Weakly nonlinear multi-mode Bell–Plesset growth in cylindrical geometry

Evolution of interfacial profiles for two-mode perturbations at C_r = 1 (solid), 1.5 (dashed), 3 (dot-dashed) and 5 (dashed) with A = 1.0. The mode numbers are m₁ = 4 (a), 5 (b), 6 (c) and 7 (d), m₂ = 4. The initial velocity perturbations are ${\dot{\alpha }}_{m_1}={\dot{\alpha }}_{m_2}=0.01V_0$.

Normalized perturbation amplitudes for the initial two-mode perturbation at convergent ratio C_r = 3.5. The equivalent wavenumber is k = m/R, the other parameters are A = 1.0, m₁ = 20, m₂ = 8 and ${\dot{\alpha }}_{m_1}={\dot{\alpha }}_{m_2}=0.01V_0$.

Temporal evolution of mode m = 60 from initial two-mode perturbation (m₁ = 40, m₂ = 20). The solid and dashed lines are WN solutions to Eq. (16), the markers are the corresponding simulation results in Ref. [26] for α_m1 = ± 0.2 mm and α_m2 = 1.5 mm.

Perturbation amplitude of mode m = 20 versus time for initial two-mode perturbation (m₁ = 40, m₂ = 20) with α_m1 = –0.2 mm and α_m2 = 1.5 mm. The solid line denotes our WN result, the dotted line is the numerical result taken from Ref. [26].

The normalized amplitudes η₊/λ₁ (solid) and η_–/λ₁ (dashed) versus convergence ratio C_r for two nearby modes (m₁ = 24, m₁ = 20) (a) and dissimilar modes (m₁ = 50, m₁ = 20) (b) perturbations at the initial stage. The initial perturbations are ${\dot{\alpha }}_{m_1}={\dot{\alpha }}_{m_2}=0.01V_0$ and the Atwood number A = 1.0.

The normalized amplitudes η₊/λ₁ (a) and η_–/λ₁ (b) for the Atwood numbers A = 1.0, 0.5, –0.5 and –1.0 with the initial perturbation ${\dot{\alpha }}_{m_1}={\dot{\alpha }}_{m_2}=0.01V_0$. The perturbation mode numbers are m₁ = 40 and m₂ = 20.