Rodolfo R. Rosales

ROTATING SPIRAL WAVE SOLUTIONS OF REACTION-DIFFUSION EQUATIONS*

We resolve the question of existence of regular rotating spiral waves as a consequence of only the processes of chemical reaction and molecular diffusion. We prove rigorously the existence of these waves as solutions of reaction-diffusion equations, and we exhibit them by means of numerical computations in several concrete cases. Existence is proved via the Schauder fixed point theorem applied to a class of functions with sufficient structure that, in fact, important constructive properties such as asymptotic representations and frequency of rotation are obtained.

WEAKLY NONLINEAR DETONATION WAVES.

The authors develop a simplified asymptotic model for studying nonlinear detonation waves in chemically reacting fluids which propagate with wave speed close to the acoustical sound speed. In this regime the fluid mechanical and chemical phenomena interact substantially with each other. The model provides simplified equations for describing this interaction. When the model is specialized to unidirectional combustion waves advancing into a region of uniform flow, the travelling waves of this model system moving with positive speed coincide with the travelling waves in a qualitative model for such effects introduced previously by the second author. Some interesting new combustion waves with partial burning are also analyzed. The two main assumptions in deriving the asymptotic model are weak nonlinearity and a sufficiently high activation energy for the chemical kinetics.

Diffusion and mixing in gravity-driven dense granular flows

We study the transport properties of particles draining from a silo using imaging and direct particle tracking. The particle displacements show a universal transition from superdiffusion to normal diffusion, as a function of the distance fallen, independent of the flow speed. In the superdiffusive (but sub-ballistic) regime, which occurs before a particle falls through its diameter, the displacements have fat-tailed and anisotropic distributions. In the diffusive regime, we observe very slow cage breaking and Péclet numbers of order 100, contrary to the only previous microscopic model (based on diffusing voids). Overall, our experiments show that diffusion and mixing are dominated by geometry, consistent with long-lasting contacts but not thermal collisions, as in normal fluids.

A Theory for Spontaneous Mach Stem Formation in Reacting Shock Fronts, I. The Basic Perturbation Analysis

A theory to explain the experimentally observed spontaneous formation of Mach stems in reacting shock fronts is developed. A linearized mechanism of instability through radiating boundary waves is analyzed. Then through weakly nonlinear asymptotics, an integro-differential scalar conservation law is derived. The asymptotic expansion relates the breakdown of solutions for this scalar equation with the spontaneous formation of the complete triple-shock, slip-line Mach stem configuration.

A gradient-augmented level set method with an optimally local, coherent advection scheme

The level set approach represents surfaces implicitly, and advects them by evolving a level set function, which is numerically defined on an Eulerian grid. Here we present an approach that augments the level set function values by gradient information, and evolves both quantities in a fully coupled fashion. This maintains the coherence between function values and derivatives, while exploiting the extra information carried by the derivatives. The method is of comparable quality to WENO schemes, but with optimally local stencils (performing updates in time by using information from only a single adjacent grid cell). In addition, structures smaller than the grid size can be located and tracked, and the extra derivative information can be employed to obtain simple and accurate approximations to the curvature. We analyze the accuracy and the stability of the new scheme, and perform benchmark tests.

Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection

Some phenomena involving intersection of weak shock waves at small angles are considered: the focusing of curved fronts at aretes, the transition between regular and irregular reflection of oblique shock waves on rigid walls and the diffraction patterns arising behind obstacles. The intersection of three shock waves plays a central role in most of these phenomena, giving rise to the von Neumann paradox of oblique shock reflection and to the curious transition between linear and fully nonlinear focusing investigated experimentally by Sturtevant and Kulkarny [J. Fluid Mech. 73, 651 (1976)]. This ‘‘triple‐point paradox’’ is studied in the context of an asymptotic model, and a solution is proposed that involves an unusual kind of singularity.

Internal hydraulic jumps and mixing in two-layer flows

Internal hydraulic jumps in two-layer flows are studied, with particular emphasis on their role in entrainment and mixing. For highly entraining internal jumps, a new closure is proposed for the jump conditions. The closure is based on two main assumptions: (i) most of the energy dissipated at the jump goes into turbulence, and (ii) the amount of turbulent energy that a stably stratified flow may contain without immediately mixing further is bounded by a measure of the stratification. As a consequence of this closure, surprising bounds emerge, for example on the amount of entrainment that may take place at the location of the jump. These bounds are probably almost achieved by highly entraining internal jumps, such as those likely to develop in dense oceanic over flows. The values obtained here are in good agreement with the existing observations of the spatial development of oceanic downslope currents, which play a crucial role in the formation of abyssal and intermediate waters in the global ocean.

Interaction of Large-Scale Equatorial Waves and Dispersion of Kelvin Waves through Topographic Resonances

A new theoretical mechanism is developed in which large-scale equatorial Kelvin waves can modify their speed through dispersion and interaction with other large-scale equatorial waves, such as Yanai or Rossby modes, through topographic resonance. This resonance mechanism can prevent the breaking of a propagating nonlinear Kelvin wave, slow down its speed, and concentrate most of its energy in large-scale zonal wavenumbers while simultaneously generating large-scale Yanai or Rossby modes with specific zonal wavelengths. Simplified reduced dynamic equations for this resonant interaction are developed here via suitable asymptotic expansions of the equatorial shallow water equations with topography. Explicit exact solutions for the reduced equations and numerical experiments are utilized to display explicitly the features of large-scale dispersion and topographic resonance for equatorial Kelvin waves mentioned earlier. Two examples of this theory, corresponding to the barotropic and first baroclinic modes of the equatorial troposphere, are emphasized.

An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary

Common efficient schemes for the incompressible Navier-Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier-Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, including semi-implicit treatments of viscosity, and in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy in the L^~ norm, for both the velocity and the pressure. The scheme has a natural extension to 3-D.

Nonlinear mean field-high frequency wave interactions in the induction zone

Simplified asymptotic equations are derived for nonlinear high frequency-mean field wave interactions in chemically reacting gases during the induction period. Rigorous proofs of both enhanced combustion and explosion occurring at times preceding the homogeneous induction time are developed. Asymptotic equations are derived for both the simpler case of a high frequency simple wave interacting with a mean field, as well as for the general resonant wave interaction of many high frequency waves and the mean field. Generally, there is a nonlinear feedback mechanism between the mean field and the high frequency waves which enhances combustion. The equations derived here include, as extremely special cases and in a unified fashion, earlier separate theories of both low frequency and pulsed high frequency wave propagation in the induction zone. In these earlier theories, the nonlinear feedback mechanism is completely absent.