Results of calculations and experiments on the cold co-axial flow presented in this paper are summalized as follows ; (1) A theoretical expression method for co-axial flow field of two dimension is investigated to estimate more
exactly the flow profile and the velocity gradient. (2) Measurements of mixing length were carried out for the confined co-axial jet flow. On the basis of measured data, inquiry is made for the propriety of the assumption proposed in the previous paper (1) that the mixing length may be expressed as a function of the minimum distance to the nearby wall. (3) Experimental inquiry was also done on the corelation between Lagrangian length scale and the mixing length

Shimamoto, Yuzuru

Tanaka, Yutaka

Okayama University Scientific Achievement Repository

Memoirs of the School of Engineering Okayama University Vol. 10, No~ 2, January 1976An Arrangement of Flow Velocities and Turbulence Profilesof Co-Axial Cold JetYuzuru SHIMAMOTO * and Yutaka TANAKA *(Received December 20, 1975)SynopsisResults of calculations and experiments on the coldco-axial flow presented in this paper are summalized asfollows ;(1) A theoretical expression method for co-axial flowfield of two dimension is investigated to estimate moreexactly the flow profile and the velocity gradient.(2) Measurements of mixing length were carried out forthe confined co-axial jet flow. On the basis of measureddata, inquiry is made for the propriety of the assump-tion proposed in Ghe previous paper (1) that the mixinglength may be expressed as a function of the minimumdistance to the nearby wall.(3) Experimental inquiry was also done on the co-relation between Lagrangian length scale and the mixinglength.1. Introduction.For the last forty years, new theories of turbulence which are fullof variety in content, form and applicability have been found in litera-ture; most of them are successful in explaining turbulence phenomena.However,recently the analytical treatment of turbulence properties suchas velocity and concentration fluctuations are being re-examined on thepurpose of application to the simulation of turbulent diffusion flame .The purpose of these re-examination is to derive the more skillfulturb-lence model on the standpoints of applicability, accuracy, economy ofcomputer time and simplicity.The mathematical modelsof turbulence are classified into three largegroups, that is , algebraic turbulence-viscosity models, differentialturbulence-viscosity models and models employing differential expressionsfor stresses of turbulences. The first two groups employ Boussinesq's orPrandtl's suggestion that the stress-strain relation for turbulent flowscould be represented in the same form as that for a Newtonian fluid inlaminar motion. The third group of model includes those that dispencewith the notion of effective turbulence properties and, instead, provide* Department of Mechanical Engineering1516 Yuzuru SHIMAMOTO and Yutaka TANAKAdifferential transport equations for the turbulent fluxes themselves.Itis difficult to prefer the best one from the above models, because theavailability of models vary with subjects of study. In order to solvethe set of differential equations govering the flow fields, even if thebest model is used for analysis, some gloval assumptions must be givenfor the factors, such as mixing length, turbulence length scale, turb-lence strength and non-homogeneity.In our last two papers on the simulation method of diffusion flameswith co-axial swirl flow (1,2), the algebraic model in which the time-mean deformation tensor is connected with the viscosity are used fromthe standpoints of both the reduction of computation time and the simp-licity. In this model the mixing length are assumed as a function 'ofthe minimum distance to solid walls. The first purpose of this reportis