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Error Analysis and Estimation for the Finite Volume Method With Applications to Fluid Flows

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... F x , F y , M z denote the longitudinal force, lateral force and yawing moment on the ship respectively. . . . . . . . 25 2.17 Classifications of ship maneuvering test methods. . . . . . . . . . . . 28 3.1 Parameter definitions in the finite volume discretization (Jasak, 1996). 4.14 (a) Convergence histories of the drag (F x ) and (b) comparison of resistance coefficients with different turbulence models. ...
... Since DNS (Direct Numerical Simulation) and LES (Large-Eddy Simulation) are computationally expensive, Reynolds decomposition is always used for the NS equations 3.1 and 3.2 to separate the velocity fluctuations from the mean flow velocity. The famous Reynolds-Averaged Navier-Stokes (RANS) equations can then be obtained (Jasak, 1996): ...
... The equation discretization is to transform the governing equations 3.1 and 3.2 into a set of algebraic equations, which is solved to get the approximate solutions at pre-defined locations and times. The continuity equation 3.1 and the momentum equation 3.2 are integrated over the CV and time: (Jasak, 1996). ...
Thesis
In this thesis, the ship hydrodynamics during inland waterway transport and ship maneuvering are investigated using CFD (Computational Fluid Dynamics) based onOpenFoam. Validation and verification studies are carried out for the mesh convergence, time step convergence, sensitivity to turbulence models and dynamic mesh techniques. A quaternion-based 6DoF motion solver is implemented for the trim and sinkage predictions. Environmental effects on several inland vessels (convoy 1, convoy 2, tanker) are studied using the validated numerical models. Three important aspects, the confinement effect of the waterway, head-on encounter and ship-bridge pile interaction are simulated. The testing conditions cover a wide range, including various channel dimensions, water depths, ship draughts and speeds. The ship resistance, wave pattern, Kelvin angle and wave elevation at specific positions are investigated as functions of these parameters. Ship maneuvering is investigated using virtual captive model tests based on the MMG (Mathematical Maneuvering Group) model. An actuator disk is implemented to replace the real propeller. Open water test, rudder force test, OTT (Oblique Towing Tank test) and CMT (Circular Motion Test) of a KVLCC2 model are carried out to obtain the hydrodynamic coefficients of the propeller, rudder and ship hull. Using the obtained coefficients, system-based maneuvering simulations are carried out and validated using the free running test data. These studies reproduce real ship tests and thus prove the validity of our numerical models. As a result, the numerical solver is promising in ship hydrodynamics and marine engineering simulations.
... In the solution of both sub-problems, we selected second-order interpolation profiles for the spatial discretization to maintain the second-order accuracy level of the FVM. The central differences scheme was used for all the Laplacian terms, with non-orthogonal and skewness corrections [63]. Particularly for the flow's momentum and pressure equations, the second-order upwind scheme was used for the advective term. ...
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Numerical simulations have been extensively used in the past two decades for the study of intracranial aneurysms (IAs), a dangerous disease that occurs in the arteries that reach the brain. They may affect up to 10 % of the world's population, with up to 50 % mortality rate, in case of rupture. Physically, the blood flow inside IAs should be modeled as a fluid-solid interaction problem. However, the large majority of those works have focused on the hemodynamics of the intra-aneurysmal flow, while ignoring the wall tissue's mechanical response entirely, through rigid-wall modeling, or using limited modeling assumptions for the tissue mechanics. One of the explanations is the scarce data on the properties of IAs walls, thus limiting the use of better modeling options. Unfortunately, this situation is still the case, thus our present study investigates the effect of different modeling approaches to simulate the motion of an IA. We used three hyperelastic laws-the Yeoh law, the three-parameter Mooney-Rivlin law, and a Fung-like law with a single parameter-and two different ways of modeling the wall thickness and tissue mechanical properties-one assumed that both were uniform while the other accounted for the heterogeneity of the wall by using a "hemodynamics-driven" approach in which both thickness and material constants varied spatially with the cardiac-cycle-averaged hemodynamics. Pulsatile numerical simulations, with patient-specific vascular geometries harboring IAs, were carried out using the one-way fluid-solid interaction solution strategy implemented in solids4foam, an extension of OpenFOAM ® , in which the blood flow is solved and applied as the driving force of the wall motion. We found that different wall morphology models yield smaller absolute differences in the mechanical response than different hyperelastic laws. Furthermore, the stretch levels of IAs walls were more sensitive to the hyperelastic and material constants than the stress. These findings could be used to guide modeling 1 decisions on IA simulations, since the computational behavior of each law was different, for example, with the Yeoh law yielding the smallest computational time.
