Talk_MR_ver_b_2

TUM - Lehrstuhl für Aerodynamik Seminar im Lecce, 23 Juni 2009

Marco Ellero , Michele Romeo

Stochastic Turbulence Modelingusing Smoothed Particle Hydrodynamics

Institute of Aerodynamics, Technical University Munich

Overview

Why studies on turbulence are important

TUM - Lehrstuhl für Aerodynamik Doktorandenseminar WS 2008 / 2009

...because almost everything is related to turbulent phenomena :

> Turbulence is a very fundamental aspect of fluid problems in a large number of physical fields, like Astrophysics, Microfluidics, Bio-Engineering, Condensed Matter, etc.

> Turbulence shows interesting analogies with other theoretical models, like spontaneous symmetry breaking in Quantum Field Theory [1].

> tipically, Turbulence is not only in physical fluids strictly intended but it is in any 'motions' system that shows instability in its behaviour under certain conditions, everytime a 'flux' of 'something' made of interacting parts can be measured, like economical systems, demographical and social sciences [2],[3],[4], data fluxes in abstract numerical schemes [5] and linguistical model of communication [6].

> Turbulence is an excellent source of problems for pure Mathematics [7].

> and so on...


Fundamentals

Turbulence from the original point of view


The views and analyses of the 1894 paper set the “w a y o f s e e in g ” turbulence for generations to come.In particular, when Reynolds studied Turbulence, he concluded that it was far too complicated ever to permit a detailed understanding, and in response to this he introduced the decomposition of flow variables into mean and fluctuating parts that bears his name, and which has resulted in a century of study in an effort to arrive at usable predictive techniques based on this viewpoint.Beginning with this work the prevailing view has been that turbulence is a random phenomenon, and as a consequence there is little to be gained by studying its details, especially in the context of engineering analyses.

Fundamentals



Following Reynolds’ introduction of the random view of turbulence and proposed use of statistics to describe turbulent flows, essentially all analyses were along these lines. The first major result was obtained by Prandtl [8] in 1925 in the form of a prediction of the eddy viscosity (introduced by Boussinesq) that took the character of a “first-principles” physical result, and as such no doubt added significant credibility to the statistical approach.

The next major steps in the analysis of turbulence were taken by G. I. Taylor during the 1930s. He was the first researcher to utilize a more advanced level of mathematical rigor, and he introduced formal statistical methods involving correlations, Fourier transforms and power spectra into the turbulence literature. In his 1935 paper [9] he very explicitly presents the assumption that turbulence is a random phenomenon and then proceeds to introduce statistical tools for the analysis of homogeneous, isotropic turbulence.

Fundamentals



In 1941 the Russian statistician A. N. Kolmogorov published three papers (in Russian) [10] that provide some of the most important and most-often quoted results of turbulence theory. These results comprise what is now referred to as the “K41 theory” (to help distinguish it from later work―the K62 theory [11]) and represent a distinct departure from the approach that had evolved from Reynolds’ statistical approach (but are nevertheless still of a statistical nature).

However, it was not until the late 20th Century that a manner for directly employing the theory in computations was discovered, and until recently the K41 (and to a lesser extent, K62) results were used mainly as tests of other theories (or calculations).

Fundamentals

Stochastic point of view


Statistical approach

Turbulent flows, with their irregular behavior, confound any simple attempts to understand them. But it seems that by a reasonable statistical approach it can be possible to have succeed in identyfying some universal properties of turbulence and relating them, for example, to broken symmetries [13].

Turbulence is mainly a phenomenon that we can describe with statisticalmeans but it present much singular simmetries in its evolution structures that can be treated by a deterministic point of view, as Lorenz showed in itsnumerical experiment using a simple form of the Navier – Stokes equations.In 1963 the MIT meteorologist published a paper [12], based mainly on machinecomputations, in which a deterministic solution to a model of the N.–S. equations(albeit, a very simple one) had been obtained which possessed severalnotable features of physical turbulence.

