chap 10 fluid mechanics

Universität Stuttgart Institut für Wasserbau, Lehrstuhl für Hydromechanik und Hydrosystemmodellierunglehre/VL-HM/E-HYDRO-LECTURE-NOTES/HYDROFOLIEN/gerin_F.tex Open channel flow

Flow in ducts and in open channels

Flow in ducts:

∆ h

Q

driving force is the pressure difference between the ends of the pipe

no free surface

Open channel flow:

vy

patm

driving force is gravity (fluid weight)

free surface with atmospheric pressure

. – p.1/28


Velocity profiles (isolines)

0.5

1.0

1.5

2.0 0.5

1.0

1.5

2.0

1.00.5

1.5

2.0

0.51.0

1.52.0

2.5

2.5

. – p.2/28


One-dimensional description

21

∆ z

v2

2g

v

h v

y

EGL

The one-dimensional analysis of open channel flows still plays an important role forpractical engineering problems.

→ Definition of cross-section-averaged parameters required.

. – p.3/28



Continuity condition: v(x)A(x) = Q = const.

x coordinate in channel flow directionv(x) average flow velocity of the channel flow at point xA(x) local cross-section area at point x

Energy balance:

H0,1 + ∆z = H0,2 + hv orv2

1

2g+ y1 + ∆z =

v2

2

2g+ y2 + hv

H0 specific energy (related to channel bottom)hv head loss between cross-sections 1 (upstream) and 2 (downstream)

(for example with the approach of Darcy and Weisbach):

hv = λx2 − x1

dhy,m

v2

m

2g= λ

x2 − x1

4rhy,m

v2

m

2g

. – p.4/28



y

b 0

P

hydraulic radius: rhy = A/Pwetted perimeter: P (e.g., = b + 2y for rectangular cross-section)equivalent water depth: y = A/b0

specific energy: H0 = y + v2

2g= y + Q2

2gy2b20

. – p.5/28


Froude number - flow regimes

��

vδ

yδ

gyρ g(y+ y)ρ

vδy

c

stillwater

controlvolume

yδ

δ

fixedwave

c c −

b)a)

Propagation velocity of a small surface wave: c =√

gy

The Froude number Fr describes the ratio of the channel flow velocity to the propaga-tion velocity of an infinitesimal shallow-water surface wave:

Fr =v

c=

v√

gy

Thus, Fr is a measure for the influence of inertia in comparison to gravity.

. – p.6/28


Froude number - flow regimes

Three different flow regimes can be distinguished:

Fr < 1.0 subcritical flow

Fr > 1.0 supercritical flow

Fr = 1.0 critical flow

Different flow regimes in a flow over a weir:

supercriticalflow

criticalflow

subcriticalflow

hydraulicjump

energygrade line

specificenergy

lake

. – p.7/28


Specific energy and discharge diagrams

Specific energy diagram for q = const.

0

y = H

Hmin0

H 0

y

ycr

Specific discharge diagram for H0 =const.

qmax

y

q

= 2 H_

0H

ycr 3 0

. – p.8/28


Frictionless flow over a bump

��

��

y1,sub

y2,sup

y2,sub

h∆y

1,sup

subcritical flow

supercritical flow

H 20H min0

ycr

y 2,sub

y 1,sup

y 2,sup

y 1,sub

H 10

h∆

. – p.9/28


Hydraulic jump

Discontinuous transition from supercritical to subcritical flow.

Strong turbulence, dissipation of flow energy→ e.g., purposeful construction of stilling basins

Behavior of the hydraulic jump is predominantly affected by the upstream Froudenumber (in any cases: Fr > 1 !!).

��

y1

y2

v1

v2

F2F1

hydraulic jump

Fr > 1supercritical

subcriticalFr < 1

control volume

. – p.10/28


Hydraulic jump - upstream and downstream depths

Determination of the conjugated upstream and downstream depths for constantchannel width b and horizontal bottom level (this assumption can also be taken as anapproximation for channels with not too steep bottom inclination):

Continuity (mass conservation):

y1v1b1 = y2v2b2 .

Momentum balance:1

2gb(y2

1 − y2

2) = v1y1b(v2 − v1) .

