Dynamics of flame/vortex interactions

A direct spatial image correlation technique is presented for estimating the Lagrangian displacement vector from image pairs based on molecular tagging diagnostics. The procedure provides significant improvement in measurement accuracy compared to existing approaches for molecular tagging velocimetry (MTV) analysis. Furthermore, this technique is of more general utility in that it is able to accommodate other laser tagging patterns besides the usual grid arrangement. Simulations are performed to determine the effects of many experimental and processing issues on the sub-pixel accuracy of displacement estimates. The results provide guidelines for optimizing the implementation of MTV. Experimental data in support of this processing technique are provided.

A spatial correlation technique for estimating velocity fields using molecular tagging velocimetry (MTV)

The development and applications of a new class of water-soluble compounds suitable for molecular tagging diagnostics are described. These molecular complexes are formed by mixing a lumophore, an appropriate alcohol, and cyclodextrin. Using 1-BrNp as the lumophore, cyclohexanol is determined to be the most effective overall among the alcohols for which data are currently available. Information is provided for the design of experiments based on these complexes along with a less complex method for generating the grid patterns typically used for velocimetry. Implementation of a two-detector system is described which, in combination with a spatial correlation technique for determining velocities, relaxes the requirement that the initial tagging pattern be known a priori, eliminates errors in velocity estimates caused by variations in the grid pattern during an experiment, and makes it possible to study flows with non-uniform mixtures. This detection and analysis combination also solves one of the problems associated with using caged fluorescein to study high-speed flows. In addition to the traditional implementation for velocimetry, novel applications for studying the Lagrangian evolution of both reacting and non-reacting interfaces and obtaining combined passive scalar/velocity measurements are demonstrated.

https://www.egr.msu.edu/tmual/Papers_PDF/1997_MTV_Supramolecules-EIF1997.pdf

Molecular tagging velocimetry and other novel applications of a new phosphorescent supramolecule

A large-Reynolds-number asymptotic solution of the Navier–Stokes equations is sought for the motion of an axisymmetric vortex ring of small cross-section embedded in a viscous incompressible fluid. In order to take account of the influence of elliptical deformation of the core due to the self-induced strain, the method of matched of matched asymptotic expansions is extended to a higher order in a small parameter e = (v/Γ)1/2, where v is the kinematic viscosity of fluid and Γ is the circulation. Alternatively, e is regarded as a measure of the ratio of the core radius to the ring radius, and our scheme is applicable also to the steady inviscid dynamics.We establish a general formula for the translation speed of the ring valid up to third order in e. This is a natural extension of Fraenkel–Saffman's first-order formula, and reduces, if specialized to a particular distribution of vorticity in an inviscid fluid, to Dyson's third-order formula. Moreover, it is demonstrated, for a ring starting from an infinitely thin circular loop of radius R0, that viscosity acts, at third order, to expand the circles of stagnation points of radii Rs(t) and R˜s(t) relative to the laboratory frame and a comoving frame respectively, and that of peak vorticity of radius R˜p(t) as Rs ≈ R0 + [2 log(4R0/√vt) + 1.4743424] vt/R0, R˜s ≈ R0 + 2.5902739 vt/R0, and Rp ≈ R0 + 4.5902739 vt/R0. The growth of the radial centroid of vorticity, linear in time, is also deduced. The results are compatible with the experimental results of Sallet & Widmayer (1974) and Weigand & Gharib (1997).The procedure of pursuing the higher-order asymptotics provides a clear picture of the dynamics of a curved vortex tube; a vortex ring may be locally regarded as a line of dipoles along the core centreline, with their axes in the propagating direction, subjected to the self-induced flow field. The strength of the dipole depends not only on the curvature but also on the location of the core centre, and therefore should be specified at the initial instant. This specification removes an indeterminacy of the first-order theory. We derive a new asymptotic development of the Biot-Savart law for an arbitrary distribution of vorticity, which makes the non-local induction velocity from the dipoles calculable at third order.

Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity

We numerically study a vortex ring impacting a flat wall with an angle of incidence θ ≥ 0°) in three dimensions by using the lattice Boltzmann equation. The hydrodynamic behaviour of the ring–wall interacting flow is investigated by systematically varying the angle of incidence θ in the range of 0° ≤ θ ≤ 40° and the Reynolds number in the range of 100 ≤ Re ≤ 1000, where the Reynolds number Re is based on the translational speed and initial diameter of the vortex ring. We quantify the effects of θ and Re on the evolution of the vortex structure in three dimensions and other flow fields in two dimensions. We observe three distinctive flow regions in the θ–Re parameter space. First, in the low-Reynolds-number region, the ring–wall interaction dissipates the ring without generating any secondary rings. Second, with a moderate Reynolds number Re and a small angle of incidence θ, the ring–wall interaction generates a complete secondary vortex ring, and even a tertiary ring at higher Reynolds numbers. The secondary vortex ring is convected to the centre region of the primary ring and develops azimuthal instabilities, which eventually lead to the development of hairpin-like small vortices through ring–ring interaction. And finally, with a moderate Reynolds number and a sufficiently large angle of incidence θ, only a secondary vortex ring is generated. The secondary vortex wraps around the primary ring and propagates from the near end of the primary ring, which touches the wall first, to the far end, which touches the wall last. The rings develop a helical structure. Our results from the present study confirm some existing experimental observations made in the previous studies.

Numerical study of a vortex ring impacting a flat wall

Laminar vortex rings approaching a solid planar surface and a free surface have been investigated experimentally over the range of Reynolds numbers based on the circulation, ReΓ, from 900 to 2350. The emphasis is on the process of vortex stretching, induction of boundary layer on the surface, formation of secondary vortices, and rebounding of the primary vortex ring. Detailed measurements were made using laser‐induced photochemical anemometry (LIPA), a nonintrusive visualization technique which enables multipoint simultaneous measurement of the unsteady velocity field. Results show that enstrophy, rather than circulation, clearly signifies three stages of the behavior of the primary vortex ring—free‐traveling stage, vortex stretching stage, and vortex rebounding stage. Although the flow phenomena are very similar when a vortex ring approaches a solid surface or a slightly contaminated free surface, the difference between the induced boundary layers under two surface conditions is obvious in terms of the circulation and the enstrophy. In addition, the surface deformation and subsequent surface waves resulting from the impacting vortex ring on the contaminated free surface are also discussed. The present study provides a basis for validation and comparison with numerical simulations.

An experimental investigation of vortex motions near surfaces

The behaviour of a viscous vortex ring is examined by a matched asymptotic analysis up to three orders. This study aims at investigating how much the location of maximum vorticity deviates from the centroid of the vortex ring, defined by P. G. Saffman (1970). All the results are presented in dimensionless form, as indicated in the following context. Let Γ be the initial circulation of the vortex ring, and R denote the ring radius normalized by its initial radius R i . For the asymptotic analysis, a small parameter ∊ = ( t / Re ) ½ is introduced, where t denotes time normalized by R 2 i / Γ , and Re = Γ/v is the Reynolds number defined with Γ and the kinematic viscosity v . Our analysis shows that the trajectory of maximum vorticity moves with the velocity (normalized by Γ/R i ) U m = – 1/4π R {ln 4 R /∊ + H m } + O (∊ ln ∊), where H m = H m ( Re, t ) depends on the Reynolds number Re and may change slightly with time t for the initial motion. For the centroid of the vortex ring, we obtain the velocity U c by merely replacing H m by H c , which is a constant –0.558 for all values of the Reynolds number. Only in the limit of Re → ∞, the values of H m and H c are found to coincide with each other, while the deviation of H m from the constant H c is getting significant with decreasing the Reynolds number. Also of interest is that the radial motion is shown to exist for the trajectory of maximum vorticity at finite Reynolds numbers. Furthermore, the present analysis clarifies the earlier discrepancy between Saffman’s result and that obtained by C. Tung and L. Ting (1967).

Chin-Chou Chu

Papers

An experimental investigation of vortex motions near surfaces

Initial motion of a viscous vortex ring