Hayashi, Tsutomu

Tabe, Koshi

Waka, Ryoji

Yoshino, Fumio

Tottori University Research Result Repository

The Measurement of Reynolds Stresseswith a HOt―Wire AnemOmeterbyRyoji VAKA*,Fumio YosHINO*,Tsutomu HAYAsHIキ,and Koshi TABE**(Recdved May 31,1978)This Paper deals with the measurement of the mean velocity and the Re,喧nolds stresses in a turbulent flow by using a linearized constant temperaturehot‐wire anemometer.  A system of analytical response equations was derived,in this study, to determine all the compOnents of mean velocity and the Rey―nolds stresses in a three_dimensional fユo17 field by using a single slanted wirein combination with a straight wire. Here the direction of the flow itself isinitially unknown. This method of measurement with twO types of hot‐wiresis tested experimentally in the initial Tegion of a round free iet・ The r sultsobtained by this methOd are compared with experimental results in the noise‐producing FegiOn OF a circular iet by P`Bradshaw and thOse in the flow‐establishment regions Of a round iet by S.sami,and found satisfactory exceptthe radial and tangential components Of the mean velocity.l  lntroductiOnDespite the li:nitatiOn, the hot‐wire anemometer is still an indispensable instru―ment for investigating instantaneously unsteady flow such as turbulence, Themeasurements Of mean velocity and the Reynolds stresses have been carried out bymany wOrkers, c.g。, IIinze,1)Champagne and Sleicher,2)Kovasznay and Fujitaa).AlthOugh the hot‐wire echnique may now appearto have been、vell establis ed, itis noticeable that theェmajori y of published works on this subject is concerned onlywith one or t、vo―dilnensional flow field and has sOme restrictive assumptions.Hence this paper describes the rnethod of the measurement of the three componentsof mean velocity and the six components of Reynolds stresses in a three‐dimensio aturbulent flo、v ield of unknO、7n direction、vith the least assumptions.For this purpose, the hot‐wire response equations 、vere derived for evaluating thethree components of mean velocity and the s? comp nents of Reynolds stresses,These measurements are carried out with two types of hot‐、vire probes, ユ.e., the*Department of Mechanical Engineering**Craduated Student of DePartment of Mechanical Engineering14     RyotiヽVAICA,Fumio YosHINO,TSutomu HAYASHI,and Koshi TABE:The Measurement of Reynolds Stresses with a Hot―Wire An mometerslanted wとe and the sttaight wire probes, at a given positiOn in a kコown flo、v fi ld,i.e.,the initial region of a round free iet.The results obtained by this method arecompared with the cxperilnental results in the noise‐producing region of a circulariet ineasured with the x‐probe by P.Bradshaw4)and those in the flow establishmentregion of a round iet measured with four types of probes by S. Sa“li。).SymbolsDhku,v,wueFFx,y,zSuperscriptsSubscriptsαl,歩れ9Fig。l shows the velocity vector compo―nents,un,ub and ut at the hot‐wire sensorwhich are the components in the directionsof the axis xa perpendicular tO the sensorprong plane,  of the probe axis xb andof the sensor xt, respectively.If the wire、ve  infinitely long,only thenormal components un and ub would con―tribute to the cooling, In realtty, the wireis about 200 diameters lo4g and must besupported on the prong for the measure―ment. In consequence ut contributes to the2  NomenclaturesNozzle diameter,Pitch factor,Ya、v factor;Velocity components in a fixed reference syste■1,Effective cooling velocity,Coordinate axes,Indicates tiine average,Indicates the fluctuating component,Indicate the pOints at which velocity deficit is O。1,half,0,9 of the maxilnum velocityこ。, resp ctivel .??????????〓??〓3  Derivatio■of the hot‐wire equationsted end//XbSensOr ProngFig。 l Attached reFereace systeコn of thehot‐wire.cooling of the wとe and ub acts differently from un because of the blockage effectRepOrts Of the Faculty of Enginee■ing;「Γottori U■とversity,Voユ. 9             15of the prong. The effective cooling velocity uerF aCting on the hot‐wire sellso  c nbe expressed as :u&f=て十五2ゞ+k2ギ (1),where h is the pttch factor and k the yaw factor. The estiinate of the pitch factorand the ya、7 faCtOr is based on the calibration curve of the probe obtained fromthe experiinent. The electrical signal through a linearizer is proportional to theeffective cooling velocity.As Eq。