harp pwn ege of he roof, he bonary layer eparae from he blng. The eparaon rel n a regon wh low veloce, a hgh rblence level an recrclaon of he flow a he roof an e of a blng. Th recrclaon regon hol be avoe for ng of a wn rbne. I h mporan o know he ze of he recrclaon regon

44 3 Wn characerc 3.3 Wn a blng Fgre FD calclaon of he velocy vecor aron a D blng (lef) an zoome n a he roof (rgh). The flow come from he lef. Becae of he eparaon a he pwn roof ege, he velocy vecor oe he recrclaon regon no parallel o he roof. The angle beween roof an velocy vecor oe he recrclaon regon calle he kew angle o ngh from he yaw angle n he horzonal plane. The kew angle vare wh: poon on he roof, roghne of he pwn area, ze of he blng, pwn ege ronng an yaw of he free ream wn o he blng. The epenence on he poon on he roof eay o neran. The veloce cloe o he blng are hgh an ecreae wh ncreang ance o he blng. The prere cloe o he blng are coneqenly low an ncreae wh ncreang ance o he blng. Th prere graen force he flow n a crve pah. On he crve pah, he ra of he crvare efne by eqlbrm beween he force cae by he prere graen an he cenrfgal force cae by he crvare (Sm & Scanlan [67]). The effec of roghne an ze of he blng on he kew angle can be neroo from he amon of laeral momenm cae by blockage of he blng. ompare o a low roghne, a hgh roghne rel n a maller ma flow owar he blng (rece veloce n he bonary layer for hgh roghne). A hgh roghne h rel n le laeral momenm becae a maller ma-flow ha o move aron he blng. The ame argmenaon val for mall blng ze. Therefore, he flow ay cloer o he blng for hgh roghne or mall blng ze compare o low roghne an large blng ze Wake an recrclaon zone moel Lerare oe prove heory, moel an mearemen ha can be e o fn he ze of a wake or recrclaon zone on a roof of olae blng. The moel on he ze of a wake or recrclaon zone are compare an ce n he followng paragraph. Trblen growh of a 3D an D wake A moel for he ze of a wake e he bonary layer eqaon erve from he NaverSoke eqaon by amng mall laeral lengh cale compare o he axal lengh cale. From hee bonary layer eqaon an elf-preervaon of he rcre n he rblen wake (ee for nance Tenneke & Lmley [69]), he growh of a wake of a blff boy rea y x n for x >> y (66) where y he laeral ze an x he axal ze of he wake. The orgn of he coornae yem locae a he pwn ege of he boy ha cae he wake. Depenng on he menon of he wake, he exponen n n (66) rea

45 3 Wn characerc 3.3 Wn a blng n = for a D wake n = for a 3D wake 3 (67) Emprcal moel for he growh of a 3D recrclaon zone A moel for he ze of a 3D recrclaon zone ownwn of he leang ege of a harpege moel blng gven n Wlon [83]. The moel a f of a menonal analy on nmero mearemen for we a well a all moel blng. The moel bae on a characerc ze of he pwn blng façae D fon by 3 3 mall D large D = D (68) D enoe he malle crown ze of he blng an mall D enoe he large large crown ze. We a one mple change n h eqaon o accon for a large placemen hegh. A = H, we have kmmng flow (econ 3..) an he flow experence no aonal blockage of he blng. The blng hegh n D or mall D large herefore efne a H. For = H h efnon rel n D =, whch mean ha he flow experence no blng o ha moohly flow acro he roof nea of bmpng no he blng. For a 3D wake Wlon [83] fon a goo correlaon wh mearemen a varo blng hape f x 3 3 y =.8D x for.< <.4 (69) D where D fon by (68). The x-exponen n agreemen wh (66) an (67),.e. can be concle ha he growh rae of he recrclaon zone eqal o he growh rae of a rblen wake. I frhermore nereng o ee he ame conan.8 n boh (69) an ( ) =.8. (3) for he growh of he nernal bonary layer hegh: ( )..8 h x z x Accorng o (67), n = expece n cae of a D wake, ye (3) gve.8. Th llrae ha he nernal bonary layer nee no wake. Frhermore, (69) an (3) ffer n menon of he flow. Eqaon (69) fon for 3D recrclaon zone whle (3) fon for he growh of a D nernal bonary layer. oneqenly, here eem o be no phycal orgn for he mlary of (69) an (3) an rprng ha a conan.8 how p n boh moel. Free Sreamlne Theory for he growh of a D wake A econ moel for a D wake of a blff boy fon n Bachelor [] an bae on Free Sreamlne Theory (FST). In FST, he pream conon of a hgh Re nmber flow are ame rroaonal everywhere excep near he bonare of he flow. The moel bae on he followng aonal conon: a backprere eqal o he free ream prere nform pwn pee Becae of he neceary conon of a backprere eqal o he free ream prere, FST ceae o be val cloe o he eparaon pon where he prere are conerable lower han free ream prere. Therefore, FST nhab he ame far fel ampon x >> y a n (66). Whn he frame of he ampon, Bachelor [] gve he relng eqaon for he hape of he free reamlne of a D wake n paramerc form ( + k) k,max x = k ln + + (7) k k

46 3 Wn characerc 3.3 Wn a blng ( k + k ) πk y = + (7) where he ance along he free reamlne an k may be eermne from b k = (7) π + 4 wh b a he ance from agnaon pon o he pon of eparaon. The orgn of he coornae yem locae a he agnaon pon.e. bon of = rel n ( x, y) = (, b). In cae of eparaon a he roof of a blng, we e he flow mlary wh he FST o erve he hape of he eparaon reamlne a he roof. Boh flow confgraon are hown n Fgre 3. From he analogy of boh flow, clear ha b = H (73) H can be aken, where H he blng hegh an fon wh (65). H he hegh of he agnaon pon y y x b x H b Fgre 3 onfgraon of he D flow n he ervaon of free reamlne heory (lef), an analoge confgraon for he prpoe of eparaon on he roof of a blng (rgh). For large ance ownwn of he eparaon pon, we fn he followng aympoc expanon of FST. x lm = ln (74) k k ( + k) k + + = lm y = lm ( k k ) + + k = k π (75) lm o ha bon of (7) wh b = H H rel n ( ).5 y k x.8 H H x.5.5 = = for y x >> (76)

47 3 Wn characerc 3.3 Wn a blng where H can be fon from (65). The x-exponen n (76) n agreemen wh he D rel (67) of Tenneke & Lmley [69]. Th agreemen omewha rprng becae FST lack he rblen mxng on whch (67) bae. The agreemen can herefore be characerze a a lcky concence. For frher calclaon on he eparaon reamlne, we hall e he moel of Wlon (66) for an olae 3D blng an he aympoc expanon of FST (76) for an olae D blng. Verfcaon of he moel The moel for he ze of he recrclaon regon of olae blng are compare wh FD calclaon. For h verfcaon, a D an 3D blng wh a hegh of m n an amopherc bonary layer wh z =. 3m are choen. Table 7 Blng ze for he verfcaon. Dmenon eph:wh:hegh 3D :3: D : : The FD calclaon (Appenx ) howe ha he recrclaon regon maller a he corner of he roof. The eparaon reamlne aron he blng are epce n Fgre 4. The large expanon ha were fon a he roof cenre an m above he earh rface were compare wh he recrclaon regon moel. Fgre 4 FD calclaon of he eparaon reamlne on he e (lef) an roof (rgh) of a 3D blng. 3D blng The rel for he eparaon reamlne a boh e of he 3D blng m above he earh rface are hown n Fgre

48 3 Wn characerc 3.3 Wn a blng Fgre 5 FD calclaon of he eparaon reamlne a he e of a 3D blng (eph:wh:hegh=:3:) a m hegh above he earh rface wh z =. 3m compare wh he moel of Wlon [83]. The rel for he eparaon reamlne above he cenre lne of he roof are hown n Fgre 6. Fgre 6 FD calclaon of he eparaon reamlne above he cenrelne on he roof of a 3D blng (eph:wh:hegh=:3:) wh rronng rface roghne z =.3m compare wh he moel of Wlon [83]. A he ownwn roof ege, he FD calclaon how a change n orenaon of he eparaon reamlne (Fgre 6). Th cae by he p-wah a he ownwn face of he blng (ee Fgre ). The p-wah phe he eparaon reamlne pwar n he wake. Wlon [83] gve a varaon of he mearemen rel aron he moel locaon of he eparaon reamlne of approxmaely 5%. Some of he ee blng have a me hgher wh compare o he hegh. In h econ, hown ha he growh of a D wake fferen from a 3D wake. oneqenly, par of he fference beween he mearemen rel an he moel cae by he fac ha ome of he ee blng canno be clafe a real 3D blng. Frhermore, Wlon moel fe on mearemen a one z. Oher z can gve a fferen f. The verfcaon n Fgre 5 an Fgre 6 alo how a fference beween moel an FD rel. Accrae precon of he (me-average) locaon of he eparaon reamlne for a 3D blng coneqenly nlkely. Ye, he f on nmero mearemen a we an all blng n Wlon [83] how ha he moel efl. The overall conclon ha he moel of Wlon efl b no able o gve accrae precon of he eparaon reamlne

49 3 Wn characerc 3.3 Wn a blng D blng The rel for he eparaon reamlne above he roof of a D blng are hown n Fgre 7. Fgre 7 FD calclaon of he eparaon reamlne above he roof of a D blng (eph:wh:hegh=: :) wh rronng rface roghne z =. 3 m compare wh FST. Accorng o he nce f of he aympoc expanon of FST wh he FD rel, lkely ha he eparaon rajecory can be accraely prece for D blng. Th no rprng becae he characerc blng ze of a D blng nambgo H, whch make he flow confgraon for D blng well efne Reaachmen ance Downwn of he eparaon pon, reaachmen of he bonary layer can occr for large enogh ownwn ance. Bae on nmero mearemen a we an all blng, Wlon [83] gve he lengh of he recrclaon regon L c n erm of he characerc blng ze D fon wh (68): W L c =. 9D for < < H (77) where W he wh of he blng. The mearemen were carre o a Re nmber p o a facor maller compare o he acal Re nmber wh fll-cale blng. Frhermore, he mearemen are carre o for a ngle z. I herefore expece ha he acal L c fferen a a fferen z an oher Re nmber. For he common plae-lke blng, reaachmen can only occr for flow parallel o he long e of he roof. For h flow recon, D mall an a large lengh ownwn of he eparaon pon avalable for reaachmen The cavy moel Th econ ce he orgn of he backprere of bln an blff boe. The cavy moel We wll now y he relaon beween backprere, rag an velocy a he eparaon pon. Bachelor [] how a mple approach ha cople of a boy mmerge n waer wh he velocy a he eparaon pon. Le he velocy a he wee rface of he blff boy be enoe by an he velocy a he pon of eparaon by (ee Fgre 8)

50 3 Wn characerc 3.3 Wn a blng y avy R x Fgre 8 Flow aron a blff boy. Accorng o poenal heory, he veloce a he rface of a phere or cylner are fon by (37) an (38). A he rface of hee boe poenal heory gve θ ( θ, R) = n( θ ) θ ( θ = 9, R). Wh = θ ( θ, R), = θ ( θ = 9, R) an n( θ ) = y R h rel n y = wh y R (78) R I ame ha he lnear epenence y n (78) val for all bln an blff boe. Wh pream flow conon ch ha he flow rroaonal everywhere excep a he bonare, Bernoll heorem gve he ac prere on he free reamlne aron he blff boy. Wh he aonal ampon of a cavy or wake wh conan prere, he ac prere on he free reamlne eqal o he ac prere n he cavy. Then, he prere fference p acro he boy a locaon y rea p = ρ y R (79) The rag coeffcen of he boy hen follow from y A = A R (8) If we efne a conan,, y = A A R we have, = (8) - 4 -

51 3 Wn characerc 3.3 Wn a blng where -accorng o he efnon-, ecrbe he nflence of he hape of he wnwar face on of he blff boy. Bachelor [] how ha (8) n goo agreemen wh mearemen for everal 3D bln an blff boe mmere n waer. In oher wor, he lnear epenence n (78) an he ame cavy are goo approxmaon for he acal flow aron hee boe n waer. In waer, he neracon of he ar n he wake wh he velocy a he free reamlne can be neglece becae here a bg fference n eny beween boh fl. oneqenly, n waer, well efne wh he velocy on he free reamlne. For boe n ar, h eny fference aben an he neracon of he ar n he wake wh he ar aron he wake canno be neglece. are hol be aken wh he efnon of. We efne for blff an bln boe n ar a he velocy ha reproce he meare average backprere p, b a he boy. Wh (4) an he ampon of a cavy wh average cavy prere p, b, h efnon rel n p, b = (8) Th enfe one fxe, whch eay o work wh. The acal flow oe no prove ch ragh efnon becae nclear where hol be meare. Sbon of (8) n (8) gve ( ) = (83), p, b We wll frhermore refer o (8), (8) or (83) an he nerlyng ampon wh he cavy moel. Verfcaon of he cavy moel The an p, b vale n Table 8 are meare an hown n Hoerner [6]. The, vale gven n Table 8 f mearemen hown n Bachelor []. The vale of are calclae from (8) an he meare, an fon n Bachelor [] an (8) an he meare p, b fon n Hoerner [6]. The rel of he foremenone wo way o calclae are hown n Table 8. Table 8 alclae vale of fon wh he cavy moel an meare,, Boy, an p, b vale. from,, an (8) p, b from p, b an (8) Dk Flle half phere Thee rel how ha: = for he flle half-phere approxmaely eqal o =.5, whch expece from (37) accorng o poenal heory an ha - 4 -

52 3 Wn characerc 3.3 Wn a blng he vale fon wh (8) an (8) an nepenen mearemen are approxmaely eqal. Frhermore he vale =. for a k alo confrme by nepenen mearemen n econ 7..3 carre o for h he. Hence, concle ha: he rec coplng beween, an p, b a ggee by he cavy moel work well an he cavy moel very efl o reveal an compare he bac propere of he concenraor blng. We wll herefore make ample e of he cavy moel. I hol however be kep n mn ha he flow fel of blff an bln boe ha are calclae by amng poenal flow an wake abence are fferen from he acal flow fel becae of he omeme large wake of bln an blff boe. Thee wake mooh he conor of he boe an vrally elongae hem. By ong o hey ecreae he veloce pwn of he wake. The poenal flow ampon ogeher wh he ame wake abence herefore overemae he veloce compare o he acal veloce. Secon 3.. howe ha blng ha cae a D flow are preferre for BAWT becae of her hgh. From h paragraph, frhermore clear ha he mo promng blng for BAWT are characerze a bln Inflencng he eparaon velocy an backprere Eqaon (8) how ha he acceleraon gnfcanly nflence he backprere p, b. For he wn rbne n a plae concenraor (chaper 7), a large negave p, b preferre b for normal blng, a le negave p, b an accorngly lower rag of he blng preferable. Eqpmen o change can con of plae or arfol cloe o he pwn ege of he blng a epce n Fgre 9. The arfol change by mean of he nclnaon angle of he arfol ϕ. For large ϕ, ncreae, whle a mall ϕ ecreae. ϕ Fgre 9 Top vew of a blng wh arfol cloe o he pwn ege o nflence he eparaon velocy. Srangely enogh, h meho o rece he rag of a blng no ofen een. The GSW blng n Berln, Germany how ch arfol-lke rcre on he roof alhogh he ahor oe no know f h mean o rece he rag of he blng. Frhermore, a Delf Unvery blng n he Neherlan rece he eparaon velocy by a creen-lke rcre on he roof

53 3 Wn characerc 3.3 Wn a blng Fgre 3 Inflencng of he rag of a blng by changng he eparaon velocy. Lef: blng of Delf Unvery of Technology, The Neherlan. Rgh: blng of GSW, Berln, Germany

54 4 Analy ool 4 Analy ool Aeroynamc reearch can be bae on hree man pllar: mahemacal moel, mearemen an mlaon or ompaonal Fl Dynamc (FD) calclaon. They form he fonaon of he aeroynamcal knowlege we have oay an are e a analy ool for aeroynamc phenomenon n h he. The eparae analy ool have pecfc avanage an rawback ha efne he ably of a ool for a ceran analy. The mearemen an FD calclaon prove nmber. For he coherence beween hoe nmber an a phycal explanaon of he obervaon, for egn prpoe, we nee a mahemacal moel. The mahemacal moel oe alo nee he mearemen an FD calclaon. They hol verfy he moel ampon. Accrae mearemen of a ceran qany are omeme ffcl o oban. In ha cae, he mearemen can be ppore by FD calclaon. 4. Mahemacal The Naver-Soke eqaon (ee for nance Bachelor []) ecrbe he flow behavor b ffcl o e becae of complex nare. Fornaely, here are a nmber of flow ha can be approxmae wh a pecal cae of he Naver-Soke eqaon. If we ame ha he flow can be characerze a homogeneo an nvc, he Naver-Soke eqaon mplfe o he o calle Eler Eqaon. A vergence-free flow who vorcy gve a non-rval olon of he Eler eqaon. Th olon he cla of poenal flow. In pe of hee rercon of he Naver-Soke eqaon, poenal flow ecrbe a conerable range of acal flow. Th range of flow wll be characerze n he nex econ. 4.. Poenal flow Sppoe we have a flow wh hgh Re nmber b wh a velocy maller han 3% of he pee of on (n ar: on 34 [m/]). Then, he flow can be reae a ncompreble (Bachelor []) an approxmae wh he Eler eqaon excep n a mall bonary layer cloe o he boy (Pranl [59]). Accorng o Kelvn [34], a olon of he Eler eqaon crl-free f ar crl-free an he volme force n he Eler eqaon are conervave 7. An mporan cla of crl-free flow forme by all flow ha ar a zero velocy wh a conervave velocy fel. oneqenly, all flow wh velocy oe he bonary layer, a hgh Re, wh <.3 on, ha are a = an have conervave volme force, can be approxmae wh poenal heory. A large nmber of flow can h be approxmae wh he relavely mple mahemacal ecrpon of poenal heory. 7.e. he volme force can be fon from he graen of a one-vale force poenal lke for nance he gravy fel. In ch conervave fel, he work one by akng a boy from one place o anoher nepenen of he pah aken beween hoe pon

55 4 Analy ool 4. Mahemacal 4.. Vorex hee Whn he ampon of poenal heory, can be hown ha he effec of force on he flow fel connece o vorce (ee for nance Saffman [6]). An example of h he bon vorex of an arfol on whch a lf force ac. Küchemann & Weber [36] prove anoher example: he wake of a loae acaor. The laer e n more eal n h econ. Smplfe vorex heory Sppoe, we have wo ragh nfne long vorex hee (D) or an nfne long vorex be (3D) a hown n Fgre 3. I can be hown wh Bo-Savar ha he velocy beween he vorex hee or ne he vorex be nform (Bachelor []). Le h nform nce velocy a locaon ( x, y) = ( a, b) beween he vorex hee or ne he vorex be be =., =, ( x, y) = ( a, b) Fgre 3 Two ragh nfne vorex hee (D) or an nfne vorex be (3D). Removng he pwn par of he vorex hee or vorex be gve he followng half-nfne confgraon. A rel of removng he pwn half ha he vorex hee or he vorex be have a prere rop acro her rface. =, ( x, y) = ( a, b) =, Fgre 3 Two ragh half-nfne vorex hee (D) or a half-nfne vorex be (3D). Only half he nform nce velocy, can be generae a ( x, y) = ( a, b) now becae only half he poron of vorex hee or half he vorex be nce a velocy a (a,b). The half-nfne confgraon hown n Fgre 3 can be e a a moel for he wake of a D or 3D acaor. In ha cae, he free reamlne a he acal wenng wake bonary of he acaor are exchange by he hown ragh bonary. Th approxmae moel for he nce veloce of a ragh wake wll be referre o a mplfe vorex heory. Le compare he rel of mplfe vorex heory wh momenm heory for an acaor (econ.). Accorng o mplfe vorex heory, he ncon facor a he acaor eqal o.5 f =,, whch gve a relan velocy eqal o zero a nfne ance ownwn of he acaor. Th aon canno be moelle wh ragh vorex hee or a vorex be. Hence, we fn he breakown of mplfe vorex heory f he ncon facor

56 4 Analy ool 4. Mahemacal a he acaor excee.5, whch analogo o momenm heory for an acaor (econ.). D Vorex heory Sppoe we have a D loae acaor an we moel he velocy conny a he wake bonary wh ragh half-nfne vorex hee (Fgre 33). The orgnally free vorex hee a he expanng wake are h -lke n mplfe vorex heory- exchange by ragh vorex hee wh a prere rop acro he vorex hee. y, x (a,b) r θ x e R, r θ Fgre 33 Wake of a loae D acaor (hae lne) moelle by ragh vorex hee. Bo-Savar (ee for nance Bachelor []) gve he nce veloce a coornae (a,b). The calclaon carre o n Appenx E where fon ha he relan nce veloce n x an y-recon repecvely an v a coornae ( x, y) = ( a, b) are γ a a = π arcan arcan π R b R + b (84) an v γ a = ln + ( R + b) ( R ) 4π a + b (85) where γ enoe he vorex rengh per mere vorex hee. Some manplaon wh (84) an (85) how he rel gven n Table 9. Table 9 Ince veloce a ( x, y) = ( a, b) by half nfne ragh vorex hee. a >>, < b < R a =, b = a =, < b < R = γ = γ = γ v = v = v A a >>, he velocy e fon from mere vorex hee rea e = = γ o ha he vorex rengh per γ = e (86)

57 4 Analy ool 4. Mahemacal omparon of Table 9 an he rel of mplfe vorex heory (Fgre 3) how ha mplfe vorex heory exac for a he ymmery plane or acaor ( a = ) Momenm heorem The eqaon of conervaon of momenm n negral form ha ome nereng applcaon where recly reveal reqre flow propere who he nee o know eal of he flow ne he area of negraon (Bachelor []). Th make he momenm heorem a powerfl ool. onervaon of axal momenm n negral form for a aonary, ncompreble fl whn a conrol area A rea (Pranl & Tejen [6]) F, ax = I, o, ax I, n, ax (87) where F, ax a m of he exernal axal force acng on he fl whn A an I, o, ax, n, ax I he fference n axal momenm beween he ole an nle of A. The ffcly an challenge wh h mple eqaon o fn a able conrol area ha reveal he reqre flow propere. Ofen a conrol area bone by reamlne a goo choce ha allow parclarly mple algebrac hanlng. 4. Expermenal Th econ eal wh wn nnel mearemen. In orer o avo error n he mearemen, he ze of ee boe n he wn nnel lme becae of blockage of he flow of he wn nnel. Unfornaely, here are nmero oher effec ha nflence he meare rel, whch hol be careflly conere. Th chaper how ome major effec ha nflence he wn nnel mearemen carre o for h he. 4.. The open je wn nnel The mearemen for h he were carre o n he open je wn nnel of Delf Unvery of Technology. The open je wn nnel mae of a large crclar be (Fgre 34) ha encloe a fan a he con e of he be. There no collecor for he flow an he hall e a rern channel for he flow. Fgre 34 Overvew of he open je wn nnel of Delf Unvery of Technology rng he expermen wh a mall H-Darre n kewe flow (Saro [6])

58 4 Analy ool 4. Expermenal The fan pee p he ar o he mearemen econ cloe behn he ex of he be. A he nnel ex, for po be meare he nnel velocy. The velocy fon by he for po be average. Th average velocy conere o be he nnel velocy. Screen an raghener n he be ake care of he elvere flow qaly. They mnme he effec of he flow roaon cae by he roang fan, ncreae he nformy of he velocy fel an ecreae he rblence level 8. Some characerc of he open je wn nnel of Delf Unvery of Technology are fon n Toe & Vermeer [7] an mmarze n Table. Table haracerc of he open je wn nnel of Delf Unvery of Technology Qany vale Tnnel ameer.4 m Tnnel lengh.85 m Maxmm nnel pee 4 m/ Non-nformy of velocy 9 7% Trblence level 9.8 % The non-nformy nroce an mporan mearemen error ha ecrbe n econ Tnnel correcon A boy ownwn of he nnel ex block par of he flow of he wn nnel. Th blockage nflence he flow an correcon o accon for h are neee. The bonare of he e econ of open je wn nnel are no fxe o ha he flow can ealy move aron a ee boy. Open je wn nnel are herefore le enve o blockage han wn nnel wh cloe e econ. In orer o rece he effec of blockage, he e for h he are carre o wh a mall blockage of approxmaely 5% of he nnel ex area. Mearemen n Ewal [5] how neverhele ha a fla plae wh 5% blockage ha a conerable 6% lower compare o he a zero blockage. Two major error orce for an open je wn nnel who collecor can be enfe:. he je over expanon cae by blockage of he je an. he hgher velocy a he bonare of he je a he ee boy approache he nnel ex. There are mahemacal moel o calclae correcon ha accon for hee error orce, b he blockage correcon for wn nnel wh open e econ are no prece known (Ewal [5]). We herefore avo he e of hee mahemacal correcon an nea e an overall correcon bae on mearemen carre o for h he. The free ream velocy experence by a k a he e econ of he wn nnel, calclae from he meare rag of he k F. Hoerner [6] how ha =.7 f Re > 5 3 a he rblence level of or wn nnel (.8%, ee Table ). Wh a meare F D,, can be calclae from D 8 Deal of he effec of confgraon change n he open je wn nnel are fon n Toe & Vermeer [7]. 9 A a e econ of.9. m,. m ownwn of he nnel ex wh a nnel pee of 6 m/. a zero blockage fon by exrapolaon of mearemen a varo nonzero blockage rao

59 4 Analy ool 4. Expermenal FD, = (88) ρa Sacal error The, fon wh (88) compare wh he meare velocy a he nnel ex by po be, p. By ong ch, wh: a k area of A =.96 m, whch eqal o a nnel blockage of 5% an he k place.5 m behn he nnel ex, Fokkema [8] obane he followng rel ,p /, [.] ,p [m/] Fgre 35 Rao of he nnel velocy meare wh po be, p an he experence free ream velocy a he e econ bae on he rag of a k fncon of, p wh a nnel blockage of 5%., a a A acal analy of,, wh he meare anar evaon an he en p rbon (ee for nance Taylor [68]) how ha,, ha a probably of 95% o be whn he nerval ( ),, = 94,8 ±.5 (89) p p alclae error The error n (89) wll be verfe wh a calclaon bae on he phycal qane n he mearemen of,,. The mnmm error n, fon from (88) wh =. 7, p δ =. an all oher error n he qane n (88) aben. The error n, hen rea (Taylor [68])

