058:0160
Professor Fred Stern
Chapters 3 & 4
1
Fall 2004
Chapter 6: Viscous Flow in Ducts
Entrance, developing, and fully developed flow
L  f ( D, v,  ,  )
L
 theorem 
 f (Re)
D
e
e
i
Laminar Flow: Recrit ~ 2000
L / D  .06 Re
Re < Recrit
Re > Recrit
e
L
e max
 .06 Re D ~ 138D
crit
Max Le for laminar flow
laminar
turb.
058:0160
Professor Fred Stern
Chapters 3 & 4
2
Fall 2004
Turbulent flow:
Re
4000
104
105
106
107
108
Le/D
18
20
30
44
65
95
1/ 6
L / D ~ 4.4 Re
e
(Relatively shorter than for
laminar flow)
Laminar vs. Turbulent Flow
058:0160
Professor Fred Stern
Chapters 3 & 4
3
Fall 2004
Laminar pipe flow:
1. CV Analysis
Continuity:
0    v  dA  Q  Q  const.
1
2
CS
due to A  A ,   const., and v  v
i.e. v  v
1
2
1
2
ave
Momentum:

p  p R   2RL  R L sin
  m( B v  B v )
F  
  



z
/
L
W
0
p
2
x
1
2
2
w
pR   2RL  R z  0
2
2
w
2
2
2
2
058:0160
Professor Fred Stern
p  z 
Chapters 3 & 4
4
Fall 2004
2 L
R
w
h  h  h  ( p /   z ) 
1
 
w
Or
2
2 L
8 R
w
R h
R dh

2 L
2 dx
R d

( p  z )
2 dx
Since CV can be for r = R, can be generalized to (where
τ(R) = - τw i.e. τ(r) positive)

r d
( p  z )
2 dx
i.e. shear stress varies linearly in r across pipe for either
laminar or turbulent flow
Energy:
p

1


1
2g
p
v z 
1

1
2
h  h 
L


2
2g
v z h
2
2
L
2 L
 R
w
 once τw is known, we can determine pressure drop
058:0160
Professor Fred Stern
Chapters 3 & 4
5
Fall 2004
In general,
roughness
   (  ,V ,  , D,  )
w
w
Πi Theorem
8
 f  friction factor  f (Re ,  / D)
v
w
D
2
where
Re 
vD

D
2
L v
h  h  f
D 2g
L
Darcy-Weisbach Equation
f (ReD, ε/D) still needs to be determined. For laminar
flow, there is an exact solution for f since laminar pipe
flow has an exact solution. For turbulent flow,
approximate the solution for f using log-log as per Moody
diagrams and discussed later.
2. Differential Analysis
Continuity:
v  0
Use cylindrical coordinates (r, θ, z) where z replaces x in previous
CV analysis
058:0160
Professor Fred Stern
Chapters 3 & 4
6
Fall 2004
1
1
v
(rv ) 
(v ) 
0
r r
r r
z
z
r
^
^
^
where v  v e  v e  v e
r
r


z
z
Assume vφ = 0 i.e. no swirl and fully developed flow
v
 0 , which shows vr = constant = 0 since vr(R)=0
z
z
^
^
z
z
 v  v e  u (r ) e
z
Momentum:

Dv
 ( p  z )   v
Dt
2
only z equation:
u


 v  u    ( p  z )   u
z
 t

 
2

1   u 
0   ( p  z )  
r 
z  r
r  

r


f (z)
f (r )
 both terms must be constant
058:0160
Professor Fred Stern
Chapters 3 & 4
7
Fall 2004
^
u
2
^
r dp
 A ln r  B
4 dz
u ( r  0) finite
p  p  z

A=0
^
B

u (r = 0) = 0
2
R dp
4  dz
^
u (r ) 
^
r R d p
4 dz
2
2
v
u
u
max
R dp
4 dz
u
       
r
 z r 
r
 u (0)  
2
As per CV analysis
^

u
 
y
w
rR
rp
2 z
u
 
r
^
r R
Rp

2 z
^
R
 R d p
8 dz
1
 u R
2
4
Q   u (r )2r dr 
0
2
max
(y = R-r)
058:0160
Professor Fred Stern
Chapters 3 & 4
8
Fall 2004
^
v 
ave
Q
1
 u
R 2
2

max
f 
8
v
f 
64
64

Dv Re
R d p
8 dz
2
R 8v
4 v
8v


2 R
R
D
  
w
w
2
ave
ave
2
D
L v
64 L v
32Lv
h  h  f

 

