An implicit three‐dimensional fully non‐hydrostatic model for free‐surface flows

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.     

An efficient curvilinear non‐hydrostatic model for simulating surface water waves

An efficient curvilinear non‐hydrostatic free surface model is developed to simulate surface water waves in horizontally curved boundaries. The generalized curvilinear governing equations are solved by a fractional step method on a rectangular transformed domain. Of importance is to employ a higher order (either quadratic or cubic spline function) integral method for the top‐layer non‐hydrostatic pressure under a staggered grid framework. Model accuracy and efficiency, in terms of required vertical layers, are critically examined on a linear progressive wave case. The model is then applied to simulate waves propagating in a canal with variable widths, cnoidal wave runup around a circular cylinder, and three‐dimensional wave transformation in a circular channel. Overall the results show that the curvilinear non‐hydrostatic model using a few, e.g. 2–4, vertical layers is capable of simulating wave dispersion, diffraction, and reflection due to curved sidewalls. Copyright © 2010 John Wiley & Sons, Ltd.     

Comparative study on multibody vehicle dynamics models based on subsystem synthesis method using Cartesian and joint coordinates

The subsystem synthesis method has been developed in order to improve computational efficiency for a multibody vehicle dynamics model. Using the subsystem synthesis method, equations of motion of the base body and each subsystem can be solved separately. In the subsystem synthesis method, various coordinate systems can be used and various integration methods can be applied in each subsystem, as long as the effective mass matrix and the effective force vector are properly produced. In this paper, comparative study has been carried out for the subsystem synthesis method with Cartesian coordinates and with joint relative coordinates. Two different integration methods such as an explicit integrator and an explicit implicit integrator are employed. In order to see the accuracy and computational efficiency from the different models based on the different coordinate systems and different integration methods, a rough terrain run simulations has been carried out with a 6 × 6 off-road multibody vehicle model.     

动态混合网格生成及隐式非定常计算方法

建立了一种基于动态混合网格的非定常数值计算方法. 混合网格由贴体的四边形网格、外场 的多层次矩形网格和中间的三角形网格构成. 当物体运动时，贴体四边形网格随物体运动而 运动，而外场的矩形网格保持静止，中间的三角形网格随之变形；当物体运动位移较大，导 致三角形网格的质量降低，甚至导致网格相交时，在局部重新生成网格. 新网格上的物理量 由旧网格上的物理量插值而得. 为了提高计算效率，采用了双时间步和子迭代相结合的隐式 有限体积格式计算非定常Navier-Stokes方程. 子迭代采用高效的块LU-SGS方法. 利用该 方法数值模拟了NACA0012振荡翼型的无黏和黏性绕流，得到了与实验和他人计算相当一致 的结果.     

An Adaptive Cartesian Mesh Based Method to Simulate Turbulent Flows of Multiple Rotating Surfaces

An efficient Cartesian cut-cell/level-set method based on a multiple grid approach to simulate turbulent turbomachinery flows is presented. The finite-volume approach in an unstructured hierarchical Cartesian setup with a sharp representation of the complex moving boundaries embedded into the computational domain, which are described by multiple level-sets, ensures a strict conservation of mass, momentum, and energy. Furthermore, an efficient kinematic motion level-set interface method for the rotation of embedded boundaries described by multiple level-set fields on a computational domain distributed over several processors is introduced. This method allows the simulation of multiple boundaries rotating relatively to each other in a fixed frame of reference. To demonstrate the efficiency of the numerical method and the quality of the computed findings the generic test problem of a rotating cylinder surrounded by a stationary hull and the flow over a ducted rotating axial fan with a stationary turbulence generating grid at the inflow are simulated. The computational results of the axial fan show a good agreement with the experimental data.     

An unstructured/structured multi‐layer hybrid grid method and its application

A multi‐layer hybrid grid method is constructed to simulate complex flow field around 2‐D and 3‐D configuration. The method combines Cartesian grids with structured grids and triangular meshes to provide great flexibility in discretizing a domain. We generate the body‐fitted structured grids near the wall surface and the Cartesian grids for the far field. In addition, we regard the triangular meshes as an adhesive to link each grid part. Coupled with a tree data structure, the Cartesian grid is generated automatically through a cell‐cutting algorithm. The grid merging methodology is discussed, which can smooth hybrid grids and improve the quality of the grids. A cell‐centred finite volume flow solver has been developed in combination with a dual‐time stepping scheme. The flow solver supports arbitrary control volume cells. Both inviscid and viscous flows are computed by solving the Euler and Navier–Stokes equations. The above methods and algorithms have been validated on some test cases. Computed results are presented and compared with experimental data. Copyright © 2006 John Wiley & Sons, Ltd.     