... The spatially-filtered transport equations, discussed in the previous chapter, employing a top-hat kernel [75] are integrated in time using a second-order, fully implicit Crank-Nicolson method with a maximum Courant number of 0.3. The convective term present in the momentum equations (Equation (9)) are discretised using the second-order central-difference scheme, while a normalised variable diagram (NVD) based differencing scheme [34] is employed to avoid unbounded values of scalars. The particle composition and position (Equations (58) and (59)) are solved by the fractional steps for advection, mixing and reaction [69]. ...
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Numerical simulations of a partially-premixed, turbulent jet diffusion flame stabilised in a hot vitiated co-flow are performed. For auto-igniting flames, an accurate prediction of flame stabilisation, which depends on a delicate balance between turbulent transport and chemical kinetics at the flame base, poses an enormous challenge to conventional turbulent combustion models. Multiple mapping conditioning/large eddy simulation (MMC-LES), a promising tool for modelling turbulence-chemistry interactions, has been successfully applied to simulate a variety of combustion applications involving gaseous, liquid, and solid fuels. MMC-LES is a full \gls{PDF} method where MMC plays the role of the mixing model, emulating molecular mixing phenomenon. MMC attempts to produce accurate molecular mixing by localising mixing in an independent, composition-like reference space. Due to this enforced localised mixing, a sparse distribution of stochastic Lagrangian particles for may be used for the Monte-Carlo simulation of the sub-grid joint-composition PDF equation. In the present study, we employ MMC-LES to investigate the auto-igniting methane/air flames of UC Berkeley. A sparse resolution of 1 particle per 10 Eulerian finite-volume cells is used in this study, which offers much cheaper computing expenses in comparison to the conventional transported PDF approach. A skeletal chemical mechanism, based on GRI 3.0, containing 30 species and 184 reactions, represents the oxidation of methane. The time-scale of molecular mixing is modelled using the recently published \textit{dyn-aISO} model to assess its performance in an auto-igniting configuration.
... A pressure based segregated algorithm is adopted and extended to two-phase compressible flows. The segregated projection algorithm used for a compressible solver is the PIMPLE algorithm (Jasak, 1996;Rusche, 2002;Miller et al., 2013). Equation (15) Table 1. ...
Article
Sloshing-induced impulsive pressures may be extremely high and of short rise time duration. A compressible Volume of Fluid (VoF) method was adopted to study these pressures in a partially filled three-dimensional tank with various oblique tank ceiling angles. Comparative available experimental model test measurements of free surface profiles and velocity and pressure time histories validated this numerical method. To obtain a sufficiently fine grid to yield converging predictions, grid sensitivity studies were performed. The influence of the oblique tank ceiling angle on sloshing-induced impulsive pressures was examined. Results demonstrated that the VoF method obtained accurate predictions of sloshing phenomena. The oblique tank ceiling angle dampened impulsive pressure and, therefore, this angle had to be accounted for in the design of the tank.
... The PIMPLE algorithm is based on a blend of the SIMPLE and PISO algorithms and is used for pressure-velocity decoupling. By performing two loops for the pressure-velocity decoupling (outer loop) and the pressure and velocity corrections (inner loop) [26]. This algorithm is capable of ensuring numerical stability even at a large Courant number. ...
