09 external flows

7/28/2019 09 External Flows

1/50

Monroe L. Weber-ShirkSchool ofCivil and

Environmental Engineering

External Flows

CEE 331

May 30, 2013
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2/50

Overview

The Fss connection to Drag

Boundary Layer Concepts

DragShear Drag

Pressure Drag

Pressure Gradients: Separation and Wakes

Drag coefficients

Vortex Shedding


3/50

Fss: 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 forcespressure drag or form drag

caused by _____________from the body

function of area normal to the flow

surface area length

flow separation

UU

UU

Major losses in pipes

Flowexpansion

losses

Projected area


4/50

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 flowdisturbance)

In external flows the boundary layer (the flow

influenced by the solid object) is always growing

and the flow is non-uniform

We need to calculate shear in this non-uniform

flow!


5/50

Boundary Layer Concepts

Two flow regimes

Laminar boundary layer

Turbulent boundary layerwith laminar sub-layer

Calculations of

boundary layer thickness

Shear (as a function of location on the surface)

Viscous Drag (by integrating the shear over the entire

surface)


6/50

Flat Plate: Parallel to Flow

Ux

y

U U U

d

to

Why is shear maximum at the leading edge ofthe plate?

boundary

layer

thickness

shear

du

dyis maximum


7/50

Laminar Boundary Layer:

Shear and Drag Force

5

Rexx

d Re

x

Ux

Boundary Layer thickness increases with the _____________ of the distance from the leading edge of the plate

x

U3

0 332.0

t

ll

d dxx

UwdxwF

0

3

0

0 332.0

t

lUwFd3

664.0

5x

U

d

On one side of the plate!

Based on momentum and mass

conservation and assumed

velocity distribution

squareroot

Integrate along length of plate


8/50

Laminar Boundary Layer:

Coefficient of Drag

2

2FC (Re)dd f

U A lUwFd

3664.0

lwU

lUwd

2

3)664.0(2C

lwU

lUwd

2

3328.1C

Uld

328.1C

1.328C

Red

l

Rel

Ul

Dimensional analysis


9/50

Transition to Turbulence

The boundary layer becomes turbulent when

the Reynolds number is approximately

500,000 (based on length of the plate)

The length scale that really controls the

transition to turbulence is the

_________________________

5

Rexx

dRex

Ux

Re

Ud

d

Re

Rex x

d d Re 5 Rexd

boundary layer thickness

Red = 3500

=


10/50

Transition to Turbulence

Ux

y

U Ud

to

U

turbulent

Viscous

sublayer

This slope (du/dy) controls t0.

Transition (analogy to pipe flow)


11/50

more rapidly

Turbulent Boundary Layer:

(Smooth Plates)

1/ 5

0.37

Rexx

d

5/1

2

0 029.0

UxU

t

5/1

2

0

0 036.0

UlwlUdxwF

l

dt

1/5C 0.072Red l

22F

C Re,dd fU A l

RexUx

5/1

5/437.0

Ux

d

Derived from momentum conservation

and assumed velocity distribution

Integrate shear over plate

Grows ____________ than laminar

5 x 105 < Rel< 107

x 5/4


12/50

Boundary Layer Thickness

Water flows over a flat plate at 1 m/s. How long is the laminar region?

RexUx

RexxU

smsmxx/1

)000,500(/101

26

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 0.5 1 1.5 2

length along plate (m)

boundarylayerthickness(

m)

.

-

200,000

400,000

600,000

800,000

1,000,000

1,200,000

1,400,000

1,600,000

1,800,000

2,000,000

ReynoldsNumber

laminarturbulentReynolds Number

Grand Coulee

x = 0.5 m

5/1

5/437.0

Ux

d

5x

U

d


13/50

Flat Plate Drag Coefficients

0.001

0.01

1000

0

10000

0

1000

000

10000

000

1000

000

00

1000

000

000

1000

000

0000

Rel Ul

le

fdC

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 10-55 x 10-62 x 10-61 x 10-6

