058:0160
Professor Fred Stern
Fall 2010
Chapter 7
1
Chapter 7: Boundary Layer Theory
7.1. Introduction:
Boundary layer flows: External flows around streamlined bodies at
high Re have viscous (shear and no-slip) effects confined close to
the body surfaces and its wake, but are nearly inviscid far from the
body.
Applications of BL theory: aerodynamics (airplanes, rockets,
projectiles), hydrodynamics (ships, submarines, torpedoes),
transportation (automobiles, trucks, cycles), wind engineering
(buildings, bridges, water towers), and ocean engineering (buoys,
breakwaters, cables).
7.2 Flat-Plate Momentum Integral Analysis & Laminar approximate
solution
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 .
058:0160
Professor Fred Stern
Chapter 7
2
Fall 2010
Conservation of mass:
H  
H
 V ndA=0= 0 Udy  0
udy
CS
Assuming incompressible flow (constant density), this relation
simplifies to
UH   udy   U  u  U dy  UY   u  U dy
Note:
Y
Y
Y
0
0
0
Y  H   * , we get the definition of displacement thickness:
u
 *  0Y 1  dy
 U
 * ( a function of x only) is an important measure of effect of BL on
external flow. To see this more clearly, consider an alternate derivation
based on an equivalent discharge/flow rate argument:
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 = equivalent 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.
058:0160
Professor Fred Stern
Chapter 7
3
Fall 2010
Conservation of x-momentum:
 Fx   D 

CS
H
Y
0
0
 uV  ndA    U Udy     u  udy 
Y
Drag  D  U 2 H  0 u 2 dy = Fluid force on plate = - Plate
force on CV (fluid)
Y u
H

Again assuming constant density and using continuity:
0 U dy
Y
x
2 Y
D  U 0 u / Udy   u 2 dy  0 w dx
0
D

2
U
u
Y u
 0 1 
U
U


dy

where,  is the momentum thickness (a function of x only), an
important measure of the drag.
2D
2 1
CD 


C dx Per unit span
U 2 x x x 0 f
x
Cf 
w
1
U 2
2
d C f

dx
2
 Cf 
d
xCD   2 d
dx
dx
 w  U 2
d
dx
Special case 2D
momentum integral
equation for px = 0
058:0160
Professor Fred Stern
Chapter 7
4
Fall 2010
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
dx
dx
1 / 2 U
15dx
d 
U
30dx
2 
U
 / x  5.5 / Re1x/ 2
Re x  Ux /  ;
 * / x  1.83 / Re1x/ 2
 / x  0.73 / Re1x/ 2
C D  1.46 / Re1L/ 2  2C f ( L)
10% error, cf. Blasius
058:0160
Professor Fred Stern
Chapter 7
5
Fall 2010
7.3. Boundary layer approximations, equations and comments
U, ,
y
x
u=v=0
2D NS, =constant, neglect g
ux  vy  0
1 p
  (u xx  u yy )
 x
1 p
vt  uv x  vv y  
  (v xx  v yy )
 y
ut  uu x  vu y  
Introduce non-dimensional variables that includes scales such that all
variables are of O(1):
x*  x / L
y
y* 
Re
L
t *  tU / L
u*  u /U
v* 

