Vehicle aerodynamics

6. - external flows ; flows around the object which is completely
surrounded by the fluid
- aerodynamics ; External flows involving air are often termed
aerodynamics in response to the important external flows when
an object such as an airplane flies through the atmosphere.
- ship
* surface vessels ; surrounded by two fluids, air and water
* submersible vessels ; surrounded completely by water
-Marked Difference in the Results Sought in Analysis of External
Flows and Internal Flows
1. External Flows
One seeks ;
1) the flow pattern around an object immersed in the fluid
CH. 9 Flow over Immersed Bodies

7. 2) the lift(揚力) and drag(抗力) on the object
3) the pattern of viscous action in the fluid as it passes around a
body
4) In this case, energy or work is used typically to move the object
through the fluid.
2. Internal Flows
One seeks ;
1) the flow pattern in an object through which the fluid flows
2) The focus is often not on lift and drag, but on energy or head
losses, pressure drops, and cavitation where energy is dissipated,
because in internal flow, energy or work is used to move fluid
through passages.
3) the pattern of viscous action in the fluid as it passes in a body

8. Classification of Body Shape
* 3 general categories of bodies (see p553 Fig. 9.2)
1) 2-dimensional objects (infinitely long and of constant cross-
sectional size and shape)
2) axisymmetric bodies (formed by rotating their cross-sectional
shape about the axis of symmetry)
3) 3-dimensional bodies that may or may not possess a line or
plane of symmetry
* Streamlined Bodies (airfoils, racing cars) and
Blunt Bodies (parachutes, buildings)
9.1 General External Flow Characteristics

10. 1. Forces from the Surrounding Fluid on a Two-Dimensional
Object (see p554 Fig. 9.3)
- A body interacts with the surrounding fluid through pressure
and shear stresses.
- Drag and Lift
* drag ; the resultant force in the direction of the upstream
velocity
* lift ; the resultant force normal to the upstream velocity
2. Lift and Drag from Pressure and Shear Stress Distribution (Fig.
9.4)
- x and y components of the fluid force on the small element dA


 sin
)
(
cos
)
( dA
pdA
dF w
x 

9.1.1 Lift and Drag Concepts


 cos
)
(
sin
)
( dA
pdA
dF w
y 



11. net x and y components of the force on the object :
3. Lift Coefficient and Drag Coefficient
where ; characteristic area of the object
4. Frontal Area and Planform Area
- frontal area ; the projected area seen by a person looking
toward the object from a direction parallel to the upstream velocity.
cos sin (9.1)
x w
D dF p dA dA
  
  
  
sin cos (9.2)
y w
L dF p dA dA
  
   
  
2 2
1/ 2 1/ 2
L D
L D
C C
U A U A
 
 
A

12. - planform area ; the projected area seen by an observer
looking toward the object from a direction normal to the
upstream velocity

15. Example 9.1
<Sol>
?
?, 
 D
L
surface
bottom
the
on
surface
top
the
on
a 


 270
,
90
) 

o
pdA
pdA
L
bottom
top



 



 


top
w
bottom
w
top
w dA
dA
dA
D 

 2

16. lb
dx
x
ft
o
x
0992
.
0
)
10
(
10
24
.
1
2
4
2
/
1
3








 
 

back
the
on
front
the
on
b 


 180
,
0
) 

o
dA
dA
L
back
w
front
w 

 
 


 

back
front
pdA
pdA
D
 
  lb
dy
y
y
6
.
55
)
10
(
)
0893
.
0
(
4
/
1
774
.
0
2
2
2




  


17. 1. Character of the Flow Field
= function of the shape of the body
2. Important Dimensionless Parameters for Typical External Flows
* Reynolds number
* Mach number
* Froude number (for flows with a free surface)
3. External Flow (Lift and Drag) - Reynolds number
1] Value of Reynolds number
- in the absence of all viscous effects (=inviscid flow);
- in the absence of all inertial effects ( or )
- actual flow :
2] Rule of Thumb
- flow dominated by inertial effects :


