upscaling primer by b. d. wood

7/30/2019 Upscaling Primer by B. D. Wood

1/65

+

A Primer on

Upscaling Transport

Phenomena in

Complex Media

Brian Wood


2/65

+Perspective

This talk is meant to be an introduction to some ofthe essential features of upscaling transport

behavior in complex mediaThe focus is meant to favor the big picture over

details

I will favor discussing results over derivationsHistorical context will be introduced where relevant The history of science is ultimately critical to understanding

the scientific process!


3/65

+Outline Overview: Models of physical systems What are models? Entropy, uncertainty, and Occhams razor Examples: The ideal gas law

Kinds of Complexity Degrees of freedom- Ideal Gas Nonlinear behavior- Turbulence

The role of Upscaling What is upscaling?

How is it useful?

Scaling Laws The ontological state of randomness Filtering via measurement


4/65

+Outline (continued) Local and Nonlocal Models History of nonlocal models

Some examples Diffusion in porous media Diffusion in biofilms Reaction-diffusion with effectiveness factors


5/65

+

Motivation for UpscalingThe problem of complex systems


6/65

+Complexity

In the scientific context, a complexsystemrefers to two possible ideas

A system that, because of nonlinearity, exhibitssubstantial sensitivity to initial conditions

e.g., the Lorenz equations This is the idea behind chaos

A system that requires an enormous amountof information to characterize and describe

For a discrete system, this can be though ofas the degrees of freedom of the system

For continuous fields, this is more difficult Notions of information or entropy are

more suitable


7/65

+Complexity Reduction

Upscalingusually deals with the second kind ofcomplexity

The goals of upscaling are to Develop an effective model that reduces the information

content of the system being modeled

Applies at an integrated space or time scale of the system This includes developing representations of the system that

include the influence ofmeasurement by instruments

The effective parameters in the effective model dependexplicitly on the microscale physics


8/65

+Complex Media

In this talk, complex media arephysical systems that have a largeamount of small scale structure

They would, in principle, take anenormous amount of data to

characterizeThey are defined by being multi-

scale

In its simplest form, details are naturallyarranged by (at least) two scales of

featuresMicroscale (small scale)Macroscale (large scale)

A biofilm in a porous mediumResearch collaboration withD. Wildenschild, OSU


9/65

+

1 Mole of Ideal Gas, specified V:Newtons Laws (6.02 x 1023 particles)

Equate this with KE

Sum over all molecules Then, we are stuck

F =d(mv)

dtq "FEnergy

!= q "

d(mv)

dt

q ! F = 12

mv ! v

q !F = 1N

1

2mv

i! v

i

i=1

N

"

Example of Complexity

Reduction via Upscaling


10/65

+Example of Complexity


And it follows directly that

However, if we now assume that the velocities aredistributed by the (maximum entropy) Maxwell-

Boltzmann distribution, it is easy to show

10

q!F = 3kT

3PV = 3NkT Ideal Gas Law

Results from the

assumedScaling Law and

integral of the

measure (p)

q !F = 3PV (from Newton's third law andthe definition of pressure)

1

N

1

2mv

i!v

i

i=1

N

" = 3NkT

(left side)

(right side)


11/65

+Example of Complexity


The result of upscaling wasA dramatic reduction of the degrees of freedom of the problem Initially: 6x1023 x 3 position x 3 momentum = 5 x 1024 DOF Upscaled version: Only T is variable: 1 DOF

The other helpful thing that happened was that we obtained aresult that has much more utility for practical problems!

Even if you knew the entire position/momentum history of a mole ofmolecules, what would you do with it?

