

16.3.2 Laminar Flow with Roughness

e1

e1

60f = for a 90 V shape channel. Roughness 0.3023 mm R33f = R

←

10

Laminar Transitional Turbulent

102 103 104 105 1062 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8

0.004

0.006

0.008

0.01

0.02

0.04

0.06

0.08

0.1

0.2

0.4

0.6

0.8

1.0

Re1

f

f = 14___Re1

f = 24___Re1

Reference:"Chow Ven Te- Open ChannelHydraulics", Mc Graw Hill Company, International student edition, 1959, page - 10

Variation of friction coefficient f with Reynolds number Re1

in smooth channels=vR__

υ( )



f = 14___Re1

f = 33___Re1

f = 60Re1

___

10 102 103 104 105 1062 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8

Re1

Variation of friction coefficient f with Reynolds number Re1

in rough channels=vR__

υ( )

1072 4 6 80.006

0.008

0.01

0.02

0.04

0.06

0.08

0.1

0.2

0.4

0.6

0.8

1.0

f

37.5 cm

25 cmVarwick

Varwick

11

20 cm

Laminar Transitional Turbulent

103 104 105 1062 4 6 8 2 4 6 8 2 4 6 8 1072 4 6 8

0.02

0.04

0.06

0.08

0.1

Reference:"Chow Ven Te- Open Channel Hydraulics",Mc Graw Hill Company, Internationalstudent edition, 1959, page - 11

Rectangular Channel - Rough flow (Roughness = 0.7188)

Bazin conducted experiment using (500 measurements were made at greatest care)

(1) Gravel embedded in cement.

(2) Unpolished wood roughened by transverse wooden strip

(i) 27 mm long * 10 mm high * 10 mm spacing.

(ii) 27 mm * 10 mm at 50 mm spacing.

3) Cement lining

4) Unpolished wood

If the behavior of n and C is to be investigated then a number of basic definitions

regarding the types of hydrodynamic flow must be recalled.

Flow can be divided into



(i) Hydro dynamically smooth turbulent flow

(ii) Hydro dynamically Rough turbulent flow

(iii) Hydro dynamically transition turbulent flow.

1 2o

1 57o

e

The boundary layer δ for flow past a flat plate is given by

V xδ 5 Laminarx υ

V xδ 0 38 turbulent R 2 10 logarthmic velocity law holdsx υ

/

/. *

−

−

⎛ ⎞= ⎜ ⎟⎝ ⎠

⎛ ⎞= >⎜ ⎟⎝ ⎠

Velocity

V

99% V

y

δ

Velocity distribution

δ∗

δ0 Viscous sub layer

Transitional region

TurbulentPseudo boundary

y



δ

δ0 δ0 δ0k

kc kc

k

kc

Different surface roughness(c) rough

k

kc = υ

v*

__100

kc = 5υ

v*

__Smooth

for average condition

kc is critical roughness heightk is roughness height

(b) wavy(a) Smooth



Viscous sublayer

ks

(i) Hydrodynamically smooth turbulent flow f = f(Re)

Viscous sublayer

ks

Viscous sublayer

ks

(ii) Hydrodynamically transition flow f = f (Re, ks/y)

(iii) Hydrodynamically rough turbulent flow f = f (ks/y)

For hydro dynamically smooth condition, viscous sub layer submerges the roughness

elements.

For hydro dynamically transitional case the roughness element are partly exposed with

reference to viscous sub layer.

For hydro dynamically rough turbulent flow the roughness elements are completely

exposed above the viscous sub layer.

For hydro dynamically rough turbulent flow resistance is a function of Reynolds number

and the roughness height.

If we define e*R = shear Reynolds number * sv Kυ

. ; and o* f

τv gRSρ

= = .



