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0 OUvrE)eltJLT1~ nvrLoT6oEro~ C ElYal nOA6nAOK~ OUvapshyTTJOV 1tUlCTtOOY napay6YTOOY PITOI TOO npl8~oCi Reynolds Re ~~ jlOPCP~~ Kal toO llpoundyt90~ tfj~ t-yKapoiou uypd SLatOIJij~ TOI) ayooyoO Tfje crxpoundTICf~ TPa1Utll0~ f(bV 01shyXOOjlatOOV TOO ay(l)1oiJ tit~ m)lcy6TTJT~ TOO ptOVTOlt 6ypoO Kat ~~ o1tOAutOU OUYtKUK6t11tltgtlt oUTOU HtOl penSl va gXOOjltY OUvltiPTTlOlV tij Ilopqgtfi~

C = F (Re p R K p ~) (I) c51tou P napaYMY Ot oli xapaKTllPlCttat 11 jlopcp1 Crypftlt otaTojlfie R i igtOpaUA(Kt ani XP11oLjlOnOlOUjltYll np6e poundK1tPOOOraquon)shyO1Y TOO 1l6ye90ue tfj OLatOIlil~

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p t 7tUKy6TTl~ TOO ptOVTO uypol) Kal IJ 0 OUvaIJIK6~ ouVT6AeOtt~ OUWKTLKOtTlTOe

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I TIoluuo M1xavlK~ BMO (L939) AyjlovOIlO at T Olloy pashycpOIO MTl1avlKO EMn (936)

2 Elllt)iI~ T~ middotYlipoulllt1 Ittll TIIV Y6puu)IIt(llv -EpYQ)v Iv Tlil B M nOAutexVe(tp (1943-1941)

3 oI61i1C1WP Mll XOlKo EMn Bt~ trv IUPIOVlV 1ft Y6pou)I(it Ita 1il1 Y6pOUAIK(LlV BPtwv (1955)

4 Texvlxov ott)poundXOt 10D Yh8 a1l0 100 1936 J~ TnV oKOAOUOOV lCAIJDK(i)U1

0 An6 10V 1936 IIhpl 94 al TOlloypaltpUl-f)v middotYnTlPpoundolQvIi 1945 - 1965 ysptlI)AIlCQ epyo ell nPO(01a~SvOIO 1ft Y6po)o-

1IKfIO YnT]peolw l(bV KaTaaltliU(bv Y6p06uvq~Hltibv Ep-yQ)v libv KOTOOl(elGlv Y8jlou)u6)v epTCDV 0 aoooAlo Kal AVa loAlshyKft EAMSo 6000 Kat W hljlA61lQlV KId tnOltII~ olaltpopCDv ir8 pouJlltillv ltalotnCpoundwv

y A1T6 10il ltoUl 1966 bIEOOUVT-f) JT(tjlQ 1ft rtvlalI EIl19ecop-f)oel bl~oolQ)v 8plQ)v

s tUYTpultflX nAtov t(bv Altutov bnOtTlJovlltctlv tpyoolllv ampI tllv IlliptOXrV 111 yopau) 1CI Y5p0Ao1loIO Kal Y~paulI1Clilvmiddot8jl(tOV EK t06tWv 10 OI1X6vouv 01lY1paJIW1Cl Kal t5 6vOItOl~I til JlaTKlolllu Ouvtl3pa Wip aU AIK lil v pIiUlLIV

6 eltpauphll 3laqlOjXOv tllol1l10VlIt(bv ~PyooTPloIt4lv opyQVOW 6lI 11 1 lOCI t))e-tPOJflYVT)lIKOpoundl Ilulhnau (AE 24164) fOCI -f)JIOojlCIlPIItOO 1-Ipound1jl11100 tPlool(016rou poll (oE 24165) 100 clIotiltoO pcupa10Jhpou (6B 24166) Itd

1 Yn01poltp0 11l EUllvLK1 Ital A116ptlCClvlKlI KuJkpvllo8M (l950-S) tOO rolluoo Kpa1oW (l95S56) 11110 raJJItIl~ KIgtshytEpvllOpoundIll (1963) Ito TOO AIJEpUCavucOD Eltn(1lkUfIXOO UiDil 1-1(110 (1965-66)

8 middotBtlpatlitel Itat $t 1Jr911 o11Joo[w ()rro laO navPI0111Jlol) 910shyoaAoKTl Itol Tile Axaolwi oe AthlllIv T~lIt tvv trVaTO MtAo olacenttgtltO ilIliOrTlIJOVlltlw OpyaV1o~uiJv ff dUoliolllI

9 ~poundrrpoaQIllOc ftApoundlCf1Utl 16 YnoupYIlov 41100(0)v EpyGIlI Kat 10 TBlVlItOv middotSltlJsAI1pIOV pound)-600t II~ Sui4JOpa 6Ovtj ouvss pla 1t09 (Ii lIapouolacrev tpltlOIOIO TOU

10 BUhKamp(911 Kal tfpoundllt0l01911 B11O liba dVIpa oopoullC4Iv pe~ v(j)v Itat pound1 oha ttooopo l-iflaAa tjll01~I(l 05poullKIw KafGshyOICelGlv tft EupwlIT] Col fi)V H nA

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12 ElvOl l AO ibv thOAO(J9Oll b10111l0V1Kflv bpyavo~v 1ft clliIK0111(h TOU

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11 E)JnEtpIK tampo 504 UO)0YIO)lOO

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(2)

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L Vmiddot H - f4it 2g (3)

