Statistical studies of turbulent flows: self-similarity, intermittency, and structure visualization

Transcription

1 Statistical studies of turbulent flows: self-similarity, intermittency, and structure visualization P.D. Mininni Departamento de Física, FCEyN, UBA and CONICET, Argentina and National Center for Atmospheric Research, Boulder, CO, USA.

2 Outline Turbulence and the Navier-Stokes equations. Simulations, 3D visualization, and rendering. Isotropic and homogeneous turbulence: K41 theory. Hydrodynamic simulations at large resolution. Intermittency and high order statistics. Rotating turbulence. Some results for conducting fluids. Alexakis, Mininni, & Pouquet, PRL 95, Alexakis, Mininni, & Pouquet, PRE 72, and Martin & Mininni, PRE 81, Rodriguez Imazio & Mininni, JFM 651, 241 Mininni & Pouquet, PRL 99, and PRE 80, Mininni & Pouquet, PoF 22, and

3 Turbulence A turbulent flow is a strongly nonlinear, dissipative system with an extremely large number of degrees of freedom. Turbulence is characterized by: Strong and impulsive events (except in inverse cascades) Wide range of strongly interacting scales Sensitivity to initial conditions (but stable statistical properties; universality?) Highly dissipative, statistically irreversible Strongly diffusive (enhanced transport) Non-gaussian statistics Due to the large number of degrees of freedom, tools are more likely to be probabilistic than geometric (as opposed to low dimensional chaos).

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5 20483

6 Multi-scale structures ( ) Self-similarity?

7 How do we simulate turbulence? GHOST: Geophysical High Order Suite for Turbulence Aim: to develop a code that scales from desktop computers to local clusters, and to state of the art supercomputers (collaborations!). Pseudo-espectral, modular, portable, parallel, and scales up to tens of thousands of processors. General PDE solver: solves the equations for neutral fluids, conducting fluids, rotating flows, stratified flows, several LES models, etc. For production runs, used with resolutions from 64 3 to (over data points), and in over cores.

8 Big datasets pose a problem for scientific discovery. Can we recover interactivity? Can we overcome HPC limitations? Visualization: VAPOR

9 VAPOR: Visualization and data analysis Interactive visualization and data analysis for terascale and petascale datasets. Runs in desktop computers (even in this laptop!). 3D rendering technology plus hierarchical representation of data with lossless compression. Led by John Clyne and Alan Norton (NCAR).

10 VAPOR: Visualization and data analysis Data analysis tools designed keeping the scientific problems in mind.

11 Turbulence Two approaches: A statistical theory of turbulence Statistics used to characterize general properties of small scale turbulent flows: Scaling with Reynolds number (Landau-Hopf theory of turbulence) What is the limit of large Reynolds number? How do we extrapolate from simulations and experiments to geophysical flows? Scaling with scales at constant (but large) Reynolds number (Kolmogorov theory of turbulence) Self-similarity Universality

12 Turbulence I soon understood that there was little hope of developing a pure, closed theory, and because of absence of such a theory the investigation must be based on hypotheses obtained on processing experimental data. (Kolmogorov) Statistics used to process experimental data: First studies focused on mean quantities (velocity), energy spectrum, and probability density functions (PDFs). Then multi-point correlations were considered (high order correlation functions and structure functions). More recently, conditional statistics were introduced. A need for structure dependent statistical quantities, or a link between statistics and structural properties of turbulent flows became apparent.

13 The Navier-Stokes equations Momentum equation v t 2 + P is the pressure, F an external force, ν the kinematic viscosity, and v the velocity; incompressibility is assumed. Quadratic invariants (F = 0, ν =0): E = v 2 d 3 x H = v ω d 3 x Reynolds numbers: ( v ) v= P+ ν v+ F Re = UL / ν ω = v R λ = Uλ / ν v where L is the integral scale andλthe Taylor scale. = 0

14 Starting from v= The energy cascade sin( k0x)cos( k0 y)cos( k0z) cos( k0x)sin( k0 y)cos( k0z) 0 as initial condition, and replacing in the Navier-Stokes equation t v x = k sin(2k 8 x) 2 [ cos(2k z) cos(2k y) ] 3k ν cos( k x)sin( k y)sin( k ) 0 0 z This process can be repeated, and smaller eddies are created until reaching the scale where the dissipative term dominates! Taylor & Green, Proc. Roy. Soc. A 151, 421 (1935)

15 Isotropic and homogeneous turbulence A uniform velocity field produces no distortion of small scale structures. Distortion is controlled by shear. Assuming predominant distortion comes from scales l ~ l (local interactions), for an eddy of size l distortion takes place in an inverse time 1/τ l = u l /l. The energy flux ε is constant in the inertial range. As a result, from the nonlinear term in the momentum equation ε ~ u l2 /τ l ~ u l 3 /l, and u l2 ~ l 2/3. Then the energy spectrum is E(k) = de/dk ~ E/k ~ k -2/3 /k ~ k -5/3.

16 TG hydrodynamic simulations sin( k0x)cos( k0 y)cos( k0z) F= cos( k0x)sin( k0 y)cos( k0z) 0 Taylor-Green forcing up to Non-helical force; helical fluctuations are generated locally by the nonlinear terms in the NS equation. Proposed as a paradigm of turbulence: eddies at the scale 1/k 0 cascade down to smaller eddies until reaching the viscous scale. Taylor & Green, Proc. Roy. Soc. A 151, 421 (1937) Another simulation with the same resolution using ABC forcing (helical force). z

17 Large and small scale structures in TG 3D visualization of vorticity intensity in the tail of the PDF in the TG simulation. Turbulent fluctuations and a large scale pattern. High density of vortex tubes in regions with F = 0.