60 4 Analy ool 4. Expermenal δ, δ >, =.43 (9) The veloce, p an, are calclae from ochacally nepenen mearemen. The error n,, p can h be fon by a mmaon of he qarac error, p an, (Taylor [68]). Wh an nfne nmber of mearemen, he acal error n, p vanhe. Wh (9), hen fon ha δ ( p ),,,, p >.43 (9) Wh a large nmber of mearemen, he acal analy (89) wol gve an error below ha gven by (9), whch mpoble becae of he lme accracy of =. 7. The error n (89) lghly hgher han he error gven by (9). oneqenly, he overall nnel correcon,, p for a nnel blockage of 5% ffcenly accraely known by (89). We wll e (89) whenever a boy a he e econ ha a nnel blockage of approxmaely 5%. A ae n he begnnng of h econ, Ewal [5] howe ha a fla plae wh 5% blockage ha a, whch 6% oo mall becae of he blockage. Becae of he qarac epenence of oo mall. on he velocy, he velocy a he e econ a facor.6.97 Accorng o (89), we fon a.95 me oo mall velocy. Or % ( =. ) lower velocy compare o Ewal [5] probably a rel of he velocy non-nformy or more pecfc he recon n velocy cae by he hb of he fan Error n power coeffcen from mearemen The power coeffcen fon wh () P 3 ( ρ ) = P A () The qane n () are nepenenly meare. Taylor [68] how ha he error n hen be fon from he mmaon of he qare relave error P can P P δ δ A 3 ρ δ P δ δρ = P A (9) Now ppoe ha all error n (9) are mall excep he error n. Th gve a mnmm error of δ = 3δ. Inclng of he oher error n (9) herefore gve P P δ P P δ 3 > (93) The free ream velocy n (93) he free ream velocy experence a he e econ. For a k wh 5% blockage of he nnel area, (89) gve he free ream velocy. For he - 5 -

61 4 Analy ool 4. Expermenal fferen operang ae of a wn rbne or for a wn rbne n yaw, (89) can only gve an approxmaon of. The flow aron an hrogh a wn rbne a fferen loa n general no comparable wh he flow aron a k. The emaon of he error δ n (93) herefore ffcl. A he en of econ 4.., wa fon ha he non-nformy nroce a % recon n free ream velocy for a k a he e econ. In general, for oher ee boe, he effec of he non-nformy nknown. I herefore ame ha δ =. a 5% blockage of he nnel area. Bae on h error, (93) gve δ P P >.6 (94) Th error canno be avoe n general. Therefore, P ffer from a gnfcan error. alclaon of P from mearemen are herefore avoe a far a poble an nea FD calclaon ha are verfe on qane ha col be meare more accraely are e o calclae. P 4..4 Scalng rle Some major error n he mearemen are nroce n he prevo econ. Unfornaely, more effec nflence he qaly of he mearemen. The flow propere are moly meare on cale veron of he fll-cale moel. The effec of h calng eerve or aenon. The meare flow propere a he cale moel hol repreen he flow propere a he fll-cale moel. Th reqre ome mlary rle beween cale an fll-cale moel o be flflle. Wh he ame fl a fll-cale an cale moel, he Re nmber (4) change proporonal o he characerc velocy me he characerc ze. For a cale veron of a blng, he characerc ze can be a facor maller. In orer o manan he ame Re nmber, he velocy hol hen be an mpoble facor hgher. The Re nmber a he cale moel wll herefore ffer from he fll-cale moel. Th Re change can only be gnore f he meare propere are Re-nepenen. Sch a Re-nepenen propery for nance of ome blff boe. I concle ha he effec of calng hol be conere careflly becae hey can preven a (rec) ranlaon o he fll-cale propere. FD calclaon o no have h calng problem becae he flow can be mlae a fll-cale moel. They form he alernave n cae of nevable calng problem n mearemen Wn rbne power Mearemen of he performance of he roor of a wn rbne can be carre o on: a (cale) veron of he roor f he geomery of he roor o be e an he ee roor no oo mall or wh a rec ranlaon of he acaor concep: wre creen wh fferen poroy, f he roor mall or f he operaon of a wn rbne n a ceran envronmen e. Boh mearemen are carre o for h he. Power of a (cale) wn rbne The power of a roor can be calclae from he meare orqe of he roor ax a a ceran meare nmber of revolon. The power P hen fon by mlplcaon of he orqe Q an he roaonal velocy ω : P = Qω. A Prony brake can be e o meare Q. The Prony brake con of a wheel on he ax of he roor wh a plley aron he wheel ha brake he wheel. On one e of he plley, a wegh cae a enon F n he plley whle w - 5 -

62 4 Analy ool 4. Expermenal on he oher e of he plley, he enon F meare wh a enor. The mearemen confgraon wh he Prony brake epce n Fgre 36. ω R F w F Fgre 36 Prony brake. In he eay aon he force are n eqlbrm an he oal orqe aron he ax zero where ( R + ) + F R + F ( R + ) = Fw f (95) Ff he frcon force a he crcmference of he wheel cae by he plley. The aborbe power P( ω ) by he Prony brake fon from (95) ( )( ) w P = Qω = F Rω, whch gve wh P = F F R + ω (96) Wh (96) an mearemen of F an ω a a known loa w can be calclae. F he power ( ) f P ω of he roor Power pae by a wre creen The large nflence of Re effec (econ..) make mearemen wh mall roor ameer no convenen. For mall roor ameer, mearemen wh a wre creen ha mlae he roor prove rel ha are eaer o nerpre. The hr force of an acaor F can be wren a F = pa, where p enoe he prere rop acro he acaor an A enoe he area of he acaor. Wh () h how ha P 3 ρ = p. Now ppoe a wre creen p n a parallel flow. If he wre ameer of he wre creen are mall compare o he creen ameer, he wre creen how an abrp rop n prere acro he creen p, combne wh a velocy hrogh he creen (War-Smh [78]). A wre creen can herefore be e o mlae an acaor or a In Appenx F hown ha a roaonal recon oppoe o he recon hown n Fgre 36 no able for mearemen

63 4 Analy ool 4. Expermenal roor of a wn rbne a a ceran loa. coneqenly fon from where P P of he acaor or mlae roor a a ceran loa p = (97) ρ p ogeher wh are meare a a free ream wn pee. The prere rop acro a wre creen moly efne a 3 p = K ρ (98) where K he reance facor of a ceran wre creen. The power coeffcen of he mlae roor hen fon a K 3 P = (99) Wre creen wh fferen poroy an accorngly fferen K mlae he fferen loa of he acaor. P,max fon on he crve of P vale for fferen K. Th how ha mearemen wh a e of creen are neceary o oban P,max (De Vre [76]). The prere rop p acro he creen can be meare wh ac prere be ome wre ameer ownwn of he creen. A ha ownwn ance o he wre creen, he local effec of he nval wre are aben (Yma [9]). The creen n a rblen flow how mall axal movemen ha make mpoble o meare he velocy recly ownwn of he creen (collon of he mearemen eqpmen wh he creen). The velocy hrogh he creen can herefore only be approxmae wh he velocy meare ownwn of he wre creen ũ. For mall creen ameer, h nroce an mporan ncerany n becae of he wenng of he wake ownwn of he creen. FD calclaon can be e o fn from ũ. In ha cae, he mearemen prove ũ, whch alo calclae wh a FD calclaon. If boh ũ vale agree, he FD calclaon e o fn. Th approach e n he verfcaon of a moel n econ ompaonal Fl Dynamc ompaonal Fl Dynamc (FD) calclaon are e o verfy he rel of he mahemacal moel an wn nnel e. Thee FD calclaon evalae he governng flow eqaon n fne volme or cell efne by a gr. Dfferen approxmaon of he flow eqaon or rblence moel ecreae he compaonal effor o olve he flow eqaon. The conran le n he choce of he rblence moel an he gr confgraon The rblence moel Trblen flow how very mall a well a large rcre. The large rcre have he ze of he characerc lengh n he oman. The mall rcre ze wh he Kolmogorov lengh (ee for nance Tenneke & Lmley [69] or Newa [5]). In amopherc flow, he large rcre cale wh he bonary layer hegh δ an he mall rcre cale wh he roghne hegh z. In a neral amophere for gralan roghne we have δ m

64 4 Analy ool 4.3 ompaonal Fl Dynamc an z.3 m. In orer o mlae boh he large an he mall rcre, h neceary o have large oman O( δ ) wh a reolon ha allow he malle rcre o be repreene O( z ). In oher wor, we nee o mlae he flow n a very we oman wh a very hgh accracy. Even for he fa comper oay h oo me-conmng. Approxmaon of he governng flow eqaon are neee n orer o rece he compaonal effor. The approxmae flow eqaon are fon wh he followng procere. The governng flow eqaon (he Naver Soke eqaon an eqaon of ae) are rewren wh e of a ecompoon n an average componen an a flcang componen. For nance, he nananeo velocy componen ecompoe by = + () where he enemble average of an he flcang componen of. Th ecompoon e for he velocy, he prere an he emperare an calle he Reynol ecompoon. The ecompoe flow eqaon are Reynol or enemble average. For he Naver Soke eqaon, h rel n he Reynol Average Naver Soke eqaon (RANS). The RANS eqaon how ome new nknown erm lke.e. hey are no cloe. Thee j j erm are calle he Reynol ree an hey m be moelle wh o-calle rblence moel n orer o cloe he RANS eqaon. Several rblence moel are avalable n lerare an ce hereafer n orer o how he pro an con of he moel. The anar k ε moel The k ε rblence moel he mo wely e an verfe moel. I nee a mall compaonal effor, b he moel ha lme phycal backgron. In he k ε moel, moelle wh e of he Boneq clore j hypohe. Analoge o he conve relaon 3, h clore hypohe calclae from he local graen of he average velocy (ee for nance Tenneke & Lmley [69] or Newa [5]) j j ρ j = µ + 3 ρ k + µ δ j x j x x () where he Enen Smmaon convenon e an µ he ey vcoy or rblen vcoy, k he rblen knec energy an δ he Kronecker-ela 4. The ey vcoy µ ame o be oropc an calclae wh j k = ρ () ε µ µ The Boneq clore hypohe alo known a K-heory. 3 The relaon ha cople ree n he fl wh velocy graen. 4 δ = f = j, whle δ = f j j j

65 4 Analy ool 4.3 ompaonal Fl Dynamc where µ =.9 ame conan an ε he rblen paon rae. The k ε moel h forme he mple wo-eqaon moel. I e one moel ranpor eqaon for k an one moel ranpor eqaon for ε. Newa [5] how ha he Boneq clore hypohe () can only gve afacory rel f he characerc rblen lengh cale l mch maller han he characerc lengh cale of he geomery of he flow D. In general, h no he cae becae l = O( D). The ampon n he Boneq hypohe gve re o he followng propere of he anar k ε moel: a lme applcably rerce o flly rblen wall bone an free hear flow (becae l O( D) ), no enve o free ream rblence (becae j calclae from local flow propere) canno be re n flow ha nvolve rong reamlne crvare (becae µ no oropc n ch flow). a proly large generaon of k an a relng overprecon of µ aron a agnaon pon (Km & Boyan [35]). Th large k ranpore ownwn where cae: o eparaon of he bonary layer o be ppree, o he veloce oe recrclaon regon o be oo mall an accorngly o he prere coeffcen n he recrclaon regon o be oo hgh, a non-phycal behavor a large ran rae. The la propery nee ome more explanaon. The Boneq hypohe () for ncompreble flow ( x = ) gve µ k 3 ρ x ( ) = + (3) Wh () h how ha ( ) < f k > 3.7 ε x 3 µ (4).e. a large ran rae x he anar ε k moel gve ( ) <, whch phycally mpoble or nrealzable becae he normal re or rblen prere re pove by efnon. Smlarly, can be hown ha he Schwarz neqaly for hear ree ( ( α β ) ( α ) ( β ) (Flen [6]). who mmaon over α an β ) can be volae a large ran rae The anar k ε moel no e becae he moel ampon an weaknee make nable for he flow of nere n h he. The realzable k ε moel In orer o rece ome horcomng of he anar k ε moel, he FD package Flen 5 ha e for he calclaon n h he, prove a more elaborae k ε moel: he 5 Flen

66 4 Analy ool 4.3 ompaonal Fl Dynamc realzable k ε moel 6. Th moel ha aonal erm an fncon n he ranpor eqaon for k an ε, whch make he moel applcable o a wer range of flow. A n he anar k ε moel, µ calclae wh () b wh he mprovemen ha µ no longer conan b ene o graen n he man flow. The varable µ well banae by expermenal evence an ggee by many moeller nclng Reynol (Flen [6]). I rece he proly large generaon of k aron he agnaon pon fon wh he anar k ε moel. Frhermore, he realzable k ε moel gve phycal or realzable rel a large ran rae whle he oher k ε moel gve non-phycal or nrealzable rel (4). Th explan prefx, realzable. The aonal nformaon n he realzable k ε moel rel n ome mporan avanage compare o he anar k ε moel. Accorng o he mprovemen, he realzable k ε moel hol gve beer rel for (Flen [6]): he preang rae of planar an ron je an bonary layer ner rong prere graen, eparaon an recrclaon. Th make he realzable k ε moel a promng canae for or FD calclaon. The Reynol Sre moel The Reynol Sre (RS) moel he mo avance rblence moel n Flen. I e he exac ranpor eqaon for n he RANS eqaon an cloe he exac ranpor j eqaon wh emprcal moel. The moel h nvolve mo phyc of all rblence moel. oneqenly, offer a greaer poenal of all rblence moel o gve accrae precon for complex flow. Ye, he clore ll bae on emprcal moel, whch are conere reponble for omeme compromng rel of he RS moel (Flen [6]). The RS moel le able han he k ε moel. I herefore nee ophcae gr 7 an converge rel of oher le avance rblence moel. A goo calclaon procere for he RS moel:. ar wh a converge olon of a k ε moel hen. change he rblence moel o he efal 8 RS moel an 3. lowly a complexy o he mple RS moel wh a cloe wach on he convergence. The RS moel e abo 5-6% more PU me per eraon compare o he k ε moel (Flen [6]) an nee more eraon o converge. The fac ha he moel: nable, e more PU me for a converge olon an no alway proce beer rel, make he RS moel no very belove. Mo FD er ck o he k ε moel. However, he RS moel a m when anoropy ha a omnan effec on he mean flow or where feare of nere are he rel of anoropy. Wh a foc on he opc of he he, ch flow are characerze by (Flen [6]): re rven econary flow or 6 The oher more elaborae k ε moel n Flen: he RNG k ε moel, no ce becae nal e how ha he realzable moel gve he be performance of all k ε moel (Flen [6]). 7 Gr wh a mall growh rae an mall kewne of he cell. The moel oberve o verge n he oer flow regon only f he gr n he oer regon ha mem growh rae alhogh he flow graen n he oer regon were mall. 8 There are everal oponal complex moel n Flen

67 4 Analy ool 4.3 ompaonal Fl Dynamc flow nvolvng eparaon, flow nvolvng rong avere prere graen. The RS moel lkely o gve beer rel compare o he nvolve (Flen [6]): rap change of ran rae, complex flow. k ε moel n flow ha An oponal qarac prere ran moel can be e becae of mprove accracy for complex flow. Parclarly bac hear flow are emonrae o be more accraely prece wh h opon (Flen [6]). Unfornaely, h opon ofen rel n vergence of he olon. onclng remark The RS moel ha he greae poenal o accraely prec an even neceary for everal flow of nere for h he (ee con of he RS moel). Ye, rel are ll comprome by moel ampon an he e of he RS moel oe no jfy he exra compaonal effor for mple flow. For he FD calclaon n h he he e of a ceran rblence moel fone on verfcaon of he FD calclaon wh mearemen. The pro an con of he rblence moel are e for an nal ge of he mo able rblence moel for a ceran flow Near-wall regon There are wo way o eal wh he near-wall regon: he wall fncon approach an he near-wall moel approach. The wall fncon approach brge he vcoy affece regon (bffer an vco b layer, ee for nance Newa [5]) wh a wall fncon. Th wall fncon he log-law ha we alreay know from econ 3... The near-wall moel approach calclae he flow all he way own o he wall. oneqenly, he wall fncon approach nee few cell cloe o he wall whle he near-wall moel nee many cell (becae of he large flow graen cloe o he wall) all he way own o he wall. Wall fncon The wall fncon or log-law meare o be val for y * > 3 ~ 6 (ee for nance Newa [5]), where * y a non-menonal ance efne by 4 ρ µ k p y p y * = (5) µ In (5), he qany ρ enoe he eny, k p he rblen knec energy a pon p, he ance from he wall o a pon p an µ he ynamc vcoy of he fl. In Flen, he efal opon apple (5) f * y >.5. Near-wall moel The wall fncon or log-law ceae o be val f he flow conon epar oo mch from he ame (ee econ 3..): conan hear re an flly rblen near-wall flow. Accorng o he ampon of he log-law, he wall fncon nrelable for flow wh: a low Re nmber (no flly rblen flow near he wall), evere prere graen leang o eparaon (hear re no conan). y p

68 4 Analy ool 4.3 ompaonal Fl Dynamc In hoe cae, he near-wall moel neee,.e. he flow nee o be olve all he way own o he wall n pe of he exra compaonal effor..5, he efal opon n Flen apple he lamnar re- For a fne meh, where ran relaonhp * y < * * = y (6) wh * y fon by (5) an hear re a he wall. pµ k p * =, where p he velocy a pon p an τ w he τ ρ w The gr The e of an avance rblence moel can be oally cancelle o by a ba gr. The gr confgraon h nee a lo of aenon. A goo gr alo enhance he convergence. The major gr egn gelne are ha: regon wh large flow graen nee many (mall) cell, he flow hrogh all cell hol -a mch a poble- be normal o he nle a well a normal o he ole e of he cell, non-nform gr hol have a mall growh rae (<.). In orer o rece he compaonal effor, he nmber of cell n he oman ha o be kep whn lm. Non-nform gr are nroce o flfl he gr egn gelne of many mall cell a regon wh large flow graen an a few a poble cell whn he oal oman. The gr egn mplfe by he nrocon of block-rcre gr. Th nvolve a oman ve no everal maller oman or block, whch are eaer o fll wh cell. The block hol alreay ake care of he gr gelne ha he man flow hrogh all cell hol be normal o he nle a well a he ole e of he cell. Flen offer he pobly o e a rcre or an nrcre gr. The rcre gr ha o be efne by he egner whle he nrcre gr largely aomacally generae. Becae manfacrng of a rcre gr me-conmng, he aomac generaon of an nrcre gr an nereng opon. However, he nrcre gr can cae conerable angle beween he flow recon an he normal on he nle an ole of he cell. Th generae an error. In every cell, he componen of he velocy hrogh he cell rface are calclae an repreene wh a fne precon. The effec of h fne precon omeme referre o a nmercal vcoy becae ha he ame effec a vcoy. ompare o a fr orer calclaon cheme, a econ orer calclaon cheme can rece he nmercal vcoy. Unfornaely, he econ orer cheme make he calclaon nable an nee more compaonal effor. A olon of a FD calclaon nee o be gr-nepenen. The gr ze hol be mall enogh o acheve h. The nepenency of he gr can be checke by comparng wo converge olon for wo fferen gr: a fne an a core gr. Gr-nepenency reache a he olon are approxmaely he ame. Th of core me-conmng. Gelne, for nance for he near-wall regon, can prove ome of he neceary nformaon for he gr ze. However, gr-nepenence only are f checke wh he above ecrbe meho of refnemen of he gr. Tme-conmng b neceary! For all gr e for he FD calclaon n h he, gr-nepenency are by he foremenone meho

69 4 Analy ool 4.3 ompaonal Fl Dynamc The amopherc bonary layer Th econ ecrbe he mlaon of a neral amopherc bonary layer n Flen 5. Log-law nle parameer The rblence parameer, of a neral amopherc bonary layer (ee econ 3..) can be efne by (Wlcox [8]) he rblen knec energy k * k = 3. 3* (7) wh k ε moel parameer =. 9 µ µ, an he rblen paon rae ε ( z) ε ( z) = κ 3 * ( z + z ) (8) Log-law n Flen For he mlaon of an amopherc bonary layer, he wall fncon approach (econ 4.3.3) can be e o avo a large compaonal effor. By ong ch, ha o be kep n mn ha we mplcly ame a conan hear re an flly rblen flow near he earh rface (econ 4.3.). I no phycally meanngfl o have a fr cell a he rface wh a cenro hegh below he average hegh of he roghne elemen. The example n econ frhermore how ha alo proce valele rel. For he bl envronmen he average hegh of he roghne elemen he average blng hegh H. We h eman h cell H for he fr cell hegh h cell ajacen o he rface. oneqenly, h cell oo large o be able o calclae eal of he flow aron blng wh a hegh lghly hgher han H. Th obervaon analoge o he obervaon ha lea o (3): zmn =.5, or n wor he velocy cloe o he roghne elemen eermne by he local roghne an can no be fon from he average roghne. The velocy cloe o he roghne elemen can only be obane f he nval roghne elemen n he oman are moelle nea of he average roghne ha hey cae. Th me-conmng job no carre o for he mlaon n h he. A hgh z, he flow a he earh rface omnae by eparaon on he nval roghne elemen an vco effec are neglgble. oneqenly, he hegh of he roghne elemen H mch hgher han he hegh of he vco blayer ρ * µ. The regme of he bonary layer hen characerze a flly rogh (ee for nance Newa [5]). In a flly rogh regme, Flen e he law of he wall mofe for roghne (Flen [6]) * Eρ* z * + * ( ) = ( + ) = z z ln ln ln hbh + ( + ) (9) κ µ κ κ µ hbh Eρ* where E = 9.8 for he efal wall fncon, hb a conan ha moel he ype of roghne an + ρ H* H () µ

70 4 Analy ool 4.3 ompaonal Fl Dynamc where H he average phycal roghne hegh 9 an µ he ynamc vcoy (Flen [6]). In a flly rogh regme, H >> µ ρ o ha H +. For nform roghne, he * roghne conan can be aken a =. 5, whle for non-nform roghne a hgher vale hb hb =.5 ~ more approprae (Flen [6]). We wll ame hb =. 75 for non-nform roghne. Who he roghne elemen beng nvally moelle, m be ame ha = n he log-law for he neral amopherc bonary layer (7). Eqang of (9) an (7) hen gve he conon where boh log-law moel have eqal velocy profle + µ ( + hbh ) z = () Eρ * Becae H + wh hb =. 75 (Flen [6]) we have hbh + an h wh bon of () hb z H () E Wh E = 9.8 an =. 75 we h have hb E H z 3z for non-nform roghne hb E H z z for nform roghne K (3) H gven by (3) he np for he roghne of he mlae amopherc bonary layer n Flen. Eqaon (3) can be compare wh he rle of hmb (3): z.9a H H or = 4z (econ 3..). Apparenly, he law of he wall mofe for roghne ame H A H a ceran eny of he roghne a he rface. If A. 4, he rle of hmb gve H 7z. The amopherc bonary layer profle an rblence parameer a he bonare of he oman n Flen can be pecfe wh a Uer Defne Fncon (UDF). Th UDF a mall c-program, whch rea by Flen. The UDF for he bonary layer mlaon for h he pecfe ( z) wh (7), k( z ) wh (7) an ε ( z) wh (8). They were pecfe a hree bonare of he oman: he nle, he op of he bonary layer an he ole. A he op of he mlae bonary layer, he velocy pecfe becae h gve he be mlary wh he acal neral amopherc bonary layer where mo energy pple by he pper layer an pae n he lower par of he bonary layer. I emphaze ha (3) rerce o mall z ay z <.5 becae ha o be ame ha = Some nal verfcaon The flow n he bl envronmen can be characerze a flow aron blng n an amopherc bonary layer. Thoe blng can have a blff, bln or aeroynamc hape (econ 3..). The flow n he bl envronmen herefore nvolve eparaon an large ran rae an can be qalfe a complex flow. Mrakam [5] how ha he κ ε moel H 9 For he bl envronmen, we have an average phycal roghne hegh eqal o he average blng hegh H. We herefore be H nea of he ymbol e n Flen [6]

71 4 Analy ool 4.3 ompaonal Fl Dynamc a well a he RS moel have ffcle o prec ch complex flow, alhogh hey o no qalfy he precon a poor excep for he anar κ ε moel n rafe flow. Km & Boyan [35] how ha he realzable κ ε moel an he RS moel can acheve accrae rel for flow ha nvolve eparaon. Some nal qanavely verfe FD calclaon of flow characerc of nere for h he hol prove he ably o ppor he mearemen n h he. For h prpoe, he followng apec of he flow n he bl envronmen are mlae. Table Inal verfcaon Apec Verfcaon wh Flow aron a blff boy Fla plae Flow aron an aeroynamc boy NAA 8 Flow n he amopherc bonary layer z =. 3 m Fla plae A an example of he flow aron a blff boy, he flow aron a D an 3D fla plae mlae. The rag coeffcen, he backprere p, b an Srohal nmber S of he fla plae were calclae from he mlaon an compare o mearemen. The 3D calclaon carre o wh an ax-ymmerc gr an a eay formlaon of he rblence moel becae mearemen o no how reglar vorex heng for a k. The D calclaon carre o wh a fll D gr an an neay formlaon of he olver o allow he oberve vorex heng n mearemen. The rel of mearemen ogeher wh he rel of he mlaon are gven n Table. Table, S an p, b of a fla plae, calclae wh he realzable k ε moel an RS moel (ee Appenx ) compare o he mearemen (Hoerner [6]). Sorce Hoerner [6] 3D ax-ymmerc eay D neay p, b p, b.7 Re > < Re < 6 S Realzable.5 k ε moel Re = 8 RS moel.8 Re = Re = Re = Table clearly how ha he RS moel oe no alway gve he be rel n pe of he more avance moellng compare o k ε moel. Several oher reearcher compare mearemen an mlaon wh varo rblence moel. Km & Boyan [35] fon ha he avance (non-anar) k ε moel gve o hgh (le negave) backprere a boe wh eparae bonary layer. Th n agreemen wh he rel n Table where frhermore hown ha he RS moel proce a more accrae backprere. Th more accrae backprere he reaon for aopon of he RS moel n chaper 7 n pe of he oher le accrae rel wh he RS moel (Table ). Amopherc bonary layer A D amopherc bonary layer mlae wh he nle conon gven n econ (rblence moel ee Appenx ). The oman for he mlaon ha a hegh of 5 m an a lengh of 4km an he mlae roghne z =. 3 m. The nle profle of velocy - 6 -

