058:0160
Professor Fred Stern
Fall 2010
Chapter 6-part3
1
Chapter 6: Viscous Flow in Ducts
6.3 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 flows 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 quantities can be modeled and used in
conjunction with the averaged equations for solving
practical engineering problems.
Turbulent motions range in size from the width in the
flow δ to much smaller scales, which become
progressively smaller as the Re = Uδ/υ increases.
058:0160
Professor Fred Stern
Fall 2010
Chapter 6-part3
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Professor Fred Stern
Chapter 6-part3
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Fall 2010
Physical description:
(1) Randomness and fluctuations:
Turbulence is irregular, chaotic, and unpredictable.
However, for statistically 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
t0 T
t 0 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 oscillation
(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
substitution of Reynolds decomposition into NS equations
and time averaging.
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Professor Fred Stern
Chapter 6-part3
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Fall 2010
(3) Diffusion
Large scale mixing of fluid particles greatly enhances
diffusion of momentum (and heat), i.e.,
viscous stress
Reynolds Stresses:



  u ' u '    
Isotropic eddy viscosity:
2
 u 'i u ' j   t  ij   ij k
3
i
j
ij
ij
(4) Vorticity/eddies/energy cascade
Turbulence is characterized by flow visualization as
eddies, which varies in size from the largest Lδ (width of
flow) to the smallest. The largest eddies have velocity
scale U and time scale Lδ/U. The orders of magnitude of
the smallest eddies (Kolmogorov scale or inner scale) are:
1
4
 3
LK = Kolmogorov micro-scale =  3 
U 
LK = O(mm) >> Lmean free path = 6 x 10-8 m
Velocity scale = (νε)1/4= O(10-2m/s)
Time scale = (ν/ε)1/2= O(10-2s)
Largest eddies contain most of energy, which break up
into successively smaller eddies with energy transfer to
yet smaller eddies until LK is reached and energy is
dissipated by molecular viscosity.
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Professor Fred Stern
Chapter 6-part3
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Fall 2010
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
 0  L
u0  k
k  u '  v'  w'
2
2
Energy comes from
largest scales and
fed by mean motion
2
 0 (U )
Re  u 0  0 /   big
ε = rate of dissipation = energy/time

=
u02
u03
o
l0
o 
0
u0
independent υ
Dissipation rate is
determined by the
inviscid large scale
dynamics.
Dissipation
occurs at
smallest
scales
1
 3  4
LK   
 
Decrease in  decreases
scale of dissipation LK not
rate of dissipation .
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Professor Fred Stern
Chapter 6-part3
6
Fall 2010
Fig. below shows measurements of turbulence for
Rex=107.
Note the following mean-flow features:
(1) Fluctuations are large ~ 11% U∞
(2) Presence of wall causes anisotropy, i.e., the
fluctuations differ in magnitude due to geometric and
physical reasons. u ' is largest, v ' is smallest and reaches
its maximum much further out than u ' or w' . w' is
intermediate in value.
2
2
2
2
2
(3) u ' v'  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.
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 ~ 0.78 δ.
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Professor Fred Stern
Fall 2010
Chapter 6-part3
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(6) Near wall turbulent wave number spectra have more
energy, i.e. small λ, whereas near δ large eddies dominate.
058:0160
Professor Fred Stern
Fall 2010
Chapter 6-part3
8
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 t0  t
u
 u (t ) dt T > any significant period of u '  u  u
T t0
(e.g. 1 sec. for wind tunnel and 20 min. for ocean)
Ensemble Averaging
For non-stationary flow, the mean is a function of time
and ensemble averaging is used
1 N i
u (t )   u (t ) N is large enough that u independent
N i 1
ui(t) = collection of experiments performed under
identical conditions (also can be phase aligned
for same t=o).
058:0160
Professor Fred Stern
Fall 2010
Chapter 6-part3
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Professor Fred Stern
Chapter 6-part3
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Fall 2010
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
oscillation.
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  f

s s
s = x or t
fg fg
f 'g  0
fg  f g  f ' g '
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Professor Fred Stern
Chapter 6-part3
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Fall 2010
Reynolds-Averaged Navier-Stokes Equations
For convenience of notation use uppercase for mean and
lowercase for fluctuation in Reynolds decomposition.
~
u i  U i  ui
~
p  P p
~
 ui
0
xi
~
~
~
~
2
~
 ui
 ui
1 p
 ui
 ui


 g i 3
t
xi
 xi
x j x j
NS
equation
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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Professor Fred Stern
Chapter 6-part3
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Fall 2010
Mean Momentum Equation