 /
/
Re Ul
Ul 

c
U
Ma /

gl
U /


Re
o

 o

Re


 Re
o
100
Re 
9.1.2 Characteristics of Flow past an Object

18. - flow dominated by viscous effects :
Hence, most familiar external flows are dominated by inertia.
4. Flows past a Flat Plate (Streamlined Body) of Length
with (low, moderate, and large
Reynolds number) (see p559 Fig. 9.5)
- boundary layer : a thin region on the surface of a body in
which viscous effects are very important and outside of which
the fluid behaves essentially as if it were inviscid.
- Clearly the actual fluid viscosity is the same throughout; only
the relative importance of the viscous effects (due to the
velocity gradients) is different within or outside of the boundary
layer.
5. Flow past a Circular Cylinder (Blunt Object or Body) (see Fig.
9.6)
1
Re 
l
7
10
,
10
,
1
.
0
/
Re 
 
Ul

19. -In general, the larger the Reynolds number, the smaller the
region of the flow field in which viscous effects are important.
-Flow Separation
- Most familiar flows are similar to the large Reynolds number
flows depicted in Figs. 9.5c and 9.6c.

24. Example 9.2
) 20 / , , model; 34 , 100 , 40
a U mm s flow of glycerin h mm l mm b mm
   
) 20 / , , model; 34 , 100 , 40
b U mm s flow of air h mm l mm b mm
   
) 25 / , , model; 1.7 , 5 , 2
c U m s flow of air h m l m b m
   
Re / , Re / , Re /
h b l
Uh Ub Ul
  
  
5 2 3 2
1.46 10 / , 1.19 10 /
air glycerin
m s m s
 
 
   
various characteristics of flow past a car
<Sol.>

25. 9.2 Boundary Layer Characteristics
- treatment of flow past an object = combination of viscous flow
in the boundary layer and inviscid flow elsewhere.
- a necessary condition for this section is that the Reynolds
number be large.
- viscous, incompressible flow
1.Characteristic Length
- for a finite length plate : plate length ,
- for an infinitely long plate : x coordinate distance along the
plate from the leading edge ,
2. Flow Characteristics
l 
/
Re Ul


/
Re Ux

9.2.1. Boundary Layer Structure and Thickness on a Flat Plate

26. - The presence of the plate is felt only in the relatively thin
boundary layer and wake region.
- flow outside the boundary layer : irrotational flow
flow within the boundary layer : rotational flow
-The transition from laminar to turbulent flow occurs at critical
value of the Reynolds number, , on the order of 2x105 to
3X106, depending on the roughness of the surface and the
amount of turbulence in the upstream flow.
3. Boundary Layer Thickness
- velocity field in boundary layer on flat plate
- We define the boundary layer thickness as that distance from
the plate at which the fluid velocity is within some arbitrary value
of the upstream velocity. Typically, as indicated in Fig. 9.8a,
xcr
Re

)
,
( y
x
u




 y
at
i
U
V
and
o
y
at
o
V ˆ



27. 4. Boundary Layer Displacement Thickness (see Fig. 9.8)
- definition
-The displacement thickness represents the amount that the
thickness of the body must be increased so that the fictitious
uniform inviscid flow has the same mass flow rate properties as the
actual viscous flow.
5. Boundary Layer Momentum Thickness (see Fig.9.8)
-Definition
U
u
where
y 99
.
0



*

 




o
dy
b
u
U
U
b
*
 *
1 (9.3)
o
u
dy
U

  
 
 
 


 dy
u
U
u
b
dA
u
U
u
o
 



 
 )
(  dy
u
U
u
b
U
b
o



 

 2
1 (9.4)
o
u u
dy
U U

  
 
 
 


31. Paul Richard Heinrich Blasius (1883-1970) One of Prandtl's
students who provided an analytical solution to the boundary layer
equations. Also, demonstrated that pipe resistance was related to
the Reynolds number
1. Navier-Stokes eq. (Eqs. 6.127 a, b, c) for steady, two-
dimensional laminar flows with negligible gravitational effects :
1] Equations : (9.5), (9.6), (9.7)
2 2
2 2
2 2
2 2
1
(9.5)
1
(9.6)
(9.7)
u u p u u
u v
x y x x y
v v p v v
u v
x y y x y
u v
o
x y




 
    
    
 
    
 
 
    
    
 
    
 
 
 
 
9.2.2 Prandtl/Blasius Boundary Layer Solution

32. 2 2 2
2 2 2
(6.127 )
x
u u u u p u u u
u v w g a
t x y z x x y z
  
 
 
       
        
 
 
       
   
2 2 2
2 2 2
(6.127 )
y
v v v v p v v v
u v w g b
t x y z y x y z
  
 
 
       
        
 
 
       
   
2 2 2
2 2 2
(6.127 )
z
w w w w p w w w
u v w g c
t x y z z x y z
  
 
 
       
        
 
 
       
   
2 2
2 2
2 2
2 2
1
(9.5)
1
(9.6)
(9.7)
u u p u u
u v
x y x x y
v v p v v
u v
x y y x y
u v
o
x y




 
    
    
 
    
 
 
    
    
 
    
 
 
 
 
Navier-Stokes Eq.  Boundary Layer Eq.