Re
U
p  p0
p* 
U 2
Re  UL / 
058:0160
Professor Fred Stern
Fall 2010
Chapter 7
6
The NS equations become (drop *)
ux  vy  0
1
u xx  u yy
Re
1
1
1
(vt  uvx  vv y )   p y  2 vxx 
v yy
Re
Re
Re
ut  uu x  vu y   px 
For large Re (BL assumptions) the underlined terms drop out and the BL
equations are obtained.
Therefore, y-momentum equation reduces to
py  0
i.e. p  p ( x, t )
 px    (U t  UU x )
From Euler/Bernoulli equation for
external flow
2D BL equations:
u x  v y  0;
ut  uu x  vu y  (U t  UU x )  u yy
Note:
(1)
(2)
(3)
(4)
U(x,t), p(x,t) impressed on BL by the external flow.
2
 0 : i.e. longitudinal (or stream-wise) diffusion is
2
x
neglected.
Due to (2), the equations are parabolic in x. Physically, this
means all downstream influences are lost other than that
contained in external flow. A marching solution is possible.
Boundary conditions
058:0160
Professor Fred Stern
Chapter 7
7
Fall 2010
matching
inlet
δ
Solution by
marching
y
x
X0
No slip
No slip: u  x,0, t   v x,0, t   0
Initial condition: u  x, y,0 known
Inlet condition: u  x0 , y, t  given at x0
Matching with outer flow: u  x, , t   U  x, t 
(5)
When applying the boundary layer equations one must keep in
mind the restrictions imposed on them due to the basic BL
assumptions
→ not applicable for thick BL or separated flows (although
they can be used to estimate occurrence of separation).
(6)
Curvilinear coordinates
058:0160
Professor Fred Stern
Chapter 7
8
Fall 2010
Although BL equations have been written in Cartesian
Coordinates, they apply to curved surfaces provided δ << R and
x, y are curvilinear coordinates measured along and normal to
the surface, respectively. In such a system we would find under
the BL assumptions
py 
u 2
R
Assume u is a linear function of y: u  Uy
dp U 2 y 2

dy
R 2
p( )  p(0) 
U 2
3R
Or
p

;

U 2 3R therefore, we require δ << R

058:0160
Professor Fred Stern
Chapter 7
9
Fall 2010
(7)
Practical use of the BL theory
For a given body geometry:
(a) Inviscid theory gives p(x) → integration gives L,D = 0
(b) BL theory gives → δ*(x), τw(x), θ(x),etc. and predicts
separation if any
(c) If separation present then no further information → must
use inviscid models, BL equation in inverse mode, or NS
equation.
(d) If separation is absent, integration of τw(x) → frictional
resistance body + δ* , inviscid theory gives → p(x), can go
back to (2) for more accurate BL calculation including
viscous – inviscid interaction
(8)
Separation and shear stress
At the wall, u = v = 0 → u yy 
1

px
1st derivative u gives τw →  w  u y
τw = 0 separation
2nd derivative u depends on
px
w
058:0160
Professor Fred Stern
Fall 2010
Chapter 7
10
Inflection point
7.4. Laminar Boundary Layer - Similarity solutions (2D, steady,
incompressible): method of reducing PDE to ODE by appropriate
similarity transformation
ux  vy  0
uu x  vu y  UU x  u yy
BCs:
u  x,0  v x,0  0
u  x,    U  x 
+ inlet condition
058:0160
Professor Fred Stern
Chapter 7
11
Fall 2010
 y 
u  x, y 

 F 
g  x  related to   x 
For Similarity U  x 
 g  x   expect
Or in terms of stream function  :
For similarity
u   y v   x
  U  x g  x  f  
u   y  Uf ' v   x
  y g x 
 (U x gf  Ug x f  Ug x f ' )
BC:
u  x,0  0  U ( x) f (0)  0  f (0)  0
v x,0  0  U x ( x) g ( x) f (0)  U ( x) g x ( x) f (0)
 U ( x) g x ( x)  0  f (0)  0
 U x ( x) g ( x)  U ( x) g x ( x)  f (0)  0
 f (0)  0
u  x,    U  x   U ( x) f ()  U  x   f ()  1
Write boundary layer equations in terms of 
 y yx   x yy  UU x   yyy
Substitute
 yy  Uf '' g
 yyy  Uf
'''
g
2
 xy  U x f '  Uf ''g x / g
Assemble them together:
Uf ' U x f '  Uf '' ggx   U x gf  Ug x f  Ug xf ' Uf '' g 


 UU x   U f ''' g 2 
058:0160
Professor Fred Stern
Chapter 7
12
Fall 2010
U
UU x f '2  UU x ff ''  U 2 g x g ff ''  UU x   2 f '''
g
U
U
UU x f '2  Ug x ff ''  UU x   2 f '''
g
g
f 
'''
g

