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# ch76a - Chapter 7 Flow Past Immerses Bodies 7.1 7.2 7.3 7.4...

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Chapter 7: Flow Past Immerses Bodies 7.1 Reynolds Number and Geometry Effects 7.2 Momenum Integral Estimates 7.3 The Boundary Layer Equations 7.4 The Flat Plate Boundary Layer 7.5 Boundary Layers with Pressure Gradients (skipped Laminar Integral Theory) 7.6 Experimental External Flows

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p = ? Boundary layer theory can usually predict separation point but not pressure distribution in separated region “There is at present no satisfactory theory for the forces on an arbitrary body immersed in a stream flowing at an arbitrary Reynolds number.” White – Fluid Mechanics
C D = F/[½ ρ U 2 A] f = (dp/dx)D/( ½ ρ U avg 2 ) Re 10 4

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FLUID FLOW ABOUT IMMERSED BODIES U p 4 p 1 p 2 p 3 p 6 p 5 p 7 p 8 p 9 p 10 p 11 p 13 p p 12 τ 10 τ 9 τ 8 τ 7 τ 6 τ 5 τ 4 τ 3 τ 2 τ 1 τ Drag due to surface stresses composed of normal (pressure) and tangential (viscous) stresses. All we need to know is p and τ on body to calculate drag. Could do for flat plate with zero pressure gradient because U and p, which were constant, we knew everywhere. If μ = 0 then pressure distribution is symmetric, so no net pressure force ( D’Alembert’s Paradox - 1744) DRAG LIFT
Forces and Moments on Body Immersed in a Uniform Flow (LIFT: perpendicular to free stream velocity, not necessarily body, also perpendicular to drag) (DRAG: parallel to free stream velocity)

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If body has planes of symmetry around lift-drag axis and horizontal axis and the free stream is parallel to the intersection of these two planes then the body only experiences drag. Drag = force parallel to V Lift = force to V & no camber
Characteristic Area – May Differ Frontal area (as seen from free stream) if “stubby” spheres, cylinders, trucks … Planform area (as seen from above) suitable for wide flat bodies like wings and hydrofoils. Wetted area, customary for surface ships and barges Same body – different A - different C D C D = D/( ½ ρ U 2 A ) Re c = 10 6 C Dplate ~ 0.002 – 0.004

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VISCOUS DOMINATED 2-D
Flat Plate C D (like pipe Moody diagram) Constant U ext No Pressure Gradient Laminar: C D =1.328/Re L 1/2 Turbulent (smooth): C D =0.031/Re L 1/7 Turbulent (fully rough) C D ~ (1.89 + 1.62log[L/ ε ]) -2.5

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PRESSURE DOMINATED (Re 10 4 ) 2-D Newton (Proposition 23 of the Principia July 5 th 1687)
Drag Force = Δ p/ Δ t = Δ (mv)/ Δ t m ~ ρ UA f = mass per second passing through area Δ v ~ U-0 = U (Newton) (fixed) C D = D/( ½ ρ U 2 A f ) ~ ρ UA f U /( 1 / 2 ρ U 2 A f ) C D ~ 2 Newton predicted correctly: dependence of D on U 2 and A, C D ~ 2, and insensitivity of C D on Re.

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Flow parallel to plate – viscous forces dominate and Re dependence Flow perpendicular to plate – pressure forces dominate and no strong Re dependence What about Re dependence for flow around sphere?
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