Adaptive Runge–Kutta discontinuous Galerkin method for complex geometry problems on Cartesian grid

A Cartesian grid method using immersed boundary technique to simulate the impact of body in fluid has become an important research topic in computational fluid dynamics because of its simplification, automation of grid generation, and accuracy of results. In the frame of Cartesian grid, one often uses finite volume method with second order accuracy or finite difference method. In this paper, an h‐adaptive Runge–Kutta discontinuous Galerkin (RKDG) method on Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is developed. A ghost cell immersed boundary treatment with the modification of normal velocity is presented. The method is validated versus well documented test problems involving both steady and unsteady compressible flows through complex bodies over a wide range of Mach numbers. The numerical results show that the present boundary treatment to some extent reduces the error of entropy and demonstrate the efficiency, robustness, and versatility of the proposed approach. Copyright © 2013 John Wiley & Sons, Ltd.     

复杂无粘流场数值模拟的矩形/三角形混合网格技术

建立了一套模拟复杂无粘流场的矩形／三角形混合网格技术，其中三角形仅限于物面附近，发挥非结构网格的几何灵活性，以少量的网格模拟复杂外型；同时在以外的区域采用矩形结构网格，发挥矩形网格计算简单快速的优势，有效地克服全非结构网格计算方法需要较大内存量和较长ＣＰＵ时间的不足．混合网格系统由修正的四分树方法生成．将ＮＮＤ有限差分与ＮＮＤ有限体积格式有机地融合于混合网格计算，消除了全矩形网格模拟曲边界的台阶效应，同时保证了网格间的通量守恒．数值实验表明本方法在模拟复杂无粘流场方面的灵活性和高效性．     

A simplified adaptive Cartesian grid system for solving the 2D shallow water equations

This paper presents a new simplified grid system that provides local refinement and dynamic adaptation for solving the 2D shallow water equations (SWEs). Local refinement is realized by simply specifying different subdivision levels to the cells on a background uniform coarse grid that covers the computational domain. On such a non‐uniform grid, the structured property of a regular Cartesian mesh is maintained and neighbor information is determined by simple algebraic relationships, i.e. data structure becomes unnecessary. Dynamic grid adaptation is achieved by changing the subdivision level of a background cell. Therefore, grid generation and adaptation is greatly simplified and straightforward to implement. The new adaptive grid‐based SWE solver is tested by applying it to simulate three idealized test cases and promising results are obtained. The new grid system offers a simplified alternative to the existing approaches for providing adaptive mesh refinement in computational fluid dynamics. Copyright © 2011 John Wiley & Sons, Ltd.     

Hybrid grid generation for viscous flow analysis

Cartesian grid with cut‐cell method has drawn attention of CFD researchers owing to its simplicity. However, it suffers from the accuracy near the boundary of objects especially when applied to viscous flow analysis. Hybrid grid consisting of Cartesian grid in the background, body‐fitted layer near the object, and transition layer connecting the two is an interesting alternative. In this paper, we propose a robust method to generate hybrid grid in two‐dimensional (2D) and three‐dimensional (3D) space for viscous flow analysis. In the first step, body‐fitted layer made of quadrangles (in 2D) or prisms (in 3D) is created near the object's boundary by extruding front nodes with a speed function depending on the minimum normal curvature obtained by quadric surface fitting. To solve global interferences effectively, a level set method is used to find candidates of colliding cells. Then, axis‐aligned Cartesian grid (quadtree in 2D or octree in 3D) is filled in the rest of the domain. Finally, the gap between body‐fitted layer and Cartesian grid is connected by transition layer composed of triangles (in 2D) or tetrahedrons (in 3D). Mesh in transition layer is initially generated by constrained Delaunay triangulation from sampled points based on size function and is further optimized to provide smooth connection. Our approach to automatic hybrid grid generation has been tested with many models including complex geometry and multi‐body cases, showing robust results in reasonable time. Copyright © 2012 John Wiley & Sons, Ltd.     

