Lecture 4.Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels1 ENGR 691 – 73:...

Lecture 4. Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels 1

ENGR 691 – 73: Introduction to Free-Surface Hydraulics in Open Channels

Lecture 04: Nonuniform Flow

Course Notes by: Mustafa S. Altinakar and Yan Ding

Yan Ding, Ph.D. Research Assistant Professor, National Center for Computational

Hydroscience and Engineering (NCCHE), The University of Mississippi, Old Chemistry 335, University, MS 38677

Phone: 915-8969; Email: [email protected]


Outline

• Transition Between Subcritical and Critical Flow• Introduction to Hydraulic Jump• Gradually Varied Flow (Governing Equations)• Forms of water surface (Channels on Mild Slope, Critical Slope,

Steep Slope, Adverse Slope, Horizontal Slope)• Control Points• Computation of Water Surface (Method of successive

Approximations; Method of Direct Integration; Method of Graphical Integration)

• Rapidly Varied Flow (Weirs; Spillways; Hydraulic Drop; Underflow Gates; Hydraulic Jump)

• Transitions (Channel with variable Bed Floor; Channel of variable Width; Oblique Jump)

• Lateral Inflow


co SS

co SS

Transitionfrom subcritical to supercritical flow

When the flow changes from subcritical to supercritical the water surface lowers gradually from a higher depth to a lower depth by passing through critical depth.

In the region where the flow changes from subcritical to critical flow, a gradually varied flow takes place.

Transitionfrom supercritical to subcritical flow

When the flow changes from supercritical to subcritical the water surface rapidly increases from a supercritical depth to subcritical depth. This sudden increase is called a rapidly varied flow.

The rapidly varied flow may be preceded by a gradually varied flow region where the flow depth rises but stays below critical depth.

co SS

co SS

Transitions between subcritical and critical flow


1sH

h

Specific Energyg

VhH s 2

2

1h

2h

ch

2sH hjh

conjugate depthsorsequent depths

sHM

h

21 MM

Specific Momentum

2

22 h

gh

qM

1h

2h

ch

q

1h 2h1V

2V

2211 hVhVq Equation of continuity

1221VVQFFF pp Momentum equation

with Bh

Fp 2

21

1 B

hFp 2

22

2

alternate depths

hjh

Things to remember:• Conjugate depths or

sequent depths (on Specific Momentum Curve)

• Alternate depths (on Specific Energy Curve)

Introduction to Hydraulic Jump


12

22

21

22 h

q

h

qQB

hB

h

12

22

21

22 h

q

h

qq

hh

1

2

2

222

21

22 gh

q

gh

qhh

2

222

1

221

22 gh

qh

gh

qh 21 MM

By combining momentum equation and continuity equation, on gets:

181

2

1 21

1

2 Frh

hor

181

2

1 22

2

1 Frh

h

where1

11

gh

VFr

2

22

gh

VFr and

Introduction to Hydraulic Jump


Q1h

1z

gU 2/21

2h

2z

gU 2/22

ref. line

fh

L

Consider the steady non uniform flow in a channel. We wish to develop an equation for the variation of the water surface h(x), i.e. longitudinal water surface profile.

For this, we will consider the equation of energy:

H

g

AQhz

g

Uhz

2

/

2

22

and the equation of continuity: UAQ

h

z

gU 2/2

H

Differentiate the energy equation with respect to x to get:

dx

dH

g

AQ

dx

d

dx

dh

dx

dz

2

/ 2

oS eS

Assuming that the head loss can be expressed using Chezy equation, we have:

he RC

AQS

2

2/

h

o RC

AQS

dx

dh

g

AQ

dx

d2

22 /

2

/

Gradually Varied Flow Equation


Q1h

1z

gU 2/21

2h

2z

gU 2/22

ref. line

fh

L

Consider the steady non uniform flow in a channel. We wish to develop an equation for the variation of the water surface h(x), i.e. longitudinal water surface profile.

For this, we will consider the equation of energy:

H

g

AQhz

g

Uhz

2

/

2

22

and the equation of continuity: UAQ

h

z

gU 2/2

H

Differentiate the energy equation with respect to x to get:

dx

dH

g

AQ

dx

d

dx

dh

dx

dz

2

/ 2

oS eS

Assuming that the head loss can be expressed using Chezy equation, we have:

he RC

AQS

2

2/

h

o RC

AQS

dx

dh

g

AQ

dx

d2

22 /

2

/



Note that for a prismatic channel the flow area is only a function of the flow depth, A = f(h) :

We can, therefore, write:

dx

dh

gA

BQ

dx

dhB

gA

Q

dx

dh

dh

dA

gA

Q

dx

dA

Ag

Q

g

AQ

dx

d3

2

3

2

3

2

3

22 2

22

/

B

h

o RC

AQS

dx

dh

dx

dhB

gA

AQ2

22 //Substitute this expression back into the previous equation to get:

By rearranging the terms, we obtain a differential equation describing the variation of flow depth with distance, i.e. the equation for longitudinal water surface profile:

BgA

AQ

SRCAQ

Sdx

dh oho

//

1

/1

2

2

2

It is important to note that when: 0dx

dh

the water surface profile equation reduces to oh SRCUAQ 222/ oh SRCU i.e. Chezy equation for uniform flow



Consider again the equation for longitudinal water surface profile:

BgA

AQ

SRCAQ

Sdx

dh oho

//

1

/1

2

2

2

For

BgA

AQ

/

/1

2

the denominator becomes zero and we have: dx

dh

hgD

U 2

1 21 Fr

We can, therefore conclude that, at critical flow (Fr = 1 and h = hc), the water surface profile is perpendicular to bed.

