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Velocity Boundary Layers
Prandtl first observed that the equations (A) and (B) were not useable inside the
boundary layer. Although the majority of the fluid flow is without the thin boundary
layer region around the body, he observed that whatever happens in this region
(causing flow separation, as we shall see later) has profound effects on the flow
measurements. To probe this region let us consider the steady, two-dimensional,
incompressible flows:
Continuity:
x:
y:
u v

0
x y
u
u
p
1   2u  2u 


u v
 

x
y
x Re L  x 2 y 2 
v
v
p
1  2v 2v 


u v  

x
y
y Re L  x 2 y 2 
Note: The above equations are written in the non-dimensional form with the use of the
primes. We need to discuss the “order-analysis” preformed by Prandtl. It is better
understood if we keep the equations in the non-dimensional form. After reaching the
conclusions we’ll return to the dimensional form again.
Order Analysis by Prandtl
Prandtl hypothesized that use of the same length scale in x and y is incorrect if we
need to probe inside the thin boundary layer. Thus he devised a new coordinate
system in the y direction.
If
Then,
x ~ 1 (i.e. x ~ L) (Read as: “x is of the order of L”)
y ~  (i.e. y ~ L)
u ~ 1 (i.e. u ~ U)
u  v

0
x  y

 To be able to cancel the

u  1
~
x  1
u 
term,
x 
v 
~ ~1
y 
v 
must be of the order 1 also.
y 
 v ~ 
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In other words, by choosing x, y and u as the above order, the continuity equation
yields the order of v as .
Here the variable  is chosen to represent the size of the boundary layer and is a nondimensional quantity (not a length). For example, if the chord of the airfoil is 1 meter,
the boundary layer thickness on the airfoil (which may be a few millimeters depending on
the speed of the flow) must be of order 10-2 (<< 1). Typically a  of ~10-3 or 10-2 means
2 is even a smaller quantity.
To perform the order analysis on the momentum equations, we note that we’ll not
normally neglect the pressure terms. Since pressure is a variable which stays
significant even in static fluids. However the pressure derivatives may be of
different order of magnitudes. Also we have a new variable, ReL in these equations,
which we must assess. We first observe that while density is a significant quantity (e.g.
1.23 kg/sec for air) viscosity is a small quantity. Thus
Re L 
UL UL



will
depend mainly on the size of  (if U ~ 1 and L ~ 1). Experiments show that  ~ 2 [e.g.
air  1.5 x 10-5 m2/s at 20 C].
Therefore,
Re L ~
1
1
or,
~ 2
2
Re L

Based upon our observations and the result of the continuity equation we perform the
order analysis of the x and y momentum equations as given below:
x:
u 
u 
p
1   2 u  2 u 


u
 v



x 
y
x  Re L  x2 y2 
(orders )
(1)
y:
v
v
p
1   2 v  2 v 
 2 

u
 v


2 





x
y
x Re L  x
y 
(orders )
(1)
(1)
(1)
( )
(1)
( )
( )
()
( )
(1)
()
?
?
 (1)
( 2 )
 (1)
(1)
( 2 )
 (1) (1)
( 2 )
2
 (1) ( )






In the y-momentum above we notice that all the terms are order () or, smaller
Ghosh - 550
[e.g.,
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p 
1  v
]. Therefore the
must be order  (or, smaller). This is an
~
y 
Re x
2
3
2
L
important consideration which says although pressure is significant in the boundary
layer, the pressure gradient in the direction transverse to the body is significantly
small.
i.e.,.
p
~  . Similarly, from the x-component we note that all the terms are order 1 or
y
smaller. Specifically the term
compared to the other
 2u
x
2
~  2 , which means, this term is much smaller
 2u 
term
2
. These conclusions due to the order analysis performed
y
by Prandtl are summarized below:
(i)
(ii)
(iii)
v ~ 
p
~
y
 v << u
p
p


y
x
1  2u
~2
2
Re L x 

 2u
x 2

(since, u ~ 1)
p 
~ 1)
(since,
x 
 2u
y 2
(since,
1  2u
~ 1)
Re L y  2
Also, since all terms in the y-momentum equation were of  or smaller, Prandtl
neglected the importance of this equation in his analysis.
Boundary Layer Equations
(Velocity Boundary Layers)
Based upon the foregoing analysis, Prandtl gave the following two equations in the
boundary layer over any object oriented in the x-direction:
u v

0
x y
 u
u 
dp
 2u
  u  v      2
y 
dx
y
 x
(continuity)
(x-momentum)
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Note that the y-momentum is dropped out from the formulation,
 2u
term is dropped out
x 2
from the x-momentum and the partial derivatives on the right hand side of the xmomentum are re-written to reflect the single variable dependencies p(x) and u(y). The
above two equations have 3-unknowns u, v, and p. However, note that p(x) only. This
pressure solution may be known from the solution of ideal flows, solved at the edge
of the boundary layer, the solution of the Euler’s equations may be computed using
a CFD codes to provide the p(x) necessary at the edge of a boundary layer on
arbitrarily shaped objects.
Ideal flow solutions hold
Boundary layer equations hold
Flow
L
dp
dp
0.
is a known parameter. For a flat plate,
dx
dx
Boundary layer equations over a flat plate are normally solved following two approaches:
For our analysis, we will assume that
(i)
(ii)
Exact solution method (Blasius solution)
Approximate solution method (Karman-Pohlhausen method or,
Momentum Integral Method)
Both are discussed in different sections. But before the solution methodology, let us
explore further physical concepts.
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