Here, we introduce a method for calculating the accurate time evolution of solutions to nonlinear PDEs, while using an order-of-magnitude coarser grid than is traditionally required for the same accuracy. This is mainly due to the first-order dependence of LES on the subgrid-scale (SGS) model, especially for flows whose rate controlling scale is unresolved ( 20). Despite promising progress in LES over the last two decades, there are severe limits to what can be accurately simulated. For example, current state-of-the-art LES with mesh sizes of O ( 10 ) to O ( 100 ) million has been used in the design of internal combustion engines ( 16), gas turbine engines ( 17, 18), and turbomachinery ( 19). Instead, it is traditional to use smoothed versions of the Navier–Stokes equations ( 10, 11) that allow coarser grids while sacrificing accuracy, such as Reynolds averaged Navier–Stokes ( 12, 13) and large-eddy simulation (LES) ( 14, 15). Consequently, direct numerical simulation (DNS) for, e.g., climate and weather, is impossible. A 10-fold increase in R e leads to a thousandfold increase in the computational cost. For a turbulent fluid flow, the requirement to resolve the smallest flow features implies a computational cost scaling like R e 3, where R e = U L / ν, with U and L the typical velocity and length scales and ν the kinematic viscosity. Classical methods for computational fluid dynamics (CFD), such as finite differences, finite volumes, finite elements, and pseudo-spectral methods, are only accurate if fields vary smoothly on the mesh, and hence meshes must resolve the smallest features to guarantee convergence. The size of the smallest eddy is tiny: For an airplane with chord length of 2 m, the smallest length scale (the Kolomogorov scale) ( 9) is O ( 1 0 − 6 ) m. Our approach exemplifies how scientific computing can leverage machine learning and hardware accelerators to improve simulations without sacrificing accuracy or generalization.Ī paradigmatic example is turbulent fluid flow ( 8), underlying simulations of weather, climate, and aerodynamics. Our method remains stable during long simulations and generalizes to forcing functions and Reynolds numbers outside of the flows where it is trained, in contrast to black-box machine-learning approaches. For both direct numerical simulation of turbulence and large-eddy simulation, our results are as accurate as baseline solvers with 8 to 10× finer resolution in each spatial dimension, resulting in 40- to 80-fold computational speedups. Here we use end-to-end deep learning to improve approximations inside computational fluid dynamics for modeling two-dimensional turbulent flows. This leads to unfavorable trade-offs between accuracy and tractability. Fluids are well described by the Navier–Stokes equations, but solving these equations at scale remains daunting, limited by the computational cost of resolving the smallest spatiotemporal features. The journal publishes only selective manuscripts that are accepted by eminent experienced world class experts after closely evaluating the article in terms of quality, scientific validity and significance of the work.Numerical simulation of fluids plays an essential role in modeling many physical phenomena, such as weather, climate, aerodynamics, and plasma physics. All the published content are made available to the readers to access and use limitlessly when cited under the terms of Creative Commons Attribution License. The journal provides a forum for researchers to share, contribute and promote various forms of manuscripts such as original, reviews, mini reviews, rapid communication, perspectives, opinion, letters and other short articles. The journal encourages researchers to discuss upon problems including but not limited to calculate various properties of the fluid, such as velocity, pressure, density, and temperature as functions of space and time. Journal of Fluid Dynamics is an open access, international, peer reviewed journal dedicated to publish novel research insights in the branch of continuum mechanics that deals with physics of continuous materials which deform when subjected to a force.
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