Rectangular Grid System Research Articles

Discrete Vortex Prediction of Flows around a Cylinder Near a Wall Using Overlapping Grid System

Modelling of unidirectional and oscillatory flows around a cylinder near a wall using an overlapping grid system is carried out. The circular grid system of the cylinder was overlapped with the rectangular grid system of the wall. The use of such an overlapping grid system is intended to reduce the CPU time compared to the cloud scheme in which vortex-to-vortex interaction is used, i.e., especially in calculating the shedding vortex velocity, since calculating the vortices velocity takes the longest CPU time. This method is not only time efficient, but also gives a better distribution of surface vorticity as the scattered vortices around the body are now concentrated on a grid point. Therefore, grid-to-grid interaction is used instead of vortex-to-vortex interaction. Velocity calculation was also carried out using this overlapping grid in which the new incremental shift position was summed up to obtain the total new vortices position. The engineering applications of this topic are to simulate the loading of submarine pipeline placed close to the seabed or to simulate the flow as a result of the scouring process below the cylinder since there is space for the fluid to flow beneath it. The in-line and transverse force coefficients are found by integrating the pressure around the cylinder surface. The flow patterns are then obtained and presented. The comparison of the results with experimental evidence is presented and the range of good results is discussed.

Discrete Vortex Method Of Flow Around A Cylinder In A Channel Using Simple Grid System

Modelling of unidirectional and oscillatory flows around a cylinder in a channel using a simple overlapping grid system are carried out. The importance of this cylinder-wall configuration is the effect of blockage which suppress the development of the vortex shedding The polar grid system of the cylinder is then overlapped with the rectangular grid system of the wall. The length of rectangular grid element is about the same as the length of the polar grid system in the cylinder surface. The use of such overlapping grid system is for reducing the CPU time, i.e. in calculating the vortex velocity since the CPU time in calculating the vortices velocity takes the longest time. This method is not only time efficient, but also gives a better distribution of surface vorticity as the scattered vortices around the body are now concentrated on grid point. In this study there is no vortex-to-vortex interaction, but instead it uses node-to-node interactions. Velocity calculation also uses this overlapping grid in which the new incremental shift position then summed up to get the total new vortices position. In this overlapping system the grid can be either off or on depend on process involved to get rid of the nodes not being used. The engineering applications of this topic is to simulate the loading pipeline placed in the channel such as in the heat exchanger or simulation of U-tube experiment or other system. The in-line and transverse force coefficients are found by integrating the pressure around the cylinder surface. The flow patterns are then can be obtained and presented. The comparison of the results with experimental evidence is presented and also the range of good results is discussed.

A Method to Solve the Reaction-Diffusion-Chemotaxis System

Abstract The objective of this article is to use a recent developed scheme to simulate reaction-diffusion-chemotaxis equations. The solution gradient required for an accurate discretization is computed directly as an additional variable rather than interpolated from solution values around neighboring computational nodes. To achieve this goal, a supplementary equation and its associated control volume are introduced to retain a compact and accurate discretization. Scheme essentials are exposed by the numerical analysis on two-dimensional chemotaxis problems to reveal its formal accuracy. Due to its highly comprehensible and practical features, this formulation can be easily extended to solve problems for other two-dimensional rectangular grid systems. One- and two-dimensional problems are solved to verify its simulation accuracy and to study the possible formation of bacteria bands. We further perform the linearization technique to the reaction term to increase the stability of the current scheme. From the numerical analysis and computational results, it is found that the present formulation is a useful tool to solve reaction-diffusion-chemotaxis equations.

What is the shape of a bundle? An analysis of Rosen's fibrous composites experiments using the chain‐of‐bundles model

AbstractIn the 1960s, W. B. Rosen conducted some remarkable experiments on unidirectional fibrous composites that gave seminal insights into their failure under increasing tensile load. These insights led him to a grid system where the nodes in the grid were ineffective length fibers and to model the composite as something he called a chain‐of‐bundles model (i.e., a series system of parallel subsystems of horizontal nodes that he referred to as bundles), where the chain fails when one of the bundles fails. A load‐sharing rule was used to quantify how the load is borne among the nodes. Here, Rosen's experiments are analyzed to determine the shape of a bundle. The analysis suggests that the bundles are not horizontal collection of nodes but rather small rectangular grid systems of nodes where the load‐sharing between nodes is local in its form. In addition, a Gibbs measure representation for the joint distribution of binary random variables is given. This is used to show how the system reliability for a reliability structure can be obtained from the partition function for the Gibbs measure and to illustrate how to assess the risk of failure of a bundle in the chain‐of‐bundle model.

