Session GC: General Fluid Dynamics I

Incompressible flow around small obstacles

In recent years, the author and his research team have obtained several results concerning the limiting behavior of incompressible flow around small obstacles, both in the inviscid and viscous cases. These results showcase the difficulties and mathematical issues surrounding the description of fluid-solid interaction at large Reynolds number. In this talk, we will present the main results obtained, focusing especially of the joint small viscosity/small obstacle limit, see [1], ongoing research on homogenization, and open problems. \\[4pt] [1] Iftimie, D., Lopes Filho, M.C. and Nussenzveig Lopes, H.J., \textit{Incompressible flow around a small obstacle and the vanishing viscosity limit}, Commun. Math. Phys. V. 287 (2009), 99-115   

Stochastic Modeling of Turbulence-Driven Systems: Application to Wind Energy

The recent increase in the exploitation of the wind energy resource stresses the need for fundamental research in fluid dynamics. The complex wind inflows that drive wind turbines affect their availability in terms of electric power production, as well as in operation lifetime. Short-scale turbulent effects in the wind such as intermittency, as well as large-scale atmospheric non-stationarity lead to ever-changing power signals fed into the electric grid. This calls for a theoretical classification of wind energy phenomena into complex, turbulence-driven systems. Our raising dependence on wind energy requires a better understanding of these phenomena, as well as reliable models. A stochastic model is proposed as an alternative to standard wind energy models that often neglect turbulent effects or CFD models that cannot decribe large wind turbines yet. This model is based on the stochastic equation of Langevin that can simulate these complex systems after their proper characterization. This stochastic model can be applied separately on both atmospheric wind speed signals as well as wind turbine power production signals, after the wind turbine was characterized properly. The signals generated display the proper statistics and represent fast and flexible models for wind energy applications such as monitoring, availability prediction or grid integration. A future analysis of fatigue loads is also under development.   

Variance and Skewness Evolution in Transient Taylor Dispersion

This talk will report on a combined numerical, experimental, and theoretical study of the variance and skewness evolution for passive scalar particles transported in pipe and channel flows, with explicit differences between these two cases illustrated and explained. We will investigate the dependence of initial conditions and Peclet number in effecting evolution dynamics within the first diffusive timescale. Questions concerning how the properties of a given initial condition determine the time required for asymptotic theory to become valid will also be addressed.   

Passive scalar advection in parallel shear flows: WKBJ mode sorting on intermediate times

The evolution of a passive scalar diffusing in simple parallel shear flows is a problem with a long history. In 1953, GI Taylor showed theoretically and experimentally that on long times, the passive scalar experiences an enhanced diffusion in the longitudinal direction. On shorter times the scalar evolution is anomalous, characterized by second moments growing faster than linear in time as we show by analysis of the stochastic differential equations underlying the passive scalar equation. The spatial structures associated with this intermediate time evolution are predicted using WKBJ analysis of an associated non-self adjoint eigenvalue problem. This analysis predicts a sorting of wall modes and interior modes with specific predictions of the decay and propagation rates as a function of the Peclet number.   

Structure from the critical layer framework in turbulent flow

We extend the critical layer framework for turbulent pipe flow proposed by McKeon \& Sharma (\textit{J. Fluid Mech, 2010}) to investigate vortical structure generated at particular streamwise/azimuthal wavenumber and frequency combinations, $(k,n,\omega)$. This framework utilizes an input-output formulation of the Navier-Stokes equations in a divergence-free basis to analyze the transfer function (the ``resolvent'') and identify the dominant forcing and response mode shapes at each $(k,n,\omega)$ combination relevant to experimental spectra. It is shown that the hairpin vortex is a natural constituent of the velocity field associated with so-called wall modes, such that our model gives important predictive information about both the statistical and structural make-up of wall turbulence. Thus the dominant response mode shapes form a suitable basis by which to decompose the full turbulent velocity field. \textbf{Acknowledgements:} This research is sponsored by an Imperial College Junior Research Fellowship and the AFOSR (program manager J. Schmisseur).   

A topological approach to three-dimensional laminar mixing

Research into laminar mixing has enjoyed a renaissance in the last decade since the realisation that the Thurston--Nielsen (TN) theory of surface homeomorphisms can assist in designing efficient ``topologically chaotic'' mixers. However, published results to date have been limited to 2D flows and quasi-3D protocols. Motivated by a simple stretching and folding argument used to derive stretching bounds in 2D flows (what Thurston describes as the iterate-and-guess method for constructing invariant train-tracks), we propose a topological approach to fully 3D fluid mixing. We consider periodic braiding of fluid in two orthogonal directions by inducing a flow with strategically placed ghost rods. The action of this braiding may be encoded by a transition matrix describing how certain area elements are mapped onto each other. The spectral radius of this matrix then furnishes an estimate of large-time asymptotic area growth rate. While this approach to mixing does not sit within the rigorous setting for TN theory, we find nonetheless that the predicted area stretch rates are very sharp for some model flows. Furthermore, we find that certain braids that are topologically trivial in 2D are quite effective in 3D.   

Optimal Mixing, Part I

We investigate optimal incompressible stirring to mix an initially inhomogeneous distribution of diffusionless passive tracers. The $H^{-1}$ Sobolev norm is adopted as the quantitative mixing measure of the tracer concentration field: its vanishing as $t \rightarrow \infty$ is equivalent to the stirring flow's mixing property in the sense of ergodic theory. We derive rigorous bounds on the rate of mixing by flows with fixed energy or energy dissipation rate constraints, and determine the flow field that instantaneously maximizes the decay of the mixing measure -- when such a flow exists -- by solving a variational problem. When no such `steepest descent' flow exists (a possible but non-generic situation) we determine the flow that maximizes the growth rate of the scalar's $H^{-1}$ norm's decay rate. This optimal stirring strategy is implemented numerically on a benchmark problem and compared to the rigorous bounds as well as an optimal control approach using a restricted set of flows.   

Optimal Mixing, Part II

We investigate optimal incompressible stirring to mix an initially inhomogeneous distribution of diffusionless passive tracers. The $H^{-1}$ Sobolev norm is adopted as the quantitative mixing measure of the tracer concentration field: its vanishing as $t \rightarrow \infty$ is equivalent to the stirring flow's mixing property in the sense of ergodic theory. We derive rigorous bounds on the rate of mixing by flows with fixed energy or energy dissipation rate constraints, and determine the flow field that instantaneously maximizes the decay of the mixing measure -- when such a flow exists -- by solving a variational problem. When no such `steepest descent' flow exists (a possible but non-generic situation) we determine the flow that maximizes the growth rate of the scalar's $H^{-1}$ norm's decay rate. This optimal stirring strategy is implemented numerically on a benchmark problem and compared to the rigorous bounds as well as an optimal control approach using a restricted set of flows.