The normal is equal to critical depth, hn = hc , when:

BgA

AQ

SRC

AQ

oh /

/10

/1

2

2

2

BgASRC oh /2

0dx

dhUniform flow

1Fr

0dx

dh

0dx

dh

0dx

dh

The flow depth remains constant and is equal to normal depth (uniform flow)

The flow depth increases in the direction of flow

The flow depth decreases in the direction of flow



Consider the flow cases below (for all cases channel cross section characteristics are the same):

chnh

ch

nh

nc hh

co SS co SS co SS

cn hh cn hh cn hh

1Fr 1Fr 1Fr

Critical slope is the bed slope when normal depth, hn, is equal to critical depth, hc.

When flow is critical, we have: 1hgD

UFr 1

BA

gA

QFr 1

3

2

gA

BQ

B

gAQ

32

Since the flow is also uniform, Chezy equation holds:2/12/1

oh SRCAQ

Equating two expressions, we have:B

gASRACQ ch

3222 note that we have changed So to Sc.

hc BRC

gAS

2The expression for critical discharge is obtained as:

If Manning-Strickler is used:B

gASR

n

AQ ch

33/4

2

22 3/4

2

h

cBR

gAnS

Review of the Notion of Critical Flow


The gradually varied flow equation can therefore be written as:

c

on

n

o

SS

KK

KK

Sdx

dh2

2

1

1

The equation for gradually varied flow can also be written using the notion of conveyance:

Remember the definition of conveyance:

3/2)( hRn

AhK

2/1)( hRAChK

when using Manning Strickler

when using Chezy

2/1)( on ShKQ

2

2

2

2

22/1

2

22

2

2

2/

K

K

SK

SK

SCAR

Q

SRAC

Q

SRC

AQ n

o

on

ohohoh

3

22

/

/

gA

BQ

BgA

AQ

Consider the term in the denominator of gradually varied flow equation: 3

222

222

2

3

2

gA

SRAC

SRAC

BQ

gA

BQ oh

oh

oh

ho

SgA

RBC

CARS

Q

gA

BQ 22

22/1

2

3

2 1

when the flow is uniform, in either case we can write:

cS

1

or

2/1)(o

nS

QhK

2)(hKn )(hK

c

on

S

S

K

K

gA

BQ2

2

3

2

Now consider the term in the nominator of gradually varied flow equation:

Gradually Varied Flow Equation in Terms of Conveyance


3

3

1

1

hh

hh

Sdx

dh

c

n

o

Let us now consider a wide rectangular channel. The Chezy equation can be written as:

2/12/12/12/1onoh ShACSRACQ

on SC

qh

2

23

The critical depth in a rectangular channel is given by:g

qhc

23

Using these expressions and assuming that the Chezy coefficient C does not depend on depth h, the gradually varied flow equation can be written as:

This equation is known as equation of Bresse

If we use Manning-Strickler, we have:n

ShBSh

n

BhSh

n

AQ o

onn

on

2/13/52/13/22/13/2 2/12/1

3/5

oo S

qn

BS

Qnh

on ShBCQ 3222

In this case equation of Bresse becomes: 3

3/10

1

1

hh

hh

Sdx

dh

c

n

o

named after the French scientist J.A.C. BRESSE (1822-1883), who developed it first.

Special forms of Gradually Varied Flow Equation: Wide Channel


Before we present all possible gradually varied flow profiles, let us take a look at the general properties of such curves:• The water surface profile approaches asymptotically to uniform depth hn.

• The water surface profile is orthogonal to the critical depth line, when h = hc.

Water surface profiles are classified according to the bed slope.

0oS

0oS

0oS

co SS

co SS

co SS

Channel on Mild slope M

Channel on Steep slope S

Channel on Critical slope C

Channel on Horizontal slope H

Channel on Adverse slope A

type profile

type profile

type profile

type profile

type profile

For each profile type several possibilities are distinguished. These are called branches.

In studying gradually varied water surface profiles we should also keep in mind that:

• In subcritical flow (Fr < 1), the perturbations travel both upstream and downstream. The water surface profiles for subcritical flow are controlled by a downstream control section.

• In supercritical flow (Fr > 1), the perturbations travel only downstream. The water surface profiles for supercritical flow are controlled by an upstream control section.

Gradually Varied Flow: Forms of Water Surface


Convention for numbering branches:

• When the water surface profile is higher than both the normal depth and the critical depth, the branch is numbered as type 1,

• the water surface profile is between the normal and critical depths, the branch is numbered as type 2,

• the water surface profile is lower than both the normal depth and the critical depth, the branch is numbered as type 3,



0oS co SS

Channel on Mild slope

M-type profiles

and

cn hhh

1Fr

0dx

dh

Branch M1

cn hhh

1Fr

0dx

dh

Branch M2

hhh cn

1Fr

0dx

dh

Branch M3

Towards upstream the profile approaches asymptotically

normal depth, towards downstream the curve tends to

become horizontal.

