Chapter9.pdf

57:020 Mechanics of Fluids and Transport Processes Chapter 9
Professor Fred Stern Fall 2014 1
Chapter 9 Flow over Immersed Bodies
Fluid flows are broadly categorized:
1. Internal flows such as ducts/pipes, turbomachinery, open
channel/river, which are bounded by walls or fluid interfaces:
Chapter 8.
2. External flows such as flow around vehicles and structures,
which are characterized by unbounded or partially bounded
domains and flow field decomposition into viscous and
inviscid regions: Chapter 9.
a. Boundary layer flow: high Reynolds number flow
around streamlines bodies without flow separation.
Re ≤ 1: low Re flow (creeping or Stokes flow)
Re > ∼ 1,000: Laminar BL
Re > ∼ 5×105
(Recrit): Turbulent BL
b. Bluff body flow: flow around bluff bodies with flow
separation.

3. Free Shear flows such as jets, wakes, and mixing layers,
which are also characterized by absence of walls and
development and spreading in an unbounded or partially
bounded ambient domain: advanced topic, which also uses
boundary layer theory.
Basic Considerations
Drag is decomposed into form and skin-friction
contributions:
( )






∫ ⋅
τ
+
∫ ⋅
−
ρ
= ∞
S
w
S
2
D dA
î
τ
dA
î
n
p
p
A
V
2
1
1
C
CDp Cf

( )






⋅
∫ −
ρ
= ∞ dA
ĵ
n
p
p
A
V
2
1
1
C
S
2
L
c
t
<< 1 Cf > > CDp streamlined body
c
t
∼ 1 CDp > > Cf bluff body
Streamlining: One way to reduce the drag
 reduce the flow separationreduce the pressure drag
 increase the surface area  increase the friction drag
 Trade-off relationship between pressure drag and friction drag
Trade-off relationship between pressure drag and friction drag
Benefit of streamlining: reducing vibration and noise

Qualitative Description of the Boundary Layer
Flow-field regions for high Re flow about slender bodies:

τw = shear stress
τw ∝ rate of strain (velocity gradient)
=
0
y
y
u
=
∂
∂
µ
large near the surface where
fluid undergoes large changes to
satisfy the no-slip condition
Boundary layer theory and equations are a simplified form
of the complete NS equations and provides τw as well as a
means of estimating Cform. Formally, boundary-layer
theory represents the asymptotic form of the Navier-Stokes
equations for high Re flow about slender bodies. The NS
equations are 2nd
order nonlinear PDE and their solutions
represent a formidable challenge. Thus, simplified forms
have proven to be very useful.
Near the turn of the last century (1904), Prandtl put forth
boundary-layer theory, which resolved D’Alembert’s
paradox: for inviscid flow drag is zero. The theory is
restricted to unseparated flow. The boundary-layer
equations are singular at separation, and thus, provide no
information at or beyond separation. However, the
requirements of the theory are met in many practical
situations and the theory has many times over proven to be
invaluable to modern engineering.

The assumptions of the theory are as follows:
Variable order of magnitude
u U O(1)
v δ<<L O(ε) ε = δ/L
x
∂
∂
1/L O(1)
y
∂
∂
1/δ O(ε-1
)
ν δ2
ε2
The theory assumes that viscous effects are confined to a
thin layer close to the surface within which there is a
dominant flow direction (x) such that u ∼ U and v << u.
However, gradients across δ are very large in order to
satisfy the no slip condition; thus,
y
∂
∂
>>
x
∂
∂
.
Next, we apply the above order of magnitude estimates to
the NS equations.

2 2
2 2
u u p u u
u v
x y x x y
ν
 
∂ ∂ ∂ ∂ ∂
+ =
− + +
 
∂ ∂ ∂ ∂ ∂
 
1 1 ε ε-1
ε2
1 ε-2
2 2
2 2
v v p v v
u v
x y y x y
ν
 
∂ ∂ ∂ ∂ ∂
+ =
− + +
 
∂ ∂ ∂ ∂ ∂
 
1 ε ε 1 ε2
ε ε-1
0
y
v
x
u
=
∂
∂
+
∂
∂
1 1
Retaining terms of O(1) only results in the celebrated
boundary-layer equations
2
2
u u p u
u v
x y x y
ν
∂ ∂ ∂ ∂
+ =
− +
∂ ∂ ∂ ∂
0
y
p
=
∂
∂
0
y
v
x
u
=
∂
∂
+
∂
∂
elliptic
parabolic

Some important aspects of the boundary-layer equations:
1) the y-momentum equation reduces to
0
y
p
=
∂
∂
i.e., p = pe = constant across the boundary layer
from the Bernoulli equation:
=
ρ
+ 2
e
e U
2
1
p constant
i.e.,
x
U
U
x
p e
e
e
∂
∂
ρ
−
=
∂
∂
Thus, the boundary-layer equations are solved subject to
a specified inviscid pressure distribution
2) continuity equation is unaffected
3) Although NS equations are fully elliptic, the
boundary-layer equations are parabolic and can be
solved using marching techniques
4) Boundary conditions
u = v = 0 y = 0
u = Ue y = δ
+ appropriate initial conditions @ xi
edge value, i.e.,
inviscid flow value!