D 2 g Dv D 2 g gD
2
2
L
2
v
for z  0  p  v
or
C 
f


w
1
v
2
2
16
Re
D
Both f and Cf based on v2 normalization, which is
appropriate for turbulent but not laminar flow. The more
appropriate case for laminar flow is:
Poiseuille #  P  C Re  16
0
f
for pipe flow
Compare with previous solution for flow between parallel
^
plates with p .
x
058:0160
Professor Fred Stern
Fall 2004
  y
u  u 1   
 h
2
max
4
q  hu
3
Chapters 3 & 4
9



h ^

p
2
2
u
max
x
2h  ^ 

 p 
3 

3
max
x
q
h  ^  2
v

 p   u
2h 3 
 3
2
x
max
  3v h
f 
w
24
48

vh Re
2h
C  12
f
Re
Po =12
2h
Some as pipe other than constants!
Exact laminar solutions are available for any “arbitrary”
cross section for laminar steady fully developed duct flow
058:0160
Professor Fred Stern
Chapters 3 & 4
10
Fall 2004
BVP
u 0
x
^
0   p   (u  u )
x
YY
ZZ
u ( h)  0
y  y/h
*
z  z/h
*
u  u /U
*
U
h
2

^
( p )
x
Related umax
Re only
enters
through
stability
and
transition
 u  1
Poisson equation
u(y) = 0
Dirichlet boundary condition
2
Can be solved by many methods such as complex
variables and conformed mapping, transformation into
Laplace equation by redefinition of dependent variables,
and numerical methods.
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
11
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
12
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
13
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
14
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
15
058:0160
Professor Fred Stern
Chapters 3 & 4
16
Fall 2004
Stability and Transition
Stability: can a physical state withstand a disturbance and
still return to its original state.
In fluid mechanics, there are two problems of particular
interest: change in flow conditions resulting in (1)
transition from one to another laminar flow; and (2)
transition from laminar to turbulent flow.
(1) Transition from one to another laminar flow
(a) Thermal instability: Bernard Problem
A layer of fluid heated from below is top heavy,
but only undergoes convective “cellular” motion:
Raleigh #:
gd
Ra 
h
4
>
Racr
bouyancy force
viscous force
w
1   
α = coefficient of thermal expansion =   
d
  T 
   dT ,    1  T 
dz
k / d k



d = depth of layer

d
2
P
0
k, ν =thermal, viscous diffusivities
058:0160
Professor Fred Stern
Chapters 3 & 4
17
Fall 2004
Solution for two rigid plates:
Racr = 1708
for disturbance
^
   ei ( x it )
αcr/d = 3.12
λcr = 2π/α = 2d
α = αr
αr = 2π/λ
cr = wave speed
ci : > 0
=0
<0
unstable
neutral
stable
^
i ( x it )
T Te
c=cr + ici
wavelength
For temporal stability
Ra > 5 x 104 transition
to turbulent flow
Thumb curve: stable for low Ra < 1708 and very long or
short λ.
058:0160
Professor Fred Stern
Chapters 3 & 4
18
Fall 2004
(b) finger/oscillatory instability: hot/salty over
cold/fresh water and vise versa.
Rs  gd ds dz  /k
4
(Rs – Ra)cr = 657
S
   (1  T  S )
0
058:0160
Professor Fred Stern
Chapters 3 & 4
19
Fall 2004
(c) Centrifugal instability: Taylor Problem
Bernard Instability: buoyant force > viscous force
Taylor Instability: Couette flow between two
rotating cylinders where centrifugal force (outward
from center opposed to centripetal force) > viscous
force.
Ta 
r c  
i
2
2
i
o


2
c  r  r  r
0
i
i
 centrifugal force / viscous force
Tacr
Tatrans
= 1708
= 160,000
αcrc = 3.12  λcr = 2c
Square counter rotating vortex
pairs with helix streamlines
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
20
058:0160
Professor Fred Stern
Chapters 3 & 4
21
Fall 2004
(d) Gortler Vortices
Longitudinal vortices in concave curved wall
boundary layer induced by centrifugal force and
related to swirling flow in curved pipe or channel
induced by radial pressure gradient and dismissed
later with regard to minor losses.
For δ/R > .02~.1
and
Reδ = Uδ/υ > 5
058:0160
Professor Fred Stern
Chapters 3 & 4
22
Fall 2004
(e) Kelvin-Helmholtz instability
Instability at interface between two horizontal
parallel streams of different density and velocity with
heavier fluid on bottom, or more generally ρ = cons.
and U = continuous (i.e. shear larger instability e.g.
as per plow separation). Former case, viscous force
overcomes stability (density) stratification.
g  p  p     U  U
U U
1
 
1
2
2
2
2
1
1
2
1

2
2
 c 0
i
(unstable)
large α i.e. short λ
2
1
 c  (U  U ) > 0
2
Vortex Sheet
i
1
1
always unstable
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
23
058:0160
Professor Fred Stern
Chapters 3 & 4
24
Fall 2004
(2) Transition from laminar to turbulent flow
Not all laminar flows have different equilibrium states,
but all laminar flow for subsequently large Re becomes
unstable and undergoes transition to turbulence.
Transition: change over space and time at Re range of
laminar flow into a turbulent flow.
Re 
cr
U