Encountered:• Upstream of a weir or a dam• Upstream of a pier• Upstream of certain bed

slope changes points


normal depth, towards downstream the curve

decreasingly tends to critical depth.

Encountered:• Upstream of an increase in

bed slope• Upstream of a free drop

structure

Towards downstream the profile approaches increasingly

to critical depth.

Encountered:• When a supercritical flow

enters a mild channel• After a change in slope from

steep to mild

cn hh



0oS co SS

Channel on Steep slope

S-type profiles

and

nc hhh

1Fr

0dx

dh

Branch S1

nc hhh

1Fr

0dx

dh

Branch S2

hhh nc

1Fr

0dx

dh

Branch S3


normal depth, towards downstream the curve tends to

become horizontal.

Encountered:• Upstream of a weir or a dam• Upstream of a pier• Upstream of certain bed

slope changes points


normal depth, towards downstream the curve

decreasingly tends to critical depth.

Encountered:• Upstream of an increase in

bed slope• Upstream of a free drop

structure

Towards downstream the profile approaches increasingly

to critical depth.



steep to mild

cn hh



0oS co SS

Channel on Critical slope

C-type profiles

and

nc hhh

1Fr

0dx

dh

Branch C1 Branch C2

nc hhh

1Fr

0dx

dh

Branch C3

The water surface profile is horizontal, when Chezy

equation is used.

Encountered:• Upstream of a dam/weir• At certain bed slope change

locations

There is no physically possible C2 profile.

The water surface profile is horizontal, when Chezy

equation is used.



steep to mild

cn hh



0oS

Channel on Horizontal slope

H-type profiles

Branch H1

chh

1Fr

0dx

dh

Branch H2

hhc

1Fr

0dx

dh

Branch H3

Normal depth becomes infinite and is meaningless.

Consequently, H1 profile is not possible.

Similar to M2 profile

Encountered:• Upstream of a free drop

structure

Similar to M3 profile


enters a horizontal channel

nh



0oS

Channel on Adverse slope

H-type profiles

Branch A1 Branch A2 Branch A3

Normal depth becomes infinite and is meaningless.

Consequently, A1 profile is not possible.

Similar to H2 profile (parabolic)

Encountered:• Upstream of a certain bed

slope change location

Similar to H3 profile(parabolic)


enters a channel with adverse slope

nh

chh

1Fr

0dx

dh

hhc

1Fr

0dx

dh



Note that the passage from subcritical flow to supercritical flow occurs with a smooth surface.

On the other hand, when the flow passes from supercritical flow to subcritical flow, a sudden increase in the water depth is observed. On the figure this is indicated by HJ, which means hydraulic jump. We will study hydraulic jump in more detail later.

http://www.lmnoeng.com/Channels/HydraulicJump.htm

Photograph from Ohio University's Fluid Mechanics Laboratory. Athens, Ohio USA

Control point, as the name implies, is the point that controls the water surface profile. At a control point we can generally write an expression between discharge and depth. Thus, it can be used as boundary condition for calculating the water surface profile.

Gradually Varied Flow: Notion of Control Section


Changing from a mild slope to a steep slope (passage from subcritical flow to supercritical flow).

Subcritical flow at a free overfall.

In open channel flow critical section is a valuable tool because, knowing the geometry of the section, one can write the relationship between flow depth and discharge.

Due to this property, critical condition is sometimes forced at a point in the channel. Then the discharge can be obtained by measuring the flow depth.

cb hh 71.0

In open channel flow locally a critical flow situation may exist for certain situations, such as slope change from mild to steep, free fall (drop structure), and excessive contraction, etc.

In fact, the critical depth takes place about 3 to 4 times hc upstream of the brink (due to curvature of streamlines). The depth at the brink is approximately equal to:

The cases of critical flow due to excessive contraction and a high positive step will be studied later.

Critical Depth as Control Section and Other Uses of Critical Depth


x

c

n

o hxf

hh

hh

Sdx

dh,

1

1

3

3

Several methods are available for computing gradually varied water surface profiles:

1. The most obvious is to solve the differential equation of gradually varied flow, equation of Bresse, using a numerical method, such as 4th order Runge-Kutta method. This method is called method of direct integration.

Equation of Bresse using Chezy equation:

Equation of Bresse using Manning-Strickler equation:

4th order Runge-Kutta method formula can be written as: 4321 226

kkkkx

hh xxx

where: x Coordinate along the channel length. The origin can be arbitrarily placed at any location.

xh Flow depth at location x. All flow parameters at this location are known.

xxh Flow depth at location x+Dx. This is the unknown flow depth we are calculating.

x

c

n

o hxf

hh

hh

Sdx

dh,

1

1

3

3/10

xhxfk ,1

12 2,

2k

xh

xxfk x

23 2,

2k

xh

xxfk x

34 , kxhxxfk x

Computations should start from a point where all flow parameters are known (such as a control point) and proceed upstream if the flow is subcritical and downstream if the flow is supercritical.

Computation of Gradually Varied Flow


Several methods are available for computing gradually varied water surface profiles:

2. The second possibility is to use directly the energy equation to compute the water surface profile by employing an iterative procedure. This approach is called method of successive approximations.

This method can be applied in two ways:

2.1 The open channel reach under study is divided into sub-reaches at known intervals starting from a control point where all the hydraulic parameters are known. Based on the depth at the known point the depth at the next station is computed. This method is called method of reaches (Stand Step Method in Open-Channel Flow, MH Chaudhry).

