Pure single-mode Rayleigh-Taylor instability for arbitrary Atwood numbers

Wanhai Liu,Xiang Wang,Changping Yu,Ming Fang,Wenhua Ye,Xingxia Liu

doi:10.1038/s41598-020-60207-y

Abstract

The validity of theoretical investigation on Rayleigh-Taylor instability (RTI) with nonlinearity is quite important, especially for the simplest and the commonest case of a pure single-mode RTI, while its previous explicit solution in weakly nonlinear scheme is found to have several defections. In this paper, this RTI is strictly solved by the method of the potential functions up to the third order at the weakly nonlinear stage for arbitrary Atwood numbers. It is found that the potential solution includes terms of both the stimulating and inhibiting RTI, while the terms of the decreasing RTI are omitted in the classical solution of the weakly nonlinear scheme, resulting in a big difference between these two results. For the pure single-mode cosine perturbation, comparisons among the classical result, the present potential result and numerical simulations, in which the two dimensional Euler equations are used, are carefully performed. Our result is in a better agreement with the numerical simulations than the classical one before the saturation time. To avoid the tedious expressions and improve a larger valid range of the solution, the method of the Taylor expansion is employed and the velocities of the bubble and spike are, respectively, obtained. Comparisons between the improved and the simulation results show that the improved theory can better predict the evolution of the interface from the linear to weakly nonlinear, even to later of the nonlinear stages.

Highlights

The validity of theoretical investigation on Rayleigh-Taylor instability (RTI) with nonlinearity is quite important, especially for the simplest and the commonest case of a pure single-mode RTI, while its previous explicit solution in weakly nonlinear scheme is found to have several defections
Based on the pure single-mode RTI, our work presents the third-order theory of the weakly nonlinearity with the role of the whole terms, and predicts the better results from the initial linear stage to the later stage of the nonlinearity
Considering the tedious expressions of the amplitude and the velocity of the bubble and spike, together with the larger range of the valid time in the weakly nonlinear stage predicted by present potential theory, the www.nature.com/scientificreports among the present potential theory, the present improved theory and the numerical simulation for Atwood numbers A = 0.2 (a,c) and A = 0.8 (b,d) vs dimensionless time kg t

Summary

Theory Framework and Explicit Solutions

The velocity potentials of the fluids, satisfying their Laplace equations and infinity conditions, can be expressed as φh(x, y, t) = [φ1h,1(t) + φ3h,1(t)]e−kycos(kx) + φ2h(t)e−2kycos(2kx) + φ3h(t)e−3kycos(3kx). (6b) and (6c); (ii) replace y in the new equations with function η(x, t) [see Eq (4)]; (iii) move the non-zero terms to the left hand side from the right of the each equation; (iv) expand the expressions into the third-order (ε3) Taylor series at ε = 0; (v) construct an ordinary differential equation for the first-order, second-order, and third-order terms, respectively, consisting of the coefficients of ε, ε2, and ε3; (vi) eliminate the unknown coefficients of the velocity potential, and obtain the corresponding deference equations on amplitude coefficients of the evolution interface. If the inhibiting term in the linear solution is deleted, many terms will disappear

Comparison between the Results of the Theory and the Numerical Simulations

Conclusion

Author contributions

Additional information