Chapter 9
Flow over Immersed
Bodies
z
z
We consider flows over
bodies that are immersed in
a fluid and the flows are
termed external flows.
We are interested in the fluid
force (lift and drag) over the
bodies. For example,
correct design of cars,
trucks, ships, and airplanes,
etc. can greatly decrease
the fuel consumption and
improve the handling
characteristics of the vehicle.
9.1 General External Flow Characteristics
z
z
z
A body immersed in a moving fluid experiences a
resultant force due to the interaction between the body
and the fluid surrounding it.
Object and flow relation:
stationary air with moving object or
flowing air with stationary object
Flow classification:
Figure 9.2
Flow classification: (a) two-dimensional, (b) axisymmetric, (c) three-dimensional.
z
Body shape:
Streamlined bodies or blunt bodies
9.1.1 Lift and Drag Concepts
z
z
z
z
When any body moves through a fluid, an interaction
between the body and the fluid occurs
This can be described in terms of the stresses-wall
shear stresses due to viscous effect and normal
stresses due to the pressure P
Drag, D
the resultant force in the
direction of the upstream
velocity
Lift, L
the resultant force normal to
the upstream velocity
Resultant force--lift and drag
z
Resultant force
dFx = ( pdA) cos θ + (τ w dA) sin θ
and
dFy = −( pdA) sin θ + (τ w dA) cos θ
The net x and y components of the force on the object are,
D = ∫ dFx = ∫ p cos dA + ∫ τ w sin θ dA
L = ∫ dFy = − ∫ p sin θ dA + ∫ τ w cos dA
z
Nondimensional lift coeff. CL and drag coeff. CD
CL =
L
D
C
,
=
D
1 ρU 2 A
1 ρU 2 A
2
2
A: charateristic area, frontal area or
planform area?
Note: When CD or CL=1, then D or L equals the dynamic pressure on A.
Velocity and pressure distribution around an air foil
(From S.R. Turns, Thermal-Fluid Sciences, Cambridge Univ. Press, 2006)
Drag and lift due to pressure and friction: example of an air foil
(From S.R. Turns, Thermal-Fluid Sciences, Cambridge Univ. Press, 2006)
9.1.2 Characteristics of Flow Past an Object
z
For typical external flows the most important of
parameters are
the Reynolds number
ρUl
Re =
μ
the Mach number
Ma = U
c
the Froude number, for flow with a free surface
Fr = U
z
gl
Flows with Re>100 are dominated by inertia effects, whereas
flows with Re<1 are dominated by viscous effects.
Flow past a flat plate (streamlined body)
Figure 9.5
Character of the steady, viscous flow past
a flat plate parallel to the upstream
velocity: (a) low Reynolds number flow, (b)
moderate Reynolds number flow, (c) large
Reynolds number flow.
Steady flow past a circular cylinder (blunt body)
z
z
The velocity gradients
within the boundary
layer and wake regions
are much larger than
those in the remainder
of the flow fluid.
Therefore, viscous
effects are confined to
the boundary layer and
wake regions.
V9.2 Streamlined and blunt bodies
9.2 Boundary Layer Characteristics
9.2.1 Boundary Layer Structure and Thickness on a Flat Plate
Figure 9.7
Distortion of a fluid particle as it flows within the boundary layer.
z
Boundary layer thickness, δ
δ =y where u=0.99U
Boundary Layer Structure and Thickness on a Flat Plate
Transition occurs at
Re xcr =
ρUx
= 2 × 105 ~ 3 × 106
μ
depending on surface roughness and
amount of turbulence in upstream flow
Figure 9.9
Typical characteristics of boundary layer thickness and wall
shear stress for laminar and turbulent boundary layers.
V9.3 Laminar boundary layer
V9.4 Laminar/turbulent transition
Boundary layer displacement thickness
δ *bU =
∞ or δ
∫ (U − u ) bdy
0
∞
u
⎛
δ = ∫ ⎜1 −
U
0⎝
*
⎞
⎟ dy
⎠
(9.3)
Figure 9.8 (p. 472)
Boundary layer thickness: (a) standard boundary layer thickness, (b) boundary layer
displacement thickness.
