Vermelding onderdeel organisatie

October 24, 2011

1

Flow, Stability

chapter 3

ct4310 Bed, Bank and Shoreline protection

H.J. Verhagen

Faculty of Civil Engineering and Geosciences Section Hydraulic Engineering

October 24, 2011 2

Introduction

• focus on non-cohesive grains

• grains may vary in size from microns to tons

• basic principle not very different

• always turbulent

October 24, 2011 3

forces on a grain in flow

October 24, 2011 4

forces on a stone

1

2

1

2

1

2

2D DD w

2 2 2S SF w w

2L LL w

Drag force : = C uF A

Shear force : = F C u u dF A

Lift force : = C uF A

October 24, 2011 5

load and strength relationship

2 2s wc c

w

- g d = g d = K g du u

October 24, 2011 6

Izbash (1930)

2c c

c

u u = 1.2 2 g d or = 1.7 or d = 0.7 u

2 g g d

•no waterdepth

•no good definition of uc and d

October 24, 2011 7

Approach of Shields (1936)

• Stability of stones depends on (generalized) friction force

• The force of flowing water on bed is:

F= Area * ghi (or t = ghi)

• Make stability number based on t and d

• Make this number dimensionless by dividing by g and (s-w)

• So:

c w

c

s w s w

ghi

gd gd

t

No velocity in

equation

No need to measure

velocity

October 24, 2011 8

Comparison of Shields and Izbash

• Both are formulas with stability as function of u2

• Izbash focuses on the force action on one single grain

• Shields focuses on the average shear stress on the bed

• Shields does not consider individual grains

• Izbash explicitly looks to individual rocks

October 24, 2011 9

Shields (1936)

2

* **Rec c c

c

s w

u u df f

gd gd

t

*

gu u

C

2

2c

u

C d

Only valid assuming the

Chezy equation is valid

v C hi

October 24, 2011 10

original Shields (1936)

October 24, 2011 11

example :

waterdepth 10 m, u= 1m/s

D50 = 8 cm, k= 25 cm

z0 = 8 mm

U* = 0.063 m/s

Velocity at a certain height

October 24, 2011 12

critical shear stress according to Shields Van Rijn

2

* **Rec c c

c

s w

u u df f

g d g d

t

October 24, 2011 13

Example for determination of d* (Shields)

What is u*c for sand of 2 mm??

Wild guess: u*c is 1 m/s

*

* 6

1 0.002Re 1500

1.33 10

cu d

Thus: c = 0.055 2*

*

0.055 1.65 9.8 0.002

0.042 /

ccc c

uu gd

gd

m s

* 6

0.042 0.002: Re 63 0.04

1.33 10cThus

*

0.04 1.65 9.81 0.002

0.036 /

c cu gd

m s

October 24, 2011 14

Example for determination of d* (Van Rijn)

d = 2 mm

3

* 2 236

1.65 9.810.002 42

1.33 10

gd d

0.04c

*

0.04 1.65 9.81 0.002

0.036 /

c cu gd

m s

October 24, 2011 15

relative protrusion of a grain

October 24, 2011 16

Load and strength distribution

0 no movement at all

1 occasional movement at some locations

2 frequent movement at some locations

3 frequent movement at several locations

4 frequent movement at many locations

5 frequent movement at all locations

6 continuous movement at all locations

7 general transport of the grains Shields

October 24, 2011 17

Incipient motion according to Shields

U = 0.70 m/s, =0.04

U = 0.83 m/s, =0.05

U = 0.90 m/s, =0.055

U = 0.92 m/s, =0.06

U = 0.97 m/s, =0.07

U = 0.60 m/s, =0.03

ct4310/1 u=xxenPsi=xx

bb 4310-3: Shieldsxxx

October 24, 2011 18

Threshold of motion

extrapolation to zero Paintal

(Shields)

is a stability parameter is a mobility parameter

October 24, 2011 19

Paintal

* 18 16

*

* 2.5 3

6.56 10 (for 0.05)with

13 (for 0.05)

s ss

s

q qq

q g d

October 24, 2011 20

example

• For c = 0.03 is considered a safe choice

• Assume dd = 0.4 m

• qs = 6.56 *1018 16 (d3) =3*10-6 m3/m/s

• This is equivalent to 4 stones per day per m width

• Design velocity occurs only exceptional (1 % per year)

• Note: This loss is per m width and not per m2

De Boer, 1998

After some time transport stops

in case < 0.06

In case > 0.06 transport never

stops

October 24, 2011 21

nominal diameter

33 /nd V M

50 50nd d

usually d50 = 1.2 dn50

for a sphere d50 = 1.24 dn50

October 24, 2011 22

influence of waterdepth

50

cc

n

Cu

g d g

1.7icu

gd

Isbash:

1218log

r

hC

k

roughness

kr = 2*d50

or kr = 3*d50

Attention:

c icu u

October 24, 2011 23

influence water depth on critical velocity

Shields is valid in deep water (h/d>100)

Ashida found a larger for shallow water (h/d <5)

It is obvious that Izbash gives a horizontal line

October 24, 2011 24

practical application

2

50 2

50

cc cn

cn

Cu ud

Cg d g

October 24, 2011 25

Roughness and threshold of motion

Note the plating-effect (pantsering)

Problem with the choice of :

do we select on the safe side or do we

use the expected value of ??