δ = transverse viscous thickness
~ 1000
Retrans > Recr
with
xtrans ~ 10-20 xcr
Small-disturbance (linear) stability theory can be used
with some success for parallel viscous flow such as plane
Couette, plane or pipe Poiseuille, boundary layers without
or with pressure gradient, and free shear flows (jet, wake,
mixing layer).
Note: No theory for transition, but recent DNS helpful.
058:0160
Professor Fred Stern
Chapters 3 & 4
25
Fall 2004
Outline linearized stability theory for parallel viscous
flows:
u, v  mean flow, which is solution steady NS
^
u uu
^ ^
u,v  small 2D oscillating in time disturbance is s
^
vv
NS
^
p p p
^
x
v  vv
t
^
y
^
^
^ ^ ^
p  
u
Linear PDE for u , v , p for ( u , v , p ) known.

^
1 ^
  p   u
u  uu  v u  
t
1
2
x
2

x
x
^
u v 0
x
x
Assume disturbance of form of sinusoidal waves
propagating in x direction: Tollman-Schlicting waves.
^
x ( x, y , t )   ( y )e
i ( xit )
Stream function
^
^
u
x
i ( xit )
 'e
y
^
^
v
x
i ( xit )
 ie
x
y =distance across
shear layer
058:0160
Professor Fred Stern
^
Fall 2004
^
u v 0
x
Chapters 3 & 4
26
Identically, as per used of stream
function for 2D flow.
y
^
x corresponds to progressive wave traveling in x
direction at speed c.
    i = wave number = 2 
c  c  ic = wave speed = 

r
r
i
i
Where λ = wave length and ω =wave frequency
Temporal stability:
Disturbance (α = αr only at cr)
ci
>0
=0
<0
unstable
neutral
stable
Spatial stability:
Disturbance (c, α = real only)
αi
<0
=0
>0
unstable
neutral
stable
058:0160
Professor Fred Stern
Chapters 3 & 4
27
Fall 2004
Inserting u , v into small disturbance equations and
^
eliminating p results in Orr-Sommerfeld equation:
inviscid Raleigh equation




i
(u  c)( ' '  )  u ' '  
(  2  ' '  )
 Re
2
u  u /U
IV
Re=UL/υ
2
4
y=y/L
4th order linear homogeneous equation with
homogenous boundary conditions (not discussed here) i.e.
eigen-value problem, which can be solved albeit not
easily for specified geometry and ( u , v , p ) solution to
steady NS.
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
28
Although difficult, methods are now available for the
solution of the O-S equation. Typical results are shown in
Figures 5-6 and 5-7 of the text.
(1) Flat Plate BL:
U
Re 
 520


crit
(2) αδ* = 0.35
 λmin = 18 δ* = 6 δ (smallest unstable λ)
 unstable T-S waves are quite large
(3) ci = const. represent constant rates of damping (ci < 0)
~ amplification (ci > 0). ci max = .0136 is small
compared with inviscid rates indicating a gradual
evolution of transition.
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
29
(4) (cr/U0)max = 0.4  unstable wave travel at average
velocity.
(5) Reδ*crit = 520  Rex crit ~ 91,000
Exp: Rex crit ~ 2.8 x 106 (Reδ*crit = 2,400) if care taken
058:0160
Professor Fred Stern
Chapters 3 & 4
30
Fall 2004
Falkner-Skan Profiles:
(1) strong influence of B
Recrit  B > 0

fpg
Recrit  B < 0

apg
Reδ*crit :
67
520
12,490
Table
5.1
Separating flat plate
stagnation point
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
31
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
32
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
33
Extent and details of theses processes depends
on Re and many other factors (geometry, pg, free-stream,
turbulence, roughness, etc).
058:0160
Professor Fred Stern
Fall 2004
Rapid development of spanwise flow, and interaction of
nonlinear processes
- stretched vortices disintrigrate
- cascading breakdown into
families of smaller and smaller
vortices
- onset of turbulence
Note: apg may undergo
much more abrupt
transition. However, in
general, pg effects less
on transition then on
stability
Chapters 3 & 4
34
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
35
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
36
Some recent work concerns recovering distance:
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
37
Turbulent Flow
Most flows in engineering are turbulent: flows over
vehicles (airplane, ship, train, car), internal flows (heating
and ventilation, turbo-machinery), and geophysical flows
(atmosphere, ocean).
v (x, t) and p(x, t) are random functions of space and time,
but statistically stationary flaws such as steady and forced
or dominant frequency unsteady flows display coherent
features and are amendable to statistical analysis, i.e. time
and place (conditional) averaging. RMS and other loworder statistical quantification can be modeled and used in
conjunction with averaged equations for solving practical
engineering problems.
Turbulent motions range in size from the width in the
flow δ to much smaller scales, which come progressively
smaller as the Re = Uδ/υ increases.
058:0160
Professor Fred Stern
Fall 2004
Chapters 3 & 4
38
058:0160
Professor Fred Stern
Chapters 3 & 4
39
Fall 2004
Physical description:
(1) Randomness and fluctuations:
Turbulence ins irregular, chaotic, and unpredictable.
However, structurally stationary flows, such as steady
flows, can be analyzed using Reynolds decomposition.
u  u  u'
1
u   u dT
T
1
u '   u ' dT
T
t 0 T
t0 T
2
u ' 0
t0
2
etc.
t0
u = mean motion
u ' = superimposed random fluctuation
u ' = Reynolds stresses; RMS = u '
2
2
Triple decomposition is used for forced or dominant
frequency flows
u  u  u ' 'u '
Where u ' ' = organized fluctuation
(2) Nonlinearity
Reynolds stresses and 3D vortex stretching are direct
result of nonlinear nature of turbulence. In fact, Reynolds
stresses arise from nonlinear convection term after
058:0160
Professor Fred Stern
Chapters 3 & 4
40
Fall 2004
substitution of Reynolds decomposition into NS equations
and time averaging.
(3) Diffusion
Large scale mixing of fluid particles greatly enhances
diffusion of momentum (and heat), i.e.,
viscous stress
Reynolds Stresses:
Isotropic eddy viscosity:



