Transcript Slide 1

Fluid Mechanics
Lecture 6
The boundary-layer equations
1
The need for the boundary-layer model
• While the flow past a streamlined body may be
well described by the inviscid flow (and even the
potential flow) equations over almost all the flow
region, those equations do not satisfy the fact that
– because of finite viscosity of real fluids – the
flow velocity at the wall itself must vanish.
• So, we need a flow model that uses the simplest
possible form of the Navier Stokes equation but
which does enable the no-slip condition to be
satisfied.
• Such a model was first developed by Ludwig
Prandtl in 1904.
2
Objectives of this lecture
• To explore the simplification of the Navier
Stokes equations to obtain the boundary
layer equations for steady 2D laminar flow.
• To understand the assumptions used in
deriving these equations.
• To understand the conditions in which the
boundary-layer equations can be used
reliably.
3
The governing equations
• Navier-Stokes equations:
Continuity:
U
x

y-momentum:
y
0
2
  2U
 U
U
 V



2
2
x
y
x
 x

y





2
  2V
 V
U
 V



2
2
x
y
y
 x
y





U
x-momentum:
V
V
U
V
P
P
• We seek to simplify these equations by
neglecting terms which are less important
under particular circumstances.
• Key assumptions: the thickness of the region
where viscous effects are significant,δ, is very
thin , i.e. d << L and ReL >>1.
4
Non-dimensionalized form of N-S Equations
• Non-dimensionalize equations using
V, a constant
(approach) velocity,
L ,an overall
dimension i.e.
• U*= U/ V;
V*=V/V; x*=x/L;
y*=y/L
• P*=???
( A question for you)
U
x



V

y

 0
   2U 
2  
 U 
P
1 
 U
 U
U
V









2
2  
R e L  x
x
y
x
y

 

   2V 


2  


V

V

V 
P
1 


U
V









2
2  
R e L  x
x
y
y
y

 




L
L
5
Non-dimensionalized N-S equations
• Since 0  x  L ,
0  x* 
x
L
1.
x has a magnitude
comparable to L
x* has an order of
magnitude of 1.

Hence we write x  O (1)

y  O (d / L )  O (d )

u  O (1)

  O (1)
L

p  O (1)

  O (1)

k  O (1)
L
6
Non-dimensionalized N-S equations
• Since
0 yd
and
d  L
,
0  y* 
y
L
Hence we write
y*= O(d).


d
. 1
L
y* is at least an order of
magnitude smaller than 1.
x  O ( x / L )  O (1)
• Also we have
y  O (d / L )  O (d )
U

 O (1)
L

P  O (1)

  O (1)
k

 O (1)

C p  O (1)
L
7
Continuity equation

U
x



V

y

x  O (1)

0
y  O (d )
U
O (1)
O (1)




[V ]
O (d )
 O (1)
0
O (1) 
[V ]
O (d )
0
[V*] has to be of order O(d) to satisfy continuity, i.e..
*
V  O (d )
No term can be omitted hence the continuity equation
remains as it is, i.e.
U
x

V
y
0
8
x-momentum equation

O ((dx /) L )
xy  O

U
U
O (1
O (1),
 U

x

 U

x

V
 O (1)
y

x

2

 U
2


O (1)
P

1
 O (1),
V
O (1)
 U
P
x
V
  2 U   2U 







2
2
R e L  x
y
x

y

 U


 O (d )
O (1)
O (1)
O (d )
 u

y

 O (1),

x


*
Vy  OO((dd /)L )
U

 O (1)
P
O (1)

O
 O (1),



O (d )
P  Ox(1) O
 O (d )
p





O (1)
  O (1)
 O (1)
O (1)
k  O (1)


C p  O (1)
 O (1)
O (1)
2

 U
y

2

O (1)
O (1)

O (1)
2
O (d )
 O (1)  O (
1
d
2
)
9
x-momentum equation
U
 U

x

V
  2 U   2U 




2



2

R e L  x
y
x
y

 U

O (1)  O (1)  O (1) 
P
1
Re
L

1



1 

 O (1)  O ( d 2 ) 


To make the above equation valid, we must have:
Re
L
 O(
1
d
2
)
ReL has to be large and x-gradients in the viscous term can be dropped
in comparison with y-gradients. The dimensional form of the equation
thus becomes:
U
V
x
 V
U
y

P
x
2

 U
y
2
10
y-momentum equation

U
 V

x

V
  2V   2V 




2



2

R e L  x
y
y
y

 V

P

1
y  O (d )



V
To do an order of magnitude analysis for each


P
term and estimate the order of magnitude for
y
*
 O (d )
[Re]  O (
1
d
2
)

11
y-momentum equation

U
 V
x


V
  2V   2V 




2



2

R e L  x
y
y
y

 V

P

1



y  O (d )
V
*
 O (d )
[Re]  O (

1
d
2
)
 O (d )
O (d ) 
O [1]
 O [d ]






2
1
O [1]
O [d ]
O
(1)
y
O (d ) 
O(
) 
O (d )
O (d )
P
1
d
P

2
1 
O (d )  O (d )  O [
]  O (d )  O (d )  O ( ) 

d 

y
 Hence
P

y

 O [d ]
2 
at most
12
y-momentum equation
P

y

 O (d )
P
y
0
The pressure can be assumed to be constant
across the boundary layer over a flat plate. Hence
the pressure only varies in the x-direction and the
pressure at the wall is equal to that at the edge of the
layer, i.e. P(x,y)=P(x).
L
L
13
Two qualifiers
• If the surface has substantial longitudinal
curvature (d/R >0.1)
it may not be adequate to
assume constant pressure across boundary layer.
Then one needs to apply radial equilibrium to
compute P (see Slide 16)
• In 3D boundary layers (not covered in this course
but very important in the industrial world) one
needs to be able to work out the presssure
variations in the y-z plane (normal to the mean
flow) to compute the secondary velocities .
14
Summary of assumptions
• Basic assumption:
• Derived results
– V is small, i.e.
– Re must be large:
d  L
V  O (d )
[Re]  O (
1
d
2
)
and then only velocity gradients normal to the wall are
significant in the viscous term
– The pressure is constant across the boundary layer
(for 2D nearly straight) flows, i.e.
P
y
0
P(x,y)=P(x).
15
Boundary layer equations
Continuity
x-momentum
U
x

U
V
y
0
U
x
 V
U
y

dP
dx
2

 U
y
2
2
 U
• Since  x 2 disappears, the equations become of
parabolic type which can be solved by knowing
only the inlet and boundary conditions... i.e. no
feedback from downstream back upstream.
• Unknowns: U and V; (P may be assumed known)
• Boundary conditions:
y = 0; U = V = 0
At wall :
Free stream: y = d; U = V U
U(x0,y), V(x0,y)
Inlet:
L
L
16
Boundary layer over a curved surface
Pressure gradient
across boundary layer:
P
y

U
2
R
Assume a linear velocity distribution, i.e.U
integrating from y=0 to d gives
2
P (d )  P (0) 
U d
3R

P 

U y
P (d )  P (0)
2
U 
d

d
3R
Hence pressure variations
across the boundary layer
are negligible when d  R
17
Limitations
• Large Reynolds number, typically Re >1000
• Boundary-layer approximations inaccurate
beyond the point of separation.
• The flow becomes turbulent when Re > 500,000.
In that case the averaged equations may be
describable by an adapted for of momentum
equation – to be treated later.
• Applies to boundary layers over surfaces with
large radius of curvature.
18