This paper contributes to the state of the art in Cartesian-grid methods through development of new advection and reconstruction Volume-of-Fluid (VOF) algorithms which are applicable to two and three-dimensional flows. A computationally efficient and secondorder VOF reconstruction method is presented which uses no inversions to determine the interface normal direction. Next, the lack of conservation of fluid volume in previous VOF advection methods are shown to be due to improper treatment of one-dimensional stretching in the velocity field. This paper uses simple explicit time stepping and a cell-center estimate of the volume fraction in the dilatation term to achieve a completely conservative advection method. The new methods are simple, robust and shown to out perform existing approaches for canonical test problems relevant to breaking wave flows.@en

We present the concepts of physics-based learning models (PBLM) and their relevance and application to the field of ship hydrodynamics. The utility of physics-based learning is motivated by contrasting generic learning models for regression predictions, which do not presume any knowledge of the system other than the training data provided with methods such as semi-empirical models, which incorporate physical insights along with data-fitting. PBLM provides a framework wherein intermediate models, which capture (some) physical aspects of the problem, are incorporated into modern generic learning tools to substantially improve the predictions of the latter, minimizing the reliance on costly experimental measurements or high-resolution highfidelity numerical solutions. To illustrate the versatility and efficacy of PBLM, we present three wave-ship interaction problems: 1) at speed waterline profiles; 2) ship motions in head seas; and 3) three-dimensional breaking bow waves. PBLM is shown to be robust and produce error rates at or below the uncertainty in the generated data at a small fraction of the expense of high-resolution numerical predictions.@en

A new robust and accurate Cartesian-grid treatment for the immersion of solid bodies within a fluid with general boundary conditions is described. The new approach, the Boundary Data Immersion Method (BDIM), is derived based on a general integration kernel formulation which allows the field equations of each domain and the interfacial conditions to be combined analytically. The resulting governing equation for the complete domain preserves the behavior of the original system in an efficient Cartesian-grid method, including stable and accurate pressure values on the solid boundary. The kernel formulation allows a detailed analysis of the method, and it is demonstrated that BDIM is consistent, obtains second-order convergence relative to the kernel width, and is robust with respect to the grid and boundary alignment. Formulation for no-slip and free slip boundary conditions are derived and numerical results are obtained for the flow past a cylinder and the impact of blunt bodies through a free surface. The BDIM predictions are compared to analytic, experimental and previous numerical results confirming the properties, efficiency and efficacy of this new boundary treatment for Cartesian grid methods.@en

The connected component labeling technique (CCL), which labels regions of connected Eulerian field data, will inaccurately identify closely spaced components when applied to the volume-of-fluid function. We present two modifications to the CCL that improve its robustness and accuracy. This Informed Component Labeling algorithm (ICL) incorporates the normal and uses multilevel thresholding to improve and refine connectivity decisions for components with spacing just larger than the grid size. Through detailed verification and validation using synthetic volume fraction data, we show that the ICL algorithm removes the bias to larger components, provide guidelines for its use, and estimate its error bounds for the smallest components. The ICL produces zero standard deviation in the number of components identified for those with radius larger than twice the grid size and can reduce it by ∼ 38% for smaller components. The modifications that comprise the ICL can be applied to any existing CCL algorithm with a known increase in computational cost. It enables robust identification of connected components for accurate transfer of information in mixed Eulerian-Lagrangian methods and statistical analysis that use the volume-of-fluid function.@en

We present high-resolution implicit large eddy simulation (iLES) of the turbulent air-entraining flow in the wake of three-dimensional rectangular dry transom sterns with varying speeds and half-beam-to-draft ratios $B/D$. We employ two-phase (air/water), time-dependent simulations utilizing conservative volume-of-fluid (cVOF) and boundary data immersion (BDIM) methods to obtain the flow structure and large-scale air entrainment in the wake. We confirm that the convergent-corner-wave region that forms immediately aft of the stern wake is ballistic, thus predictable only by the speed and (rectangular) geometry of the ship. We show that the flow structure in the air-water mixed region contains a shear layer with a streamwise jet and secondary vortex structures due to the presence of the quasi-steady, three-dimensional breaking waves. We apply a Lagrangian cavity identification technique to quantify the air entrainment in the wake and show that the strongest entrainment is where wave breaking occurs. We identify an inverse dependence of the maximum average void fraction and total volume entrained with $B/D$. We determine that the average surface entrainment rate initially peaks at a location that scales with draft Froude number and that the normalized average air cavity density spectrum has a consistent value providing there is active air entrainment. A small parametric study of the rectangular geometry and stern speed establishes and confirms the scaling of the interface characteristics with draft Froude number and geometry. In Part 2 (Hendrikson & Yue, J. Fluid Mech., vol. 875, 2019, pp. 884-913) we examine the incompressible highly variable density turbulence characteristics and turbulence closure modelling.@en

We present the concepts of physics-based learning models (PBLM) and their relevance and application to the field of ship hydrodynamics. The utility of physics-based learning is motivated by contrasting generic learning models for regression predictions, which do not presume any knowledge of the system other than the training data provided with methods such as semi-empirical models, which incorporate physical insights along with data-fitting. PBLM provides a framework wherein intermediate models, which capture (some) physical aspects of the problem, are incorporated into modem generic learning tools to substantially improve the predictions of the latter, minimizing the reliance on costly experimental measurements or high-resolution highfidelity numerical solutions. To illustrate the versatility and efficacy of PBLM, we present three wave-ship interaction problems: 1) at speed waterline profiles; 2) ship motions in head seas; and 3) three-dimensional breaking bow waves. PBLM is shown to be robust and produce error rates at or below the uncertainty in the generated data at a small fraction of the expense of high-resolution numerical predictions.@en