Boundary layer displacement thickness
z
z
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.
It represents the outward displacement of the
streamlines caused by the viscous effects on the
plate.
Ex 9.3 Determine the velocity U=U(x) of the air within the duct
but outside of the boundary layer with δ * = 0.004( x)1 2
Boundary layer momentum thickness
z
Boundary layer momentum thickness θ
∞
2
ρ
u
U
−
u
dA
=
ρ
b
u
(
U
−
u
)
dy
=
ρ
bU
θ
(
)
∫
∫
0
∞
u
U
0
θ =∫
u
⎛
1
−
⎜
⎝ U
⎞
⎟ dy
⎠
(9.4)
9.2.2 Prandtl/Blasius Boundary Layer Solution
z
The Navier-Stokes equations are too complicated that
no analytical solution is available. However, for large
Re, simplified boundary layer equations can be
derived.
z
2-D Navier-Stokes Equations
⎛ ∂ 2u ∂ 2u ⎞
1 ∂p
∂u
∂u
u +v
=−
+ν ⎜ 2 + 2 ⎟
ρ ∂x
∂x
∂y
⎝ ∂x ∂y ⎠
⎛ ∂ 2v ∂ 2v ⎞
1 ∂p
∂v
∂v
u +v = −
+ν ⎜ 2 + 2 ⎟
ρ ∂y
∂x
∂y
⎝ ∂x ∂y ⎠
∂u ∂v
+ =0
∂x ∂y
(9.5)
Elliptic equation
(9.6)
Boundary layer assumption
z
The boundary layer assumption is based on the fact
that the boundary layer is thin.
δ << x so that v << u and
Thus the equations become,
∂u ∂v
+
=0
∂x ∂y
(9.8)
1 ∂p
∂u
∂u
∂u
u
+v
=−
+ν 2
ρ ∂x
∂x
∂y
∂y
2
z
z
∂
∂
<<
∂x
∂y
parabolic equation
Detailed derivation is given in
Cengel & Cimbala, Fluid
Mechanics, 2006, pp. 516-519.
(9.9)
Physically, the flow is parallel to the plate and any fluid is
convected downstream much more quickly than it is diffused
across the streamlines.
For boundary layer flow over a flat plate the pressure is
constant. The flow represents a balance between viscous
and inertial effects, with pressure playing no role.
Boundary conditions
BCs: u = v = 0 , y = 0
u →U , y → ∞
It can be argued that the velocity profile should be similar →
In addition, order of magnitude analysis leads to:
∂u
∂u
∂ 2u uu ν u
~ 2 ,
u
+v
=ν 2 ,
x δ
∂x
∂y
∂y
δ ~
2
νx
U
,
⎛ν x ⎞
δ ~⎜ ⎟
⎝U ⎠
1
2
u
⎛ y⎞
=g ⎜ ⎟
U
⎝δ ⎠
Similarity variables
Define the dimensionless similarity variable
1
y
⎛U ⎞ 2
η =⎜ ⎟ y~ ,
δ
⎝ν x ⎠
and stream function
ψ = (ν xU )
1
2
f (η ) where f (η ) is an unknown function
Thus
1
⎛
2
1
∂ψ
U
⎛
⎞
2
⎜u =
= (ν xU ) f ′ (η ) ⎜ ⎟ = Uf ′ (η )
∂y
⎜
⎝ν x ⎠
⎜
1
ψ
ν
U
∂
⎛
⎞ 2
⎜
′ d
⎜ v = − ∂x = ⎜ 4 x ⎟ (η f ′ − f ) where ( ) = dη
⎝
⎠
⎝
Then the parabolic equation (9.9) becomes
2 f ′′′ + ff ′′ = 0
with the boundary conditions
f (0) = f ′(0) = 0 at η = 0, f ′(∞) → 1 as η → ∞
(9.14a)
(9.14b)
Blasius solutions
u
= f ′ (η )
U
from the numerical solution
u
≈ 0.99 η = 5.0 (Table 9.1)
U
1
νx
⎛U ⎞ 2
η = 5 = ⎜ ⎟ δ, δ = 5
U
⎝ν x ⎠
Ux
δ
5
or
Re x =
=
x
ν
Re x
HW: Derive (9.14&9.18) and
solve using Matlab to get
Table 9.1
Blasius solution
Figure 9.10
Blasius boundary layer profile: (a) boundary layer profile in dimensionless form using
the similarity variable ŋ. (b) similar boundary layer profiles at different locations along
the flat plate.
