^{1,a)}and Rama Govindarajan

^{2,b)}

### Abstract

We study the effect of density stratification in the plane on the merging of two equal vortices. Direct numerical simulations are performed for a wide range of parameters. Boussinesq and non-Boussinesq effects are considered separately. With the Boussinesq approximation, moderate to high Prandtl number and Froude number close to unity, there is a monotonic drifting away of the vortices from each other, and merger is completely prevented. Among non-Boussinesq effects, the inertial effects of density stratification are highlighted. These give rise to a breaking of symmetry, and consequently, the vorticity centroid is found to drift significantly from its initial position. Using an idealized model, we explore the role of baroclinic vorticity in determining these features of the merger process.

A large part of this work was carried out when H.N.D. was a Ph.D. student at JNCASR, Bangalore. H.N.D. wishes to thank the Department of Science and Technology, Government of India for financial support during his Ph.D., and Professor Homsy (UBC) for providing infrastructure to complete this work.

I. INTRODUCTION

II. GOVERNING EQUATIONS AND NUMERICAL METHOD

III. EFFECT OF STRATIFICATION UNDER THE BOUSSINESQ APPROXIMATION

A. Effect of diffusivity and of low Froude number

IV. NON-BOUSSINESQ EQUATIONS

V. MERGER MECHANISM IN A STRATIFIED FLUID

VI. SUMMARY AND DISCUSSION

### Key Topics

- Rotating flows
- 129.0
- Vortex dynamics
- 47.0
- Viscosity
- 12.0
- Stratified flows
- 11.0
- Reynolds stress modeling
- 10.0

## Figures

Schematic of a two-dimensional initial vortex configuration, which will lead to merger. The arrows indicate the sense of vorticity, such that the two vortices initially describe an anti-clockwise revolution of period , where the subscript 0 stands for initial conditions, about the vorticity centroid “O.” The 2πΓ is the circulation of each vortex.

Schematic of a two-dimensional initial vortex configuration, which will lead to merger. The arrows indicate the sense of vorticity, such that the two vortices initially describe an anti-clockwise revolution of period , where the subscript 0 stands for initial conditions, about the vorticity centroid “O.” The 2πΓ is the circulation of each vortex.

Time evolution of vorticity (upper panel) and density (lower panel) contours for merger at higher levels of stratification than in Figure 4 with Re = 5000, Fr = 1, and Pr = 1. Significant generation of small scales can be observed. This figure is not to the same scale as the previous figure.

Time evolution of vorticity (upper panel) and density (lower panel) contours for merger at higher levels of stratification than in Figure 4 with Re = 5000, Fr = 1, and Pr = 1. Significant generation of small scales can be observed. This figure is not to the same scale as the previous figure.

Evolution of kinetic (K.E. - solid), potential (P.E. - dashed), and total (E tot - dashed-dotted lines) for Re = 5000 and Pr = 1 for four different Froude numbers indicated with different colors. The kinetic and potential energies are always in anti-phase with each other.

Evolution of kinetic (K.E. - solid), potential (P.E. - dashed), and total (E tot - dashed-dotted lines) for Re = 5000 and Pr = 1 for four different Froude numbers indicated with different colors. The kinetic and potential energies are always in anti-phase with each other.

Time evolution of vorticity (color) and density (gray scale) contours for merger in the presence of density stratification with Re = 5000, Fr = 2, and Pr = 10.

Time evolution of vorticity (color) and density (gray scale) contours for merger in the presence of density stratification with Re = 5000, Fr = 2, and Pr = 10.

Effect of Prandtl number, shown in the legend, on the separation distance for a fixed Re = 1000 and different Froude numbers, (a) Fr = 2 and (b) Fr = 1.

Effect of Prandtl number, shown in the legend, on the separation distance for a fixed Re = 1000 and different Froude numbers, (a) Fr = 2 and (b) Fr = 1.

Variation of separation distance with time for Re = 5000 with a mean non-dimensional density gradient of At = 0.0954. For the purely Boussinesq flow, gravity is chosen to give Fr = 3. In the purely inertial case, gravity is neglected and baroclinic torque is generated only due to the nonlinear terms in the governing equation.

Variation of separation distance with time for Re = 5000 with a mean non-dimensional density gradient of At = 0.0954. For the purely Boussinesq flow, gravity is chosen to give Fr = 3. In the purely inertial case, gravity is neglected and baroclinic torque is generated only due to the nonlinear terms in the governing equation.

Variation of separation distance with time for Re = 3000 by varying the Atwood number from 0.0106 to 0.0954 in nine equal steps of 0.0106. All the nine curves are clearly identical to each other.

Variation of separation distance with time for Re = 3000 by varying the Atwood number from 0.0106 to 0.0954 in nine equal steps of 0.0106. All the nine curves are clearly identical to each other.

