Nonlinear stabilitty for steady vortex pairs

In this article, we prove nonlinear orbital stability for steadily translating vortex pairs, a family of nonlinear waves that are exact solutions of the incompressible, two-dimensional Euler equations. We use an adaptation of Kelvin's variational principle, maximizing kinetic energy penalised by a multiple of momentum among mirror-symmetric isovortical rearrangements. This formulation has the advantage that the functional to be maximized and the constraint set are both invariant under the flow of the time-dependent Euler equations, and this observation is used strongly in the analysis. Previous work on existence yields a wide class of examples to which our result applies.Comment: 25 page

High-order accurate Lagrange-remap hydrodynamic schemes on staggered Cartesian grids

International audienceWe consider a class of staggered grid schemes for solving the 1D Euler equations in internal energy formulation. The proposed schemes are applicable to arbitrary equations of state and high-order accurate in both space and time on smooth flows. Adding a discretization of the kinetic energy equation, a high-order kinetic energy synchronization procedure is introduced, preserving globally total energy and enabling proper shock capturing. Extension to 2D Cartesian grids is done via C-type staggering and high-order dimensional splitting. Numerical results are provided up to 8th order accuracy

Recursive Newton-Euler dynamics and sensitivity analysis for dynamic motion planning

In this study, sensitivity equations are derived for recursive Newton-Euler dynamics using Denavit-Hartenberg (DH) moving coordinates. The dynamics and sensitivity equations depend on the 3x3 DH rotation matrices. Compared to recursive Lagrangian formulation, which depends on 4x4 DH transformation matrices, the recursive Newton-Euler formulation requires less computational effort. In addition, recursive Newton-Euler dynamics explicitly calculates internal joint forces that are not available in recursive Lagrangian formulation. The proposed formulation can handle both prismatic joints and revolute joints. More importantly, analytical sensitivities provide aid to dynamic motion predictions. Three numerical examples are presented to verify the efficacy of the proposed algorithms including two time-minimization trajectory-planning problems and one gait prediction problem. The example model setup, optimization formulation, and simulation results are presented. All three examples are successfully optimized using the proposed dynamics and sensitivity equations. The predicted kinematic and kinetic profiles are partially validated with the data in the literature. Finally, the recursive Newton-Euler dynamics and sensitivity equations are programmed using C++ with the latest math library (Eigen) for developing a general purpose motion prediction software

Kinetic schemes and initial boundary value problems for the Euler system

We study kinetic solutions, including shocks, of initial and boundary value problems for the Euler equations of gases. In particular we consider moving adiabatic boundaries, which may be driven either by a given path or because they are subjected to forces. In the latter case we consider a gas in a cylinder, and the boundary may represent a piston that suffers forces by the incoming and outgoing gas particles. Moreover, we will study periodic boundary conditions. A kinetic scheme consists of three ingredients: (i) There are periods of free flight of duration τM, where the gas particles move according to the free transport equation. (ii) It is assumed that the distribution of the gas particles at the beginning of each of these periods is given by a MAXWELLian. (iii) The interaction of gas particles with a boundary is described by a so called extension law, that determines the phase density at the boundary, and provides additionally continuity conditions for the the fields at the boundary in order to achieve convergence. The EULER equations result in the limit τM → 0. We prove rigorous results for these kinetic schemes concerning (i) regularity, (ii) weak conservation laws, (iii) entropy inequality and (iv) continuity conditions for the fields at the boundaries. The study is supplemented by some numerical examples. This approach is by no mean restricted to EULER equations or to adiabatic boundaries, but it holds also for other hyperbolic systems, namely those that rely on a kinetic formulation

Rotation Prevents Finite-Time Breakdown

We consider a two-dimensional convection model augmented with the rotational Coriolis forcing, $U_t + U\cdot\nabla_x U = 2k U^\perp$, with a fixed $2k$ being the inverse Rossby number. We ask whether the action of dispersive rotational forcing alone, $U^\perp$, prevents the generic finite time breakdown of the free nonlinear convection. The answer provided in this work is a conditional yes. Namely, we show that the rotating Euler equations admit global smooth solutions for a subset of generic initial configurations. With other configurations, however, finite time breakdown of solutions may and actually does occur. Thus, global regularity depends on whether the initial configuration crosses an intrinsic, ${\mathcal O}(1)$ critical threshold, which is quantified in terms of the initial vorticity, $\omega_0=\nabla \times U_0$, and the initial spectral gap associated with the $2\times 2$ initial velocity gradient, $\eta_0:=\lambda_2(0)-\lambda_1(0), \lambda_j(0)= \lambda_j(\nabla U_0)$. Specifically, global regularity of the rotational Euler equation is ensured if and only if $4k \omega_0(\alpha) +\eta^2_0(\alpha) <4k^2, \forall \alpha \in \R^2$ . We also prove that the velocity field remains smooth if and only if it is periodic. We observe yet another remarkable periodic behavior exhibited by the {\em gradient} of the velocity field. The spectral dynamics of the Eulerian formulation reveals that the vorticity and the eigenvalues (and hence the divergence) of the flow evolve with their own path-dependent period. We conclude with a kinetic formulation of the rotating Euler equation

Variational approach to low-frequency kinetic-MHD in the current coupling scheme

Hybrid kinetic-MHD models describe the interaction of an MHD bulk fluid with an ensemble of hot particles, which is described by a kinetic equation. When the Vlasov description is adopted for the energetic particles, different Vlasov-MHD models have been shown to lack an exact energy balance, which was recently recovered by the introduction of non-inertial force terms in the kinetic equation. These force terms arise from fundamental approaches based on Hamiltonian and variational methods. In this work we apply Hamilton's variational principle to formulate new current-coupling kinetic-MHD models in the low-frequency approximation (i.e. large Larmor frequency limit). More particularly, we formulate current-coupling hybrid schemes, in which energetic particle dynamics are expressed in either guiding-center or gyrocenter coordinates.Comment: v3.0. 30 page

Finite element formulation of general boundary conditions for incompressible flows

We study the finite element formulation of general boundary conditions for incompressible flow problems. Distinguishing between the contributions from the inviscid and viscid parts of the equations, we use Nitsche's method to develop a discrete weighted weak formulation valid for all values of the viscosity parameter, including the limit case of the Euler equations. In order to control the discrete kinetic energy, additional consistent terms are introduced. We treat the limit case as a (degenerate) system of hyperbolic equations, using a balanced spectral decomposition of the flux Jacobian matrix, in analogy with compressible flows. Then, following the theory of Friedrich's systems, the natural characteristic boundary condition is generalized to the considered physical boundary conditions. Several numerical experiments, including standard benchmarks for viscous flows as well as inviscid flows are presented