Mu M u ul lt l ti t im i ma m at a te t e er ri r - - PowerPoint PPT Presentation

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LA-UR-02-5575

M M M Mu u u ul l l lt t t ti i i im m m ma a a at t t te e e er r r ri i i ia a a al l l l E E E Ef f f ff f f fe e e ec c c ct t t ts s s s i i i in n n n C C C Co

• m

m m mp p p pu u u ua a a at t t ti i i io

• n

n n na a a al l l l S S S Si i i im m m mu u u ul l l la a a at t t ti i i io

• n

n n ns s s s

• f

f f f R R R Ri i i ic c c ch h h ht t t tm m m my y y ye e e er r r r-

• M

M M Me e e es s s sh h h hk k k ko

• v

v v v E E E Ex x x xp p p pe e e er r r ri i i im m m me e e en n n nt t t ts s s s Bill Rider Jim Kamm, Raul Coral-Pinto (MIT), Chris Tomkins and Mark Marr-Lyon

Los Alamos National Laboratory

23 September 2002

Workshop on Numerical methods for multimaterial compressible fluid flows Paris, France

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Outline

Motivation: Multimaterial

Fluid instabilities

Richtmyer-Meshkov experiments Numerical

simulation techniques

Results & Analysis Concluding remarks

U U F F

i n i n t x i n i n + + +

• +

=

• Ê

Ë Á ˆ ¯ ˜

1 1 2 1 2 1 2 1 2 / / / / D D

Ut F U x ( ) + = 0

Validation: perspective & motivation

SGI Bluemountain Compaq Q

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LA-UR-02-5575 Richtmyer-Meshkov – Impulsively accelerated interface

between two fluids

Idealized Fluid Interface Instabilities

Velocity Jump

Kelvin-Helmholtz – Jump in tangential velocity at the

fluid interface

Rayleigh-Taylor – Uniformly accelerated interface

between disparate fluids

Exponential growth rate

= =

È Î Í Í ˘ ˚ ˙ ˙

||

sKH U k

1 2

Uniform Accel’n Heavy Light

1

=

• +

=

Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜

r r r r

1 1

A

Atwood Number

= = sRT Agk

Exponential growth rate Linear growth rate

= =

+

sRM Ak a D

— ¥— p r acts as vorticity source Impulsive Accel’n

1

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Motivation: Experimental Validation of Computations

Use experimental data to assess the validity of

numerical methods and models

Richtmyer-Meshkov-driven material mixing Good agreement on an “integral” scale Experiments typically image large scales Some detailed spatial-temporal data are available

Such validation is an essential activity! Validation bottom line: is the model appropriate

for the given physical circumstances?

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A Starting Point: 1997

RAGE FronTier

PROMETHEUS

Nova/LLNL

Viewgraph Norm Study by Holmes et al.* on RM instability growth:

*Holmes et al., Richtmyer-Meshkov instability growth: experiment, simulation and

theory, J. Fluid Mech. , 389 , pp. 55–79, 1999

Comparison of

NOVA laser experiments with three different codes

The integral

scale compares well— but how do the details compare?

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What About The Small Scales?

120∆x 960∆x Calculation by B. Fryxell RAGE FronTier

PROMETHEUS

R-T

Later time behavior suggested

differences among the methods… …and structural variations that depend

• n the

mesh resolution

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Validation Perspective

Validation is the process of determining that the

equations in a simulation code accurately model the physics of interest

Visualization is helpful for understanding

flows, but quantification is required to obtain credible, defensible simulations

Viewgraph Norm Validation can only be done in reference to

experimental data

Code-to-code comparison (“benchmarking”)

does not constitute validation

We want to get the right answers for the

right reasons

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Bigger computers

must provide better answers

Validation Motivation

Both (1) compressible flow features (e.g., shocks)

and (2) the presence of large-scale, dynamically evolving structures (e.g., flow instabilities) are readily computed with today’s codes

Many calculations may “look good”, but they can

be inaccurate — a challenging situation in an era

• f increasing reliance on simulation

We must get the

right answers for the right reasons

SGI Bluemountain Compaq Q

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Shock Tube Experiments at LANL (DX-3)

Low-speed (M≤2) shock tube facility where SF6 +

glycol fog form diffuse “target” struck by shocked air

Snapshots at ∆t ≈ 100 µs of the SF6 curtain impacted by M=1.2 shock

Driver Section Driven Section Diaphragm Shock Laser Sheet Camera Test Section End Section Nozzle 1.28 m 3.20 m 0.23 m 0.69 m Gas Curtain

Downstream “mushrooms” Modal coupling

Analysis indicates that the glycol fog accurately follows the SF6

Side View

Rightley et al. “Evolution of a shock-accelerated thin fluid layer”,

• Phys. Fluids. 9, 1770–1792, 1997.

