Journal of Hydraulic Research Vol. **, No. * (****), pp. 493 501

**** ************* *********** ** ********* Engineering and Research

Modi ed log-wake law for turbulent ow in smooth pipes

Loi log-train e modi e pour coulement turbulent en conduite paroi lisse

JUNKE GUO, Assistant Professor, Department of Civil Engineering, National University of Singapore, 10 Kent Ridge Crescent,

Singapore 119260. Email: abqp36@r.postjobfree.com

PIERRE Y. JULIEN, Professor, Engineering Research Center, Department of Civil Engineering, Colorado State University,

Fort Collins, CO 80523, USA. Email: abqp36@r.postjobfree.com

ABSTRACT

A modi ed log-wake law for turbulent ow in smooth pipes is developed and tested with laboratory data. The law consists of three terms: a log term,

a sine-square term and a cubic term. The log term re ects the restriction of the wall, the sine-square term expresses the contribution of the pressure

gradient, and the cubic term makes the standard log-wake law satisfy the axial symmetrical condition. The last two terms de ne the modi ed wake

law. The proposed velocity pro le model not only improves the standard log-wake law near the pipe axis but also provides a better eddy viscosity

model for turbulent mixing studies. An explicit friction factor is also presented for practical applications. The velocity pro le model and the friction

factor equation agree very well with Nikuradse and other recent data. The eddy viscosity model is consistent with Laufer s, Nunner s and Reichardt s

experimental data. Finally, an equivalent polynomial version of the modi ed log-wake law is presented.

R SUM

Une loi log-tra n e modi e est d velopp e pour d crire les coulements turbulents dans les conduites parois lisses. La loi modi e consiste en trois

termes: un terme logarithmique, un terme de tra n e, et un terme cubique de condition limite axiale. Le terme logarithmique repr sente les conditions

de la paroi, le terme de tra n e d nit le gradient de pression de l coulement, et le dernier terme satisfait la condition limite au centre de la conduite.

Le terme de tra n e d crit physiquement les conditions de m lange turbulent caus es par le gradient de pression. Le coef cient de frottement et le

coef cient de viscosit tourbillonnaire sont d crits de fa on analytique. Les quations de pro ls de vitesse et de coef cient de frottement sont en

accord avec les donn es exp rimentales de Nikuradse et autres donn es r centes. Le mod le de viscosit tourbillonnaire est consistent avec les donn es

exp rimentales de Laufer, Nunner et Reichardt. Finalement, une version polynomiale de la loi log-tra n e modi e est galement pr sent e.

Keywords: Pipe ow; turbulence; logarithmic matching; log law; log-wake law; velocity pro le; velocity distribution; eddy viscosity;

friction factor.

1 Introduction This study emphasizes the velocity pro le away from the pipe

wall where yu / > 30 given = uid kinematic viscosity. The

Fully developed turbulent ow in circular pipes has been inves- law of the wall or the logarithmic law proposed by von Karman

tigated extensively not only because of its practical importance, and Prandtl (Schlichting, 1979, p. 603) is widely used in this

but also for the extension of the results to open-channel ows and region. For smooth pipe ow, it is written as

boundary layer ows. The rst systematic study of the velocity

u 1 yu

= ln +A (2a)

pro le in turbulent pipe ows may be credited to Darcy in 1855

u

(Schlichting, 1979, p. 608) who deduced a 3/2-nd-power velocity

where = the von Karman constant. Nikuradse (1932) has

defect law from his careful laboratory measurements, i.e.

veri ed (2a) with his classical experiments. When (2a) was

umax u y 3/2

= 5.08 1 (1) rst proposed, the value of was suggested to be 0.4. Most

u R

researchers subsequently preferred the value of 0.41 (Nezu and

in which umax = the maximum velocity at the pipe axis, u =

Nakagawa, 1993, p. 51). Recently, Barenblatt (1996, p. 278)

the time-averaged velocity at a distance y from the pipe wall,

showed that for very large Reynolds number, the value of von

u = the shear velocity, and R = the pipe radius. Equation (1)

Karman constant tends to

is seldom used because it is invalid near the pipe wall with =

2

y/R

3e

(White, 1991, p. 415) or O Connor s law (1995) may be applied.

Revision received February 17, 2003. Open for discussion till February 29, 2004.

