Turbulent Scalar Mixing Revisiting the classical paradigm in variable diffusivity medium
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Turbulent Scalar MixingRevisiting the classical paradigm in variable diffusivity medium
Gaurav KumarAdvisor: Prof. S. S. Girimaji
Turbulence Research Group @ A&M
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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
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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.
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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'
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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.
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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.
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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
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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
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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,
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Scalar dissipation
Scalar dissipation: rate at which scalar variance is dissipated. It is most direct measure of
rate of mixing.
2
si ix x
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CASE-A: [Baseline case] vs. x, ,i i yz
Evolution of scalar dissipation for single species (case A), ,i i yz
2l h
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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.
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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
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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
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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
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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
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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.
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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
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Evolution of velocity to scalar timescale ratio
Evolution of velocity-to-scalar timescale ratio (r) with time: (a) case B (b) caseC
r
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Velocity-to-scalar timescale ratio
Result:Heterogeneity of the medium does not affect the relation between scalar and velocity timescales.
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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
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Conditional scalar dissipation
Conditional scalar dissipation vs. normalized scalar value (case C): (a) time t’=0.45 (b) time t’=0.54
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Conditional scalar dissipation
Conditional scalar dissipation vs. normalized scalar value (case E): (a) time t’=0.45 (b) time t’=0.54
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Conditional scalar dissipation
Result: Normalized conditional scalar dissipation is
nearly unity in the interval indicating a nearly Gaussian of the scalar field.
* 2,2
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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.