72 4 Analy ool 4.3 ompaonal Fl Dynamc ( z ), rblen knec energy k( z ) an rblen paon ε ( z) gve he followng agreemen wh he profle a he ole of he oman. z [m] (z) [m/] Fgre 37 ( z ) a nle an ole of he mlae flow oman calclae wh he realzable nle real. k-e ole RS ole k ε moel an he RS moel for z =. 3 m. z [m] nle real. k-e ole RS ole z [m] nle real. k-e ole RS ole k [m / ] e [m/3] Fgre 38 k( z ) profle (lef) an ε ( z) profle (rgh) a nle an ole of he mlae flow oman calclae wh he realzable k ε moel an he RS moel for a roghne z =. 3. From he fgre, clear ha he RS moel gve a beer performance han he realzable k ε moel for all profle. oneqenly, n h he he RS moel e f he flow n amopherc bonary layer nee o be mlae. Frhermore, he goo agreemen of he nle an ole profle for he RS moel (excep for k( z ) ) prove ha he mlaon ep n econ aeqae f he RS moel e. Two mporan gelne for he mlaon of an amopherc bonary layer are: h cell H an (econ 4.3.4). The effec of obeyng he gelne hown by mlang a bonary layer wh z = m wh h cell fng a mlaon for z =. 3 m.e oo mall cell a he rface. The rel epce n Fgre

73 4 Analy ool 4.3 ompaonal Fl Dynamc z [m] nle real. k-e ole RS ole (z) [m/] Fgre 39 ( z ) a nle an ole of he mlae flow oman calclae wh he realzable k ε moel an he RS moel for z = wh oo mall h cell a he earh rface. I clear from Fgre 39 ha h gve very poor rel. NAA 8 Arfol Arfol-hape blng ex. Sch arfol-hape blng form par of he opc of chaper 6. oneqenly, an nal exploraon of FD for aeroynamc boe neceary an carre o hereafer. Several rblence moel are e o calclae he lf of a NAA 8 (Jacob [3]) arfol a 8 o angle of aack. The mlaon rel are compare wh rel of calclaon gven n Parachvo [87]. The near-wall moel approach aope for he gr of he FD mlaon. The mlaon how ha he realzable k ε moel wh a n orer pwn cheme gve he be rel for h confgraon combne wh a fa convergence (Appenx H). The RS moel verge a he bonare of he oman becae of he nonnform gr (alhogh he growh facor of he gr wa only.). The realzable k ε moel wh a n orer pwn cheme (Appenx ) e n a frher verfcaon of l an for fferen α p o angle above all. Thee FD mlaon rel are agan compare wh rel of calclaon gven n Parachvo [87] an calclaon wh a commercal panel coe (RFOIL) an hown n Fgre 4. l [.] Fgre l an alfa [ o ] Parachvo ke-real r-fol [.] of a NAA 8 arfol a Parachvo ke-real r-fol alfa [ o ] 6 Re =. a a fncon of α. From Fgre 4, we concle ha he realzable k ε moel gve: a goo performance concernng ren precon,

74 4 Analy ool 4.3 ompaonal Fl Dynamc a ffcen qanave agreemen for l below all an a poor qanave agreemen for. Overall conclon concernng nal verfcaon From he hown nal verfcaon, clear ha FD mlaon accorng o he gelne n econ 4.3 gve: goo qalave rel an accepable-o-goo qanave rel. The mearemen for h he are herefore ppore by FD calclaon wh: he RS moel for mlaon of bonary layer an flow aron blff boe an he realzable k ε moel wh a n orer pwn cheme for he mlaon of flow aron aeroynamc boe. In orer o be able o rely on he qanave rel of he mlaon, backng by mearemen hown o be neceary. The FD calclaon for h he are herefore verfe wh mearemen or oher qanave nformaon

75 5 Wn rbne cloe o blng 5 Wn rbne cloe o blng Wn rbne cloe o (bee or on op of) a blng operae n he flow regon nflence by he blng. If he whole ream be, wh an approxmae lengh of x roor ameer (econ..), le whn he regon wh accelerae wn, he power can be calclae from he cbe of he accelerae wn pee. For a cylnrcal or phercal blng hape, he acceleraon fon n econ 3.. hen gve he followng power agmenaon compare o he power n he free ream wn pee. Table 3 Maxmm power agmenaon for mall wn rbne cloe o blng. Blng hape Acceleraon Power agmenaon (econ 3..) ylner (D) 8 Sphere (3D) Th chaper eal wh he performance of he wn rbne cloe o blng an prove he energy yel of a wn rbne above he roof of a harp-ege blng. Wn rbne hgh enogh above he roof o no ffer from he low veloce n he recrclaon regon (econ 3.3.) an operae n accelerae wn for all wn recon whle wn rbne bee blng are e n he wake of he blng for ceran wn recon. Th an mporan avanage for he energy yel of wn rbne a he roof b he acceleraon bee D blng (for nance blng wh cylnrcal hape) can be hgher an rece h avanage. haper 8 gve example energy yel for a cylnrcal an phercal blng hape. 5. The wn rbne performance cloe o a blng I clear ha a hgh acceleraon of he free ream wn pee wane for he wn rbne cloe o he blng. B more eale flow nformaon neceary for aonal neranng of he performance. Th econ herefore analye he flow cloe o a blng. 5.. Performance n parly accelerae flow Wn rbne wh a large roor compare o he characerc blng ze o no olely perform n he accelerae wn cloe o he blng. The power agmenaon for hoe wn rbne can herefore no be fon wh he cbe of he acceleraon. Defnon large roor Le examne he flow confgraon for a wn rbne wh a large roor ameer D compare o he characerc blng ze. The vral nle of he ream be of he roor locae approxmaely.5d pwn of he roor (econ.., Appenx A) n parly accelerae or ecelerae flow pwn of he blng façae. The ownwn par of he ream be wh approxmae lengh 3.5D (econ.., Appenx A) locae n he accelerae wn pee r wh r > a he e of he blng. The flow confgraon hown n Fgre 4 for a wn rbne above he roof an a wn rbne bee a blng. Locaon on he vral nle an ole of he ream be are marke wh a o

76 5 Wn rbne cloe o blng 5. The wn rbne performance cloe o a blng Fgre 4 Sreamlne hrogh he roor of a wn rbne on he roof of a blng (lef) an bee a blng (rgh). Very approxmaely, he wn pee aron he nle of he ream be whle he velocy aron he roor an ole of he ream be r. Therefore, he velocy aron he pwn par of he ream be wh lengh.5d accelerae from O.5 ance of ( ) O H H for he a he roof confgraon an ( ) o r n a W for he bee he blng confgraon (ee Fgre 4). Accorngly, he power agmenaon wll be maller han he cbe of he acceleraon f.5d > O( H H ), where H fon by (63). Th gve D > O(.) (4) H for he wn rbne a he roof. A mlar reaonng gve D O (.) W > (5) for he wn rbne bee he blng. Eqaon (4) an (5) once more llrae ha he BAWT ha a mall ze compare o he characerc ze of he blng. Performance large roor D H O. Sppoe > ( ) or D W O(.) > o ha he large roor perform n parly accelerae flow. The complex confgraon of he wn rbne n he crve ream be n he accelerang flow (Fgre 4) mplfe o an acaor n accelerang parallel flow a hown n Fgre 4. Locaon on he vral nle an ole of he ream be are agan hown wh o

77 5 Wn rbne cloe o blng 5. The wn rbne performance cloe o a blng r r Fgre 4 Performance of an acaor (hae lne) n accelerang parallel flow from aron he vral ream be nle o r aron he vral ream be ole. The accompanyng large loa of relavely large roor wll affec he pee p of he flow o ha epen on he loa. I however ame ha he loa an he roor ze are r r ch ha h neracon can be neglece.e. ame ha r = con. Smplfe vorex heory (ee econ 4..) can be e o calclae he power coeffcen of a wn rbne n ch confgraon. The moel for he flow n Fgre 4 hown n Fgre 43. r p+ p e r Fgre 43 Vorex hee moel of an acaor (haowe lne) n parly accelerae flow. The oer vorex hee are e o moel he ncreae of he velocy from o r. The nner vorex hee are he bonary of he wake cae by he acaor, a n Fgre 33. The confgraon hown n Fgre 43 wll be referre o a he half-ream-be confgraon becae approxmaely half he ream be of he acaor le whn accelerae flow. If he half-nfne oer hee n Fgre 43 are conne o an nfne ance pwn of he acaor, he whole ream be le whn he accelerae flow. The power agmenaon 6 3 hen fon from he cbe of r o ha P,max = 7 r where P,max efne on. Th confgraon wll be referre o a whole-ream-be confgraon. From mplfe vorex heory (econ 4..), he velocy hrogh he acaor n he halfream-be confgraon rea ( ) ( ) = + (6) r r e

78 5 Wn rbne cloe o blng 5. The wn rbne performance cloe o a blng whch proce exacly he ame eqaon a fon for an acaor n nform flow (econ.., (8)): ( ) + (7) = e Wh Bernoll heorem an ampon of eqal ac prere n he wake an he locaon wh velocy r, he prere fference acro he acaor rea (( ) ) p p = ρ (8) + r e h gve for he power coeffcen of he acaor n parly accelerae flow P e e = r + (9) The maxmm power coeffcen P,max fon by fferenaon of (9) wh repec o. The opmal en velocy n he ream be rea e e, op 3 3 3r = + + () whch gve ( ) ( ) = () P,max 7 r r r Th rel applcable for: D an 3D acaor (he e mplfe vorex hee heory val for D an 3D confgraon) n he half-ream-be confgraon

79 5 Wn rbne cloe o blng 5. The wn rbne performance cloe o a blng Fgre 44 how ha he for he whole-ream-be confgraon ( P,max P,max n he half-ream-be confgraon () le han he 6 P,max 7 = ). 3 r 8 P,max/P,Bez [.] pmax,half/pbez [.] Pmax,whole/PBez [.] Fgre 44,,4,6,8 r [.] fon for he half- an whole-ream-be confgraon a a fncon P,max of r. An example of he performance n he half-ream-be confgraon gve aonal nformaon. A r =., () gve 34 % le power for he half-ream-be confgraon compare o he whole-ream-be confgraon. I clear, alhogh a b remarkable, ha he acaor n he half-ream-be confgraon proce only lghly le power han he whole-ream-be confgraon wh j half he ream be whn accelerae flow. Wh Bernoll heorem, he prere coeffcen a large ance ownwn of he acaor rea = () p, e r The Bez lm (econ..), () an () prove he followng nformaon on he operaon a r =.. Table 4 Dfference beween he performance n he whole- an half-ream-be confgraon wh r =.. Qany e, op Whole-reambe confgraon Half-reambe confgraon p, e Table 4 how ha he fference n e, op for he half- an whole-ream-be confgraon mall whle he fference n, p e large. Dmpng of he ecelerae ar a he low prere n he regon wh accelerae flow clearly he orgn of he mall power rop n he half-ream-be confgraon compare o he whole-ream-be confgraon

80 5 Wn rbne cloe o blng 5. The wn rbne performance cloe o a blng 5.. The acceleraon a he roof In econ 3..3, he wn pee cloe o he blng efne n erm of he free ream wn pee, a a ceran reference hegh (for nance roof hegh) an wn recon. = r,, The facor r, ame Re-nepenen. Frhermore, r, a fncon of: he (local) rface roghne aron he blng, he blng hape, he wn recon, he hegh above he roof. A FD calclaon (Appenx ) prove he Table 5 onfgraon for he FD calclaon of hegh (Meren [49]). eph:wh:hegh :3: Blng hegh [m] Srronng roghne [m] z =. 3 r, vale for he followng confgraon. r, where, efne on he roof The wn pee wa calclae wh a FD calclaon on hree locaon above he roof ha are marke wh a o (Fgre 45). Fgre 45 Locaon where he wn pee calclae wh a FD calclaon. For hoe locaon above he roof he calclae r, vale for fferen wn recon ϕ are gven n Table 6. The hegh above he roof H H were choen becae hey howe he hghe acceleraon for a wn recon perpenclar o he large facae of he blng

81 5 Wn rbne cloe o blng 5. The wn rbne performance cloe o a blng Table 6 r, efne on a roof hegh for a hegh rronng roghne of z =. 3 m (Meren [49]). H H above he roof an a A he ege an corner locaon he r, enre Ege orner ϕ H H =. 5 H H =. 5 H H = r, vale a ϕ = 8 o are very low, whch pon o preence n or ownwn of he recrclaon regon. Th alo clear from he reamlne n Fgre 46 fon wh he FD calclaon. Fgre 46 Flow confgraon a ϕ = 8 o. The r, vale for he cenre locaon how he malle nflence from he recrclaon regon. For he cenre locaon, he hegh above he roof aken a H H =. 5 (Table 6). The hegh of he eparaon reamlne above he roof of h 3D blng a ϕ = o fon wh (69) o be H H =.95. Oher ϕ wll gve maller recrclaon regon. Hence he cenre locaon above he recrclaon regon for all ϕ. omparon of cenre- wh he ege- an corner locaon how ha: he meare for he energy eny a he cenre locaon fon wh (57) 4% hgher han he ege- an corner locaon, he angle of he flow wh he roof a he cenre locaon how le varaon han he ege- an corner locaon, he angle of he flow wh he roof a he ege- an corner locaon can be large a ϕ = o an Table 6 how ha he velocy varaon for fferen wn recon a he cenre locaon are maller han a he ege- an corner locaon

82 5 Wn rbne cloe o blng 5. The wn rbne performance cloe o a blng The maller velocy varaon for fferen wn recon a he cenre locaon are alo clear from he probably rbon (55) ha gve he rel hown n Fgre 47. Fgre 47 Wn pee probably rbon on he roof (black crve) compare wh he free ream wn pee probably rbon (grey crve); for he cenre (lef), ege (mle) an corner locaon (rgh). The cenre locaon coneqenly rongly favorable. 5. The lf-rven HAWT n kewe flow Many blng have a cbc hape: he roof fla an harp-ege. In econ 3.3. wa hown ha he flow eparae a ch harp ege. oneqenly, he flow make an angle wh he roof ha vare wh he poon on he roof. Th angle wll be calle he kew angle o ngh from he yaw angle n he horzonal plane. Smlar kewe flow fon bee a blng. Wn rbne above he roof or bee a blng have o operae n he kewe flow (Fgre 48). A l mechanm a he ma ha creae normal flow for he roor oo expenve an ncreae he probably of falre. Therefore, he roor operae n kewe flow. Fgre 48 A HAWT n he flow above he recrclaon regon on he roof (reamlne fon by a FD calclaon). The behavor of a HAWT n kewe flow can be nvegae wh Glaer momenm heory, whch how goo agreemen wh mearemen (Maen e al. [39]). Glaer momenm heory a mx of momenm heory an engneerng ampon. I bae on - 7 -

83 5 Wn rbne cloe o blng 5. The lf-rven HAWT n kewe flow wo fferen ervaon for he nce velocy. One ervaon val for large kew angle an one ervaon val for zero kew angle. Becae boh ervaon gve he ame rel for he nce velocy, Glaer ame ha he rel hol alo be applcable o nermeae kew angle. The hr- an power coeffcen fon wh Glaer momenm heory for a lf-rven HAWT n kewe flow rea (Bron e al. [8]) an P ( γ ) = 4a a co a (3) ( coγ a)( co a) = 4 a a γ (4) where a efne on he oal free ream velocy. P,max fon from fferenaon of (4) wh repec o a ecreae wh γ a hown n Fgre 49.. P,max /P,max, [.] kew angle [ o ] Fgre 49 6 of a HAWT a a fncon of he kew angle γ ( = ). P,max P,max,

84 5 Wn rbne cloe o blng 5. The lf-rven HAWT n kewe flow 5.3 The H-Darre n kewe flow Fgre 5 An H-Darre (Trby) n he flow above he recrclaon regon on he roof (reamlne fon by a FD calclaon). In econ.4 wa hown ha an H-Darre very able for operaon n he bl envronmen. However, he performance of an H-Darre n kewe flow nknown. The prooype of or aeroynamc H-Darre egn, Trby, wa herefore ee n kewe flow n he open je wn nnel (Meren [44], Saro [6]). Srprngly, operaon n kewe flow howe an apprecable ncreae power op compare o normal flow. Moellng he operaon of an H-Darre an even more an H-Darre n kewe flow wh low λ (egn gelne bl envronmen, econ.4 ) very ffcl, an nmero complcae e nee o be aree:. fne apec rao effec of he H-Darre blae n kewe flow.e. for wep blae,. behavor of local hr n kewe flow a large ncon facor, 3. neracon of he fferen flow regme hrogh he H-Darre, 4. behavor of he blae a low Re nmber, 5. behavor of he blae n ynamc all. There a varey of moel avalable o calclae he performance of an H-Darre (Parachvo [53]). Three of hoe moel are bae pon blae elemen an momenm heory. The mple a moel ha ame a ngle ream be hrogh he whole roor, moelle by a ngle acaor. A more avance moel he ngle acaor mlple ream be moel. Th moel bae on mlple non-neracng ream be hrogh he roor. The mo avance moel n h row he oble acaor mlple ream be moel, whch eparaely moel he pwn an ownwn roor par of he H-Darre by wo acaor nea of he ngle acaor n he foremenone moel. Frhermore, an avance moel bae on vorex heory avalable. For or goal, we chooe he ngle acaor mlple ream be moel a a comprome beween he mple an avance moel Incon facor a mall loa The ngle acaor mlple ream be moel fr evelope by Wlon & Laman [84] wll be e. The bac ampon, mporan for he moel of an H-Darre n kewe flow, are ha: he eceleraon of he ar by he pwn an ownwn par of he roor can be moelle wh one acaor beween he pwn an ownwn roor par,

85 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow he flow hrogh he roor can be moelle a flow hrogh mlple ream be ha o no nerac wh each oher, he chor lengh of he blae c mch maller han he ra R of he H-Darre. The fr moel of an H-Darre n kewe flow wa pblhe by Meren e al.. [48]. A econ -more elaborae- moel ha form he ba of he moel preene n h econ wa alo pblhe by Meren e al.. [47]. In kewe flow wh kew angle γ o he horzonal, he roor can be ve no wo par a hown n Fgre 5. One par of he roor ecelerae he flow wh only one roor par: he pwn or ownwn roor par (bcrp ) an he oher par of he roor ecelerae he flow wh wo roor par: he pwn an ownwn roor par (bcrp ). γ H h h θ D Fgre 5 Deceleraon of he flow by a ngle- or oble-roor par of an H-Darre. A cro-econ of he roor a he lef e hown a he rgh e. The vercal lengh of he blae h or h ha operae n repecvely he ngle- or obleroor par confgraon epen on: he kew angle of he flow, he ncon facor, he angle θ an he rao H D. The la nflence mmeaely reveal an mporan operaon qaly of he H-Darre n kewe flow. For a large kew angle an a we roor of he H-Darre, almo he whole roor operae n he ngle-roor par confgraon whle a lener roor almo enrely operae n he oble-roor par confgraon. An elegan ran of hogh ecrbe n Hoerner [7] can be e o fn he lf of an arfol n kewe flow. oner he e-p of a non-wep arfol n poenal parallel flow. The lf force of ha arfol eermne by he angle of aack α of he flow an he free ream velocy. Now ranlae ha arfol perpenclar o he flow recon. The poenal flow experence he ame arfol bonare. Hence, he lf of he ranlae non-wep arfol wll be nchange an eermne by he ame α an velocy of he parallel flow. I concle ha, a long a kn frcon of no mporance, he normal velocy componen,, eermne he lf an he velocy componen parallel o he arfol no mporan.,, Accorng o h o-calle cro-flow prncple, he lf force of a wep arfol or an arfol n kewe flow eermne by he componen perpenclar o he arfol. l,max Lehman [38] how ome mearemen, whch make clear ha above all, l a well a α a largely ncreae wh ncreang weep angle. Above all, he e of he croflow prncple h no allowe

86 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow Fr, he oble-roor par operaon analye. Accorng o he aope ngle acaor moel of he H-Darre, he oble-roor par moelle wh a ngle acaor. By efnon n he acaor concep, he prere force an coneqenly he ncon facor pon perpenclar o he acaor. We h fn he velocy perpenclar o he obleroor par of he acaor a ( coγ ) = (5),, a where he kewe velocy, γ he kew angle an a he local ncon facor of he oble-roor par. The aon hown n Fgre 5. Acaor k repreenaon of he obleroor par F a,, γ,, a γ γ Fgre 5 The flow hrogh he acaor ha moel he oble-roor par n kewe flow. There no nee o moel he local flow angle γ for HAWT becae an P can be fon a a fncon of γ (econ 5., (3) an (4)). For an H-Darre however, he local flow angle γ nee o be moelle becae: γ eermne he blae lengh h or h (Fgre 5) an wh h he performance of he H-Darre an γ can be very fferen from γ becae of he omeme very large a compare o he local ncon facor of a HAWT. Accorng o Fgre 5, γ can be fon from γ nγ = arcan (6) coγ a The horzonal componen of he velocy nfnely ownwn of he acaor (ee Fgre 5): ( coγ a ), eqal zero or become negave for oo large ncon facor. Th cae In econ.4, hown by () ha a vare acro he roor a a nθ, where θ enoe he roaonal angle θ of he roor blae. A moerae a a mall θ coneqenly o cople wh a very hgh a a θ =

87 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow he flow ampon n momenm heory o be volae. We herefore reqre ha co γ a >, whch gve he rercon a < coγ (7) Blae elemen heory By efnon, he magne of he lf force L per n blae lengh can be fon from L = ρ c l, r, (8) Th gve for he rengh of he bon vorex a he blae (ee for nance Kaz & Plokn [33]) where ρ he ar eny, Fgre 53), blae, an Γ = c l, r, (9) r, he magne of he relan velocy on he blae (ee Γ he rengh of he bon vorex of he blae, c he chor lengh of he l, he lf coeffcen of he blae of he oble-roor par. y Γ R θ r, ω x,. α ωr L,, c Fgre 53 Top vew of he H-Darre n kewe flow. The l an of he arfol hol be fon from mearemen or mlaon/ calclaon. The arfol can be rppe a % chor lengh becae h evoke a fxe locaon of he recrclaon regon a he arfol. Who rppng, he recrclaon regon can move ownwn of he ralng ege, whch ncreae he noe emon of he arfol. In Appenx I, calclae - aa for a rppe NAA 8 arfol a fferen Re nmber are hown. Parachvo [53] gve calclae l - aa from Shelahl & Klma [85], l

88 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow whch are -bae on x-fol calclaon- for an nrppe NAA 8 arfol. They how a lower lf a low Re nmber an faer all compare o he rppe NAA 8 arfol. oneqenly, he rppng eem o have a pove nflence on he l an wh h on he performance of he roor a well. alclae rel ece from mearemen on a NAA 8 arfol for low Re nmber are fon n Eaman e al. [3]. The lf force of he blae per n blae lengh can be fon from L = ρ Γ r, (3) wh r, = r,, x r,, y an Γ = Γ (3) Th rel n a lf force of L = Γ Γ (,, ) (,, ) r y x r x y (3) Sbon of r, y, (ee Fgre 53) gve L, x = ωr n θ Γ (33) Wh (9) h gve he lf force n he x-recon a he oble-roor par a = (34) L, x l, ρωr nθ cr, wh ra of he H-Darre R, roaonal pee ω, an roaonal angle θ, where θ = correpon o he locaon where he blae move parallel o he wn recon (ee Fgre 53). The ream be wh eqal o,, y = R θ nθ (35) Becae of perocy, he analy on L, x lme o one revolon. In oble-roor par operaon, he blae pae he flow n he ream be wce n one revolon. The blae h θ π of he oal me of one revolon. So, ay n he ream be for a fracon of ( ) (34) hol be mlple wh θ π o fn he average L, x n he ream be. For he ngle-roor par, h averagng facor half h vale becae he blae pae he flow n he ream be j one me. From (34) an he foregong averagng con, he average force F, for B blae n he recon of he ream be a operaon n he oble-roor Bl par now follow from L, x a F θ = Bl, ρω R nθ cr (36) π Bl,,

89 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow where he nex Bl a F Bl, refer o he o-calle blae elemen heory. Momenm heory The hr force fon wh (36) can alo be fon wh momenm heory, whch hen enoe a F M,. Glaer momenm heory how goo agreemen wh mearemen (Maen, e al. [39]) an wll herefore be e for he calclaon of F M,. I gve (ee for nance Bron e al.. [8]) F M, ρr θ nθ w = a (37) where w he magne of he relan velocy owar he acaor (ee Fgre 5) w n + co ( a ) = γ γ (38) Eqang of F Bl, (36) an F M, (37), gve an mplc relaon for a for an H-Darre wh B blae for operaon n he oble-roor par confgraon n kewe flow a Bc r, = l, λ (39) 4π R w where λ fon from λ = ωr. The ame ervaon for (39) carre o for he ngle-roor par operaon. However, he blae move only one me hrogh he flow n he ream be o ha he averagng facor whn blae elemen heory fon by θ / π. The rel a Bc r, = l, λ (4) 8π R w The local flow angle hen fon a nγ γ = arcan (4) coγ a Noe he fferen ncon facor n he oble- an ngle-roor par operaon an coneqenly he fferen flow angle n he oble- an ngle-roor par operaon (ee alo Fgre 5, lef). To llrae he fference, we be l, = π nα, l, = π nα an γ = an fn afer ome manplaon ha a = a. Th hol nee be he ocome becae only half he nmber of blae ecelerae he ar n he ngle-roor par operaon compare o oble-roor par operaon Incon facor a hgh loa For an ncon facor ha oe no flfl (7).e. f a co γ, momenm heory for an acaor n kewe flow brake own. The velocy far ownwn of he acaor approache zero or become negave an he ame flow paern volae. The (local) a vale can be well above he vale gven by (7) o ha he e of momenm heory no allowe an

90 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow a ha o be eermne by oher mean. Hereafer from mearemen an raher cre econ. a for a co γ wll be eermne Le fr look a he oble-roor par operaon a hgh ncon facor. A a = coγ, he roor block all he flow hrogh he roor an he flow experence a ol plae. The hr force of a ol plae F, p, whch pon normal o he plae rea F = ρ A ol plae (4), p, p where A enoe he plae area. The hr coeffcen of a ol plae, p a a fncon of γ can be fon from mearemen (ee Fgre 54). Fgre 54, p a a fncon of γ (from of a plae wh apec rao 5 gven n Sm & Scanlan [67] an, p of a qare plae or k gven n Hoerner [6]). Mearemen of he hr force of a HAWT n normal flow prove an exrapolae hr coeffcen of, b =.6 ~ or, b =.8 a a = (Bron e al.[8]). Hoerner [6] an Sm & Scanlan [67] gve, p. (ee alo Fgre 54) for a roang or non-roang k n normal flow. Apparenly,, b a facor.8. =. 5 me, p. Hence, =.5 a a = coγ. (43), b, p where, p fon from Fgre 54 for he approprae hape of he H-Darre roor. We wll ame ha (43) can be e for all wn rbne a a = coγ. The hr coeffcen fon wh Glaer momenm heory n he oble-roor par operaon (nex ) - 8 -