  u ' u '    
2
 u ' u '      k
3
i
i
j
ij
j
t
ij
ij
ij
(4) Vorticity/eddies/energy cascade
Turbulence is characterized by flow visualization as
eddies, which vary in size from the largest Lδ (width of
flow) to the smallest.

LK = Kohnogorov micro-scale =  
3
1
4
U 
3
LK = 0 (mm) >> Lmean free path = 6 x 10-8 m
Largest eddies contain most of energy, which break up
into successively smaller eddies with energy transfer to
yet smaller eddies until LK reached and energy dissipated
by molecular viscosity (i.e. viscous diffusion).
058:0160
Professor Fred Stern
Chapters 3 & 4
41
Fall 2004
Richardson (1922):
Lδ Big whorls have little whorls
Which feed on their velocity;
And little whorls have lesser whorls,
LK And so on to viscosity (in the molecular sense).
(5) Dissipation
  L
0
u  k
k  u '  v'  w'
2
0
2
Energy comes from
largest scales and
fed by mean motion
2
 0 (U )
Re  u  /   big
0
0
ε = rate of dissipation = energy/time

u
2
0

3
=u l
0
0
 
o
o

Dissipation
occurs at
smallest
scales
0
u
0
independent υ
 
L  
 
3
1
4
K
It is generally assumed (following Reynolds) that the
motion can be separated into a mean ( u , v , w , p ) and a
superimposed turbulent fluctuating (u’,v’,w’,p’)
components, the mean values of the latter of which are 0.
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Chapters 3 & 4
42
Fall 2004
1
u   u dt
T
t 0 T
u  u  u'
to
T > any significant period u’
Wind tunnel T ~ 1 second
Ocean-wave T ~ 20 minutes
u ' 0
The measure of u’ is found to be of its greatest value
when it is the mean-square value.
1
u '   u ' dt
T
t 0 T
2
2
related to important statistical properties
t0
such as standard deviation and probability density distribution.
Fig. 6-4 shows measurements of turbulence for Rex=107.
Note the following mean-flow features:
(1) Fluctuations are large ~ 11% U∞
(2) Presence of wall cause an isotropy, i.e., the
fluctuations differ magnitude due to geometric and
physical reasons. u ' is largest, v ' is smallest and reaches
2
2
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Professor Fred Stern
Chapters 3 & 4
43
Fall 2004
its maximum much further out then u ' or w' . w' is
intermediate in value.
2
2
2
(3) u ' u '  0 and, as will be discussed, plays a very
important role in the analysis of turbulent shear flows.
(4) Although u u  0 at the wall, it maintains large values
right up to the wall
i
j
(5) Turbulence extends to y > δ due to intermittency (Fig
6-5). The interface at the edge of the boundary layer is
called the superlayer. This interface undulates randomly
between fully turbulent and non-turbulent flow regions.
The mean position is at y ~ .78 δ.
(6) Turbulent wave number spectra are shown in Fig 6-6.
Near wall more energy, i.e. small λ, whereas near δ large
eddies dominate.
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Fall 2004
Chapters 3 & 4
44
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Fall 2004
Chapters 3 & 4
45
Averages:
For turbulent flow v (x,t), p(x,t) are random functions of
time and must be evaluated statistically using averaging
techniques: time, ensemble, phase, or conditional.
Time Averaging
For stationary flow, the mean is not a function of time and
we can use time averaging.
1
u   u (t ) dt
T
t 0 t
T > any significant period of u '  u  u
t0
(e.g. 1 sec. for wind tunnel and 20 min. for ocean u’)
Ensemble Averaging
For non-stationary flow, the mean is a function of time
and ensemble averaging is used
u (t ) 
1
 u (t )
N
N
i 1
i
N is large enough that u independent
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Fall 2004
Chapters 3 & 4
46
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Professor Fred Stern
ui(t) =
Chapters 3 & 4
47
Fall 2004
collection of experiments performed under
identical conditions.
Phase and Conditional Averaging
Similar to ensemble averaging, but for flows with
dominant frequency content or other condition, which is
used to align time series for some phase/condition. In this
case triple velocity decomposition is used: u  u  u ' 'u '
where u’’ is called organized fluctuation.
Phase/conditional averaging extracts all three
components.
Averaging Rules:
f  f  f'
f ' 0
f g f g
 f ds   f ds
g  g  g'
f  f
f  s

