57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
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.
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
2
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:
CD 




p

p
n

î
dA


t

î
dA




w
1 2 S

S
V A
2
CDp
Cf
1
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
CL 
Chapter 9
3




p

p
n

ĵ
dA


1 2 S

V A
2
1
t
<< 1
c
Cf > > CDp
streamlined body
t
1
c
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
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
4
Qualitative Description of the Boundary Layer
Flow-field regions for high Re flow about slender bodies:
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
5
w = shear stress
w  rate of strain (velocity gradient)
=
u
y
y 0
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 w 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.
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
6
The assumptions of the theory are as follows:
Variable
u
v

x

y

order of magnitude
U
<<L
O(1)
O()
1/L
O(1)
1/
O(-1)
2
2
 = /L
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  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.
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
7
  2u  2u 
u
u
p
u  v     2  2 
x
y
x
 x y 
1 1
 -1
2
1
-2
  2v  2v 
v
v
p
u  v     2  2 
x
y
y
 x y 
1 
 1
2 
elliptic
-1
u v
 0
x y
1
1
Retaining terms of O(1) only results in the celebrated
boundary-layer equations
u
u
p
 2u
u v
   2
x
y
x
y
p
0
y
u v
 0
x y
parabolic
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
8
Some important aspects of the boundary-layer equations:
1) the y-momentum equation reduces to
p
0
y
i.e.,
p = pe = constant across the boundary layer
edge value, i.e.,
from the Bernoulli equation:
inviscid flow value!
1 2
p e  U e  constant
2
p e
U e
 U e
i.e.,
x
x
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=
+ appropriate initial conditions @ xi
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
9
There are quite a few analytic solutions to the boundarylayer equations. Also numerical techniques are available
for arbitrary geometries, including both two- and threedimensional 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
u v
 0
x y
Note:
p
=0
x
for a flat plate
u
u
 2u
u v  2
x
y
y
u=v=0 @y=0
u = U
@y=
We now introduce a dimensionless transverse coordinate
and a stream function, i.e.,
 y
U y

x

  xU  f 
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
u
  

 U  f 
y  y
v
Chapter 9
10
f   u / U
 1 U 
f   f 

x 2
x
Substitution into the boundary-layer equations yields
ff   2f   0
Blasius Equation
f  f  0 @  = 0
f  1 @  = 1
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.  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
vs. y
U
v
U  vs. y
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
5x
Re x
U x
Re x  


Chapter 9
11
value of y where u/U = .99
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
12
U f (0)

w 
2x / U

cf 
i.e.,
see below
2 w 0.664 


2
Re x x
U 
: Local friction coeff.
L
b
Cf 
c f dx  2c f ( L)

bL 0
1.328
Note:
=
𝑏 = plate width
Re L
𝐿 = plate length
Wall shear stress:  w  0.332U 
32

x
: Friction drag coeff.
UL

or  w  0.332 U  x 
Re x
Other:

u 
x
*
dy  1.7208
displacement thickness
   1 
U
Re
0

x
measure of displacement of inviscid flow due to
boundary layer

u  u
x

   1 
dy  0.664
U  U
Re x
0
momentum thickness
measure of loss of momentum due to boundary layer
*
H = shape parameter = =2.5916

57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
13
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
14
Quantitative Relations for the Turbulent
Boundary Layer
2-D Boundary-layer Form of RANS equations
u v
 0
x y
u
u
  pe 
 2u 
u  v       2  u v
x
y
x   
y y
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
 For flat plate or  for general case
 momentum equation  (u  v) continuity dy
y 0
w
1
d
 dU



