Constrained pseudo‐transient continuation

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/111771/1/nme4858.pd

Black box probabilistic numerics

Peer reviewe

Black box probabilistic numerics

Peer reviewe

Essentially Analytical Theory Closure for Space Filtered Thermal-Incompressible Navier-Stokes Partial Differential Equation System on Bounded Domains

Numerical simulation of turbulent flows is identified as one of the grand challenges in high-performance computing. The straight forward approach of solving the Navier-Stokes (NS) equations is termed Direct Numerical Simulation (DNS). In DNS the majority of computational effort is spent on resolving the smallest scales of turbulence, which makes this approach impractical for most industrial applications even on present-day supercomputers. A more feasible approach termed Large Eddy Simulation (LES) has evolved over the last five decades to facilitate turbulent flow predictions for reasonable Reynolds (Re) numbers and domain sizes. LES theory uses the concept of convolution with a spatial filter, which allows it to compute only the major scales of turbulence as determined by the diameter of the filter. The rest of the length scales are not resolved posing the so-called closure problem of LES. For bounded domains, besides the closure problem, an equally challenging issue of LES is that of prescribing the suitable boundary conditions for the resolved-scale state variables. Additional problems arise because the convolution operation does not generally commute with differentiation in the presence of boundaries. This dissertation details derivation of an essentially analytical closure theory for the unsteady three-dimensional space filtered thermal-incompressible NS partial differential equation (PDE) system on bounded domains. This is accomplished by the union of rational LES theory, Galdi and Layton, with modified continuous Galerkin theory of Kolesnikov with specific focus on correct adaptation of a constant measure filter near the Dirichlet type boundary. The analytical closure theory state variable organization is guided by classic fluid mechanics perturbation theory. Derivation and implementation of suitable boundary conditions (BCs) as well as the boundary commutation error (BCE) integral is accomplished using the ideas of approximate deconvolution (AD) theory. Non-homogeneous BCs for the auxiliary problem of arLES theory are derived

POROUS MEDIUM CONVECTION AT LARGE RAYLEIGH NUMBER: STUDIES OF COHERENT STRUCTURE, TRANSPORT, AND REDUCED DYNAMICS

Buoyancy-driven convection in fluid-saturated porous media is a key environmental and technological process, with applications ranging from carbon dioxide storage in terrestrial aquifers to the design of compact heat exchangers. Porous medium convection is also a paradigm for forced-dissipative infinite-dimensional dynamical systems, exhibiting spatiotemporally chaotic dynamics if not ``true turbulence. The objective of this dissertation research is to quantitatively characterize the dynamics and heat transport in two-dimensional horizontal and inclined porous medium convection between isothermal plane parallel boundaries at asymptotically large values of the Rayleigh number $Ra$ by investigating the emergent, quasi-coherent flow. This investigation employs a complement of direct numerical simulations (DNS), secondary stability and dynamical systems theory, and variational analysis. The DNS confirm the remarkable tendency for the interior flow to self-organize into closely-spaced columnar plumes at sufficiently large $Ra$ (up to $Ra \simeq 10^5$), with more complex spatiotemporal features being confined to boundary layers near the heated and cooled walls. The relatively simple form of the interior flow motivates investigation of unstable steady and time-periodic convective states at large $Ra$ as a function of the domain aspect ratio $L$. To gain insight into the development of spatiotemporally chaotic convection, the (secondary) stability of these fully nonlinear states to small-amplitude disturbances is investigated using a spatial Floquet analysis. The results indicate that there exist two distinct modes of instability at large $Ra$: a bulk instability mode and a wall instability mode. The former usually is excited by long-wavelength disturbances and is generally much weaker than the latter. DNS, strategically initialized to investigate the fully nonlinear evolution of the most dangerous secondary instability modes, suggest that the (long time) mean inter-plume spacing in statistically-steady porous medium convection results from an interplay between the competing effects of these two types of instability. Upper bound analysis is then employed to investigate the dependence of the heat transport enhancement factor, i.e. the Nusselt number $Nu$, on $Ra$ and $L$. To solve the optimization problems arising from the ``background field upper-bound variational analysis, a novel two-step algorithm in which time is introduced into the formulation is developed. The new algorithm obviates the need for numerical continuation, thereby enabling the best available bounds to be computed up to $Ra\approx 2.65\times 10^4$. A mathematical proof is given to demonstrate that the only steady state to which this numerical algorithm can converge is the required global optimal of the variational problem. Using this algorithm, the dependence of the bounds on $L(Ra)$ is explored, and a ``minimal flow unit is identified. Finally, the upper bound variational methodology is also shown to yield quantitatively useful predictions of $Nu$ and to furnish a functional basis that is naturally adapted to the boundary layer dynamics at large $Ra$

Reactive Flow and Transport Through Complex Systems

The meeting focused on mathematical aspects of reactive flow, diffusion and transport through complex systems. The research interest of the participants varied from physical modeling using PDEs, mathematical modeling using upscaling and homogenization, numerical analysis of PDEs describing reactive transport, PDEs from fluid mechanics, computational methods for random media and computational multiscale methods

The role of advection in phase-separating binary liquids

Using the advective Cahn-Hilliard equation as a model, we illuminate the role of advection in phase-separating binary liquids. The advecting velocity is either prescribed, or is determined by an evolution equation that accounts for the feedback of concentration gradients into the flow. Here, we focus on passive advection by a chaotic flow, and coupled Navier-Stokes Cahn-Hilliard flow in a thin geometry. Our approach is based on a combination of functional-analytic techniques, and numerical analysis. Additionally, we compare and contrast the Cahn-Hilliard equation with other models of aggregation; this leads us to investigate the orientational Holm-Putkaradze model. We demonstrate the emergence of singular solutions in this system, which we interpret as the formation of magnetic particles. Using elementary dynamical systems arguments, we classify the interactions of these particles.Comment: Ph.D. Thesis, Imperial College London, February 200