The near wall behavior of an impinging jet - UFRJ

D.R.S. Guerra et al. / International Journal of Heat and Mass Transfer 48 (2005) 2829–2840 283750y[mm]40r= 80mm r= 85mm r=90mm r= 95mm r= 100mm3020100T w T w T w T w T w(a)26 30 34 38 26 30 34 38 26 30 34 38 26 30 34 38 26 30 34T[ºC]3850y[mm]40r= 105mm r= 110mm r= 115mm r= 120mm r= 130mm302010T w T w T wT wT w026 30 34 38 26 30 34 38 26 30 34 38 26 30 34 38 26 30 34 38(b)T[ºC]50y[mm]40r= 130mm r= 135mm r= 140mm r= 145mm r=150mm302010T w T w T w T wT w026 30 34 38 26 30 34 38 26 30 34 38 26 30 34 38 26 30 34 38(c)T[ºC]Fig. 5. Mean temperature profiles. (a) Stations 80–100 mm, (b) stations 105–125 mm, and (c) stations 130–150 mm.or Preston tubes. Since these devices are calibrated takingas reference the universal law of the wall, they cannotbe reliably used in regions where the existence ofthe law of the wall can be questioned. Wygnanskiet al. estimated the skin-friction through three differenttechniques: a momentum integral method, the meanvelocity gradient in the viscous sub-layer, and by useof a Preston tube.The establishment of the above concepts for thevelocity field clearly raises some questions for the temperaturefield. An immediate question concerns the existenceof an appropriate temperature scale at the outeredge of the equilibrium layer. At the point of velocitymaximum, Fig. 5 shows that the temperature profilesreach a minimum. Thus, drawing an analogy to thevelocity analyses of Narasimha et al. [6] and of Özdemirand Whitelaw [5], one would expect the appropriatescaling temperature parameter to be this minimumtemperature.The law of the wall for the temperature profile can bewritten asT w T¼ 1 ln yu sþ Bð6Þt s k t mwhere t s is the friction temperature and j t is the vonKarman constant for the temperature field.The expected parametric behavior of B is then to berepresented byB ¼ g T w T mð7Þt swhere T w represents the wall temperature, T m the minimumtemperature in a given profile and t s is the frictiontemperature.To find the values of A and of B, the graphical methodof Coles [42] was used. Here, we must point out that thethickness of the inner turbulent region for an impingingjet is very thin, so that the fitting of a straight line to the

2838 D.R.S. Guerra et al. / International Journal of Heat and Mass Transfer 48 (2005) 2829–2840fully turbulent region is a difficult affair. For a plane walljet, the fully turbulent region is rather arbitrary [8], normallybeing located within the interval 70 < yu s /m < 170.Since the analysis of Wygnanski et al. [7] suggests thatvon KarmanÕs parameter can be considered constantand that A varies from 5.5 to 9.5, the fitting of a straightline to the velocity and to the temperature data in semilogplots in the region 70 < yu s /m < 170 can then be usedto find u s , A, t s and B in Eqs. (4)–(7). The graphicalmethod used for the determination of parameters Aand B is illustrated in Fig. 6.The resulting linear behavior of parameters A and Bis shown in Fig. 7. This figure indicates that both A andB increase as the maximum jet velocity increases and theminimum jet temperature decreases, respectively. Specifically,the following equations result:A ¼ 1.124 u M27.538; ð8Þu sB ¼ 1.031 T wt sT m25.869. ð9ÞThus, the trend observed by Özdemir and Whitelaw [5]is confirmed here. Furthermore, the present analysisgives us a strong hint that a possible linear behavior ofA and of B as a function of the maximum jet velocityand of the minimum jet temperature would be in order.When the profile-shift parameters A and B are subtractedfrom the velocity and the temperature profiles,the resulting curves exhibit the behavior of equilibriumlayers that extends to the locations of the velocity maximumand the temperature minimum, respectively. Thisis shown in Fig. 8.Despite our brief account of the problem of anorthogonal jet impinging on a wall, the following findingsare remarkable: (1) the variation of both A and Bis well defined and is in accordance with the accountof other authors, (2) the level in the logarithmic expressionsfor the laws of the wall have a tendency to in-6.4u[m/s]6H/D= 2.011T w-T H/D= 2.0105.695.28(a)4.84.4r= 120mm-0.8 -0.4 0 0.4 0.8 1.2ln(y)(b)76r= 150mm-1 0 1 2 3 4 5ln(y)Fig. 6. Graphical method for the determination of parameters A and B. (a) Determination of A and (b) determination of B.15A10H/D= 2.016B12H/D= 2.05804-50-10-4B = 1.031* ((Tw-Tm)/Ttau)-25.869A= 1.124*(UM/Utau)-27.538-15-810 15 20 25 30 35 40 20 24 28 32 36 40(a) U m /u (b) T w -T m /tFig. 7. Deviation function for the (a) velocity and the (b) temperature profiles.

