10/25/2003 AMS Sectional Conference1

The Old Well

Continuum Fluid Simulations Using Microscopically Polymer Computed Constitutive Laws

Sorin Mitranmitran@amath.unc.edu

http://www.amath.unc.edu/Faculty/mitran

Effect of polymer additive on 2D soap film flow

(Walter Goldburg, Univ. of Pittsburgh)

Cardoso, Marteau, Tabeling experiments on 2D stratified

flows

Applied Mathematics Programhttp://www.amath.unc.edu

The University of North Carolina

at Chapel Hill

10/25/2003 AMS Sectional Conference2

"Computation and control of 2D dilute polymer flows"

Small amounts of polymer additives have been shown to affect overall flow characteristics

of 2D flows in addition to the well documented influence they have on 3D turbulence. In this

talk the issue of how control of microscopic motions of the polymers can be used to control

the overall flow is investigated computationally. The computational method involves simultaneous computation of the continuum flow through an adaptive mesh, finite volume

method and of the microscopic dynamics of the polymers. Imposed and controlled motions of the polymer molecules are sought that bring about desired continuum level changes in the flow.

10/25/2003 AMS Sectional Conference3

The Old Well

Overview of drag reduction by polymers

cba

Large change in instantaneous wall-friction due to small PEO concentrations (Stanford U, Reynolds Lab.)

History:

• Experimental observations, Toms (1948, Proc. Intl. Congress on Rheology)

• Lumley qualitative analysis of behavior (Ann. Rev. Fl. Mech. 1:367, 1969, Phys. Fl. 20:S64, 1977)

- polymers stretched by turbulent fluctuations in mid-channel

- polymers remain coiled in boundary layer shear flow

- turbulent bulk viscosity increases, viscosity in boundary layer remains the same

10/25/2003 AMS Sectional Conference4

The Old Well

Overview of drag reduction by polymers

Modification of wall flow structure (Stanford U, Reynolds Lab.)

Drag mechanism unclear:

• Experimental measurements shows increase of viscous sublayer thickness expected by Lumley model does not occur

• The buffer layer in the boundary layer undergoes significant change

• Visualization of wall flow shows overall dampening of small structures upon polymer addition

10/25/2003 AMS Sectional Conference5

The Old Well

Experimental Observations in 2D

Bullet 3

Soap film experiments (Goldberg, U. Pittsburgh)

No polymers 25 ppm PEO

Amarouchene & Kellay (Phys. Rev. Lett. 89(10), 2002)

10/25/2003 AMS Sectional Conference6

The Old Well

Standard Computation of Non-Newtonian Fluid

Assume some model for viscoelastic fluid at microscopic level (dumbbell, FENE, FENE-P) Work out constitutive law analytically Solve continuum equations

fuupuuut 2

)(

2/120

20

21,

2)()()(

RRuRR

R

RR

luuu

jiij

Tt

Momentum equation with additional polymer stress

Continuum equation for polymer conformation field

Critique:

• Approach is analytically tractable only for relatively simple polymer models

• Approximations in deriving continuum equations from microscopic, polymer model

• Homogeneity of polymer additive is often invoked

• Approach is very useful for applications but limited for understanding basic questions such as drag reduction mechanism

10/25/2003 AMS Sectional Conference7

The Old Well

Adaptive Computation

For continuum levels Trial step on coarse grid determines placement of finer grids Boundary conditions for finer grids from space-time interpolation

Time subcycling: more time steps (of smaller increments) are taken on fine grids Finer grid values are obtained by interpolation from coarser grid values Coarser grid values are updated by averaging over embedded fine grids Conservation ensured at coarse-fine interfaces (conservative fixups)

t

ionInterpolat

Inject

Averaging

Restrict

t

2/t 2/t

4/t 4/t 4/t 4/t

10/25/2003 AMS Sectional Conference8

The Old Well

Extension of AMR to Microscopic Computation

Maintain idea of embedded grids

Establish a cutoff length at which microscopic computation is employed

Redefine injection/prolongation operators

Redefine error criterion for grid refinement

Redefine time subcycling

10/25/2003 AMS Sectional Conference9

The Old Well

A Simple Microscopic Model

vectorrandom a

2

1

w

x

uS

wtk

kTtRtRSR

j

iij

HelasticH

Discrete time evolution of individual dumbbell

model of polymer molecule

CFLtt

Vastly different time scales

10/25/2003 AMS Sectional Conference10

The Old Well

Additional Stress Induced by Polymers

Kramers form stress tensor

QQnknkT elasticp

Main ideas: Estimate on a cell by cell basis the statistical certainty of the stress tensor

10/25/2003 AMS Sectional Conference11

The Old Well

Molecular-Continuum Interaction

Prolongation operator from continuum to microscopic levels instantiates a statistical distribution of dumbbell configurations (e.g. Maxwell-Boltzmann) Prolongation operator from microscopic to microscopic levels is a finer sampling operation Restriction operator from microscopic to continuum level is a smoothing of the additional stress tensor (avoid microscopic noise in the continuum simulation) Time subcycling determined by desired statistical certainty in the stress tensor

10/25/2003 AMS Sectional Conference12

The Old Well

Results – No polymer additive

10/25/2003 AMS Sectional Conference13

The Old Well

Results – 25 ppm polymer additive

10/25/2003 AMS Sectional Conference14

The Old Well

Further Work

More realistic polymer models FENE-P More realistic molecular dynamics, especially MD of water polymer interaction

10/25/2003 AMS Sectional Conference15

The Old Well

Conclusions

Extensions of AMR framework to microscopic levels of simulation

Redefinition of restriction, prolongation, time subcycling concepts

Promising approach for flows with strong inhomogeneities