Modified log-wake laws for turbulent flow of the outer and inner regions in smooth pipes

188

Ser.B, 2007,19(2):188-194

MODIFIED LOG-WAKE LAWS FOR TURBULENT FLOW OF THE OUTER AND INNER REGIONS IN SMOOTH PIPES* LIU Ya-kun, NI Han-gen State key Laboratory of Coastal and offshore Engineering, Dalian University of Technology, Dalian 116024, china, E-mail: [email protected] (Received February 6, 2006; Revised May 17, 2006) ABSTRACT: Based on theoretical analysis two modified log-wake laws for turbulent flow in smooth pipes are obtained, one is applicable to the outer region and other one to inner region, the new law for the outer region fits the velocity profile measured in smooth pipes by Zagarola very well and the effect of Reynolds number can be taken into consideration, the velocity profile for inner region satisfies the wall boundary conditions ,equals zero at the pipe wall and smoothly joins up with the velocity profile in outer region, the adopted eddy viscosity model is consistent with Laufer’s, Nunner’s and Reichardt’s experimental data.

engineering view, its shortages are also quite clear. Such as the log law does not satisfy the boundary conditions at the wall surface and at the pipe center line, i.e., the velocity and velocity gradient are nonzero at the boundaries mentioned above, respectively. Besides these, the results of Nikuradse’s experiments already displayed that the measured velocity profile deviated from the

KEY WORDS: turbulent pipe flow, eddy viscosity, log-wake law, outer and inner regions

distance from the pipe wall, r is the pipe radius ) about larger than 0.15. Over the past twenty years, extensive experiment research with high accuracy had been performed on the mean flow and turbulence structure of open channels [1-4] and pipes[5-8]. All these experimental results confirmed the deviation of log law from that measured, further. Therefore, it is quite natural to correct the standard log law or to work out a new law. In 1956, Coles originally suggested to correct log law by use of the wake function. Since then, Dean[9], [10] [1] [2] Hinze ,Cardoso and Graf , Li and dong Guo[11], Guo[12] and others continuously examined and improved the characteristics of the wake function and the modified log-wake law. On the other side. Driest (1956) ,Gosse(1961), Stablewske(1968), Dou[13] and others adopted different eddy viscosity or mixing length models to establish new velocity distribution law. But their results were not to be popularized. This article is a new try along the

1. INTRODUCTION Despite the turbulent flows in the pipes has been studied by many researchers for a long time, there is still considerable debate on the existence of a log law. Prandtl (1925) with his mixing length hypothesis obtained the log law of mean velocity distribution in the boundary layer of plate. Some years later, Nikuradse (1932) completed a lot of measurements of velocity distribution in pipes and his results showed that the log law is able to describe the mean velocity profile in the pipes well, too. In 1938 Keulegan suggested that log law may describe the velocity profile for the entire flow depth of channel, also. Since then, the log law has been widely used to depict the velocity distribution in boundary layers, circular pipes and open channels. Though the log law is satisfactory to describe the velocity profile from the point of

standard log law in the region of ratio

* Biography: LIU Ya-kun (1968-), Female, Ph. D., Associate Professor

y (y is the r

189

Gosse’s way. A simple eddy viscosity model which is consistent with the experimental data is adopted and two new modified log-wake laws are obtained. One new law is applicable to the outer region and other one to inner region. The new law for the outer region fits the velocity profile measured in smooth pipes by Zagarola very well and the effect of Reynolds number can be taken into consideration. The velocity profile in inner region corresponding to the new law satisfies the wall boundary conditions, equals zero at the wall surface and smoothly joins up with the velocity profile in outer region.

−u'xu'y = ν e

du dy

(6)

where ν e is the eddy kinematical viscosity coefficient. From expressions (5) and (6) we obtain

y⎞ du du ⎛ u*2 ⎜ 1 − ⎟ = (ν m + ν e ) =νt dy dy ⎝ R⎠

(7)

ν t is the total kinematical viscosity coefficient. Near the wall ν t ≈ ν m and near the pipe axis ν t ≈ ν e . where

2. FUNCTIONAL FORM OF THE NEW MODIFIED LOG-WAKE LAWS The governing equation of the time-averaged velocity u in circular pipe may be written as .