2.5

1.89 1.62log /fd

C l

0.5

1.328

Refd

l

C

2.58

0.455 1700

Relog Refd

ll

C

2.58

0.455

log Refd

l

C

0.2

0.072Refd lC

laminar

transitional

Turbulent boundary

rough


14/50

Example: Solar Car

Solar cars need to be as efficient as

possible. They also need a large surface

area for the (smooth) solar array. Estimatethe power required to counteract the

viscous drag on the solar panel at 40 mph

Dimensions: L: 5.9 m W: 2 m H: 1 m

Max. speed: 40 mph on solar power alone

Solar Array: 1200 W peak

air= 14.6 x10-6 m2/s air= 1.22 kg/m

3


15/50

Viscous Drag on Ships

The viscous drag on ships can be calculated

by assuming a flat plate with the wetted

area and length of the ship

f wavdd eF F F

AU

dd

2

F2C

RelUl

Lr3Fwave scales with ____ (based on _______ similarity)

2CF2f

dd

U A0.001

0.01

1000

0

1000

00

1000

000

1000

0000

1000

0000

0

1000

0000

00

1000

0000

000

Froude


16/50

Separation and Wakes

Separation often occurs at sharp corners

fluid cant 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

UU


17/50

Pressure Gradients: Separation

and Wakes

Van Dyke, M. 1982. An Album of Fluid Motion. Stanford:

Parabolic Press.

Diverging streamlines


18/50

Adverse Pressure Gradients

Increasing pressure in direction of flow

Fluid is being decelerated

Fluid in boundary layer has less ______

than the main flow and may be completely

stopped.

If boundary layer stops flowing then

separation occurs

inertia

Streamlines diverge behind object

2

2

p Vz C

g


19/50

Point of Separation

Predicting the point of separation on smooth

bodies is beyond the scope of this course.

Expect separation to occur wherestreamlines are diverging (flow is slowing

down)

Separation can be expected to occur aroundany sharp corners

(where streamlines diverge rapidly)


20/50

Flat Plate:

Streamlines

U

0 1

2

3

4

2

0

2

2

21U

ppUvCp

Point v Cp p

1 ______ ________ ____

2______ ________ ____3______ ________ ____

4______ ________ ____

0 Cp = 1

U Cp < 0

>p0

>p0


21/50

Application of Bernoulli

Equation

g

vp

g

Up

22

22

0

2

0

2

2

21U

pp

U

v

In air pressure change due to

elevation is small

U = velocity of body relative to fluid

2 2

1 1 2 21 2

2 2

p v p vz z

g g

0

22

22

pp

g

v

g

U

pC


22/50

Flat Plate:

Pressure Distribution

2

0

2

2

21U

pp

U

vCp

1 0 -1 -1.2

rearfront dddFFF

AppF rearfrontd

AUCCFrearfront ppd

2

2

Cp

2

02U

ppCp

0

2

2pp

CU p

0.8

AU

Fd2

2.18.0

2

Cd = 2

0

U

1

2

3

Front of plate

Back of plate


23/50

Bicycle page at Princeton

Drag Coefficient of Blunt and

Streamlined Bodies

Drag dominated by viscousdrag, the body is __________.

Drag dominated by pressure

drag, the body is _______. Whether the flow is viscous-

drag dominated or pressure-drag dominated dependsentirely on the shape of the

body.

This drag coefficient iscalculated from a measuredvalue of ____

streamlined

bluff

Flat plate

AU

dd

2

F2C

Fss
http://www.princeton.edu/~asmits/Bicycle_web/blunt.htmlhttp://www.princeton.edu/~asmits/Bicycle_web/blunt.html


24/50

Drag Coefficient at High

Reynolds Numbers

Figures 9.28-9.30 bodies with drag

coefficients on p 593-595 in text.

hemispherical shell 0.38

hemispherical shell 1.42

cube 1.1

parachute 1.4

Why?

Vs

?

Velocity at

separation point

determines pressurein wake.

The same!!!


25/50

SUVs have got Drag

Ford Explorer 2002 Cd = 0.41

2

2Cd

Drag

U A

2C

2

d U ADrag


26/50

Automobile Drag Coefficients

(High Reynolds Number)

Cd= 0.32

Height = 1.539 m

Width = 1.775 m

Length = 4.351 mGround clearance = 15 cm

100 kW at 6000 rpm

Max speed is 124 mph

Calculate the power required to overcome drag at 60 mph and 120 mph.