1 
( P  p) 
U i  ui   (U j  u j ) (U i  ui )  
t
x j
 xi
2
(U i  ui )  g i 3
x j x j

U u U
(U  u ) 


t
t
t
t
i
i
i
(U  u )
j
i
i
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
j
i
j
i
j
Since
u

u
u
uu  u
u
u
x
x
x
x
j
i
j
i
i
j
j
j
i
i
 g   g
i3
i
i
i
j
j

P  p P
( P  p) 


x
x x x
i3
j
j
j
i
j
j
058:0160
Professor Fred Stern

2
x j
2
Chapter 6-part3
13
Fall 2010
(U i  ui )  
 2U i
x 2j
 2U i
2 ui
 2  
x j
x 2j
U i
U i  (ui u j )
1 P
2
U j


  2 U i  g i 3
t
x j
x j
 xi
x j
DU i
1 P


 g i 3 
Or
Dt
 xi
x j
 U i

 ui u j 

 x j

DU i
1 


g


 ij
Or
i3
Dt
 xi
 U U
   P   


x
x

U
0
with
x
i

  u u


j
ij
j
i
i
RANS
Equations
j
i
i
The difference between the NS and RANS equations is
the Reynolds stresses   u u , which acts like additional
stress.
i
j
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Professor Fred Stern
Chapter 6-part3
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Fall 2010
 u u =  u u
(i.e. Reynolds stresses are symmetric)
   u 2   uv   uw


2
    uv   v
  vw 
  uw   vw   w 2 


u are normal stresses
uu i  j
are shear stresses
i
j
j
i
2
i
i
j
For homogeneous/isotropic turbulence u u i  j = 0 and
u  v  w  constant; however, turbulence is generally
non-isotropic.
i
2
2
j
2
For example, consider shear flow with
y
dU
 0 as below,
dy
U(y)
u<0
y+dy
v>0
fluid
particle
y
v<0
y-dy
u>0
The fluid velocity is: V  (U  u , v, w)
U
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Professor Fred Stern
Chapter 6-part3
15
Fall 2010
Assuming that fluid particle retains its velocity V from y
to ydy gives,
v0  u0
v0  u0
uv  0
x-momentum tends towards
decreasing y as turbulence
diffuses gradients and
decreases
dU
dy
x-momentum transport in y direction, i.e., across y =
constant AA per unit area
𝑀𝑥𝑦 = ∫ 𝜌𝑢̃𝑉 ∙ 𝑛 𝑑𝐴
𝑑𝑀𝑥𝑦
𝑑𝐴
=  (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
058:0160
Professor Fred Stern
Fall 2010
Chapter 6-part3
16
Closure Problem:
1. RANS equations differ from the NS equations due to
the Reynolds stress terms
2. RANS equations are for the mean flow (U i , P) ; thus,
represent 4 equations with 10 ten unknowns due to
the additional 6 unknown Reynolds stresses ui u j
3. Equations can be derived for uiu j by summing
products of velocity and momentum components
and time averaging, but these include additionally 10
triple product ui u j ul unknowns. Triple products
represent Reynolds stress transport.
4. Again equations for triple products can be derived
that involve higher order correlations leading to fact
that RANS equations are inherently nondeterministic, which requires turbulence modeling.
5. Turbulence closure models render deterministic
RANS solutions.
6. The NS and RANS equations have paradox that NS
equations are deterministic but have
nondeterministic solutions for turbulent flow due to
inherent stochastic nature of turbulence, whereas the
RANS equations are nondeterministic, but have
deterministic solutions due to turbulence closure
models.
058:0160
Professor Fred Stern
Chapter 6-part3
17
Fall 2010
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 i and RANS equation for Ui
results in equation for ui:
ui
ui
U i
ui
 2 ui

