An efficient finite difference scheme for free‐surface flows in narrow rivers and estuaries

This paper presents a free‐surface correction (FSC) method for solving laterally averaged, 2‐D momentum and continuity equations. The FSC method is a predictor–corrector scheme, in which an intermediate free surface elevation is first calculated from the vertically integrated continuity equation after an intermediate, longitudinal velocity distribution is determined from the momentum equation. In the finite difference equation for the intermediate velocity, the vertical eddy viscosity term and the bottom‐ and sidewall friction terms are discretized implicitly, while the pressure gradient term, convection terms, and the horizontal eddy viscosity term are discretized explicitly. The intermediate free surface elevation is then adjusted by solving a FSC equation before the intermediate velocity field is corrected. The finite difference scheme is simple and can be easily implemented in existing laterally averaged 2‐D models. It is unconditionally stable with respect to gravitational waves, shear stresses on the bottom and side walls, and the vertical eddy viscosity term. It has been tested and validated with analytical solutions and field data measured in a narrow, riverine estuary in southwest Florida. Model simulations show that this numerical scheme is very efficient and normally can be run with a Courant number larger than 10. It can be used for rivers where the upstream bed elevation is higher than the downstream water surface elevation without any problem. Copyright © 2003 John Wiley & Sons, Ltd.     

A Godunov method for the computation of erosional shallow water transients

A Godunov method is proposed for the computation of open‐channel flows in conditions of rapid bed erosion and intense sediment transport. Generalized shallow water equations govern the evolution of three distinct interfaces: the water free‐surface, the boundary between pure water and a sediment transport layer, and the morphodynamic bottom profile. Based on the HLL scheme of Harten, Lax and Van Leer (1983), a finite volume numerical solver is constructed, then extended to second‐order accuracy using Strang splitting and MUSCL extrapolation. Lateralisation of the momentum flux is adopted to handle the non‐conservative product associated with bottom slope. Computational results for erosional dam‐break waves are compared with experimental measurements and semi‐analytical Riemann solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

Development of self-aeration process for supercritical chute flows

When a high-velocity flow discharges into a chute, air is entrained through the free surface. This is relevant to the development of self-aeration for mixture flow. In this study, the air concentration was measured in the self-aerated developing region for various initial flow velocities, depths, and chute slopes. The effect of hydraulic conditions on the bottom self-aeration process was analyzed. Increasing the initial flow velocity and depth was found to increase the rate of air diffusion into the water flow. This positive correlation indicates that flow turbulence is a key factor for the self-aeration development process. The Reynolds number of the flow was found to be an appropriate hydraulic condition for describing self-aeration development. In addition, the constraint of buoyancy on air bubble diffusion into the chute bottom decreased as the chute slope was increased, which made the development process for bottom self-aeration more pronounced. A new empirical equation is presented for predicting the development process of bottom self-aeration in open channel flows.     

Higher order Boussinesq-type equations for water waves on uneven bottom

Higher order Boussinesq-type equations for wave propagation over variable bathymetry were derived. The time dependent free surface boundary conditions were used to compute the change of the free surface in time domain. The free surface velocities and the bottom velocities were connected by the exact solution of the Laplace equation. Taking the velocities on half relative water depth as the fundamental unknowns, terms relating to the gradient of the water depth were retained in the inverse series expansion of the exact solution, with which the problem was closed. With enhancements of the finite order Taylor expansion for the velocity field, the application range of the present model was extended to the slope bottom which is not so mild. For linear properties, some validation computations of linear shoaling and Booij' s tests were carried out. The problems of wave-current interactions were also studied numerically to test the performance of the enhanced Boussinesq equations associated with the effect of currents. All these computational results confirm perfectly to the theoretical solution as well as other numerical solutions of the full potential problem available.     

细颗粒泥沙净冲刷和输移的大涡模拟研究

在传统水沙输移数值模拟研究中一般采用雷诺时均模拟技术(Reynolds-averaged simulation,RANS).与RANS相比,大涡模拟技术(large eddy simulation,LES)能够更加精确反映细部流动结构,计算机的发展使得采用LES探讨水流和泥沙运动规律成为可能.本文尝试给出净冲刷条件下悬沙计算的边界条件,采用动态亚格子模式对循环槽道和长槽道中的水流运动和泥沙输移进行了三维大涡模拟研究.利用直接数值模拟(direct numerical simulation,DNS)结果对LES模型进行了率定,计算结果符合良好,在此基础上初步探讨了泥沙浓度、湍动强度和湍动通量等的分布特征.结果表明,净冲刷条件下输沙平衡时泥沙浓度符合Rouse公式分布,单向流动中泥沙浓度沿着流向逐渐增大.泥沙浓度湍动强度和湍动通量都在近底部达到最大值,沿着垂向迅速减小.湍动黏性系数和扩散系数基本上在水深中间处达到最大.湍动Schmidt数沿着水深方向不是常数,在近底部和自由水面附近较大,在水深中间处较小.     