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Shock Tube Experiments: Gas Cylinders

Recent experiments have used

gas cylinders as the target

Shock Laser sheet Gas cylinder

Suction

Air Air

DYN

PIV

Fog generator Fog Generator

SF chamber

6

IC

Particle Image Velocimetry data have been collected

Single cylinder D = 3mm Double cylinder D = 3mm, S/D = 1.5

• Zoldi. “A numerical and experimental study of a shock-accelerated

heavy gas cylinder”, Ph.D. Thesis, SUNY Stony Brook, 2002. Tomkins et al.. “Flow morphologies of two shock-accelerated gas cylinders”, J. Visualization, to appear, 2002.

Experiments conducted by

Benjamin, Prestridge, Rightley, Vorobieff (UNM), and Zoldi.

D S

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Shock Tube Experiments: Gas Cylinders

Experiments are quite

repeatable

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PIV Data and Analysis

Analysis: two-frame cross-correlation

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Results: A Wide Variety of Modern Methods

MUSCL split*, MUSCL unsplit, ENO**, WENO**,

DG***, PPM****, TVD,…

All numerical results are more-or-less self-

consistent and not consistent with the experimental data at late time

*Gittings & Zoldi (LANL), ** Aslam (LANL), *** Lowrie (LANL), **** Flash Code (UChicago)

• Vol. Fraction

Image

• Vol. Fraction

Smooth

Initial Grid Level

Import

Density+Velocity

Compute

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Shocks and rarefactions leading to mixing Typically modeled by the compressible Euler

equations (or Navier-Stokes in Re ) limit

Physical Situation of Interest Æ •

2.5µs

r —◊ r u

35µs shock 200µs 400µs 600µs Shock (M=1.2) accelerated Shock (M=1.2) accelerated double SF double SF6

6 cylinders

cylinders Simulation of experiments at LANL Simulation of experiments at LANL Acoustic waves weak shocks strong expansion Shocks are “easy,” Mixing is “hard”

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Turbulent Mixing?

Flowfield characterization -Is it turbulent?

Curtain Single cylinder (5 mm) Double cylinder(3mm,S/D=1.5)

Re , = ª G n 70 000 ReT

T

u = ª

• l

n 800 2500 lT u u x mm mm = ∂ ∂

( ) ª

• /

2 5 Re , = ª G n 10 000 ReT

T

u = ª

• l

n 150 600 lT u u x mm mm = ∂ ∂

( ) ª

• /

. 1 2 5 Re , = ª G n 30 000 ReT

T

u = ª

• l

n 600 1500 lT u u x mm mm = ∂ ∂

( ) ª

• /

2 4

Certainly not fully developed

• r isotropic

h m = Ï Ì Ó 0 01 1 . mm m

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LA-UR-02-5575 Fractal dimension measures the complexity of an

interface and is computed as:

Quantitative Analysis Methods

Wavelet coefficients provide a local, scale dependent

“lens” into the data: a n

• 2·

Ê Ë Á ˆ ¯ ˜

y x a b , , , f x

( )Ò

Wavelet coefficient =

Wavelet Data Scale

D N r r

f r

=

Æ

( )

È Î Í Í ˘ ˚ ˙ ˙

lim log log 1

• 10
• 8
• 6
• 4
• 2

1 2 3 4 5

log(r) log[1/(N(r)]

Df =

slope

Volume Fraction Wavelet Coeff’t

Structure function of order p is defined as:

S x x

p p

( ) ( ) ( ) l l ∫ +

• Ê

Ë Á ˆ ¯ ˜

x x

pth correlation at length scale

l The exponent a characterizes the (local) scaling behavior ~ la r = box size N(r) = min # boxes

