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Cartesian Grid (cartesian + grid)
Selected AbstractsA versatile software tool for the numerical simulation of fluid flow and heat transfer in simple geometriesCOMPUTER APPLICATIONS IN ENGINEERING EDUCATION, Issue 1 2010A. M. G. Lopes Abstract The present work describes a software tool aimed at the simulation of fluid flow and heat transfer for two-dimensional problems in a structured Cartesian grid. The software deals with laminar and turbulent situations in steady-state or transient regime. An overview is given on the theoretical principles and on the utilization of the program. Results for some test cases are presented and compared with benchmarking solutions. Although EasyCFD is mainly oriented for educational purposes, it may be a valuable tool for a first analysis of practical situations. EasyCFD is available at www.easycfd.net. © 2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 18: 14,27, 2010; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/cae.20230 [source] Simulating Seepage into Mine Shafts and Tunnels with MODFLOWGROUND WATER, Issue 3 2010Jacob Zaidel In cases when an equivalent porous medium assumption is suitable for simulating groundwater flow in bedrock aquifers, estimation of seepage into underground mine workings (UMWs) can be achieved by specifying MODFLOW drain nodes at the contact between water bearing rock and dewatered mine openings. However, this approach results in significant numerical problems when applied to simulate seepage into an extensive network of UMWs, which often exist at the mine sites. Numerical simulations conducted for individual UMWs, such as a vertical shaft or a horizontal drift, showed that accurate prediction of seepage rates can be achieved by either applying grid spacing that is much finer than the diameter/width of the simulated openings (explicit modeling) or using coarser grid with cell sizes exceeding the characteristic width of shafts or drifts by a factor of 3. Theoretical insight into this phenomenon is presented, based on the so-called well-index theory. It is demonstrated that applying this theory allows to minimize numerical errors associated with MODFLOW simulation of seepage into UMWs on a relatively coarse Cartesian grid. Presented examples include simulated steady-state groundwater flow from homogeneous, heterogeneous, and/or anisotropic rock into a vertical shaft, a horizontal drift/cross-cut, a ramp, two parallel drifts, and a combined system of a vertical shaft connected to a horizontal drift. [source] Numerical simulation of drop deformation and breakup in shear flowHEAT TRANSFER - ASIAN RESEARCH (FORMERLY HEAT TRANSFER-JAPANESE RESEARCH), Issue 5 2007Lin Chang-Zhi Abstract Three-dimensional numerical simulation of the deformation and breakup of an isolated liquid drop suspended in immiscible viscous fluid under shear flow was performed with diffuse interface method. The governing equations of the model were described by Navier, Stokes, Cahn, Hilliard equations. The surface tension was treated as a modified stress. In this paper, a uniform staggered Cartesian grid was used. The transient Navier, Stokes equations were solved by an approximation projection method based on pressure increment formulation, while the Cahn, Hilliard equations were solved by a nonlinear full approximation multigrid method. The numerical results of the drop deformation and breakup were in good agreement with the experimental measurements. Therefore, the present model could be perfectly applied to study the mechanism of drop deformation and breakup. © 2007 Wiley Periodicals, Inc. Heat Trans Asian Res, 36(5): 286, 294, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20160 [source] Numerical simulation of free-surface flow using the level-set method with global mass correctionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2010Yali Zhang Abstract A new numerical method that couples the incompressible Navier,Stokes equations with the global mass correction level-set method for simulating fluid problems with free surfaces and interfaces is presented in this paper. The finite volume method is used to discretize Navier,Stokes equations with the two-step projection method on a staggered Cartesian grid. The free-surface flow problem is solved on a fixed grid in which the free surface is captured by the zero level set. Mass conservation is improved significantly by applying a global mass correction scheme, in a novel combination with third-order essentially non-oscillatory schemes and a five stage Runge,Kutta method, to accomplish advection and re-distancing of the level-set function. The coupled solver is applied to simulate interface change and flow field in four benchmark test cases: (1) shear flow; (2) dam break; (3) travelling and reflection of solitary wave and (4) solitary wave over a submerged object. The computational results are in excellent agreement with theoretical predictions, experimental data and previous numerical simulations using a RANS-VOF method. The simulations reveal some interesting free-surface phenomena such as the free-surface vortices, air entrapment and wave deformation over a submerged object. Copyright © 2009 John Wiley & Sons, Ltd. [source] A level set-based immersed interface method for solving incompressible viscous flows with the prescribed velocity at the boundaryINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2010Zhijun