Simulation of interface and free surface flows in a viscous fluid using adapting quadtree grids

A new adaptive quadtree method for simulating laminar viscous fluid problems with free surfaces and interfaces is presented in this paper. The Navier–Stokes equations are solved with a SIMPLE‐type scheme coupled with the Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) (Numerical prediction of two fluid systems with sharp interfaces, Ph.D. Thesis, Imperial College of Science, Technology and Medicine, London, 1997) volume of fluid (VoF) method and PLIC reconstruction of the volume fraction field during refinement and derefinement processes. The method is demonstrated for interface advection cases in translating and shearing flow fields and found to provide high interface resolution at low computational cost. The new method is also applied to simulation of the collapse of a water column and the results are in excellent agreement with other published data. The quadtree grids adapt to follow the movement of the free surface, whilst maintaining a band of the smallest cells surrounding the surface. The calculation is made on uniform and adapting quadtree grids and the accuracy of the quadtree calculation is shown to be the same as that made on the equivalent uniform grid. Copyright © 2004 John Wiley & Sons, Ltd.     

Viscous waves and wave-structure interaction in a tank using adapting quadtree grids

Viscous waves and waves over a submerged cylinder in a stationary tank are simulated using a volume-of-fluid numerical scheme on adaptive hierarchical grids. A high resolution interface-capturing method is used to advect the free surface interface and the Navier–Stokes equations are discretised using finite volumes with collocated primitive variables and solved using a Pressure Implicit with Splitting of Operators (PISO) algorithm. The cylinder is modelled by using the technique of Cartesian cut cells. Results of flow of a single fluid past a cylinder at Reynolds number Re=100 are presented and found to agree well with experimental and other numerical data. Viscous free surface waves in a tank are simulated using uniform and quadtree grids for Reynolds numbers in the range from 2 to 2000, and the results compared against analytical solutions where available. The quadtree-based results are of the same accuracy as those on the equivalent uniform grids, and retain a sharp interface at the free surface while leading to considerable savings in both storage and CPU requirements. The nonlinearity in the wave is investigated for a selection of initial wave amplitudes. A submerged cylinder is positioned in the tank and its influence on the waves as well as the hydrodynamic loading on the cylinder is investigated.     

A numerical method for 3D viscous incompressible flows using non-orthogonal grids

This paper presents a numerical method for fluid flow in complex three-dimensional geometries using a body-fitted co-ordinate system. A new second-order-accurate scheme for the cross-derivative terms is proposed to describe the non-orthogonal components, allowing parts of these terms to be treated implicitly without increasing the number of computational molecules. The physical tangential velocity components resulting from the velocity expansion in the unit tangent vector basis are used as dependent variables in the momentum equations. A coupled equation solver is used in place of the complicated pressure correction equation associated with grid non-orthogonality. The co-ordinate-invariant conservation equations and the physical geometric quantities of control cells are used directly to formulate the numerical scheme, without reference to the co-ordinate derivatives of transformation. Several two- and three-dimensional laminar flows are computed and compared with other numerical, experimental and analytical results to validate the solution method. Good agreement is obtained in all cases.     

Calculation of viscous and inviscid gas flows on unstructured grids using the AUSM scheme

An approach to the solution of the two-dimensional Navier-Stokes equations on triangular unstructured grids is considered. The method is based on the key idea of the Godunov scheme, namely, the advisability of solving the Riemann problem of arbitrary discontinuity breakdown. In the calculations the derivatives with respect to space are approximated with both the first and the second order. However, as distinct from the conventional Godunov method, in calculating the fluxes across the cell boundaries the Riemann problem is solved using the Advection Upstream Splitting Method (AUSM). The concepts involved in the AUSM scheme are discussed. The solution of the discontinuity breakdown problem obtained within the framework of this approach is compared with the results obtained using the Godunov method. Numerical solutions of some problems of viscous and inviscid perfect-gas flows obtained on unstructured grids of different fineness and those obtained on structured grids are also compared. The effect of the spatial approximation order on the accuracy of numerical solutions is studied.     

