Dynamics of Fluid Flows

7/21/2019 Dynamics of Fluid Flows

http://slidepdf.com/reader/full/dynamics-of-fluid-flows 1/13

1

DYM MICS OF FLUID FLOW

SPECIFIC ENERGY AND CRITICAL DEPTH

In open channel flows one of the important parameter is “specific energy” and is defined

as,

2

2

V E y

g (1)

where y is the water depth and V is the flow velocity. It is also called as energy grade

line (EGL). For a given flow rate, there are two possible states for the same specific

energy as shown in Fig. 1.

h f

Velocity head E

y

Horizontal

EGL

Fig. 1: Illustrative sketch for specific energy.

In a simpler case, consider two possible states of specific energy in a rectangular

channel of width b . The discharge per unit width for this channel is given by,

.Q

q V yb

Eq. (1) can now be written as,

2

22

q E y

gy (2)

For a given channel of constant width, the value of q remains constant along the channel

although the depth y may vary. The variation of q with y is plotted in specific energy

diagram (Fig. 2). From this curve, it is clear that specific energy attains to a minimum



2

value at certain depth for a given q . This depth is known as critical depth and it can be

obtained by setting 0dE

dy in Eq. (2).

yc

y

EEmin

Sub-critical

Super-critical

0

Critical

Constant q

Fig. 2: Depth verses specific energy curve.

Minimum specific energy occurs at,

1 12 23 3

min2

3;

. 2c c

q Q y y E y

g b g

(3)

The velocity of flow at “critical depth” is known as “critical velocity” cV and the

corresponding discharge is .c c cq V y

.

Referring to Fig. 2, for min E E , no solution exists and thus the flow is unrealistic.

For min E E , there are two possible solutions;

Large depth withc

V V sub-critical flow.

Small depth with cV V s called super-critical flow.

Frictionless flow over a bump

Consider an open channel flow over a bump as shown in Fig. 3. The behavior of the free

surface is sharply different based on the approach of the flow; i.e. sub-critical or super-

critical. For frictionless two-dimensional flow, sections ‘1’ and ‘2’ are related by

continuity and momentum equation;



3

2 2

1 21 1 2 2 1 2;

2 2

V V y V y V y y h

g g

Eliminating 2V between the above cubic polynomial equations for water depth 2

y over

the bump:

2 2 23 2 1 1 12 2 2 2 1

.0 where

2 2

V y V y E y E y h

g g (4)

Supercritical approach flow

V1 V2y1 y2

hBump

Subcritical approach flow

Fig. 3: Frictionless two-dimensional flow over a bump.

yc

y

h

Ec

Sub-critical

bump

Super-critical

bump

(1)

(2)

y1

y2

hmax

EE2 E1

Fig. 4: Specific energy plot for flow over a bump.

Following points may be noted down from the above analysis;

The Eq. (4) has one negative and two positive solutions if h is not too large. Its

behavior is illustrated in Fig. 4 and depends upon whether the point ‘1’ is on the

upper or lower leg of the energy curve.



4

In sub-critical approach 1 1r F , the water level will decrease at the bump

whereas in super-critical approach flows 1 1r F , the water level will increase

over the bump.

If the bump height reaches max 1 ch E E , the flow at the crest will be critical

1 1r F . In this case, no physical solution is possible i.e. too large bump will

‘choke’ the channel and cause frictional effect, called “hydraulic jump”.

HYDRAULIC JUMP

A “hydraulic jump” is a discontinuity when there is a conflict between the upstream and

downstream control parameters in an open channel flow. It occurs, when the upstream

flow is fast and shallow, and the downstream flow is slow and deep and thus provides a

mechanism to make a transition between two types of flow. One such example is flow

under the sluice gate where the downstream portion of the gate (upstream of the channel)

is super-critical flow, while the flow is sub-critical in upstream side (downstream of the

channel).

Consider a simplest type of hydraulic jump that occurs in a horizontal, rectangular

channel as shown in Fig. 5. Take two sections ‘1’ and ‘2’ in upstream and down

streamside where the flow is nearly uniform, steady and one-dimensional. Neglecting thewall shear stress, the momentum equation can be written as,

1 2 2 1 1 1 2 1. . .F F Q V V V y b V V (5)

where b is the channel width. The pressure force at either side is hydrostatic and acts at

the channel cross-sections i.e.

2

1 11 1 1 1

2

2 22 2 2 2

. . . . .. . .

2 2

. . . . .. . .2 2

c

c

g y g b yF p A y b

g y g b yF p A y b

(6)

Using Eq. (6) in Eq. (5), we get,

2 2

1 2 1 12 1

.