Blasius solution
displacement thickness: from (9.3) →
momentum thickness:
from (9.4) →
boundary layer Shear stress:
∂u
τw = μ
∂y
= 0.332U
y =0
3
2
δ*
x
θ
=
1.721
Re x
0.664
=
x
Re x
(9.16)
(9.17)
ρμ
x
(9.18)
Note: For fully developed pipe flow shear stress, τ w ∝ U (why?)
9.2.3 Momentum-Integral Boundary Layer
Equation for a Flat Plate
z
z
One of the important aspects of boundary layer theory
is the determination of the drag caused by shear forces
on a body.
Consider a uniform flow past a flat plate
Figure 9.11
Control volume used in the derivation of the momentum integral equation for boundary
layer flow.
x-component of momentum equation:
uv v
uv v
∑ Fx = ρ ∫ uV ndA + ρ ∫ uV ndA
(1)
∑F
x
(2)
∫
= −D = −
τ w dA = −b
plate
∫
τ w dx or
plate
δ
− D = ρ ∫ U (−U ) dA +ρ ∫ u dA → D = ρU bh − ρ b ∫ u 2 dy
2
(1)
2
(2)
δ
0
δ
δ
0
0
Since Uh = ∫ udy, ρU bh = ρ bU ∫ udy = ρ b ∫ Uudy
2
0
δ
δ
δ
δ
D = ρU bh − ρ b ∫ u du = ρ b ∫ Uudy − ρ b ∫ u dy = ρ b ∫ u (U − u )dy
2
2
0
δ
2
0
0
u
u
(1 − )dy = ρ bU 2θ , and
U
U
0
D = ρ bU 2 ∫
0
dD
dθ
= ρ bU 2
dx
dx
The increase in drag per length of the plate occurs at the expense of an
increase of the momentum boundary layer thickness, which represents
a decrease in the momentum of the fluid.
Q dD = τ wbdx →
dD
dθ
= bτ w ∴τ w = ρU 2
→ momentum integral equation by von Karman
dx
dx
Momentum integral equation
dθ
τ w = ρU
dx
2
z
z
→ momentum integral equation
If we know the velocity distributions, we can obtain
drag or shear stress.
The accuracy of these results depends on how closely
the shape of the assumed velocity profile approximates
the actual profile.
Example 9.4
u=
Uy
0 ≤ y ≤δ u =U y >δ
δ
dθ
τ w = ρU
dx
U
τw = μ
2
δ
∞
u
⎛
1
−
⎜
⎝ U
u
θ =∫
U
0
δ
u
⎞
dy
=
⎟
∫0 U
⎠
δ
u
⎛
1
−
⎜
⎝ U
⎞
⎟ dy
⎠
1
⎛y
y⎛
y⎞
y ⎞
δ
2
dy
y
y
dy
1
−
=
δ
−
=
δ
−
=
⎜
⎟
⎜
⎟
∫
2
3
δ
δ
⎝
⎠
⎝
⎠0 6
0
0
1
=∫
(
)
2
3
6μ
μU ρU 2 dδ
dx
, δ dδ =
=
6 dx
δ
ρU
δ2
2
=
6μ
12 μ
x ⇒ δ2 =
x
ρU
ρU
or δ = 3.46
δ
x
= 3.46
θ=
δ
6
μx
ρU
⇔ compare with Blasius solution
x
=5
μ
ρUx
= 0.576
μx
ρU
3
dθ
ρμ
2
= 0.289U
τ w = ρU
dx
x
2
δ
⇔ τ w = 0.332U
3
2
ρμ
x
μ
ρUx
Figure 9.12
Typical approximate boundary layer
profiles used in the momentum
integral equation.
