Curvilinear Coordinate Grid Research Articles

2-D poroelastic wave modelling with a topographic free surface by the curvilinear grid finite-difference method

The curvilinear grid finite-difference method, which is suitable for accurate modelling of seismic waves in the presence of topographic free surfaces, is applied in this study to model poroelastic waves. Poroelastic wave equations of Biot’s theory are expressed in curvilinear coordinates and solved on a collocated grid by utilizing the fourth-order Runge–Kutta time integral scheme with alternative applications of forward and backward MacCormack finite differences. Discretization of the irregular free surface with the curvilinear coordinate grid enables a good fit of the topographic surface, and this good fit results in higher numerical accuracy in comparison with stair-case approximation. The formulas of the free-surface boundary conditions for poroelastic media are also derived for the curvilinear coordinates and subsequently implemented with the traction-imaging method, which is a generalization of the stress-imaging method and can maintain high accuracy in curvilinear coordinates. To verify the performance of the proposed algorithm and determine its accuracy, we conduct numerical simulations on several different models, and compare our numerical solutions with other solutions that are calculated by analytical expressions, generalized reflection/tranmission method and spectral-element method. Besides, the fluid–solid dynamic compatibility of porous media is demonstrated using a three-layered half-space model, and effects of the irregular free-surface on poroelastic waves are investigated for a four-layered model with strong irregular free-surface topography. The good performance of the proposed algorithm suggests that the curvilinear grid finite-differece method is a useful tool for the study of poroelastic wave propagation.

Application of a Steady Meandering River with Piers Using a Lattice Boltzmann Sub-Grid Model in Curvilinear Coordinate Grid

A sub-grid multiple relaxation time (MRT) lattice Boltzmann model with curvilinear coordinates is applied to simulate an artificial meandering river. The method is based on the D2Q9 model and standard Smagorinsky sub-grid scale (SGS) model is introduced to simulate meandering flows. The interpolation supplemented lattice Boltzmann method (ISLBM) and the non-equilibrium extrapolation method are used for second-order accuracy and boundary conditions. The proposed model was validated by a meandering channel with a 180° bend and applied to a steady curved river with piers. Excellent agreement between the simulated results and previous computational and experimental data was found, showing that MRT-LBM (MRT lattice Boltzmann method) coupled with a Smagorinsky sub-grid scale (SGS) model in a curvilinear coordinates grid is capable of simulating practical meandering flows.

Lattice Boltzmann model for shallow water in curvilinear coordinate grid

In this study, a multi-relaxation time lattice Boltzmann model for shallow water in a curvilinear coordinate grid is developed using the generalized form of the interpolation supplemented lattice Boltzmann method. The Taylor-Colette flow tests show that the proposed model enjoys a second order accuracy in space. The proposed model is applied to three types of meandering channels with, and consecutive bends. The numerical results demonstrate that the simulated results agree well with previous computational and experimental data. In addition, the model can achieve the acceptable accuracy in terms of the water depth and the depth-averaged velocities for shallow water flows in curved and meandering channels over a wide range of bend angles.

Numerical Simulation of Free Surface in the Case of Plane Turbulent Wall Jets in Shallow Tailwater

Wall-jet flow is an important flow field in hydraulic engineering, and its applications include flow from the bottom outlet of dams and sluice gates. In this paper, the plane turbulent wall jet in shallow tailwater is simulated by solving the Reynolds Averaged Navier-Stokes equations using the standard turbulence closure model. This study aims to explore the ability of a time splitting method on a non-staggered grid in curvilinear coordinates for simulation of two-dimensional (2D) plane turbulent wall jets with finite tailwater depth. In the developed model, the kinematic free-surface boundary condition is solved simultaneously with the momentum and continuity equations, so that the water surface elevation can be obtained along with the velocity and pressure fields as part of the solution. 2D simulations are carried out for plane turbulent wall jets free surface in shallow tailwater. The comparison undertaken between numerical results and experimental measurements show that the numerical model can capture the velocity field and the drop in the water surface elevation at the gate with reasonable accuracy.

