Yoshino, Fumio

Tottori University Research Result Repository

18A Proposition of a Universal Coefficient of Growthof a Turbulent,Curved Wall一JetbyFunlio YosHINO(ReceiVed September 30,1971)Geometrical consideration of a sPace Curve and fluidkinematical consideratiOnof a curved wall jet with a similar velocity profile led to a cOnclusion that acoefficient of growth of a iet iS a function of a ratio of width of a jet to a ra―dius of curvature of trajectory of a iet, The effect of the past history oF a jetseems to be negligible.  A felT experimental d2ta on different geometries confirmthe theory justitiable. Thus we propose a universal coefficient of growth of acurved wall jet.1. IntroductiOnln case of calculating the characteristics of flo、v with tangential injection of airover a curved wall, it is in general necessary to predict the growth of a iet a10ngthe wall.The prediction is made using its own experi14ental data which are deter■linedfor specific geometries such as a circular cylinder, logarith■lic spiral,stra ght walland so on. The data of a circular cylinder for instance is, however, applicableonly to the flo、v over the 、vall with a sixnilar geometry. This inconvenience seemsto cbme mainly from facts that firstly the coordinate systems to express the floヽvare different frOm each other, just convenient to a specific problenl, secondly it iSnot obviously known which one among many variables is essentially of vital impo‐rtance and thirdly it is difficult to experiinentally obtain a jet with an arbitrarytraieCtOry.However the first two problems are readily solved when the concept of naturalequation in the differential geometry is applied to the prOblems.The first problenl iS replaced、vith the on  of finding a geometrical relation oftrajectory of a tet With the wall. Considering the second problem geometrically,we deduced a concept of a universal coefficient of grO、th of a wall ie unde  theassumptions of a small ratio of a、vidth of iet to a radius of curvature of the walland a siinilar velocity profile.This coefficient is a function of a ratiO of aure of a representative trajectory of iet,)牲/ρ,、vidth of jet to a radius of curvat‐、vhich was confirmed true by a fe、vRoports of the Faculty oだEngineering, Tottori University, Vol.2 No.2         19experiinental data. The lnixing length theory in a centrifugal field by Stratford,JaWOr and Golesworthy seems to give an insight into the quantitative representationof that coefficient, It is desired to confirm our theory by a systematic experilnentand find out a more reliable quantitative relation of the coefficient to the ratioノ指チ。・2. TheoryThe fundamental theore■l of the differential geometry requires only a curvature,torsion and arc length of a space curve to express itself, in which we need onlytwo variables, i.ρ。, curvature and arc length、vhen the space is t、vo d inensional.Then the natural equation Of a curve is given as follows,ρ=ρ(∫)                     (1)where P and∫are a radius of curvature and arc length respectively.By definition,where σ is an angle between twO tangents at two points apartThus we may use σ instead of ρ as an independent variable.(2)by 'F on a curve,Therefore the total differential of the、vidth of a jetノ″ iSV｀rit en as fo119ws,2??〓???