Momentum integral equation for general profile of u/U
u
u
y
= g (Y ) for 0 ≤ Y ≤ 1,
= 1 for Y > 1 , where Y =
U
U
δ
B.C.: g (0) = 0 , g (1) = 1
Let
δ
1
u
u
D = ρ bU ∫ (1 − )dy = ρ bU 2δ ∫ g (Y )[1 − g (Y )]dY =ρ bU 2δ C1
U
U
0
0
2
τw = μ
∂u
∂y
=
y =0
μU dg
δ dY
=
Y =0
μU
C
δ 2
1
C1 = ∫ g (Y )[1 − g (Y )]dY
we can obtain that
δ
x
2C2 / C1
=
τw =
cf =
Re x
(Blasius solution:
C1C2
μU
C2 =
U
δ
2
τw
1
ρU 2
2
= 2C1C2
3
2
δ
x
0
5
)
Re x
=
C2 =
ρμ
dg
dY
Y =0
x
2C1C2
μ
=
ρUx
Re x
(Blasius solution: c f =
0.664
)
Re x
l
CDf =
Df
1
ρU 2bl
2
=
b ∫ τ w dx
0
1
ρU 2bl
2
=
8C1C2
Rel
, where Rel =
1.328
ρU l
(Blasius solution: CDf =
)
μ
Rel
9.2.4 Transition from Laminar to Turbulent Flow
z
z
The analytical results are restricted to laminar
boundary layer along a flat plate with zero pressure
gradient.
Transition to turbulent boundary layer occurs at
Re x ,cr = 2 ×105
3 ×106
Example 9.5 Boundary layer
transition
Figure 9.9
Typical characteristics of boundary layer
thickness and wall shear stress for laminar
and turbulent boundary layers.
Figure 9.13 (p. 483)
Turbulent spots and the transition from laminar
to turbulent boundary layer flow on a flat plate.
Flow from left to right. (Photograph courtesy of B.
Cantwell, Stanford University.
V9.5 Transition on flat plate
Figure 9.14 (p. 484)
Typical boundary layer profiles on a
flat plate for laminar, transitional, and
turbulent flow (Ref. 1).
9.2.5 Turbulent Boundary Layer Flow
z
z
The structure of turbulent boundary layer
flow is very complex, random, and irregular,
with significant cross-stream mixing.
Empirical power-law velocity profile is a
reasonable approximation.
Example 9.6
To begin with the momentum integral equaion:
2 dθ
τ w = ρU
dx
1
y
⎧
1
Y
=
,Y < 1
u ⎛ y⎞ 7
7 ⎪
=⎜ ⎟ =Y
Assume
δ
⎨
U ⎝δ ⎠
⎪⎩u = U ,Y > 1
and the experimentally determined formula:
⎛ ν ⎞
τ w = 0.0225ρU ⎜
⎟
U
δ
⎝
⎠
2
1
4
(1)
dθ
τ w = ρU
dx
∞
u⎛
u
θ = ∫ ⎜1 −
U⎝ U
0
2
u
⎞
=
δ
dy
⎟
∫0 U
⎠
where δ is still unknown.
