A factor of safety method for quantitative estimates of grid-spacing and time-step uncertainties for solution verification is developed. It removes the two deficiencies of the grid convergence index and correction factor methods, namely, unreasonably small uncertainty when the estimated order of accuracy using the Richardson extrapolation method is greater than the theoretical order of accuracy and lack of statistical evidence that the interval of uncertainty at the 95% confidence level bounds the comparison error. Different error estimates are evaluated using the effectivity index. The uncertainty estimate builds on the correction factor method, but with significant improvements. The ratio of the estimated order of accuracy and theoretical order of accuracy $P$ instead of the correction factor is used as the distance metric to the asymptotic range. The best error estimate is used to construct the uncertainty estimate. The assumption that the factor of safety is symmetric with respect to the asymptotic range was removed through the use of three instead of two factor of safety coefficients. The factor of safety method is validated using statistical analysis of 25 samples with different sizes based on 17 studies covering fluids, thermal, and structure disciplines. Only the factor of safety method, compared with the grid convergence index and correction factor methods, provides a reliability larger than 95% and a lower confidence limit greater than or equal to 1.2 at the 95% confidence level for the true mean of the parent population of the actual factor of safety. This conclusion is true for different studies, variables, ranges of $P$ values, and single $P$ values where multiple actual factors of safety are available. The number of samples is large and the range of $P$ values is wide such that the factor of safety method is also valid for other applications including results not in the asymptotic range, which is typical in industrial and fluid engineering applications. An example for ship hydrodynamics is provided.

High-fidelity simulations of wave breaking processes are performed with a focus on the small-scale structures of breaking waves, such as bubble/droplet size distributions. Very large grids (up to 12 billion grid points) are used in order to resolve the bubbles/droplets in breaking waves at the scale of hundreds of micrometres. Wave breaking processes and spanwise three-dimensional interface structures are identified. It is speculated that the Görtler type centrifugal instability is likely more relevant to the plunging wave breaking instabilities. Detailed air entrainment and spray formation processes are shown. The bubble size distribution shows power-law scaling with two different slopes which are separated by the Hinze scale. The droplet size distribution also shows power-law scaling. The computational results compare well with the available experimental and computational data in the literature. Computational difficulties and challenges for large grid simulations are addressed.

Advances in high-fidelity shape optimization for industrial problems are presented, based on geometric variability assessment and design-space dimensionality reduction by Karhunen–Loève expansion, metamodels and deterministic particle swarm optimization (PSO). Hull-form optimization is performed for resistance reduction of the high-speed Delft catamaran, advancing in calm water at a given speed, and free to sink and trim. Two feasible sets (A and B) are assessed, using different geometric constraints. Dimensionality reduction for 95% confidence is applied to high-dimensional free-form deformation. Metamodels are trained by design of experiments with URANS; multiple deterministic PSOs achieve a resistance reduction of 9.63% for A and 6.89% for B. Deterministic PSO is found to be effective and efficient, as shown by comparison with stochastic PSO. The optimum for A has the best overall performance over a wide range of speed. Compared with earlier optimization, the present studies provide an additional resistance reduction of 6.6% at 1/10 of the computational cost.