493

494 Junke Guo and Pierre Y. Julien

deviation from the log law, de ne a law of the wake. It is noted

that although (5a) or (5b) satis es the axial symmetrical condi-

tion, the constant derivative of the linear correction term brings

an additional shear stress in the near wall region, which slightly

perturbs the law of the wall. It is therefore concluded that the

linear function is not the best axial boundary correction.

The purpose of this paper is to develop a physically-based

velocity pro le model for turbulent pipe ows called the modi ed

log-wake law. It is proposed to improve upon Eq. (5b) in the light

of a better physical interpretation of the wake law. The proposed

modi ed log-wake law is also tested with experimental velocity

Figure 1 The law of the wall or the logarithmic law.

measurements in pipes. Analytical relationships for the turbulent

eddy viscosity and for the friction factor are then compared with

The value of A varies in literature; it is about 5.29 0.47 accord-

experimental data.

ing to Nezu and Nakagawa (1993, p. 51). Equation (2b) will be

examined later in this paper.

Equation (2a) is applicable away from the wall where yu / > 2 Development of the modi ed log-wake law

30, as shown in Fig. 1. It is not only valid for steady ow, but

is also frequently used as a reference condition in unsteady ow This section formulates the modi ed log-wake law in smooth

simulations (Ferziger and Peric, 1997, p. 277). This is because pipes. First a theoretical analysis is considered. Section 2.2

even in unsteady ows, the wall shear stress predominates in the proposes the basic structure of the modi ed log-wake law.

near-wall ow, and the in uence of inertial forces and pressure Section 2.3 de nes the axial boundary correction function. The

gradient are vanishingly small. nal formulation of the modi ed log-wake law is written in

Laufer (1954) found that the logarithmic law (2a) actually velocity-defect form in Section 2.4.

deviates from experimental data when = y/R > 0.1 0.2.

Coles (1956) further con rmed this nding and claimed that

the deviation has a wake-like shape when viewed from the 2.1 Theoretical analysis

freestream. Thus, he called the deviation the law of the wake.

Consider fully developed turbulent ow with homogeneous den-

Based on Coles digital data, Hinze (1975, p. 98) proposed the

sity through a pipe of radius R . For convenience, cylindrical

following expression for the wake function, i.e.

coordinates are used with the x -axis coinciding with the axis of

2

W = sin2 the pipe, as shown in Fig. 2(a). One can show that the continuity

(3)

2 equation is automatically satis ed and none of the ow variables

in which is Coles wake strength. Finally, one can modify the depend on . The momentum equation in the r -direction gives

logarithmic law by adding the wake function, i.e. that the pressure p is a function of x alone, i.e.

u 1 yu 2 p

= +A + sin2

ln (4) =0 (6)

u 2 r

This is called the log-wake law. When applied to pipe ows, in which p = dynamic pressure. The only nonzero component

the reader can easily show that the log-wake law (4) does not of velocity is the axial velocity u(r) or u(y) where y = distance

satisfy the axial symmetrical condition, i.e., the velocity gradient from the wall, i.e. y = R r . With reference to Fig. 2(b),

is nonzero at the pipe axis. Besides, the physical interpretation for steady and incompressible ow, the force balance in the x -

of the wake function is not clear in pipe ows. direction gives

Similar to ows in narrow open-channels (Guo and Julien,

2 (R y)dx + ( + d ) 2 (R y dy)dx

2001), to correct the velocity gradient at the pipe axis, Guo (1998)

dp [(R y)2 (R y dy)2 ] = 0

proposed the following velocity pro le model for pipe ows,

u 1 yu 2

in which = local shear stress. Neglecting the 2nd order terms,

= ln +A+

sin2 (5a)

u 2

the above equation reduces to

The law of the wall Law of the wake

2 dxdy + d 2 (R y)dx dp 2 (R y)dy = 0

or

umax u 1 2

= (ln + 1 ) + cos2 Dividing by 2 dxdy, the above becomes

(5b)

u 2

d dp

+ (R y) (R y) = 0

in which the Coles wake strength is due to the pressure-

dy dx

gradient. The log term in (5a) expresses the restriction of the

This can be rearranged as

wall, the sine-square term indicates the effect of the pressure-

d [(R y) ]

gradient, and the last term re ects the axial boundary correction. dp

(R y) + =0 (7)

According to Coles, the last two terms in (5a), which are the dx dy

Modi ed log-wake law for turbulent ow in smooth pipes 495

Figure 2 Scheme of a developed turbulent pipe ow.

