CHAPTER 5: FREE SHEAR FLOWS Turbulent Flows · 2019. 2. 7. · CHAPTER 5: FREE SHEAR FLOWS...
Transcript of CHAPTER 5: FREE SHEAR FLOWS Turbulent Flows · 2019. 2. 7. · CHAPTER 5: FREE SHEAR FLOWS...
CHAPTER 5: FREE SHEAR FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
fluidsupply
nozzled
U θ
r
WU
V
xquiescentsurroundings
J
Figure 5.1: Sketch of a round jet experiment, showing the polar-cylindrical coordinate system employed.
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CHAPTER 5: FREE SHEAR FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0 10 200.0
0.1
0.2
〈U〉/UJ
x/d = 30
x/d = 60
x/d = 100
r/dFigure 5.2: Radial profiles of mean axial velocity in a turbulent roundjet, Re = 95, 500. The dashed lines indicate the half-width, r 1
2(x),
of the profiles. (Adapted from the data of Hussein et al. (1994).)
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CHAPTER 5: FREE SHEAR FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.5
1.0
0.00.0 1.0 2.0
r/r1/2
UU0
Figure 5.3: Mean axial velocity against radial distance in a turbulentround jet, Re ≈ 105; measurements of Wygnanski and Fiedler(1969). Symbols: ◦, x/d = 40; △, 50; !, 60; ♦, 75; •, 97.5.
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0 40 80 1200
5
10
15
20
25
x/d
UJU0(x)
Figure 5.4: Centerline mean velocity variation with axial distance in aturbulent round jet, Re = 95, 500: symbols, experimental data ofHussein et al. (1994), line—Eq. (5.6) with x0/d = 4, B = 5.8.
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.0 1.0 2.00.0
0.5
1.0
f
ξ ≡ r/r1/2
0.0 0.1 0.2
η ≡ r/(x-x0)
Figure 5.5: Self-similar profile of the mean axial velocity in the self-similar round jet. Curve fit to the LDA data of Hussein et al.(1994).
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CHAPTER 5: FREE SHEAR FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
1.0 2.0
-0.02
0.00
0.02
〈V〉/Uo
ξ ≡ r/r1/2
Figure 5.6: Mean lateral velocity in the self-similar round jet. Fromthe LDA data of Hussein et al. (1994).
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CHAPTER 5: FREE SHEAR FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.0 1.0 2.0 3.00.00
0.02
0.04
0.06
0.08
0.10
〈uiuj〉
U02
r/r1/2
〈u2〉
〈v2〉
〈w2〉
〈uv〉
0.0 0.1 0.2 0.3η
Figure 5.7: Profiles of Reynolds stresses in the self-similar round jet.Curve fit to the LDA data of Hussein et al. (1994).
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CHAPTER 5: FREE SHEAR FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.0 1.0 2.0 3.00.0
1.0
2.0
3.0
4.0
5.0
〈U〉
r/r1/2
0.0 0.1 0.2 0.3η
Figure 5.8: Profile of the local turbulence intensity—⟨u2⟩12/⟨U⟩—in
the self-similar round jet. From the curve fit to the experimentaldata of Hussein et al. (1994).
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CHAPTER 5: FREE SHEAR FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.0 1.0 2.00.0
0.1
0.2
0.3
0.4
0.5
r/r1/2
ρuv
〈uv〉/k
0.0 0.1 0.2η
Figure 5.9: Profiles of ⟨uv⟩/k and the u-v correlation coefficient ρuv inthe self-similar round jet. From the curve fit to the experimentaldata of Hussein et al. (1994).
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CHAPTER 5: FREE SHEAR FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.0 1.0 2.00.00
0.01
0.02
0.03
r/r1/2
νT^
0.0 0.1 0.2η
Figure 5.10: Normalized turbulent diffusivity ν̂T (Eq. (5.34)) in theself-similar round jet. From the curve fit to the experimental dataof Hussein et al. (1994).
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CHAPTER 5: FREE SHEAR FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.0 1.0 2.00.00
0.05
0.10
0.15
r/r1/2
r1/2
l
0.0 0.1 0.2η
Figure 5.11: Profile of the lengthscale defined by (Eq. (5.35)) in theself-similar round jet. From the curve fit to the experimental dataof Hussein et al. (1994).