Tan Abstract A second-order accurate immersed interface method (IIM) is presented for solving the incompressible Navier,Stokes equations with the prescribed velocity at the boundary, which is an extension of the IIM of Le et al. (J. Comput. Phys. 2006; 220:109,138) to a level set representation of the boundary in place of the Lagrangian representation of the boundary using control points on a uniform Cartesian grid. In order to enforce the prescribed velocity boundary condition, the singular forces at the immersed boundary are applied on the fluid. These forces are related to the jump in pressure and the jumps in the derivatives of both the pressure and velocity, and are approximated via using the local Hermite cubic spline interpolation. The strength of singular forces is determined by solving a small system of equations at each time step. The Navier,Stokes equations are discretized via using finite difference method with the incorporation of jump conditions on a staggered Cartesian grid and solved by a second-order accurate projection method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate the viscous flows in irregular domains. Copyright © 2009 John Wiley & Sons, Ltd. [source] A Cartesian grid technique based on one-dimensional integrated radial basis function networks for natural convection in concentric annuliINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2008N. Mai-Duy Abstract This paper reports a radial basis function (RBF)-based Cartesian grid technique for the simulation of two-dimensional buoyancy-driven flow in concentric annuli. The continuity and momentum equations are represented in the equivalent stream function formulation that reduces the number of equations from three to one, but involves higher-order derivatives. The present technique uses a Cartesian grid to discretize the problem domain. Along a grid line, one-dimensional integrated RBF networks (1D-IRBFNs) are employed to represent the field variables. The capability of 1D-IRBFNs to handle unstructured points with accuracy is exploited to describe non-rectangular boundaries in a Cartesian grid, while the method's ability to avoid the reduction of convergence rate caused by differentiation is instrumental in improving the quality of the approximation of higher-order derivatives. The method is applied to simulate thermally driven flows in annuli between two circular cylinders and between an outer square cylinder and an inner circular cylinder. High Rayleigh number solutions are achieved and they are in good agreement with previously published numerical data. Copyright © 2007 John Wiley & Sons, Ltd. [source] Large eddy simulation of turbulent flows in complex and moving rigid geometries using the immersed boundary methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2005Mayank Tyagi Abstract A large eddy simulation (LES) methodology for turbulent flows in complex rigid geometries is developed using the immersed boundary method (IBM). In the IBM body force terms are added to the momentum equations to represent a complex rigid geometry on a fixed Cartesian mesh. IBM combines the efficiency inherent in using a fixed Cartesian grid and the ease of tracking the immersed boundary at a set of moving Lagrangian points. Specific implementation strategies for the IBM are described in this paper. A two-sided forcing scheme is presented and shown to work well for moving rigid boundary problems. Turbulence and flow unsteadiness are addressed by LES using higher order numerical schemes with an accurate and robust subgrid scale (SGS) stress model. The combined LES,IBM methodology is computationally cost-effective for turbulent flows in moving geometries with prescribed surface trajectories. Several example problems are solved to illustrate the capability of the IBM and LES methodologies. The IBM is validated for the laminar flow past a heated cylinder in a channel and the combined LES,IBM methodology is validated for turbulent film-cooling flows involving heat transfer. In both cases predictions are in good agreement with measurements. LES,IBM is then used to study turbulent fluid mixing inside the complex geometry of a trapped vortex combustor. Finally, to demonstrate the full potential of LES,IBM, a complex moving geometry problem of stator,rotor interaction is solved. Copyright © 2005 John Wiley & Sons, Ltd. [source] Large eddy simulations of turbulent swirling flows in a dump combustor: a sensitivity studyINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2005P. Wang Abstract Large eddy simulations (LES) of confined turbulent swirling flows in a model dump combustor are carried out. The simulations are based on a high-order finite difference method on a Cartesian grid, with the sub-grid scale stress tensor modelled using a scale-similarity model. The aims of this work are to study the physics of the flow and to evaluate the performance of LES method for simulation of the major features of turbulent swirling flows,the vortex breakdown, the highly anisotropic and fast-decaying turbulence structure. Influences of inflow/outflow conditions, combustor geometry, inlet swirl profile and Reynolds numbers on the vortex breakdown and turbulence structures are investigated. At very high swirl levels, the influence of the outflow conditions and the outlet geometry is fairly significant, not only at downstream near the outlet, but also at far upstream. At