Application of hybrid Cartesian grid and gridless approach to moving boundary flow problems

The paper presents a hybrid Cartesian grid and gridless approach to solve unsteady moving boundary flow problems. Unlike the Chimera clouds of points approach, the hybrid approach uses a Cartesian grid to cover most of the computational domain and a gridless method to calculate a relatively small region adjacent to the body surface, making use of the flexibility of the gridless method in handling surface grid with complicated geometry and the computational efficiency of the Cartesian grid. Four cases were conducted to examine the applicability, accuracy and robustness of the hybrid approach. Steady flows over a single NACA0012 airfoil and dual NACA0012 airfoils at different Mach numbers and angles of attack were simulated. Moreover, by implementing a dynamic hole cutting, node identification and information communication between the Cartesian grid and the gridless regions, unsteady flows over a pitching NACA0012 airfoil (small displacement) and two‐dimensional airfoil/store separation (large displacement) were performed. The computational results were found to agree well with earlier experimental data as well as computational results. Shock waves were accurately captured. The computational results show that the hybrid approach is of potential to solve the moving boundary flow problems. Copyright © 2013 John Wiley & Sons, Ltd.     

Stretched Cartesian grids for solution of the incompressible Navier–Stokes equations

Two Cartesian grid stretching functions are investigated for solving the unsteady incompressible Navier–Stokes equations using the pressure–velocity formulation. The first function is developed for the Fourier method and is a generalization of earlier work. This function concentrates more points at the centre of the computational box while allowing the box to remain finite. The second stretching function is for the second‐order central finite difference scheme, which uses a staggered grid in the computational domain. This function is derived to allow a direct discretization of the Laplacian operator in the pressure equation while preserving the consistent behaviour exhibited by the uniform grid scheme. Both functions are analysed for their effects on the matrix of the discretized pressure equation. It is shown that while the second function does not spoil the matrix diagonal dominance, the first one can. Limits to stretching of the first method are derived for the cases of mappings in one and two directions. A limit is also derived for the second function in order to prevent a strong distortion of a sine wave. The performances of the two types of stretching are examined in simulations of periodic co‐flowing jets and a time developing boundary layer. Copyright © 2000 John Wiley & Sons, Ltd.     

A two‐dimensional vertical non‐hydrostatic σ model with an implicit method for free‐surface flows

An implicit finite difference model in the σ co‐ordinate system is developed for non‐hydrostatic, two‐dimensional vertical plane free‐surface flows. To accurately simulate interaction of free‐surface flows with uneven bottoms, the unsteady Navier–Stokes equations and the free‐surface boundary condition are solved simultaneously in a regular transformed σ domain using a fully implicit method in two steps. First, the vertical velocity and pressure are expressed as functions of horizontal velocity. Second, substituting these relationship into the horizontal momentum equation provides a block tri‐diagonal matrix system with the unknown of horizontal velocity, which can be solved by a direct matrix solver without iteration. A new treatment of non‐hydrostatic pressure condition at the top‐layer cell is developed and found to be important for resolving the phase of wave propagation. Additional terms introduced by the σ co‐ordinate transformation are discretized appropriately in order to obtain accurate and stable numerical results. The developed model has been validated by several tests involving free‐surface flows with strong vertical accelerations and non‐linear waves interacting with uneven bottoms. Comparisons among numerical results, analytical solutions and experimental data show the capability of the model to simulate free‐surface flow problems. Copyright © 2004 John Wiley & Sons, Ltd.     

A three‐dimensional hydrodynamic model for shallow waters using unstructured Cartesian grids

This paper presents a three‐dimensional unstructured Cartesian grid model for simulating shallow water hydrodynamics in lakes, rivers, estuaries, and coastal waters. It is a flux‐based finite difference model that uses a cut‐cell approach to fit the bottom topography and shorelines and, at the same time, has the flexibility of discretizing complex geometries with Cartesian grids that can be arbitrarily downsized in the two horizontal directions simultaneously. Because of the use of Cartesian grids, the grid generation is very simple and does not suffer the grid generation headache often seen in many other unstructured models, as the unstructured Cartesian grid model does not have any requirements on the orthogonality of the grids. The newly developed unstructured Cartesian grid model was validated against analytical solutions for a three‐dimensional seiching case in a rectangular basin, before it was compared with another three‐dimensional model named LESS3D for circulations and salinity transport processes in an idealized embayment that is driven by tides and freshwater inflows. Model tests show that the numerical procedure used in the unstructured Cartesian grid model is robust. Similar to other unstructured models, a variable grid size has resulted in a smaller number of grids required for a reasonable model simulation, which in turn reduces the CPU time used in the model run. Copyright © 2010 John Wiley & Sons, Ltd.     