Construction methodology for lip surface of a submerged inlet

The design of submerged inlet remains a challenge as the design involves both the fuselage and the inlet internal duct. The inlet lip surface blending needs to meet stringent conditions, which vary considerably from fore lip to aft lip. The need for high aerodynamics performance also adds to the difficulty of the design process. In this paper a methodology is presented by which the inlet lip surface is mapped to a rectangular grid system which is topologically similar to the surface to be blended. A number of key grid lines can be identified, through the construction of the key curves, which are outer profile curve, inner profile curve, ridge curve and multiple blending curves. Furthermore, lip surface can be blended through the application of sweeping algorithm. A detailed example as presented in this paper illustrates the whole design procedure and technique to transform the design of a spatial curve into that of two planar curves on two developable surfaces. The above method facilitates the design of spatial curves for outer profile curve and ridge curve. The lip surface blending leaves a number of design parameters which can be manipulated to offer the scope of optimization of the aerodynamic performance in the subsequent CFD phase of the development. The numerical simulation conducted indicates that actual performances meet the design objectives well, which demonstrates the effectiveness of this design methodology.

Finite analytic numerical method for three-dimensional fluid flow in heterogeneous porous media

Understanding fluid flows in heterogeneous porous media is fundamental to applied geosciences. The wide connectivity variations in the natural aquifer or oil reservoirs make the equivalent permeability have strong spatial variations. When performing the simulations for subsurface flows, the permeabilities may have strong discontinuities across the interfaces between different grid cells. Utilizing the traditional numerical schemes to simulate flows in strong heterogeneous media, the refinement ratio for the grid cell needs to increase dramatically to get an accurate result. Recently, we proposed a finite analytic numerical scheme to solve the two-dimensional fluid flows in heterogeneous porous media. With only 2×2 or 3×3 subdivisions, this scheme can provide rather accurate solutions. In this paper, we develop the finite analytic numerical method for solving the three-dimensional fluid flows in heterogeneous porous media. For the rectangular grid system, it is generally proposed that the pressure gradient in a plane normal to the edge joining different permeability regions will tend to infinite as approaching the edge according to a typical power-law solution and the tangential derivate of the pressure along the edge must be of limited value due to the pressure continuity. Consequently, the three-dimensional flow will reduce to the two-dimensional one in the neighborhood around each edge. Such quasi-two-dimensional behavior is then applied to construct a finite analytic numerical scheme. Numerical examples show that the proposed scheme can provide rather accurate solutions with only 2×2×2 or 3×3×3 subdivisions and the convergent speed is independent of the permeability heterogeneity. Due to its high calculation efficiency, the proposed scheme is utilized to test the well known LLM (Landau, Lifshitz and Matheron) conjecture, which provides keq/kG=exp⁡(16σln⁡k2) for the isotropic log-normal porous medium. The numerical results do not support this conjecture for large σln⁡k, but strongly suggest the linear relation keq/kG=1+16σln⁡k2.

Modified classification system for estimating the CO2 storage capacity of saline formations

In this study, a modified classification system is proposed for estimating the CO2 storage capacity of saline formations. This system seeks to further integrate existing CO2 storage capacity classification systems with the Petroleum Resources Management System. The synthesis is a rectangular grid system that includes smaller rectangles, which represent the different classes. Three sides of the diagram of the classification system correspond to three information axes: the CO2 storage resources/capacities of the saline formations (bottom), estimation methods (left), and the project maturity (right). Estimating CO2 storage capacity using this system begins by evaluating the total storage amount, which is defined at the bottom of the grid system. When more data become available as the project matures, the different classes of storage capacity (Total Storage Resource, Characterized Storage Resource, Effective Storage Resource, etc.) are sequentially obtained by applying different methods, criteria, and constraints, and the evaluation stage moves from one stage to the next. A case study is done to estimate the different classes of CO2 storage capacity shown on the modified classification system.

Nonlinear ship waves and computational fluid dynamics.

Research works undertaken in the first author’s laboratory at the University of Tokyo over the past 30 years are highlighted. Finding of the occurrence of nonlinear waves (named Free-Surface Shock Waves) in the vicinity of a ship advancing at constant speed provided the start-line for the progress of innovative technologies in the ship hull-form design. Based on these findings, a multitude of the Computational Fluid Dynamic (CFD) techniques have been developed over this period, and are highlighted in this paper. The TUMMAC code has been developed for wave problems, based on a rectangular grid system, while the WISDAM code treats both wave and viscous flow problems in the framework of a boundary-fitted grid system. These two techniques are able to cope with almost all fluid dynamical problems relating to ships, including the resistance, ship’s motion and ride-comfort issues. Consequently, the two codes have contributed significantly to the progress in the technology of ship design, and now form an integral part of the ship-designing process.