low Reynolds numbers, the onset of vortex breakdown is fairly sensitive to the change of Reynolds number; however, at high Reynolds numbers it is rather insensitive to the Reynolds number. Comparisons of LES results with experimental data are made. The LES results are shown to be in reasonably good agreement with the experimental data if appropriate inflow and outflow boundary conditions are imposed. Copyright © 2004 John Wiley & Sons, Ltd. [source] An implicit three-dimensional fully non-hydrostatic model for free-surface flowsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2004Hengliang Yuan Abstract An implicit method is developed for solving the complete three-dimensional (3D) Navier,Stokes equations. The algorithm is based upon a staggered finite difference Crank-Nicholson scheme on a Cartesian grid. A new top-layer pressure treatment and a partial cell bottom treatment are introduced so that the 3D model is fully non-hydrostatic and is free of any hydrostatic assumption. A domain decomposition method is used to segregate the resulting 3D matrix system into a series of two-dimensional vertical plane problems, for each of which a block tri-diagonal system can be directly solved for the unknown horizontal velocity. Numerical tests including linear standing waves, nonlinear sloshing motions, and progressive wave interactions with uneven bottoms are performed. It is found that the model is capable to simulate accurately a range of free-surface flow problems using a very small number of vertical layers (e.g. two,four layers). The developed model is second-order accuracy in time and space and is unconditionally stable; and it can be effectively used to model 3D surface wave motions. Copyright © 2004 John Wiley & Sons, Ltd. [source] Modelling of lossy curved surfaces in the 3-D frequency-domain finite-difference methodsINTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 5 2006Riku M. Mäkinen Abstract A conformal first-order or Leontovic surface-impedance boundary condition (SIBC) for the modelling of fully three-dimensional (3-D) lossy curved surfaces in a Cartesian grid is presented for the frequency-domain finite-difference (FD) methods. The impedance boundary condition is applied to auxiliary tangential electric and magnetic field components defined at the curved surface. The auxiliary components are subsequently eliminated from the formulation resulting in a modification of the local permeability value at boundary cells, allowing the curved 3-D surface to be described in terms of Cartesian grid components. The proposed formulation can be applied to model skin-effect loss in time-harmonic driven problems. In addition, the impedance matrix can be used as a post-processor for the eigenmode solver to calculate the wall loss. The validity of the proposed model is evaluated by investigating the quality factors of cylindrical and spherical cavity resonators. The results are compared with analytic solutions and numerical reference data calculated with the commercial software package CST Microwave StudioÔ (MWS). The convergence rate of the results is shown to be of second-order for smooth curved metal surfaces. The overall accuracy of the approach is comparable to that of CST MWSÔ. Copyright © 2006 John Wiley & Sons, Ltd. [source] Gamma-SLAM: Visual SLAM in unstructured environments using variance grid mapsJOURNAL OF FIELD ROBOTICS (FORMERLY JOURNAL OF ROBOTIC SYSTEMS), Issue 1 2009Tim K. Marks This paper describes an online stereo visual simultaneous localization and mapping (SLAM) algorithm developed for the Learning Applied to Ground Robotics (LAGR) program. The Gamma-SLAM algorithm uses a Rao,Blackwellized particle filter to obtain a joint posterior over poses and maps: the pose distribution is estimated using a particle filter, and each particle has its own map that is obtained through exact filtering conditioned on the particle's pose. Visual odometry is used to provide good proposal distributions for the particle filter, and maps are represented using a Cartesian grid. Unlike previous grid-based SLAM algorithms, however, the Gamma-SLAM map maintains a posterior distribution over the elevation variance in each cell. This variance grid map can capture rocks, vegetation, and other objects that are typically found in unstructured environments but are not well modeled by traditional occupancy or elevation grid maps. The algorithm runs in real time on conventional processors and has been evaluated for both qualitative and quantitative accuracy in three outdoor environments over trajectories totaling 1,600 m in length. © 2008 Wiley Periodicals, Inc. [source] Deterministic and statistical methods for reconstructing multidimensional NMR spectra,MAGNETIC RESONANCE IN CHEMISTRY, Issue 3 2006Ji Won Yoon Abstract Reconstruction of an image from a set of projections is a well-established science, successfully exploited in X-ray tomography and magnetic resonance imaging. This principle has been adapted to generate multidimensional NMR spectra, with the key difference that, instead of continuous density functions, high-resolution NMR spectra comprise discrete features, relatively sparsely distributed in space. For this reason, a reliable reconstruction can be made from a small number of projections. This speeds the measurements by orders of magnitude compared to the traditional methodology, which explores all evolution space on a Cartesian