Keywords: Zero-pressure-gradient boundary layers, turbulence, logarithmic law, wake law, velocity profile, velocity distribution, skin friction.

Transcription

1 Jornal of Hydralic Research Vol. 43, No. 4 5, pp International Association of Hydralic Engineering and Research Modified log wake law for zero-pressre-gradient trblent bondary layers Loi log-trainée modifiée por coche limite trblente sans gradient de pression JUNKE GUO, Assistant Professor, Department of Civil Engineering, University of Nebraska Lincoln, Omaha, NE 6818, USA. jnkego@mail.nomaha.ed; Affiliate Faclty, State Key Lab for Water Resorces and Hydropower Engineering, Whan University, Whan, Hbei 437, PRC. PIERRE Y. JULIEN, Professor, Engineering Research Center, Department of Civil Engineering, Colorado State University, Fort Collins, CO 853, USA. pierre@engr.colostate.ed ROBERT N. MERONEY, Professor, Engineering Research Center, Department of Civil Engineering, Colorado State University, Fort Collins, CO 853, USA. meroney@engr.colostate.ed ABSTRACT This paper shows that the trblent velocity profile for zero-pressre-gradient bondary layers is affected by the wall shear stress and convective inertia. The effect of the wall shear stress is dominant in the so-called overlap region and can be described by a logarithmic law in which the von Karman constant is abot.4 while the additive constant depends on a Reynolds nmber. The effect of the convective inertia can be described by the Coles wake law with a constant wake strength abot.76. A cbic correction term is introdced to satisfy the zero velocity gradient reqirement at the bondary layer edge. Combining the logarithmic law, the wake law and the cbic correction prodces a modified log wake law, which is in excellent agreement with experimental profiles. The proposed velocity profile law is independent of Reynolds nmber in terms of its defect form, while it is Reynolds nmber dependent in terms of the inner variables. The modified log wake law can also provide an accrate eqation for skin friction in terms of the momentm thickness. Finally, by replacing the logarithmic law with van Driest s mixing-length model in which the damping factor varies with Reynolds nmber, the modified log wake law can be extended to the entire bondary layer flow. RÉSUMÉ Le profil de vitesse por ne coche limite trblente sans gradient de pression dépend de la contrainte de cisaillement à la paroi et de l inertie convective. L effet de cisaillement est dominant dans la zone de transition décrite par la loi logarithmiqe avec constante de von Karman d environ.4. L effet d inertie convective est décrit par la loi de trainée de Coles avec n coeffcient de.76. Un terme de correction cbiqe est introdit por satisfaire la condition limite spériere sans gradient de vitesse. La loi logarithmiqe-trainée modifiée qi en réslte se compare très bien avec les profils de vitesse expérimentax. Sos forme de déviation de vitesse, le profil de vitesse proposé devient indépendent d nombre de Reynolds. La loi proposée prodit des éqations exactes d coefficient de frottement et d épaisser d film de qantité de movement. Finalement, en remplacant la loi logarithmiqe par la longer de mélange de van Driest avec coefficient d amortissement fonction d nombre de Reynolds, la loi log-trainée modifiée devient applicable à la coche limite tote entière. Keywords: Zero-pressre-gradient bondary layers, trblence, logarithmic law, wake law, velocity profile, velocity distribtion, skin friction. 1 Introdction Trblent flows in pipes, zero-pressre-gradient ZPG flat plate bondary layers and open-channels are not only three fndamental bondary shear flows bt also important in mechanical, aeronatic and hydralic engineering. These three types of flows have similarities. The flow near the wall can be described by the law of the wall. The flow near the pipe axis, the bondary layer edge and the free srface can be described by the velocity defect law. In terms of the momentm eqation in the primary flow direction, steady pipe flows are the simplest for which differential shear stress is balanced by a constant pressre gradient; ZPG bondary layer flows are more complicated