2.2 A control point where all the hydraulic parameters are known is identified. The depth at that station, h, is known. We choose another depth h+Dh, and compute where this depth will be along the channel. This method is called method of depth variation (Direct-Step Method, MH Chaudhry).

In this course, we will study only the method of reaches.

Please refer to the textbook and other references for more information on other methods that can be used for computation of water surfaces.



Consider the gradually varied flow shown in the figure.

eSx

We have divided the reach under study into smaller sub reaches of length Dx. We also define the cross sections i, i+1, i+2, ….. etc.

We will assume that the geometric properties of the channel (A, P, B, Rh, Dh) at each cross section can be calculated by knowing the depth.

We will also assume that the depth at cross section i is known. We would like to calculate the depth at cross section i+1.

Let us write the equation of energy Bernoulli equation) between two cross sections i and i+1 :

ei

iii

ii Sxg

Uhz

g

Uhz

22

21

11

2

ei

iii

ii Sxg

Uhz

g

Uhz

22

21

11

2

essii SxHHzzii

11

e

ssii Sx

HH

x

zzii

11

essii S

x

HH

x

zzii

11

es S

dx

dH

dx

dz e

so S

dx

dHS eo

s SSdx

dH

Computation of Gradually Varied Flow: Method of Reaches


Therefore, when using the method of reaches, we will be solving this ordinary differential equation:

eSx

The basic equation we are using is:

ei

iii

ii Sxg

Uhz

g

Uhz

22

21

11

2

Since depth hi , invert elevation zi and the discharge Q are known, we can calculate the left side of the equation, i.e. the total energy head, Hi directly.

eii SxHH 1

Let us now assume a depth hi+1 . Since the invert elevation zi and the discharge Q are known, we can also calculate the total energy head, Hi+1 directly.

Now the question is weather the assumed that is the correct depth. This can be easily done. If the assumed depth hi+1 is correct, then, the difference between the total heads Hi and Hi+1 should be equal to Dx Se.

The energy gradient can be calculated using either the equation of Chezy or Manning Strickler:

h

e RC

AQS

2

2/Chezy equation: Manning Strickler equation:

3/4

22/

h

eR

nAQS



Since the hydraulic parameters are varying from cross section i to i+1, we may want to use the average value of the energy gradient:

21

ii ee

e

SSS

Note also that : ii xxx 1

We should therefore check that: eii SxHH 1 or

21

11ii ee

iiii

SSxxHH is satisfied.

If the above equation is not satisfied, a new value should be assumed for hi+1 and the computations must be carried out again.

All these calculation can easily be carried out on a spread sheet.

If there are singular losses between the two cross sections i and i +1, this should also be taken into account. Then the equation becomes:

g

AQKSx

g

UKSxHH eeii 2

/

2

22

1

Again considering average values we can write:

2

//

2

1

2

21

2

111

iiee

iiii

AQAQ

gK

SSxxHH ii

2

//

2

1

2

/ 21

22

ii AQAQ

gg

AQ

Consequently:



The computation of gradually varied flow equations can be easily carried out on a spreadsheet using Goal Seek function

A trapezoidal channel having a bottom width of b = 7.0m and side slopes of m = 1.5, conveys a discharge of Q = 28m3/s. The channel has a constant bed slope of So = 0.001. The Manning friction coefficient for the channel is n = 0.025m-1/3s. The channel terminates by a sudden drop of the bed.1. Determine the type of water surface profile to be expected.2. Calculate the water surface profile for a reach length of 3200m.



A transition is a change in the channel geometry over a relatively short distance. The change can be contraction or expansion of the section, or a change in the section cross section geometry (say from rectangular to trapezoidal), or an abrupt rise or drop of the channel bed. In designing transition, the attention must be paid to create minimum amount of disturbance to the flow.

https://www.fhwa.dot.gov/engineering/hydraulics/pubs/06086/hec14ch06.cfm#fig096

The figure shows, typical designs for channel transition from a rectangular cross section to a trapezoidal cross section.

Rapidly Varied Flow at Channel Transitions


ss HE

gU 2/22

chh 1

gU c 2/2ch

gU 2/21

1h

ch

chh 2

Alternate depths

ss HorE

h

Specific Energy

Curve plotted for a constant Q

Subcrit

ical f

low

Supercritical flow

2

22

22 gA

Qh

g

UhH s

Specific energy curve is an extremely useful tool for analyzing various flow situations. In the following slides we will learn how the specific energy curve can be used to analyze various flow situations in channel transitions (flow over a positive or negative step, flow through a contraction or expansion).

2h

Use of Specific Energy to study Rapidly Varied Flow at Channel Transitions


In the following pages we will study in detail the rapid change of water surface at four types of channel transitions under both subcritical and supercritical conditions:

1. Subcritical flow over a positive step

2. Supercritical flow over a positive step

3. Subcritical flow over a negative step

4. Supercritical flow over a negative step

5. Subcritical flow through a contraction

6. Supercritical flow through a contraction

7. Subcritical flow through an expansion

8. Supercritical flow through an expansion

z

z

Side view

Side view

Top view

Top view

Q

Q

Q

Q

Rapidly Varied Flow at Channel Transitions


zzH s 1

1sH

z2h

1h

h

sH

2sH

gU 2/21

zH s 1

2sH

2h

h

sH

gU 2/22

./ constBQq

./ constBQq

ch

Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed).

