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.
0.2 x/d = 30 U /U J 0.1 x/d = 60 x/d = 100 0.0 0 10 20 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).)
1.0 U U 0 0.5 0.0 0.0 1.0 2.0 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.
25 U J U 0 (x) 20 15 10 5 0 0 40 80 120 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.
1.0 f 0.5 0.0 0.0 1.0 2.0 ξ r/r 1/2 0.0 0.1 0.2 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).
0.02 V /U o 0.00 1.0 2.0 ξ r/r 1/2-0.02 Figure 5.6: Mean lateral velocity in the self-similar round jet. From the LDA data of Hussein et al. (1994).
0.10 u i u j 0.08 2 U 0 0.06 0.04 u 2 w 2 v 2 0.02 uv 0.00 0.0 1.0 2.0 r/r 1/2 3.0 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).
5.0 4.0 U 3.0 2.0 1.0 0.0 0.0 1.0 2.0 r/r 1/2 3.0 0.0 0.1 0.2 0.3 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).
0.5 0.4 ρ uv 0.3 0.2 uv /k 0.1 0.0 0.0 1.0 r/r 1/2 2.0 0.0 0.1 0.2 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).
ν ^ T 0.03 0.02 0.01 0.00 0.0 1.0 r/r 1/2 2.0 0.0 0.1 0.2 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).
l r 1/2 0.15 0.10 0.05 0.00 0.0 1.0 r/r 1/2 2.0 0.0 0.1 0.2 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).
1.0 0.8 L 11 /r 1/2 0.6 0.4 0.2 L 22 /r 1/2 0.0 0.0 1.0 r/r 1/2 2.0 0.0 0.1 0.2 η Figure 5.12: Self-similar profiles of the integral lengthscales in the turbulent round jet. From Wygnanski and Fiedler (1969).
R (x,0,s) 11 1.0 0.5 0.0 0 1 2 3 4 5 s/l 11 Figure 5.13: Longitudinal autocorrelation of the axial velocity in the self-similar round jet. From Wygnanski and Fiedler (1969).
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.
1.0 U U 0 0.8 0.6 0.4 0.2 0.0 1.0 2.0 r/r 1/2 0.0 0.1 0.2 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).
Gain 0.02 0.01 Production Mean-flow convection 0.00 Loss 0.01 Turbulent transport Dissipation 0.02 0.0 1.0 2.0 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).)
f(ξ) U h U l +0.9(U h -U l ) 0.5 0.4 U c 0.0 U l +0.1(U h -U l ) U l δ -0.4-0.5 y 0.1 y 0.9 y -1.0-0.5 0.0 0.5 1.0 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.
f (ξ) 0.5 x = 39.5 cm. x = 49.5 cm. x = 59.5 cm. 0.0-0.5-1.0-0.5 0.0 0.5 1.0 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.
y(cm) 4 y 0.95 0 y 0.5 y 0.1 0 20 40 60 80 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).
0.5 U U s 0.0-0.5-1 0 1 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.
u i u j U s 2 0.030 u 2 /U s 2 0.015 v 2 /U s 2 0.000 1 0 1 n u i u j U s 2 w 2 /U s 2 0.015 0.000 0.015 1 0 1 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).
1 f (ξ) 0-3 -2-1 0 1 2 3 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).
1.0 f(ξ) 0.5 0.0-2 -1 0 1 2 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)
1.0 0.5 u /U 2 1/2 S v /U 2 1/2 S w /U 2 1/2 S 0.0-4.0-2.0 0.0 2.0 4.0 ξ Figure 5.28: R.m.s. velocity profiles in a self-similar axisymmetric wake. Experimental data of Uberoi and Freymuth (1970).
Gain Mean Flow Convection Production 0.0 Dissipation Loss Turbulent Transport -3.0 0.0 3.0 ξ Figure 5.29: Turbulent kinetic energy budget in a self-similar axisymmetric wake. Experimental data of Uberoi and Freymuth (1970).
y h 0 U c U Figure 5.30: Sketch of the mean velocity profile in homogeneous shear flow.
u i u j U c 2 0.003 0.002 0.001 0 4 5 6 7 8 9 10 11 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.
M M d Figure 5.32: Sketch of a turbulence generating grid composed of bars of diameter d, with mesh spacing M.
10 3 u 2 2 U 0 10 4 10 500 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).)
1.0 f(ξ) ϕ(ξ) 0.5 0.0 1.0 2.0 3.0 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
0.3 φ 2 1/2 φ y=0 0.2 0.1 0.0 0.0 1.0 r/r 1/2 2.0 0.0 0.1 0.2 η Figure 5.35: Normalized r.m.s. scalar fluctuations in a round jet. From the experimental data of Panchapakesan and Lumley (1993b).
Gain 0.02 0.01 Production Mean-flow convection 0.00 Loss 0.01 Turbulent transport Dissipation 0.02 0.0 1.0 2.0 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).)
2.0 τ/τ φ 1.5 1.0 0.5 0.0 1.0 2.0 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).)
φ 2 φ 2 0.20 0.15 0.10 0.05 0.00 10 3 10 4 10 5 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).)
I(t) I(t) 1 0 1 0 (a) (b) γ=0.0 γ=0.25 t t I(t) I(t) 1 0 1 0 (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.
1.0 0.5 φ γ 0 0 1 2 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).
1.0 (a) 1.5 (b) φ' T 1.0 0.5 φ T φ 0.5 φ' 2 1/2 0.0 1.0 2.0 0.0 y/y φ 0.0 0.0 1.0 2.0 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).
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.
γ 1.0 U U S 0.5 U γ U T U N 0-1 0 1 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).
u 2 0.03 U S 2 0.02 0.01 T 2 (u' ) u 2 N 2 (u' ) 0-1 0 1 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).
f^u f^w (a) (c) 0.3 0.3 0.2 0.1 0.2 0.1-3 -2-1 0 1 2 3 f^v 0.3 (b) 0.2 0.1 V 1-3 -2-1 0 1 2 3 f^φ V 3 0.3 (d) 0.2 0.1-3 -2-1 0 1 2 3 V 2-3 -2-1 0 1 2 3 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).)
(a) V 2 2 1-3 -2-1 0 1 2 3-1 V 1 (b) -2 V 2 2 1-3 -2-1 0 1 2 3-1 ψ (c) ψ -2 2 1-3 -2-1 0 1 2 3-1 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 0.01. Dashed lines are corresponding contours for joint-normal distributions with the same correlation coefficients. (From Tavoularis and Corrsin (1981).)
f (ψ) φ 2 ξ = 0 ξ = 0.22 1 ξ = 0.43 ξ = 0.65 0 ξ = 0.86 0.0 0.2 0.4 0.6 0.8 1.0 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).
4.0 3.0 2.0 S φ S φ 9.0 7.0 K φ K φ 1.0 5.0 0.0 S φt 3.0 K φt -1.0 1.0 0.0 1.0 y/y 2.0 φ 0.0 1.0 2.0 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).
10 ξ=0.66 U s f u (V) 5 ξ=0.44 ξ=0 ξ=0.22 0-0.5 0.0 0.5 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).
S u 4.0 20 K u 2.0 10 3 1.0 2.0 3.0 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).
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).)
20 15 10 5 f (Hz) 0 θ (mm) 60 50 40 30 500 1000 1500 x (mm) Figure 5.53: Mixing layer thickness, θ, against axial distance, x, for different forcing frequencies, f. (From Oster and Wygnanski (1982).)