Remark:

Remark

Computational approach


Chapman and Tobak, in [14], conclude the paper by expressing the belief that future directions in the study of turbulence will reflect developments of the deterministic movement, but that they will undoubtedly incorporate some aspects of both the statistical and structural movements.

Numerical approach is substantial in this sense but it is not the only important aspect about the fundamental problem of Turbulence, because all tests (numerical and experimental) have shown that understanding of mechanisms in turbulent flows needs necessarily of a mathematical 'symbiosis' of both computational and theoretical developments.

This is basically the reason for which we retain a stochastic physical schemelinked to an SPH model for discretization as a good architecture for a PDF development of the Turbulence problem in a LES scenario.

Applications

Mesoscopic Engineering and Aerodynamic Science

TUM - Lehrstuhl für Aerodynamik Doktorandenseminar WS 2008 / 2009 TUM - Lehrstuhl für Aerodynamik Seminar im Lecce, 23 Juni 2009

Multiphase fluids

(on mesoscopic scales)

Turbulent phenomena

(in generic mesoscopic frameworks)

Aerodynamic design

(i.e. aerospace resources)

Applications

Rheology and Medical Science


Rheology and General Fluidics Bio-implatations and diagnostics

Applications

Astrophysical Science


Astrophysical jets andaccretion Turbulence

Star flares andMagnetic Reconnection

phenomenaRelativistic correctionsin Fluid Dynamics

lead to..

Applications

... and many other general fundamentals


Diffusion mechanisms Aerodynamics

Theoretical assumptions

PDF methods – Fokker - Planck equation and stochastic approach


Stochastic models for motions are the better way to deal with chaotic phenomena

Consider the Itō stochastic differential equation

where is the state and is a standard M-dimensional Wiener process.

If it is , then the probability density of the state is given by the

Fokker–Planck equation with the drift and diffusion terms




The Fokker–Planck equation describes the time evolution of the probability density function of the position of a particle, and can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck and is also known as the Kolmogorov forward equation. The first use of the Fokker–Planck equation was for the statistical description of Brownian motion of a particle in a fluid. The first consistent microscopic derivation of the Fokker-Planck equation in the single scheme of classical and quantum mechanics was performed by Nikolay Bogoliubov and Nikolay Krylov

More generally, the time-dependent probability distribution may depend on a set of N macrovariables xi. The general form of the Fokker–Planck equation is then


PDF methods – BBGKY hierarchy


In statistical physics, the BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, sometimes called Bogoliubov hierarchy) is a set of equations describing the dynamics of a system of a large number of interacting particles. The equation for an s-particle distribution function (probability density function) in the BBGKY hierarchy includes the (s+1)-particle distribution function thus forming a coupled chain of equations. This formal theoretic result is named after Bogoliubov, Born, Green, Kirkwood, and Yvon.

The evolution of an N-particle system is given by the Liouville equation for the probability density function in 6N phase space

Here are the coordinates and momentum for ith particle, is the external field potential, and is the pair potential for interaction between paticles. The equation above for s-particle distribution function is obtained by integration of the Liouville equation over the variables .




By integration over part of the variables, the Liouville equation can be transformed into a chain of equations

where the first equation connects the evolution of one-particle density probability with the two-particle density

probability function, second equation connects the two-particle density probability function with the three-particle

probability function, and generally the s-th equation connects the s-particle density probability function

and (s+1)-particle density probability function:




The problem of solving the BBGKY hierarchy of equations is as hard as solving the original Liouville equation, but approximations for the BBGKY hierarchy which allow to truncate the chain into a finite system of equations can readily be made. Truncation of the BBGKY chain is a common starting point for many applications of kinetic theory that can be used for derivation of classical or quantum kinetic equations. In particular, truncation at the first equation or the first two equations can be used to derive classical and quantum and the first order corrections to the Boltzmann equations. Other approximations, such as the assumption that the density robability function depends only on the relative distance between the particles or the assumption of the hydrodynamic egime, can also render the BBGKY chain accessible to solution.




