Analysis and mitigation of numerical dissipation in inviscid and viscid computation of vortex-dominated flows

The conservative unsteady Euler equations for the flow relative motion in the moving frame of reference are used to solve for the steady and unsteady flows around sharp-edged delta wings. The resulting equations are solved by using an implicit approximately-factored finite volume scheme. Implicit second-order and explicit second- and fourth-order dissipations are added to the scheme. The boundary conditions are explicitly satisfied. The grid is generated by locally using a modified Joukowski transformation in cross flow planes at the grid chord stations. The computational applications cover a steady flow around a delta wing whose results serve as the initial conditions for the unsteady flow around a pitching delta wing about a large angle of attack. The steady results are compared with the experimental data and the periodic solution is achieved within the third cycle of oscillation

Transonic Flows with Shocks Past Curved Wedges for the Full Euler Equations

We establish the existence, stability, and asymptotic behavior of transonic flows with a transonic shock past a curved wedge for the steady full Euler equations in an important physical regime, which form a nonlinear system of mixed-composite hyperbolic-elliptic type. To achieve this, we first employ the coordinate transformation of Euler-Lagrange type and then exploit one of the new equations to identify a potential function in Lagrangian coordinates. By capturing the conservation properties of the Euler system, we derive a single second-order nonlinear elliptic equation for the potential function in the subsonic region so that the transonic shock problem is reformulated as a one-phase free boundary problem for a second-order nonlinear elliptic equation with the shock-front as a free boundary. One of the advantages of this approach is that, given the shock location or quivalently the entropy function along the shock-front downstream, all the physical variables can expressed as functions of the gradient of the potential function, and the downstream asymptotic behavior of the potential function at the infinite exit can be uniquely determined with uniform decay rate. To solve the free boundary problem, we employ the hodograph transformation to transfer the free boundary to a fixed boundary, while keeping the ellipticity of the second-order equations, and then update the entropy function to prove that it has a fixed point. Another advantage in our analysis here is in the context of the real full Euler equations so that the solutions do not necessarily obey Bernoulli's law with a uniform Bernoulli constant, that is, the Bernoulli constant is allowed to change for different fluid trajectories.Comment: 35 pages, 2 figures in Discrete and Continuous Dynamical Systems, 36 (2016

Euler potentials for the MHD Kamchatnov-Hopf soliton solution

In the MHD description of plasma phenomena the concept of magnetic helicity turns out to be very useful. We present here an example of introducing Euler potentials into a topological MHD soliton which has non-trivial helicity. The MHD soliton solution (Kamchatnov, 1982) is based on the Hopf invariant of the mapping of a 3D sphere into a 2D sphere; it can have arbitrary helicity depending on control parameters. It is shown how to define Euler potentials globally. The singular curve of the Euler potential plays the key role in computing helicity. With the introduction of Euler potentials, the helicity can be calculated as an integral over the surface bounded by this singular curve. A special programme for visualization is worked out. Helicity coordinates are introduced which can be useful for numerical simulations where helicity control is needed.Comment: 15 pages, 12 figure

Stability of Attached Transonic Shocks in Steady Potential Flow past Three-Dimensional Wedges

We develop a new approach and employ it to establish the global existence and nonlinear structural stability of attached weak transonic shocks in steady potential flow past three-dimensional wedges; in particular, the restriction that the perturbation is away from the wedge edge in the previous results is removed. One of the key ingredients is to identify a "good" direction of the boundary operator of a boundary condition of the shock along the wedge edge, based on the non-obliqueness of the boundary condition for the weak shock on the edge. With the identification of this direction, an additional boundary condition on the wedge edge can be assigned to make sure that the shock is attached on the edge and linearly stable under small perturbation. Based on the linear stability, we introduce an iteration scheme and prove that there exists a unique fixed point of the iteration scheme, which leads to the global existence and nonlinear structural stability of the attached weak transonic shock. This approach is based on neither the hodograph transformation nor the spectrum analysis, and should be useful for other problems with similar difficulties.Comment: 28 Pages; 2 figure

Tippe Top Equations and Equations for the Related Mechanical Systems

The equations of motion for the rolling and gliding Tippe Top (TT) are nonintegrable and difficult to analyze. The only existing arguments about TT inversion are based on analysis of stability of asymptotic solutions and the LaSalle type theorem. They do not explain the dynamics of inversion. To approach this problem we review and analyze here the equations of motion for the rolling and gliding TT in three equivalent forms, each one providing different bits of information about motion of TT. They lead to the main equation for the TT, which describes well the oscillatory character of motion of the symmetry axis $\mathbf{\hat{3}}$ during the inversion. We show also that the equations of motion of TT give rise to equations of motion for two other simpler mechanical systems: the gliding heavy symmetric top and the gliding eccentric cylinder. These systems can be of aid in understanding the dynamics of the inverting TT

The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions

We consider the problem of integrability of the Poisson equations describing spatial motion of a rigid body in the classical nonholonomic Suslov problem. We obtain necessary conditions for their solutions to be meromorphic and show that under some further restrictions these conditions are also sufficient. The latter lead to a family of explicit meromorphic solutions, which correspond to rather special motions of the body in space. We also give explicit extra polynomial integrals in this case. In the more general case (but under one restriction), the Poisson equations are transformed into a generalized third order hypergeometric equation. A study of its monodromy group allows us also to calculate the "scattering" angle: the angle between the axes of limit permanent rotations of the body in space