Chapter 4 Flowing Fluids Pressure Variations (1).pptx

1
Flowing Fluids &
Pressure Variation
Chapter 4
CE319F: Elementary Mechanics of Fluids

2
Velocity
• Momentum = p = mv
• Specific momentum = p/m = v = velocity
• Velocity related to pressure/forces on/by fluids
– Measure velocity and can get pressure/forces
– Measure changes in pressure and can get velocity
– Very powerful concept!
• Other important issues related to velocity (a few examples)
– Scouring of river beds
– Re-suspension of particles in an HVAC duct
– Pollution transport
– Transport of thermal energy

3
Lagrangian & Eulerian
Descriptions of Fluid Motion
Figure 4.2

4
Fluid Motion: Lagrangian Approach
 Follow motion of individual particles
 Position vector of particle: r(t) = xi + yj + zk
 i, j, k = unit vectors in x, y, z directions
 But, for entire flow field would need to follow every particle!
k
j
i
V
dt
dz
dt
dy
dt
dx



k
j
i
V w
v
u 


Fig 4.1
here

5
Fluid Motion: Eulerian Approach
 Watch fluid pass by point in space (fixed viewpoint)
 Easier than tracking individual particles
 For entire flow field, must know at all points in space
u = f1(x, y, z, t)
v = f2(x, y, z, t)
w =f3(x, y, z, t)
Also, can define velocity along a “streamline”
V = V(s,t)

6
Streamlines
Line drawn in flow field so
velocity vector is tangent at every
point along the line (at any instant)
Speed of flow (V) inversely
proportional to distance between
streamlines

7
Simulated Streamline Pattern over a Volvo ECC
Prototype

8
Flow Patterns: Uniform vs. Non-Uniform Flow
• Uniform flow
– V = velocity at any point, s =
distance along streamline
– Velocity does not change from
point to point along a streamline
– Streamline is straight and parallel
– Not straight = directional change
– Not parallel = change of speed
• Non-uniform flow
0



s
V
0



s
V

9
Flow Patterns: Steady vs. Unsteady Flow
• Steady flow
– If at any point the velocity does not vary in
magnitude or direction with time, flow = steady
• Unsteady flow
• Example of steady and unsteady flow?
Flow patterns = uniform vs. non-uniform?
steady vs. unsteady?
0



t
V
0



t
V

10
Example: Textbook Problem 4.1
• Valve C is slowly opened
• Classify the flow at B
while valve is opened
• Classify the flow at A

11
Laminar vs. Turbulent Flow
• Turbulent flow
– Eddies (3D) of varying size (mm to km) cause intense mixing
– Example = gusts of wind, plume from stack, candle/cigarette
• Laminar flow
– No eddies = “smooth” flow
– Example = pour syrup or oil from a bottle/can
• Distinguishing between = very important
– Modeling flow fields
– Drag on objects
– Scouring of pipes, etc.
– Pollutant transport

13
Laminar vs. Turbulent Flow
• Laminar • Turbulent

14
Pathline: Follow particles, photo, light
mark during exposure time
Streakline: Release tracer at point
continuously and follow travel

15
Acceleration
• Acceleration = rate of
change of particle velocity
with time
• Components:
– Normal – changing direction
– Tangential – changing speed
dt
d
V
dt
dV
dt
d
t
s
V
t
t
t
e
e
V
a
e
V









 )
,
(
n
t
n
t
r
V
t
V
s
V
V
r
V
dt
d
t
V
s
V
V
dt
dV
e
e
a
e
e





2
)
( 













et = unit vector
Chain rule for function
of 2 variable

16
Acceleration – Cartesian Components
• Cartesian
coordinates
k
j
i
V




w
v
u 


Convective Local
t
w
w
z
w
v
y
w
u
x
w
t
w
dt
dz
z
w
dt
dy
y
w
dt
dx
x
w
dt
dw
a
t
v
w
z
v
v
y
v
u
x
v
t
v
dt
dz
z
v
dt
dy
y
v
dt
dx
x
v
dt
dv
a
t
u
w
z
u
v
y
u
u
x
u
t
u
dt
dz
z
u
dt
dy
y
u
dt
dx
x
u
dt
du
a
z
y
x











































