The flow is classified as follows:

* s

* s

* s

v K 4 Hydrodynamically smoothυ

v K4 100 Hydrodynamically transitionυ

v K 100 Hydrodynamically fully developed turbulent flow υ

<

< <

>

Summary of Velocity-Profile Equations for Boundary layers with dp 0dx =

Zone Smooth Walls Rough Walls Law of the wall

Universal equations

Laminar sub layer ( y δ≤ )

*v y 4υ

< *

*

v yvv υ

= -

Buffer zone

*v y4 30 to 70υ

< <

-

-

Logarithmic zone (also called turbulent layer)

*v y 30 to 70

y 0.15

υ

δ

>

<

*

*

*

*

v yv A log Bv

v yv 5.6 log 4.9v

υ

υ

= +

= +

*

*

v kA log Bv y

v k5.6 log Bv yB f

= +

= − +

=

(roughness size, shape and distribution)

Velocity-defect law Inner region (overlaps with logarithmic wall law)

y 0.15δ

<

Outer region (approximate formula)

y 0.15δ

<

*

*

V v yA log Bv

V v y5.6 log 2.5v

δ

δ

−= − +

−= − +

*

*

V v yA logv

V v y8.6 logv

δ

δ

−= −

−= −

(3000 < eR < 70,000)

outer region

-

Power Law 1

7*

*

v v y8.74v υ

⎛ ⎞= ⎜ ⎟⎝ ⎠

-

A and B are constants.



Table shows velocity distributions for different conditions

Pipe flow equation eVRRυ

= Open channel flow eVRR 4υ

=

Blasius equation for smooth flow

5e0.25

e

e

5e

0.3164f = upto R <10R

R f1 =2log 2 51f

R 10

.

>

e

e

1/8 2

0.25

e

e

e

C=18.755 R mks units for g = 9.806 m/sec

0 223fR

R 8C = 4 2g log

2 51 CR 8g

C = 17.72 log 2 51 C3.5294RC = 17.72 log

C

.

g.

.

⎡ ⎤⎣ ⎦

=

⎡ ⎤⎢ ⎥⎣ ⎦

Smooth pipe flow Nikuradse Rough pipe Nikuradse

( )

o

1 = 0.86 ln Re f - 0.8f

1 = 1.14 - 0.86 lndf∈

e

s

R 8gC = 2 log C 2 518g

C 12R = 2 log k8g

*

.

⎡ ⎤⎢ ⎥⎣ ⎦

White and Colebrook formula

o/d1 2.51= 0.86 ln3 7f Re f.

∈⎡ ⎤+⎢ ⎥⎣ ⎦ s

e

2.52 8gkC = -2 log 14.83R8g R f

⎡ ⎤+⎢ ⎥

⎢ ⎥⎣ ⎦

Suggested modification to equation is

s

e

kC 2.5 = -2 log 12R8g R f

⎡ ⎤+⎢ ⎥

⎣ ⎦

[ASCE Task Force Committee 1963]. R is hydraulic mean radius, 4R = Diameter of

pipe.

In open channel flow following aspects come into picture

( )ef = f R K, C,N, F,U (1) (2) (3)

,

In which Re is the Reynolds number, K is the Relative Roughness, C Shape factor of the

cross-section, N is the Non- uniformity of the channel both in profile and in plan, F is the

Froude number, U is the degree of unsteadiness.

In the above equation, the first term corresponds to, Surface Resistance (Friction), the

second term corresponds to wave resistance and the third term corresponds to Non

uniformity due to acceleration/ deceleration in flow.



Surface Resistance: To be accounted based on Karman - Prandtl - velocity distribution.

The constant in resistance equation is due to the numerical integration, and is a function

of shape of the cross-section.

C 1 R=A log +By'2g f

For circular section A = 2.0, B = -0.62For rectangular section: A = 2, B = -0.79 (for large ratio of width/depth)

=

It has remained customary to delineate roughness in terms of the equivalent sand grain

dimensions ks. For its proper description, however, a statistical characteristic such as

surface texture requires a series of lengths or length derivatives, though the significance

of successive terms in the series rapidly approach a minimum. Morris classified the flow

into three categories namely (1) isolated roughness flow, (2) Wake interference flow,

and (3) Quasi smooth flow. The figure provides the necessary details.

s

y

k

Isolated - roughness flow (k/s) - Form drag dominates

s

The wake and the vortex are dissipated before the next elementis reached. The ratio of (k/s) is a significant parameter forthis type of flow



s s

Wake interference flow (y/s)

j j j

Quasi smooth flow - k/s or j/s becomes significant acts as Pseudo wall

s

y

k

y

k

s s s

j

k is surface roughness heights is the spacing of the elementsj is the groove widthy is the depth of flow

Concept of three basic types of rough surface flow

When the roughness elements are placed closer, the wake and the vortexat each element will interfere with those developed by the following element and results in complex vorticity and turbulent mixing. The height of the roughness is not important, but the spacing becomes an importantparameter. The depth 'y' controls the vertical extent of the surface region ofhigh level turbulence. (y/s) is an important correlating parameter.

