^{1,a)}, Hu Li

^{1,b)}, Xuliang Liu

^{1,c)}, Hanxin Zhang

^{2}and Chi-Wang Shu

^{3,d)}

### Abstract

Two-dimensional interaction between two Taylor vortices is simulated systematically through solving the two-dimensional, unsteady compressible Navier-Stokes equations using a fifth order weighted essentially nonoscillatory finite difference scheme. The main purpose of this study is to reveal the mechanism of sound generation in two-dimensional interaction of two Taylor vortices. Based on an extensive parameter study on the evolution of the vorticity field, we classify the interaction of two Taylor vortices into four types. The first type is the interaction of two counter-rotating vortices with similar strengths. The second type is the interaction of two co-rotating vortices without merging. The third type is the merging of two co-rotating vortices. The fourth type is the interaction of two vortices with a large difference in their strengths or scales. The mechanism of sound generation is analyzed.

Research of the first author is supported by the Chinese National Natural Science Foundation (Grant Nos. 11172317, 91016001, and 973 program 2009CB724104). Research of the fourth author is supported by the Chinese National Natural Science Foundation (Grant No. 91016001). Research of the fifth author is partially supported by ARO Grant No. W911NF-11-1-0091 and NSF Grant No. DMS-1112700.

I. INTRODUCTION

II. THE PHYSICAL MODEL FOR THE INTERACTION OF TWO TAYLOR VORTICES

A. The numerical method

B. The physical model for the interaction of two Taylor vortices

III. CLASSIFICATION AND SOUND GENERATION OF THE TWO-DIMENSIONAL INTERACTION OF TWO TAYLOR VORTICES

A. Classification of the two-dimensional interaction of two Taylor vortices

1. Type I: The interaction of two counter-rotating vortices with similar strengths

2. Type II: The interaction of two co-rotating vortices without merging

3. Type III: The merging of two co-rotating vortices

4. Type IV: The interaction of two vortices with a large difference in their strengths or scales

B. Sound generation in the two-dimensional interaction of two Taylor vortices

1. The mechanism of sound generation by the first type of interaction

2. The mechanism of sound generation by the other types of interaction

IV. REVISIT THE INTERACTION OF TWO CO-ROTATING GAUSSIAN VORTICES

A. Revisit the interaction of two co-rotating Gaussian vortices

B. Discussion on the differences between Gaussian vortex and Taylor vortex and their effect on the sound generation

V. DISCUSSION ON THE INTERACTION OF TWO VORTICES IN A COMPLEX FLOW

VI. CONCLUDING REMARKS

### Key Topics

- Rotating flows
- 336.0
- Sound generation
- 61.0
- Vortex dynamics
- 35.0
- Vortex interactions
- 29.0
- Turbulent flows
- 18.0

## Figures

Schematic diagram of the flow model for the interaction of two Taylor vortices.

Schematic diagram of the flow model for the interaction of two Taylor vortices.

The evolution of the vorticity field in the interaction of two counter-rotating vortices in the case of M υu = −0.5, M υd = 0.5, d = 4 and R cu = R cd = 1.

The evolution of the vorticity field in the interaction of two counter-rotating vortices in the case of M υu = −0.5, M υd = 0.5, d = 4 and R cu = R cd = 1.

The evolution of the vorticity field in the interaction of two counter-rotating vortices in the case of M υu = −0.5, M υd = 0.45, d = 4, and R cu = R cd = 1.

The evolution of the vorticity field in the interaction of two counter-rotating vortices in the case of M υu = −0.5, M υd = 0.45, d = 4, and R cu = R cd = 1.

The evolution of the vorticity field in the interaction of two co-rotating vortices in the case of M υu = M υd = 0.5, d = 4 and R cu = R cd = 1.

The evolution of the vorticity field in the interaction of two co-rotating vortices in the case of M υu = M υd = 0.5, d = 4 and R cu = R cd = 1.

The evolution of the vorticity field in the merging process of two co-rotating Taylor vortices in the case of M υu = M υd = 0.5, d = 2, and R cu = R cd = 1.

The evolution of the vorticity field in the merging process of two co-rotating Taylor vortices in the case of M υu = M υd = 0.5, d = 2, and R cu = R cd = 1.

The evolution of the vorticity field in the interaction of two counter-rotating vortices with a large difference in their strengths in the case of M υu = −0.8, M υd = 0.25, d = 4, and R cu = R cd = 1.

The evolution of the vorticity field in the interaction of two counter-rotating vortices with a large difference in their strengths in the case of M υu = −0.8, M υd = 0.25, d = 4, and R cu = R cd = 1.

The evolution of the vorticity field in the interaction of two co-rotating vortices with a large difference in their strengths in the case of M υu = 0.8, M υd = 0.25 and d = 4, and R cu = R cd = 1.

The evolution of the vorticity field in the interaction of two co-rotating vortices with a large difference in their strengths in the case of M υu = 0.8, M υd = 0.25 and d = 4, and R cu = R cd = 1.

The evolution of the vorticity field in the interaction of two counter-rotating vortices with a large difference in their spatial scales in the case of M υu = −0.5, M υd = 0.5, d = 2.2, R cu = 1, and R cd = 0.2.

The evolution of the vorticity field in the interaction of two counter-rotating vortices with a large difference in their spatial scales in the case of M υu = −0.5, M υd = 0.5, d = 2.2, R cu = 1, and R cd = 0.2.

The instantaneous contours (left) and radial (middle) distributions (right) of the sound pressure at the typical time t = 200 in the two-dimensional interaction of two Taylor vortices. Solid lines in the contours represent Δp > 0 while dashed lines represent Δp < 0.