Simulation of shallow flows over variable topographies using unstructured grids

Simulation of shallow flows over variable topographies is a challenging case for most available shock‐capturing schemes. This problem arises because the source terms and flux gradients are not balanced in the numerical computations. Treatments for this problem generally work well on structured grids, but they are usually too expensive, and most of them are not directly applicable to unstructured grids. In this paper we propose two efficient methods to treat the source terms without upwinding and to satisfy the compatibility condition on unstructured grids. In the first method, the calculation of the bed slope source term is performed by employing a compatible approximation of water depth at the cell interfaces. In the second one, different components of the bed slope term are considered separately and a compatible discretization of the components is proposed. The present treatments are applicable for most schemes including the Roe's method without changing the performance of the original scheme for smooth topographies. Copyright © 2006 John Wiley & Sons, Ltd.     

Parallel computation of viscous incompressible flows using Godunov‐projection method on overlapping grids

The Godunov‐projection method is implemented on a system of overlapping structured grids for solving the time‐dependent incompressible Navier–Stokes equations. This projection method uses a second‐order fractional step scheme in which the momentum equation is solved to obtain the intermediate velocity field which is then projected on to the space of divergence‐free vector fields. The Godunov procedure is applied to estimate the non‐linear convective term in order to provide a robust discretization of this terms at high Reynolds number. In order to obtain the pressure field, a separate procedure is applied in this modified Godunov‐projection method, where the pressure Poisson equation is solved. Overlapping grids are used to discretize the flow domain, as they offer the flexibility of simplifying the grid generation around complex geometrical domains. This combination of projection method and overlapping grid is also parallelized and reasonable parallel efficiency is achieved. Numerical results are presented to demonstrate the performance of this combination of the Godunov‐projection method and the overlapping grid. Copyright © 2002 John Wiley & Sons, Ltd.     

颗粒材料柱体崩塌物质点法数值模拟

为了研究颗粒材料崩塌的运动规律和堆积特性,采用物质点法对颗粒材料柱体崩塌试验进行数值模拟,并将模拟结果与试验结果进行对比验证。对颗粒材料柱体崩塌过程中颗粒的流动特性(滑动距离、堆积高度、速度、能量和动能通量的演化)进行了分析。进一步探究了颗粒材料柱体高宽比对颗粒流动能通量的影响,从而反应颗粒材料柱体崩塌过程中颗粒流的破坏能力。颗粒材料柱体高宽比越大,颗粒材料柱体外侧边缘颗粒速度越大,其溃散的程度更加强烈,并且滑动距离和动能均在增大。对于动能通量的分布,水平方向越靠近初始颗粒材料柱体,动能通量越大。     

Simulation seismic wave propagation in topographic structures using asymmetric staggered grids

A new 3 D finite- difference ( FD ) method of spatially asymmetric staggered grids was presented to simulate elastic wave propagation in topographic structures. The method approximated the first-order elastic wave equations by irregular grids finite difference operator with second-order time precise and fourth-order spatial precise. Additional introduced finite difference formula solved the asymmetric problem arisen in non-uniform staggered grid scheme, The method had no interpolation between the fine and coarse grids. All grids were computed at the same spatial iteration. Complicated geometrical structures like rough submarine interface, fault and nonplanar interfaces were treated with fine irregular grids. Theoretical analysis and numerical simulations show that this method saves considerable memory and computing time, at the same time, has satisfactory stability and accuracy.     

Octree partitioning of hybrid grids for parallel adaptive viscous flow simulations

A parallel finite volume method for the Navier–Stokes equations with adaptive hybrid prismatic/tetrahedral grids is presented and evaluated in terms of parallel performance. A new method of domain partitioning for complex 3D hybrid meshes is also presented. It is based on orthogonal bisection of a special octree corresponding to the hybrid mesh. The octree is generated automatically and can handle any type of 3D geometry and domain connectivity. One important property of the octree-based partitioning that is exploited is the octree's ability to yield load-balanced partitions that follow the shape of the geometry. This biasing of the octree results in a reduced number of grid elements on the interpartition boundaries and thus fewer data to communicate among processors. Furthermore, the octree-based partitioning gives similar quality of partitions for very different geometries, while requiring minimal user interaction and little computational time. The partitioning method is evaluated in terms of quality of the subdomains as well as execution time. Viscous flow simulations for different geometries are employed to examine the effectiveness of the octree-based partitioning and to test the scalability of parallel execution of the Navier–Stokes solver and hybrid grid adapter on two different parallel systems, the Intel Paragon and the IBM SP2. © 1998 John Wiley & Sons, Ltd.     