Ug x
ff 
''
C1
g2


U x 1 f
'2
 0
C2
Where for similarity C1 and C2 are constant or function  only
 i.e. for a chosen pair of C1 and C2 U
(Potential flow is NOT known a priori)
 Then solution of
x  , g x  can be found


f '''  C1 ff ''  C2 1  f '2  0 gives
f    u x, y  ,
u
w  
y

Uf ''  0 
g
w
, , *,, H, Cf, CD
The Blasius Solution for Flat-Plate Flow
U=constant U x
U
Then C1  gg x
 0  C2  0

d 2 2C1
g 
dx
U
 
Let
g  x   2C1x U 1 2
C1  1 , then g  x   2x
U
y
U
2x
058:0160
Professor Fred Stern
Chapter 7
13
Fall 2010
Blasius equations
for Flat Plate
Boundary Layer
f '''  ff ''  0
f 0  f ' 0  0, f '    1
Solutions by series technique or numerical

u

 0.99 when   3.5 
x
U
*
 

u

0 1  U

u


1

0
U


5
Re x
Re x 

2x


'
dy  0 1  f d
U


Uf '' 0 
u
w  

y w
2x U

CD 
D
1
U 2 L
2
L
  Cf
0
v f '  f

 1
U
2 Re x
Cf 

dx 1.328

L
Re L ;
for
*
w
Re L 
UL

x


1
U 2
2
Re x  1


u

' ' 2x
d
 dy  0 1  f f
U
U
*
 H  2.59
So,

Ux
;

x

1.7208
Re x

0.664
Re x
0.664 

Re x x
058:0160
Professor Fred Stern
Oseen
Blasius
Chapter 7
14
Fall 2010
CD
3-226 (3rd
edition,vicous
flows)
ReL
<1
100<Re<Retr~3
×106
LE Higher
order
correction
C D  1.328 / Re L  2.3 / Re L
Rex small therefore local breakdown
of BL approximation
Similar breakdown occurs at Trailing edge.
From triple – deck theory the correction is
+2.661/ Re7L/ 8
058:0160
Professor Fred Stern
Fall 2010
Chapter 7
15
058:0160
Professor Fred Stern
Chapter 7
16
Fall 2010
Falkner-Skan Wedge Flows

f  f  

f '''  C1 ff ''  C2 1  f '2  0
f 0  f ' 0  0, f '    1
C1 
g

Consider
Ug x
Ug 
2
x
C2 
g2

  y g x 
u U  f '  
Ux
(Blasius Solution: C2=0, C1=1)
 2Ugg x  g 2U x
 2Ugg x  2 g 2U x  g 2U x
 2 g Ug x  g 2U x
 2C1  C2
Hence 
Ug x   2C1  C2 ,
2
C2 
Choose C1=1 and C2 arbitrary=C,
Integrate
Combine
2
C  g Ux 
Ug 2   2  C x
Ux
C 1

U 2C x
ln U 
Then
C
ln x  k
2C
U  x   kxC 2 C 
g x 
Similarity
form of BL
eq.
 2  C 
k
1C
x 2 C
g2

Ux
058:0160
Professor Fred Stern
Chapter 7
17
Fall 2010
Change constants
U  x   kx m


f  ff   1  f
'''
''
y
m 1 U
y
g
2 x
'2
 0 ,

f 0  f ' 0  0

2m
m

2
m 1 ,
f '    1
Solutions for  0.19884    1.0
Separation (  w  0 )
Solutions show many commonly observed characteristics of BL flow:
 The parameter  is a measure of the pressure gradient, dp dx .
For   0 , dp dx  0 and the pressure gradient is favorable. For
  0 , the dp dx  0 and the pressure gradient is adverse.
 Negative  solutions drop away from Blasius profiles as separation
approached
 Positive  solutions squeeze closer to wall due to flow acceleration
 Accelerated flow:  max near wall
 Decelerated flow:  max moves toward
 2
058:0160
Professor Fred Stern
Fall 2010
Chapter 7
18
058:0160
Professor Fred Stern
Fall 2010
Chapter 7
19
7.5. Momentum Integral Equation
Historically similarity and other AFD methods used for idealized flows
and momentum integral methods for practical applications, including
pressure gradients.
Momentum integral equation, which is valid for both laminar and
turbulent flow:

 BL form of momentum equation  u  U continuity dy
y 0
d
w
 dU
1


C
H



2

f
dx
U dx
U 2 2
dU
0
For flat plate equation
dx
 u

u
1  dy;
U
0U 

*
H ;