33. 2] Appropriate Boundary Conditions
- far-field condition : The fluid velocity far from the body is the
upstream velocity.
- no-slip condition : The fluid sticks to the solid body surface.
2. Boundary Layer Flow Equation for a Flat Plate parallel to the
Flow
1] Assumptions
: Physically, the flow is primarily parallel to the plate and any fluid
property is convected downstream much more quickly than it is
diffused across the stream.
2]
y
x
and
u
v






2
2
(9.8) (9.9)
u v u u u
o u v
x y x y y

    
   
    
2 2
2 2
1
(9.5)
u u p u u
u v
x y x x y


 
    
     
 
    
 

34. 3] Boundary Conditions
and
3. Solution for a Flat Plate parallel to the Flow : Blasius Solution
-
-
As postulated, the boundary layer is thin provided that
Is large. (i.e., )
-
Note that the shear stress decreases with increasing x because
of the increasing thickness of the boundary layer-the velocity
gradient at the wall decreases with increasing x. Also, varies as
, not as as it does for fully developed laminar pipe flow.
o
y
on
o
v
u 

 

 y
as
U
u
5 (9.15)
x
U

 
5
Rex
x


*
1.721
(9.16)
Rex
x


0.664
(9.17)
Rex
x


x
Re


 x
as
o
x Re
/

3/2
0.332 / (9.18)
w U x
 

w

2
/
3
U U

35. 1. Drag
- derivation ; see p570-572
- Result
where
; width of plate
- Equation (9.22) points out the important fact that boundary
layer on a flat plate is governed by a balance between shear
drag and a decrease in the momentum of the fluid.
2. Momentum Integral Equation for the Boundary Layer on a Flat
Plate
  2
(9.22, 23)
o
D b u U u dy bU

  
  

 
2
/ (9.26)
w U d dx
  

9.2.3 Momentum-Integral Boundary Layer Equation
for a Flat Plate
b

36. -The usefulness of this equation lies in the ability to obtain
approximate boundary layer results easily by using rather crude
assumptions.
- example ; see p576 Table 9.2

37. Theodore von Karman
http://www.ceemast.csupomona.edu/nova/vonK.html
http://www.arnold.af.mil/aedc/highmach/stories/vonkarman.htm

38. Born: 11 May 1881 in Budapest, Hungary
Died: 6 May 1963 in Aachen, Germany
Theodore Von Kármán was a mathematical prodigy(경이) and his father,
fearing that his son would become a freak, steered him towards
engineering. He graduated in 1902 from Budapest and from 1903 to
1906 he worked at the Technical University of Budapest.
He left Budapest to study at Göttingen, where he was greatly
influenced by Klein, and Paris where he watched some pioneering
aviation flights which turned his interest to apply mathematics to
aeronautics. In 1911 he made an analysis of the alternating double row
of vortices behind a flat body in a fluid flow which is now known as
Kármán's Vortex Street.
The following year Kármán accepted a post as director of the
Aeronautical Institute at Aachen in Germany. He visited the USA in 1926
and four years later he was offered the post of director of the
Aeronautical Laboratory at California Institute of Technology. Despite his
love for Aachen, the political events in Germany persuaded him to
accept.
In 1933 he founded the U.S. Institute of Aeronautical Sciences where
he continued his research on fluid mechanics, turbulence theory and
supersonic flight. He studied applications of mathematics to engineering,
aircraft structures and soil erosion.