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KoO 6)~ toUc ~lttampmL ttl Xp6VOlll 01 Oxc((tt 2 Kcd 3 QItcCcirouv pamlut ftl(1ppaopoundu XPTal)J0lt0ITl9tiocli nJlshyCflUllt uno taW hmiddotaltJTotp jlt)eTfJ1WY ttpOC OuJlllo)T1v Eit 6 tv )6yl np6~1~10()

61tOU~ to f1 JlfOTl tam OlOlllllOpoundooc Kala tue 1tapEU~e tol) dyooyoD Koi

- - V_ Vi J shyVo - tu p = rgRS - - g - V r - O35V[

tl )Jt1oll laquomxurl TpI6flQ) alteuro ciJJi~ laquon taXUlTle Olatlll1shyCE(j)tI)

At KOtcl T6 napc)96v npOOnUOeuroIUl KaOO()(JJO) til~ tlJl~~ toO O1lVttiAtCftOV (JVtlOtucrtcor C dxov we nnotJ tOIlU lTlV 1tPOtOOIV OlUqlll plaquoJV axemiddottnN CIJ1tpoundIPLKill lKIlj)UOlCaw (I) 0 JltYIl~ lOiTrruV apI9J16lt tOlljHUl)flY1lcrl tti C17topiov 1toto~ [ I( l OUTO) 86 fSCI va npOtI~Ll10ei 1XetlKU ttpoundlPfi~tUtu Q1ti8fn~av Ott 6 nU ov t 1tltUxilc t(mOl (m)o)1 de poundIHPcishyV101V 6)Mi Kal haYOTtOU1Tl~ ~t1ro nJGupGc up I Pdct~ Sui npolmue UpaP~lO)ot d UI 6 tmOUtrn r6w Mann irta w

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I C = - R I (6)

11 tv9u Tl 6 Ouvtcleuromnlt lpaXlrrllro~ lCOtcl Manning H tJ11tSlP1Kl) ~Kltpplaquo(JIO (6) EXpound l fOll -rOxpound1 lilEOVOCC oVUYVCl)Ptshy

1 Oldovq Oxnului -n1IlPoqtopiot lluPt(OMll I oi l 16 ltOYfllUI Tl-ltIv ~8WflltlllCi1 Kul l~tpllflJloOLliVll middotY6pol~u(II bull bull ~EfLIL AlJ ~~v Aovshy

11 it)fgt2 O~l 2511middot263

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2 H ~CPQP~OYli Kat tv ytm XpqOllJ07toiTl01C toO ux l5f1w tunol) (h poaC IJqaAillv tlUI)tilfOOV d Ql aJlqgti~oAoC (3) -ai

3 A1to IIIl6uJ)O OUJotaUcll c1VClAUrn(t n ltTX~ (J(~ (6)oe clVOl apOq rrpri)~ l ~cp6oov 6 CfUVTCAccrtlle C elVUl uouiOtcn~ Kai b8p(tl))IKll nKt l R dvaa Jltypound90~ YPOW jJLK6v 6 CNvtEAampotli tpaxmlt~ fJ 1tppound7tamp1 va tilU lhocrtashyC8l~ L I E[~ ttiv npaypauK61llfO 6J1cot euro(p6aov

_ I _ I =r r RS = - RI YRS == - RJ8 SI (7)

~ ~ 00 l1ffiIJ6V

~ = C VL ( i) - C L

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2V = g l c middot g ill Rl ll 8 1 = -Vci YgR 101 C o ygRS (8)

middotHml V = Co Vo (80)

on()u Co yC _ = Crl2 = -I Rill g_l 11 Kail1li

S ~

V = YiRS = yiJi KaT6mv Tillv rot uvro (JXtoEOlV e1tiCT 96 sxw)JEV

_1_ _ _ C___ Co ~

yf - YBYg - IS - ~ li8 (9)

middotH CX~ (9) 6scoprTtat PUCHKOU 1tpoundPUXOIlpoundOll me (Juv5eshyouoa To ()8pUUllIU OTolXpoundIa f Kat T]

III C)EWprtT11C Jlt8oaoC loAoYICJlO~

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A 0 Q ltlH dF - P = yRS R = yAS = yS Y = - y V dx (10)

dH Mv Q 1tapoxi S = - dx f) 6v6 IJoyooa JlTl Kout tOll ayooyoO

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dV = (~ + ~) ct (II)

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KClm tOY Llldwig Prandtl 01 olo OllItOtl (Jll~triov Otpoalshyln po~

to = p( (~r (12)

lvOu I iiv XOPOKtllPlltTTIKOV Ili)KOlt YVO)(fto 01lt laquojfwe ~i~pound~

Tt)~ Cuni tov Thcachu von KarmanX) (ltIV d)

to = t (1 - ro = pKr (dt tVclX1 (13)

M Y rn ft dK~ CHiraquotlVOt -cd KT 65ulcr[(lt~ (ipIO)JoC iaolt 1tpO~ 040 uTonlv c[crCl(IDYII4 lIi t ttlv CJXtow 13 Tfl t CTpOshyfh Af tI1Xinrrro tn 6f1lt Yu Kai tilPIO)Jt vltOV iiAflDV ~mshypOA-mv il 6 ot()-pu)(H 11r t )6 y(p (Jxtmro~ 1tOptXct to (lY~6JouOov vUJlOV lhuvo~L~ ttexOtntQlrv

V= Vm -r ~ [1 r + In ( I - V- )middotj (14)K II V -

TPOPU nUIIlIWpnXUVlKWV

Tpnpo nollTIKWV pnXOVlKWV

610 t1jc poundlao)ooyfjc SE tf1c hvolru tftc 6P1CtKf1C crnpaoOc 11 aXEOle 14 SUlllEV )elolX aooAfllCll OtE x=R yIVEtCtl

