Abstract
In this research, the temporal evolution of the bubble tip velocity in RayleighTaylor instability (RTI) at arbitrary Atwood numbers and different initial perturbation velocities with a discontinuous profile in irrotational, incompressible, and inviscid fluids (i.e., classical RTI) is investigated. Potential models from Layzer [Astrophys. J. 122, 1 (1955)] and perturbation velocity potentials from Goncharov [Phys. Rev. Lett. 88, 134502 (2002)] are introduced. It is found that the temporal evolution of bubble tip velocity [u(t)] depends essentially on the initial perturbation velocity [u(0)]. First, when the , the bubble tip velocity increases smoothly up to the asymptotic velocity ( ) or terminal velocity. Second, when , the bubble tip velocity increases quickly, reaching a maximum velocity and then drops slowly to the . Third, when , the bubble tip velocity decays rapidly to a minimum velocity and then increases gradually toward the . Finally, when , the bubble tip velocity decays monotonically to the . Here, the critical coefficients , and , which depend sensitively on the Atwood number (A) and the initial perturbation amplitude of the bubble tip [h(0)], are determined by a numerical approach. The model proposed here agrees with hydrodynamic simulations. Thus, it should be included in applications where the bubble tip velocity plays an important role, such as the design of the ignition target of inertial confinement fusion where the RichtmyerMeshkov instability (RMI) can create the seed of RTI with , and stellar formation and evolution in astrophysics where the deflagration wave front propagating outwardly from the star is subject to the combined RMI and RTI.
The author would like to thank the anonymous referee for suggestions that have considerably improved the paper. This work was supported by the National Basic Research Program of China (Grant No. 10835003) and the National Natural Science Foundation of China (Grant No. 11274026).
I. INTRODUCTION
II. POTENTIAL MODEL: MATHEMATICAL FORMULATION
A. 2D geometry
B. 3D geometry
III. BUBBLE TIP VELOCITY AT DIFFERENT INITIAL PERTURBATION VELOCITIES
A. Critical coefficients
B. Physical explanations
IV. NUMERICAL SIMULATIONS
V. CONCLUSIONS
Key Topics
 Rayleigh Taylor instabilities
 30.0
 Richtmyer Meshkov instabilities
 15.0
 Inertial confinement
 13.0
 Bubble dynamics
 8.0
 Astrophysics
 6.0
Figures
Temporal evolutions of the normalized bubble tip velocity at different initial perturbation velocities , where (line with squares), 2.9 (line with circles), 14.8 (linewith triangles), and 61.7 (line with stars) with (a), (c) and 0.3 (b), (d) for the 2D geometry. The initial perturbation amplitude of the bubble tip is (a), (b) and (c), (d).
Temporal evolutions of the normalized bubble tip velocity at different initial perturbation velocities , where (line with squares), 2.9 (line with circles), 14.8 (linewith triangles), and 61.7 (line with stars) with (a), (c) and 0.3 (b), (d) for the 2D geometry. The initial perturbation amplitude of the bubble tip is (a), (b) and (c), (d).
Normalized bubble tip velocity versus the normalized time , at different initial velocities , where C = 0.04 (line with squares), 2.9 (line with circles), 14.8 (line with triangles), and 61.7 (line with stars) with A = 0.9 (a), (c) and 0.3 (b), (d) for the 2D geometry. The initial perturbation amplitude of the bubble tip is (a), (b) and (c), (d).
Normalized bubble tip velocity versus the normalized time , at different initial velocities , where C = 0.04 (line with squares), 2.9 (line with circles), 14.8 (line with triangles), and 61.7 (line with stars) with A = 0.9 (a), (c) and 0.3 (b), (d) for the 2D geometry. The initial perturbation amplitude of the bubble tip is (a), (b) and (c), (d).
Critical coefficients for the 2D (left) and 3D (right) geometries versus the A and the .
Critical coefficients for the 2D (left) and 3D (right) geometries versus the A and the .
Critical coefficients for the2D (left) and 3D (right) geometries versus the A and the .
Critical coefficients for the2D (left) and 3D (right) geometries versus the A and the .
Critical coefficients for the 2D (left) and 3D (right) geometries versus the A and the h(0).
Critical coefficients for the 2D (left) and 3D (right) geometries versus the A and the h(0).
Critical coefficients (line with squares), (line with circles), and (line with stars) for the 2D (left) and 3D (right) geometries versus with the fixed A = 0.8.
Critical coefficients (line with squares), (line with circles), and (line with stars) for the 2D (left) and 3D (right) geometries versus with the fixed A = 0.8.
Equivalent driving force (the dashed lines), the equivalent resistance force (the dotted lines), and the resultant force (the solid lines) versus the normalized time , with for A = 0.10, 0.40, 0.75, and 1.0 in the 2D geometry. The initial perturbation velocity is .
Equivalent driving force (the dashed lines), the equivalent resistance force (the dotted lines), and the resultant force (the solid lines) versus the normalized time , with for A = 0.10, 0.40, 0.75, and 1.0 in the 2D geometry. The initial perturbation velocity is .
Equivalent driving force (the dashed lines), the equivalent resistance force (the dotted lines), and the resultant force (the solid lines) versus the normalized time with A = 0.9 and (a), A = 0.3 and (b), A = 0.9 and (c), and A=0.3 and (d). The initial perturbation velocity is in the 2D geometry.
Equivalent driving force (the dashed lines), the equivalent resistance force (the dotted lines), and the resultant force (the solid lines) versus the normalized time with A = 0.9 and (a), A = 0.3 and (b), A = 0.9 and (c), and A=0.3 and (d). The initial perturbation velocity is in the 2D geometry.
Basic flows for the Atwood numbers A = 0.3 (a) and 0.8 (b) cases.
Basic flows for the Atwood numbers A = 0.3 (a) and 0.8 (b) cases.
Comparisons of the normalized bubble tip velocity, , from the potential model (solid and dashed lines) and from the simulations (triangles and stars) versus the normalized time . For A = 0.8 (a), the case of and is traced with the solid line and triangles; the case of and is drawn with dashed line and stars; for A = 0.9 (b), the case of and is traced with the solid line and stars.
Comparisons of the normalized bubble tip velocity, , from the potential model (solid and dashed lines) and from the simulations (triangles and stars) versus the normalized time . For A = 0.8 (a), the case of and is traced with the solid line and triangles; the case of and is drawn with dashed line and stars; for A = 0.9 (b), the case of and is traced with the solid line and stars.
Comparisons of the normalized bubble tip amplitude from the potential model and that from the numerical simulations for the 2D geometry. With , and , the results from the potential model (solid line) and from the simulation (triangles) are shown; with , and , the results from the potential model (dashed line) and from the simulation (stars) are illustrated. The uniform parameters are selected as and A = 0.3.
Comparisons of the normalized bubble tip amplitude from the potential model and that from the numerical simulations for the 2D geometry. With , and , the results from the potential model (solid line) and from the simulation (triangles) are shown; with , and , the results from the potential model (dashed line) and from the simulation (stars) are illustrated. The uniform parameters are selected as and A = 0.3.
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