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Register a new userPartial Optimal Transport for a Constant-Volume Lagrangian Mesh with Free Boundaries
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This article introduces a representation of dynamic meshes, adapted to some numerical simulations that require controlling the volume of objects with free boundaries, such as incompressible fluid simulation, some astrophysical simulations at cosmological scale, and shape/topology optimization. The algorithm decomposes the simulated object into a set of convex cells called a Laguerre diagram, parameterized by the position of $N$ points in 3D and $N$ additional parameters that control the volumes of the cells. These parameters are found as the (unique) solution of a convex optimization problem -- semi-discrete Monge-Amp`ere equation -- stemming from optimal transport theory. In this article, this setting is extended to objects with free boundaries and arbitrary topology, evolving in a domain of arbitrary shape, by solving a partial optimal transport problem. The resulting Lagrangian scheme makes it possible to accurately control the volume of the object, while precisely tracking interfaces, interactions, collisions, and topology changes.
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A Lagrangian experimental study of an axisymmetric turbulent water jet is performed to investigate the highly anisotropic and inhomogeneous flow field. The measurements were conducted within a Lagrangian exploration module, an icosahedron apparatus, to facilitate optical access of three cameras. The stereoscopic particle tracking velocimetry results in three component tracks of position, velocity and acceleration of the tracer particles within the vertically-oriented jet with a Taylor-based Reynolds number $mathcal R_lambda simeq 230$. Analysis is performed at seven locations from 15 diameters up to 45 diameters downstream. Eulerian analysis is first carried out to obtain critical parameters of the jet and relevant scales, namely the Kolmogorov and large turnover (integral) scales as well as the energy dissipation rate. Lagrangian statistical analysis is then performed on velocity components stationarised following methods inspired by Batchelor (textit{J. Fluid Mech.}, vol. 3, 1957, pp. 67-80) which aim to extend stationary Lagrangian theory of turbulent diffusion by Taylor to the case of self-similar flows. The evolution of typical Lagrangian scaling parameters as a function of the developing jet is explored and results show validation of the proposed stationarisation. The universal scaling constant $C_0$ (for the Lagrangian second-order structure function), as well as Eulerian and Lagrangian integral time scales are discussed in this context. $C_0$ is found to converge to a constant value (of the order of $C_0 = 3$) within 30 diameters downstream of the nozzle. Finally, the existence of finite particle size effects are investigated through consideration of acceleration dependent quantities.
A mesh is a graph that divides physical space into regularly-shaped regions. Meshes computations form the basis of many applications, e.g. finite-element methods, image rendering, and collision detection. In one important mesh primitive, called a mesh update, each mesh vertex stores a value and repeatedly updates this value based on the values stored in all neighboring vertices. The performance of a mesh update depends on the layout of the mesh in memory. This paper shows how to find a memory layout that guarantees that the mesh update has asymptotically optimal memory performance for any set of memory parameters. Such a memory layout is called cache-oblivious. Formally, for a $d$-dimensional mesh $G$, block size $B$, and cache size $M$ (where $M=Omega(B^d)$), the mesh update of $G$ uses $O(1+|G|/B)$ memory transfers. The paper also shows how the mesh-update performance degrades for smaller caches, where $M=o(B^d)$. The paper then gives two algorithms for finding cache-oblivious mesh layouts. The first layout algorithm runs in time $O(|G|log^2|G|)$ both in expectation and with high probability on a RAM. It uses $O(1+|G|log^2(|G|/M)/B)$ memory transfers in expectation and $O(1+(|G|/B)(log^2(|G|/M) + log|G|))$ memory transfers with high probability in the cache-oblivious and disk-access machine (DAM) models. The layout is obtained by finding a fully balanced decomposition tree of $G$ and then performing an in-order traversal of the leaves of the tree. The second algorithm runs faster by almost a $log|G|/loglog|G|$ factor in all three memory models, both in expectation and with high probability. The layout obtained by finding a relax-balanced decomposition tree of $G$ and then performing an in-order traversal of the leaves of the tree.
The present study deals with the finite element discretization of nanofluid convective transport in an enclosure with variable properties. We study the Buongiorno model, which couples the Navier-Stokes equations for the base fluid, an advective-diffusion equation for the heat transfer, and an advection dominated nanoparticle fraction concentration subject to thermophoresis and Brownian motion forces. We develop an iterative numerical scheme that combines Newtons method (dedicated to the resolution of the momentum and energy equations) with the transport equation that governs the nanoparticles concentration in the enclosure. We show that Stream Upwind Petrov-Galerkin regularization approach is required to solve properly the ill-posed Buongiorno transport model being tackled as a variational problem under mean value constraint. Non-trivial numerical computations are reported to show the effectiveness of our proposed numerical approach in its ability to provide reasonably good agreement with the experimental results available in the literature. The numerical experiments demonstrate that by accounting for only the thermophoresis and Brownian motion forces in the concentration transport equation, the model is not able to reproduce the heat transfer impairment due to the presence of suspended nanoparticles in the base fluid. It reveals, however, the significant role that these two terms play in the vicinity of the hot and cold walls.
Transport and mixing of scalar quantities in fluid flows is ubiquitous in industry and Nature. Turbulent flows promote efficient transport and mixing by their inherent randomness. Laminar flows lack such a natural mixing mechanism and efficient transport is far more challenging. However, laminar flow is essential to many problems and insight into its transport characteristics of great importance. Laminar transport, arguably, is best described by the Lagrangian fluid motion (`advection) and the geometry, topology and coherence of fluid trajectories. Efficient laminar transport being equivalent to `chaotic advection is a key finding of this approach. The Lagrangian framework enables systematic analysis and design of laminar flows. However, the gap between scientific insights into Lagrangian transport and technological applications is formidable primarily for two reasons. First, many studies concern two-dimensional (2D) flows yet the real world is three dimensional (3D). Second, Lagrangian transport is typically investigated for idealised flows yet practical relevance requires studies on realistic 3D flows. The present review aims to stimulate further development and utilisation of know-how on 3D Lagrangian transport and its dissemination to practice. To this end 3D practical flows are categorised into canonical problems. First, to expose the diversity of Lagrangian transport and create awareness of its broad relevance. Second, to enable knowledge transfer both within and between scientific disciplines. Third, to reconcile practical flows with fundamentals on Lagrangian transport and chaotic advection. This may be a first incentive to structurally integrate the `Lagrangian mindset into the analysis and design of 3D practical flows.
Steady flows that optimize heat transport are obtained for two-dimensional Rayleigh-Benard convection with no-slip horizontal walls for a variety of Prandtl numbers $Pr$ and Rayleigh number up to $Rasim 10^9$. Power law scalings of $Nusim Ra^{gamma}$ are observed with $gammaapprox 0.31$, where the Nusselt number $Nu$ is a non-dimensional measure of the vertical heat transport. Any dependence of the scaling exponent on $Pr$ is found to be extremely weak. On the other hand, the presence of two local maxima of $Nu$ with different horizontal wavenumbers at the same $Ra$ leads to the emergence of two different flow structures as candidates for optimizing the heat transport. For $Pr lesssim 7$, optimal transport is achieved at the smaller maximal wavenumber. In these fluids, the optimal structure is a plume of warm rising fluid which spawns left/right horizontal arms near the top of the channel, leading to downdrafts adjacent to the central updraft. For $Pr > 7$ at high-enough Ra, the optimal structure is a single updraft absent significant horizontal structure, and characterized by the larger maximal wavenumber.
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