QBUhBUh 2211

1sH

21 22

22

2

21

1 ss Hg

Uhz

g

UhzH

Rapidly Varied Flow: Subcritical Flow over a Positive Step


zzH s 1

1sH

z2h1h

h

sH

1sH

gU 2/21

./ constBQq

ch


QBUhBUh 2211

zH s 1

2sH

2h

h

sH

2sH

gU 2/22

./ constBQq

21 22

22

2

21

1 ss Hg

Uhz

g

UhzH

Rapidly Varied Flow: Supercritical Flow over a Positive Step


zz

zHH ss 12

1sH

1h

2h

h

sH

gU 2/21

./ constBQq

1sH

2sH

2h

h

sH

gU 2/22

./ constBQq

2sH


21 22

22

2

21

1 ss Hg

Uhz

g

UhzH QBUhBUh 2211

Rapidly Varied Flow: Subcritical Flow over a Negative Step


zz

zHH ss 12

1sH

1h2h

h

sH

gU 2/21

./ constBQq

1sH

2sH

2h

h

sH

gU 2/22

./ constBQq

2sH


21 22

22

2

21

1 ss Hg

Uhz

g

UhzH QBUhBUh 2211

Rapidly Varied Flow: Supercritical Flow over a Negative Step


1sH

1ch

1h

h

sH

gU 2/21

11 / BQq

Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed).

QBUhBUh 222111

1sH

21 22

22

2

21

1 ss Hg

Uh

g

UhH

2h

h

sH

gU 2/2222 / BQq

2sH

1B 2B

Top

view

Side

vie

w

12 ss HH

2ch

Rapidly Varied Flow: Subcritical Flow through a Contraction


1sH

1ch1h

h

sH

gU 2/21

11 / BQq


QBUhBUh 222111

1sH

21 22

22

2

21

1 ss Hg

Uh

g

UhH

12 ss HH

2h

h

sH

gU 2/2222 / BQq

2sH

1B 2B

Top

view

Side

vie

w

2ch

Rapidly Varied Flow: Supercritical Flow through a Contraction


1sH

1ch

1h

h

sH

gU 2/21

22 / BQq


QBUhBUh 222111

1sH

21 22

22

2

21

1 ss Hg

Uh

g

UhH

2h

h

sH

gU 2/2211 / BQq

2sH

1B 2B

Top

view

Side

vie

w

12 ss HH

2ch

Rapidly Varied Flow: Subcritical Flow through an expansion


Rapidly Varied Flow: Supercritical Flow through an Expansion

1sH

1ch1h

h

sH

gU 2/21

22 / BQq


QBUhBUh 222111

1sH

21 22

22

2

21

1 ss Hg

Uh

g

UhH

2h

h

sH

gU 2/2211 / BQq

2sH

1B 2B

Top

view

Side

vie

w

2ch

12 ss HH


Rapidly Varied Flow: Special Case of Choked Flow due to a High Positive Step

z

1sH

z

22 chh

fh1

h

sH

zH s 1

22 css HH

h

sH

./ constBQq

./ constBQq

If the step is too high, subtracting Dz from Hs1, we cannot fall back onto the specific energy curve. The flow is said to be choked. The step is too high. The water accumulates upstream of the step until it can pass over it by going through critical flow over the step. Same equations hold. However, now h1 is also an unknown. Condition of critical flow over the step provides the third equation needed for the analysis.

QBUhBUhff

2211

1sH

21 22

22

2

21

1 ss Hg

Uhz

g

UhzH f

ff

By subtracting Dz from Hs1, we cannot fall back onto the specific

energy curve

initial water surface

final water surface

ih1

At one point on the step the flow goes through the critical depth of

the cross section on the step.

22 chh

1ch

initial energy line

final energy line


Rapidly Varied Flow: Special Case of Choked Flow due to too much Contraction

1sH

1ch

fh1

h

sH11 / BQq

1sH

h

sH

22 / BQq

2sH

1B 2B

Top

view

Side

vie

w initial water surface

final water surface

ih1

At one point on the contracted section the flow goes through the critical depth of that cross section.

22 chh

With the available energy Hs1, we cannot cut the specific energy curve of the contracted section

22 chh

If the step is contracted too much, with the specific energy Hs1 we cannot cut the specific energy curve of the contracted section. The flow is said to be choked. The section is contracted too much. The water accumulates upstream of the contraction until it can pass a discharge of Q to the downstream by going through critical flow of the contracted section. Same equations hold. However, now h1 is also an unknown. Condition of critical flow at the contracted section provides the third equation needed for the analysis.

QBUhBUhff

221121 22

22

2

21

1 ss Hg

Uhz

g

UhzH f

ff

22 css HH

initial energy line

final energy line


Energy Losses For Subritical Flow in Open Channel Transitions

Taken from USACE (1994)

U.S. Army Corps of Engineers, 1994. “Hydraulic Design of Flood Control Channels,” Engineering and Design Manual, EM 1110-2-1601, July 1991, Change 1 (June 1994).

Head losses at contractions and expansions can be calculated using the following expressions:

g

U

g

UCH cLc 22

21

22

g

U

g

UCH eLe 22

21

22

U1 U2 U1 U2

Contraction Expansion


Rapidly Varied Flow: Hydraulic Jump

Hydraulic jump is a natural phenomenon that occurs when supercritical flow is forced to become subcritical.