since differential shear stress is balanced by nonlinear convective inertia; and open-channel flows are the most complex becase both constant pressre gradient gravity term and nonlinear convective inertia secondary crrents exist. A new trblent velocity profile model is sally first applied to pipes, then ZPG bondary layers and finally open-channels. For example, the classic logarithmic law proposed by Prandtl and von Karman was first developed for pipes and ZPG bondary layers in the early 193s Schlichting 1979, p It was then applied to open-channels by Kelegan Lafer 1954 first pointed ot that experimental data deviate from the logarithmic law away from the pipe wall. Sbseqently Coles 1956 confirmed this behavior for bondary layers and sggested the law of the wake. Dring the 198s, the Revision received September 7, 4 / Open for discssion ntil May 31, 6. 41

2 4 Go et al. law of the wake was compared with open-channel profiles by Coleman 1981, 1986, Graf 1984 and Nez and Rodi This paper is a contination of Go and Jlien 3, where a modified log wake law was proposed for smooth pipe trblence, by considering the vale of the same concepts for ZPG bondary layers. The application of the modified log wake law to open-channel trblence will be considered separately becase of the complexity of secondary crrents and free srface. The following backgrond is closely related to the development of this stdy. Combining the logarithmic law and the wake law prodces the log wake law Coles 1956, = 1 κ ln y + B + Wξ 1 in which the terms in the parentheses are the logarithmic law while the last term is called the law of the wake, which defines the deviation from the logarithmic law away from the wall, in which = time-averaged velocity along the wall, = shear velocity, κ = von Karman constant, y = distance from the wall, = flid kinematic viscosity, B = additive constant, ξ = y/δ the relative distance from the wall, and δ = the bondary layer thickness, which is defined by y = δ =.999U in this paper. Coles 1956 described the wake fnction Wξ in an empirical table. Hinze 1975, p. 698 fit the Coles data analytically, Wξ = πξ sin κ in which = Coles wake strength, which acconts for the effects of Reynolds nmber in ZPG bondary layers. Inclding Hinze s eqation the log wake law 1 is often written as 1 = κ ln y + B + πξ sin 3 κ Unfortnately comparison of Eq. 3 with experimental data Coles, 1969; Hinze, 1975, p. 699 showed that 3 is invalid near the bondary layer edge where the zero velocity gradient reqirement is not satisfied. Except for the above classic reslts, niversal relations for near-wall flows have been extensively debated since the early 199s Afzal, 1997, 1a,b; Barenblatt, 1993; Barenblatt et al., a,b; Bschmann, 1; Bschmann and Gad-el-Hak, 3a,b; George and Castillo, 1997; Osterlnd et al., ; Wosnik et al., ; Zagarola, 1996; Zagarola and Smits, 1998; Zagarola et al., 1997; and others. Most of them conclded that the velocity profile for near-wall flows is Reynolds nmber dependent. In particlar, the velocity in the overlap layer follows a power law where the parameters vary with Reynolds nmber. This paper tries to incorporate the above state-of-the-art knowledge into the modified log wake law for ZPG trblent bondary layers. It will: a examine the bondary layer eqations and formlate a hypothesis of the modified log wake law; b validate the modified log wake law by comparing with recent experimental velocity profiles in ZPG bondary layers; c derive a skin friction formla in terms of the momentm thickness Reynolds nmber; and d extend the modified log wake law to the entire bondary layer by applying van Driest s mixing-length model where the damping factor varies with Reynolds nmber. Hypothesis of the modified log wake law This section examines the shear stress distribtion in ZPG bondary layers and formlates a hypothesis of the modified log wake law..1 Shear stress distribtion Consider a steady two-dimensional incompressible viscos flow over a ZPG flat plate where the x direction is along the wall and y normal to the wall. Neglecting gravity, Prandtl s bondary layer eqations are Schlichting, 1979, p. 563 x + v y = 4 x + v y = 1 τ 5 ρ y where = time-averaged velocity in the x direction, v = timeaveraged velocity in the y direction, ρ = flid density, and τ = local shear stress that incldes viscos and trblent shear stresses. Eqation 4 is the continity eqation, and 5 is the momentm eqation along the wall. To obtain an expression for the shear stress distribtion one may combine the continity eqation and momentm eqation and integrate across the bondary layer to develop y τ = τ w + ρ v y + v dy 6 y where τ = τ w at the wall y =. One can realize that the shear stress in ZPG bondary layers incldes the contribtions of the wall shear stress and convective inertia.. Dimensional analysis of velocity distribtion In the oter region inclding the overlap layer, the viscos shear stress can be neglected. Applying the eddy viscosity model, d τ t = ρ t 7 dy in which τ t = trblent shear stress and t = eddy viscosity, to 6 gives ρ t y = τ w + ρ y v y + v dy 8 y Considering that the eddy viscosity can be expressed by Hinze, 1975, p. 645 y t = δ f 9 δ in which f is an nknown fnction, and applying the definitions ξ = y/δ and τ w = ρ to 8, one obtains fξ ξ [ = 1 + v + v ] dξ ξ ξ ξ 1 Except for the complicated integrodifferential form in the above, the eddy viscosity fnction fξ is not really specified. Ths, it is impossible to get an analytical soltion for. However,

3 Modified log wake law 43 the preceding eqation sggests the following dimensionless soltion form: = F ξ, v 11 in which v = ηξ 1 and F and η are velocity distribtion fnctions in the x and y directions. Sbstitting 1 into 11 gives = Fξ, η 13 Since the transverse velocity v or η is very small compared with the primary velocity or F in the oter region, one can approximate the primary velocity / by expanding 13 at η =, Fξ, = Fξ, + η + η Fξ, + 14 η! η Taking the first two-term approximation, one has Fξ, = Fξ, + η 15 η Note that the above analysis is eqivalent to a small pertrbation introdced by the transverse velocity fnction ηξ. In fact, the classical Blasis soltion Schlichting, 1979, p. 136; Knd, 199, p. 31 can also be expressed in the above dimensionless form when one expresses the velocity distribtion in terms of the transverse velocity v. In the next sbsection, the primary fnction Fξ,, the transverse velocity distribtion fnction ηξ and the derivative fnction Fξ, / η will be specified..3 Approximation of the velocity distribtion.3.1 The primary fnction Fξ, The fnctions Fξ,, ηξ and Fξ, / η are approximated asymptotically and empirically. First, consider the overlap layer where the effect of the transverse velocity v or η can be neglected and Fξ, / η is finite. One can conclde that the primary fnction Fξ, is the law of the wall, which is often described by the classical logarithmic law or a power law. Recently based on experimental velocity profiles and asymptotic analysis, many researchers Afzal, 1997; Barenblatt et al., a,b; Bschmann and Gad-el-Hak, 3a,b; George and Castillo, 1997; Osterlnd et al., ; Wosnik et al., ; Zagarola, 1996; Zagarola and Smits, 1998; Zagarola et al., 1997 showed that a Reynolds nmber dependent power law can even better represent the velocity profile in the overlap layer. Ths, this paper assmes the following law of the wall: y c/ln Reδ = a ln Re δ + b 16 in which a, b and c are positive constants and the Reynolds nmber Re δ is defined as Re δ = δ 17 Since the power exponent c/ln Re δ in 16 is sally very small say.1.15 Barenblatt et al., a,b, one can rewrite 16 as { c = a ln Re δ + b exp ln ln Re δ { = a ln Re δ + b 1 + c ln ln Re δ = a ln Re δ + b + ac + bc ln Re δ y } y ln } + y + 18 In the overlap layer, one has y δ or lny / ln Re δ, the above eqation can then be approximated by = a ln Re δ + b + ac + bc y ln ln Re δ Comparing it with the classical logarithmic law, one has 19 1 κ = ac + bc ac ln Re δ for large Reynolds nmber, and B = a ln Re δ + b 1 where B = additive constant. Note that Eqs and 1 show that: a the von Karman constant κ increases with Reynolds nmber; b a niversal von Karman constant κ may exist only for large Reynolds nmber; and c the additive constant B increases with Reynolds nmber even for large Reynolds nmber. The dependence of Reynolds