grid, one step at a time. Speed is of crucial importance for structural investigations of biomolecules such as proteins and for the investigation of time-dependent phenomena. Whereas the recording of a suitable set of projections is a straightforward process, the reconstruction stage can be more problematic. Several practical reconstruction schemes are explored. The deterministic methods,additive back-projection and the lowest-value algorithm,derive the multidimensional spectrum directly from the experimental projections. The statistical search methods include iterative least-squares fitting, maximum entropy, and model-fitting schemes based on Bayesian analysis, particularly the reversible-jump Markov chain Monte Carlo procedure. These competing reconstruction schemes are tested on a set of six projections derived from the three-dimensional 700-MHz HNCO spectrum of a 187-residue protein (HasA) and compared in terms of reliability, absence of artifacts, sensitivity to noise, and speed of computation. Copyright © 2006 John Wiley & Sons, Ltd. [source] Matched interface and boundary (MIB) method for the vibration analysis of platesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 9 2009S. N. Yu Abstract This paper proposes a novel approach, the matched interface and boundary (MIB) method, for the vibration analysis of rectangular plates with simply supported, clamped and free edges, and their arbitrary combinations. In previous work, the MIB method was developed for three-dimensional elliptic equations with arbitrarily complex material interfaces and geometric shapes. The present work generalizes the MIB method for eigenvalue problems in structural analysis with complex boundary conditions. The MIB method utilizes both uniform and non-uniform Cartesian grids. Fictitious values are utilized to facilitate the central finite difference schemes throughout the entire computational domain. Boundary conditions are enforced with fictitious values,a common practice used in the previous discrete singular convolution algorithm. An essential idea of the MIB method is to repeatedly use the boundary conditions to achieve arbitrarily high-order accuracy. A new feature in the proposed approach is the implementation of the cross derivatives in the free boundary conditions. The proposed method has a banded matrix. Nine different plates, particularly those with free edges and free corners, are employed to validate the proposed method. The performance of the proposed method is compared with that of other established methods. Convergence and comparison studies indicate that the proposed MIB method works very well for the vibration analysis of plates. In particular, modal bending moments and shear forces predicted by the proposed method vanish at boundaries for free edges. Copyright © 2008 John Wiley & Sons, Ltd. [source] Semi-coupled air/water immersed boundary approach for curvilinear dynamic overset grids with application to ship hydrodynamicsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2008Juntao Huang Abstract For many problems in ship hydrodynamics, the effects of air flow on the water flow are negligible (the frequently called free surface conditions), but the air flow around the ship is still of interest. A method is presented where the water flow is decoupled from the air solution, but the air flow uses the unsteady water flow as a boundary condition. The authors call this a semi-coupled air/water flow approach. The method can be divided into two steps. At each time step the free surface water flow is computed first with a single-phase method assuming constant pressure and zero stress on the interface. The second step is to compute the air flow assuming the free surface as a moving immersed boundary (IB). The IB method developed for Cartesian grids (Annu. Rev. Fluid Mech. 2005; 37:239,261) is extended to curvilinear grids, where no-slip and continuity conditions are used to enforce velocity and pressure boundary conditions for the air flow. The forcing points close to the IB can be computed and corrected under a sharp interface condition, which makes the computation very stable. The overset implementation is similar to that of the single-phase solver (Comput. Fluids 2007; 36:1415,1433), with the difference that points in water are set as IB points even if they are fringe points. Pressure,velocity coupling through pressure implicit with splitting of operators or projection methods is used for water computations, and a projection method is used for the air. The method on each fluid is a single-phase method, thus avoiding ill-conditioned numerical systems caused by large differences of fluid properties between air and water. The computation is only slightly slower than the single-phase version, with complete absence of spurious velocity oscillations near the free surface, frequently present in fully coupled approaches. Validations are performed for laminar Couette flow over a wavy boundary by comparing with the analytical solution, and for the surface combatant model David Taylor Model Basin (DTMB) 5512 by comparing with Experimental Fluid Dynamics (EFD) and the results of two-phase level set computations. Complex flow computations are demonstrated for the ONR Tumblehome DTMB 5613 with superstructure subject to waves and wind, including 6DOF motions and broaching in SS7 irregular waves and wind. Copyright © 2008 John Wiley & Sons, Ltd. [source] High