s s
s = x or t
fg fg
f 'g  0
fg  f g  f ' g '
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Chapters 3 & 4
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Fall 2004
Reynolds-Averaged Navier-Stokes Equations
For convenience of notation used uppercase for mean
lowercase for fluctuation in Reynolds decomposition.
~
u U u
~
pP p
i
i
u
~
u
0
x
~
~
~
~
~
u
u
1p
 u
u


 g
t
x
 x
x x
i
NS
Equations
i
2
i
i
i
i
i
i
i
i3
j
Mean Continuity Equation

U u U
(U  u ) 


0
x
x
x
x
~
 u U u
u


0 
0
x
x x
x
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
Both mean and fluctuation satisfy divergence = 0
condition.
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Chapters 3 & 4
49
Fall 2004
Mean Momentum Equation

U  u   (U  u )  (U  u )   1  ( P  p)   (U  u )  g
t
x
 x
x x
2
i
i
j
j
i
i
i
j
i

U u U
(U  u ) 


t
t
t
t
i
i
i
i
i
(U  u )
j
j

U
u
U
u
(U  u )  U
U
u
u
x
x
x
x
x
U

U

uu
x
x
i
i
i
i
j
i
j
j
j
j
j
i
j
j
j
u

u
u
uu  u
u
u
x
x
x
x
Since
j
i
j
i
i
i
j
j
j
j
j
j

P  p P
( P  p) 


x
x x x
i
i
 g   g
i3
i
i
i3

U
u
U

(U  u )  


x
x
x
x
2
2
2
2
i
i
j
i
i
2
2
2
j
j
j
i
i
j
j
i
j
2
i
j
j
i
i3
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Professor Fred Stern
Chapters 3 & 4
50
Fall 2004
U
U
1 P

U
(u u )  

U  g
t
x
 x
x
2
i
i
j
i
j
i
2
j
i
i3
j

DU
1 P
  U

 g 
uu 
Or

Dt
 x
x  x

i
i
i3
i
i
j
DU
1 P
1 

 g 

Dt
 x
 x
j
j
i
i3
Or
i
ij
i
 U
U 
  u u
   P   

x 
 x
U
with
0
x
i
j
ij
i
j
RANS
Equations
j
i
i
i
The difference between the NS and RANS equations is
the Reynolds stresses   u u , which acts like additional
stress.
i
 u u =  u u
i
j
j
i
j
(i.e. Reynolds stresses are symmetric)
   u   uv   uw


    uv   v   vw
  uw   vw   w 
2
2
2
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Chapters 3 & 4
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Fall 2004
u are normal stresses
uu i  j
are shear stresses
2
i
i
j
For isotropic turbulence u u i  j = 0 and u  v  w 
constant; however, turbulence is generally non-isotropic.
2
i
For example, consider shear flow with
v  (U  u, v, w) :
v0  u0
v0  u0
2
2
j
uv  0
dU
 0,
dy
x-momentum tends
towards decreasing y as
turbulence diffuses
gradients and decreases
dU
dy
x-momentum transport across y = constant AA per unit
area
 (U  u )v  U v   uv   uv
i.e  u u
i
j
=
average flux of j-momentum in
i-direction = average flux of
i-momentum in j-direction
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Chapters 3 & 4
52
Fall 2004
Turbulent Kinetic Energy Equation
1
1
k  u  u  v  w  = turbulent kinetic energy
2
2
2
2
2
2
i
~
Subtracting NS equation for u and RANS equation for Ui
i
results in equation for ui:
u
u
U
u

1 p
u
U
u
u

(u u )  

t
x
x
x x
 x
x
2
i
i
j
i
j
j
i
j
j
Multiply by ui average
i
j
j
j
i
2
i
j
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Professor Fred Stern
Chapters 3 & 4
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Fall 2004
Dk
 1
1
U


