Adnan Khan Lahore University of Management Sciences Peter Kramer Rensselaer Polytechnic Institute

 To study turbulent transport  Transport occurs via two mechanisms  Advection  Diffusion  The advective field being turbulent has waves at a continuum of wavelengths  We study simplified models with widely separated scales to understand these issues

 We study the simplest case of two scales with periodic fluctuations and a mean flow  The case of weak and equal strength mean flows has been well studied  For the strong mean flow case standard homogenization theory seems to break down

 We study the transport using Monte Carlo Simulations for tracer trajectories  We compare our MC results to numerics obtained by extrapolating homogenization code  We develop a non standard homogenization theory to explain our results

 Transport is governed by the following non dimensionalized Advection Diffusion Equation  There are different distinguished limits Weak Mean Flow Equal Strength Mean Flow Strong Mean Flow

 For the first two cases we obtain a coarse grained effective equation  is the effective diffusivity given by  is the solution to the ‘cell problem’  The goal is to try an obtain a similar effective equation for the strong mean flow case

 We use Monte Carlo Simulations for the particle paths to study the problem  The equations of motion are given by  The enhanced diffusivity is given by

 We use the CS flow as the fluctuation and different mean flows  Changing the parameter gives different flow topologies  We run Monte Carlo simulation of the tracer trajectories with this flow

 We note that the Monte Carlo Simulations in the Strong Mean Flow case also seem to agree with the homogenization numerics  This indicates that homogenization does take place in this case as well  Standard derivation of homogenization theory leads to ill posed equations  We develop a Non Standard homogenization theory to explain our numerical results

 We consider one distinguished limit where we take  We develop a Multiple Scales calculation for the strong mean flow case in this limit  We get a hierarchy of equations (as in standard Multiple Scales Expansion) of the form  is the advection operator, is a smooth function with mean zero over a cell

 We develop the correct solvability condition for this case  We want to see if becomes large on time scales  This is equivalent to estimating the following integral  The magnitude of this integral will determine the solvability condition

 Analysis of the integral gives the following  Hence the magnitude of the integral depends on the ratio of and  For low order rational ratio the integral gets in time  For higher order rational ratio the integral stays small over time

 We develop the asymptotic expansion in both the cases  We have the following multiple scales hierarchy  We derive the effective equation for the quantity

 For the low order rational case we get  Where the operators are given by

 For the high order rational ratio case we get the following homogenized equation