3D Large-Scale DNS of Weakly-Compressible Homogeneous Isotropic Turbulence with Lagrangian Tracer Particles

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1 D Large-Scale DNS of Weakly-Compressible Homogeneous Isotropic Turbulence with Lagrangian Tracer Particles Robert Fisher 1, F. Catteneo 1, P. Constantin 1, L. Kadanoff 1, D. Lamb 1, and T. Plewa 1 University of Chicago, 540 S. Ellis Ave., Chicago, Il., USA 07 rfisher@flash.uchicago.edu We present the results of a large-scale weakly-compressible homogeneous isotropic turbulence simulation. Analysis of the anomalous scaling exponents derived from the simulation are in excellent agreement with both model prediction and experimental results, and strongly suggest that the inertial range properties are insensitive to the detailed dissipation mechanism. In addition, we present the foundations of a rigorous determination of the errors of model prediction at finite sample size. 1 Introduction During December 005 through January 00, the FLASH team executed a large-scale homogeneous isotropic compressible turbulence simulation with Lagrangian tracer particles on the Lawrence Livermore National Laboratory IBM BG/L using the fluid dynamics code FLASH. Although FLASH has been developed primarily for the purpose of simulating astrophysical fluid flows, and therefore includes appropriate features such as adaptive mesh refinement and multiple physics solvers (self-gravity, nuclear networks, etc., this simulation included only a single uniform, periodic domain mesh and compressible hydrodynamics solver. The FLASH BG/L simulation was performed on a 185 base Eulerian mesh, with 5 Lagrangian tracer particles. The hydrodynamics was evolved using a MILES-based approach solving the Euler equations with the Piecewise Parabolic Method (PPM. The initial conditions consisted of the fluid initially at rest, and subsequently driven using a stochastic driver. The D turbulent RMS Mach number of the flow in steady-state was 0. (.17 in 1D. The estimated Taylor-microscale Reynolds number Re λ is in the range

2 Fisher et al 1.1 Analysis of Simulation Results We consider the scaling properties of the p-th order structure functions S p (r of turbulent flows: S p (r = v(x + r v(x p r ζp (1 where v is the velocity field, and denotes an average of spatial locations over all spatial locations x. The proportionality applies in the inertial regime, and is a direct consequence of self-similarity at high Reynolds numbers. If one further assumes homogeneous local dissipation [8], one obtains the Kolmogorov (K41 scaling ζ p = p/. However, while both experiment and simulation [, 7] do exhibit power-law behavior, they do not follow the K41 scaling law, and are said to exhibit anomalous scaling. Various theories [9, 4, 1, 1, 11] have been advanced to explain anomalous scaling as the result of dissipative intermittency. Experiment [], first-principles theory [5], shell models [10], and numerical simulation [7] all provide useful constraints on the scaling exponents ζ p, and can begin to exclude some models, but it remains unclear which theories can be definitively ruled out by both experiment and simulation, and which remain as viable theories of turbulence. We have completed an analysis on our dataset, stored over roughly three hundred evenly spaced intervals in one eddy turnover time. In order to maximize the information stored in the inertial range, we have coarsened each cube by a factor of two in linear dimension, though at the expense of eliminating some of the stored dissipation range quantities. However, by time-averaging our results, this coarsening technique dramatically improves the overall statistics of the computed structured function exponents. The results of this analysis are shown in figure 1, alongside both the K41 and She-Leveque model predictions, as well as experimental wind tunnel results []. The agreement between our simulation and both the She-Leveque model and wind tunnel results is excellent through these orders. We infer both that the effects of compressibility at this low Mach number are negligible and that the inertial range scaling is insensitive to the details of the dissipation mechanism. We have also examined higher-order moments of our derived structure functions against model prediction and previous simulation. In figure, we show our results along those of a previous simulation by Gotoh and collaborators [7]. Both simulations show a systematic departure from model prediction at higher-order moments. The key question which then arises is whether such deviations are to be expected, or whether it is a signature of the breakdown of the model, or possibly of universality. This is a highly delicate issue, and requires a careful attention paid to statistical errors.

3 D Large-Scale DNS of Homogeneous Isotropic Turbulence Fig. 1. A plot of the anomalous scaling exponents ζp versus order p for K41 (solid red line, She-Leveque (solid purple line, experiment [] (circles, and our simulation (crosses with error estimates. The error bars correspond to estimates of systematic errors stemming from lack of perfect isotropy. Fig.. Our simulation (blue circles shown along with previous simulation results [7] (red croses. Solid green line shows the She-Leveque model. Error Analysis To date, attention has primarily focused on the origin of errors in actual turbulence data gathered from either numerical simulation or experiment. However, this approach has significant drawbacks. Real data is itself subject to multiple sources of both systematic and random errors, making it difficult to disentangle these effects. Some progress on the issue of statistical convergence has been made by in various contexts by examining the sensitivity of results to data sample size [10], by extrapolating the tails of the velocity increment PDF [1], and by determination of the power law of order-ranked velocity increments []. Working from these methods, various authors have reached widely-varying conclusions with regard to the maximum structure function moment order pmax which can be meaningfully determined. Crucially, there exists an intrinsic uncertainty related to the model prediction of the higher-order moment exponents ζp for a finite, as opposed to infinite, sample size. Specifically, by treating the moment estimates as a random variable, we develop an asymptotic expansion in 1/N of the moment

