Finite difference solution of a Newtonian jet swell problem

A finite difference technique has been developed to study the Newtonian jet swell problem. The streamfunction and vorticity were used as dependent variables to describe the jet flow. The boundary-fitted co-ordinate transformation method was adopted to map the flow geometry into a rectangular domain. The standard finite difference method was then applied for solving the flow equations. The location of the jet free surface was updated by the kinematic boundary condition, and an adjustable parameter was included in the free-surface iteration. We could obtain numerical solutions for the Reynolds number as high as 100, and the differences between the present study and previous finite element simulations on the jet swell ratio are less than 5%.     

A fully coupled numerical technique for 2D bed morphology calculations

A fully coupled two‐dimensional subcritical and/or supercritical, viscous, free‐surface flow numerical model is developed to calculate bed variations in alluvial channels. Vertically averaged free‐surface flow equations in conjunction with sediment transport equation are numerically solved using an explicit finite‐volume scheme using transformed grid in order to handle complex geometry fluvial problems. Convergence is accelerated with use of a multi‐grid technique. Firstly the capabilities of the proposed method are demonstrated by analyzing subcritical and supercritical hydrodynamic flows. Thereafter, an analysis of one‐ and two‐dimensional flows is performed referring to aggradation and scouring. For all reported test cases the computed results compare reasonably well with measurements as well as with other numerical solutions. The method is stable, reliable and accurate handling a variety of sediment transport equations with rapid changes of sediment transport at the boundaries. Copyright © 2002 John Wiley & Sons, Ltd.     

A study on the extension of a VOF/PLIC based method to a curvilinear co-ordinate system

The conventional volume-of-fluid method has the potential to deal with large free surface deformation on a fixed Cartesian grid. However, when free-surface flows are within or over complex geometries of industrial relevance, such as free-surface flows over offshore oil platforms, it is advantageous to extend the method originally written in Cartesian forms into non-Cartesian forms. In the present study, an algorithm similar to the algorithm described by Rudman in 1997 is proposed for use with curvilinear co-ordinates. This extension results in the ability to model complex geometries which could not be modelled using the original algorithm. Excellent agreement between the solutions obtained on both orthogonal and non-orthogonal meshes is achieved, although in general the L ₁ error, based on the exact solution, on the non-orthogonal mesh is slightly higher than that on the orthogonal mesh. The extended fluid flow solving capacity of the present method is demonstrated through its application to a non-orthogonal Rayleigh–Taylor instability problem.     

Modeling Three-Dimensional Groundwater Flows by the Body-Fitted Coordinate (BFC) Method: II. Free and Moving Boundary Problems

This paper presents a methodology and solution procedure of the time-dependent body-fitted coordinate (BFC) method for the analysis of transient, three-dimensional groundwater flow problems characterized by free and moving boundaries. The technique consists of numerical grid generation, time-dependent body-fitted coordinate transformation, and application of the finite difference method (FDM) to the transformed partial differential equations. Based on the time-dependent BFC method, a three-dimensional finite-difference computer code, BFC3DGW, was developed and used to solve two unconfined flow problems. The code was verified by comparing numerical results with analytical solutions for a steady-state seepage problem. In order to demonstrate capability of the method in dealing with flow problems with irregular and moving boundary surfaces, an unconfined well-flow problem was solved by the developed code. Difficulties associated with the free and moving irregular boundary have been successfully overcome by employing this method.     

Three-dimensional isothermal lava flows over a non-axisymmetric conical surface

Within the thin-layer approximation for a highly-viscous heavy incompressible fluid, a hydrodynamicmodel of a 3D isothermal lava flow over a non-axisymmetric conical surface is constructed. Using analytical methods, a self-similar solution for the law of leading-edge propagation is obtained in the case of a flow from a non-axisymmetric source located at the apex of a conical surface with smoothly varying properties. In the case of a flow over a substantially non-axisymmetric surface, it is shown that there exists a self-similar solution for the free-surface shape and the law of leading-edge motion. This solution is studied numerically for particular examples of the substrate surface and the source. In the general case of a non-self-similar flow over a substantially non-axisymmetric conical surface, a local analytical solution is obtained for the free-surface shape and the velocity field near the leading flow front.     

A high-resolution Godunov-type scheme in finite volumes for the 2D shallow-water equations

A high-order Godunov-type scheme based on MUSCL variable extrapolation and slope limiters is presented for the resolution of 2D free-surface flow equations. In order to apply a finite volume technique of integration over body-fitted grids, the construction of an approximate Jacobian (Roe type) of the normal flux function is proposed. This procedure allows conservative upwind discretization of the equations for arbitrary cell shapes. The main advantage of the model stems from the adaptability of the grid to the geometry of the problem and the subsequent ability to produce correct results near the boundaries. Verification of the technique is made by comparison with analytical solutions and very good agreement is found. Three cases of rapidly varying two-dimensional flows are presented to show the efficiency and stability of this method, which contains no terms depending on adjustable parameters. It can be considered well suited for computation of rather complex free-surface two-dimensional problems.     