nmber acconts for the effect of the viscos sperlayer Hinze, 1975, pp. 567, 571, 68, which is near the bondary layer edge where Kolmogoroff length scale energy dissipation exists. In fact, Hinze 1975, p. 68 has noticed that the von Karman constant κ varies slightly abot.4 whereas the additive constant B corresponds with mch greater variations, say, from 4 to 1 George and Castillo, 1997, which may be explained by and 1. For simplicity, this paper concentrates on large Reynolds nmber and assmes κ = 1 =.4 ac Frthermore, the primary fnction Fξ, can be approximated by 19, i.e., Fξ, = 1 κ ln y + B 3 in which κ =.4 and B is estimated by 1 where the constants a and b will be specified in Section The transverse velocity distribtion fnction ηξ It is assmed that the shape of the fnction ηξ is similar to its conterpart in laminar flows. Inspired by the Blasis classical

4 44 Go et al. soltion Schlichting, 1979, p. 137; Knd, 199, p. 311 and the conventional sine-sqare wake fnction, one may assme v = V sin πξ 4 in which V is the transverse velocity at the bondary layer edge. Comparing 4 with 1, one mst have V = χ 5 in which χ is a proportional constant. With 5 Eq. 4 can be rewritten as v = ηξ = χ sin πξ 6 The constant χ will be considered together with the derivative fnction Fξ, / η..3.3 The derivative fnction Fξ, / η With 6 one can write the second term in 15 as Fξ, χ sin πξ 7 η which is the same as Hinze s version of the law of the wake assming Fξ, χ = 8 η κ and Fξ, / η is independent of ξ. In other words, the second term in 15 can be approximated by the conventional law of the wake, Fξ, η = η κ sin πξ 9 According to Coles Fernholz and Finley, 1996, the wake strength increases with Reynolds nmber and tends to a constant for large Reynolds nmber. To be consistent with where an assmption of large Reynolds nmber is employed, one can assme = constant 3 in this paper. Sbstitting 3 and 9 into 15 prodces the conventional log wake law 3 except that the additive constant B varies with Reynolds nmber..3.4 Bondary correction Strictly speaking, bondary layers do not have edges; the mean velocity is only asymptotic to the free stream velocity at the socalled bondary layer edges, i.e., U at ξ = y/δ = 1. However, in practice the assmptions of ξ = 1 = U 31 and d dξ = 3 ξ=1 are good approximations. To meet the zero velocity gradient reqirement 3, one mst modify 3 by adding a bondary correction fnction. Go and Jlien 3 have shown that a cbic correction is a good approximation for pipe axis. Similarly, this paper modifies 3 by adding the same correction fnction, ξ The modified log wake law and its defect form Combining 3 and 33 leads to the following velocity profile model: = 1 κ ln y + B + κ sin πξ ξ3 34 Similar to pipe flows Go and Jlien, 3, this paper calls the above eqation the modified log wake law MLWL, which shold be valid from the overlap region till the bondary layer edge. Eqation 34 is different from the conventional log wake law in two aspects: it meets the zero velocity gradient at the bondary layer edge; and the additive constant B acconts for the effect of the Reynolds nmber. To eliminate the effect of Reynolds nmber in 34, one can introdce the freestream velocity U at ξ = 1 to the modified log wake law. From 34, one obtains U = 1 κ ln δ + B + κ 1 35 Sbtracting 34 from 35 gives the velocity defect form of the modified log wake law U = 1 κ ln ξ cos πξ + 1 ξ As aforementioned, both κ and shold be niversal constants nder the assmption of large Reynolds nmber. 3 Test of the modified log wake law This section first examines the niversality of the modified log wake law by plotting all data points according to the defect form 36. It then tests the applicability of 36 to describe individal velocity profiles in terms of the inner variable y /. The 7 experimental velocity profiles by Osterlnd 1999 in ZPG bondary layers with Reynolds nmbers 9 Re δ 1, will be sed in this test. The complete descriptions of the experimental apparats and measred velocity profiles can be fond on the web site jens/zpg/. Note that Osterlnd 1999 measred the wall shear stress by sing an oil film interferometry, which is independent of the logarithmic law. In the data of Osterlnd 1999, the freestream velocity U, the shear velocity and the measred velocity profile ξ are given. The bondary layer thickness δ in this paper is interpolated by y = δ =.999U. According to the defect form, all 7 measred velocity profiles are plotted in Fig. 1 where except for the viscos sblayer and the bffer layer, all data points fall almost on a single crve down into the overlap layer. This reveals that the velocity defect in the oter region inclding the overlap region is independent of Reynolds nmber. Frthermore, it