In this case, the microscale details have low utility for many practicalproblems of interest


12/65

+

I. The Structure of Complex MultiscaleSystems

Biofilm

Tissue Scaffold


13/65

+Motivation: Jean Baptiste Perrin (Nobel

Laureate*): From his 1913 bookLes

Atomes

My eye seeks in vain for a practically homogeneousregion in the soilthat I can see from my windowIhave only to go close to it to distinguish details and to beled to suspect the existence of others. Fresh details willbe reached at each stage In fact, as is well known, aliving cell is far from homogeneous, and within it we areable to recognise the existence of a complex organisationof fine threads and granules immersed in an irregularplasma

Thus, the portion of matter that to begin with we had

expected to find almost homogeneous appears to beindefinitely diverse, and we have no right to assume thaton going far enough we should ultimately reachhomogeneity, or even matter having properties that varyregularly from point to point.

*1926, For his work on the discontinuous structure of matter.


14/65

+Discontinuous media Continuum mechanics is The study of the balance of mass, momentum, and energy For matter treated as if it were a continuum On the basis of a kinematics

Many interesting media,however, are not continuous!

A question of longstandinginterest is the question:

Can discontinuous media be

homogenized so that it maybe treated as continuous?


15/65

+Some Examples of Discontinuous

Media

structured catalysts

beadpacks

composite materialsmetal foams


16/65

+Multiscale Media

The interest in representing physics in discontinuous, multi-scale engineered and natural media has been formally

recognized since at least 1850

This was problem of broad interest to a variety of disciplines The earliest efforts were by some of the most famous names in

science!

Simon Denis Poisson (1824) James Clerk Maxwell (1873) Rudolph Clausius (1879) Lord Rayleigh (1892) Heindrikk Antoon Lorentz (1909)


17/65

+Types of Multiscale Structure

There are really three basic types of multiscalestructures. These will be discussed in more detail

later...

Discretely Hierarchical Media This is the classical picture of multiscale media that has

been adopted for about 200 years now

Includes spatially-nonstationary media Such Media have a finite variance in the field structure

Continuously Evolving Media More recently, it has been observed that some media do

not have inherent length scales

These systems have unbounded variance Fractal media (or Power Law media) are one example Such models can explain some interesting new kinds of

behavior, but also creates some substantial difficulties!

Benot Mandelbrot


18/65

+

Three Types of StableDistributions

Platykurtic (e.g.,Laplace)

Mesokurtic (e.g.,Gaussian)

Leptokurtic, or heavy-tailed (e.g., Lvy /

Inverse Gamma

For continua, the field properties are usually assumed tofollow a stable distribution (P. Lvy)

The Gaussian is a well-known example of a stabledistribution, but there are others

Power law

tails,

infinite

variance

Finite

variance

Lvy

Laplace

Primary Statistical Features of the

Two Kinds of Structure


19/65

+Primary Statistical Features of

the Two Kinds of Structure Platykurtic and Mesokurtic Gaussian distribution

is one example (mesokurtic)

Laplace distribution is another(platykurtic)

Finite variance Finite correlation length

Laplace

Finite

variance


20/65

+Primary Statistical Features of

the Two Kinds of Structure

Leptokurtic Often known as

LvyorInverse Gamma

distributions

Infinite variance Infinite correlation structure

Lvy

Power law

tails,

infinite

variance


21/65

+

Discretely HierarchicalStructures


22/65

+ Example of Hierarchical StructureIdealized and Observed Biofilms


23/65

+ The Structure of Discretely HierarchalSystems Defined by a stable, non-leptokurtic distribution

Correlation structure has hierarchical structure withstationary intervals!Separation of length scales

!


24/65

+The Structure of Discretely Hierarchal

Systems Observations

Data from Zhang et al. (2000, Geophysical Research Letters,27: 1195-1198)


25/65

+

ContinuouslyEvolving StructuresThe fractal / power law revolution


26/65

+The Structure of Continuously

Evolving Media No characteristic lengthscales (& no separation oflength scales)

Correlation structure hasinfinite range!