→ Conjugated upstream and downstream depths:

y2

y1

=1

2

„q

1 + 8Fr21 − 1

«

y1

y2

=1

2

„q

1 + 8Fr22 − 1

«

Loss of energy in the hydraulic jump:

∆H = H01 − H02 =(y2 − y1)

3

4y1y2

. – p.11/28


Discharge control

The fluid follows the principle of less constraint.

For a given channel geometry and a given level of the available energy headthe maximal possible discharge occurs.The cross-section which can deliver the smallest Qmax (at critical flowconditions) for a given level of the available energy, controls the discharge.

For a given discharge the minimal required energy level adapts in such a waythat the discharge can be delivered.The discharge is controlled in that cross-section where the maximum absoluteenergy head (H0+ distance to the datum line) is required to let the given dis-charge pass through.

The locus of discharge control is always characterized by critical flow conditions.

It is further valid:

Supercritical flow is controlled upstream.

Subcritical flow is controlled downstream.

. – p.12/28


Uniform flow with friction

In long undisturbed channels with constant bottom inclination and constant cross-section the flow conditions are uniform (normal-depth conditions).

If normal-depth conditions are given, then there is an equilibrium between the com-

ponent of the gravity in flow direction and the friction-induced forces due to shear

stress.

Dependent on the flow regime at normal-depth conditions, the channel slope can beclassified:

mild: subcritical normal-depth conditionsyN > ycr

steep: supercritical normal-depth conditionsyN < ycr

critical: critical normal-depth conditionsyN = ycr

. – p.13/28


Normal-depth conditions

Chézy approach:For normal-depth conditions the slope of the energy grade line IE is equal to the bottomslope I0. Then it follows:

hv = ∆z = I0L

Head loss according to Darcy and Weisbach

hv = λL

4rhy

v2

2gwith rhy =

A

P

By combinations of both equations and algebraic transformation it follows

v0 =

„

8g

λ

«

1/2

r1/2

hy I1/2

0.

Problem:Determination of the Chézy coefficient C = (8g/λ)1/2, which is a function of the chan-nel geometry and the bottom roughness.

. – p.14/28



Approach of Gauckler, Manning, and StricklerFrom the empirical conclusion that the Chézy coefficient increases approximately asthe sixth root of the hydraulic radius (channel size), we can introduce with the help of

C =

„

8g

λ

«

1/2

≈ r1/6

hy kst

the Strickler coefficient kst.

kst is a roughness coefficient (dimension [m1/3

s]).

Gauckler-Manning-Strickler equation:

v0 = kst r2/3

hy I1/2

0

. – p.15/28



Roughness coefficients kst for open channel flow:

kst [m1/3

s]

River bed with firm bottom, no irregularities 40Neckar near Wendlingen 35River bed, covered with aquatic plants 30-35River bed with boulders and irregularities 30River with high bed load 28River bed with sand and gravel, plastered banks 40-50River bed outlayed with large stones 25-30Smooth cement surface 100Concrete, constructed with wooden formwork, without plaster 65-70

. – p.16/28


Normal-depth estimation

Graphical evaluationof the GMS equationfor given discharge Qand trapezoidalcross-section withdifferent bankinclinations:

yNb

yNb

Sb

Sb

Sb kst IE1/2 b8/3. .=Q

yN

b 0

5

0.5

1

10.1 10 100

1

10 10

0.250m = 0.5 1 24

0.1

0.21 2 40m =

0.05

0.020.10.010.001

b1:m

Normal depth for trapezoidal cross−sections

. – p.17/28


Gradually varied flow

Assumptions:

Slowly changing bottom slope, cross-section and water depth (no hydraulicjump).

One-dimensional velocity distribution.

Pressure distribution approximately hydrostatic.

Differential equation:

dy

dx=

I0 − IE,m

1 − Fr2.

��

��

E,mI

0I dx

v 2

2g

v 2

2gv 2

2g+ d

E,mI dx

dxx x+dx

0I

α

y

y+dyv+dv

v

. – p.18/28


Gradually varied flow

For non-uniform flows the bottom slope I0 and the slope of the energy grade line IE

differ.

y < yN :The flow velocity is larger than at normal-depth conditions. Since the lossesincrease with increasing velocities, it follows that the slope of the EGL is largerthan the bottom slope.IE > I0

y > yN :The opposite case holds accordingly.IE < I0

y = yN :Normal-depth conditions, IE = I0 !!

