8 results
On asymmetric vortex pair interactions in shear
- Patrick J.R. Folz, Keiko K. Nomura
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- Journal:
- Journal of Fluid Mechanics / Volume 969 / 25 August 2023
- Published online by Cambridge University Press:
- 16 August 2023, A21
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This study examines the two-dimensional interaction of two unequal co-rotating viscous vortices in uniform background shear. Numerical simulations are performed for vortex pairs having various circulation ratios $\varLambda _0 = \varGamma _{1,0}/\varGamma _{2,0} = (\omega _{1,0}/\omega _{2,0})(a^2_{1,0}/a^2_{2,0}) \leqslant 1$, corresponding to different initial characteristic radii $a_{i,0}$ and peak vorticities $\omega _{i,0}$ of each vortex $i=1,2$, in shears of various strengths $\zeta _0 = \omega _S/\omega _{2,0}$, where $\omega _S$ is the constant vorticity of the shear. Two primary flow regimes are observed: separations ($\zeta _0 < \zeta _{sep} < 0$), in which the vortices move apart continuously, and henditions ($\zeta _0 > \zeta _{sep}$), in which the interaction results in a single vortex (where $\zeta _{sep}$ is the adverse shear strength beyond which separation occurs). Vortex motion and values of $\zeta _{sep}(\varLambda _0)$ are well-predicted by a point-vortex model for unequal vortices. In vortex-dominated henditions, shear varies the peak–peak distance $b$, and vortex deformation. The main convective interaction begins when core detrainment of one vortex is established, and proceeds similarly to the no-shear ($\zeta _0 = 0$) case: merger occurs if the second vortex also detrains, engendering mutual entrainment; otherwise straining out occurs. Detrainment requires persistence of straining of both sufficient magnitude, as indicated by relative straining above a consistent critical value, $(S/\omega )_i > (S/\omega )_{cr}$, where $S$ is the strain rate magnitude at the vorticity peak, and conducive direction. Hendition outcomes are assessed in terms of an enhancement factor $\varepsilon \equiv \varGamma _{end}/\varGamma _{2,start}$. Although $\varepsilon$ generally varies with $\zeta _0$, $(a^2_{1,0} /a^2_{2,0} )$ and $(\omega _{1,0}/\omega _{2,0})$ in a complicated manner, this variation is well-characterized by the pair's starting enstrophy ratio, $Z_2/Z_1$. Within a transition region between merger and straining out (approximately $1.65 < Z_2/Z_1 < 1.9$), shear of either sense may increase $\varepsilon$.
Vortex breakdown in variable-density gaseous swirling jets
- Benjamin W. Keeton, Jaime Carpio, Keiko K. Nomura, Antonio L. Sánchez, Forman A. Williams
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- Journal:
- Journal of Fluid Mechanics / Volume 936 / 10 April 2022
- Published online by Cambridge University Press:
- 07 February 2022, A1
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Theoretical predictions and numerical simulations are used to determine the transition to bubble and conical vortex breakdown in low-Mach-number laminar axisymmetric variable-density swirling jets. A critical value of the swirl number $S$ for the onset of the bubble ($S^*_B$) and the cone ($S^*_C$) is determined as the jet-to-ambient density ratio $\varLambda$ is varied, with the temperature dependence of the gas density and viscosity appropriate to that of air. The criterion of failure of the slender quasi-cylindrical approximation predicts $S^*_B$ that decreases with increasing values of $\varLambda$ for a jet in solid-body rotation emerging sharply into a quiescent atmosphere. In addition, a new criterion for the onset of conical breakdown is derived from divergence of the initial value of the radial spreading rate of the jet occurring at $S^*_C$, found to be independent of $\varLambda$, in an asymptotic analysis for small distances from the inlet plane. To maintain stable flow in the unsteady numerical simulations, an effective Reynolds number $Re_{eff}$, defined employing the geometric mean of the viscosity in the jet and ambient atmosphere, is fixed at $Re_{eff}=200$ for all $\varLambda$. Similar to the theoretical predictions, numerical calculations of $S^*_B$ decrease monotonically as $\varLambda$ is increased. The critical swirl numbers for the cone, $S^*_C$, are found to depend strongly on viscous effects; for $\varLambda =1/5$, the low jet Reynolds number (51) at $Re_{eff}=200$ delays the transition to the cone, while for $\varLambda =5$ at $Re_{eff}=200$, the large increase in kinematic viscosity in the external fluid produces a similar trend, significantly increasing $S^*_C$.
