Georgia Institute of Technology College of Sciences

I'll discuss two methods for finding bounds on sums of graph eigenvalues (variously for the Laplacian, the renormalized Laplacian, or the adjacency matrix). One of these relies on a Chebyshev-type estimate of the statistics of a subsample of an ordered sequence, and the other is an adaptation of a variational argument used by P. Kröger for Neumann Laplacians. Some of the inequalities are sharp in suitable senses. This is ongoing work with J. Stubbe of EPFL

More than 125 years ago Osborne Reynolds launched the quantitative study of turbulent transition as he sought to understand the conditions under which fluid flowing through a pipe would be laminar or turbulent. Since laminar and turbulent flow have vastly different drag laws, this question is as important now as it was in Reynolds' day. Reynolds understood how one should define "the real critical value'' for the fluid velocity beyond which turbulence can persist indefinitely. He also appreciated the difficulty in obtaining this value. For years this critical Reynolds number, as we now call it, has been the subject of study, controversy, and uncertainty. Now, more than a century after Reynolds pioneering work, we know that the onset of turbulence in shear flows is properly understood as a statistical phase transition. How turbulence first develops in these flows is more closely related to the onset of an infectious disease than to, for example, the onset of oscillation in the flow past a body or the onset of motion in a fluid layer heated from below. Through the statistical analysis of large samples of individual decay and proliferation events, we at last have an accurate estimate of the real critical Reynolds number for the onset of turbulence in pipe flow, and with it, an understanding of the nature of transitional turbulence. This work is joint with: K. Avila, D. Moxey, M. Avila, A. de Lozar, and B. Hof.

The phenomenon of wave run-up has the capital importance for the beach erosion, coastal protection and flood hazard estimation. In the present talk we will discuss two particular aspects of the wave run-up problem. In this talk we focus on the wave run-up phenomena on a sloping beach. In the first part of the talk we present a simple stochastic model of the bottom roughness. Then, we quantify the roughness effect onto the maximal run-up height using Monte-Carlo simulations. A critical comparison with more conventional approaches is also performed.In the second part of the talk we study the run-up of simple wave groups on beaches of various geometries. Some resonant amplification phenomena are unveiled. The maximal run-up height in resonant cases can be 20 times higher than in regular situations. Thus, this work can provide a possible mechanism of extreme tsunami run-up conventionally ascribed to "local site effects".References:Dutykh, D., Labart, C., & Mitsotakis, D. (2011). Long wave run-up on random beaches. Phys. Rev. Lett, 107, 184504.Stefanakis, T., Dias, F., & Dutykh, D. (2011). Local Runup Amplification by Resonant Wave Interactions. Phys. Rev. Lett., 107, 124502.

An implicit method [1, 2], TARDIS (Transient Advection Reaction Diffusion Implicit Simulations), has been developed that successfully couples the compressible flow to the comprehensive chemistry and multi-component transport properties. TARDIS has been demonstrated in application to two fundamental combustion problems of great interest. First, TARDIS was used to investigate stretched laminar flame velocities in eight flame configurations: outwardly and inwardly propagating H2/air and CH4/air in cylindrical and spherical geometries. Fractional power laws are observed between the velocity deficit and the flame curvature Second, the response of transient outwardly propagating premixed H2/air and CH4/air flames subjected to joint pressure and equivalence ratio oscillations were investigated. A fuller version of the abstract can be obtained from http://www.math.gatech.edu/~rll6/malik_abstract-Apr-2012.docx [1] Malik, N.A. and Lindstedt, R.P. The response of transient inhomogeneous flames to pressure fluctuations and stretch: planar and outwardly propagating hydrogen/air flames. Combust. Sci. Tech. 82(9), 2010. [2] Malik, N. A. “Fractional powers laws in stretched flame velocities in finite thickness flames: a numerical study using realistic chemistry”. Under review, (2012). [3] Markstein, G.H. Non-steady Flame Propagation. Pergamon Press, 1964. [4] Weis,M., Zarzalis, N., and Suntz, R. Experimental study of markstein number effects on laminar flamelet velocity in turbulent premixed flames. Combust. Flame, 154:671--691, 2008.