The effect of using non-orthogonal boundary-fitted grids for solving the shallow water equations

The purpose of this paper is to examine the effects of using non-orthogonal boundary-fitted grids for the numerical solution of the shallow water equations. Two geometries with well known analytical solutions are introduced in order to investigate the accuracy of the numerical solutions. The results verify that a reasonable departure from orthogonality can be allowed when the rate of change of cell areas is kept sufficiently small (i.e. it is not necessary to create a strictly orthogonal grid when the grid is sufficiently smooth).     

Two-fluid smoothed particle hydrodynamics simulation of submerged granular column collapse

In this paper, a two-fluid smoothed particle hydrodynamics (SPH) model, based on the mixture theory, is employed to investigate the complex interactions between the solid particles and the ambient water during the process of submerged granular column collapse. From the simulation, two regimes of the collapse, one being quick and the other being slow, are identified and the reasons of formation are analyzed. It is found that, a large internal friction angle of the granular phase, representing large drag force between solid particles, helps form the slow regime. Small hydraulic conductivity, representing large inter-phase drag force, also retards the collapse dramatically. Good agreements between our numerical results and other researchers’ numerical and experimental results are observed, which demonstrates the capability of the proposed two-fluid SPH approach in dealing with saturated water–soil mixture flows.     

用鲁棒Riemann求解器和运动重叠网格计算旋翼粘性绕流

发展了一种基于鲁棒Riemann求解器和运动重叠网格技术计算直升机悬停旋翼流场的方法。基于惯性坐标系,悬停旋翼流场是非定常流场,控制方程为可压缩Reynolds平均Navier-Stoke方程,其对流项采用Roe近似Reimann求解器离散,使用改进的五阶加权基本无振荡格式进行高阶重构,非定常时间推进采用含牛顿型LUSGS子迭代的全隐式双时间步方法。为实施旋转运动和便于捕捉尾迹,计算采用运动重叠网格技术。计算得到的桨叶表面压力分布及桨尖涡涡核位置都与实验结果吻合较好。数值结果表明:所发展方法对桨尖涡具有较高的分辨率,对激波具有较好的捕捉能力,该方法可进一步推广到前飞旋翼粘性绕流的计算。     

Generation of nearly isotropic turbulence using two oscillating grids

Isotropic turbulence has unique properties and is impracticable to realize experimentally. Past experiments in this context have been performed by passing a uniform mean flow through a grid, which yields approximately isotropic decaying turbulence. Here an alternative approach of obtaining approximately isotropic stationary turbulence is described, which utilizes two monoplanar grids oscillating in a homogeneous fluid. It was found that the central region between the grids has certain properties similar to that of isotropic turbulence.The funding for the particle dispersion studies at Arizona State University (ASU) is provided by the Environmental Protection Agency (Office of Exploratory Research). The stratified and rotating flow research at ASU is funded by the Office of Naval Research and the National Science Foundation.     

Simulation of 2D external viscous flows by means of a domain decomposition method using an influence matrix technique

Two-dimensional external viscous flows are numerically approximated by means of a domain decomposition method which combines a vortex method and a finite difference method. The vortex method is used in the flow region which is dominated by convective effects, whereas the finite difference method is used in the flow region where viscous diffusion effects are dominant. An influence matrix technique combined with the uniformity condition of the pressure is used to enforce the tangential velocity boundary condition. Comparisons between numerical and experimental data show that the method is well adapted for simulating two-dimensional flows.     

The motion of viscous liquid column with finite length in a vertical straight capillary tube

This paper deals with the motion of viscous liquid column with finite length and two free surfaces in a vertical straight capillary tube. It is assumed that fluid is Newtonian. Linearizing the boundary conditions, analytic expressions in the form of infinite series have been obtained for velocity, piessure and free surface at low Reynolds number. The numerical calculation is carried out for a set of cylinder’s length of water and blood. It has been revealed that there are considerable circulating currents at the upper and lower meniscuses. Its maximum velocity is about 57% of the average velocity of the mainstream. Iner-tial effect is also studied in this paper. Using the time-dependent method in finite difference techniques, numerical solution of the corresponding nonlinear equation at Re<24.5 is computed. Comparing it with analytic exact solution at low Reynolds number shows that inertial effect is negligible provided Re<24.5.     