This is the momentum equation in the x -direction. Integrating the second term, one can apply (9) to (13) and eliminate the

the above with respect to y and applying the wall shear stress w pressure-gradient. Integrating (13) gives

at y = 0 gives ( 2)d

u d

= + (14)

(1 )f (1 )f u

R dp (2R y)y

R w

= + (8)

R y 2 dx (R y)R Clearly, the solution of the above equation requires the knowledge

of f that is complicated and unknown. Therefore, this paper

Although the relation tries to construct an approximate velocity pro le model, based

on a physical and mathematical reasoning.

R dp

w = (9)

2 dx

2.2 Approximation of the velocity pro le model

which can be found from ( = 1) = 0, can further reduce (8)

to a linear function, the current form in (8) has a clear physical The effect of the wall shear stress is often expressed by the law

interpretation. Its rst term expresses the effect of the wall shear of the wall (2a). This implies that the rst integral of (14) must

stress, and the second term re ects the effect of the pressure- reduce to the logarithmic law (2a) near the wall. It is then assumed

gradient. Near the wall, the effect of the pressure gradient can be that the rst integral can be approximated by

neglected, and the uid shear stress is balanced by the wall shear

d 1 yu

= + A + F1 stress. ln (15)

(1 )f Applying the eddy viscosity model,

in which F1 is a correction function of the effect of the wall on

du the core ow region. Obviously, F1 must satisfy

t = t (10)

dy

F1 ( = 0) = 0 (16a)

in which t = turbulent shear stress, = uid density, and

and

t = eddy viscosity, to (8) and neglecting the viscous shear

F1 ( = 0) = 0

stress gives (16b)

R dp (2R y)y Condition (16a) keeps F1 negligible near the wall, and condition

du R w

= +

t (11)

R y 2 dx (R y)R (16b) guarantees F1 does not bring a shear stress to the near wall

dy

region.

According to previous experience (Hinze, 1975, p. 730), one can For simplicity, one can de ne the second integral of (14) as

assume an eddy viscosity as

( 2)d

F2 = (17)

(1 )f w y

t = R f (12)

R Like F1, F2 must be negligible and not bring a shear stress near

the wall, i.e.

in which f is an unknown function. Applying the de nition of

F2 ( = 0) = 0

the shear velocity u = w / and the normalized distance (18a)

= y/R into (11) and (12) leads to

and

(2 )

1 du 1 R dp 1 F2 ( = 0) = 0 (18b)

= + (13)

(1 )f 2 dx u2 (1 )f u d

This is equivalent to removing the wall restriction in the ow

This shows that the pipe velocity pro le is a result of the effects direction. Therefore, the effect of the pressure gradient can be

of the wall shear stress (the rst term) and the pressure-gradient considered a wall-free shear, like a jet, except that the pressure

(the second term). After making clear the physical meaning of gradient is the driving force in pipe ows while the inertia is

496 Junke Guo and Pierre Y. Julien

the axis = 1. In terms of Taylor series, all odd derivatives at

the driving force in a developed jet. Furthermore, the pipe ow

= 1 must be zero, i.e.

can be considered a superposition of a wall-bounded shear and a

wall-free shear. Because of the symmetrical condition, F2 must

1 d 2u 1 d 4u

u = umax + (1 )2 + (1 )4 +

reach its maximum value at the pipe axis and satisfy

2! d 2 4! d 4

=1 =1

F2 ( = 1) = 0 (18c)

To correct the modi ed log-wake law to the third order term,

According to (18b) and (18c), one can approximate the derivative letting

of F2 as

d 3u

=0

F2 sin (19) d 3 =1

The integration of (19) with the boundary condition (18a) gives in (21) where (25) gets applied, one can show that

2

n=3

F2 = sin2 (26)

(20a)

2

or

2.4 The modi ed log-wake law and its defect form

( 2)d 2

= sin2 (20b)

(1 )f 2 Combining (21), (22), (25) and (26) leads to the following

in which the constant 2/ gets buried in in the integration modi ed log-wake law:

and is introduced as per Coles wake function. Clearly, the

3e 3

u 3e yu 2

= + A + 2 sin

sine-square function in pipe ows is due to the effect of pressure- ln (27)

u 2 2 23

gradient. The value of might vary with a Reynolds number

the modi ed law of the wake

the law of the wall

slightly, but a universal constant might be good enough for large

Since the last two terms in the above equation express the devia-

Reynolds number ows.

tion from the law of the wall, following Coles (1956), they de ne

Substituting (15) and (20b) into (14) gives

the law of the wake in this paper, i.e.

