resistance in open channel hydraulics -...
TRANSCRIPT
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
16.3.1 Resistance in Open Channel Hydraulics
If Manning and Chezy equations are compared
23
1
1
1 112 220 0
2 1 1-3 2 6
16
e
e
2 2
e12 2 2
2e1
e1
1 R S CR Sn
R RC=n n
RC=n
For laminar flow:Kf =
R
VRRυ
VRK= fυ
8gSR 8gVR S 8gR S 8gBut f = K= = RυVυV CV
8gC RK
VRIf R υ
=
=
=
∴ =
⎡ ⎤= ⎢ ⎥⎣ ⎦
= 28g f = C
14f = For triangular Smooth Channel (Refer: Chow)Re124f = For Rectangular Smooth Channel (Refer: Chow)
Re1
Sand Roughness Fixed to Flume Bed (Photograph - Thandaveswara)
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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( )
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
(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
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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
= = .
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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.
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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.
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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).
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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)
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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
. * −≥
Hydraulics Prof. B.S. Thandaveswara
Indian Institute of Technology Madras
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
=