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Fluid Mechanics

- EIT Review

Shear Stress

Tangential force per unit area

change in velocity with respect to distance

rate of shear

Manometers for High Pressures

- Find the gage pressure in the center of the

sphere. The sphere contains fluid with g1 and the

manometer contains fluid with g2. - What do you know? _____
- Use statics to find other pressures.

g2

h1

P1 0

g1

h2

P3

h1g2

- h2g1

P1

Mercury!

For small h1 use fluid with high density.

Differential Manometers

Water

p2

p1

h3

orifice

h1

h2

Mercury

p2

p1

h1gw

- h2gHg

- h3gw

Find the drop in pressure between point 1 and

point 2.

p1 - p2 (h3-h1)gw h2gHg

p1 - p2 h2(gHg - gw)

Forces on Plane Areas Inclined Surfaces

Free surface

O

q

A

B

O

The origin of the y axis is on the free surface

Statics

- Fundamental Equations
- Sum of the forces 0
- Sum of the moments 0

centroid of the area

pc is the pressure at the __________________

Line of action is below the centroid

Properties of Areas

a

Ixc

yc

b

d

Properties of Areas

Inclined Surface Summary

- The horizontal center of pressure and the

horizontal centroid ________ when the surface has

either a horizontal or vertical axis of symmetry - The center of pressure is always _______ the

centroid - The vertical distance between the centroid and

the center of pressure _________ as the surface

is lowered deeper into the liquid - What do you do if there isnt a free surface?

coincide

below

decreases

Example using Moments

An elliptical gate covers the end of a pipe 4 m

in diameter. If the gate is hinged at the top,

what normal force F applied at the bottom of the

gate is required to open the gate when water is 8

m deep above the top of the pipe and the pipe is

open to the atmosphere on the other side? Neglect

the weight of the gate.

Solution Scheme

Magnitude of the force applied by the water

Location of the resultant force

Find F using moments about hinge

Magnitude of the Force

h _____

10 m

Depth to the centroid

pc ___

Fr ________

1.54 MN

Location of Resultant Force

Slant distance to surface

12.5 m

0.125 m

Force Required to Open Gate

- How do we find the required force?

Moments about the hinge

Fltot - Frlcp

2.5 m

ltot

b 2 m

F ______

809 kN

Example Forces on Curved Surfaces

Find the resultant force (magnitude and location)

on a 1 m wide section of the circular arc.

water

W1 W2

FV

W1

3 m

(3 m)(2 m)(1 m)g p/4(2 m)2(1 m)g

2 m

58.9 kN 30.8 kN

W2

89.7 kN

2 m

FH

x

g(4 m)(2 m)(1 m)

78.5 kN

y

Example Forces on Curved Surfaces

The vertical component line of action goes

through the centroid of the volume of water above

the surface.

A

water

Take moments about a vertical axis through A.

W1

3 m

2 m

W2

2 m

0.948 m (measured from A) with magnitude of

89.7 kN

Example Forces on Curved Surfaces

The location of the line of action of the

horizontal component is given by

A

water

W1

3 m

2 m

W2

2 m

(1 m)(2 m)3/12 0.667 m4

4 m

x

y

Example Forces on Curved Surfaces

78.5 kN

horizontal

0.948 m

4.083 m

89.7 kN

vertical

119.2 kN

resultant

Cylindrical Surface Force Check

89.7kN

0.948 m

- All pressure forces pass through point C.
- The pressure force applies no moment about point

C. - The resultant must pass through point C.

C

1.083 m

78.5kN

(78.5kN)(1.083m) - (89.7kN)(0.948m) ___

0

Curved Surface Trick

- Find force F required to open the gate.
- The pressure forces and force F pass through O.

Thus the hinge force must pass through O! - All the horizontal force is carried by the hinge
- Hinge carries only horizontal forces! (F

________)

A

water

W1

3 m

2 m

O

F

W2

W1 W2

11.23

Dimensionless parameters

- Reynolds Number
- Froude Number
- Weber Number
- Mach Number
- Pressure Coefficient
- (the dependent variable that we measure

experimentally)

Model Studies and SimilitudeScaling Requirements

- dynamic similitude
- geometric similitude
- all linear dimensions must be scaled identically
- roughness must scale
- kinematic similitude
- constant ratio of dynamic pressures at

corresponding points - streamlines must be geometrically similar
- _______, __________, _________, and _________

numbers must be the same

Mach

Reynolds

Froude

Weber

Froude similarity

- Froude number the same in model and prototype
- ________________________
- define length ratio (usually larger than 1)
- velocity ratio
- time ratio
- discharge ratio
- force ratio

difficult to change g

11.33

Control Volume Equations

- Mass
- Linear Momentum
- Moment of Momentum
- Energy

Conservation of Mass

2

If mass in cv is constant

1

v1

A1

Area vector is normal to surface and pointed out

of cv

V spatial average of v

M/t

If density is constant L3/t

Conservation of Momentum

Energy Equation

laminar

turbulent

Moody Diagram

Example HGL and EGL

velocity head

pressure head

energy grade line

hydraulic grade line

elevation

z

pump

z 0

datum

Smooth, Transition, Rough Turbulent Flow

- Hydraulically smooth pipe law (von Karman, 1930)
- Rough pipe law (von Karman, 1930)
- Transition function for both smooth and rough

pipe laws (Colebrook)

(used to draw the Moody diagram)

Moody Diagram

0.10

0.08

0.05

0.04

0.06

0.03

0.05

0.02

0.015

0.04

0.01

0.008

friction factor

0.006

0.03

0.004

laminar

0.002

0.02

0.001

0.0008

0.0004

0.0002

0.0001

0.00005

0.01

smooth

1E03

1E04

1E05

1E06

1E07

1E08

R

Solution Techniques

- find head loss given (D, type of pipe, Q)

- find flow rate given (head, D, L, type of pipe)

- find pipe size given (head, type of pipe,L, Q)

Power and Efficiencies

- Electrical power
- Shaft power
- Impeller power
- Fluid power

Motor losses

IE

bearing losses

Tw

pump losses

Tw

gQHp

Manning Formula

The Manning n is a function of the boundary

roughness as well as other geometric parameters

in some unknown way...

Hydraulic radius for wide channels

Drag Coefficient on a Sphere

1000

100

Stokes Law

Drag Coefficient

10

1

0.1

0.1

1

10

102

103

104

105

106

107

Reynolds Number

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