SPECIFIC FORCE CONCEPT

Momentum Eq. x-direction

� Pressure Force substituting into Eq 1

� for horizontal channel & neglecting friciton

� Rearranging and Q=VA

(1) ... 1221 QVQVFFPPF HfT ρρ −=+−−=

⇒= AyP γ

0FF Hf ==

222

111 A

QQAy

A

QQAy ργργ +=+

2

2

22

1

2

11 ←+=+=gA

QyA

gA

QyAF Specific Force

Specific force for any channel section

For a given Q, F = F (y)The plot of F vs y gives specific force curve

gA

QAyF

2

++++==== y = depth of centroid of a channel section.

Variation of the momentum function with depth.

Specific Force

Minimum Moment

is at critical depth

←+=+=2

2

22

1

2

11gA

QyA

gA

QyAF

T

A

g

Q32

=

Tdy

dA

dy

dF== , 0

3

22

gA

TQFr =

Characteristics of the Specific Force Curve:

•Specific force curve has two limps

•Specific force becomes minimum when the flow is critical

•Lower limp corresponds to supercritical flow

•Upper limp corresponds to subcritical flow.

•For a given F and Q, there are two possible flow regimes represented by the depths y1 and y2 which are called conjugate or sequent depths, corresponding to supercritical and subcritical flows, respectively.

USE OF F-y AND E-y CURVES: EXAMPLES

� HYDRAULIC JUMP

y y

y2

ycy1

E2≠E1

E

F1=F2

F

EGL.

21

USE OF F-y AND E-y CURVES: EXAMPLES

� SLUICE GATE

y y

y1

y2

E1=E2

E

F2≠F1

F

e.g.l.

21

Fb/γ

FbyC

USE OF F-y AND E-y CURVES: EXAMPLES

� SLUICE BLOCKS - CREATE HYDRAULIC JUMP

y2

yC

y1

y

E2≠E1

E

y

F2≠F1

F

e.g.l.

21

Fb/γFb

Rectangular Channels

2

2

22

1

2

1121gA

QAy

gA

QAyFF +=+→=

( )1812

1 21

1

2 −+= Fy

y

21

3

12

4

)(

yy

yyhloss

−=

For rectangular channels, the specific force can be written as specific force per unit width as

gy

qy

b

F 22

2

1++++====

HYDRAULIC JUMP

� Hydraulic jump is a rapidly varied flow in which flow changes abruptly from supercritical state to a subcritical state accompanied by considerable turbulence and energy loss.

21FF =

lhEE ++++====

21

2

2

22

1

2

11gA

QAy

gA

QAy +=+

PRACTICAL APPLICATIONS OFHYDRAULIC JUMP

Hydraulic jump occurs when flow transitions from supercritical to subcritical for ex. base of spillway

It can be used to

�dissipate energy

�recover head or raise the water level

�increase weight on apron

�mix chemicals used for water purification

�aerate water

We would like to know depth of water downstream from jump as well as the location of the jump

�Which equation, Energy or Momentum?

EXAMPLE 5 : HYDRAULIC JUMP

Water flows in a rectangular horizontal open channel at a depth of 0.6m; the flow rate is 3.7 m3/s per meter of width. If a hydraulic jump is possible, calculate the depth just downstream from the jump and the power dissipated in it.

V1 = 3.7/0.6 = 6.17 m/s

OKgy

VFr 154.2

6.081.9

17.6

1

11 >=

×==

( )1812

1 21

1

2 −+= Fy

yy2 = 1.88 m

From continuity i.e.,

V2 = 1.97 m/s

myy

yyhloss 46.0

4

)(

21

3

12 =−

=

lhEE += 21

Rapidly Varied Flow and Hydraulic Jump

� Flow is called rapidly varied flow (RVF) if the flow depth has a large change over a short distance– Sluice gates– Weirs– Waterfalls– Abrupt changes in cross section

� Often characterized by significant 3D and transient effects– Backflows– Separations

Hydraulic Jump!

Rapidly Varied Flow and Hydraulic Jump

� Consider the control volumesurrounding the hydraulic jump

� Assumptions

1. V is constant at sections (1) and (2), and β1 and β2 ≈ 1

2. P = ρgy

3. τw is negligible relative to the losses that occur during the hydraulic jump

4. Channel is horizontal

5. No external body forces other than gravity

Rapidly Varied Flow and Hydraulic Jump � Continuity equation (rectangular channel)

� X momentum equation (horizontal channel, β1 and β2 ≈ 1)

� Substituting and simplifying

Quadratic equation for y2 / y1

a

Rapidly Varied Flow and Hydraulic Jump

� Solving the quadratic equation and keeping only the positive root leads to the depth ratio

� Energy equation for this section can be written as

� Head loss associated with hydraulic jump

Rapidly Varied Flow and Hydraulic Jump

� Experimental studies indicate that hydraulic jumps can be classified into 5 categories, depending upon the upstream Fr

GRADUALLY VARIED FLOW

In a channel, the flow will not be uniform in the vicinity of � Channel transition