.
.
.
k
j
i
a




z
y
x a
a
a 


Local acceleration = 0 if flow is steady
Convective acceleration = 0 if flow is uniform

17
Example
2
2
3
on,
Accelerati
:
Find
3
:
Given
ty
w
xz
v
t
u
ty
xz
t






a
k
j
i
V





0
)
(
)
(
0
)
3
(
0
0
3
0
0
0















tyy
x
xz
t
z
dt
dv
a
xw
v
zu
dt
dv
a
w
v
u
dt
du
a
y
y
x

18

20
Euler’s Equation
• Fluid element accelerating in l direction
• Acted on by pressure and weight forces only (no friction)
• Newton’s 2nd
Law

21
Euler’s Equation (continued)
S Fl = m al
p DA -(p + Dp) DA – DW sin(a) = r DA Dl al
-(Dp) DA – g Dl DA sin(a) = r DA Dl al
-(Dp) – g Dl sin(a) = r Dl al
-(Dp / Dl ) – g sin(a) = r al
-(Dp/Dl ) – g (Dz /Dl ) = r al
- (D(p + g z) /Dl ) = r al

22
Euler’s Equation (continued)
l
a
g
z
p
dl
d 1
)
( 



For an incompressible fluid
g = constant
Consider free open surface
(P=0, constant)
Consider constantly rotating

24
Bernoulli Equation
• Consider steady flow along
streamline
• s is along streamline, and t is
tangent to streamline
Constant
2
0
2
2
1
1
)
(
2
2
2



























g
V
z
p
g
V
z
p
ds
d
g
V
ds
d
ds
dV
V
g
a
g
z
p
ds
d
t



head
dynamic
Velocity
g
V
head
c
Piezometri
z
p
)
(
2
2




g
V
z
p
g
V
z
p
2
2
2
2
2
2
2
1
1
1 






Note similarity with relationship in
fluid statics for incompressible fluid!

25
Textbook Problem 4.103
• Given: Velocity in outlet pipe from
reservoir is 6 m/s and h = 15 m.
• Find: Pressure at A.
Point 1
Point A

26
Stagnation Tube
gl
V
d
d
l
p
p
V
p
g
V
p
g
V
z
p
g
V
z
p
2
)
)
(
(
2
)
(
2
2
2
2
1
1
2
2
1
2
2
1
1
2
2
2
2
2
1
1
1





















Method for relating pressure measurement to velocity

27
Stagnation Tube in a Pipe

p
g
V
2
2
z
Flow
Pipe
0

z
g
V
z
p
H
2
2




1
2

28
Pitot Tube
)
(
2
)
(
)
[(
2
2
2
2
2
2
1
1
1
1
1
2
2
2
2
2
1
1
2
2
2
2
2
1
1
1
h
h
g
V
z
p
z
p
g
V
g
V
p
g
V
p
g
V
z
p
g
V
z
p





















29
Textbook Example 4.4
1
2
V
y
l
z1-z2
s
ft
gy
V
y
h
h
y
z
z
p
p
z
z
y
p
p
p
y
l
y
l
z
z
p
k
Hg
k
Hg
k
k
Hg
k
k
k
Hg
k
Hg
k
k
/
3
.
24
)
1
/
(
2
)
1
/
(
)
(
)
(
)
(
)
(
)
(
2
1
2
1
2
1
2
1
2
1
2
2
1
1







































30
A tube with a 2 mm diameter is
mounted at the center of a duct
conveying air. The well of
manometer fluid is large enough so
that level changes in the well are
negligible. With no flow in the
duct, the level of the slant
manometer is 2.3 cm. With flow in
the duct it moves to 6.7 cm on the
slant scale.
Find the velocity of air in the duct.

Chapter 4 Flowing Fluids Pressure Variations (1).pptx

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