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A Recent Comparison to Current Experiments

Curtain Cylinder Double Cylinder Experiment Calculation Comparison

0.01 0.02 0.03 0.5 1 CWT Spectrum Scale (cm) 2.0 2.2 2.4 2.6 2.8 0.0 0.5 1.0 Local Df Scale (cm) 0.01 0.02 0.03 0.5 1 CWT Spectrum Scale (cm) 1.7 1.9 2.1 2.3 2.5 0.0 0.5 1.0 Local Df Scale (cm) FWHM IC=0.5 cm S/D=1.2 S/D=1.5 S/D=2.0

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Improved Characterization of Initial Conditions

Use Rayleigh scattering to see if the fog and SF6 diffuse from each other. They do!

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Improved Characterization of Initial Conditions

Use Rayleigh scattering to see if the fog and SF6 diffuse from each other. For smaller cylinders a greater difference between the fog and SF6.

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• Comp. N-S or Euler Equations:

Godunov Method p p e

n n n + + +

= (

)

1 1 1

r , V x V V x x

n n n

( ) =

+ —

• (

)

r V x V x tV

n n t +

( ) = ( ) +

1 2

2 D U U t F F x

j n j n j n j n + + +

• +

= +

• (

)

1 1 2 1 2 1 2 1 2

D D F U R U U

( ) = (

)

• +

,

1 Compute limited spatial gradients: 3 Solve the Riemann problem

Primitive-form equations Cell-averaged conserved quantities Evolve solution by 1D cell-edge shock tube problems

2 Advance data in time 4 Advance conservation laws

Time-center information for consistent 2nd order Limiters ensure solution monotonicity

5 Update constitutive laws

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Control Volumes in Physics

Observations are essentially made the same

way (either camera or the human eye).

If reality is imaged, the light is integrated over a

region of space (pixel size) and for a period of time (exposure time)

This is analogous to numerics with control

volumes! Control volumes capture experimental conditions

LANL Double Cylinder Experiment: Tompkins, Marr-Lyon, Benjamin

D D x m t s = m = m 76 10 ;

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Connecting Physics and Numerics via Control Volumes

This implies that the evolution of control

volumes has an intimate connection with what is

• bserved experimentally

We want to model what we can observe!

The evolution of the quantities on the control

volumes is also natural for large scale physics and the basis of non-oscillatory methods

Together this provides a rationale basis for

LES* via numerical methods - MILES**

*Margolin and Rider, “A Rationale For Implicit Turbulence Modeling,” to be Published, International Journal of Numerical Methods in Fluids ** J. P. Boris coined this term for numerical methods mimic LES modeling

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Euler Equation Simulations

Experiment Computation Previous simulations showed qualitative agree-

ment in large-scale structures for “regular” gas curtain morphologies

Not explicitly modeled

3-Dimensional Real fluid effects 2-Dimensional

• Initial conditions visually 2D
• This does constrain dynamics

“Numerical fluid” effects

• Viscosity
• Material diffusion
• Tracer fidelity
• Numerical Viscosity
• Numerical diffusion/dispersion
• Good tracer fidelity assumed

A “first principles” Direct Numerical Simulation

(DNS) — even in 2D — is not feasible with today’s simulation capability

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Ran out

• f

machine!

Aside: Why Work on Methods? Just Crush the Problem?

Why not just add a viscosity and refine…

∆x=31.25µm 15.625µm 7.8125µm ∆x=3.90625µm Results w/Raptor: 2nd Order Godunov AMR code Inviscid Viscous, still 4-8 times above Kolmogorov length scale

SGI Bluemountain Compaq Q

With Greenough (LLNL) and Zoldi (LANL), Presented APS DFD Dec. 2001

Re µ

• M

1

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A Parametric Study of Initial Conditions

We choose a set of initial conditions to span the

expected range

Diffusion (50-200 on a 200x200 grid) iterations of

a diffusion operator because the physical or effective diffusivity is not known.

Initial peak concentration of SF6 (0.5-1.0)

We will assess and account for numerical error

using calculation verification and adjust our results according to these estimates.

