open-channel hydraulics -...

Open-channel hydraulics

K141 HYAE Open-channel hydraulics 1

STEADY FLOW IN OPEN CHANNELS

→ constant discharge, other geometric and flow characteristicsdepended only on position

Uniform flow Non-uniform flow

S; y; v = const.i = i0 = iE

y1 ≠ y2; v1 ≠ v2i ≠ i0 ≠ iE


0,5 11 1v Ri, C m s

a a

v C Ri

− = =

=

Equation of uniform flow

ρgSdLG =weight of water

slope of bottom sinαtgαdLdZ

i0 ≈==

2

pressure forces F1 = F2

⇒ force in direction of motion ρgSdLiGsinαG ==′against motion – friction force t 0F OdL= τ

Equilibrium of forces ⇒ G’ = Ft ⇒ 0ρgSdLi OdL= τ ⇒

0 ρgRi (R S/O)τ = = →

friction loss b0tZ avρgτ

= = in quadratic zone b = 2 ⇒

- Chézy equation

0*

gRi vρ

τ= = - friction velocity

dL

GF1

F2

S,O

Ft

G’

dZ

1

y1

y2


SOLUTION OF CHANNELS

1. Chézy equation (1768)

C – velocity coefficient, K - conveyance (m3⋅s-1)

2. Manning equation (1889)n - roughness coefficient

0v C R i= ⋅ 0 0Q C S R i K i⋅= =

2 13 21

v R in

=

validity: n > 0,011, 0,3m < R < 5m

Pavlovskij (1925):

validity: 0,011 < n < 0,04 , 0,1m < R < 3m

P,

1C R

n⋅= ( )P 2,5 n 0,13 0,75 R n 0,1− −= −

Bretting (1948):

comparison of both equations

+= 1,171

dR

log17,72Ce

161

C = Rn

⇒ ⋅


Grain-size curve- screen analysis (fine-grained)- random sample (course-grained)- .....

- formulas in dependency on di

Strickler (1923) validity: 4,3 < R/de < 27616

e

1 21,1n d

=

- tables –values 0,008 ÷ 0,150 (÷ 0,500):

0,0450,0400,033c) clean winding, some pools and shoals0,0400,0350,030b) same as above but more stones and weeds0,0330,0300,025a) clean, straight, full stage, no rifts or deep pools

Streams on plainn max.n nor.n min.Type of channel and description

- photographic method

Determination of n:


weighted average

Horton, Einstein, Banks

O ni inO

∑= 2

332O ni i

nO

∑=

16R8 C

g n gλ= =

different roughness on wetted perimeter→ equivalent roughness coefficient

Relation among C, n and λ:

3. Darcy-Weisbach equation2

tL v

Z v4R 2g

= λ ⇒

Hey (1979):84

1 aR2,03 log

3,5d=

λa = 11,1 ÷ 13,6 … coefficient of channel shape

validity: R/d84 > 4

O1,n1

O3,n3O2,n2


- calculation of channel width b → similarly as determination of depth Compound channels

! velocities, roughness coefficient, discharge Q = ∑Qi

CHANNEL DESIGN

- calculation of velocity and discharge Q → basic equations- calculation of bottom slope i0 → basic equations- calculation of depth y0 → semi-graphically y = f(Q) (rating curve)

→ by numerical approximation yi → Qi ; Q → y0

S2 S3S1

S2S1

S3

O1O2 O3


Part-full circular pipes

max

max D

yv for 0,813

Dy

Q 1,087Q for 0,9495D

→ =

= → =


CRITICAL, SUB-CRITICAL AND SUPERCRITICAL FLOW

Ed – energy head of cross section (specific energy)

Ed = f (y) → for Q = const.

