Talk_MR_ver_b_2

TUM - Lehrstuhl für Aerodynamik Seminar im Lecce, 23 Juni 2009
Marco Ellero , Michele Romeo
Stochastic Turbulence Modeling
using 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 ing ”
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 statistical
means but it present much singular simmetries in its evolution structures that
can be treated by a deterministic point of view, as Lorenz showed in its
numerical experiment using a simple form of the Navier – Stokes equations.
In 1963 the MIT meteorologist published a paper [12], based mainly on machine
computations, in which a deterministic solution to a model of the N.–S. equations
(albeit, a very simple one) had been obtained which possessed several
notable 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 scheme
linked 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 / 2009TUM - 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 and
accretion Turbulence
Star flares and
Magnetic Reconnection
phenomena
Relativistic corrections
in 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 system
we can start almost always from the Probability Transition approach
provided 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 case
of Fine – Grained PDF in fact we have

In this way, averaging the above transport equation we obtain Fokker – Planck evolution
equation 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 useful
in 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 scheme
from 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 it
can be directly related to Lagrangian statistics obtained from direct numerical
simulations, 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 ?
?Theoretical
model
(PDE)
Numerical
model
(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

Talk_MR_ver_b_2

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