A quantitative assessment of viscous asymmetric vortex pair interactions
- Patrick J. R. Folz, Keiko K. Nomura
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- Journal of Fluid Mechanics / Volume 829 / 25 October 2017
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- 14 September 2017, pp. 1-30
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The interactions of two like-signed vortices in viscous fluid are investigated using two-dimensional numerical simulations performed across a range of vortex strength ratios, $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D6E4}_{1}/\unicode[STIX]{x1D6E4}_{2}\leqslant 1$, corresponding to vortices of circulation, $\unicode[STIX]{x1D6E4}_{i}$, with differing initial size and/or peak vorticity. In all cases, the vortices evolve by viscous diffusion before undergoing a primary convective interaction, which ultimately results in a single vortex. The post-interaction vortex is quantitatively evaluated in terms of an enhancement factor, $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D6E4}_{end}/\unicode[STIX]{x1D6E4}_{2,start}$, which compares its circulation, $\unicode[STIX]{x1D6E4}_{end}$, to that of the stronger starting vortex, $\unicode[STIX]{x1D6E4}_{2,start}$. Results are effectively characterized by a mutuality parameter, $MP\equiv (S/\unicode[STIX]{x1D714})_{1}/(S/\unicode[STIX]{x1D714})_{2}$, where the ratio of induced strain rate, $S$, to peak vorticity, $\unicode[STIX]{x1D714}$, for each vortex, $(S/\unicode[STIX]{x1D714})_{i}$, is found to have a critical value, $(S/\unicode[STIX]{x1D714})_{cr}\approx 0.135$, above which core detrainment occurs. If $MP$ is sufficiently close to unity, both vortices detrain and a two-way mutual entrainment process leads to $\unicode[STIX]{x1D700}>1$, i.e. merger. In asymmetric interactions and mergers, generally one vortex dominates; the weak/no/strong vortex winner regimes correspond to $MP<,=,>1$, respectively. As $MP$ deviates from unity, $\unicode[STIX]{x1D700}$ decreases until a critical value, $MP_{cr}$ is reached, beyond which there is only a one-way interaction; one vortex detrains and is destroyed by the other, which dominates and survives. There is no entrainment and $\unicode[STIX]{x1D700}\sim 1$, i.e. only a straining out occurs. Although $(S/\unicode[STIX]{x1D714})_{cr}$ appears to be independent of Reynolds number, $MP_{cr}$ shows a dependence. Comparisons are made with available experimental data from Meunier (2001, PhD thesis, Université de Provence-Aix-Marseille I).
Characterization of the interactions of two unequal co-rotating vortices
- LAURA K. BRANDT, KEIKO K. NOMURA
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- Journal of Fluid Mechanics / Volume 646 / 10 March 2010
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- 08 March 2010, pp. 233-253
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The interactions and merging of two unequal co-rotating vortices in a viscous fluid are investigated. Two-dimensional numerical simulations of initially equal-sized vortices with differing relative strengths are performed. In the case of equal-strength vortices, i.e. symmetric vortex pairs (Brandt & Nomura, J. Fluid Mech., vol. 592, 2007, pp. 413–446), the mutually induced strain deforms and tilts the vortices, which leads to a core detrainment process. The weakened vortices are mutually entrained and rapidly move towards each other as they intertwine and destruct. The flow thereby develops into a single compound vortex. With unequal strengths, i.e. asymmetric pairs, the disparity of the vortices alters the interaction. Merger may result from reciprocal but unequal entrainment, which yields a compound vortex; however other outcomes are possible. The various interactions are classified based on the relative timing of core detrainment and core destruction of the vortices. Through scaling analysis and simulation results, a critical strain rate parameter which characterizes the establishment of core detrainment is identified and determined. The onset of merging is associated with the achievement of the critical strain rate by ‘both’ vortices, and a merging criterion is thereby developed. In the case of symmetric pairs, the critical strain rate parameter is shown to be related to the critical aspect ratio. In contrast with symmetric merger, which is in essence a flow transformation, asymmetric merger may result in the domination of the stronger vortex because of the unequal deformation rates. If the disparity of the vortex strengths is sufficiently large, the critical strain rate is not attained by the stronger vortex before destruction of the weaker vortex, and the vortices do not merge.
