^{1,a)}

### Abstract

We study Dhanak's model [J. Fluid Mech.269, 265 (Year: 1994)]10.1017/S0022112094001552 of a viscous vortex sheet in the sharp limit, to investigate singularity formations and present nonlinear evolutions of the sheets. The finite-time singularity does not disappear by giving viscosity to the vortex sheet, but is delayed. The singularity in the sharp viscous vortex sheet is found to be different from that of the inviscid sheet in several features. A discontinuity in the curvature is formed in the viscous sheet, similarly as the inviscid sheet, but a cusp in the vortex sheet strength is less sharpened by viscosity. Exponential decay of the Fourier amplitudes is lost by the formation of the singularity, and the amplitudes of high wavenumbers exhibit an algebraic decay, while in the inviscid vortex sheet, the algebraic decay of the Fourier amplitudes is valid from fairly small wavenumbers. The algebraic decay rate of the viscous vortex sheet is approximately −2.5, independent of viscosity, which is the same rate as the asymptotic analysis of the inviscid sheet. Results for evolutions of the regularized vortex sheets show that the roll-up is weakened by viscosity, and the regularization parameter has more significant effects on the fine-structure of the core than does viscosity.

The author would like to thank the anonymous referees for their valuable comments and suggestions which led to improve the paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2012-0002995).

I. INTRODUCTION

II. MODEL DESCRIPTIONS

III. NUMERICAL METHODS

IV. SINGULARITY FORMATIONS

V. EVOLUTIONS OF REGULARIZED SHEETS

VI. DISCUSSION AND CONCLUSIONS

### Key Topics

- Rotating flows
- 110.0
- Viscosity
- 83.0
- Reynolds stress modeling
- 31.0
- Interface diffusion
- 13.0
- Numerical modeling
- 13.0

## Figures

Evolution of the sheet for Re = ∞. Times are t = 0, 0.1, 0.2, 0.3, and 0.363. The number of points is N = 512.

Evolution of the sheet for Re = ∞. Times are t = 0, 0.1, 0.2, 0.3, and 0.363. The number of points is N = 512.

Curvature and vortex sheet strength for Re = ∞. The curvature is plotted from t = 0.351 to 0.363 in the increment of 0.003. The vortex sheet strength is plotted at t = 0.32, 0.34, 0.352, 0.36, and 0.363.

Curvature and vortex sheet strength for Re = ∞. The curvature is plotted from t = 0.351 to 0.363 in the increment of 0.003. The vortex sheet strength is plotted at t = 0.32, 0.34, 0.352, 0.36, and 0.363.

Evolution of the sheets for Re = 1000 and Re = 200. Times are t = 0, 0.1, 0.2, 0.3, 0.389 for Re = 1000, and t = 0, 0.1, 0.2, 0.3, 0.438 for Re = 200. The number of points is N = 1024.

Evolution of the sheets for Re = 1000 and Re = 200. Times are t = 0, 0.1, 0.2, 0.3, 0.389 for Re = 1000, and t = 0, 0.1, 0.2, 0.3, 0.438 for Re = 200. The number of points is N = 1024.

Curvature and vortex sheet strength of the sheets for Re = 1000 and Re = 200. (a) Curvature for Re = 1000 from t = 0.377 to 0.389 in the increment of 0.003. (b) Curvature for Re = 200 from t = 0.418 to 0.438 in the increment of 0.004. (c) Vortex sheet strength for Re = 1000 at t = 0.34, 0.36, 0.37, 0.38, 0.386 and 0.389. (d) Vortex sheet strength for Re = 200 at t = 0.38, 0.4, 0.416, 0.428, 0.434, and 0.438.

Curvature and vortex sheet strength of the sheets for Re = 1000 and Re = 200. (a) Curvature for Re = 1000 from t = 0.377 to 0.389 in the increment of 0.003. (b) Curvature for Re = 200 from t = 0.418 to 0.438 in the increment of 0.004. (c) Vortex sheet strength for Re = 1000 at t = 0.34, 0.36, 0.37, 0.38, 0.386 and 0.389. (d) Vortex sheet strength for Re = 200 at t = 0.38, 0.4, 0.416, 0.428, 0.434, and 0.438.

Log-linear plot of the Fourier amplitudes for Re = 1000.

Log-linear plot of the Fourier amplitudes for Re = 1000.

Fits to the exponents α and β using Ansatz (16) near the singularity times for Re = ∞, 1000, and 200. The dashed lines in the β plots are at 2.57 for Re = ∞, and are at 2.5 for Re = 1000 and 200. The number of points is N = 512 for Re = ∞, and N = 1024 for Re = 1000 and 200.