1 du grJ + ν m − u'xu'r = 0 2 dr J =−

d ⎛ p ⎞ ⎜ ⎟ dx ⎝ ρ g ⎠

(1)

(2)

where g is the gravity acceleration, J is the hydraulic slope, ν m molecular kinematical viscosity coefficient, x and r is the cylindrical coordinates, u ′x ur′ is the Reynolds shear stress, p is the time-average pressure, ρ is the water density. There is a relationship between the shear velocity u* and hydraulic slope J

u* = gJRh

(3)

where Rh is the hydraulic radius. A variable y is introduced as

y = R−r

νt =

Ru* a (ξ + δ )

ξ + b (ξ + 1) 2

,ξ=

y R

(8)

Upon letting

u+ =

u u∗

(9)

and substituting Eq. (8) into Eq. (7) we get

(

2 du + (1 − ξ ) ξ + bξ + b = dξ a (ξ + δ )

)

(10)

where a, b and δ are coefficients waited to determine Integrating Eq.(10) the following expression can be obtained

(1 − b ) ξ + δ ξ 2 − 1 u + = [b ln (ξ + δ ) + ( ) a 2 (4)

where R is the radius of the pipe. By use of Eqs. (3) and (4), Eq. (1) becomes

y⎞ du ⎛ u* 2 ⎜ 1 − ⎟ = ν m − u'x u'y dy ⎝ R⎠

After comparisons we choose

( ξ + δ )3 ] + C 3

(5)

According to Boussinesq’s hypothesis we have

(11)

where C is the integration constant. Because δ is a very small quantity, i.e. δ ≈ 0 , so that Eq. (11) can be approximately simplified to

190

(1 − b ) ξ 2 − ξ 3 ] + C 1 u + = [b ln (ξ + δ ) + ξ a 2 From the condition of u +

ξ =1

(12)

+ = uCL , the value

of C may be determined immediately

1 ⎛ 1− b 1 ⎞ + C = uCL − ⎜ − ⎟ a⎝ 2 3⎠

(13)

+ where uCL is the non-dimensional velocity at the pipe axis. Therefore, the functional form of the new modified log-wake law will be

+

+ uCL

u =

1 1 + [b ln (ξ + δ ) + (1 − ξ 3 ) − a 3

(1 − b ) (1 − ξ 2 )] 2

(14)

It must be noticed that this modified log-wake law is not only for the outer region but the inner region and the parameters a and δ are adjustable for to satisfy with the different boundary conditions in the two regions.

3. DIVISION OF FLOW REGION AND SOME RELATIONSHIPS 3.1 Division of flow region It is still difficult to make Eq.(14) well fit the measured velocity distributions over the entire section by use of the constant parameters a , b and δ . According to the conventional way this paper divides the flow region into two regions, i.e., the outer region or core region and the inner region consisted of the viscous sublayer and the transition region. Referred to results of Zagarola’s experiment[6] performed in a fully developed, smooth pipe flow for Reynolds numbers from 3.1×104to 3.5 ×107 we adopt the inner region-outer region interface at

y+ =

yu∗

νm

= 25, it means that y + = 25 is the lower

bound of the outer region or the upper bound the inner region. By use of the relationship

u*2 =

λu m2

(15)

8

the non-dimensional lower bound of the outer region, ξ ol ,may be determined

ξ ol =

y0l 100 = R ReD

ReD =

2 λ

(16)

um D

(17)

νm

where λ is the drag coefficient, um is the mean velocity in section, D is the pipe diameter. It is clear from above mentioned that the inner region is from ξ = 0 to ξ ol and the outer region is from ξ ol to ξ = 1 . 3.2 Some relationships Let us now discuss the relationships between + + .From Eq.(15) we have u m , λ and u CL

8

u m+ =

(18)

λ

Based on the experimental results for Reynolds numbers between 3.1 × 104 and 3.5×107 , Zagarola proposed a new friction factor relation[6]. 1 ⎞ ⎛ = 1.884 log10 ⎜ Re p λ 2 ⎟ − 0.331 1 ⎜ ⎟ ⎝ ⎠ λ2

