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