91 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow or wh (37) F M,, M, = (44) ρ R θ θ n, M, 4a w = (45) coγ The fr orer approxmaon of a ragh lne from, b o a pon angen o, M,, how a goo f wh mearemen on a HAWT n normal flow (ee for nance Bron e al.[8]). We wll e a ragh-lne, angen o, M, a γ = an eqal o, b a a = coγ. = a + ( ), M,, b, ln,, b at for a > a (46) T where he o calle ranon ncon facor a T mark he change from momenm heory o he emprcal ragh lne. In normal flow, a angen ragh lne acheve wh a = ( γ = ) (Bron e al. [8]). Bae on an approxmae rao of T, b, b, ( a = at ).5 for all γ, we fn at n kewe flow a T 3 (, b ( ) ) co a = γ = γ (47) I hol be noe (ee Fgre 55 a γ = 5 or γ = 6 ) ha a T mark a change n he, ervave a γ >. Th eem nphycal b neglece a a rec rel of he a fr orer approxmaon of a lnear relaon (46) for a > at

92 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow Fgre 55 Example of, n kewe flow wh, b calclae wh (43) from a plae wh apec rao 5 (Fgre 54). In agreemen wh he al ampon n normal flow, we wll ame he e of eqaon for a > at o be applcable o local conon of he roor of he H-Darre n kewe flow. The ame ervaon carre o for he ngle-roor par, whch rel n, M, 4aw = (48) coγ an = a + ( ), M,, b, ln,, b at for a > a (49) T Power coeffcen In econ 5.3., wa hown ha he power of he H-Darre n kewe flow cople wh he flow angle hrogh he roor an he blae lengh n he ngle- an oble-roor par operaon. Frly, h econ how he moel for he nflence of he flow angle. Seconly, he power coeffcen of he H-Darre n kewe flow calclae. The flow angle In econ 5.3., he flow angle n he oble-roor par operaon fon o be fferen from he flow angle n he ngle-roor par operaon. In oher wor, he flow hrogh one roor - 8 -

93 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow par phe ae he flow hrogh he oher roor par (ee alo Fgre 5). I ame ha he flow o no mx b nea have an average flow angle γ ~. Accorng o Fgre 5, f he blae lengh h for he ngle-roor par operaon, rea H γ arcan (5) R h = R an ~ γ nθ (5) For he oble-roor par operaon we have coneqenly h = H h = H R an ~ γ nθ (5) If H γ > arcan (53) R an H H arcn θ π arcn R an ~ γ ~ R anγ (54) h lme o h = H (55) an h = (56) H For γ > arcan an θ oe he bonare efne by (54), h an h can agan be R fon wh (5) an (5). I now poble o fn ~ γ from he mplc weghe average of h an h efne by ~ h H h γ = γ + γ H + h H + h (57) whch gve γ ( h = H ) = γ an γ ( h = H ) = γ. Local power coeffcen Wh ampon of he cro-flow prncple (econ 5.3.), he exrace power P, per n blae lengh, n a ream be wh wh y, can be fon from blae elemen heory. Bae on he rel of blae elemen heory n De Vre [76], we have

94 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow B cλ h r P l α α θ πr H,,, =, n co (58) l, for he oble-roor par operaon an B cλ h r P l α α θ πr H,,, =, n co (59) 4 l, for he ngle-roor par operaon. The lf- an rag coeffcen of he blae can be fon from calclaon or mearemen (ee for nance Appenx I). Fne apec rao effec A HAWTS, fne A r effec are ally moelle wh a p lo facor ha correc he local ncon facor a he blae for he ownwah of he p vorex of he blae. Th approach rel n a pan-we varaon of he ncon facor. For an H-Darre, he ncorporaon of fne A r effec can be moelle wh he negral effec of he p vorex on blae rag an lf by amng ellpc pan we loang. Sch approach no able for a HAWT becae he reqre nrealc ampon of ellpc pan we loang oe no make ene for a HAWT. For ellpc pan-we loae blae, he effec of a fne Abbo & von Doenhoff []). A r gven by (ee for nance l ˆ = + (6) πa r an where ^ refer o fne ˆ = l l + (6) A r A r an abence of ^ refer o nfne Decreae n power coeffcen cae by rag of bearng an ro The oal ecreae n power cae by frcon n he bearng an rag of he ro calclae wh a o-calle eqvalen rag coeffcen of he blae. We fn he power lo from h eqvalen rag coeffcen a eq P A r., eq = F Rω, where he eqvalen rag force lo, eq F, of a blae area A calclae wh (ee Fgre 53) F, eq, eq ρ r, From he geomery n Fgre 53 we ee ha he velocy = A (6) r, can be fon from ( λ θ ) ( θ ) a a r, = + ( ) n + ( ) co (63) For hgh λ, h can be approxmae wh

95 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow ( λ ) r, Wh Rω = λ an (64) P lo fon a (64) P lo ( ) 3 A ρ (65), eq λ For B blae h gve c θ 3 P, lo = B, eqλ (66) R π For an H-Darre n normal flow, wh λop 3, can be hown ha (64) rel n a fference n oal P of 4% compare o he rel fon wh (63). Whn he moel, he approxmaon (64) herefore no rerce o hgh λ. Inegrae power coeffcen The oal power coeffcen P of a roor par whn a ream be wh wh can be fon by ang procon an lo n power P P P, P, lo R θ nθ, =, + (67) We cone he power procon of he ngle-roor par wce becae here one ngleroor par a he wnwar e of he roor an one ngle-roor par a he leewar e of he roor. The oal power coeffcen of he H-Darre n kewe flow negraon of P fon from (68) P, can be fon from θ = π P = P θ (68) θ = We now have a cloe e of eqaon o calclae he performance of an H-Darre n kewe flow. Maple prove an envronmen for explorng an applyng mahemac, whch allow he e of eqaon for he H-Darre n kewe flow o be eravely olve. The op of he Maple program gven n he nex econ Performance The fference n a an a fon n econ 5.3. mporan for he performance of he H- Darre n kewe flow. In orer o acheve P,max of boh he ngle- an he oble-roor par, op, a an a op, hol be eqal, whch of core mpoble. Therefore, P,max of he whole roor acheve a non-opmal λ op, an λ op,. In orer o acheve P,max,, he ngle-roor par enforce he oble-roor par o operae wh λ op,. On he oher han, he oble-roor par enforce he ngle-roor par o operae wh λ op, fng P,max, comprome vale λ op ha rel n P,max. The of he whole roor epen on he relave poron of roor area ha operae n ngle- or oble-roor par operaon. Becae aop, > aop,, we know ha λop, > λop, o ha λop, < λop < λop,, whch how ha λ op

96 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow ncreae wh ncreang roor area n ngle-roor par operaon. Hence, λ op ncreae wh ncreang γ. Above a ceran γ where he whole roor operae n ngle-roor par operaon, λop = λop, an λ op ecreae agan wh ncreang γ becae le ma flow hrogh he roor ha o be ecelerae by he roor. A γ ncreae, an ncreae roor area experence he free ream velocy. To be more prece, he roor area ha experence he free ream velocy ncreae wh he roor par operang n ngle-roor par operaon a he ownwn e of he roor. Th gve a poble power ncreae n kewe flow. Poble an no ceran becae he effec of he non-opmal λ op, an λ op, can cancel he effec of he ncreae roor area ha experence he free ream velocy Verfcaon of he moel The moel for he H-Darre n kewe flow, preene n econ 5.3. o 5.3.3, fr verfe wh mearemen n normal flow. Seconly, he moel verfe wh mearemen n kewe flow. The rel are hen nerpree an ce. Moel np A mall H-Darre ee n he open je wn nnel o verfy he moel. The characerc of he mall e H-Darre are gven n Table 7. Table 7 haracerc of he mall e H-Darre B Arfol NAA 8 rppe a.c c [m].8 H [m].5 D [m].755 Bc [.].457 R The l an vale of he rppe NAA 8 arfol are gven n Appenx I. Verfcaon of he moel n normal flow Saro [6] carre o wo mearemen of he power of he H-Darre a a fncon of λ, n normal flow (Fgre 56). The mearemen howe ha P,max =.5, a λ op, low =.9, where he nex low ae o Darre operae a mch hgher λ op. λ op o ncae he relavely low vale of λ op a mo A hgh λ, (ynamc) all a he H-Darre blae avoe. Snce h f he moel ampon be, he moel fe by nng, eq on he mearemen a hgh λ. Sch f of he moel wh A r an l, from Appenx I rele n, =. 4. The mearemen an moel rel are hown n Fgre 56. By amng A r, he fne A r effec were mplcly ame o be repreene by, eq. Th can nee no moel he fne A r effec. The ba agreemen wh he rel erve from mearemen a low λ herefore no rprng. eq

97 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow A f of he moel a hgh λ wh he acal A r = 6.5 of he ee H-Darre (Table 7), wh l an from Appenx I an correcon for fne A r rele n, eq =.5. The vale nee how an apprecable ecreae compare o, eq =. 4 a A, eq becae par of he loe are now moelle wh fne A r nea of amng all loe o be repreene by, eq. Fgre 56 how ha he rel wh fne A r effec ncorporae lghly cloer o he mearemen alhogh he mprovemen mall. r P [.] mearemen mearemen moel nf. Ar moel fn. Ar Fgre 56 P a a fncon of λ (lamba) n normal flow, a fon from mearemen an he moel wh:, eq =. 4 an nfne A r (ol lne) an wh, eq =.5 an fne..5 A r (oe lne) lamba [.] The mprovemen by moellng fne A r effec n normal flow mall an he ffcle enconere n moellng fne A r effec n kewe flow are nmero. Fne A r effec n kewe flow were herefore no aken no accon. Verfcaon of he moel n kewe flow Accorng o he verfcaon n normal flow,, eq =. 4 hol be e f A ame. The moel wh, eq =. 4 compare wh he rel erve from mearemen n kewe flow (Saro [6]). The rel are hown n Fgre 57, where of he oal roor wh he accompanyng op r P,max λ a fferen kew angle are gven. Whn he Maple coe, α an α are calclae an repore. Th how alle blae of he H- o o Darre a all γ excep for < γ < 5. For an H-Darre wh alle blae, he rel of he moel canno be re becae ynamc all, whch preen (ee econ 5.3.6), gve a fferen lf of he blae an accorngly a fferen performance

98 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow Fgre 57 P,max an λ op erve from mearemen n kewe flow compare wh he o o moel rel for. 4. The moel applcably rerce o < γ < 5. The nex refer o, = γ = o. The moel eem able o prec he o o n kewe flow for < γ < 5 wh a P,max crepancy of only % compare o he e rel. The moel rel of o o < γ < 5 gve a le afyng crepancy of 9% compare o he e rel. λ op for A ce n econ 5.3., expece ha λ op ncreae wh ncreang γ p o a ceran γ where he whole roor operae n he ngle-roor par operaon an hen ecreae. The moel how a qalave agreemen wh he mearemen concernng h behavor Dcon on he valy of he moel I nereng o compare wo H-Darre wh he ame hegh an ameer b fferen λ op, repecvely λ op, low an λ op, hgh where λop, hgh > λop, low. The vco power loe (65) 3 are fon o be proporonal o λ. A mall ncreae n λ op, whch expece n kewe flow, wll herefore cae a mch hgher ncreae n vco loe for he H-Darre operang aron λ op, hgh compare o he H-Darre ha operae aron λ op, low. Ye, for boh H-Darre, he poble power ncreae n kewe flow a fncon of he exra roor λ op

99 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow area n kewe flow, whch eqal for boh H-Darre a hey have he ame hegh an ameer. Hence, he power ncreae wll be mch maller for an H-Darre a λ op, hgh compare o he H-Darre a λ, op low. Th llrae wh e of he moel hereafer. Performance n kewe flow a hgh λ op Sppoe we have an H-Darre n kewe flow wh ¼ of he oly of he H-Darre n Table 7. Table 8 haracerc of a mall H-Darre for a moel calclaon B Arfol NAA 8 rppe a.c c [m]. H [m].5 D [m].755 Bc [.].4 R The moel how ha λ op, hgh > 4.9 for he H-Darre of Table 8 compare o λ op, low =.9 of he H-Darre of Table 7. The low oly an accorngly hgh λ op, hgh make re ha all of he blae avoe n boh he ngle- an he oble-roor par operaon. o o Therefore, he oal kew angle banwh γ 6 covere wh he preene moel for he H-Darre n kewe flow. The moel rel are hown n Fgre 58. P,max / P,max, [.] laba op /laba op, [.] gamma [ o ] 3 4 kew angle [ o ] Fgre 58 Opmal power coeffcen ver opmal λ n kewe flow an. 4 a low oly., =

100 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow The moel h how ha P,max ecreae wh ncreang γ whle λ op only lghly ncreae wh γ. In conra o he H-Darre wh low λ op (Table 7), he ncreae of P,max n kewe flow aben a he H-Darre (Table 8) wh hgh λ op. Th n agreemen wh he con a he begnnng of he econ. We h know ha he ncreae n power of an H-Darre n kewe flow rerce o low λ op H-Darre. Re effec Fgre 59 how he calclae vale of ac NAA 8 arfol rppe a % erve from Appenx I. l,max / l,max (Re=6) ( l,max )/ ( l,max,re=6) l, max an a he ame angle of aack for a Re/ 5 Fgre 59 alclae normalze lf an rag coeffcen ver Re nmber of a NAA 8 arfol rppe a %. Fgre 59 how a banwh Re = 6 ~ 5 where l, max ncreae an ecreae wh ncreang Re nmber or λ op. We know ha λ op ncreae n kewe flow. oneqenly, an H-Darre wh blae ha operae a for example Re = 6 a γ = o experence an ncreang l, max an ecreang a γ ncreae. Th very profable for he power ncreae n kewe flow. In h conex, mporan o noe ha he blae of he hgh oly H-Darre operae a Re=7 a γ = o, whle he blae of he low oly H-Darre operae aron Re=7 a γ = o. Hence, he hgh oly H-Darre prof from he ecreae rag whle he low oly H-Darre ffer from ncreae rag. Dynamc all Dynamc all can occr f he lf change raply a he cae wh he lf of he H-Darre blae. An mporan parameer for ynamc all he rece freqency of he moon ω c efne a k = where ω he roaonal pee of he H-Darre, c he chor lengh of he blae an he free ream velocy. Seo an Galbrah [64] meare ha ynamc all occr f k >. 4. For he H-Darre efne by Table 7 an Table 8, k mch hgher o ha ynamc all occr for boh H-Darre n cae of alle blae. In orer o f he moel ampon, all of he blae hol h be avoe. Sall aben a a hgh egn λ op of he H-Darre. However, emonrae ha he power ncreae n cl c 6 The mearemen were carre o a Re=.5 an he blae wa bjece o pch rae moon aron he ¼-chor pon

101 5 Wn rbne cloe o blng 5.3 The H-Darre n kewe flow kewe flow aben for an H-Darre wh hgh λ op. The verfcaon of he preene moel for he H-Darre n kewe flow herefore navoable wh low egn λ op an herefore rerce o a mall banwh of γ. The moel of an H-Darre n kewe flow epen on np of everal -very-ffcl-omeare- or relae -no-accraely-known- qane. For nance he np of a local a hgh a n kewe flow (econ 5.3.) an l an a low Re nmber are mpoble o meare accraely. However, n orer o prof from a maxmm ncreae n n kewe flow, a moel o egn he H-Darre for kewe flow npenable. P,max We h look forwar o a large reearch effor cople wh he generaon of accrae np aa for he moel of he H-Darre n kewe flow. 5.4 The energy yel a he roof Th econ ce he power op of a wn rbne on a roof of a blng. Bae on he knowlege hown n he prevo econ, poble o gve emae of he power op a all flow angle. Th allow a fr orer calclaon of he energy yel of a wn rbne on a roof of a blng n omnreconal wn. A a harp-ege blng rrone by low roghne Sppoe we have a.5 kw H-Darre, whch wll be nalle above he cenre of a harpege roof of a m hgh blng wh wh 3 m an eph m. The Darre ha he followng characerc. Table 9 Moel H-Darre aa. Rae power.5 kw n wn pee 4 m/ Rae wn pee 3 m/ o wn pee m/ Roor area 6 m Varable λ.3 P A characerc ze of he roor fon from he roor area: D = 6.5 m. The blng ha H = m o ha D H =. an he H-Darre can be characerze a mall (econ 5..). The power of he H-Darre can h be fon from he cbe of he local wn pee. The hegh of he eparaon rajecory a he roof can be calclae wh (68) an (69) an fon a 4 m above he roof cenre. We e a 5 m above he roof cenre. Then, he acceleraon of he free ream wn pee by he harp-ege roof gven n econ 5.., Table 6. The kew angle above he cenre of he roof can be fon from he ervave of he eparaon rajecory (69) an fon o be 4 o. Fgre 57 gve a % ncreae of he power op of he H-Darre a 4 o kew angle. For flow parallel o he long e of he blng o or ϕ = 9, (77) n econ 3.3.4, gve reaachmen a m. Hence, he kew angle a he cenre of he roof for o ϕ = 9 mall an he power ncreae of he H-Darre The power ncreae n kewe flow rerce o he parclar H-Darre e n he mearemen o ha ame ha he H-Darre n h calclaon ha he ame propere

102 5 Wn rbne cloe o blng 5.4 The energy yel a he roof approxmae by zero. The followng behavor of he power ncreae of he H-Darre n kewe flow a oher ϕ ame.,5, P / P, [.],5,, ph [ o ] Fgre 6 Increae n power for fferen wn recon ϕ. Sppoe ha he parclar blng locae n a regon wh a poenal wn pee of 4.5 m/ wh a rronng roghne of gralan or z =. 3 m. The free ream wn pee a roof hegh for z =. 3 m fon wh (6) o be 5. m/. The change from free ream wn pee o he local wn pee a he roof r, gven n econ 5.., Table 6 an he velocy for he hroe wn rbne fon by = r,. The energy yel of he H- Darre above he roof cenre can be calclae wh (6), erve n econ Th gve an energy yel of he H-Darre of 377 year kwh m year. kwh or 6 ( ) A a harp-ege blng rrone by hgh roghne Now ppoe ha he ame confgraon of H-Darre an blng a e n he prevo energy yel calclaon locae a he followng e. Table Locaon of he blng n a cy. z Upwn cy borer a m 7.5 m 3 km Wh (3), he ep n roghne 3 km pwn of he blng rel n h k = 69 m. The blng hegh of m le above zmn =.5 m o ha he log-law can be e (ee econ 3..) o calclae he average free ream wn pee a roof hegh. Sbon of h = 69 m an he aa n Table n (33) h meanngfl an a free ream wn pee a k roof hegh of 3.3 m/ fon. Sppoe ha r, oe no epen on z b only on blng hape. r, hen eqal o he mlae vale gven n econ 5.., Table 6. The kew angle wll be maller han he kew angle wh he ame blng a gralan roghne a he effecve hegh of he blng ecreae wh. The ervave of he eparaon rajecory (69) gve a kew angle of o. Th only o le han he kew angle wh z =.3 m n he prevo example. I herefore ame ha Fgre 6 can be e o - 9 -

103 5 Wn rbne cloe o blng 5.4 The energy yel a he roof approxmae he power ncreae of he H-Darre n kewe flow. Accorng o (4) n econ 5.. an he aa n Table, he relave ze of he H-Darre hol obey D H < O(.) n orer o be reae a mall, whle he acal ze of he Darre gve D H =.. The Darre ze accorngly a he ege of he creron. If he Darre ame o be mall, he energy yel of he H-Darre above he roof cenre can be calclae wh (6), erve n econ Th gve an energy yel of he H-Darre of 46 kwh year or 74 kwh ( m year). Becae of he relavely large H-Darre ze, h an pper lm for he energy yel a h e. A a phercal blng Sphercal blng cae a mch hgher acceleraon han harp-ege blng. On op of a phere n poenal parallel flow fon ha, =.5 (econ 3..). r Fgre 6 Ar mpreon of a phercal blng wh BAWT. Wh h, he energy yel can hen be calclae wh (6). Th gve an pper lm for he energy yel of a mall BAWT a a phercal blng n an amopherc bonary layer becae he blockage of a phere n an amopherc bonary layer maller compare o a phere n parallel flow. I ame ha = 5. m/ a z =.3 an = 3.3m/ a z = (econ 3..). Th gve he followng energy yel. Table Upper lm of he energy yel of a.5 kw BAWT a he roof of a phercal blng. e Energy yel kwh ( m year) = 5. m/ a z =.3 m 3 = 3.3m/ a z = m The energy yel a he e of a blng ylner-lke D blng cae a hgh acceleraon of he free ream wn pee

104 5 Wn rbne cloe o blng 5.4 The energy yel a he roof Fgre 6 Ar mpreon of cylnrcal blng wh BAWT. Sppoe we have he followng confgraon of a mall acaor cloe o he rface of a cylner. ϕ Fgre 63 Top vew of a mall BAWT a he e of a cylner. Poenal heory prove an approxmaon of he change n wn pee r, aron a cylner n parallel flow (econ 3..). A he rface of he cylner, poenal heory gve he followng r, vale a he e of he mall acaor cloe o he cylner rface. Table r, a he e of a cylner. ϕ r, o o,8 o 9 o o 45,35.4 The e of r, from poenal heory a an approxmaon of r, a a cylnrcal blng o o o o allowe excep a ϕ 5 35 (econ 3..). A ϕ 5 35, he acaor locae n he wake of he cylner o ha r, = can be ame. r, for a BAWT a he e of a cylnrcal blng can h be approxmae wh he followng vale

105 5 Wn rbne cloe o blng 5.4 The energy yel a he roof,5 r, [.],5, ph [ o ] Fgre 64 Ame r, for a BAWT a he e of a cylnrcal blng a a fncon of ϕ (ph). Wh he ame wn conon a e for Table, (6) an Fgre 64, gve he followng energy yel. Table 3 Energy yel of a.5 kw BAWT a he e of a cylnrcal blng. e Energy yel kwh ( m year) = 5. m/ a 67 z =.3 m = 3.3m/ a 68 z = m

106 6 Wn rbne beween arfol-hape blng 6 Wn rbne beween arfol-hape blng Th chaper eal wh he performance of a wn rbne beween arfol-hape blng. Th BAWT opon nroce n econ.3, where ecrbe a one of he hree bac aeroynamc confgraon for concenraor n he bl envronmen. ompare o he cloe o a blng concenraor (chaper 5), he egn of h confgraon reqre more effor. No only he BAWT nee o be egne b alo he blng an moreover he neracon of boh. Th egn herefore no one of he fr concenraor for BAWT ha wll be lze. Solary all blng wh an arfol hape or aeroynamc hape ex (Fgre 65). Th ngreen of he concenraor ha ce n h chaper herefore no oo far from everyay pracce. Fgre 65 Example of ngle blng wh aeroynamc hape. From lef o rgh: EDF an egeel blng La Defene, France, Hoforen he Hage, he Neherlan. The confgraon of BAWT beween all aeroynamc blng chemacally hown n Fgre

107 6 Wn rbne beween arfol-hape blng Fgre 66 Schemac confgraon of BAWT beween aeroynamc blng. In h confgraon, wo ngle aeroynamc blng are ymmercally poone aron he ax of he BAWT. In h way, he aeroynamc blng are able o accelerae he nrbe wn pee for ceran wn recon. The hegh of he blng n Fgre 66 mch hgher han he oher ze of he blng. The confgraon herefore how omnan D effec (econ 3..). In wn energy lerare, he concep of a wn rbne beween rng arfol (3D/ ax-ymmerc) well known. Whn he conex of he BAWT, ch confgraon reqre a blng egne a a rng arfol. Th eem no convenen for he laon of he blng elf an complex o bl. A large par of he rng-hape blng frhermore locae cloe o he earh rface o ha ha o perform a low wn pee. Th lea o a low energy yel of he BAWT. The reearch n h chaper herefore foce on he D confgraon of a wn rbne beween arfol-hape boe a howe n Fgre 66. Lerare e everal name for he confgraon wh aeroynamc boe aron a wn rbne. Th confgraon ofen calle: a ce wn rbne, a hroe wn rbne or a ffer-agmene wn rbne. In h he, he generc erm ce wn rbne e o enoe he man prncple of power agmenaon wh aeroynamc boe aron a wn rbne. We wll ngh hroe wn rbne from ffer-agmene wn rbne by he hape of he c a hown n Fgre 67. The hroe confgraon on he lef of Fgre 67 bae on an arfol hape of he c whle he c of he ffer-agmene confgraon on he rgh of Fgre 67 bacally a wenng be or ffer

108 6 Wn rbne beween arfol-hape blng Fgre 67 Dc confgraon calle hroe (lef) an ffer-agmene (rgh). The flow come from he lef e an he reamlne are hown a hn lne. The wo c hape have an horcal orgn. One col ay ha he hroe wn rbne npre by aeropace engneerng whle he ffer-agmene wn rbne npre by mechancal engneerng. The fr expermen wh ce wn rbne were carre o wh ffer (Igra [9], Foreman e al. [85]). Follow p expermen of Igra [9] howe ha rng arfol a he ffer ex cae a hgher concenraor effec of he ffer. Th marke he evelopmen owar he mch horer hroe wn rbne confgraon. Oher expermen an mlaon concernng he performance of he ce wn rbne are fon n De Vre [76], Phllp e al. [57] an Hanen [3]. The horcal orgn of he wo c hape reflece n he evelopmen of wo fferen mahemacal moel: hoe ha moel he performance bae on ffer propere (Wlon & Laman [84], De Vre [76] an Van Bel [9]) an hoe ha moel he performance wh arfol characerc (De Vre [76] an Van Holen [5]). The moel wll be ce ogeher wh mprovemen n econ 6. an econ 6.. Mo mahemacal moel for he ce wn rbne n lerare are no cloe n he ene ha he e of eqaon no enogh o calclae he performance of he ce wn rbne. Aonal np reqre o cloe he moel. Sch aonal nformaon ofen obane by expermen or by herc argmen aken from mlar flow moellng effor. The moel of Van Holen [5] e he lf coeffcen of he arfol of he c a clore. Th appealng a here ample heory an mearemen on lf of arfol. The moel erve for lghly loae ce wn rbne a neglec wake expanon. Th chaper how ha ch rercon nneceary by ervaon of an analoge moel, b wh wake expanon, whch how ha wake expanon apprecable. The clore alo acheve wh he lf coeffcen of he arfol ha form he c. Hence, he moel clafe a a moel for a hroe wn rbne. 6. Momenm heory Th econ ecrbe mahemacal moel for he ce wn rbne ha are erve wh he eqaon of conervaon of momenm n negral form (econ 4..3). The arng pon of all moel n h econ D flow hrogh he ce wn rbne.e. a nform flow ame a all area perpenclar o he ax of he ce wn rbne an varaon of he flow qale only poble parallel o he ax of he ce wn rbne