27/65

+What is a Fractal?The coastline of Britain (1967, Mandelbrot, Science, 156:636-638)

L() = 2300 km

D = 1.05

L() = 2800 km

D = 1.1

L() = 3500 km

D = 1.25

L the measured lengthF a constant! size of the measuring stickD the fractal dimension

L(!) = F!1"D A power

law!


28/65

+Power law Behavior

For the coastline example, Mandelbrot found

L() = F1D

log[L()] = log[F]+ (1 D)log[]

1 < D < 1.25

D = slope+1


29/65

+ How Does this Affect Upscaling inHeterogeneous Media?

(Photo Courtesy of Scott Tyler)


30/65

+ How Does this Affect Upscaling inHeterogeneous Media?

(Photo Courtesy of Scott Tyler)


31/65

+ A Power Law Transport Example

In fractal fields, the correlation structure is infinite This means that properties that depend on the correlation

structure increase with scale in a power law form

Apparent Dispersivity, (from Neuman, 1995)


32/65

+The Occurrence of Fractal Media

Fractal ideas presented a new way of examining complexmedia

These revolutionary ideas can help explain behavior that wasnot previously well-understood

However, despite claims to the contrary, not all complex mediaare fractal!

This remains a hotly contested area of continuing research There appears to be support forboth kinds of ideas when one

looks at various kinds of data sets.

As I have stressed, fractals are not a panacea; they

are not everywhere.

Benot Mandelbrot (1998, Science, 279: 783)


33/65

+

UpscalingMethods for rational information reduction


34/65

+

Brief discussion ofinformation reduction


35/65

+ Information Theory

Organized = Low entropy, H1 Few such states

Initial

Final

Information Gain =

Disorganized = High entropy, H0 Many such states

Before discussing information reduction, we need to have a shortdiscussion about information D Imagine two states of a system: An initial state and a final state Information is always the decrease in uncertainty (entropy)

associated with such a change of state

H0 H

1> 0

H0= p

iln p

i

states i

> 0

H1= pi ln pi

states i

< H0


36/65

+ Information Reduction Now, lets use the example of the porosity of a spatially-stationary

medium (i.e., the average is a constant in space)

Consider the two states of a system as described by(1) the microscale state, and(2) the post-averaging (macroscale) state

H0 = pi ln pi states i

= ln(

) (1

)ln(1

) > 0

Microscale Macroscale

H1 = pi ln pi states i

= 1ln(1) = 0

Information Change: I = 0 + ln(

)+ (1

)ln(1

) < 0

The result is that upscaling results in a decrease in the information content.


37/65

+Information Reduction

The goal of upscaling is to reduce the information content ofa system

The information that we wish to reduce has low value The value of information has to be assigned by practical

considerations (e.g., what is possible to measure!)

The upscaling process adds constraints consistent with thephysical system to obtain a result

These are sometimes called scaling postulates


38/65

+The Upscaling Process

The classical problem of the Reynolds averaged Navier-Stokesequations (RANS) provide a familiar archetype for theupscaling process

In short, the process is roughly as follows1.

Identification of the microscale balance equations

2. Averaging (volume or ensemble) of these balance equations

a) Yields an expression with integrals of the microscale dependentvariable

3. Information reduction (simplification) via imposing one ore morescaling postulates

4. Prediction of the integral quantities (which we can think of asstatistics) of the microscale variables

a) This is known as closureb) Generates the effective parameters associated with the

upscaled model


39/65

+Scaling Postulates

Scaling Postulates are strong statements about the thephysical structure of the underlying fields

The concept of scaling postulates in science has been aroundfor a hundreds of years

E.g., 1638, Galileo Galilei, Two New Sciences, Elzevier These statements are axiomatic They have to be consistent with what is known about the physical

system, but they can not be proven (they can, however, be

refuted...)

Examples: Spatially homogenous field statistics, separation oflength scales...

These statements are essentially what is responsible for theinformation reduction

In the context of upscaling, we have discussed this in detail inseveral papers. The concept will be used more loosely for the

remainder of the talk...