to examine the propriety of this assumption through the measurementof the mixing length variations in the regions where the flow fieldsvary remarkably. Since the turbulence measurement is not easy for aco-axial flow with swirl and combustion, the experiment has been carri-ed out for iso-thermal co-axial flows without swirl.The second purpose is to express the steady field obtained forconfined flow as functions of axial and radial distances. This expr-ession is necessary to treat theoretically the profile of the flow,inwhich the deformation tensors can not be simply expressed as a radialgradient of axial velocity.The third purpose is to study whether Lagrangian scale of turbulencemay be estimated from the mixing length or not. Although both moment ofturbulence and time-mean velocity profile must be measured to obtainthe mixing length, only the moment is required for the determination ofthe Lagrangian scale of turbulence. If this estimation is applicable,the troublesome task for measuring the mixing length may be largelyreduced. However the measured data revealed that the distribution tend-ency of Lagrangian scale is different from that of mixing length.2. Nomenclature.turbulence componenttime - mean componentdensitystressdirection for x and r respec-tivelyfor Lagrangian scale of turb-lenceprimary flowsecondary flowLpscentercoefficients of Fourier ptransforms Tflow distance required toreach ultimate velocity pro- Subscriptsfile ( entrance length)mixing length or length scale 1, 2of turbulencepipe radiuscoefficients of correlationat the time difference ytimeflow velocityminimum distance to solid wall Superscriptsaxial distance from nozzleexitradial distance fromline of jetrt, Tu, UzxRR('[)L3. Apparatus and procedures.The basic structure of test apparatus consists of two co-axial circ-lar pipes. The outer side pipe has inner diameter (=2R) of 98 mm, theinner side one has inner diameter of 16 mm and thickness of 3 mm atthe outlet. Hereafter, the properties of inner side pipe ( nozzle) andannular parts between two pipes are expressed as primary and secondaryAn Arrangement of Co-Axial Cold Jet 17Fig. 1 . Static pressure tube.~~ones, respectively.For the measurement of time - t . Imean velocity, a set of dynamic and _ _ ~j ..static pressure tubes were used. -=- --- -- - - _n~~" "1The former outer and inner dia- .meters are 2 mrn and 3 mrn respect i- ....1'1vely. The static pressure tube is 050.5illustrated in Fig. 1. The pressuredifference of these tubes was meas-ured with a X-type hot wire probe,and a set of anemometers and linealizers. The outputs of these instru-ments are recorded by a data recorder, then the recorded tape is pro-cessed by a high speed data analizer of our School of Engineering.4. Procedures for analysis.4-1. Fully developed velocity profile.In any case of entrance profile, the flow of co-axial jet in a con-fined fields are developed, reaching an ultimately developed profile.If the pressure and density changes between the entrance and the fUllydeveloped section are negligible, the fully developed velocity profileis supposedly expressed by the following approximation with modifiedlin - th power law,U1,00U1,00Umax[1-(r/R)2],U [1-(ro/R)2][(R-r)/(R-ro)]1/n.max(1)Denoting the ratio of sectional-mean to maximum velocity by Ru ,Ru = U I U .mean max(2)From Eq.(l) , Ru and radial distance ratio rO/R become functions of theindex n ,Ru 64 n6- 128 n 5 + 112 n 4 - 56 n 3 + 18 n 2 - 5 n + 12( n + 1)(2n + 1)(2n _ 1)4ro/R = II ( 2n - 1 ) .(3)(4)From the data obtained by Nikuradse (3) , the index n is expressedas a function of the Reynolds number Re .For turbulent region of Re ~ 5xl0 3 , n = 4 ( Re/l0 3 )0.13 86 ,for transition region of 5xl03 ~Re > 103 n Re/l0 3for laminar region of 103 > Re > 0 n = 1 (5)Entrance length L is ~xpressed usually as followsfor laminar flowfor turbulent flowLL0.106 Re R60 R (6)The Fourier series of order n 2 of Eq.