Hydraulics of structures - cvut.czhydraulika.fsv.cvut.cz/.../_old/2005/_06_Hydraulics_of_structures.pdf · Bazin spillway – rectangle without side contraction 3 Q=mb 2gh 2 2 0,003

Hydraulics of structures

K141 HYAE Hydraulics of structures 1

OUTFLOW FROM ORIFICE


TYPES OF OUTFLOW

Outflow

Outflow

steady: z = const, hE = const(H = const, HE = const) Qp = Q

quasi-steady: z ~ const., phenomenon of large reservoirunsteady: z ≠ const (H ≠ const)

Qp ≠ Q, filling and drawdown of tank (reservoir)

free (a) → free outlet jetsubmerged (b) → submerged outlet jetpartly submerged, e.g. outflow from large orifices at the bottom (slide gate)


STEADY FREE OUTFLOW (SFO) OF IDEAL LIQUID

BE surface – outflow:

Torricelli (1608 - 1647) equation for outlet velocity of ideal liquid vi

For large reservoirs with free level:

outlet discharge of ideal liquid Qi:

for small orifice (bottom and wall):

2 20 iv vp

h+ + =g 2g 2g

∆ρ

2i

E

i E

vh =

2 g

v = 2gh

iv = 2gh

i i iS S

Q = dQ = u dS∫ ∫

i iS

Q =v dS∫ i iQ =v S=S 2gh

ui,vi,dQi,Qi

pa(=0)

S orificesection

overpressure

i iu v≈h… depth of centoid of orifice

hh,0g2

v,0p E

20 =≈=∆

STEADY OUTFLOW FROM ORIFICE


Hydraulic lossesoutlet loss ζv ... depends on shape, setup and

size of orifice (structure), Re

CONTRACTION OF OUTLET JET

Strip area Sc < S, Sc = ε · S, contraction coefficient ε ≤ 1

well mouthed orifice

partial contraction

re-entrant streamlined mouthpiece

external mouthpiece ∅D

sharp edged orifice

TAB.

imperfect contraction

2

vvcZ=2g

ζ


SFO OF REAL LIQUID FROM ORIFICE AT THE BOTTOM OF TANK

g2v

g2v

gp

gp

g2v

lh2

c2

ca0s2

0c

α⋅ζ+α+ρ

=ρ

+α++

BE 0 - 1

lc ~ 0,5·D

ρ−

ρ+α++⋅

ζ+=

gp

gp

g2v

lhg211

v a0s2

0cc

φ ... velocity coefficientα

c c c vQ v S , S S, ... orifice discharge

coefficient= ⋅ = ε ⋅ ε ⋅ ϕ = µ

contraction coefficientφ, µv, ε ... TAB.

Simplification:free level → ps0 = pa →

S0 >> S → v0 ~ 0lc << hE → lc ~ 0

0g

pp a0s =ρ−

hg2SQ

,hg2v

v

c

⋅⋅µ=

⋅ϕ=⇒


- Large orifice hT < (2 - 3)·a ⇒ change of outlet velocity u

with height of orifice

- Open reservoir and large rectangular orifice in vertical wall:

for large tank:

Eu= 2ghϕ1/2

v EQ= 2g h dSS

µ ∫

E2

E1

h1/2

v E Eh

Q= b 2g h dhµ ∫( )( )

3/2 3/2v E2 E1

3/2 3/2v 2 1

2Q= b 2g h -h

32

Q= b 2g h -h3

µ

µ

EdS=bdh S=ba

hh02gv

E

20 =⇒≈

SFO OF REAL LIQUID FROM ORIFICE IN VERTICAL WALL OF TANK

- Small orifice hT > (2 - 3)afor S0 >> S → v0 ~ 0 tv

tc

hg2SQ

,hg2v

⋅⋅µ=

⋅ϕ=⇒


Coefficients for discharge determination

- small sharp-edged orificewith full contraction 0,97 0,63 0,61

- external cylindrical mouthpiece L/D = 2 ÷ 4 0,81 1,00 0,81- streamlined mouth piece jet tube 0,95 1,00 0,95- large orifices at the bottom with significant 0,65 až 0,85

or continuous side contraction

- outlet tube of diameter D and length L with free outflow

v

i

1=

L1+ +

D

µλ ζ∑

ϕ ε µv

Note:special application of outflow through mouthpiece -- Mariotte vessel - with function of solution dosing,

Q = const.