 2 e e
 u  u u  2 u e   u u

Dt
x  
2

x
 


III
I
II
2
j
i
i
j
i
j
ij
i
j
ij
ji
j
u
Dk k

U
Dt t
x
Where
j
j
i
I = diffusive turbulence transport due to turbulence
modeling.
II = shear production (usually > 0) represents loss of mean
kinetic energy and gain of turbulent kinetic energy due to
interaction u u and
i
j
U
.
x
i
j
III = viscous dissipation = ε
Recall previous discussions of energy cascade and
dissipation:
Energy fed from mean flow to largest eddies and cascades
to smallest eddies where dissipation takes place
Kinetic energy = k = uo2
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Professor Fred Stern
u
2
3
0
0
0
0
u
 

l
Chapters 3 & 4
54
Fall 2004
l0 = Lδ = width of flow (i.e. size of
largest eddy)
Kolmogorov Hypothesis:
(1) local isotropy: for large Re, micro-scale ℓ << ℓ0
and turbulence is isotropic.
(2) first similarity: for large Re, micro-scale has
universal form uniquely determined by υ and ε.
   /  
length
 / l  Re
u   
velocity
u / u  Re
time
 /   Re
1/ 4
3
1/ 4
   /  
1/ 2

3 / 4
0
1 / 4
0
1 / 2




0
Microscale<<large scale
Also shows that as Re increases, the range of scales
increases as well.
(3) second similarity: for large Re, intermediate scale
has a universal form uniquely determined by ε and
independent of υ.
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Fall 2004
(2) and (3) are called universal equilibrium range in
distinction from non-isotropic energy-containing range.
(2) is called the dissipation range and (3) is the inertial
subrange.

u   S (k ) dk
2
k = wave number in inertial subrange.
0
S  A k
2/3
A ~ 1.5
5 / 3
l  k  y
1
0
1
(used on dimensional analysis)
Called Kolmogorov k-5/3 law
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Chapters 3 & 4
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Fall 2004
6-4.2 Velocity Profiles: Inner, Outer, and Overlap Layers
Detailed examination of turbulent-flow velocity
profiles (Fig 6-8) indicates the existence of a three-layer
structure:
(1) a thin inner layer close to the wall where
turbulence is inhibited by the pressure of the solid
boundary, i.e. u u are negligibly small (=0 at the
i
j
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Chapters 3 & 4
57
Fall 2004
wall) and the flow is controlled by molecular
viscosity.
(2) An outer layer where turbulent shear dominates.
(3) An overlap layer where both types of shear are
important.
Considerable more information is obtained from
dimensional analysis and confirmed by experiment.
Inner law:
u  f ( ,  ,  , y)
w
 yu 
u  u /u  f 

  
*

u   /
*
*
w
Wall shear
velocity
 f (y )

u+, y+ are called inner-wall variables
Note that the inner law is independent of δ or r0, for
boundary layer and pipe flow, respectively.
Outer Law:
U  u
 g ( ,  , y,  )

velocity defect
w
for px = 0
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Chapters 3 & 4
58
Fall 2004
U  u
 g ( )
u
*
where   y / 
Note that the outer wall is independent of μ.
Overlap law: both laws valid
It is not that difficult to show (see AMF chapter VII) that
for both laws to overlap, f and g are logarithmic
functions:
1
u  ln y  B
k
inner variables
U  u
1
  ln( y /  )  A
u
k
outer variables



k, a, and B are pure dimensionless constants
Values vary
somewhat
depending on
different exp.
arrangements
k
=
0.41
B
=
5.5
A
=
=
2.35
0.65
von Karmen constant
BL flow
pipe flow
(A = 0 means small
outer layer)
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Chapters 3 & 4
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Fall 2004
The validity of these laws has been established
experimentally as shown in Fig. 6-9, which shows the
profiles of Fig 6-8 in inner-law variable format. All the
profiles, with the exception of the one for separated flow,
are seen to follow the expected behavior. In the case of
separated flow, scaling the profile with u* is inappropriate
since u* ~ 0.
Details of Inner Law
Very very near the wall,   
w
 u
u

y
u
~ constant
y
i.e. varies linearly
Or u+ = y+
This is the linear sub-layer which must merge smoothly
with the logarithmic overlap law. This takes place in the
blending zones. Evaluating the momentum equation near
the wall shows that:
μt ~ y3
y  0
Several expressions which satisfy this requirement have
been derived and are commonly used in turbulent-flow
analysis. That is:
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Professor Fred Stern
Chapters 3 & 4
60
Fall 2004
ku  

  ke e  1  ku 
2 


 kB
ku

2

t
And equations (6-61, 6-63a – 6-63e) of the text.
Assuming the total shear is constant (6-61) can be
integrated to obtain a composite formula which is valid in
the sub-layer, blending layer, and logarithmic-overlap
regions.
ku   ku  

y  u  e e  1  ku 
2
6 




 kB
ku


2

3
(6-41)
Fig. 6-11 shows a comparison of (6-41) with experimental
data obtained very close to the wall. The agreement is
excellent. It should be recognized that obtaining data this
close to the wall is very difficult.
Details of the Outer Law
With pressure gradient included, the outer law
becomes:
U  u
 g ( , B )
u
*
  y /
 dp
= Clausers Equilibrium
B
parameter
 dx
*
c
w
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Fall 2004
Clauser (1954,1956):
BL’s with px but constant B are in equilibrium, i.e., can be
scaled with a single parameter:
U  u
u
*
vs. y / 

  defect thickness  
0
  2/C
U  u
dy   
u
*
*
f
Also, G = Clauser Shape parameter
1 U  u 
  
 dy
  u 
2

0
*

6
.1B

1.
81

1
.7


Curve fit by Mach
Which is related to the usual shape parameter by
H  1  G /    const. due to    ( x)
1
Finally, Clauser showed that the outer layer has a wakelike structure such that
  0.016 U 
t