� Channel bed slope change location

� Fall

� etc

� The change in flow depth is small and this type flow called as Gradually Varied Flow

Non-uniform gradually varied flow.

xSh fl ∆=Sf

Sf≠Sw ≠So

GRADUALLY VARIED FLOW (dy/dx

2

2

2gA

QyzH ++= T

dy

dA=TdyA=A(y)

θ

θ

(1)(2)

x

d=ycosθy Sw

Sf

Datum S

o

dcosθ = ycos2θ

−+=

3

2

gA

TQ1

dx

dy

dx

dz

dx

dH

2g

VαθycoszH

22 ++=

dx

dy

dy

dA

A

2)(

2g

Q

dx

dy

dx

dz

dx

dH3

2 −++=

( )2rF1dx

dy

dx

dz

dx

dH−+=

velocity uniform and small θ

Assumptions:

� 1) Small channel bottom slope;

� 2) θ

Numerator 34

22

R

VnS f =

If y > yn , V < Vn , R > Rn therefore Sf < So

If y < yn , V > Vn , R < Rn therefore Sf > So

Denominator

3

22

Ag

TQFr ====

If y = yc Fr2 = 1

If y < yc Fr2 > 1

If y > yc Fr2 < 1

ofn SSyy =⇒=

2

r

fo

F1

SS

dx

dy

−

−=

TYPES OF SLOPES(For a Given Q,n, Cross-section, So)

� Calculate Critical Depth of the Channel (Fr=1)

� Calculate the critical slope, Sc of the channel by using Critical depth in the Manning Equation.

� Compare So and Sc

a) If SoSc the state of the flow in the channel under the uniform flow conditions is supercritical flow. Thus, slope of the channel is called STEEP SLOPE and the channel is called as STEEP Channel

a) If So=Sc the state of the flow in the channel under the uniform flow conditions is critical flow. Thus, slope of the channel is called CRITICAL SLOPE.

Gradually Varied Flow� This result is important. It

permits classification of liquid surface profiles as a function of Fr, S0, Sf, and initial conditions.

� Bed slope S0 is classifed as– Steep : yn < yc– Critical : yn = yc– Mild : yn > yc– Horizontal : S0 = 0

– Adverse : S0 < 0

� Initial depth is given a number– 1 : y > yn– 2 : yn < y < yc– 3 : y < yc

EXAMPLES OF WATER PROFILES

� Mild Slope (yn>yc)

yc

ZONE 1; M1 y > yn > yc

ZONE 2; M2 yn > y >yc

ZONE 3; M3 yn > yc >y

yn

y

E

Mild Slope (yn>yc)

01dx

dy 1 Fr y y S S yy

2cofnM1 >

−

−=

Fr

SS fo

yc

ZONE 1; M1 y > yn > yc

ZONE 2; M2 yn > y >yc

ZONE 3; M3 yn > yc >y

yn

y

E

)(0 ofn SSdxdy

yy →→→

)0,0( →→→∞→ fo SFrSdxdy

y

)( line.horizontalatoaproachallyasymptotic surfaceWater

yc

Steep Slope yn⇒ yc > yn

ZONE 2; S2 yc > y > ynZONE 3; S3 yc > yn > y

yn

y

E

)1( →∞→→ Frdx

dyyy c

)(0 ofn SSdxdy

yy →→→

Gradually Varied Flow

Gradually Varied Flow

Gradually Varied Flow12 distinct configurations for surface profiles in GVF.

The Occurence of Critical Flow ConditionsIn an open channel for a given discharge at any section thefollowing equations can be written

zEg

qzyH +=++=

2

2

y 2

( )=− 21 rFdx

dy

Consider the special case S0 – Sf =0It is possible for uniform flow where, dy/dx=0or when the second term is zero.Around pt A dy/dx not zero thus, Fr =1S0 < Sf on upstream and S0 > Sf on downstream thusmust be equal each other somewhere between two.

S0 –Sf

.

Outflow from a reservoir with critical flow at the channel entrance.

1

11

gy

VFr =

Representative controls: (a)sluice gate; (b)change in slope from mild (S01)to steep (S02);(c)entrance to a steep channel; (d)free outfall.

Flow Control and Measurement

� Flow rate in pipes and ducts is controlled by various kinds of valves

� In Open Channel flows, flow rate is controlled by partially blocking the channel.– Weir : liquid flows over device

– Underflow gate : liquid flows under device

� These devices can be used to control the flow rate, and to measure it.

Flow Control and MeasurementUnderflow Gate

� Located at the bottom of a wall, dam, or open channel

� Outflow can be either free or drowned

� In free outflow, downstream flow is supercritical

� In the drowned outflow, the liquid jet undergoes a hydraulic jump. Downstream flow is subcritical.

Free outflowDrowned outflow

Flow Control and MeasurementUnderflow Gate

� Es remains constant for idealized gates with negligible frictional effects

� Es decreases for real gates

� Downstream is supercritical for free outflow (2b)

� Downstream is subcritical for drowned outflow (2c)

Schematic of flow depth-specific

energy diagram for flow through

underflow gates