*


0
   1 
u
dy
U
  p  1 
 
Momentum: uu x  vu y  
x     y
The pressure gradient is evaluated form the outer potential flow using
Bernoulli equation
p
1
U 2  constant
2
1
px   2UU x  0
2
 p x  UU x
058:0160
Professor Fred Stern
Chapter 7
20
Fall 2010
u  U u x  v y   uu x  uv y  Uu x  Uv y ,
Continuity
uu x  vu y  UU x 

1

1

 y  uu x  uv y  Uu x  Uv y  0
0
0
 y  2uu x  vu y  UU x  uv y  Uu x  Uv y 




uU  u 2  U  u U x  vU  vu 
x
y

0
0

 

   y dy  (    w ) /    u U  u dy  U x  U  u dy  vU  vu  0
x 0
0 
0
1

w   2 u  u 

 U  1  dy  U x  U  u dy  
 x  0 U  U 
0

U 2 x  2UU x  U x *
Cf
2



1 dU
d
 2   *
dx
U dx
Cf
d
 dU
*
,H

 2  H 
2
dx
U dx

w

1



C



2

H
Ux
f
x
2
2
U
U
058:0160
Professor Fred Stern
Chapter 7
21
Fall 2010
Historically two approaches for solving the momentum integral equation
for specified potential flow U(x):
1. Guessed Profiles
2. Empirical Correlations
Best approach is to use empirical correlations to get integral parameters
(, *,, H, Cf, CD) after which use these to get velocity profile u/U
Thwaites Method
Multiply momentum integral equation by

U
 w U d  2 dU


2  H 
U  dx  dx
LHS and H are dimensionless and can be correlated with pressure
 2 dU
as shear and shape-factor correlations
gradient parameter  
 dx
 w
 S     (  0.09)0.62
U
5
H   * /   H      ai (0.25   )i
i 0
ai = (2, 4.14, -83.5, 854, -3337, 4576)
Note
U d 1 d   2 
 U  
 dx 2 dx   
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Fall 2010
Substitute above into momentum integral equation
1 d  2 
S ( )  U     2  H 
2 dx   
d  / U x 
U
 2S   2  H    F  
dx
F     0.45  6 based on AFD and EFD
dU
2
Define z 
so that   z
dx

dz
dU
 0.45  6  0.45  6 z
dx
dx
dz
dU
U
 6z
 0.45
dx
dx
1 d
6
i.e.
zU
 0.45
5 dx
U
U

6

x
zU  0.45  U 5 dx  C
0
0.45
5




U
dx

6

U 0
x
2
2
0
 0 ( x  0)  0 and U(x) known from potential flow solution
Complete solution:
 2 dU
     
 dx
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 w
 S  
U
 *  H  
Accuracy: mild px  5% and strong adverse px (w near 0)  15%
i. Pohlhausen Velocity Profile:
u
 f    a  b 2  c 3  d 4 with   y

U
a, b, c, d determined from boundary conditions
U
1)
y  0  u = 0, u yy  
2)
y    u  U , u y  0 , u yy  0

Ux
No slip is automatically satisfied.
F    2  2 3   4
G   

6
1   3

separation
u
 F    G  ,  12    12
U
 2 dU
2

  px
 dx
U
pressure gradient parameter related to
(experiment:  separation = -5)
2 
 37



      


 315 945 9072 


Profiles are fairly realistic, except near separation. In guessed profile
methods u/U directly used to solve momentum integral equation
numerically, but accuracy not as good as empirical correlation methods;
therefore, use Thwaites method to get  etc., and then use  to get and
plot u/U.
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ii. Howarth linearly decelerating flow (example of exact solution
of steady state 2D boundary layer)
Howarth proposed a linearly decelerating external velocity distribution
x

U ( x)  U 0 1   as a theoretical model for laminar boundary layer study.

