We discuss the recent use of large-eddy simulations (LES) for the analysis of the flow over dunes. In large-eddy simulations the governing equations of fluid motion are solved on a grid sufficiently fine to resolve the largest eddies, while the effect of the smallest ones is modelled. Two-dimensional and three-dimensional dunes in rivers, as well as Barchan dunes in deserts are considered. Examples are shown to demonstrate how LES can be used to complement experiments, by exploiting its ability to calculate full-field information, and to highlight the temporal development of the flow. The main challenges slowing a more widespread application of this method are discussed.

we will present our most recent results, obtained using very-large-scale, data-intensive, direct numerical simulation (VLS DNS), on the Osborne Reynolds’ smooth pipe flow. For the smooth pipe flow, we consider the cases of Reynolds number at 6000 and at 8000, based on bulk velocity and pipe diameter. The pipe inlet condition is fully-developed laminar profile superimposed with finite-level but weak perturbations. The flow downstream of the inlet gradually transitions to turbulent, finally arriving at a state of fully-developed turbulent pipe flow. This is the first time in fluid mechanics research history that the Osborne Reynolds’ pipe flow has been accurately simulated from laminar through transition all the way to fully-developed turbulent state in a spatial DNS.

The motion of a fluid trapped between two parallel, moving walls, otherwise known as Couette flow, is known to be laminar at small forcing and turbulent at large forcing. However, the laminar state is a linearly stable equilibrium at all Reynolds numbers. There is no generally accepted theory for the transition to turbulence in the absence of a critical Reynolds number, at which travelling waves or periodic solutions branch off the laminar state. In this presentation I will review some proposed theories, focussing on the "edge state" hypothesis, which asserts that the transition is regulated by simple, saddle-type invariant solutions. In particular, I will focus on the recent discovery of solutions homoclinic to edge states. Such solutions might explain irregular turbulent bursting near the transition threshold and the complicated dependence on initial conditions of the flow. I will discuss how the homoclinic tangle depends on the Reynolds number and how it is tied in with recently uncovered routes to transient chaos in Couette flow.

The seismic imaging problem entails recovering an image of the earth's subsurface from data that is recorded on the surface of the earth, determined by the propagation of seismic (vibrational) waves through the body of the earth. The 2D or 3D acoustic wave equation is commonly used as a simplified mathematical model for this seismic wave propagation. The full waveform inverse problem aims to deduce the physical parameters of the (acoustic) medium of propagation from recorded data of impulsive waves that are transmitted or reflected through the medium, and thus form an image of the subsurface.

In this study, we demonstrate a numerical algorithm that uses factorization in the PDE solver of the 2D acoustic wave model, a multi-scale approach to the inverse solution, and a projection-based linearization search for the solution to the inverse problem. The multi-scale approach is used to decrease the rank of the inverse problem, thus decreasing the ill-posedness and under-determinedness of the solution. With a few examples, we show the robust properties of the inversion algorithm, a fast numerical convergence rate, and the advantages of multi-scaling.