The document discusses the differences between internal and external fluid flows, focusing on the concept of boundary layers which form in such flows. It defines key terms related to boundary layer characteristics, including boundary layer thickness, displacement thickness, and momentum thickness, alongside their implications for drag force and wall shear stress. Finally, the document highlights the growth behavior of laminar boundary layers, indicating how thickness and shear stress vary with distance from the leading edge of a flat plate.

1. External Flows
An internal flow is surrounded by solid boundaries that can restrict the
development of its boundary layer, for example, a pipe flow. An external flow,
on the other hand, are flows over bodies immersed in an unbounded fluid so that
the flow boundary layer can grow freely in one direction. Examples include the
flows over airfoils, ship hulls, turbine blades, etc.
One of the most important concepts in understanding the external flows is the
boundary layer development. For simplicity, we are going to analyze a boundary
layer flow over a flat plate with no curvature and no external pressure variation.
laminar turbulent
transition
Dye streak
U U U
U

2. Boundary Layer Definition
 Boundary layer thickness (d): defined as the distance away from the surface
where the local velocity reaches to 99% of the free-stream velocity, that is
u(y=d)=0.99U. Somewhat an easy to understand but arbitrary definition.
 Displacement thickness (d*): Since the viscous force slows down the
boundary layer flow, as a result, certain amount of the mass has been displaced
(ejected) by the presence of the boundary layer (to satisfy the mass
conservation requirement). Image that if we displace the uniform flow away
from the solid surface by an amount d*, such that the flow rate with the
uniform velocity will be the same as the flow rate being displaced by the
presence of the boundary layer.
0 0
* ( ) , or * (1 )
u
U w U u wdy dy
U
d d
 
 

   
 
Amount of fluid
being displaced
outward
d*
U-u
equals

3. Momentum Balance
Example: Determine the drag force acting on a flat plate when a uniform flow past
over it. Relate the drag to the surface shear stress.
tw: wall shear stresses
s
all surfaces surface(1) surface(2)
2
2 2
(1) (2) 0
Net Force change of linear momentum
F ( ) ( ) ( )
( ) . (Assume unit width)
From mass conservation: U
V V dA V V dA V V dA
U U dA u dA U h u dy
h u
d
  
   
  


     
    

  
  
2
0 0
s
0
,
( ) . Force F is the surface acting on the fluid
s
dy U h U udy
F u u U dy
d d
d
 

 


 
 

h d
(1)
(2)

4. Skin Friction
s
0
The force acting on the plate is called the friction drag (D)
(due to the presence of the skin friction).
D -F ( )
The drag is related to the deficit of momentum flux across the boundary laye
u U u dy
d
 
  

w
r.
It can also be directly determined by the integration of the wall shear stress
over the entire plate surface:
D
Define momentum thickness ( ) : thickness of a layer of fluid with
a
w
plate plate
dA dx
t t

 
 
2
0 0
uniform velocity U and its momentum flux is equal to the deficit
of boundary layer momentum flux.
( ) , (1 )
u u
U u U u dy dy
U U
   

 
 
 
   
 

5. Wall Shear Stress and Momentum Thickness
2
Therfore, the drag force can be related to the momentum thickness as
D U , for a unit width boundary layer and this relation is valid
for laminar or turbulent flows.
It is also known that D w
plate
dx
 
t


 
2
w
,
U
Shear stress can be directly related to the gradient of
d
the momentum thickness along the streamwise direction .
dx
Recall that, for laminar flow, the wall shear stress is defined a
w
dD d
dx dx

t 
t


 
w 0
s:
( )y
u
y
t  




6. Example
Assume a laminar boundary layer has a velocity profile as u(y)=U(y/d) for
0yd and u=U for y>d, as shown. Determine the shear stress and the
boundary layer growth as a function of the distance x measured from the
leading edge of the flat plate.
u(y)=U(y/d)
u=U 
y
x
2
w 0
0 0
U
For a laminar flow ( ) from the profile.
Substitute into the definition of the momentum thickness:
U y
(1 ) (1 ) , since u
.
6
w
y
d
dx
U
u
y
u u y y
dy dy
U U
d

t 
t  
d

d d d
d






 


 

    

 

7. Example (cont.)
2 2
2 2
x
3 2
1
U , U
6
6 12
Separation of variables: , integrate 12( ) ,
U U U
1
3.46 3.46 , where Re
Re
3.46 ,
0.289 1
0.289 ,
Re
w
x
w w
x
U
d d
dx dx
d x x
x
U x
x U x
x
x
U
U U U
x x
 d
t   
d
  
d d d
  
d 


d d
 
t  t
d

 
  



  
 
  
  
 
   
Note: In general, the velocity distribution is not a straight line. A laminar flat-
plate boundary layer assumes a Blasius profile (chapter 9.3). The boundary
layer thickness d and the wall shear stress tw behave as:
(9.14).
,
Re
332
.
0
.
(9.13)
,
Re
0
.
5
0
.
5 2
x
w
x
U
x
x
U






t

d

8. Laminar Boundary Layer Development
d x
( )
x
0 0.5 1
0
0.5
1
• Boundary layer growth: d  x
• Initial growth is fast
• Growth rate dd/dx  1/x,
decreasing downstream.
t w x
( )
x
0 0.5 1
0
5
10
• Wall shear stress: tw  1/x
• As the boundary layer grows, the
wall shear stress decreases as the
velocity gradient at the wall becomes
less steep.