Thermodynamically coupled heat and mass flows in a reaction-transport system with external
09 External Flows
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Monroe L. Weber-ShirkSchool ofCivil and
Environmental Engineering
External Flows
CEE 331
May 30, 2013
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Overview
The Fss connection to Drag
Boundary Layer Concepts
DragShear Drag
Pressure Drag
Pressure Gradients: Separation and Wakes
Drag coefficients
Vortex Shedding
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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
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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!
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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)
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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
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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
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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
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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
=
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Transition to Turbulence
Ux
y
U Ud
to
U
turbulent
Viscous
sublayer
This slope (du/dy) controls t0.
Transition (analogy to pipe flow)
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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
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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
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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
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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
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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
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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
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Pressure Gradients: Separation
and Wakes
Van Dyke, M. 1982. An Album of Fluid Motion. Stanford:
Parabolic Press.
Diverging streamlines
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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
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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)
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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
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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
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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
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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 -
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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!!!
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SUVs have got Drag
Ford Explorer 2002 Cd = 0.41
2
2Cd
Drag
U A
2C
2
d U ADrag
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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
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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?
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Spinning Spheres
What happens to the separation points ifwe start spinning the sphere?
LIFT!
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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
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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
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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
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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
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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
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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
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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
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Reynolds Number Check
R
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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
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Grand Coulee Dam
Turbulent boundary layer reaches surface!
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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
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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