Hydrodynamic sensing is instrumental to fish and some amphibians. It also represents, for underwater vehicles, an alternative way of sensing the fluid environment when visual and acoustic sensing are limited. To assess the effectiveness of hydrodynamic sensing and gain insight into its capabilities and limitations, we investigated the forward and inverse problem of detection and identification, using the hydrodynamic pressure in the neighbourhood, of a stationary obstacle described using a general shape representation. Based on conformal mapping and a general normalization procedure, our obstacle representation accounts for all specific features of progressive perceptual hydrodynamic imaging reported experimentally. Size, location and shape are encoded separately. The shape representation rests upon an asymptotic series which embodies the progressive character of hydrodynamic imaging through pressure sensing. A dynamic filtering method is used to invert noisy nonlinear pressure signals for the shape parameters. The results highlight the dependence of the sensitivity of hydrodynamic sensing not only on the relative distance to the disturbance but also its bearing.@en

One of the remaining challenges involved in modern naval ship design and analysis is to account for the effects of breaking waves, spray and air entrainment on the performance and non-acoustical signature of a surface ship. The near field flow about a surface ship is characterized by complex physical processes such as: (i) spray sheet and jet formation; (ii) strong free-surface turbulence interactions with (large-amplitude) breaking waves; (iii) air entrainment and bubble generation; and (iv) post-breaking turbulence and dissipation. The challenges associated with this task are twofold. The first is robustly simulating the large-scale problem which involves the flow about an entire surface ship. The second is the development of physics-based closure models for steep breaking waves in the presence of turbulence. To wit, a two-pronged approach consisting of developing an understanding for closure model development and applying cutting-edge computational capabilities has been developed to accurately simulate the free-surface flow around naval combatants. Using high-resolution direct numerical simulation of the Navier-Stokes equations employeeing the level set method, we have successfully simulated an ensemble of unsteady breaking waves at Reynolds numbers O(103-4). This includes steady and unsteady as well as spilling and plunging events. This dataset is continually being improved upon in terms of depth and breadth as a direct result of this Challenge Project. The goal of this core research area is to develop understanding of the physics of breaking waves to help guide the development of physics-based breaking wave modes. The dataset is being used for the evaluation of closure models for inclusion in current larger scale simulations such as large eddy simulation and Reynolds-Averaged Navier-Stokes. Robustly simulating the near-field flow of a surface ship requires the development of new models and numerical techniques suitable for use in large scale applications. We have performed more moderate-scale simulations to design, verify, and validate these capabilities before their implementation on the largescale simulations. Using Numerical Flow Analysis (NFA), simulations of several naval combatants were performed at a range of speeds. The numerical results show wave overturning at the bow and flow separation at the transom. Air is entrained along the side of the hull and in the rooster-tail region behind the stern. In both regions, numerical predictions agree well with experimental measurements. This work marks the first time that NFA has been used to simulate an entire ship hull. The numerical simulations were performed on the Engineer Research and Development Center (ERDC) Cray XT3 using 128-256 processors. Approximately, 90 million grid points were used in the simulations.@en

A combination of cartesian-grid methods and volume-offluid methods is used to simulate breaking waves around ships and the resulting hydrodynamic forces. A surface panelization of a ship hull is used as input to automatically generate an immersed-boundary representation of the geometry on a cartesian grid. No additional gridding beyond what is already used in potential-flow methods and hydrostatics calculations is required. The volumeof- fluid portion of the numerical algorithm is used to capture the free-surface interface, including the breaking of waves, the formation of spray, and the entrainment of air. The numerical scheme is implemented on a parallel computer. The numerical simulations are compared to analytical solutions and experimental measurements. Together, the ease of input and usage, the ability to model and resolve complex free-surface phenomena, and the speed of the numerical algorithm provide a robust capability for simulating the free-surface disturbances near a ship.@en

The primary purpose of our research efforts is to improve naval design and detection capabilities. Our current research efforts leverage high performance computing (HPC) resources to perform high-resolution numerical simulations with hundreds-of-millions to billions of unknowns to study wave breaking behind a transom stern, wave-impact loading, the generation of spray by high-speed planing craft, air entrainment by plunging breaking waves, forced-motion, and storm seas. This paper focuses on the air entrainment and free-surface turbulence in the flow behind a transom-stern and wave-impact loading on marine platforms. Two codes, Numerical Flow Analysis (NFA) and Boundary Data Immersion Method (BDIM), are used in this study. Both codes are Cartesian-based Large-Eddy Simulation (LES) formulations, and use either Volume-of-Fluid (VOF) (NFA) or conservative Volume-of-Fluid (cVOF) BDIM treatments to track the free-surface interface. The first project area discussed is the flow behind the transom stern. BDIM simulations are used to study the volume of entrained air behind the stern. The application of a Lagrangian bubble-extraction algorithm elucidates the location of air cavities in the wake and the bubble-size distribution for a flow that has over 10 percent void fraction. NFA simulations of the transom-stern flow are validated by comparing the numerical simulations to experiments performed at the Naval Surface Warfare Center, Carderock Division (NSWCCD), where good agreement between simulations and experiments is obtained for mean elevations and regions of white water in the wake. The second project area discussed is wave impact loading, a topic driven by recent structural failures of high-speed planing vessels and other advanced vehicles, as well as the devastation caused by Tsunamis impacting low-lying coastal areas. NFA simulations of wave breaking events are compared to the NSWCCD cube impact experiments and the Oregon State University, O.H. Hinsdale Wave Research Laboratories Tsunami experiments, and it is shown that NFA is able to accurately simulate the propagation of waves over long distances after which it also accurately predicts highly-energetic impact events.@en