Quasi smooth flow is also known as skimming flow. The roughness elementsare so closed placed. The fluid that fills in the groove acts as a pseudo walland hence flow essentially skims the surface of roughness elements. In sucha flow (k/s) or (j/s) play a significant role.

k, j, s should describe the characteristics of roughness in one dimensional situations is

Areal concentration of or density distribution of roughness elements. (after Moris).



16.3.3 Areal concentration or Density Distribution Roughness

Elements

SpheresSpatial distribution of roughnessSchlichting, 1936

Koloseus (1958) and Koloseus and Davidian (1965)conducted experiments using Cubical Roughness Symmetrical diamond shaped pattern.

O'Loughlin and Mcdonald (1964) Cubes arrangedas in (1) abd (2) also sand grains (2.5 mm dia)cementedto the bed .

1 2



Logarithmic plot of data from figure at lowconcentration

Effective roughness as a function of form pattern, andconcentration of roughness elements. (Assuming highReynolds number)

Open channel resistance (after H. Rouse, 1965)

1.00.10.010.0010.1

1

10

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1.0

Spheres

SandCubes

Nikuradse

Sand

Schlichting (1936) - Sphere spacing

Koloseus (1958)Koloseus and Davidian (1965)

Cubical RoughnessSymmetrical diamond shaped pattern

O'Loughlin and Mcdonald (1964)

ks___y

λ − Areal concentration

Cubes arranged as in 1 and in 2.Also sand grains centered to the sand grains(2.5 m diameter)

λ − Areal concentration

ks___y



Resistance of a bridge pier in a wide channel, after Kobus and Newsham

F = 1.51.0

0.5

b

3b

d = 3bV

0 0.5 1.0 1.5 2.0 2.50

0.5

1.0

1.5

Froude number, F

Variation of pier resistance with lateral spacing "S"

0 0.5 1.0 1.5 2.0Froude number, F

0

0.5

1.0

1.5

CD

SD__ = 5 7.5

30

D

SDd = 30

V d



Loss at one of a series of channel bends after Hayet

0.2 0.4 0.6 0.8 1.0 2.0 4.0

0.1

0.2

0.4

0.6

y/b = 1/16

y/b = 1/8

y/b = 1/4

2b

4b

900

y

b

ζ

Froude number, F

Some of the important References:

(i) Task force on friction factors in open channels Proc. ASCE JI. of Hyd. Dn. Vol. 89.,

No. Hy2, March 1963, pp 97 - 143.

(ii) Rouse Hunter, "Critical analysis of open channel resistance" , Proceedings of ASCE

Journal of Hydraulic division, Vol.91, Hyd 4, pp 1 - 25, July 1965 and discussion pp 247

- 248, Nov. 1965, March 1966, pp 387 to 409.

Schlichting, "Boundary layer theory", Mc Graw Hill Publication.

16.3.4 Open Channel Resistance

There is an optimal area concentration 15% to 25% which produces greater relative

resistance.

1 R A log BDhSf

= +

h is the roughness height , S is the areal concentration (<15%), D is the constant which

depends on shape and arrangement of the roughness elements.

For sanded surface: D = 21 and B = 2.17

The existence of free surface makes it difficult to assume logarthmic velocity distribution

and to integrate over the entire area of flow for different cross-sectional shapes. The



lograthmic velocity distribution can be integrated only for the wide rectangular and

circular sections.

Effect of boundary non-uniformity is normally ignored and particularly so for gradually

varied flow profile computation.

The dependence on Froude number is clearly seen in case of pier.