1
u
⎛
−
1
⎜
⎝ U
)
(
1
1
7
⎞
7
7
=
−
=
1
δ
δ
dY
Y
Y
dY
⎟
∫
72
⎠
0
1
(2)
To determine δ , we combine Eq. 1 with Eq. 2 to get the ODE:
1
1
1
7
⎛ ν ⎞
⎛ν ⎞ 4
2 dδ
4
or δ d δ = 0.231⎜ ⎟ dx
ρU
0.0225ρU ⎜
⎟ =
72
dx
⎝ Uδ ⎠
⎝U ⎠
with BC: δ = 0 at x = 0, we obtain
2
⎛ν ⎞
δ = 0.370 ⎜ ⎟
⎝U ⎠
∞
u
⎛
δ = ∫ ⎜1 −
U
0⎝
*
δ
1
5
4
δ
4
x 5 , or
x
u
⎞
⎛
dy
1
=
−
δ
⎟
∫0 ⎝⎜ U
⎠
⎛ν ⎞
= 0.036 ⎜ ⎟
θ=
72
⎝U ⎠
1
1
5
x
4
5
,
=
0.370
Re x
1
5
(
⎞
⎟ dY = δ ∫ 1 − Y
⎠
0
1
θ <δ* <δ
1
7
)
δ
⎛ν ⎞
dY = = 0.0463 ⎜ ⎟
8
⎝U ⎠
1
5
x
4
5
⎡
⎤
ν
⎥
τ w = 0.0225ρU 2 ⎢
1
4
⎢ U 0.37 (ν / U ) 5 x 5 ⎥
⎣
⎦
1
4
=
0.0288 ρU 2
Re x
1
5
1
A
⎛ν ⎞ 5
2
D f = ∫ bτ w dx = b0.0288ρU ∫ ⎜ ⎟ dx = 0.036 ρU
,
1
Ux ⎠
Rel 5
0
0⎝
l
l
2
CDf =
Df
=
0.072
1
1
2
ρU A Rel 5
2
Note: The above results are valid for smooth flat plates with
5 × 105 <Re x < 107
Turbulent B.L.: δ
Laminar B.L.: δ
4
x ,τ w
1
5
x ,τ w
2
x
x
−1
−1
5
2
A = bl
Friction drag coefficient for a flat plate
Although Fig. 9.15 is similar
with Moody diagram (pipe
flow) of Fig. 8.23, the
mechanisms are quite
different.
For fully developed pipe flow,
fluid inertia remains constant
and the flow is balanced
between pressure forces and
viscous forces.
For flat plate boundary layer
flow, pressure remains
constant and the flow is
balanced between inertia
effects and viscous forces.
z
Figure 9.15
Friction drag coefficient for a flat plate parallel to the upstream flow.
Empirical equations for CDf
Example 9.7
9.2.6 Effects of Pressure Gradient
Inviscid flow over a cylinder
Viscous flow over a cylinder
z
Viscous flow
Viscous flow over cylinder: velocity-pressure
z
z
Separated flow
No matter how
small the viscosity,
provided it is not
zero, there will be a
boundary layer that
separates from the
surface, giving a
drag that is, for the
most part,
independent of the
value of μ.
9.2.7 Momentum-Integral Boundary Layer
Equation with Nonzero Pressure Gradient
z
Outside the boundary layer
p+
ρU fs 2
2
= constant
dU fs
dp
= − ρU fs
dx
dx
z
(9.34)
Momentum integral boundary layer equation
dU fs
d
dθ
2
*
τ w = −ρ
⇔ (for U = C , τ w = ρU 2
) (9.35)
U fs θ + ρδ U fs
dx
dx
dx
(
z
z
)
This equation represents a balance between viscous forces
(τ w ), pressure forces( dp = − ρU dU fs ), and the fluid
fs
dx
dx
momentum( θ ).
Eq. 9.35 can be used to provide information about the
boundary layer thickness, wall shear stress, etc,
Derivation of equation (9.35)
∂u ∂v
+
= 0 (a)
∂x ∂y
∂u
∂u
∂U 1 ∂τ
+v
=U
+
u
∂x
∂y
∂x ρ ∂y
(b)
(b)- ( u − U ) (a)
−
δ
∂
∂U ∂
1 ∂τ
=
+ ( vU − vu )
uU − u 2 + (U − u )
∂x ∂y
ρ ∂y ∂x
(
∫−
0
)
δ
∂
∂U
1 ∂τ
dy = ∫ u (U − u )dy +
∂x 0
∂x
ρ ∂y
δ
δ
0
δ
δ
∂ ⎡
u
= ⎢U 2 ∫
∂x ⎣ 0 U
=
u
⎛
−
1
⎜
⎝ U
)
δ
∫ (U − u )dy
0
δ
∫ (U − u )dy
0
δ
∂U
u
⎞
⎛
+
−
1
dy
U
⎟
⎜
∂x ∫0 ⎝ U
⎠
∂
∂U
U 2θ + U δ *
∂x
∂x
(
∂
( vU − vu )dy
∂
y
0
v=0 at y=0,δ
∫ (U − u )dy + ∫
1 ∂τ
∂
∂U
−
=
−
+
dy
u
U
u
dy
(
)
∫0 ρ ∂y
∂x ∫0
∂x
δ
τ
∂
∂U
= ∫ u (U − u )dy +
ρ ∂x 0
∂x
δ
⎞ ⎤
⎟dy ⎥
⎠ ⎦
(9.35)
9.3 Drag
z
Any object moving through a fluid will
experience a drag D -- a net force in the
direction of flow due to pressure (pressure
drag) and shear stress (friction drag) on the
surface of the object.