呼=隼み+隼みwllere d is a traiectOry ofノ歩・From equations(2)and(3)ヽVe haVe呼=(+鶏無+等)あ鏡●ら甲=■隼 キ準equation(4)iS reduced toウ生 =Cl(∫,σ,ノ4)ぬThe first term in the parentheses is an increment of width of a jetノ多per unit Va_riation Of tangential direction multiplied by a curvaⅢure, the second term being anincrement of r/V per unit arc length.2Denoting the quantity in the parentheses by Cl(∫,σ,ノ争)Such that(4a)(4b)where Cl(∫,σ,ノ多)iS a fmction of σ,∫andノ生or a radius of curvature p,arc len‐2gth∫andノ多only.The Q(∫,σ,ノ生)in the form of equation(4a)seemS tO be con‐2A Proposition of a Universal Coefficient of Crolvth of a Turbulent, Curved wa11‐JetVenient tO experimentally find Out the separate effect of each variable Onノ生・2If the quantity Cl(∫, cr,)生)iS, hoWever, to be handled as a coefficient in pract‐ical sense Jttke a practical ttlculation of grOwth Of a iet,then the trajectory ofノ生,2i.e., ∫in Cl (∫, σ,ノ聖生) Vヽill be a draw‐back of this theory as the relation of∫to awall surface ′ is in 2general not simply expressed.So we use a variable ρ mstead of σ,then we have働(∫,P,j光竺1)fOr Q(∫,σ,多)・If the velocity prOfile is si■lilar, the ratio between characteristic linear diinens‐ions must be constant so that∫ェ/ρl=S2/P2,ノ生 /P]=ノ生 /P2and SO on when two21      22traiectories∫l and∫2 and tWo ietsノ生 and 7孝2 are geOmetrically similar.ThereforeQ(d/P,ノ争力)muSt be cOnstant fOFと。nstant d/P and∞nst t 7p/P.∫力 h銑 (■/P,2ノ空u4二箋江打灘こ1lC猪盈オ孔露薫亀経ettth稲彗をち監子打塩an ttciS that the history of a wall jet is calculable ifノ争/P iS Small.Furthermore,ifノ多 /P′Sin two different ietS are equal each other at a point of interest, 91(d/P,ノ聖_/ρ)'Stoo seena equal each other at the point no matter what kind Of path two flowscame up to that point thrOugh. This is seen in Fig。 1み(S)FIg.1There are shown two wall jets,SW along a straight wan to pOint A and CWover a curved、vall in the figure. Both ietS have、vans w th equal curvature after4報♂it'逮:;盈丈晶1:塩e]話イ免(Ъ)そat pOht A,then粉(∫)亀at pdnt Bfar as velocity profiles at point A aresiinilar in both flows, ignoring a discontinuous change of curvature at point A inthe jet SW. This is due to a si=ndar prOfile assumption.The above diScussion means that銑(∫/ρ,ノ多/P)iS a local quantity and a functiony号(S。)NOZZle cRoPortS Of the Faculty of Engineering, TOttori University, VoI.2 No.2         21of a variableノ生/P Only。Thus、ve have instead of equation (4b)の多=Q(ノ多/p)ぬ                      (4c)whereノ生/ρ in the parentheses means tt is a function ofノ多/p.StratfBrd et al(1)showed the ratio,π,of mi?ng length in a iet Of a centrifugalfield to that in a straight、vall iet in still air vas approxil■ately related to う/psuch that万=___1(5)1-÷十where bis a width Of half jet.Ifノ多 is  diStance,perpendicular to the wall, me‐asured from the 、vall to the point at which the velocity is half the maxiinuni vel‐ocity of jet,then we have b=2夕多andL=`6)???????22   A ProPoSitiOn oF a Universal Ccefficient of Growth of a Turbulent, C,rved Wa11‐Jetlt is known from the theory of iet that a 、vidth of iet is prOpOrtinal to the mixi‐ng length.Therefore吼(辛)=C万=1_+芋     mwhere C is a constant of a straight iet.Now the remaining problem is to relate a curve d to a certain fixed curve, ofcourse in this case, to a wall J.All the symbols necessary are shown in Fig.2, in which(″,θ)and (資,0)arecylindrical coordinates of the  curves ∫ andど respectively.  The origin of themmust be appropriately fixed somewhere on the body, for instance to a center of acircle in case of a circular cylinder. β is an angle of the radius of curvature mea‐sured from χ axis, The symbols 、vith the suffixes∫and ′mean to be those of thecurves s and′respectively. Since、ve havお 言 (r2+″′2)J/2′θ               (8)ρS~~下宴絡響年下          働a=o―肋‐+        硼″σο∫θ = ■ω∫o+ノ生ιο'βr                    O〕2″∫ゲ2θ=盈ど20+ノ歩訪 β′                 l121手=(寺ソ(半)    ⑬、There the pril■s on r and n indicate the derivatives of them 、vith respect t  θand O respectively,we can expressあin equation(4c)by■,O andノ子.We will not make an involved calculation for a general case, but、vill deal 、viththe cases of a logarithH?c spiral, circular cylinder and a straight、vall. Comparingthe resuits 、v h the data by the other researchers, we、vill show our theory justi―fiable.3. Logarithmic SPiral WallA logarith■l c spiral wall is described by the following equation,£ =諺′θ                                  uawhere々and,are constants。