arXiv subjects

Keaton Burns

Publications and source records attributed to Keaton Burns.

4 recordsLinked to original sources

arXiv2025

Steady advection-diffusion in multiply-connected potential flows

We consider the steady heat transfer between a collection of impermeable obstacles immersed in an incompressible 2D potential flow, when each obstacle has a prescribed boundary temperature distribution. Inside the fluid, the temperature satisfies a variable-coefficient elliptic partial differential equation (PDE), the solution of which usually requires expensive techniques. To solve this problem efficiently, we construct multiply-connected conformal maps under which both the domain and governing equation are greatly simplified. In particular, each obstacle is mapped to a horizontal slit and the governing equation becomes a constant-coefficient elliptic PDE. We then develop a boundary integral approach in the mapped domain to solve for the temperature field when arbitrary Dirichlet temperature data is specified on the obstacles. The inverse conformal map is then used to compute the temperature field in the physical domain. We construct our multiply-connected conformal maps by exploiting the flexible and highly accurate AAA-LS algorithm. In multiply-connected domains and domains with non-constant boundary temperature data, we note similarities and key differences in the temperature fields and Nusselt number scalings as compared to the isothermal simply-connected problem analyzed by Choi et al. (2005). In particular, we derive new asymptotic expressions for the Nusselt number in the case of arbitrary non-constant temperature data in singly connected domains at low Péclet number, and verify these scalings numerically. While our language focuses on the problem of conjugate heat transfer, our methods and findings are equally applicable to the advection-diffusion of any passive scalar in a potential flow.

physics.flu-dyn↗

arXiv2025

The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning

Machine learning based surrogate models offer researchers powerful tools for accelerating simulation-based workflows. However, as standard datasets in this space often cover small classes of physical behavior, it can be difficult to evaluate the efficacy of new approaches. To address this gap, we introduce the Well: a large-scale collection of datasets containing numerical simulations of a wide variety of spatiotemporal physical systems. The Well draws from domain experts and numerical software developers to provide 15TB of data across 16 datasets covering diverse domains such as biological systems, fluid dynamics, acoustic scattering, as well as magneto-hydrodynamic simulations of extra-galactic fluids or supernova explosions. These datasets can be used individually or as part of a broader benchmark suite. To facilitate usage of the Well, we provide a unified PyTorch interface for training and evaluating models. We demonstrate the function of this library by introducing example baselines that highlight the new challenges posed by the complex dynamics of the Well. The code and data is available at https://github.com/PolymathicAI/the_well.

cs.LG↗

arXiv2024

Formulation and calibration of CATKE, a one-equation parameterization for microscale ocean mixing

We describe CATKE, a parameterization for fluxes associated with small-scale or "microscale" ocean turbulent mixing on scales between 1 and 100 meters. CATKE uses a downgradient formulation that depends on a prognostic turbulent kinetic energy (TKE) variable and a diagnostic mixing length scale that includes a dynamic convective adjustment (CA) component. With its dynamic convective mixing length, CATKE predicts not just the depth spanned by convective plumes but also the characteristic convective mixing timescale, an important aspect of turbulent convection not captured by simpler static convective adjustment schemes. As a result, CATKE can describe the competition between convection and other processes such as shear-driven mixing and baroclinic restratification. To calibrate CATKE, we use Ensemble Kalman Inversion to minimize the error between 21 large eddy simulations (LES) and predictions of the LES data by CATKE-parameterized single column simulations at three different vertical resolutions. We find that CATKE makes accurate predictions of both idealized and realistic LES compared to microscale turbulence parameterizations commonly used in climate models.

physics.ao-ph↗

arXiv2018

Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: Mathematical analysis and derivations

This paper presents a method for the accurate and efficient computations on scalar, vector and tensor fields in three-dimensional spherical polar coordinates. The methods uses spin-weighted spherical harmonics in the angular directions and rescaled Jacobi polynomials in the radial direction. For the 2-sphere, spin-weighted harmonics allow for automating calculations in a fashion as similar to Fourier series as possible. Derivative operators act as wavenumber multiplication on a set of spectral coefficients. After transforming the angular directions, a set of orthogonal tensor rotations put the radially dependent spectral coefficients into individual spaces each obeying a particular regularity condition at the origin. These regularity spaces have remarkably simple properties under standard vector-calculus operations, such as \textit{grad} and \textit{div}. We use a hierarchy of rescaled Jacobi polynomials for a basis on these regularity spaces. It is possible to select the Jacobi-polynomial parameters such that all relevant operators act in a minimally banded way. Altogether, the geometric structure allows for the accurate and efficient solution of general partial differential equations in the unit ball.

math.NA↗