In order to study evolution for a generic stochastic systemwe can start almost always from the Probability Transition approachprovided by microscopic Fine Grained Probability Density Function




Remark: The Fine - Grained PDF is very useful in obtaining and manipulating PDF equations,because of the following two properties:

Transport equation related to probability transition current is straightforward; in the caseof Fine – Grained PDF in fact we have




In this way, averaging the above transport equation we obtain Fokker – Planck evolutionequation for the one-point, one-time Probabilty Density Function f

According with the total density force portion in the Navier – Stokes equations,Lagrangian derivative for particle velocity yields what follows


Lagrangian Methods and Particle Dynamics


Lagrangian approach to fluid problems is very primitve from the point of view of Dynamics

A Lagrangian viewpoint is useful when modelling, interpreting and solving pdf evolution equations: the behaviour of fluid particles in a turbulent flow provides a complete description of the turbulence. According to the perturbed Navier-Stokes equations, at time t, infinitesimal variations for position and velocity of a fluid particle are denoted by


Lagrangian Methods and Particle Dynamics


In place of the exact expression, in the same way as before we model the Lagrangian velocity increment by the stochastic equation (Langevin equation)

Where

The form of this term is consistent with Kolmogorov inertial-range scaling and the Kolmogorov constant C0 has been determined to be positive

Theoretical model

A stochastic approach to the problem


An interesting stochastic approach to turbulent structures comes from S.B.Pope [15]:

Theoretical model



We have a good instrument to deal with turbulence by a stochastic way

> the prototypical Langevin stochastic motion model is mathematically consistent with the Kolmogorov hypotheses for turbulent motions in his K41 theory

> the Generalized Langevin Model is developed for inhomogeneous flows that take place in small-time scale processes (from which we can assume local isotropicity for high Reynolds numbers in respect to Kolmogorov hypotheses)

> there is conservation of momentum and energy

> we have a stochastic model for lagrangian particles that leads naturally to a 'smoothed particle discretization' in a DNS numerical scheme (Alias-DNS)

> Lévy process in the starting equation is a much powerful instrument for theoretical turbulence modeling [16].

Theoretical model



Lévy processes are implicit in stochastic modelling and they are always usefulin order to study stochastic particle motions

(Lévy processes are Càdlàg stochastic processes with stationary independent increments)

This is true also for Langevin equation in which we have the Wiener process

Theoretical model



The Wiener process in the basic Langevin model for stochastic motion is strictly related to the Kolmogorov Universality in his theory of Turbulence

In fact, (in a one-dimensional process, for instance) for a process W(t) we have the following properties (related to self-similarity):

Brownian scaling For every c>0 the process is another Wiener process.

Time reversal The process V(t) = W(1) − W(1 − t) for 0 ≤ t ≤ 1 is distributed like W(t) for 0 ≤ t ≤ 1.

Time inversion The process V(t) = tW(1 / t) is another Wiener process.

Numerical assumptions

SPH – Smoothed Particle Hydrodynamics


LLet us assume the original position of Lucy [17]:







Expressions (18) and (15) are the 'core' of any SPH numerical scheme

Numerical model

Smoothed Particle Hydrodynamics methods


In a natural way, an SPH discretization takes place in the stochastic model:

Further theoretical developments

Refinements


Subsequent extensions of the model are possible:

i.e. differential analysis of the acceleration leads to another stochastic schemefrom Sawford (1991)

Further theoretical developments

Refinements


There are several possible refinements for the stochastic model of Pope:

> Generalized Refined Langevin Models (named RLM) from Pope & Chen (1990)

> other stochastic mixing models which are fluid-dependent

this model for acceleration comes from the same Langevin motion model and itcan be directly related to Lagrangian statistics obtained from direct numericalsimulations, which are found to depend strongly on Reynolds number;

in the limit of infinite Reynolds number, the model reverts to the Langevin equation( as showed from Krasnoff and Peskin (1991) );

it is strongly related to Kolmogorov hypotheses in his K62 theory [15].

Remarks:

About coherence

Internal coherence of the numerical models


What is numerical choerence ?