Using the relations(8)and(9)a10ng with l141,We getIp/=考(1+,2)2/θ ????RoPortS Of the Faculty Of Engineering, Tottori University, Vol.2 No.2          23′=+。キ″清/θand+=+=半 =F=ω務筋    側for a given r where F is a constant used by Kamemoto in his report。(3)。温挙盈.1鳥)l:域itiとギ翠:手lf帯彗atiott proportional to′,Щノ争/ρ)? the form鏡(埼)=蒋        硼、vhere C′is a proportiOnal conStant。We empirically know thatノ争/P iS always non‐zero and is a constant from theorigin of the iet on fOr a given F. Therefore C′, the prOportional constant ofフ聖 ,is always a constant in a constant centrifugal field and has never been equal to Cfroni the beginning of a iet for a given F,1.e.,C′:>C。Equation(4c)iS reWritten below by using equation l181,ど吟 =1_÷『 十ぬ         側・From equations llol, llll and l121,ヽVe have″=■2+ノ∵+2■特♂ο,α汚        卿whereα力=肋」キ =物″1÷        側・Experience for a logarithnlic spiral wall indicates牛=F字=O=ω溶肋      閉for a given r whereッ竺/′ iS equal to C′in equation 1481。Introducing thiS relation and equations l141 and lla into 9111 giVes the fo1lowing eq‐uation for T,1231″=々Q(r)′¢οwhereユC4(r)=(1+2C21/1+'2仰,αを+軽( α2)}7    9い。From Fig.2,we obtainノ∠=″2+■2_2濯ω∫(θ-0)               90,224   A ProPosition of a Universal Coefficient of Gro、vth oF a Turbulent, Curved 再7a11‐Jettherefore♂ο∫(θ -0)= 1+C2R/1+'2 θο∫α々 = constant硫 (K)for given F.Consequently that′θ 室 ″O gives/=季=+So cquation 1231 yieldS″′=考α亀 (F)″0EhHlinating r and″′f om equation(8), We haVe1261Inasmuch aswe haveゐ =た1/1+,2Q(F)′′0′o"= 1々/1+,2♂,0″oお =C4(r)"lpl.?‐???〔???13111??1321.T4en equation t9 iS Witten as follows,ウ件"儀 (r)1-÷ C′F4. Circular C,1二nder and Striight wa趾。Equation (4c)and(8)yield隼=鏡(幸)(″+中庫IntrOducing the relations for a circular cylinderθ″=ァぎ+ノ多   }into equation 1331, We haVe????Picking values of F andノ多/J Off the figure 7 of the reference(1),We can makethe Table I.Table I C=0.128 was used to make Table I.ノ争〃iS Va‐lue from the referttce偲),特材being by equation1321.0.500.751.00。1550.200.29Q④l力7'0.1550.1870.2761341Roports of the Faculty of Engineering, Tottori University, Vol,2 No。2        25筈=Q(率)R(1+幸)|い去終緋〒|;芯酢貴″O>ウ争and aSSumingノ作/R≪1,We Can ignore the second term in the bracesof the above equation and we can approxilnateノ多/ρs t°」′些生/A・ Then we hav箸=Q(拳)・(1+幸)where Ci(ノ多/R)iS giVen by equation(7),that iS,CI(学)=1-+幸t働(拳)=下与丁Finany we get肇=五=罰£(1+争)   倒.Newman(4)empirically obtained a relation forノ与and O aS below,ま = C・ ( 1 + 考〒≒1生 )                                     14111.Differentiating the above equation with respect to O, 、ve get a ve y silnilar equa―tion to 1391,牛 三百寺 y貴(1+準々 )1361)多°n a Circular cylinder is, as the first apprOxilnation, proportional to the arclength of the surface■0。In this case, different frokn the case of a logarith■lic spiral wan,ノ生 IR continu…ously changes its value starting from zero at the origin of a jet. Therefore theproportional constant as the first apprOximation is taken equal to that on a strai‐ght wall iet SO thatノ多■ CR O・Equation 1371 becomeswhere C is a constant of growth of a jet and々a constant of order of unity.Newman gave to theln,'巨!