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CHAPTER 5: FREE SHEAR FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.0 1.0 2.00.0
0.2
0.4
0.6
0.8
1.0
r/r1/2
L22/r1/2
L11/r1/2
0.0 0.1 0.2η
Figure 5.12: Self-similar profiles of the integral lengthscales in the tur-bulent round jet. From Wygnanski and Fiedler (1969).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
1.0
0.5
R (x,0,s)11
1 2 3 4 5s/L11
00.0
Figure 5.13: Longitudinal autocorrelation of the axial velocity in theself-similar round jet. From Wygnanski and Fiedler (1969).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
(a)
(b)
(c)
(d)
<U>plane jet
y
U = Uc s
xδ
<U>
plane mixing layer
y
U c
x
δ
U s
<U>
plane wake
y
U c
xδ
U s
boundary layer
y
xδ
U = Uc s
Figure 5.14: Sketch of plane two-dimensional shear flows showing thecharacteristic flow width δ(x), the characteristic convective veloc-ity Uc, and the characteristic velocity difference Us.
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.0 0.1 0.2η
0.0 1.0 2.0
0.2
0.4
0.6
0.8
1.0
〈U〉U0
r/r1/2
Figure 5.15: Mean velocity profile in the self-similar round jet: solidline, curve fit to the experimental data of Hussein et al. (1994);dashed line, uniform turbulent viscosity solution (Eq. 5.82).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
Production
Mean-flowconvection
Turbulenttransport
Dissipation
0.0 1.0 2.0r/r1/2
0.00
0.01
–0.01
–0.02
0.02
Gain
Loss
Figure 5.16: Turbulent kinetic energy budget in the self-similar roundjet. Quantities are normalized by U0 and r1
2. (From Panchapake-
san and Lumley (1993a).)
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CHAPTER 5: FREE SHEAR FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
-1.0 -0.5 0.0 0.5 1.0
δ
Ul+0.1(Uh-Ul)
y0.1 y0.9
ξ
Uh
Uc
Ul
Ul+0.9(Uh-Ul)
y
0.40.5
0.0
-0.5-0.4
f(ξ)
Figure 5.21: Sketch of the mean velocity ⟨U⟩ against y, and ofthe scaled mean velocity profile f(ξ), showing the definitions ofy0.1, y0.9 and δ.
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CHAPTER 5: FREE SHEAR FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
x = 39.5 cm.x = 49.5 cm.x = 59.5 cm.
-1.0 -0.5 0.0 0.5 1.0
-0.5
0.0
0.5
f (ξ)
ξ
Figure 5.22: Scaled velocity profiles in a plane mixing layer. Symbols,experimental data of Champagne et al. (1976); line, error-functionprofile (Eq. (5.224)) shown for reference.
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0 20 40 60 80
0
4y(cm)
x(cm)
y0.5
y0.95
y0.1
Figure 5.23: Axial variation of y0.1, y0.5 and y0.95 in the plane mixinglayer, showing the linear spreading. Experimental data of Cham-pagne et al. (1976).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
-1 0 1
-0.5
0.0
0.5
ξ
〈U〉Us
Figure 5.24: Scaled mean velocity profile in self-similar plane mixinglayers. Symbols, experiment of Bell and Mehta (1990) (Uℓ/Uh =0.6); solid line, DNS data for the temporal mixing layer (Rogersand Moser 1994); dashed line error-function profile with widthchosen to match data in the center of the layer.
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CHAPTER 5: FREE SHEAR FLOWS
Turbulent FlowsStephen B. Pope
Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
–1 0 10.000
0.015
0.030
n
〈uiuj〉
Us 2
〈u 2〉/Us 2
〈v 2〉/Us 2
–1 0 1
– 0.015
0.000
0.015
n
〈w 2〉/Us 2
〈uv〉/Us 2
〈uiuj〉
Us 2
Figure 5.25: Scaled Reynolds stress profiles in self-similar plane mixinglayers. Symbols, experiment of Bell and Mehta (1990) (Uℓ/Uh =0.6); solid line, DNS data for the temporal mixing layer (Rogersand Moser 1994).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
-3 -2 -1 0 1 2 30
1
f (ξ)
ξ
Figure 5.26: Normalized velocity defect profile in the self-similarplane wake. Solid line, from experimental data of Wygnanskiet al. (1986); dashed line, constant-turbulent-viscosity solution,Eq. (5.240).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
-2 -1 0 1 20.0
0.5
1.0
f(ξ)
ξ
Figure 5.27: Mean velocity deficit profiles in a self-similar axisymmetricwake. Symbols, experimental data of Uberoi and Freymuth (1970);line, constant-turbulent-viscosity solution f(ξ) = exp (−ξ2 ln 2)
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.0-2.0-4.0 2.0 4.0
0.5
1.0
ξ
u /U2 1/2S
w /U 2 1/2S
v /U2 1/2S
0.0
Figure 5.28: R.m.s. velocity profiles in a self-similar axisymmetricwake. Experimental data of Uberoi and Freymuth (1970).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.0-3.0 3.0ξ
0.0
Gai
nLo
ss
Mean Flow Convection
Production
Dissipation
Turbulent Transport
Figure 5.29: Turbulent kinetic energy budget in a self-similar axisym-metric wake. Experimental data of Uberoi and Freymuth (1970).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
