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1. Notches:
A notch may be defined as an opening in one side of a tank or a reservoir, like a large orifice, with the
upstream liquid level below the top edge of the opening. Since the top edge of the notch above the liquid
level serves no purpose, therefore a notch may have only the bottom edge and sides.
The bottom edge, over which the liquid flows, is known as sill or crest of the notch and the sheet of liquid
flowing over a notch (or a weir) is known as nappe or vein. A notch is, usually made of a metallic plate
and is used to measure the discharge of liquids.
Types Of Notches
There are many types of notches, depending upon their shapes. But the following are important from the
subject point of view.
 Rectangular notch
 Triangular notch
 Trapezoidal notch
 Stepped notch
 Rectangularnotch
Consider a rectangular notch in one side of a tank over which water is flowing as shown in figure.

2.  H = Height of water above sill of notch
 b = Width or length of the notch
 Cd = Coefficient of discharge
 Triangular notch
A triangular notch is also called a V-notch. Consider a triangular notch, in one side of the tank, over
which water is flowing as shown in figure.
 H = Height of the liquid above the apex of the notch
 θ = Angle of the notch
 Cd = Coefficient of discharge
Trapezoidal Notch
A trapezoidal notch is a combination of a rectangular notch and two triangular notches as shown in figure.
It is, thus obvious that the discharge over such a notch will be the sum of the discharge over the
rectangular and triangular notches.
 = Height of the liquid above the sill of the notch
 = Coefficient of discharge for the rectangular portion

3.  = Coefficient of discharge for the triangular portion
 = Breadth of the rectangular portion of the notch
 = Angle, which the sides make with the vertical
 Stepped Notch
A stepped notch is a combination of rectangular notches as shown in figure. It is thus obvious that the
discharge over such a notch will be the sum of the discharges over the different rectangular notches.
 weirs
A structure, used to dam up a stream or river, over which the water flows, is called a weir. The conditions
of flow, in the case of a weir, are practically the same as those of a rectangular notch. That is why, a
notch is, sometimes, called as a weir and vice versa The only difference between a notch and a weir is
that the notch of a small size and the weir is of a bigger one
 ADANTAGES OF WEIRS
 Capable of accuratelymeasuringawide range of flows
 tendsto provide more accurate discharge ratings thanflumesandorifices
 easyto construct
 can be usedincombinationwithturnoutanddivisionstructure
 can be bothportablesandadjustable
 Most floating debris tends topass over the structure
 weirs used in open channel flow
 weirare overflowstructuresbuiltacrossopenchannel tomeasure the volumetricrate of water
flow.
 the crest of a measurementweirisusuallyperpendiculartothe directionof flow
 if thisis notthe case special calibrationsmustbe made todevelopstage dischargerelationship

4.  oblique andduckbillweirsare sometimesusedtoprovide nearlyconstantupstreamwater
depth,buttheycanbe calibreatedasmeasuremnt device
 notch...the openingwhichwaterflowsthrough
 crest...the edge whichwaterflowsover
 nappe...the overflowingsheetof water
 length...thewidthof the weirnotch
 Archimedes’Principle
 Whena body is completelyor partiallyimmersedin a fluid,the fluidexertsan upward
force on the body equal to the weightof the fluid displacedby the body.
 Buoyancy
 Buoyancy: The decrease in weight(gravitational force) causedby the buoyant force.
(example:floatingina swimmingpool)
 Buoyant force: The upward force on an object producedby the surrounding fluid.
 centre of bouyancy
An object whose center of mass is lower than its center of buoyancy will float
stably, while an object whose center of mass is higher than its center of
buoyancy will tend to be unstable and have a tendency to flip over in the fluid that

5. is buoying it up. The centre of buoyancy is the centre of gravity of the displaced
fluid.
 Stable equilibrium
When the center of gravity of a body lies below point of suspension or
support, the body is said to be in STABLE EQUILIBRIUM. For example a book
lying on a table is in stable equilibrium.
 Unstable equilibrium
When the center of gravity of a body lies above the point of suspension or
support, the body is said to be in unstable equilibrium
Example
example of unstable equilibrium are vertically standing cylinder and funnel
etc.
 Neutral equilibrium
When the center of gravity of a body lies at the point of suspension or
support, the body is said to be in neutral equilibrium. Example: rolling ball.

6.  METACENTER:
 Meta center(M): The pointaboutwhicha body in stable equilibriumstarttooscillate
whengivenasmall angulardisplacementiscalledmetacenter.
 Meta centric height:
 Meta centricheight(GM):The distance betweenthe centerof gravity(G) of floating
bodyand the metacenter(M) iscalledmetacentricheight.
 GM=BM-BG
 Darcys formula:
Consider a uniform long pipe through which water is flowing at a uniform rate as shown in figure.