Drag coefficient:
D
CD = 1
= φ ( shape, Re, Ma, Fr, ε / l )
2
2 ρU A
These are determined experimentally, and very
few can be obtained analytically.
9.3.1 Friction Drag
z
z
z
z
Drag due to the shear stress τ w , on the object.
For large Re, the friction drag is generally smaller
than pressure drag.
However, for highly streamlined bodies or for low
Reynolds number flow, most of the drag may be
due to friction drag.
Drag on a plate of width b and length l
1
D f = ρU 2blCDf
2
where CDf is the friction drag coefficient.
b
L
Friction Drag
For laminar flow, CDf is independent of
For turbulent flow, CDf is function of
CDf
ε
ε
D
D
along the surface of a curved body is
quite difficult to obtain
Example 9.8 Friction drag coeff. for a cylinder
9.3.2 Pressure Drag
z
z
Drag due to pressure on the object.
Pressure drag – also called form drag because of its
strong dependency on the shape or form of the object.
p − p0
: pressure coefficient
1
ρU 2
2
p cos θ dA ∫ C p cos θ dA
Dp
∫
p0 : reference pressure
=
=
=
1
1
A
ρU 2 A
ρU 2 A
2
2
D p = ∫ p cos θ dA C p =
CDp
Pressure drag--Large Reynolds number
For large Reynolds number, ( inertial effect
viscous effect )
1
p − p0 ∝ ρU 2 ( dynamic pressure )
2
p − p0
: pressure coefficient
D p = ∫ p cos θ dA C p =
2
ρU / 2
CDp =
Dp
p cos θ dA ∫ C
∫
=
=
p
cos θ dA
1
1
A
2
ρU A
ρU 2 A
2
2
Therefore, C p is independent of Reynolds number,
CDp is also independent of Reynolds number ( Re
1)
Pressure drag--low Reynolds number
For very small Reynolds number ( inertial effect
U
⎧
∝
p
μ
⎪⎪
l
⎨
⎪τ ∝ μ U
⎪⎩ w
l
( viscous stress )
CDp
μ
Dp
1
ρU 2 A
2
viscous effect )
U
l
1
ρU 2
2
Comparisons:
1
Re
For large Reynolds number f constant ( pipe flow )
For laminar pipe flow f
Example 9.9 Pressure drag coeff. for a cylinder
1
μ
=
ρUl Re
Viscosity Dependence
Inviscid, Viscous flows over object
For μ = 0, the pressure drag on any shaped object (symmetrical
or not) in a steady flow would be zero.
For μ ≠ 0, the net pressure drag
may be non-zero because of
boundary layer separation.
9.3.3 Drag Coefficient Data and
Examples
ε⎞
⎛
CD = φ ⎜ shape, Re, Ma, Fr, ⎟
l⎠
⎝
Shape Dependence
Figure 9.20
Two objects of considerably different size that have the same drag force:
(a) circular cylinder CD = 1.2; (b) streamlined strut CD = 0.12.
Shape Dependence
Figure 9.19
Drag coefficient for an ellipse with the characteristic area either the frontal area, A = bD,
or the planform area, A = bl (Ref. 5).
Reynolds Number Dependence
Low Reynolds number, Re<1
viscous force ≈ pressure force
D = f (U , l , μ )
From dimensional analysis
D = C μ lU [= C ( μU / l ) ⋅ l 2 ]
CD =
D
1
ρU 2l 2
2
=
2C μ lU 2C
ρUl
,
Re
=
=
ρU 2l 2 Re
μ
Note: For Re<1,
streamlining
increases the drag
due to an increase
in the area on
which shear force
act.
