Nonlinear interaction between bulk point vortices and an unstable interface with nonuniform velocity shear such as Richtmyer–Meshkov instability

Abstract

The nonlinear interaction between bulk point vortices and a vortex sheet with initially nonuniform velocity shear is investigated theoretically and numerically by use of the vortex method, taking the incompressible Richtmyer-Meshkov instability as an example. As the point vortices approach the interface, i.e., a nonuniform vortex sheet, they increase the local sheet strength of the vortex sheet, which causes different types of interface deformation depending on the sign of their circulation of point vortices. For example, when the circulation of a point vortex is the opposite sign of the local sheet strength, it induces a new type of vortex pair with an local enhanced sheet vortex. We refer to that as a pseudo-vortex pair in the current study. The pseudo-vortex pair creates a local satellite mushroom at the fully nonlinear stage. The obtained results indicate that the complexity of the interface structure is enhanced if the bulk vortices exist.

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

“…17,34,67 There is another numerical method to avoid the clustering, which is known as the point insertion scheme. 49,58,59,68 This method can capture the complicated structure of the vortex sheet at the late stage by inserting grid points successively; however, the roll-up of the vortex sheet is too strong, and the asymptotic growth rate of the bubble and spike deviates 68 from the value obtained by the theoretical prediction. 40,69 In addition to that, the point insertion scheme is unsuitable for the calculation of high Atwood numbers.…”

“…while we select αi as αi = −A 2 i when δ ≠ 0 34,58,59 throughout this paper, by which we can suppress the rapid increase in the tangential velocity in rolling-up of the vortex sheet at the late stage. 61 Rewriting the Bernoulli equation (3) using the velocity (7), we obtain the following equation at interface Ii:…”

“…( 16) by the wavenumber k and the initial velocity shear of a single interface v lin , which corresponds to the linear growth rate in the single-layer RMI. 21,24,25,58,59 From now on, the dimensionless variables for space kx, time kv lin t, circulation of vortex sheet ikΓi/v lin (i = 1, 2), and vortex sheet strength γi/v lin are used as x, t, Γi, and γi. The dimensionless distance kd between the interfaces is replaced with d below.…”

“…where Eq. ( 24) provides the initial limit sheet strength of the conventional single-layer RMI: γ = lim d→0 (γ 1 + γ 2 ) t=0 = −2 sin e 21,[34][35][36]58,59 when a 1 = a 2 = 1, while γ = lim d→0 (γ 1 + γ 2 ) t=0 = 0 for Eq. ( 25).…”

“…49 It is known that the regularized vortex motion is not an exact solution to the Euler equation; 56 however, the model equation effectively describes real interfacial motion such as roll-ups 49,57 together with the velocity field in bulk. 17,58,59 Using the VSM, we regard the double-layer interfacial motion in RMI as the motion of two vortex sheets. In the VSM, the interfacial velocity is described by a Lagrangian parameter that parameterizes the interface.…”