41. 1. Value of the Reynolds number at the Transition Location =
function (roughness of the surface, curvature of surface, measure
of disturbances etc)
2. Critical Reynolds Number =
in our calculation
3. Effect of Small Disturbance
- If these disturbances occur at a location with :
they will die out, and the boundary layer will return to laminar flow
at that location.
- If these disturbances occur at a location with :
they will grow and transform the boundary layer flow downstream
of this location into turbulence.
4. Typical Boundary Layer Profiles on a Flat Plate for Laminar,
Transitional, and Turbulent Flow (see p578, Fig. 9.14)
6
5
10
3
Re
10
2 


 xcr
5
10
5
Re 

xcr
xcr
x Re
Re 
xcr
x Re
Re 
9.2.4 Transition from Laminar to Turbulent Flow

43. Example 9.5
, 10 / , ?, ?:
cr cr
flat plate U ft s x 
  
) 60 , ) standard , ) 68
a water at F b air c glycerin at F
 
.
Sol
 
5
5 4
Re 5 10
: Re 5 10 5 10
10
xcr
cr xcr cr
x x
U
 


      
1/2
4
5 5 5 (5 10 ) 354
10
cr
cr
x
x
U U

 
   
 
     
 
 

44. 1. Friction Drag Coefficient for Flat Plate parallel to the Upstream
Flow (see p583 Fig. 9.15)
; This drag coefficient diagram shares many characteristics in
common with the familiar Moody diagram (pipe flow) of Fig.
8.23, even though the mechanisms governing the flow are
quite different.
2. Empirical Equations for the Flat Plate Drag Coefficient (see
p584, Table 9.3)
9.2.5 Turbulent Boundary Layer Flow

47. Pressure Gradient within Thin Boundary Layer for Curved Body
- : The pressure does vary in the direction along the
body surface if the body is curved. (=streamwise direction)
- : The pressure does not vary in the direction from
the body surface to the boundary layer edge.(=normal to the
streamwise direction)
- The variation in the free-stream velocity, , the fluid velocity
at the edge of the boundary layer, is the cause of the pressure
gradient in the boundary layer.
2. Upstream Velocity , Free Stream Velocity
- upstream velocity : fluid velocity far ahead of the plate
free-stream velocity : fluid velocity at the edge of the boundary
layer
o
s
p 

 /
o
n
p 

 /
fs
U
U fs
U
9.2.6 Effects of Pressure Gradient

48. - : for body of negligible thickness (e.g., flat plate )
: for bodies of nonzero thickness (e.g., cylinder)
3. Inviscid Flow past a Circular Cylinder ; see p586 Fig. 9.16
4. D’Alembert Paradox
- statement : The drag on an object in an inviscid fluid is zero,
but the drag on an object in a fluid with vanishingly small (but
nonzero) viscosity is not zero.
- reason for the d’Alembert paradox : see p 587
5. Favorable Pressure Gradient and Adverse Pressure Gradient
- favorable pressure gradient :
decrease in pressure in the direction of flow
- adverse pressure gradient :
increase in pressure in the direction of flow
6. Boundary Layer Separation (see p588 Fig. 9.17)
fs
U
U 
fs
U
U 
o
s
p 

 /
o
s
p 

 /

51. Jean Le Rond d'Alembert
Born: 17 Nov 1717 in Paris, France
Died: 29 Oct 1783 in Paris, France
http://www-groups.dcs.st-
andrews.ac.uk/~history/Mathematicians/D'Alembert.html

52. Frenchman d’Alembert was abondoned by his parents as a baby and
lived with his adopted parents as a child. He attended the Collège de
Quatre-Nations to study the classics, law, and medicine. Later he
studied mathematics on his own. He appeared on the scientific scene in
1739, when he sent his first communication to the Académie des
sciences. During the next two years he sent the academy five more
papers dealing with methods of integrating differential equations and
with the motion of bodies in a resisting media. Although he had
received little formal scientific education, it is clear that he had become
familiar with the works of Newton, L’Hospital, and the Bernoullis.
D’Alembert continued to produce advanced research and published
many works on mathematics and mathematical physics. His major work
was Traité de dynamique (1743). D'Alembert's work made partial
differential equations a part of calculus. He considered the derivative as
a limit of difference quotients, which put him ahead of his peers in
understanding calculus. He also contributed major results in geometry,
complex numbers, and probability.
Major publication: Traité de dynamique
Quotation:
"Algebra is so generous, she often gives more than is asked of her."