The passage from supercritical flow to subcritical takes place with a sudden rise of the flow depth accompanied by a very turbulent motion that may entrain air into the flow.

To derive the equation governing hydraulic jump in a channel (see figure above), we will make use of momentum and continuity equations simultaneously.

12sin21

UUQFWFFF fPPx

Consider a control volume, which comprises the hydraulic jump. The upstream cross section of the control volume is in supercritical flow and the downstream section is in subcritical flow. Forces acting on this control volume are the weight of the fluid, W, the upstream and downstream pressure forces, FP1 and FP2 respectively, and the friction force, Ff. The momentum equation can be written as:

Assuming a rectangular channel, we have: BhA 11 BhA 22 BQq / 11

21A

hFP 2

1

22A

hFP



Using these expressions and neglecting the component of weight and friction forces, the momentum equation becomes:

12

22

21

22 h

q

h

qQB

hB

h

12

22

21

22 h

q

h

qq

hh

1

2

2

222

21

22 gh

q

gh

qhh

2

222

1

221

22 gh

qh

gh

qh

Note that the left and right hand side of the equation represent the specific momentum, which is defined as: 2

22 h

gh

qM

Let us now make use of equation of continuity to write the momentum equation as:

2

11

212

12

2 122

1

h

hh

Uhh

Divide both sides by (h2 – h1) to get: 022

1112

22

g

Uhhhh

Only the positive root of the above quadratic equation is physically meaningful:

181

2

1 21

1

2 Frh

h

181

2

1 22

2

1 Frh

h

g

Uh

hhh

21

1

21

12 242

1

Written in dimensionless form, the above equation becomes:

or

where1

212

1 gh

UFr

2

222

2 gh

UFr and This equation is called the equation of Bélanger in honor

of the French scientist who developed it for the first time.



Note that for a hydraulic jump on larger slopes, the weight of the fluid cannot be neglected. In this case, the equation of Bélanger for hydraulic jump becomes:

181

2

1 21

1

2 Frh

hHJ where 027.010HJ

as given by Rajaratnam.

https://www.fhwa.dot.gov/engineering/hydraulics/pubs/06086/hec14ch06.cfm#fig096

Hydraulic jumps are classified according to the approach flow Froude number.

a is in degrees



21

312

22

2

21

1 42221 hh

hh

g

Uh

g

UhHHh sshj

Energy loss across the hydraulic jump:

Length of the hydraulic jump: 7512

hh

Lhj

Classification of hydraulic jumps:

7.11 Fr Undular jump

5.27.1 1 Fr Weak jump

5.45.2 1 Fr Oscillating jump

Strong jump

0.95.4 1 Fr Steady jumpjump type generally preferred in engineering applications

6.11 Fr

0.91 Fr

Photos from (Dr. H. Chanson): http://www.uq.edu.au/~e2hchans/undular.html


Use of Hydraulic Jump in Hydraulic Engineering

Hydraulic jump is used for dissipating the energy of high speed flow which may harm the environment if released in an uncontrolled way.

The hydraulic jump, should take place in a area where the bottom is protected (for example by a concrete slab or large size rocks). If the jump takes place on erodible material the formation of the erosion hole may endanger even the foundations of the structure.

In real engineering projects measures are taken to ensure that the hydraulic jump takes place in the area with a protected bottom. This is achieved by creating a stilling basin with the use of chute blocks, baffle piers, and end sill, etc.


Examples of the Use of Hydraulic Jump in Hydraulic Engineering

http://www.engineering.uiowa.edu/~cfd/gallery/images/hyd8.jpg


Design of Stilling Basins

USBR Type I Stilling Basin USBR Type II Stilling Basin

USBR Type IV Stilling Basin

USBR Type III Stilling Basin

SAF Stilling Basin Pillari’s Stilling Basin


Books on Design of Stilling Basins

Hydraulic Design of Stilling Basins and Energy Dissipators

by A. J. Peterka, U.S. Department of the Interior,

Bureau of Reclamation

Energy Dissipators and Hydraulic Jump

by Willi H. Hager


Positioning of a Hydraulic Jump

Draw the upstream supercritical flow profile starting from a control section at the upstream.

Draw the downstream subcritical flow profile starting from a control section at the downstream.

Draw the conjugate depth curve for the upstream supercritical flow profile.

For a hydraulic jump with zero length the jump is a vertical water surface between A’ and Z’.

If we wish to take into account the length of the jump for each point on the conjugate depth curve, draw a line parallel to the bed. The length of the line should be equal to the length of the jump, i.e. 3 to 5 times the height difference between the conjugate depth and the water depth. The tips of these lines are joined to obtain a translated conjugate depth curve which takes into account the length of the jump. The intersection of the downstream profile with the translated conjugate depth gives the downstream end of the jump.

Thus, the jump takes place between A and Z.


Oblique Hydraulic Jump

Consider again the specific energy curve.

ss HE

gU 2/22

chh 1

gU c 2/2ch

gU 2/21

1h

ch

chh 2

Alternate depths

ss HorE

h

Specific Energy

g

UhH s 2

2

Curve plotted for a constant Q

Subcrit

ical f

low

Supercritical flow2h

It can be seen that when the flow is supercritical, a small variation in depth (say Dh) causes a large variation in kinetic energy and, thus the specific energy (DEs or DHs).

h

ss HorE

Therefore, in supercritical flow, a transition, such as a change in width or a change in direction, will provoke an abrupt variation of flow depth and stationary, stable gravity waves will appear on the free surface.