φ, ε, µv for imperfect and partial contraction > φ, ε, µv for full contraction

empirical formulas


OUTLET FROM SUBMERGED ORIFICE

For both small and large orifices ofwhatever shape

for small orifice

Note:resolution for partial submergence: Q = Q1 + Q2(Q1 outflow from free part of orifice, Q2 outflow from submerged part of orifice).

for large reservoirH = H0

02gHvu ϕ==

2gHSµQ

2gHSµQ

v

0v

=

=


OUTFLOW JETS

Free outflow jet

Supported outflow jet Submerged outflow jet

- different functions of jet → requirements for outlet equipment and outlet velocity

- free jets – cutting, drilling, hydro-mechanization (unlinking), extinguishing, irrigation jets …

- submerged jets - dosing, mixing, rectifying, …

type water - air

type water – air –solid surface type water - water

jet core with constant velocity

pulsating margin of boundary layer (mixing regions)

theoretical trajectory (parabola 2°)

decay of jet, aeration, drops

connected part


hd

Theoretical shape of outflow jet (projection at an angle)arcing distance of jet

maximum height

20

p0 dv

L = sin2 =2h sin2g

δ δ

22 20

0 dv

y = sin =h sin2g

δ δ

20

dv

=h2g

energetic head of jet

For δ = 45° → Lp0max = v02/g = 2hd, y0 = 0,5 hd

For δ = X° a δ = 90 -X° → same arcing distanceFor δ = 90° vertical jet → y0max = v0

2/2g = hd

For δ = 0° horizontal jet (horizontal projection)

real liquid, large reservoirp d TL =2 h y

p T TL =2 h yϕ

0

2

x =v t

1y = gt

2

theoretical

20

0

gt21

sinδtvy

cosδtvx

−=

=

δv0cosδ

v 0si

nδ

v0


UNSTEADY OUTFLOW FROM ORIFICE

Differential equation of unsteady flow

Qp < Q0 drawdown, Qp > Q0 filling

0 p 0

p 0 0

Q dt -Q dt =-S dh

Q dt -Q dt =S dh (filling: t1 ↔ t2, h1 ↔ h2)

0 0

0 p p 0

S dh S dhdt =- =

Q -Q Q -Q

the same equation for drawdown and filling

1 1

2 2

h h0 0

2 1h h0 p p 0

S dh S dht = t - t = =

Q -Q Q -Q∫ ∫

For Qp ≠ const., S0 ≠ const. , irregular reservoir ⇒⇒ numerical solution in intervals ∆t

(drawdown)


Drawdown of prismatic tank (S0 = const.), at Qp= 0

Assumptions:- outflow from small orifice, mouthpiece, tube- free level- S0 >>S → v0 ~ 0

Time of total emptying (h2 = 0):

1

2

h-1/20

hv

St = h dh

S 2gµ∫ ( )0

1 2v

2St = h - h

S 2gµ

0 1 0 1 1

01v v 1

2S h 2S h 2 VT= = =

QS 2g S 2ghµ µ

0 vQ = S 2ghµ


OVERFALL


TYPES OF SPILLWAYS- sharp-crested, t < 2/3 h, measuring spillways (a)- weir, spillway of practical profile (b,c), streamlined spillway (d)- broad-crested, 2 h < t < 10 h (e)- special - shaft (f), side (g), bed drop (h) ...

(a)

(g)(f) (h)

(c)(b)

(e)

(d)streamlined weir body

h

- overfall: hydraulic phenomenon- spillway: structure, weir


FLOW OVER WEIRS

• Q… discharge• h… overflow head

(measurements of non-reducedlevel in distance OP = (3 ÷ 4) h)

• v0…approach velocity v0= Q / S0• h0…overfall energetic head

h0 = h + h0d = h + α v02 / 2 g

• b… spillway width, length of crest• b0…active spillway width

(side contraction of overfall jet, b0 < b)• H… gradient head in levels UW and DW• s, sd… spillway height in UW, DW• µp,m… coefficient of discharge (describes hydraulic losses, depends

on type, shape and arrangement of weir and also on further characteristics experiments → TAB.


• hσ… height of overfall submergence (effect of DW)• σz… coefficient of submergence

- free overfall σz = 1, no effect of DW- submerged overfall σz < 1,

DW → decreases Q → enlarges h

(for submergence of overfall, H < h, i.e. hσ > 0, is not sufficient)


EQUATION OF OVERFALL

Weisbach equation

Bazin equation

Steady flow, rectangular overflowing jet-width b,

BE for O-P, analogy of outflow from large rectangular orifice in vertical wall.

( ) ( )

( )

1/ 20d 0 p 0 0d

h 1/ 2p 0 0d

0

u= 2g z+h dQ=u b dz= b 2g z+h dz

Q= b 2g z+h dz

ϕ ε µ

µ ∫

free overfall2

300

23

d023

00p

hg2bmQ

hhg2b32

Q

=

−µ=

without side contraction: b0 = b


⇒ side contraction:

i

n

0 i 01

b= b

b =b-0,1 hξ

∑

∑

n ... number of contractionsξi… pier coefficient (shape, location)

circle-curved intermediate pier ξ = 0,5circle-curved bank pier ξ = 1tapered or streamlined pier with forward nosing ξ ∼ 0

TAB.

