Nonlinear bubble behaviours of compressible Rayleigh-Taylor instability with isothermal stratification in cylindrical geometry

Nonlinear evolutions of two-dimensional single-mode compressible Rayleigh--Taylor instability (RTI) with isothermal stratification are investigated in cylindrical geometry via direct numerical simulation for different Atwood numbers ($A_T=0.1-0.9$) and Mach numbers ($Ma=0.1-0.9$). It is found that the nonlinear bubble growth involves the effects of density stratification, vorticity accumulation and flow compressibility and shows considerable differences between convergent (acceleration acting radially inward) and divergent (acceleration acting radially outward) cases. Specifically, the density stratification leads to non-acceleration at low $A_T$ and high $Ma$. The accelerations in convergent cases are dominated by vorticity accumulation at low $A_T$ and low $Ma$ and by flow compressibility at high $A_T$ and high $Ma$ whereas the accelerations in divergent cases are purely induced by flow compressibility at high $A_T$ and high $Ma$. Based on the nonlinear theory of incompressible cylindrical RTI with uniform-density background~(Zhao et al., J. Fluid Mech., vol. 900, 2020, A24), an improved model is proposed by taking the density variation, vorticity accumulation and flow compressibility into consideration. This model is verified by numerical results and well reproduces the bubble evolution for different $A_T$ and $Ma$ from linear to highly nonlinear regimes.