Method of Assumed Profiles
Here are the basic steps:
1. Assume some basic boundary velocity profile for u ( x, y ) . For example, this is a
crude approach but illustrates the ideas:
 y
,0 ≤ y < δ ( x )
u( x, y ) 
=  δ ( x)
ue ( x ) 
1, y ≥ δ ( x )

where δ (x) is the single unknown describing the velocity distribution.
2. Calculate δ * , θ (or H ) , and C f for the assumed profile:
δ
δ


u
y
1 y2 

δ = ∫  1 −  dy = ∫  1 −  dy =  y −

δ
2 δ 0
ue 

0
0
∞
*
1
⇒ δ* = δ
2
δ
δ
 1 y2 1 y3 
u
u
y
y
−
θ = ∫  1 −  dy = ∫  1 −  dy = 
2 
δ δ
u
ue 
 2 δ 3 δ 0
0 e 
0
∞
1
⇒θ= δ
6
1
δ
δ
= 2 =3
H=
θ 1δ
6
*
Note:
Finally, to find C f we need τ w = µ
∂u
∂y
y =0
∂u
∂  y u
= ue   = e , for 0 ≤ y < δ
∂y
∂y  δ  δ
Method of Assumed Profiles
τw
⇒ Cf =
1
ρ eue2
2
=
µ
ue
δ
1
ρ eue2
2
=
2µ
ρ eueδ
3. Plug results from step 2 into integral b.l. equation:
Cf
=
2
du
dθ θ
+ (2 + H ) e
dx u e
dx
So, for our assumed linear profile:
µ
ρ eueδ
=
1 d δ 5δ due
+
6 dx 6ue dx
(1)
where δ (x) is the only unknown. We can solve this by specifying u e (x) , setting
an initial value for δ at x = 0 (i.e. the leading edge) and then integrate in x .
Note: in many cases, this integration will need to be done numerically.
Example: Flat Plate
8
Ue(x)=U
ρe=ρ
8 8
U
ρ
8
y
δ*(x)
x
Since u e ( x) = u ∞ is a constant, the governing equation (1) becomes:
µ
ρ ∞ u ∞δ
=
1 dδ
6 dx
Re-arrange and integrate:
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2
Method of Assumed Profiles
6µ
dδ
=δ
ρ ∞U ∞
dx
6µ
1 d (δ 2 )
=
ρ ∞U ∞ 2 dx
6µ
1 d (δ 2 )
dx
dx
=
∫0 ρ ∞U ∞
2 ∫0 dx
x
x
6µ
1
x = δ 2 ( x ) − δ 2 (0) 
ρ ∞U ∞
2
But, our initial condition is δ (0) = 0 .
6µ
1
x = δ 2 ( x)
2
ρ ∞U ∞
δ
⇒
⇒
x
δ*
δ*
x
=
x
=
=
12 µ
2 3
3.464
=
=
ρ ∞U ∞ x
Re x
Re x
1
2
δ
x
3
Re x
=
=
1 2 3
2 Re x
1.732
Re x
and C f :
Re x
2µ
2µ
=
ρ eueδ ρ ∞u∞ x 2 3
1
Cf =
3 Re x
Cf =
Cf =
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0.577
Re x
3
Method of Assumed Profiles
Comparison with Blasius solution:
Blasius
δ*
Int. Method with linear velocity
Re x
1.720
1.732
C f Re x
0.664
0.577
x
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