Ex 9.10: particle in the water dragged up by upward
motion
D = 0.1mm, SG = 2.3,
W = D + FB
W =γ sand ∀ = SGγ H 2O
CD =
24
Re
( For
π
6
V9.7 Skydiving practice
to determine U
FB = γ H 2O ∀ = γ H 2O
D3 ,
π
6
D3 ,
Re < 1)
⎛
1
1
24
2 π
2
2 π
2
D = ρ H 2OU
D CD = ρ H 2OU
D ⎜
⎜ ρ H OUD / μ H O
2
4
2
4
2
⎝ 2
π
π
∴ SGγ H O D 3 = 3πμ H OUD + γ H O D 3
2
6
2
2
6
γ = ρg
SG ρ
(
U=
H 2O
)
− ρ H 2O gD 2
18μ
For 15.6°C , ρ H 2O = 999 kg
Re =
ρUD
= 0.564
μ
= 6.32 ×10−3 m
m
3
( Re < 1)
s
, μ H 2O = 1.12 × 10−3 N S
m2
⎞
⎟ = 3πμ H 2OUD
⎟
⎠
Moderate Reynolds Number
CD of cylinder and sphere
Re < 103 CD
Re −1/ 2
103 < Re < 105 CD
[Re ↑ , CD ↓ (not D)]
constant
Flow over cylinder at various Reynolds
number
Re < 1 , no separation
Re ≈ 10, steady separation bubble
V9.8 Karman vortex street
Re ≈ 100, vortex shedding starts at Re ≈ 47
Flow over cylinder at various Reynolds number
Re ≈ 5 × 104
Re ≈ 4 × 105
from M. Van Dyke, An
Album of Fluid Motion
Laminar and Turbulent Boundary Layer
Separation
from M. Van Dyke, An Album of Fluid Motion
Drag of streamlined and blunt bodies
Drag coefficients
The different relations of CD with Re for objects with various
streamlining are illustrated in Fig. 9.22.
For streamlined bodies, CD when the boundary layer becomes turbulent.
For blunt objects, CD when the boundary layer becomes turbulent.
Note: In a portion of
the range 105<Re<106,
the actual drag (not
just CD) decreases with
increasing speed.
V9.9 Oscillating sign
V9.10 Flow past a flat plate
V9.11 Flow past an ellipse
Example 9.11 Terminal velocity of a falling object
W = D + FB
W = γ ice V , FB = γ air V
1
π
ρ airU 2 D 2CD = W − FB
2
4
γ ice γ air ∴W FB
1
2 π
D 2CD = W = γ ice V
ρ airU
2
4
⎛ 4 ρice gD ⎞
U =⎜
⎟
3
C
ρ
air
D ⎠
⎝
1
2
Compressibility Effects
When the compressibility effects become important
CD = φ (Re, Ma)
Notes: 1. Compressibility effect is negligible for Ma<0.5.
2. CD increases dramatically near Ma=1, due to
the existence of shock wave.
Surface Roughness Effects
z
z
Surface roughness influences drag when the boundary
layer is turbulent, because it protrudes through the
laminar sublayer and alters the wall shear stress.
In addition, surface roughness can alter the transitional
critical Re and change the net drag.
In general,
Streamlines bodies:
Blunt bodies:
ε
ε
D
↑→ CD ↑
↑→ CD
constant (pressure drag)
D
until transition to turbulence occurs and the wake region becomes
considerably narrower so that the pressure drag drops. (Fig. 9.25)
Surface Roughness
A well-hit golf ball has Re of O(105 ), and the dimpled golf ball
has a critical Reynolds number 4 ×104 → dimples reduce CD
Table tennis Reynolds number is less than 4 ×104 → no need of dimples
Example 9.12 Effect of Surface Roughness for golf ball
and table tennis ball
dimpled
CD ,rough
CD ,smooth
=
0.25
= 0.5
0.5
Froude Number Effects (flows with free
surface)
When the free surface is present, the wave-making effects
becomes important, so that
CD = φ (Re, Fr)
Figure 9.26 (p. 507)
Typical drag coefficient
data as a function of
Froude number and hull
characteristics for that
portion of the drag due to
the generation of waves.
Composite body drag
z
z
Approximate drag calculations for a complex body by
treating it as composite collection of its various parts.
e.g., drag on an airplane or an automobile.