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The process of pollutant mixing is significantly influenced by secondary flow and turbulence in meandering rivers. To investigate the influence of different point source release positions on the pollutant mixing process in sinuous open channel flows, a 3D large-eddy simulation (LES) model based on OpenFOAM was established to simulate the process of passive scalar transport in a sinuous channel with a rectangular cross-section. After verification by a flume experiment, two sets of cases in which the point sources were arranged at identical intervals in spanwise and streamwise directions were configured to evaluate the mixing efficiency. The effect of flow structure, secondary motion, and the turbulent viscosity on the scalar transport and mixing was discussed. The distribution of scalar as well as the scalar flux was analyzed in detail, and the fluctuation characteristics were also described. The results demonstrate that due to the existence of secondary flow in the sinuous channel, different transverse and streamwise release positions of the point source have significant influence on mixing efficiency and spatial distribution of the pollutant. The point source placed near the center of the cross-section in transverse or near the apex of the bend in streamwise result in higher mixing efficiency. Mixing efficiency calculated by different indices can be different, which requires comprehensive assessment.
... However, the pressure and velocity solutions can be obtained through numerical algorithms. In this study, the PISO algorithm is applied [122]. The algorithm can be summed up as follows: ...
Thesis
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Global climate change is presenting opportunities for new networks of maritime transportation through the Arctic. However, these sea routes are often infested by floating sea ice, which brings uncertainties to shipping operators, designers and builders. This work aimed to develop reliable simulation approaches for shipping scenarios in the presence of sea ice and investigate the associated changes to ship calm water resistance. For this purpose, computational fluid dynamics and ice solid mechanics were combined to model the potential ship-wave-ice interactions. Specifically, models were developed to simulate the two primary scenarios of a cargo ship operating in the Arctic, respectively a waterway with floating ice floes and an open-water channel created by icebreakers. Additionally, to build understanding of the Arctic sea condition, two other models were developed simulating the interaction of ocean waves with a rigid ice floe and then an elastic ice sheet, which provided a new solver capable of modelling hydroelastic fluid-structure interactions. Based on validation against experiments, these models provided the ability to accurately predict the ship-wave-ice interactions and the ice-induced resistance changes. Through conducting a systematic series of simulations, it was found that ice floes can increase the ship resistance by the same order of magnitude as the open water resistance, but this is strongly dictated by the ship beam, ice concentration, ice thickness and floe diameter. An open-water ice channel was found to increase the ship resistance by up to 15% compared to the situation without ice, particularly when the channel width is less than 2.5 times the ship beam and the ice thickness is greater than 5% of the ship draught. Moreover, this work developed a procedure to derive simple ice-resistance equations from the simulation results, enabling fast prediction of ship fuel consumption in sea ice fields and incorporation into a new Arctic Voyage Planning Tool.
... Further details can be found in e.g. Jasak [73]. ...
Article
To simulate the interaction of ocean waves with marine structures, coupling approaches between a potential flow model and a viscous model are investigated. The first model is a full y nonlinear potential flow (FNPF) model based on the Harmonic Polynomial Cell (HPC) method, which is highly accurate and best suited for representing long distance wave propagation. The second model is a CFD code, solving the Reynolds-Averaged Navier–Stokes (RANS) equations within the OpenFOAM® toolkit, more suited to represent viscous and turbulent effects at local scale in the body vicinity. Two one-way coupling strategies are developed and compared in two dimensions, considering fully submerged and fixed structures. A domain decomposition (DD) strategy is first considered, introducing a refined mesh in the body vicinity on which the RANS equations are solved. Boundary conditions and interpolation operators from the FNPF results are developed in order to enforce values at its outer boundary. The second coupling strategy considers a decomposition of variables (functional decomposition, FD) on the local grid. As the FNPF simulation provides fields of variables satisfying the irrotational Euler equations, complementary velocity and pressure components are introduced as the difference between the total flow variables and the potential ones. Those complementary variables are solutions of modified RANS equations. Extensive comparisons are presented for nonlinear waves interacting with a horizontal cylinder of rectangular cross-section. The loads exerted on the body computed from the four simulation methods (standalone FNPF, standalone CFD, DD and FD coupling schemes) are compared with experimental data. It is shown that both coupling approaches produce an accurate representation of the loads and associated hydrodynamic coefficients (inertia and drag) over a large range of incident wave steepness and Keulegan–Carpenter number, for a small fraction of the computational time needed by the complete CFD simulation.