2 2

y y V yV V

g (7)



5

Control volume

Energy line

(1)

(2)

Q, F1

F2

y2

y1

hL

V2

V1

Shear stress = 0

Fig. 5: Hydraulic jump geometry.

Similarly, the mass and energy conservation equations can be written for section ‘1’ and

‘2’ respectively i.e.

1 1 2 2. . . . y bV y bV Q (8)

2 2

1 21 2

2 2 L

V V y y h

g g (9)

The head loss Lh is due to violent turbulent mixing and dissipation that occurs within the

jump itself. All Eqs. (7 to 9), are satisfied for 1 2 1 2; and 0 L y y V V h . It represents a

trivial case when there is no jump. The other possible solution can be obtained by

combining Eqs. (7 and 8) to eliminate 2V i.e.

2 2 2

1 2 1 1 1 1 1 11 1 2

2 2

. . .

2 2 .

y y V y V y V yV y y

g y g y

(10)

Simplification of this equation gives,

2

22 21

1 1

2 0r

y yF

y y

(11)

where1

11

r

V

F gy is the upstream Froude number. The possible solution for Eq. (11) is

221

1

11 1 8

2 r

yF

y (12)

The dimensionless head loss,1

Lh

y

can then be obtained from Eq. (9) as,



6

22

2 1 2

1 1 1

1 12

L r h y F y

y y y

(13)

For a given values of 1r F , the depth ratios 2

1

y

y

are obtained from Eq. (12) and then

the head loss1

Lh

y

is calculated from Eq. (13). It will be negative if 1 1r F (since the

negative head loss violates second law of thermodynamics). The flow must be super-

critical 1 1r F to produce the discontinuity called as “hydraulic jump” and there is a

considerable energy loss across the hydraulic jump. This in fact is extremely useful in

many situations; e.g. the relatively large amount of energy contained in the fluid flowing

down the spillway of dam causes damage to the channel below the dam. By placing

suitable flow control objects in the channel downstream spillway, it is possible to produce

hydraulic jump on the apron of the spillway and thereby dissipate a considerable portion

of the energy of the flow i.e. the dam spillway produces super-critical flow and the

channel downstream of the dam requires sub-critical flow. Hence, the hydraulic jump

provides a means to change the character of the flow.

Classification of hydraulic jumpThe principal parameter affecting hydraulic-jump performance is Froude number. The

Reynolds number and the channel geometry have the secondary effect. Based on the

Froude number, the hydraulic jumps are classified as;

1 1r F : Jump is impossible as it violates second law of thermodynamics.

1 1 to 1.7r

F : Standing wave or undular jump; low dissipation less than 5%.

1 1.7 to 2.5r F : Smooth surface rise known as “weak jump”; dissipation is 5 to

15%.

1 2.5 to 4.5r F : Unstable, Oscillating jump; each irregular pulsation creates a

large wave which can travel downstream for miles, damaging earth banks and

other structures. Dissipation is 15 to 45%.



7

1 4.5 to 9r

F : Stable, well balanced, steady jump; best performance and action,

insensitive to downstream conditions. Dissipation is 45 to 70%.

1 9r

F : Rough, intermittent strong jump but good performance. Dissipation is 70

to 85%.

UNDERFLOW GATES

These gates are typical structures constructed at the crest of an overflow spillway, or at

the entrance of an irrigation canal/river for controlling the flow rate. Some of the typical

structures are vertical gates (commonly called sluice gate), radial gates, and drum gates.

y1

Water level

y2

a

(a) (b) (c)

Water level

Water level

Fig. 6: Underflow gates; (a) vertical gate, (b) radial gate, (c) drum gate.

The flow under the gate is said to be free flow when the fluid issues as a jet of

supercritical flow with free surface open to atmosphere as shown in the Fig. 6. The

discharge per unit width of the gate can be expressed as,

1. . 2 .d q C a g y (14)

The discharge coefficient d C is a function of the contraction coefficient 2c

C y a and

depth ratio 1 y a . The typical values of discharge coefficient from a vertical sluice gate

with free out flow are of the order of 0.55 to 0.6.

There are certain situations where the depth downstream of the gate is controlled by

some downstream obstacle. The jet of water issuing from underflow gate is overlaid by

mass of water that is quite turbulent (Fig. 7). Such a gate is known as

drowned/submerged gate. The flow rate can be obtained from the same equation (Eq. 14)

with appropriate modification in d C .



