Open channel hydraulics N - K141hydraulika.fsv.cvut.cz/Hydraulika/Hydraulika/Predmety/HyaE/... ·...

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


Equation of uniform flow

ρgSdLG =weight of water

slope of bottom sinαtgαdLdZi0 ≈==

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 = = → 0*gRi v

ρ

τ= = - friction velocity

dL

GF1F2

S,O

Ft

G’

dZ

1

y1y2


expressing ⇒ρgRiτ0 =ρ⋅τ⋅

=f

0f2v

iRf

g2f

iRg2vff

⋅⋅⋅

=ρ⋅

⋅⋅⋅ρ⋅=

introducingffg2C ⋅

=

iRCv ⋅⋅= - Chézy equation

[m0,5s-1]

Relation for τ0 and v

2

00f

v21volumeelementaryinenergykinetic

stresssheerf⋅ρ⋅

τ=

τ=

ρ⋅τ⋅

=⇒⋅ρ⋅⋅=τf

02f0 f

2vvf21

friction coefficient:


SOLUTION OF CHANNELS1. 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 21v R i

n=

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

Pavlovskij (1925):

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

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

Bretting (1948):

comparison of both equations

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 1,171

dRlog17,72C

e

161C = R

n⇒ ⋅


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 vZ v

4R 2g= λ ⇒

Hey (1979):84

1 aR2,03 log3,5d

=λ

a = 11,1 ÷ 13,6 … coefficient of channel shapevalidity: 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

S2 S1

S3

O1O2 O3


Part-full circular pipes

max

max D

yv for 0,813D

yQ 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

3dE Q dS1 0dy dygS

α= − =

determination of minimum Ed = f (y)

S = f (y) → dS = Bdy2

3Q B1- = 0g S

α

2 2

d 2v QE y y

2 g 2 gSα α

= + = +

yk

Q = const.y

sub-criticalflow

supercritical flowcritical flow

EdEdmin


a) from Ed = f(y) ⇒ Edmin ⇒ ykb) from general condition - analytically

– possible only exceptionally: S = f (y), B = f (y) for rectangle: B = b, Sk = b ⋅ yk, specific discharge Qq

b=

322 3k

kk

SQ b yg B

α= =

223 3k 2

Qy qggb

α α= =⇒

- general condition of critical flow ⇒ yk

2 3Q S=g B

α

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

BSys =

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

3Q B =1g S

α

Fr2

Frgyv

gyv

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 FLOWin direction of flow: depth increases → backwater curve

depth decreases → drawdown curve

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

i0 < iki0 < ik

i0 < ik

backwater – supercritical flow hydraulic jump

subcritical flow


sub-critical flow - backwatersupercritical flow - drawdown

sub-critical flow - drawdownsupercritical flow - backwater


Bernoulli equation 1 – 2:

Expression of iE from Chézy equation:2 2

E E 2 2 2v 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)

⇒ ΔLΔL

ΔZ

1

y1y2

i0

i

iE

2

v2

v1

Determination of free surface profile

( ) ( ) ΔLi2g

vvαyyΔLi

ΔZ2gαvy

2gαvyΔLi

E

21

22

120

22

2

21

10

+−

=−−

++=++


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 N - K141hydraulika.fsv.cvut.cz/Hydraulika/Hydraulika/Predmety/HyaE/... ·...

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