c


2

H
f
dx
U dx
U 2 2
dU
0
flat plate equation
dx

dp
dU
 U
dx
dx
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
15

u u
1  dy
U
 U
0

momentum thickness
*
H

shape parameter

 u
   1  dy
U
0
*
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 Uiˆ .
Boundary-layer thickness arbitrarily defined by y =  99% (where,
 99% is the value of y at u = 0.99U). Streamlines outside  99% will
deflect an amount  * (the displacement thickness). Thus the
streamlines move outward from y  H at x  0 to
y  Y    H   * at x  x1 .
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
16
Conservation of mass:
H  
H
 V  ndA =0= 0 Udy  0
udy
CS
Assume incompressible flow (constant density):
UH   udy   U  u  U dy  UY   u  U dy
Substituting
Y
Y
Y
0
0
0
Y  H   * defines displacement thickness:
u
 *  0Y 1  dy
 U
 * is an important measure of effect of BL on external flow.
Consider alternate derivation based on equivalent flow rate:
δ
δ* Lam=/3
δ* Turb=/8


*
0
 Udy  udy
Inviscid flow about δ* body
Flowrate between  and  of inviscid flow=actual flowrate, i.e., inviscid flow rate
about displacement body = viscous flow rate about actual body
*

*


u

Udy

Udy

udy



1


0
0
0
0  U dy
*
w/o BL - displacement effect=actual discharge
For 3D flow, in addition it must also be explicitly required that  *
is a stream surface of the inviscid flow continued from outside of
the BL.
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
17
Conservation of x-momentum:
F
x

 D 
uV  ndA    U Udy    u  udy 
H
Y
0
0
CS
Y
Drag  D  U 2 H  0 u 2 dy
= Fluid force on plate = - Plate force on CV (fluid)
Again assuming constant density and using continuity:
Y
H 
0
u
dy
U
D  U
2

Y
0
Y
x
u / Udy    u dy    w dx
2
0
0
u
Y u



1

0 U  U dy
2


U
D
where,  is the momentum thickness (a function of x only), an
important measure of the drag.
2D
2 1
CD 

  c f dx
2
U x x x 0
x
cf 
w
1
U 2
2
d c f

dx 2
 cf 
Per unit span
d
d
 xCD   2
dx
dx
 w  U 2
d
dx
Special case 2D
momentum integral
equation for dp/dx = 0
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
18
Simple velocity profile approximations:
u  U (2 y /   y 2 /  2 )
u(0) = 0
u(δ) = U
uy(δ)=0
no slip
matching with outer flow
Use velocity profile to get Cf() and () and then integrate
momentum integral equation to get (Rex)
δ* = δ/3
θ = 2δ/15
H= δ*/θ= 5/2
 w  2U / 
2 U / 
d
d
 cf 
2
 2 (2 /15)
2
1/ 2 U
dx
dx
15 dx
 d 
U
30  dx
2 
U
 / x  5.5 / Re1/2
x
Re x  Ux /  ;
 * / x  1.83 / Re1/2
x
 / x  0.73 / Re1/2
x
CD  1.46 / Re1/2
L  2C f ( L )
10% error, cf. Blasius
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
19
Approximate solution Turbulent Boundary-Layer
Ret = 5×105 3×106 for a flat plate boundary layer
Recrit  100,000
c f d

2 dx
as was done for the approximate laminar flat plate
boundary-layer analysis, solve by expressing cf = cf () and
 = () and integrate, i.e. assume log-law valid across
entire turbulent boundary-layer
u 1 yu *
 ln
B

u* 
neglect laminar sub layer and
velocity defect region
at y = , u = U
U 1 u *
 ln
B

u* 
1/ 2
c 
Re   f 
2
or
2

 cf
1/ 2



1/ 2

 cf  
 2.44 ln Re      5
 2  

c f  .02 Re 
1 / 6
power-law fit
cf ()
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
20
Next, evaluate
d d  u  u 

 1  dy
dx dx 0 U  U 
can use log-law or more simply a power law fit
1/ 7
u  y
Note: cannot be used to
 