1

(19)

which has the same functional form as the one proposed by Prandtl but has different values for the coefficients. +

So that, if given ReD the value of um can be calculated using Eqs.(18) and (19). On the other

um+ may be determined from Eq. (14), too. Because of δ << 0 the + expression for um can be simplified as. hand, the mean velocity

0

+ = 2 u + 1 − ξ dξ = u + − 1 ⎛ 13 b + 7 ⎞ um CL ⎜ ⎟ ∫ ( ) 1

or

a ⎝ 12

60 ⎠

(20)

191

8

+ = u CL

λ

1 ⎛ 13

7 ⎞ ⎜ b+ ⎟ a ⎝ 12 60 ⎠

+

(21)

The data shown in table indicate that Eqs. (21) and (24) are consistent each other. In what follows Eq. (24) is used to determine the maximum + non-dimensional velocity u CL .

It is evident that if given ReD the value of

u

+ CL

can be obtained from Eqs. (18),(19) and (21). The parameters b and a are adopted as

b = 0.1255 + 0.0059 log10 ReD

(22)

a = 0.42b

(23)

+ u CL

Re* ⎞ ⎛ = 9.9 Re ⎜1 + ⎟ ⎝ 3720 ⎠

Re* =

RU*

νm

=

−

νt Ru*

1 16

(24)

λ ReD 32

(25)

3.5×107

3×106

3×105

3.1×104

λ

0.00711

0.01002 0.01466

0.02304

521708.43 53086.02` 6421.16

831.82

aξ

ξ + b (ξ + 1) 2

(26)

ξ =0 →

aξ = 0.42ξ b

(27)

which is consistent with the mixing length model. Near the pipe axis, ξ → 1 ,one has

νt Ru*

ReD

=

Near the pipe wall, ξ → 0 ,Eq.(26) reduces to

curve-fitting

Table 1 comparison between Eq.(21)and Eq.(24)

Re*

νt Ru*

Guo proposed an accurate + [14] empirical formula for u CL 1 8 *

4. VELOCITY PROFILE IN OUTER REGION In outer region, δ = 0 , Eq.(8) is simplified to

ξ →1 →

a 1 + 2b

(28)

The maximum value locates at ξ = b and the corresponding value is

νt Ru*

max =

0.42 b 2+ b

(29)

+ uCL 1 (Eq.24)

37.67

32.53

27.82

22.65

The comparison of Eq. (26) with the experimental data is shown in Fig. 1, the upper and

b

0.17

0.1637

0.1578

0.152

lower curve correspond to ReD = 3.5 × 107 and

a

0.0714

0.0688

0.0663

0.0638

+ uCL 1 (Eq.24)

37.76

32.53

27.7

23.04

-0.2

0

0.4

-1.7

+ + uCL 1 − uCL 2 + uCL 1

(%)

Now, there is a question that whether the Eqs. (19),(24) and (25) are consistent with Eqs. (19),(21),(22) and (23). In order to answer it, some calculated results are shown in Table 1 to make a comparison.

ReD = 3.1× 104 respectively, the eddy viscosity model proposed by Guo[12] are shown in Fig. 1, too. Corresponding to the eddy viscosity model (26) the velocity profile in the outer region may be directly obtained from Eq. (14)

1 1− b 1 + uCL (1 − ξ 2 ) − (1 − ξ 3 ) − − u+ = [ a 2 3 1− b blnξ ] = 2.381[ (1 − ξ 2 ) − 2b

1− ξ 3 − lnξ ], ξ ≥ ξol 3b

(30)

192

b =0.170, the comparison between them is shown in Table 2. Table 2

Comparison between Eq.(30) and Eq.(31)

u

+ CL

−u

for

+

Eq.(30) ξ

Eq.(31) ReD = 3.5 × 107

Fig.1 Comparison of the eddy viscosity model with experimental data [10,15]

Fig.2 Comparison between the calculated and measured velocity profiles for Reynolds numbers between 3.1× 104 and 3.5×107 (Data source: Zagarola and Smits [6])

Because parameter b is slightly dependent on Reynolds number so is the right side of Eq.(30). The calculated velocity profile for ReD = 3.5 × 107 and 3.1× 10 4 are shown in Fig. 2 and compared with the experimental data[12], they are in good agreement. Reference [12] obtained a velocity profile which satisfied the boundary condition at pipe axis.

