External FlowsExternal Flows

OverviewOverview

Non-Uniform Flow Boundary Layer Concepts Viscous Drag Pressure Gradients: Separation and Wakes Pressure Drag Shear and Pressure Forces Vortex Shedding

Non-Uniform Flow Boundary Layer Concepts Viscous Drag Pressure Gradients: Separation and Wakes Pressure Drag Shear and Pressure Forces Vortex Shedding

Non-Uniform FlowNon-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

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

Boundary Layer ConceptsBoundary 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)

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 FlowFlat Plate: Parallel to Flow

Ux

y

U U U

Why is shear maximum at the leading edge of the plate?

boundary layer thickness

shear

dudy

is maximum

Flat Plate Drag CoefficientsFlat Plate Drag Coefficients

0.001

0.01

1e+04

1e+05

1e+06

1e+07

1e+08

1e+09

1e+10

RelUln

=RelUln

=

lele

DfCDfC

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

( )[ ] 2.51.89 1.62log /DfC le

-= - ( )[ ] 2.5

1.89 1.62log /DfC le-

= -

( )0.5

1.328

ReDf

l

C =( )0.5

1.328

ReDf

l

C =( )[ ]2.58

0.455 1700Relog Re

Dfll

C = -( )[ ]2.58

0.455 1700Relog Re

Dfll

C = -

( )[ ]2.58

0.455

log ReDf

l

C =( )[ ]2.58

0.455

log ReDf

l

C =

0.20.072ReDf lC -= 0.20.072ReDf lC -=

Separation and WakesSeparation 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)

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

Flat Plate:StreamlinesFlat Plate:

Streamlines

U

0 1

2

3

4

2

0

2

2

21U

pp

U

vC p

2

0

2

2

21U

pp

U

vC p

Point v Cp p1234

0 1<U >0>U <0

>p0

>p0

<p0

<p0

Points outside boundary layer!

Bicycle page at Princeton

Drag of Blunt Bodies and Streamlined Bodies

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.

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

Drag Coefficient on a Sphere Drag Coefficient on a Sphere

0.10.1

11

1010

100100

10001000

0.10.1 11 1010 102102 103103 104104 105105 106106 107107

Reynolds NumberReynolds Number

Dra

g C

oeff

icie

ntD

rag

Coe

ffic

ient Stokes Law

24ReDC =24ReDC =

Shear and Pressure Forces: Horizontal and Vertical Components

Shear and Pressure Forces: Horizontal and Vertical Components

dApD cossinF 0 dApD cossinF 0

dApL sincosF 0 dApL sincosF 0

lift

drag

U

Parallel to the approach velocity

Normal to the approach velocity

2

2UACF dd

2

2UACF dd

2

2UACF LL

2

2UACF LL

A defined as projected area _______ to force!normalnormal

drag

liftp < p0

negative pressure

p < p0

negative pressure

p > p0 positive pressurep > p0 positive pressure

Shear and Pressure ForcesShear 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

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

Example: Beetle PowerExample: Beetle Power

Cd = 0.38Height = 1.511 mWidth = 1.724 mLength = 4.089 mGround clearance = 15 cm?85 kW at 5200 rpm

Calculate the power required to overcome drag at 60 mph and 120 mph.Is the new beetle streamlined?

Where does separation occur?

Solution: Power a New Beetle at 60 mph

Solution: Power a New Beetle at 60 mph

)(F2

C2

RfAU

Dd

)(

F2C

2Rf

AU

Dd

2

CF

2AUdD

2

CF

2AUdD

2

C 3AUP d

2

C 3AUP d

2

)0.2()/82.26)(/2.1)(38.0( 233 msmmkgP

2

)0.2()/82.26)(/2.1)(38.0( 233 msmmkgP

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

Drag on a Golf BallDrag 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.

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.

inertiainertia

Why not use this for aircraft or cars?

Effect of Turbulence Levels on Drag

Effect of Turbulence Levels on Drag

Flow over a sphere: (a) Reynolds number = 15,000; (b) Reynolds number = 30,000, with trip wire.

Flow over a sphere: (a) Reynolds number = 15,000; (b) Reynolds number = 30,000, with trip wire.

Point of separationPoint of separation

Causes boundary layer to become turbulentCauses boundary layer to become turbulent

Effect of Boundary Layer Transition

Effect of Boundary Layer Transition

Ideal (non viscous) fluid

Real (viscous) fluid: laminar

boundary layer

Real (viscous) fluid: turbulent boundary layer

No shear!No shear!

Spinning SpheresSpinning Spheres

What happens to the separation points if we start spinning the sphere?

What happens to the separation points if we start spinning the sphere?

LIFT!LIFT!

Vortex SheddingVortex 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)

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)

U

ndS

U

ndS