Heat Transfer and Particle Behaviours in Dispersed Two-Phase Flow with Different Heat Conductivities for Liquid and Solid

Liquid-solid two-phase flow with heat transfer is directly simulated, to investigate the effects of the ratios of heat conductivities (solid to liquid) and bulk solid volume fraction from dense to dilute situations. The interaction between fluid and particles is solved by our original immersed solid approach on a rectangular grid system. A discrete element method with a soft-sphere collision model is applied for particle-particle and particle-wall interactions. Governing equation of temperature is time-updated with an implicit treatment for the diffusion term, which enables robust simulation with particles of very high/low ratios of heat conductivities (from 1/1000 to 1000) to the fluid. The local heat flux at the fluid-solid interface is modelled by a new flux decomposition technique, and incorporated into the implicit scheme of the temperature. The method is applied to a 2-D particulate flow in a natural convection in a square domain at a relatively low Rayleigh number. In the dense condition, for the cases with high ratios of heat conductivity, the heat transfer is promoted by strong convection, while the particles of low ratios of heat conductivity tend to hinder the development of the temperature rise in the flow field, causing a weak convection and low Nusselt number. Under a condition of relatively low solid volume fraction, fixed particles only depress the heat convection as the number of particles and heat conductivity ratio increase. For the cases with freely-moving particles, on the other hand, heat conductivity of particles has a stronger influence on the heat transfer of the system than the number of particles. The above simulation results highlight the effect of temperature distributions within the particles and liquid.

Effect of temperature gradient within a solid particle on the rotation and oscillation modes in solid-dispersed two-phase flows

Liquid–solid two-phase flow with heat transfer is simulated, and the effect of temperature gradient within a solid particle on the particle behaviour and heat transfer is studied. The interaction between fluid and particles is considered with our original immersed solid approach on a rectangular grid system. The local heat flux at the fluid–solid interface is described with an anisotropic heat conductivity matrix, and the governing equation of temperature is time-updated with an implicit treatment for the diffusion term. The method is applied to a 2-D natural convection flow of a relatively low Rayleigh number including multiple particles. Heat transfer and particle behaviours are studied for different solid heat conductivities (ratio to the fluid conductivity ranging between 10−3 and 103) and solid volume fractions. Under a condition of relatively low heat conductivity ratio, the particles show a simple circulating flow. By increasing the heat conductivity ratio, a transition of the particulate flow is observed to oscillation mode around the domain centre due to the buoyancy force as a restitution force. The oscillation period is found to vary with the heat conductivity ratio, and it is related to the time scales for the heat transfer via fluid and solid.

Internal generation of waves on an arced band

We develop a technique of generating waves internally on an arced band in a rectangular grid system using a Gaussian-shaped source function. Tests are made for four different types of wave generation layouts. Numerical experiments are conducted under the following conditions: the propagation of linear and cnoidal waves on a flat bottom, and the refraction and shoaling of cnoidal waves on a planar slope. Numerical experiments are conducted using the linear extended mild-slope equations of Suh et al. (1997) [Coastal Eng. 32, 91–117], and nonlinear extended Boussinesq equations of Nwogu (1993) [J. Waterw. Port Coastal Ocean Eng. 119, 618–638]. The third layout type consisting of two orthogonal bands connected to a quadrant band show the best solutions especially for a fine grid size. Use of the present method on a source band gives smaller errors than Lee and Yoon's (2007) [Coastal Eng. 54, 357–368], method on a source line for a coarse grid size but the two methods yield similar results for a fine grid size. Generation of cnoidal waves gives larger errors than linear waves. To reduce these errors, it is necessary to use a fine grid size and nonlinear wave numbers for incident waves.