D.R.S. Guerra et al. / International Journal of Heat and Mass Transfer 48 (2005) 2829–2840 2839Fig. 8. Velocity and temperature profiles in inner variables, y + = yu s /m, with subtraction of the profile-shift parameters. (a) Velocityprofiles and (b) temperature profiles.creased with increasing maximum jet velocity and withdecreasing minimum temperature.The relations for A and for B derived here are particularlyimportant for the determination of the skin-frictioncoefficient and of the heat transfer coefficient.This issue will be dealt with in separate article.5. ConclusionThe present work has described the behavior of asemi-confined impinging jet over a heated flat plate.Experimental data for the pressure distribution, velocityand temperature fields were obtained. The heat transferdata confirmed the existence of a minimum in temperatureprofile away from the wall. The existence of a velocityand a temperature equilibrium layer was alsoinvestigated. The results found at this investigation indicatethat the level of the logarithmic portion of thevelocity and the temperature laws of the wall increaseswith increasing maximum jet velocity and decreasingminimum temperature. This fact, for the temperatureprofiles, has been observed for the first time in the courseof the present research.The present research is particularly relevant due to itsapplication for the development of methods that can beused for the determination of the local skin-friction andof the local heat transfer coefficient.AcknowledgementsDRSG is grateful to CAPES (Brazilian Ministry ofEducation) for the award of a D.Sc. scholarship in thecourse of the research. APSF is grateful to the BrazilianNational Research Council (CNPq) for the award of aresearch fellowship (Grant No. 304919/2003-9). Thework was financially supported by CNPq through GrantNo. 472215/2003-5 and by FAPERJ through GrantsE-26/171.198/2003 and E-26/152.368/2002. JS has alsobenefited from a CNPq research fellowship (Grant No.550780/2002-5).References[1] R.P. Patel, Self preserving two dimensional turbulent jetsand wall jets in a moving stream, M.Sc. Thesis, McGillUniversity, Montreal, 1962.[2] A. Tailland, J. Mathieu, Jet parietal, J. Mecanique 6 (1967)1.[3] V. Ozarapoglu, Measurements in incompressible turbulentflows. D.Sc Thesis, Laval University, Quebec, 1973.[4] H.P.A.H. Irwin, Measurements in a self-preserving planewall jet in a positive pressure gradient, J. Fluid Mech. 61(1973) 33–63.[5] I.B. Ozdemir, J.H. Whitelaw, Impingement of an axisymmetricjet on unheated and heated flat plates, J. FluidMech. 240 (1992) 503–532.[6] R. Narasimha, K.Y. Narayan, S.P. Pathasarathy, Parametricanalysis of turbulent wall jets in still air, Aeronaut.J. 77 (1973) 335.[7] I. Wygnanski, Y. Katz, Horev, On the applicability ofvarious scaling laws to the turbulent wall jet, J. FluidMech. 234 (1992) 669–690.[8] G.P. Hammond, Complete velocity profile and ‘‘optimum’’skin-friction formulas for the plane wall-jet, J. Fluids Eng.104 (1982) 59–66.[9] D.B. Spalding, A single formula for the law of the wall,ASME J. Appl. Mech. 28 (1961) 455–458.[10] M. Behnia, S. Parneix, P.A. Durbin, Prediction of heattransfer in an axisymmetric turbulent jet impinging on aflat plate, Int. J. Heat Mass Transfer 41 (1998) 1845–1855.[11] M. Behnia, S. Parneix, Y. Shabany, P.A. Durbin, Numericalstudy of turbulent heat transfer in confined andunconfined impinging jets, Int. J. Heat Fluid Flow 20(1999) 1–9.

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