⎛ 1− ξ 3 ⎞ πξ u − u = −2.354 ⎜ lnξ + 3 ⎟ + 2cos 2 ξ ⎠ 2 ⎝ + CL

+

(31) the right side of Eq.(31) is independent of Reynolds number, by which the calculation value is consistent very well with that one from Eq. (30) for

0.005

13.76

13.687

0.01

12.108

12.055

0.05

8.263

8.255

0.1

6.573

6.587

0.2

4.781

4.819

0.3

3.614

3.658

0.4

2.695

2.732

0.5

1.925

1.945

0.6

1.277

1.278

0.7

0.747

0.736

0.8

0.346

0.333

0.9

0.09

0.084

1

0

0

5.VELOCITY PROFILE IN THE INNER REGION Using subscript i to denote the quantity of the inner region, the parameters ai and δ i are determined as follows

ai = aw +

⎛

a − aw

ξ ol

δ i = δ w ⎜1 − ⎝

ξi , ξ ≤ ξ ol

ξ ⎞ ⎟ , ξ ≤ ξol ξol ⎠

(32)

(33)

it is clear that aw = ai |ξ = 0 and δ w = δ i |ξ = 0 , which are determined by the boundary conditions at the pipe wall

193

ui+ |ξ = 0 = 0

(34)

dui+ |ξ = 0 = Re* dξ

(35)

Reynolds numbers between 3.5 × 107 and 3.1 × 104,and it smoothly joins up with the velocity + profile in outer region at y =25.

substituting Eq.(34) into Eq.(14) and Eq.(35) into Eq.(8) and completing some simplification we get the Eqs. for δ w and aw +

δw +

1 ⎛ 1− b 1 ⎞ = ⎜ − δ w Re* b ⎝ 2 3 ⎟⎠

(36)

aw =

b Re*δ w

(37)

uCL

At last, substituting Eqs. (32) and (33) into Eq. (14) is obtained the velocity profile in the inner region. + u + = uCL +

ξol i ξol aw + ( a − aw ) ξ

{bln[ξ (1 −

In 1956, von Driest obtained a law described by an integrating form for the turbulent flow near a wall, but up to the date a corresponding explicit one still lacks for the pipe flow. The formula (38) may be the first suggested to depict the velocity profile for the pipe turbulent flow in the inner region.

δw

1 ) + δ w ] + (1 − ξ 3 ) − 3 ξ ol

1− b (1 − ξ 2 )}, ξ ≤ ξol 2

Fig. 3 A comparison of the calculated and measured velocity profiles normalized using inner scaling variables for different Reynolds numbers between 3.1×104 and 3.5 ×107

(38)

It still seems as a modified log-wake law. Because the inner region is consisted of the viscous sublayer and the transition region, so Eq. (38) applicable to the two regions. The calculated results from Eq. (38) for different Reynolds numbers are shown in Table 3. A comparison of the calculated velocity profiles for inner and outer regions and the measured ones only in the outer region normalized using inner scaling variables for different Reynolds numbers from 3.1×104 to 3.5×107 is shown in Fig.3. It can be seen from Table 3 and Fig. 3 that + + + the calculated value of u nears u =y when y+ ≤ 5. In the transition region from y+=5 to y+=25, the calculated value of u+ only slightly varies for

6. CONCLUSIONS (1) Based on an eddy viscosity model consistent with the experimental data and the theoretical considerations two modified log-wake laws are obtained. One is applicable to the outer region and other one to the inner region. (2) The new law for the outer region fits the velocity profiles measured in smooth pipes by Zagalora very well and the effect of Reynolds number can be taken into consideration. The calculated results show that Reynolds number only slightly affects the velocity profile normalized by shear velocity in the outer region. (3) The proposed velocity profile for the inner region satisfies the pipe wall boundary conditions, equals zero at the pipe wall and smoothly joins up with the velocity profile in outer region. The