8

y2

y1

a

q

Water level

Fig. 7: Drowned outflow from a sluice gate.

Flow under a sluice gate

The flow pattern under a sluice gate is shown in Fig. 8. If the flow is allowed free

discharge (Fig. 8-a), then it smoothly accelerates from “sub-critical (upstream) to critical

(near the gap) to super-critical (downstream)”. For “free discharge” the friction may be

neglected. Applying continuity and momentum equation,

2 2

1 21 1 2 2 1 2. . and

2 2

V V V y V y y y

g g (15)

Eliminating 2V , we get

2 2 23 21 1 12 1 2

.0

2 2

V V y y y y

g g

(16)

Thus, for a given sub-critical upstream flow 1 1,V y , there is only one real solution

i.e. super-critical flow at the same specific energy as shown in Fig. 8-b. The flow rate

varies with the ratio 2

1

y

y

and reaches to a maximum when 2

1

2

3

y

y

. When the depth

2 y contracts to 40% less than the gate’s gap height, the flow pattern is similar to that of a

free orifice discharge and can be approximated in the range1

0.5 H

y as,



9

1

1

0.61. . 2 where

1 0.61

d d Q C H b gy C

H

y

(17)

If the tail-water is high as in the case of Fig. 8-c, the free discharge is not possible.

The sluice gate is said to be drowned or partially drowned . There will be energy

dissipation in exit flow, in the form of drowned hydraulic jump and the downstream flow

will return to sub-critical. Hence, Eqs (16 and 17) will not be applicable for such

situations and experimental correlations are necessary.

(a)(b) (c)

Vena contracta

Water level

V1, y1

V2, y2

V1, y1

Water level Dissipation

V2, y2

High tail water

y1

y2

y

EE1 = E2

(1)

(2)

Sub-critical

Super-critical

Fig. 8: Flow under a sluice gate; (a) Free discharge; (b) Specific energy for free

discharge; (c) Dissipative flow under a drowned gate.

Example 1

A rectangular channel 6m wide carries 168 lits/min at a depth of 0.9m. What is the height

of a rectangular weir which must be installed to double the depth? Discharge coefficient

of weir may be taken as 0.85.

Solution

The discharge for a broad crested weir is given by,

32 2

1.7 .

2

ad w

V Q C L H

g

Here, 3 3168m min 2.8m sQ ; 6mw L ; 0.85d

C

Then,

2 22 3 32.8

0.47m2 1.7 1.7 0.85 6

a

d w

V Q H

g C L



10

The depth of the flow required = 2 0.9 = 1.8m

The velocity of approach is given by,

2

2.80.26m s

6 1.8 6 1.8

0.0034m2

0.47 0.0034 0.4666m

a

aa

QV

V h

g

H

Height of the broad crested weir = 1.8 – 0.4666 = 1.3334m.

Example 2

Water flow in a wide channel approaches a 10cm high bump at 1.5m/s. and a depth of

1m. Estimate: (a) the water depth over the bump; (b) the bump height that will cause the

crest flow to be critical. Take the head loss as 0.1m and the flow is frictionless.

Solution

(a) First, the Froude number is calculated as,

11

1

1.50.48

2 2 9.81 1r

V F

gy

It means that the flow is sub-critical. Take two sections ‘1’ and ‘2’ in the entire length of

the flow (Fig 8).

Specific energy of the flow is,

22

11 1

2 1

1.51.0 1.115m

2 2 9.81

1.115 0.1 1.015m

V E y

g

E E h

Applying continuity and energy equations,

2 2

1 21 1 2 2 1 2. . and

2 2

V V V y V y y y h

g g

Eliminating 2V ,

2 2 23 2 1 1 12 2 2 2 1

.0 where

2 2

V y V y E y E y h

g g

3 2

2 2or, 1.015 0.115 0 y y



11

Three real roots of the above equation are, 2 0.86m, 0.45m and -0.29m y . The

negative root is physically impossible. For sub-critical conditions, 2 0.86m y and for

super-critical condition, 2 0.45m y .

The surface level has dropped by 1 2 1 0.86 0.1 0.04m y y h .

The crest velocity, 1 12

2

. 1.5 11.745m s

0.86

V yV

y

The Froude number at the crest is, 22

2

1.7450.424

2 2 9.81 0.86r

V F

gy

i.e. the flow

downstream of the bump is sub-critical.