There are quite a few analytic solutions to the boundary-
layer equations. Also numerical techniques are available
for arbitrary geometries, including both two- and three-
dimensional flows. Here, as an example, we consider the
simple, but extremely important case of the boundary layer
development over a flat plate.
Quantitative Relations for the Laminar Boundary
Layer
Laminar boundary-layer over a flat plate: Blasius solution
(1908) student of Prandtl
0
y
v
x
u
=
∂
∂
+
∂
∂
2
2
y
u
y
u
v
x
u
u
∂
∂
ν
=
∂
∂
+
∂
∂
u = v = 0 @ y = 0 u = U∞ @ y = δ
We now introduce a dimensionless transverse coordinate
and a stream function, i.e.,
δ
∝
ν
=
η ∞ y
x
U
y
( )
η
ν
=
ψ ∞ f
xU
Note:
x
p
∂
∂
= 0
for a flat plate

( )
η
′
=
∂
η
∂
η
∂
ψ
∂
=
∂
ψ
∂
= ∞f
U
ψ
ψ
u ∞
=
′ U
/
u
f
( )
f
f
x
U
2
1
x
v −
′
η
ν
=
∂
ψ
∂
−
= ∞
Substitution into the boundary-layer equations yields
0
f
2
f
f =
′
′
′
+
′
′ Blasius Equation
0
f
f =
′
= @ η = 0 1
f =
′ @ η → ∞
The Blasius equation is a 3rd
order ODE which can be
solved by standard methods (Runge-Kutta). Also, series
solutions are possible. Interestingly, although simple in
appearance no analytic solution has yet been found.
Finally, it should be recognized that the Blasius solution is
a similarity solution, i.e., the non-dimensional velocity
profile f′ vs. η is independent of x. That is, by suitably
scaling all the velocity profiles have neatly collapsed onto a
single curve.
Now, lets consider the characteristics of the Blasius
solution:
∞
U
u
vs. y
∞
U
v
vs. y

5
Rex
x
δ =
value of y where u/U∞ = .99
Rex
U x
ν
∞
=

∞
ν
′
′
∞
µ
=
τ
U
/
x
2
)
0
(
f
U
w
see below
i.e.,
x
Re
664
.
0
U
2
c
x
2
w
f
θ
=
=
ρ
τ
=
∞
: Local friction coeff.
∫ =
=
L
f
f
f L
c
dx
c
bL
b
C
0
)
(
2 : Friction drag coeff.
=
L
Re
328
.
1
ν
∞L
U
Wall shear stress:
3 2
0.332
w U
x
ρµ
τ ∞
= or ( )
0.332 Re
w x
U x
τ µ ∞
=
Other:
∫ =








−
=
δ
δ
∞
0 x
*
Re
x
7208
.
1
δy
U
u
1 displacement thickness
measure of displacement of inviscid flow due to
boundary layer
∫ =








−
=
θ
δ
∞
∞
0 x
Re
x
664
.
0
δy
U
u
U
u
1 momentum thickness
measure of loss of momentum due to boundary layer
H = shape parameter =
θ
δ*
=2.5916
Note:
𝑏𝑏 = plate width
𝐿𝐿 = plate length

For flat plate or δ for general case
Quantitative Relations for the Turbulent
Boundary Layer
2-D Boundary-layer Form of RANS equations
0
y
v
x
u
=
∂
∂
+
∂
∂
( )
v
u
y
y
u
p
x
y
u
v
x
u
u 2
2
e
′
′
∂
∂
−
∂
∂
ν
+






ρ
∂
∂
−
=
∂
∂
+
∂
∂
requires modeling
Momentum Integral Analysis
Historically similarity and AFD methods used for idealized
flows and momentum integral methods for practical
applications, including pressure gradients. Modern
approach: CFD.
To obtain general momentum integral relation which is
valid for both laminar and turbulent flow
( )dy
continuity
)
v
u
(
equation
momentum
0
y
∫ −
+
∞
=
( )
dx
dU
U
H
2
dx
d
c
2
1
U
f
2
w θ
+
+
θ
=
=
ρ
τ
dx
dU
U
dx
dp
ρ
=
−
flat plate equation 0
dx
dU
=

