Theoretical Study Of Weakly Nonlinear Rayleigh-Taylor Instability In Cylindrical Implosions

Inertial confinement fusion(ICF)ignition requires a high implosion converge-nce-ratio(the convergence-ratio is defined as the ratio of the initial outer target radius to the hot-spot radius)design,to achieve the central hot-spot ignition condition.However,the implosion shell interface will exhibit Rayleigh-Taylor instability(RTI)during acceleration and deceleration stage.The nonlinear RTI will destroy the symmetry of the implosion shell,causing the implosion shell to rupture and leading to the ignition failure.Therefore,investigating the nonlinear development of the Rayleigh-Taylor instability in the convergent geometry and the effects on the center ignition are the core issues of the ICF center ignition implosion.Due to the limitation of the driving energy and the requirement of high implosion speed,the implosion needs to accelerate the thin shell to achieve a high convergent process.The high convergent-ratio design converts the kinetic energy of the imploded thin shell into the internal energy of the hot-spot at the stagnation,in order to achieve the maximum implosion pressure.The increase of fluid instability at multiple interfaces can easily lead to the generation of strong nonlinear flow fields and nonlinear vortices,destroying the formation of ignition hot-spot.The RTI growth of perturbation at the multiple interfaces of the im-plosion shell in the convergent geometry will be more complicated.At present,the applied basic research in this area is relatively small in the world,and the physical understanding of the nonlinear RTI growth in the convergent geometry is unclear.Therefore,many problems still require deeply analysis and research.In this paper,we use velocity potential theory to analyze the multi-interface of planar and cylindrical RTI growth and the RTI growth in cylindrical geometry.Based on the principle from simple to complex physical model,this paper,re-spectively,studies the linear growth of RTI at multiple interfaces in different geometries,the weakly nonlinear growth of RTI at the single cylindrical interface,the weak nonlinear growth of RTI at the finite-thickness fluid in the planar and cylindrical geometry,and the weakly nonlinear growth of RTI in cylindri-cally convergent geometry.The physical understanding and the further research results are as follows:(1).According to the velocity potential theory,the coupled governing equation of RTI growth at multiple interfaces was obtained considering the continuous velocity and continuous pressure conditions across the interface.The analytical results show that the finite-thickness layer fluid reduces the intrinsic growth of the perturbation.However,finite-thickness layer fluid aggravates the feedback effect of the perturbation at the adjacent interfaces and accelerates the develop-ment of RTI.The feedback factor of the perturbation from outer interface to the inner interface of the shell in cylindrical and spherical geometry is larger than that in the plane,the feedback factor from the inner interface to the outer in-terface is smaller than the corresponding result in the planar.The RTI growth in cylindrical and spherical geometry is larger than that in planar for smaller radius.When the mode number l<20,RTI growth in spherical geometry is larger than that in cylindrical geometry.When the mode number l>30,the RTI in three different geometries are approximately the same.(2).The weakly nonlinear solution for the coupled RTI growth of a stable in-terface and an unstable interface is obtained.Our analytical theory can be used to describe the entire process of the perturbation from linear growwth to weakly nonlinear evolution at the two interfaces of the shell.The RTI growth of finite-thickness layer is much greater than the RTI growth at a single interface.The finite-thickness shell causes the nonlinear coupling feedback effect between the interfaces.With the development of time,the amplitude of the perturbation amplitude will increase at the initial stable interface.The nonlinear saturation amplitude of the finite-thickness RTI is less than the classical value～0.1λ(λis the perturbation wavelength).Considering the high-order(9th order)correc-tions of the perturbations,the convergence of the weak nonlinear solutions at the two interfaces are obtained.(3).The weakly nonlinear results of finite-thickness layer in the planar geom-etry are generalized to the cylindrical geometry.The linear growth of the RTI at the two interfaces of the cylindrical shell layer is obtained.If we retain the dominate term in each harmonic of the perturbation,weakly nonlinear solutions at the cylindrical interfaces can be obtained.The finite-thickness RTI growth in cylindrical geometry is greater than that of the single interface.When the shell thickness in the cylindrical geometry is much smaller than the inner radius of the shell,the results of the cylindrical shell are the same as those in plane geometry.The perturbation saturation amplitude at the double shell interface increases with the shell thickness.(4).Considering the radial motion of a single interface of two fluid in the cylin-drical convergent geometry,the perturbation velocity potential is expanded to the third-order.The governing equations of the fundamental mode growth,the second harmonic generation and third harmonic generation are derived.In the uniform converging process of the interface,the weakly nonlinear growth of the perturbation at the interface caused by the radial movement of the interface is obtained for the first time.The inward-going and outward-going movement of the fluid causes the shape of the interface to change during the convergent process.The nonlinear saturation amplitude of the fundamental mode is approximately to～0.2-0.6λ for m<100.(5).The third-order weakly nonlinear solution for the convergent RTI growth is obtained.It’s found that the Bell-Plesset effect(BP)are strongly coupled with RTI growth.BP effects further amplify the RTI growth.The amplitude of the spike is much larger than the amplitude of the bubble during the high convergence-ratio implosion.During the implosion process,the volume of the hot-spot will decrease dramatically because of the RTI growth.The low-order mode perturbation growth of the convergent RTI is difficult to reach saturation.The convergent geometry effect during the implosion exacerbates the nonlinear development of RTI.The fluid field of high convergence-ratio implosion is difficult to control because of the nonlinear RTI growth.