*
(**)
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Fall 2004
Mellor and Gibson (1966) continued (*) and (**) into a
theory for equilibrium outer law profiles with excellent
agreement with experimental data: Fig. 6-12
Coles (1956):
A weakness of the Clauser approach is that the
equilibrium profiles do not have any recognizable shape.
This was resolved by Coles who showed that:
Deviations above log-overlap layer
u  2.5 ln y  5.5 1
 W ( y / )
U   2.5 ln y  5.5 2



(***)

Max deviation at δ
Single wake-like function of
g/δ
 y 
W  wake function  2 sin 
  B 2

 2

2
2
3
  y /
curve fit
Thus, from (***), it is possible to derive a composite
which covers both the overlap and outer layers.
1

u  ln y  B  W ( y /  )
k
k


  wake parameter = π(B)
 .8( B  .5)
.75
(curve fit for data)
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Fall 2004
Note the agreement of Coles’ wake law even for B 
Constant. Bl’s is quite good.
We see that the behavior in the outer layer is more
complex that that of the inner layer due to pressure
gradient effects. In general, the above velocity profile
correlations are extremely valuable both in providing
physical insight and, as we shall see, in providing
approximate solution for simple geometries: pipe and
channel flow and flat plate bl. Furthermore, such
correlations have been extended through the use of
additional parameters to provide velocity formulas for use
with integral methods for solving the b.l. equations for
arbitrary px.
6-4 Summary of Inner, Outer, and Overlap Layers
Mean velocity correlations
Inner layer:
U  U /U

*
Sub-layer: 0  y  5
Buffer layer: 5  y  30


y  y /

U   /
*
w
U+ = y+
where sub-layer
merges smoothly
with log law
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Chapters 3 & 4
64
Fall 2004
Outer Layer:
U U
 g ( , B )
U
*

B p

*
  y / ,
x
w
Overlap layer (log region):
1
U  ln y  B
k
inner variables
U  Y
1
  ln   A
U
k
outer variables


*
Composite Inner/Overlap layer correlation
1
2
U  ln y  B  W ( )
k
k


 
W  sin     3  2
2 
2
Equilibrium Boundary Layers
U U
U
*
vs. y /  ,
(shows similarity)
  0.8( B  0.5)
0.75
2
(B = constant)
  ,
*
  2/C
f
2
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Chapters 3 & 4
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Fall 2004
Reynolds Number Dependence of Mean-Velocity
correlations and Reynolds stresses
1. inner/overlap U+ scaling shows similarity; extent of
overlap region (i.e. similarity) increases with Reθ.
2. outer layer may asymptotically approach similarity
for large Re as shown by U ( 2 / k ) vs. Reθ, but
controversial due to lack of data for Reθ 5 x 104.

3. streamwise turbulence intensity u  u U vs. y+
shows similarity for 0  y  15 (i.e., just beyond the
point of kmax, y+ = 12)

2
*

u+max vs. Reθ increases with Reθ.
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Fall 2004
Chapters 3 & 4
66
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Fall 2004
Chapters 3 & 4
67
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Fall 2004
Chapters 3 & 4
68
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Professor Fred Stern
Fall 2004
Chapters 3 & 4
69
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Professor Fred Stern
Fall 2004
Chapters 3 & 4
70
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Professor Fred Stern
Fall 2004
Chapters 3 & 4
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Chapters 3 & 4
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Fall 2004

v+ and uv do not show similarity: peak values increase
with Reθ and move away from wall.
6-5 Turbulent Flow in Pipes and Channels using meanvelocity correlations
1. Circular pipe

8
 64 / Re
u
Re 
w
2
d
d
ave
u d
ave

 2000
A turbulent-flow “approximate” solution can
obtained simply by computing uave based on log law.
a
y ar
0
dy  dr
Q
1
u  
 U 2r dr
A a
ave
2
3
1
u  U  ln a  B  
2k 
k

*
ave
 
1 / 2
Adjusted constants for
better fit at low Re
 1.99 log[Re  ]  1.02
1/ 2
d
 2 log[Re  ]  0.8
1/ 2
d
2. Turbulent Flow in Rough Pipes
U  f (y ,k )



  f (Re , k / d )
d
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Chapters 3 & 4
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Fall 2004
1
U  ln y  B  B(k )
k



Log law shifts downward
which leads to three roughness regimes:
1. k+ < 4
2. 4 < k+ < 60
3. k+ > 60