. – p.19/28


Gradually varied flow - basic solution curves

yN ycr

3S

S 2

1SyN ycr

I0 Icr

ycrycr

yNyNyN

S

ycr

I0 Icr

Steep (S)

<>

y

> y

<

<

y:

:

:

1

2

3

S

S

>< <

>

��

��

yN ycr

yN

ycr

M 1

M 2

M 3

ycrycr

yNyNyN

M 1

M 2

M 3

I0 Icr

I0 Icr

Mild (M)

>

> y >

> y

<

<

y:

:

:<

<

��

��

. – p.20/28


Gradually varied flow - basic solution curves

yNyNC y

< y:

:1C

3 >

I0 Icr

1C3C

yN ycr

I0 Icr

ycryN

=

Critical (C)

=

==

��

��

I0= 0

ycr

2H

H 3

yN

8

I0= 0

yNyN

ycr

ycr

2H

H 3

Horizontal (H)

:

:

>

< ��

��

. – p.21/28


Gradually varied flow - integration of the diff. equation

Given discharge Q:

dy

dx=

I0 −Q2

k2str

4/3hy

A2

1 − Q2bgA3

Choose the discretizationlength ∆x or ∆y

Numerical integration

Computation from upstreamto downstream forsupercritical flow

Computation fromdownstream to upstream forsubcritical flow

Single-step approximation:e.g., for the calculation of a backwatercurve

v2

1

2g+ y1 + ∆xI0 =

v2

2

2g+ y2 + ∆xIE,m

IE,m =v2

m

k2

str4/3

hy,m

∆x =

“

y2 +v22

2g

”

−“

y1 +v21

2g

”

I0 −v2

m

k2str

4/3hy,m

=H02 − H01

I0 − IE,m

. – p.22/28


Gradually varied flow - composite flow profiles

Changing water depth in channel regions with different roughness (constant bottomslope)

��

��

��

��

yN,1

yN,3

yN,2

flowcritical

jumphydraulic

kst,2 kst,1 kst,3> >

S 2

S 1

ycr

M 2

mild

steepmild

. – p.23/28


Gradually varied flow - composite flow profiles

Changing water depth in channel regions with different bottom slope (constant rough-ness)

��

��

yN,1

S 2

M 2

yN,2

yN,3

flowcritical

M 1

I 0,3 I 0,1 I 0,2> >

ycr

mildmilder

steep

. – p.24/28


Broad-crested weirs

��

��

ycr

w0

h1H1

L

boundary−layer

ventilation

Critical flow occurs on the weir crest (position of discharge control)

Specific discharge over a broad-crested weir

q =1√

3

2

3

p

2gh3/2

1

General weir formula:

q = µ2

3

p

2gh3/2

1

In general, the discharge coefficient µ must be determined by experiments.

. – p.25/28


Sharp-crested weirs

��

w0

h1 h123

ventilation1

2

Formulation of the Bernoulli equation along the streamline 1–2

Specific discharge over a sharp-crested weir:

q = µ2

3

p

2gh3/2

1

Rough estimation µ =p

2/3 ≈ 0.81

Poleni equation for h1/w0 < 6:

µ = 0.611 + 0.075h1/w0

. – p.26/28


Round-crested weirs

��

h1

w0

h1 h1a

w0

h1a

h1

w0

: head on the weir

: design head on the weir

: height of weir

round−crested weirsharp−crested, ventilated weir

Atmospheric conditions on the weir crest for design water level h1a

h1 > h1a: Under-pressure on the weir crest, the nappe (water overflow) issucked to the weir crest. The discharge coefficient is increased.

h1 < h1a: Over-pressure on the weir crest, decreased discharge coefficient.

. – p.27/28


Round-crested weirs - discharge coefficient

Discharge coefficient µ for round-crested weirs at design head on the weir (h1 = h1a)

0 2 3 4 51

0.8

0.9

0.7

1.0

1.1

1.2

1ah /w0

µ

. – p.28/28

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chap 10 fluid mechanics