∫ 





−
=
θ
δ
0
δy
U
u
1
U
u
momentum thickness
θ
δ
=
*
H shape parameter
dy
U
u
1
0
*
∫ 





−
=
d
d
displacement thickness
Can also be derived by CV analysis as shown next for flat
plate boundary layer.
Momentum Equation Applied to the Boundary Layer
Consider flow of a viscous fluid at high Re past a flat plate, i.e.,
flat plate fixed in a uniform stream of velocity ˆ
Ui .
Boundary-layer thickness arbitrarily defined by y = %
99
δ (where,
%
99
δ is the value of y at u = 0.99U). Streamlines outside %
99
δ will
deflect an amount *
δ (the displacement thickness). Thus the
streamlines move outward from H
y = at 0
=
x to
*
δ
δ +
=
=
= H
Y
y at 1
x
x = .

Conservation of mass:
CS
V ndA
ρ •
∫ =0= 0 0
H H
Udy udy
d
ρ ρ
∗
+
− +
∫ ∫
Assume incompressible flow (constant density):
( ) ( )
∫ ∫ ∫ −
+
=
−
+
=
=
Y Y Y
dy
U
u
UY
dy
U
u
U
udy
UH
0 0 0
Substituting
*
δ
+
= H
Y defines displacement thickness:
dy
U
u
Y
∫ 





−
= 0
*
1
d
*
δ is an important measure of effect of BL on external flow.
Consider alternate derivation based on equivalent flow rate:
∫
∫ =
δ
δ
δ 0
*
uδy
Uδy
Flowrate between
*
δ and δ of inviscid flow=actual flowrate, i.e., inviscid flow rate
about displacement body = viscous flow rate about actual body
∫
∫
∫
∫ 





−
=
⇒
=
−
δ
δ
δ
δ
δ
0
*
0
0
0
1
*
δy
U
u
uδy
Uδy
Uδy
w/o BL - displacement effect=actual discharge
For 3D flow, in addition it must also be explicitly required that *
δ
is a stream surface of the inviscid flow continued from outside of
the BL.
δ* Lam=δ/3
δ
δ* Turb=δ/8
Inviscid flow about δ* body

Conservation of x-momentum:
( ) ( )
0 0
H Y
x
CS
F D uV ndA U Udy u udy
ρ ρ ρ
=
− = • =
− +
∑ ∫ ∫ ∫
dy
u
H
U
D
Drag
Y
∫
−
=
= 0
2
2
r
r
= Fluid force on plate = - Plate force on CV (fluid)
Again assuming constant density and using continuity:
∫
=
Y
dy
U
u
H
0
2 2
0 0
0
/
Y
Y x
w
D U u Udy u dy dx
ρ ρ τ
= − =
∫ ∫ ∫
dy
U
u
U
u
U
D Y






−
=
= ∫ 1
0
2
θ
ρ
where, θ is the momentum thickness (a function of x only), an
important measure of the drag.
2
0
2 2 1
x
D f
D
C c dx
U x x x
θ
ρ
= = = ∫
( )
2
2
1
2
w
f f D
d d
c c xC
dx dx
U
τ θ
ρ
= ⇒= =
2
f
c
d
dx
θ
=
dx
d
U
w
θ
ρ
τ 2
=
Per unit span
Special case 2D
momentum integral
equation for dp/dx = 0

Simple velocity profile approximations:
)
/
/
2
( 2
2
δ
δ y
y
U
u −
=
u(0) = 0 no slip
u(δ) = U matching with outer flow
uy(δ)=0
Use velocity profile to get Cf(δ) and θ(δ) and then integrate
momentum integral equation to get δ(Rex)
δ* = δ/3
θ = 2δ/15
H= δ*/θ= 5/2
2
2
1/2
* 1/2
1/2
1/2
2 /
2 /
2 2 (2 /15)
1/ 2
15
30
/ 5.5 / Re
Re / ;
/ 1.83/ Re
/ 0.73/ Re
1.46 / Re 2 ( )
w
f
x
x
x
x
D L f
U
U d d
c
U dx dx
dx
d
U
dx
U
x
Ux
x
x
C C L
τ µ d
µ d θ
d
ρ
µ
dd
ρ
µ
d
ρ
d
ν
d
θ
=
⇒ = = =
∴ =
=
=
=
=
=
= =
10% error, cf. Blasius