109 6 Wn rbne beween arfol-hape blng 6. Momenm heory 6.. Shroe wn rbne Sppoe we have a hroe acaor a hown n Fgre 68. y e A Acaor x A A ex A e Shro Fgre 68 Top vew of a hroe acaor. Whn he al ampon of momenm heory, conervaon of axal momenm yel ( ) F + F = ρ A (69) S e where flow. F he hr force of he acaor an F S he axal force of he arfol on he The hr force of he acaor fon wh Bernoll heorem rea ( ) F = ρ A (7) e ombnaon of (69) an (7) ng wn pee hrogh he acaor., S FS ρ = A gve an expreon for, he ( ) = + + e, S ( ) e (7) Accorng o (), mlplyng wh F gve he aborbe power of an acaor. ce acaor coneqenly fon a P of he P e e e, S = (7) omparng P of he hroe roor (7), wh P of a bare wn rbne (), how ha a >. All mearemen an mlaon of reference power ncreae only poble f, S menone n he nrocon of chaper 6 how a power ncreae of he ce wn rbne

110 6 Wn rbne beween arfol-hape blng 6. Momenm heory Eqaon () how ha h only poble wh, S >. In poenal flow, he rag force of an arfol zero (Paraox of D Alember). Hence,, S > how ha he lf force of he arfol ha an axal componen. Th make frhermore clear ha arfol wh hgh-lf -ch a cambere arfol- are wane for he hroe wn rbne a hey cae a hgh power op. By ampon of, S = conan, P,max can be fon from (7) by fferenaon of P wh repec o e (De Vre [76]). Ye,, S pobly a fncon of e o ha P,max ay nknown who amng, S = conan. Momenm heory for he bare wn rbne (econ..) prove help. I how ha acheve a e = nepenen of he P,max energy exracon proce pwn of an area wh axal velocy ( ) 3 e nrbe flow. I herefore ame ha a hroe wn rbne alo acheve e = a long a an area wh velocy ( ) 3 fon a or ownwn of he c. e If ame 3 ha here an area wh axal velocy ( ) + bone by P,max a + bone by nrbe flow can be e 3 + bone by nrbe flow a or ownwn of he c ex, bon of e = n (7) gve P,max of he hroe wn rbne a 6 P, max 7 + 3, S = (73) Whn he conor of momenm heory,, S ay nknown (he clore problem) an ffcl o emae, S by calclaon a h nvolve negraon of he (complex) prere rbon on he arfol of he loae hroe acaor (De Vre [76]). Mearemen can be e o emae, S b n ha cae he mearemen are alo able o reveal P,max. omparng (7) an (7) a e = 3 how ha F eqal for he hroe wn rbne an bare wn rbne. A he exrace power of boh he bare an ce acaor fon by mlplyng F wh, he rao of P,max of he hroe wn rbne an bare wn rbne fon a P,max, hro, hro P,max, bare, bare = (74) In oher wor, he power ncreae of he ce wn rbne proporonal o he ncreae of he velocy hrogh he acaor. For a bare wn rbne, he power can be fon by 3 P = P ρ A, where he free ream velocy. A velocy ncreae ' h 3 rel n P P = ( ). In oher wor, he power ncreae of a bare wn rbne cae by a velocy ncreae proporonal o he cbe of he velocy ncreae. The fference beween h rel an (74) how ha he velocy n he c, canno be reae a free ream velocy for he ce wn rbne. hro 3 Th ampon reae n more eal n he exene ffer moel n econ 6.., where FD calclaon are e o how he valy of he ampon. - -

111 6 Wn rbne beween arfol-hape blng 6. Momenm heory 6.. Dffer-agmene wn rbne Wlon & Laman [84] nroce a moel for a ce wn rbne ha bae on he ampon ha he c operae a a ffer or be wenng ownwn of he wn rbne. Van Bel [9] nroce wake expanon n h moel. Th econ preen he moel of Van Bel wh a pl p n a mplfe ffer moel an an exene ffer moel. Smplfe ffer moel Accorng o Van Bel [9] ame ha bare acaor wake expanon (ee econ.) ake place from he ffer ex o a ance nfnely ownwn of he ex. Wh h ampon, he e of eqaon on he performance of he ffer-agmene wn rbne cloe.e. no frher np reqre o olve he e of eqaon an nknown. We wll refer o h ffer-agmene wn rbne moel, whch hown n Fgre 69, by he mplfe ffer moel. ( ) e = ( a) a A Aex Ae Fgre 69 Operaon of a hroe roor accorng o he mplfe ffer moel. Bernoll heorem gve he prere rop acro he acaor a ( ) p = ρ (75) Accorng o he hypohe above (73), he maxmm power exrace by he acaor fon a a =. Ma conervaon beween A 3 ex an A e hen gve e A = (76) ex, op 3 A The power pae by he acaor fon by mlplcaon of he hr force of he acaor ( = pa ) wh. Th gve P e = (77) Wh bon of e = an (76) n (77), P,max fon a 3 A 6 ex P,max = 7 (78) A 6 By bon of he rel of ma conervaon n he ffer an P, max, bare = 7 n (74), clear ha (78) eqal o (74). Th no rprng a he e of bare acaor wake expanon n fac eqal o ng momenm heory. Boh moel for he ce acaor - -

112 6 Wn rbne beween arfol-hape blng 6. Momenm heory h how ha he power agmenaon cae by he ffer proporonal o he area rao of he ffer A A. ex Exene ffer moel Le rern o he ampon of bare acaor expanon. Mearemen (Igra [9], Foreman e al. [85], De Vre [76]) how ha he prere a he ffer ex below amben prere. Th rel n an exra expanon compare o bare acaor expanon. Van Bel [9] nroce h exra expanon by movng he area wh velocy ( a) ownwn of he ffer ex area A ex where he area hen enoe by wake can hen be realze beween he area ( ) A ex an A ~ ex where A ~ ex A ~ ex. The exra expanon of he ha he ame velocy a. The moel hown n Fgre 7 an wll be referre o a he exene ffer moel. The acaor ae c ownwn of he ffer enrance4 o allow eay 4 comparon wh he vorex moel n econ 6.. I however no relevan for he moel ecrbe here. c 4 A A ex ( ) e = ( a) a A ~ ex Ae c Fgre 7 Operaon of a ffer-agmene roor accorng o exene ffer heory. The Lanceer-Bez lm agan prove a = 3 (ee explanaon above (73)) o ha ma conervaon ownwn of he ffer ex rel n Inea of (78), P,max now fon a ~ A ex, op = 3 (79) A = 6 P,max 7 A A ex (8) lore of exene ffer moel wh he vorex moel for a hroe roor The vorex moel for he hroe roor (econ 6.) able o cloe he exene ffer moel. I gve n (99) a = ( + e )( + ξ ), where ξ gven by (98) ecrbe he 4 The choce for c bae on arfol heory a brefly ce n econ 6. an canno 4 be fon whn he ffer-agmene wn rbne moel. - -

113 6 Wn rbne beween arfol-hape blng 6. Momenm heory concenraor effec of he c fon wh he vorex moel. Sbon of e = from he Lanceer-Bez lm n ( )( ξ ) = + + rel n e ( ) = +ξ (8) 3 where he average gn on kppe a = whn he exene ffer moel. Th gve wh (79) ha ~ A A ex ex A = ( + ξ ) A ex 3 (8) If A = D h, where D he acaor ameer an h he hegh of he arfol, from he geomery of he ffer. 3 Aex D h ch α A ex fon = + n (83) where α half he cone angle of he ffer. Sbon of (83) n (8) gve P,max a A ~ ex A ex a A A ex ex + ξ = 3 c + nα D (84) In orer o calclae ome vale for he rao Table 6. Th gve he followng rao ex A ~ A we e he rel from econ 6..5, ex ex A ~ A. Table 4 Rao A ~ ex Aex erve from a clore of he exene ffer moel wh he vorex moel (Table 6). D c ξ A ~ ex A ex ~ Apparenly, he ampon A ex A ex = n he mplfe ffer moel can be a cre approxmaon. 6. D Vorex moel for a hroe roor A fr ecrpon of a hroe wn rbne moelle wh vorce fon n Van Holen [5]. Sch a moel wll be referre o a vorex moel. The vorex moel erve n h econ ffer from Van Holen moel by wake expanon an coneqenly by an nce velocy of he arfol ha epen on he velocy n he wake of he hroe wn rbne. ompare o oher moel, he vorex moel prove aonal nformaon on he operaon of a hroe wn rbne. The vorex moel ncle a non-nform axal velocy rbon hrogh he acaor an able o hanle fne apec rao effec of he arfol. ex - 3 -

114 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor 6.. Se-p of he moel A repreenaon of he force of a loae hroe acaor by vorce n poenal flow hown n Fgre 7. The acaor locae beween he ¼ chor pon on he arfol a ha parclar locaon gve he hghe nce veloce accorng o a ngle-bon vorex repreenaon of arfol. c 4 D D e Fgre 7 Vorex repreenaon of a loae hroe wn rbne n poenal flow. The arfol are repreene by a vorex rbon ha accelerae beween he arfol. The eceleraon of cae by he acaor repreene by crve vorex hee aron he wake of he acaor. e efne a he velocy a nfne ance ownwn of he hroe acaor. Hence, he vorex rbon a he enrance of he vorex hee ha repreen he wake, have a neglgble effec on e. In oher wor, e m be nform. The nknown ameer D e an hape of he reamlne aron he wake are a fncon of he performance of he hroe acaor, whch a fncon of he wake hape a he wake nce a velocy a he arfol. A very complex erave procere neceary o calclae he performance of he hroe wn rbne from h nenve feeback yem. An approxmae vorex repreenaon hown n Fgre 7 rel n a moel of he hroe roor ha mch eaer o hanle b ha ll capable of moellng he man feare of he hroe roor

115 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor y c 4 3 c 4 w e D e D x Fgre 7 Vorex moel of D loae hroe acaor n poenal flow. The hroe wn rbne wll be moelle wh he followng ampon (ee Fgre 7):. he hr force of he acaor F repreene by half nfne ragh vorex hee an a en wake expanon, wh De D, veloce n y-recon can be neglece, he angle of aack on he arfol α conan,. he arfol wh ame α are moelle by a ranglar vorex rbon, 3. he velocy experence by he arfol moelle by a ngle axal velocy w, whch ac on he ¾-chor pon of he arfol, 4. he HAWT moelle by an acaor, 5. he flow fel ecrbe by poenal heory. Th moel, ee Fgre 7, wll be referre o a he vorex moel of he hroe wn rbne. The ampon are ce an banae hereafer. A horogh banaon of ampon popone o econ 6..6 a h reqre he rel of he vorex moel. On ampon The ampon of ragh vorex hee an De D allow eermnaon of he nce veloce of he wake on he hroe acaor by mplfe vorex heory (econ 4..), o ha he veloce n y-recon can be neglece an he approxmaon α = conan l = conan jfe. l hol be fon from exernal orce lke mearemen, mlaon or mahemacal moel. Th he clore of he vorex moel. On ampon A bon ranglar chor-we vorex rbon a fr orer approxmaon of he bon vorex rbon a an arfol fon n poenal flow calclaon (ee for nance Kaz & Plokn [33]). The ampon α, enre aache flow on he arfol an largely mplfe he calclaon of he nce veloce. On ampon 3 Whn he ngle-bon vorex moel of a hn ymmercal arfol, Kaz & Plokn [33] how ha he bonary conon of zero normal velocy a he rface of an arfol hol be pecfe a one pon. Th pon calle: he ¾-chor pon or collocaon pon. Whn - 5 -

116 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor he hn arfol ngle-bon vorex approxmaon, he ¾ chor apparenly he locaon where he arfol experence he flow. The ¾-chor pon h he approprae locaon for he velocy w o ac on he arfol. On ampon 4 Th HAWT moel nroce by Froe [] an ce n econ... On ampon 5 Becae of he large characerc ze of he blng ha form he arfol, obvo ha Re. oneqenly, he flow aron he hroe roor can be approxmae wh poenal flow (econ 4..). 6.. Ince veloce Th econ gve he ervaon of he nce veloce of wake an arfol. Ince veloce of he wake an arfol A explane n on ampon, he nce axal velocy of he wake, wake on boh he acaor an he arfol profle can be approxmae wh mplfe vorex heory (econ 4..) ( ), wake = e (85) The nce velocy of he ranglar vorex rbon of one arfol on he ¾-chor pon of he oher arfol, w, hro calclae a f a ngle-bon vorex a he ¼-chor pon of one arfol nce an axal velocy a he ¾ -chor pon of he oher arfol. The aon hown n Fgre 73. c 4 c Γ w D ϕ, w, hro Fgre 73 Ince velocy a he ¾-chor pon cae by a ngle-bon vorex a he ¼-chor pon., w, hro hen rea Γ = χ c π D + D, w, hro (86) - 6 -

117 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor where Γ = c (87) l w he rengh of a ngle-bon vorex a he arfol (ee for nance Bachelor [] or Kaz & Plokn [33]), he conan lf coeffcen of he arfol (ampon ) n parallel l flow wh velocy w an χ a correcon facor. Th correcon neee a he ranglar vorex rbon wren a a ngle-bon vorex a he ¼-chor pon of he arfol. In Appenx G, he complex eqaon for χ fon o be well fe by he mch mpler eqaon χ = D.5 c.6 D f.3 c (88) A e nepenen of y (nrocon of econ 6.), he local hr force F fon by (7) alo nepenen of y. The local power of he hroe acaor P( y ) fon by mlplyng F wh he local velocy hrogh he acaor ( y ) (econ..). The oal power P exrace by he acaor h rea y= D y= D P = F ( y) y = F ( y) y = F (89) D y= y= where can be fon from he average nce velocy hrogh he acaor by he arfol,, hro. Th average nce velocy,, hro calclae a f a ngle-bon vorex on he ¼-chor pon of he arfol nce a velocy a he ymmery ax. Γ = χ (9),, hro a π ( D ) The facor χ a accon for he non-nform velocy hrogh he acaor an he fac ha he ranglar vorex rbon calclae a a ngle-bon vorex on he ¼-chor pon of he arfol ha nce a velocy a he ax. The complex eqaon for χ a well fe by he mch mpler eqaon (Appenx G) R χa =.55 ln c R for. 5 (9) c 6..3 Relng veloce The relng veloce w en can be fon from w +, w, hro, w, wake = (9) an - 7 -

118 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor = + (93),, hro,, wake Sbon of (85), (86) an (87) n (9) gve he velocy a he ¾-chor pon of he arfol c c π D + D w = l w + χ e ( ) (94) Rewrng gve w a an explc fncon of he propere of he hroe acaor w ( + ) e = χ lc c 4π D + D (95) fon from bon of (85) an (9) n (93), whch gve c = + χ ( ) (96) π D l w a e Wh bon of (95) n (96) h gve = ( + e ) + χalc π D χ lc c 4π D + D (97) We efne an agmenaon facor ξ ξ = χalc π D χ lc c 4π D + D (98) Then (97) rea omparng (99) wh ( ) ( )( ) = + + ξ (99) e = + for he bare acaor (econ..) how ha ξ enoe he velocy ncreae a he acaor cae by he arfol. e - 8 -

119 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor 6..4 Power coeffcen Wh (89), (7) an (99), P of he hroe roor rea = e + e P ( +ξ ) () P,max can be fon by fferenang P wh repec o nepenen of e, P,max acheve a e. A (98) how ha ξ, op 3 e = () Th n agreemen wh he Lanceer-Bez lm (ee explanaon above (73)). Sbon of () n () gve he fnal eqaon for of he hroe acaor where ξ fon from (98). P,max ( ) 6 = +ξ () P,max I clear ha a hgh ξ wane n orer o acheve a hgh 7 P,max. By conlng (98) clear ha a hgh ξ acheve a a hgh lf coeffcen. The praccal lmaon of he agmenaon of he power by he hroe acaor herefore forme by all of he arfol Verfcaon of he moel In expermen wh a hroe wn rbne n or open je wn nnel eher cale effec or blockage effec wol gve large error n he mearemen (econ 4.). FD calclaon have neher of he problem. They were herefore e nea of he mearemen. In orer o pck he rgh rblence moel, ome nal FD calclaon on l an of an arfol were carre o. They howe ha he k ε -realzable rblence moel prove (econ 4.3.5): a goo performance concernng ren precon of l an, a ffcen qanave agreemen wh oher reference on l below all an a poor qanave agreemen wh oher reference on. Excep for, h a goo ba o rely on he k ε -realzable rblence moel n a performance calclaon of he hroe wn rbne. The combnaon of momenm heory an he vorex moel for he hroe roor help o jge he mporance of an naccrae. Wh (),, S a 8 e, op = fon a S = ξ. For arfol below all, o 3, 9 ha, S a ξ >. Hence, ha a neglgble effec on, S an, S manly eermne by he axal componen of l. We can herefore rely on he k ε -realzable rblence moel o verfy he vorex moel for he hroe wn rbne. The verfcaon wll be carre o wh wo arfol hape: cambere arfol an ymmercal arfol. A of he e cambere arfol mch hgher han of he ymmercal l arfol, expece ha De D n ampon beer approxmae for he cambere arfol. The vorex moel herefore expece o proce more accrae rel wh cambere arfol. Th verfe by a performance calclaon of he hroe roor wh boh arfol hape. l - 9 -

120 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor Verfcaon hroe acaor wh cambere arfol A cambere hgh-lf arfol evelope a Delf Unvery, econ wn energy, e for he arfol (Appenx D). The performance of a D hroe acaor eermne wh an ax-ymmerc FD calclaon. The rel for he velocy fel a 74. hown n Fgre P,max Fgre 74 FD calclaon of he hroe acaor wh cambere arfol a zero loa wh D c =. (low veloce: ark grey, hgh veloce lgh grey). The confgraon eal of he hroe acaor are gven n Table 5. Table 5 onfgraon eal hroe acaor wh cambere arfol. l a ambere arfol Da-999 α o o α = (Appenx D).8 Acaor locaon beween c 4 pon arfol D c Dfferen rao Re 6. The rel of he FD calclaon are compare wh he vorex moel for he hroe roor (econ 6., (99) an ()) an momenm heory for he hroe roor (econ o 6.., (7), (7)). The vorex moel wa cloe wh l =.8 a α = for he arfol an neee no more np. The moel obane wh momenm heory wa cloe by he - -

121 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor FD rel for, S a he fferen F vale. Wh (7), a gven F of he acaor n he FD calclaon prove he np for n (7) an (7). Table 6 mmarze he rel for P,max an, op of he vorex moel an momenm heory. e Table 6 omparon of he moel an FD rel for he hroe acaor wh cambere arfol. Vorex moel Momenm heory D c ξ, op, vor, op, cf P,max, vor P,max, cf, op, mom, op, cf P,max, mom P,max, cf I can be concle from Table 6 ha he vorex moel how a remarkable goo agreemen wh he FD calclaon excep for, op a mall D c. The momenm heory moel, cloe wh FD rel for, S, how a hgher crepancy wh he FD rel n boh, op an P,max. The rel of P,max from he FD calclaon are hown n Fgre 75.,5 P,max /(6/7) [.],5,5,5,5 D /c [.] Fgre 75 P,max of he hroe acaor a a fncon of D c a fon by he FD calclaon. Verfcaon of hroe acaor wh ymmercal arfol A Naca 8 arfol (Jacob [3]) e for he c. The performance of a D hroe acaor eermne wh an ax-ymmerc FD calclaon. The rel for he velocy fel a of he acaor are hown n Fgre 76. P,max - -

122 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor Fgre 76 FD calclaon of he hroe acaor wh ymmercal arfol a zero loa wh D c =.84 (low veloce: ark grey, hgh veloce lgh grey). The confgraon eal of he hroe acaor are gven n Table 7. Table 7 onfgraon eal hroe acaor wh ymmerc arfol. ymmercal arfol Naca 8 α o o l a α = (Parachvo [53]).3 Acaor locaon Beween c pon arfol 4 D c.84 Re 6. Table 8 mmarze he rel for P,max an, op of he vorex moel an momenm heory. Table 8 omparon of he moel an FD rel for he hroe acaor wh cambere arfol. Vorex moel Momenm heory D c ξ, op, vor, op, cf P,max, vor P,max, cf, op, mom, op, cf P,max, mom P,max, cf

123 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor Table 8 how ha he rel of he vorex moel for he ymmercal arfol le accrae han he rel for he cambere arfol (Table 6). Neverhele, he rel of he vorex moel ll goo a comparng wh he FD rel how 3% fference n an only 3% fference n, op! 6..6 Dcon on he valy of he moel Le look n more eal a he ampon D D n he vorex moel. From ma conervaon n he ream be ownwn of he acaor, bon of (99) for fon by he vorex moel h gve e P,max D e D rea De D e =. Wh D D e = + e ( + ξ ) (3) Wh = a P,max of he acaor hen fon ha e, op 3 D D e op = + ξ (4) Th he approxmaon [ De D ] jfe for ξ, whch acheve wh hgh-lf op arfol. Sbon of ξ gven n Table 6 an Table 8 n (4) gve [ De D ] =.6 ~ 4.. The goo agreemen of he vorex moel wh he FD calclaon op even a mall ξ lea o he conclon ha [ De D ] >.6 eem alreay enogh o jfy op he approxmaon[ De D ]. op An mporan rel of he ampon [ ] e op D D = conan. Becae = conan, mplcly ame ha α = conan. A long a he laeral velocy a he l ¾-chor pon of he arfol v mall compare o he axal velocy w, he ampon α = conan jfe. Whn he vorex moel ampon of a ragh wake an a en wake expanon (Fgre 7), vorex heory (85) prove v an he vorex moel (95) prove w. Table 9 mmarze he np neee for a calclaon of v an w a =. e, op 3 l Table 9 Inp for a calclaon of v an w a =. e, op 3 a b R = D e, op c 3 e D ( + ξ ) D Wh (85) an Table 9 v rea v = ln 4π 3 ( ξ 3 ) ( ξ ) 4 c + + D 4 c + + D (5) - 3 -

124 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor Wh (95) an Table 9, w rea w 3 = χ lc c 4π D + D (6) The rao v w gve he change n angle of aack α cae by he laeral velocy. v α = arcan w (7) For he hroe acaor confgraon wh cambere an ymmercal arfol gven n Table 5 an Table 7, α α = ( α α) ( α) ( α) are calclae an an [ ] l l l l l hown n Fgre 77.,6,6 Dalfa/alfa [ o ],4, D l / l [.],4, Dl/l amb. hro Dl/l Symm. hro,5,5 x [.],5,5 x [.] Fgre 77 α α an l l a = a a fncon of ξ. e, op 3 Fgre 77 confrm ha he approxmaon [ ] e op = conan allowe for ξ or D D. I aonally how ha <. or conan for cambere arfol wh ξ >.8. l l for he ymmercal arfol oe no eem o f he crve hrogh he l l vale for he cambere arfol. The reaon fon n l of he arfol, whch can be obane from poenal heory (ee for nance Bachelor []) for he cambere arfol an l, camb l ( β ) l l n( β + α) = π (8) co, = π n( α) (9) l ymm for he ymmercal arfol. Takng he ervave of l, camb an l, ymm wh repec o α wh bon of o α = gve o β = for he camber angle of he arfol (Appenx D) an l l, camb l, camb = co( β + α) α =.4α () - 4 -

125 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor l, ymm l, ymm = co( α) α = 4.7α Hence, cambere arfol how a mch maller nflence of change n α. Frhermore, he nflence of change n α are mall for arfol operae aron all a α = α = a alle arfol. Whn he aache flow ecrpon on he l, camb l, ymm arfol n he vorex moel, he mo accrae rel are h expece for cambere arfol operae j before all occr. 6.3 Applcaon of vorex moel rel o momenm heory omparon of from he vorex moel for he hroe roor (99) wh he rel of momenm heory for he hroe roor (7) gve e, = ξ () Hence,, a fncon of e an one may no ame, o be nepenen of e n momenm heory (ee con below (7)). 6.4 Fne apec rao effec In a confgraon where blng are e a arfol (Fgre 66), he fne hegh of he blng nroce fne A r effec n he performance of he hroe wn rbne. Th paragraph gve an approxmaon of he evaon on he D vorex moel for he hroe wn rbne calclae by nclng he 3D effec of fne A of he arfol. Sppoe ha he arfol are locae n a nform parallel flow (ch flow approxmae n an amopherc bonary layer evelope over low roghne) an have ellpcal pan-we loang. For ch loang, he effec of fne A gven by (6) r r ˆ l l + A = r where he refer o fne A r an abence of refer o nfne A r. The arfol are e on he earh rface,.e. he earh rface a ymmery plane for he flow aron he arfol. The characerc lengh of he arfol for he calclaon of A r hol h be oble he hegh of he arfol h. A r h = () c Becae of he fne A r, ξ fon by (98) ecreae o - 5 -

126 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor ˆ ˆ χ alc ξ = πd ˆ χ lc c 4πD + D (3) where Ĉ l fon by (6). An example how he mporance of he effec of fne acaor wh he followng characerzaon. A r. Sppoe we have a hroe Table 3 Shroe acaor characeraon propery α l D c Vale o π nα Wh (3), ξˆ how he followng behavor a a fncon of A r..8 G~/G [.] Ar [.] Fgre 78 Example of ξˆ of a hroe acaor a a fncon of A r. The ecreae n ξˆ for blng ha normally (ee Fgre 65) have A r = 4 ~ 8, a conerable % ~ 35%. En plae on he arfol can be e o rece he effec of fne A r an ncreae ξˆ. 6.5 The ce mall wn rbne A ce mall wn rbne efne a a wn rbne ha experence he local velocy n he c a a free ream velocy. Th econ gve he ervaon of he relave ze he mall wn rbne compare o he ze of he c. Frhermore, h econ gve he ervaon of he free ream velocy ha experence by he ce mall wn rbne

