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Professor Fred Stern
Fall 2009
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 low-order statistical quantities 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 become
progressively smaller as the Re = Uδ/υ increases.
058:0160
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Fall 2009
Chapter 6-part3
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Chapter 6-part3
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Fall 2009
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
t 0 +T
1
u ' = ∫ u ' dT
T
u '= 0
2
t0
t 0 +T
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
results 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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Fall 2009
(3) Diffusion
Large scale mixing of fluid particles greatly enhances
diffusion of momentum (and heat), i.e.,
viscous stress
Reynolds Stresses:
6
474
8
− ρ u ' u ' >> τ = με
Isotropic eddy viscosity:
2
− u 'i u ' j >> υ t ε ij − δ ij k
3
i
j
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. 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
⎡υ 3δ
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 (i.e. viscous diffusion).
058:0160
Professor Fred Stern
Chapter 6-part3
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Fall 2009
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 = Lδ
2
u0 = k
2
k = u ' + v' + w'
Energy comes from
largest scales and
fed by mean motion
2
= 0 (U )
Reδ = u 0 l 0 / υ = big
ε = rate of dissipation = energy/time
=
=
u02
u03
τo
l0
τo =
l0
Dissipation
occurs at
smallest
scales
u0
independent υ
1
⎡υ 3 ⎤ 4
LK = ⎢ ⎥
⎣ε ⎦
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Professor Fred Stern
Chapter 6-part3
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Fall 2009
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 cause 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
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Fall 2009
Chapter 6-part3
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between fully turbulent and non-turbulent flow regions.
The mean position is at y ~ 0.78 δ.
(6) Near wall turbulent wave number spectra have more
energy, i.e. small λ, whereas near δ large eddies dominate.
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Fall 2009
Chapter 6-part3
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Fall 2009
Chapter 6-part3
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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).
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Fall 2009
Chapter 6-part3
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Chapter 6-part3
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Fall 2009
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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Chapter 6-part3
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Fall 2009
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
~
~
~
~
∂ ui ~ ∂ ui
∂2 u i
1∂p
+ ui
=−
+υ
− gδ i 3
ρ ∂xi
∂t
∂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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Chapter 6-part3
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Fall 2009
Mean Momentum Equation
∂
∂
1 ∂
( P + p) +
(U i + ui ) + (U j + u j ) (U i + ui ) = −
ρ ∂xi
∂t
∂x j
∂2
(U i + ui ) − gδ 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
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
− gδ = − gδ
i3
i
i
i
j
j
∂
∂P ∂ p ∂P
+
=
( P + p) =
∂x
∂x ∂x ∂x
i3
j
j
j
i
j
j
j
058:0160
Professor Fred Stern
υ
∂2
∂x j
2
Chapter 6-part3
14
Fall 2009
(U i + ui ) = υ
∂ 2U i
∂x 2j
∂ 2U i
∂2 ui
+υ 2 = υ
∂x j
∂x 2j
∂U i ∂ (ui u j )
∂U i
∂2
1 ∂P
+U j
+
=−
+ υ 2 U i − gδ i 3
∂t
∂x j
∂x j
ρ ∂xi
∂x j
DU i
∂
1 ∂P
=−
Or
− gδ i 3 +
Dt
ρ ∂xi
∂x j
⎤
⎡ ∂U i
− ui u j ⎥
⎢υ
⎣ ∂x j
⎦
DU i
1 ∂
=
−
δ
+
σ ij
g
Or
i3
ρ ∂xi
Dt
⎛ ∂U ∂U ⎞
⎟ − ρu u
σ = − Pδ + μ ⎜⎜
+
∂x ⎟⎠
⎝ ∂x
∂U
=0
with
∂x
j
i
ij
i
j
RANS
Equations
j
i
i
i
The difference between the NS and RANS equations is
the Reynolds stresses − ρ u u , which acts like additional
stress.