u 1 yu 2

= +A + + F1 sin2

ln (21)

3e 3

2

u 2

W = 2 sin (28)

2 23

The last two terms disappear near the wall, thus the values of

To distinguish it from the standard sine-square wake law, this

and A should be the same as those in the law of the wall. It is

then suggested = 2/( 3e) 0.42. The value of A will be paper calls (28) the modi ed wake law. Furthermore, (27) is

called the modi ed log-wake law that is graphically represented

replaced with the maximum velocity umax by using the velocity

by Fig. 3. Equation (27) has at least two advantages over the

defect formulation in Section 2.4. Besides, it is shown later that

= ts the modi ed log-wake law well with standard log-wake law. It shows that the law of the wake in pipes

the value of

results from the pressure-gradient and the axial symmetrical con-

experimental data. Therefore, this paper assumes

dition. It meets the symmetrical condition at the pipe axis where

2

= = 0.42 (22) the standard log-wake law fails.

3e

To eliminate A from the modi ed log-wake law (27), one can

introduce the maximum velocity umax at the axis = 1 to the

2.3 The axial boundary correction modi ed log-wake law. From (27), one obtains

Because of the axial symmetry, the velocity gradient must be zero umax 3e Ru 1

= +A+2

ln (29)

at = 1. From (21), one has u 2 3

1 du 1

= + F1 (1) = 0 Furthermore, eliminating A from (27) and (29) gives the velocity

u d

=1 defect form of the modi ed log-wake law

which gives

umax u 1 3

3e

= ln + + 2 cos2 (30)

1 2 3 2

u

F1 (1) = (23)

This is the most important result of this study.

From (16b) and (23), one can assume

n 1

F1 = (24)

3 Determination of the maximum velocity and comparison

with recent experiments

in which n > 1. Integrating the above equation and applying

(16a) gives

3.1 Determination of the maximum velocity

n

F1 = (25)

n The maximum velocity in (30) plays the role of wall friction that

Since pipe ows are completely symmetrical about the axis, will be discussed in Section 5. This section directly correlates the

mathematically, the pipe velocity pro le is an even function about maximum velocity umax with Reynolds number Ru / . A plot of

Modi ed log-wake law for turbulent ow in smooth pipes 497

Near wall region

R

Core region

=1

Pipe

axis

y

Near wall region

(a) The effect of (b) The effect of the (c) The boundary (d) The composite

the wall pressure-gradient correction velocity profile

Figure 3 Components of the modi ed log-wake law.

3.2 Comparison with recent experiments

This section rst examines the applicability of (30) to describe

individual velocity pro les. The universality of the parameters

is then tested by plotting all data points according to the defect

form (30). Zagarola (1996) at Princeton University measured 26

mean velocity pro les with different Reynolds numbers between

3.1 104 and 3.5 107 . The test pipe was smooth and had a

nominal diameter of 129 mm. The complete descriptions of the

experimental apparatus and experimental data can be found on

the web site http://www.princeton.edu/ gasdyn/ or in Zagarola

(1996).

To illustrate the procedures of analysis, take Run 16 for an

example where R = 6.47 cm, u = 0.7 m/s and = 1.07

10 6 m2 /s. One can calculate that

Ru

= 45290

Figure 4 The relation of umax /u versus Ru / in loglog coordinates.

and from (31c) one gets

Nikurasde s (1932) classical data and the recent Princeton Uni-

versity experiments (Zagarola, 1996) is shown in Fig. 4. One can umax

= 31.2

see that for Ru / 2 104, the data ts the following power law

well, Using the above procedures, all 26 pro les are obtained and plot-

ted in Fig. 5. One can conclude that: (i) the basic structure of the

1/16

modi ed log-wake law is correct; (ii) Eq. (30) can replicate the

umax R u

= 16.55 (31b)

experimental data very well; (iii) the empirical Eq. (31c) for the

u

maximum velocity works very well; (iv) the modi ed log-wake

According to Guo s (2002) logarithmic matching, an accurate law tends to a straight line in the semilog plot near the wall and

curve- tting equation is then obtained for any Reynolds number, then coincides with the log law there; and (v) the zero velocity

gradient at the pipe axis can be clearly seen from all the experi-

1/16

1/8

mental pro les which show that the axial boundary correction is

umax R u 1 Ru

= 9.9 1+ (31c)

necessary.

u 3720

Besides, according to the defect form, all 26 pro les including

where the shape transition parameter = 1 in Guo s (2002) 1040 data points are also plotted in Fig. 6 where all data points

method provides excellent agreement with experimental data, as fall in a narrow band. This shows that the model parameters,

shown in Fig. 4. and n are universal constants.