(l) become,18 Yuzuru SHIMAMOTO and Yutaka TANAKAn 2Ul oo( r ) = - a l !2 + L a. Cos[(j-l)n r!R] ,, . j=l J (7)where coefficients a j are represented bya j = ( 2!R ) f~ Ul,oo(r) cos[(j-l) nr!R] (8 )4-2. Velocity profile.Denoting the difference of the measured axial velocity profiles Ul(x,r) and the fully developed velocity profiles Ul oo(r) by a symbolF(x,r) , then the axial velocity equation of two dlmension can be ob-tained by the double integral procedure of F(x,r) as follows ;+n 2I: (c l ' - 2aJ,) cos[(j-l)1fr / R]}j=l ' J +n 2 n lI: I:j=l i=lc, .l, J-where c ..l,J· cos[(j-l)1fr!R] cos[(i-l)1fx!L] ,L R(2!LR) f O f O F(x,r) cos[(j-l)nr!R] cos[(i-l)nx!L] dr.dx( 9)The homogeneous density approximation gives the next equation ofcontinuity,dUl +dU2+u2 0-dX dr rThe boundary conditions of u2 (x,r) aredU 2 0 at r = 0dru 2 0 at r = R(10)( ll)From the Eqs.(9),(10) and (11) , the radial flow velocity u2(x,r)is obtained asn l 2I: c. l' (i-I) sin[(i-l)nx!L] + (R !Li=2 l,n 1 2I: c .. [(i-l)!(j-l) ] sin[(i-l)nx!L]i=2 l,J• {[ cos[(j-l)nr/R] - 1 ] ! r + [(j-l)n!R]· sin[(j-l)nr!R]} (12 )4-3. Turbulence properties.Turbulence components ui and u2 of velocity are obtained from theAn Arrangement of Co-Axial Cold Jet 19sum and the difference of two AC outputs of X-type hot wires respecti-vely. By numerical computation of these measured analogous data, thestresses TIl' T22 and t 12 are obtained as follows;TIl = - pu'u'1 1T22 pu'u'2 2T12 pu'u' (13)1 2If d represents the deformation velocity tensor, the square of mix-ing length from the expansion of Prandtl's hypothesis is expressed as,(14)Since there is no tangential components of velocity, Eq.(14) can besimplified as,The mlxlng length can be calculated by using Ul ' U2 and T12 obta-ined from Eq.(9), (12) and (13) respectively.Coefficients of self correlation Rl(L) and R2 ( T) are representedby the equationwhere k represents directional index for 1 or 2 and T representstime interval which is long enough with respect to the time of turbu-lence oscillation. Then the Lagrangian scale of turbulence £k L canbe estimated by the next numerical integration , ,5. Results and discussions.5-1. Time-mean velocity profile.There exist two types of flow pattern which are characterized by theratio of the primary nominal flow velocity u l to the secondary nomi-nal flow velocity uls . One is a jet type flowPand other is a "wak~ typeflow.Figure 2 shows the ul - profiles for the velocity groups (8,4),(2"",4) and (0,4). Two figures in the parentheses mean a group of u l . _ andu l as (ul ,u l ). The plotted points in Fig.2 indicate the measBredda~a. The Burv~s obtained by means of Fourier transform coincide withthe curve which pass smoothly through these plotted points. It is knownfrom the figures that extreme values in the ul-profiles are classifiedinto three kinds. The one is a maximum or a mInimum value at the centeraxis [C.A.L] , the second is seen in the secondary flow field [S.F.L],and the third is a extreme minimum value in the mixing region whichlies between the primary and the secondary flow fields [M.R.L].In Figure 3 the extreme values measured and obtained from Fourier20 Yuzuru SHIMAMOTO and Yutaka TANAKAHo E F G(b)UIP=O. U2s=4 m,sCDEFG H( C)A~0,.-.--:----:-----.,.-....,..----.--.,.--,.----:-::0.;A4040EE20x romo305580105130155250500..SymbolABCDEFGHI:5 11II1Fig.2. Flow profiles .JEE 20~40A B...............".-----rUIP=IOnvsUI$=8 • CAL.o S.F. L.AM.R.L.I~ .---.-I--r --UIP=8nvsUIS=8RVS • C.A.L.o SF L.A M.R.L.IoilE5,05o 100 200 300X mm( a)400 50010III'"E5o 100 200 300X mm(b)400 500I• CAL.~ o S.F.L.~ A M.R.L.r---t----r IuIP=8nvslJlS=4v- I---( UIP=Om,sUrs=8/• C.A.L.I o SF~.105o 100 200 300 400 50010III'"E5o 100 200 300 400 500X mm(c)X mm(d)Fig.3. Longitudinal changes of extreme velocities.21An Arrangement of Co-Axial Cold Jet403055•o•o• X= 5 mm20r mmUlp=8,UI.