57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
23
9.5 Quantitative Relations for the Turbulent Boundary
Layer
Description of Turbulent Flow
V and p are random functions of time in a turbulent flow
The mathematical complexity of turbulence entirely
precludes any exact analysis. A statistical theory is well
developed; however, it is both beyond the scope of this
course and not generally useful as a predictive tool. Since
the time of Reynolds (1883) turbulent flows have been
analyzed by considering the mean (time averaged) motion
and the influence of turbulence on it; that is, we separate
the velocity and pressure fields into mean and fluctuating
components
u  u  u
v  v  v
w  w  w
p  p  p
and for compressible flow
     and T  T  T
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
24
where (for example)
and t1sufficiently large
that the average is
independent of time
1 t 0  t1
u
 udt
t1 t 0
Thus by definition u   0 , etc. Also, note the following
rules which apply to two dependent variables f and g
f g f g
f f
f g  f g
f  f

s s
f = (u, v, w, p)
s = (x, y, z, t)
 fds   f ds
The most important influence of turbulence on the mean
motion is an increase in the fluid stress due to what are
called the apparent stresses. Also known as Reynolds
stresses:
ij  u i u j
=
 u  2
 u v
 u w 
 u v
  v 2
 vw 
 u w 
 vw 
 w  2
Symmetric
2nd order
tensor
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
25
The mean-flow equations for turbulent flow are derived by
substituting V  V  V into the Navier-Stokes equations
and averaging. The resulting equations, which are called
the Reynolds-averaged Navier-Stokes (RANS) equations
are:
Continuity
  V  0 i.e.   V  0 and   V  0
Momentum

DV

ui uj   gk̂  p   2 V

Dt
x j
or

DV
 gk̂  p    ij
Dt
 u u j 
ij    i 
  u i u j
 x j x i 
ij
u1 = u
u2 = v
u3 = w
Comments:
1) equations are for the mean flow
2) differ from laminar equations by Reynolds stress
terms = u i u j
3) influence of turbulence is to transport momentum
from one point to another in a similar manner as
viscosity
4) since u i u j are unknown, the problem is
indeterminate: the central problem of turbulent flow
analysis is closure!
4 equations and 4 + 6 = 10 unknowns
x1 = x
x2 = y
x3 = z
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
26
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
27
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
Turbulence Modeling
Closure of the turbulent RANS equations require the
determination of  u v , etc. Historically, two approaches
were developed: (a) eddy viscosity theories in which the
Reynolds stresses are modeled directly as a function of
local geometry and flow conditions; and (b) mean-flow
velocity profile correlations which model the mean-flow
profile itself. The modern approaches, which are beyond
the scope of this class, involve the solution for transport
PDE’s for the Reynolds stresses which are solved in
conjunction with the momentum equations.
(a) eddy-viscosity: theories
(mainly used with differential methods)
u
In analogy with the laminar viscous
 u v   t
stress, i.e., t  mean-flow rate of strain
y
The problem is reduced to modeling t, i.e.,
t = t(x, flow at hand)
Various levels of sophistication presently exist in
modeling t
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
 t  Vt L t
turbulent
length scale
turbulent
velocity scale
Chapter 9
28
where Vt and Lt are
based an large scale
turbulent motion
The total stress is
 total     t 
u
y
eddy viscosity
(for high Re flow t >> )
molecular
viscosity
Mixing-length theory (Prandtl, 1920)
 u v  c u 
2
u  1
2
v   2

2
v
u
y
based on kinetic
theory of gases
 1 and  2 are mixing lengths
which are analogous to
molecular mean free path,
but much larger
u
y
 u v   2
2
u u
y y
distance across shear layer
Known as a zero
equation model since
no additional PDE’s
are solved, only an
algebraic relation
   y 
= f(boundary layer, jet, wake, etc.)
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
29
Although mixing-length theory has provided a very useful
tool for engineering analysis, it lacks generality. Therefore,
more general methods have been developed.
One and two equation models
Ck 2
t 

C = constant
k2 = turbulent kinetic energy
= u  2  u  2  v 2  w  2
 = turbulent dissipation rate
Governing PDE’s are derived for k and  which contain
terms that require additional modeling. Although more
general then the zero-equation models, the k- model also
has definite limitation; therefore, recent work involves the
solution of PDE’s for the Reynolds stresses themselves.
Difficulty is that these contain triple correlations that are
very difficult to model.
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
30
(b) mean-flow velocity profile correlations
(mainly used with integral methods)
As an alternative to modeling the Reynolds stresses one can
model mean flow profile directly. For simple 2-D flows
this approach is quite food and will be used in this course.
For complex and 3-D flows generally not successful.
Consider the shape of turbulent velocity profiles.
Note that very near the wall laminar must dominate since
 u i u j = 0 at the wall (y = 0) and in the outer part
turbulent stress will dominate. This leads to the three layer
concept:
Inner layer:
viscous stress dominates
Outer layer:
turbulent stress dominates
Overlap layer: both types of stress important
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
31
1) Inner layer (Prandtl, 1930)
u = f(, w, , y)
From dimensional
analysis
note: not f()
 