The physics of vortex merger and the effects of ambient stable stratification
- LAURA K. BRANDT, KEIKO K. NOMURA
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- Journal:
- Journal of Fluid Mechanics / Volume 592 / 10 December 2007
- Published online by Cambridge University Press:
- 14 November 2007, pp. 413-446
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The merging of a pair of symmetric, horizontally oriented vortices in unstratified and stably stratified viscous fluid is investigated. Two-dimensional numerical simulations are performed for a range of flow conditions. The merging process is resolved into four phases of development and key underlying physics are identified. In particular, the deformation of the vortices, explained in terms of the interaction of vorticity gradient, ∇ω, and rate of strain, S, leads to a tilt in ω contours in the vicinity of the center of rotation (a hyperbolic point). In the diffusive/deformation phase, diffusion of the vortices establishes the interaction between ∇ω and mutually induced S. During the convective/deformation phase, induced flow by filaments and, in stratified flow, baroclinically generated vorticity (BV), advects the vortices thereby modifying S, which, in general, may enhance or hinder the development of the tilt. The tilting and diffusion of ω near the center hyperbolic point causes ω from the core region to enter the exchange band where it is entrained. In the convective/entrainment phase, the vortex cores are thereby eroded and ultimately entrained into the exchange band, whose induced flow becomes dominant and transforms the flow into a single vortex. The critical aspect ratio, associated with the start of the convective/entrainment phase, is found to be the same for both the unstratified and stratified flows. In the final diffusive/axisymmetrization phase, the flow evolves towards axisymmetry by diffusion. In general, the effects of stratification depend on the ratio of the diffusive time scale (growth of cores) to the turnover time (establishment of BV), i.e. the Reynolds number. A crossover Reynolds number is found, above which convective merging is accelerated with respect to unstratified flow and below which it is delayed.
Short-wavelength instability and decay of a vortex pair in a stratified fluid
- KEIKO K. NOMURA, HIDEAKI TSUTSUI, DANIEL MAHONEY, JAMES W. ROTTMAN
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- Journal:
- Journal of Fluid Mechanics / Volume 553 / 25 April 2006
- Published online by Cambridge University Press:
- 06 April 2006, pp. 283-322
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The evolution of a counter-rotating vortex pair in a stably stratified fluid is investigated using direct numerical simulations. The study focuses on the short-wavelength elliptic instability occurring in this flow and the subsequent decay of the vortices. Depending on the level of stratification, as characterized by the Froude number which indicates the time scale of buoyancy to that of the instability, and the stage of evolution, stratification effects may significantly alter the behaviour of the flow. In the case of weak to moderate stratification, the elliptic instability develops qualitatively in the same manner as in unstratified fluid. The primary effect of stratification is to reduce the vortex separation distance which enhances the mutually induced strain. Consequently, the instability has an earlier onset and higher growth rate with increasing stratification. The behaviour is essentially described by linear stability theory for unstratified flow if the varying separation distance is taken into account. On the other hand, the final breakdown and decay of the flow may be greatly modified by stratification since buoyancy effects eventually emerge after sufficient time has elapsed. The decay is enhanced owing to additional mechanisms not present in unstratified flow. Secondary vertical vortex structures form between the primary vortices promoting fluid exchange in the transverse direction. Detrainment of fluid from the primary vortices by the generated baroclinic torque also contributes to the more rapid breakdown of the flow. In the case of strong stratification, in which the time scale of buoyancy is comparable to that of the instability, the flow is significantly altered. As a result of strong baroclinic torque, the primary vortices are brought together and detrainment occurs earlier. The associated reduction in radii of the vortices results in a higher axial wave mode and a more complex radial structure of the instability. Detrainment and mixing accelerate their decay. Late time evolution is dominated by the successive generation of alternate signed baroclinic torque which results in an oscillation of the total flow circulation at the buoyancy frequency.
