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16100lectre23

# 16100lectre23 - Correlation Methods for Integral Boundary...

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Unformatted text preview: 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 2 ...
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16100lectre23 - Correlation Methods for Integral Boundary...

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