(b) For critical flow, Froude’s number is unity at the crest and

1 112 23 331 1.3 3 3 3 1.5

0.918m2 2 2 2 9.81

c c

V yq E y

g g

Maximum height for the bump, 1 1.115 0.918 0.197mch E E

The cubic polynomial equation becomes,

2 23 2 1 12 2

3 2

2 2

.0

2

or, 0.918 0.115 0

c

V y y E y

g

y y

The solution is 2 0.61mc

y y

The surface level has dropped by 1 1 0.61 0.197 0.193mc y y h

Example 3

Water flows under a sluice gate on a horizontal bed at the inlet to a flume. The water

depth is 50cm in the upstream of the gate and the speed is negligible. At the vena-

contracta downstream of the gate, the flow streamlines are straight and the depth is 6cm.

Determine the flow speed downstream from the gate and discharge per unit width.

Solution

Referring to the Fig. 13-a, Bernoulli’s equation can be applied upstream and at the vena-

contracta of the flow field as,



12

2 2

1 21 2

2 2

V V gz gz

Solving for 2V ,

2

2 1 2 12 2 9.81 0.5 0.06 0 2.94m sV g z z V

The discharge per unit width is given by,

3

2 2. 2.94 0.06 0.1764m sQ

q V Db

EXERCISES

1. Water flows freely under a sluice gate with upstream depth of 5m and gate opening of

1.5m. Determine: (i) the discharge per unit width; (ii) water depth just upstream of the

gate; (iii) what will be the discharge if the water depth immediately downstream of the

gate is 2m; (iv) compare this value with estimation of discharge under submerged

condition assuming the flow immediately downstream of the gate to be unaffected by

submergence.

2. Water in an open channel flows under a sluice gate. The flow is incompressible and

uniform at two sections ‘1’ and ‘2’ upstream and downstream of the flow respectively.

The depth of water and velocity at the section ‘1’ are 1.5m and 2m/s respectively. The

corresponding values in section ‘2’ are 0.05m and 5m/s respectively. Determine the

direction and magnitude of the hydrostatic force per unit width exerted on the gate by the

flow.

3. Water flows at a rate 10m3/s.m (per unit width) in a wide channel with upstream depth

of 1.25m. If the water undergoes a hydraulic jump, compute the following parameters in

the downstream of the gate: (i) depth of the water; (ii) velocity of the flow; (iii) Froude

number; (iv) head loss; (v) percentage dissipation; (vi) the power dissipated per unit

width; (vii) temperature rise due to dissipation if pc =4.2kJ/kg. K.

4. A rectangular channel with a bottom slope of 1:150 carries water at a rate 20m3/s.

Determine the width of the channel when the flow is in critical condition. Take

Manning’s coefficient as 0.016.



13

5. In a rectangular canal 3.2m wide is laid with a slope of 0.004. The uniform flow occurs

at a depth of 2m. How much height can the hump be raised without causing transition? If

the upstream depth of flow is to be raised to 2.5m, what should be the height of the

hump? Assume Manning’s coefficient as 0.016.

6. Water flows steadily in a rectangular channel laid with a slope of 0.001. The base

width of the channel is 5m and depth of flow is 2m. It is desired to obtain a critical flow

in the channel by providing a hump in the bed. Sketch the flow profile and calculate the

height of the hump required. Take Manning’s coefficient as 0.016 for channel surface.

7. A 3.6m wide rectangular channel conveys 10m3/s of water with a velocity of 6m/s.

(i) Is there a condition for hydraulic jump to occur? If so, calculate the height, length and

strength of the jump?

(ii) What is the loss of energy per kg of water?

8. In a rectangular canal of 0.5m width, a hydraulic jump occurs at a point where the

depth of flow is 0.12m and Froude number is 2.5. Determine,

(i) the specific energy; (ii) the critical and subsequent depths; (iii) loss of head; (iv)

energy dissipated.

9. A hydraulic jump occurs in a V-shaped channel with side slope of 450. Derive the

expression for flow rate in terms of upstream and downstream depth. If the depths of flow

before and after the jump are 0.4m and 0.8m, determine the flow rate and Froude number

before and after the jump.

10. The depth and velocity of the flow in a rectangular channel are 0.9m and 1.5m/s

respectively. If a gate at the downstream of the channel is abruptly closed, what will be

the height and absolute velocity of the resulting surge? If the channel is 1000m long, how

much time will be required for the surge to reach the upstream end of the channel.

11. A rectangular canal carries a discharge of 1.8m3/s per meter width of the canal. The

energy loss due to a hydraulic jump is found to be 3m. Determine the conjugate depths

before and after the jump.

Dynamics of Fluid Flows

Documents

Transcript of Dynamics of Fluid Flows