Lagrangian evolution of non-gaussianity in a restricted Euler type model of rotating turbulence

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1 Lagrangian evolution of non-gaussianity in a restricted Euler type model of rotating turbulence School of Mathematics and Statistics University of Sheffield INI Workshop, Cambridge, September 2008

2 Non-Gaussian statistics/lagrangian approaches Falkovich et al 01. Chertkov et al 99, Pumir et al 00, Pumir et al 01, Naso & Pumir 05, Naso et al 06, Naso et al 07. Advected delta-vee system: Li & Meneveau 05, Li & Meneveau 06 Restricted Euler approximation (Leorat 75, Vieillefosse 82, Vieillefosse 84, Cantwell 92) Jeong & Girimaji 03, Chevillard & Meneveau 06

3 Advected delta-vee system (N spatial dimensions) Definitions of increments: δu l Ãrr l Ãki ˆr k ˆr i, δv l P ij (r) Ãkj ˆr k = l à ij ˆr i ê j, δt l G k ˆr k Advected delta-vee system: D t δu = δu 2 l 1 + δv 2 l 1 + DlN 1 + Y, D t δv = 2 δu δv l 1 + Z, D t δt = δu δt l 1 + W Unclosed terms: Y,Z,W and D ÃijÃji

4 Restricted Euler approximations Equation for velocity gradient: D t à ij = ÃikÃkj + 1 N Dδ ij + H ij. where H ij ( 2 ij p 2 ik τ kj + ν 2 à ij ) d Y, Z are projections of H ij : Y = H ijˆr iˆr j, Z = H ijˆr i ê j Restricted Euler approximation: H ij = 0 (Leorat 75, Vieillefosse 82 84, Cantwell 92). Truncation in the advected delta-vee system: Y = Z = W = 0 Note: information on Y δu,δv, Z δu,δv etc is needed for more sensible models. (Li et al, JoT, in press)

5 Effects of incompressibility Effects of incompressibility: D = ÃijÃji à rr à re à rn à = à er à ee à en à nr à ne (Ãrr + Ãee) Note Ãrr = δu/l: D = 2Ã2 rr +... = 2l 2 δu Truncated system: (Andreotti 97, Galanti et al 97, Hou & Li 07) D t δu = (1 2/N) δu 2 l 1 + δv 2 l 1, D t δv = 2 δu δv l 1, D t δt = δu δt l 1,

6 Results: velocity increments in 3D & 2D PDFs of δu and δv in 3D 10 0 (a) 10 0 (b) P u P c v (δu-<δu>)/σ δu PDFs of δu and δv in 2D (δv c -<δv c >)/σ δv c 10 0 (a) 10 0 (b) P u P c v (δu-<δu>)/σ δu (δv c -<δv c >)/σ δv c

7 Results: passive scalars in 2D 10 0 (b) P T Summary: (δt-<δt>)/σ δt Several qualitative trends are reproduced. The effects of incompressibility constraint are important: (D ÃijÃji = 2Ã2 rr +...). Coupling between the two components of velocity increments leads to different behaviors in scalar and transverse velocity increments.

8 Delta-vee system with rotation Lagrangian Coordinate Frame ˆk = Direction of rotation axis. ˆr(t) = Direction of a material line. ŝ(t) = Ω ˆr/ Ω ˆr ˆt(t) = ˆr ŝ Velocity increments over distance l: δu(lˆr,t) u(x + lˆr,t) u(x,t) = Uˆr + V ŝ + Wˆt. Approximation (Ãij: filtered velocity gradient): U = lãjiˆr jˆr i, V = lãijˆr i ŝ j, W = lãijˆr iˆt j,

9 Delta-vee system with rotation (N = 3). Equations for velocity gradients: D t à ij = ÃikÃkj 2ε jmn Ω m à in Dδ ij 2 3 Ω m ω m δ ij + H ij. where H ij ( ij p 2 2 ik τ kj + ν 2 à ij ) d Delta-vee type system in rotating frame D t U = 1 3 U2 l 1 + V 2 l 1 + W 2 l 1 + 2ΩV sin θ D t V = 2UV l 1 + WV l 1 cot θ 2Ω(U sin θ W cos θ) D t W = 2UWl 1 V 2 l 1 cot θ 2V Ωcos θ. D t θ = W l 1 (θ: the angle between ˆk and ˆr).