... The interFoam solver in OpenFOAM is an air-water two-phase flow solver and is the basic free surface flow solver used in this study. It solves the RANS equations for two incompressible phases using a finite volume discretization and the Volume of Fluid (VOF) surface capturing method (Jasak, 1996;Rusche, 2002). The RANS equations describing mass continuity and conservation of momentum for an incompressible fluid are given by ...
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It is increasingly popular to use Computational Fluid Dynamics (CFD) models to study floating structures subjected to ocean waves, especially when it comes to applications of floating offshore wind turbines and Wave Energy Converters (WECs). Mooring dynamics are currently lacking in most of these applications. This paper presents a coupled simulation study of moored floating body motion by coupling two open-source libraries: a finite volume CFD toolbox, OpenFOAM, and a lumped-mass mooring model, MoorDyn. The instantaneous floating body position and velocity are passed from the body motion solver in the CFD model to the mooring model to calculate the fairlead kinematics. The mooring reaction forces, which are calculated by MoorDyn after updating the mooring system states, are then returned to the body motion solver to update the floating body motion. Both mesh deformation and overset mesh methods are used as the mesh motion solver in the CFD model to account for the floating body motion. The coupled model was validated against experimental measurements for a floating box moored with four catenary lines under the action of regular waves, which came from a preliminary test campaign for WECs. Apart from the lumped-mass mooring model, the present work also coupled a quasi-static mooring model and a finite element model with the floating body motion solver in OpenFOAM. The mooring line tensions predicted by these models were compared. The coupled model equipped with three mooring line codes may be further used to carry out survivability studies of FOWTs and WECs subject to severe sea states.
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The behaviour of submerged granular flow is strongly dependent on the solid volume fraction and the viscosity discontinuity over a wide range of flow regimes. To obtain a general description of this type of flow, this study proposes a new model to compute solid effective stresses of submerged granular materials across multiple flow regimes. Here, based on the critical state soil mechanics framework, a new equation is proposed to describe the evolution of elastic reference of materials caused by elastoplastic deformation. The evolution of elastic reference subsequently informs the development of static pressure, and together with the dynamic pressure computed using a well-established blended model, resulting in a new approach to compute the solid pressure induced by both dynamic and static effects. The proposed model is then implemented in the Eulerian-Eulerian approach using the finite volume method to simulate the collapses of submerged granular columns, covering different flow regimes from quasi-static to viscous depositions. Simulation results agreeing well with experimental and numerical data in the literature are a testament to the performance of a well-developed constitutive law. Besides, the simulation results comprehensibly demonstrate the important role of interstitial fluid flow as well as the initial solid volume fraction in the collapsing process across different flow regimes with different packing densities. Furthermore, the effects of initial volume fraction, fluid pressure and phase interaction forces on the flow responses are also discussed.
Chapter
Concrete gravity dams are usually designed to be non-overflow sections. However, due to global warming, flood events have become more frequent and more intense, possibly surpassing the amount of flow for which old structures were designed. Estimating the hydrodynamic forces acting on a dam or spillway during a flood is challenging. Typically, either simplified analytical solutions or complex and expensive physical models have been used. However, Computational Fluid Dynamics (CFD) is now an attractive alternative that can yield more accurate results than simplified analytical solutions while being cheaper and faster to implement than physical models. This paper presents a back analysis of the Chute Garneau concrete spillway during the Saguenay flood of 1996. During this event, the spillway bridge, which is 6.35 m high, was overtopped by about 2 m. There was a significant accumulation of floating debris that got stuck on the gates lifting structure. Some gates could not be opened because of ongoing rehabilitation work. Despite the severity of the flood, the structure survived the event. Herein, a CFD analysis is performed to obtain the hydrodynamic pressure fields on the structure under different scenarios, such as with gates open, closed, partially closed and with accumulation of floating debris. Then, the hydrodynamic pressure is integrated to obtain the resultant forces which are used as input to a stability analysis using the gravity method. It is shown that a small amount of cohesion and tensile strength on the rock-concrete interface, both very uncertain parameters, were mobilized to keep the structure stable.
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