18 Energy spectrum and flux (TG) Well developed (compensated) energy spectrum. Wide range of wave numbers with constant energy flux. Large resolution is needed to observe a Kolmogorov spectrum. At lower resolution mostly the bottleneck is observed. The bottleneck seems to follow a k -4/3 power law. Kurien, Taylor, & Matsumoto69, (2004).

19 Transverse velocity increments PDFs of velocity increments δv = v (x+l)-v (x)] TG (right) and ABC (left).

20 Intermittency: time series Observations in the PBL Kholmyansky and Tsinober (2000) Note the intermittent nature of the signals associated with velocity derivatives. Peaks with are hundreds times larger than the mean are not unusual.

21 Velocity increments and intermittency The non-gaussian tails in the PDFs of velocity increments are often associated to intermittency. The closer the two points (the smaller the scale), the more the PDFs deviate from a Gaussian distribution, both in the center and in the tails. However, PDFs of velocity increments contain no information of the structure of the underlying strong and weak events. Similar PDFs can have different underlying structures! What is the origin of intermittency? Direct coupling between large and small scales (goes away for infinite Reynolds number Mininni, Alexakis, & Pouquet 2008; Aluie & Eyink 2010 Singular structures Multiplicative noise

22 Intermittency: structure functions For a component of a field f we define the longitudinal structure functions of order p as where the longitudinal increment is given by where f is in the direction of the increment l. If the flow is self-similar we expect a behavior with the exponents linear in p. K41 predicts In practice departures from K41 are observed, and the anomalous scaling observed in the data is the result of intermittency.

23 Intermittency: structure exponents Scaling exponents determined from increments in the x, y, and z directions, using ESS. The data seems slightly more intermittent than the SL model. Small differences are observed between the ABC and TG simulations. Are differences related to helicity, or are due to large scale anisotropies?

24 Intermittency: SO(3) Structure functions were computed for increments in 146 directions, and averaged to obtain the isotropic component (Taylor, Kurien, and Eyink, 2003). Exponents for the ABC and TG runs are within error bars for all orders computed. The exponents are more intermittent than what is predicted by most models. Martin & Mininni, PRE 81,

25 Intermittency: Reynolds dependence Deviations from S p (l) ~ l p/3 and from the SL model increase with Re, with or without ESS or SO(3). The intermittency exponent increases with Re, and at large resolution is consistent with experimental values (Meneveau and Sreenivasan 1991).

26 Structure exponents and intermittency As in the PDFs, qualitatively different phenomena can possess the same set of scaling exponents. It is believed vortex tubes are associated to the intermittency. However, it is not clear what is the contribution of these structures to the exponents, or to the anomalous scaling. Are these quantities universal? The origin of intermittency in turbulent flows remains open. How to characterize intermittency is also an open problem. The existence of scaling exponents is a problem: if the Euler (or Navier-Stokes) equations have scaling symmetries, should statistical quantities have such symmetries too?

27 Why are structure functions important? Since S 3 (l) ~ ε l p/3, the sign of S 3 can be used to discriminate between a inverse and direct cascades of energy Lindborg (1999)

28 Why are structure functions important? If isotropy is recovered in the small scales (local interactions between scales), then odd moments of the velocity increments should be zero. However, a persistence of the anisotropy is observed in the small scales Staicu and van de Water (2003)

29 Momentum equation Rotating flows P is the pressure, F an external force, ν the kinematic viscosity, Ω the angular velocity, and u the velocity; incompressibility is assumed. Quadratic invariants (F = 0, ν =0): E = u 2 d 3 x H = u ω d 3 x ω = u Reynolds, Rossby, and Ekman numbers where L is the forcing scale.

30 Turbulence in rotating flows spatial resolution. Column-like structures develop in the flow. Structures are helical and stable. Applications to the study of supercell storms.

31 Turbulence in rotating flows The flow becomes anisotropic. An inverse cascade of energy develops, and in helical flows the dyrect cascade of energy becomes sub-dominant to the helicity cascade. Rossby waves slow down the direct cascade giving as a result a steeper spectrum.

32 Intermittency in rotating turbulence The direct cascade of energy (in the presence of helicity and rotation) becomes self-similar! The direct cascade of helicity may be intermittent. There are no obvious differences in the structures besides the development of large-scale columnar structures.

33 The MHD equations Momentum equation v + v v= P t Induction equation 2 t ( ) + j B+ ν v+ F B 2 + v B = B v + η B c j= B 4π j= σ ( E+ v B / c) 1 B = E c t P is the pressure, j is the current, F an external force, ν the viscosity, η the resistivity, v the velocity, and B the induction (in Alfven velocity units); incompressibility is assumed.

34 Turbulence in conducting fluids spatial resolution. Regions with strong gradients (current sheets) are spontaneously generated in the flow. Current sheets separate regions with different magnetic field line topology.

35 MHD turbulence: structures Orszag-Tang

36 MHD turbulence: structures ABC flow in the large scales (helical) plus random initial conditions in the small scales. A small region in a simulation is shown.

37 Intermittency in MHD turbulence MHD turbulence is more intermittent than HD turbulence: current sheets! The magnetic field is more intermittent than the velocity field: vortex sheets are thicker than current sheets. Similar results are observed in the solar wind.

38 Conclusions A major problem of turbulence is to characterize the statistical properties of the flows. Progress can be made using a variety of techniques and contrasting them: theory, models, experiments, observations, and numerical simulations. We discussed results high-resolution DNS and models of threedimensional Navier-Stokes and MHD turbulence, and posed several questions that may be relevant from a physical point of view. These problems include: The problem of universality of statistical properties of turbulence, at either small or large scales. What is the origin of intermittency in isotropic and homogeneous turbulence? Is rotating turbulence scale invariant? Can the enhanced intermittency of MHD turbulence explain solar wind observations?