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CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
h=2d
b z
y
flow
(a)
L
z
y
(c)
U0
flowy
rR=dD
(b)
x
x
x
Figure 7.1: Sketch of (a) channel flow (b) pipe flow and (c) flat-plate
boundary layer.
1
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
1.2
y/δ
U<U>
Figure 7.2: Mean velocity profiles in fully-developed turbulent channel
flow from the DNS of Kim et al. (1987): dashed line, Re = 5, 600;
solid line, Re = 13, 750
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CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
viscousstress
τ/τw
y/δ0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Reynoldsstress
τ/τw
y/δ
Figure 7.3: Profiles of the viscous shear stress, and the Reynolds shear
stress in turbulent channel flow: DNS data of Kim et al. (1987):
dashed line, Re = 5, 600; solid line, Re = 13, 750.
3
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0 10 20 30 40 50 600.0
0.2
0.4
0.6
0.8
1.0
y+
τ(y)ρν d<U>
dy
-ρ<uv>τ(y)
Figure 7.4: Profiles of the fractional contributions of the viscous
and Reynolds stresses to the total stress. DNS data of Kim et
al. (1987): dashed lines, Re = 5, 600; solid lines, Re = 13, 750.
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CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0 5 10 15 200
5
10
15
u+
y+
Figure 7.5: Near-wall profiles of mean velocity from the DNS data of
Kim et al.: dashed line, Re = 5, 600; solid line, Re = 13, 750;
dot-dashed line, u+ = y+.
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CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0 10 20 30 40 50 60 70 800
5
10
15
y+
u+
Figure 7.6: Near-wall profiles of mean velocity: solid line, DNS data of
Kim et al.: Re = 13, 750; dot-dashed line, u+ = y+; dashed line,
the log law, Eqs. (7.43)–(7.44).
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CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
100 101 102 1030
5
10
15
20
25
y+
u+
Figure 7.7: Mean velocity profiles in fully-developed turbulent channel
flow measured by Wei and Willmarth (1989): ◦,Re0 = 2, 970; ¤,
Re0 = 14, 914; ∆, Re0 = 22, 776; ∇, Re0 = 39, 582; line, the log
law, Eqs. (7.43)–(7.44).
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CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
10-1 1000
2
4
6
8
10
U0-<U>
y/δ
uτ
Figure 7.9: Mean velocity defect in turbulent channel flow. Solid line,
DNS of Kim et al. (1987), Re = 13, 750; dashed line, log law,
Eqs. (7.43)–(7.44).
8
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
102 103 104 105 1060.000
0.002
0.004
0.006
0.008
0.010
Re
cf
Figure 7.10: Skin friction coefficient cf ≡ τw/(12ρU 2
0 ) against Reynolds
number (Re = 2Uδ/ν) for channel flow: symbols, experimental
data compiled by Dean (1978); solid line, from Eq. (7.55); dashed
line, laminar friction law cf = 16/(3Re).
9
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
103 104 105 106101
102
103
104
105
Reτ
Re
Figure 7.11: Outer-to-inner lengthscale ratio δ/δν = Reτ for turbulent
channel flow as a function of Reynolds number (obtained from
Eq. 7.55).
10
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
103 104 105 106
15
20
25
30
Re
U/uτ
U/uτ
U0 /uτ
Figure 7.12: Outer-to-inner velocity scale ratios for turbulent channel
flow as functions of Reynolds number (obtained from Eq. 7.55):
solid line, U/uτ ; dashed line U0/uτ .
11
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
103 104 105 10610-4
10-3
10-2
10-1
100
y/δ
Re
bufferlayer
sublayerviscous
log lawregion
innerlayer
overlapregion
layerouter
y/δ=0.1
y/δ=0.3
y+=50
y+=30
y+=5
Figure 7.13: Regions and layers in turbulent channel flow as functions
of Reynolds number.
