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External Flows Overview Non-Uniform Flow Boundary Layer Concepts Viscous Drag Pressure Gradients: Separation and Wakes Pressure Drag Shear and Pressure Forces Vortex Shedding Non-Uniform Flow In pipes and channels the velocity distribution was uniform (beyond a few pipe diameters or hydraulic radii from the entrance or any flow disturbance) In external flows the boundary layer is always growing and the flow is nonuniform Boundary Layer Concepts Two flow regimes Laminar boundary layer Turbulent boundary layer with laminar sub-layer Calculations of boundary layer thickness Shear (as a function of location on the surface) Drag (by integrating the shear over the entire surface) Flat Plate: Parallel to Flow U y U U U boundary layer thickness x shear Why is shear maximum at the leading edge of the plate? du dy is maximum Flat Plate Drag Coefficients 0.01 CDf = [1.89 - 1.62log (e / l )] 1 x 10-3 5 x 10-4 2 x 10-4 1 x 10-4 5 x 10-5 2 x 10-5 1 x 10-5 5 x 10-6 2 x 10-6 1 x 10-6 - 2.5 CDf CDf = 1.328 (Rel )0.5 CDf = 0.455 [log (Re )] 2.58 l - e l CDf = 0.072Rel- 0.2 1700 Rel CDf = 0.001 0.455 [log (Re )] Rel = Ul n +1 0 1e +0 9 1e +0 8 1e +0 7 1e +0 6 1e +0 5 1e 1e +0 4 l 2.58 Separation and Wakes Separation often occurs at sharp corners fluid can’t accelerate to go around a sharp corner Velocities in the Wake are ______ small (relative to the free stream velocity) Pressure in the Wake is relatively ________ constant (determined by the pressure in the adjacent flow) Flat Plate: Streamlines 3 U 0 p p0 Cp 1 2 2 2 U U v Cp p >p0 0 1 <U >0 >p0 <p0 >U <0 <p0 v2 2 1 4 Point 1 2 3 4 Points outside boundary layer! Drag of Blunt Bodies and Streamlined Bodies Drag dominated by viscous streamlined drag, the body is __________. Drag dominated by pressure drag, the body is _______. bluff Whether the flow is viscousdrag dominated or pressuredrag dominated depends entirely on the shape of the body. Bicycle page at Princeton Drag Coefficient on a Sphere Drag Coefficient 1000 100 Stokes Law 10 1 0.1 0.1 CD = 1 24 Re 10 102 103 104 Reynolds Number 105 106 107 Shear and Pressure Forces: Horizontal and Vertical Components FD p sin 0 cos dA drag Parallel to the approach velocity FL p cos 0 sin dA lift Normal to the approach velocity p < p0 U 2 negative pressure Fd Cd A 2 A defined as projected normal to force! area _______ U 2 FL CL A 2 lift q U drag p > p0 positive pressure Shear and Pressure Forces Shear forces viscous drag, frictional drag, or skin friction caused by shear between the fluid and the solid surface length object function of ___________and ______of surface area Pressure forces pressure drag or form drag caused by _____________from the body flow separation function of area normal to the flow Example: Beetle Power Cd = 0.38 Height = 1.511 m Width = 1.724 m Length = 4.089 m Ground clearance = 15 cm? 85 kW at 5200 rpm Where does separation occur? Calculate the power required to overcome drag at 60 mph and 120 mph. Is the new beetle streamlined? Solution: Power a New Beetle at 60 mph Cd FD 2FD U 2 A f (R ) C d U 2 A 2 P C d U 3 A 2 P (0.38)(1.2kg / m3 )(26.82m / s)3 (2.0m 2 ) 2 P = 8.8 kW at 60 mph P = 70 kW at 120 mph Drag on a Golf Ball DRAG ON A GOLF BALL comes mainly from pressure drag. The only practical way of reducing pressure drag is to design the ball so that the point of separation moves back further on the ball. The golf ball's dimples increase the turbulence in the inertia of the boundary layer, increase the _______ boundary layer, and delay the onset of separation. The effect is plotted in the chart, which shows that for Reynolds numbers achievable by hitting the ball with a club, the coefficient of drag is much lower for the dimpled ball. Why not use this for aircraft or cars? Effect of Turbulence Levels on Drag Flow over a sphere: (a) Reynolds number = 15,000; (b) Reynolds number = 30,000, with trip wire. Causes boundary layer to become turbulent Point of separation Effect of Boundary Layer Transition Ideal (non viscous) fluid No shear! Real (viscous) fluid: laminar boundary layer Real (viscous) fluid: turbulent boundary layer Spinning Spheres What happens to the separation points if we start spinning the sphere? LIFT! Vortex Shedding Vortices are shed alternately from each side of a cylinder The separation point and thus the resultant drag force oscillate Dimensionless frequency of shedding given by Strouhal number S S is approximately 0.2 over a wide range of Reynolds numbers (100 - 1,000,000) S nd U