Nce403 mod unit3

Equation of gradually varied flow and its limitations, Flow classification and surface profiles, Integration of varied flow equation by analytical, graphical

and numerical methods, Flow in channels of non-linear alignment specifically for the

case of a bend.

UNIT 3

05/03/2023 1MODASSAR ANSARI

BY MODASSAR ANSARI 2nd Year Department of civil Engineering SUBJECT- HYDRAULICS & HYDRAULIC

MACHINES SUBJECT CODE-NCE 403

HYDRAULIC & HYDRAULIC MACHINES


The flow in an open-channel is termed as gradually varied flow (GVF) when the depth of flow varies gradually with longitudinal distance. Such flows are encountered both on upstream and downstream sides of control sections. Analysis and computation of gradually varied flow profiles in open-channels are important from the point of view of safe and optimal design and operation of any hydraulic structure.

Gradually varied flow


Find Change in Depth wrt x

Gradually Varied Flow

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2 21 2

1 22 2o fV Vy S x y S xg g

energy equation for non-uniform, steady flow

12 yydy

2

2 f oVdy d S dx S dxg

PA

T

dyy

dydxS

dydxS

gV

dyd

dydy

of

2

2

2 2

2 12 1 2 2o f

V VS dx y y S dxg g

Shrink control volume

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Governing equation

So and Sf are positive when sloping down in direction of flow

y is measured from channel bottomdy/dx =0 means water depth is _______

Gradually Varied Flow:

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21 Fr

SS

dx

dy fo

yn is when o fS S=constant

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Mild slope (yn>yc)◦ in a long channel subcritical flow will occur

Steep slope (yn<yc)◦ in a long channel supercritical flow will occur

Critical slope (yn=yc)◦ in a long channel unstable flow will occur

Horizontal slope (So=0)◦ yn undefined

Adverse slope (So<0)◦ yn undefined

Surface Profiles


Normal depthSteep slope (S2)Hydraulic Jump

Sluice gateSteep slope

Obstruction

Surface Profiles

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21 Fr

SS

dx

dy fo

S0 - Sf 1 - Fr2 dy/dx

+ + +

- + -

- - + 0

1

2

3

4

0 1 2 3 4

E

y

ynyc

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More Surface Profiles

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S0 - Sf 1 - Fr2 dy/dx

1 + + +

2 + - -

3 - - +

21 Fr

SS

dx

dy fo

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Direct Step Method

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xSg

VyxS

g

Vy fo

22

22

2

21

1

of SS

g

V

g

Vyy

x

22

22

21

21

energy equation

solve for x

1

1y

qV

2

2y

qV

2

2A

QV

1

1A

QV

rectangular channel prismatic channel

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Direct Step Method Friction Slope

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2 2

4 /3fh

n VSR

=2 2

4/32.22fh

n VSR

=

2

f8f

h

VSgR

=

Manning Darcy-Weisbach

Si units

english units

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Limitation: channel must be PRISMATIC (channel geometry is independent of x so that velocity is a function of depth only and not a function of x)

Methodidentify type of profile (determines whether Dy is + or

-)choose Dy and thus yi+1calculate hydraulic radius and velocity at yi and yi+1calculate friction slope given yi and yi+1 calculate average friction slope calculate Dx

Direct Step


Given a depth at one location, determine the depth at a second given location

Step size (x) must be small enough so that changes in water depth aren’t very large. Otherwise estimates of the friction slope and the velocity head are inaccurate

Can solve in upstream or downstream directionUsually solved upstream for subcriticalUsually solved downstream for supercritical

Find a depth that satisfies the energy equation

Standard Step

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xSg

VyxS

g

Vy fo

22

22

2

21

1

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0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

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elev

atio

n (m

)

distance upstream (m)

bottom

surface

yc

yn

What curves are available?Steep Slope

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S1

S3

is there a curve between yc and yn that increases in depth in the downstream direction? 13MODASSAR ANSARI

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0510152025303540

elev

atio

n (m

)

distance upstream (m)

bottom

surface

yc

yn

If the slope is mild, the depth is less than the critical depth, and a hydraulic jump occurs, what happens next?

Mild Slope

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Rapidly varied flow!

When dy/dx is large then V isn’t normal to csHydraulic jump!

Check conjugate depths

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All the complications of pipe flow plus additional parameter... _________________

Various descriptions of energy loss◦ Chezy, Manning, Darcy-Weisbach

importance of Froude Number◦ Fr>1 decrease in e gives increase in y◦ Fr<1 decrease in e gives decrease in y◦ Fr=1 standing waves (also min e given Q)

Summary

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free surface location

0

1

2

3

4

0 1 2 3 4

E

y

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Methods of calculating location of free surface (Gradually varying)◦ Direct step (prismatic channel)◦ Standard step (iterative)◦ Differential equation

Rapidly varying◦ Hydraulic jump

Summary

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21 Fr

SS

dx

dy fo

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Specific energy

Two depths with same energy◦ How do we know which depth is the right one?◦ Is the path to the new depth possible?

Energy equation

05/03/2023

2 21 2

1 22 2o fV Vy S x y S xg g

+ + D = + + D

2

22qygy

= +g

VyE

2

2

2

22QygA

= +

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Specific Energy-Depth graph

05/03/2023

0

1

2

3

4

0 1 2 3 4

E

y

18MODASSAR ANSARI

Nce403 mod unit3

Engineering

Transcript of Nce403 mod unit3