CHAPTER 7: WALL FLOWS Turbulent Flows - Cornell … · CHAPTER 7: WALL FLOWS Turbulent Flows...

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





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

2





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





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.

4





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+.

5





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).

6





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).

7





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





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





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





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





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





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





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





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





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





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





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





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





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





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





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





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





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





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





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





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





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





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





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





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





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


32





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


33





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


34





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





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





z

x

Figure 7.42: Sketch of counter-rotating rolls in the near-wall region.

(From Holmes et al. 1996.)

37





z

y

x

Figure 7.43: Sketch of counter-rotating rolls in the near-wall region.

(From Holmes et al. 1996.)

38





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

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