A normal ray refinement technique for Cartesian-grid based Navier–Stokes solvers

A hybrid Cartesian grid and gridless method for compressible flows

The added resistance of a ship in waves can be related to ship-generated unsteady waves. In the present study, the unsteady wave-pattern analysis is applied to calculate the added resistance in waves for two modified Wigley models using a Cartesian-grid method. In the present numerical method, a first-order fractional-step method is applied to the velocity-pressure coupling in the fluid domain, and one of volume-of-fluid (VOF) methods is adopted to capture the fluid interface. A ship is embedded in a Cartesian grid, and the volume fraction of the ship inside the grid is calculated by identifying whether each grid is occupied by liquid, gas, and solid body. The sensitivity to the location of measuring position of unsteady waves as well as the number of solution grids is examined. The added resistance computed by direct pressure integration and wave pattern analysis is compared with experimental data. In addition, nonlinear characteristics of the added resistance in waves are investigated by detailed analyses of unsteady flow field and resulting wave pattern.

Normal ray coordinates are conventionally constructed from ray tracing, which inherently requires smooth velocity profiles. To use rays as coordinates, the velocities have to be smoothed further to avoid caustics, which is detrimental to the mapping process. Solving the eikonal equation numerically for a line source at the surface provides a platform to map normal rays in complex unsmoothed velocity models and avoid caustics. We implement reverse-time migration (RTM) and downward continuation in the new ray coordinate system, which allows us to obtain efficient images and avoid some of the dip limitations of downward continuation.

The Cartesian gird has its unique advantages in computational fluid dynamics, especially for complicated boundary cases. However, the boundary layer structures can not be resolved efficiently and effectively using the Cartesian mesh. To overcome such a problem, a new boundary layer structure resolving (BLSR) algorithm is proposed on the basis of the boundary layer physics and force balance analysis. For the present two‐dimensional test cases, numerical results justify that the surface friction and drag force can be more accurately calculated without refining the near‐wall resolution. In principle this BLSR algorithm is easy to implement with negligible increase of the computational cost.

In this study, the added resistance of ships in short waves is systematically studied by using two different numerical methods - Rankine panel method and Cartesian grid method – and existing asymptotic and empirical formulae. Analysis of added resistance in short waves has been preconceived as a shortcoming of numerical computation. This study aims to observe such preconception by comparing the computational results, particularly based on two representative three-dimensional methods, and with the existing formulae and experimental data. In the Rankine panel method, a near-field method based on direct pressure integration is adopted. In the Cartesian grid method, the wave-body interaction problem is considered as a multiphase problem, and volume fraction functions are defined in order to identify each phase in a Cartesian grid. The computational results of added resistance in short waves using the two methods are systematically compared with experimental data for several ship models, including S175 containership, KVLCC2 and Series 60 hulls (C_B = 0.7, 0.8). The present study includes the comparison with the established asymptotic and empirical formulae in short waves.

Cartesian grid method for gas kinetic scheme on irregular geometries

The Euler equations describe the ow phenomena of compressible inviscid gas dynamics We simulate such ows using a higher order Cartesian grid method together with a special treatment for the cells cut by the boundary of a body We describe a new method for the treatment of the boundary where these cut boundary cells are maintained as whole cells rather than as cut cells thus avoiding stability problems The method is second order accurate but not strictly conservative but we can show that this error in the conservation does not lead to spurious phenomena on some representative test calculations The advantages of the new boundary treatment are that it is second order accurate that it is independent of the applied method and that it can easily be extended to three dimensional calculations

Development of Cartesian grid method for simulation of violent ship-wave interactions

Recently, the Cartesian grid method was used on simulations of fluid flow around the complex geometry problems. In this paper, the Cartesian grid method with cut cells are used to simulate incompressible viscous flows around moving objects. In this study, the small cut cells are merged with neighboring cells. The code was developed based on Fractional Step Method on Finite Volume Method. In time integrate, the second order explicit Adams-Bashforth method was used for the convective terms and second order implicit Crank-Nicholson method was used for the diffusive terms. Some moving boundary problem are simulated using the presented code and compared with other previous numerical and experimental results.

