Turbulent Flows. quiescent surroundings W U V. r U. nozzle. fluid supply CHAPTER 5: FREE SHEAR FLOWS

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1 quiescent surroundings x W U V r U J θ d nozzle fluid supply Figure 5.1: Sketch of a round jet experiment, showing the polarcylindrical coordinate system employed.

2 0.2 x/d = 30 U /U J 0.1 x/d = 60 x/d = 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).)

3 1.0 U U r/r 1/2 Figure 5.3: Mean axial velocity against radial distance in a turbulent round jet, Re 10 5 ; measurements of Wygnanski and Fiedler (1969). Symbols:, x/d = 40;, 50;, 60;, 75;, 97.5.

4 25 U J U 0 (x) x/d 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 x 0 /d = 4, B = 5.8.

5 1.0 f ξ r/r 1/ η r/(x-x 0 ) Figure 5.5: Self-similar profile of the mean axial velocity in the selfsimilar round jet. Curve fit to the LDA data of Hussein et al. (1994).

6 0.02 V /U o ξ r/r 1/ Figure 5.6: Mean lateral velocity in the self-similar round jet. From the LDA data of Hussein et al. (1994).

7 0.10 u i u j U u 2 w 2 v uv r/r 1/ 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).

8 U r/r 1/ η Figure 5.8: Profile of the local turbulence intensity u 2 1 2/ U in the self-similar round jet. From the curve fit to the experimental data of Hussein et al. (1994).

9 ρ uv uv /k r/r 1/ η Figure 5.9: Profiles of uv /k and the u-v correlation coefficient ρ uv in the self-similar round jet. From the curve fit to the experimental data of Hussein et al. (1994).

10 ν ^ T r/r 1/ η Figure 5.10: Normalized turbulent diffusivity ˆν T (Eq. (5.34)) in the self-similar round jet. From the curve fit to the experimental data of Hussein et al. (1994).

11 l r 1/ r/r 1/ η 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).

12 L 11 /r 1/ L 22 /r 1/ r/r 1/ η Figure 5.12: Self-similar profiles of the integral lengthscales in the turbulent round jet. From Wygnanski and Fiedler (1969).

13 R (x,0,s) s/l 11 Figure 5.13: Longitudinal autocorrelation of the axial velocity in the self-similar round jet. From Wygnanski and Fiedler (1969).

14 y U = U c s (a) δ x plane jet <U> y U s U c (b) plane mixing layer δ x <U> y U s U c (c) δ x plane wake <U> y U c = U s (d) δ x boundary layer Figure 5.14: Sketch of plane two-dimensional shear flows showing the characteristic flow width δ(x), the characteristic convective velocity U c, and the characteristic velocity difference U s.

15 1.0 U U r/r 1/ η 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).

16 Gain Production Mean-flow convection 0.00 Loss 0.01 Turbulent transport Dissipation r/r 1/2 Figure 5.16: Turbulent kinetic energy budget in the self-similar round jet. Quantities are normalized by U 0 and r1 2. (From Panchapakesan and Lumley (1993a).)

17 f(ξ) U h U l +0.9(U h -U l ) U c 0.0 U l +0.1(U h -U l ) U l δ y 0.1 y 0.9 y 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 y 0.1, y 0.9 and δ.

18 f (ξ) 0.5 x = 39.5 cm. x = 49.5 cm. x = 59.5 cm 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.

19 y(cm) 4 y y 0.5 y x(cm) Figure 5.23: Axial variation of y 0.1, y 0.5 and y 0.95 in the plane mixing layer, showing the linear spreading. Experimental data of Champagne et al. (1976).

20 0.5 U U s ξ Figure 5.24: Scaled mean velocity profile in self-similar plane mixing layers. Symbols, experiment of Bell and Mehta (1990) (U l /U h = 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 center of the layer.

21 u i u j U s u 2 /U s v 2 /U s n u i u j U s 2 w 2 /U s uv /U s 2 n Figure 5.25: Scaled Reynolds stress profiles in self-similar plane mixing layers. Symbols, experiment of Bell and Mehta (1990) (U l /U h = 0.6); solid line, DNS data for the temporal mixing layer (Rogers and Moser 1994).

22 1 f (ξ) ξ Figure 5.26: Normalized velocity defect profile in the self-similar plane wake. Solid line, from experimental data of Wygnanski et al. (1986); dashed line, constant-turbulent-viscosity solution, Eq. (5.240).

23 1.0 f(ξ) ξ Figure 5.27: Mean velocity deficit profiles in a self-similar axisymmetric wake. Symbols, experimental data of Uberoi and Freymuth (1970); line, constant-turbulent-viscosity solution f(ξ) = exp ( ξ 2 ln 2)

24 u /U 2 1/2 S v /U 2 1/2 S w /U 2 1/2 S ξ Figure 5.28: R.m.s. velocity profiles in a self-similar axisymmetric wake. Experimental data of Uberoi and Freymuth (1970).

25 Gain Mean Flow Convection Production 0.0 Dissipation Loss Turbulent Transport ξ Figure 5.29: Turbulent kinetic energy budget in a self-similar axisymmetric wake. Experimental data of Uberoi and Freymuth (1970).

26 y h 0 U c U Figure 5.30: Sketch of the mean velocity profile in homogeneous shear flow.

27 u i u j U c x / h Figure 5.31: Reynolds stresses against axial distance in the homogeneous shear flow experiment of Tavoularis and Corrsin (1981):, u 2 ;, v 2 ;, w 2.