53. - pressure distribution ; see Fig. 9.17 ©
-The location of separation, the width of the wake region behind
the object, and the pressure distribution on the surface depend
on the nature of the boundary layer flow, I.e., whether laminar
flow or turbulent flow.
7. Flow past an Airfoil (see p589 Fig. 9.18)
8. Streamlined Body :
Streamlined bodies are generally those designed to eliminate
(or at least to reduce) the effects of separation, whereas
nonstreamlined bodies generally have relatively large drag due to
the low pressure in the separated regions (the wake).

54. 9.3 Drag
Character of as a Function of Dimensionless Parameters
1. Definition :
Friction drag, , is that part of the drag that is due directly
to the shear stress, , on the object.
It is a function of not only the magnitude of the wall shear
stress, but also of the orientation of the surface on which it acts.
2. Portion of Friction Drag to the Overall Drag
- Because the Reynolds number of most familiar flows is quite
2
( , Re, , , / ) /(1/ 2 ) (9.36)
D
C shape Ma Fr l D U A
  
 
D
C
f
D
w

9.3.1 Friction Drag
sin
f w
D dA
 
 

55. large, the percent of the drag caused directly by the shear stress
is often quite small.
- For highly streamlined bodies or for low Reynolds number flow,
however, most of the drag may be due to friction drag
3. Friction Drag on a Flat Plate of width and length oriented
parallel to the upstream flow
where
(see p583 Fig. 9.15 and p584 Table 9.3)
- The drag coefficient is not a function of the plate roughness if
the flow is laminar. However, for turbulent flow the roughness
does considerably affect the value of .
b l
Df
f blC
U
D 2
2
/
1 

(Re , / )
Df Df l
C C l friction drag coefficient

 
Df
C

56. 1. Definition
Pressure drag is that part of the drag that is due directly to the
pressure on an object. It is often referred to as form drag
because of its strong dependency on the shape or form of the
object.
2. Pressure Drag and Pressure Drag Coefficient
Here :
pressure coefficient :
p
D
Dp
C

 dA
p
Dp 
cos
2 2
cos cos
(9.37)
1/ 2 1/ 2
p
p
Dp
p dA C dA
D
C
U A U A A
 
 
  
 
)
2
/
/(
)
( 2
U
p
p
C o
p 


9.3.2 Pressure Drag

57. 3. Reynolds Number Dependency (see p594)
- Large Reynolds Number Flow :
- Very Small Reynolds Number Flow :
- Nonviscous Flow ( ) :
(Re)
p
p C
C 
Re
/
1

p
C
o
CDp 
o



58. 1./ Shape Dependence
- Clearly the drag coefficient for an object depends on the
shape of the object, with shapes ranging from those that are
streamlined to those that are blunt.
- For extremely thin bodies (e.g., a flat plate, or very thin airfoils)
it is customary to use the planform area in defining the drag
coefficient.
- Drag coefficient for an ellipse : see p597, Fig. 9.19
- Effect of the amount of streamlining on the drag : see p597 Fig.
9.20
2./ Reynolds Number Dependence
1. Main categories of Reynolds number dependence :
1) very low Reynolds number flow
2) moderate Reynolds number flow
9.3.3 Drag Coefficient Data and Examples

59. 3) very large Reynolds number flow
2. Low Reynolds number flows :
- Low Reynolds number flow are governed by a balance between
viscous and pressure forces. Inertia effects are negligibly small.
-
- Low Reynolds number drag coefficients (see p598 Table 9.4)
3. Moderate Reynolds number flows
- Moderate Reynolds number flows tend to take on a boundary
layer flow structure.
- For such flows past streamlined bodies, the drag coefficient
tends to decreases slightly with Reynolds number.
-Moderate Reynolds number flows past blunt bodies generally
produce drag coefficients that are relatively constant.
4. Drag Coefficient as a Function of Reynolds Number for a
Smooth Circular Cylinder and a Smooth Sphere (see p600 Fig. 9.21)
1
Re 
Re
/
2C
CD 

60. 5. Typical Flow Patterns for Flow past a Circular Cylinder at
Various Reynolds Numbers (see p601 Fig. 9.21)
6. Character of the Drag Coefficient as a Function of Reynolds
Number for Objects with Various Degrees of Streamlining (see
p601 Fig. 9.22)
3./ Compressibility Effects
- The introduction of Mach number effects complicates matters
because the drag coefficient for a given object is then a function
of Reynolds number and Mach number. The Mach number and
Reynolds number effects are often closely connected because
both are directly proportional to the upstream velocity. (see p604
Fig. 9.23)
- For low Mach numbers, the drag coefficient is essentially
independent of Ma. For this situation, if Ma<0.5 or so,
compressibility effects are unimportant. On the other hand, for
larger Mach number flows, the drag coefficient can be strongly
dependent on Ma, with only secondary Reynolds number effects.