Referring to the figure on the left, consider the case of a supercritical deflected by a side wall making an angle q with the approach channel.

A standing wave front forms making an angle b with the approach channel direction. This is called an oblique (hydraulic) jump. Note that this is somewhat different than a classical hydraulic jump due to the fact that the flow is still subcritical downstream of the wave front:

and

11

1

1

11

c

U

gh

UFr

The continuity equation in the direction normal to the wave front gives:

2211 UhUh

Neglecting the bottom friction, the momentum equation in the direction normal to the wave front gives:

nnn UUq

hhF 12

22

21

22

Froude numbers using the velocity components normal to the oblique wave front are defined as: 1

11

gh

UFr

nn

2

22

gh

UFr

nn

12

2

2

22

c

U

gh

UFrand



Note that in the direction tangent to the wave front no momentum change takes place. The equation of momentum in tangential direction becomes:

This clearly shows that:

Combining continuity and momentum equations, one obtains the equation for change of depth across an oblique jump:

In which the Froude number normal to the wave front is defined as: sin

sin1

1

1

1

11 Fr

gh

U

gh

UFr

nn

ttt UUqF 120

tt UU 21

Geometric considerations allow us to write:

sin11 UU n sin22 UU n

tan1

1

nt U

U

tan2

2

nt U

U

181

2

1 2

11

2 nFrh

h


The angle b of the wave front can be expressed as:

Note that, for small variations of depth, thus for gradual transitions, one gets:

Combining two equations for change of depth across an oblique jump, we can write:

Using equation of continuity and geometric relationships, the equation for the oblique jump can also be written as:

tan

tan

2

1

1

2n

n

U

U

h

h

181tan2

381tantan

2

12

2

1

n

n

Fr

Fr

1

2

11sin

1

2

1

2

1 h

h

h

h

Fr

1

1

1

1sin

U

c

Fr

These derivations were originally carried out by Ippen (1949). He also experimentally verified the relationship above which gives the relationship between q and b for contracting channels (only).




The relation ship between q and b is plotted on the left. Following observations can be made:

• For all Froude numbers there exists a maximum value for the angle of deflection, qmax.

• For all values q smaller than qmax, two values of b are possible. However, since the analysis is made for the case the flow remains supercritical after the jump, i.e. Fr2 > 1, we should consider only the values on the left side (solid lines).

It is important to note that, any perturbation created by one wall will be reflected by the other wall and so on. To study this behavior, we will consider two cases:• Asymmetrically converging channel, and• Symmetrically converging channel.


Reflection of Oblique Jumps in an Asymmetrical Channel Contraction

Consider the channel on the left. The left wall is deflected into the flow by an angle q, while the right wall remains straight.

1Fr 181tan2

381tantan

2

112

2

11

n

n

Fr

Fr

1

1

1

1

2

tan

tan

h

h 12 hh

121 FrFr

2Fr 181tan2

381tantan

2

222

2

22

n

n

Fr

Fr

12

2

2

2

3

tan

tan

h

h 23 hh

132 FrFr

3Fr 181tan2

381tantan

2

332

2

33

n

n

Fr

Fr

23

3

3

2

3

tan

tan

h

h 23 hh

143 FrFr

… and so on. If the contracting channel is sufficiently long, the wave reflection continues until the flow finally becomes subcritical.


Reflection of Oblique Jumps in an Symmetrical Channel Contraction

Causon D. M., C. G. Mingham and D. M. Ingram (1999), Advances in Calculation Methods for Supercritical Flow in Spillway Channels, ASCE, Journal of Hydraulic Engineering, Vol. 125, No. 10, pp. 1039-1050.

View of Supercritical Flow in Curved Transition:(a) Experimental Visualization of Standing Wave Patterns, after Ippen and Dawson (1951);(b) Computer Visualization Based on 2D, Shock-Capturing, Numerical-Model Predictions


Designing a Symmetrical Channel Contraction for Supercritical Flow

A good channel contraction design for supercritical flow should reduce or eliminate the undesirable cross wave pattern. This can be achieved by choosing a linear contraction length LT, thus by choosing a contraction angle q’, such that the positive waves emanating from points A and A’, due to converging walls, arrive directly at points D and D’, where negative waves are generated due to diverging walls. Such a design is shown in the figure on the left.

The choice of the angle q’ depends on the approach Froude number, Fr1, and the contraction ratio B3/B1.

Based on continuity equation, and assuming that the flow remains supercritical in the contracted section, i.e. Fr3 > 1, we can write: 3

1

2/3

3

1

33

11

1

3

Fr

Fr

h

h

Uh

Uh

B

B

From geometric considerations , we also have the relationship:tan2

31 BBLT

The angle q’ to satisfy these two equations is calculated using an iterative procedure.


Gradually Varied Flow with Lateral Inflow

xqdx

dQ

We will now consider the case of gradually varied flow with lateral inflow. In the most general case, the discharge added subtracted laterally affects both to the mass and the momentum of the flow.