Free-surface fluctuations in hydraulic jumps: Experimental observations

A hydraulic jump is the rapid and sudden transition from a high-velocity supercritical open channel flow to a subcritical flow. It is characterised by the dynamic interactions of the large-scale eddies with the free-surface. New series of experimental measurements were conducted in hydraulic jumps with Froude numbers between 3.1 and 8.5 to investigate these interactions. The dynamic free surface measurements were performed with a non-intrusive technique while the two-phase flow properties were recorded with a phase-detection probe. The shape of the mean free surface profile was well defined and the turbulent fluctuation profiles highlighted a distinct peak of turbulent intensity in the first part of the jump roller, with free-surface fluctuation levels increasing with increasing Froude number. The dominant free-surface fluctuation frequencies were typically between 1 and 4 Hz. A comparison between the acoustic sensor signals and conductivity probe data suggested that the air–water “free-surface” detected by the acoustic sensor corresponded to about the boundary between the turbulent shear layer and the upper free-surface layer. Simultaneous measurements of free surface and bubbly flow fluctuations for Fr = 5.1 indicated that the frequency ranges of both sensors were similar (F < 5 Hz) whatever the position downstream of the toe. The present results highlighted that the dynamic free-surface measurements can be conducted successfully using acoustic displacement meters, and the time-averaged depth measurements was a physical measure of the free-surface location in hydraulic jumps.     

Interaction of viscous wakes with a free surface

The interaction of laminar wakes with.free-surface waves generated by a moving body beneath the surface of an incompressible viscous fluid of infinite depth was investigated analytically. The analysis was based on the steady Oseen equations for disturbed flows.The kinematic and dynamic boundary conditions were linearized for the small-amplitude free-surface waves. The effect of the moving body was mathematically modeled as an Oseenlet.The disturbed flow was regarded as the sum of an unbounded singular Oseen flow which represents the effect of the viscous wake and a bounded regular Oseen flow which represents the influence of the free surface. The exact solution for the free-surface waves was obtained by the method of integral transforms. The asymptotic representation with additive corrections for the free-surface waves was derived by means of Lighthill‘s two-stage scheme. The symmetric solution obtained shows that the amplitudes of the free-surface waves are exponentially damped by the presences of viscosity and submergence depth.     

Partitioned solution to fluid–structure interaction problem in application to free-surface flows

In this work we discuss a way to compute the impact of free-surface flow on nonlinear structures. The approach chosen relies on a partitioned strategy that allows us to solve the strongly coupled fluid–structure interaction problem. It is then possible to re-use the existing and validated strategy for each sub-problem. The structure is formulated in a Lagrangian way and solved by the finite element method. The free-surface flow approach considers a Volume-Of-Fluid (VOF) strategy formulated in an Arbitrary Lagrangian–Eulerian (ALE) framework, and the finite volume is used to discrete and solve this problem. The software coupling is ensured in an efficient way using the Communication Template Library (CTL). Numerical examples presented herein concern the 2D validation case but also 3D problems with a large number of equations to be solved.     

The development of Marangoni convection induced by local addition of a surfactant

The development of concentration convection induced by local addition of a surfactant solution onto a horizontal free surface of water is studied experimentally and theoretically. The experiment revealed that the capillary motion develops in a threshold manner, with the threshold value depending on the degree of purification of the fluid, the initial concentration of the surfactant, and the area of the free surface. To describe the threshold mechanism of the concentration convection, a number of theoretical models is considered. Different rheological properties of the surface phase, including the nonlinear dependence of the surface shear stress on the surface velocity, are examined. In the numerical experiment, the convective-flow patterns are calculated for different free-surface boundary conditions, and the time dependence of the flow intensity is investigated.     

Boundary conformed co-ordinate systems for selected two-dimensional fluid flow problems. Part I: Generation of BFGs

In this paper the generation of general curvilinear co-ordinate systems for use in selected two-dimensional fluid flow problems is presented. The curvilinear co-ordinate systems are obtained from the numerical solution of a system of Poisson equations. The computational grids obtained by this technique allow for curved grid lines such that the boundary of the solution domain coincides with a grid line. Hence, these meshes are called boundary fitted grids (BFG). The physical solution area is mapped onto a set of connected rectangles in the transformed (computational) plane which form a composite mesh. All numerical calculations are performed in the transformed plane. Since the computational domain is a rectangle and a uniform grid with mesh spacings Δξ = Δη = 1 (in two-dimensions) is used, the computer programming is substantially facilitated. By means of control functions, which form the r.h.s. of the Poisson equations, the clustering of grid lines or grid points is governed. This allows a very fine resolution at certain specified locations and includes adaptive grid generation. The first two sections outline the general features of BFGs, and in section 3 the general transformation rules along with the necessary concepts of differential geometry are given. In section 4 the transformed grid generation equations are derived and control functions are specified. Expressions for grid adaptation arc also presented. Section 5 briefly discusses the numerical solution of the transformed grid generation equations using sucessive overrelaxation and shows a sample calculation where the FAS (full approximation scheme) multigrid technique was employed. In the companion paper (Part II), the application of the BFG method to selected fluid flow problems is addressed.     

