Pharos UniversityME 253 Fluid Mechanics II

Flow over bodies;Lift and Drag

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.

External External Flows

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 Dragshear stress and pressure integrated over body surfacedrag: force component in the direction of upstream velocitylift: force normal to upstream velocity

212

212

cos sin

sin cos

x w D

y w L

dF p dA dA CU A

dF p dA dA CU A

DD

LL

AIRFOIL NOMENCLATURE

Mean Chamber Line: Points halfway between upper and lower surfaces

Leading Edge: Forward point of mean chamber lineTrailing Edge: Most reward point of mean chamber lineChord Line: Straight line connecting the leading and trailing edgesChord, c: Distance along the chord line from leading to trailing edgeChamber: Maximum distance between mean chamber line

and chord line

AERODYNAMIC FORCERelative Wind: Direction of V∞

We used subscript ∞ to indicate far upstream conditionsAngle of Attack, Angle between relative wind (V∞) and chord line

Total aerodynamic force, R, can be resolved into two force componentsLift, L: Component of aerodynamic force perpendicular to relative windDrag, D: Component of aerodynamic force parallel to relative wind

Pressure Forces acting on the Airfoil

High PressureLow velocity

High PressureLow velocity

Low PressureHigh velocity

Low PressureHigh velocity

Bernoulli’s equation says where pressure is high, velocity will below and vice versa.

Relationship between L´ and p

EdgeTrailing

EdgeLeading

sideupper sidelower

EdgeTrailing

EdgeLeading

sideupper

EdgeTrailing

EdgeLeading

sidelower

pp

pp

sideupper on Force-sidelower on the acting Forces

direction wind the tonormal Force

dx

dxdx

L

V

Relationship between L´ and p(Continued)

EdgeTrailing

EdgeLeading

sideupper sidelower

EdgeTrailing

EdgeLeading

sideupper sidelower

pp

pp

dxpp

dxL

Divide left and right sides by cV 2

2

1

EdgeTrailing

EdgeLeading

upperlower

c

xd

V

pp

V

pp

cV

L

222

21

21

21

We get:

Pressure Coefficient CpFrom the previous slide,

EdgeTrailing

EdgeLeading

upperlower

c

xd

V

pp

V

pp

cV

L

222

21

21

21

The left side was previously defined as the sectional liftcoefficient Cl.

The pressure coefficient is defined as:

2

21

V

ppC p

Thus, edgeTrailing

edgeLeading

upperplowerpl c

xdCCC ,,

Fluid dynamic forces are due to pressure and viscous forces.

Drag: component parallel to flow direction.

Lift: component normal to flow direction.

Drag and LiftLift and drag forces can be found by integrating pressure and wall-shear stress.

Drag and LiftLift 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

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 GeometriesFor 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

Laminar:

Turbulent:

Cylinder and Sphere Drag

Cylinder and Sphere DragFlow 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.

Angle of Attack, in degrees or radians

Cl

Slope= 2 if is in radians.

= 0

Angle ofzero lift

Stall

30

EXAMPLE: AIRFOIL STALLLi

ft

Angle of Attack,

Effect of Angle of Attack

CL≈2 for < stall

Lift increases linearly with Objective:Maximum CL/CD

CL/CD increases until stall.

Effect of Foil ShapeThickness and camber affects pressure distribution andlocation of flow separation.

End Effects of Wing TipsTip 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

Supercritical flowturbulent B.L.

Stokes’ Flow, Re<1

Relatively constant CD

Drag

Drag Coefficient

with

or

DRAG FORCE

Friction has two effects:Skin friction due to shear stress at wallPressure drag due to flow separation

pressurefriction DDD Total drag due toviscous effectsCalled Profile Drag

Drag due toskin friction

Drag due toseparation= +

Less for laminarMore for turbulent

More for laminarLess for turbulent

38

COMPARISON OF DRAG FORCES

d

d

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

bluffFlat plate

AU

dd

2

F2C

Drag

Pure Friction Drag: Flat Plate Parallel to the FlowPure Pressure Drag: Flat Plate Perpendicular to the FlowFriction and Pressure Drag: Flow over a Sphere and CylinderStreamlining

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.

DragFlow over a Sphere and Cylinder: Friction and Pressure Drag (Continued)

StreamliningUsed 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 airfoilPressure 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 = LiftForces parallel to free stream = Drag Top of Airfoil:- The velocity of the flow is greater than the free-stream.- The pressure is negativeUnderside 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 forceThe 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

cU

dspp

C sL

2

21

sin

2

21

U

ppC i

p

ppU stagnation2

bcU

LCL 2

2

dsppLs

sin

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 forceThe drag force D = integration

of the momentum lossThe 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 dyuUu

cUC i

y

y

iD

U

L

2

2

pypyu stagnation )(2)(

bcU

DCD 2

2

dyyuUyuDU

L

y

y

)()(

Velocity and Drag: Spheres

C ,Re, , ,d f shape orientationD

M

2

2FC d

d U A

2

2FC Red

d fU A

2C

F2

dd

U A

Spheres only have one shape and orientation!

General relationship for submerged objects

Where Cd is a function of Re

Sphere Terminal Fall Velocity

maF

2

2FC d

d U A

0 WFF bd

gW pp

2

2t

d d P w

VF C A

3

3

4rp 2rAp

W

dF

bF

gF wpb

velocity terminalparticle

tcoefficien drag

gravity todueon accelerati

densitywater

density particle

area sectional cross particle

volumeparticle

t

D

w

p

p

p

V

C

g

ρ

ρ

A

Sphere Terminal Fall Velocity (continued)

bd FWF

2

( )2t

d P w p p w

VC A g

2 2 ( ) p p w

td P w

gV

C A

dAp

p

3

2

2 4

3p w

td w

gdV

C

4

3

p w

td w

gdV

C

General equation for falling objects

Relationship valid for spheres

Drag Coefficient on a Sphere

0.1

1

10

100

1000

0.1 1 10 102 103 104 105 106 107

Reynolds Number

Dra

g Co

effici

ent

Stokes Law

24

RedC Re=500000

Turbulent Boundary Layer

Drag Coefficient for a Sphere:Terminal Velocity Equations

Laminar flow R < 1

Transitional flow 1 < R < 104

Fully turbulent flow R > 104

24

RedC

Re tV d

18

2wp

t

gdV

0.3p w

tw

gdV

0.4dC

4

3p w

td w

gdV

C

Valid for laminar and turbulent

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.

ms

kg1.14x1018

kg/m 999kg/m 1040m/s 189.m 4x10

3

33226

tV

18

2wp

t

gdV

ms

kg1.14x10

m 4x10

m/s 189.

kg/m 999

kg/m 1040

3

6

2

3

3

d

g

ρ

ρ

w

p

m/s1014.3 7 xVt

cm/day 7.2 tVReynolds