40/65

+

Examples of Upscaling via Volume Averaging:Local ModelsThe role ofscaling postulates in upscaling theories


41/65

+

Upscaling Example 1The classical problem of diffusion in

porous media & extensions to diffusion

in biofilms


42/65

+ Classical problem of diffusion inmultiscale systems

Diffusion-like problems in hierarchicaldiscontinuous media are one of the oldest, and most

well-trodden examples of upscaling

Ottaviano Mossotti (1791-1863). His paper of1850 outlined a mean field theory approach to

the problem. Made explicit by Clausius (1879)

James Clerk Maxwell. A Treatise on Electricityand Magnatism (1873), Chapter IX Conductionthrough heterogeneous media

John William Strutt (a.k.a. Lord Rayleigh-1842-1919), On the influence of obstacles

arranged in rectangular order upon theproperties of a medium (1892)

Zvi Hashin (1929- ) Developed upper andlower bounds for the problem

This is far from an exhaustive list!


43/65

+

Maxwell (1873) Noninteracting spheres Examined at a large distance

from perturbation

Rayleigh* (1892) Periodic array of spheres

Problem: Find the (macroscale) effective diffusion tensor of the form

* Nobel Prize in physics, 1904

Diffusion in a Dilute System of

Particles


44/65

+Scaling Postulates

For this classical problem, the scaling postulates can be(roughly) identified as

Spatially stationary structure Separation of length and time scales Existence of a representative volume


45/65

+ Representative Volume The idea behind separation of length scales

and the representative volume has beenknown in physics for a long time

Poisson (1829) The molecules are so small and so close to one

another that a portion of the body containing anextremely large number of them can still besupposed to be extremely small...

Lorentz (1909) As to the sphere it must be chosen neither too

small nor too large. Since our purpose is to get ridof the irregularities... the sphere must contain alarge number of particles. On the other hand, wemust be careful not to obliterate the changes frompoint to point... both conditions can be satisfied atthe same time.

!


46/65

+Solution to the Diffusion Problem

Both Maxwell and Rayleigh approached the problem via anexpansion in harmonic functions

Implicit in both of their analyses are the scaling postulates thatmake such a solution possible!

The first approximation to their series solutions gives the followingwell known (and identical!) result


47/65

+Biofilms as Porous Media

A. Ochoa-Tapia (1986, 1994, Chemical EngineeringScience volumes 41 and 49) examined extensions

to the diffusion problem via volume averaging

Two materials with different diffusion coefficients Interfacial resistance to mass transfer between

materials

Their results extended the Maxwell-Mosotti-Clausiusresults for this more general case


48/65

+ Biofilms as Porous Media

Representative Volume of Biofilm Based on observed biofilm structure 11.25 x 11.25 x 11.25 m cube Over 2000 individual bacterial cells within

the volume

Grid resolution: 150 x 150 x 150 (75 nm spacing)

(3 million nodes)

Numerical Solution to Closure Problem Finite difference scheme Solved on a massively parallel computer

Wood and Whitaker (1998, 2000, Chemical Engineering

Science, vols 53, 55) revisited the Ochoa-Tapia problem in thecontext of diffusion in biofilms


49/65

+

Upscaling Example 2Chemotaxis in porous media


50/65

+Chemotaxis

Chemotaxis is the process by whichmotile bacterial move along a

concentration gradient toward a substrate

source

This can influence how bacteria move inporous media systems

Applications to bioremediation &biotechnology

Collaboration with Rosanne Ford ofUniversity of Virginia


51/65

+Scaling Postulates

The scaling postulates imposed are similar to the case for thediffusion problem

Spatially stationary structure Separation of length and time scales Existence of a representative volume The upscaled model contained effective parameters that were

predictable from the solution of the ancillary closure problem over a

representative volume


52/65

+Results

The computation ofthe effective

parameters over a

representative volume

lead to a model that

was able to describe

some interestingfeatures of the

experimental data

with

chemotaxis

without

chemotaxis

Breakthrough

curvesx= 4 cm


53/65

+

Nonlocal Transport ModelsProblems where separation of time and length scales is not possible.