L
Use Thwaites’s method to compute:
a) Xsep
x
b) C f   0.1

L
Note Ux = -U0/L
Solution
0.45
2
 
x
5
U
1 

0
6
x 0 

U 06 1  
 L
L  x 
x

0
.
075
dx
1  

L
U 0  L 
5
can be evaluated for given L, ReL
(Note:
  0  x  0,
)
 xL
 x   6 
 2 dU
 0.0751    1

 dx
 L 

6

 1

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 sep  0.09 
Chapter 7
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Fall 2010
X sep
L
 0.123
3% higher than exact solution =0.1199
x

C f   0.1 i.e. just before separation

L
  0.0661
1
S     0.099  C f Re
2
2(0.099)
Cf 
Re
Compute Re in terms if ReL
 2  0.075
L
U0
1  0.1
6

 1  0.0661
L
U0
2
L 0.0661

0
.
0661

2
Re L
U
L
0
 0.257

L Re 12
L
Re 

1
Re L  0.257 Re L 2
L
1
20.099 
Cf 
Re L 2  0.77 Re L1/ 2
0.257
To complete
solution must
specify ReL
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Fall 2010
Consider the complex potential
a
a
F  z   z 2  r 2 e 2 i
2
2
a
2
a
  ImF  z   r 2 sin 2
2
  ReF  z   r 2 cos 2
Orthogonal rectangular hyperbolas
 : asymptotes y = ± x
 : asymptotes x=0, y=0
1
V     r eˆr   eˆ
r

v r  ar cos 2
   0 (flow direction as shown)
v  ar sin 2
2
V  vr cos iˆ  sin ˆj   v  sin iˆ  cos ˆj  
vr cos  v sin  iˆ  vr sin   v cos  ˆj
Potential flow slips along surface: (consider 
1) determine a such that vr  U 0 at r=L, 
 90  )
 90 
vr  aL cos(2  90)  U 0  aL  U 0 , i.e. a  
2) let U  x   vr at x=L-r:
 vr  aL  x cos(2  90)  U ( x)
U
x
Or : U ( x)  a( L  x)  0 ( L  x)  U 0 (1  )
L
L
U0
L
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Chapter 7
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Fall 2010
Chapter 7
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7.6. Turbulent Boundary Layer
1. Introduction: Transition to Turbulence
Chapter 6 described the transition process as a succession of TollmienSchlichting waves, development of Λ - structures, vortex decay and
formation of turbulent spots as preliminary stages to fully turbulent
boundary-layer flow.
The phenomena observed during the transition process are similar for
the flat plate boundary layer and for the plane channel flow, as shown in
the following figure based on measurements by M. Nishioka et al.
(1975). Periodic initial perturbations were generated in the BL using an
oscillating cord.
For typical commercial surfaces transition occurs at Re x ,tr  5  10 5 .
However, the transition can be delayed to Re x ,tr  3 10 6 by different ways
such as having very smooth walls and/or very low turbulent wind tunnel.
2. Reynolds Average of 2D boundary layer equations
u  u  u ;
v  v  v;
w  w  w;
p  p  p ;
Substituting u, v and w into continuity equation and taking the time
average we obtain,
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u v w


0
x y z
u ' v ' w'


0
x y z
Similarly for the momentum equations and using continuity (neglecting
g),

DV
 p    ij
Dt
Where
 ui u j 
   ui' u 'j

 ij   
 x j xi 


Laminar
Turbulent
Assume
a.
  x   x which means v  u ,



x
y
b. mean flow structure is two-dimensional:

w  0 , z  0
'2
Note the mean lateral turbulence is actually not zero, w  0 , but its z
derivative is assumed to vanish.
Then, we get the following Reynolds averaged BL equations for 2D
incompressible steady flow:
u v

0
x y
u
dU e 1 
u
u
v
 Ue

dx
 y
x
y
p
v '2
 
y
y
Continuity
x-momentum
y-momentum
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Where U e is the free-stream velocity and:
 
u
 u 'v '
y
Note:
 The equations are solved for the time averages u and v
 The shear stress now consists of two parts: 1. first part is due to
the molecular exchange and is computed from the time-averaged
field as in the laminar case; 2. The second part appears
additionally and is due to turbulent motions.
 The additional term is new unknown for which a relation with
the average field of the velocity must be constructed via a
turbulence model.
Integrate y- momentum equation across the boundary layer
p  p e  x   v ' 2
So, unlike laminar BL, there is a slight variation of pressure across the
turbulent BL due to velocity fluctuations normal to the wall, which is no
more than 4% of the stream velocity and thus can be neglected. The
Bernoulli relation is assumed to hold in the inviscid free-stream:
dpe / dx   U e dU e / dx
Assume the free stream conditions, U e  x  is known, the boundary
conditions:
No slip:
u  x ,0   v  x , 0   0
u  x,    U e  x 
Free stream matching:
3. Momentum Integral Equations valid for BL solutions
The momentum integral equation has the identical form as the
laminar-flow relation:
C