V RVo - = 550 + 575 log - - (15)Yo v

~lG) SUI tpaXElw ~1wpCtVElac poTit Y RV- ~ 848 + 575 log K (16)

EK t(i)v axecrwv (15) Ked (l6) Daov 01 h TOO SUVCtjlEVOU

Vll rcpoocUVU laquoou9voCllt v61lou OICtV01-lljc TooV -ruXunrroovraquo -rGJ Prandtl ai Von Karman

Y x Vo = 25ln Co (17)

bnou Co t1 CftaSpoundpa 6~odllproCJEOlC npmCU1rtEL Otl n Stetmiddot VOIl tic CtxutlltOC 1]1 Tn ~oovnv crtpoPtilC po11c Elva Mgtyapt9Il llC1 At 9EUlPlltiKol epEuval oov L Prandtl Von Karman lCat ) Njkuradse aTHIOVlLKWC l)1tEpofl9llaav de tTl ElaayooYliv Kai Ka611prooIV aVlloHKGlv jlE66Boov UltOAO(lOlloC tilc po~C ~V16r HtO nlfOLv uyooyfhv R KOa TO ~to 19041tposhyaBEie-a 1tO TOO L Prandti 9Eoopla rcspt 6plaK~C aneaoegtlt (5) cd SeooplltlKol ~PEuvat w(rrou Qaov Kai ToD T Von Ka rman (1930 - 35) Kot to nEl)1UI10tO to) 1 Nikuradse t933) aoeinoto ~nllptacav fCu JUXPl Tfjlt tnoxflt tKE1Vl1lt o~acralt Kat aVtlArIVW tGJI Sv npOKEljlEVCP ESlKGV Kata ta jlEdnEltO CTll poundltpoTu81lcrav vEal avaAutiKoI axtshyOW bnoAOYlajlOO trlt TptPiilt (6) tnt t1amplt jlOpltpfe tampV 6nooov crllllOvlUcOv pOAOV SVpoundlXOV KOT ~oxilv OOPlCjltVO tmanwovLKCt aUjlTCEpOCjlUra tOO T Nikuradse cOlt nmiddotxmiddot on sL UjITlwlt tjlCu ToO Re Kat tpOxeuroa toIXOOjlatO to f dnojo(vEl uvelaptrltov f0) apL9jlo) to) Reynolds pound~apshytOOjlEvOV jlOvOV ono Tv CXETllCt]1 tPOXUTlltO 1lt ToiJvavtfov 01 O)(yov tpoXtTlt tnupovtlolt pof]lt KOt aXEtKwlt XajlT)Ctt tljlW TOn Re a aUV1EAecrtrlt tpLIl~lt f ai~6veL jl6 till OUshyInaw ToD Re

Ta npoovacpEpStvra SEOOPlltlKa Kat nElpajlOtLKCt atljIshynEp~ajlato Tl)v L Prandtl T Von Kannan Kat 1 Nikumiddot radse iyivoVto OleZ flv 1tI~pintroalv uYWYellv uno nlEow napa -roOta it XPllmp01tOtllallt TGlv tv A6yQ) tpyocruuv KOt de ayooyoult t4u8imiddotpat pof1lt ElVOL OUVIlr Kat toOtO 0101 laquoaaallcGJe) ( jlllxaVlajlOlt po)lt Elt t00c uyrorOUC im6 nlEOlv daov Kat Ellt tOtoumuc tAEo8tpOC pone SuvaTal va SEOOPTletj dN nopEjllPtiprK lI)l)DtKa E( taUlt ayroyolx t)cu8tpolt PO~lt UqliOra[al Ida t Aw Atpo EnL(p6VElll imOKElshyjlevll de ttv lijJEcrOV (nlopaClv tflr atjlOacpmplKfjlt 1tlSshyotroe KatOnlV Se tou[ou~ AlryC)) tnnrpfe QEPOt KOt fiOotolt ava1ttUaaOvtat uvrlat altTpoundI C jlt crovaVtWjJpoundVfU err taU one n(EaLv 6YW100lt nAilv OJl~ TotJtOOV Ii uypa f4u9tpCl tm~ q1C1VEltl tmOKEI[ol de OllllCUllCzVaEte paSoult YpoundYOVOt on-cp jlEta tellV fjJqKlvllOllEVCOV OCJCpEll6vrov PCUjllltooV fm~ ltptpel jlctapoAalt dlt t1)V [PIPV KO tT)v tpaxulltll Kom atlVfnElOvoLa SUo OYCOyou tva ono nlEatv Kal EtEPOV 6no eAEuSsPQV htCPOVElOV jleuro tilv outv tlll1)V 00 OUV1E~ 1t0100 tPLP~lt otv npt1tpoundl v6 6vajJtVlltOl Kal 1 auT tlJlTt to) Re n)Tlv l)jlooe napa taC EV AY((9 oloqJopae ano YEVlmiddot 1(ootEPac CfKomot KPLVOjlEVTI n toto Tflc XPTlcrllJononiaCUlt Tampv nopLaJ16trov [Illv npOaVuqlEpSEVTCW tnLatTljl6voov de dV01KtOigtc oycuyoOc Elval OXI 1J6vov AOY1Kt 6)A0 Kat oKOtljlOC