The structure and dynamics of overturns in stably stratified homogeneous turbulence
- PETER J. DIAMESSIS, KEIKO K. NOMURA
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- Journal of Fluid Mechanics / Volume 499 / 25 January 2004
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- 27 January 2004, pp. 197-229
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Direct numerical simulations of stably stratified homogeneous turbulence, with and without mean shear, are used to investigate the three-dimensional structure, evolution and energetic significance of density overturns. Although the flow conditions are idealized, examination of the full-field simulation data provides insight into flow energetics and mixing which may assist in the interpretation of physical measurements, typically limited to one-dimensional vertical profiles. Overturns, defined here through the density field as contiguous regions of non-zero Thorpe displacement, are initially generated by the stirring action of coherent vortex structures present in the flow and further develop through merging with adjacent overturns. During this growth phase, overturns exhibit irregular spatial structure in unsheared flow and elongated structure with distinct orientation in shear flow. Although most of the available potential energy (APE) and buoyancy flux are associated with stable (non-overturning) regions in the flow, young overturns actively contribute to the flow energetics. In particular, overturn peripheries are sites of high levels of APE, buoyancy flux and diapycnal mixing. A collapse phase may follow the growth phase in the absence of adequately strong mean shear. During this phase, buoyancy gradually assumes control of the overturns and their vertical scale steadily decreases. The energetic significance of the overturns diminishes, although high APE and diapycnal mixing continue to occur near their boundaries. In the final phase of their evolution, overturns contribute negligibly to the energetics. The remaining overturns are characterized by a viscous–buoyant balance which maintains their vertical scale. The overturns eventually vanish due to homogenization of their internal density distribution by diffusion. Activity diagrams, sampled at different points of flow evolution, show significant variation in overturn Reynolds and Froude numbers which may have implications for vertical sampling of a turbulent event.
The structure and dynamics of vorticity and rate of strain in incompressible homogeneous turbulence
- KEIKO K. NOMURA, GARY K. POST
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- Journal of Fluid Mechanics / Volume 377 / 25 December 1998
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- 25 December 1998, pp. 65-97
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The structure and dynamics of vorticity ω and rate of strain S are studied using direct numerical simulations (DNS) of incompressible homogeneous isotropic turbulence. In particular, characteristics of the pressure Hessian Π, which describe non-local interaction of ω and S, are presented. Conditional Lagrangian statistics which distinguish high-amplitude events in both space and time are used to investigate the physical processes associated with their evolution. The dynamics are examined on the principal strain basis which distinguishes vortex stretching and induced rotation of the principal axes of S. The latter mechanism is associated with misaligned ω with respect to S, a condition which predominates in isotropic turbulence and is dynamically significant, particularly in rotation-dominated regions of the flow. Locally-induced rotation of the principal axes acts to orient ω towards the direction of either the intermediate or most compressive principal strain. The tendency towards compressive straining of ω is manifested at the termini of the high-amplitude tube-like structures in the flow. Non-locally-induced rotation, associated with Π, tends to counteract the locally-induced rotation. This is due to the strong alignment between ω and the eigenvector of Π corresponding to its smallest eigenvalue and is indicative of the controlling influence of the proximate structure on the dynamics. High-amplitude rotation-dominated regions deviate from Burgers vortices due to the misalignment of ω. Although high-amplitude strain-dominated regions are promoted primarily by local dynamics, the associated spatial structure is less organized and more discontinuous than that of rotation-dominated regions.