Local flow structure and Reynolds number dependence of Lagrangian statistics in DNS of homogeneous turbulence. P. K. Yeung

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1 Local flow structure and Reynolds number dependence of Lagrangian statistics in DNS of homogeneous turbulence P. K. Yeung Georgia Tech, USA; B.L. Sawford (Monash, Australia); S.B. Pope (Cornell, USA) Thanks to: Workshop Organizers; NSF (Fluid Dynamics) Pittsburgh & San Diego Supercomputing Centers (USA) Workshop on Inertial-Range Dynamics and Mixing (HRT Programme) Isaac Newton Institute for Mathematical Sciences Cambridge, UK; October 2, 2008 PK Yeung, INI-HPC Workshop, Cambridge p.1/17

2 Lagrangian statistics from DNS Principle is straightforward: integrate equation of motion with Lagrangian velocity interpolated from Eulerian fields Multiple research groups have contributed recently to DNS for Lagrangian statistics; notable advances in experiments also Very useful for stochastic modeling [based on (a) inertial range behavior is not well captured (b) intermittency strong but not well characterized?], but Connection to local flow structure is the key... PK Yeung, INI-HPC Workshop, Cambridge p.2/17

3 Local flow structure Fluid particles moving in regions of large fluctuating velocity gradients will experience a rapid change in velocity, i.e., a large acceleration Local straining, rotation, or combination of effects dissipation: enstrophy: pseudo-dissipation: Strain-dominated vs rotation-dominated regions strain is very important in dispersion of particle pairs rotation can cause frequent changes in direction Need to know statistics and time scales of, and particle trajectories, as function of Reynolds no., along fluid PK Yeung, INI-HPC Workshop, Cambridge p.3/17

4 Acceleration: Eulerian view From the Navier-Stokes equations: analysis of archived velocity fields, pressure via Poisson eqn. highly intermittent, dominated by pressure gradient local and convective contributions are in strong mutual cancellation, re: random sweeping hypothesis (Tennekes 1975, DNS data in Tsinober, Vedula & Yeung Phys. Fluids 2001) Conditional sampling and intermittency:, where,, or if intermittency in acceleration were entirely due to conditional PDF of given would be Gaussian, then PK Yeung, INI-HPC Workshop, Cambridge p.4/17

5 Lagrangian conditional statistics Conditional sampling based on trajectories, e.g.:,, or along particle with in logarithmic intervals Lagrangian time series of, and can be obtained by high-order interpolation in DNS dependence expected to last for time lags comparable to integral time scales of,, Stochastic model development: e.g., conditional cross-correlation between velocity and acceleration, given pseudo-dissipation, to characterize a joint stochastic process (More details in 2 JFM papers, 2007) PK Yeung, INI-HPC Workshop, Cambridge p.5/17

6 Results in isotropic turbulence DNS to, Pseudo-spectral, cubic-spline interpolation Eulerian statistics of acceleration unconditional and conditional, resolution effects Lagrangian statistics of the conditioning variables Lagrangian conditional statistics Other aspects and future plans (towards Petascale) PK Yeung, INI-HPC Workshop, Cambridge p.6/17

7 Scaling of Acceleration Variance Is universal at high? : is irrotational is solenoidal not yet resolved, but weak intermittency correction likely higher still needed : skewness of Resolution effects may be present, but seems sufficient PK Yeung, INI-HPC Workshop, Cambridge p.7/17

8 Eulerian Conditional Accel. Statistics Conditional variances Conditional flatness factors e.g. K62 Large tends to give larger large from K62), does (and latter deviates, than, Modeling: Approx conditional Gaussanity for, given intermittency of both, reflecting and PK Yeung, INI-HPC Workshop, Cambridge p.8/17

9 Resolution Effects on Conditional Stats. Acceleration is highly intermittent and hence sensitive to resolution (Yakhot & Sreeni, 2005) Acceleration PDF, At, flatness is 20, 27 and 26 for simulations with However conditional statistics are much less sensitive: is not very far from Gaussian Model accel PDF as where is better known Normalized fluctuation Outer lines: Unconditional PDF; Inner lines: Conditional PDF PK Yeung, INI-HPC Workshop, Cambridge p.9/17

10 Integral time scales of,, (Yeung, Pope, Sawford, J. Turb., 2006)) Grid As increases, drops relative to but rises relative to Time scales of and becoming similar consistent with trends towards same scaling in Eulerian data (Donzis, Yeung & Sreenivasan, Phys. Fluids, Apr. 2008) PK Yeung, INI-HPC Workshop, Cambridge p.10/17

11 Dissipation-enstrophy cross-correlation vs. : narrower at high Stronger for : high followed by high is more likely than high followed by high events of high more intermittent and last shorter Degree of asymmetry weakens as peak value ( ) rises Eulerian: extreme and more nearly coincident. Lines A-E: PK Yeung, INI-HPC Workshop, Cambridge p.11/17

12 Conditional Velocity Autocorrelations is different... PK Yeung, INI-HPC Workshop, Cambridge p.12/17 and low at high

13 Motion in local coordinate axes Velocity and accel. and to vorticity vector: larger than accel. in fixed frame because vorticity orientation changes rapidly (in time ), esp. at high Reynolds no. Signatures of vortex-trapping may include: rapid change of compared to large centripetal acceleration ( to ) PK Yeung, INI-HPC Workshop, Cambridge p.13/17

14 Unconditional autocorrelations:,, Reynolds number dependence of mean-squares: PK Yeung, INI-HPC Workshop, Cambridge p.14/17

15 Conditional Accel. Autocorrelations Stronger dependence than for velocity autocorrelation (as expected), especially on pseudo-dissipation Result for moderately large, or autocorrelaton: i.e. events of large, unconditional acceleration statistics closest to unconditional or dominate the PK Yeung, INI-HPC Workshop, Cambridge p.15/17

16 What Else Two-particle dispersion, accumulated data over a wide range of Reynolds numbers and initial separations, for theory and modeling (Brian Sawford s talk, after lunch) Three and four particle statistics: measures of size (increase strongly with ) measures of shape (asymptotic distributions) Higher-order Lagrangian velocity structure functions need large numbers of Lagrangian trajectories sampled at intervals much shorter than Kolmogorov time scale Full analysis of high-resolution datasets (Eulerian) PK Yeung, INI-HPC Workshop, Cambridge p.16/17

17 What s Next simulation in the next few months, using 16K processors (cores) on 170-Teraflop machine (NSF Track 2B, U. Tennessee, to reach 1 Petaflop/s in 2009) More complex flows: stratified, including horizontal versus vertical dispersion (collaboration with J.J. Riley, U. Washington, Petascale computing applications project) Lunch... PK Yeung, INI-HPC Workshop, Cambridge p.17/17