Modified Log–Wake Law for Zero-Pressure-Gradient Turbulent

Modiﬁed log–wake law for zero-pressure-gradient turbulent boundary layers Loi log-trainée modiﬁée pour couche limite turbulente sans gradient de pression

JUNKE GUO, Assistant Professor, Department of Civil Engineering, University of Nebraska - Lincoln, Omaha, NE 68182, USA. E-mail: [email protected]; Afﬁliate Faculty, State Key Lab for Water Resources and Hydropower Engineering, Wuhan University, Wuhan, Hubei 430072, PRC.

PIERRE Y. JULIEN, Professor, Engineering Research Center, Department of Civil Engineering, Colorado State University, Fort Collins, CO 80523, USA. E-mail: [email protected]

ROBERT N. MERONEY, Professor, Engineering Research Center, Department of Civil Engineering, Colorado State University, Fort Collins, CO 80523, USA. E-mail: [email protected]

ABSTRACT This paper shows that the turbulent velocity profile for zero-pressure-gradient boundary 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 about 0.4 while the additive constant depends on a Reynolds number. The effect of the convective inertia can be described by the Coles wake law with a constant wake strength about 0.76. A cubic correction term is introduced to satisfy the zero velocity gradient requirement at the boundary layer edge. Combining the logarithmic law, the wake law and the cubic correction produces a modified log–wake law, which is in excellent agreement with experimental profiles. The proposed velocity profile law is independent of Reynolds number in terms of its defect form, while it is Reynolds number dependent in terms of the inner variables. The modified log–wake law can also provide an accurate equation for skin friction in terms of the momentum thickness. Finally, by replacing the logarithmic law with van Driest’s mixing-length model in which the damping factor varies with Reynolds number, the modified log–wake law can be extended to the entire boundary layer flow.

RÉSUMÉ Le profil de vitesse pour une couche limite turbulente 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 logarithmique avec constante de von Karman d’environ 0.4. L’effet d’inertie convective est décrit par la loi de trainée de Coles avec un coeffcient de 0.76. Un terme de correction cubique est introduit pour satisfaire la condition limite supérieure sans gradient de vitesse. La loi logarithmique-trainée modifiée qui en résulte se compare très bien avec les profils de vitesse expérimentaux. Sous forme de déviation de vitesse, le profil de vitesse proposé devient indépendent du nombre de Reynolds. La loi proposée produit des équations exactes du coefficient de frottement et d’épaisseur du film de quantité de mouvement. Finalement, en remplacant la loi logarithmique par la longueur de mélange de van Driest avec coefficient d’amortissement fonction du nombre de Reynolds, la loi log-trainée modifiée devient applicable à la couche limite toute entière.

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

1 Introduction shear stress is balanced by nonlinear convective inertia; and open-channel flows are the most complex because both constant Turbulent flows in pipes, zero-pressure-gradient (ZPG) flat plate pressure gradient (gravity term) and nonlinear convective iner- boundary layers and open-channels are not only three fundamen- tia (secondary currents) exist. A new turbulent velocity profile tal boundary shear flows but also important in mechanical, aero- model is usually first applied to pipes, then ZPG boundary layers nautic and hydraulic engineering. These three types of flows have and finally open-channels. For example, the classic logarithmic similarities. The flow near the wall can be described by the law law proposed by Prandtl and von Karman was first developed of the wall. The flow near the pipe axis, the boundary layer edge for pipes and ZPG boundary layers in the early 1930s (Schlicht- and the free surface can be described by the velocity defect law. ing 1979, p. 578–665). It was then applied to open-channels by In terms of the momentum equation in the primary flow direc- Keulegan (1938). Laufer (1954) first pointed out that experimen- tion, steady pipe flows are the simplest for which differential tal data deviate from the logarithmic law away from the pipe wall. shear stress is balanced by a constant pressure gradient; ZPG Subsequently Coles (1956) confirmed this behavior for boundary boundary layer flows are more complicated since differential layers and suggested the law of the wake. During the 1980s, the

