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Unformatted text preview: Correlation Methods for Integral Boundary Layers
We will look at one particularly wellknown and easy method due to Thwaites in
1949.
First, start by slightly rewriting 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 2 ...
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 Fall '03
 willcox
 Dynamics, Aerodynamics, Trigraph, dx, Thwaites, θ du e

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