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01(( t(l)v 9EroP11tlctlv tlmrov oaov Kut ola t(l)v ~JlrtpoundlP1KlUv 447 tOLOUtrov 08v poundtval tnopK1ie OMd tOUtrov nopEXpoundt ttv UATjS t(~lflv avnatamro )OYC) -rPlPGJV Kai )ot1wiv altcuv H poundv AOYCampl laquo(lililSel(lraquo unExpecooc tbv ypacpoVta va OtPQCJltI np6e 6))ue Kotpoundu8uvOEle Nit uvaCTi-rno1] Jls8000V K08 1Iv va urtpoundlaEPxcuvtal Kol va tftT]pE01ouv tflv P0TtV 5Oov lCal -rae avrm8poundjltvoc dlt tofrrrtv OUVaIJCIe dnavtEc ot qnr alKol nopUYOvtEe OUjlcpoovOOlt ltPQ( Ttllt O1wap-rnast 1 Kat la

IV to 114ampoioC urroAoYIGiJoG Tille po

EtauyovrElt dC ttlv oxampatv (1 0) ~puca Jtopayffiyoolt oUvOjJEOa va ampXSi)jJEV On avo jJovaoa JlflKOUe toO clymshyyo) OUVaJlle uvnataaEOllt

ill Q ilH ax = -Y ilx (18)

T6aov 11 nelpa Oaov Kat n tv t4l ~PyootllP(qgt nuP0nlPfmiddot 01lt JlfiC olaacpowov tin f tv )6YC1 OUvajJl elval auvupshyTllcrl~ ouXi povov ti)v napoy6vtrov R f 0 tfic TOXUT11tOlt V tflC PQXl)tT]tOC TI [ot) napctyoVTOC ~ tfjc mlKVOTntoc TO) ptoVtOC ilypo) p toO d01KOD popoult y KOt TOO Oly

TcAtatOn KtVl1~IUtlKfjlt OUI6KllK6tTjtOr v OAMl Kat tfje ovOJlOlojlopcpiot fv nopoucrtUEI etc qlOOlKOe ayroy6( ~)EueEpac poiC Toaov tv KIlf6+El Sta taC OQ(popooe am JlfjKoe StcrEte outoO Oaov Kai ~v 0Jlf t EnOTlt ~mpponv ht tfle tlJlmiddot~e TT)tOUVOjlECOe uvnatacrerolt np~nel va txu a KLVflJatlK6~ auvrfAeatl)lt pone A amprov at a joe~oe jlEtaaOA~~ u8 uVOt Tfe ~4uetpalt bupavdac pofjr 6shy)oYQ) olaKujJitvOEOOV Kota atlVEnUOV SU li1gtpound1 vel fXCUlEV

~p

ox = P (R Y ~ B p y v T 1 IJ) (19)

ad fvOo ~ = at nouvuptrau toO xp6voo t Ka9 (hllQC

jJeta3oAl tfc tAeu8tplllt 6ypW poundtlqJuvcioC toO ayuryo) Pli9oult d

MeAttGlvtflt lCat uva)orov Tp6nov0 8poundjlo art6 nAeu~ crovtwatoO Tpl[Hte ItXOjlEV

f = F (R K B R T 1 1) (20) (1( EK elIV atlVaptroErov 19 Kal 20 ~aYEtQl 6 aVaAUTlICOl EftE Kal tll1UQ)lKOe npooS10plOJlOt tfie poundJtUKpLaolJlt Tljll) tGlv aVnuSClJeVCOV ampIt [Ill poilv tampv ir(p(i)v ouV6jlECOV oaov Kat t1C 1tPOPOAAo~tVTlC tv YEVEl UVrtGTaaEOllt c1vm noU~ 1tAOKOC Kai aUOXtpnc Kat tOuto o6xl ~IOVOV AOYCil Tiilv npoavuCj)Ep8ElOli)v 65uvaptliv 111 M091lJ1ntmiddotIKijc A vaAampshyaeuroCUlt Kat -roO A011(fjlOU ulu Kot tol1 ycyowhoCmiddot6u datn Stv ltatt~ ijuvltm] 11 t~ltpl ~~ t~~ tmppal iv etG bcuOlO( t(i)v npoovocptpepoundVtrov napay6vno ~I toU t-upou np6~ )1to)oytCJJl0v jlCy tOOlX Othro llX nt1pa~Uv8l poundlattl liyvltiK)t~ 6 pOAOC Kal a Tp6ftolt tmppofe tol) oxnJloT~ Kol tOO llEytOOJe tfjlt 01UfOjJfjt tot 6YWyoO btl to) cruVtEmiddot AEatOO tPl3ftlt dtE tOn TOI00tou clVttO[aaEcoe napa to tv AOYC) yeyovoC linavut o( dgtlt 611(0 nop6YOvtpounde t7tTlpco~ouv tTlv POllv ano KAEUPOC uopauAucfle clvtIGTaOEmlt O[ 0)[0[ Enlafe nOpoYOvtElt (meLatpxovtllt aUTOjlutOllt Kata tflv 6vannXlv Kai -rnv AttToupyluv [00 UOPOUAlKO) IDjlaoc OnEP KAV rampv yvcuOtellV jlampxPl 0ME tqaplloyiv taU dtto TCAtupfiC XPllcrLp01tOlTiapoundroe 8a lltO ouva6v wi uXU xpnshyaUDlt np6e Ko90pla~ov KQt npoaolOp1crlOV GT01Xe(rov axe~ tt~ojlfvwv ~t [ilv oopauALKTtV avtlcrtOatv Koi OUtV tautfv tv nopoxv trov 6yroyG)v

