Correlation Methods for Integral Boundary Layers
We will look at one particularly well-known and easy method due to Thwaites in
1949.
First, start by slightly re-writing the integral b.l. equation. We had:
τw
dθ
θ du e
=
+ (2 + H )
2
u e dx
ρ e u e dx
Multiply by
u eθ
:
v
τ wθ u eθ dθ θ 2 du e
=
+
(2 + H )
v dx
v dx
µu e
Then define λ =
ue
θ 2 du e
d
λ
(
dx du e
v dx
dx
and this equation gives:
⎡τ θ
⎤
) = 2⎢ w − λ (2 + H )⎥
⎣ µu e
⎦
Thwaites then assumes a correlation exists which only depends on λ.
Specifically:
H = H (λ ) and
τ wθ
= S (λ )
µu e
shape factor
correlation
shear correlation
λ
d
(
dx du e
) ≅ 2[S (λ ) − λ (2 + H (λ )]
⇒ ue
dx
now this is an approximation
Correlation Methods for Integral Boundary Layers
In a stroke of genius and/or luck, Thwaites looked at data from experiments and
known analytic solutions and found that
ue
d
x
(
dx du e
≈ 0.45 − 6λ !!
dx
This can actually be integrated to find:
θ2 =
0.45v
ue
6
x
∫u
5
e
dx
o
where we have assumed θ ( x = 0) = 0 for this.
16.100 2002
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