近水面水下爆炸的数值研究

近水面水下爆炸后形成的冲击波很快与空气一水的自由界面相互作用，产生折射激波和反射稀疏波，这些波在形成初期是规则的，但很快转变为不规则波。伴随冲击波与自由界面的相互作用，水面要上升直至被炸开。这些复杂的物理现象对数值计算提出了很高的要求。本文采用近期发展起来的位标函数方法以及高精度的ＮＮＤ格式来数值模拟冲击波、自由界面的运动及其相互作用，获得了成功，近水面水下爆炸时的一系列复杂的物理现象均得到了合理的再现。与现有的跟踪自由界面运动的方法相比较，本文方法更简洁，更实用。     

A high-order compact scheme for solving the 2D steady incompressible Navier-Stokes equations in general curvilinear coordinates

In this paper, a high-order compact finite difference algorithm is established for the stream function-velocity formulation of the two-dimensional steady incompressible Navier-Stokes equations in general curvilinear coordinates. Different from the previous work, not only the stream function and its first-order partial derivatives but also the second-order mixed partial derivative is treated as unknown variable in this work. Numerical examples, including a test problem with an analytical solution, three types of lid-driven cavity flow problems with unusual shapes and steady flow past a circular cylinder as well as an elliptic cylinder with angle of attack, are solved numerically by the newly proposed scheme. For two types of the lid-driven trapezoidal cavity flow, we provide the detailed data using the fine grid sizes, which can be considered the benchmark solutions. The results obtained prove that the present numerical method has the ability to solve the incompressible flow for complex geometry in engineering applications, especially by using a nonorthogonal coordinate transformation, with high accuracy.     

Parameter estimation for a three‐dimensional numerical barotropic tidal model with adjoint method

The parameters of a three‐dimensional (3‐D) barotropic tidal model are estimated using the adjoint method. The mode splitting technique is employed in both forward and adjoint models. In the external mode, the alternating direction implicit method is used to discretize the two‐dimensional depth‐averaged equations and a semi‐implicit scheme is used for the 3‐D internal mode computations. In this model the bottom friction is expressed in terms of bottom velocity which is different from the previous works. Besides, the bottom friction coefficients (BFCs) are supposed to be spatially varying, i.e. the BFC at some grid points are selected as the independent BFC, while the BFC at the other grid points can be obtained through linear interpolation with these independent BFCs. On the basis of the simulation of M₂ tide in the Bohai and North Yellow Seas (BNYS), twin experiments are carried out to invert the prescribed distributions of model parameters. The parameters inverted are the Fourier coefficients of open boundary conditions (OBCs), the BFC and the vertical eddy viscosity profiles. In these twin experiments, the real topography of BNYS is installed. The ‘observations’ are produced by the tidal model and recorded at the position of TOPEX/Poseidon altimeter data, tidal gauge data and current data. The experiments discuss the influence of initial guesses, model errors and data number. The inversion has obtained satisfactory results and the prescribed distributions have been successfully inverted. The results indicate that the inversion of BFC is more sensitive to data error than that of OBC and the vertical eddy viscosity profiles. Copyright © 2007 John Wiley & Sons, Ltd.     

A Numerical Method for 2-D Bed Morphology Calculations

A fully coupled two-dimensional sub-critical and/or supercritical, free-surface flow numerical model is developed to calculate bed variations in alluvial channels. Vertically averaged free-surface flow equations in conjunction with sediment transport equation are numerically solved using an explicit finite-volume scheme in integral form. The capabilities of the proposed method are first demonstrated by analyzing supercritical flow in an expansion channel. Thereafter, one and two-dimensional applications referring to aggradation and scouring are reported. For each of these test cases, computed results compare satisfactorily with measurements as well as with other numerical solutions. The method is stable, reliable and accurate, although time consuming, handling a variety of sediment transport equations with rapid changes of sediment transport at the boundaries.     

三维非均匀介质中弹性波传播的数值模拟

提出了一种三维非均匀介质中弹性波传播数值模拟的方法,文中称为三维格子法。该算法是二维格子法（一种二维非均匀介质中P－SV波传播的数值模拟算法）向三维非均匀介质情况的推广。在空间离散上该文方法与有限元方法类似,容许根据连续体的形状和介质分界面任意剖面网格,且自然满足自由表面边界条件。不同于常规有限差分法在各个节点上满足动力学微分方程,该算法通过满足各节点周围格子的整体平衡（积分平衡方程）来对问题进行求解,三维格子法所需的计算机内存及计算耗时与同阶精度的规则网格有限差分法相当。算例表明,该文提出的三维格子法具有较高的精度且可很好地模拟三维复杂形状地表对弹性波的反射和绕射。