2-5.30pm, Friday 2nd August

L5, Mathematical Institute, Andrew Wiles Building


A half-day workshop bringing together leading researchers to discuss advances in the field.  The workshop is supported by the EPSRC Centre for Doctoral Training in Partial Differential Equations, and the Oxford Fluids Network.



2:00 - 2:45   Prof James Glimm (Stony Brook University, USA)

                     Title: Admissibility as a Physics Principle defining Euler Equation Turbulent Flow

2:50 – 3:10   Dr. Tao Wang (Wuhan University-China & University of Oxford, UK)

                     Title:  Nonlinear Stability of Relativistic Vortex Sheets in Three-Dimensional Minkowski Spacetime

3:10 – 3:30   Dr. Srikanth Toppaladoddi (University of Oxford, UK)

                     Title:  Effects of rough boundaries on heat transport in turbulent thermal convection

3:30 – 4:00    Refreshments

4:00 – 4:45    Prof Shijun Liao (Shanghai Jiao Tong University, China)

                      Title: On the Method of Directly Defining Inverse Mapping (MDDiM) for Nonlinear Equations

4:50 -  5:10   Prof Ton van den Bremer (University of Oxford, UK)

                      Title:  Dispersion and Modulational Instability in Crossing Waves

5:10 – 5:30    Prof Paul Taylor (University of Western Australia)

                      Title:  Random Wave Reflections from a Box - Strong but Local 4-Wave Effects

5:30               Drink Reception



James Glimm: Admissibility as a physics principle defining Euler equation turbulent flow

Abstract: Solutions of Euler equation turbulence are known to be nonunique. Popular algorithms, applied to a classical problem of Rayleigh-Taylor turbulent mixing, give widely divergent answers in numerical simulations for basic observables to characterize the mixing rate. We trace the difference as due to different models for the LES defined Reynolds stress. Resolution of nonuniqueness requires an admissibility condition, to select the relevant from the physically meaningless solutions. We find that a maximum energy dissipation rate is a necessary admissibility condition.

We then comment on the controversy between Prigogine and Ziegler on the related issue of entropy production as a dynamic refinement of the 2nd law of thermodynamics. Prigogine prefers minimum rates, Ziegler maximal rates. The distinction is not fundamental, but rather a point of modeling preference. For an open system, with turbulent entropy not present in the model, Prigogine is applicable, while for a closed system, Ziegler is applicable. Thus the irreversible dropping of stones from the tower of Pisa induces a maximum rate of entropy production and associated turbulent drag in the atmosphere but no entropy increase in the stone dropped.


Tao Wang:  Nonlinear Stability of Relativistic Vortex Sheets in Three-Dimensional Minkowski Spacetime

Abstract: I will present our recent progress on relativistic vortex sheets in three-dimensional Minkowski spacetime. This talk is based on a joint work with Prof. Gui-Qiang G. Chen and Prof. Paolo Secchi.


Srikanth Toppaladoddi: Effects of rough boundaries on heat transport in turbulent thermal convection

Abstract: Turbulent flows over rough boundaries are ubiquitous in nature, and have important implications in many geophysical, astrophysical, and engineering settings. In this talk, I will consider the classical problem of Rayleigh-Bénard convection and systematically explore the effects of sinusoidal and fractal rough boundaries on heat transport in it using direct numerical simulations. 


Shijun Liao: On the Method of Directly Defining Inverse Mapping (MDDiM) for Nonlinear Equations

Abstract:  Generally speaking, it is time-consuming to gain an inverse operator/mapping of an equation, even if it is linear.  In this talk, we describe the basic ideas of the method of directly defining inverse mapping (MDDiM) for nonlinear differential equation.  Some examples are used to illustrate its validity.  First, based on the homotopy analysis method (HAM), the convergence of the MDDiM is guaranteed.  Besides, since the inverse mapping is directly defined, the MDDiM is rather efficient.  It might suggest a new way for solving differential equations, which could be quite different from traditional ones. 



Ton van den Bremer: Dispersion and Modulational Instability in Crossing Waves
Abstract: Crossing seas, in which waves travel in multiple directions, have been identified as an important challenge to offshore operations, linked to an increased probability of extreme waves. In addition to specific environmental forcing such as wind or (sudden) changes in bathymetry, two important mechanisms play a role in the formation of so-called rogue waves in the ocean, namely random dispersive focusing enhanced by weak bound-wave nonlinearity and modulational instability. Herein, experimental results obtained in the FloWave Ocean Energy Research Facility at the University of Edinburgh are presented that confirm aspects of the dispersive and unstable behaviour of crossing waves predicted by the 2D+1 nonlinear Schrödinger equation (2D+1NLSE) and the crossing nonlinear Schrödinger equation (CNLSE).


Paul Taylor: Random Wave Reflections from a Box - Strong but Local 4-Wave Effects

Abstract: Wave run-up is investigated on the exposed side of a fixed rectangular box on deep water. Experiments are carried out in uni-directional waves with normal incidence. Significant wave run-ups featuring tertiary interactions, similar to those of Molin et al. (J. Fluid Mech. 528, 2005, 323–354) for a fixed vertical plate, are observed in regular wave tests. Even in irregular waves the wave elevations at the centre of the front face of the box can reach 3-4× the incident waves, much larger than the ∼2× from linear theory and seen for transient groups. The extra amplification results from interactions between the incident and reflected wave fields upstream. This builds up slowly and is localized on the weather side of the box.



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