.4 g u.3.2.1 t. 6 4 2 2 4 6 Figure 12.1: Effect of diffusion on PDF shape: solution to Eq. (12.29) for Dt =,.2,.2, 1. The dashed line is the Gaussian with the same mean () and variance (3) as the PDF at Dt = 1. v
1. g u.5 t. 3 2 1 1 2 3 Figure 12.2: Solutions (Eq. 12.34) to Eq.(12.32) for t/t L =, 1 2, 1. v
.8 g u.6 t.4.2. 4 2 2 4 Figure 12.3: PDF g u (v; t) evolving according to the simplified Langevin model, Eq. (12.36). The PDF is shown at times at which the standard deviations are 1,.99,.9,.75 and.5. (The constant C is taken to be 2.1.) v
U * (t) σ 1-1 1-1 1-1 1-1 1-1 1 2 3 4 t/t L Figure 12.6: Sample paths of the Ornstein-Uhlenbeck process generated by the Langevin equation, Eq. (12.89)
ρ(s) 1..8.6.4.2. 1 2 3 4 5 s/t L Figure 12.7: Lagrangian velocity autocorrelation function. Line, Langevin model ρ(s) = exp( s/t L ); solid symbols, experimental data of Sato and Yamamoto (1987) R λ = 46, R λ = 66; open symbols, DNS data of Yeung and Pope (1989), R λ = 9.
(a) Y X + ( t,y ) (b) Y + ( t, x) x t t Figure 12.9: Sketches of forward (a) and backward (b) fluid particle trajectories (on different realizations of the turbulent flow). (a) Forward trajectories fluid particle paths originating at Y at time t. (b) Backward trajectories fluid particle paths that reach x at time t.
1 1 slope 1/2 σ X u T L 1 1-1 slope 1 1-1 1 1 1 1 2 t/t L Figure 12.1: Standard deviation σ X of dispersion from a point source given by the Langevin model (Eq. 12.159).
(a) 2. X * (t) u«t L 1.. 1. 2...2.4.6.8 1. t/t L (b) 4. X * (t) u«t L 3. 2. 1.. 1. 2. 3. 4. 2 4 6 8 1 Figure 12.11: Samples of fluid particle paths given by the Langevin model, shown for (a) moderate times (b) long times. The dashed lines show ±σ X (t). t/t L
Grid x x w U Source wire y x y= M φ(x,y) x Figure 12.12: Sketch of a thermal line source experiment, showing a heated wire downstream of a turbulence generating grid.
σ y L 1.34 x w 1-1 x w 1-2.5 x w 1-3 1-2 1-1 1 1 1 x w /x Figure 12.13: Thermal wake thickness σ Y (normalized by the turbulence lengthscale L ) as a function of the distance x w downstream of the wire (normalized by the distance from the grid to the wire x ). Line, Langevin model Eq. (12.168); symbols, experimental data of Warhaft (1984), x /M = 2 ( ), 52 ( ), 6 ( ).
5 ω*(t) ω 5 5 5 5 1 2 3 4 5 Figure 12.14: Sample paths of the lognormal stochastic model for turbulent frequency, Eq. (12.181). t/τ
4. 2. < v 3 >/< v 2 > 3/2 2. 15. < v 4 >/< v 2 > 2. 1. -2. 5. -4. -1. -.5..5. 1. -1. -.5..5 1. 4. 2. < u 3 >/< u 2 > 3/2 < u 4 >/< u 2 > 2 2. 15.. 1. -2. 5. -4.. -1. -.5..5 1. -1. -.5..5 1. Figure 12.15: Profiles of skewness and flatness of the axial (u) and lateral (v) velocities in the self-similar plane mixing layer. Lines, calculations by Minier and Pozorski (1995) based on the lognormal/refined Langevin model of Pope (1991a); symbols, experimental data of Wygnanski and Fiedler (197) ( ) and of Champagne et al. (1976) ( ). The abscissa is a normalized cross-stream coordinate. (From Minier and Pozorski (1995).)
ω*(t) ω 2 2 2 2 2 1 2 3 4 5 t/τ Figure 12.16: Sample paths of the gamma-distribution model for turbulent frequency, Eq. (12.191).
f ω (θ) 1.2 1..8.6.4.2. 1 2 3 4 5 θ/ ω Figure 12.17: Stationary PDF s of turbulent frequency given by the lognormal model (dashed line) and the gamma-distribution model (solid line).
ω * ω 1 1 1 1-1 1-2 1-3 1-4 -1. -.5..5 1. ξ Figure 12.18: Scatter plot of turbulence frequency ω (normalized by ω at ξ = ) against normalized lateral distance in the self-similar temporal shear layer. The dashed line is the unconditional mean, ω. The solid line is the conditional mean, Ω, Eq. (12.193). (From Van Slooten Jayesh, and Pope (1998).)
y y U R y p V I U I y p V R U R U I x x Incident Reflected Figure 12.19: Incident and reflected particle velocities for wall functions imposed at y = y p.
5 k u τ 2 4 3 2 1 1 2 3 4 y + Figure 12.2: Turbulent kinetic energy profile (in wall units) for fully developed channel flow at Re = 13,75. Symbols, DNS data of Kim et al. (1987); line, velocity-frequency joint PDF calculation using wall functions (from Dreeben and Pope 1997b).
.8 c f.6.4.2 1 3 1 4 1 5 Re Figure 12.21: Skin friction coefficient c f τ w /( 1 2 ρu 2 ) against Reynolds number (Re = 2U δ/ν) for channel flow: symbols, experimental data compiled by Dean (1978); solid line, velocity-frequency joint PDF calculations using wall functions (Dreeben and Pope 1997a); dashed line, near-wall joint PDF calculations using elliptic relaxation (Section 12.7.2, Dreeben and Pope 1998).
.4 y +.3 y p +.2.1...2.4.6.8 t + d t + t + u Figure 12.22: Distance Y + (t + ) of a particle from the wall (in wall units) as a function of time: sample path of reflected Brownian motion, Eq. (12.293). For the given level y + p, there is a down-crossing at t + d and the subsequent up-crossing is at t+ u.
8 u i u j u τ 2 7 6 5 4 3 2 1 1 2 3 4 y + Figure 12.23: Reynolds stresses in fully-developed turbulent channel flow at Re = 13,75. Symbols, DNS data of Kim et al. (1987), u 2, v 2, w 2, k; lines, near-wall velocity-frequency joint PDF calculations (from Dreeben and Pope 1998).
.3.2.1 b ij..11.22. 5. 1. 15. Figure 12.24: Evolution of Reynolds-stress anisotropies in homogeneous shear flow with (Sk/ε) = 2.36. Velocity-wavevector PDF model calculations of Van Slooten and Pope (1997) (lines) compared to the DNS data of Rogers and Moin (1987) (symbols): (, ), b 11 ; (---, ), b 12 ; (- -, ), b 22 ; (, ), b 33. t
2. f φ (ψ;t) 1.5 2. f φ (ψ;t) 1.5 1. 1..5.5. -2-1 1 2 ψ. -2-1 1 2 ψ Figure 12.25: Evolution of the PDF f φ (ψ; t) of a conserved passive scalar in isotropic turbulence from a double-delta-function initial condition: (a) DNS of Eswaran and Pope (1988a); (b) calculated from the mapping closure (Pope 1991b).