Falkner-Skan Flows
For the family of flows, we assume that the edge velocity, u e (x) is of the
following form:
u e ( x) = Kx m
K = arbitrary constant
The pressure can be calculated from the Bernoulli in the outer, inviscid flow:
1 2
ρu e = const.
2
dp
du
⇒ e = − ρu e e
dx
dx
dp
⇒ e = − ρK m2 x 2 m −1
dx
pe +
if m > 0 then
dp e
< 0 ⇒ favorable pressure gradient
dx
if m < 0 then
dpe
> 0 ⇒ adverse pressure gradient
dx
These edge velocities result from the following inviscid flows:
β≡
y
2m
1+ m
x
u e (x)
u e ( x)
x=0
β
π
y
β
2
Flow around a corner (diffusion)
−2≤ β ≤0
Wedge flow
0≤β ≤2
π
2
Falkner-Skan Flows
Some important cases:
β = 0, m = 0 : flat plat (Blasius)
β = 1, m = 1 : plane stagnant point
The boundary layer independent variable η from the Blasius solution generalizes
to:
η≡y
m + 1 u e ( x)
and u ( x, y ) = u e ( x ) f 1 ( m)
2
vx
An interesting case in β = 1, m = 1 , i.e. stagnation point flow:
x
u e (x)
ue = K x
inviscid flow velocity
increases away from
stag. pt. at x = 0
y
⇒
η=y
1+1 Kx
2 vx
η=y
K
v
⇒ η is independent of x
⇒ Boundary layer at a stagnation point does not
grow with x !
The skin friction can be found from:
τw = µ
∂u
∂y
= µu e ( x)
y =0
d2 f
dη
∂η
∂y
η =0
y =0
f 11 (o)
16.100 2002
2
Falkner-Skan Flows
Since η = y
⇒
m + 1 u e ( x)
∂η
⇒
=
vx
2
∂y
m + 1 u e ( x)
vx
2
m + 1 u e ( x) 11
f ( o)
vx
2
τ w = µu e ( x)
tabulated
The skin friction coefficient is normalized by
C f ( x) ≡
τw
1 2
ρu e ( x )
2
2
⇒
Cf =
Re x ≡
1 2
ρu e ( x ) :
2
m +1 v
f 11 (o)
2 u e ( x) x
=2
m + 1 11
f (o)
2
Re x
u e ( x) x
v
Note: separation occurs when C f = 0 which means f 11 (o) = 0 . From the table,
this occurs for β = −0.19884
o
18
16.100 2002
⇒ This is only an angle of 18o!
3