4 4 Fisher et al estimator of a known model PDF at finite sample size N, and thereby construct the distribution of moment estimates of the underlying known PDF. Our formulation is entirely general with regard to any underlying PDF; in the context of the analysis of turbulence datasets it is applied to the velocity increment PDF P ( v r. Knowledge of this distribution of moment estimates at finite sample size lays the foundation for a rigorous statistical comparison of turbulence datasets against model predictions. The p th order algebraic moment estimator for a sample of size N is Y = 1 N N X p i ( i=1 where the X i are random variables drawn from a known parent distribution f X (X. We wish to obtain the distribution of algebraic moment estimates of sample size N drawn over a full ensemble. Formally, this is given by the probability distribution function of the random variable Y. Note that the algebraic moment estimator is always unbiased for any N, because its expectation value is trivially the same as the true moment by linearity of the expectation operator E : ( 1 N E(Y = E X p i = E(X p i N ( i=1 The method we will employ to determine this distribution is an asymptotic expansion in 1/N for the characteristic function φ Y (t, defined by the Fourier transform of f Y (Y φ Y (t = e ity f Y (Y dy (4 where f Y (Y is the PDF from which Y is drawn. Therefore, knowing the characteristic function, we can invert the transform to determine the PDF. f Y (Y = 1 π e ity φ Y (tdt (5 Using a multiplicative identity for sums of random variables, Z = i X i ( where Z and the X i are all random variables, then φ Z (t = i φ Xi (t (7 and another identity for multiplication of a random variable by a constant a,

5 We have D Large-Scale DNS of Homogeneous Isotropic Turbulence 5 φ Y (t = Z = ax (8 φ Z = φ X (t/a (9 N i=1 Taking the log of both sides : φ X p (t/n = φn i Xp(t/N (10 log φ Y (t = N log φ X p(t/n (11 The characteristic function φ X p(t/n can be expanded in terms of the algebraic moments of the PDF of X p, φ X p(t/n = (it/n r µ r (1 r! r=0 where µ r is the r th order algebraic moment of X p. We can rewrite these moments of the distribution over X p in terms of the moments of the underlying F X (X distribution. We may considering the random variable X p as the result of a transformation of the random variable X Z, with Z = X p. Writing the PDF over Z as G(Z, we may express the conservation of probability between the two spaces as : G(ZdZ = F (XdX (1 The r th order algebraic moment over Z may be found by multiplying through by Z r and integrating both sides : µ r = Z r G(ZdZ = X rp F (XdX (14 Therefore µ r, the rth order moment of X p, is identical to the rpth moment of the parent F (X distribution. µ 0 = 1 by normalization of the PDF, so : φ X p(t/n = N itµ 1 1 N µ i 1 N µ +... (15 If N >> 1, we may carry out an asymptotic expansion in 1/N, using the identity log(1 + x x x / + x / +..., where x is the infinite series x = 1 N itµ 1 1 ( t N i 1 ( t N µ +... (1 We therefore have

6 Fisher et al log φ X p(t = 1 N itµ 1 1 N Therefore, [ φ Y (t = exp itµ 1 1 N (µ µ 1 i 1 N (µ µ 1 i 1 N f Y (Y = 1 [ e ity exp itµ 1 1 π N ( 1 (µ µ µ 1 +µ 1+O (17 N 4 ( 1 (µ µ µ 1 + µ 1 + O (µ µ 1 i 1 N (19 When we retain only the leading terms in this expansion to order 1/N, and compute the distribution function of moments f Y (Y, we recover the familiar central limit theorem result : f Y (Y 1 πσ exp ( (Y µ 1 σ N 4 ] (18 (µ µ µ 1 + µ 1 + O (0 This proves that the moment estimator (eqn. is consistent. However, while the unbiased property ensures that the expectation value of the moment estimator f Y (Y is identical to the true value E[f Y (Y ], it does not guarantee that the underlying shape of the moment estimate PDF is Gaussian. In general, for finite and diminishing N, the distribution f Y (Y will become more highly skewed towards low values of Y, and increasingly non-gaussian. We believe this effect explains the systematic deviation seen in figure, and are currently extending this result to a rigorous statistical comparison of the data with model prediction. ( ] 1 N 4 dt References 1. Benzi, R., Biferale, L., Paladin, G., Vulpiani, A., & Vergassola, M. 1991, Physical Review Letters, 7, 99. Benzi, R., Ciliberto, S., Tripiccione, R., Baudet, C., Massaioli, F., & Succi, S. 199, Physical Review E, 48, 9. Dudok de Wit, T. 004, Physical Review E, 70, Frisch, U., Sulem, P.-L., & Nelkin, M. 1978, Journal of Fluid Mechanics, 87, Frisch, U. 1995, Cambridge: Cambridge University Press, c1995,. Fryxell, B., et al. 000, Astrophysical Journal Supplement, 11, 7 7. Gotoh, T., Fukayama, D., & Nakano, T. 00, Physics of Fluids, 14, Kolmogorov, A. N. 1991, Royal Society of London Proceedings Series A, 44, 9 9. Kolmogorov, A. N. 19, Journal of Fluid Mechanics, 1, Leveque, E., & She, Z.-S. 1997, Physical Review E, 55, Meneveau, C., & Sreenivasan, K. R. 1987, Physical Review Letters, 59, She, Z.-S., & Leveque, E. 1994, Physical Review Letters, 7, 1. van de Water, W., & Herweijer, J. A. 1999, Journal of Fluid Mechanics, 87,