Journal of Engineering Mathematics

, Volume 100, Issue 1, pp 81–94

# Numerical simulation of the Rayleigh–Taylor instability of a miscible slice in a porous medium

• Min Chan Kim
Article

## Abstract

Convective instability in miscible slices is important in understanding the contaminant spreading in groundwater and sample dispersion in a chromatographic column. In the present study, the temporal evolution of the Rayleigh–Taylor instability of a miscible high-density slice in a porous medium is analyzed theoretically using nonlinear numerical simulations. Nonlinear governing equations are derived and solved with the Fourier spectral method. To connect the previous linear stability analysis and the present nonlinear simulation, the most unstable disturbance which was identified in the linear analysis is employed as an initial condition for the nonlinear study. In contrast to the fingering between two semi-infinite regions, the nonlinear fingering of a finite slice is influenced by the depth of the high-density region. The present nonlinear analysis shows that there exists a critical depth below which the system is linearly unstable, but nonlinear phenomena cannot be expected. The nonlinear simulation results show that nonlinear competition yields a series of cells of slightly different widths and amplitudes. Also, it is found that the upper stable region hinders the development of instability motions and stabilizes the system.

## Keywords

Fourier spectral method Miscible slice Nonlinear simulation Rayleigh–Taylor instability

## Mathematics Subject Classification

76E06 76E30 76S99

## Notes

### Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2015R1D1A3A01015798).

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