The Old Well 10/25/2003 AMS Sectional Conference 1 Continuum Fluid Simulations Using Microscopically Polymer Computed Constitutive Laws Sorin 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 The University of North Carolina at Chapel Hill

The Old Well 10/25/2003 AMS Sectional Conference 2 Overview of drag reduction by polymers 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

The Old Well 10/25/2003 AMS Sectional Conference 3 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

The Old Well 10/25/2003 AMS Sectional Conference 4 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)

The Old Well 10/25/2003 AMS Sectional Conference 5 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 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

The Old Well 10/25/2003 AMS Sectional Conference 6 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)

The Old Well 10/25/2003 AMS Sectional Conference 7 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

The Old Well 10/25/2003 AMS Sectional Conference 8 A Simple Microscopic Model Discrete time evolution of individual dumbbell model of polymer molecule Vastly different time scales

The Old Well 10/25/2003 AMS Sectional Conference 9 Additional Stress Induced by Polymers Kramers form stress tensor Main ideas: Estimate on a cell by cell basis the statistical certainty of the stress tensor

The Old Well 10/25/2003 AMS Sectional Conference 10 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

The Old Well 10/25/2003 AMS Sectional Conference 11 Results – No polymer additive

The Old Well 10/25/2003 AMS Sectional Conference 12 Results – 25 ppm polymer additive

The Old Well 10/25/2003 AMS Sectional Conference 13 Further Work More realistic polymer models FENE-P More realistic molecular dynamics, especially MD of water polymer interaction

The Old Well 10/25/2003 AMS Sectional Conference 14 Conclusions Extensions of AMR framework to microscopic levels of simulation Redefinition of restriction, prolongation, time subcycling concepts Promising approach for flows with strong inhomogeneities