The contributions of the drag due to various portions of
car (i.e., front end, windshield, roof, rear end, etc.) have
been determined. As a result it is possible to predict the
aerodynamic drag on cars of a wide variety ouf body
styles (Fig. 9.27).
V9.14 Automobile
streamlining
Example 9.13 Drag on a composite body
9.4 Lift (L)
9.4.1 Surface Pressure Distribution
z
Lift -- a force that is normal to the free stream.
For aircraft, L ↑
For car, L ↓ for better traction and cornering ability
CL =
ε⎞
⎛
, CL = φ ⎜ shape, Re, Ma, Fr, ⎟
1
l⎠
⎝
ρU 2 A
2
L
Froude no. – free surface present
ε is relatively unimportant
Ma is important for Ma > 0.8
Re effect is not great
The most important parameter that affects the lift coefficient
is the shape of the object.
Surface Pressure Distribution
z
z
Common lift-generating devices (airfoils, fans,…)
operate at large Re. Most of the lift comes from
the surface pressure distribution.
For Re<1, viscous effect and pressure are equally
important to the lift (for minute insects and the
swimming of microscopic organisms).
Figure 9.31
Pressure distribution
on the surface of an
automobile.
Example 9.14 Lift from pressure and shear distributions
Airfoil
z
z
For airfoils, the characteristic area A is the planform
area in the definition of both CLand CD. For a
rectangular planform wing, A=bc, where c is the
chord length, b is the length of the airfoil.
Typical CL ~O(1), i.e., L~(ρU2/2)A,
Wright Flyer: 1.5 lb / ft 2
and the wing loading L/A~(ρU2)/2:
Boeing 747: 150 lb / ft 2
z
Aspect ratio A = b2/A (=b/c if c is constant)
In general, A ↑⇒ CL ↑, CD ↓
Large A : long wing, soaring
airplane, albatross, etc.
Small A : short wing, highly
maneuverable fighter, falcon
Lift and drag coefficient data as a function of
angle of attack and aspect ratio
Figure 9.33
Typical lift and drag coefficient data as a function of angle of attack and the aspect ratio
of the airfoil: (a) lift coefficient, (b) drag coefficient.
Ratio of lift to drag/Lift-drag polar
V9.15 Stalled airfoil
z
Although viscous effects contributes little to the direct generation
of lift, viscosity-induced boundary layer separation can occur
when α is too large to lead to stall.
Lift of airfoil with flap
z
Lift can be increased by
adding flap
V9.17 Trailing edge flap
V9.18 Leading edge flap
Ex. 9.15
Figure 9.35
Typical lift and drag alterations
possible with the use of various
types of flap designs (Ref. 21).
From Cengel and Cimbala, Fluid Mechanics, McGraw Hill, 2006
9.4.2 Circulation
2-D symmetric air foil
Kutta condition—The flow over
both the topside and the underside
join up at the trailing edge and leave
the airfoil travelling parallel to one
another.
--by inviscid theory
--adding circulation
Note: The circulation needed is
a function of airfoil size and
shape and can be calculated
from potential flow theory.
(Section 6.6.3)
L= -ρUΓ (Kutta–Joukowski Law)
From Cengel and Cimbala, Fluid Mechanics, McGraw Hill, 2006
Circulation
Flow past finite length wing
Figure 9.37 (p. 546)
Flow past a finite length
wing: (a) the horseshoe
vortex system produced by
the bound vortex and the
trailing vortices: (b) the
leakage of air around the
wing tips produces the
trailing vortices.
V4.6 Flow past a wing
V9.19 Wing tip vortex
z
Vee-formation of bird -- 25 birds fly 70% farther than 1 bird
Circulation
Flow past a circular cylinder
A rotating cylinder in a stationary real fluid can produce
circulation and generate a lift—Magnus effect
EXAMPLE 9.16 Lift on a rotating table tennis ball
z
Figure 9.38
Inviscid flow past a circular cylinder: (a) uniform upstream flow without circulation. (b) free
vortex at the center of the cylinder, (c) combination of free vortex and uniform flow past a
circular cylinder giving nonsymmetric flow and a lift.