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

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

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    fluid supply

    nozzled

    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.

    1

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    0 10 20 0.0

    0.1

    0.2

    〈U〉/UJ

    x/d = 30

    x/d = 60

    x/d = 100

    r/d Figure 5.2: Radial profiles of mean axial velocity in a turbulent round jet, 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).)

    2

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    0.5

    1.0

    0.0 0.0 1.0 2.0

    r/r1/2

    U U0

    Figure 5.3: Mean axial velocity against radial distance in a turbulent round jet, Re ≈ 105; measurements of Wygnanski and Fiedler (1969). Symbols: ◦, x/d = 40; △, 50; !, 60; ♦, 75; •, 97.5.

    3

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    0 40 80 120 0

    5

    10

    15

    20

    25

    x/d

    UJ U0(x)

    Figure 5.4: Centerline mean velocity variation with axial distance in a turbulent round jet, Re = 95, 500: symbols, experimental data of Hussein et al. (1994), line—Eq. (5.6) with x0/d = 4, B = 5.8.

    4

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    0.0 1.0 2.0 0.0

    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).

    5

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen 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. From the LDA data of Hussein et al. (1994).

    6

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    0.0 1.0 2.0 3.0 0.00

    0.02

    0.04

    0.06

    0.08

    0.10

    〈uiuj〉

    U0 2

    r/r1/2

    〈u2〉

    〈v2〉

    〈w2〉

    〈uv〉

    0.0 0.1 0.2 0.3 η

    Figure 5.7: Profiles of Reynolds stresses in the self-similar round jet. Curve fit to the LDA data of Hussein et al. (1994).

    7

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    0.0 1.0 2.0 3.0 0.0

    1.0

    2.0

    3.0

    4.0

    5.0

    〈U〉

    r/r1/2

    0.0 0.1 0.2 0.3 η

    Figure 5.8: Profile of the local turbulence intensity—⟨u2⟩12/⟨U⟩—in the self-similar round jet. From the curve fit to the experimental data of Hussein et al. (1994).

    8

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    0.0 1.0 2.0 0.0

    0.1

    0.2

    0.3

    0.4

    0.5

    r/r1/2

    ρuv

    〈uv〉/k

    0.0 0.1 0.2 η

    Figure 5.9: Profiles of ⟨uv⟩/k and the u-v correlation coefficient ρuv in the self-similar round jet. From the curve fit to the experimental data of Hussein et al. (1994).

    9

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    0.0 1.0 2.0 0.00

    0.01

    0.02

    0.03

    r/r1/2

    νT ^

    0.0 0.1 0.2 η

    Figure 5.10: Normalized turbulent diffusivity ν̂T (Eq. (5.34)) in the self-similar round jet. From the curve fit to the experimental data of Hussein et al. (1994).

    10

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    0.0 1.0 2.0 0.00

    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 the self-similar round jet. From the curve fit to the experimental data of Hussein et al. (1994).

    11

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    0.0 1.0 2.0 0.0

    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).

    12

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    1.0

    0.5

    R (x,0,s)11

    1 2 3 4 5 s/L11

    0 0.0

    Figure 5.13: Longitudinal autocorrelation of the axial velocity in the self-similar round jet. From Wygnanski and Fiedler (1969).

    13

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    (a)

    (b)

    (c)

    (d)

    plane jet

    y

    U = Uc s

    plane mixing layer

    y

    U c

    x

    δ

    U s

    plane wake

    y

    U c

    U s

    boundary layer

    y

    U = Uc s

    Figure 5.14: Sketch of plane two-dimensional shear flows showing the characteristic flow width δ(x), the characteristic convective veloc- ity Uc, and the characteristic velocity difference Us.

    14

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    0.0 0.1 0.2 η

    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: solid line, curve fit to the experimental data of Hussein et al. (1994); dashed line, uniform turbulent viscosity solution (Eq. 5.82).

    15

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    Production

    Mean-flow convection

    Turbulent transport

    Dissipation

    0.0 1.0 2.0 r/r1/2

    0.00

    0.01

    –0.01

    –0.02

    0.02

    G ai n

    Lo ss

    Figure 5.16: Turbulent kinetic energy budget in the self-similar round jet. Quantities are normalized by U0 and r1

    2 . (From Panchapake-

    san and Lumley (1993a).)

    16

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen 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

    0.4 0.5

    0.0

    -0.5 -0.4

    f(ξ)

    Figure 5.21: Sketch of the mean velocity ⟨U⟩ against y, and of the scaled mean velocity profile f(ξ), showing the definitions of y0.1, y0.9 and δ.

    17

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen 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-function profile (Eq. (5.224)) shown for reference.

    18

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    Cambridge University Press, 2000

    c⃝Stephen B. Pope 2000

    0 20 40 60 80

    0

    4 y(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 mixing layer, showing the linear spreading. Experimental data of Cham- pagne et al. (1976).

    19

  • CHAPTER 5: FREE SHEAR FLOWS

    Turbulent Flows Stephen B. Pope

    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 mixing layers. Symbols, experiment of Bell and Mehta (1990) (Uℓ/Uh = 0.6); solid line, DNS data for the temporal mixing layer (Rogers and Moser 1994); dashed line error-function profile with width chosen to match data in the c