Nce 403 mod unit1

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

MACHINES SUBJECT CODE-NCE 403

01/07/17MODASSAR ANSARI 1

Difference between open channel flow and pipe flow, geometrical parameters of a channel.

Continuity equation for steady and unsteady flow. Critical depth, concepts of specific energy and

specific force, application of specific energy principle for

interpretation of open channel phenomena, flow through vertical and horizontal contractions.


Liquids are transported from one location to another using natural or constructed conveyance structures. The cross section of these structures may be open or closed at the top. The structures with closed tops are referred to as closed conduits and those with the top open are called open channels. For example, tunnels and pipes are closed conduits whereas rivers, streams, estuaries etc. are open channels. The flow in an open channel or in a closed conduit having a free surface is referred to as free-surface flow or open-channel flow.


Liquid (water) flow with a Free surface (interface between water and air)

relevant for◦ natural channels: rivers, streams◦ engineered channels: canals, sewer

lines or culverts (partially full), storm drains of interest to hydraulic engineers◦ location of free surface◦ velocity distribution◦ discharge – Depth relationships◦ optimal channel design


Based on their existence, an open channel can be Natural channels such as streams, rivers, valleys , etc. These

are generally irregular in shape, alignment and roughness of the surface.

Artificial channels are built for some specific purpose, such as irrigation, water supply, wastewater, water power development, and rain collection channels. These are regular in shape and alignment with uniform roughness of the boundary surface.


Based on their shape, an open channel can be prismatic or non-prismatic:

Prismatic channels: a channel is said to be prismatic when the cross section is uniform and the bed slop is constant.

Non-prismatic channels: when either the cross section or the slope (or both) change, the channel is referred to as non-prismatic. It is obvious that only artificial channel can be prismatic.

The most common shapes of prismatic channels are rectangular, parabolic, triangular, trapezoidal and circular.



Uniform Flow◦ Discharge-Depth relationships

Channel transitions◦ Control structures (sluice gates, weirs…)◦ Rapid changes in bottom elevation or cross section

Critical, Subcritical and Supercritical Flow Hydraulic Jump Gradually Varied Flow◦ Classification of flows◦ Surface profiles


Steady and Unsteady◦ Steady: velocity at a given point does not change with time◦ Uniform, Gradually Varied, and Rapidly VariedUniform: velocity at a given time does not change within a given

length of a channelGradually varied: gradual changes in velocity with distanceLaminar and TurbulentLaminar: flow appears to be as a movement of thin layers on top

of each otherTurbulent: packets of liquid move in irregular paths


Steady: A steady flow is one in which the conditions (velocity, pressure and cross-section) may differ from point to point but DO NOT change with time.

unsteady: If at any point in the fluid, the conditions change with time, the flow is described as unsteady. (In practise there is always slight variations in velocity and pressure, but if the average values are constant, the flow is considered steady.



Channel Geometry Characteristics• Depth, y• Area, A• Wetted perimeter, P• Top width, T Hydraulic Radius, Rh = Area / Wetted perimeter Hydraulic Depth, Dh = Area / Top width


Geometric parameters


P

ARh =

Hydraulic radius (Rh)

Channel length (l)

Roughness (ε)

This principle is know as the conservation of mass and we use it in the analysis of flowing fluids. The principle is applied to fixed volumes, known as control volumes (or surfaces), like that in the figure below.

For unsteady flow Mass entering per unit time = Mass leaving per unit time + Increase of mass in the control

volume per unit time For steady flow Mass entering per unit time = Mass leaving per unit time


Conservation of energy◦ losses due to conversion of turbulence to heat◦ useful when energy losses are known or small◦ Must account for losses if applied over long distances

Conservation of Momentum◦ “losses” due to shear at the boundaries◦ useful when energy losses are unknown


Given a long channel of constant slope and cross section find the relationship between discharge and depth

AssumeSteady Uniform Flow prismatic channel (no change in with distance)

Use energy, Momentum, empirical or Dimensional Analysis?

What controls depth given a discharge?Why doesn’t the flow accelerate?