Multielectron effects in high harmonic generation in N2 and benzene: Simulation using a non-adiabatic quantum molecular dynamics approach for laser-molecule interactions

A mixed quantum-classical approach is introduced which allows the dynamical response of molecules driven far from equilibrium to be modeled. This method is applied to the interaction of molecules with intense, short-duration laser pulses. The electronic response of the molecule is described using time-dependent density functional theory (TDDFT) and the resulting Kohn-Sham equations are solved numerically using finite difference techniques in conjunction with local and global adaptations of an underlying grid in curvilinear coordinates. Using this approach, simulations can be carried out for a wide range of molecules and both all-electron and pseudopotential calculations are possible. The approach is applied to the study of high harmonic generation in N2 and benzene using linearly polarized laser pulses and, to the best of our knowledge, the results for benzene represent the first TDDFT calculations of high harmonic generation in benzene using linearly polarized laser pulses. For N2 an enhancement of the cut-off harmonics is observed whenever the laser polarization is aligned perpendicular to the molecular axis. This enhancement is attributed to the symmetry properties of the Kohn-Sham orbital that responds predominantly to the pulse. In benzene we predict that a suppression in the cut-off harmonics occurs whenever the laser polarization is aligned parallel to the molecular plane. We attribute this suppression to the symmetry-induced response of the highest-occupied molecular orbital.

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.

BiGlobal stability analysis in curvilinear coordinates of massively separated lifting bodies

A methodology based on spectral collocation numerical methods for global flow stability analysis of incompressible external flows is presented. A potential shortcoming of spectral methods, namely the handling of the complex geometries encountered in global stability analysis, has been dealt with successfully in past works by the development of spectral-element methods on unstructured meshes. The present contribution shows that a certain degree of regularity of the geometry may be exploited in order to build a global stability analysis approach based on a regular spectral rectangular grid in curvilinear coordinates and conformal mappings. The derivation of the stability linear operator in curvilinear coordinates is presented along with the discretisation method. Unlike common practice to the solution of the same problem, the matrix discretising the eigenvalue problem is formed and stored. Subspace iteration and massive parallelisation are used in order to recover a wide window of its leading Ritz system. The method is applied to two external flows, both of which are lifting bodies with separation occurring just downstream of the leading edge. Specifically the flow configurations are a NACA 0015 airfoil, and an ellipse of aspect ratio 8 chosen to closely approximate the geometry of the airfoil. Both flow configurations are at an angle of attack of 18° with a Reynolds number based on the chord length of 200. The results of the stability analysis for both geometries are presented and illustrate analogous features.

A time-splitting method on a nonstaggered grid in curvilinear coordinates for implicit simulation of non-hydrostatic free-surface flows

The numerical solution of flows with a freely moving boundary is of great importance in practical application such as ship hydrodynamics. Details are given of the development of a two-dimensional vertical numerical model for simulating unsteady and steady free-surface flows on a nonstaggered grid in curvilinear coordinates, using a non-hydrostatic pressure distribution. In this model, Reynolds equation and the kinematic free-surface boundary condition are solved simultaneously, so that the water-surface elevation can be integrated into the solution and solved for, together with the velocity and pressure field. In computational space, the Cartesian velocity components and the pressure are defined at the center of a control volume, while the volume fluxes are defined at the midpoint on their corresponding cell faces. Detailed numerical results are presented for the wave generation above an obstacle and resonant motion standing wave. The results show that the numerical algorithm described is able to produce accurate predictions and is also easy to apply.Key words: free-surface simulation, nonstaggered grid, time-splitting method, unsteady flow, turbulent flow.

Use of a hybrid code for global-scale simulation of the Earth's magnetosphere

An understanding of the Earth's magnetosphere and the magnetospheric substorm requires an ability to simulate kinetic processes in a global context. To this end we have developed a hybrid code with a number of innovative features: (1) The code uses a general curvilinear coordinate grid enabling it to accommodate disparate size scales. (2) The cold, dense plasma inside the plasmapause, where kinetic effects are unimportant, is treated in the fluid approximation. There is a seamless interface between the fluid and kinetic descriptions. (3) The field update is subcycled to the particle update to circumvent the Courant condition with respect to the propagation of Alfvén and whistler mode waves. A two-dimensional version of the code has established a causal link between events in the plasma sheet and the filamentary currents carried by auroral electrons. We are now embarking on the development of a full three-dimensional code for the coupled ionosphere–magnetosphere–magnetosheath system. The simulation domain is comprised of seven discontinuously joined coordinate patches. The Earth's ionospheric shell constitutes one of the boundary surfaces.

Numerically Consistent Strong Conservation Grid Motion for Finite Difference Schemes

An investigation into the numerical issues of curvilinear coordinate grid motion for e nite difference schemes is performed. Several formulations of the time-dependent curvilinear coordinate equations are compared using unsteady benchmark test problems. A new, numerically consistent method of computing theunsteady grid metrics for the strong conservation equations is developed and demonstrated.

