CHAPTER 11: REYNOLDS-STRESS AND RELATED MODELS. Turbulent Flows. Stephen B. Pope Cambridge University Press, 2000 c Stephen B. Pope y + < 1.

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1 1/3 η 1C 2C, axi 1/6 2C y + < 1 axi, ξ > 0 y + 7 axi, ξ < 0 log-law region iso ξ -1/6 0 1/6 1/3 Figure 11.1: The Lumley triangle on the plane of the invariants ξ and η of the Reynolds-stress anisotropy tensor. The lines and vertices correspond to special states (see Table 11.1). Circles: from DNS of channel flow (Kim et al. 1987). Squares: from experiments on a turbulent mixing layer (Bell and Mehta 1990). 1C, one-component; 2C, two-component.

2 1/3 η ξ -1/6 0 1/6 1/3 Figure 11.2: Trajectories on the ξ-η plane given by the model of Sarkar and Speziale (1990) (Eqs and 11.57).

3 1/3 η F=0 F<0 (a) (b) (c) F>0 ξ -1/6 0 1/6 1/3 Figure 11.3: The Lumley triangle showing trajectories of three types: (a) violates realizability; (b) satisfies weak realizability; (c) satisfies strong realizability. (Note: other types of trajectories are possible.)

4 1/3 η 1/6 A O B -1/6 0 1/6 ξ 1/3 Figure 11.4: Sketch of trajectories (A and B) on the ξ-η plane for two experiments (or DNS) in which the initial spectra are different, but the initial values of b are the same. A Reynolds-stress model yields a unique trajectory from initial point O.

5 (a) x 2 x 1 (b) (c) Figure 11.5: Crests of the fields φ(x, t) evolving by Dφ/ Dt = 0 (a) initial condition, φe iκo x, κ o 1 = κ o 2 > 0, κ o 3 = 0 (b) after plane straining ( S 11 = S 22 > 0) (c) after shearing U 1 / x 2 > 0.

6 (a) (b) (c) (d) Figure 11.6: Trajectories of the unit wavevector ê(t) on the unit sphere from random initial conditions for (a) axisymmetric contraction (b) axisymmetric expansion (c) plane strain (d) shear. The ê 1 direction is horizontal, the ê 2 direction is vertical, and the ê 3 direction is into the page. The symbols mark the ends of the trajectories after distortion.

7 u (t) e(t) Figure 11.7: Sketch of the unit sphere showing the unit wavevector ê(t). The Fourier component of velocity û(t) is orthogonal to ê(t), and so it is in the tangent plane of the unit sphere at ê(t).

8 4.0 u i u j k(0) u 2 2 = u u S λ t Figure 11.8: Sketch of the unit sphere showing the unit wavevector ê(t). The Fourier component of velocity û(t) is orthogonal to ê(t), and so it is in the tangent plane of the unit sphere at ê(t).

9 u 2 2 k(0) S λ t Figure 11.9: Evolution of u 2 2 (on a log scale) for axisymmetric contraction rapid distortion (solid line). The dashed line is 1 2 exp(s λt) indicating the asymptotic growth rate.

10 3.0 u i u j k(0) u 1 2 u 2 2 = u S λ t Figure 11.10: Evolution of Reynolds stresses for axisymmetric expansion rapid distortion. The dashed lines show the asymptotic growth as exp(s λ t).

11 5.0 u i u j k(0) u u u S λ t Figure 11.11: Evolution of Reynolds stresses for plane strain rapid distortion. The dashed line is 1 2 exp(s λt).

12 0.4 b 11 b ij b 33 b b Figure 11.12: Evolution of Reynolds-stress anisotropies for shear rapid distortion.

13 1/3 η 1/6 ξ -1/6 0 1/6 1/3 Figure 11.13: Evolution of the Reynolds-stress invariants for shear rapid distortion. Starting from the origin (corresponding to isotropy), each symbol gives the state after an amount of shear St = 0.5.

14 10 k k(0) Figure 11.14: Evolution of the turbulent kinetic energy for shear rapid distortion.

15 0.2 b 11 b ij b 12 b 22, b Figure 11.15: Reynolds-stress anisotropies in homogeneous shear flow. Comparison of LRR-IP model calculations (lines) with the DNS data of Rogers Moin (1987) (symbols):, b 11 ;, b 12 ; squares, b 22 ; triangles, b 33.

16 0.010 (a) production production (b) turbulent transport pressure transport turbulent transport pressure turbulent production transport transport pressure transport dk/dt -dk/dt -dk/dt dissipation y/δ dissipation dissipation y/δ y/δ Figure 11.16: Kinetic energy budget in the temporal mixing layer from the DNS data of Rogers and Moin (1994): (a) across the whole flow (b) an expanded view of the edge of the layer. The contributions to the budget are: production P; dissipation ε; rate of change dk/dt; turbulent transport; pressure transport (dashed line). All quantities are normalized by the velocity difference and the layer thickness δ (see Fig. 5.21).

17 ν T /ν y + Figure 11.17: Turbulent viscosity against y + for channel flow at Re = 13, 750. Symbols, DNS data of Kim et al. (1987); solid line, 0.09k 2 /ε; dashed line, 0.22 v 2 k/ε.

18 ε ij 2 ε ε ε 33 ε ε y + Figure 11.18: Normalized dissipation components in a turbulent boundary layer at Re θ = 1, 410: symbols, DNS data of Spalart (1988); dashed lines, Rotta s model, Eq. (11.167); solid lines, Eq. (11.169).

19 x r r x wall, y=0 x Figure 11.19: Sketch of the point x and its image x, showing the vectors r and r that appear in the Green s function solutions, Eqs. (11.181) and (11.182).

20 0.3 b ij 0.2 b b 22 =b b 12 b 12 (k-ε) P/ε Figure 11.20: Reynolds-stress anisotropies as functions of P/ε according to the LRR-IP algebraic stress model. The dashed line shows b 12 according to the k-ε model.

21 0.30 C µ P/ε Figure 11.21: The value of C µ as a function of P/ε given by the LRR-IP algebraic stress model (Eq ).

22 Ωk/ε (a) 0.03 Ωk/ε (b) Sk/ε Sk/ε Figure 11.22: Contour plots of (a) C µ = G (1), and (b) G (2), for the LRR-IP nonlinear viscosity model (Eqs ).