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Example Results S/D=1.5 at 750µs

0.5 0.6 0.7 0.8 0.9 1 0.8 1 1.2 1.4 1.6 1.8 2 2.2

Initial Concentratio n

Height Width

50 100 150 200

width height

Experiment @750µs

200x200 grid ∆x=0.025 cm

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An Estimate of Numerical Error

Use the following error assumption This requires solutions on three grids to solve (we use

100 diffusion passes & an initial concentration of 0.8)

Simple to solve if the grids are factors of two

difference in ∆x (100x100, ∆x=0.05cm, 200x200 , ∆x=0.025cm &

400x400 , ∆x=0.0125cm)

For S/D=1.5

Height, A=1.68, C=-290, a=2.48 Width, A=0.86, C=-1.10, a=0.82

A S C x a = + (

)

D

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0.5 0.6 0.7 0.8 0.9 1 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Initial Concentratio n

200x200 mesh Converged Value

cm

Error’s Impact on Initial Estimates

Widths for S/D=1.5 indicates lower concentrations than originally believed, but a little more diffusion than experimental estimates

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Single Cylinder with New ICs

1.0 10 3.0 10 5.0 10 7.0 10 1.1 10 1.0 10 2.0 10 3.0 10 4.0 10 5.0 10 5.6 10

Cuervo 2 D Single Cylinder Simulation

time (s)

1.0 10 3.0 10 5.0 10 7.0 10 1000 2000 3000 4000 5000 6000 7000

Cuervo 2 D Single Cylinder Simulation

time (s)

• Comp. N-S w/∆x=0.01 cm

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Evolution of the Taylor Microscale

1.0 10 2.0 10 3.0 10 4.0 10 0.1357 0.2 0.3 0.4 0.4357

Cuervo 2 D Single Cylinder Simulation

time

L_xx L_xy L_yy L_yx

1.0 10 3.0 10 5.0 10 7.0 10 1000 2000 3000 3500

Cuervo 2 D Single Cylinder Simulation

time(s)

• Comp. N-S w/∆x=0.01 cm

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Comparison with PIV

• 0.02

0.02 0.04 0.06 0.08 0.1 5 15 25 35 45

Experiment (PIV) Rage Cuervo

PDF Velocity (m/s)

• 0.02

0.02 0.04 0.06 0.08 0.1 5 15 25 35 45

Experiment (PIV) Rage Cuervo

PDF Velocity (m/s)

• Comp. N-S w/∆x=0.01 cm

Vortex induced V=30m/s Vortex induced V=27m/s

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Double Cylinder with New ICs

S/D=1.2 S/D=1.5 S/D=2.0

• Comp. N-S w/∆x=0.01 cm

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Integral Scale Comparison

2.37 2.50 2.45 0.84 0.92 0.72 1.99 2.30 1.77 S/D=2.0 2.38 2.32 2.26 0.80 0.92 0.76 1.90 2.13 1.72 S/D=1.5 1.99 1.74 1.46 0.82 1.01 0.87 1.63 1.76 1.28 S/D=1.2 1.32 1.16 1.17 1.11 1.30 1.37 1.47 1.51 1.60 Cylinder New Old Exp. New Old Exp. New Old Exp. Geometry Height (cm) Width (cm) Aspect Ratio

• Comp. N-S w/∆x=0.01 cm

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Gas Curtain with New ICs

1.0 10 3.0 10 5.0 10 100 200 300 400 500 600 700 800 900 945.6

Cuervo 2 D Gas Curtain Simulation

time (s)

1.0 10 3.0 10 5.0 10 3.0 10 1.0 10 2.0 10 3.0 10 4.0 10 5.0 10 5.7 10

Cuervo 2 D Gas Curtain Simulation

time (s)

• Comp. N-S w/∆x=0.01 cm

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Acknowledgements

Experimentalists

Bob Benjamin, Kathy Prestridge, Paul Rightley,

Peter Vorobieff (UNM),

Computational Physicists

Mike Gittings (SAIC), Jeff Greenough

(LLNL), Cindy Zoldi, Tariq Aslam, Rob Lowrie, Mark Taylor, Rich Holmes

Theoretical Physicists/App. Mathematicians

Len Margolin, Darryl Holm, Tim Clark

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Summary

Experimental validation provide an essential

test of numerical methods

F i n !

