Nonlinear dynamics of double-layer unstable interfaces with non-uniform velocity shear

Abstract

The interaction of double-layer density stratified interfaces with initial non-uniform velocity shear is investigated theoretically and numerically, taking the incompressible Richtmyer-Meshkov instability as an example. The linear analysis for providing the initial conditions in numerical calculations is performed, and some numerical examples of vortex double layers are presented using the vortex sheet model. We show that the density stratifications (Atwood numbers), the initial distance between two interfaces, and the distribution of the initial velocity shear determine the evolution of vortex double layers. When the Atwood numbers are large, the deformation of interfaces is small, and the distance between the two interfaces is almost unchanged. On the other hand, when the Atwood numbers are small and the initial distance between two interfaces is sufficiently close (less than or equal to the half of the wavelength of the initial disturbance), the two interfaces get closer to each other and merge at the last computed stage due to the incompressibility. We confirm this theoretically expected fact numerically.

Motion of unstable two interfaces in a three-layer fluid with a non-zero uniform current

“…The governing equations adopted here [Eqs. (3) and ( 4) below] for the linear analysis are the same as those used in the reference [35] except the gravity term.…”

“…The density ratios in (12) are selected as values close to those adopted by Craik and Adam [20], and Chen and Forbes [15]. The typical values of the density ratios are 0 < 1 − ρ0 ≪ 1 and 0 < ρ2 − 1 ≪ 1 in geophysical fluid dynamics or internal gravity waves; however, any values are possible in plasma physics [43,44,45,46,33,35]. We use the parameters (12) in the numerical calculations in Section 4 except Section 4.3.2, where smaller values of |ρ 0 | are adopted.…”

“…The subscript e denotes the differentiation with respect to e and s i,e = √ X 2 i,e + Y 2 i,e . The integral (15) becomes the principal value integral in case that the point x = (x, y) is on the interface I i ; i.e., (x, y) = (X i , Y i ) [35].…”

“…It should be noted that the second term on the right-hand side of ( 17) only contributes to the tangential component of the velocity u + i . The interfacial velocity (17) gives a weighted average between the fluid velocity u i and u i−1 [35]. Supposing the temporal evolution of X i is provided by u + i in the Lagrange frame, we obtain the relation…”

“…Interfacial dynamics in a layered fluid with density stratification is important in various areas such as geophysical fluid dynamics [21,60], internal gravity waves [19,61], and plasma physics [63,35]. In geophysical fluid dynamics and internal gravity waves, the continuous density profile of an ocean or atmosphere is often approximated to planar multi-component fluid layers of constant densities.…”