# New studies in vortex dynamics: Incompressible and compressible vortex reconnection, core dynamics, and coupling between large and small scales

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## Abstract

Coherent structure dynamics in turbulent flows are explored by direct numerical simulations of the Navier-Stokes equations for idealized vortex configurations. For this purpose, two dynamically significant coherent structure interactions are examined: (i) incompressible and compressible vortex reconnection and (ii) core dynamics (with and without superimposed small-scale turbulence). Reconnection is studied for two antiparallel vortex tubes at a Reynolds number (Re) of 10^{3}. Incompressible reconnection consists of three distinct phases: inviscid advection, bridging and threading. The key mechanism, bridging, involves the ‘cutting’ of vortex lines by viscous cross diffusion and their subsequent reconnection in front of the advancing vortex dipole. We conjecture that reconnection occurs in successive bursts and is a physical mechanism of cascade to smaller scales.

Compressible reconnection is seen to be significantly affected by the choice of pressure and density initial conditions. We propose a polytropic initial condition which is consistent with experimental results and low-Mach number asymptotic theories. We also explain how compressibility initiates an early reconnection due to shocklet formation, but slows down the circulation transfer at late times. Thus, the reconnection timescale increases with increasing Mach number.

Motivated by the important role of helical vortex lines in the reconnected vortices (bridges), we focus our attention on the dynamics of an axisymmetric vortex column with axial variation of core size. The resulting core dynamics is first explained via coupling between swirl and meridional flows. We then show that core dynamics can be better understood by applying a powerful analytical tool —*helical wave decomposition* — which extracts vorticity wave packets, thereby providing a simple explanation of the dynamics. The increase in core size variation with increasing Re in such a vortex demonstrates the limitation of the prevalent vortex filament models which assume constant core size. By studying the columnar vortex with superimposed small-scale, homogeneous, isotropic turbulence, we address the mutual interactions between large and small scales in turbulent flows. At its boundary the columnar vortex organizes the small scales, which, if Re is sufficiently high, induce bending waves on the vortex which further organize the small scales. Such backscatter from small scales cannot be modelled by an eddy viscosity. Based on the observation of such close coupling between large and small scales, we question the local isotropy assumption and conjecture a fractal vortex model for high Re turbulent flows.

## Keywords

Vortex dynamics vortex reconnection compressible vortex dynamics core dynamics helical wave decomposition coherent structures large-scale/small-scale interaction coherent structure/small-scale turbulence coupling## Preview