order interpolation methods for semi-Lagrangian models of mobile-bed hydrodynamics on Cartesian grids with cut cellsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10-11 2005Giorgio Rosatti Abstract High order approximation methods based on radial basis functions are applied to the extension of semi-Lagrangian shallow water models to staggered Cartesian meshes with cut boundary cells. The accuracy and efficiency of the resulting semi-Lagrangian method is demonstrated by test cases simulating open channel flow. The derivative reconstruction provided by radial basis function interpolators is also employed successfully in the discretization of sediment transport models for mobile bed river flow. Copyright © 2005 John Wiley & Sons, Ltd. [source] Semi-analytical method for departure point determinationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2005Nick Martin Abstract A new method for departure point determination on Cartesian grids, the semi-analytical upwind path line tracing (SUT) method, is presented and compared to two typical departure point determination methods used in semi-Lagrangian advection schemes, the Euler method and the four-step Runge,Kutta method. Rigorous comparisons of the three methods were conducted for a severely curving hypothetical flow field and for advective transport in the rotation of a Gaussian concentration hill. The SUT method was shown to have equivalent accuracy to the Runge,Kutta method but with significantly improved computational efficiency. Depending on the case being simulated, the SUT method provides either far greater or equivalent computational efficiency and more certain accuracy than the Euler method. Copyright © 2004 John Wiley & Sons, Ltd. [source] A cascadic conjugate gradient algorithm for mass conservative, semi-implicit discretization of the shallow water equations on locally refined structured gridsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1-2 2002Luca Bonaventura Abstract A semi-implicit, mass conservative discretization scheme is applied to the two-dimensional shallow water equations on a hierarchy of structured, locally refined Cartesian grids. Different resolution grids are fully interacting and the discrete Helmholtz equation obtained from the semi-implicit discretization is solved by the cascadic conjugate gradient method. A flux correction is applied at the interface between the coarser and finer discretization grids, so as to ensure discrete mass conservation, along with symmetry and diagonal dominance of the resulting matrix. Two-dimensional idealized simulations are presented, showing the accuracy and the efficiency of the resulting method. Copyright © 2002 John Wiley & Sons, Ltd. [source] A uniformly stable conformal FDTD-method in Cartesian gridsINTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 2 2003I.A. Zagorodnov Abstract A conformal finite-difference time-domain algorithm for the solution of electrodynamic problems in general perfectly conducting 3D geometries is presented. Unlike previous conformal approaches it has the second-order convergence without the need to reduce the maximal stable time step of conventional staircase approach. A novel proof for the local error rate for general geometries is given, and the method is verified and compared to other approaches by means of several numerical 2D examples. Copyright © 2003 John Wiley & Sons, Ltd. [source] Quasi optimal finite difference method for Helmholtz problem on unstructured gridsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010Daniel T. Fernandes Abstract A quasi optimal finite difference method (QOFD) is proposed for the Helmholtz problem. The stencils' coefficients are obtained numerically by minimizing a least-squares functional of the local truncation error for plane wave solutions in any direction. In one dimension this approach leads to a nodally exact scheme, with no truncation error, for uniform or non-uniform meshes. In two dimensions, when applied to a uniform cartesian grid, a 9-point sixth-order scheme is derived with the same truncation error of the quasi-stabilized finite element method (QSFEM) introduced by Babu,ka et al. (Comp. Meth. Appl. Mech. Eng. 1995; 128:325,359). Similarly, a 27-point sixth-order stencil is derived in three dimensions. The QOFD formulation, proposed here, is naturally applied on uniform, non-uniform and unstructured meshes in any dimension. Numerical results are presented showing optimal rates of convergence and reduced pollution effects for large values of the wave number. Copyright © 2009 John Wiley & Sons, Ltd. [source] Local defect correction with different grid typesNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2002V. Nefedov Abstract For a Poisson problem with a solution having large gradients in (nearly) circular subregions a local defect correction method is considered. The problem on the global domain is discretized on a cartesian grid, whereas the restriction of the problem to a circular subdomain is discretized on a polar grid. The two discretizations are then combined in an iterative way. We show that LDC can be viewed as an iterative method for the Poisson equation on a single composite cartesian-polar grid. The efficiency of methods is illustrated by numerical examples. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 454,468, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10018 [source] | |||||||||||||||||||||||||