127 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor 6.5. Allowe ream be ze A wn rbne wh ameer D ha a ream be wh a lengh of approxmaely 6D (ee econ.). If he velocy aron ha ream be conan, he power P of ha wn 3 rbne fon wh () a P = P ρ A. Le a wn rbne wh ameer D an power P be e a (, ) ( x, y) ( a, b).5 ( a, b) f wh a b beween he arfol. Ame ha ( x, y) ( a, b) = ±. I hen follow ha ( ) 3 P ρ, = f P = a b A (4) (, ) = (, ) ±.5 (, ) x y a b a b a.5d x a + 3.5D an b D y b + D The wn rbne wll hen be calle mall compare o he arfol. A FD calclaon of he x, y = conan hown n Fgre 79. conor lne ( ) Fgre 79 FD calclaon of ( x, y) = conan beween cambere arfol (confgraon Table 5) an maxmm ream be ze of mall wn rbne. Accorng o Fgre 79, he maxmm lengh of he ream be hen vare from 6D.7c D c =. a he ax o D c cloe o he arfol, he velocy cloe o he arfol hghe. The rao D c for whch (4) apple vare wh hroe wn rbne confgraon

128 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor 6.5. Velocy beween he ¼-chor pon Sppoe, ch a mall wn rbne e beween he ¼-chor pon: he locaon where he velocy approxmaely he hghe. Frhermore, ame flow parallel o he ax beween he arfol. Then, h mall wn rbne experence he velocy fon wh mmaon of he nce velocy of he arfol ((66) n Appenx G) an he free ream velocy : Γ 3c c ( y) = + 3arcan 3arcan 4πc + 4( R y) 4( R y) ( ) ( ) ln ( ) Γ R y 6 R y + c + + 4π c c 6 R y + 9c Γ 3c c 3arcan + 3arcan 4π c 4( R + y) 4( R + y) ( ) ( ) ln ( ) Γ R + y 6 R + y + c + 4π c c 6 R + y + 9c (5) In h eqaon c he chor lengh of he arfol, R half he ance beween he ¼- chor pon on he arfol ( R = D ), y he ance o he ymmery ax beween he arfol an Γ fon wh (87): Γ = lwc The velocy w n (87) can be fon wh an analoge ervaon ha lea o (94) wh abence of he nce velocy from he wake of he roor (85) a he wake of he mall wn rbne oe no nce a velocy on he arfol. w hen fon a w c l w χ = + c π D + D (6) Afer gropng w h rel n w = χ lc c 4π D + D (7) ( ) y can now be calclae an compare o a FD calclaon. The rel hown n Fgre

129 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor.75 y/r [.] (y)/ [.] Fgre 8 Velocy ( y ) for a mall hroe wn rbne (confgraon Table 5). The marker how he FD rel whle he ol lne how he rel of (5). Fgre 8 how a maxmm of 4 % fference beween he FD rel an ( y ) from (5). 6.6 The hroe H-Darre The confgraon of he hroe H-Darre hown n Fgre 8 (H-Darre keche a a be). Fgre 8 onfgraon of he hroe H-Darre. Wlon & Laman [84] gve a mple moel for an H-Darre. The moel how ha he H- Darre exrac he energy where he blae move crown an canno exrac energy where he blae move along-wn. Hence, he energy exracng par of he H-Darre le n he locaon wh he lowe velocy aron he ymmery ax beween he arfol (econ 6.5). Th rel n a maller power agmenaon compare o a hroe HAWT. The hroe H-Darre herefore no e. 6.7 Yawe an oppoe flow Whn he conex of he he, blng form he concenraor of he wn rbne. The concenraor coneqenly oo heavy o yaw no he recon of he wn. Operaon of he hroe wn rbne a varo yaw angle nevable. I herefore nereng o y he behavor of he hroe wn rbne n yawe flow. Sch one n One of yawe flow conon flow from he back of he hroe wn rbne or oppoe flow. The behavor n oppoe flow reae n

130 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor 6.7. Yawe flow The confgraon of a hroe roor n yawe flow hown n Fgre 8. A a rel of he yawe flow, he angle of aack of he ownwn arfol ecreae α an he angle of aack of he pwn arfol ncreae α α +. α Γ ϕ Γ + Fgre 8 Shroe roor n yawe flow. If he arfol are operae well below all, α α wll lea o l ( α ) < l ( α) whle α α + lea o l ( α+ ) > l ( α). A ξ of he hroe roor n yawe flow con of a conrbon from boh arfol, he overall effec of hee change on wll be mall. If he arfol are operae j below all, α α wll lea o l ( α ) < l ( α) b α α + lea o l ( α+ ) l ( α). Th how ha he relave change n P of a hroe roor n yaw hgher f he arfol have α = αall compare o a confgraon wh α αall. Mearemen of P,max of a hroe roor n yaw, wh cambere arfol a α are fon n Phllp e al. [6]. Ther mearemen how he followng ecreae of a fncon of he yaw angle. P =, o 5 αall P,max a Normalze P [.] Fgre 83 Decreae of yaw [ o ] P,max of a hroe roor n yaw wh α =. o 5 αall - -

131 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor Th how a lghly lower P,max n yawe flow han P,max erve by Glaer momenm heory (Fgre 49) Oppoe flow of a HAWT n yawe flow There a weprea mneranng of he operaon of a hroe roor n oppoe flow o ( ϕ = 8, ee Fgre 8 for he efnon of ϕ ) where he large area of he hroe roor rece pwn. Many non-aeroynamc expec a large concenraor effec an he confgraon hogh o be an ar cacher. Th paragraph he ome lgh on he performance of he hroe roor n oppoe flow. The confgraon hown n Fgre 84. avy e A A e Fgre 84 Shroe roor n oppoe flow, wh lf of he arfol moelle by bon vorce hown a crclar arrow. In conra o he acceleraon of he flow cae by he bon vorce n normal flow, he bon vorce of he arfol now nce a velocy ha ecelerae he flow hrogh he enrance of he c. The harp leang ege of he arfol cae eparaon an a cavy forme ownwn of he arfol. Vorex hee canno moel he cavy ownwn of he arfol. A ervaon of a vorex moel for he hroe wn rbne n oppoe flow herefore no poble. I convenen o look for oher way o moel he hroe roor n oppoe flow. Becae of he analoge flow conon, a hroe roor n oppoe flow, a hown n Fgre 84, can be conere a a ce roor. A moel for wn rbne n c hrogh blng evelope n chaper 7. In econ 7.3, an approxmae plae concenraor moel preene, val for well-rone c nle. Accorng o h moel, he performance of a concenraor n a confgraon wh a cavy yel P,max β = 3 3 where β he area conracon behn he acaor an calle he eparaon velocy, whch a meare for he velocy aron he cavy. nknown b accorng o he mlary wh he flow aron he plae concenraor n econ 7.3, can be aken eqal o fon for he plae concenraor. For ymmercal arfol, he wake of he acaor wll conrac a a coneqence of he c hape (econ 7..). Wh cambere arfol, he wake o 3 - -

132 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor wll how approxmaely zero conracon, a he flow n he c parallel ownwn of he acaor. I can herefore be ame ha β = for cambere arfol. From hee coneraon (35) lea o (36) an (37), whch rea P , max for a D hroe roor wh cambere arfol n oppoe flow an P , max for a 3D hroe roor wh cambere arfol n oppoe flow. Hence, he concenraor effec mall n 3D (3%) b conerable n D (43%). A mall varaon aron o ϕ = 8 (ee Fgre 8 for he efnon of ϕ ), he ecreae n P,max mall a aron he cavy oe no change mch (econ 7.5) Zero power op There a range of ϕ wh o o ϕ 9 < < ha cae a cavy a he nle an ole of he o o hroe roor. A one parclar < ϕ < 9, he cavy prere a nle an ole of he hroe wn rbne are eqal o ha he power of he hroe wn rbne zero (Fgre 85). I lkely ha h parclar ϕ approxmaely perpenclar o he lne ha connec he leang an ralng ege of he arfol becae n ha cae he cave a nle an ole approxmaely experence eqal eparaon conon. Hence, he power op of o he hroe wn rbne lkely o be approxmaely zero a ϕ 9 α. cavy cavy α ϕ Fgre 85 Flow conon a zero power op of he hroe wn rbne. 6.8 The energy yel of he hroe roor Th econ ce he power op of a hroe roor n omnreconal wn. Bae on he knowlege hown n he prevo econ, poble o gve emae of he power op a all flow angle. Th allow a fr orer calclaon of he energy yel of he hroe roor n omnreconal wn. - -

133 6 Wn rbne beween arfol-hape blng 6. D Vorex moel for a hroe roor The energy yel calclaon wll be bae on he D confgraon of he hroe roor wh cambere arfol a hown n Table 5, wh D c =.. Accorng o Table 6 an (), o he arfol gve a power agmenaon of.76 a ϕ = (ee Fgre 8 for he efnon of ϕ ). The con an mearemen rel n econ 6.7 make clear ha he power op lowly ecreae wh ncreang yaw angle. The ecreae n power wh ncreang yaw angle approxmae wh he mearemen rel hown n Fgre 83. Zero power op o o o expece a ϕ 9 α = 78. A ϕ = 8, he mearemen an moel n econ 6.7 gve a power agmenaon of.43. Th wll lowly ecreae wh ncreang varaon from o ϕ = 8. Wh he prevo coneraon, r, can be calclae by (6). A mooh crve o hrogh he r, aa, gve r,.3 a ϕ = 9. The rel hown n Fgre 86.,4, r, [.],8,6,4, ph [ o ] Fgre 86 Ame r, for a BAWT beween D arfol a a fncon of ϕ (ph). In orer o allow comparon of he energy yel of varo BAWT n econ 8.., he arng pon for all BAWT eqal,.e. ame ha = 5. m/ a z =.3 an = 3.3m/ a z = (econ 3..). Wh (6) an Fgre 86, h gve he followng energy yel. Table 3 Energy yel of a ce.5 kw BAWT. e Energy yel kwh ( m year) = 5. m/ a z =.3 m 36 = 3.3m/ a z = m

134 7 Wn rbne n c hrogh blng 7 Wn rbne n c hrogh blng The blockage of he blng cae low veloce a he pwn façae of he blng an hgh veloce a he e of he blng. Th rel n a hgh prere a he pwn façae an a low prere a he e an ownwn façae of he blng (Bernoll heorem). Thee prere fference nce a flow n a connecon beween hgh- an low prere. In everyay lfe, h nce arflow known a ragh when one open a wnow on a wny ay. A wn rbne can perhap operae n h ragh. In h chaper, ch a concep e n bac confgraon: a wn rbne n a c hrogh a k or plae, whch wll be calle he plae concenraor. 7. The plae concenraor moel The moel for he performance of he wn rbne n he plae concenraor, wh area (c area ncle), bae on he followng ampon:. he flow aron he plae concenraor characerze by Re an can h be approxmae wh poenal flow (ee econ 3..),. he wake of he plae concenraor can be approxmae by a cavy.e. he veloce n he wake are ame o be zero, 3. he c area A Ap o ha he nflence of he ma flow hrogh he c on he ma flow aron he plae an wake can be neglece, 4. he wn rbne n he plae concenraor can be moelle by an acaor. The flow aron he ax-ymmerc plae concenraor (D or 3D) hown n Fgre 87 an characerze a follow. A he pwn ege of he plae, he bonary layer eparae () an a cavy ownwn of he plae forme (v). A je (v) ha orgnae from he prere fference beween he pwn an ownwn e of he plae ven no he cavy (v). A p v v e βa A Ap plae concenraor acaor eparaon v cavy v je Fgre 87 Flow aron he plae concenraor n parallel flow perpenclar o he plae. Lef: FD calclaon of he reamlne, rgh: flow characerzaon

135 7 Wn rbne n c hrogh blng 7. The plae concenraor moel In econ 7.., hown ha he je conracon epen on he nle ronng of he acaor. 7.. Power coeffcen Accorng o Bernoll heorem, he local ac prere a he pwn e of he acaor p rea + where he local velocy a he acaor. ( ) p+ = p + ρ (8) Wh Bernoll heorem an he cavy moel (econ 3.3.5), he nform prere n he cavy p fon a c p c ( ) = p + ρ (9) where he o calle eparaon velocy ha reproce he meare n (9). p c by bon The acaor n he plae concenraor aborb maxmm power f all ar hrogh he acaor ecelerae wh he ame opmal amon. A maxmm power op we h have a nform velocy e a je area β A wh parallel reamlne (ee Fgre 87). In orer o fn he maxmm power aborbe by he acaor, h ame ha e nform. Ma conervaon n he je ownwn of he acaor hen gve A, x eβ =, x e = () β A A where he bar above, x enoe averagng over he acaor area A. A je area β A, he reamlne are parallel an he prere n he je eqal o he cavy prere p. Hence, Bernoll heorem on a reamlne n he je ownwn of he acaor c gve he ownwn prere on he acaor p a p = p + Th gve a prere rop acro he acaor ( ) c ρ e () p of ( ) p = p+ p = ρ e () - 5 -

136 7 Wn rbne n c hrogh blng 7. The plae concenraor moel Noe ha p n () oe no epen on an ha a nform e rel n a nform p acro he acaor. A p =, () how ha = e. FD calclaon n Boema [] ppor he rel of () ha = e a p = for blng hape mlar o he plae concenraor an for a cro-hape blng arrangemen wh he c a he cenre of he cro 5. Sbon of () n () gve ( ) =, x p ρ β (3) For nform e an accorng o () coneqenly nform acaor P can be fon from p, he power aborbe by he A wh () an (3) h gve P = p A = p A = pa A (4), x, x, x P of he acaor a, x, x P = β (5) Accorng o (5) he hghe P acheve wh β a large a poble. A h momen, P,max canno be fon by fferenaon of P n (5) wh repec o, x a β nknown an pobly a fncon of, x. The nare of β herefore analye n he nex econ. 7.. Je conracon Momenm conervaon coneraon n he area behn he acaor gve he neceary backgron for β. The conrol volme aken a hown n Fgre 88. v cavy v je w onrol volme r v e βa A acaor v 5 In econ 7.7, h blng arrangemen hown an calle he combne plae concenraor

137 7 Wn rbne n c hrogh blng 7. The plae concenraor moel Fgre 88 onrol volme for momenm conervaon n he je. onervaon of axal momenm n a conrol volme rea (econ 4..3) F, ax = I, o, ax I, n, ax (6) Wh he prere an veloce n he conrol volme, (6) rea p A c + A A e e p A = ρ A ρ A (7) Sbon of (9), () an () n (7) gve a qarac eqaon n β, whch prove wh (, x ) (, x, r ) γ = an β = γ ± γ, x, r Eqaon (8) recly reveal ha β no fncon of je., x (8) = + where, r enoe he local raal velocy. A.e. β eqal for a D an 3D When ng he mn gn n (8) an png rea β = γ.e. γ, he je conracon, x, r (9) whch he well known rel for Bora mohpece (ee for nance Bachelor []). Fgre 89 Bora mohpece The pl gn n (8) gve β =, whch oe no e he ame flow paern. Hence, γ only he mn gn n (8) gve meanngfl rel for ce plae. For ( ) = an, x, x =, γ = o ha (8) gve, r β = γ (3) whch he well known rel for a lowly convergng mohpece (ee for nance Bachelor [])

138 7 Wn rbne n c hrogh blng 7. The plae concenraor moel Fgre 9 Slowly convergng mohpece We h know ha β wh he accompanyng c hape an velocy rbon ha lea o h rel. By oher mean, can be hown ha he ce plae how β =.6 (ee for nance Bachelor []). The eqaon (5) an (8) allow one o calclae P of he plae concenraor from he velocy rbon a he acaor. They wll be referre o a he plae concenraor moel. canno recly be calclae wh he plae concenraor moel b hol be P,max emae from nerpolaon beween he P vale a fferen loa of he acaor. 7. Verfcaon of he plae concenraor moel The verfcaon of he plae concenraor moel bae on mearemen an FD calclaon a a plae concenraor wh he followng confgraon eal. Table 3 onfgraon eal verfe plae concenraor. 3D plae D plae concenraor concenraor A A 3.4% 8.5% p r w The FD calclaon are fr compare wh mearemen n orer o verfy he FD calclaon. Seconly, he eale op of he FD calclaon, ha col no be obane wh mearemen, e o verfy he plae concenraor moel. The mearemen are ecrbe n Yma [9]. A e of creen wh varo poroy mlae he acaor a varo loa. For all creen, he local prere rop acro he creen ogeher wh he local oal velocy j ownwn of he creen were meare. The prere rop wa meare wh an array of ac po be, whle he local oal velocy wa meare wh a howre. Deal of he FD calclaon are gven n Appenx. I wa hown n econ 4.3. ha he κ ε moel gve a proly large generaon of rblen knec energy aron he agnaon pon. Th rblen knec energy ranpore ownwn, where nflence by nene rblen mxng of he low velocy n he wake wh he hgh velocy j oe he wake. A he flow how a agnaon conon pwn of he plae concenraor, he κ ε moel gve an naccrae mlaon of he acal flow. The RSM oe have h problem. I herefore e o mlae he flow aron he plae concenraor. 7.. Verfcaon of he je velocy Fgre 9 how he meare an mlae velocy profle a a ance ownwn of he acaor wh zero prere rop acro he acaor..rc - 8 -

139 7 Wn rbne n c hrogh blng 7. Verfcaon of he plae concenraor.4. / [.].8.6 meare cf r/r c [.] Fgre 9 Meare an FD (rblence moel ee Appenx ) rel of he velocy profle a an axal ance.rc ownwn of he acaor a zero prere rop acro he acaor for a k plae concenraor. The mlae oal velocy how a crepancy of le han % compare o he meare velocy. The error n he mearemen (ee econ 4..) make no worhwhle o ry o ge a beer agreemen by ng he varo opon n he RS moel. 7.. Verfcaon of he je conracon The je conracon β a fferen loa of he acaor calclae by (8) from he mlaon of a D an 3D (k) plae concenraor. Frhermore, β wa calclae from he rao of he je ameer a he acaor an he je ameer a he plane wh parallel reamlne. An example of he mlae reamlne conracon hown n Fgre 9. The je conracon for he D plae concenraor col no be aken from he reamlne conracon becae he D je wa no aonary an moreover no ymmercal (Fgre 9). Fgre 9 FD rel of he je conracon for he 3D (lef) an D (rgh) plae concenraor wh zero acaor loa

140 7 Wn rbne n c hrogh blng 7. Verfcaon of he plae concenraor The rel of β calclae by (8) an β aken from he reamlne conracon are hown n Fgre 93. The ncon facor a a he x-ax calclae from ( ) = a., x bea [.],95,9 D plae concenraor 3D plae concenraor 3D f D f aken from reamlne,85,8,,,3,4,5,6,7,8,9 a [.] Fgre 93 FD rel of he je conracon ver ncon facor calclae wh (8) an je conracon aken from he reamlne conracon n he FD rel for a 3D plae concenraor. The je conracon aken from he reamlne conracon how a crepancy of 4% for a.5. The crepancy ncreae wh ncreang a. Th manly cae by he fac ha he je vanhe a hgh a a he c blocke a hgh a. In econ 7.., wa hown ha β oe no epen on he flow menon (D or 3D). Th confrme by he mall fference of only.6% beween he D an 3D vale for β fon by (8) an hown n Fgre 93. A goo f hrogh hoe β vale acheve wh β =.8 +., 8 x + (3) wh ( ) = a. Th f e for frher verfcaon of he plae concenraor moel., x For a <.8, he crepancy beween f an β =.8 le han %. Hence, β =.8 = conan a goo approxmaon for a < Verfcaon of he prere rop acro he acaor p acro he acaor a a fncon of he velocy a he ax ownwn of he creen can be accraely meare a a 3D plae concenraor an compare wh a FD calclaon. The ame mearemen a a D plae concenraor le relable a ffcl o eablh a D flow n he crclar open je wn nnel e for he mearemen. The rel are hown n Fgre 94. The mearemen are f wh (3) an (3), whch gve

141 7 Wn rbne n c hrogh blng 7. Verfcaon of he plae concenraor p [.] -, -,4 -,6 -,8 -,,4,6,8, 3D mearemen 3D FD moel -, -,4 -,6 Fgre 94 p,ax / [.] acro he acaor of a 3D plae concenraor a a fncon of,. ax The rel how ha he FD calclaon gve a oo mall prere rop acro he acaor. Th n agreemen wh he rel ploe n Fgre 9 where he calclae velocy profle aron he ax of he acaor lghly lower han he meare velocy. Fng of from he FD calclaon wh (3) an (3) gve p. = (3) for a 3D plae concenraor. Th n agreemen wh he rel n econ 3.3.5, Table 8. alclaon of he flow velocy aron a harp ege n poenal flow gve an nfne hgh flow velocy a he ege. The fne fon n (3) a rel of vco effec. In oher wor, he orgn of he ame for all harp-ege boe an coneqenly hol be roghly eqal for all harp-ege boe. The D confgraon of he plae concenraor how he followng agreemen beween mearemen an FD calclaon

142 7 Wn rbne n c hrogh blng 7. Verfcaon of he plae concenraor p [.] Fgre 95 Fng of -, -,4 -,6 -,8 - -, -,4 -,6 -,8 p,,4,6,8,,4 D mearemen D FD moel,ax / [.] acro he acaor of a D plae concenraor a a fncon of,. p from he FD calclaon wh (3) an (3) gve ax for a D plae concenraor. Fgre 95 how ha he preang n he meare hgher han he preang n he meare.3 = (33) p a he D plae concenraor p a he 3D plae concenraor hown n Fgre 94. Frhermore, he FD calclaon gve a generally hgher p compare wh he mearemen. No conclon are rawn from hee obervaon a a real D flow ffclly acheve n a crclar open je wn nnel Verfcaon of he power coeffcen The la verfcaon concern P a a fncon of a calclae from he mearemen an FD calclaon. Wh from (3) an (33), β from (3) an P from (5), he followng rel are calclae from he FD calclaon

143 7 Wn rbne n c hrogh blng 7. Verfcaon of he plae concenraor,8,6 rm 3D rm D moel P [.],4, Fgre 96 Fgre 96 how ha P of he acaor n he plae concenraor calclae from FD calclaon an plae concenraor moel wh ( ) = a., x P accraely prece by he plae concenraor moel. 7.3 Approxmae plae concenraor moel The verfcaon of he je conracon moel n econ 7.. make clear ha β.8 = conan for a <.8. Frhermore, he accracy of he plae concenraor moel confrm ha one may p of P gven by (5) wh repec o Sbon of h n (5) gve -,,,4,6,8 = conan. If β conan an = conan, fferenaon, gve x a [.] β, x, op = (34) 3 P,max β = 3 3 o 3 (35) where can be fon wh (3) or (33). A P,max proporonal o he hr power of ) rel n a large, a change n (ee econ 3.., an for backgron on ncreae of P,max. The e of eqaon (34), (35) an (3) or (33) wll be referre o by he approxmae plae concenraor moel. Accorng o he approxmae plae concenraor moel, a well-rone c nle ( β = ) gve for a D plae concenraor an P,max.85 (36)

144 7 Wn rbne n c hrogh blng 7.3 Approxmae plae concenraor moel for a 3D plae concenraor. P,max.67 (37) I nereng o compare he hr force of a ol plae F T, plae wh he m of he hr force of he ce plae an acaor F T, plae conc a P,max of he acaor. Ame agnaon prere a he oal pwn area of he ol plae an he ce plae. Wh (3) an (34) hen follow ha F T, plae conc F T, plae A = (38) 3 A p where A p enoe he projece area of he ol plae an A enoe he c area. Th how ha F T, plae conc a P,max maller han F T, plae of a ol plae, nepenen of β or. 7.4 Dcon on he valy of he moel The rel n econ 7., jfy he conclon ha he performance of he verfe plae concenraor confgraon accraely moelle by he plae concenraor moel. Wh β =.8, he approxmae plae concenraor moel gve Table 33, x, op an P,max accorng o he approxmae plae concenraor moel. Dmenon, x, op P,max D D.6.68 omparon wh Fgre 96 how ha he approxmae plae concenraor moel gve a goo approxmaon for, x, op an P,max f β =.8. Weelberger e al. [85] fon by mearemen ha of a ce k change le han 9% f A A 5%. The cavy moel (8), e for he ervaon of he plae concenraor p moel, how ha. We h fn le han 3% change n for A Ap 5%. Wh (3), he fnal conclon rea ha =. = conan a goo approxmaon f A A 5%. p Accorng o Hoerner [6], a plae ha a Srohal nmber of S =.3. Wh (43), he S heng freqency of he vorce a one e of a plae f rea f = =. 3. For D D normal blng ze D an moerae wn veloce, follow ha f <. Hence, he ac performance calclaon of a wn rbne n a plae concenraor a hown n h chaper normally allowe. 7.5 Yawe flow Mearemen of he hr coeffcen, p of a plae an k n yawe flow wh yaw angle ϕ how ha, p conan f ϕ φcr o. In econ 5.3., hown ha ϕ = 35 for a 3D cr

145 7Wn rbne n c hrogh blng 7.5Yawe flow o plae concenraor an ϕ = 5 for a D plae concenraor wh A = 5. Hence, p acro cr a plae or k n yaw expece o be roghly conan for ϕ cr ϕ ϕcr. For ϕ well above ϕ cr,, p an accorngly p ecreae wh ncreang ϕ (ee econ 5.3.). Yaw alo change he flow phenomena a he nle of he c. A ϕ ncreae, he local crown veloce a he pwn e of he plae concenraor wll ncreae oo. Th lea o local eparaon a he c nle an a ecreae n β wh ncreang ϕ (Fgre 97). r Fgre 97 Plae concenraor n yawe flow wh eparaon a he c nle. The roghly conan p for ϕ cr ϕ ϕcr n combnaon wh a ecreang β a ncreang ϕ rel n a ecreae n P,max a ncreang ϕ. For ϕ well above ϕ cr, where p a well a β ecreae wh ncreang ϕ, he ecreae of P,max wh ncreang ϕ wll be hgher. The power op a concenraor zero a o ϕ = 9. o ϕ = 9 zero a he prere fference acro he plae 7.6 The energy yel of a plae concenraor Th econ ce he power op of a wn rbne n a plae concenraor wh wellrone c enrance n omnreconal wn. Bae on he knowlege hown n he prevo econ, poble o gve emae of he power op a all flow angle. Th allow a fr orer calclaon of he energy yel of a wn rbne n a plae concenraor n omnreconal wn. P,max of a plae concenraor hghe for wn perpenclar o he plae rface an ecreae wh ncreang yaw angle o he ax normal o he plae rface (econ 7.5). There no prere fference acro he c for wn perpenclar o he c ax. Hence, P,max a wn perpenclar o he c ax zero. P,max for a well-rone c enrance o an wn perpenclar o he plae rface fon by (36) an (37). A ϕ = 45, ame ha he je conracon β.7, whle p acro he plae eqal o p a flow perpenclar o he plae rface. Hence, P,max a o ϕ = 45 7% of P,max calclae by