12
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0 50 100 150 200 250 300 350 400-1
0
1
2
3
4
5
6
7
8
⟨uiuj⟩
uτ2
⟨w2⟩
⟨v2⟩ ⟨uv⟩
k
⟨u2⟩
y+
Figure 7.14: Reynolds stresses and kinetic energy normalized by friction
velocity against y+ from DNS of channel flow at Re = 13, 750 (Kim
et al. 1987).
13
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0 50 100 150 200 250 300 350 400-0.5
0.0
0.5
1.0
1.5
2.0
⟨uiuj⟩k
⟨u2⟩
⟨w2⟩
⟨uv⟩
⟨v2⟩
y+
Figure 7.15: Profiles of Reynolds stresses normalized by turbulent ki-
netic energy from DNS of channel flow at Re = 13, 750 (Kim et
al. 1987).
14
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0 50 100 150 200 250 300 350 400-0.5
0.0
0.5
1.0
1.5
2.0
-5
0
5
10
15
20
y+
ε
εSk
ρuv
PP/ε
ρuv
Sk/ε
Figure 7.16: Profiles of the ratio of production to dissipation (P/ε),
normalized mean shear rate (Sk/ε), and shear stress correlation
coefficient (ρuv) from DNS of channel flow at Re = 13, 750 (Kim
et al. 1987).
15
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0 5 10 15 20 25 30 35 40 45 50-1
0
1
2
3
4
5
6
7
8
⟨uiuj⟩
uτ2
y+
⟨u2⟩
⟨w2⟩
⟨uv⟩⟨v2⟩
k
Figure 7.17: Profiles of Reynolds stresses and kinetic energy normalized
by friction velocity in the viscous wall region of turbulent channel
flow: DNS data of Kim et al. (1987) Re = 13, 750.
16
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0 5 10 15 20 25 30 35 40 45 50
–0.20
–0.10
0.00
0.10
0.20
y+
Gain
Loss
Production
Dissipation
Viscousdiffusion
Turbulent convection
Pressure transport
Figure 7.18: Turbulent kinetic energy budget in the viscous wall region
of channel flow: terms in Eq. (7.64) normalized by viscous scales.
From the DNS data of Kim et al. (1987) Re = 13, 750.
17
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
100 101 102 1030.0
0.5
1.0
1.5
2.0
2.5
3.0
y+
u’ v’,uτ uτ
Figure 7.19: Profiles of r.m.s. velocity measured in channel flow at dif-
ferent Reynolds numbers by Wei and Willmarth (1989). Open
symbols: u′/uτ = 〈u2〉12/uτ ; ©,Re0 = 2, 970; ¤,Re0 =
14, 914; 4,Re0 = 22, 776; 5,Re0 = 39, 582. Solid symbols:
v′/uτ = 〈v2〉12/uτ at the same Reynolds numbers.
18
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
100 101 102 103 104 105 106
10
20
30
40
y+
u+
Figure 7.20: Mean velocity profiles in fully-developed turbulent pipe
flow. Symbols, experimental data of Zagarola and Smits (1997) at
six Reynolds numbers (Re≈ 32×103, 99×103, 409×103, 1.79×106,
7.71 × 106, 29.9 × 106). Solid line, log law with κ = 0.436 and
B = 6.13; dashed line, log law with κ = 0.41, B = 5.2.
19
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
100 101 102 103 104 105
5
10
15
20
25
30
35
y+
u+
Figure 7.21: Mean velocity profiles in fully-developed turbulent pipe
flow. Symbols, experimental data of Zagarola and Smits (1997)
for y/R < 0.1, for the same values of Re as in Fig. 7.20. Line, log
law with κ = 0.436 and B = 6.13.
20
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0 1 2 3 4 5 6 70.0
0.2
0.4
0.6
0.8
1.0
U / Uo
τ / τw
y / δx
Figure 7.25: Normalized velocity and shear stress profiles from the Bla-
sius solution for the zero-pressure-gradient laminar boundary layer
on a flat plate: y is normalized by δx ≡ x/Re12x = (xν/U0)
12 .