7. Let,
 = Velocityof waterinthe pipe
 = Frictional resistance perunitareaatunitvelocity
Considersections(1-1) and(2-2) of the pipe Let,
 = Intensityof pressure atsection(1-1)
 = Intensityof pressure atsection(2-2)
A little considerationwillshowthatp1 and p2 wouldhave beenequal,if there wouldhave beenno
frictional resistance.Nowconsideringhorizontal forcesonwaterbetweensections(1-1) and(2-2) and
equatingthe same,
Dividingbothsidesby -
But

8. We knowthatas perFroude'sexperiment,frictional resistance
Substitutingthe value of frictional resistance inthe above equation,
Let usintroduce anothercoefficient( ) suchthat,
(1)
We knowthatthe discharge,
Substitutingthe value of inequation(1)
 Chezy’sFormulafor Lossof Headdueto Friction

9. Consider uniform horizontal pipe as shown in and Equation derived in Darcy-
Weisbach equation.
Where is, hf = f’/ γ. P/A X L V2
We know, hydraulic radius is the ratio of area of flow to wetted perimeter. It is
denoted by ‘m’.
m = A/P = π/4d2
/ πd = d/4
P/A = 1/m
put value of P/A in Equation,
hf = f’/ γ. 1/m. L V2
V2
= hf. γ.m / f’.L
V = √ γ / f'. hf / L. m
Consider √ γ / f' = C i.e. Chezy’s constant and
hf / L = I i.e. loss of head per unit length of pipe.
Put the above value in Equation,
V = C√m i
This is known as Chezy’s formula.
Relation between the friction factor f and the chezy’s constant C:

10. Head loss due to Darcy Equation,
hf = fLV2
/ 2gd
From Chezy’s Equation, V = C√m i
Where, m = d/4, I = hf/L put in Equation,
V = C √m i
Squaring both side, V2
= C2
x d/4 x hf / L
hf = 4V2
L / C2
d
Equate Equation
f LV2
/2gd = 4V2
L/C2
d

f = 8g / C2
 BOUNDARY LAYER THEORY
L.PRANDTL..
A boundary layer is a thin layer of viscous fluid close to the solid surface of a wall in
contact with a moving stream in which (within its thickness δ) the flow velocity varies from
zero at the wall (where the flow “sticks” to the wall because of its viscosity) up to Ue at the
boundary, which approximately (within 1% error) corresponds to the free stream velocity
(see Figure 1). Strictly speaking, the value of δ is an arbitrary value because the friction
force, depending on the molecular interaction between fluid and the solid body, decreases
with the distance from the wall and becomes equal to zero at infinity.

11. The fundamental concept of the boundary layer was suggested by L. Prandtl (1904), it
defines the boundary layer as a layer of fluid developing in flows with very high Reynolds
Numbers Re, that is with relatively low viscosity as compared with inertia forces. This is
observed when bodies are exposed to high velocity air stream or when bodies are very
large and the air stream velocity is moderate. In this case, in a relatively thin boundary
layer, friction Shear Stress (viscous shearing force): τ = η[∂u/∂y] (where η is the dynamic
viscosity; u = u(y) – “profile” of the boundary layer longitudinal velocity component,
see Figure 1)....
 Separation of boundary layer
• As the flow proceed over a soil surface , the boundary layer thickness increases .
• The velocity profile change from parabolic to logarithmic .
• The fluid layer adjacent to the solid surface has to do work against surface friction by
consuming some kinetic energy. This loss of kinetic energy recovered from adjacent fluid
layer through momentum exchange process.
• Thus the velocity of the layer goes on decreasing.
• Along the length of solid body, at a certain point a stage may come when the boundary
layer may not be able to keep sticking to the solid body .
• In other words , the boundary layer will be separated from the surface . This
phenomenon is called the boundary layer separation.
• The point on the body at which the boundary layer is on the verge of separation from
the surface is called point of separation......

12. Different parameter used in boundary layer
Three main parameters that are used to characterize the size and shape of a
boundary layer are the boundary layer thickness, the displacement thickness, and
the momentum thickness.
1. Boundary Layer Thickness
 δ(x) is the boundary layer thickness when u(y) =0.99V
 V is the free-stream velocity
 The purpose of the boundary layer is to allow the fluid to change its velocity from the
upstream value of V to zero on the surface
Displacement Thickness
 There is a reduction in the flow rate due to the presence of the boundary layer
 This is equivalent to having a theoretical boundary layer with zero flow

13. Momentum Thickness
 Momentum thickness is a measure of the boundary layer thickness.
 It is defined as the distance by which the boundary should be displaced to
compensate for the reduction in momentum of the flowing fluid on account of
boundary layer formation
 The momentum thickness, symbolized by Ө is the distance that, when multiplied by
the square of the free-stream velocity, equals the integral of the momentum defect,
across the boundary layer.
 It is often used when determining the drag on an object. Again because of the
velocity deficit U-u, in the boundary layer, the momentum flux across section b–b in
Fig. 9.8 is less than that across section a–a. This deficit in momentum flux for the
actual boundary layer flow on a plate of width b is given by

14.  characteristic of boundary layer
 influence of surface
 friction ,shear, turbulence
 strong vertical gradients
 vertical fluxes of momentum heat
 Turbulence
 turbulent eddies are generated mechanically by strong shear as flow adjust to
condition at surface
 thermal generation of turbulence through buoyancy by destabilized stratification
thermal stability
 governing quantities
 wind speed driving large scale wind field
 surface roughness
 thermal stability