A hybrid building‐block and gridless method for compressible flows

A hybrid building‐block Cartesian grid and gridless method is presented to compute unsteady compressible flows for complex geometries. In this method, a Cartesian mesh based on a building‐block grid is used as a baseline mesh to cover the computational domain, while the boundary surfaces are represented using a set of gridless points. This hybrid method combines the efficiency of a Cartesian grid method and the flexibility of a gridless method for the complex geometries. The developed method is used to compute a number of test cases to validate the accuracy and efficiency of the method. The numerical results obtained indicate that the use of this hybrid method leads to a significant improvement in performance over its unstructured grid counterpart for the time‐accurate solution of the compressible Euler equations. An overall speed‐up factor from six to more than one order of magnitude and a saving in storage requirements up to one order of magnitude for all test cases in comparison with the unstructured grid method are demonstrated. Copyright © 2008 John Wiley & Sons, Ltd.     

基于复合叉树的自适应笛卡尔网格应用研究

采用复合叉树自适应笛卡尔网格和有限体积法求解三维Euler方程,在网格生成过程中,以模型几何外形、模型表面曲率为基础,构建了基于复合叉树的网格生成和加密方法。在流场计算过程中,又针对流场变化特征,建立了基于复合叉树的网格各向异性拆分模式,同时采用以中心差分为基础的Jameson有限体积法。通过对M6机翼在跨音速情况下的数值仿真,表明计算结果与风洞实验结果符合良好,同时也表明本算法具有高分辨率、节省机时,提高计算效率等特点。     

An implicit three‐dimensional numerical model to simulate transport processes in coastal water bodies

A three‐dimensional baroclinic numerical model has been developed to compute water levels and water particle velocity distributions in coastal waters. The numerical model consists of hydrodynamic, transport and turbulence model components. In the hydrodynamic model component, the Navier–Stokes equations are solved with the hydrostatic pressure distribution assumption and the Boussinesq approximation. The transport model component consists of the pollutant transport model and the water temperature and salinity transport models. In this component, the three‐dimensional convective diffusion equations are solved for each of the three quantities. In the turbulence model, a two‐equation k–ϵ formulation is solved to calculate the kinetic energy of the turbulence and its rate of dissipation, which provides the variable vertical turbulent eddy viscosity. Horizontal eddy viscosities can be simulated by the Smagorinsky algebraic sub grid scale turbulence model. The solution method is a composite finite difference–finite element method. In the horizontal plane, finite difference approximations, and in the vertical plane, finite element shape functions are used. The governing equations are solved implicitly in the Cartesian co‐ordinate system. The horizontal mesh sizes can be variable. To increase the vertical resolution, grid clustering can be applied. In the treatment of coastal land boundaries, the flooding and drying processes can be considered. The developed numerical model predictions are compared with the analytical solutions of the steady wind driven circulatory flow in a closed basin and of the uni‐nodal standing oscillation. Furthermore, model predictions are verified by the experiments performed on the wind driven turbulent flow of an homogeneous fluid and by the hydraulic model studies conducted on the forced flushing of marinas in enclosed seas. Copyright © 2000 John Wiley & Sons, Ltd.     

Euler solution using adaptive Cartesian grid with a gridless boundary treatment

A quadtree-based adaptive Cartesian grid generator and flow solver were developed. The grid adaptation based on pressure or density gradient was performed and a gridless method based on the least-square fashion was used to treat the wall surface boundary condition, which is generally difficult to be handled for the common Cartesian grid. First, to validate the technique of grid adaptation, the benchmarks over a forward-facing step and double Mach reflection were computed. Second, the flows over the NACA 0012 airfoil and a two-element airfoil were calculated to validate the developed gridless method. The computational results indi- cate the developed method is reasonable for complex flows.     

Ground thermal response to heat conduction in a power transmission tower foundation

An analytical formulation is developed to predict transient heat conduction in a semi-infinite medium with a vertical finite line heat source, which represents a buried tower of a power transmission line foundation. Unlike past studies with a constant line heat source, the current model develops a time-dependent variable heating strength, as well as a time varying surface temperature of the ground. An approximate VHS model (variable heating strength) is developed for sinusoidal variations of the line source strength and surface temperature, in order to simulate seasonal variations of ground temperatures. The VHS model reduces computational time and exhibits good accuracy, when compared against a full exact solution. The model is applied to heat conduction in a tower foundation, with time-varying ground surface temperatures. Effects of ground thermal conductivity and diffusivity, as well as variations of the line source strength, are investigated in this article.