^{1,a)}, Shufen Jiang

^{2}, Yong-Tao Zhang

^{3,b)}and Chi-Wang Shu

^{4,c)}

### Abstract

The interaction between a shock wave and two counter-rotating 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 the interaction between a shock wave and two counter-rotating vortices. It is found that there are two regimes of sound generation in this interaction. The first regime corresponds to the shock interaction with two isolated vortices, in which the sound wave generated by the interaction between the shock wave and two counter-rotating vortices equals to the linear combination of the sound waves generated by the interactions between the same shock wave and each vortex. The second regime corresponds to the shock interaction with a coupled vortex pair, in which the sound wave comes from two processes. One is the vortex coupling, and the second is the interaction between the shock wave and the coupled vortex pair.

The research by Shuhai Zhang is supported by Chinese National Natural Science Foundation Grant Nos. 10572146 and 10772193 and 973 Program No. 2009CB724104. The research by Yong-Tao Zhang is partially supported by NSF Grant No. DMS-0810413 and Oak Ridge Associated Universities (ORAU) Ralph E. Powe Junior Faculty Enhancement Award. The research by Chi-Wang Shu is partially supported by NSF Grant Nos. DMS-0510345 and DMS-0809086 and by ARO Grant No. W911NF-08-1-0520.

I. INTRODUCTION

II. THE PHYSICAL MODEL

III. SIMULATION RESULTS

A. Shock interaction with two isolated vortices

B. Shock interaction with coupled vortices

1. The free evolution of vortex coupling

2. Interaction between a shock and a vortex dipole

3. Interaction between a shock and a coupling vortex pair

IV. CONCLUDING REMARKS

### Key Topics

- Rotating flows
- 236.0
- Sound generation
- 79.0
- Shock waves
- 71.0
- Shock wave interactions
- 56.0
- Vortex interactions
- 37.0

## Figures

Schematic diagram of the flow model. (a) shock interaction with two counter-rotating vortices; (b) shock interaction with a single vortex.

Schematic diagram of the flow model. (a) shock interaction with two counter-rotating vortices; (b) shock interaction with a single vortex.

The comparison of the sound pressure generated by the shock interaction with two counter-rotating vortices with the linear combination of the sound waves generated by the two interactions between the same shock and each vortex. , , , , and . Left: contours of the sound pressure. Solid line: vortex pair; dashed line: linear combination of two isolated vortices. Right: circumferential distribution of sound pressure. Lines: vortex pair; symbols: linear combination of two isolated vortices. (a) , (b) , (c) .

The comparison of the sound pressure generated by the shock interaction with two counter-rotating vortices with the linear combination of the sound waves generated by the two interactions between the same shock and each vortex. , , , , and . Left: contours of the sound pressure. Solid line: vortex pair; dashed line: linear combination of two isolated vortices. Right: circumferential distribution of sound pressure. Lines: vortex pair; symbols: linear combination of two isolated vortices. (a) , (b) , (c) .

The sound pressure along the radial direction from the middle of the two vortex centers to the point of the minimum value of the precursor in the negative region at , , , , and .

The sound pressure along the radial direction from the middle of the two vortex centers to the point of the minimum value of the precursor in the negative region at , , , , and .

The comparison of the sound pressure generated by the shock interaction with two counter-rotating vortices with the linear combination of the sound waves generated by the two interactions between the same shock and each vortex. , , , , and . Left: contours of the sound pressure. Solid line: vortex pair; dashed line: linear combination of two isolated vortices. Right: circumferential distribution of sound pressure. Lines: vortex pair; symbols: linear combination of two isolated vortices. (a) , (b) , (c) .

The comparison of the circumferential distribution of the sound pressure generated by the shock interaction with two counter-rotating vortices with the linear combination of the sound waves generated by the two interactions between the same shock and each vortex. , , , , and . Lines: vortex pair; symbols: linear combination of two isolated vortices. (a) , (b) , (c) .

The comparison of the circumferential distribution of the sound pressure generated by the shock interaction with two counter-rotating vortices with the linear combination of the sound waves generated by the two interactions between the same shock and each vortex. , , , , and . Lines: vortex pair; symbols: linear combination of two isolated vortices. (a) , (b) , (c) .

