A topology-based approach towards understanding mixing in high-speed flows

1. A topology-based approach towards understanding mixing in high-speed flows Sawan Suman Post-doc Turbulence Research Group Texas A&M University

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 Mixing in high speed environment DNS LES RANS

3. 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

4. Structure of velocity-grad field: Topology • Local flow-field topology: Visual, intuitive and physically sensible way to study velocity-gradient structure • 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 Reference: Strain-dominated topology, real eigenvalues Rotation-dominated topology, complex eigenvalues Stable-node Stable-focus

5. 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

6. Velocity-gradients & mixing • 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 gradient tensor “Incompressible” mixing “compressible” mixing

7. 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) vorticity vector Plane of Scalar gradient

8. Mixing efficiency definitions • 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

9. Incompressible Turbulence

10. Incompressible turbulence: Stable node/Saddle/Saddle Unstable node/Saddle/Saddle Stable-focus Stretching Unstable-focus Compressing UNSS best mixer

11. Validation Scatter plot of scalar dissipation in DNS of incompressible turbulence Stable-focus Stretching Unstable-focus Compressing Stable node/Saddle/Saddle Unstable node/Saddle/Saddle Stable node/Saddle/Saddle O’Neill & Soria (Fluid Dynamics Research, 2004) • DNS of scalar field shows UNSS has highest scalar dissipation • Our approach – despite being based on only velocity-field information – reaches the same conclusion

12. Compressible Turbulence • Using DNS results of compressible turbulence • Only velocity-field available • No scalar

13. Contracting fluid elements Unstable node /Saddle/Saddle Stable node/Saddle/Saddle Stable node/Stable node /Stable node Stable-focus compressing Stable-focus Stretching Unstable-focus Compressing SN/SN/SN • SN/SN/SN (isotropic contraction) best mixer • All contraction topologies better mixers than the incompressible ones, dilatational shrinking favors mixing

14. Incompressible turbulence:

15. Action of velocity field on scalar field • Compressive strain pushes iso-scalar surfaces closer, increasing scalar dissipation in that direction only Iso-scalar surfaces • Negative dilatation (volume contraction) amplifies this process in all directions – possible only in compressible flows

16. Expanding fluid elements Stable-focus Stretching Unstable-focus Compressing Stable node/Saddle/Saddle Unstable node /Saddle/Saddle Unstable node/Unstable node /Unstable node Unstable-focus Stretching UN/UN/UN • B (UNSS) best mixer, UN/UN/UN (isotropic expansion) worst mixer • All topologies less efficient than incompressible topologies; negative contribution from dilatation

17. Incompressible turbulence:

18. Conclusions • 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 Isotropic contraction Best mixers