1 p
U j
uj
uj

(ui u j )  
 2
t
x j
x j
x j x j
 xi
x j
Multiply by ui and average
U i
Dk
1 
1  2


pu j 
ui u j  2
ui eij  ui u j
 2 eij e ji
Dt
 x j
2 x j
x j
x j
V
I
II
III
IV
Where
Dk k
k

U j
Dt t
x j
VI
I =pressure transport
II= turbulent transport
1 ui u j
e 
and ij 2 x j xi
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Professor Fred Stern
Chapter 6-part3
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Fall 2010
III=viscous diffusion
IV = shear production (usually > 0) represents loss of
mean kinetic energy and gain of turbulent kinetic energy
due to interactions of u u and
i
j
U
.
x
i
j
V = viscous dissipation = ε
VI= turbulent convection
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
u
2
3
0
0
0
0
u
 

l
l0 = Lδ = width of flow (i.e. size of
largest eddy)
Kolmogorov Hypothesis:
(1) local isotropy: for large Re, micro-scale ℓ << ℓ0
turbulence structures are isotropic.
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Professor Fred Stern
Chapter 6-part3
19
Fall 2010
(2) first similarity: for large Re, micro-scale has
universal form uniquely determined by υ and ε.

  3 / 
1/ 4
u   1 / 4
    /  1/ 2
length
 / l0  Re  3 / 4
velocity
u / u0  Re 1 / 4
time
 /  0  Re 1 / 2

Micro-scale<<large scale
Also shows that as Re increases, the range of scales
increase.
(3) second similarity: for large Re, intermediate scale
has a universal form uniquely determined by ε and
independent of υ.
(2) and (3) are called universal equilibrium range in
distinction from non-isotropic energy-containing range.
(2) is the dissipation range and (3) is the inertial subrange.
Spectrum of turbulence in the inertial subrange

u   S (k ) dk
2
0
k = wave number in inertial subrange.
S  A 2 / 3k  5 / 3
For l  k   (based on dimensional analysis)
1
0
1
058:0160
Professor Fred Stern
A ~ 1.5
Fall 2010
Called Kolmogorov k-5/3 law
Chapter 6-part3
20
058:0160
Professor Fred Stern
Fall 2010
Chapter 6-part3
21
Velocity Profiles: Inner, Outer, and Overlap Layers
Detailed examination of turbulent boundary layer
velocity profiles indicates the existence of a three-layer
structure:
Figure: Pope (2000, Fig. 7.8)
(1) A thin inner layer close to the wall, which is
governed by molecular viscous scales, and
independent of boundary layer thickness , freestream velocity Ue and pressure gradient.
(2) An outer layer where the flow is governed by
turbulent shear stresses,, Ue and pressure gradient,
but independent of .
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Professor Fred Stern
Chapter 6-part3
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Fall 2010
(3) An overlap layer which smoothly connects inner
and outer regions. In this region both molecular
and turbulent stresses and pressure gradient are
important.
Considerable more information is obtained from the
dimensional analysis and confirmed by experiment.
Inner layer: 𝑈 = 𝑓(𝜏𝑤 , 𝜌, 𝜇, 𝑦)
+
𝑈 =
𝑈
𝑢∗
𝑦𝑢∗
= 𝑓(
𝜈
)
u*   w / 
Wall shear
velocity
 f ( y )
U+, y+ are called inner-wall variables
Note that the inner layer is independent of δ or r0, for
boundary layer and pipe flow, respectively.
Outer Layer:
𝑈
⏟𝑒 − 𝑈
= 𝑔(𝜏𝑤 , 𝜌, 𝑦, 𝛿) for px = 0
𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑑𝑒𝑓𝑒𝑐𝑡
𝑈𝑒 −𝑈
𝑢∗
= 𝑔(𝜂)
where   y / 
Note that the outer wall is independent of μ.
058:0160
Professor Fred Stern
Chapter 6-part3
23
Fall 2010
Overlap layer: both laws are valid
In this region both log-law and outer layer is valid.
It is not that difficult to show that for both laws to
overlap, f and g are logarithmic functions.
Inner region:
2
dU u
df

dy
 dy 
Outer region:
dU u dg

dy  d
2
y u dg
 
; valid at large y+ and small η.
 

