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

Lower surface (underside of wing): high pressure 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

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 High Pressure Low velocity Bernoulli’s equation says where pressure is high, velocity will be low and vice versa.

Relationship between L´ and p V

Relationship between L´ and p (Continued) Divide left and right sides by We get:

Pressure Coefficient Cp From the previous slide, The left side was previously defined as the sectional lift coefficient Cl. The pressure coefficient is defined as: Thus,

Drag: component parallel to flow direction. 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 integrating pressure and wall-shear stress. Drag and Lift

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=1/2V2(CDA) will be ~ 10 times larger for Scion XB Source is large CD and large projected area Power consumption P = FDV =1/2V3(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 Fluid dynamic forces: pressure and friction effects. FD = FD,friction + FD,pressure CD = CD,friction + CD,pressure Friction drag 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 of Common Geometries For many shapes, total drag CD is constant for Re > 104

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

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

EXAMPLE: AIRFOIL STALL Lift Angle of Attack, a

Effect of Angle of Attack CL≈2 for  < stall Lift increases linearly with  Objective:Maximum CL/CD CL/CD increases until stall.

Thickness and camber affects pressure distribution and location of flow separation. Effect of Foil Shape

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

Friction has two effects: DRAG FORCE Friction has two effects: Skin friction due to shear stress at wall Pressure drag due to flow separation Total drag due to viscous effects Called Profile Drag Drag due to skin friction Drag due to separation = + Less for laminar More for turbulent More for laminar Less for turbulent

COMPARISON OF DRAG FORCES Same total drag as airfoil

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, the body is __________. Drag dominated by pressure drag, the body is _______. streamlined bluff Flat plate

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

Airfoil Temp. sensor Pitot tubes Pressure sensors Data acquisition 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

measured pressure distribution over the Airfoil’s surface. 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

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 The drag force D = integration of the momentum loss 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

Velocity and Drag: Spheres General relationship for submerged objects Spheres only have one shape and orientation! Where Cd is a function of Re

Sphere Terminal Fall Velocity

Sphere Terminal Fall Velocity (continued) General equation for falling objects Relationship valid for spheres

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 Re=500000 Reynolds Number Turbulent Boundary Layer

Drag Coefficient for a Sphere: Terminal Velocity Equations Valid for laminar and turbulent Laminar flow R < 1 Transitional flow 1 < R < 104 Fully turbulent flow R > 104

Example Calculation of Terminal Velocity Determine the terminal settling velocity of a cryptosporidium oocyst having a diameter of 4 mm and a density of 1.04 g/cm3 in water at 15°C. Reynolds