MECH 221 FLUID MECHANICS
(Fall 06/07)
REVIEW
1
MECH 221 – Review
What Have You Learnt?
1. Fluid Statics
2. Fluids in Motions
3. Kinematics of Fluid Motion
4. Integral and Differential Forms of Equations of Motion
5. Dimensional Analysis
6. Inviscid Flows
7. Boundary Layer Flows
8. Flows in Pipes
9. Open Channel Flows
On coming week lectures
2
MECH 221 – Review
Fluid Statics

It is to calculate the fluid pressure when the
fluid is no moving

Shear stress is due to relative motion of fluid,
so no shear stress and only normal stress
(Pressure) acting on the fluid

The fluid pressure is only due to body force,
Gravitational Force
3
MECH 221 – Review
Fluid Statics

Fluid pressure will increase when the position
of the fluid become deeper, we have following
equation:
z
dp
  g
dz
g
y
0
x
4
MECH 221 – Review
Fluid Statics

Total force acting on the surface become:
F   pdA  patm A  g  hdA
5
MECH 221 – Review
Fluid In Motion (Inviscid Flow)

2 sets equations for solving fluid motion problems

Conservation of Mass

V ( t ) t dV   S  v  ds  0

Conservation of Momentum

V ( t ) t ( v )dV   S vv  ds   S pds  V ( t )  gdV
6
MECH 221 – Review
Fluid In Motion (Inviscid Flow)

By invoking the continuity equation, the
momentum equation becomes Euler’s
equation of motion

Bernoulli equation is a special form of the
Euler’s equation along a streamline
v2
  gz  constant
 2
p
Along streamline incompressible flow
7
MECH 221 – Review
Fluid In Motion (Inviscid Flow)

A conical plug is used to regulate the air flow
from the pipe. The air leaves the edge of the
cone with a uniform thickness of 0.02m. If
viscous effects are negligible and the flowrate
is 0.05m3/s, determine the pressure within the
pipe.
8
MECH 221 – Review
Fluid In Motion (Inviscid Flow)

Procedure:


Choose the reference point
From the Bernoulli equation



P, V, Z all are unknowns
For same horizontal level, Z1=Z2
Flowrate conservation

Q=AV
9
MECH 221 – Review
Fluid In Motion (Inviscid Flow)

From the Bernoulli equation,
2
2
p1 v1
p 2 v2

 z1 

 z2
g 2 g
g 2 g
Since,
at the same horizontia l level, z1  z 2
2
2
p1 v1
p 2 v2



g 2 g g 2 g
p1  p2 

2
(v2  v1 )
2
2
10
MECH 221 – Review
Fluid In Motion (Inviscid Flow)

From flowrate conservation,
Q  A1v1  A2 v2
3
m
Given Q  0.5
, D  0.23m, t  0.02m, r  0.2m
s
D 2  0.232
A1 

 0.0415m 2
4
4
A2  2rt  2 (0.2)(0.02)  0.0251m 2
Therefore,
v1  0.5 / 0.0415  12.034 m
s
v2  0.5 / 0.0251  19.894 m
s
11
MECH 221 – Review
Fluid In Motion (Inviscid Flow)

Sub. into the Bernoulli equation,
p1  p2 

(v2  v1 )
2
2
2
v1  12.034 m , v2  19.894 m
s
s
For standard air@1 atm, 25C,   1.184 kg
m3
Set p 2 becomes reference point, p 2  0
1.184
p1  0 
(19.894 2  12.034 2 )
2
p1  148.565 N 2
m
12
MECH 221 – Review
Fluid In Motion (Viscous Flow)

In the mentioned fluid motion is inviscid
flows, only pressure forces act on the fluid
since the viscous forces (stress) were
neglected

With the viscous stress, the total stress on
the fluid is the sum of pressure stress ( σ p)
and viscous stress ( τ ) given by:
σ σ p  τ
13
MECH 221 – Review
Fluid In Motion (Viscous Flow)