10 Comparison with DNS Forced DNS: resolution with Ro = 0.02 (Yeung & Zhou 98). Pseudospectral simulation. AB2 in time. Use helical decomposition to diagonalize the Coriolis force. Deterministic forcing to maintain constant energy dissipation: ˆf(k) = A û(k) when k < 2.

11 Comparison with DNS Inset: E(k) k 2. Solid: with rotation; dashed: without rotation. 1 Ratios of RMS values: Ratios E(k) k k k = π / (D t U) RMS,model (D t U) RMS,DNS for l in perpendicular plane. Squares: U; gradients: V ; circules: W. Considerable contributions are observed, especially for U and V. The contribution for W component decreases when displacement l increases. Indirect effects of rotation are not captured.

12 Evolution starting from Gaussian: no rotation Gaussian initial conditions: U rms = V rms = W rms = 1. P (U cosθ = 0) 10 0 (a) (U - < U > ) / σ U P (V cosθ = 0), P ( W cosθ = 0) 10 0 (b) (V - < V > ) / σ V, (W - < W > ) / σ W Dotted: t = 0 (Gaussian); dashed: t = 0.06; dash-dotted: t = 0.12; long-dashed: t = 0.18; dash-double-dotted: t = Reproduce the results obtained from equations for δu and δv.

13 Ro = 0.025: PDFs of U and V P (U cosθ = 0), P (V cosθ = 0) 10 0 (a) - S U t ( U - < U > ) / σ U, ( V - < V > ) / σ V 0 Reduced skewness in U (Cambon et al 97, Morize et al 05). Quasi-two-dimensionalization Explanation from the model: nonlinear term negligible D t U = 2ΩV D t V = 2ΩU At strong rotation, the PDFs of U and V stay close to Gaussian. Energy conservation: U 2 + V 2 = const

14 Ro = 0.025: PDF of W P (W cosθ = 0) 10 0 (b) F t W decoupled from U, similar to a passive scalar: D t U = 2 Ω V D t V = 2 Ω U D t W = 2 U W (W - < W > ) / σ W PDF of W develops non-gaussian tails. Consistent with the 2D3C state (Chen et al 05). P (W cosθ = 0 ) (W - < W > ) / σ W

15 Positive skewness in P(V ): Ro = 0.1 Experiments Morize et al 05 Model results P (V cosθ = 0) S V 10-1 Ω t / 2 π ( V - < V >) / σ V Explanations (Lesieur et al 91, Bartello et al 94): prevalence of cyclonic vortices. Destablization of anti-cyclonic vortices when Ω not very large. Model: From negative skewness in U (absence for strong rotation). Negative V fluctuations are self-destroying (destablization of anti-cyclonic vortices). D t U = U 2 /3 + V 2 + W 2 + 2ΩV, D t V = 2UV 2ΩU

16 Results for Ro = PDF of U PDF of V P (U cosθ = 0 ) P (V cos θ = 0 ) (U - < U > ) / σ U P (W cos θ = 0 ) PDF of W ( V - < V >) / σ V Reduced non-gaussianity for U and V. Small positive skewness in V. Non-Gaussian tails in W (W - < W > ) / σ W

17 Summary The interaction of the nonlinear terms and the Coriolios force is studied using a restricted-euler-type model for velocity increments. Several trends related to the evolution of non-gaussianity are qualitatively reproduced, including reduced skewness in longitudinal velocity increments quasi-two-dimensionalization positive skewness in transvers increment in the perpendicular plane.