Revision received

1 2 Guo et al. law of the wake was compared with open-channel profiles by 2 Hypothesis of the modified log–wake law Coleman (1981, 1986), Graf (1984) and Nezu and Rodi (1986). This paper is a continuation of Guo and Julien (2003), where This section examines the shear stress distribution in ZPG a modified log–wake law was proposed for smooth pipe turbu- boundary layers and formulates a hypothesis of the modified lence, by considering the value of the same concepts for ZPG log–wake law. boundary layers. The application of the modified log–wake law to open-channel turbulence will be considered separately because of the complexity of secondary currents and free surface. 2.1 Shear stress distribution The following background is closely related to the develop- Consider a steady two-dimensional incompressible viscous flow ment of this study. Combining the logarithmic law and the wake over a ZPG flat plate where the x direction is along the wall and y law produces the log–wake law (Coles 1956), normal to the wall. Neglecting gravity, Prandtl’s boundary layer u yu equations are (Schlichting, 1979, p. 563) = 1 ∗ + B + W(ξ) u κ ln ν (1) ∂u ∂v ∗ + = ∂x ∂y 0 (4) in which the terms in the parentheses are the logarithmic law while the last term is called the law of the wake, which defines ∂u ∂u 1 ∂τ u + v = (5) the deviation from the logarithmic law away from the wall, in ∂x ∂y ρ ∂y which u = time-averaged velocity along the wall, u∗ = shear where u = time-averaged velocity in the x direction, v = time- velocity, κ = von Karman constant, y = distance from the wall, averaged velocity in the y direction, ρ = fluid density, and ν = fluid kinematic viscosity, B = additive constant, ξ = y/δ τ = local shear stress that includes viscous and turbulent shear the relative distance from the wall, and δ = the boundary layer stresses. Equation (4) is the continuity equation, and (5) is the thickness, which is defined by u(y = δ) = 0.999U in this paper. momentum equation along the wall. Coles (1956) described the wake function W(ξ) in an empirical To obtain an expression for the shear stress distribution one table. Hinze (1975, p. 698) fit the Coles data analytically, may combine the continuity equation and momentum equation 2 πξ and integrate across the boundary layer to develop W(ξ) = sin2 (2) κ 2 y ∂v ∂u τ = τ + ρ −u + v y = w d (6) in which Coles’ wake strength, which accounts for the 0 ∂y ∂y effects of Reynolds number in ZPG boundary layers. Including where τ = τw at the wall y = 0. One can realize that the shear Hinze’s equation (2) the log–wake law (1) is often written as stress in ZPG boundary layers includes the contributions of the u 1 yu∗ 2 πξ wall shear stress and convective inertia. = ln + B + sin2 (3) u∗ κ ν κ 2 Unfortunately comparison of Eq. (3) with experimental data 2.2 Dimensional analysis of velocity distribution (Coles, 1969; Hinze, 1975, p. 699) showed that (3) is invalid near the boundary layer edge where the zero velocity gradient In the outer region including the overlap layer, the viscous shear requirement is not satisfied. stress can be neglected. Applying the eddy viscosity model, Except for the above classic results, universal relations for du τt = ρνt (7) near-wall flows have been extensively debated since the early dy