Flow visualization of shock/water column interactions

The breakup of a liquid droplet induced by a high speed gas stream is a typical multiphase flow problem. The shock/droplet interaction is the beginning stage of the droplet breakup. Therefore, investigation of the shock/droplet interactions would be a milestone for interpreting the mechanism of the droplet breakup. In this study, a compressible multiphase solver with a five-equation model is successfully developed to study shock/water column interactions. For code validation, interface-only, gas–gas shock tube, and gas–liquid shock tube problems are first computed. Subsequently, a planar shock wave interacting with a water column is simulated. The transmitted wave and the alternative appearances of local high- and low-pressure regions inside the water column are observed clearly. Finally, a planar shock wave interacting with two water columns is investigated. In this work, both horizontal and vertical arrangements of two water columns are studied. It is found that different arrangements can result in the diversity of the interacting process. The complex flow structures generated by shock/water column interactions are presented by flow-visualization techniques.      

Efficient computation of unsteady vortical flow using flow-adaptive time-dependent grids

The objective of this study is to efficiently simulate vortex-dominated highly unsteady flows. In such flows, the locations as well as the extent of the regions requiring fine-mesh resolution vary with time. A technique has been developed to simulate these flows on a temporally adapting grid in which the adaption is based on the evolving flow solution. The flow in an axisymmetric constriction has been selected as an illustrative problem. The multiple and disparate length scales inherent in this complex flow make this problem ideally suited for evaluating the adaptive-grid technique. Adaption is based on the equidistribution of a weight function, through the use of forcing functions. The significance of this is that the method can be implemented into existing flow-analysis systems with minimal changes. The grid-generation equations developed are viewed as grid-transport equations. The time-dependent control functions perform the role of the convective speed in this transport mechanism. The equations provide the efficiency and flow tracking capability of parabolic equations, while maintaining the smoothness of computationally expensive elliptic equations. The efficiency and flow tracking capability of the approach is demonstrated for both steady and unsteady flows.     

Godunov-type solution of the shallow water equations on adaptive unstructured triangular grids

A Godunov-type upwind finite volume solver of the non-linear shallow water equations is described. The shallow water equations are expressed in a hyperbolic conservation law formulation for application to cases where the bed topography is spatially variable. Inviscid fluxes at cell interfaces are computed using Roe's approximate Riemann solver. Second-order accurate spatial calculations of the fluxes are achieved by enhancing the polynomial approximation of the gradients of conserved variables within each cell. Numerical oscillations are curbed by means of a non-linear slope limiter. Time integration is second-order accurate and implicit. The numerical model is based on dynamically adaptive unstructured triangular grids. Test cases include an oblique hydraulic jump, jet-forced flow in a flat-bottomed circular reservoir, wind-induced circulation in a circular basin of non-uniform bed topography and the collapse of a circular dam. The model is found to give accurate results in comparison with published analytical and alternative numerical solutions. Dynamic grid adaptation and the use of a second-order implicit time integration scheme are found to enhance the computational efficiency of the model.     

Free vibration of non-uniform column using DQM

Most of engineering problems are governed by a set of partial differential equations with proper boundary conditions. The present work is concerned with free vibration analysis of non-uniform column resting on elastic foundation and subjected to follower force. The used method of solution is the differential quadrature method (DQM). Formulation of the problem is introduced. The results obtained and compared with the exact solution and traditional numerical techniques such as finite element method. The parametric study is used to investigate the effect of column geometry on the natural frequencies, the mode shapes and the critical load.     

Numerical simulations of viscous flows using a meshless method

This paper uses the element‐free Galerkin (EFG) method to simulate 2D, viscous, incompressible flows. The control equations are discretized with the standard Galerkin method in space and a fractional step finite element scheme in time. Regular background cells are used for the quadrature. Several classical fluid mechanics problems were analyzed including flow in a pipe, flow past a step and flow in a driven cavity. The flow field computed with the EFG method compared well with those calculated using the finite element method (FEM) and finite difference method. The simulations show that although EFG is more expensive computationally than FEM, it is capable of dealing with cases where the nodes are poorly distributed or even overlap with each other; hence, it may be used to resolve remeshing problems in direct numerical simulations. Flows around a cylinder for different Reynolds numbers are also simulated to study the flow patterns for various conditions and the drag and lift forces exerted by the fluid on the cylinder. These forces are calculated by integrating the pressure and shear forces over the cylinder surface. The results show how the drag and lift forces oscillate for high Reynolds numbers. The calculated Strouhal number agrees well with previous results. Copyright © 2008 John Wiley & Sons, Ltd.