The substitution of the viscous stress into the
momentum equations leads to:

( v )    ( vv)  p    (τ )  b
t

These equations are also named as the
Navier-Stokes equations
14
MECH 221 – Review
Dimensional Analysis

The objective of dimensional analysis is to obtain
the key non-dimensional parameters that govern
the physical phenomena of flows

After the dimensional analysis or normalization of
the complicated Navier-Stokes equations (steady
flow), the non-dimensional parameters are
identified

The equations are reduced to simple equation
and solvable analytically under certain conditions
15
MECH 221 – Review
Dimensional Analysis

By using proper scales, the variables, velocity
(u), pressure (p) and length (L) are
normalized to obtain the non-dimensional
variables, which are order one
 v*    v*   P  p*   2 v*  gL i*
g


U 2
UL
U2



v  v/U p  p/P   L
i*  unit vecto r in gravitatio nal direction
g
16
MECH 221 – Review
Dimensional Analysis

For simplicity consider the case where the
gravitational force has no consequence to the
dynamic of the flow, the Navier-Stokes
equations becomes
 v*    v*   P  p*  1 2 v* , Re  
UL


U 2
Re
When Re >> 1
 v*    v*   p*  1 2 v* , P  U 2 as pressure scale


Re
17
MECH 221 – Review
Inviscid Flow Vs. Boundary Layer Flow
L2
L2
Re 


 2

viscous force L / U  v
UL
inertia force
where  is the viscous diffusion length in an
advection time interval of L / U.
  L / U  τ a

Here, τ a  L / U measures the time required for
fluid travel a distance L.
18
MECH 221 – Review
Inviscid Flow Vs. Boundary Layer Flow

When Re  1, inertia force is much greater than
viscous force, i.e., the viscous diffusion distance is
much less than the length L.

Viscous force is unimportant in the flow region of
O( L ), but can become very important in the region of
O(  )near the solid boundary.

This flow region near the solid boundary is called an
boundary layer as first illustrated by Prandtl.
19
MECH 221 – Review
Inviscid Flow Vs. Boundary Layer Flow

Flow in the region outside the boundary
layer where viscous force is negligible is
inviscid. The inviscid flow is also called the
potential flow.

Potential flow
Boundary layer flow
U
20
MECH 221 – Review
Inviscid Flow

Inviscid flow implies that the viscous effect is
negligible. The governing equations are
Continuity equation and Euler equation.

We introduce a potential function, which is
automatically satisfy the continuity equation
v  
2
2
2






2
   2  2  2 0
x
y
z
21
MECH 221 – Review
Inviscid Flow


The continuity equation becomes Laplace
equation. The flow is described by Laplace
equation is called potential flow
For 2D potential flows, a stream function (x,y)
can also be defined together with (x,y)
 

x y
and



y
x
22
MECH 221 – Review
Inviscid Flow


If 1 and 2 are two potential flows, the sum
=(1+2) also constitutes a potential flow
We can combine certain basic solutions to
obtain more complicated solution
+
=
23
MECH 221 – Review
Inviscid Flow
Basic Potential Flows
Uniform Flow
Stagnation Flow
Source (Sink)
Free Vortex
Combined Potential Flows
Source
and Sink
Doublet
Source in
Uniform Stream
2-D Rankine
Ovals
Flows Around a
Circular Cylinder
24
MECH 221 – Review
Inviscid Flow
For stagnation flow,
25
MECH 221 – Review
Boundary Layer Flow

The thin layer adjacent to a solid boundary is
called the boundary layer and the flow inside
the layer is called the boundary layer flow

Inside the thin layer the velocity of the fluid
increases from zero at the wall (no slip) to the
full value of corresponding potential flow.
26
MECH 221 – Review
Boundary Layer Flow

There exists a leading edge for all external
flows. The boundary layer flow developing from
leading edge is laminar
27
MECH 221 – Review
Boundary Layer Flow