Approximate solution Turbulent Boundary-Layer
Ret = 5×105
∼ 3×106
for a flat plate boundary layer
Recrit ∼ 100,000
dx
d
2
cf θ
=
as was done for the approximate laminar flat plate
boundary-layer analysis, solve by expressing cf = cf (δ) and
θ = θ(δ) and integrate, i.e. assume log-law valid across
entire turbulent boundary-layer
B
yu
ln
1
u
u *
*
+
n
κ
=
at y = δ, u = U
B
u
ln
1
u
U *
*
+
n
δ
κ
=
2
/
1
f
2
c
Re 





δ
or 5
2
c
Re
ln
44
.
2
c
2
2
/
1
f
2
/
1
f
+














=








δ
6
/
1
f Re
02
.
c −
δ
≅ power-law fit
neglect laminar sub layer and
velocity defect region
cf (δ)

Next, evaluate
∫ 





−
=
θ δ
0
δy
U
u
1
U
u
δx
δ
δx
δ
can use log-law or more simply a power law fit
7
/
1
y
U
u






δ
=
( )
δ
θ
=
δ
=
θ
72
7
⇒
dx
d
U
72
7
dx
d
U
U
2
1
c 2
2
2
f
w
d
ρ
=
θ
ρ
=
ρ
=
τ
dx
d
72
.
9
Re 6
/
1 d
=
−
d
or
1/7
0.16Rex
x
δ −
=
7
/
6
x
∝
δ almost linear
1/7
0.027
Re
f
x
c =
( )
1/7
0.031 7
Re 6
f f
L
C c L
= =
These formulas are for a fully turbulent flow over a smooth
flat plate from the leading edge; in general, give better
results for sufficiently large Reynolds number ReL > 107
.
Note: cannot be used to
obtain cf (δ) since τw → ∞
i.e., much faster
growth rate than
laminar
boundary layer

Comparison of dimensionless laminar and turbulent flat-plate velocity profiles (Ref:
White, F. M., Fluid Mechanics, 7th
Ed., McGraw-Hill)
Alternate forms by using the same velocity profile u/U =
(y/δ)1/7
assumption but using an experimentally determined
shear stress formula τw = 0.0225ρU2
(ν/Uδ)1/4
are:
1/5
0.37Rex
x
δ −
= 1/5
0.058
Re
f
x
c = 1/5
0.074
Re
f
L
C =
shear stress:
2
1/5
0.029
Re
w
x
U
ρ
τ =
These formulas are valid only in the range of the
experimental data, which covers ReL = 5 × 105
∼ 107
for
smooth flat plates.
Other empirical formulas are by using the logarithmic
velocity-profile instead of the 1/7-power law:
𝑢𝑢
𝑈𝑈
≈ �
𝑦𝑦
𝛿𝛿
�
1
7
𝑢𝑢
𝑈𝑈
≈ 2 �
𝑦𝑦
𝛿𝛿
� − �
𝑦𝑦
𝛿𝛿
�
2
(See Table 4-1 on
page 13 of this
lecture note)

𝛿𝛿
𝐿𝐿
= 𝑐𝑐𝑓𝑓(0.98 log 𝑅𝑅𝑅𝑅𝐿𝐿 − 0.732)
𝑐𝑐𝑓𝑓 = (2 log 𝑅𝑅𝑅𝑅𝑥𝑥 − 0.65)−2.3
𝐶𝐶𝑓𝑓 =
0.455
(log10 𝑅𝑅𝑒𝑒𝐿𝐿)2.58
These formulas are also called as the Prandtl-Schlichting skin-
friction formula and valid in the whole range of ReL ≤ 109
.
For these experimental/empirical formulas, the boundary layer
is usually “tripped” by some roughness or leading edge
disturbance, to make the boundary layer turbulent from the
leading edge.
No definitive values for turbulent conditions since depend on
empirical data and turbulence modeling.
Finally, composite formulas that take into account both the
initial laminar boundary layer and subsequent turbulent
boundary layer, i.e. in the transition region (5 × 105
< ReL < 8
× 107
) where the laminar drag at the leading edge is an
appreciable fraction of the total drag:
𝐶𝐶𝑓𝑓 =
0.031
𝑅𝑅𝑒𝑒𝐿𝐿
1
7
−
1440
𝐶𝐶𝑓𝑓 =
0.074
1
5
−
1700

𝐶𝐶𝑓𝑓 =
0.455
(log10 𝑅𝑅𝑒𝑒𝐿𝐿)2.58
−
1700
with transitions at Ret = 5 × 105
for all cases.
Local friction coefficient 𝑐𝑐𝑓𝑓 (top) and friction drag coefficient
𝐶𝐶𝑓𝑓(bottom) for a flat plate parallel to the upstream flow.