Fits to the exponents α and β using Ansatz (16) near the singularity times for Re = ∞, 1000, and 200. The dashed lines in the β plots are at 2.57 for Re = ∞, and are at 2.5 for Re = 1000 and 200. The number of points is N = 512 for Re = ∞, and N = 1024 for Re = 1000 and 200.

Estimation of the singularity time from the fits to α for Re = ∞ and Re = 1000. The circles are the values of α at k = 30 for Re = ∞, and at k = 200 for Re = 1000. The curves represent the quadratic polynomial fittings, which give t c = 0.364 for Re = ∞, and t c = 0.391 for Re = 1000.

Estimation of the singularity time from the fits to α for Re = ∞ and Re = 1000. The circles are the values of α at k = 30 for Re = ∞, and at k = 200 for Re = 1000. The curves represent the quadratic polynomial fittings, which give t c = 0.364 for Re = ∞, and t c = 0.391 for Re = 1000.

Fits to α and β for several values of N near the singularity times, for Re = 3000, 1000, and 200. The time for each plot is t = 0.376 for α and t = 0.374 for β for Re = 3000, t = 0.389 for α and t = 0.386 for β for Re = 1000, and t = 0.438 for α and t = 0.435 for β for Re = 200. The dashed lines in the β plots are at 2.5.

Fits to α and β for several values of N near the singularity times, for Re = 3000, 1000, and 200. The time for each plot is t = 0.376 for α and t = 0.374 for β for Re = 3000, t = 0.389 for α and t = 0.386 for β for Re = 1000, and t = 0.438 for α and t = 0.435 for β for Re = 200. The dashed lines in the β plots are at 2.5.

Fits to α and β for several values of the filter level ξ near the singularity time, for Re = 1000. (a) α for ξ = 10−14, (b) α for ξ = 10−20, (c) α for ξ = 10−25, and (d) β for ξ = 10−14, 10−20, and 10−25, at t = 0.386. The number of points is N = 1024 for all the cases.

Fits to α and β for several values of the filter level ξ near the singularity time, for Re = 1000. (a) α for ξ = 10−14, (b) α for ξ = 10−20, (c) α for ξ = 10−25, and (d) β for ξ = 10−14, 10−20, and 10−25, at t = 0.386. The number of points is N = 1024 for all the cases.

Dependence of the singularity time on (a) the Reynolds number and (b) the initial perturbation amplitude for Re = ∞, 1000, and 200. The dashed line in (a) is the singularity time for the inviscid case.

Dependence of the singularity time on (a) the Reynolds number and (b) the initial perturbation amplitude for Re = ∞, 1000, and 200. The dashed line in (a) is the singularity time for the inviscid case.

(a) for N = 128, 256, and 512, near the singularity time for Re = 1000. The solid lines are at t = 0.384, 0.386, 0.388, and 0.390, and the dashed line is at t c = 0.391. (b) Close-up of the sheets around the center at t c = 0.391 for N = 256, 512, and 1024.

(a) for N = 128, 256, and 512, near the singularity time for Re = 1000. The solid lines are at t = 0.384, 0.386, 0.388, and 0.390, and the dashed line is at t c = 0.391. (b) Close-up of the sheets around the center at t c = 0.391 for N = 256, 512, and 1024.

Evolution of the sheets for (a) Re = 1000 and (b) Re = 200, with δ = 0.1. Times are t = 0, 0.5, 1, 1.5, and 2 for both cases.

Evolution of the sheets for (a) Re = 1000 and (b) Re = 200, with δ = 0.1. Times are t = 0, 0.5, 1, 1.5, and 2 for both cases.

Comparison of the sheets at t = 1.5, for varying Reynolds number and δ. The left plots correspond to δ = 0.15, and the right plots to δ = 0.1.

Comparison of the sheets at t = 1.5, for varying Reynolds number and δ. The left plots correspond to δ = 0.15, and the right plots to δ = 0.1.

Profiles of the sheets at t = 2 for several values of the regularization parameter and the Reynolds number, for the initial amplitude ε = 0.05. The upper row corresponds to Re =1000, and the lower row to Re =10000. From the left column to the right, δ = 0.316, 0.158, and 0.079.

Profiles of the sheets at t = 2 for several values of the regularization parameter and the Reynolds number, for the initial amplitude ε = 0.05. The upper row corresponds to Re =1000, and the lower row to Re =10000. From the left column to the right, δ = 0.316, 0.158, and 0.079.

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