28 M M d Figure 5.32: Sketch of a turbulence generating grid composed of bars of diameter d, with mesh spacing M.

29 10 3 u 2 2 U x/m Figure 5.33: Decay of Reynolds stresses in grid turbulence: squares u 2 /U 2 0 ; circles v 2 /U 2 0 ; triangles k/u 2 0 ; lines, proportional to (x/m) 1.3. (From Comte-Bellot and Corrsin (1966).)

30 1.0 f(ξ) ϕ(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( ξ 2 ln 2); dashed line, ϕ(ξ) = exp( ξ 2 σ T ln 2) with σ T = 0.7. ξ

31 0.3 φ 2 1/2 φ y= r/r 1/ η Figure 5.35: Normalized r.m.s. scalar fluctuations in a round jet. From the experimental data of Panchapakesan and Lumley (1993b).

32 Gain Production Mean-flow convection 0.00 Loss 0.01 Turbulent transport Dissipation r/r 1/2 Figure 5.36: Scalar variance budget in a round jet: terms in Eq. (5.281) normalized by φ y=0, U s and r1 2. (From the experiment data of Panchapakesan and Lumley (1993b).)

33 2.0 τ/τ φ r/r 1/2 Figure 5.37: Scalar-to-velocity timescale ratio in a round jet. (From the experimental data of Panchapakesan and Lumley (1993b).)

34 φ 2 φ Re Figure 5.38: Normalized scalar variance on the axis of self-similar round jets at different Reynolds numbers. Triangles, air jets (experiments of Dowling and Dimotakis (1990); circles, water jets (experiments of Miller (1991)). (From Miller (1991).)

35 I(t) I(t) (a) (b) γ=0.0 γ=0.25 t t I(t) I(t) (c) (d) γ=0.75 t γ=0.95 t Figure 5.39: Sketch of the intermittency function vs. time in a free shear flow (a) in the irrotational non-turbulent surroundings (b) in the outer part of the intermittent region (c) in the inner part of the intermittent region and (d) close to the center of the flow.

36 φ γ y / y φ Figure 5.40: Profile of the intermittency factor in the self-similar plane wake. The mean scalar profile shown for comparison: y φ is the half-width. From the experimental data of LaRue and Libby (1974, 1976).

37 1.0 (a) 1.5 (b) φ' T φ T φ 0.5 φ' 2 1/ y/y φ y/y φ Figure 5.41: Comparison of unconditional and turbulent mean (a) and r.m.s. (b) scalar profiles in the self-similar plane wake. From the experimental data of LaRue and Libby (1974).

38 s O E s=0 δ U s viscous superlayer Figure 5.42: Sketch of a turbulent round jet showing the viscous superlayer, and the path of a fluid particle from a point in the quiescent ambient, O, to the superlayer, E.

39 γ 1.0 U U S 0.5 U γ U T U N ξ Figure 5.43: Profiles of the intermittency factor γ, the unconditional mean 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).

40 u U S T 2 (u' ) u 2 N 2 (u' ) ξ Figure 5.44: Profiles of unconditional u 2, turbulent (u T )2 and nonturbulent (u N )2 variances of axial velocity in a self-similar mixing layer. From the experimental data of Wygnanski and Fiedler (1970).

41 f^u f^w (a) (c) f^v 0.3 (b) V f^φ V (d) V ψ Figure 5.45: Standardized PDF s of (a) u, (b) v, (c) w and (d) φ in homogeneous shear flow. Dashed lines are standardized Gaussians. (From Tavoularis and Corrsin (1981).)

42 (a) V V 1 (b) -2 V ψ (c) ψ V 1-2 Figure 5.46: Contour plots of joint PDF s of standardized variables measured 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 Dashed lines are corresponding contours for joint-normal distributions with the same correlation coefficients. (From Tavoularis and Corrsin (1981).)

43 f (ψ) φ 2 ξ = 0 ξ = ξ = 0.43 ξ = ξ = ψ Figure 5.47: PDF s of a conserved passive scalar in the self-similar temporal mixing layer at different lateral positions. From direct numerical simulations of Rogers and Moser (1994).

44 S φ S φ K φ K φ S φt 3.0 K φt y/y 2.0 φ y/y φ Figure 5.48: Profiles of unconditional (S φ, K φ ) and conditional turbulent (S φt, K φt ) skewness and kurtosis of a conserved passive scalar in the self-similar plane wake. From the experimental data of LaRue and Libby (1974).

45 10 ξ=0.66 U s f u (V) 5 ξ=0.44 ξ=0 ξ= V / U s Figure 5.49: PDF s of axial velocity in a temporal mixing layer. The distance from the center of the layer is ξ = y/δ. The dashed line corresponds to the freestream velocity U h. From the DNS data of Rogers and Moser (1994).

46 S u K u r/r 1/2 Figure 5.50: Profiles of skewness (solid line) and kurtosis (dashed line) in the self-similar round jet. From the experimental data of Wygnanski and Fiedler (1969).

47 Figure 5.51: Flow visualization of a plane mixing layer. Spark shadow graph of a mixing layer between helium (upper) U h = 10.1m/s and nitrogen (lower) U l = 3.8m/s at a pressure of 8 atm. (From Brown and Roshko (1974).)

48 f (Hz) 0 θ (mm) x (mm) Figure 5.53: Mixing layer thickness, θ, against axial distance, x, for different forcing frequencies, f. (From Oster and Wygnanski (1982).)

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