61. (see p604 Fig. 9.24)
4./ Surface Roughness
- The drag on a flat plate parallel to the flow is quite dependent
on the surface roughness, provided the boundary layer flow is
turbulent.
- For streamlined bodies, the drag increases with increasing
surface roughness. For blunt bodies like a circular cylinder or
sphere, an increase in surface roughness can actually cause a
decrease in drag. (see p605 Fig. 9.25)
5./ Composite Body Drag
- Drag of Car : p610 Fig. 9.27
- Typical drag coefficients for regular two-dimensional objects
(p611, Fig. 9.28)
- Typical drag coefficients for regular three-dimensional objects
(p612, Fig. 9.29)
- Typical drag coefficients for object of interest (p613, Fig. 9.30)

62. small grain of sand :
<Sol>
0.10 , 2.3, ?
D mm SG U
  
2
2 2
3
3
1) From Free-Body Diagram :
( / 6) (1)
( / 6) (2)
B
sans H O
B H O H O
W D F
where W V SG D
and F V D
  
  
 
 
 
2
2 2 2
2
2 2
2 2
2) Assume Creeping flow (Re 1)
with 24/ Re ( 9.4)
1/ 2 ( / 4)
1/ 2 ( / 4) [24/( / )]
D=3 (3)
; Stokes Law
D
H O D
H O H O H O
H O
C Table
D U D C
U D UD
UD
 
   






Example 9.10

63. 2 2 2
2 2
3 3
2
Eqs. 1, 2, and 3
( / 6) 3 ( / 6)
( ) /18 (4)
H O H O H O
H O H O
From
SG D UD D
U SG gD
    
  
 
  
2 2
3 3 2
3 2 3
3
From Table 1.6 for water at 15.6 C
: 999 / , 1.12 10 /
(2.3 1)(999)(9.81)(01 10 ) /[18(1.12 10 )]
6.32 10 /
H O H O
kg m N s m
U
U m s
  
 


   
   
  

64. frontal area
planform area

65. (b)
(a)

68. http://green.caltech.edu/~colonius/me101/ho5/ho5.html

72. Example 9.12

73. * : 1.69 , 0.0992 , 200 /
* : 1.50 , 0.00551 , 60 /
standard , , ?
well hit golf ball D in W lb U ft s
well hit table tennis ball D in W lb U ft s
Drag on a golf ball a smooth golf ball and a table tennis ball
Deceler
   
   
?
ation of each ball
2 2
4 5
4 4
1/ 2 ( / 4) (1)
: Re / (200)(1.69/12)/(1.57 10 ) 1.79 10
: Re / (60)(1.50/12)/(1.57 10 ) 4.78 10
D
Sol
D U D C
for golf ball UD
for table tennis ball UD
 




 
 
     
    
:
- standard : 0.25
- : 0.51
- : 0.50
D
D
D
Drag coefficient
for golf ball C
for smooth golf ball C
for table tennis ball C





74. -3 2 2
-3 2 2
-3 2
;
* standard : 1/ 2(2.38 10 )(200) ( / 4)(1.69/12) (0.25) 0.185
* : 1/ 2(2.38 10 )(200) ( / 4)(1.69/12) (0.51) 0.378
* : 1/ 2(2.38 10 )(60) (
Drag
golf ball D lb
smooth golf ball D lb
table tennis ball D




  
  
  2
/ 4)(1.50/12) (0.50) 0.0263lb

/ / / /
* standard : / 0.185/ 0.0992 1.86
* : / 0.378/ 0.0992 3.81
* : / 0.0263/ 0.00551 4.77
a D m gD W a g D W
golf ball a g
smooth golf ball a g
table tennis ball a g
    
 
 
 

83. 9.4 Lift
1. Functional Relationship for the Lift Coefficient
- The most important parameter that affects the lift coefficient
is the shape of the object.
- Most of the lift comes from the surface pressure distribution.
- For large Reynolds number flows these pressure
distributions are usually directly proportional to the dynamic
pressure, with viscous effects being of secondary
importance.
2. Airfoil Geometry and Its Performance (see p616 Fig. 9.32,
p617 Fig. 9.33)
2
( , Re, , , / ) /(1/ 2 ) (9.39)
L
C shape Ma Fr l L U A
  
