Our interest in fluid dynamics is in problems with evolving free boundaries, such as shock/detonation waves and their stability, hydraulic jumps and their stability, non-linear waves in free-surface fluid flows.
Employing asymptotic approximations of small shock curvature and slow time variation, we develop reduced theories of detonation dynamics
Stability of a circular hydraulic jump
When a jet of fluid from your kitchen tap strikes a plate, the radial flow on the plate is initially very shallow, but at some distance away from the impact point, the depth suddenly increases, forming a ring - the hydraulic jump. Despite its apparent simplicity, the hydraulic jump is a source of many fascinating phenomena. For example, one observes that the hydraulic jump can lose stability under certain conditions, resulting in a transition from a circular to polygonal shapes.
Waves on an oscillating free suface
This topic involves the study of free-surface waves on a fluid which is being accelerated by a periodic forcing (Faraday waves) when other factors are also involved (such as rotation).
Our primary focus is on developing high-order shock-fitting methods for problems of multi-dimensional detonation propagation. The phenomenon of detonation poses significant challenges for numerical methods due to very complex instabilities that are inherent to detonation shocks. Very careful resolution of shock propagation is necessary to capture these instabilities. In addition, we also work on algorithms and numerical simulations of systems of reaction-diffusion equations and dispersive PDE.
Shock-fitting simulation of two-dimensional detonation instability
We use high-order WENO schemes combined with high-order Runge-Kutta method to integrate the reactive Euler equations in our shock-fitting method. The movie below shows the evolution of two-dimensional detonation in an ideal gas in a channel from a steady steady 1D ZND solution subject to a small initial perturbation. Due to inherent instability of the steady solution, we observe the growth of instability and subsequent formation of transverse shock waves propagating between the walls of the channel along the lead shock. The simulation algorithm is described in Taylor, Kasimov, and Stewart, Comb. Theo. Modelling (2009).
Obstacle-stabilized detonation in radial outflow
A. Kasimov, S. Korneev, Detonation in supersonic radial outflow, J. Fluid Mech., 760, 313-341, 2014 .
Numerical solution of dispersive PDE
With the goal of accurate numerical solution of the multi-dimensional nonlinear Schroedinger equation arising in the study of Bose-Einstein condensate, we develop algorithms for obtaining the steady-state solutions, calculating linear stability properties, and for full time-integration of the PDE. The movie below shows formation of vortices in a numerical simulation of the complex Gross-Pitaevskii equation for exciton-polariton condensate starting from a steady-state radially symmetric solution. This result is part of an MS thesis by Jesus Sierra, KAUST, 2011. For details, see J. Sierra, A. Kasimov, P. Markowich, R.-M. Weishäupl, On the Gross-Pitaevskii equation with pumping and decay: stationary states and their stability, submitted, 2013, arXiv preprint.
Applied Partial Differential Equations
Our group's efforts are focused on the study of partial differential equations with particular emphasis on traveling wave solutions, their stability and nonlinear dynamics, asymptotic decay of stable solutions, and existence of attractors. Our primary interest is in hyperbolic PDE and shock waves.
Stability and nonlinear dynamics of shock waves
We investigate the stability of shock-wave solutions in hyperbolic balance laws such as those arising in detonation theory, shallow-water flows, and continuum models of traffic (with L. Faria, R. Rosales, B. Seibold, R. Semenko, Y. Trakhinin.
We investigate the existence of steady solutions of the Gross-Pitaevskii equation in the presence of sources and sinks, such as those present in modeling exciton-polariton condensates (with J. Sierra, P. Markowich, R. Weishaupl)
Decay of solutions of hyperbolic and hyperbolic/parabolic systems
We analyze the rates of decays of dissipative hyperbolic and hyperbolic/parabolic systems arizing in elastic wave propagation using energy methods (with B. Said-Houari, R. Racke)
Blow-up of solutions of nonlinear wave equations
We analyze the blow-up of solutions in nonlinear wave equations (with B. Said-Houari)
We investigate the existence of attractors for systems of reaction-diffusion equations arising in population biology and investigate stability of stationary solutions (with R. Parshad)