θW

θ

W sin θ

∆x

a

bc

dShear force

energy grade lineHydraulic grade line

Shear force =________

0sin =∆−∆ xPxA oτθγ

θγτ sin P

Ao =

hR = P

A

θθθ

sincos

sin ≅=S

W cos θ

g

V

2

2

Wetted perimeter = __

Gravitational force = ________

Hydraulic radius

τoP ∆ x

P

γA ∆x sinθ

o hR Sτ γ=

The sum of the depth of flow and the velocity head is the specific energy:


g

VyE

2

2

+=

if channel bottom is horizontal and no head loss

21 EE =

y - _______ energy

g

V

2

2

- _______ energy

For a change in bottom elevation

1 2E y E− ∆ =

xSExSE o ∆+=∆+ f21

y

potential

kinetic


in a channel with constant discharge, Q

2211 VAVAQ ==

2

2

2gA

QyE +=

g

VyE

2

2

+= where A=f(y)

Consider rectangular channel (A = By) and Q = qB

2

2

2gy

qyE +=

A

B

y

3 roots (one is negative)

q is the discharge per unit width of channel

How many possible depths given a specific energy? _____2


Specific force is the sum of the pressure force and momentum flux per unit weight of the fluid at a section.

Specific force is constant in a horizontal frictionless channel. By plotting the depth against the specific force for a given

channel section and discharge, a specific force curve is obtained.


0

1

2

3

4

0 1 2 3 4

E

y


2

2

2gy

qyE +=

1 2

E1 = E2

sluice gate

y1

y2

eGL

as sluice gate is raised y1 approaches y2 and e is minimized: Maximum discharge for given energy.

PA


T

dy

y

T=surface width

Find critical depth, yc

2

2

2gA

QyE +=

0=dy

dE

dA =0dEdy

= =

3

2

1c

c

gA

TQ=

Arbitrary cross-section

A=f(y)

2

3

2

FrgA

TQ = 22

FrgA

TV =

dA

AD

T= Hydraulic Depth

2

31Q dAgA dy

− Tdy

More general definition of Fr


2 21 1 2 2

1 1 2 22 2 L

p V p Vz z h

g gα α

γ γ+ + = + + +

2 21 2

1 22 2o f

V Vy S x y S x

g g+ ∆ + = + + ∆

Turbulent flow (α ≅ 1)

z - measured from horizontal datum

y - depth of flow

Pipe flow

energy equation for Open Channel Flow

2 21 2

1 22 2o f

V Vy S x y S x

g g+ + ∆ = + + ∆

From diagram on previous slide...


( ) 0

11 ln

yv y V gdS

dκ = + + ÷

1 lnyd

− =

At what elevation does the velocity equal the average velocity?

For channels wider than 10d

0.4κ ≈ Von Kármán constantV = average velocityd = channel depth

1y de

= 0.368dV


2g

V21

1α

2g

V22

2α

xSo∆

2y1y

x∆

L fh S x= ∆______ grade line

_______ grade line

velocity head

Bottom slope (So) not necessarily equal to eGL slope (Sf)

hydraulic

energy


yc

T

Ac

3

2

1c

c

gA

TQ=

qTQ = TyA cc =

3

2

33

32

1cc gy

q

Tgy

Tq==

3/12

=

g

qyc

3cgyq =

Only for rectangular channels!

cTT =

Given the depth we can find the flow!

Rectangular Channels


3/12

=

g

qyc cc yVq =

=

g

yVy ccc

223

g

Vy cc

2

=

1=gy

V

c

c Froude number

velocity head =

because

g

Vy cc

22

2

=

2

cc

yyE += Eyc

3

2=

forcegravity

force inertial

0.5 (depth)

g

VyE

2

2

+=

Kinetic energyPotential energy

Minimum energy for a given q

◦ Occurs when =

◦ Fr=1 Critical flow

◦ Fr>1 = super critical◦ Fr<1 = sub critical


dE

dy

0

1

2

3

4

0 1 2 3 4

E

y

2

2 2c cV yg

=

3

TQgA

=3c

q

gy=c

c

VFr

y g=

Characteristics◦ Unstable surface◦ Series of standing waves

Occurrence◦ Broad crested weir (and other weirs)◦ Channel Controls (rapid changes in cross-section)◦ Over falls◦ Changes in channel slope from mild to steep

Used for flow measurements◦ ___________________________________________


The channel contraction may comprise of a reduction in the channel width, raising the channel bottom, or a combination of the two An abrupt change in the cross section is called a sudden contraction, whereas if the change occurs over a distance, it is called a gradual contraction.



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Nce 403 mod unit1