nEplYPQ(J)tKO crtOlXeto ELtt Ull01 tmonljlOvlKol ApoundntOjlpound~ pCLal 1tcpl toCgt MPOllAKOI) tu~otoc otv 80 ooSoCv evtaOSa TOlOUtae auVllTOt 6 dvuyvroaTTt va CtveuPlJ etc novitjlatQ middotyopUllALKijlt 0) caov KOt tIt Ent IltpolX slOlKde ii1tt foU autoO 9EJlUtCX 6Pyacr(ru (91415) np6e unAortohlalV tOU 5)ou 8t~o0e tlHtQlOjlSV T1)V ftCpimroalv OXTlj1QftO~oU ilSpaUALKO) tullOt~ KataVTll 8uPOQIJ)QYJlU10lt H sv A6yq) ftEpntWCle emtptn-El TtlV ~EAtTnv tAEDOtpOO Q)jlUTOC dc oLuqgt6pout KUta jJflKOlt 9tOCle tot) 6yroyotl Bla 1fl amp1l~10Upshyyiac vallaTrov 1toldAllt tVtaaECI)lt Edv h TO i51pOt ToCgt 6pSOYCtlVLKOO 9upoqgtp6Y~OTOC Kat Yo t6 tYrpov 1680e d~ tl)v StatO~TtV OualOAflc (EX~jla I) t6tC yo = Coh (21) 6noJ Co 6 auv[eAtgtGtfC cruatoAfjc dl0 [ljlac tot) A6you h j H U1tO 0 jlhPI 020 iJ tLjlf tOn tv A6yctJ auVftAfalOil Elval KOcl jltoov Opov 061 AqI ~TEpOJ oLa crtaeep6v qgtOptlov H nopo1il

Q = Cy C A Y2g (H - Yo) = c A Y2g(H - Yo) (22) ev8a C1 CCO 6 auvTcAtattic tKPOIC H tLjltl tOlhou KUjlaivc1l1 (Jn6 062 ~XPI 083 Te4utoicue 6 W Wunmiddot perlich (10) E~pe nElpQJIaTlKffilt nAneOr [l~(i)V TOn C L gt

44B

Et

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Q = C h V28 (H + ~ - y)

6noo 10 6 nAQroc toO 90po~pa)1oTOlt 01 npoc 16 lJltKOlt La Tic SeerEOgt( ~VeQ Ml)Jaavel xmpav it 6taton croO[OAIlt 6opt6T] (II) (12) lSn

L = h (23) degH ilpaplJorT) toO eeropiHlotOlt tvlPYtf~ )Jemiddott(l~iJ tow 51(l~ (oui 0 - 0 Kol I - I fine Kat oujJ1tnTeI ~pound Tn Sto[o~rnv npOOoYl1l(l)C toO 6SpauAuwtl liAfIQtOC ~c 6(pound1 e~ ltTXampOLV lt~ Opltp~

d ( Zo ~+ do cruvcp + 00 ~~o) = - Sdx - rVt dx (24)

6J(oo Zo ~ 6UVlTllCl) ~VEpyela rl tvtpyrIQ OtOpoundmt de Til ~latOIllV 0 - Oyo = doOovcp [6 uyp6v plIeo tIc T~ at6 011shyjJEfov 110 6 tilpound[ ouvtp4CJT11lt 5tavoJ1~C TOXUtJl[OlV fJ cruvtpoundshyAoot11lt Coriolis Va f J1taTl tQllYrl1t de 0 -0 Kat rV1dxfl dnmALLCl tvcpyr[~ )6ycp atpoplAIOjuiw lhvlbv TUx6v jJEtUshyPOAIt tf)r ltaroJ1ii t KcIi AOLnWV 1WpqUptpwV altirov (13) H ~h04Popll(Tt axtO~ (24) rpaqgtpoundtOI

(Zo + d crov~ + ~ ) = (S + eY)- (246)

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dZ d~dd d(Y)di + ltroVcp dX - do I1jlCP dx + 00 ~21f +

+ (S + rV) = 0 (25) H Kuta ~ftKOt 1IAIOu to) ciYOYyoCl Sb 8EU1PEltOl dpvrrcunl noo6TTJC K(ltU v cpopav QuETtOpoundIDI TaOTTJ~ WHtOl 96

dZo fXrojlEV Sb = - dx AUl )rovlav ICAICJElttX IP CJtuOcpQV Kal U~ATJTta~ QrtroAE(~ til lhv6lv ICal a1PO~lA1C)lmV (Saov Kol Aoml)v nappoundl1qrcptilv atdrov ti tEicr(J)(JIC (25) y(vCTal

dd d(V)crovltp d + dx 2g + S - S = 0 (25)

EI~ tOlK Q)CO)OOC tApoundl)epoundpltl~ ro~c (Jovne~ 11 tWf Tft Ycoviac cp Elval nOA1 jllICpU Aui IPO CJ)q) 1 Kai ShO E1tlmlC

ddo dyo-r6u do Y teat JX - dX KOT61tlv 10UtOO eti Elval

dyo + ~(Vo) + S = 0 (26)d x d)( 2g EavQo ~J = 100 en Elval

d (Y ) d (Vo) d ( Q )Godx 28 = dx 28 = dx 2gA2

Acp h Apoo btsl61t eft mplmcoolV 1[p)()~anKrov oYOIKtGlV

1

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A = f (ygt Stc jl6vt~v porv

dyd(VI) d ( Q) 2f(y) Q d 28 = d 19f (Y) = - f (y) 28 d

R axtaL~ ollt1 MvataL en(011~ va ypacp(

~ (VI) _ _f (y) ~ ~ (27)dx 28 - f (y) 8 dx

dA EnUi~ 5t ltIgt ampyplgtv tle68cpov nMlt T = dY = f (y)

YvtKa ~ t~(aoxnlt (26) ypoltjleuroT dy f (y) Q dy d - f (y) 8 dx + S = 0 (28)

stu 6J(61l~

(28laquo)

