10/25/2003 AMS Sectional Conference 1 The Old Well Continuum Fluid Simulations Using Microscopically Polymer Computed Constitutive Laws Sorin Mitran [email protected]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 Program http://www.amath.unc.edu The University of North Carolina at Chapel Hill
15
Embed
The Old Well 10/25/2003 AMS Sectional Conference 1 Continuum Fluid Simulations Using Microscopically Polymer Computed Constitutive Laws Sorin Mitran [email protected].
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
10/25/2003 AMS Sectional Conference1
The Old Well
Continuum Fluid Simulations Using Microscopically Polymer Computed Constitutive Laws
"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)
- 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