Critical flow:→ Q = const. → Edmin (Ed= const. → Qmax)

2d

3

dE Q dS1 0

dy dygS

α= − =

determination of minimum Ed = f (y)

S = f (y) → dS = Bdy2

3

Q B1- = 0

g S

α

2 2

d 2

v QE y y

2 g 2 gS

α α= + = +

yk

Q = const.y

sub-criticalflow

supercritical flow

critical flow

EdEdmin


a) from Ed = f(y) ⇒ Edmin ⇒ yk

b) from general condition - analytically– possible only exceptionally: S = f (y), B = f (y)

for rectangle: B = b, Sk = b ⋅ yk, specific dischargeQ

qb

=32

2 3kk

k

SQb y

g Bα = =

223 3k 2

Qy q

ggb

α α= =⇒

- general condition of critical flow ⇒⇒⇒⇒ yk

2 3Q S=

g Bα

Determination of critical depth yk

d) iteratively (approximation)

e) empirical formulas

c) from general condition - graph.–numer.


Froud number - from general condition of critical flow

BS

ys =

s kg y v≅

Transition through critical depth

Q → yk → ik … e.g. from Chézy equation

Fr = 1 - critical flow

→ velocity of wave front on water level

- meandepth

application of continuity equation Q = B ys v, α ≈ 1 :

2

3

Q B=1

g S

α

Fr2

Frgy

vgyv

BgyByv

gSBQ

ss

2

33s

32s

2

3

2

====


Determination of type of flow (regime of flow)

Flow Fr y v i

critical Fr = 1 y = yk v = vk i = ik sub-critical Fr < 1 y > yk v < vk i < ik

supercritical Fr > 1 y < yk v > vk i > ik


NON-UNIFORM FLOW

in direction of flow : depth increases → backwater curvedepth decreases → drawdown curve

Profile of free surface - exampledrawdown – subcritical flowbackwater – subcritical flow

i0 < iki0 < ik

i0 < ik

backwater – supercritical flow hydraulic jump

subcritical flow


Bernoulli equation 1 – 2:

Expression of iE from Chézy equation:2 2

E E 2 2 2

v Qv C R i i

C R C S R= ⋅ ⇒ = =

⋅ ⋅ ⋅p p p p p

index p → values calculated from depth yp= 0,5(y1+y2)(event. average of values in pf. 1 and 2)

⇒ ∆L

∆L

∆Z

1

y1y2

i0

i

iE

2

v2

v1

Determination of free surface profile

( ) ( )∆Li

2gvvα

yy∆Li

∆Z2gαv

y2gαv

y∆Li

E

21

22

120

22

2

21

10

+−=−−

++=++


22dh

h dvv

h + = h + + Z2 g 2 g

αα

( )2 2d h

h d

v - vh - h = z = + Z

2g

α∆

Bernoulli equation 1– 2:

general method – „step method“, both for regular and natural channelsprinciple: utilization of BE

Z = Zt + Zm: LRSC

QZ

p2p

2p

2

t ∆=

( )2g

vvαξZ

2h

2d

m

−= m

∆LZ

S1

S1

2gαQ

∆z 2h

2d

2

+

−=

⇒

( )

+

−=

p2p

2p

2h

2d

2

RSC∆L

S1

S1

ξ12gα

Q∆z m

backwater – vd < vh ⇒ - ; drawdown – vd > vh ⇒ +


vd < vh → sub-critical flow - backwatersupercritical flow - drawdown

vd > vh → sub-critical flow - drawdownsupercritical flow - backwater

gradual contraction of channel: ξ = 0,0 ÷ 0,1gradual widening of channel : ξ = 0,2 ÷ 1,0sudden widening, contraction: ξ = 0,5 ÷ 1,0

Calculation for known Q:- sub-critical flow–against flow direction; supercritical–in flow dir.- known profiles ⇒ ∆Li,i+1, Ci=f(h), Si=f(h) (⇒ ξ), Ri=f(h)

- known initial level (pf. 1) + estimate in pf. 2 → Cp, Sp, Rp

- calculation of ∆z ⇒ improved estimate of level in pf. 2- when improved estimate = previous estimate ⇒ further reach


HYDRAULIC JUMP

– transition from supercritical to sub-critical flow

direct (with bottom regime) undularFr1 ≤ 2

yk

practical significance: kinetic energy dissipation bellow spillways, weirs, dams ... → stilling pool

Ls Ls

open-channel hydraulics -...

Documents