1. Turbulent Scalar Mixing Revisiting the classical paradigm in variable diffusivity medium Gaurav Kumar Advisor: Prof. S. S. Girimaji Turbulence Research Group @ A&M

2. ARSM reduction RANSLESDNS 2-eqn. RANS Averaging Invariance Application DNS 7-eqn. RANS Body force effects Linear Theories: RDT Realizability, Consistency Spectral and non- linear theories 2-eqn. PANS Near-wall treatment, limiters, realizability correction Numerical methods and grid issues Navier-Stokes Equations

3. Motivation: Why study scalar mixing ?  Classical understanding of mixing Constant transport properties – viscosity, diffusivity.  Hypersonic boundary layers, high speed combustion Large variations in molecular transport properties  5 times  Classical understanding may fail. New terms due to large spatio-temporal variations.  Development of better scaling laws and turbulence closure models.  Important in many other fields including: Energy, environment, manufacturing, combustion, chemical processing, dispersion.

4. Classical mixing paradigm 1.Scalar cascade rate is determined by variance and scalar timescale: cascade rate 2.Scalar analogue of Taylor’s viscosity dissipation postulate: scalar dissipation is independent of diffusivity. 3.Since the scalar field is advected by the velocity field: scalar timescale  velocity timescale 4.Conditional scalar dissipation is insensitive to diffusivity:

5. Classical mixing paradigm Validated in constant diffusivity medium. Validity in inhomogeneous media not excluded, but remains dubious due to: Rapid spatio-temporal changes in scalar diffusivity. - Scalar gradients may not adapt to local transport properties. New transport terms in scalar dissipation evolution equation.

6. Objective of the study To examine the validity of “the classical mixing paradigm” in heterogeneous media. To study the behavior of conditional scalar dissipation and timescale ratios. Benefits Confidence in applying scaling laws and closure models developed for uniform diffusivity media in inhomogeneous media.

7. Governing equations Mass conservation: Momentum consv: Mixture fraction evolution: Scalar evolution:

8. Numerical setup DNS using Gas Kinetic Methods. Domain: 256 3 box with periodic boundaries. N x = 256, N y = 256, N z = 256 Initial condition: statistically homogenous, isotropic and divergence free velocity field. = 2 x 10 -5, 1 ≤ i ≤ 128 = 1 x 10 -4, 129 ≤ i ≤ 256

9. Cases Linear mixing law: Wilkes formula: LeftRightLeftRightLeftRight CaseRe Pr Sc Mixing Formula A64.49 1.0 Premixed B64.49 3.00.61.0 Linear C64.49 3.00.61.0 Wilkes D64.49 3.00.61/35/3Wilkes E193.4738.691.0 3.00.6Wilkes where,

10. Scalar dissipation Scalar dissipation: rate at which scalar variance is dissipated. It is most direct measure of rate of mixing.

11. CASE-A: [Baseline case] vs. x Evolution of scalar dissipation for single species (case A)

12. CASE-B,C: vs. x Evolution of scalar dissipation for two species case: case B (left), case C (right) In 1/3 eddy turnover time, scalar dissipation is uniform across the box. Linear mixing law Wilkes formula Choice of mixing formula does not affect the result.

13. CASE-B,C: vs. x Evolution of conductivity for two species case: case B (left), case C (right) Still, a large disparity in diffusivity in left and right halves of the box persists.

14. CASE-B,C: vs. x Evolution of scalar dissipation for two species case: case B (left), case C (right) Scalar gradient is large in smaller conductivity region and small in higher side.

15. Case C: Evolution of planar spectra Evolution of planar spectra for two species case (case C): [left] low conductivity plane (n x =64), [right] high conductivity plane (n x =192) Less scales More scales

16. Case C: Iso-surfaces of scalar gradient (a) time t’=0.00 (b) time t’=0.36 (c) time t’=0.54 Iso-surfaces of scalar gradient for two species case (case C) Smaller scales / higher gradients t

17. Scalar dissipation Result: 1.Within 1/3 eddy turnover time scalar dissipation becomes independent of diffusivity, despite large initial disparity. 2.Scalar gradient adjusts itself inversely proportional to diffusivity. 3.Mixing formula does not affect the results.

18. Velocity-to-scalar timescale ratio Velocity to scalar timescale ratio: An important scalar mixing modeling assumption: Scalar mixing timescale  velocity field timescale Proportionality constant is dependent on -Initial velocity-to-scalar length scale ratio.

19. Evolution of velocity to scalar timescale ratio Evolution of velocity-to-scalar timescale ratio (r) with time: (a) case B (b) caseC r

20. Velocity-to-scalar timescale ratio Result: Heterogeneity of the medium does not affect the relation between scalar and velocity timescales.

21. Conditional scalar dissipation Normalized conditional scalar dissipation: -determines the rate of evolution of pdf of scalar field.

22. Conditional scalar dissipation Conditional scalar dissipation vs. normalized scalar value (case C): (a) time t’=0.45 (b) time t’=0.54

23. Conditional scalar dissipation Conditional scalar dissipation vs. normalized scalar value (case E): (a) time t’=0.45 (b) time t’=0.54

24. Conditional scalar dissipation Result: Normalized conditional scalar dissipation is nearly unity in the interval indicating a nearly Gaussian of the scalar field.

25. Conclusions 1.Scalar gradients adapt rapidly to diffusivity variations − renders scalar dissipation independent of diffusivity 2.Normalized conditional scalar dissipation is independent of diffusivity. 3.Scalar-to-velocity timescale ratio also independent of: (i) viscosity (ii) diffusivity 4.Findings confirm the applicability of Taylor’s postulate to heterogeneous media.