i
− ρu u = − ρu u
i
j
j
i
j
(i.e. Reynolds stresses are symmetric)
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Chapter 6-part3
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Fall 2009
⎡ − ρ u 2 − ρ uv − ρ uw ⎤
⎢
⎥
2
= ⎢ − ρ uv − ρ v
− ρ vw ⎥
⎢− ρ uw − ρ vw − ρ w 2 ⎥
⎣
⎦
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
ρ (U + u )v = ρU v + ρ uv = ρ uv
i.e ρ u u
i
j
=
average flux of j-momentum in
i-direction = average flux of
i-momentum in j-direction
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Professor Fred Stern
Chapter 6-part3
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Fall 2009
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:
∂ 2 ui
∂ui
∂U i
∂ui
∂ui
1 ∂p
∂
(ui u j ) = −
+υ 2
−
+uj
+uj
+U j
ρ ∂xi
∂x j ∂x j
∂x j
∂x j
∂t
∂x j
Multiply by ui and average
∂U i
1 ∂
1 ∂ 2
Dk
∂
pu j −
ui u j + 2υ
ui eij − ui u j
=−
− 2υ eij e ji
2 ∂x j
Dt
ρ ∂x j
∂x j
∂x j 123
14243 14243 14243 1
424
3
V
I
II
III
IV
Where
Dk ∂k
∂k
1 ∂ui ∂u j
=
+U j
e =
Dt ∂t
∂x j and ij 2 ∂x ∂x
j
i
123
VI
I =pressure transport
II= turbulent transport
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Fall 2009
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
u
ε= =
τ
l
2
3
0
0
0
0
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.
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Fall 2009
(2) first similarity: for large Re, micro-scale has
universal form uniquely determined by υ and ε.
(
3
η = υ /ε
)
1/ 4
uη = (ευ )1 / 4
τ η = (υ / ε )1/ 2
length
η / l0 = Re − 3 / 4
velocity
uη / u0 = Re −1 / 4
time
τη / τ 0 = Re −1 / 2
1442443
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
2
∞
u = ∫ S (k ) dk
0
S = Aε 2 / 3k − 5 / 3
k = wave number in inertial subrange.
058:0160
Professor Fred Stern
Fall 2009
For
(based on dimensional analysis)
l0−1 << k << η −1
A ~ 1.5
Called Kolmogorov k-5/3 law
Chapter 6-part3
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Fall 2009
Chapter 6-part3
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Velocity Profiles: Inner, Outer, and Overlap Layers
Detailed examination of turbulent-flow velocity
profiles (Fig 6-9) 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
wall) and the flow is controlled by molecular
viscosity.
i
j
(2) An outer layer where turbulent shear dominates.
(3) An overlap layer where both types of shear are
important.
Considerably more information is obtained from
dimensional analysis and confirmed by experiment.
u = f (τ , ρ , μ , y )
Inner law:
w
⎛ yu * ⎞
⎟
u = u / u = f ⎜⎜
⎟
⎝ υ ⎠
+
*
u* = τ w / ρ
Wall shear
velocity
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Fall 2009
= 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:
Ue − u
= g (τ w , ρ , y , δ )
123
velocity defect
Ue − u
u
*
= g (η )
for px = 0
where η = y / δ
Note that the outer wall is independent of μ.
Overlap law: both laws are valid
It is not that difficult to show (see AMF chapter VII) that
for both laws to overlap, f and g are logarithmic
functions:
Inner region:
∗2
dU u
df
=
ν dy +
dy
Outer region:
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Fall 2009
dU u ∗ dg
=
dy δ dη
∗2
y u∗ dg
= ∗
; valid at large y+ and small η.
∗ ν
+
u
dy
u δ dη
y u
df
f(y+)
g(η)
Therefore, both sides must equal universal constant, κ −1
+
f (y ) =
g (η ) =
1
κ
1
κ
ln y + + B = u / u ∗ (inner variables)
lnη + A =
Ue − u
u∗
(outer variables)
κ , A, and B are pure dimensionless constants
Values vary
somewhat
depending on
different exp.
arrangements
κ =
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
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Professor Fred Stern
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Fall 2009
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 near the wall, τ = μ
w
Æ u=
τw
y
μ
Or u+ = y+
∂u
~ constant
∂y
i.e. varies linearly
y+ ≤ 5
This is the linear sub-layer which must merge smoothly
with the logarithmic overlap law. This takes place in the
blending zone. Evaluating the momentum equation near
the wall shows that:
μt ~ y3
y Æ 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
Chapter 6-part3
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Fall 2009
( )⎥
⎡
κu +
−κB ⎢ κu +
+
μt = μκe
e
− 1 − κu −
2
⎢
⎣
2⎤
⎥
⎦
(6.61)
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.
