Boundary Layer on a Flat Plate: Blasius Solution
from Kundu’s book
H
z
Assuming displacement of streamlines is negligible →u = U = constant
everywhere
u
u
u
1 p

u

w


The irrotational flow, according to Euler’s equation: t
x
z
 x
0
p
x
The complete set of equations for Boundary Layer are:
p
g
z
u
u
 2u
u
w

x
z
z 2
uz   U
@x  0
u w

0
x
z
all z
u w 0
@z  0 0  x  L
u U
z
@


from Kundu’s book
H
z
0 x L
 
 x  ~
L
U
x
U
The velocity profile in the boundary layer can be obtained with a SIMILARITY
SOLUTION – following Blasius, a student of Prandtl
Velocity distributions at various x can collapse into a single curve if the
solution has the form u
z
x
     



x
~
U
 x 
U
from Kundu’s book
For similarity solution, use streamfunction:
u
H
z

z
w 

x
Using similarity form above:
z


0
0
0
   udz    ud    U  d
 U f  
Using the definition:
df
   
d
Applying streamfunction to:
u

z
w 

U
z

x
@x 0
u
u
 2u
u
w

x
z
z 2
u w

0
x
z
  2
  2
 3


2
z xz
x z
z 3

  0
z
@z  0

U
z
@
z

  U f   
d
f 
 d
f  f  

U
 U f


dx
x
x 
 dx
 2
d 
f  f     Uf  d
U
xz
dx z
 dx

 Uf 
z
 2
Uf 

z 2

 3
Uf 

z 3
2
f 

df
d
 d U 

f  f   f 
 dx  
 d U 

f  f   f 
 dx  

f and its derivatives do not explicitly depend on x : f
    d 
0
Can be valid only if:
d U
1
 constant 
dx 
2
1
 f  f   f 
2
1
f   f  f   0
2
1
dx
2U
1 2 1 x
 
2
2 U
d 3f
1 d 2f
 f
0
3
2
d
2 d
Blasius equation
initial and boundary conditions: f   df d  1
 d 
z

 x 

x
U
f 0  f 0  0
d 3f
1 d 2f
 f
0
3
2
d
2 d
f    1 f 0  f 0  0
f    u U
u U vs   z 
% uses Matlab ODE45 - Runge-Kutta method
ti = 0.0; % start of integration
tf = 7.0; % final value of integration
bcinit = [0.0 0.0 0.33206]; % initial values
[eta f] = ode45('state',[ti tf],bcinit);
==================
function stst = state(eta,f)
stst = [ f(2) , f(3) , -0.5*f(1)*f(3)]';
Boundary Layer Thickness
Distance η where u = 0.99 U
η = 4.9
 99  4.9
 99  4.9
x
U
 x2
4.9 x

Ux
Ux

 99
x

4.9
Re x
Rex
 99  4.9
x
U
ν = 1×10-6 m2/s; U = 1 m/s


0 
 *   1
u
x
dz  1.72
U
U
displacement thickness
 

u
0 U
u

1
U

x

dz  0.664
U

momentum thickness
Local wall shear stress
  2 
 u 

 0  
   
2 
 z  0
 z  0
u
0  

z
0.332 U
 

x
U
0.332  U 2
0 
Re x
Re x 
Ux

Skin Friction
using:
 2
Uf 

2
z

0  
Uf 0 

Skin Friction
Local wall shear stress
Re x 
Ux

0.332  U 2
0 
Re x
Wall shear stress then changes as x -½ , i.e., decreases with increasing x
τ decreases because of thickening of δ
Local shear stress at wall can be expressed in terms of the local drag coefficient
0.332  U 2
1
0 
 Cf  U 2
2
Re x
Cf 
0
0.664

1 U 2
Re x
2
and the drag force per unit width of plate of length L
0.664  U 2 L
D    0dx 
ReL
0
L
ReL 
So the drag force is proportional to the 3/2 power of velocity (low Re)
For high Re the drag force is proportional to the square of velocity
Now, the overall drag coefficient is defined as:
D
1.33
CD 

1  U 2L
ReL
2
L
1
CD   Cf dx
L0
overall drag coefficient is average of local drag coefficient
UL

http://www.symscape.com/node/447
 99  4.9
x
U
Breakdown of Blasius solution
Transition from laminar to turbulent region occurs at Recr (~106)
Transition depends on a) surface roughness and b) shape of leading edge
Boundary layer grows faster in the turbulent region because of macroscopic
eddies