146 7Wn rbne n c hrogh blng 7.6The energy yel of a plae concenraor (36) an (37). Wh he prevo coneraon, r, can be calclae by (6). The rel hown n Fgre 98., r, [.],8,6,4, 3D D ph [ o ] Fgre 98 Ame r, for a BAWT n a plae concenraor a a fncon of ϕ (ph). In orer o allow comparon of he energy yel of varo BAWT n econ 8.., ame ha = 5. m/ a z =.3 an = 3.3m/ a z = (econ 3..). Wh (6) an Fgre 98, h gve he followng energy yel. Table 34 Energy yel of a.5 kw BAWT n a plae concenraor. e Energy yel kwh ( m year) = 5. m/ a z =.3 m = 3.3m/ a z = m kwh m year D plae concenraor 3D plae concenraor 3 47 Energy yel ( ) Energy yel of a combne plae concenraor In he prevo econ hown ha he concenraor effec of he plae concenraor o o hghe a flow parallel o he normal vecor on he large blng façae ( ϕ =, ϕ = 8 ). A flow recon parallel o he large façae of he blng rece he power op of he o o acaor o zero ( ϕ = 9, ϕ = 7 ). For omnreconal wn, h gnfcanly rece he concenraor effec of he plae concenraor average over all ϕ. Two plae concenraor n o o a cro-confgraon cae an aonal concenraor effec a ϕ = 9 an ϕ = 7, whch ncreae he concenraor effec average over all ϕ. The cro-confgraon hown a he lef of Fgre 99. The acaor n ch confgraon ffer from he wake of he plae a o o o ϕ =,9,8,7. An aeroynamc hape of he plae help o preven h 6. Sch 6 The rong avere prere graen owar he combne plae concenraor make ffcl o compleely preven eparaon

147 7Wn rbne n c hrogh blng 7.7Energy yel of a combne plae concenraor confgraon hown a he rgh of Fgre 99. I wll be referre o a he combne plae ϕ concenraor. For Fgre 99 Top vew of a combne plae concenraor. o o o ϕ =,9,8,7, he c enrance ha harp ege o ha he je conracon β o o o o eqal o β =.6 of a ce plae (econ 7..). For ϕ = 45,35,5,35, he c enrance can be characerze a well-rone whle he c ownwn of he acaor ac a a ffer (ee Fgre ). A βa A p Fgre Top vew of a combne plae concenraor n ffer operaon. o o o o Accorng o econ 7.., h expece ha β > a ϕ = 45,35,5,35. Dffer have aache flow for ffer cone angle p o approxmaely o (ee Paeron e al. [4]). The geomery of he ffer wh bonary conon of eparaon a cone angle o lea o β. For h example calclaon wll be ame ha β =. a o o o o ϕ = 45,35,5,35. The flow conon cloe o he eparaon pon are mlar o he conon a he plae concenraor an ame ha A A 5% o ha he flow hrogh he c oe no nflence he eparaon velocy. Uner ch conon, one may e he rel obane for he plae concenraor. can be approxmae by (3) an (33) an P,max fon by (35). Analoge o he plae concenraor, he fference n concenraor effec a mall yaw angle o p o o o o o o o ϕ =,9,8,7 an ϕ = 45,35,5,

148 7Wn rbne n c hrogh blng 7.7Energy yel of a combne plae concenraor expece o ncreae wh ncreang yaw angle. Wh he prevo coneraon, r, can be calclae by (6). The rel hown n Fgre.,4, r, [.],8,6,4, 3D D ph [ o ] Fgre Ame r, for a BAWT n a combne plae concenraor a a fncon of ϕ (ph). Wh he ame wn conon a e for Table 34, (6) an Fgre, gve he followng energy yel. Table 35 Energy yel of a.5 kw BAWT n a combne plae concenraor. e Energy yel kwh ( m year) = 5. m/ a z =.3 m = 3.3m/ a z = m kwh m year D plae concenraor 3D plae concenraor Energy yel ( )

149 8Rel 8 Rel an conclon The he foce on he aeroynamc of wn rbne cloe o blng. Whn ha foc, he aeroynamc of he flow aron blng an he aeroynamc of he neracon of ha flow wh he wn rbne are analye. The he make clear ha he performance of a Blng-Agmene Wn Trbne (BAWT) a a parclar locaon epen on many e. Moel ha allow calclaon of eale nformaon on hee e are preene n h he. B he rel are alo helpfl n (reearch) area oher han wn energy n he bl envronmen. In he fr econ of h chaper, he applcably of he rel n (reearch) area oher han wn energy n he bl envronmen ce. The econ econ ce he achevemen of he he for wn energy n he bl envronmen. Bae on hee achevemen, he la econ of h chaper ce he fre of wn energy n he bl envronmen an reearch on he aeroynamc of wn energy n he bl envronmen. 8. Smmary of rel for general e Apar from he npenable nformaon on he analy ool (chaper 4), he mo efl nformaon for h he fon n orce concernng: wn energy, wn engneerng, meeorology, wn rven venlaon, aeropace engneerng, exha ack egn. The overlap beween hee area an he opc of he he apparenly apprecable. Hence, hee area are mo lkely o benef from he achevemen of he he. They herefore form he heang of he followng paragraph f here are poble benef o menon. Wn Energy Se of low roghne are preferre for wn rbne becae of her hgh wn pee an low rblence level compare o e wh a hgh roghne. Mo wn rbne are herefore fon a e wh a low roghne. B he nmber of onhore low roghne e for wn energy ecreae becae wn rbne alreay occpy an apprecable nmber of e an becae he poplaon ncreae. For onhore ng n he fre, coneqenly navoable o look for e wh a hgher roghne. Example of ch e are: nral area, e cloe o work of nfrarcre (brge, ke. ec), e a harp monan or clff, e n woo. The coneraon an moel for hgh roghne e n h he are moly recly applcable o hee e. oncernng fnamenal reearch on wn energy, h he how he vale of ng varo moelng ool. Some rel ha are obane wh momenm heory col never be obane wh (mplfe) vorex heory an va vera. A ecae e of he ool neceary. The followng moel on he aeroynamc of wn energy n he bl envronmen are fr 7 nroce n h he: 7 Alhogh omeme parly bae on exng moel

150 8Rel 8.Smmary of rel for general e he performance of a wn rbne n parly accelerae flow, he performance of an H-Darre n kewe flow, he vorex moel for he performance of a hroe wn rbne, he performance of a wn rbne n a plae concenraor. Wn engneerng/ Wn hnrance A lo of he reearch n wn engneerng or n he wn hnrance area eal wh mearemen or mlaon of wn prere aron blng. Sch approach generae nformaon on parclar aon b fal o explan he phyc. I eem profable o rec more effor on moel o ecrbe he phyc of ceran phenomena. The foc of he he rece on wn energy n he bl envronmen b here are ome moel ha may be nereng for wn engneerng or wn hnrance colleage. Thee moel are: he vorex moel for he hroe wn rbne, he plae concenraor moel, he probably rbon of he wn pee cloe o boe, he hegh of he recrclaon regon on he fla roof of harp-ege olae blng, Th he conan everal moel on he acceleraon of he free ream wn pee by a blng. Th acceleraon cople wh he wn loa of a blng an he wn hnrance for peeran aron he blng. The moel on he phyc of he acceleraon by blng are herefore efl o avo wn hnrance an prec he wn loa of a blng. Moreover hey allow calclaon of he wn loa an wn hnrance rng he egn of he blng. Sch pracce avo he problem relae o ealng wh wn loa an wn hnrance n a laer age. Meeorology The roghne lengh of a parclar e moly emae by comparng a mall nmber of reference pcre of e wh a known roghne wh he e of nere. For he bl envronmen, ch reference pcre o no ex. alclaon of he roghne lengh from mearemen of he velocy a a fncon of he hegh above ce how large -no ye neroo- varaon n he roghne lengh. I eem profable o evelop a moel for he roghne lengh. Th he nroce he anar evaon n he hegh of he blng a a moel parameer for he roghne lengh of ce. Fre reearch neceary for a verfcaon of h ea. Exha ack egn/ Wn rven venlaon The pwn ege or leang ege of a harp-ege fla roof cae eparaon of he bonary layer. Th cae a recrclaon regon a he roof ha encloe by he eparaon rajecory. An exha below he eparaon rajecory cae he hazaro exha gae o be rbe o all blng facae ownwn of he leang ege. The exha gae are hen cke no he blng by he venlaon nake an cae a ero healh hazar. The hegh of he eparaon rajecory above he roof h fxe he mnmm hegh of an exha ack a he roof. Th he ce exng moel an erve complemenary moel o calclae he hegh of he eparaon rajecory a he e or roof of harp-ege D or 3D 8 blng. I how ha he eparaon rajecory a he D blng ha a hgher ance o he blng o ha he exha ack a he roof of D blng nee o be hgher han a 3D blng. 8 The 3D moel no evelope n h he b fon n lerare

151 8Rel 8.Smmary of rel for general e Wn rven venlaon e he local prere fference aron a blng o venlae he blng. I nee an ole of he venlaon channel n a regon wh low prere or hgh veloce, b he exhae ar hol no be aken n agan. Smlar o he exha ack, he ole of he venlaon channel hol be locae oe he recrclaon regon. The ar nake hol be locae n a regon wh a hgh local prere, a he agnaon pon f poble. Th he oe alo preen nformaon on he locaon of he agnaon pon. 8. Smmary of rel for wn energy n he bl envronmen Th econ mmarze he achevemen of he he concernng wn energy n he bl envronmen. The achevemen are alo compare n h econ an fnally h econ gve he conclon ha are rawn from h comparon. 8.. Inegraon of wn rbne an blng The average wn pee n he bl envronmen low b on ceran locaon n he bl envronmen cloe o blng he wn pee can be apprecable. A wn rbne a ch locaon prof from he acceleraon of he free ream wn pee by he blng. Sch a wn rbne calle a Blng-Agmene Wn Trbne (BAWT). Aeroynamcal negraon A BAWT experence he acceleraon of he free ream wn pee by he blng an herefore aeroynamcally negrae wh he blng. The area aron a blng wh an apprecable acceleraon of he free ream wn pee a fracon of he blng area ha cae he acceleraon. Hence, he roor area of he BAWT hol be mall compare o ha area n orer o prof from he acceleraon. omparon of he acceleraon facor a he along-wn-e of a cylner, a D plae, a phere an a k how ha bln D boe cae he hghe acceleraon of he free ream wn pee. Hence, bln D blng are mo promng for BAWT. Mechancal negraon The mechancal negraon of wn rbne an blng eem a logcal coneqence of he aeroynamcal negraon. Ye, eerve mch aenon a can lea o vbraon n he blng. A wn rbne a roang evce. Apar from he occrrence of aeroynamc vbraon ha rel from he neracon of he roor blae wh he wn, mechancal vbraon cople wh he navoable ma nbalance of he roor wll how p. Vbraon of he wn rbne can cae vbraon n he blng f he wn rbne mechancally cople wh he blng. The vbraon of he wn rbne can h he egenfreqency of a blng or par of a blng. In ha cae, he connecon of he wn rbne wh he blng hol be ampe. I eem we o avo mechancal coplng of wn rbne an blng f poble. Th poble for he hroe wn rbne an he wn rbne n he (combne) plae concenraor. Boh confgraon can explo a wn rbne wh a ma fone a he earh rface

152 8Rel 8. Smmary of rel for wn energy n he bl envronmen 8.. Energy yel example of Blng-Agmene Wn Trbne Th econ mmarze he arng pon an energy yel of example BAWT gven a he en of he chaper 5, 6 an 7. The mmary hen allow an eay comparng all example BAWT on he energy yel. Reference aa an arng pon All energy yel calclaon n h econ are fon wh a reference wn rbne wh he followng qale. Table 36 Daa of he reference wn rbne. P r =.5 kw = 4 m/ c = r co 3 m/ = m/ A = 6 m λ = Varable =.3 P The example energy yel calclaon frhermore e he followng arng pon for he rronng area of he BAWT. Table 37 Sarng pon example calclaon. ρ =. kg/m 3 p = 4.5 m/ omnreconal m an aken a z of a cy z = m = 3.3 m/ z of gralan z =.3 m = 5 m/ Energy yel bare wn rbne The energy yel of he reference Wn Trbne (WT) n he free ream wn pee can hen be fon from he probably rbon (econ 3..3) wh =. r, Table 38 Energy yel of a.5 kw bare WT. kwh m year z =.3 m 364 z = m 89 e Energy yel ( ) omparon of he energy yel of he example BAWT wh he bare WT The energy yel of he BAWT, calclae n he chaper 5, 6 an 7 are mmarze n Table 39. The able oe alo prove he oncenraor Rao (R), whch efne a he rao of he energy yel of he BAWT ve by he energy yel of he bare wn rbne

153 8Rel 8. Smmary of rel for wn energy n he bl envronmen z.3 m m Table 39 Energy yel an oncenraor Rao (R ) of example BAWT n omnreconal wn a m hegh wh poenal velocy p = 4.5 m/. kwh Bare WT Sharpege, roof cenre loe o a blng On op of a phere Bee a cylner Shroe m year D R kwh m year R Plae conc. 3D (D) 47 (3).7 (.9) 59 (79).7 (.9) omb. plae conc. 3D (D) 369 (46). (.3) 9 (). (.4) Table 39 how ha he cloe-o-a-blng confgraon an he D combne plae concenraor confgraon have R> whle he hroe an plae concenraor confgraon approxmaely have R. R a rel of wo propere of he concenraor: he acceleraon of he free ream wn pee an he recon envy. A omnreconal wn, he concenraor effec average over all flow angle ha hol be large o acheve a hgh energy yel. Apparenly, he confgraon wh R are no ne concenraor n omnreconal wn! Accorng o Table 39, a bare WT n a rral area ( z =.3 m) ha an energy yel of 364 ( m year) kwh. Th energy yel moly elvere o he gr. In conra, a BAWT elver energy o he local gr n he blng.e. elver he energy behn he mere. The energy yel of he BAWT h fel a an energy avng of he blng an accorngly reflece n energy avng on he comer bll from he ly company. Th an approxmaely hree me hgher rembremen compare o wn rbne n rral area ha elver her energy o he gr. In orer o have comparable or beer economc propec, he BAWT h nee o generae more han kwh m year. Accorng o Table 39, he BAWT n he cloe-o-a-blng confgraon an he D combne plae concenraor have ch energy yel. Hence, her economc propec can be comparable wh wn rbne n rral area f her co are comparable. Th eem he cae for he cloe-o-a-blng BAWT where no exra effor ha o be mae for he ng of he wn rbne. ( ) The cloe-o-a-blng confgraon how he hghe R. I moreover he opon ha can be realze fa an cheap, a he blng ex. Th confgraon herefore very aracve. onfgraon wh a hgh acceleraon for few wn recon, lke for nance wh he cylner confgraon, are le aracve. Sch confgraon cae a low operaonal me of he BAWT

154 8Rel 8. Smmary of rel for wn energy n he bl envronmen 8..3 Wn Trbne for he bl envronmen Th econ mmarze he qale of a wn rbne neee o f he bl envronmen. General ompare o he lf-rven wn rbne, he rag-rven wn rbne ha a hgh maeral age n combnaon wh low converon effcency of wn energy no mechancal power. Th make he rag-rven wn rbne oo expenve for he generaon of elecrcy. Wn rbne for he bl envronmen hol be relave mall n orer o harve he wn energy from freqenly changng wn recon an prof from he mall regon near blng where he wn pee accelerae by he blng. A Darre preferable for ch freqenly changng wn recon, a oe no nee a yaw mechanm. A p pee rao λ 3 eem o be a goo gelne. A lower λ rel n a roor wh hgh maeral age an low energy yel. A hgher λ rel n an ncreae noe emon. A 5 he noe emon proporonal o λ, he noe emon qckly ncreae wh λ o naccepable level. loe o a blng The flow on he roof of harp-ege blng characerze by early eparaon on he pwn roof ege an accorngly kewe flow for he wn rbne above he roof. The H- Darre wh low λ how an ncreae power op n ch kewe flow whle he lfrven HAWT ha a ecreae power op n kewe flow. oneqenly, he H-Darre no only favorable n he bl envronmen n general b epecally n he kewe flow above he roof of harp-ege blng. Wn rbne cloe o blng are lme n ze o approxmaely % of he characerc blng ze. Sch a wn rbne can opmally prof from he mall regon wh concenrae wn energy aron a blng. Hgher wn rbne e h concenrae wn energy le effcenly. For large wn rbne, only a fracon of he ream be whn he accelerae wn pee. A wn rbne wh only he wake n. me accelerae free ream wn pee proce approxmaely 9% of he power of a wn rbne wh he whole ream be whn. me accelerae free ream wn pee. Plae concenraor Th concenraor operae n wo oppoe wn recon. There are wo poble roor ha can cope wh h: a Darre roor an a lf-rven HAWT o wh varable pch of he roor blae of 8 egree an a generaor ha able o operae n oppoe recon or o wh a yaw mechanm of he roor. Excep for he pch mechanm, he HAWT eqal o he common wn rbne n rral area. No large egn change are neee. Frhermore, he Darre ha a lghly lower power coeffcen han he lf-rven HAWT. The lf-rven HAWT coneqenly favorable for e n he plae concenraor. ombne plae concenraor Th concenraor operae n all wn recon. The poble roor for h confgraon are: he Darre or he lf-rven HAWT wh yaw mechanm. The conor of he H-Darre eem mo favorable becae f well n he c of he combne plae concenraor. Ye, he plae concenraor can be egne wh foobrge ha

155 8Rel 8. Smmary of rel for wn energy n he bl envronmen prove he rgh conor for a HAWT. The lf-rven HAWT ha a lghly hgher power coeffcen han he H-Darre. The lf-rven HAWT roor herefore preferre for he combne plae concenraor. The hroe wn rbne Th concenraor operae n wo oppoe wn recon. There are wo poble roor for h BAWT: a Darre roor an a lf-rven HAWT wh varable pch of he roor blae of 8 egree wh a generaor ha able o operae n oppoe recon. The arfol aron he roor cae he hghe veloce n her proxmy. Unfornaely, he Darre no able o harve he energy n he area wh he hgh wn pee cloe o he arfol a he Darre blae move parallel o he wn a ha locaon. oneqenly, he lf-rven HAWT favorable for he hroe wn rbne confgraon. 8.3 Expecaon Th econ keche he expecaon for wn energy n he bl envronmen an reearch on wn energy converon n he bl envronmen Expecaon for wn energy n he bl envronmen BAWT are locae a e where he econ o nflence he lancape alreay aken. In conra o wn rbne n rral area, he BAWT cae coneqenly no exra mparmen of he lancape. are hol be aken o avo he poble hnrance of BAWT (noe emon, nce vbraon, haow flckerng, ec.). Thee echncal problem eem echncal olvable. Overall, here eem o be enogh force n favor of he BAWT o enre ha hey become one of he ero local renewable energy orce n he bl envronmen. The BAWT n he cloe-o-a-blng confgraon he fr BAWT ha lze a he blng ex an only he BAWT nee o be evelope. The relave mple confgraon an effcen e of he acceleraon by he blng gve re o he expecaon ha h BAWT wll be one of he mo weprea. The choce of a concenraor confgraon for a BAWT epen on he recon probably of he wn pee. A BAWT n nreconal wn (rae wn aron he eqaor) be e by a nreconal concenraor. In ha cae, all he maeral of he concenraor e o concenrae he wn energy for a ngle wn recon. Th no poble wh an omnreconal concenraor n nreconal wn. A BAWT n omnreconal wn reqre an omnreconal concenraor. Th enre an apprecable concenraor effec average over all wn recon wh a mall varaon a a fncon of he wn recon, whch enable he e of a relave mall generaor Reearch on aeroynamc of Blng-Agmene Wn Trbne Th he gve he man e an fr orer gelne for wn energy converon n he bl envronmen. In oher wor, aonal reearch welcome on mo opc. Some of he reearch opc flfll a key role n many oher e an reearch opc. Thoe opc are mmarze hereafer. Behavor of an H-Darre n kewe flow Verfcaon of he moel for he H-Darre n kewe flow wh mearemen very complcae. Th parly cae by he fac ha he moel nee np from mearemen ha lkely o be naccrae. Sch np for nance he an of he Darre blae l

156 8Rel 8.3Expecaon a low Re. The complcaon wh mearemen can be avoe by mlang he H-Darre wh known an l vale ha are alo e a np n he moel. Th eem o be a fr valable reearch effor. Accraely moelng of he behavor of an H-Darre n kewe flow alo very complcae. Th cae by he many complex phenomena ha are reponble for he behavor of he H-Darre n kewe flow. A beer neranng of hee phenomena reqre aonal reearch on: T of an acaor a hgh a, large γ an mall λ, mearemen or mlaon of H-Darre wh varo geomere, he neracon of he flow hrogh he ngle- an oble-roor par of he H- Darre. Th a remeno reearch effor. Roghne lengh An accrae precon of he wn pee eenal for he precon of he energy yel of a wn rbne. B he wn pee a fncon of z, whch no ofen accraely known. Who mearemen of he velocy profle above an area, he emaon of z moly one wh reference pcre. Sch an approach enve for nerpreaon. There even le known concernng z for ce. The moel for z bae on area occpaon by roghne obacle eem oo mple o be able o moel he many qane ha lea o z. Syemac mearemen combne wh a goo ecrpon of he oberve cy roghne are neceary. Wh ch mearemen perhap poble o verfy ome fr orer moel for z. Acceleraon aron blng In orer o be able o calclae he energy yel of a wn rbne cloe o a blng, neceary o know he acceleraon of he free ream wn pee by he blng for varo blng, z an angle of he wn o he blng. I laboro o generae ch nformaon b mporan for accrae precon of he energy yel of BAWT

157 9 Reference 9 Reference [] Abbo, I.H., Von Doenhoff, A.E. Theory of wn econ, Dover Pblcaon Inc., New York, 959 [] Bachelor, G.K., An Inrocon o Fl Dynamc, ambrge Unvery Pre, 988 [3] Bane, W.D., Effec of velocy rbon on wn loa an flow paern on blng, Procere of Sympom No 6 on Wn Effec on Blng an Srcre, hel a he Naonal Phycal Laboraory, Tengon Englan, 965, pp 98-5 [4] Bez, A., Da Maxmm er heorech möglchen anüzng e Wne rch Wnmooren, Zechrf für a geame Trbnenween, Volme 6, 9, p. 37. [5] Blackwell, B.F., Shelahl, R.E., Felz, L.V., Wn Tnnel Performance Daa for Two-an Three-Bcke Savon Roor, Sana repor 76-3, Jly 977 [6] Boema, M., Wn lmae an Urban Geomery, The, Enhoven Unvery of Technology, ISBN , 993 [7] Blng Energy Daabook, November 3 [8] Bron, T., Sharpe, S., Jenkn, N., Boany, E., Wn Energy Hanbook, John Wley & Son L, hcheer, [9] Bel, G.J.W., An Aemen of he performance of ffer-agmene wn rbne, proceeng of he 3r ASME/JSME Jon Fl Engneerng onference, San Francco, alforna, Jly 8-3, 999 [] ampbell, N.S., Sankovc, S., Wn energy for he bl envronmen, projec web, In he framework of he Non-nclear energy programme Jolle III, onrac JOR3- T98-7, Sepember [] hanler, H., (Eor), Wn Energy he Fac, an analy of wn energy n he EU- 5, EWEA, 4 [] Ell, B.R., UR rappor, Dynamch gerag van hoge gebowen, rappor 97-5, Okober 997 [3] Energy Informaon Amnraon Offce of Inegrae Analy an Forecang, U.S. Deparmen of Energy, repor #DOE/EIA-484(3), November 3, Wahngon, D 585, May 3 [4] ESDU, Srong wn n he amopherc bonary layer, ESDU 86, par : meanhorly wn pee, Sepember 98 [5] Ewal, B.F.R., Wn Tnnel Wall orrecon, AGARDograph 336, RTO/ NATO, ISBN , 998 [6] Flen Inc., Flen 5 er ge, manal Jly 6, 998 [7] Fokkema, E., Mengen aan een moel gebow n een grenlaag n e open raal blaapjp, Sageverlag,. [8] Fokkema, E., Vergeljk van nnelmengen aan een vlakke plaa concenraor, Afeerverlag, ecember [9] Foreman, K.M., Gber, B., Oman, R.A., Dffer agmenaon of wn rbne, Jornal of Solar Energy, Vol., 978, pp

158 9 Reference [] Froe, R.E., On he par playe n proplon by fference of fl prere, Tranacon of he Ine of Naval Archec, Volme 3, 889, p 39 [] Garra, J.R., The nernal bonary layer, a revew, Bonary layer Meeorology, 5, 99, pp 7-3 [] Glaer, K.H., The Lanceer-Bez lm, Jornal of Energy, Volme 3, No 6, 935 [3] Hanen, M.O., Effec of placng a ffer aron a wn rbne, Jornal of Wn Energy, 3, DOI:./we.37,, pp 7-3 [4] Ha, E., e al., The nex generaon of large wn rbne, mmary of he fnal repor, Mnch, ETAPlan, May 99 [5] Holen, van Th., Performance analy of a wnmll wh ncreae power op becae of p vane nce ffon of he ar ream, Memoranm M-4, Delf Unvery of echnology, November 974 [6] Hoerner, S.F., Fl ynamc rag, L.A. Hoerner, 985 Hoerner [7] Hoerner, S.F., Fl ynamc lf, L.A. Hoerner, 975 [8] Han, M., A y of he wn force on low re blng an her applcaon o naral venlaon meho, PhD The, Dep. of Bl. Sc. Unv. of Sheffel, 978 [9] Igra, O., Shro for aerogeneraor, 4 h AIAA aeropace cence meeng, AIAA paper no Janary 6-8, 976 [3] Jacob, E.N., Sherman, A., Arfol econ characerc a affece by varaon of he Re nmber, NAA repor No. 586, 937 [3] Jacob, E.N., Arfol wh A r = 6 n NAA varable eny nnel, 78 relae e econ, echncal repor 46, 933 [3] Jenen, M., The moel law for phenomena n naral wn, Ingenoeren, 958 [33] Kaz J., Plokn, A., Low-pee aeroynamc, from wn heory o panel meho, McGraw-Hll, Inc, Inernaonal eon, 99 [34] Kelvn, L., Mahemacal an Phycl Paper 4, Trancrpon of he Royal Socey of Enbrgh, No. 5, 869 [35] Km, S.E., Boyan, F., Applcaon of FD o envronmenal flow, Jornal of Wn Engneerng an Inral Aeroynamc, No. 8, 999, pp [36] Küchemann, D., Weber, J., Aeroynamc of proplon, McGraw-Hll book company, 953 [37] Lanceer, F.W., A conrbon o he heory of proplon an he crew propeller, Tranacon of he Inon of Naval Archec, Volme 45, 95, p. 87 [38] Lehman, J.G., Prncple of Helcoper Aeroynamc, ambrge Unvery Pre, [39] Maen, H.A., Sorenen, N.N., Schreck, S., Yaw aeroynamc analye wh hree coe n comparon wh expermen, ASME Wn Energy Sympom, conrbon nr. AIAA-3-59, 3. [4] Mele, E., Drzame Archecr, reven naar een conrarjke omgevng, NA Ugever, Roeram, 999 [4] Mlne-Thomon, L.M., Theorecal aeroynamc, Dover pblcaon, Inc., New York, 958 [4] Meren, S., Wn energy n rban area, concenraor effec for wn rbne cloe o blng, Refoc, March/ Aprl