〈U〉
h
y
0 UcFigure 5.30: Sketch of the mean velocity profile in homogeneous shearflow.
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
4 5 6 7 8 9 10 11
0.003
0.002
0.001
0
x / h
u ui j
Uc2
Figure 5.31: Reynolds stresses against axial distance in the homoge-neous shear flow experiment of Tavoularis and Corrsin (1981): ⃝,⟨u2⟩; !, ⟨v2⟩; △, ⟨w2⟩.
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
M
M
d
Figure 5.32: Sketch of a turbulence generating grid composed of barsof diameter d, with mesh spacing M .
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
10– 4
10– 3
10 500x/M
U02
〈u2〉
Figure 5.33: Decay of Reynolds stresses in grid turbulence: squares⟨u2⟩/U 2
0 ; circles ⟨v2⟩/U 20 ; triangles k/U 2
0 ; lines, proportional to(x/M)−1.3. (From Comte-Bellot and Corrsin (1966).)
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.0 1.0 2.0 3.0
0.5
1.0
ξ
ϕ(ξ)f(ξ)
Figure 5.34: Normalized mean velocity deficit f(ξ) and scalar, ϕ(ξ) =⟨φ⟩/⟨φ⟩y=0 in the self-similar plane wake. Symbols (solid f ,open ϕ) experimental data of Fabris (1979); solid line, f(ξ) =exp(−ξ2 ln 2); dashed line, ϕ(ξ) = exp(−ξ2σT ln 2) with σT = 0.7.
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.0 1.0 2.00.0
0.1
0.2
0.3
〈φ〉y=0
〈φʹ2〉1/2
r/r1/2
0.0 0.1 0.2η
Figure 5.35: Normalized r.m.s. scalar fluctuations in a round jet. Fromthe experimental data of Panchapakesan and Lumley (1993b).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
–0.02
0.02
0.01
0.00
–0.01
Gain
Loss
r/r1/2
0.0 1.0 2.0
Production
Mean-flowconvection
Turbulenttransport
Dissipation
Figure 5.36: Scalar variance budget in a round jet: terms in Eq. (5.281)normalized by ⟨φ⟩y=0, Us and r1
2. (From the experiment data of
Panchapakesan and Lumley (1993b).)
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
τ/τφ
1.0 2.00.0
0.5
1.0
1.5
2.0
r/r1/2
Figure 5.37: Scalar-to-velocity timescale ratio in a round jet. (Fromthe experimental data of Panchapakesan and Lumley (1993b).)
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
103 104 1050.00
0.05
0.10
0.15
0.20
Re
〈φ〉2〈φʹ2〉
Figure 5.38: Normalized scalar variance on the axis of self-similar roundjets at different Reynolds numbers. Triangles, air jets (experimentsof Dowling and Dimotakis (1990); circles, water jets (experimentsof Miller (1991)). (From Miller (1991).)
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0
0
0
01
1
1
1
γ=0.0
γ=0.25
γ=0.75
γ=0.95t
t
t
tI(t)
I(t)
I(t)
I(t)
(a)
(b)
(c)
(d)
Figure 5.39: Sketch of the intermittency function vs. time in a freeshear flow (a) in the irrotational non-turbulent surroundings (b)in the outer part of the intermittent region (c) in the inner part ofthe intermittent region and (d) close to the center of the flow.
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
γφ
1.0
0.5
00 1 2y / yφ
Figure 5.40: Profile of the intermittency factor in the self-similar planewake. The mean scalar profile shown for comparison: yφ is thehalf-width. From the experimental data of LaRue and Libby (1974,1976).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
1.0
0.5
0.0
φT
φ
1.0 2.00.0 y/yφ
(a) (b)
y/yφ
φ' 1/22
φ'T
0.0 1.0 2.00.0
0.5
1.0
1.5
Figure 5.41: Comparison of unconditional and turbulent mean (a) andr.m.s. (b) scalar profiles in the self-similar plane wake. From theexperimental data of LaRue and Libby (1974).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
viscous superlayer
Us
•O
s ∞
δ
s=0
E
Figure 5.42: Sketch of a turbulent round jet showing the viscous super-layer, and the path of a fluid particle from a point in the quiescentambient, O, to the superlayer, E.