1990s (Afzal, 1997, 2001a,b; Barenblatt, 1993; Barenblatt in which τt = turbulent shear stress and νt = eddy viscosity, to et al., 2000a,b; Buschmann, 2001; Buschmann and Gad-el-Hak, (6) gives 2003a,b; George and Castillo, 1997; Osterlund et al., 2000; ∂u y ∂v ∂u ρν = τ + ρ −u + v y Wosnik et al., 2000; Zagarola, 1996; Zagarola and Smits, 1998; t ∂y w ∂y ∂y d (8) Zagarola et al., 1997; and others). Most of them concluded 0 that the velocity profile for near-wall flows is Reynolds number Considering that the eddy viscosity can be expressed by (Hinze, 1975, p. 645) dependent. In particular, the velocity in the overlap layer follows a power law where the parameters vary with Reynolds number. y νt = δu∗f (9) This paper tries to incorporate the above state-of-the-art δ knowledge into the modified log–wake law for ZPG turbulent in which f is an unknown function, and applying the definitions 2 boundary layers. It will: (a) examine the boundary layer equa- ξ = y/δ and τw = ρu∗ to (8), one obtains tions and formulate a hypothesis of the modified log–wake law; ∂ u ξ u ∂ v v ∂ u (b) validate the modified log–wake law by comparing with recent f(ξ) = 1 + − + dξ ∂ξ u∗ 0 u∗ ∂ξ u∗ u∗ ∂ξ u∗ experimental velocity profiles in ZPG boundary layers; (c) derive (10) a skin friction formula in terms of the momentum thickness Reynolds number; and (d) extend the modified log–wake law to Except for the complicated integrodifferential form in the above, the entire boundary layer by applying van Driest’s mixing-length the eddy viscosity function f(ξ) is not really specified. Thus, model where the damping factor varies with Reynolds number. it is impossible to get an analytical solution for u. However, Modified log–wake law 3 the preceding equation suggests the following dimensionless Since the power exponent c/ln Reδ in (16) is usually very small solution form: say 0.1–0.15 (Barenblatt et al., 2000a,b), one can rewrite (16) as u v = F ξ, u u (11) u c yu∗ ∗ ∗ = (a ln Reδ + b) exp ln u∗ ln Reδ ν in which c yu v = (a + b) + ∗ +··· = η(ξ) ln Reδ 1 ln ν u (12) ln Reδ ∗ bc yu∗ and F and η are velocity distribution functions in the x and y = (a ln Reδ + b) + ac + ln +··· (18) ln Reδ ν directions. Substituting (12) into (11) gives u = F(ξ,η) In the overlap layer, one has y δ or ln(yu∗/ν) ln Reδ, the u (13) ∗ above equation can then be approximated by Since the transverse velocity v or η is very small compared with the primary velocity u or F in the outer region, one can u bc yu∗ = (a ln Reδ + b) + ac + ln (19) approximate the primary velocity u/u∗ by expanding (13) at u∗ ln Reδ ν η = 0, Comparing it with the classical logarithmic law, one has u ∂F(ξ, 0) η2 ∂2F(ξ,0) = F(ξ,0) + η + +··· (14) u∗ ∂η 2! ∂η2 1 bc = ac + → ac (20) Taking the first two-term approximation, one has κ ln Reδ u ∂F(ξ, 0) = F(ξ,0) + η (15) for large Reynolds number, and u∗ ∂η

