Turbulent Scalar Mixing Revisiting the classical paradigm in variable diffusivity medium

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

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

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 • Scalar cascade rate is determined by variance and scalar timescale: cascade rate • Scalar analogue of Taylor’s viscosity dissipation postulate: scalar dissipation is independent of diffusivity. • Since the scalar field is advected by the velocity field: scalar timescale  velocity timescale • 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: 2563 box with periodic boundaries. • Nx = 256, Ny = 256, Nz = 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: where, Wilkes formula:

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 Linear mixing law Wilkes formula 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. 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 More scales Less scales Evolution of planar spectra for two species case (case C): [left] low conductivity plane (nx=64), [right] high conductivity plane (nx=192)

16. Case C: Iso-surfaces of scalar gradient Smaller scales / higher gradients t (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)

17. Scalar dissipation • Result: • Within 1/3 eddy turnover time scalar dissipation becomes independent of diffusivity, despite large initial disparity. • Scalar gradient adjusts itself inversely proportional to diffusivity. • 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 r Evolution of velocity-to-scalar timescale ratio (r) with time: (a) case B (b) caseC

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