Modeling multi-band effects of hot electron transport in silicon by self-consistent solution of the Boltzmann transport and Poisson equations

This paper presents a self-consistent numerical technique for the solution of the multi-band Boltzmann transport equation (BTE) and the Poisson equation in silicon. The effects of high energy bands (≳ 3 eV) are modeled in the formulation. The numerical technique utilizes a new curvilinear boundary-fitted coordinate grid which is tailored for self-consistent calculations. A new Scharfetter-Gummel like discretization of the BTE is presented. The numerical algorithm is tested on a n + − n − n + device structure.

Finite Difference Scheme for Solving Incompressible Viscous Flow around an Oscillating Body.

This paper presents a finite difference scheme for solving incompressible viscous flow around an arbitrary oscillating body. This scheme is an extension of the SMAC method to a moving curvilinear coordinate grid. The scheme comprises two stages. At stage one, the new time step velocity is calculated in the old time step coordinates. At stage two, this calculated velocity is re-observed in terms of the new time step coordinates (rezoning). This scheme is applied to calculation of two-dimensional flow around an oscillating flat plate, which is a fundamental model of the fin movement of certain kinds of fish or cetaceans or the flutter of an airplane wing. The solution is compared with that from the discrete vortex method. In particular,d the characteristics of thrust and propulsive efficiency of the oscillating flat plate as a propulsion mechanism show good agreement with those from the discrete vortex method.

Some numerical schemes using curvilinear coordinate grids for incompressible and compressible Navier-Stokes equations

In this review paper some numerical schemes recently developed by the authors and their coworkers for analysing the cascade flows of turbomachinery are described. These schemes use the curvilinear coordinate grid and solve the momentum equations of contravariant velocities (volume flux). The compressible flow schemes are based on the delta-form approximate-factorization finite-difference scheme, and are improved by using the diagonalization, the flux difference splitting and thetvd schemes to save computational effort and to increase stability and resolvability. Furthermore, using higher-order compacttvd muscl schemes, we can capture not only shock waves but also contact surfaces very sharply. On the other hand, the incompressible flow schemes are based on the well-knownSMAC scheme, and are extended to the curvilinear coordinate grid and further to the implicit scheme to reduce computations. These schemes, like thesmac scheme, satisfy the continuity condition identically, and suppress the occurrence of spurious errors. In both the compressible and incompressible schemes, for the turbulent flow thek-ɛ turbulence model with the law of the wall or considering the low Reynolds number effects is employed, and for the unsteady flow the Crank-Nicholson method is employed and the solution at each time step is obtained by the Newton iteration. Use of the volume flux instead of the physical velocity is inevitable for theMAC type schemes, and makes it easy to impose boundary conditions. Finally, some calculated results using the present schemes are shown.

Thermophoretic transport in the outside vapor deposition process

A study has been made of the flow, heat transfer and thermophoretic deposition to a circular boule in respect to the outside vapor deposition (OVD) process. Numerical solutions of the governing conservation equations have been obtained utilizing a non-orthogonal curvilinear coordinate grid. The particle deposition rate and the variation of the deposition efficiency are investigated for a range of parameters relative to the OVD process. The interaction of the jet-like torch flow with the target and the surroundings results in a flow, heat transfer and particle transport to the circular target that are quite different from that for uniform flow. The increase in the concentration prior to the target is an important factor in determining the particle deposition rate. The effects of buoyancy augment the deposition rate. Larger values of the average Nusselt number and the efficiency are obtained for a cold target. For a hot target, the local Nusselt number becomes negative over the leeward side and a particle-free layer is formed. For the particle jet, the concentration and the deposition rate rapidly decrease downstream from the stagnation point. The deposition efficiency is greatly reduced when the torch is not aligned with respect to the target. Increasing the speed of the gas flow increases the average Nusselt number, but reduces the efficiency. Higher efficiency is obtained for a higher torch temperature. The efficiency is slightly increased for a larger target. Increasing the distance between the torch and the target and increasing the rotational speed of the boule slightly reduce the efficiency.