A numerical simulation method for free surface flows near a two-dimensional moving body in fixed rectangular grid system

A numerical simulation method is developed for the description of a two-dimensional body moving in free surface flows using a fixed rectangular grid system. The nonlinear free surface is determined by the modified Marker-density method. The body boundary is defined by the line segments connecting the points where the body surface and grid lines meet. The Navier–Stokes equations and the continuity equation are used as the governing equations, which are coupled with a two-step projection method described in detail in this research. The velocities and pressures in the body boundary cells as well as the free surface cells are calculated using a simultaneous iterative method. For the verification of the present numerical method, four kinds of simulations have been conducted. The first case is to investigate the vortex shedding of an advancing cylinder, the second case is about the forced oscillation of a box, the third case is to clarify the sloshing phenomenon in a horizontally excited tank, and the last case is waves generation by a plunging-type wave maker. The simulation results are confirmed with corresponding experimental results or analytical solutions.

Algorithmic Improvements and PDE Decoupling, for the Simulation of Porous Electrode Cells

A reformulation of porous electrode theory is presented which allows efficient simulation of numerous battery systems. An efficient finite-difference rectangular grid system is revealed which for 1D systems eliminates banded matrices and reduces all relevant implicit solutions to much simpler tridiagonal systems. Use of Greens function methods to decouple the liquid phase diffusion from the electrode potentials, and a similar decoupling of potentials from the solid phase diffusion are described. The so called hot loop in most full physics simulations of porous electrode batteries results from solving the spherical diffusion equation for concentration of active species in cathode and anode particles. A non linear grid is described which results in very accurate results for the solid phase surface concentrations, from a surprisingly small number of grid points.

Development of a Compact and Accurate Discretization for Incompressible Navier-Stokes Equations Based on an Equation-Solving Solution Gradient, Part I: Kernel Scheme Fundamentals

This is the first part of our attempt to develop a compact and accurate numerical scheme for incompressible Navier-Stokes equations in complicated domains. The objective of this article is to unveil the kernel scheme fundamentals in our formulation, where the solution gradient required for an accurate discretization is computed directly as an additional variable rather than interpolated from solution values at neighboring computational nodes. To achieve this goal, a supplementary equation and its associated control volume are introduced to retain a compact and accurate discretization. Scheme essentials are exposed by numerical analyses on simple one-dimensional modeled problems to reveal its formal accuracy. Due to its highly comprehensible and practical features, this formulation can be easily extended to solve problems in two-dimensional rectangular grid systems. Several one- and two-dimensional problems are solved to verify its simulation accuracy. From the numerical analyses and computational results of test problems, it is found that the present formulation is a useful tool to solve convection-diffusion equations and can be employed as the kernel scheme for fluid flow simulations.

Optimization of Bathymetry Database for Coastal Areas

Problem statement: A database containing gridded bathymetries with different coverages and resolutions has been established. Approach: The interpolation techniques of the gridded bathymetries have been developed to define the spherical, rectangular and curvilinear coordinate grid systems. Results: Subsequently, merging of the bathymetry results has also been developed to define the optimal bathymetry. Furthermore, the bathymetries were established in order to apply for the Gulf of Thailand (GoT) and also for the storm wave and surge models of the Coasts of Thailand (CoT). Conclusion: The bathymetry covering the coastal waters was produced to extend the applicability of the operational 2D and 3D ocean models. In addition, the data sets used in this study were integrated and collected in order to be the database and post processing tools to convert the output to the specific formats required for various operational models at the CoT.

LAMINAR FLOW PAST AN OSCILLATING CIRCULAR CYLINDER IN CROSS FLOW

The present study numerically investigates the twodimensional laminar flow past a circular cylinder forced to oscillate transverse to the free-stream. The numerical simulations are performed at a various range of cylinder oscillation frequency ranged between 0.8 and 1.2 of the natural vortex shedding frequency, and the oscillation amplitude extended up to 50% of the cylinder diameter at one Reynolds number of 185 showing the typical two-dimensional vortex shedding. The immersed boundary method is used to handle the oscillating cylinder in a rectangular grid system using the finite volume method. The primary vortex shedding frequency has the same value with the exciting frequency. When the exciting frequency exceeds the natural vortex shedding frequency, the secondary vortex shedding frequency appeared with the value less than the natural shedding frequency. The time sequence of the wake structures near the cylinder at the extreme upper position reveals that a single vortex or a pair of vortices appears according to the time. A pair of vortices is composed of two saddle points and two centers of vortices, showing the occurrence of vortex switching phenomenon. The quantitative information about the flow variables such as the distribution of wall vorticity on the cylinder surface, drag and lift coefficients is highlighted.