Where does separation occur?

What is the projected area? A H G W

21.539 0.15 1.775 2.5A m m m m


27/50

Electric Vehicles

Electric vehicles are designed to minimize drag.

Typical cars have a coefficient drag of 0.30-0.40.

The EV1 has a drag coefficient of 0.19.Smooth connection to windshield

Plan view of car?


28/50

Velocity and Drag: Spheres

C ,Re, , ,d f shape orientationD

M

2

2FC dd

U A

2

2FC Red d f

U A

2CF

2

dd

U A

Spheres only have one shape and orientation!

General relationship for

submerged objects

Where Cdis a function of Re


29/50

Sphere Terminal Fall Velocity

maF

2

2FC dd

U A

0 WFF bdgW pp

2

2t

d d P w VF C A

3

3

4rp

2rAp W

dF

bF

gF wpb

velocityterminalparticle

tcoefficiendrag

gravitytodueonaccelerati

densitywater

densityparticle

areasectionalcrossparticle

volumeparticle

t

D

w

p

p

p

V

C

g

A


30/50

Sphere Terminal Fall Velocity

(continued)

bd FWF

2

( )

2

td P w p p w

VC A g

22 ( )

p p w

t

d P w

gV

C A

dAp

p

32

2 4

3

p w

t

d w

gdV

C

4

3

p w

t

d w

gdV

C

General equation for falling objects

Relationship valid for spheres


31/50

Drag Coefficient on a Sphere

0.1

1

10

100

1000

0.1 1 10 102 103 104 105 106 107

Reynolds Number

DragCoefficient

Stokes Law

24

Re

dC Re=500000

Turbulent Boundary Layer


32/50

Drag Coefficient for a Sphere:

Terminal Velocity Equations

Laminar flow R < 1

Transitional flow 1 < R < 104

Fully turbulent flow R > 104

24

Red

C

Re tV d

18

2 wp

t

gdV

0.3

p w

t

w

gdV

0.4

dC

4

3

p w

t

d w

gdV

C

Valid for laminar and turbulent


33/50

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 15C.

ms

kg1.14x1018

kg/m999kg/m1040m/s189.m4x10

3

33226

tV

18

2wp

t

gdV

ms

kg1.14x10

m4x10

m/s189.

kg/m999

kg/m1040

3

6

2

3

3

d

g

w

p

m/s1014.37 xVt

cm/day7.2tV Reynolds


34/50

Drag on a Golf Ball

Drag on a golf ball comes mainly from pressuredrag. The only practical way of reducing

pressure drag is to design the ball so that thepoint of separation moves back further on the

ball. The golf ball's dimples increase the turbulence

in the boundary layer, increase the _______ ofthe boundary layer, and delay the onset ofseparation.

What is the Reynolds number where theboundary layer begins to become turbulent witha golf ball? _________

Why not use this for aircraft or cars?

inertia

40,000

Boundary layer is already turbulent


35/50

At what velocity is the boundary

layer laminar for an automobile?

RelUl

251.5 10air

mx

s

RelU l

4l m

Re 500,000l

2

51.5 10 500000

1.9 / 6.8 /4

mxs

U m s km hr m


36/50

Effect of Turbulence Levels on

Drag

Flow over a sphere with a trip wire.

Point of separation

Causes boundary layer to become turbulent

Re=15,000 Re=30,000


37/50

Effect of Boundary Layer

Transition

Ideal (non

viscous) fluid

Real (viscous)

fluid: laminar

boundary layer

Real (viscous)

fluid: turbulent

boundary layer

No shear!Increased inertia in

boundary layer


38/50

Spinning Spheres

What happens to the separation points ifwe start spinning the sphere?

LIFT!