Note that the above analysis is equivalent to a small perturbation B = a ln Reδ + b (21) introduced by the transverse velocity function η(ξ). In fact, the classical Blasius solution (Schlichting, 1979, p. 136; Kundu, where B = additive constant. Note that Eqs (20) and (21) show 1990, p. 310) can also be expressed in the above dimensionless that: (a) the von Karman constant κ increases with Reynolds num- u form when one expresses the velocity distribution in terms of the ber; (b) a universal von Karman constant κ may exist only for v transverse velocity . In the next subsection, the primary function large Reynolds number; and (c) the additive constant B increases F(ξ, ) η(ξ) 0 , the transverse velocity distribution function and the with Reynolds number even for large Reynolds number. The ∂F(ξ, )/∂η derivative function 0 will be specified. dependence of Reynolds number accounts for the effect of the “viscous superlayer” (Hinze, 1975, pp. 567, 571, 628), which is near the boundary layer edge where Kolmogoroff length scale 2.3 Approximation of the velocity distribution energy dissipation exists. In fact, Hinze (1975, p. 628) has 2.3.1 The primary function F(ξ,0) noticed that the von Karman constant κ varies slightly about The functions F(ξ,0), η(ξ) and ∂F(ξ, 0)/∂η are approximated 0.4 whereas the additive constant B corresponds with much asymptotically and empirically. First, consider the overlap layer greater variations, say, from 4 to 12 (George and Castillo, where the effect of the transverse velocity v or η can be neglected 1997), which may be explained by (20) and (21). For sim- and ∂F(ξ, 0)/∂η is finite. One can conclude that the primary plicity, this paper concentrates on large Reynolds number and function F(ξ,0) is the law of the wall, which is often described assumes by the classical logarithmic law or a power law. Recently based on experimental velocity profiles and asymptotic analysis, many κ = 1 = . ac 0 4 (22) researchers (Afzal, 1997; Barenblatt et al., 2000a,b; Buschmann and Gad-el-Hak, 2003a,b; George and Castillo, 1997; Furthermore, the primary function F(ξ,0) can be approximated Osterlund et al., 2000; Wosnik et al., 2000; Zagarola, 1996; by (19), i.e., Zagarola and Smits, 1998; Zagarola et al., 1997) showed that a Reynolds number dependent power law can even better represent yu F(ξ, ) = 1 ∗ + B the velocity profile in the overlap layer. Thus, this paper assumes 0 κ ln ν (23) the following law of the wall: κ = . B u yu∗ c/ln Reδ in which 0 4 and is estimated by (21) where the constants = (a ln Reδ + b) (16) a b u∗ ν and will be specified in Section 4. in which a,b and c are positive constants and the Reynolds number Reδ is defined as 2.3.2 The transverse velocity distribution function η(ξ) δu It is assumed that the shape of the function η(ξ) is similar to its = ∗ Reδ ν (17) counterpart in laminar flows. Inspired by the Blasius classical 4 Guo et al. solution (Schlichting, 1979, p. 137; Kundu, 1990, p. 311) and correction is a good approximation for pipe axis. Similarly, this the conventional sine-square wake function, one may assume paper modifies (3) by adding the same correction function, πξ v = V 2 ξ 3 sin (24) − (33) 2 3κ in which V is the transverse velocity at the boundary layer edge. Comparing (24) with (12), one must have 2.3.5 The modified log–wake law and its defect form

V = χu∗ (25) Combining (3) and (33) leads to the following velocity profile model: in which χ is a proportional constant. With (25) Eq. (24) can be 3 u 1 yu∗ 2 πξ ξ rewritten as = ln + B + sin2 − (34) v πξ u∗ κ ν κ 2 3κ = η(ξ) = χ sin2 (26) u∗ 2 Similar to pipe flows (Guo and Julien, 2003), this paper calls The constant χ will be considered together with the derivative the above equation the modified log–wake law (MLWL), which function ∂F(ξ, 0)/∂η. should be valid from the overlap region till the boundary layer edge. Equation (34) is different from the conventional log–wake law in two aspects: it meets the zero velocity gradient at the 2.3.3 The derivative function ∂F(ξ, 0)/∂η boundary layer edge; and the additive constant B accounts for With (26) one can write the second term in (15) as the effect of the Reynolds number. ∂F (ξ, 0) πξ χ 2 To eliminate the effect of Reynolds number in (34), one can ∂η sin (27) 2 introduce the freestream velocity U at ξ = 1 to the modified which is the same as Hinze’s version of the law of the wake log–wake law. From (34), one obtains assuming U δu = 1 ∗ + B + 2 − 1 ∂F (ξ, 0) 2 ln (35) χ = u∗ κ ν κ 3κ ∂η κ (28) Subtracting (34) from (35) gives the velocity defect form of the and ∂F (ξ, 0)/∂η is independent of ξ. In other words, the second modified log–wake law term in (15) can be approximated by the conventional law of the wake, U − u 1 πξ 1 − ξ 3 =− ln ξ − 2 cos2 + (36) ∂F(ξ, 0) 2 πξ u∗ κ 2 3 η = 2 ∂η κ sin (29) 2 As aforementioned, both κ and should be universal constants According to Coles (Fernholz and Finley, 1996), the wake under the assumption of large Reynolds number. strength increases with Reynolds number and tends to a constant for large Reynolds number. To be consistent with (22) where an assumption of large Reynolds number is employed, one can 3 Test of the modified log–wake law assume This section first examines the universality of the modified = constant (30) log–wake law by plotting all data points according to the defect in this paper. Substituting (23) and (29) into (15) produces the form (36). It then tests the applicability of (36) to describe indi- conventional log–wake law (3) except that the additive constant B vidual velocity profiles in terms of the inner variable yu∗/ν. The varies with Reynolds number. 70 experimental velocity profiles by Osterlund (1999) in ZPG boundary layers with Reynolds numbers 900 ≤ Reδ ≤ 10,000 2.3.4 Boundary correction will be used in this test. The complete descriptions of the exper- Strictly speaking, boundary layers do not have edges; the mean imental apparatus and measured velocity profiles can be found ∼ velocity is only asymptotic to the free stream velocity at the so- on the web site http://www2.mech.kth.se/ jens/zpg/. Note that called boundary layer edges, i.e., u → U at ξ = y/δ = 1. Osterlund (1999) measured the wall shear stress by using an oil However, in practice the assumptions of film interferometry, which is independent of the logarithmic law. In the data of Osterlund (1999), the freestream velocity U, u(ξ = ) = U 1 (31) the shear velocity u∗ and the measured velocity profile u(ξ) are δ and given. The boundary layer thickness in this paper is interpo- u(y = δ) = . U lated by 0 999 . According to the defect form, all du = 0 (32) 70 measured velocity profiles are plotted in Fig. 1 where except dξ ξ=1 for the viscous sublayer and the buffer layer, all data points fall are good approximations. To meet the zero velocity gradient almost on a single curve down into the overlap layer. This reveals requirement (32), one must modify (3) by adding a boundary cor- that the velocity defect in the outer region including the over- rection function. Guo and Julien (2003) have shown that a cubic lap region is independent of Reynolds number. Furthermore, it Modified log–wake law 5

