Turbulent Scalar MixingRevisiting the classical paradigm in variable diffusivity medium

Gaurav KumarAdvisor: Prof. S. S. Girimaji

Turbulence Research Group @ A&M

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

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.

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:

( , )si i

fx x

2'

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.

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.

Governing equations

• Mass conservation:

• Momentum consv:

• Mixture fraction evolution:

• Scalar evolution:

( )( )j

j j j

fuf fD ft x x x

( )( )j

j j j

uf

t x x x

, 0i iu

( , )i ji i

j i j j

u uu up x tt x x x x

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 ≤ 256l

h

24( , 0) , 1 8BkE k t Ak e k

l h

Cases

Linear mixing law:

Wilkes formula:

Left Right Left Right Left Right

Case Re Re Pr Pr Sc Sc Mixing Formula

A 64.49 64.49 1.0 1.0 1.0 1.0 Premixed

B 64.49 64.49 3.0 0.6 1.0 1.0 Linear

C 64.49 64.49 3.0 0.6 1.0 1.0 Wilkes

D 64.49 64.49 3.0 0.6 1/3 5/3 Wilkes

E 193.47 38.69 1.0 1.0 3.0 0.6 Wilkes

( ) (1 )h lf f f

(1 )( )

(1 ) (1 ) l

h

h lf ff

f f f f

21/21 14

h

l

where,

Scalar dissipation

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

rate of mixing.

2

si ix x

CASE-A: [Baseline case] vs. x, ,i i yz

Evolution of scalar dissipation for single species (case A), ,i i yz

2l h

CASE-B,C: vs. x, ,i i yz

Evolution of scalar dissipation for two species case: case B (left), case C (right), ,i i yz

In 1/3 eddy turnover time, scalar dissipation is uniform across the box.

l h

Linear mixing law Wilkes formula

Choice of mixing formula does not affect the result.

CASE-B,C: vs. x

Evolution of conductivity for two species case: case B (left), case C (right)yz

Still, a large disparity in diffusivity in left and right halves of the box persists.

l h

CASE-B,C: vs. x, ,i i yz

Evolution of scalar dissipation for two species case: case B (left), case C (right), ,i i yz

Scalar gradient is large in smaller conductivity region and small in higher side.

l h

Case C: Evolution of planar spectra

Evolution of planar spectra for two species case (case C): [left] low conductivity plane (nx=64), [right] high conductivity plane (nx=192)

Less scalesMore scales

l h

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

l h

t

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.

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.

2

2

3u

s

ur

Evolution of velocity to scalar timescale ratio

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

r

Velocity-to-scalar timescale ratio

Result:Heterogeneity of the medium does not affect the relation between scalar and velocity timescales.

Conditional scalar dissipation

Normalized conditional scalar dissipation:

- determines the rate of evolution of pdf of scalar

field.

i i i i yzyzx x x x

Conditional scalar dissipation

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

Conditional scalar dissipation

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

Conditional scalar dissipation

Result: Normalized conditional scalar dissipation is

nearly unity in the interval indicating a nearly Gaussian of the scalar field.

* 2,2

Conclusions1. 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.