Effect of wave amplitude on turbulent flow in a wavy channel by direct numerical simulation

The present study numerically investigates the characteristics of three-dimensional turbulent flow in a wavy channel. For the purpose of a careful observation of the effect of the wave amplitude on the turbulent flow, numerical simulations are performed at a various range of the wave amplitude to wavelength ratio (0.01⩽ α/ λ⩽0.05), where the wavelength is fixed with the same value of the mean channel height ( H). The immersed boundary method is used to handle the wavy surface in a rectangular grid system, using the finite volume method. The Reynolds number ( Re= U bH / ν) based on the bulk velocity ( U b ) is fixed at 6760. The present computational results for a wavy surface are well compared with those of references. When α/ λ=0.02, the small recirculating flow occurs near the trough at the instant, but the mean reverse flow is not observed. In the mean flow field, the reverse flow appears from α/ λ=0.03 among the wave amplitude considered in this study. The domain of the mean reverse flow defined by the locations of separation and reattachment depends strongly on the wave amplitude. The pressure drag coefficient augments with increasing the wave amplitude. The friction drag coefficient shows the increase and decrease behavior according to the wave amplitude. The quantitative information about the flow variables such as the distribution of pressure and shear stress on the wavy surface is highlighted.

Waviness effect of a wavy circular cylinder on the heat transfer at a Reynolds number of 300

The present study investigates three-dimensional characteristics of fluid flow and heat transfer around a wavy cylinder which has the sinusoidal variation in the cross sectional area along the spanwise direction. The three different wavelengths of π/4, π/3 and π/2 at the fixed wavy amplitude of 0.1 have been considered to investigate the effect of waviness on especially the forced convection heat transfer around a wavy cylinder when the Reynolds and Prandtl numbers are 300 and 0.71, respectively. The numerical solution for unsteady forced convective heat transfer is obtained using the finite volume method. The immersed boundary method is used to handle the wavy cylinder in a rectangular grid system. The present computational results for a wavy cylinder are compared with those for a smooth cylinder. The fluid flow and heat transfer around the wavy cylinder depends on both the location along the spanwise direction and the wavelength. The time- and total surface-averaged Nusselt number for a wavy cylinder with λ = π/2 is larger than that for a smooth cylinder, whereas that with λ = π/4 and π/3 is smaller than that for a smooth cylinder. However, because the surface area exposed to heat transfer for a wavy cylinder is larger than that for a smooth cylinder, the total heat transfer rate for a wavy cylinder with different wavelengths of λ = π/4, π/3 and π/2 is larger than that for a smooth cylinder.

Internal generation of waves on an arc in a rectangular grid system

This paper presents a technique to generate waves at oblique angles in finite difference numerical models in a rectangular grid system by using internal generation technique [Lee, C., Suh, K.D., 1998. Internal generation of waves for time-dependent mild-slope equations. Coast. Eng. 34, 35–57.] along an arc-shaped line source. Tests were made for four different types of wave generation layouts. Quantitative experiments were conducted under the following conditions: the propagation of waves on a flat bottom, the refraction and shoaling of waves on a planar slope, and the diffraction of waves to a semi-infinite breakwater. Numerical experiments were conducted using the extended mild-slope equations of Suh et al. [Suh, K.D., Lee, C., Park, W.S., 1997. Time-dependent equations for wave propagation on rapidly varying topography. Coast. Eng. 32, 91–117.]. The fourth layout type consisting of two parallel lines connected to a semicircle showed the best solutions, especially for a small grid size. This technique is useful for the numerical simulation of irregular waves with broad-banded directional spectrum using conventional spectral wave models for the reasonable estimation of bottom friction and wave-breaking.

Improved numerical modelling of estuarine flows

Three key improvements to a widely used numerical model for predicting depth-averaged shallow water flows in estuarine and coastal waters are outlined herein. They include the adoption of a new grid layout for calculating friction and viscosity, a new method for simulating flooding and drying processes and a better representation of irregular wall boundaries. Although these improvements are currently implemented in the Depth Integrated Velocities And Solute Transport (DIVAST) model, they can easily be extended to other numerical models in the framework of a finite-difference or finite-volume method on a space-staggered rectangular grid system. Much effort has been made to keep the improvements simple and efficient so that they can be used in long-term and large-area simulations of estuarine and coastal flows. Numerical tests have been undertaken to verify the performance of these refinements for idealised bed forms, different beach configurations and irregular shoreline boundaries. The results show that for all conditions these refinements produce either the same or better results than the original model. Finally, the refined numerical model was used to simulate the tidal flow in a natural coastal basin. On the whole, the predicted variations in the inundation areas and the velocity fields were reproduced more accurately for different stages of the tide. At field measurement sites, the predicted water levels and velocities agreed favourably with the measured data.