30 25 Modified log-wake law (36) Modified log-wake law (36) 25 Data of Osterlund (1999) Data of Osterlund (1999) 20

* 15

/ )

) 15

U ( ( 10 10

5 5

-3 -2 -1 0 1 0 10 10 10 10 10 0 0.2 0.4 0.6 0.8 1 ξ=y/δξ=y/δ Figure 1 Veriﬁcation of the universality of the velocity defect law.

1.2 boundary layer edge, say 30 ≤ yu∗/ν and y/δ ≤ 1; (c) the mod- 1 iﬁed log–wake law tends to a straight line in a semilog plot in the overlap region and then coincides with the logarithmic law; and 0.8 Π= 0.7577 ±0.07577 Π (d) the zero velocity gradient at the boundary layer edge can be 0.6 clearly seen from all proﬁles in Fig. 3(a–g) which imply that the 0.4 boundary correction is necessary.

0.2

0 3 4 4 Skin friction and the additive constant in the 10 10 Re =δu /ν logarithmic law δ * Figure 2 Verification of the universal constant . One can compute the velocity profile by using the velocity defect law (36), which does not require the additive constant B.Nev- κ implies that the model parameters and in the modified log– ertheless, if the modified log–wake law (34) is preferred, the wake law (36) are universal constants. A preliminary analysis additive constant B can be defined by studying the skin friction suggests that factor cf , which is defined as