A))Q tIC rl~ aXpoundCfCOlt 7 fXO~EV 6n 111 VI lltl QI

S = R4JI = At Roe

Kat6nw tOUTOO it Y(cro)(n~ 28 Aajlj6Vpound1 nv JlOpptv

TQ dy ~Q (29)( g A- J) dX = AiRJ$

I (TR AIRtl)Ka[ dx = --A- - - Q dy (30)

~ g AaJampPavoflpoundvtt 6n OjJw tfJ~ OXpoundOEOOlt 6 KaI6AoK1lpoOvr~ 11tv crytcn v (30) jJe-rnsu x= 0 Kal )( -= Xt 61[00 T6 6yp6v paeo~ ovtlltJtoiXCl)( elVQl yo Kat Y pound6p(mcOjlEV

f X fX (TR AR)dx = C J - - - dy (31) 8 Ao 0 Q

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I 1 TR AR ]Ci = L-h

[fY8A dy -

f YQI dy (32)

yo yo

lt1 f = i~b [t ~ dy - f ~~~ dy](320) Yo Yo

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449 V Etpapoyl Tft~ vitae na6lioll 11e ncplnT(JOIV a9~ 6 ouvap-rlJaeox ~ IDpcp~ y=f(x) 6Tpnpo tKaCTIICWVU TUnoU auoJpuyuv

A (34)= PdX = fi(X) dxnollTllwv ELl 6YClYYour tK6ElUOO t(rnou

A=f(y)= Kym (33) o 0InJOVlIWV Onou y t6 ~p6v pa~ polK Ex til axtaEOl( 33 ~Xov

6nou K ata9EpO ampUVOIJ6Vf1 va npoa5lOpla611 tlVaAUtuc(b( dA y rnA T y leal m ~K6etK1t 1tap6Ju~tp~ -d =T~Ktmym-I=Ktm- =-Ko m=-A (35)ampP Ooov [6 axfUIQ tyxapo(oo lhatoJ1fl~ fi6votot v6 h Qlpa- y y y

y if---- - T --t--T -I--T

y

1

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(b) (c) lOx 1

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P = N yS (36) eurov9a N Ot09Ep6 liuV04JeVTl va npoolhoplo9t Qva)uUKtigtlt Kat P toctltt~ Elt til axtaugtlt 36 ixov

d d d S t dP dy (loP) = dY (IoN) + S dy (loy) = y = p dy

y dP Kal S = P dy (37)

rpoundVlf((i)( euroxo~v T

(38) o o

P=-VI+ Y dx = ~TI-+f-(x) dx

Ex ti~ axt~ 38 EopioK8lQt 1) nil toO P 600y Kal lOO

dP I bulldy Ev (JI61poundtQ Q1t6 TtV OX~OlY 37 eop OKpoundta I t41Tt

t oll AK9ttou s A Kym (K)

bullAlp tdpou R = P = - ys- = N ymmiddots = Z yt

K c5nou Z = N )Caf t = m - S Kat6Jrlv tGw ~ uvco a( OXtshy

Otl~ J2 Kal 320 En OtootO)( IhcgettKOll rigt1[OuAalla~VOUV nlv jlopqniv

I 1 [mZ ZK 1 c~ Il L-h tg(Ylt _ YOI) _ PQ (YIP-yOP) (39)

4gtOJl 1

450

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1 1 [I J J(5 - L-h g-(Y - yo) -4qt (Y14 _yo) (40)

6nou q~ ~ fl ava j1Ovaoa dUTOOt Trapaft middot

aui TPlYCIlYKUC oltilpo)~ eilplcncojJSv

r 1 TIIJI [1 1 cpo ~ = L- h i(y -yo) - 6Qpound~ TCY-Yo)(4J)

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J J [I 64Po - = [i1 gCY-Yo) - 135Q ( y-Yo) (42)

amp100 Po 1l fl ~nTtapUjLe1poc t1c Un OflW (11 2c) 1topaPOA1shyK1 BIQtOlrile

VI ETfaAI8ruci Tft~ Eamp615ou iSla Tft4i XP~QE(U nElpaJlClTIKQv alEaoJlfvwv

np~ tTtaAiIgeoow Tt~ KpmEaVO~EVTllt J1e06lioo poundytVEtO xpflau 1UlpaWtlKWV oBlio~VO)v ava)byou np6t ntv 1tupoOshyoav 9coopTtucytv 1ll90Sov 1tEPUrTOOOEIDlt TIpuYJ1Qn KaTU tCu ono TOO ypl1qgtOVt~ )fVOl-tvw Av t~ EpyaltTtllP(ql middotYOpoUAlKi)C toO E MfloAuTpoundxveioo rtpdtrw ~Pl (olJOs tv -EllUlh AmCtTlJlovu(W tpellVW t~8T911 10 UOpaUAIshyKOY UAIla el4i 1tnpaaOAIKOY aymyov (14) t6 hot 1954 -cut tv cruVEXCiQ Eti tPlYQ)Vu(6v TOlOutOV (IS) (6 ttOC 1955 EytvCtO xPilCI~ ilSpaOAllcoO lil4wt~ tIC Bopomiddot ~paYJllnoe Ikrl9EIIJ tYKUtclCtCwECLM 11IIKOOe 735 Jl UHW ll~ampv t1tlVO~Ceroc ltH fy l6ycp MpaUAlli) tYKatucrtcwu tlJqgtatve1UI s~ tic sh6yw 1 Kai II

0 mlVtuAmt~ 1paxvTT)to~ lCam Manning 1ItO 11=0015 Av) ~ ~~ t1j ~~apalttpo napa~ol~ Po =0039