u
dy
u  d
y u
df
g(η)
f(y+)
Therefore, both sides must equal universal constant,  1
f ( y ) 
g ( ) 
1

1

ln y   B  U / u 
ln   A 
Ue U
u
(inner variables)
(outer variables)
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Professor Fred Stern
Chapter 6-part3
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Fall 2010
 , A, and B are pure dimensionless constants
Values vary
somewhat
depending on
different exp.
arrangements
 =
0.41
B
=
5.5
A
=
=
2.35
0.65
Von Karman constant
BL flow The difference is due to
pipe flow loss of intermittency in
duct flow. A = 0 means
small outer layer
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.
058:0160
Professor Fred Stern
Fall 2010
Chapter 6-part3
25
----------------------------------------------------------------------
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Professor Fred Stern
Chapter 6-part3
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Fall 2010
Details of Inner Layer
Neglecting inertia and pressure forces in the 2D turbulent
boundary layer equation we get:
𝑑
𝑑𝑦
(𝜇 (
𝑑𝑈
𝑑𝑦
) − 𝜌𝑢𝑣 ) = 0

𝜇(
𝑑𝑈
𝑑𝑦
) − 𝜌𝑢𝑣 = 𝜏𝑡
The total shear stress is the sum of viscous and turbulent
stresses. Very near the wall y0, the turbulent stress
vanishes
lim 𝜇 (
𝑦→0

𝑑𝑈
𝑑𝑈
) − 𝜌𝑢𝑣 = 𝜇 ( )
= 𝜏𝑤 = 𝜏𝑡
𝑑𝑦
𝑑𝑦 𝑦=0
𝜇(
𝑑𝑈
𝑑𝑦
) − 𝜌𝑢𝑣 = 𝜏𝑤
(1)
Sublayer region: Very near the wall y0,
𝜏𝑤 = 𝜇 (
𝑑𝑈
)
𝑑𝑦 𝑦=0
From the inner layer velocity profile:
(2)
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Professor Fred Stern
Chapter 6-part3
27
Fall 2010
(
𝑑𝑈
2
)
𝑑𝑦 𝑦=0
=
𝑢∗ 𝑑𝑓(𝑦 + )
𝜈
(3)
𝑑𝑦 +
Comparing equations (2) and (3) we get,
𝑑𝑓(𝑦 + )
𝑑𝑦 +
= 1  𝑓 (𝑦 + ) = 𝑦 + + 𝐶
(4)
No slip condition at y = 0 requires 𝐶 = 0.
Sublayer: U+ = y+
valid for
y+ ≤ 5
(5)
Log-layer region: The molecular stresses are negligible
compared to turbulent stresses, thus
𝜏𝑤 = −𝜌𝑢𝑣 = 𝜇𝑡
𝑑𝑈
𝑑𝑦
(6)
𝜇𝑡 is modeled using velocity gradient for time scale and
distance from wall as length scale (𝜅𝑦),
𝜇𝑡 = (𝜅𝑦)2
𝑑𝑈
𝑑𝑦
which gives:
(7)
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𝑑𝑈 2
2
(𝜅𝑦) ( )
𝑑𝑦
(8)
From inner layer velocity profile and equations (6) and
(8) we get,
𝑑𝑓(𝑦 + )
𝑑𝑦 +
=
𝜈
𝜅𝑦𝑢∗
=
1
𝜅𝑦 +
1
 𝑓 (𝑦 + ) = ln(𝑦 + ) + 𝐶
𝜅
1
Log-law: U+ = ln(𝑦 + ) + 𝐵 valid for y+ > 30
𝜅
(9)
(10)
(11)
Buffer layer: Merges smoothly the viscosity-dominated
subb-layer and turbulence-dominated log-layer in the
region 5< y+ ≤ 30.
Unified Inner layer: There are several ways to obtain
composite of sub-/buffer and log-layers.
Evaluating the RANS equation near the wall using μt
turbulence model 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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
 U 
U 
B

t  e e  1  U 
2

 
2

(12)
Assuming the total shear is constant very near to the wall
as shown in equation (1), equation (12) can be integrated
to obtain a composite formula which is valid in the sublayer, blending layer, and logarithmic-overlap regions.


