A Framework of Runge–Kutta, Discontinuous Galerkin, Level Set and Direct Ghost Fluid Methods for the Multi-Dimensional Simulation of Underwater Explosions

This work describes the development of a hybrid framework of Runge–Kutta (RK), discontinuous Galerkin (DG), level set (LS) and direct ghost fluid (DGFM) methods for the simulation of near-field and early-time underwater explosions (UNDEX) in early-stage ship design. UNDEX problems provide a series of challenging issues to be solved. The multi-dimensional, multi-phase, compressible and inviscid fluid-governing equations must be solved numerically. The shock front in the solution field must be captured accurately while maintaining the total variation diminishing (TVD) properties. The interface between the explosive gas and water must be tracked without letting the numerical diffusion across the material interface lead to spurious pressure oscillations and thus the failure of the simulation. The non-reflecting boundary condition (NRBC) must effectively absorb the wave and prevent it from reflecting back into the fluid. Furthermore, the CFD solver must have the capability of dealing with fluid–structure interactions (FSI) where both the fluid and structural domains respond with significant deformation. These issues necessitate a hybrid model. In-house CFD solvers (UNDEXVT) are developed to test the applicability of this framework. In this development, code verification and validation are performed. Different methods of implementing non-reflecting boundary conditions (NRBCs) are compared. The simulation results of single and multi-dimensional cases that possess near-field and early-time UNDEX features—such as shock and rarefaction waves in the fluid, the explosion bubble, and the variation of its radius over time—are presented. Continuing research on two-way coupled FSI with large deformation is introduced, and together with a more complete description of the direct ghost fluid method (DGFM) in this framework will be described in subsequent papers.

Introductory Incompressible Fluid Mechanics

This introduction to the mathematics of incompressible fluid mechanics and its applications keeps prerequisites to a minimum – only a background knowledge in multivariable calculus and differential equations is required. Part One covers inviscid fluid mechanics, guiding readers from the very basics of how to represent fluid flows through to the incompressible Euler equations and many real-world applications. Part Two covers viscous fluid mechanics, from the stress/rate of strain relation to deriving the incompressible Navier-Stokes equations, through to Beltrami flows, the Reynolds number, Stokes flows, lubrication theory and boundary layers. Also included is a self-contained guide on the global existence of solutions to the incompressible Navier-Stokes equations. Students can test their understanding on 100 progressively structured exercises and look beyond the scope of the text with carefully selected mini-projects. Based on the authors' extensive teaching experience, this is a valuable resource for undergraduate and graduate students across mathematics, science, and engineering.

Inviscid Modes within the Boundary-Layer Flow of a Rotating Disk with Wall Suction and in an External Free-Stream

The purpose of this paper is to investigate the linear stability analysis for the laminar-turbulent transition region of the high-Reynolds-number instabilities for the boundary layer flow on a rotating disk. This investigation considers axial flow along the surface-normal direction, by studying analytical expressions for the steady solution, laminar, incompressible and inviscid fluid of the boundary layer flow due to a rotating disk in the presence of a uniform injection and suction. Essentially, the physical problem represents flow entrainment into the boundary layer from the axial flow, which is transferred by the spinning disk surface into flow in the azimuthal and radial directions. In addition, through the formation of spiral vortices, the boundary layer instability is visualised which develops along the surface in spiral nature. To this end, this study illustrates that combining axial flow and suction together may act to stabilize the boundary layer flow for inviscid modes.

Lie symmetry analysis and invariant solutions of 3D Euler equations for axisymmetric, incompressible, and inviscid flow in the cylindrical coordinates

Through the Lie symmetry analysis method, the axisymmetric, incompressible, and inviscid fluid is studied. The governing equations that describe the flow are the Euler equations. Under intensive observation, these equations do not have a certain solution localized in all directions $(r,t,z)$ ( r , t , z ) due to the presence of the term $\frac{1}{r}$ 1 r , which leads to the singularity cases. The researchers avoid this problem by truncating this term or solving the equations in the Cartesian plane. However, the Euler equations have an infinite number of Lie infinitesimals; we utilize the commutative product between these Lie vectors. The specialization process procures a nonlinear system of ODEs. Manual calculations have been done to solve this system. The investigated Lie vectors have been used to generate new solutions for the Euler equations. Some solutions are selected and plotted as two-dimensional plots.

A model of an inflatable elastic aerofoil

A novel method is presented to calculate the deformation of a simple elastic aerofoil with a view to determining its aerodynamic viability. The aerofoil is modelled as a thin two-dimensional elastic sheet whose ends are joined together to form a corner of prescribed angle, with a simple support included to constrain the shape to resemble that of a classical aerofoil. The weight of the aerofoil is counterbalanced exactly by the lift force due to a circulation set according to the Kutta condition. An iterative process based on a boundary integral method is used to compute the deformation of the aerofoil in response to an inviscid fluid flow, and a range of flow speeds is determined for which the aerofoil maintains an aerodynamic shape. As the flow speed is increased the aerofoil deforms significantly around its trailing edge, resulting in a negative camber and a loss of lift. The loss of lift is ameliorated by increasing the inflation pressure but at the expense of an increase in drag as the aerofoil bulges into a less aerodynamic shape. Boundary layer calculations and nonlinear unsteady viscous simulations are used to analyse the aerodynamic characteristics of the deformed aerofoil in a viscous flow. By tailoring the internal support the viscous boundary layer separation can be delayed and the lift-to-drag ratio of the aerofoil can be substantially increased.