These efforts help bring credibility to methods High quality experimental data are essential Shock driven mixing has served as a catalyst

for new approaches to numerics & models

This is ongoing work: additional experiments,

experimental & code refinements, and new methods are actively being pursued

Turbulence & mixing transition a current focus

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Fractals: Box Counting

The fractal dimension measures the complexity of

an interface and can be computed as: Image

How this applies to mixing flows:

Quantifies differences in numerical schemes Provides a metric for comparison with experiment

Contour Box Count

• 10
• 8
• 6
• 4
• 2

1 2 3 4 5

log(1/r) log[N(r)]

Dimension

Df =

slope

Box counting fractal dimension varies with:

Contour value Box size range Linear fitting

Gonzato “A Practical Implementation of the Box Counting Algorithm”, Comp. & Geosci. 24, 85–100, 1998.

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Var( ) var( , ) e e =Údy y

W

var( , ) sup ( ) inf ( ) y u x u x

x y x y

e

e e

=

• <
• <

Fractals: Variation Dimension

Use the whole image to compute the variation:

The fractal dimension is: Pyramid Algorithm: makes evaluation recursive and

fast — O(N log(N)) Df

d d

n n

=

( )

( ) ( )

• ln

ln Var e e e

n = 2 (2D)

• r 3 (3D)

“Spectrum” is given by the local dimension:

Local variation: Integrate:

Use central differences to

approximate derivatives:

Gives a local, scale-dependent spectral measure

Df =

• h

h x x

2 1 2 1

Dubuc et al. “Evaluating the fractal dimension of surfaces” , Proc. Roy. Soc. Lond. A 425,113–127, 1989.

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Fractal Analysis Verification

Box Counting Variation Method

Exact Fractal Dimension = 1.58 Computed Fractal Dimension = 1.53—1.58 Exact Fractal Dimension = 2.60

Techniques used to validate code results must

themselves be verified

0.0 0.5 1.0 Y 0.0 0.5 1.0 X

• 2.5

0.0 2.5

* Höfer, Die statistiche Analyse medizinischer Ultraschallbilder unter Berücksichtigung der physicaklischen Effekte der Ultraschallbildgebung sowie der fraktalen Struktur der abgebildeten Organe, Verlag Shaker, Aachen, 1995.

Sierpinski Gasket Discrete Fractional

Brownian Motion*

Computed Fractal Dimension = 2.45—2.64

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Continuous Wavelet Analysis

• CWT varies with choice of “mother wavelet” y

Image FFT Convolve

FFTf (k) x FFTy

y y y(k)

CWT

“Energy spectrum” is a scale-dependent measure

E a C db a W a b

f f f Rn

( ) =

• Ú

( , )

, ,

1

2 2 y y

r r

Wavelet Scale

Continuous Wavelet Transform = CWT = a dx f x x b a

n n

R

• ( )
• (

)

( )

Ú

2

r r r r y W a b

f y, ( , )

r

= Normalization Constant Data

Farge, “Wavelet Transforms and Their Applications to Turbulence ”, Ann. Rev. Fluid Mech. 24, 395–457, 1992.

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a dx f x x b a

n n

R

• ( )
• (

)

( )

Ú

2

r r r r y

Wavelet Analysis Verification

Wavelet coefficients provide a scale-by-scale measure

relative to a localized function y

Wavelet Data Continuous Wavelet Transform = CWT = Scale

Verify the wavelet code against the exact solution

for the chosen “mother” wavelet

Comparison of exact and computed spectra

Scale Wavelet “Energy”

Computed – Exact

Relative Error Scale

• Conv. Rate n=0.7

Marr Wavelet

3 1

• 1
• 3

3 1 -1 -3

0.3 0.2 0.1

• 0.1

1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 0.1 1 10 1E-6 1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 0.1 1 10

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LA-UR-02-5575 ||- and ^

^ ^ ^-structure functions of vector ~

u

Structure function of scalar quantity x

x x x

Structure Functions I

There are theoretical results, assuming :

S x x N x x

p p p

n N n n

( ) ( ) ( ) ( ) ( ) l l l ∫ +

• =

+

• Ê

Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜

=

x x x x 1

1

S

Sp

p

||( )

( ) ( )