21
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0 10
1
ττw
γ
⟨U⟩/Uo
y / δ
Figure 7.26: Profiles of mean velocity, shear stress and intermittency
factor in a zero-pressure gradient turbulent boundary layer, Reθ =
8, 000. From the experimental data of Klebanoff (1954).
22
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
100 101 102 103 1040
5
10
15
20
25
30
u+
y+
Figure 7.27: Mean velocity profiles in wall units. Circles, boundary-
layer experiments of Klebanoff (1954), Reθ = 8, 000; dashed line,
boundary-layer DNS of Spalart (1988), Reθ = 1, 410; dot-dashed
line, channel flow DNS of Kim et al. (1987), Re = 13, 750; solid
line, van Driest’s law of the wall, Eqs. (7.144)–(7.145).
23
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0.0 0.2 0.4 0.6 0.8 1.0 1.20
5
10
15
20
25
30
⟨ U ⟩uτ
wake contribution
log law
y / δ
Figure 7.28: Mean velocity profile in a turbulent boundary layer show-
ing the law of the wake. Symbols, experimental data of Kle-
banoff (1954); dashed line, log law (κ = 0.41, B = 5.2); dot-dashed
line, wake contribution Πw(y/δ)/κ (Π = 0.5); solid line, sum of
log law and wake contribution (Eq. 7.148).
24
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
10-3 10-2 10-1 1000
5
10
15
20
25
y / δ
uτ
Uo- ⟨U⟩
Figure 7.29: Velocity defect law. Symbols, experimental data of Kle-
banoff (1954); dashed line, log law; solid line, sum of log law and
wake contribution Πw(y/δ)/κ.
25
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0.0 0.2 0.4 0.6 0.8 1.00.00
0.04
0.08
0.12
0.16
νT
lm
y / δ
δlm
νT
uτδ
Figure 7.30: Turbulent viscosity and mixing length deduced from
direct numerical simulations of a turbulent boundary layer
(Spalart 1988). Solid line, νT from DNS; dot-dash line, `m from
DNS; dashed line `m and νT according to van Driest’s specification
(Eq. 7.145).
26
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
102 103 104 105
15
20
y+
u+
Figure 7.31: Log-log plot of mean velocity profiles in turbulent pipe
flow at six Reynolds number (from left to right: Re ≈ 32 × 103,
99×103, 409×103, 1.79×106, 7.71×106, 29.9×106). The scale for
u+ pertains to the lowest Reynolds number: subsequent profiles
are shifted down successively by a factor of 1.1. The range shown is
the overlap region, 50δν < y < 0.1R. Symbols, experimental data
of Zagarola and Smits (1997); dashed lines, log law with κ = 0.436
and B = 6.13; solid lines, power law (Eq. 7.157) with the power α
determined by the best fit to the data.
27
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
104 105 106 107 108
0.10
0.15
0.20
10
15
5
α n
Re
Figure 7.32: Exponent α = 1/n (Eq. 7.158) in the power-law u+ =
C(y+)α = C(y+)1/n for pipe flow as a function of Reynolds num-
ber.
28
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0.0 0.5 1.0
0
2
4
6
8
k
<u2>
<v2>
<w2>
<uv>
y/δ
<uiuj>
uτ2
(a)
0 10 20 30 40 50
0
2
4
6
8
<u2>
<v2>
<w2>
k
<uv>
y+
<uiuj>
uτ2
(b)
Figure 7.33: Profiles of Reynolds stresses and kinetic energy normalized
by the friction velocity in a turbulent boundary layer at Reθ =
1, 410: (a) across the boundary layer (b) in the viscous near-wall
region. From the DNS data of Spalart (1988).
29
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0 10 20 30 40 50
-0.20
-0.10
0.00
0.10
0.20
gain
loss
y+
production
pressure transport
meanconvection
turbulent convection
dissipation
viscousdiffusion
(b)
0.0 0.2 0.4 0.6 0.8 1.0-1.0
-0.5
0.0
0.5
1.0
gain
loss
productionconvectionturbulent
dissipation convectionmean
(a)
viscousdiffusion
pressure transport
y/δ
Figure 7.34: Turbulent kinetic energy budget in a turbulent boundary
layer at Reθ = 1, 410: terms in Eq. (7.177) (a) normalized as
a function of y so that the sum of the squares of the terms is
unity (b) normalized by the viscous scales. From the DNS data of
Spalart (1988).