κ = 0.4 and = 0.7577 (37) c U f 2 2 τw = ρU or = (40) allows Eq. (36) to fit the experimental profiles shown in Fig. 1 2 cf u∗ very well. Figure 2 further confirms Eq. (37) by plotting the wake Substituting (21) into (35) gives strength versus Reynolds number Reδ, in which the values of U for individual profiles are estimated by the measured value of = 1 + a + b + 2 − 1 u κ ln Reδ κ κ (41) u/u∗ at yu∗/ν = 100 at assuming κ = 0.4. ∗ 3 In terms of the inner variable yu∗/ν, one can rewrite (36) as which can be rearranged as u U πξ − ξ 3 1 2 1 U = + ln ξ − 2 cos + (38) 2 = = 1 + B u∗ u∗ κ 2 3 ln Reδ 1 (42) cf u∗ κ1 in which where yu∗/ν ξ = (39) 1 1 2 1 Reδ = + a and B1 = b + − (43) κ1 κ κ 3κ Obviously, the Reynolds number Reδ = δu∗/ν is a profile param- are determined experimentally. Figure 4 displays the experi- eter, which suggests that the modified log–wake law is Reynolds mental skin friction factor cf of Osterlund (1999) versus the number dependent in terms of the inner variables. Figure 3(a–g) Reynolds number Reδ according to (42). A least-squares curve compare (38), in which the constants in (37) are used, with all 70 fitting reveals that (42) with the constants experimental profiles individually and display excellent agreement for almost all profiles. Note that the dotted lines (MLWL) κ1 = 0.3820 and B1 = 6.6040 (44) are covered with the solid when yu∗/ν ≥ 30. These figures lead can represent the experimental data with a correlation coeffcient to the following conclusions: (a) the basic structure of the modi- 0.999. Substituting (37) and (44) into (43) produces fied log–wake law is correct; (b) the modified log–wake law can replicate the experimental data from the overlap region till the a = 0.1176 and b = 3.6544 (45) 6 Guo et al.

80 80 Data of Osterlund (1999) SW981008A Data of Osterlund (1999) SW981008B Modified log-wake law (38) Modified log-wake law (38) Logarithmic law (23) Logarithmic law (23) SW981006C Extension of MLWL (49) Extension of MLWL (49) SW981008C 70 or potential flow 70 or potential flow SW981006B SW981008D

SW981006A SW981012A 60 60

SW981005F SW981012B

50 SW981005E 50 SW981012C

* *

SW981005A SW981013A

u/u u/u

40 SW981005B 40 SW981013B

SW981005C SW981013D

30 30 SW981005D SW981013C

20 20

potential flow potential flow shift by 5 by shift 10 (a) 10 5 by shift (b)

0 0 0 1 2 3 4 5 0 1 2 3 4 5 10 10 10 10 10 10 10 10 10 10 10 10 yu / ν yu / ν * *

80 80 Data of Osterlund (1999) SW981113F Data of Osterlund (1999) SW981127G Modified log-wake law (38) Modified log-wake law (38) Logarithmic law (23) Logarithmic law (23) Extension of MLWL (49) SW981113E SW981127F 70 Extension of MLWL (49) or potential flow 70 or potential flow SW981113D SW981127E

SW981113C 60 60 SW981127D

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40 40 SW981112C SW981126E

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potential flow potential flow shift by 5 by shift 10 (c) 10 5 by shift (d)

0 0 0 1 2 3 4 5 0 1 2 3 4 5 10 10 10 10 10 10 10 10 10 10 10 10 yu /ν yu /ν * * Figure 3 Comparison of modified log–wake law with individual experimental velocity profiles. Modified log–wake law 7

80 80 Data of Osterlund (1999) Data of Osterlund (1999) SW981127Q Modified log-wake law (38) Modified log-wake law (38) SW981128J Logarithmic law (23) Logarithmic law (23) Extension of MLWL (49) SW981127P Extension of MLWL (49) 70 SW981128I or potential flow 70 or ptential flow

SW981127O SW981128H

SW981127N 60 60 SW981128G

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50 50

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SW981127K u/u u/u SW981128D 40 40 SW981127J SW981128C

SW981127I SW981128B 30 30 SW981127H SW981128A

20 20

potential flow potential flow shift by 5 by shift 10 (e) 10 5 by shift (f)

0 0 0 1 2 3 4 5 0 1 2 3 4 5 10 10 10 10 10 10 10 10 10 10 10 10 yu /ν yu /ν * *

80 Data of Osterlund (1999) Modified log-wake law (38) SW981129J Logarithmic law (23) Extension of MLWL (49) 70 or potential flow SW981129I

SW981129H

60 SW981129G

SW981129F 50

SW981129E *

u/u SW981129D 40 SW981129C

SW981129B 30

SW981129A

20 potential flow

10 5 by shift (g)

0 0 1 2 3 4 5 10 10 10 10 10 10 yu /ν * Figure 3 (Continued) 8 Guo et al.

