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Numerical simulation of fluids plays an essential role in modeling many physical phenomena, such as weather, climate, aerodynamics and plasma physics. 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. This leads to unfavorable trade-offs between accuracy and tractability. Here we use end-to-end deep learning to improve approximations inside computational fluid dynamics for modeling two-dimensional turbulent flows. For both direct numerical simulation of turbulence and large eddy simulation, our results are as accurate as baseline solvers with 8-10x finer resolution in each spatial dimension, resulting in 40-80x fold computational speedups. 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. Our approach exemplifies how scientific computing can leverage machine learning and hardware accelerators to improve simulations without sacrificing accuracy or generalization.
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Computational Fluid Dynamics (CFD) is a major sub-field of engineering. Corresponding flow simulations are typically characterized by heavy computational resource requirements. Often, very fine and complex meshes are required to resolve physical effects in an appropriate manner. Since all CFD algorithms scale at least linearly with the size of the underlying mesh discretization, finding an optimal mesh is key for computational efficiency. One methodology used to find optimal meshes is goal-oriented adaptive mesh refinement. However, this is typically computationally demanding and only available in a limited number of tools. Within this contribution, we adopt a machine learning approach to identify optimal mesh densities. We generate optimized meshes using classical methodologies and propose to train a convolutional network predicting optimal mesh densities given arbitrary geometries. The proposed concept is validated along 2d wind tunnel simulations with more than 60,000 simulations. Using a training set of 20,000 simulations we achieve accuracies of more than 98.7%. Corresponding predictions of optimal meshes can be used as input for any mesh generation and CFD tool. Thus without complex computations, any CFD engineer can start his predictions from a high quality mesh.
We apply supervised machine learning techniques to a number of regression problems in fluid dynamics. Four machine learning architectures are examined in terms of their characteristics, accuracy, computational cost, and robustness for canonical flow problems. We consider the estimation of force coefficients and wakes from a limited number of sensors on the surface for flows over a cylinder and NACA0012 airfoil with a Gurney flap. The influence of the temporal density of the training data is also examined. Furthermore, we consider the use of convolutional neural network in the context of super-resolution analysis of two-dimensional cylinder wake, two-dimensional decaying isotropic turbulence, and three-dimensional turbulent channel flow. In the concluding remarks, we summarize on findings from a range of regression type problems considered herein.
An autoencoder is used to compress and then reconstruct three-dimensional stratified turbulence data in order to better understand fluid dynamics by studying the errors in the reconstruction. The original single data set is resolved on approximately $6.9times10^{10}$ grid points, and 15 fluid variables in three spatial dimensions are used, for a total of about $10^{12}$ input quantities in three dimensions. The objective is to understand which of the input variables contains the most relevant information about the local turbulence regimes in stably stratified turbulence (SST). This is accomplished by observing flow features that appear in one input variable but then `bleed over to multiple output variables. The bleed over is shown to be robust with respect to the number of layers in the autoencoder. In this proof of concept, the errors in the reconstruction include information about the spatial variation of vertical velocity in most of the components of the reconstructed rate-of-strain tensor and density gradient, which suggests that vertical velocity is an important marker for turbulence features of interest in SST. This result is consistent with what fluid dynamicists already understand about SST and, therefore, suggests an approach to understanding turbulence based on more detailed analyses of the reconstruction on errors in an autoencoding algorithm.
Computational fluid dynamics is a direct modeling of physical laws in a discretized space. The basic physical laws include the mass, momentum and energy conservations, physically consistent transport process, and similar domain of dependence and influence between the physical reality and the numerical representation. Therefore, a physically soundable numerical scheme must be a compact one which involves the closest neighboring cells within the domain of dependence for the solution update under a CFL number $(sim 1 )$. In the construction of explicit high-order compact scheme, subcell flow distributions or the equivalent degree of freedoms beyond the cell averaged flow variables must be evolved and updated, such as the gradients of the flow variables inside each control volume. The direct modeling of flow evolution under generalized initial condition will be developed in this paper. The direct modeling will provide the updates of flow variables differently on both sides of a cell interface and limit high-order time derivatives of the flux function nonlinearly in case of discontinuity in time, such as a shock wave moving across a cell interface within a time step. The direct modeling unifies the nonlinear limiters in both space for the data reconstruction and time for the time-dependent flux transport. Under the direct modeling framework, as an example, the high-order compact gas-kinetic scheme (GKS) will be constructed. The scheme shows significant improvement in terms of robustness, accuracy, and efficiency in comparison with the previous high-order compact GKS.
Diving induces large pressures during water entry, accompanied by the creation of cavity and water splash ejected from the free water surface. To minimize impact forces, divers streamline their shape at impact. Here, we investigate the impact forces and splash evolution of diving wedges as a function of the wedge opening angle. A gradual transition from impactful to smooth entry is observed as the wedge angle decreases. After submersion, diving wedges experience significantly smaller drag forces (two-fold smaller) than immersed wedges. Our experimental findings compare favorably with existing force models upon the introduction of empirically-based corrections. We experimentally characterize the shapes of the cavity and splash created by the wedge and find that they are independent of the entry velocity at short times, but that the splash exhibits distinct variations in shape at later times. We propose a one-dimensional model of the splash that takes into account gravity, surface tension and aerodynamics forces. The model shows, in conjunction with experimental data, that the splash shape is dominated by the interplay between a destabilizing Venturi-suction force due to air rushing between the splash and the water surface and a stabilizing force due to surface tension. Taken together, these findings could direct future research aimed at understanding and combining the mechanisms underlying all stages of water entry in application to engineering and bio-related problems, including naval engineering, disease spreading or platform diving.
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