u   f y
law-of-the-wall
u+ = y+
where:
u 
u
u*
u* = friction velocity =
w / 
yu*
y 


very near the wall:
  w  constant = 
du
dy

u  cy
or
u+ = y+
2) Outer layer (Karmen, 1933)
U e  u outer
 g,  w , , y 
note: independent of  and actually also depends on
From dimensional
analysis
Ue  u
 y

f
 
u*

velocity defect law
dp
dx
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
32
3) Overlap layer (Milliken, 1937)
In order for the inner and outer layers to merge smooth
u 1


ln
y
B
*

u
log-law
.41
5
 and B from experiments and independent of dp/dx
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
33
Note that the y+ scale is logarithmic and thus the inner
law only extends over a very small portion of 
Inner law region < .2
And the log law encompasses most of the boundary-layer.
Thus as an approximation one can simply assume
u 1
 ln y  B
*

u
u   w / 
yu *
y 


is valid all across the shear layer. This is the approach used
in this course for turbulent flow analysis. The approach is a
good approximation for simple and 2-D flows (pipe and flat
plate), but does not work for complex and 3-D flows.
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
34
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
35
Momentum Integral Analysis
Background: History and Modern Approach: FD
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
flat plate equation
0
dx

dp
dU
 U
dx
dx

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.
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
36
Momentum Equation Applied to the Boundary Layer
y = h + δ*= streamline
starts in uniform flow
merges with  at 3
Steady
 = constant
neglect g
v << u = uo  p = constant
i.e., -p = 0
CV = 1, 2, 3, 4
x
 D  drag  b   w dx
pressure force = 0 for v << Uo
0
force on CV
wall shear stress
 Fx  D   uV  dA    uV  dA 
1

3

=   U o2 bh  b  u 2 dy
3
D( x )

2
 U o bh  b  u 2 dy
0
u  Uo
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
37
next eliminate h using continuity
0    V  dA    V  dA
1
3

U o bh  b  udy
0
depends on u(y)

U o h   udy
0


0
0
Dx   bU o  udy  b  u 2 dy

= b  u U o  u dy
0
CD 
CD 
D
1 2
U o bL
2
2
L
2 u 
u 
1 
dy
 
L 0 U o  U o 
 = momentum thickness
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
38
x
CD 
D
1 2
U o A
2
x
w
0
U o2
1
2

b   w dx
0
1 2
U o bL
2

2
L
x dx  2x 



1  w  d


2  1 2  dx
 U o 
2

c f d

2 dx
cf = local skin friction coefficient
momentum integral relation for
flat plate boundary layer
u 
u 
1  dy
uo 
0 uo 


57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
39
Approximate solution for a laminar boundary-layer
Assume cubic polynomial for u(y)
u
 A  By  Cy 2  Dy 3
U
 2u
u 2 0
y
u
u  U ;
0
y
y=0
A=0
y=
C=0
u 3 y 1 y
i.e.,

  
U 2  2 
3
3

2
1
D =  3
2
B=
 3 3 y2 
U3

u y  U 


 2 2   y  0 2 
w
1
d
dp

c

momentum
integral
equation
for
0
f
2
2
dx
dx
U
1 
3
d

U

.
139
dx
U 2  2 
w  
du
dy

u u
1  dy
U
 U
0

57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
i.e.,

4.65x
Re x
.323V 2
w 
Re x
Cf 
Chapter 9
40
Compare with
Exact Blassius
5x
7% 
Re x
.332U 2
Re x
cf 
.646
Re x
.664
Re x
Cf 
1.29
Re L
1.33
Re L
1
L
 x dx
1 2 0 w
U bL
2
span length
total skin-friction drag coefficient
3%
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
41
Approximate solution Turbulent Boundary-Layer
Ret  3 X 106 for a flat plate boundary layer
Recrit  500,000
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 Typed by Stephanie Schrader Fall 1999
Chapter 9
42
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: can not be
 
U 
used to obtain cf ()
since w  
7
     
72

w  cf
1 2
d 7
d
U  U 2
 U 2
2
dx 72
dx
Re  1/ 6  9.72
or
d
dx

 .16 Re x1/ 7
x
  x 6 / 7 almost linear
cf 
.027
Re1x/ 7
Cf 
.031 7
 C f L 
1/ 7
6
Re L
i.e., much faster
growth rate than
laminar
boundary layer
57:020 Mechanics of Fluids and Transport Processes
Professor Fred Stern Typed by Stephanie Schrader Fall 1999
Chapter 9
43
Alternate forms given in text depending on experimental
information and power-law fit used, etc. (i.e., dependent on
Re range.)
Some additional relations given in texts for larger Re are as
follows:
Total
.455
 1700
7
Re
>
10
C

shear-stress
f
2.58
Re L


log
Re
10
L
coefficient

 c f .98 log Re L  .732
L
Local
shear-stress
coefficient
c f  2 log Re x  .65
 2.3
Finally, a composite formula that takes into account both
the initial laminar boundary-layer (with translation at
ReCR = 500,000) and subsequent turbulent boundary layer
.074 1700
is C f  1/ 5 
105 < Re < 107
Re L
Re L