Transcript external flows

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.
U
U
U
U
Dye streak
turbulent
laminar
transition
Boundary Layer Parameters
 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). Imagine that 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.


d * U  w   (U   u ) wdy , or d *   (1 
0
0
Amount of fluid
being displaced
outward
equals
U-u
u
)dy
U
d*
Momentum Balance (on a flat plate B. L.)
Example: Determine the drag force acting on a flat plate when a uniform flow past
over it. Also relate the drag to the surface shear stress.
(2)
(1)
d
h
tw: wall shear stresses
Net Force  change of linear momentum

Fs 

V ( V  dA) 
all surfaces
V ( V  dA) 
surface(1)

V ( V  dA)
surface(2)
d
2
   U  ( U  )dA    u dA  U h    u 2 dy. (Assume unit width)
(1)
2

(2)
0
d
d
0
0
From mass conservation: U  h   udy , U 2 h    U  udy
d
Fs    u (u  U  )dy. Force Fs is the surface acting on the fluid
0
Momentum Thickness & Skin Friction
The force acting on the plate is called the friction drag (D)
(due to the presence of the skin friction).
d
D  -Fs    u(U   u )dy
0
The drag is related to the deficit of momentum flux across the boundary layer.
It can also be directly determined by the integration of the wall shear stress
over the entire plate surface:
D

t w dA 
plate

t w dx
plate
Define momentum thickness ( ) : thickness of a layer of fluid with
a uniform velocity U  and its momentum flux is equal to the deficit
of boundary layer momentum flux.


u
u
(1 
)dy
U
U
0
U 2    u(U   u )dy ,   
0
Wall Shear Stress and Momentum Thickness
Therfore, the drag force can be related to the momentum thickness as
D   U 2 , for a unit width boundary layer and this relation is valid
for laminar or turbulent flows.
It is also known that D 

t w dx,
plate
dD
2 d
 t w   U
dx
dx
Shear stress t w 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 as:
t w   (u y ) y 0
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=U 
y
u(y)=U(y/d)
t w   U 2
d
dx
For a laminar flow t w   (u
y
) y 0  
U
d
from the profile.
Substitute into the definition of the momentum thickness:
d

U y
u
u
y
y
 
(1 
)dy   (1  )dy, since u 
U
U
d
d
d
0
0

d
6
.
x
Example (cont.)
t w   U 2
d
,
dx

U
d
  U 2
Separation of variables:
d
x
 3.46
d  3.46

U x
x
U
,
 3.46
1 dd
6 dx
6
12 

 d d d , integrate d 2 
x  12(
) x2 ,
 U
 U
 U x
U x
1
, where Re x  

Re x
d x
U 3
0.289 U 2
1
tw  
 0.289

, tw 
d
x
Re x
x
Note: In general, the velocity distribution is not a straight line. A laminar flatplate boundary layer assumes a Blasius profile (chapter 9.3). The boundary
layer thickness d and the wall shear stress tw behave as:
U
5.0
d
U
x
5.0 x
0.332 U 2

, (9.13). t w 
, (9.14).
Re x
Re x
Laminar Boundary Layer Development
1
d( x )
• Boundary layer growth: d  x
• Initial growth is fast
• Growth rate dd/dx  1/x,
decreasing downstream.
0.5
0
0
0.5
x
1
10
t w( x )
• 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.
5
0
0
0.5
x
1
Summary of Boundary Layer2Parameters
d
t w   U
dx
• Boundary Layer Thickness, d where height, y = d when u = 0.99U
For a laminar flow t w   (u ) y 0  
y


u
d * U  w Thickness
(U   du*) wdy
, orSubstitute
d *  into
(1 the definition
)dy
• Displacement
where
of the mome
0
0


• Momentum Thickness  where
• Wall Shear Stress tw where
U

d
u
u
y
y
 
(1 
)dy   (1  )dy, si
U
U
d
d
0
0
d
 2 d.
t w   U 6 ,
dx

U
d
  U 2
1 dd
6 dx
6
Separation of variables: 2  d d d , integra
D   t w dA  U U
• Viscous or Friction Drag D where
U
d

1
 3.46
 3.46
, where Re x 
t
w
x

Re x
Cf U
 x
• Skin Friction Coefficent, Cf where
1
2

U
 x2
d  3.46
, d  x
U
t 
U
 0.289
U 3

0.289 U 2
, t