d
 dU e
 2  H 
 w2  f
dx
U e dx
2
U e
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Fall 2010
For laminar flow:
( C f , H , ) are correlated in terms of simple parameter
 2 dU e

 dx
For Turbulent flow:
( C f , H , ) cannot be correlated in terms of a single parameter.
Additional parameters and relationships are required that model the
influence of the turbulent fluctuations. There are many possibilities all of
which require a certain amount of empirical data. As an example we will
review the  method.
4. Flat plate boundary layer (zero pressure gradient)
a. Log law analysis of Smooth flat plate
Assume log-law can be used to approximate turbulent velocity profile
and use to get Cf=Cf() relationship
u /u 
*
1

ln
yu *

 B where  =0.41 and B = 5
At y=δ (edge of boundary layer)
u *
U e / u  ln
B


*
1
However:
1/ 2
 
U e / u *  U e /  w 
 
 1 / 2 U e 2 C f
 U e /



C
u * U e u *


 Re  f

 Ue
 2
 2

C
 f
1/ 2




 2

C
 f
1/ 2




1/ 2



1/ 2





Cf
 ln Re  
 
 2

1
1/ 2




B

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Following a suggestion of Prandtl, we can forget the complex log law
and simply use a power-law approximation:
C f  0.02 Re 1 / 6
b. Use
u / U e profile to get , Cf, , *, and H for smooth plate
d 1
 C f U 2
dx 2
d
or : C f  2
dx
 w  U 2
1 / 6
LHS: From Log law or C f  0.02 Re
d
u
/
U
RHS: Use
e to get
dx
Example:


7
u
u
(1  )dy  
72
Ue
Ue
0
d
7
2 d
Cf  2
 0.02 Re1/ 6  2  U e
dx
72
dx
d (Re )
d (Re )
1
 Re1/ 6  9.72


d (Re x )
1 / 6
d (Re x )
Re
9.72
u /Ue  ( y /  )
1/ 7
  
Assuming that: =0 at x=0 or Re=0 at Rex=0:
 / x  0.16 / Re1x/ 7 or :   x 6 / 7
Turbulent BL has almost linear growth rate which is much faster than
laminar BL which is proportional to x1/2.
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Fall 2010
Other properties:
C f  0.027 / Re1/x 7
0.0135 1/ 7  6 / 7U 13/ 7
 w,turb 
x1/ 7
7
CD  0.031/ Re1/L 7  C f ( L)
6
1
*  
8
H   * /   1.3
w,turb decreases slowly with x, increases with  and U2 and insensitive to

c. Influence of roughness
The influence of roughness can be analyzed in an exactly analogous
manner as done for pipe flow i.e.

u 
1

ln
yu *

B (  )  
1
 B  B (  )
×
ln(1  0.3  )

i.e. rough wall velocity profile shifts downward by a constant amount
B(  ) which, increases with
   u* / 
A complete rough-wall analysis can be done using the composite loglaw in a similar manner as done for a smooth wall i.e. determine Cf(δ)
and θ(δ) from × and equate using momentum integral equation
C f ( )  2
d
 ( )
dx
Then eliminate δ to get C f ( x,  / x)
However, analysis is complicated: solution is Fig. 7.6. For fully roughflow a curve fit to the Cf and CD equations is given by,
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Fall 2010
Fig. 7.6 Drag coefficient of laminar and turbulent boundary layers on
smooth and rough flat plates.
x
C f  (2.87  1.58 log )  2.5