=4 m/S•oo",0",'•o ~If-0.10.01NE......"".o40.o.; X= 5 mm203055Ulp s 8, UI.=4 m/so~ 0~..o v,.I0.10.01f-r mmE...........transform curves are shown for the nominal velocity groups of (10,8) ,(8,8) , (8,4) and (0,3). For 00,8) and (8,8), it is known that thevelocity at centerline (C.A.L) increases initially and then decreasesgradually with increase in axial distance. Thereafter it is mixed withM.R.L. and reaches gently to S.F.L. As is illustrated in these figures,there apparently exist some regions where the velocity at C.A.L. be-comes smallest in the range of about x=lOO rnrn to 200 rnrn. The extremevalues for (8,4) show more complex feature. S.F.L. is begin to composewith M.R.L. at the nozzle exit, and then they disappear in the regionsfrom x = 200 rnrn to x = 300 rnrn. For the case of no primary flow ( 0.8 ),C.A.L. approximately coincide with M.R.L. and the flow profile becomestypical wake type. Forthe reason of these comp-lex feature, the velocityprofiles in the vicinityof the nozzle exit cannot be simply expressedby a half value width ofthe jet and the differencebetween C.A.L. and S.F.L.This mentions that thevelocity profile withconfined secondary flowhas features essentiallydifferent from the velo-city profile without thesecondary flow such asfree jet. The reasons forthe use of two dimension-al expression by Fouriertransformation stand main-lyon these experimentalfacts.r mm r mm(c) (d)Fig.4. Turbulence stresses Tll,T22 and T12 .• x- 5 mmo 3055Ulp=O. UI.=4 m/S(b)•0.01f-o.oOIO'--'~o?r--'-----'--'-----'20 40NE.........o40N~ 0.1f-o v.t 20NE......""0.0.1,...• x- 5 mm3055U'p=8,u,.=4 m/S(a).'5-2. Turbulence properti-es.Figure 4-(a), (b), (c)and (d) indicate the tur~bulence stresses Tll' T22and T12 for (8,4) and T12for (0,4) respectively.These value become max-imum at the maximum vel-ocity gradient in the mix-ing region , as was shownby Tani and Kobashi (4).They take high value inthe regions near outersolid wall, however, theyexhibit contrast tendencyat the nearby regions ofprimary nozzle port. While,the longitudinal changeof T in the secondaryfield is quite small.Figure 5 indicates themixing length which wascalculated from the meas-ured data of the shearstress T12and the time22 Yuzuru SHIMAMOTO and Yutaka TANAKAIi 0 AIi A~~ 0• ~oA 0Fig.5. Mixing length asa function of radialdistance r .... "'""Z mm20 4010..Fig.G. fuixing length asa function of z .EE~X. 5nm3055•UIP= 8 mlSoUls=4m/IAA A51-~~ A 0 • •• A A4 ,• tIOO • • aA0• • •~oo 00•••li· o • • \.-I I0 VA 20 40r mm10-mean velocitiesUl and u2' ~ takeslarge values inthe vicinity ofcenter line and inthe region of sec-ondary flow field.As was describedin the introduction,this tendency showsthat the mixing lengthbecomes largerwith decrease inthe distance fromthe nearby solidwall. In figure 6,the relation betw-een the minimumdistamce to theneaby solid wall zand the mixinglength ~ are re-illustrated formany groups of ulp1.0.1.0U,p= 2 ,u,s= 4 m/sX=5 ,r=O mm r=2 mm0.5 "-~·:..Rr('Z')0.5 ......o "001\''-''......... 0.02,...... '. . l: ( •• e.),' ..."....".. .....~:.---'"'"'·.l( a) (b)1.0 1.0EII: r =6 mmEII::r =14 mma5 0.5.. 7: (UcJ~:\, ,.. f\('l:). .. . : .........\ / : /,...~~...( •...Ol---...-~:----:::-'-=+=-:~.....,..._....:....-(c)Fig;7. Coefficients of self correlation Rl and R2 for thevelocity group of (2,4),An Arrangement of Co-Axial Cold Jet 23and Uls.. Solid curve inthe figure indicatethe following exp-erimental equationobtained byNikuradse;30 ..."~~20 - 0• X- 5nmo 30" 5 5ee...~ "10-• x- 5nvno 30" 55UIP= 8 m/SU,s=4I40r mmo••o 0t. 0 o·o "" t "."• 0 0 0, 0rllA 20o5 - "oo•..o." 0 o.8 0 ." ,,, 40o 00 I 0 .0 Io ......-vnrl'//A--:::2."O----:4.."o-lr mm0"•••10 ...