Bacrtl t(lIv 1tElpa+tQttKUlY OTOlXeloov tie tv l6ycp apyQaTIlshypLaKi1e 1tpoO1togeioC bf-teux61 1t1ftpr]C btatft98UCLC Kai 8loO1aUpcooU Jll ta irYtLcrtOlxa 8eSolltvo Sui tfle XPllm-1J01tOlitOetOI tflc tlJ1telplKfly axtOErot toD Manning 6 At EOP IC pyai tIJlal tIC tfle Ytac tOOT1lC BECOpac amptval 6)iyoY Katompa~ t1Jv tOlOlrUOY ~K tfc crxe~ 6 T6 JOt ouroY 9EoopeiJat eGMryov Kui )OYlK6y tqJ6croy KatQ t1)V aUvmftv tfc ev A6YQgt CxsOampooc Stv tA1cp6TJ bn ~VlV to crUVOAOY tav t1tTJpe~6Vtooy tfty 68pcrO)lK~IV dvr(OtuOlv napuy6vt(ov

(I) Diflt middotA)UQ-nploll 6eOlptlmdl Kal middot ~llPIIOGA~ 1l (6poU)1shy RK6001 A5~(Jv A1DVlfl al) )51-J~9 1962

(2) P ArgyroJxtJlo~ tClIlclI1 de JgtEcollhmcnl en condulte SOli p-on OU l Ill-facc Ilb-c Ed~Dunod Po-is 1957

(1) Mso~li HfJm-Na lt111uld esitanL1 in at now o f 111gb ll rllll ilc numbcr l~rllc of Ibe fl1d J ~l)nn Nallonal Conshyfe rtcc 011 AppUfds Mec buDlcs 1952

(4J v_ T~ Chow OpeICh nnoel HydrluCSo Me Cra w-Ifill Compn ) Ille G~) 200-201 199

(S) n~ Af7I (I(IJV)ul w H 1I1XIlV1Ki) Tellll peuntlov Ed Aamp~~v HOYT1 (Jul 100-117 1962

(6) IJllt 4f711IO nUIgt ~9~ItIPlJnKJi KuL EfIJIlPIJOG~lIvT (6pouAshyIt EdCG1~ A~~(Jv InovnI G~A 167-272shy

(1) n 1J01 middotAelQo oulo cTo p6In lJ~u TOO middot J1tOU ~H1VO~f T(bV TUXUTl T(lJV 1 OyoayoOt Ueuetp~ pai MtOo6oo poa810plshyGI10 0 Tal) v611( Hl Ilnupo) TampV aBATCDv TOXUTTtTOlV T8XVUt Xpovuch 1 JHS-3Bi 1956

(8) PPw~I und C PJ4_Y cR ellstanc( Elperl)lHntll in a Thisnshygulltr Cllauuel_ Proc ASCE vol 85 11)5 MlI) 1)i9

(9) P A9yOPfJldD4 tGener a l sOlution or lhe bydriiulic jurop I s lo pinl[ cJu uuI$_ pOQellding ASCII July 1952

(10) IY 11undllJlIch Hie G rundabl ll$lle ID talsperrcn Olt WasshyserlrtB~~hraft Mopno bull bull Anpl)to 1903

(I i)J AgfJJ iin G Dimtev F PiltoJOTJ rGldftlvlkt MosCOW lcn in~rad u~5~

(12)J HiJtlcl bull bull Ole ExperLme ntnl RydrauUk Frelbtrg 1955 (ll) P ArgyrtljXJlItlJ watcr Su rra~c Profile IrTCglllar n8lural

Strcnllo [roc ASCF July 1)6 gY4 p 2849 (14) p ArgyogttlulJJ ~7beoretkal lind Ibperimtntl1l Analysis

of t h~ HydIlI1IiC JU)ll) Ie a Parobolic 11uwe bullbull Commuu lcashytlon VIr JAIIN CQI Bfel ~ Iishoo 1951

CI P ArgyrojXJulfAS 7ht Hydraulic Jump aDd tlw lllhct Of Turhlllcllcc 01 Hydraulic Siructurt- ContributioD to tbe Releulcb of tbe lhenoJncrlon Proc JAHR tg6

(16) P Agytlpulo~ ltfodero Co osid eraLlool nod New lQto 00 River Bcd RougbntHs Proc JAHR bull USA 1967

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450

At 6peOy(Ovlt~ 5ullpuyac m=J K - T t = 1 Kai Z= I iltpound p= 2+ t+I = 2+ 1+- I=4 0 ~ 101 39 AojlpaVEl tnv wpql1lv

1 1 [I J J(5 - L-h g-(Y - yo) -4qt (Y14 _yo) (40)

6nou q~ ~ fl ava j1Ovaoa dUTOOt Trapaft middot

aui TPlYCIlYKUC oltilpo)~ eilplcncojJSv

r 1 TIIJI [1 1 cpo ~ = L- h i(y -yo) - 6Qpound~ TCY-Yo)(4J)

throu ql ~ t aootcplIcTj )Olvla lCOPOrpf Wi dyroyoO ~x 2b) TUoC Sui TnV 1Wp[ttTClJCW TOO rrnpapo)u(oO oyooyoO (EX_ k) ElvUl