L
CD  (1.89  1.62 log )  2.5
Fully rough flow

Again, shown on Fig. 7.6. along with transition region curves developed
by Schlichting which depend on Ret = 5×105
3×106
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Fall 2010
Chapter 7
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5. Boundary layer with pressure gradient
ux  vy  0
1 

uu x  vu y   ( p /  ) 
 y
x
u
     u v
y
The pressure gradient term has a large influence on the solution. In
particular, adverse pressure gradient (i.e. increasing pressure) can cause
flow separation. Recall that the y momentum equation subject to the
boundary layer assumptions reduced to
py= 0 i.e. p = pe = constant across BL.
That is, pressure (which drives BL equations) is given by external
inviscid flow solution which in many cases is also irrotational. Consider
a typical inviscid flow solution (chapter 8)
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Chapter 7
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Even without solving the BL equations we can deduce information about
the shape of the velocity profiles just by evaluating the BL equations at
the wall (y = 0)
 2u pe

y 2 x
pe
dU e
 - Ue
where
x
dx

which, shows that the curvature of the velocity profile at the wall is
related to the pressure gradient.
Effect of Pressure Gradient on Velocity Profiles
Point of inflection: a point where a graph changes between concave
upward and concave downward.
The point of inflection is basically the location where second derivative
 2u
of u is zero, i.e. y 2  0
(a) favorable gradient ( px<0, Ux>0)
at y ≥ 0: uy>0, uyy<0
No point of inflection i.e. curvature is negative all across the BL and BL
is very resistant to separation. Note uyy()<0 in order for u to merge
smoothly with U.
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(b) zero gradient (px = 0, Ux =0)
at y = yPI=0: uy>0, uyy=0
at y>0: uy>0, uyy<0
(c) weak adverse gradient ( px>0, Ux<0)
at y < yPI: uy>0, uyy>0
at y = yPI: uy>0, uyy=0
at y > yPI: uy>0, uyy<0
Chapter 7
38
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Chapter 7
39
(d) critical adverse gradient ( px>0, Ux<0)
Note that τw = 0 since uy=0.
at y =0 : uy=0, uyy>0
at y < yPI: uy>0, uyy>0
at y = yPI: uy>0, uyy=0
at y > yPI: uy>0, uyy<0
(e) excessive adverse gradient ( px>0, Ux<0)
at y =0 : uy<0, uyy>0
at y < yPI: uy increases gradually to positive value, uyy>0
at y = yPI: uy>0, uyy=0
at y > yPI: uy>0, uyy<0
PI in flow, backflow near wall i.e. separated flow region
i.e. main flow breaks away or separates from the wall: large increase in
drag and loss of performance:
Hseparation = 3.5 laminar
= 2.4 turbulent
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Fall 2010
6. -Method
As mentioned earlier, the momentum integral equation for turbulent
flow has the identical form as the laminar-flow relation:
 dU e
d C f

 2  H 
2
dx
U e dx
(I)
With U(x) assumed known, there are three unknown C f , H , for
turbulent flow. Thus, at least two additional relations are needed to find
unknowns. There are many possibilities for additional relations all of
which require a certain amount of empirical data. As an example we will
review the  method.
Cole’s law of the wake:
By adding the wake to the log-law, the velocity profile for both overlap
and outer layers can be written as:
u 
1

where
ln y   B 
2

f ( )
  y /

f ( )  sin 2 (  )  3 2  2 3
2
 A/ 2
The quantity  is called Coles' wake parameter.
By integrating wall-wake law across the boundary layer:
H
H 1
2  3.179  1.5 2
a ( ) 
 (1   )
U 1  
Re 

exp(   B  2 )

H
  a ( )
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Fall 2010
If we eliminate  between these formulas, we obtain a unique relation
2
among C f  2 /  , H and  :
H 2

2



2
/

2
/
[
(
)
]
C
a
 f
H 1

2  3.179  1.5 2

 a ( ) 
 (1   )


U 1  
exp(   B  2 )

Re 

H

(II)
Clauser's equilibrium parameter :
For outer layer,
U e  u  f ( w ,  , y,  ,
dp
)
dx
Using dimensional analysis:
Ue  u
y  dp
 g( ,
)
1/ 2
  w dx
( w /  )
Clauser (1954) replaced by displacement thickness :
Ue  u
y
 g( ,  )
1/ 2
( w /  )

 * dp
 dU e
  2 H

U e dx
 w dx
 is called Clauser's equilibrium parameter.
Das (1987) showed that EFD data points fit into the following
polynomial correlation:
  0.4  0.76  0.42 2
Therefore:
 dU e
 2 H
 0.4  0.76  0.42 2
U e dx
(III)
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If we eliminate  using that
Re 
U


