Transcript Chapter 3

MECH 221 FLUID MECHANICS
(Fall 06/07)
Chapter 3: FLUID IN MOTIONS
Instructor: Professor C. T. HSU
1
MECH 221 – Chapter 3
3.1. Newton’s Second Law

The net force F acting on a matter of mass
m leads to an acceleration a following the
linear relation: F = ma

For a solid body of fixed shape, m is a
constant and a is described along the
trajectory of motion.
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MECH 221 – Chapter 3
3.1. Newton’s Second Law

If r(t) represents the particle trajectory, the
velocity v(t) and acceleration are then given
by:
v(t) = dr/dt ; a(t) = dv/dt = d2r/dt2

Therefore, F = mdv/dt

If m = m(t), then
F = (mdv/dt)+(vdm/dt)=d(mv)/dt
= dM/dt.
where M = mv is the momentum.
3
MECH 221 – Chapter 3
3.1. Newton’s Second Law


For fluids enclosed in a control volume V(t)
which may deform with time along the
trajectory of motion, it is only correct to
use the fluid momentum M to describe the
Newton’s second law
For M= V(t) vdV , we have
d 


v
dV
F = dM/dt =  V (t )

dt
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MECH 221 – Chapter 3
3.2. Description of Fluid Flow

Fluid dynamics is the mechanics to study
the evolution of fluid particles in a space
domain (flow field). There are two ways to
describe the flow field

Lagrangian description

Eulerian description
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MECH 221 – Chapter 3
3.2.1 Lagrangian Description

Given initially the locations of all the fluid
particles, a Lagrangian description is to
follow historically each particle motion by
finding the particle locations and
properties at every time instant.
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MECH 221 – Chapter 3
3.2.1 Lagrangian Description

Therefore, if a specific fluid particle is initially
(t=t0) located at (x0, y0, z0), the Lagragian
description is to determine (x(t), y(t), z(t)) and
the fluid properties, such as
f(t) = f [x(t), y(t), z(t); t], (for any function f)
v(t) = v [x(t), y(t), z(t); t] , etc.
given f0 = f (x0, y0, z0; t0),
v0 = v (x0, y0, z0; t0) etc.

Note that (x(t), y(t), z(t)) is a function of time.
7
MECH 221 – Chapter 3
3.2.2. Eulerian Description

An Eulerian description of fluid flow is simply to
state the evolution of fluid properties at a fixed
point (x, y, z) with time. Here (x, y, z) is
independent of time. Hence,
f(t) = f[x, y, z, t] , v = v [x, y ,z, t] etc.

Eulerian description is to observe the fluid
properties of different fluid particles passing
through the same fixed location at different time
instant, while Lagrangian description is to
observe the fluid properties at different locations
following the same particle
8
MECH 221 – Chapter 3
3.2.3. Relation between Lagrangian and
Eulerian description

It is important to note that there is only one flow
property at the same location with respect to the
same time, i.e., f(x(t),y(t),z(t),t) = f(x,y,z,t).
Therefore, the Lagrange differential with respective
to dt,
Df
df 
dt
Dt
which is equal to the total Eulerian differential:
f
f
f
f
df  dx  dy  dz  dt
x
y
z
t
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MECH 221 – Chapter 3
3.2.3. Relation between Lagrangian and
Eulerian description

Hence, the total derivative is given by:
Df f dx f dy f dz f




Dt x dt y dt z dt t
f
f
f f
u v w 
x
y
z t
f

 v  f
t
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MECH 221 – Chapter 3
3.3. Equations of Motion for Inviscid Flow

Conservation of Mass

Conservation of Momentum
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MECH 221 – Chapter 3
3.3.1. Conservation of Mass

Mass in fluid flows must conserve. The total mass in
V(t) is given by:
m  
V (t )

 dV
Therefore, the conservation of mass requires that
dm/dt = 0.
d 

dm / dt 

dV

dt  V (t )

 dV 
 
dV   

V ( t ) t
dt

S

 
dV    v  ds
V ( t ) t
S
where the Leibniz rule was invoked.
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MECH 221 – Chapter 3
3.3.1. Conservation of Mass

Hence:

V ( t ) t dV   S  v  ds  0
This is the Integral Form of mass conservation
equation.
13
MECH 221 – Chapter 3
3.3.1. Conservation of Mass

Integral form of mass conservation equation

V (t ) t dV   S  v  ds  0

By Divergence theorem:

S

Hence:
v  ds  
V (t )
  ( v )dV

=0
[



(

v
)]
dV
V (t ) t
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MECH 221 – Chapter 3
3.3.1. Conservation of Mass

As V(t)→0, the integrand is independent of V(t)
and therefore,

   ( v )  0
t
This is the Differential Form of mass
conservation and also called as continuity
equation.
15
MECH 221 – Chapter 3
3.3.2. Conservation of Momentum

The Newton’s second law,
dM
F
dt
is Lagrangian in a description of momentum
conservation. For motion of fluid particles that
have no rotation, the flow is termed irrotational.
An irrotational flow does not subject to shear force,
i.e., pressure force only. Because the shear force is
only caused by fluid viscosity, the irrotational flow
is also called as “inviscid” flow
16
MECH 221 – Chapter 3
3.3.2. Conservation of Momentum

For fluid subjecting to earth gravitational acceleration,
the net force on fluids in the control volume V enclosed
by a control surface S is:
F    pd s  
S
V(t)
ρgdV
where s is out-normal to S from V and the divergence
theorem is applied for the second equality.

This force applied on the fluid body will leads to the
acceleration which is described as the rate of change in
momentum.
17
MECH 221 – Chapter 3
3.3.2. Conservation of Momentum
d 

dM / dt 

v
dV

dt  V ( t )

dV 

 
( v )dV  ( v ) 
V ( t ) t
dt  S


 
( v )dV   vv  ds
V ( t ) t
S
where the Leibniz rule was invoked.
18
MECH 221 – Chapter 3
3.3.2. Conservation of Momentum

Hence:

V ( t ) t ( v )dV   S vv  ds   S pds  V ( t )  gdV
This is the Integral Form of momentum
conservation equation.
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MECH 221 – Chapter 3
3.3.2. Conservation of Momentum

Integral form of momentum conservation equation

V ( t ) t ( v )dV   S vv  ds   S pds  V ( t )  gdV

By Divergence theorem:

S
vv  ds  
V(t )
 pds  
s

  ( vv )dV
V(t )
pdV
Hence:

[
V ( t ) t ( v )    ( vv)]dV  V ( t ) [p  g ]dV
20
MECH 221 – Chapter 3
3.3.2. Conservation of Momentum

As V→0, the integrands are independent of V.
Therefore,
This is the Differential Form of momentum
conservation equation for inviscid flows.
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MECH 221 – Chapter 3
3.3.2. Conservation of Momentum

By invoking the continuity equation,

   ( v )  0
t

The momentum equation can take the following
alternative form:

 ( v )   ( v   )v  p  g
t
which is commonly referred to as Euler’s equation
of motion.
22
MECH 221 – Chapter 3
3.4. Bernoulli Equation for Steady Flows

Bernoulli equation is a special form of the Euler’s
equation along a streamline. For a first look, we
restrict our discussion to steady flow so that the Euler’s
equation becomes:
 ( v   )v  p  g

Assuming that g is in the negative z direction, i.e.,
g =- gz and using the following vector identity,
1
( v   )v  ( v  v )  v  (   v )
2
the Euler’s equation for steady flows becomes
p
1
 ( v 2 )  gz  v  (   v )
 2
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MECH 221 – Chapter 3
3.4. Bernoulli Equation for Steady Flows

We now take the scalar product to the above equation
by the position increment vector dr along a streamline
and observe that
f  dr  df

;
[ v  (  v )]  dr  0
Thus, the result leads to
dp


(for any function f)
 d ( v 2 /2)  gdz  0
The above equation now can be integrated to give

dp
2
v

 gz  constant(along streamline)

2
24
MECH 221 – Chapter 3
3.4. Bernoulli Equation for Steady Flows

For incompressible fluids where ρ = constant, we
have
v2
  gz  constant
 2
p

For irrotational flows,
flow domain and
(along streamline)
  v  0 everywhere in the
p
1
 ( v 2 )  gz  0
 2
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MECH 221 – Chapter 3
3.4. Bernoulli Equation for Steady Flows