|:1  }141)1421.A ProposJとion of a Universal Coeだficient of Crowth of a Turbulent, Curved WalI・JetThe relation 1401 tOgether with the values of C andたwere later confirmed by Fur‐uya.(5)In case of a straight、7hll iet,ρsis infinity and the first terttl on the right handSide of equatiOn(4a)vaniShes and equation(7)rnerely gives a constant to Cュ(ノ生/r)s),2仇(牛)=QThus we get与 弁=C5. Discussion and Concluding RemarksComparingゥ多/"Withノ多/′ in Table I,we see that our theory seems to under‐estimate the value of乳(ノ聖生/P)When r is large,This takes place also in case2of a circular cylinder.Integrating equation 1391 With respect to O, 、ve get買~「―+θOF~10n the other hand, rearranging equation 1401 yieldS辛 =下棉 下 キ1431.Letting C=0.1l andた = 1.5,we computedノ生/翼 frOm bOth equaitons and showthenl in Table II.Table II Valuc of)」 Д2 Table II  indicates  thatbased on the■lixing length theory by Stratford et al, the model of 、vhich seemsto be over siinplified. The discrepancy of the value of C′∫froni that of a straight、vall iet may originate in the mechanism Of turbulence itself or in thc way of anexpressiOn Ofじ1(ノ生/ρ), eqn(7)Or in a difference of Reynolds number.2Though we ignored an effect on Cl(∫/P,ノ4/ρ)Of he past history of a iet, thiSmust have some influence on Cl(J/P,ノ坐/ρ)b cause turbulence dOes not instantlydecay, nOr is developed instantly.Nevertheless, we believe that the data discussed in the sections 3 and 4 confirmsour theory justifiable,that is,ノ多/p playS a role Of vital importance in Cl(ノ多/ρ)RoPortS Of the Faculty oE Engineering, Tottori University, VoI.2 No.2          27and other apparent factors in many an experiinental equation come merely frOmdiffeFenCe Of coordinate systems between each experiinent, Stratford et al showedthat a■?xing length in a centrifugal field is a function ofノ多/P only but nevershowed what else was geometrically ignored. On the other hand, we showed thatthere are two variablesノ多/P and d/P and the former is much more important thanthe latter also in case、vhere we consider the accumulated effect of a centrifugalfield on the growth of a jet by applying Stratford et al アs theory a d a fe、v exis―ting experil■ental data,  The advantage of our theory is that Ⅲve can predict ho、va tet grOMrs along its own traiectOry.The theory assumes the smallノ多/P and a similar velocity prOfile of a turbulentiet. Thus we propose the universal coefficient of growth of a curved、vall―jet,In order to find out the unforseen effect on a constant C and the more rehableform Of Gl(ノ多/p),it iS desired to carry out a systematic experiment on this su…biect.The author is indebted to Proffessor Furuya of NagOya UniverSity for his encOu…ragement and his showing us the data of the reference(5).Referencesl) StratFord,B,S,,Jawor,Z.M.and Goleswonhy,G.T,,The mixing with ambient air Of acold air strea■l in a centritugal tield, June, 1962, A. Ri C, C, P. 6872) Sasa ti,S,,The differential geometry,VoI.16 of the lecture on the fund2mental of mathem―atics, June, 1965, Kyoritsu Shuppan3) Kamemoto,K.,Research of a turbulent wall iet, Prepint of Japan Soc. Mech. Engrs, (inJapanese),No。 7132,Mar.,1971,4) Lachmann,G,V., 1961(ed。), BOundary Layer and Flow Control, pp 232-264, Oxtord, Perga―mon Press.5) Furuya,Y.,Two ditnensional jet alOng a circular cylnder, Preprint of Japan Soc. Mech.Engrs.(in Japanese),No.95,Oct,,1963

A Proposition of a Universal Coefficient of Growth of a Turbulent, Curved Wall-Jet

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A Proposition of a Universal Coefficient of Growth of a Turbulent, Curved Wall-Jet

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