~ ~ ~ ~ ~ ~

l l l ∫ +

• ◊

Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙

Ÿ

u u x x S m

p p

^ Ÿ

∫ +

• ◊

Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙

( ) ( ) ( )

~ ~ ~ ~ ~ ~

l l u u x x

where and are unit vectors with ^ ^ ^ ^ ~

Ÿ

m

~

Ÿ

m

~

Ÿ

l

~

Ÿ

l

~(~) u x

• ~(~)

u x ~(~ ~) u x + l ~ Ÿ l ~ l ~ Ÿ m ~(~ ~) ~(~) u u x x +

• l

Infinite Re:

Advection-dominated

Homogeneous:

Invariant under translations

Isotropic:

Invariant under rotations

Incompressible:

“Independent of thermodynamics”

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Structure Functions II

The theoretical results (K41 theory) include: But these assumptions do not hold...

In the inertial range: l l > > > > h

Dissipat’n scale Integral scale

S3

4 5

||( )

l l = - e

Kolmogorov’s 4/5-law

S2

2 3 ||( ) ~

l l

Kolmogorov’s 2/3-law

E k k ( ) ~ e2 3

5 3

• Kolmogorov’s 5/3-law

S C

p p p p ||( )

l l = e

3 3

General relation Finite Re:

Large variations in Re among experim’ts

Inhomogeneous: Statistics vary with location Non-isotropic:

Statistics not rotation-independent

Compressible:

Thermodynamics can play a pivotal role in flow dynamics and evolution

SLIDE 44

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44

LA-UR-02-5575

Techniques for Data Analysis

Codes compute the integral scale reliably

(mixing width), statistical details are not reliable!

Image statistics:

Fractals Wavelets Correlations

S x x

p p

( ) ( ) ( ) l l ∫ +

• Ê

Ë Á ˆ ¯ ˜

x x ~ la

a n

• 2·

Ê Ë Á ˆ ¯ ˜

y x a b , , , f x

( )Ò

Wavelet coefficient =

Volume Fraction Wavelet Coeff’t

Var e e

( ) = ( )

Ú var , x dx

W

var , sup inf x u x u x

x x x x

e

e e

( ) =

( )

( ) -

( )

( )

• <
• <

Surface, not contour Surface, not contour

Df

d d

n n

=

( )

( ) ( )

• ln

ln Var e e e

SLIDE 45

Bill Rider rider@lanl.gov www.ccs.lanl.gov

45

LA-UR-02-5575

0.5 0.6 0.7 0.8 0.9 1 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Initial Concentratio n

200x200 mesh Converged Value

cm

Error’s Impact on Initial Estimates

Widths for S/D=1.5 indicates lower concentrations than originally believed, but a little more diffusion than experimental estimates

SLIDE 46

Bill Rider rider@lanl.gov www.ccs.lanl.gov

46

LA-UR-02-5575

0.5 0.6 0.7 0.8 0.9 1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

Initial Concentratio n

200x200 mesh Converged Value

Error’s Impact on Initial Estimates

Heights for S/D=1.5 indicate lower concentrations than originally believed, and much more diffusion.

SLIDE 47

Bill Rider rider@lanl.gov www.ccs.lanl.gov

47

LA-UR-02-5575

0.5 0.6 0.7 0.8 0.9 1 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 200x200 mesh Converged Value

Initial Concentratio n

Error’s Impact on Initial Estimates

Widths for S/D=1.2 are consistent with S/D=1.5 widths with respect to diffusion and concentration

SLIDE 48

Bill Rider rider@lanl.gov www.ccs.lanl.gov

48

LA-UR-02-5575

0.5 0.6 0.7 0.8 0.9 1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 200x200 mesh Converged Value

Initial Concentratio n

Error’s Impact on Initial Estimates

Heights for S/D=1.2 are consistent with S/D=1.5 widths with respect to diffusion and concentration

SLIDE 49

Bill Rider rider@lanl.gov www.ccs.lanl.gov

49

LA-UR-02-5575

0.5 0.6 0.7 0.8 0.9 1 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 200x200 mesh Converged Value

Initial Concentratio n

Error’s Impact on Initial Estimates

Widths for S/D=2.0 indicate much more diffusion and lower concentrations than either of the

• ther separations.