When we normalize the governing equations with
Re underneath the viscous term and resolve the
variables of y and v inside the boundary flow, the
non-dimensional normalized variables are selected:
x  y  u  v
x  , y  ,u  ,v 
L
L
U
V

V be the scale of v in the boundary layer
L is viscous diffusion layer near the wall (boundary layer)
28
MECH 221 – Review
Boundary Layer Flow

These results in the boundary layer equations that
in dimensional form are given by:
Continuity:
u v

0
x y
X-momentum:
 u
u 
p
 2u
  u  v      2
y 
x
y
 x
Y-momentum:
p
0
y
29
MECH 221 – Review
Boundary Layer Flow

A boundary layer flow is similar and its velocity
profile as normalized by U depends only on the
normalized distance from the wall:
1/ 2
 U 
   
 x  x 
y
y
(*)
i.e.,
u
 g  
U
30
MECH 221 – Review
Boundary Layer Flow

By introduce a stream function   U  xv f  
1
2

u
 U  f ' ( )
y

(**)
The boundary layer equation in term of the
similarity variables becomes:
2 f  ff  0
'' '
''
f  f  0 at   0 and f  1 as   
'
'
31
MECH 221 – Review
Boundary Layer Flow



After we solve this ordinary equation, we obtain a
'
solution of f ( )
We first find the value of  by Equ. (*) based on
'
coordinate of x and y, then find out the value of f ( )
by checking the solution table in the reference. Finally
the u at x and y is calculated by Equ. (**)
Therefore, we obtain following results:
vx
 5
U
0.332 U 2
w 
Re x
0.664
Cf 
Re x
32
MECH 221 – Review
Boundary Layer Flow

Laminar boundary layer flow can become
unstable and evolve to turbulent boundary
layer flow at down stream. This process is
called transition
33
MECH 221 – Review
Boundary Layer Flow

Under typical flow conditions, transition usually
occurs at a Reynolds number of 5 x 105

Velocity profile of turbulent boundary layer flows is
unsteady

A good approximation to the mean velocity profile
for turbulent boundary layer is the empirical 1/7
power-law profile given by
u  y
 
U   
1
7
34
MECH 221 – Review
Boundary Layer Flow

For turbulent boundary layer, empirically we
have

x

0.37
Re x  5
1
2 v
 w  0.00225U  
 U 
Cf 
w
U  / 2
2




1
4
0.0577
Re x
1
5
35
MECH 221 – Review
Boundary Layer Flow

The net force, F, acting on the body
F
 dF   dF
pressure
b. s .

b. s .


dFshear
b. s .
The resultant force, F, can be decomposed into
parallel and perpendicular components. The
component parallel to the direction of motion is
called the drag, D, and the component
perpendicular to the direction of motion is called
the lift, L.
36
MECH 221 – Review
Boundary Layer Flow

If i is the unit vector in the body motion direction,
then magnitude of drag FD becomes:
Friction Drag
FD  F  i 
 ( pdA n
b. s .

s
  wdA t s )  i
Pressure Drag
For two-dimensional flows, we can denotes j as
the unit vector normal to the flow direction, FL is
the magnitude of lift and is determined by:
FL  F  j   ( pdA n s   w dA t s )  j
b. s .
37
MECH 221 – Review
Boundary Layer Flow

The drag coefficient defined as
FD
CD 
2
U A / 2

For uniform flow passing a flat plate and no
pressure gradient is zero and no flow separation, :
Laminar Friction Drag
Turbulent Friction Drag
CD 
CD 
1.328
Re L
where
Re 
L
UL
v
0.072
Re L
1
5
38
MECH 221 – Review
Boundary Layer Flow

The pressure drag is usually associated with
flow separation which provide the pressure
difference between the front and rear faces of
the body

For low velocity flows passing a sphere of
diameter D, the drag coefficient then is expressed
as:
FD
24
CD 

2
U A / 2 Re D
where A  πD2 / 4 is the projected area of the sphere in the flow direction
39