A topology-based approach towards

understanding mixing in high-speed flows

Sawan SumanPost-docTurbulence Research GroupTexas A&M University

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

Mixing in high speed environment

Introduction Enhanced scalar mixing: essential in ramjet/scramjet combustors

Compressibility reduces KE: reduces turbulent mixing

Understanding/modeling/improvement at two levels:

(a) Macro stirring:

(b) Micro mixing: Scalar dissipation must be enhanced

Requires understanding of small-scale structures: scalar- and velocity-gradients

Aims: -to understand the role played by the structure of velocity gradient in shaping the behaviour of this term? -to quantify the mixing capability of various possible structures of velocity gradient

' '

i ix x

' ' ' 'i i j

j

u u ux

' ' '' '

2 ...j i ji

j

i

D

Dt xx x

u

x x

Structure of velocity-grad field: Topology

0

0 ii i j

j

VV x x

x

Topology= Local streamline pattern within a fluid element/Exact deformation pattern of a fluid element

Pattern of streamlines = Nature of eigen values of the tensor

Local flow-field topology: Visual, intuitive and physically sensible way to study

velocity-gradient structure

Reference:

Strain-dominated topology, real eigenvalues Rotation-dominated topology, complex eigenvalues Stable-node Stable-focus

Introduction 3-D flows have more complex topologies Unstable node/saddle/saddle (UNSS), Stable focus stretching (SFS), etc.

Compressible flows have more possible topologies compared to incomp. flows (Chong & Perry, 1990, POF, Suman & Girimaji, 2010, JoT)

Unstable focus stretching (UFS), Stable focus compressing (SFC), etc

Which topologies in compressible turbulence are more efficient in mixing?

CFD analysis and design can aim to maximize the population of efficient topologies

How can velocity gradient maximize production of scalar dissipation?

Normalized evolution equation: time normalized by velocity gradient magnitude

Decomposition of velocity gradient: strain-rate and dilatation and rotation

Simpler form of evolution equation:

What do we know about the “incompressible” mixing?

Velocity-gradients & mixing

' '2 ' '2 ' '21 2 3

'2 '2 '21 2 3'

...k kD

dt

11 12 13 3 1

21 22 23 3 2

31 32 33 1 2

/ 3 0 0

0 / 3 0

0

0 0

0 0

0 0

0

0

1 1

3

0

0 /

kk

a a a

a a a

a a a

roDilatatiAnisotro tationon

s

pi ratec Strain

i ik kx x

“Incompressible” mixing “compressible” mixing

' ' '' ' ' ''

'... : ( )2 ... 2 ji

ji ji jii j i j ii i i k ki i

AA a a structureof velocity grad tensor

x x x x

D D

Dt x x Dt x x A A

iij

j

VA

x

Velocity gradient tensor

Known incomp. behaviour

Scalar dissipation maximum when scalar grad. aligned with large, -ve strain-rate

Scalar gradient is found to be aligned with large negative strain-rate

Vorticity mis-aligns scalar grad., reduces dissipation

Ashurt et al. (POF,1987), Brethouwer et al, (JFM, 2003), O’Neill et al (Fluid Dynamics Research, 2004)

large, +ve

small, +ve

large, ve

Plane of &

vorticity vector

Scalar gradient

Mixing efficiency definitions

max algebriacincompressible

max algebraiccompressible

incompressible compressiba el ltot

i ik kx x

' '2 ' '2 ' '2

1 2 3'2 '2 '2

1 2 3'...

k kD

dt

Definitions take into account the role of velocity field only, scalar field is not accounted for

Will check the validity of this approach

Will compute volume averaged values of efficiency in decaying turbulence

Incompressible Turbulence

incompressible compressible total

UNSS best mixer

Incompressible turbulence:

Stable node/Saddle/Saddle Unstable node/Saddle/Saddle Stable-focus Stretching Unstable-focus Compressing

max algebriacincompressible

max algebraiccompressible

Validation

DNS of scalar field shows UNSS has highest scalar dissipation

Our approach – despite being based on only velocity-field information – reaches the same conclusion

O’Neill & Soria (Fluid Dynamics Research, 2004)

Scatter plot of scalar dissipation in DNS of incompressible turbulence

Unstable-focus Compressing

Stable node/Saddle/Saddle

Stable-focus Stretching

Unstable node/Saddle/Saddle

Compressible Turbulence

Using DNS results of compressible turbulence Only velocity-field available No scalar

SN/SN/SN

incompressible compressible total

Contracting fluid elements

SN/SN/SN (isotropic contraction) best mixerAll contraction topologies better mixers than the incompressible ones, dilatational shrinking favors mixing

Stable node/Saddle/Saddle Stable node/Stable node/Stable node

Unstable node/Saddle/Saddle Stable-focus

compressingStable-focus Stretching

Unstable-focus Compressing

max algebriacincompressible

max algebraiccompressible

incompressible compressible total

Incompressible turbulence:

max algebriacincompressible

max algebraiccompressible

Action of velocity field on scalar field

Negative dilatation (volume contraction) amplifies this process in all directions – possible only in compressible flows

Iso-scalar surfaces

Compressive strain pushes iso-scalar surfaces closer, increasing scalar dissipation in that direction only

UN/UN/UN

incompressible compressible total

Expanding fluid elements

B (UNSS) best mixer, UN/UN/UN (isotropic expansion) worst mixerAll topologies less efficient than incompressible topologies; negative contribution from dilatation

Stable node/Saddle/Saddle Unstable node/Unstable node/Unstable node

Unstable node/Saddle/Saddle

Stable-focus Stretching Unstable-focus

Stretching

Unstable-focus Compressing

max algebriacincompressible

max algebraiccompressible

incompressible compressible total

Incompressible turbulence:

max algebriacincompressible

max algebraiccompressible

A topology–based approach proposed to study the association of velocity field and scalar mixing

Method reproduces major conclusion from DNS of incomp. turbulence with scalar mixing

Preliminary predictions for compressible flows:

-Mixing efficiencies: Contracting > Incompressible > Expanding fluid elements

-SN/SN/SN: isotropic contraction is the best mixer in compressible turbulence

-UNSS best mixer in incompressible and expanding fluid elements

Future work:

-Needs further validation with DNS of canonical compressible flows with scalars

-Combustor design: maximize isotropic contraction

Conclusions

Isotropic contraction Best mixers