Lammers, 1997

October 24, 2011 26

demo influence

run demo Cress

River structures

Bed protections

Influence of the parameter on rock size

v vary from 1…3 m/s

4 lines 0.03…0.06

h 6 lines 1 ….11

choice 2 lines 1 and 2

Fi 3 lines 30…40

October 24, 2011 27

Variation of (output D50)

October 24, 2011 28

Variation of (output W50)

October 24, 2011 29

Variation of h

October 24, 2011 30

Comparison Shields and Pilarczyk

Shields

Pilarczyk

October 24, 2011 31

Movement of marine gravel (tests by Delft Hydraulics)

Flow velocity 0.65 m/s

Flow velocity 1.05 m/s

Flow velocity 1.35 m/s

Flow velocity 1.43 m/s

Flow velocity 1.53 m/s

Flow velocity 1.70 m/s

Flow velocity 1.80 m/s

Flow velocity 2.10 m/s

ct4310/05

October 24, 2011 32

angles of repose for non-cohesive materials

1:1

1:1.5

1:2

1:3

October 24, 2011 33

influence of slope on stability

b = parallel flow

c = perpendicular flow

October 24, 2011 34

slope parallel to current

////

cos tan sin sin cos cos sin sin

tan sin sin

F( ) W - W ( - )K( ) = = =

F(0) W

////

cos tan sin sin cos cos sin sin

tan sin sin

F( ) W - W ( - )K( ) = = =

F(0) W

October 24, 2011 35

slope perpendicular to current

cos tan sin tan sincos

tan tan sin

2 2 2 2 2

2 2 2

F( ) - K( ) = = = 1 - = 1 -

F(0)

cos tan sin tan sincos

tan tan sin

2 2 2 2 2

2 2 2

F( ) - K( ) = = = 1 - = 1 -

F(0)

paral n from 1:20.. 1:100

perp. nfrom 1:2 .. 1:10

3 lines fi 30..40

October 24, 2011 36

Variation of bed slope (parallel flow)

= 30 º

= 40 º

October 24, 2011 37

Variation of bed slope (perpendicular)

= 30 º

= 40 º

October 24, 2011 38

stability on top of sill

use velocity above threshold

Important aspect for closure works

much research has been done in the framework of the

Deltaworks

October 24, 2011 39

stability and head difference

Shields is useless here

because Shields contains

waterdepth

waterlevel downstream is below

the top of the dam

October 24, 2011 40

shields for flow over sill

50

22 d1 u d u d

n

h = 2 g (h - h ) = (0.5 + 0.04 ) 2 g (h - h )u

d

discharge coefficient

October 24, 2011 41

stability flow under weir

Note that this cross section is less

than the gap width

October 24, 2011 42

Shields in horizontal constriction

c

n

42

2c

n

gap

d

h - 1

g C =

gd

u

5050

3log5.4

sin

sin

General formula

(horizontal closure with trucks)

Correction for horizontal closure

slope of construction

(see also next slide)

angle of repose (internal stability)

October 24, 2011 43

stability on head of dam

from equation in book: h/d

c = 0.04

October 24, 2011 44

deceleration

cv

c

uniform flowu = K

with load increase u

October 24, 2011 46

relation between K and turbulence level

1 3

(1 3 ) 1 31 3

cu cscu cu cs cs v

cs cu

u rr u r u K

u r

ucu = vertically averaged critical velocity in uniform flow

ucs = velocity in case with a structure

rcu = turbulence intensity in uniform flow

rcs = vertically averaged turbulence intensity

October 24, 2011 47

stability downstream of a sill

high dam

no dam

October 24, 2011 48

damage after some time

October 24, 2011 49

K in vertical constriction

2 21 2 2 2 1 2

2 2

h h ( - D) = = K u h u h u u

- D - Dh h

October 24, 2011 50

stone stability downstream of a hydraulic jump

October 24, 2011 51

peak velocities and incipient motion in horizontal constriction

October 24, 2011 52

damage after constriction

zuurveld,

1998

ct4310/1 strommstenen

bb 4310-2 Expansion sbability

October 24, 2011 53

definition of velocities

groyne vertical pole

October 24, 2011 54

Kv - factors for various structures Structure Shape Kv0 KvG KvM

Rect-

angular

b0*KvG/bG 1.3 - 1.7 1.1 - 1.2 Groyne

Trape-

zoidal

b0*KvbG 1.2 1

Rect-

Angular

b0*Kv/bG 1.3 - 1.7 1.2

Round

b0*Kv/bG 1.2 - 1.3 1.2

Abutment

Stream

Lined

b0*Kv/bG 1 - 1.1 1 - 1.1

Round

b0*Kv/bG

2*Kv

1.2 - 1.4 1 - 1.1 Pier

Rect-

Angular

b0*Kv/bG

2*Kv

1.4 - 1.6 1.2 - 1.3

Abruptly

-- 1 -- Outflow

Stream

Lined

-- 0.9 --

Top

Section

3.6.1

Section

3.6.1

Section

3.6.1

Sill

Down

Stream

Fig 3.13 Fig 3.13 Fig 3.13

For many piers in a river the first expression for Kv is appropriate. The second

is valid for a detached pier in an infinitely wide flow, where KG is not defined.

October 24, 2011 55

combined equation

22v c

2s c

uKd =

CK

Kv = reduction for constriction, etc.

Ks = reduction for slope (parallel, perpendicular)

October 24, 2011 56

practical application (1)

22v c

2s c

uKd =

CK

However, in practice Kv2 / Ks 1

D A un c 2

October 24, 2011 57

practical application (2)

October 24, 2011 58

placed blocks