Non-Boussinesq effects on the time evolution of vorticity contours for merger in the presence of density stratification with Re = 5000, At = 0.0954, and Pr = 1. The black line shows the drift of the vorticity centroid.

Non-Boussinesq effects on the time evolution of vorticity contours for merger in the presence of density stratification with Re = 5000, At = 0.0954, and Pr = 1. The black line shows the drift of the vorticity centroid.

Trajectory of a single vortex with Re = 10 000 (a) no stratification, (b) with Boussinesq approximation at Fr = 3, (c) with only inertial effects of stratification at At = 0.0954. For visual clarity, the curves are shifted vertically from each other.

Trajectory of a single vortex with Re = 10 000 (a) no stratification, (b) with Boussinesq approximation at Fr = 3, (c) with only inertial effects of stratification at At = 0.0954. For visual clarity, the curves are shifted vertically from each other.

Same as Figure 9 , but showing the x-coordinate of both the vortices as a function of time. (a) Unstratified fluid, (b) Boussinesq fluid, (c) non-Boussinesq fluid.

Same as Figure 9 , but showing the x-coordinate of both the vortices as a function of time. (a) Unstratified fluid, (b) Boussinesq fluid, (c) non-Boussinesq fluid.

Drift of the vortex centroid as a function of Atwood number for four different Reynolds numbers with Pr = 1.

Drift of the vortex centroid as a function of Atwood number for four different Reynolds numbers with Pr = 1.

Variation of drift velocity as a function of Atwood number for four different Reynolds numbers with Pr = 1.

Variation of drift velocity as a function of Atwood number for four different Reynolds numbers with Pr = 1.

Baroclinic torque produced due to centrifugal effects and gravity. Shown here is the locus at a given time of an initially horizontal interface separating light and heavy fluid. The solid arrow represents the normal to this line at one point. The effect of gravity (centrifugal acceleration) produces the same (opposite) sign of vorticity along diametrically opposite points as shown in the figure. The sign of the vorticity produced depends on the orientation of the normal vector to the interface with respect to gravity and to the radial vector. The first in the pair of signs shown indicates the sense of torque produced by gravity effects, while the second shows the sign of torque production due to centrifugal effects. The small dashed circle shows the locus of the two point vortices.

Baroclinic torque produced due to centrifugal effects and gravity. Shown here is the locus at a given time of an initially horizontal interface separating light and heavy fluid. The solid arrow represents the normal to this line at one point. The effect of gravity (centrifugal acceleration) produces the same (opposite) sign of vorticity along diametrically opposite points as shown in the figure. The sign of the vorticity produced depends on the orientation of the normal vector to the interface with respect to gravity and to the radial vector. The first in the pair of signs shown indicates the sense of torque produced by gravity effects, while the second shows the sign of torque production due to centrifugal effects. The small dashed circle shows the locus of the two point vortices.

A schematic of the dominant vorticity due to gravity alone, based on Figure 13 . The two primary vortices are shown in solid circles, and marked “P,” and the “clubbed” baroclinic vortices are marked “B” and shown by open circles (not to any scale). In (a), the primary vortices are at such a phase with respect to the baroclinic vortices causing them to move away from each other. In (b), the net effect on the primary vortices is to push them towards each other leading to accelerated merger.

A schematic of the dominant vorticity due to gravity alone, based on Figure 13 . The two primary vortices are shown in solid circles, and marked “P,” and the “clubbed” baroclinic vortices are marked “B” and shown by open circles (not to any scale). In (a), the primary vortices are at such a phase with respect to the baroclinic vortices causing them to move away from each other. In (b), the net effect on the primary vortices is to push them towards each other leading to accelerated merger.

Vorticity (lines) and density (gray scale) contours for Re = 5000, Pr = 1 at time t* = 0.5 for various Fr: (a) Fr = ∞, (b) Fr = 3, (c) Fr = 2, (d) Fr = 1. Solid and dashed lines represent positive and negative vorticity levels. Note that in (a), the density field is a passive scalar.

Vorticity (lines) and density (gray scale) contours for Re = 5000, Pr = 1 at time t* = 0.5 for various Fr: (a) Fr = ∞, (b) Fr = 3, (c) Fr = 2, (d) Fr = 1. Solid and dashed lines represent positive and negative vorticity levels. Note that in (a), the density field is a passive scalar.

Same as Figure 14 with “B” now representing baroclinic vorticity generated from inertial effects of stratification. The baroclinic vortices act like a dipole imparting a push on the primary vortices. In both (a) and (b), the net effect on the primary vortices is to push them leftward.

Same as Figure 14 with “B” now representing baroclinic vorticity generated from inertial effects of stratification. The baroclinic vortices act like a dipole imparting a push on the primary vortices. In both (a) and (b), the net effect on the primary vortices is to push them leftward.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content