1 / 2
hydraulically smooth
transitional roughness (Re dependence)
full rough (no Re dependence)
k / d
2.51 
 2 log 


 3.7 Re  
1 / 2
d
 6.9  k / d  
~ 1.8 log 

 
Re
3
.
7

 

1.11
Moody diagram
approximate explicit
formula
d
3. Turbulent Flat Plate Boundary Layer
C  2
f
(1) momentum integral with px = 0
X
1
2
U  ln y  B  W ( )
k
k


  0.45
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Fall 2004
Chapters 3 & 4
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Professor Fred Stern
Chapters 3 & 4
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Fall 2004
Concept of hydraulic diameter for noncircular ducts
For noncircular ducts, τw=f(parameter); thus, new
2
8
definitions of f 
and C 
are required.
v
v
w
w
f
2
2
Define average wall shear stress
1
   ds
P
P
w
ds = arc length, P = perimeter
w
0
Momentum:
 z 
pA   PL  
AL    0
W  L
w
h   p /   z  
 L
A / P
w
circle: A/P = D/4
Energy:
h  h
L
^

A  d p 
 


P
dx 


w
A/P = length= Dh/4
noncircular duct
Dh = 4A/P
Hydraulic
diameter
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Chapters 3 & 4
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Fall 2004
For multiple surfaces such as concentric annulus P and A
based on wetted parameter and area
f 
8
 f (Re ,  / D )
v
w
Dh
2
h
Re 
Dh
VD

h
2
L V
h  h  f
D 2g
L
h
f
 const. / Re  16 / Re
4
P  C Re  const.
C 
f
0
Dh
f
D
Circular pipe
Dh
For laminar flow, P
varies greatly,
therefore it is better to
used the exact solution
0
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Chapters 3 & 4
77
Fall 2004
For turbulent flow, Dh works much better especially if
combined with “effective diameter” concept based on
ratio of exact laminar circular and noncircular duct P0
numbers, i.e. 16 / P .
0
First recall turbulent circular pipe solution and compare
with turbulent channel flow solution using log-law in both
cases
Circular pipe
u  u ( y ) k  .41,
*
u
1 yu
 ln
B
u k 
B 5 u   /
*
*
1
V  u  Q/ A 
R
1
 u
2
ave
w
y  Rr
dy   dr
 1 ( R  r )u

 B  2r dr
 u  ln

k

3
 2 Ru
 2R  
 ln

k
k
*
R
*
2
0
*
*
*
Or
v
Ru
 2.44 ln
 1.34
u

f  1.99 log(Re f )  1.02
*
1 / 2
1/ 2
D
f
1 / 2
 2 log(Re f )  0.8
1/ 2
D
F only drops by a factor of 5 over 104 < Re < 108
EFD adjusted
constants
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Chapters 3 & 4
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Fall 2004
Since f equation is implicit, it is not easy to see
dependency on ρ, μ, v, and D
f ( pipe)  0.316 Re
p
1 / 4
4000 < ReD < 105
Blasuis (1911) power law
curve fit to data
D
2
L v
h 
 f

D 2g
f
p  0.158L  D v
3/ 4
5 / 4
1/ 4
7/4
Near quadratic (as expected)
Nearly linear
Drops weakly with
pipe size
Only slightly
with μ
 0.241L  D v
3/ 4
4.75
1/ 4
1.75
laminar flow: p  8LQ / R
4
p (turbulent) decreases more sharply than p (laminar)
for same Q; therefore, increase D for small p . 2D
decreases p by 27 for same Q.
u
max
u
*
 u (r  0)
Combine with
*
1 Ru
 ln
B
u k

v
*
u
*
 2.44 ln
Ru

*
 1.34
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v
u
max
Chapters 3 & 4
79
Fall 2004
 1  1.3 f

1
Recall laminar flow: v / u  0.5
max
Channel Flow
C  6
Laminar Solution
1
v  u
h
Re
P0 = 6
h= half width
h
 1 (h  y)u

ln

B
 k
 dY

*
h
0
f
*
Y=h-y wall coordinate
1
 1 hu
 u  ln
B 
k
k 
*
*
Define Re 
Dh
f
1 / 2
4A
D 
P
VD

h
 2 log Re f
Dh
h
1 / 2
A=2h*w
P=4h+2w
  1.19  lim 4(2hw)  4h
w
2 w  4h
Very nearly the same as circular pipe
7% too large at Re = 105
4% too large at Re = 108
Therefore error in Dh concept relatively smaller for
turbulent flow.
058:0160
Professor Fred Stern
Note
f
Chapters 3 & 4
80
Fall 2004
1 / 2
(channel)  2 log 0.64 Re f
Dh
Before Deffective  0.64 D ~
h
1/ 2
  0.8
P (circle)  16
D
P (channel)  24
0
h
0
(improvement on Dh)
Re
Deff

VD
eff

D 
eff
16
D
P (lam)
h
0
From exact laminar solution
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