32 only for velocity proﬁles but also for the momentum thickness and Data of Osterlund (1999) the skin friction factor. Note that Osterlund et al. (2000) obtained Equation (42) from MLWL

30 a small von Karman constant from the same data set since *

u they overlooked the effect of Reynolds number on the additive / U constant B.

= 28

f C

/ Finally, combining (42) and (47) provides a method to estimate 2 √ the wall shear stress τw and the boundary layer thickness δ from 26 a measured velocity profile. That is, (a) calculate the momentum thickness θ by applying a measured velocity profile to the 24 3 4 definition of the momentum thickness; (b) estimate the friction 10 10 c τ Re =δu /ν factor f from (47); (c) compute the wall shear stress w or the δ * shear velocity u∗ from (40); and (d) obtain the boundary layer c Figure 4 Skin friction factor f versus Reynolds number Reδ. thickness δ from (42). With this method, the wall shear stress and boundary layer thickness can be defined in an experimental which specify the additive constant B through (21). Like the program. modified log–wake law, the logarithmic law (23), to which (21) and (45) are applied, also fits the experimental data in the overlap region very well for most profiles, as shown by the dashed lines 5 Extension of the modified log–wake law in Fig. 3(a–g), which validate the hypothesis of the dependence of Reynolds number in (21). From the above analysis, one can see that the modified log–wake law indeed agrees with the experimental data in the outer region. The skin friction factor cf is often correlated with the momen- It is noteworthy that the modified log–wake law can be extended tum thickness Reynolds number Reθ , to the inner region by applying van Driest’s mixing-length model θU c = f( ); = (Schlichting, 1979, p. 604) for the law of the wall or the pri- f Reθ Reθ ν (46) mary function F(ξ,0). In other words, replacing the logarithmic in which θ = the momentum thickness. Applying (38) to the law in (34) with van Driest’s mixing-length model, one can get definition of the momentum thickness θ, one can derive a relation a complete velocity profile model for the entire boundary layer, between Reδ and Reθ (see the Appendix at the end of this paper). + u y y+ Applying (A7) in the Appendix to (42) leads to = 2d u +{ + κ2y+2[ − (−y+/A)]}1/2 ∗ 0 1 1 4 1 exp c 2 = 1 − 1 α f + β + B 2 πξ ξ 3 c κ ln Reθ κ ln 1 (47) + sin2 − (49) f 1 1 2 κ 2 3κ + in which the constants in (44) are used for the values of κ1 and in which κ = 0.4,= 0.7577,y = yu∗/ν and A is a damping B1, and factor,

α =−26.538 and β = 3.7693 (48) A ≈ 5B = 5(a ln Reδ + b) (50) are obtained from (A5) and (A6) in the Appendix. Equation (47) in which (21) has been used. Note that van Driest suggested agrees with the experimental data, as shown in Fig. 5, where A = 26 to reproduce the additive constant B ≈ 5.0. In the empirical law of Osterlund et al. (2000) is also plotted. The this paper, the additive constant B is a function of Reynolds excellent agreement validates the modified log–wake law (36) not number, the damping factor A is then modified as Eq. (50). Figure 3(a–g) reveal that the extension equation (49) can indeed clone the entire boundary layer velocity profile, including the -3 inner and outer regions. In brief, the extension of the modified x 10 4 log–wake law (49) agrees with experimental profiles well, satis- Data of Osterlund (1999) fies all boundary conditions at the wall and at the boundary layer Equation (47) from MLWL 3.5 edge, and connects to the constant potential velocity smoothly, Empirical law of Osterlund et al. (2000) as indicated by the solid lines in Fig. 3(a–g).