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
U T
U N
U
γ
γ
UUS
ξ
0
1.0
1-1 0
0.5
Figure 5.43: Profiles of the intermittency factor γ, the unconditionalmean axial velocity ⟨U⟩ and the turbulent ⟨U⟩T and non-turbulent⟨U⟩N conditional mean velocities in a self-similar mixing layer.From the experimental data of Wygnanski and Fiedler (1970).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
u 2
T2(u' )
N2(u' )
ξ
u 2
US2
-1 0 1
0
0.01
0.02
0.03
Figure 5.44: Profiles of unconditional ⟨u2⟩, turbulent (u′T )2 and non-
turbulent (u′N)2 variances of axial velocity in a self-similar mix-
ing layer. From the experimental data of Wygnanski and Fiedler(1970).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
-3 -2 -1 0 1 2 3
0.1
0.2
-3 -2 -1 0 1 2 3
0.1
0.2
-3 -2 -1 0 1 2 3
0.1
0.2
-3 -2 -1 0 1 2 3
0.1
0.2
0.3
0.3 0.3
0.3
f̂u f̂w
f̂v f̂φV1 V3
ψ V2
(a) (c)
(b) (d)
Figure 5.45: Standardized PDF’s of (a) u, (b) v, (c) w and (d) φ inhomogeneous shear flow. Dashed lines are standardized Gaussians.(From Tavoularis and Corrsin (1981).)
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
-2
-1
-2
-1
-2
-1
1
0 3-2-3 -1 1 2
0 3-2-3 -1 1 2
0 3-2-3 -1 1 2
2
1
2
1
2
V1
V1
V2
V2
ψ
ψ
(a)
(b)
(c)
Figure 5.46: Contour plots of joint PDF’s of standardized variablesmeasured in homogeneous shear flow: (a) u and v, (b) φ and v,(c) u and φ. Contour values are 0.15, 0.10, 0.05 and 0.01. Dashedlines are corresponding contours for joint-normal distributions withthe same correlation coefficients. (From Tavoularis and Corrsin(1981).)
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
0.0 0.2 0.4 0.6 0.8 1.0
1
2
0
ψ
f (ψ)φ
ξ = 0 ξ = 0.22
ξ = 0.43
ξ = 0.65
ξ = 0.86
Figure 5.47: PDF’s of a conserved passive scalar in the self-similartemporal mixing layer at different lateral positions. From directnumerical simulations of Rogers and Moser (1994).
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
y/yφ1.00.0 2.0y/yφ
0.0
-1.0
1.0
2.0
3.0
4.0
Sφ
Sφ
SφT
0.0 1.0 2.0
1.0
3.0
5.0
7.0
9.0Kφ
Kφ
KφT
Figure 5.48: Profiles of unconditional (Sφ, Kφ) and conditional tur-bulent (SφT , KφT ) skewness and kurtosis of a conserved passivescalar in the self-similar plane wake. From the experimental dataof LaRue and Libby (1974).
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
-0.5 0.0 0.50
5
10
V / Us
Us fu(V)
ξ=0ξ=0.22
ξ=0.44
ξ=0.66
Figure 5.49: PDF’s of axial velocity in a temporal mixing layer. Thedistance from the center of the layer is ξ = y/δ. The dashed linecorresponds to the freestream velocity Uh. From the DNS data ofRogers and Moser (1994).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
1.0 2.0 3.0r/r1/2
20
10
Ku
2.0
4.0
Su
3
Figure 5.50: Profiles of skewness (solid line) and kurtosis (dashed line)in the self-similar round jet. From the experimental data of Wyg-nanski and Fiedler (1969).
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CHAPTER 5: FREE SHEAR FLOWS
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
Figure 5.51: Flow visualization of a plane mixing layer. Spark shadowgraph of a mixing layer between helium (upper) Uh = 10.1m/sand nitrogen (lower) Uℓ = 3.8m/s at a pressure of 8 atm. (FromBrown and Roshko (1974).)
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Cambridge University Press, 2000
c⃝Stephen B. Pope 2000
20
15
10
5
500 1000 1500
x (mm)
θ (mm) 60504030
0f (Hz)
Figure 5.53: Mixing layer thickness, θ, against axial distance, x, for dif-ferent forcing frequencies, f . (From Oster and Wygnanski (1982).)
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