 
9.4.1 Surface Pressure Distribution

87. - chord length :
- lift coefficient :
* Typical lift coefficients are on the order of unity, I.e., the lift
force is on the order of the dynamic pressure times the planform
area of the wing,
*
* Aspect Ratio (縱橫比) : ( ; wing length)
- wing loading : : increases with speed.
- Induced Drag : the increase in drag due to the finite length of
the wing ( )
-stall (失速) : If is too large, the boundary layer on the upper
surface separates , the flow over the wing develops a wide,
turbulent wake region, the lift decreases, and the drag increases.
The airfoil stalls. Such conditions are extremely dangerous if they
occur while the airplane is flying at a low altitude where there is
not sufficient time and altitude to recover from the stall.
A
U
L )
2
/
( 2


A
L/
)
,
( AR
C
C L
L 

c
b
A
b
AR /
/
2


c
b


AR



88. 3. Ratio of the Lift to Drag
- : see p618 Fig. 9.34
- : lift-drag polar
-The most efficient angle of attack (I.e., largest ) can
be found by drawing a line tangent to the curve from
the origin, as is shown in Fig. 9.34b.
-High-performance airfoils generate lift that is perhaps 100 or
more times greater than their drag. This translates into the fact
that in still air they can glide a horizontal distance of 100 m for
each 1 m drop in altitude.
4. Flap : see p619 Fig. 9.35


D
L C
C /
D
L C
C 
D
L C
C /
D
L C
C 

91. <Sol>
1) For steady flight conditions
15 / , 96 , 7.5 , 210 , 0.046( )
D
U ft s b ft c ft W lb C based on planeform area
    
0.8, ?,
L
C P required by the pilot
  
2
2
2
1/ 2 L L
W
W L U AC C
U A


   
2 3 3
(96)(7.5) 720 , 210 , 2.38 10 /
where A bc ft W lb slugs ft
 
     
3 2
2(210)
1.09 / 1.09/ 0.046 23.7
(2.38 10 )(15) (720)
L L D
C C C

     

2
2) 1/ 2 D
P DU where D U AC
 
 
2 3
1/ 2
2
D D
U AC U AC U
DU
P
 
  
   
3 3
(2.38 10 )(720)(0.046)(15)
166 / 0.302
2(0.8)
ft lb s hp


   
Example 9.15

92. Example 9.16
-2 -2
: 3.8 10 , 2.45 10 , 12 / , ?
table tennis ball D m W N U m s 
     
2
2 2
2
2 2 2
2
1/ 2
( / 4)
2(2.45 10 )
0.244
(1.23)(12) ( / 4)(3.8 10 )
L L
L
Sol
W
W L U AC C
U D
C

 



 
    

 

2
According to Fig. 9.39
0.244 / 2 0.9
2U(0.9) 2(12)(0.9)
=
(3.8 10 )
568 / 5420
L
C if D U
D
rad s rpm

 

 
 

 

95. http://unixweb.ecs.syr.edu/mame/simfluid/redder/golfballpics1.ht
ml
The effect of spin on the flow behind a sphere.
In the following photographs, you can "see" the flow behind a
non-spinning and spinning golf ball. The experiments were
performed in the small water tunnel at syracuse University, with a
mean flow velocity of 3.8 centimeters (1.5 inches) per second,
corresponding to a Reynolds number of 1.6*10^3. In the first two
pictures, the golfball isn't spinning. In the third, it's spinning with 7
revolutions per minute (rpm), and in the fourth with 12 rpm. These
spin rate correspond to the surface speed, 41% and 71% of the
flow speed, respectively

97. - Computation of Unsteady Flow Separation
Here you can observe how the separation point moves and the
velocity reverses its direction with time
- Vortex shedding as a result of boundary layer separation from a
circular cylinder at Reynolds number 100,000.
http://unixweb.ecs.syr.edu/mame/simfluid/redder/movie45.html
.

99. In photographs 3 and 4,
you can clearly see the
effect of spin. Now, the
golfball is spinning in a
clockwise direction, and
the wake is directed
downwards. In Photo 4 the
ball is spinning faster, and
the angle of the wake
deflection is steeper,
indicating a larger lift.