The objective of the Cranked-Arrow Wing Aerodynamics Project International (CAWAPI) was to allow a comprehensive validation of Computational Fluid Dynamics methods against the CAWAP flight database. A major part of this work involved the generation of high-quality computational grids. Prior to the grid generation an IGES file containing the air-tight geometry of the F-16XL aircraft was generated by a cooperation of the CAWAPI partners. Based on this geometry description both structured and unstructured grids have been generated. The baseline structured (multi-block) grid (and a family of derived grids) has been generated by the National Aerospace Laboratory NLR. Although the algorithms used by NLR had become available just before CAWAPI and thus only a limited experience with their application to such a complex configuration had been gained, a grid of good quality was generated well within four weeks. This time compared favourably with that required to produce the unstructured grids in CAWAPI. The baseline all-tetrahedral and hybrid unstructured grids has been generated at NASA Langley Research Center and the USAFA, respectively. To provide more geometrical resolution, trimmed unstructured grids have been generated at EADS-MAS, the UTSimCenter, Boeing Phantom Works and KTH/FOI. All grids generated within the framework of CAWAPI will be discussed in the article. Both results obtained on the structured grids and the unstructured grids showed a significant improvement in agreement with flight test data in comparison with those obtained on the structured multi-block grid used during CAWAP.

A hybrid Cartesian grid/gridless Euler solver has been developed for efficient prediction of store separation. In this approach, a so-called building-cube Cartesian grid method is used to solve the flows over the majority of the computational domain, taking advantage of the simplicity and efficiency of the Cartesian grid approach. Near the body surfaces, on the other hand, a body-conforming gridless method is used for accurate implementation of the boundary conditions on the body surfaces. As a result, the grid generation effort for store separation prediction is significantly alleviated without grid overlapping. The application of this algorithm to the Eglin supersonic store separation case has proven successful. Good agreement with the measured pressure distributions over the store surfaces and trajectory data has been achieved.

A nested multi-grid solution algorithm has been developed for an adaptive Cartesian/Quad grid viscous flow solver. Body-fitted adaptive Quad (quadrilateral) grids are generated around solid bodies through ‘surface extrusion’. The Quad grids are then overlapped with an adaptive Cartesian grid. Quadtree data structures are employed to record both the Quad and Cartesian grids. The Cartesian grid is generated through recursive sub-division of a single root, whereas the Quad grids start from multiple roots—a forest of Quadtrees, representing the coarsest possible Quad grids. Cell-cutting is performed at the Cartesian/Quad grid interface to merge the Cartesian and Quad grids into a single unstructured grid with arbitrary cell topologies (i.e., arbitrary polygons). Because of the hierarchical nature of the data structure, many levels of coarse grids have already been built in. The coarsening of the unstructured grid is based on the Quadtree data structure through reverse tree traversal. Issues arising from grid coarsening are discussed and solutions are developed. The flow solver is based on a cell-centered finite volume discretization, Roe's flux splitting, a least-squares linear reconstruction, and a differentiable limiter developed by Venkatakrishnan in a modified form. A local time stepping scheme is used to handle very small cut cells produced in cell-cutting. Several cycling strategies, such as the saw-tooth, W- and V-cycles, have been studies. The V-cycle has been found to be the most efficient. In general, the multi-grid solution algorithm has been shown to greatly speed up convergence to steady state—by one to two orders. Copyright © 2000 John Wiley & Sons, Ltd.

The fast multipole method and Fourier convolution for the solution of acoustic scattering on regular volumetric grids

A moving-least-square based finite-volume method is developed to simulate acoustic wave propagation and scattering from complicated solid geometries. This hybrid method solves the linearized perturbed compressible equations as the governing equations of the acoustic field. The solid boundaries are embedded in a uniform Cartesian grid and represented using level set fields. Thus, the current approach avoids unstructured grid generation for the irregular geometries. The desired boundary conditions are imposed sharply on the immersed boundaries using a ghost fluid method. The scope of the implementation of the moving moving-least-square approach in the current solver is threefold: reconstruction of the field variables on cell faces for high-order flux construction, population of the ghost cells based on the desired boundary condition, and filtering the high wave number modes near the immersed boundaries. The computational stencils away from the boundaries are identical; hence, only one moving-least-square shape-function is computed and stored with its underlying grid pattern for all the interior cells. This feature significantly reduces the memory requirement of the acoustic solver compared to similar finite-volume method on irregular unstructured mesh. The acoustic solver is validated against several benchmark problems.

Numerical simulation of the boundary-layer flow by using the boundary-layer equations and an imposed edge condition inevitably encounters a difficulty when the flow tends to separate from the surface. For avoiding this numerical difficulty and ensuring a successful computation over the recirculating region, an interacting boundary-layer edge condition may be employed to substitute that imposed one. This interacting boundary-layer edge condition is based on the pressure distribution imposed on the body surface by its outer inviscid flow and also includes a correction that properly takes the boundary-layer's own effects into account. Consequently, the boundary-layer equations and their interacting edge condition are mutually coupled and the full linearization of both of them is essential to obtain an efficient computational procedure. Therefore this study focuses on the full linearization of the two-dimensional turbulent boundary-layer equations and their interacting edge condition; some strategies associate with it are also discussed.