1 
exp(   B  2 ) ,
H
we obtain
2
another relation among C f  2 /  , H and  .
Equations (I), (II), and (III) can be solved simultaneously using say a
Runge-Kutta method to find C f , H , . Equations are solved with initial
condition for (x0) and integrated to x=x0+x iteratively. Estimated 
gives Re and ,  gives H. Lastly Cf is evaluated using Re and H.
Iterations required until all relations satisfied and then proceed to next
x.
7. 3-D Integral methods
Momentum integral methods perform well (i.e. compare well with
experimental data) for a large class of both laminar and turbulent 2D
flows. However, for 3D flows they do not, primarily due to the inability
of correlating the cross flow velocity components.
The cross flow is driven by
outer potential flow U(x,z).
p
, which is imposed on BL from the
z
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3-D boundary layer equations


( p /  )  u yy  (u v);
x
y


uwx  vw y  wwz   ( p /  )  w yy  (vw);
y
z
u x  v y  wz  0;
uu x  vu y  wu z  
 closure equations
Differential methods have been developed for this reason as well as for
extensions to more complex and non-thin boundary layer flows.
7.7 Separation
What causes separation?
The increasing downstream pressure slows down the wall flow and
can make it go backward-flow separation.
dp dx  0 adverse pressure gradient, flow separation may occur.
dp dx  0 favorable gradient, flow is very resistant to separation.
Previous analysis of BL was valid before separation.
Separation Condition
 u 
 w      0
 y  y 0
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Note: 1. Due to backflow close to the wall, a strong thickening of the
BL takes place and BL mass is transported away into the
outer flow
2. At the point of separation, the streamlines leave the wall at a
certain angle.
Separation of Boundary Layer
Notes:
1. D to E, pressure drop, pressure is transformed into kinetic energy.
2. From E to F, kinetic energy is transformed into pressure.
3. A fluid particle directly at the wall in the boundary layer is also
acted upon by the same pressure distribution as in the outer flow
(inviscid).
4. Due to the strong friction forces in the BL, a BL particle loses so
much of its kinetic energy that is cannot manage to get over the
“pressure gradient” from E to F.
5. The following figure shows the time sequence of this process:
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a. reversed motion begun at the trailing edge
b. boundary layer has been thickened, and start of the reversed
motion has moved forward considerably.
c. and d. a large vortex formed from the backflow and then soon
separates from the body.
Examples of BL Separations (two-dimensional)
Features: The entire boundary layer flow breaks away at the point of
zero wall shear stress and, having no way to diverge left or right, has to
go up and over the resulting separation bubble or wake.
1. Plane wall(s)
Thin wall
(a). Plane stagnation-point flow: no separation on the streamlines of
symmetry (no wall friction present), and no separation at the wall
(favorable pressure gradient)
(b).Flat wall with right angle to the wall: flow separate, why?
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2. Diffuser flow:
3. Turbulent Boundary Layer
(a)
(b)
Influence of a strong pressure gradient on a turbulent flow
(a) a strong negative pressure gradient may re-laminarize a flow
(b) a strong positive pressure gradient causes a strong boundary
layer top thicken. (Photograph by R.E. Falco)
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Examples of BL Separations (three-dimensional)
Features: unlike 2D separations, 3D separations allow many more
options.
There are four different special points in separation:
(1). A nodal Point, where an infinite number of surface streamlines
merged tangentially to the separation line
(2). A saddle point, where only two surface streamlines intersect and
all others divert to either side
(3). A focus, or spiral node, which forms near a saddle point and
around which an infinite number of surface streamlines swirl
(4). A three-dimensional singular point, not on the wall, generally
serving as the center for a horseshoe vortex.
1. Boundary layer separations induced by free surface (animation)
CFDSHIP-IOWA
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2. Separation regions in corner flow
3. 3D separations on a round-nosed body at angle of attack
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Video Library (animations from “Multi-media Fluid Mechanics”,
Homsy, G. M., etc.)
Conditions Producing Separation
Leading edge separation
Separations on airfoil (different attack angles)
Separations in diffuser
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Effect of body shape on separation
Flow over cylinders: effect of Re
Laminar and Turbulent separation
Flow over spheres: effect of Re
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Flow over edges and blunt bodies
Effect of separation on sports balls
Flow over a truck