J J [I 64Po - = [i1 gCY-Yo) - 135Q ( y-Yo) (42)

amp100 Po 1l fl ~nTtapUjLe1poc t1c Un OflW (11 2c) 1topaPOA1shyK1 BIQtOlrile

VI ETfaAI8ruci Tft~ Eamp615ou iSla Tft4i XP~QE(U nElpaJlClTIKQv alEaoJlfvwv

np~ tTtaAiIgeoow Tt~ KpmEaVO~EVTllt J1e06lioo poundytVEtO xpflau 1UlpaWtlKWV oBlio~VO)v ava)byou np6t ntv 1tupoOshyoav 9coopTtucytv 1ll90Sov 1tEPUrTOOOEIDlt TIpuYJ1Qn KaTU tCu ono TOO ypl1qgtOVt~ )fVOl-tvw Av t~ EpyaltTtllP(ql middotYOpoUAlKi)C toO E MfloAuTpoundxveioo rtpdtrw ~Pl (olJOs tv -EllUlh AmCtTlJlovu(W tpellVW t~8T911 10 UOpaUAIshyKOY UAIla el4i 1tnpaaOAIKOY aymyov (14) t6 hot 1954 -cut tv cruVEXCiQ Eti tPlYQ)Vu(6v TOlOutOV (IS) (6 ttOC 1955 EytvCtO xPilCI~ ilSpaOAllcoO lil4wt~ tIC Bopomiddot ~paYJllnoe Ikrl9EIIJ tYKUtclCtCwECLM 11IIKOOe 735 Jl UHW ll~ampv t1tlVO~Ceroc ltH fy l6ycp MpaUAlli) tYKatucrtcwu tlJqgtatve1UI s~ tic sh6yw 1 Kai II

0 mlVtuAmt~ 1paxvTT)to~ lCam Manning 1ItO 11=0015 Av) ~ ~~ t1j ~~apalttpo napa~ol~ Po =0039

Bacrtl t(lIv 1tElpa+tQttKUlY OTOlXeloov tie tv l6ycp apyQaTIlshypLaKi1e 1tpoO1togeioC bf-teux61 1t1ftpr]C btatft98UCLC Kai 8loO1aUpcooU Jll ta irYtLcrtOlxa 8eSolltvo Sui tfle XPllm-1J01tOlitOetOI tflc tlJ1telplKfly axtOErot toD Manning 6 At EOP IC pyai tIJlal tIC tfle Ytac tOOT1lC BECOpac amptval 6)iyoY Katompa~ t1Jv tOlOlrUOY ~K tfc crxe~ 6 T6 JOt ouroY 9EoopeiJat eGMryov Kui )OYlK6y tqJ6croy KatQ t1)V aUvmftv tfc ev A6YQgt CxsOampooc Stv tA1cp6TJ bn ~VlV to crUVOAOY tav t1tTJpe~6Vtooy tfty 68pcrO)lK~IV dvr(OtuOlv napuy6vt(ov

(I) Diflt middotA)UQ-nploll 6eOlptlmdl Kal middot ~llPIIOGA~ 1l (6poU)1shy RK6001 A5~(Jv A1DVlfl al) )51-J~9 1962

(2) P ArgyroJxtJlo~ tClIlclI1 de JgtEcollhmcnl en condulte SOli p-on OU l Ill-facc Ilb-c Ed~Dunod Po-is 1957

(1) Mso~li HfJm-Na lt111uld esitanL1 in at now o f 111gb ll rllll ilc numbcr l~rllc of Ibe fl1d J ~l)nn Nallonal Conshyfe rtcc 011 AppUfds Mec buDlcs 1952

(4J v_ T~ Chow OpeICh nnoel HydrluCSo Me Cra w-Ifill Compn ) Ille G~) 200-201 199

(S) n~ Af7I (I(IJV)ul w H 1I1XIlV1Ki) Tellll peuntlov Ed Aamp~~v HOYT1 (Jul 100-117 1962

(6) IJllt 4f711IO nUIgt ~9~ItIPlJnKJi KuL EfIJIlPIJOG~lIvT (6pouAshyIt EdCG1~ A~~(Jv InovnI G~A 167-272shy

(1) n 1J01 middotAelQo oulo cTo p6In lJ~u TOO middot J1tOU ~H1VO~f T(bV TUXUTl T(lJV 1 OyoayoOt Ueuetp~ pai MtOo6oo poa810plshyGI10 0 Tal) v611( Hl Ilnupo) TampV aBATCDv TOXUTTtTOlV T8XVUt Xpovuch 1 JHS-3Bi 1956

(8) PPw~I und C PJ4_Y cR ellstanc( Elperl)lHntll in a Thisnshygulltr Cllauuel_ Proc ASCE vol 85 11)5 MlI) 1)i9

(9) P A9yOPfJldD4 tGener a l sOlution or lhe bydriiulic jurop I s lo pinl[ cJu uuI$_ pOQellding ASCII July 1952

(10) IY 11undllJlIch Hie G rundabl ll$lle ID talsperrcn Olt WasshyserlrtB~~hraft Mopno bull bull Anpl)to 1903

(I i)J AgfJJ iin G Dimtev F PiltoJOTJ rGldftlvlkt MosCOW lcn in~rad u~5~

(12)J HiJtlcl bull bull Ole ExperLme ntnl RydrauUk Frelbtrg 1955 (ll) P ArgyrtljXJlItlJ watcr Su rra~c Profile IrTCglllar n8lural

Strcnllo [roc ASCF July 1)6 gY4 p 2849 (14) p ArgyogttlulJJ ~7beoretkal lind Ibperimtntl1l Analysis

of t h~ HydIlI1IiC JU)ll) Ie a Parobolic 11uwe bullbull Commuu lcashytlon VIr JAIIN CQI Bfel ~ Iishoo 1951

CI P ArgyrojXJulfAS 7ht Hydraulic Jump aDd tlw lllhct Of Turhlllcllcc 01 Hydraulic Siructurt- ContributioD to tbe Releulcb of tbe lhenoJncrlon Proc JAHR tg6

(16) P Agytlpulo~ ltfodero Co osid eraLlool nod New lQto 00 River Bcd RougbntHs Proc JAHR bull USA 1967

TUOIID IfDAITIKWV pnXllVIKWV