The continuity equation with lateral flow becomes:

represents lateral discharge which can be positive if a discharge is added, or negative if a discharge subtracted.

q

dx

dUA

dx

dAU

dx

dQq We can also write:

Consider the gradually varied flow with lateral inflow as shown on the left.

Equation of the momentum states that the sum of all forces is equal to the change in momentum:

QUFWFF fPx sin

Let us now analyze all the terms in this equation one by one.


AdhAdhzAzF PPP The net hydrostatic pressure force will be:

zP is the distance from the free surface to the centroid of the flow area A:

dxASdxAW oP sinsinThe weight of the water prism between two sections that are dx apart is:

The rightmost side assumes that a is small.

dxASdxPF eof The friction force can be written as: since eho SR

The change in the momentum can be written as:

cos UdxqQUdUUdQQQU

Note that dQdxq

Note also that the lateral flow is entering or leaving the channel at an angle f and with velocity Uℓ.

Let us insert above expression the equation of momentum, and simplify:

cos UdxqQUdUUdQQdxASdxASAdh eo

A

Udxq

A

QU

A

dUUdQQdxSSdh eo

cos



gAdx

Udxq

gAdx

QU

gAdx

dUUdQQSS

dx

dheo

cos

gA

Uq

gAdx

QUdQdUUdQQdUQU

gAdxSS

dx

dheo

cos1

gA

Uq

gAdx

QU

gAdx

UdQ

gAdx

QdU

gAdx

QUSS

dx

dheo

cos

gA

Uq

gAdx

UdQ

gAdx

QdUSS

dx

dheo

cos

gA

Uq

dx

dQ

Adx

dU

g

USS

dx

dheo

cos1

This is the equation of free surface for a steady gradually varied flow with lateral inflow, which is also called a steady spatially varied flow.

Recalling that: dQdxq and dAA

dQQdUU

Simplifying also second order terms AdA and dQdA, the spatially varied flow equation can be written as:

BgA

AQ

UgAgA

QqSS

dx

dhleo

//

1

cos1

2

2

2

It can be verified that, this equation reduces to gradually varied flow formula when lateral flow is zero, qℓ = 0.

Spatially varied flow equation can also be solved using the same methods for solving gradually varied flow equation.



Example of Structures for Spatially Varied Flow: Side Channel Spillway

http://www.tornatore.com/joel/pics/index.php?op=dir&directory=20040227

Side channel spillway of Hoover Dam in Nevada.

http://www.firelily.com/stuff/hoover/flood.control.html

Side channel spillway of Hoover Dam in Nevada as seen from the reservoir side.


Example of Structures for Spatially Varied Flow: Side Channel Spillway

http://www.hprcc.unl.edu/nebraska/sw-drought-2003-photos1.html

Side channel spillway of Hoover Dam in Nevada, looking downstream.


Quiz No 1(5 minutes)

A steep channel is connected to a mild channel as shown in figure. Both channels have a rectangular cross section. The following data is given:

1nh

2nh

ch1oS

2oS

mBB 0.421

01.01oS 001.0

1oS

smnn 3/121 012.0

smQ /6 3 mhc 612.0 mhn 383.01 mhn 818.0

2

The flow in steep channel is steady and uniform. Determine in which channel, steep or mild, the hydraulic jump will take place.


Quiz No 1(5 minutes): Solution

A steep channel is connected to a mild channel as shown in figure. Both channels have a rectangular cross section. The following data is given:

1nh

2nh

ch1oS

2oS

mBB 0.421 01.01oS 001.0

1oS

smnn 3/121 012.0 smQ /6 3

mhc 612.0 mhn 383.01 mhn 818.0

2

The flow in steep channel is steady and uniform. Determine in which channel, steep or mild, the hydraulic jump will take place.

Solution: Assume that the steady uniform flow continues all the way down to the point where the slope becomes mild. Let us see if there is a jump at that point what would be the conjugate depth.

181

2

1 21

1

1 Frh

hcj

181

2

1 2111 Frhhcj mhcj 92.0102.281383.0

2

1 21

smhB

QU

n

/91.3383.00.4

6

1

1

02.2383.081.9

91.3

1

1

ngh

UFr

mhmh ncj 818.092.0

21 A jump taking place at the point of slope change will be too strong. It can jump higher than the normal depth in the channel. Therefore, the flow continues into the mild channel without a jump and creates an M3 type profile up to a depth whose conjugate depth is equal to the uniform flow depth of the mild channel.


Quiz No 2 (5 minutes)

Consider the channel on the left with the following data:

Determine if the flow is choked due to the positive step. What will be the flow depth over the step?

ch

ch

1h

zmh 288.01

mhc 129.0

301.01sH

193.0csH

Q

mz 12.0


Quiz No 2 (5 minutes): Solution

Consider the channel on the left with the following data:

ch

ch

1h

zmh 288.01

mhc 129.0

301.01sH

193.0csH

Q

mz 12.0

mHmzHcss 193.0181.012.0301.0

1

Thus the flow is choked. The flow will go through the critical depth over the step, i.e. mhh c 129.02

Solution: Assuming no singular energy losses due to the step, the energy grade line remains at the same level. Over the step, the energy is reduced by an amount Dz.

Determine if the flow is choked due to the positive step. What will be the flow depth over the step?

Lecture 4.Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels1 ENGR 691 – 73:...

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Transcript of Lecture 4.Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels1 ENGR 691 – 73:...