The evolution of the vorticity field in the vortex coupling of . Solid lines represent the positive vorticity, and dashed lines represent the negative vorticity. (a) left: ; right: ; (b) left: ; right: ; (c) left: ; right: .

The evolution of the vorticity field in the vortex coupling of . Solid lines represent the positive vorticity, and dashed lines represent the negative vorticity. (a) left: ; right: ; (b) left: ; right: ; (c) left: ; right: .

The contours of the sound pressure in vortex coupling. Left: . Right: . Solid lines represent , while dashed lines represent . (a) , (b) , (c) .

The contours of the sound pressure in vortex coupling. Left: . Right: . Solid lines represent , while dashed lines represent . (a) , (b) , (c) .

The circumferential (left) and radial (right) distributions of the sound pressure in vortex coupling at . (a) , (b) , (c) .

The circumferential (left) and radial (right) distributions of the sound pressure in vortex coupling at . (a) , (b) , (c) .

The contours of the sound pressure by the interaction of a shock wave and a vortex dipole with . Solid lines represent , while dashed lines represent . (a) , (b) , (c) , (d) .

The contours of the sound pressure by the interaction of a shock wave and a vortex dipole with . Solid lines represent , while dashed lines represent . (a) , (b) , (c) , (d) .

The radial distribution of the sound pressure of shock vortex dipole interaction with .

The radial distribution of the sound pressure of shock vortex dipole interaction with .

The sound pressure of the shock interaction with two counter-rotating vortices. , , , and .

The sound pressure of the shock interaction with two counter-rotating vortices. , , , and .

The evolution of the sound pressure of the shock interaction with two counter-rotating vortices. , , , and . (a) , (b) , (c) , (d) .

The evolution of the sound pressure of the shock interaction with two counter-rotating vortices. , , , and . (a) , (b) , (c) , (d) .

The distribution of the sound pressure generated by the shock interaction with two counter-rotating vortices and a comparison with that of free vortex coupling along the line from the middle of two vortices to the point of the maximum value of the first sound wave in the positive plane. , , , and . Solid lines represent the sound pressure generated by the shock interaction with two counter-rotating vortices (SV). Dashed lines represent the sound wave generated by free vortex coupling (VC). (a) , (b) .

The distribution of the sound pressure generated by the shock interaction with two counter-rotating vortices and a comparison with that of free vortex coupling along the line from the middle of two vortices to the point of the maximum value of the first sound wave in the positive plane. , , , and . Solid lines represent the sound pressure generated by the shock interaction with two counter-rotating vortices (SV). Dashed lines represent the sound wave generated by free vortex coupling (VC). (a) , (b) .

The evolution of the sound pressure of the shock interaction with two counter-rotating vortices. , , , and . (a) , (b) , (c) , (d) .

The distribution of the sound pressure along the line from the middle of two vortices to the point of the maximum value of the first sound wave in the positive plane for the shock interaction with a vortex pair and a comparison with that of free vortex coupling. , , , and . Solid lines represent the sound pressure generated by shock vortex interaction (SV). Dashed lines represent the sound wave generated by free vortex coupling (VC). (a) , (b) .

The distribution of the sound pressure along the line from the middle of two vortices to the point of the maximum value of the first sound wave in the positive plane for the shock interaction with a vortex pair and a comparison with that of free vortex coupling. , , , and . Solid lines represent the sound pressure generated by shock vortex interaction (SV). Dashed lines represent the sound wave generated by free vortex coupling (VC). (a) , (b) .

The distribution of the sound pressure along the line from the middle of two vortices to the point of the maximum value of the first sound wave in the positive plane for the shock vortex pair interaction. , , , , and .

The distribution of the sound pressure along the line from the middle of two vortices to the point of the maximum value of the first sound wave in the positive plane for the shock vortex pair interaction. , , , , and .

## Tables

Parameters used in the simulations of shock interaction with two counter-rotating vortices and different regimes each simulation falls into.

Parameters used in the simulations of shock interaction with two counter-rotating vortices and different regimes each simulation falls into.

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