Bluff Body Drag
Drag of 2-D Bodies
First consider a flat plate
both parallel and normal to
the flow
( ) 0
î
n
p
p
A
V
2
1
1
C
S
2
Dp =
∫ ⋅
−
ρ
= ∞
∫ ⋅
τ
ρ
=
S
w
2
f dA
î
τ
A
V
2
1
1
C
= 2
/
1
L
Re
33
.
1
laminar flow
= 5
/
1
L
Re
074
.
turbulent flow
where Cp based on experimental data
vortex wake
typical of bluff body flow
flow pattern

( )
∫ ⋅
−
ρ
= ∞
S
2
Dp dA
î
n
p
p
A
V
2
1
1
C
= ∫
S
pdA
C
A
1
= 2 using numerical integration of experimental data
Cf = 0
For bluff body flow experimental data used for CD.

In general, Drag = f(V, L, ρ, µ, c, t, ε, T, etc.)
from dimensional analysis
c/L





 ε
=
ρ
= .
εtc
,
T
,
L
,
L
t
,
Aρ
Rε,
f
A
V
2
1
Dρag
C
2
D
scale factor

Potential Flow Solution: θ








−
−
=
ψ ∞ sin
r
a
r
U
2
2
2
U
2
1
p
V
2
1
p ∞
∞ ρ
+
=
ρ
+
2
2
2
r
2
p
U
u
u
1
U
2
1
p
p
C
∞
θ
∞
∞ +
−
=
r
−
=
( ) θ
−
=
= 2
p sin
4
1
a
r
C surface pressure
Flow Separation
Flow separation:
The fluid stream detaches itself from the surface of the body at
sufficiently high velocities. Only appeared in viscous flow!!
Flow separation forms the region called ‘separated region’
r
u
r
1
ur
∂
ψ
∂
−
=
θ
∂
ψ
∂
=
θ

Inside the separation region:
low-pressure, existence of recirculating/backflows
viscous and rotational effects are the most significant!
Important physics related to flow separation:
’Stall’ for airplane (Recall the movie you saw at CFD-PreLab2!)
Vortex shedding
(Recall your work at CFD-Lab2, AOA=16°! What did you see in
your velocity-vector plot at the trailing edge of the air foil?)

Terminal Velocity
Terminal velocity is the maximum velocity attained by a
falling body when the drag reaches a magnitude such that
the sum of all external forces on the body is zero. Consider
a sphere using Newton’ Second law:
d b g
F F F F ma
= + − =
∑
when terminal velocity is attained
0
F a
= =
∑ :
d b g
F F F
+ =
or
( )
2
0
1
2
D p Sphere fluid
V C A V
r γ γ
= − Sphere
For the sphere
2
4
p
A d
p
= and V 3
6
Sphere d
p
=
The terminal velocity is:
( )( )
1 2
0
4 3
sphere fluid
D fluid
d
V
C
γ γ
r
 
−
=  
 
 
Magnus effect: Lift generation by spinning
Breaking the symmetry causes the lift!
Z

Effect of the rate of rotation on the lift and drag coefficients of a
smooth sphere:
Lift acting on the airfoil
Lift force: the component of the net force (viscous+pressure) that
is perpendicular to the flow direction

Variation of the lift-to-drag ratio with angle of attack:
The minimum flight velocity:
Total weight W of the aircraft be equal to the lift
A
C
W
V
A
V
C
F
W
L
L
L
max
,
min
2
min
max
,
2
2
1
ρ
ρ =
→
=
=

< 0.3 flow is incompressible,
i.e., ρ ∼ constant
Effect of Compressibility on Drag: CD = CD(Re,
Ma)
a
U
Ma ∞
=
speed of sound = rate at which infinitesimal
disturbances are propagated from their
source into undisturbed medium
Ma < 1 subsonic
Ma ∼ 1 transonic (=1 sonic flow)
Ma > 1 supersonic
Ma >> 1 hypersonic
CD increases for Ma ∼ 1 due to shock waves and wave drag
Macritical(sphere) ∼ .6
Macritical(slender bodies) ∼ 1
For U > a: upstream flow is not warned of approaching
disturbance which results in the formation of
shock waves across which flow properties
and streamlines change discontinuously

Chapter9.pdf

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Chapter9.pdf