Since, f  dr  df for dr in any direction, we have:

For anywhere of irrotational fluids


v2

 gz  constant

2
dp
For anywhere of incompressible fluids
v2

 gz  constant
 2
p
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MECH 221 – Chapter 3
3.5. Static, Dynamic, Stagnation and Total Pressure

Consider the Bernoulli equation,
p

v 2
2
 gz  constant
(for incompressible fluid)
The static pressure ps is defined as the pressure
associated with the gravitational force when the
fluid is not in motion. If the atmospheric pressure
is used as the reference for a gage pressure at
z=0.
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MECH 221 – Chapter 3
3.5. Static, Dynamic, Stagnation and Total Pressure


Then we have ps   gz as also from chapter 2.
The dynamic pressure pd is then the pressure
deviates from the static pressure, i.e., p = pd+ps.
The substitution of p = pd+ps. into the Bernoulli
equation gives
pd 
v 2
2
 constant
28
MECH 221 – Chapter 3
3.5. Static, Dynamic, Stagnation and Total Pressure

The maximum dynamic pressure occurs at the
stagnation point where v=0 and this maximum
pressure is called as the stagnation pressure p0.
Hence,
pd 

v
2
2
 po
The total pressure pT is then the sum of the
stagnation pressure and the static pressure, i.e.,
pT= p0 - ρgz. For z = -h, the static pressure is ρgh
and the total pressure is p0 + ρgh.
29
MECH 221 – Chapter 3
3.6. Energy Line and Hydraulic Grade Line

In fact the Bernoulli equation also states that the
energy density (per unit volume) possessed by the
fluid particle is constant not only along a
streamline but also at everywhere in fluid domain
for irrotational flow. The energy consists of
pressure energy (p), kinetic energy (ρv2/2) and
gravitational potential energy (ρgz).
30
MECH 221 – Chapter 3
3.6. Energy Line and Hydraulic Grade Line

It becomes simpler if this total energy is
interpreted into a total head H (height from a
datum) by dividing the Bernoulli equation with ρg
such that
p v

z H
g 2 g
2
where p/ρg is the pressure head, v2/2g is the
velocity head and z is the elevation head.
31
MECH 221 – Chapter 3
3.6. Energy Line and Hydraulic Grade Line

A piezometric head is then defined as that consists of
only the pressure and elevation heads, i.e.,
p
Hp 
z
g

The variations of H and Hp along the path of fluid flow
can be plotted into lines and are termed as “energy
line” and “hydraulic grade line”, respectively.

It is noted that H is always higher than Hp and that a
negative pressure (below atmospheric pressure) occurs
when Hp is below the fluid streamline
32
MECH 221 – Chapter 3
3.6. Energy Line and Hydraulic Grade Line
33
MECH 221 – Chapter 3
3.7. Applications of Bernoulli Equation

Pitot-Static Tube

Free Jets

Flow Rate Meter
34
MECH 221 – Chapter 3
3.7.1. Pitot-Static Tube

Pitot-static tube is a device that measures the
difference between h1 and h2 so that the velocity of
the fluid flow at the measurement location can be
determined from
v  2 g( h2  h1 )
35
MECH 221 – Chapter 3
3.7.1. Pitot-Static Tube
36
MECH 221 – Chapter 3
3.7.2. Free Jets

Free jets are the flow from an orifice of an
apparatus that converts the total elevation head h
into velocity head, i.e.,
v  2 gh
37
MECH 221 – Chapter 3
3.7.3. Flow Rate Meter

The commonly used flow rate meter is the
Venturi meter that determines the flow rate Q
through pipes by measuring the difference of
piezometric heads at locations of different
cross-sectional areas along the pipe.
38
MECH 221 – Chapter 3
3.7.3. Flow Rate Meter

If hp1 and hp2 represent the piezometric heads at
section 1 and 2 with cross-sectional areas A1 and
A2 respectively, we have:
2
2
v1
v2
and
h p1   h p 2 
Q  A1v1  A2 v 2
2g
2g
Then, v 2 
2 g ( h p1  h p 2 )
1  ( A2 / A1 )
2
Therefore, Q  A2
2 g ( h p1  h p 2 )
1  ( A2 / A1 )2
39