(1)iS based on the reference systeEl(Called the attached syste■l from mowOn)WhiCh iS f?ed to the hot‐wire sensor as shown in Fig.1, it is necessary to ex―press Eq.(1)by using the other reference system (called the f?d ferOnce systemfrom now on)WhiCh is f?ed to the■ozzle because the fiow field is represented inthis system. The components of the mean velocity and Reynolds stresses in thef?ed reference can be deter■lined through the matrix of tralllsformation fronl theattached reference system. If the angles, for a given position of the hot‐wire, be―tween the attached system (x′′, y′r, z′′)and the fixed system (x, y,Z)are definedas in Fig。2,the■latrix of transformation can be obtained by the fonowing procedure.First, the fixed reference syste■l is ro―tated in the cou■er―clock、vise directionby an angle φ around the axis of y toyieldX θttφ―ZS力φyxsケηb tt zどθSφ(2).????〓?????比y'  「y″Fig。2 Defi=?tion of angles φ and γ.Second,(x′y′,z′)systeln is rotatedin the counter―clockwise direction byan angle γ around the axis of x′, then(X′′,y′′,z′′)syStenl, 三.e。, the hot―wire reference system   (the attachedsystem),is Obtained asXn=X′′=X′xb一y′′― y′ θttγ tt Z′Sゲ″γ (3).Xt = Z′′ = 一 yr dttγ +z′ θθsγCombining EQ′s.(2)|ユl下With(3),磁協「――――?????????????〓「――――???????????????，?源琵} ′ウri′) ・ ?????????????????????????．?????????．?????????、―――‐‐???????????????????〓???????????????????????16     Ryoti WAKA,Fumio YosHINO,TSutomu H AVASHI,and Koshi TABE:The Measurement oE Reynolds Stresses with a Hot―Wire An mometerTheル察′)iS the matrix of trallsformation.u。“in Eq.(1),Can be represented interms of u,v and w,which are the components of instantaneods velocity ill thefixed referellce system,by the help of the mttrix of traHIsformation〃ぁ(5),TFrc′)is the transpose of frc′).Taking ti:ne‐average of Eq。(5)and diViding the velocities into the mean veloc‐ities and the Reynolds stresses, we have通eff(♪-4(♪こ +B(す)▼ +C(す)帝 +D(′)五▼ +Ecヵ予掃+F(♪百五u9ff(プ)=4(ァ)u′2+】(ブ)V′2+c(′)w′2+D(ァ)uv′+2。v′w′+F(ブ)w′u′ (6).whereAの=θ鉛2φ +h2sヵ2φ sゲ″2γ+k2sゲ″φθ偽2γ, βめ =h2θθs2 γ+k2d″γ,C(の=sゲη2φ+h2θ。∫2φsヵ2 γ tt k2 σos2ぁιos2 γ,D(′)=(■2_k2)sヶηb S力彦γ,E(ァ)=(h2_k2)σttφ dゲ滋γ,長′)=―(1-L2s′″γ_k2ιθs2 γ)sヵ2b,・ where φ represents the angle of rotation of the probe arollnd its axis and γ repr←seats the angle of the inclination of the sensor to the probe axis.To solve the res■lting ix silnultaneous equations, it is necessary to take a set of           isix readings measured at the different orientatiolls. Furthermore to make the de―ter■linant of the coefficicats of Eq.(6)nonvalliShing, it is also necessary to use the          ′t vヽo types of probes which have different γ.  T er fore three readings out of sixare obtained by rotating a straight Ⅵ「ユre probe(γ = 0°), and the Others by rotatinga slanted wire probe(γ≠ O°)。                                                          .4 Experilnental equipment and procedureThe idea introduced in the preceding sectiOn is tested experilnentally in the initialregion of a round free jet,The experimental equゎ■lent iS Shown in Fig.3.The flow rate of air supplied with the air compressor is adiuSted by the pressureregulator and the atr flo、v is rectified in the settling chamber and issued into stag―/Hot wlle probeFttg. 8 Experiinental set‐up.Reports of the Faculty of Engineering,Tottori University,Vol. 9nant air froHl the llozzleo The exit ve―locity is maintained constant within lラ歩of itself and Reyコol s number is 7.5×104.The jet and the r∞m temp raturesare equal with each other during anytest runo Thё nozzle is 30脇脇 in throatdiameter, and has the contraction ratioof 9 to 100 and three static holes on theside wall of its inlet,The configurations of hot,wire probes are shown in Fig.4,(a)iS the straightwire probe and (b) the slanted wire probe.  The sensors of these are made oftungsten wire with diameter of 0100[賜v nd length of l″η,(aSpect ratio 200)withcopper plated ends. The probe diinensお■ was measured with a profile projector.The experi:nent consists of two partst First, the stralght wire traverse was madein the half of the tet a10ng the axis of y, and rotated about its axis at each points.On this traverse, the mean and the root‐=nean‐sqtlare values of the linearized slgnalare recorded agaillst the angle φ=0°,40°, -30°.Second,the same values are meas―ured with the slanted wire pЮbe in the same manner.Substituting in Eq′s。 (6)and(7)a set Of SiX signals,the wire orie?ation φ andγ, and the pitch and the yaw factors of each wire could determine three componentsof mean velocity and the six components of Reynolds stresses.5 Res■lts and Discussio■In this section, the results obtailled by our method are presented and compared(a)Straight wire probe (b)Sユanted wire probeFig. 4 COnfigurations oF the hOt・・wとe probes.18      Ryoti WAKA,Fumio YosHINo,Tsutomu HAYASHI,and Koshi TABE:The Measurement of Reynolds stresses with a Hot_wire Anemometerwith those by Bradshaw and Sami.In the follo、ving preseatatiolls,the Velocities are made di=nensionless by the center‐lille velocity五0. The radial distance y, however, Is made diinensionless in threeways,1.e.,by y1/2(the diStance at which the velocity has half its maximum value),flow width(yO,ューy09)or the nozzle diameter D.The distribution of componeats of niean velocity is show■in Fig′s.5 and 6 together、vith the profile of up/こpO°btained by means of the total pressure pitot tube. Theagreement of u/翫O Withこp/五pOiS Satisfactory except for the velocity near the edge。IEJ4彗|〕ISI>!ぢIE51S05Fig.5 Distributions ofこ/uO,v/,O and〒/ユ0at x/D=2.Re=75xI04き=4也。詭△枇ロ静。o05 2.O y塊Fig。 6 Distributions of d/■0,予/こO an 〒/こO atx/D=4.?????????????〔?????????????ュ??。????。?〓?。???，Re=7.5x10nき=2yy4が10The veloc■y profiles▼/五Oand w/乳 are Widely scattered, and seem to inchdehaccuracies in the values obtained by experiinents.In order to sLo、v the compari onof our results wtth those of Bradshaw, the distributions of normal stresses obtain―ed at twO axial stations x/D=2 and 4 are plotted in Fig′s.7 and 8, The agreementof■lagnitude of the streamwise fluctuatioIIs /ur2/uO iS g∞d but he values of themdЛ fhc樋れ出/弓乳舶d he tang繋1織整∵麗望ご評,義縦‐pared with the results of Bradsha、v. Th〈plotted to compare with the results by Bradshaw and Sa■li at the axial stations x/D=と,2,3,4,as shown in Fig。9. Although the ma?mum values of u′v′/荒 are aslarge as those of Bradshaw and Saαli,their profiles are different from ours.This method produces considerable scatter in the profiles. This isヽOns quence ofthe magnification of measurement―errors through the computation of siinuitaneousequations.If a higher accuracy of the results is required, it could be obtained、vithmore adequate measurement cOnditions and more accurate recording of the outputsignal∞rrespoュding to the fluctuating veユocity.RepOrts Of the Faculty ofEngineering,TOttori University,Voユ. 9爛遇?ぃ?ぃ??ぃ???ぃ??????????‐ ?????????????????????=4O10Q05告=2BrcdshcwO   ~~△    ~~D  ~~Fig. 7 Comparison OF our normal stresseswith other worker's at x/D=2.O.IOO05O   O.5  1.O  l.5 ±DFig. 8 cOmpa i On of our norコnal stresseswith Other worker's at x/D=4.????????O α5 LO~もき=Ioσ=3日δ=2△き=40O-2,OFig. 9 cOmparison of Our shear stresses with Otherworkers'x/D=1,2,3,4.SOmi20     Ryojl WAKA,Fumio YoSHINO,TSutomu HAYASHI,and Koshi TABE:The Measurement of Reynolds Stresses with a Hot―Wire An mometer1                                    6 Co■clusionl          The method by the combined use of a straight wire probe and a sianted wire probel         has been successfully applied to the measuremeat of the meall velocity and the Rey一i         nolds stresses in three―dilnensio al turbulent flo、v. The derived hot‐wire equationsin terms of velocity components with least assumptions can be used to investigate athree‐diinensional turbuleat flow. In order to establish more accurate measuremeat,l          some modification of the recording of the electric slgnals from the hot‐wire syst mand the accurate measurements of the orientation of the hot‐wir a e ecessary.Referencesl)HinZe,J.0,,Turbulence,(1959),123-127,McGraw―HilI,2)champagne,F.H.,SIeicher,C.A.and Wehrmann,O.H。,J.FIuid Mech.,28-1(1967),153-175.3)Kovasznay,L.S.G.and Fujita,H.,Res.Inst,Sci.,30(1968),1351.4)BradShaw,P.,Ferriss,D.H.and Johnson,R.F.,J.Fluid Mech.,19(1964),591-624.5)Sami,S・,Carmody,T.and Rouse,H。,J.Fl id Mech,,27-2(1967),231-252.

The Measurement of Reynolds Stresses with a Hot－Wire Anemometer

http://repository.lib.tottori-u.ac.jp/files/public/0/2543/20190712112611110539/rfetu9(1)_13(1).pdf

The Measurement of Reynolds Stresses with a Hot－Wire Anemometer

Abstract

Similar works

Full text

Available Versions

Tottori University Research Result Repository