498 Junke Guo and Pierre Y. Julien

1

100

1 2 3 4 5 17 6 7

0.8

10 1 0.6

RUN Re

4

3.16 10

1

4.17 104

2

0.4

4

5.67 10

3

4

7.43 10

4

4

9.88 10

10 2

5

1.46 105

6 0.2

1.85 105

7

6

3.10 10

17

12 3 4 5

5 17 6 7

0

0 5 10 15 20 25

0 5 10 15 20 25

u (m/s)

u (m/s)

100 1

18 19 8 9 10 11

0.8

10 1 0.6

RUN Re

0.4

5

2.30 10

8

5

3.10 10

9

10 2 4.09 105

10

0.2

5

5.39 10

11

4.42 106

18

18 19 8 9 10 11 6

6.07 10

19

0

0 5 **-**-**-**-*-*-** 15 20 25

u (m/s) u (m/s)

100 1

12 13 14 15 16

0.8

10 1 0.6

RUN Re

0.4

5

7.52 10

12

6

1.02 10

13

10 2 1.34 106

14

0.2

1.79 106

15

12 13 14 15 16 6

2.35 10

16

0

0 5 **-**-**-**-*-*-** 15 20 25

u (m/s) u (m/s)

100 1

**-**-**-**-**-** 26

0.8

10 1 0.6

RUN Re

0.4

7.71 106

20

7

1.02 10

21

1.36 107

22

10 2 1.82 107

23

0.2

7

2.40 10

24

2.99 107

25

**-**-**-**-**-** 26

7

3.53 10

26

0

0 *-**-**-**-**-** 35 0 *-**-**-**-**-** 35

u (m/s)

u (m/s)

Figure 5 Comparison of the modi ed log-wake law (30) with Zagarola s (1996) experimental data through individual pro les.

= w (1 )

4 Implication for eddy viscosity (33)

Neglecting the viscous shear stress, from (10) and (33), one can

Eddy viscosity is important when studying turbulent mixing.

show that the eddy viscosity can be expressed by

With the modi ed log-wake law, the eddy viscosity can now

be determined. First, applying (9) to (8) gives the shear stress 1

t

= (34)

distribution: Ru (1/u )(du/d )

Modi ed log-wake law for turbulent ow in smooth pipes 499

(b) 14

(a) 14

12

12

10

10

(umax u)/u*

u)/u

8

8

max

6

6

(u

4

4

Modified log wake law (30)

Data of Zagarola (1996) 2

2

0

03 2 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

10 10 10 10

Figure 6 Veri cation of the model universal constants with the recent Princeton University experiments.

0.1

which is consistent with the mixing length model. Near the pipe

axis, 1, one has

0.09

sin (1 )

sin

0.08

=

1 1

0.07

Substituting it into (35) yields

0.06

/(Ru )

t 1

0.05

= 0.059 (36b)

5

u = 1.05 m/s, Re = 4.1 10, Laufer

(3 3e/2) + 2

t

Ru

4

u = 0.13 m/s, Re = 4.1 10, Laufer

0.04

4

u* = 0.49 m/s, Re = 3 10, Nunner

The constant eddy viscosity near the axis corresponds to an

0.03 u* = 0.15 m/s, Re = N/A, Reichardt

asymptote of a parabolic law (Hinze, 1975, p. 732).

u = 0.20 m/s, Re = N/A, Reichardt

0.02

u = 0.45 m/s, Re = N/A, Reichardt Equation (35) may be the best result so far for the eddy vis-

The eddy viscosity model (35)

0.01

cosity model in pipe ows. It may also be used to study some

Asymptotes (36a) and (36b)

complicated turbulent ows such as a wave turbulent boundary

0

0 0.2 0.4 0.6 0.8 1

= y/R layer ow. All previous velocity pro le models, including the

log law, the log-wake law and the power law, cannot produce

Figure 7 Comparison of the eddy viscosity model with experimental

the maximum eddy viscosity at 0.3 and the constant eddy

data (Data source: Ohmi and Usui, 1976).

viscosity near the axis.

The velocity gradient from the modi ed log-wake law (30) is

5 Friction factor

1 du 3e 1

= 2 + sin

The friction factor is an essential parameter in pipe designs and

u d 2

numerical simulations. It is de ned as

Substitution into (34) gives

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