The inertia force per unit volume is equal to ρ u ( ∂ u / ∂ x ). For a distance x , the gradient ∂ u / ∂ x is proportional to U / x , where U is the velocity outside the B.L. Hence the inertia force is of the order U 2 / x . The friction force per unit volume is equal to ∂ τ / ∂ y , which for laminar flow is equal to μ ∂ 2 u / ∂ y 2 . The velocity gradient ∂ u / ∂ y perpendicular to the wall is of the order U / δ , so the friction force is of the order μ U / δ 2 . If the two forces are to balance each other, then: x U U 2 2 ~ ρ δ μ and solving for the B.L. thickness, U x ⋅ ν δ ~
89 The numerical factor is deduced from the exact solution given by Blasius, which is equal to 5. Therefore, for laminar boundary layer: U x ⋅ = ν δ 5 or x Re x 5 = δ A more accurate factor of 5.48 is found to better match experimental data, hence: x Re x 48 . 5 = δ where ν x U Re x ⋅ = 9.2. Estimation of wall shear stress The wall shear stress may be expressed as, τ w = μ ∂ u ∂ y y =0 so that, τ w = 0.365 ρ U 2 Re x and the skin friction coefficient, C f , is given by, C f = τ w 1 2 ρ U 2 = 0.730 Re x
90 The boundary layer thickness, δ , increases with x 1/2 , while the wall shear stress and the skin friction coefficient vary as x -1/2 . These are the characteristics of a laminar boundary layer over a flat plate. 9.3. Turbulent boundary layer A suitable velocity profile for turbulent B.L. is the empirical power-law profile. An exponent of 1 / 7 is used to model the velocity profile, u / U = ( y / δ ) 1/7 = η 1/7 This profile doesn't hold in the immediate vicinity of the wall, since at the wall it predicts d u /d y = ∞ . Hence, we cannot use this profile in the definition of τ w to obtain an expression in terms of δ . For turbulent boundary-layer flow we use the expressions developed for pipe flow, τ w = 0.03325 ρ V 2 ν RV 1/4 For a 1 / 7 -power profile in a pipe, V / U =0.8. So, substituting V =0.8 U and R = δ one gets,
91 τ w = 0.00225 ρ U 2 ( ν / U δ ) 1/4 or C f = 0.045( ν / U δ ) 1/4 For turbulent boundary layer, δ x = 0.382 (Re x ) 1/5 and so, C f = τ w 1 2 ρ U 2 = 0.0594 Re x 1/5 Experiment show that this eq'n predicts the turbulent skin friction on a flat plate within about 3% for 5 ⋅ 10 5 < Re x <10 7 .
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