A Non Standard Homogenization Theory for Turbulent Transport

1. A Non Standard Homogenization Theory for Turbulent Transport Adnan Khan Lahore University of Management Sciences Peter Kramer Rensselaer Polytechnic Institute

2. Introduction • 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

3. Simplified Model • 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

4. Our Work • 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

5. Problem Setup • 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

6. Previous Work • 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

7. Monte Carlo Simulations for Tracer Trajectories • 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

8. Childress Soward Flow • 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

9. Effective Diffusivity from MC and Homogenization

10. Observations • 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

11. Non Standard Homogenization Theory • 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

12. The Solvability Condition • 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

13. Characteristic Coordinates

14. Geometry of the Problem

15. Analysis • 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

16. The Asymptotic Expansion • We develop the asymptotic expansion in both the cases • We have the following multiple scales hierarchy • We derive the effective equation for the quantity

17. Homogenized Equation for the Low Order Rational Ratio Case • For the low order rational case we get • Where the operators are given by

18. Homogenized Equation for the High Order Rational Ratio Case • For the high order rational ratio case we get the following homogenized equation