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Pharos University ME 253 Fluid Mechanics II Flow over bodies; Lift and Drag External External Flows Bodies in motion, experience fluid forces and moments. Examples include: aircraft, automobiles, buildings, ships, submarines, turbo machines. Fuel economy, speed, acceleration, stability, and control are related to the forces and moments. Airplane in level steady flight: drag = thrust & lift = weight. Flow over immersed bodies flow classification: 2D, axisymmetric, 3D bodies: streamlined and blunt Airplane Upper surface (upper side of wing): low pressure Lower surface (underside of wing): high pressure Lift and Drag shear stress and pressure integrated over body surface drag: force component in the direction of upstream velocity lift: force normal to upstream velocity D dFx p cos dA w sin dA CD D 2 1 U A 2 L dFy p sin dA w cos dA CL L 2 1 U A 2 AIRFOIL NOMENCLATURE Mean Chamber Line: Points halfway between upper and lower surfaces Leading Edge: Forward point of mean chamber line Trailing Edge: Most reward point of mean chamber line Chord Line: Straight line connecting the leading and trailing edges Chord, c: Distance along the chord line from leading to trailing edge Chamber: Maximum distance between mean chamber line and chord line AERODYNAMIC FORCE Relative Wind: Direction of V∞ We used subscript ∞ to indicate far upstream conditions Angle of Attack, a: Angle between relative wind (V∞) and chord line Total aerodynamic force, R, can be resolved into two force components Lift, L: Component of aerodynamic force perpendicular to relative wind Drag, D: Component of aerodynamic force parallel to relative wind Pressure Forces acting on the Airfoil Low Pressure High velocity High Pressure Low velocity Low Pressure High velocity Bernoulli’s equation says where pressure is high, velocity will be low and vice versa. High Pressure Low velocity Relationship between L´ and p V L Force normal to the wind direction Forces acting on the lower side - Force on upper side Trailing Edge p Trailing Edge dx lower side Leading Edge Trailing Edge p Leading Edge lower side p dx upper side Leading Edge p upper side dx Relationship between L´ and p (Continued) Trailing Edge L p lower side p upper side dx Leading Edge Trailing Edge p lower side p p upper side p dx Leading Edge 1 V2 c 2 Divide left and right sides by plower p pupper p x L d 1 1 1 2 2 c V2c Leading V V Edge 2 2 2 Trailing Edge We get: Pressure Coefficient Cp From the previous slide, plower p pupper p x L d 1 1 1 2 2 c V2c Leading V V Edge 2 2 2 Trailing Edge The left side was previously defined as the sectional lift coefficient Cl. The pressure coefficient is defined as: Cp Thus, Trailing edge Cl C Leading edge p p 1 V2 2 p ,lower C p ,upper d x c Fluid dynamic forces are due to pressure and viscous forces. Drag: component parallel to flow direction. Lift: component normal to flow direction. Lift and drag forces can be found by Drag and Lift integrating pressure and wall-shear stress. Drag and Lift Lift FL and drag FD forces fn ( , A,V ) Dimensional analysis: lift and drag coefficients. Area A can be frontal area (drag applications), plan form area (wing aerodynamics). Example: Automobile Drag bile Drag CD = 1.0, A = 2.5 m2, CDA = 2.5m2 CD = 0.28, A = 1 m2, CDA = 0.28m2 • Drag force FD=1/2V2(CDA) will be ~ 10 times larger for Scion XB • Source is large CD and large projected area • Power consumption P = FDV =1/2V3(CDA) for both scales with V3! Drag and Lift If CL and CD fn of span location x. A local CL,x and CD,x are introduced. The total lift and drag is determined by integration over the span L Friction and Pressure Drag Friction drag Fluid dynamic forces: pressure and friction effects. FD = FD,friction + FD,pressure CD = CD,friction + CD,pressure Pressure drag Friction & pressure drag Flow Around Objects Streamlining Streamlining reduces drag by reducing FD,pressure, Eliminate flow separation and minimize total drag FD Streamlining CD 4 For many shapes, total drag C is constant for Re > 10 of Common Geometries D CD of Common Geometries CD of Common Geometries Flat Plate Drag Drag on flat plate is due to friction created by laminar, transitional, and turbulent boundary layers. Flat Plate Drag Local friction coefficient Laminar: Turbulent: Average friction coefficient Laminar: Turbulent: Cylinder and Sphere Drag Cylinder and Sphere Drag Flow is strong function of Re. Wake narrows for turbulent flow since turbulent boundary layer is more resistant to separation. sep, lam ≈ 80º sep,Tur ≈ 140º Lift Lift is the net force (due to pressure and viscous forces) perpendicular to flow direction. Lift coefficient A=bc is the planform area Characteristics of Cl vs. a Stall Cl Slope= 2p if a is in radians. a = a0 Angle of zero lift Angle of Attack, a in degrees or radians Lift EXAMPLE: AIRFOIL STALL Angle of Attack, a 30 Effect of Angle of Attack CL≈2pa for a < astall Lift increases linearly with a Objective:Maximum CL/CD CL/CD increases until stall. Thickness and camber affects pressure distribution and Effectofof Foil Shape location flow separation. End Effects of Wing Tips Tip vortex created by flow from high-pressure side to low-pressure side of wing. Tip vortices from heavy