5 Modified log wake law Modified log-wake law 36 Data of Osterlnd Modified log-wake law 36 Data of Osterlnd 1999 U-/ * 15 1 U-/ * ξ=y/δ Figre 1 Verification of the niversality of the velocity defect law ξ=y/δ Π Π=.7577 ±.7577 bondary layer edge, say 3 y / and y/δ 1; c the modified log wake law tends to a straight line in a semilog plot in the overlap region and then coincides with the logarithmic law; and d the zero velocity gradient at the bondary layer edge can be clearly seen from all profiles in Fig. 3a g which imply that the bondary correction is necessary Re δ =δ / * Figre Verification of the niversal constant. implies that the model parameters κ and in the modified log wake law 36 are niversal constants. A preliminary analysis sggests that κ =.4 and = allows Eq. 36 to fit the experimental profiles shown in Fig. 1 very well. Figre frther confirms Eq. 37 by plotting the wake strength verss Reynolds nmber Re δ, in which the vales of for individal profiles are estimated by the measred vale of / at y / = 1 at assming κ =.4. In terms of the inner variable y /, one can rewrite 36 as = U + 1 κ in which ln ξ cos πξ + 1 ξ ξ = y / 39 Re δ Obviosly, the Reynolds nmber Re δ = δ / is a profile parameter, which sggests that the modified log wake law is Reynolds nmber dependent in terms of the inner variables. Figre 3a g compare 38, in which the constants in 37 are sed, with all 7 experimental profiles individally and display excellent agreement for almost all profiles. Note that the dotted lines MLWL are covered with the solid when y / 3. These figres lead to the following conclsions: a the basic strctre of the modified log wake law is correct; b the modified log wake law can replicate the experimental data from the overlap region till the 4 Skin friction and the additive constant in the logarithmic law One can compte the velocity profile by sing the velocity defect law 36, which does not reqire the additive constant B. Nevertheless, if the modified log wake law 34 is preferred, the additive constant B can be defined by stdying the skin friction factor c f, which is defined as τ w = c f ρu or c f = U 4 Sbstitting 1 into 35 gives U 1 = κ + a ln Re δ + b + κ 1 41 which can be rearranged as = U = 1 ln Re δ + B 1 4 c f κ 1 where 1 = 1 κ 1 κ + a and B 1 = b + κ 1 43 are determined experimentally. Figre 4 displays the experimental skin friction factor c f of Osterlnd 1999 verss the Reynolds nmber Re δ according to 4. A least-sqares crve fitting reveals that 4 with the constants κ 1 =.38 and B 1 = can represent the experimental data with a correlation coeffcient.999. Sbstitting 37 and 44 into 43 prodces a =.1176 and b =

6 shift by 5 shift by 5 shift by 5 shift by 5 46 Go et al. 8 7 Data of Osterlnd 1999 Modified log-wake law 38 Logarithmic law 3 Extension of MLWL 49 or SW9818A SW9816C SW9816B 8 7 Data of Osterlnd 1999 Modified log-wake law 38 Logarithmic law 3 Extension of MLWL 49 or SW9818B SW9818C SW9818D 6 SW9816A 6 SW9811A SW9815F SW9811B 5 SW9815E 5 SW9811C / * SW9815A / * SW98113A 4 SW9815B 4 SW98113B SW9815C SW98113D 3 SW9815D 3 SW98113C 1 a 1 b y / * y / * 8 7 Data of Osterlnd 1999 Modified log-wake law 38 Logarithmic law 3 Extension of MLWL 49 or SW981113F SW981113E SW981113D 8 7 Data of Osterlnd 1999 Modified log-wake law 38 Logarithmic law 3 Extension of MLWL 49 or SW98117G SW98117F SW98117E 6 SW981113C 6 SW98117D SW981113B SW98117C 5 SW981113A 5 SW98117B / * SW98111D / * SW98117A 4 SW98111C 4 SW98116E SW98111B SW98116D 3 SW98111A 3 SW98116C 1 c 1 d y / * y / * Figre 3 Comparison of modified log wake law with individal experimental velocity profiles.