194

Table 3

Velocity profiles calculated in inner region for different Reyndds numbers ReD 3×106 3.5×107

λ

0.0071

8.31×102

(Eq.24)

37.67

32.53

27.82

22.65

b

0.17

0.1637

0.0663

0.0638

a

0.0714

a

y

[3]

[4] [5]

[6] [7]

3

0.0663

4.8×10

4.71×10

3.89×10

3.00×10-3

w

5.8×10-6

5.98×10-5

5.36×10-4

4.6×10-3

w

0.0563

0.0515

0.0459

0.0398

+

-4

0.0638

ol

-3

1

0.974

1.172

1.245

1.113

3

2.82

3.09

3.227

3.159

5

4.301

4.625

4.857

4.814

10

7.177

7.532

7.887

7.846

25

12.839

13.077

13.289

12.883

REFERENCES

[2]

0.0688 -5

calculated results also show that Reynolds number only slightly affects the velocity profile normalized using inner scaling variables in the inner region.

[1]

4

0.023

6.42×10

δ

u

0.0147

5.31×10

ξ

+

3.1×104

5.21×10

Re* + uCL

0.01 5

3×105

CARDOSO A. H., GRAF W. H. and GUST G. Uniform flow in a smooth open channel[J]. Journal of Hydraulic Research, 1989, 27(5): 603-616. LI Xin-yu , DONG Zeng-nan. Turbulent flows in smooth-wall open channels with different slope[J]. Journal of Hydraulic Research, 1995, 33(3): 333-348. YANG S. Q. and LIM S. Y. Boundary shear stress distributions in smooth rectangular open channel flows[J]. Proc. Inst. Civ. Engrg. Waters, Maritime Engrg., 1998, 130(3): 163-173. GUO J. and JULIEN P. Y. Shear stress in smooth rectangular open-channel flows[J]. Journal of Hydraulic Engineering, ASCE, 2005, 131(1): 30-37. TOONDRE J. M., NIEUWSTADT F. T. M. Drag reduction by polymer add-itives in a turbulent pipe flow: numerical and laboratory experiments[J]. J. Fluid Mech. , 1997, 337: 193-231. ZAGOROLA M.V. and SMITS A. J. Mean-flow scaling of turbulent pipe flow[J]. J. Fluid Mech., 1998, 373: 33-79. O’CONNOR D. J. Inner region of smooth pipes and

[8]

[9] [10] [11] [12] [13] [14] [15]

open-channels[J]. Journal of Hydraulic Engineering, ASCE, 1995, 121(7): 555-560. ZAO Jin-yin et al. Investigation of the mean-flow scaling and Tripping effect on fully developed turbulent pipe flow[J]. Journal of Hydrodynamics, Ser. B, 2003, 15(1): 14-22. DEAN R. B. and BRADSHAW P. Measurements of interacting turbulent shear layers in a duct[J]. J. Fluid Mech., 1978, 78: 641-676. HINZE J. O. Turbulence[M]. 2nd Ed., New York: McGraw-Hill, 1975. GUO J. and JULIEN P. Y. Turbulent velocity profiles in sediment-laden flows[J]. Journal of Hydraulic Research, IAHR, 2001, 39(1): 11-23. GUO. J., JULIEN D. Y. Modified log-wake law for turbulent flow in smooth pipes[J]. Journal of Hydraulic Research, 2003, 41(5): 493-501. DOU Guo-ren. An unified law for various region of turbulent flow in open channels and pipes[J]. Science in China ,Ser. A, 1980, (11): 1115-1124(in Chinese). GUO J. Logarithmic Matching and its applications in computational hydraulics and sediment transport[J]. Journal of Hydraulic Research, 2002, 40(5): 555-565. OHMI M. and USUI T. Pressur and velocity distribution in pulsating turbulent pipe flow, Part 1, Theoretical treatments[J]. Bulletin of the JSME, 1976,19(12):303-313.