159 9 Reference [43] Meren, S., Power coeffcen precon of a wn rbne negrae n a blng, The worl wn energy conference an exhbon, Berln, Jly [44] Meren, S., Trby, Unvery of Technology Delf, DUWIND repor 9, [45] Meren, S., Wn ecrpon for roof locaon of wn rbne, Global onference an exhbon on Wn Energy, Par, [46] Meren, S., Haal wnenerge je gebow, Gezon Bowen en Wonen, Nr. 3, November 3 [47] Meren, S., Kk, G. van, Bel, G. van, Performance of an H-Darre n kewe flow on a roof, pecal Wn Energy e of he Jornal of Solar Energy an Engneerng, DOI.5/.6939, November 3 [48] Meren, S., Kk, G. van, Bel, G. van, Performance of a Hgh Tp Spee Rao H- Darre n he kewe flow on a roof, ASME Wn Energy Sympom Proceeng a he 4 AIAA Aeropace Scence Meeng an Exhb, Reno, NV, 6-9 Janary 3 [49] Meren, S., The energy yel of roof mone wn rbne, Jornal of Wn Engneerng, volme 7, No. 6, 3, pp [5] Mrakam, S., rren a an fre ren n compaonal wn engneerng, Jornal of Wn Engneerng an Inral Aeroynamc, 67 & 68, 997, pp 3-34 [5] Newa, F.T.M., Trblene, Eplon gaven, Urech, 99 [5] Panofky, H.A., Don, J.A., Amopherc rblence, John Wley & Son, L, New York, 984 [53] Parachvo, I., Wn Trbne Degn, Wh Empha on Darre oncep, Ecole Polyechnqe e Monreal, [54] Paeron, G.N., Gbon, A.H., Dffer behavor, a revew of pa expermenal work, relevan oay, Jornal of Arcraf Engneerng, Aprl 974 [55] Peerka, J.A., ermak, J.E., Wn Engneerng y of one Wllam cener, Tla. Techncal repor ER7-75 J AP-JE5, Fl Dynamc Dffon laboraory, ollorao Sae Unvery, For olln, pp 88, 974 [56] Phllp, D.G., Flay, R.G.J., Nah, T.A., Aeroynamc analy an monorng of he Vorec 7 ffer-agmene wn rbne, IPENZ Tranacon, Vol. 6, No., 999 [57] Phllp, D.G., Nah T.A., Oakey A., Flay R.G.J., Rchar P.J., ompaonal Fl Dynamc an Wn Tnnel Moellng of a Dffer-agmene Wn Trbne, Jornal of Wn Engneerng, Volme 3, No., 999 [58] Plomp, H., Veo of a wn nnel y concernng he behavor of a moel wn rbne n he bl envronmen, 6 [59] Pranl, L., Tejen, O.G. Apple Hyro- & Aeromechanc, Dover Pblcaon Inc., New York, 935 [6] Pranl, L., Tejen, O.G. Fnamenal of Hyro- & Aeromechanc, Dover Pblcaon Inc., New York, 934 [6] Saffman, P.G., Vorex ynamc, ambrge Monograph on Mechanc an Apple Mahemac, ambrge Unvery Pre, 99 [6] Saro, J.A., Theory an mearemen on H-Darre rbne n kewe flow, Maer projec a he Delf Unvery of echnology, repor WE-393, Delf, The Neherlan, Ag

160 9 Reference [63] Schlchng, H., Bonary layer heory, Mc Graw-Hll, New York an Lonon, Sevenh eon, 979, pp 87 [64] Seo, L.Y., Galbrah, R.A., The effec of pch rae on he ynamc all of a NAA 3 aerofol, Elevenh Eropean Roorcraf Form, paper no. 34, 985 [65] Sharan, V.K., On he characerc of flow aron blng moel wh vew o mlang he mnmm fracon of he naral bonary layer, Inernaonal jornal of mechancal cence, 7, e 9, Sepember 975, pp [66] Shelahl, R.E., Klma, P.., Aeroynamc haracerc of Seven Symmercal Arfol Secon hrogh 8-Degree Angle of Aack for Ue n Aeroynamc Analy of Vercal Ax Wn Trbne, Sana Naonal Laboraore Techncal Repor, SAND8-4, March 98. [67] Sm, S., Scanlan, R.H., Wn effec on rcre, John Wley & Son, New York, 996 [68] Taylor, J.R., An Inrocon o Error Analye, Unvery Scence The, Mll Valley, 98 [69] Tenneke, H., Lmley, J.L., A fr core n rblence, MIT Pre, 97 [7] Thoma, D., Grnälche zr enfachen Srahlheore er Schrabe, Zechrf für Flgechnek n Moorlfchffahr, 95 [7] Throen I., Peeren, E.L., Eropean Wn Ala, RISO Naonal Laboraory, ISBN , 989 [7] Toe, S., Vermeer, N., Mofcae blaapjp In voor Wnenerge en bechrjvng hge ae, repor IW-949M, Sepember 99 [73] Verkerk, G., e al., Bna nformae boek, Woler Noorhoff Gronngen, ere rk, 99 [74] Vermeer, N.J., Wn nnel expermen on a roor moel n yaw, Proceeng of he h ympom on he aeroynamc of wn rbne n Lyngby Denmark on December , Techncal Unvery of Denmark ISSN , 998, pp. -. [75] Vre, E. onveraon wh Eze e Vre an v o a mall wn rbne on he roof of a m-re blng ha ffer from freqen yawng, Aprl 6 [76] Vre, O., Fl Dynamc apec of Wn Energy onveron, Agar-AG-43, 979 [77] VROM, mnry of, Reearch of he Eropean Unon, November 3 [78] War-Smh, A.J., Inernal Fl Flow, The fl ynamc of flow n ppe an c, larenon pre, Oxfor, 98 [79] Wernga J. & Rjkoor, P.J., Wnklmaa van Neerlan, Saageverj - Gravenhage, 983 [8] Wernga, J., Emaon of meo-cale an local-cale roghne for amopherc ranpor moellng, Proceeng NATO/ MS conference Ameram 98, 98 [8] Weelberger,., Bez, A., Pranl, L., Ergebne er Aeroynamchen Verchanal z Göngen, II Leferng, rck n verlag von R. Olenbrg, München n Berln, 93. [8] Wlcox, D.., Trblence moellng for FD, DW Inre, 993 [83] Wlon, D.J., Flow paern over fla-roofe blng an applcaon o exha ack egn, ASHRAE Tranacon, vol. 8, nr., 979, pp

161 9 Reference [84] Wlon, R.E., Laman, P.B.S., Apple aeroynamc of wn power machne, Oregon Sae Unvery, May 974 [85] W, e, M.H., Wn n e gebowe omgevng, Technche Unvere Enhoven, Facle Bowkne, apacegroep FAGO, Najaar. [86] Woo, D.H., Inernal bonary layer growh followng a ep change n rface roghne, Bonary-Layer Meeorology, No., 98, pp 4-44 [87] November 3 [88] November 3 [89] Trby, November 3 [9] Wnwall, November 3 [9] Yma, S., omparon of he precon from wo heorecal moel wh wn nnel mearemen on a fla plae concenraor, nernhp repor Delf Unvery of Technology WE-39, Jly/ Ag 3 Aonal nformaon on he phoo Fgre AES wn rbne, November 4. Th phoo how he AES near he A5 cloe o Apeloorn, The Neherlan Fgre Tlpo, November 3 Fgre 3 Wne, November 3 Fgre 4 AES an Trby ( November 3 Fgre 7 VAWT of ly ENEO calle Globan - 5 -

162 Appence Appence Appenx A Sream be lengh n vco flow The performance of a wn rbne mlae n a FD calclaon by a fan wh a prere rop acro area. The fan moel an acaor. The prere rop a maxmm energy exracon by an acaor calclae from he Lanceer-Bez lm n econ.. Th prere rop e o mlae he opmal performance of he moelle wn rbne. The area of he ream be A f (he oe area wh ameer D ) a a ance o he fan l f are fon from he FD rel. Thee area are calclae a he ownwn an pwn e of he fan. I ame ha he energy exracon ake place beween he area A an A e where he reamlne a he bonary of he mlae flow hrogh he acaor k are parallel. l f A Acaor A f A f e A e Fgre Acaor n parallel flow. The FD calclaon gve he vale of A f A an A f Ae a a fncon of l f D ha are hown n Fgre 3. Fgre 3 Area A f A an A f Ae a a fncon of l f D calclae wh FD

163 Appence From he fgre, can be ece ha n vco flow, he energy exrace n a ream l D = 6D : a fne lengh. Whle n nvc flow, he ream be wh lengh f ( ) be lengh goe o nfny (ame fgre)

164 Appence Appenx B Mearemen plae concenraor The mearemen on he plae concenraor are ecrbe n Yma [9]. For convenence, he meare confgraon are mmarze here. A D an 3D confgraon of he plae concenraor ee n he open je wn nnel of Delf Unvery of Technology (econ 4..). In orer o mnmze he error n he mearemen (econ 4..3), he e were carre o a he hghe poble wn pee of he nnel: 4 m/. The ze of he plae concenraor are mmarze n he able below. Dameer plae/ wh plae[m] Dameer c/ wh lo [m] D plae D plae.5.9. Ronng ra c/ lo [m] Sze an nnel velocy enre a Re nmber well above he vale where he rag coeffcen of plae change becae of change of he vorex paern n he wake ( Re, ee Hoerner [6]). Drng he mearemen, he wh of he D plae concenraor wa ecreae n orer o avo 3D effec an nnel blockage. I wa oberve ha he D plae pl he open je no wo je for he hgher ze (wh.5 m). The ecreae n wh of he D plae rele n a hgher relave ronng ra for he D plae (. / ) compare o he 3D plae (. /.9. ). The D flow wa prove by wo large en plae a he oer regon of he je of he nnel. The mearemen of he prere rop acro he creen wa carre o wh a oal prere rake connece o a HyScan n. Each be of he rake wa connece o wo prere enor n orer om mnmze he mearemen error. The rake ha exe of 3 be wh. m eparaon beween hem meare he ac prere.5 m ownwn he creen. Dfferen creen n he lo or c were e o mlae he fferen operang ae of he wn rbne. The veloce hrogh he creen were meare wh a ravere wh a howre approxmaely. m ownwn of he creen. The reolon of he ep of he ravere wa. m

165 Appence Appenx Ue rblence moel FD calclaon All calclaon are carre o n he eay ae wh a fr orer pwn cheme excep f ae oherwe. Topc Acaor of Appenx A Plae concenraor n chaper 7 Separaon n econ 3.3. Fla plae n econ Bonary layer n econ NAA 8 n econ Non nform Gr Sr. Qaral. Sr. Qaral. Unr. Terah. Sr. Qaral. Sr. Qaral. Sr. Qaral. Trblence Moel k-e realzable RSM, Wall bon. on. from k eqaon, wall refl. effec RSM, Wall bon. on. from k eqaon k-e realzable an RSM, Wall bon. on. from k eqaon an qarac pre. opl. k-e realzable an RSM, Wall bon. on. from k eqaon an qarac pre. opl., wall reflecon effec k-e realzable Wall fncon San. wall fnc. San. wall fnc. Non-eql. wall fnc. San. wall fnc. San. wall fnc. San. wall fnc. Doman (x,y,z) or (x,r) n erm of he characerc ze Performe a (-4~8,~4, /) Ax-ymm. (-~,~3, /) an (-~,-3~3, /) (-~,~3, /) an (-7~,~,-~) (-4~4,~4, /) an (-4~4,-4~4, /) (~8,~, /) D (-6~6,~5, /) D Ax-ymm. an D conf. D an 3D conf. Ax-ymm. (eay) an D (neay) conf

166 Appence Appenx D Daa hgh-lf cambere arfol Da-999 A hgh-lf arfol for pecal prpoe egne a Delf Unvery, econ wn energy gven below. The conor of he arfol epce n Fgre DA.999: y/c. y/c x/c Fgre 4 Da-999 hgh-lf cambere arfol. The lf crve of h arfol calclae wh r-fol. The rel are gven n Fgre cl alpha Fgre 5 Lf crve of he Da-999 hgh-lf cambere arfol. Accorng o Kaz & Plokn [33] he lf coeffcen of a hn cambere arfol n poenal flow can be fon wh ( α + β ) n l = π (39 ) co β o Eqang h lf coeffcen wh he r-fol aa a α = o β. gve he camber angle a

167 Appence Appenx E Ince veloce by a wake Wh Bo-Savar, can be hown ha an nfne long ragh vorex hee nce a nform velocy wh oppoe gn on boh e of a vorex hee (ee for nance Mlne-Thomon [4]). Sppoe we have he followng D confgraon. y, x (a,b) r θ x e R, r θ Fgre 6 onfgraon of nce veloce by a wake The velocy e a x can h be conere a nform. The vorex rengh per mere vorex hee enoe by γ. The nce veloce a coornae (a,b) can be fon wh Bo-Savar (ee for nance Bachelor []),, γx = π r γx = π r (4 ) Th rel n he followng nce veloce n x- an y-recon a (a,b) an = γ x γ x nθ + nθ (4 ), x π r π r = γ x γ x coθ coθ (4 ), y π r π r Inegraon acro he oal lengh of he vorex hee gve he oal nce velocy a coornae (a,b) a l γ R b R + b, x = lm + π l + ( x a) + ( R b) ( x a) + ( R b) x (43 ) an

168 Appence ( ) ( ) ( ) ( ) = l l y x b R a x a x b R a x a x, lm π γ (44 ) Solvng he negral gve l x x l x b R a x b R a x = = + + =, arcan arcan lm π γ = b R a b R a b R a l b R a l l arcan arcan arcan arcan lm π γ (45 ) an ( ) ( ) ( ) ( ) l x x l y b R a x b R a x = = =, ln ln lm 4π γ ( ) ( ) ( ) ( ) ( ) ( ) = ln ln lm 4 b R a b R a b R a l b R a l l π γ (46 ) Evalang he lm rel n + = b R a b R a x arcan arcan, π π γ (47 ) an ( ) ( ), ln 4 b R a b R a y = π γ (48 )

169 Appence Appenx F Prony brake F n R ω + F w F Fgre 7 Prony brake. The normal force F n on he rface of he wheel eqal o he ownwar force of he plley. F = F + F (49) n w The frcon force of he plley on he wheel relae wh h normal force on he wheel a F f F n = µ (5) where µ > he frcon coeffcen of he plley on he wheel. Th frcon coeffcen epen on he maeral propere of he wheel an he plley an he emperare of boh. The power aborbe P ( ω + ) by he wheel can be fon from P = Ff Rω + (5) Sbon of (95) an (49)-(5) n he eqaon for he aborbe power rel n ( ω+ ) = ξ+ F Rω+ wh agmenaon/ amplfcaon facor ξ + gven by P w (5) µ ξ + = µ R + R + (53) If he oppoe roaonal recon of he wheel ω e we have

170 Appence ( ω ) = ξ F Rω wh agmenaon/ amplfcaon facor ξ gven by P w (54 ) µ ξ = µ R R + (55 ) The agmenaon facor for boh confgraon wh oppoe roaonal recon ploe n Fgre 8 for << R. agmenaon facor [.] omega+ omega frcon coeffcen [.] Fgre 8 Agmenaon facor of he Prony brake for oppoe roaonal recon. Accorng o Fgre 8 here a oally fferen behavor of he power aborbe n boh confgraon for a mall change n frcon coeffcen µ. For frcon coeffcen aron, we fn ha ξ whle ξ an accorngly a change n µ rel n a mch hgher + change n he power aborbe by he Prony brake for he ω roaonal recon. oneqenly, he ω roaonal recon no able for mearemen an we can only e he ω + roaonal recon a epce n Fgre

171 Appence Appenx G Tranglar chor-we vorex rbon Th appenx gve he ervaon of ome eqaon for he vorex moel ha are no recly mporan for neranng of he vorex moel. Two approxmaon are e o avo rel ha are oo complex oo e n he vorex moel. I ame ha: he arfol have a ranglar chor-we vorex rbon an n calclaon of he arfol geomery, he angle of aack of he arfol α. A ranglar chor-we vorex rbon approxmae he vorex rbon a an arfol fon wh poenal heory (ee for nance Kaz & Plokn [33]). I avo he nglary a he leang ege of he arfol. The ampon α avo he lenghy eqaon ha rel by akng α no accon. Tranglar chor-we rbon The ranglar chor-we vorex rbon γ (x) rea for { x x c} Γ x γ ( x) = (56) c c R where γ ( x = ) = Γ c he rengh of he bon vorex a he leang ege of he arfol wh chor lengh c a epce n Fgre 9. γ x = 4 c x = c R arfol r ϕ y, y x x, x Fgre 9 Tranglar chor-we vorex rbon on an arfol wh chor lengh c. The bon vorex rengh of an arfol econ wh lengh x eqal o γ (x) x. Wh Bo- Savar, he nce velocy n x-recon, x a a ance r from he ¼-chor pon rea (ee for nance Bachelor []) x = Γ x c c co π r, ϕ x (57) From Fgre 9 can be fon ha h rel n - 6 -

172 Appence Γ x R c c, x = π ( ( ) ) x R + c x 4 (58) The oal nce velocy fon by whch gve wh (58) c = x, x, x (59) Γ 3c c R 6R + c 8πc 4R 4R c 6R + 9c, x = 3arcan + 3arcan + ln (6) A nfne ance from he arfol, where R >> c,, x ha he followng lm R>> c Γ 3c c R 8c, x = 3 + 8π c 4R 4R c 6R + 9c (6) Afer ome algebrac manplaon h rea R>> c Γ, x = (6) 4 π R Wh Bo-Savar an a ngle-bon vorex on he arfol wh rengh Γ, he nce velocy a large ance from he arfol R >> c rea We h fn ha, x R>> c Γ = (63) π R whch gve R>> c Γ = Γ (64) Γ 3c c R 6R + c = 3arcan + 3arcan + ln 4πc 4R 4R c 6R + 9c, x (65) where Γ enoe he vorex rengh of a ngle-bon vorex a he arfol. The nce velocy a he hroe acaor Ame a ance R beween he ¼-chor pon on he arfol. Wh (65), he velocy a a ance y from he ymmery ax rea - 6 -

173 Appence, x Γ 3c c ( y) = 3arcan 3arcan 4πc + 4( R y) 4( R y) ( ) ( ) ln ( ) Γ R y 6 R y + c + + 4π c c 6 R y + 9c Γ 3c c 3arcan + 3arcan 4π c 4( R + y) 4( R + y) ( ) ( ) ln ( ) Γ R + y 6 R + y + c + 4π c c 6 R + y + 9c (66) The average axal velocy beween he arfol fon from R = ( y) y (67), x, x R In orer o avo lenghy eqaon lke (66),, x n he vorex moel wren a (9) = χ,, hro a π Tha a f wo bon vorce a he ¼-chor pon of he arfol nce a velocy a he ymmery ax beween he ¼-chor pon. Th hol repreen he ranglar vorex rbon o ha,, =,. The correcon facor χ a can h be fon from hro x Γ D χ = a, x π D Γ (68) χ a calclae an ploe n Fgre. A mple power fncon gve a goo approxmaon of he lenghy eqaon for χ a R χa =.55 ln c (69) R for. 5. c

174 Appence.5 Xa [.].5 y =.55Ln(x) R /c [.] Fgre χ a a a fncon of R c for a ranglar chor-we vorex rbon on he arfol of a D hroe roor. I hol be kep n mn ha he arfol orenaon ame parallel o he flow ( α ). Fornaely, h rercon no a evere a may be hogh. The major orce of, x an locae a he pwn e of he arfol a ha par of he arfol ha he large, x, ax bone vorex rengh. Hence, he major orce of, x an,, cloe o he ¼-chor pon on he arfol an nrocon of α wll have a mall nflence on, x an,,. The nce velocy of one arfol on he ¾-chor pon on he oher arfol Ame a vorex rengh γ ( x = ) = Γ c a he leang ege of he arfol an ance D = R beween he ¼-chor pon on he arfol. The nce velocy can be fon wh he ame ervaon ha le o (65). Ye, he nce velocy a he ¾ chor pon wane o ha x ax x ax Γ x D c c = π 3 ( D + ( c x) ) 4, x x (7) Th gve Γ 3c c D 6D + 9c, x = arcan + arcan + ln 4πc 4D 4D c 6D + c (7) In (86), he velocy a he ¾ chor pon wren a

175 Appence Γ = χ c π D + D, w, hro Th hol repreen he ranglar chor-we vorex rbon o ha (7) a χ fon wh c 3c c arcan arcan ln + D D D c χ = c D 4D 4D c 6D c (7) Th relaon agan ploe an fe.6 x [.].4 Tranglar chor we vorex rbon F. Fgre The f acheve wh χ a a fncon of R /c [.] R c for ranglar chor-we vorex rbon of a D hroe roor. χ = D + m c n (73) wh m =.53, n =.69 for.3 c. D

176 Appence Appenx H Sable rblence moel for calclaon of flow aron arfol The lf of he arfol fon wh FD compare wh mearemen. The FD calclaon are carre o on a -H-gr wh qarlaeral cell. The y+ vale vare aron he chor of he arfol b ay below 3 o ha he flow propere are calclae all he way own o he vco b layer. Th calclaon e-p hol prove wh he be rel, epecally for he rag coeffcen. The calclaon conon are menone n Table 4. Table 4 calclaon conon NAA 8 hor lengh m angle of aack 8 o Re 6. T.% Trblen lengh cale y+ <5 Several rblence moel are compare wh he rel n Parachvo [53] a h angle of aack. The rel are menone n Table 4. Table 4 omparon of FD calclaon of he lf coeffcen orce Parachvo [53] Ke anar Ke-rng Ke-real Ke-real n orer pwn RSM, efal opon l Dvergng ar a he bonare becae of gr rechng /

177 Appence Appenx I alclae lf- an rag coeffcen aa for a NAA 8 arfol The arfol wa rppe a % an he calclaon wa carre o wh he panel coe RFOIL. Beyon D all, he aa are ynheze from aa n Parachvo [53] an Jacob [3]. Table 4 Wh RFOIL calclae lf coeffcen of a NAA 8 arfol rppe a % a fferen Re nmber an angle of aack (Aoa). Aoa Re nmber [ o ]

178 Appence Table 43 Wh RFOIL calclae rag coeffcen of a NAA 8 arfol a fferen Re nmber an angle of aack (aoa). Aoa Re nmber [ o ]

179 Appence Appenx J Qck ng gelne Fr, a goo e for a mall wn rbne characerze by a hgh average wn pee. In he bl envronmen, ch e are fon above he average hegh of he blng. We are herefore lookng for he hgher blng o e or wn rbne (econ.). Moreover, a wn rbne e beween blng ffer from rblence an he relng freqen change n wn recon an hgh fage loa. Sch hgh fage loa off core cople wh he probably of amagng he wn rbne an he rk for repaer ha col be h by (par of) a fracre blae (econ.). Frhermore, he freqen change n wn recon cae freqen yawng of a HAWT o ha he HAWT harly algne wh he flow. Sch average malgnmen cae a rop n energy yel (econ 5.). A VAWT beer able o perform n ch rblen envronmen, a yawng no reqre for a VAWT. B he roor ze of he VAWT hol be mall n orer o avo he neay effec cople wh freqen change n wn recon an n orer o prof from he local pee p cloe o blng (econ.). Seconly, he concenraor effec of he blng n combnaon wh he wn roe hol be conere. The concenraor effec of he blng hol f he wn roe. Acceleraon of he nrbe wn pee by he blng for only one wn recon an eceleraon for all oher wn recon n an omn-reconal wn clmae no very profable for he energy yel. Table 39 n econ 8.. gve a qck ar for he choce of he concenraor effec n combnaon wh he wn roe. Thrly, he mo able wn rbne for he e hol be choen. ompare o rag rven wn rbne, lf rven wn rbne have beer propec o become economc (econ..4). The choce beween a VAWT an a HAWT epen neverhele on everal aonal e. The freqen wn recon change n he bl envronmen are alreay menone. On roof of harp ege blng he flow angle hol alo be conere. The flow recon cloe o he ege of a harp ege blng no parallel o he roof an a a coneqence, a HAWT n ha flow from below how a power rop compare o a HAWT n horzonal flow. The power op of a mall H-Darre or lf rven VAWT ncreae n flow from below. The H-Darre herefore preferre for ha flow. For concenraor blng ch a he plae concenraor an he hroe confgraon, he HAWT preferre becae he roor of an H-Darre no able o exrac power from he oal roor rface. La b no lea, here are nmero non-aeroynamcal e ha hol be ackle for a ccefl ng of a mall wn rbne cloe o a blng. Some mporan e are: afey rk, noe emon, nrocon of vbraon no he blng an haow flcker by he roor blae (econ.). Mo of hee e can be echncally olve. Ye, hey nee a carefl y a hey can preven ccefl ng

180 rrclm Vae Appenx K rrclm vae ahor Saner Meren geboren op Me 968 n Haarlem. Hj eere van 987 o 99 Werkgbow aan e Hogechool Haarlem. Na he behalen van zjn ploma Werkgbow eere hj van 99 o 996 Technche Narkne aan e TU Delf. Al bree opgele narkng en werkgkng ngener heef hj oen een weeal jaren bj Synen al conlan gewerk op he gebe van nnovae voor he men en klen berjf. Herna maake hj e overap naar Sork Proc Engneerng al onerzoeker perone aarga rogng. Toen ook e gelroom voor onerzoek oprooge werke hj van 999 o 5 al Toegevoeg Onerzoeker bj e ece Wnenerge van e Technche Unvere Delf. Van eze fnce kon hj aan een promoe chrjven over klene wn rbne nabj gebowen. Na he beëngen van zjn onerzoek gng hj naar DHV waar hj werk al enor conlan en rekker van e Wn groep. Zjn groep Sanable Wn realeer n e vere projecen e ombgng van wnhner naar wn benen oor he onwerpen me wn bj oprachgever e nroceren