aircraft far downstream and pose danger to light aircraft. Lift Generated by Spinning Superposition of Uniform stream + Doublet + Vortex Drag Coefficient: CD Stokes’ Flow, Re<1 Supercritical flow turbulent B.L. Relatively constant CD Drag Drag Coefficient with or DRAG FORCE Friction has two effects: Skin friction due to shear stress at wall Pressure drag due to flow separation D D friction D pressure Total drag due to viscous effects Called Profile Drag = Drag due to skin friction Less for laminar More for turbulent + Drag due to separation More for laminar Less for turbulent COMPARISON OF DRAG FORCES d d Same total drag as airfoil 38 AOA = 2° AOA = 3° AOA = 6° AOA = 9° AOA = 12° AOA = 20° AOA = 60° AOA = 90° Drag Coefficient of Blunt and Streamlined Bodies Drag dominated by viscous drag, streamlined the body is __________. Drag dominated by pressure drag, bluff the body is _______. Flat plate Cd 2Fd U 2 A Drag Pure Friction Drag: Flat Plate Parallel to the Flow Pure Pressure Drag: Flat Plate Perpendicular to the Flow Friction and Pressure Drag: Flow over a Sphere and Cylinder Streamlining Drag Flow over a Flat Plate Parallel to the Flow: Friction Drag Boundary Layer can be 100% laminar, partly laminar and partly turbulent, or essentially 100% turbulent; hence several different drag coefficients are available Drag Flow over a Flat Plate Perpendicular to the Flow: Pressure Drag Drag coefficients are usually obtained empirically Flow past an object Character of the steady, viscous flow past a circular cylinder: (a) low Reynolds number flow, (b) moderate Reynolds number flow, (c) large Reynolds number flow. Drag Flow over a Sphere and Cylinder: Friction and Pressure Drag (Continued) Streamlining Used to Reduce Wake and hence Pressure Drag Lift Mostly applies to Airfoils Note: Based on planform area Ap Lift Induced Drag Experiments for Airfoil Lift & Drag Examine the surface pressure distribution and wake velocity profile on airfoil 2-D Compute the lift and drag forces acting on the airfoil Pressure coefficient Lift coefficient Test Facility: • Wind tunnel. • Airfoil • Temp. sensor • Pitot tubes • Pressure sensors • Data acquisition Test Design Airfoil in a wind tunnel with free- stream velocity of 15 m/s. This airfoil has: Forces normal to free stream = Lift Forces parallel to free stream = Drag Top of Airfoil: - The velocity of the flow is greater than the free-stream. - The pressure is negative Underside of Airfoil: - Velocity of the flow is less than the free-stream. - The pressure is positive This pressure distribution contribute to the lift & Drag Pressure taps positions The lift force, L on the Airfoil will be find by integration of the measured pressure distribution over the Airfoil’s surface. Data reduction Calculation of lift force The lift force L= Integration of the measured pressure over the airfoil’s surface. Pressure coefficient Cp where, pi = surface pressure measured, = P pressure in the free-stream U∞ = free-stream velocity, ϱ = air density pstagnation = stagnation pressure by pitot tube, L = Lift force, b = airfoil span, c = airfoil chord pi p Cp 1 U 2 2 2 pstagnation p U 2L CL U 2 bc L p p sin ds s p CL p sin ds s 1 U 2 c 2 Drag Force The drag force, D on the Airfoil = Integration of the momentum loss using the axial velocity profile in the wake of the Airfoil. Data reduction Calculation of drag force The drag force D = integration of the momentum loss The velocity profile u(y) is measured ui at predefined locations U∞ = free-stream velocity, ϱ = air density pstagnation = Stagnation pressure by Pitot tube, D = Drag force, b = airfoil span, c = airfoil chord u( y) 2 pstagnation( y) p yU D u ( y )U u ( y ) dy yL 2D CD U 2 bc yU 2 CD 2 ui U ui dy U c yL Velocity and Drag: Spheres Cd f , Re, M, shape, orientation General relationship for D submerged objects Spheres only have one shape and orientation! 2Fd Cd U 2 A 2Fd Cd f Re 2 U A Cd U 2 A Fd 2 Where Cd is a function of Re Sphere Terminal Fall Velocity p particle volume F ma Fb Fd Fb W 0 ρ p particle density Fd W ppg ρw water density g acceleration due to gravity C D drag coefficient Fb p w g Vt 2 Fd Cd AP w 2 4 p pr 3 3 Ap pr Ap particle cross sectional area Vt particle terminal velocity W 2 2Fd Cd U 2 A Sphere Terminal Fall Velocity (continued) Fd W Fb Vt 2 Cd AP w p ( p w ) g 2 Vt 2 2 p ( p w ) g Cd AP w p 2 d Ap 3 Relationship valid for spheres 4 gd p w Vt 3 Cd w 2 General equation for falling objects 4 gd p w Vt 3 Cd w Drag Coefficient on a Sphere Drag Coefficient 1000 100 Stokes Law 10 1 0.1 0.1 1 24 Cd Re 10 102 103 104 Reynolds Number 105 106 Re=500000 Turbulent Boundary Layer 107 Drag Coefficient for a Sphere: Terminal Velocity Equations 4 gd p w Vt 3 Cd w Valid for laminar and turbulent 24 Laminar flow R < 1 Cd Re Transitional flow 1 < R < 104 Fully turbulent flow R > Re Vt d 104 Cd 0.4 Vt d 2 g p w 18 gd p w Vt 0.3 w Example Calculation of Terminal Velocity Determine the terminal settling velocity of a cryptosporidium oocyst having a diameter of 4 m and a density of 1.04 g/cm3 in water at 15°C. ρ p 1040 kg/m 3 Vt ρw 999 kg/m 3 g 9.81 m/s 2 Vt d 4x10 6 m 1.14x10 3 kg sm 4x10 6 d 2 g p w 18 m 9.81 m/s 2 1040 kg/m 3 999 kg/m 3 3 kg 181.14x10 sm 2 Vt 3.14 x107 m/s Vt 2.7 cm/day Reynolds