(a) (b)Fig.8. Lagrangian lengths £1 Land £2 L asa function of radial distance r:In this equation,the minimum distancez is replaced by theradial distance r,since this equationis obtained for thefUlly developedvelocity profile ina straight pipe.Although the exp-erimental data are considerably dispersive, if we consider that the dis-persion of turbulence shear stress TIZ indicated in Fig.4 is littleenough, the origin of this dispersion seems to be mainly attributableto the estimation inaccuracies of the radial gradient of the time-meanvelocity ul . This dispersion is considerable at the region near zerovelocity gradient located in the secondary field. The secondary reasonof dispersion spreading over the whole regions is supposed to dependon the non-homogeneity of the turbulence exchange and generation.Excluding the non-homogeneity, as is usually practiced when the mixinglength model being applied, the following conclusion can be drawn ;it is seemed that the relationship between the mixing length and theminimum distance to the wall in the complex flow field near the nozzleport is identical to those for the fully developed velocity profile inthe straight pipe.Figure 7 shows the changes of the coefficients of self correlationRl and·R Z to the time interval of 1I. These figures are illustratedfor four places of r = 0 rom, 2 rom, 6 rom and 14 rom. The differences inthe value of Rl and R2 ranging from T = 0 sec to 0.01 sec for firsttwo positions of r = 0 rom and 2 mm indicate the aforementioned non-homogeneity of turbulence. On the contrary to this state, the smalldifferences of Rl and R2 for the last two positions show that theconditions of homogeneity are spreading over the secondary flow field.The Lagrangian scales of turbulence £1 L or £2 L are equal to theproduct of !uIU2 and the area bounded by the coordinates and thecurves of Rl or R2. The £1 Land £2 L may be considered as thescales for axial and radial directions respectively.In Fig.8 these scales are plotted as a function of radial position.Although each scales take large values in the vicinity of center lineand the outside wall, they become small in the middle region. It wliibe noticed that the experimental data for the mixing length ( Fig. 5.)and for the Lagrangian length scale ( Fig. 8. ) exhibit quite diff-erent behaviour, those for £ displaying comparatively smaller valuesin the secondary flow field than those for £l,L or £2,L'£/R = 0.14 - 0.08(r/R)2 0.06(r/R)~(18)246. Concluding remarks.Yuzuru SHIMAMOTO and Yutaka TANAKAvelocityeasilyBased on the results presented here, the following conclusions canbe drawn for conditions.- studied;(1) Two dimensional arrangement was done for the time-meanprofiles. By this arrangement it become possible to estimatetwo dimensional changes of mixing length.(2) The measured results of mixing length reinforce the assumptionthat the mixing length may be expressed as a function of the minimumdistance to the nearby solid wall.(3) Studies were made on the propriety of using the Lagrangian scalesof turbulence instead of the mixing length. However, on the contraryto our expectation it is found out from the comparison of measured dis-tributions that the mixing length can not be reasonably estimated fromLagrangian length scales.References.(1) Y. Tanaka and Y. Shimamoto , Preprint of Japanese Il-th Symposiumon Combustion, 11(1973), 213.(2) Y. Tanaka an~Y. Shimamoto , Preprint of Japanese 12-th Symposiumon Combustion, 12(1974), 89.(3) J. Nikuradse--,VDI Forschungsheft , 356(1932) , also appeared inthe book; S. Kitsuka, A. Hirata and A. Murakami, Heat and Mass Trans-fer for Chemical Engineers, (1973), 151 .(4) I. Tani and Y. Kobashi , Proc. 1st Japan Nat. Congr. Appl. Mech.,(1951), 223 .

An Arrangement of Flow Velocities and Turbulence Profiles of Co-Axial Cold Jet

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