39/50

Vortex Shedding

Vortices are shed alternatelyfrom each side of a cylinder

The separation point and thus theresultant drag force oscillates

Frequency of shedding (n) givenby Strouhal number S

S is approximately 0.2 over awide range of Reynolds numbers(100 - 1,000,000)

U

ndS


40/50

Summary: External Flows

Spatially varying flows

boundary layer growth

Example: Spillways

Two sources of drag (Fss)

shear (surface area of object)

pressure (projected area of object)

Separation and Wakes

Interaction of viscous drag and adverse pressure

gradient


41/50

Challenge

Im going on vacation and I cant back all

of our luggage in my Matrix. Should I put it

on the roof rack or on the hitch?

2C

2

d U ADrag


42/50

Challenges

How long would L have to be to double the

drag of a sphere?

L

V=30 m/s D = 3 m

2C

2

d U ADrag


43/50

Challenges

How long would L have to be

to double the drag of a sphere?

LV=30 m/s D = 3 m

2C

2

d U ADrag

Find drag of sphere

Guess at Re for plate

Find drag coefficient for plate(note different area)

Solve for L 0.001

0.01

100

00

100

000

1000

000

1000

0000

1000

00000

10000

0000

0

100

0000

0000

14.6106

m

2

s

Re 6.164 106

ReV D


44/50

Elongated sphere

LV=30 m/s D = 3 m

A2 Drag

C.dplate V2

Drag

L2 Drag

C.dplate V2

D

Drag

L2 C.dsphere V

2 D

2

8C.dplate V2 D

C.dsphere

LC.dsphere D

4C.dplate

C.dsphere

DragC.dsphere V

2 D

2

8

C.dsphere

C

.dsphere

0.

C.dplate 0.00

L 50 m


45/50

Solution: Solar Car

AU

dd

2

F2C

UlRx

2

CF

2AUdd

23 3 23 10 1.22 / 17.88 / 11.88

F 2 2d

x kg m m s m

U = 17.88 m/s

l= 5.9 m

air= 14.6 x 10-6 m2/s

Rel= 7.2 x 106

Cd= 3 x 10-3

air= 1.22 kg/m3

A = 5.9 m x 2 m = 11.8 m2Fd=14 N

P=F*U=250 W

2.58

0.455

log Refd

l

C

0.001

0.01

1000

0

1000

00

1000

000

1000

0000

1000

0000

0

1000

0000

00

1000

0000

000


46/50

Reynolds Number Check

R


47/50

Solution: Power a Toyota Matrix

at 60 or 120 mph

2

2FC (Re)dd f

U A

2CF 2

dd

U A

2

C 3AUP d

3 3 2(0.32)(1.2 / )(26.82 / ) (2.5 )

2

kg m m s mP

P = 9.3 kW at 60 mph

P = 74 kW at 120 mph


48/50

Grand Coulee Dam

Turbulent boundary layer reaches surface!


49/50

Reflections on Drag

What are 3 similarities with Moodydiagram?

Laminar

Smooth

Rough

Why 2 curves for smooth (red andgreen)

Fully turbulent boundary layer

Transition between laminar and turbulent onthe plate

Why more detail in transition regionhere than in Moody diagram?

Are any lines missing on the graph?

0.001

0.01

1000

0

1000

00

1000

000

1000

0000

1000

0000

0

1000

000

000

1000

0000

000

Function of conditions

at leading edge


50/50

Drexel SunDragon IV

Vehicle ID: SunDragon IV (# 76)Dimensions: L: 19.2 ft. (5.9 m) W: 6.6 ft. (2 m) H: 3.3 ft. (1 m)Weight: 550 lbs. (249 kg)Solar Array: 1200 W peak; 8 square meters terrestrial grade solar cells;

manf: ASE AmericasBatteries: 6.2 kW capacity lead-acid batteries; manf: US BatteryMotor: 10 hp (7.5 kW) brushless DC; manf: Unique MobilityRange: Approximately 200 miles (at 35 mph on batteries alone)Max. speed: 40 mph on solar power alone, 80 mph on solar and battery

power.

Chassis: Graphite monocoque (Carbon fiber, Kevlar, structural glass,Nomex)Wheels: Three 26 in (66 cm) mountain bike, custom hubsBrakes: Hydraulic disc brakes, regenerative braking (motor)

http://cbis.ece.drexel.edu/SunDragon/Cars.html

09 external flows

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