Boundary Layer on a Flat Plate: Blasius Solution
H
z
from Kundu’s book
Assuming displacement of streamlines is negligible →u = U = constant everywhere, as if the boundary didn’t existThe irrotational flow, according to Euler’s equation: x
pzuw
xuu
tu
1
xp
0 = 0 @ u = constant
2
2
zu
zuw
xuu
z
pg
0
zw
xu
zallxUzu 0@ UL
The complete set of equations for Boundary Layer are:
Lxzwu 00@0
LxzUu 0@
H
z
from Kundu’s book
Uxx
~
The velocity profile in the boundary layer can be obtained with a SIMILARITY SOLUTION – following Blasius, a student of Prandtl
H
z
from Kundu’s book
Velocity distributions at various x can collapse into a single curve if the solution has the form
Uu
xz
Uxx
~
For similarity solution, use streamfunction:
xw
zu
Using similarity form above:
00
ududzz
0
dU
fU
Uuf
ddf
'
Using the definition:
2
2
zu
zuw
xuu
0
zw
xuApplying streamfunction to:
3
3
2
22
zzxzxz
0@ xU
z 0@0
z
z
zUz
@
fU
xf
dxdfU
x ff
dxdU
ffzdx
dUzx
2
dxdfU
fUz
fUz
2
2
23
3
fUz
Uu
ddff
fffUdxd
xw
zu
xz
Uxx
~
fffUdxd
f and its derivatives do not explicitly depend on x :
0
df
Can be valid only if:21constant
U
dxd
xz
fff 21
Blasius equation
021
fff
initial and boundary conditions: 1 ddff 000 ff
dxU
d 21
Ux
21
21 2
Ux
021
2
2
3
3
d
fdfd
fd
Uuf
zvsUu
021
2
2
3
3
d
fdfd
fd
1f 000 ff
% uses Matlab ODE45 - Runge-Kutta methodti = 0.0; % start of integrationtf = 7.0; % final value of integrationbcinit = [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
Ux 9.499
xx Re9.499
Ux
xUxx 9.49.4
2
99
Rex
Uxz
9.499
Ux 9.499
ν = 1×10-6 m2/s; U = 1 m/s
x
Uxdz
Uu 72.11*
0
Uxdz
Uu
Uu
664.010
displacement thickness
momentum thickness
Skin Friction
02
2
00
zz
u
Local wall shear stress
fUz
2
2using:
00
fU
x
URe
332.0 2
0
xU
x Re
zu
U332.0
0
Ux
@z = 0
xURe
Skin FrictionLocal wall shear stress
x
URe
332.0 2
0
Wall shear stress then changes as x -½ , i.e., decreases with increasing x
xU
x Re
τ decreases because of thickening of δ
Local shear stress at wall can be expressed in terms of the local drag coefficient
xf U
CRe664.0
21 2
0 2
2
0 21
Re332.0 UCU
fx
and the drag force per unit width of plate of length L
L
dxD0
0L
LURe
664.0 2
LU
L Re
So the drag force is proportional to the 3/2 power of velocity (U 2/U 1/2)For high Re the drag force is proportional to the square of velocity
LD LU
DCRe33.1
21 2
Now, the overall drag coefficient is defined as:
L
fD dxCL
C0
1
overall drag coefficient is average of local drag coefficient
http://www.symscape.com/node/447
Breakdown of Blasius solution
Ux 9.499
Boundary layer grows faster in the turbulent region because of macroscopic eddies
Transition from laminar to turbulent region occurs at Recr (~106) Transition depends on a) surface roughness and b) shape of leading edge