Flow and heat transfer to a circular cylinder with a hot impinging air jet

A study has been made of the flow and heat transfer to a circular cylinder from a hot impinging air jet. The problem is of fundamental interest and also has important applications; e.g. in respect to the outside vapor deposition process. Numerical solutions of the governing conservation equations have been obtained utilizing a non-orthogonal curvilinear coordinate grid. The effects of the Reynolds and Grashof numbers on the flow and heat transfer, as well as those of the wall temperature, the jet width, and the distance between the nozzle and the cylinder have beeen investigated and correlation curves are presented. The flow is stable and symmetric over the range of parameters studied. The interaction of the buoyant impinging jet with the cylinder as well as with the surroundings makes the flow and heat transfer to the cylinder quite different from that for uniform flow. The effects of buoyancy, diffusion and cooling of the jet ahead of the cylinder, the development of the wall jet, and the recirculating bubble all affect the flow and heat transfer. The average Nusselt number increases with increasing Reynolds and Grashof numbers and narrow banded correlation curves are obtained by introducing an effective Reynolds number, which includes a buoyancy contribution. The width of the jet and the distance between the nozzle and the cylinder have a strong effect on the heat transfer.

Alfven wave transmission through the solar atmosphere

The transmission and reflection coefficients of shear Alfven waves in an open magnetic structure emerging from the Sun are determined. Both two-dimensional slab and axisymmetric static equilibrium configurations are solved numerically on a curvilinear coordinate grid adapted to the magnetic field geometry and to the density stratification. The equilibrium structure obtained numerically extends from the photosphere up to an altitude of 20,000 km, spanning 14 density scale heights and allowing for a magnetic flux tube expansion by a factor 40-400, corresponding to a filling factor of 2.5% to 0.25%. From 20,000 km up to 4 R ⊙ a magnetic monopole model is adopted

A Finite Difference Scheme for Solving the Incompressible Viscous Fluid Flow around a Vibrating Body.

This paper presents a finite difference scheme for solving the incompressible viscous fluid flow around an arbitrary vibrating body. This scheme is an extension of the SMAC method to a moving curvilinear coordinate grid. The scheme contains two stages. At stabe one, the new time-step velocity is calculated in the old time-step coordinate. At stage two, this calculated velocity is reobserved by the new time-step coordinate (rezoning). This scheme is applied to calculate a two-dimensinal flow around a vibrating flat plate, which is a fundamental model of the fin movement of some fishes or cetaceans and of the flutter of an airplane wing. The solution is compared with that from the discrete vortex method. In particular, the characteristics of thrust and its efficiency of the vibrating flat plate as a propelling mechanism show good agreement with those from the discrete vortex method.

A Convective-Difference Scheme Using a General Curvilinear Coordinate Grid for Steady Incompressible Viscous Flow Problems.

A numerical scheme for analyzing the steady two-dimensional incompressible viscous flow using a general curvilinear coordinate grid is proposed. In this scheme, the unsteady Navier-Stokes equations are solved by a convective-difference scheme using a staggered square grid in transformed space. An elliptic equation of pressure is solved by the Tchebyscheff SLOR method. The substantial derivative term in the convective-difference scheme is integrated along a path line and the values at the upstream end are interpolated considering a TVD concept. As numerical examples, the backward-facing step duct and U curved duct flows were calculated. The calculated results show that the scheme has good accuracy as a second-order scheme and is speedy in reaching for the steady condition.

Convective-Difference Scheme Using a General Curvilinear Coordinate Grid for Incompressible Viscous Flow Problems

A numerical scheme analyzing unsteady two-dimensional incompressible viscous flow using a general curvilinear coordinate grid is proposed. In this scheme, the unsteady Navier-Stokes equations are solved by a convective-difference scheme using a staggered square grid in transformed space and an interpolation formula considering TVD concept, and an elliptic equation of pressure is solved by the iteration scheme. The continuity condition in the scheme is identically satisfied, and the spurious errors are completely removed, in a manner similar to the MAC scheme. As numerical examples the square cavity, U-type duct and backward-facing step duct flows were calculated. The calculated results show that the scheme has a good accuracy as a second-order scheme and is stable for high Reynolds numbers flow.

A Convective-Difference Scheme Using General Curvilinear Coordinate Grid for Incompressible Viscous Flow Problems.

A numerical scheme for analyzing the unsteady two-dimensional incompressible viscous flow using a general curvilinear coordinate grid is proposed. In this scheme, the unsteady Navier-Stokes equations are solved by a convective-difference scheme using a staggered square grid in transformed space and an interpolation formula considering TVD concept. And an elliptic equation of pressure is solved by the iteration scheme. The continuity condition in the scheme is identically satisfied and the spurious errors are completely removed by the similar manner to the MAC scheme. As numerical examples square cavity, U-type duct and backward-facing step duct flows were calculated. The calculated results show that the scheme has a good accuracy as a second-order scheme and is stable for the high Reynolds numbers flows.