In case of unsteady flows such as floods, it is assumed that the inertial effects are small

in comparison with resistance. Hence, the resistance of steady uniform flow at the same

depths and velocity is taken to be valid.

Where the Froude number exceeds unity, the surface has instability in the form of roll

waves.

Earlier formulae for determining C (for details refer to Historical development of

Empirical relationships)

1. G.K. Formula (MKS)

2. Bazin’s Formula 1897 (MKS)

3. Powell Formula (1950) FPS while using Powell formula C must be multiplied by

0.5521 to get C in m1/2 s-1

4. Pavlovskii Formula (1925)

Manning equation is applicable to fully developed turbulent rough flow.

Slope of the straight line is 1:3



1/ 61/3s

2s

1/ 6s

1/3s

s1/ 3

k g gf C R kC

n k

kf = 0.113 R

If we replace k by diameter of the grain size (d)

df = 0.113 R

8g 8g RC = f 0.113 d

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

⎛= ⎜⎝

1/ 6

2

1/ 6 1/ 6

1/ 6

1/ 6

1/ 6 1/ 6

for MKS units g = 9.806 m/s

8 * 9.806 R RC = 26.34820.113 d d

Ror C = 26.34 d

Rn C1n = *d 0.0379d

26.34

⎞⎟⎠

⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠

=

A number of empirical methods to relate n diameter of the particle are advanced.

1 Strickler (1923)

[ ]1/6n = 0.02789 d d in m This is not applicable to mobile bed

2 Henderson's interpretation of Strickler's formula

[ ]1/650n = 0.034 d d in feet

3a Raudkivi (1976)

[ ]1/6n = 0.047 d d in m

3b Raudkivi (1976)

[ ]1/665n = 0.013 d d in mm d65 = 65 % of the material by

weight smaller. 3c Raudkivi

(1976) [ ]1/6

65n = 0.034 d d in feet

4 Garde and Ranga Raju

[ ]1/650n = 0.039 d d in feet ( )

( )

( )1/6

1 6

50

0 039 0 3048

0 039 0 82036 0 03199

n = 0.03199 d , d is in 'm'

/. * .

. . .= =

5 Subramanya [ ]1/650n = 0.0475 d d in m

6 Meyer and Peter and Muller

[ ]1/690n = 0.038 d d in m (Significant proportion of coarse

grained material)



7 Simons and Sentrvrk (1976)

[ ]1/6n = 0.047 d d in mm

8 Lane and Carbon (1953)

1/675n= 0.026 d (d in inches and d75 = 75% of the

material by weight is smaller)

( )

* f

* s

1/6 1/6

1/6 1/61/6 1/6

1/6

8) Consider

v g R Sυ k 4 < 100 Transition flow

vR R n = but C = 26.35

C dR d 1 n = d 0 03795 d (d in m)

26 35R 26 35Conditon for fully develop

...

=

<

∴ = =

( )

( )

68 6* s

6

6

f

-6 2 2

6f 6 6

6

ed rough flow

v k n100 d = 3 3458 10 nυ 0.03795

n 1g R S 0.03795

Assuming

= 1.01 * 10 m /s g = 9.806 m/s

9 806 1n R S 1001 01 10 0.03795

n

. *

.. *

υ

υ

−

= =

⎡ ⎤⎢ ⎥⎣ ⎦

=

14fRS 9 635 10

. * −≥



Laminar flow

Smooth surfaces

Fully rough zoneTransition zone

Commercial surfaces

Sand coated surface (Nikuradse)

Reynolds number Re = 4 V R/v

Modified Moody Diagram showing the Behavior of the Chezy C after Henderson

103 104

f = 0.316_____Re

0.25

(C = , mks)15.746 Re

1__8

1__f = 2.0 log ( )Re f_____

2.51

C = 4 2g log ( )Re 8g2.51C_____

Blasius equation ( )Re <105

______υ = 100

v* ks

1__f

= C__8g

= 2.0 log ( )12R___ks

105 106 107 1082 4 630

40

50

60

70

80

90

100

110

120130140150

180

or Manning

507

252

126

60

30.6

1510

do_____2ks

2R_____ks

=

resistance in open channel hydraulics -...

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