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External Flows

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**Overview Non-Uniform Flow Boundary Layer Concepts Viscous Drag**

Pressure Gradients: Separation and Wakes Pressure Drag Shear and Pressure Forces Vortex Shedding

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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 non-uniform

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**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)

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**Flat Plate: Parallel to Flow**

U U U boundary layer thickness U y x shear Why is shear maximum at the leading edge of the plate? is maximum

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**Flat Plate Drag Coefficients**

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

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**Separation and Wakes Separation often occurs at sharp corners**

fluid can’t accelerate to go around a sharp corner Velocities in the Wake are ______ (relative to the free stream velocity) Pressure in the Wake is relatively ________ (determined by the pressure in the adjacent flow) small constant

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**Flat Plate: Streamlines**

U 3 2 4 1 Point v Cp p 1 2 3 4 1 >p0 <U >0 >p0 >U <0 <p0 <p0 Points outside boundary layer!

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**Drag of Blunt Bodies and Streamlined Bodies**

Drag dominated by viscous drag, the body is __________. Drag dominated by pressure drag, the body is _______. Whether the flow is viscous-drag dominated or pressure-drag dominated depends entirely on the shape of the body. streamlined bluff Bicycle page at Princeton

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**Drag Coefficient on a Sphere**

1000 100 Stokes Law Drag Coefficient 10 1 0.1 0.1 1 10 102 103 104 105 106 107 Reynolds Number

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**Shear and Pressure Forces: Horizontal and Vertical Components**

drag Parallel to the approach velocity lift Normal to the approach velocity p < p0 negative pressure A defined as projected area _______ to force! U normal lift q drag p > p0 positive pressure

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**Shear and Pressure Forces**

Shear forces viscous drag, frictional drag, or skin friction caused by shear between the fluid and the solid surface function of ___________and ______of object Pressure forces pressure drag or form drag caused by _____________from the body function of area normal to the flow surface area length flow separation

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**Example: Beetle Power Cd = 0.38 Height = 1.511 m Width = 1.724 m**

Length = 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?

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**Solution: Power a New Beetle at 60 mph**

P = 8.8 kW at 60 mph P = 70 kW at 120 mph

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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 boundary layer, increase the _______ of 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. inertia Why not use this for aircraft or cars?

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**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

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**Effect of Boundary Layer Transition**

Real (viscous) fluid: laminar boundary layer Ideal (non viscous) fluid Real (viscous) fluid: turbulent boundary layer No shear!

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Spinning Spheres What happens to the separation points if we start spinning the sphere? LIFT!

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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 ( ,000,000)

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