⎡ +
+ 2
+ 3⎤
κ
u
κ
u
⎥ (6-41)
−
y + = u + + e −κB ⎢eκu − 1 − κu + −
2
6 ⎥
⎢
⎣
⎦
( ) ( )
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:
Ue − u
u
*
= g (η , β )
(*)
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Fall 2009
δ * dpc Clauser’s equilibrium
β=
τ w dx = parameter
η = y /δ
Clauser (1954,1956):
BL’s with different px but constant β are in equilibrium,
i.e., can be scaled with a single parameter:
Ue − u
u
*
vs. y / Δ
∞U
−u
Δ = defect thickness = ∫ e * dy = δ *λ
0 u
λ = 2/C
f
Also, G = Clauser Shape parameter
2
1 ∞⎛ U e − u ⎞
= ∫⎜
⎟ dy
Δ 0 ⎝ u* ⎠
= 6.1 β + 1.81 − 1.7
144
42444
3
Curve − fit
by Mach
Which is related to the usual shape parameter by
H = (1 − G / λ )−1 ≠ const. due to λ = λ ( x)
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Fall 2009
Finally, Clauser showed that the outer layer has a wakelike structure such that
μt ≈ 0.016 ρU eδ *
(**)
Mellor and Gibson (1966) combined (*) 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 /δ )
+
+
2
U e − 2.5 ln δ − 5.5
(***)
Single wake-like function of y/δ
Max deviation at δ
⎛π y ⎞
2
3
W = wake function = 2 sin ⎜
⎟ = 3η − 2η , η = y / δ
δ⎠
⎝ 243
142
2
curve fit
Thus, from (***), it is possible to derive a composite
which covers both the overlap and outer layers.
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Professor Fred Stern
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Fall 2009
u+ =
1
κ
ln y + + B +
π
W (y /δ )
κ
π = wake parameter = π(β)
= 0.8( β + 0.5)0.75
(curve fit for
data)
Note the agreement of Coles’ wake law even for β ≠
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 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 b.l. equations for
arbitrary px.
Summary of Inner, Outer, and Overlap Layers
Mean velocity correlations
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Fall 2009
+
*
Inner layer: U = U / u y + = y / u ∗ u* = τ w / ρ
U+ = y+
Sub-layer: 0 ≤ y ≤ 5
where sub-layer
Buffer layer: 5 ≤ y ≤ 30
merges smoothly
with log law
+
+
Outer Layer:
Ue − U
u
*
= g (η , B )
η = y /δ ,
δ*
B=
px
τw
Overlap layer (log region):
U+ =
1
κ
Ue − U
u
*
ln y + + B
inner variables
1
= − lnη + A
outer variables
κ
Composite Inner/Overlap layer correlation
+
+
U = y −e
−κb ⎡ κb
⎢e
⎣⎢
(κU + ) 2 (κU + )3 ⎤
− 1 − κU −
−
⎥
2
6 ⎦⎥
+
Composite Overlap/Outer layer correlation
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Fall 2009
2⎛ π ⎞
2
3
W
=
sin
U = ln y + B +
W (η )
⎜ η ⎟ = 3η − 2η
κ
κ
⎝2 ⎠
1
+
+
2π
( β = constant)
Equilibrium Boundary Layers
Ue −U
u
*
vs. y / Δ ,
Δ =δ λ,
*
λ = 2/C
f
(shows similarity)
Π = 0.8( β + 0.5)0.75
Reynolds Number Dependence of Mean-Velocity
correlations and Reynolds stresses (missing pages from
notes)
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 ΔU (= 2π / k ) vs. Reθ, but
controversial due to lack of data for Reθ 5 x 104.
+
3. 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)
+
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u+max vs. Reθ increases with Reθ.
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