The instantaneous contours (left) and radial (middle) distributions (right) of the sound pressure at the typical time t = 200 in the two-dimensional interaction of two Taylor vortices. Solid lines in the contours represent Δp > 0 while dashed lines represent Δp < 0.

The time history of the sound pressure at the monitored point (100,0) in the two dimensional interaction of two Taylor vortices. (a) The interaction of two counter-rotating vortices in the case of M υu = −0.5, M υd = 0.5, d = 4, and R cu = R cd = 1. (b) The interaction of two co-rotating vortices in the case of M υu = M υd = 0.5, d = 4, and R cu = R cd = 1. (c) The merging of two co-rotating Taylor vortices in the case of M υu = M υd = 0.5, d = 2, and R cu = R cd = 1. (d) The interaction of two counter-rotating vortices with a large difference in their strengths in the case of M υu = −0.8, M υd = 0.25, d = 4, and R cu = R cd = 1. (e) The interaction of two co-rotating vortices with a large difference in their strengths at the typical time t = 200 in the case of M υu = 0.8, M υd = 0.25, d = 4, and R cu = R cd = 1. (f) the interaction of two counter-rotating vortices in the case of M υu = −0.5, M υd = 0.5, d = 2.2, R cu = 1, and R cd = 0.2.

The time history of the sound pressure at the monitored point (100,0) in the two dimensional interaction of two Taylor vortices. (a) The interaction of two counter-rotating vortices in the case of M υu = −0.5, M υd = 0.5, d = 4, and R cu = R cd = 1. (b) The interaction of two co-rotating vortices in the case of M υu = M υd = 0.5, d = 4, and R cu = R cd = 1. (c) The merging of two co-rotating Taylor vortices in the case of M υu = M υd = 0.5, d = 2, and R cu = R cd = 1. (d) The interaction of two counter-rotating vortices with a large difference in their strengths in the case of M υu = −0.8, M υd = 0.25, d = 4, and R cu = R cd = 1. (e) The interaction of two co-rotating vortices with a large difference in their strengths at the typical time t = 200 in the case of M υu = 0.8, M υd = 0.25, d = 4, and R cu = R cd = 1. (f) the interaction of two counter-rotating vortices in the case of M υu = −0.5, M υd = 0.5, d = 2.2, R cu = 1, and R cd = 0.2.

The time history of the sound pressure at the two monitored points (solid for (100,0) and dashed for (0,100)) in the interaction of two counter-rotating vortices in the case of M υu = −0.5, M υd = 0.5, d = 4, and R cu = R cd = 1.

The time history of the sound pressure at the two monitored points (solid for (100,0) and dashed for (0,100)) in the interaction of two counter-rotating vortices in the case of M υu = −0.5, M υd = 0.5, d = 4, and R cu = R cd = 1.

The evolution of the Lamb vector in the interaction of two counter-rotating vortices in the case of M υu = −0.5, M υd = 0.5, d = 4, and R cu = R cd = 1.

The evolution of the Lamb vector in the interaction of two counter-rotating vortices in the case of M υu = −0.5, M υd = 0.5, d = 4, and R cu = R cd = 1.

The time history of the sound pressure at the two monitored points (solid for (100,0) and dashed for (0,100)) in the interaction of two co-rotating vortices in the case of M υu = 0.8, M υd = 0.25, d = 2.2, R cu = 1, and R cd = 0.2.

The time history of the sound pressure at the two monitored points (solid for (100,0) and dashed for (0,100)) in the interaction of two co-rotating vortices in the case of M υu = 0.8, M υd = 0.25, d = 2.2, R cu = 1, and R cd = 0.2.

The evolution of the Lamb vector in the merging process of two co-rotating vortices in the case of M υu = M υd = 0.5, d = 2, and R cu = R cd = 1.

The evolution of the Lamb vector in the merging process of two co-rotating vortices in the case of M υu = M υd = 0.5, d = 2, and R cu = R cd = 1.

The evolution of the vorticity field in the interaction of two co-rotating Gaussian vortices.

The evolution of the vorticity field in the interaction of two co-rotating Gaussian vortices.

The time history of dilation at the point (x, y) = (0., 1.2) and its comparison with that of Eldredge et al. 43

The time history of dilation at the point (x, y) = (0., 1.2) and its comparison with that of Eldredge et al. 43

The time history of separation distance of two co-rotating Gaussian vortices.

The time history of separation distance of two co-rotating Gaussian vortices.

The time evolution of second-order moments of vorticity and defined by Eq. (4) .

Far-field pressure traces at (left) and (right) and the comparison between our direct numerical simulation (DNS) and that by the Möhring's equation (5) .

Far-field pressure traces at (left) and (right) and the comparison between our direct numerical simulation (DNS) and that by the Möhring's equation (5) .

The comparison for the distribution of tangential velocity along radius between Taylor vortex and Gaussian vortex.

The comparison for the distribution of tangential velocity along radius between Taylor vortex and Gaussian vortex.

The comparison for the distribution of vorticity along the radius between Taylor vortex and Gaussian vortex.

The comparison for the distribution of vorticity along the radius between Taylor vortex and Gaussian vortex.

The time evolution of vorticity in two-dimensional decaying turbulence.

The time evolution of vorticity in two-dimensional decaying turbulence.

The time evolution of vorticity for vortex dipole in two-dimensional decaying turbulence.

The time evolution of vorticity for vortex dipole in two-dimensional decaying turbulence.

The time evolution of vorticity for vortex merging in two-dimensional decaying turbulence.

The time evolution of vorticity for vortex merging in two-dimensional decaying turbulence.

The time evolution of vorticity for the interaction of two vortices with large difference in their strengths in two-dimensional decaying turbulence.

The time evolution of vorticity for the interaction of two vortices with large difference in their strengths in two-dimensional decaying turbulence.

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