⇒ active width b0< clear width b

effect of intermediate piers and wing walls→separation of flow, wakes


SUBMERGENCE OF OVERFALL

small gradient H-not solved

3 32 2

z p 0 0 0d

32

z 0 0

2Q= b 2g h h

3

Q= mb 2g h

σ µ −

σ

= K,

hs

,hh

fσ0

d

0

σz

TAB., graphs

data in literature are not distinct and reliable for all cases =>submergence of overfallusually worsens accurency of calculations!


MEASURING WEIRS

Bazin spillway – rectangle without side contraction

32Q=mb 2g h

20,003 h

m= 0,405+ 1+0,55h h+s

Typical shape of Bazin free owerfall jet

air inlet into space bellow jet!

lower envelope streamlined spillway

0,2 < b [m]< 2,0 0,2 < s [m]< 2,00,1 < h [m]< 1,24

• free overfall, sharp crest• standard spillway geometry,

approach rates• analytical formulae for discharge• empirical formulae for discharge

coefficient• discharge measurement:

registration of level → h → m → Q• Q = Q(h) rating curve of spillway


BRIDGES AND CULVERTS


HYDRAULIC CALCULATION OF BRIDGE WITH ONE OPENING

yσ > κE → yσ = yd κ - TAB.

(condition for submerged entrance)

( )2gv

ζαy2gvζ

2gαv

yE2σ

σ

2σ

2σ

σ ++=++=

from BE for pf. 1 - 2:

2σ

2

2

σ S2gQ

yEϕ

+=

backwater by bridge:

σ

20 yy∆H;

2gαv

Ey −=−=

Subcritical flow → flow in opening usually affected by DW


FLOW IN CULVERTS

Flow in culvert is effected by:- geometry, arrangement and hydraulic conditions of entrance- geometry, roughness, longitudinal slope of culvert- downwater position, rates bellow outlet from culvert.

Sizing and flow resolution → simple schemesMore complicated cases → hydraulic jump, inlet vortex, ... →

→ instability of flow

culv

ert

with free level

with pressure flow

with free–surface entrance

with submergedentrance

in whole culvert in part of culvert

affected by DWnon-affected by DW

Q > QD, QD ... capacitive discharge

affected by DWnon-affected by DW


solution: inlet, flow in culvert, outlet

1) free level and free-surface inlet flow

(a) non-affected by DWyc < 1,25 yk Ø

yc < 1,1 yk □

kc yyD,βy ⋅κ=⋅<

(b) affected by DWyc > 1,25 yk Ø

yc > 1,1 yk □

HYDRAULIC CALCULATIONS OF CULVERTS

( )2

22

2222

Sg2

Q

ϕ+=++=++= 2

22

222 y

2gv

ζαy2gvζ

2gαv

yE

orig

20 yy∆H,

2gαv

Ey −=−=y2 = yc (a), y2 = yσ (b)

according to type of entrance0,9, β (1,2 1,4)Dκ ≈ = −


2) free level and submerged entrance

(a) non-affected by DWyc < 1,25 yk Ø

yc < 1,1 yk □

(b) affected by DWyc > 1,25 yk Ø

yc > 1,1 yk □

( )2

22

2222

Sg2

Q

ϕ+=++=++= 2

22

222 y

2gv

ζαy2gvζ

2gαv

yE

orig

20 yy∆H,

2gαv

Ey −=−=y2 = yc (a), y2 = yσ (b)

y > β·D, yc = 0,62·D for Ø

yc = 0,60·a for □

β (1,2 1,4)D= −according to type of entrance


Coefficients for hydraulic calculation of culverts with free level

1,10,870,75without portal, splayed

1,20,900,85common, frontal portalRectangle: width b and height a

1,40,950,95conical, skew wings

1,20,960,85common, frontal portal

Pipe culvert with inner diameter D

βκϕEntranceCulvert


3) pressure flow in culvert

(a) outlet non-affected by DW

solution:drawdown + short pipeline

Bernoulli equation- short pipeline (at the end of

culvert – „free outlet“)

(b) outlet affected by DW

( ) ( ) min

2

0E Dii ∆−∆++⋅++⋅−=2gv

ζ1LE

( )g

vvv ddmin

−=∆>∆

Hydraulics of structures - cvut.czhydraulika.fsv.cvut.cz/.../_old/2005/_06_Hydraulics_of_structures.pdf · Bazin spillway – rectangle without side contraction 3 Q=mb 2gh 2 2 0,003

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