Effects of temporal density variation and convergent geometry on nonlinear bubble evolution in classical Rayleigh-Taylor instability

Abstract

Effects of temporal density variation and spherical convergence on the nonlinear bubble evolution of single-mode, classical Rayleigh-Taylor instability are studied using an analytical model based on Layzer's theory [Astrophys. J. 122, 1 (1955)]. When the temporal density variation is included, the bubble amplitude in planar geometry is shown to asymptote to integral(t)U(L)(t')rho(t')dt'/rho(t), where U(L) = square root of (g/(C(g)k)) is the Layzer bubble velocity, rho is the fluid density, and C(g) = 3 and C(g) = 1 for the two- and three-dimensional geometries, respectively. The asymptotic bubble amplitude in a converging spherical shell is predicted to evolve as eta approximately etam(-/r0//(lU(L)sp-eta/r0), where r0 is the outer shell radius, eta(t) = integral(t)U(L)sp(t')rho(t') r(0)2(t')dt'/rho(t)r(0)2(t), U(L)(sp) = square root of (-r0(t)r0(t)/l), m(t) = rho(t)r(0)3(t), and l is the mode number.

Harmonic growth of spherical Rayleigh-Taylor instability in weakly nonlinear regime

“…When the typical perturbation amplitude is close to its wavelength, the second and third harmonics are generated successively, and then the perturbation enters the nonlinear regime. In the weakly nonlinear growth regime, [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] within the framework of the third-order weakly nonlinear theory, [24][25][26][27][28] the interface function is gðx; tÞ ¼ g 1 cosðkxÞ þ g 2 cosð2kxÞ þ g 3 cosð3kxÞ, where g n (n ¼ 1, 2, 3) is the amplitude of the nth harmonic…”

“…Weakly nonlinear behaviors in planar RTI have become a field of theoretical, [24][25][26][27][28][29][30][31][32][33][34] experimental [35][36][37] and numerical [38][39][40][41] interest. Strictly speaking, perturbation growth driven only by buoyant force is named as RTI, while the modifications of perturbation behavior by compression and geometrical a) Authors to whom correspondence should be addressed.…”

Linear and nonlinear Rayleigh-Taylor growth at strongly convergent spherical interfaces

“…In particular, attention has recently focused on the effects of nonlinearity and converging geometry in combination [15][16][17], such as occurs at the surface of an imploding inertial confinement fusion (ICF) capsule. The effect of spherical convergence on RT growth in the linear regime has long been appreciated, and its potential to enhance perturbation amplitudes over their planar analogues noted [18][19][20].…”

Effects of initial radius of the interface and Atwood number on nonlinear saturation amplitudes in cylindrical Rayleigh-Taylor instability

“…As mentioned above, the weakly nonlinear behaviors of the RTI in the Cartesian geometry have been a field of theoretical, [7][8][9][10][11][12][13][14][15][16][17][18] experimental, [19][20][21] or numerical [22][23][24][25][26][27] interest. In many applications, the RTI occurs in spherical or cylindrical geometry where the corresponding investigation has been undertaken by several authors; [28][29][30][31][32][33][34][35][36] specifically, an extra instability due to the curvature of the interface (i.e., the BellPlesset effect), the nonlinear evolution of the interface, and numerical solutions including magnetic effects have been addressed.…”

“…Before a strong nonlinear growth regime, [3][4][5][6] one has a weakly nonlinear growth regime. [7][8][9][10][11][12][13][14][15][16][17][19][20][21][22][23][24] Within the framework of the third-order weakly nonlinear theory, [7][8][9][10][11] the interface position at time t takes the form, gðx; tÞ ¼ g 1 cosðkxÞ þ g 2 cosð2kxÞ þ g 3 cosð3kxÞ, where g 1 , g 2 , and g 3 are, respectively, the amplitudes of the fundamental mode, the second harmonic, and the third harmonic…”