cf 3

6 Conclusions 2.5 Based on the boundary layer equations, dimensional analysis, 2 perturbation technique, a recent understanding of the overlap 0 0.5 1 1.5 2 2.5 3 3.5 4 layer and an analogy to the transverse velocity profile in lam- Re = θU/ν x 10 θ inar boundary layers, this paper proposes a modified log–wake c Figure 5 Skin friction factor f versus Reynolds number Reθ . law for the mean velocity profile of ZPG turbulent boundary Modified log–wake law 9 layers. The proposed law consists of three terms: a logarithmic Similarly, applying (A1) and (A2) to the definition of the term in which the von Karman constant is about 0.4 while the momentum thickness θ gives additive constant increases with Reynolds number; a sine-square θ 1 u u term with a constant wake strength about 0.76; and a cubic cor- = 1 − dξ δ 0 U U rection term. The logarithmic law reflects the effect of the wall u 1 u u 2 1 u 2 shear stress and is dominant in the overlap region; the sine-square = ∗ ξ − ∗ ξ U u d U u d function approximates the transverse velocity and then reflects 0 ∗ 0 ∗ the effect of convective inertia; and the cubic correction makes u∗ 2 u∗ = α + β (A4) the conventional log–wake law satisfy the zero velocity gradient U U requirement at the boundary layer edge. in which The proposed velocity profile law provides excellent agree- 81 3 Si(π) 1 4 2 32 ment with 70 recent experimental profiles not only for the mean α =− − + − + − 56κ2 4 π π 2 π 4 κ2 2κ2 velocity profiles but also for the skin friction factor. Specifically 1.4464 + 2.5585 + 1.52 the comparison shows that: (a) the modified log–wake law is =− =−26.538 (A5) κ2 independent of Reynolds number in terms of its defect form; that 1 3 0.75 + is, the von Karman constant and the wake strength are indeed β = + = = 3.7693 (A6) universal constants for large Reynolds number; (b) the modified κ 4 κ log–wake law is Reynolds number dependent when the inner vari- where the constants in (37) are applied. Furthermore, one can ables are used; (c) the proposed logarithmic law, with a variable show additive constant, has been validated by the data in the over- −1 δu∗ θU δ u∗ cf lap region; and (d) the friction factor derived from the modified Reδ = = = Reθ α + β (A7) ν ν θ U 2 log–wake law is accurate in terms of the momentum thickness. Finally, applying van Driest’s mixing-length model, in which the damping factor varies with Reynolds number, for the law of Notation the wall, the modified log–wake law can be extended to the entire boundary layer from the wall till the boundary layer edge. A = Van Driest’s damping factor a,b,c = Constants in the power law (16) B = Additive constant in the logarithmic law B = Appendix: The displacement thickness δ1 and the 1 Additive constant in the friction equation (42) momentum thickness θ cf = Skin friction factor F,f = Functional symbols = The displacement thickness δ1 and the momentum thickness θ Reδ Reynolds number based on the boundary layer are two important parameters in boundary layer analysis. They thickness, δu∗/ν can be estimated from the proposed modified log–wake law (38) Reθ = Reynolds number based on the momentum which gives thickness, θU/ν U = Freestream velocity 1 u 1 3 U u = Time-averaged velocity along the wall dξ =− + + (A1) 0 u∗ κ 4 u∗ u∗ = Shear velocity V = Transverse velocity at the boundary layer edge and v = Time-averaged velocity normal to the wall 1 u 2 W = Wake function dξ x = Coordinate along the wall 0 u∗ y = Coordinate normal to the wall 2 1 81 3 2 8 2Si(π) 3 + = + − + + + y = Inner variable, yu∗/ν κ2 56 2 π 2 π 4 π 2 α, β = Constants in the friction equation (47) U U 2 δ = Boundary layer thickness − 2 3 + + (A2) δ = κ 4 u∗ u∗ 1 Displacement thickness θ = Momentum thickness δ Applying (A1) to the definition of the displacement thickness 1, η = Transverse velocity distribution function one derives κ = The von Karman constant in the logarithmic law 1 κ1 = The von Karman constant in the friction law (42) δ1 u = 1 − dξ ν = Kinematic viscosity of fluid δ 0 U νt = Eddy viscosity u 1 u u = − ∗ ξ = 1 + 3 ∗ ξ = Relative distance from the wall, y/δ 1 U u d κ U (A3) 0 ∗ 4 = The Coles wake strength 10 Guo et al.

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