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## References

- Ashurst W T, Meiburg E 1988 Three-dimensional shear layers via vortex dynamics.
*J. Fluid Mech.*189: 87–116CrossRefGoogle Scholar - Batchelor G K 1967
*Introduction to fluid dynamics*(Cambridge: University Press)MATHGoogle Scholar - Bridges J, Husain H, Hussain F 1989 Whither coherent structures? In
*Whither turbulence? Turbulence at the crossroads*(ed.) J Lumley (Berlin: Springer-Verlag)Google Scholar - Brown G L, Roshko A 1974 On density effects and large structure in turbulent mixing layers.
*J. Fluid Mech.*64: 775–816CrossRefGoogle Scholar - Crow S C 1970 Stability theory of a pair of trailing vortices.
*AIAA J.*8: 2172–2179CrossRefGoogle Scholar - Crow S C, Champagne F H 1971 Orderly structure in jet turbulence.
*J. Fluid Mech.*48: 547–591CrossRefGoogle Scholar - Dosanjh D, Weeks T 1965 Interaction of a starting vortex as well as a vortex street with a travelling shock wave.
*AIAA J.*3: 216–223MATHGoogle Scholar - Erlebacher G, Hussaini M, Kreiss H, Sarkar S 1990 The analysis and simulation of compressible turbulence.icase Rep. No. 90–15Google Scholar
- Feiereisen W, Reynolds W, Ferziger J 1981 Numerical simulation of a compressible homogeneous, turbulent shear flow. Report No.tf-13, StandfordGoogle Scholar
- Greene J M 1990 Vortex nulls and magnetic nulls. In
*Topological fluid mechanics*(eds) H K Moffatt, A Tsinober (Cambridge: University Press)Google Scholar - Hussain F 1981 Role of coherent structures in turbulent shear flows.
*Proc. Indian Acad. Sci. (Eng. Sci.)*4: 129–175Google Scholar - Hussain F 1983 Coherent structure—reality and myth.
*Phys. Fluids*26: 2816–2850MATHCrossRefGoogle Scholar - Hussain F 1984 Coherent structures and incoherent turbulence. In
*Turbulence and chaotic phenomena in fluids*(ed.) T Tatsumi (Amsterdam: North-Holland)Google Scholar - Hussain F 1986 Coherent structures and turbulence.
*J. Fluid Mech.*173: 303–356CrossRefGoogle Scholar - Hussain F, Husain H 1989 Elliptic jets. Part I. Characteristics of unexcited and excited jets.
*J. Fluid Mech.*208: 257–320CrossRefGoogle Scholar - Kerr R N, Hussain F 1989 Simulation of vortex reconnection.
*Physica*D37: 474–484Google Scholar - Kida S, Takaoka M 1990 Breakdown of frozen motion of vorticity field and vortex reconnection.
*J. Phys. Soc. Jpn.*60: 2184–2196CrossRefGoogle Scholar - Kline S J, Reynolds W D, Schraub F A, Runstadler P W 1967 The structure of turbulent boundary layers.
*J. Fluid Mech.*30: 741–773CrossRefGoogle Scholar - Lee S, Lele S, Moin P 1991 Eddy shocklets in decaying compressible turbulence.
*Phys. Fluids*A3: 657–664Google Scholar - Leonard A 1985 Computing three-dimensional incompressible flows with vortex elements.
*Annu. Rev. Fluid Mech.*17: 523–559CrossRefGoogle Scholar - Lesieur M 1990
*Turbulence in fluids*2nd edn (London: Kluwer)MATHGoogle Scholar - Lundgren T S 1982 Strained spiral vortex model for turbulent fine structure.
*Phys. Fluids*25: 2193–2203MATHCrossRefGoogle Scholar - Lundgren T S, Ashurst W T 1989 Area-varying waves on curved vortex tubes with application to vortex breakdown.
*J. Fluid Mech.*200: 283–307MATHCrossRefMathSciNetGoogle Scholar - Mandella M 1987
*Experimental and analytical studies of compressible vortices*, Ph D thesis, Stanford University, StanfordGoogle Scholar - McWilliams J C 1984 The emergence of isolated vortices in turbulent flow.
*J. Fluid Mech.*146: 21–43MATHCrossRefGoogle Scholar - Meiron D I, Shelley M J, Ashurst W T, Orszag S A 1989 Numerical studies of vortex reconnection. In
*Mathematical aspects of vortex dynamics*(ed.) R Caflish (SIAM)Google Scholar - Melander M V, Hussain F 1988 Cut-and-connect of two antiparallel vortex tubes. Report-S88, 257–286, Stanford University: also, 1989
*Phys. Fluids*A1: 633–636Google Scholar - Melander M V, Hussain F 1990 Topological aspects of vortex reconnection. In
*Topological fluid mechanics*(eds) H K Moffatt, A Tsinober (Cambridge: University Press)Google Scholar - Melander M V, Hussain F 1993a Topological vortex dynamics in axisymmetriv viscous flows.
*J. Fluid Mech.*(in press)Google Scholar - Melander M V, Hussain F 1993b Polarized vorticity dynamics on a vortex column.
*Phys. Fluids*A5: 1992–2003MathSciNetGoogle Scholar - Melander M V, Hussain F, Basu A 1991 Breakdown of a circular jet into turbulence.
*Turbulent Shear Flows 8*, Munich, pp. 15.5.1–15.5.6Google Scholar - Melander M V, McWilliams J C, Zabusky N J 1987 Axisymmetrization and vorticity gradient intensification of an isolated two-dimensional vortex through filamentation.
*J. Fluid Mech.*178: 137–159MATHCrossRefGoogle Scholar - Melander M V, Zabusky N J, McWilliams J C 1988 Symmetric vortex merger in two dimensions: causes and conditions.
*J. Fluid Mech.*195: 303–340MATHCrossRefMathSciNetGoogle Scholar - Moffatt H K 1969 The degree of knottedness of tangled vortex lines.
*J. Fluid Mech.*35: 117–129MATHCrossRefGoogle Scholar - Moffatt H K, Tsinober A (eds) 1990
*Topological fluid mechanics*(Cambridge: University Press)MATHGoogle Scholar - Moses H E 1971 Eigenfunctions of the curl operator, rotationarlly invariant Helmholtz theorem, and applications to electromagnetic theory and fluid mechanics.
*SIAM J. Appl. Math.*21: 114–144CrossRefMathSciNetGoogle Scholar - Narasimha R 1989 The utility and drawbacks of traditional approaches. In
*Whither turbulence? Turbulence at the crossroads*(ed.) J Lumley (Berlin: Springer-Verlag)Google Scholar - Newcomb W A 1958 Motion of magnetic lines of force.
*Ann. Phys.*3: 347–385MATHCrossRefMathSciNetGoogle Scholar - Passot T, Pouquet A 1987 Numerical simulation of compressible homogeneous flows in turbulent regime.
*J. Fluid Mech.*181: 441–466MATHCrossRefGoogle Scholar - Schwarz K 1985 Three-dimensional vortex dynamics in superfluid
^{4}He: Line-line and line-boundary interactions.*Phys. Rev.*B31: 5782–5803Google Scholar - Shariff K, Leonard A, Zabusky N, Ferziger J 1988 Acoustics and dynamics of coaxial interacting vortex rings.
*Fluid Dyn. Res.*3: 337–343CrossRefGoogle Scholar - Siggia E D 1985 Collapse and amplification of a vortex filament.
*Phys. Fluids*28: 794–805MATHCrossRefGoogle Scholar - Stanaway S, Shariff K, Hussain F 1988 Head-on collision of viscous vortex rings.NASA reportctr S-88, 287Google Scholar
- Takaki R, Hussain F 1985 Recombination of vortex filaments and its role in aerodynamic noise.
*Turbulent Shear Flows V*, Cornell University, 3.19–3.25Google Scholar - Truesdell C 1954
*The kinematics of vorticity*, Indiana University Publications Science Series, No. 19Google Scholar - Zank G, Matthaeus W 1991 The equations of nearly incompressible fluids. I. Hydrodynamics, turbulence, and waves.
*Phys. Fluids*A3: 69–82MathSciNetGoogle Scholar