7 shift by 5 shift by 5 shift by 5 Modified log wake law Data of Osterlnd 1999 Modified log-wake law 38 Logarithmic law 3 Extension of MLWL 49 or SW98117Q SW98117P 8 7 Data of Osterlnd 1999 Modified log-wake law 38 Logarithmic law 3 Extension of MLWL 49 or ptential flow SW98118J SW98118I SW98117O SW98118H 6 SW98117N 6 SW98118G SW98117M SW98118F 5 SW98117L 5 SW98118E / * SW98117K / * SW98118D 4 SW98117J 4 SW98118C 3 SW98117I SW98117H 3 SW98118B SW98118A 1 e 1 f y / * y / * 8 7 Data of Osterlnd 1999 Modified log-wake law 38 Logarithmic law 3 Extension of MLWL 49 or SW98119J SW98119I 6 SW98119H SW98119G / * 5 4 SW98119F SW98119E SW98119D SW98119C 3 SW98119B SW98119A 1 g y / * Figre 3 Contined

8 48 Go et al. /C f = U/ * Data of Osterlnd 1999 Eqation 4 from MLWL Re δ =δ / * Figre 4 Skin friction factor c f verss Reynolds nmber Re δ. which specify the additive constant B throgh 1. Like the modified log wake law, the logarithmic law 3, to which 1 and 45 are applied, also fits the experimental data in the overlap region very well for most profiles, as shown by the dashed lines in Fig. 3a g, which validate the hypothesis of the dependence of Reynolds nmber in 1. The skin friction factor c f is often correlated with the momentm thickness Reynolds nmber Re θ, c f = fre θ ; Re θ = θu 46 in which θ = the momentm thickness. Applying 38 to the definition of the momentm thickness θ, one can derive a relation between Re δ and Re θ see the Appendix at the end of this paper. Applying A7 in the Appendix to 4 leads to = 1 ln Re θ 1 cf ln α c f κ 1 κ 1 + β + B 1 47 in which the constants in 44 are sed for the vales of κ 1 and B 1, and α = and β = are obtained from A5 and A6 in the Appendix. Eqation 47 agrees with the experimental data, as shown in Fig. 5, where the empirical law of Osterlnd et al. is also plotted. The excellent agreement validates the modified log wake law 36 not x 1-3 Data of Osterlnd 1999 Eqation 47 from MLWL Empirical law of Osterlnd et al. only for velocity profiles bt also for the momentm thickness and the skin friction factor. Note that Osterlnd et al. obtained a small von Karman constant from the same data set since they overlooked the effect of Reynolds nmber on the additive constant B. Finally, combining 4 and 47 provides a method to estimate the wall shear stress τ w and the bondary layer thickness δ from a measred velocity profile. That is, a calclate the momentm thickness θ by applying a measred velocity profile to the definition of the momentm thickness; b estimate the friction factor c f from 47; c compte the wall shear stress τ w or the shear velocity from 4; and d obtain the bondary layer thickness δ from 4. With this method, the wall shear stress and bondary layer thickness can be defined in an experimental program. 5 Extension of the modified log wake law From the above analysis, one can see that the modified log wake law indeed agrees with the experimental data in the oter region. It is noteworthy that the modified log wake law can be extended to the inner region by applying van Driest s mixing-length model Schlichting, 1979, p. 64 for the law of the wall or the primary fnction Fξ,. In other words, replacing the logarithmic law in 34 with van Driest s mixing-length model, one can get a complete velocity profile model for the entire bondary layer, = y + dy + 1 +{1 + 4κ y + [1 exp y + /A]} 1/ + πξ sin κ ξ3 49 in which κ =.4, =.7577,y + = y / and A is a damping factor, A 5B = 5a ln Re δ + b 5 in which 1 has been sed. Note that van Driest sggested A = 6 to reprodce the additive constant B 5.. In this paper, the additive constant B is a fnction of Reynolds nmber, the damping factor A is then modified as Eq. 5. Figre 3a g reveal that the extension eqation 49 can indeed clone the entire bondary layer velocity profile, inclding the inner and oter regions. In brief, the extension of the modified log wake law 49 agrees with experimental profiles well, satisfies all bondary conditions at the wall and at the bondary layer edge, and connects to the constant potential velocity smoothly, as indicated by the solid lines in Fig. 3a g. c f Re θ = θu/ x 1 4 Figre 5 Skin friction factor c f verss Reynolds nmber Re θ. 6 Conclsions Based on the bondary layer eqations, dimensional analysis, pertrbation techniqe, a recent nderstanding of the overlap layer and an analogy to the transverse velocity profile in laminar bondary layers, this paper proposes a modified log wake law for the mean velocity profile of ZPG trblent bondary