Stokes drift through corals

Motivated by shallow ocean waves propagating over coral reefs, we investigate the drift velocities due to surface wave motion in an effectively inviscid fluid that overlies a saturated porous bed of finite depth. Previous work in this area either neglects the large-scale flow between layers (Phillips in Flow and reactions in permeable rocks, Cambridge University Press, Cambridge, 1991) or only considers the drift above the porous layer (Monismith in Ann Rev Fluid Mech 39:37–55, 2007). Overcoming these limitations, we propose a model where flow is described by a velocity potential above the porous layer and by Darcy’s law in the porous bed, with derived matching conditions at the interface between the two layers. Both a horizontal and a novel vertical drift effect arise from the damping of the porous bed, which requires the use of a complex wavenumber k. This is in contrast to the purely horizontal second-order drift first derived by Stokes (Trans Camb Philos Soc 8:441–455, 1847) when working with solely a pure fluid layer. Our work provides a physical model for coral reefs in shallow seas, where fluid drift both above and within the reef is vitally important for maintaining a healthy reef ecosystem (Koehl et al. In: Proceedings of the 8th International Coral Reef Symposium, vol 2, pp 1087–1092, 1997; Monismith in Ann Rev Fluid Mech 39:37–55, 2007). We compare our model with field measurements by Koehl and Hadfield (J Mar Syst 49:75–88, 2004) and also explain the vertical drift effects as documented by Koehl et al. (Mar Ecol Prog Ser 335:1–18, 2007), who measured the exchange between a coral reef layer and the (relatively shallow) sea above.

Discretization Error Estimation in Subsonic, Transonic and Supersonic Flows of an Inviscid Fluid Over a Bump

This paper presents a solution verification exercise for the simulation of subsonic, transonic and supersonic flows of an inviscid fluid over a circular arc (bump). Numerical simulations are performed with a pressure-based, single-phase compressible flow solver. Sets of geometrically similar grids covering a wide range of refinement ratios have been generated. The goal of these grids is twofold: obtain a reference solution from power series expansion fits applied to the finest grids; check the numerical uncertainties obtained from coarse grids that do not guarantee monotonic convergence of the quantities of interest. The results show that even with very fine grids it is not straightforward to define a reference solution from power series expansions. The level of discretization errors required to obtain reliable reference solutions implies iterative errors reduced to machine accuracy, which may be extremely time consuming even in two-dimensional inviscid flows. Quantitative assessment of the estimated uncertainties for coarse grids depends on the selected reference solution.

2-D Modeling of Nonlinear Dynamics of Forcespinning Jet Formation

Forcespinning is a novel method that makes use of centrifugal forces to produce nanofibers rapidly and at high yields. To improve and enhance the forcespinning production method, a 2D computational forcespinning inviscid fluid dynamics model is developed. Two models, namely time-independent and time-dependent, are obtained in order to investigate the influence of various parameters on fiber forcespinning formation (trajectory, jet diameter, tangential velocity). The fluid dynamics equations are solved using the method of multiple scales along with the finite difference method, and including slender-jet theory assumptions. It is important to produce jets with small diameters in the micro- and nano-range. The Weber (We) and Rossby (Rb) numbers were found to both expand the jet trajectory as they increased. Increasing We and/or decreasing Rb was found to decrease the jet diameter. Also, by varying forcespinning parameters, it has been found that the jet radius can be decreased by increasing the jet exit angle in the direction of rotation, reducing the spinneret fluid level, increasing the angular velocity of the spinneret, reducing spinneret length, and/or reducing the orifice diameter. Knowing the jet trajectories is important for designing and positioning of fiber collector. It has been found that the trajectories expand out with the increase of the jet exit angle in the direction of rotation, increase of fluid level, increase of angular velocity, and/or increase of the spinneret length. Production rates and jet radii for any predetermined radial collector distance were also determined.

Hyperuniformity and phase enrichment in vortex and rotor assemblies

Ensembles of particles rotating in a two-dimensional fluid can exhibit chaotic dynamics yet develop signatures of hidden order. Such “rotors” are found in the natural world spanning vastly disparate length scales — from the rotor proteins in cellular membranes to models of atmospheric dynamics. Here we show that an initially random distribution of either ideal vortices in an inviscid fluid, or driven rotors in a viscous membrane, spontaneously self assembles. Despite arising from drastically different physics, these systems share a Hamiltonian structure that sets geometrical conservation laws resulting in distinct structural states. We find that the rotationally invariant interactions isotropically suppress long wavelength fluctuations — a hallmark of a disordered hyperuniform material. With increasing area fraction, the system orders into a hexagonal lattice. In mixtures of two co-rotating populations, the stronger population will gain order from the other and both will become phase enriched. Finally, we show that classical 2D point vortex systems arise as exact limits of the experimentally accessible microscopic membrane rotors, yielding a new system through which to study topological defects.