54/65

+Nonlocal Models

When the time and space scaling postulates are less restrictive (ofthe form)

then there is no longer a separation of scales

This yields a nonlocal macroscale equation

Transport in fractal systems are given by nonlocal models... This is an area of research for spatial averaging methods (which is what I

use...)

Nonlocal equations are integro-differential equations that containintegrals of the dependent variable

These integrals represent memory in time, and global dependenceof the field structure in space

Such models can do more, but at a cost! The information reduction for such models is smaller They are much more difficult to apply


55/65

+Nonlocal Diffusion

It is possible to develop a macroscale nonlocal diffusionequation

Wood and Valds-Parada (2012, Advances in WaterResources, in review) have proposed one such model that is

local in time but not in space

It represents an upscaled model that can account forsharpgradients in the concentration field

Here, G is a Greens function, and n

is a normal vector


56/65

+ Nonlocal Diffusion The nonlocal model predicts behavior that is different from the

local one

In particular, the early spreading is anomalous This is best evidenced by the kurtosis of an initially delta inputA standard diffusive process would have a kurtosis that is zero for all

time

The nonlocal model predicts a solute plume that is initially flatterthan pure diffusion would predict

kurtosis


57/65

+

Conclusions


58/65

+Measurements

Sir Arthur Stanley Eddington

Observation and theory get on best when they are mixed together, both

helping one another in the pursuit of truth. It is a good rule not to putovermuch confidence in a theory until it has been confirmed by

observation.


59/65

+Measurements

I hope I shall not shock the experimental physicists

too much if I add that it is also a good rule not to put

overmuch confidence in the observational results that

are put forward until they have been confirmed bytheory.

Sir Arthur Stanley Eddington


60/65

+Scientific Relativism

I am convinced that ideas such as

absolute certainty, absolute precision,

final truth etc., are phantoms whichshould be excluded from science

Max Born* (1966 Symbol and Reality,

Dialectica, 20:143-157)

* Nobel Prize in physics, 1954


61/65

+Models

The best model of a cat is another cat,and preferably the same cat

Rosenblueth and Weiner, 1945

Essentially all models are wrong; but, someare useful.

George E.P. Box, statistician,1987


62/65

+Quantitative Occam

E.T. Jaynes in 1957 developed amethod for choosing scaling lawsfor statistical thermodynamics usingmaximum entropy of information asa criterionAn extension of Laplaces principle of

insufficient reason

A powerful method that has been usedwidely Recovers Maxwells distribution for molecular

velocities

Can be used to prove the central limittheorem

Jaynes

Occam


63/65

+ Multiscale Structures-Ex. 2

Carbon Fiber Composite


64/65

+Randomness versus derminism

The Navier-Stokes equations are assumed by most to be areasonable model for the momentum balance of an

incompressible Newtonian fluid.

Turbulence is often viewed as being the chatoic behavior of afluid that occurs at sufficiently high Reynolds numbers.

However, the Navier-Stokes equations are deterministic. Sowhat gives? Why is turbulence treated as being random?


65/65

+A viewpoint on randomness

Randomness is not an inherent feature of physical systems Rather it is a statement about our inability to measure and/or predict

with sufficient resolution and detail.

Example We usually think of rolling dice as a random process However, dice are classical mechanical objects governed by (linear)

Newtons Laws!

Dice rolling is random in the sense that we are unlikely to be able tomeasure the initial conditions and subsequent interactions (e.g.,friction, elastic behavior of the dice) with sufficient resolution to

make any kind of reasonable prediction of the result

This is a DOF problem!

upscaling primer by b. d. wood

Documents