?Theoreticalmodel(PDE)

Numericalmodel(ODE)

Loss physical information

Hardware

Numerical resources – supercomputing in Leibniz Rechenzentrum


Hardware



National Supercomputer: HLRB II

Stand: 2009-06-10

The system commenced operation in Q3 2006 in the new LRZ building in Garching. It replaced the former national supercomputer system Hitachi SR8000-F1. Peak performance is more than 62 TFlop/s, which is delivered by 9,728 Intel Itanium Montecito cores. Memory size is 39 TByte. Disk capacity amounts to 660 TByte.

Hardware



Hardware Description of HLRB II

Valid as of 2007-03-29

The HLRB II is based on SGI's Altix 4700 platform. The system installed at LRZ is optimized for high application performance and high memory bandwidth.

The following table provides an overview of the hardware and characteristics of the HLRB II.

Overall Characteristics for both installation phases

Phase 1 (until 03/2007)

Phase 2 (since 04/2007)

Total number of cores 4096 9728

Peak Performance of the entire system 26.2 TFlop/s 62.3 TFlop/s

Linpack Performance 24.5 TFlop/s 56.5 TFlop/s

Total size of memory for entire system 17.5 TByte 39 TByte

Direct Attached Disks 300 TByte 600 TByte Network Attached Disks 40 TByte 60 TByte

Main targets of this research project

Ending


Final and fundamental purposes of this work are:

> Development of an essential theoretical model starting from stochastic motion structure of Pope for inhomogeneous Turbulence

> Any possible and well-posed theoretical refinement of the original model from the point of view of pure physical coherence > Implementation of a Alias-DNS numerical scheme for inhomogeneous and locally isotropic Turbulence

Essential Bibliography

References

TUM - Lehrstuhl für Aerodynamik Doktorandenseminar WS 2008 / 2009

[1],[2],[3],[4],[5],[6],[7] are related to references in: SIMPLE MATHEMATICAL MODELS WITH VERY COMPLICATED DYNAMICS by Robert M. May, published in Nature, Vol. 261, p.459, June 10, 1976

[8] L. Prandtl. Bericht über Untersuchungen zur ausgebildeten Turbulenz, Zs. agnew. Math.Mech. 5, 136-139, 1925.

[9] G. I. Taylor. Statistical theory of turbulence, Proc. Roy. Soc. London A 151, 421-478, 1935.

[10] A. N. Kolmogorov. The local structure of turbulence in incompressible viscous fluid for very large Reynolds number, Dokl. Acad. Nauk. SSSR 30, 9-13, 1941; On degeneration (decay) of isotropic turbulence in an incompressible viscous liquid, Dokl. Acad. Nauk. SSSR 31, 538-540, 1941; Dissipation of energy in locally isotropic turbulence, Dokl. Acad. Nauk. SSSR 32,16-18, 1941.

[11] A. N. Kolmogorov. A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech. 13, 82-85, 1962.


Essential Bibliography

References


[12] E. N. Lorenz. Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141, 1963.

[13] L..Ts. Adzhemyan, A.N.Vasil'ev and M. Gnatich – Turbulent Dynamo as Spontaneous Symmetry Breaking - State University, Leningrad. Translated from Teoreticheskaya i Matematicheskaya Fizika,Vol. 72, No. 3, pp. 369-385, September, 1987. Original article submitted April 14, 1986.

[14] G. T. Chapman and M. Tobak. Observations, Theoretical Ideas, and Modeling of Turbulent Flows - Past, Present and Future, in Theoretical Approaches to Turbulence, Dwoyer et al., Springer-Verlag, New York, pp. 19-49, 1985.

[15] S.B. Pope – Lagrangian PDF Methods for Turbulent Flows Annu. Rev. Fluid Mech. 1994, 26: pp. 23-63

[16] T. von K´arm´an. On the statistical theory of turbulence, Proc. Nat. Acad. Sci., Wash. 23,98, 1937.

[17] L.B. Lucy – A numerical approach to the testing of the fission hypothesis Astr. J. 1977, vol. 82, n. 12: pp. 1013-1024


Thank you all for patience

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