CHAPTER 5: FREE SHEAR FLOWS Turbulent Flows · 2019. 2. 7. · CHAPTER 5: FREE SHEAR FLOWS...

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CHAPTER 5: FREE SHEAR FLOWS Turbulent Flows Stephen B. Pope Cambridge University Press, 2000 c Stephen B. Pope 2000 fluid supply nozzle d U θ r W U V x quiescent surroundings J Figure 5.1: Sketch of a round jet experiment, showing the polar- cylindrical coordinate system employed.

Transcript of CHAPTER 5: FREE SHEAR FLOWS Turbulent Flows · 2019. 2. 7. · CHAPTER 5: FREE SHEAR FLOWS...

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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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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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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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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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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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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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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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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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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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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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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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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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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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(a)

(b)

(c)

(d)

<U>plane jet

y

U = Uc s

<U>

plane mixing layer

y

U c

x

δ

U s

<U>

plane wake

y

U c

U s

boundary layer

y

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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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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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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-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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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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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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-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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–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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-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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-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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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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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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〈U〉

h

y

0 UcFigure 5.30: Sketch of the mean velocity profile in homogeneous shearflow.

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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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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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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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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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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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–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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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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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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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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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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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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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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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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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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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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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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CHAPTER 5: FREE SHEAR FLOWS

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c⃝Stephen B. Pope 2000

y/yφ1.00.0 2.0y/yφ

0.0

-1.0

1.0

2.0

3.0

4.0

SφT

0.0 1.0 2.0

1.0

3.0

5.0

7.0

9.0Kφ

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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CHAPTER 5: FREE SHEAR FLOWS

Turbulent FlowsStephen B. Pope

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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CHAPTER 5: FREE SHEAR FLOWS

Turbulent FlowsStephen B. Pope

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).)

48