30
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0 10 20 30 40 50-0.50
-0.25
0.00
0.25
0.50
viscousdiffusion
gain
loss
y+
(b)
dissipation
productionturbulent convection
mean convection
pressure
0.0 0.2 0.4 0.6 0.8 1.0-1.0
-0.5
0.0
0.5
1.0
production turbulentconvection
viscous
dissipation
meanconvection
diffusion
gain
loss
y/δ
(a)
pressure
Figure 7.35: Budget of 〈u2〉 in a turbulent boundary layer: conditions
and normalization are the same as in Fig. 7.34.
31
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0.0 0.2 0.4 0.6 0.8 1.0-1.0
-0.5
0.0
0.5
1.0turbulent
convection
meanconvection
dissipation
viscous diffusionproduction
gain
loss
(a)
y/δ
pressure
0 10 20 30 40 50-0.04
-0.02
0.00
0.02
0.04
dissipation
gain
loss
y+
pressure
viscousdiffusion
production&
mean convection
turbulentconvection
(b)
Figure 7.36: Budget of 〈v2〉 in a turbulent boundary layer: conditions
and normalization are the same as in Fig. 7.34.
32
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0.0 0.2 0.4 0.6 0.8 1.0-1.0
-0.5
0.0
0.5
1.0
gain
loss
(a)
dissipation
meanconvection
turbulentconvection
production
viscous diffusion
y/δ
pressure
0 10 20 30 40 50-0.2
-0.1
0.0
0.1
0.2
production mean convection
loss
gain
(b)
y+
viscous diffusion
dissipation
turbulent convection
pressure
Figure 7.37: Budget of 〈w2〉 in a turbulent boundary layer: conditions
and normalization are the same as in Fig. 7.34.
33
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0 10 20 30 40 50-0.12
-0.06
0.00
0.06
0.12
dissipationturbulent convection
viscous diffusion
mean convection
gain
loss
y+
(b)
production
pressure
0.0 0.2 0.4 0.6 0.8 1.0-1.0
-0.5
0.0
0.5
1.0
production
dissipation mean convection
turbulentconvection
viscous diffusion
loss
gain
y/δ
(a)
pressure
Figure 7.38: Budget of−〈uv〉 in a turbulent boundary layer: conditions
and normalization are the same as in Fig. 7.34.
34
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
0.0 0.2 0.4 0.6 0.8 1.0-0.5
0.0
0.5
1.0
1.5
2.0
2.5
ε11
ε12
ε22
ε33
ε~23
εij
y/δ
Figure 7.39: Normalized dissipation components in a turbulent bound-
ary layer at Reθ = 1, 410: from the DNS data of Spalart (1988)
for which δ = 650δν.
35
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
x
50
100
y+
0
Figure 7.40: Dye streak in a turbulent boundary layer showing the
ejection of low-speed near-wall fluid. (From the experiment of
Kline et al. 1967.)
36
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
z
x
Figure 7.42: Sketch of counter-rotating rolls in the near-wall region.
(From Holmes et al. 1996.)
37
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
z
y
x
Figure 7.43: Sketch of counter-rotating rolls in the near-wall region.
(From Holmes et al. 1996.)
38
CHAPTER 7: WALL FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c©Stephen B. Pope 2000
25o 25
o25
oValley Valley
BackFront
2δ
δ
U0y
x
Large Eddy or bulge
Figure 7.44: The large-scale features of a turbulent boundary layer
at Reθ ≈ 4, 000. The irregular line—approximating the viscous
superlayer—is the boundary between smoke-filled turbulent fluid
and clear free-stream fluid. (From the experiment of Falco 1977.)
39