U  y e
B
2
3
 u 

U   U   


e  1  U 

2
6


(13)
Fig. 6-11 shows a comparison of equation (13) 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.
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Alternative derivation of Overlap layer:
From the log law:
𝑑𝑈
𝑢∗
=
𝑑𝑦
𝜅𝑦
From the outer layer:
dU u dg

dy  d
Comparing the log-law and outer layer, we get
𝑑g
1
=
𝑑𝜂 𝜅𝜂

g ( ) 
1

ln   A 
Ue U
u
(outer variables)
---------------------------------------------------------------------Details of the Outer Law
At the end of the overlap region the velocity defect is
given approximately by:
𝑈𝑒 −𝑈
𝑢∗
= 9.6(1 − 𝜂)2
(14)
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With pressure gradient included, the outer law becomes
(Fig. 6-10):
Ue U
 g ( ,  )
u*
  y /
(15)
 * dpc Clauser’s equilibrium

 w dx = parameter
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Clauser (1954,1956):
BL’s with different px but constant  are in equilibrium,
i.e., can be scaled with a single parameter:
Ue U
u*
vs. y / 

Ue U
dy   *
*
u
0
  defect thickness  
  2/C
f
Also, G = Clauser Shape parameter

1  U U 
   e *  dy
 0 u

2

6.1   1.81  1.7



Curve fit by Mach
Which is related to the usual shape parameter by
H  1  G /  1  const. due to    ( x)
Finally, Clauser showed that the outer layer has a wakelike structure such that
t  0.016U e *
(16)
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Mellor and Gibson (1966) combined (15) and (16) 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 e  2.5 ln    5.5 2
Max deviation at δ
(17)
Single wake-like function of y/δ
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 y 
2
3
W  wake function  2 sin 
  3  2 ,   y / 

 2

2
curve fit
Thus, from (17), it is possible to derive a composite which
covers both the overlap and outer layers, as shown in Fig.
6-13.
U 
1

ln y   B 

W(y / )

  wake parameter = π(β)
 0.8(  0.5)0.75
(curve fit for data)
Note the agreement of Coles’ wake law even for β 
constant. Bl’s is quite good.
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We see that the behavior in the outer layer is more
complex than 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 in providing approximate solutions
for simple wall bounded geometries: pipe, channel flow
and flat plate boundary layer.
Furthermore, such
correlations have been extended through the use of
additional parameters to provide velocity formulas for use
with integral methods for solving the BL equations for
arbitrary px.
Summary of Inner, Outer, and Overlap Layers
Mean velocity correlations
Inner layer:
U   f (y )
U   U / u* y   y / u 
u*   w / 
Sub-layer: U+ = y+
for 0  y  5
Buffer layer: where sub-layer merges smoothly with

log-law region for 5  y  30

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Outer Layer:
Ue  U
 g ( ,  )
u*
  y / ,

*
px
w
for  > 0.1
Overlap layer (log region):
U 
1

Ue  U
u*
ln y   B
inner variables
1
  ln  A
outer variables

for y+ > 30 and   0.3
Composite Inner/Overlap layer correlation


U  y e
b  b
e

(U  ) 2 (U  )3 
 1  U 


2
6 

+
for 0 < y  50
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Composite Overlap/Outer layer correlation
U 
1

ln y   B 
2

 
W  sin 2     3 2  2 3
2 
  0.8(  0.5) 0.75
W ( )
for y+ > 50
Reynolds Number Dependence of Mean-Velocity Profiles
and Reynolds stresses
Figure: Pope (2000, Fig. 7.13)
1. Inner/overlap U+ scaling shows similarity; extent of
overlap region (i.e. similarity) increases with Re.
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2. Outer layer for px = 0 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. The normalized Reynolds stresses ̅̅̅̅̅/𝑘,
𝑢𝑖 𝑢𝑗
production-dissipation ratio and the normalized
mean shear stress are somewhat uniform in the loglaw region. Experiments in flat plate boundary layer,
pipe and channel flow shows k = 3.34 - 3.43 u*2 in
lower part of log-law region.
4. Decay of k ~ y2 near the wall.
5. Streamwise turbulence intensity
u  u
2
u
*
vs. y+
shows similarity for 0  y  15 (i.e., just beyond the
point of kmax, y+ = 12), but u+ increases with Reθ.

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