U 
obtain cf () since w  
7
     
72
1
d 7
d
 w  c f U 2  U 2
 U 2

2
dx 72
dx
d
Re  1/ 6  9.72
dx

1/7
i.e., much faster
or x  0.16 Re x
growth rate than
6/7
almost linear
x
laminar
boundary layer
cf 
0.027
Re1/7
x
Cf 
0.031 7
 c f  L
Re1/7
6
L
These formulas are valid for a fully turbulent flow over a
smooth flat plate from the leading edge. Assuming the
transition from laminar to turbulent occurs at Re larger than
105, those formulas in general give better results for
sufficiently large Reynolds number ReL > 107.
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
21
Alternate forms by using the same velocity profile u/U =
(y/)1/7 assumption but using an experimentally determined
shear stress formula w = 0.0225U2(/U)1/4 are:

 0.37 Re x 1/5
x
shear stress:
cf 
0.058
Re1/5
x
Cf 
0.074
Re1/5
L
0.029 U 2
w 
Re1/5
x
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 for smooth flat plates are as
follows:
Total
shear-stress
coefficient
Local
shear-stress
coefficient
𝐶𝑓 = (log
0.455
10 𝑅𝑒𝐿 )
2.58

 c f .98 log Re L  .732
L
c f  2 log Re x  .65
 2.3
For the 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.
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
22
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:
𝐶𝑓 =
0.031
1
7
𝑅𝑒𝐿
𝐶𝑓 =
0.074
1
𝑅𝑒𝐿5
1440
−
𝑅𝑒𝐿
−
1700
𝑅𝑒𝐿
or
𝐶𝑓 =
0.455
1700
−
(log10 𝑅𝑒𝐿 )2.58
𝑅𝑒𝐿
with transitions at Ret = 5  105 for all cases.
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
23
Bluff Body Drag
Drag of 2-D Bodies
First consider a flat plate
both parallel and normal to
the flow
C Dp 
Cf 
1
1 2 S
V A
2
1
1 2 S
V A
2
p  p  n  î  0
 w t  îdA
=
1.33
Re1L/ 2
laminar flow
=
.074
Re1L/ 5
turbulent flow
flow pattern
vortex wake
typical of bluff body flow
where Cp based on experimental data
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
1
Chapter 9
24
p  p  n  îdA
1 2 S
V A
2
1
=  C p dA
AS
= 2 using numerical integration of experimental data
Cf = 0
C Dp 
For bluff body flow experimental data used for CD.
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
In general, Drag = f(V, L, , , c, t, , T, etc.)
from dimensional analysis
c/L
CD 
t 


 f  Re, Ar, , , T, etc.
1 2
L L


V A
2
Drag
scale factor
Chapter 9
25
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
26
 a2 
Potential Flow Solution:    U   r   sin 
r 

1
1
1 
p  V 2  p   U 2
ur 
2
2
r 
u 2r  u 2
p  p
Cp 
 1
2
1 2
U

U 
2
C p r  a   1  4 sin 2 
u  

r
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’
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
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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?)
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
33
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
34
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
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57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
37
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:
Z
F

F

F

F

ma

d
b
g
when terminal velocity is attained
F  a  0:
Fd  Fb  Fg
or
1
V02CD Ap    Sphere   fluid  V
2
For the sphere
Ap 

4
d2
and
V
Sphere


6
Sphere
d3
The terminal velocity is:
   sphere   fluid   4 3 d 
V0  

CD  fluid


12
Magnus effect: Lift generation by spinning
Breaking the symmetry causes the lift!
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
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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
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
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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
W  FL 
1
2
C L ,max Vmin
A  Vmin 
2
2W
C L ,max A
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
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Effect of Compressibility on Drag: CD = CD(Re,
Ma)
Ma 
U
a
speed of sound = rate at which infinitesimal
disturbances are propagated from their
source into undisturbed medium
Ma < 1
Ma  1
Ma > 1
Ma >> 1
< 0.3 flow is incompressible,
subsonic
i.e.,   constant
transonic (=1 sonic flow)
supersonic
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
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Fall 2012
Chapter 9
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