Fluid Mechanics

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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. 3. 1. 2. ?. Manometers for High Pressures.

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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

3

1

2

?

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

P1

+ h1g2

- h2g1

=P3

Mercury!

For small h1 use fluid with high density.

Differential Manometers

Water

p2

p1

h3

orifice

h1

h2

Mercury

p1

+ h1gw

- h2gHg

- h3gw

= p2

Find the drop in pressure between point 1 and point 2.

p1 - p2 = (h3-h1)gw + h2gHg

p1 - p2 = h2(gHg - gw)

x

centroid

center of pressure

y

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

b

a

Ixc

yc

R

Ixc

yc

Properties of Areas

a

Ixc

yc

b

d

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 isn’t a free surface?

coincide

below

decreases

hinge

8 m

water

F

4 m

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

hinge

8 m

water

Fr

F

4 m

a = 2.5 m

b = 2 m

Magnitude of the Force

h = _____

10 m

Depth to the centroid

pc = ___

Fr= ________

1.54 MN

hinge

8 m

water

Fr

F

4 m

a = 2.5 m

cp

b = 2 m

Location of Resultant Force

Slant distance to surface

12.5 m

0.125 m

hinge

8 m

water

Fr

F

4 m

lcp=2.625 m

cp

Force Required to Open Gate

How do we find the required force?

=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

b

h

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 Similitude:Scaling 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]

Energy Equation

laminar

turbulent

Moody Diagram

Example HGL and EGL

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

1E+03

1E+04

1E+05

1E+06

1E+07

1E+08

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...

Drag Coefficient on a Sphere

1000

Stokes Law

100

Drag Coefficient

10

1

0.1

0.1

1

10

102

103

104

105

106

107

Reynolds Number