9 Modified log wake law 49 layers. The proposed law consists of three terms: a logarithmic term in which the von Karman constant is abot.4 while the additive constant increases with Reynolds nmber; a sine-sqare term with a constant wake strength abot.76; and a cbic correction term. The logarithmic law reflects the effect of the wall shear stress and is dominant in the overlap region; the sine-sqare fnction approximates the transverse velocity and then reflects the effect of convective inertia; and the cbic correction makes the conventional log wake law satisfy the zero velocity gradient reqirement at the bondary layer edge. The proposed velocity profile law provides excellent agreement with 7 recent experimental profiles not only for the mean velocity profiles bt also for the skin friction factor. Specifically the comparison shows that: a the modified log wake law is independent of Reynolds nmber in terms of its defect form; that is, the von Karman constant and the wake strength are indeed niversal constants for large Reynolds nmber; b the modified log wake law is Reynolds nmber dependent when the inner variables are sed; c the proposed logarithmic law, with a variable additive constant, has been validated by the data in the overlap region; and d the friction factor derived from the modified log wake law is accrate in terms of the momentm thickness. Finally, applying van Driest s mixing-length model, in which the damping factor varies with Reynolds nmber, for the law of the wall, the modified log wake law can be extended to the entire bondary layer from the wall till the bondary layer edge. Appendix: The displacement thickness δ 1 and the momentm thickness θ The displacement thickness δ 1 and the momentm thickness θ are two important parameters in bondary layer analysis. They can be estimated from the proposed modified log wake law 38 which gives 1 dξ = U A1 κ 4 and 1 dξ = 1 81 κ π + 8 π + Siπ π 3 U U κ A Applying A1 to the definition of the displacement thickness δ 1, one derives δ 1 δ = 1 = 1 U 1 dξ U 1 dξ = κ 4 U A3 Similarly, applying A1 and A to the definition of the momentm thickness θ gives θ 1 δ = 1 dξ U U = 1 1 dξ dξ U U = α + β A4 U U in which α = κ 4 + Siπ π 1 π + 4 π 4 κ 3 κ = = A5 κ β = 1 3 κ = = A6 κ where the constants in 37 are applied. Frthermore, one can show Re δ = δ = θu 1 δ θ U = Re cf θ α + β A7 Notation A = Van Driest s damping factor a, b, c = Constants in the power law 16 B = Additive constant in the logarithmic law B 1 = Additive constant in the friction eqation 4 c f = Skin friction factor F, f = Fnctional symbols Re δ = Reynolds nmber based on the bondary layer thickness, δ / Re θ = Reynolds nmber based on the momentm thickness, θu/ U = Freestream velocity = Time-averaged velocity along the wall = Shear velocity V = Transverse velocity at the bondary layer edge v = Time-averaged velocity normal to the wall W = Wake fnction x = Coordinate along the wall y = Coordinate normal to the wall y + = Inner variable, y / α, β = Constants in the friction eqation 47 δ = Bondary layer thickness δ 1 = Displacement thickness θ = Momentm thickness η = Transverse velocity distribtion fnction κ = The von Karman constant in the logarithmic law κ 1 = The von Karman constant in the friction law 4 = Kinematic viscosity of flid t = Eddy viscosity ξ = Relative distance from the wall, y/δ = The Coles wake strength

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