彬县水帘洞煤矿工业场地西侧黄土高边坡特征及成因分析

水帘洞煤矿工业场地西侧黄土高边坡为老滑坡的后壁,在边坡的脚下有老滑坡体残留物。地表水沿黄土垂直节理、裂隙下渗,逐渐在边坡的顶部形成落水洞;边坡还受黄土层理、倾角（5°～10°）和古土壤层位置的控制,地表水下渗至古土壤层,古土壤层是良好的隔水层,加上黄土层理倾向沟谷,在古土壤层面位置（半山腰）易形成与边坡顶部落水洞贯通的落水洞出口。受长期降水或短期强降水的影响,导致边坡失稳。     

A correction for balancing discontinuous bed slopes in two‐dimensional smoothed particle hydrodynamics shallow water modeling

In this paper, a smoothed particle hydrodynamics (SPH) numerical model for the shallow water equations (SWEs) with bed slope source term balancing is presented. The solution of the SWEs using SPH is attractive being a conservative, mesh‐free, automatically adaptive method without special treatment for wet‐dry interfaces. Recently, the capability of the SPH–SWEs numerical scheme with shock capturing and general boundary conditions has been used for predicting practical flooding problems. The balance between the bed slope source term and fluxes in shallow water models is desirable for reliable simulations of flooding over bathymetries where discontinuities are present and has received some attention in the framework of Finite Volume Eulerian models. The imbalance because of the source term resulting from the calculation of the the water depth is eradicated by means of a corrected mass, which is able to remove the error introduced by a bottom discontinuity. Two different discretizations of the momentum equation are presented herein: the first one is based on the variational formulation of the SWEs in order to obtain a fully conservative formulation, whereas the second one is obtained using a non‐conservative form of the free‐surface elevation gradient. In both formulations, a variable smoothing length is considered. Results are presented demonstrating the corrections preserve still water in the vicinity of either 1D or 2D bed discontinuities and provide close agreement with 1D analytical solutions for rapidly varying flows over step changes in the bed. The method is finally applied to 2D dam break flow over a square obstacle where the balanced formulation improves the agreement with experimental measurements of the free surface. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical simulation of 3-D turbulent flows over dredged trenches

A 3-D free surface flow in open channels based on the Reynolds equations with thek-ε turbulence closure model is presented in this paper. Insted of the “rigid lid” approximation, the solution of the free surface equation is implemented in the velocity—pressure iterative procedure on the basis of the conventional SIMPLE method. This model was used to compute the flow in rectangular channels with trenches dredged across the bottom. The velocity, eddy viscosity coefficient, turbulent shear stress, turbulent kinetic energy and elevation of the free surface can be obtained. The computed results are in good agreement with previous experimental data.     

Large eddy simulation of free surface shallow‐water flow

Shallow‐water flow with free surface frequently occurs in ambient water bodies, in which the horizontal scale of motion is generally two orders of magnitude greater than the water depth. To accurately predict this flow phenomenon in more detail, a three‐dimensional numerical model incorporating the method of large eddy simulation (LES) has been developed and assessed. The governing equations are split into three parts in the finite difference solution: advection, dispersion and propagation. The advection part is solved by the QUICKEST scheme. The dispersion part is solved by the central difference method and the propagation part is solved implicitly using the Gauss–Seidel iteration method. The model has been applied to free surface channel flow for which ample experimental data are available for verification. The inflow boundary condition for turbulence is generated by a spectral line processor. The computed results compare favourably with the experimental data and those results obtained by using a periodic boundary condition. The performance of the model is also assessed for the case in which anisotropic grids and filters with horizontal grid size of the order of the water depth are used for computational efficiency. The coarse horizontal grid was found to cause a significant reduction in the large‐scale turbulent motion generated by the bottom turbulence, and the turbulent motion is predominately described by the sub‐grid scale (SGS) terms. The use of the Smagorinsky model for SGS turbulence in this situation is found inappropriate. A parabolic mixing length model, which accounts for the filtered turbulence, is then proposed. The new model can reproduce more accurately the flow quantities. Copyright © 2000 John Wiley & Sons, Ltd.     

A transport approach to the convolution method for numerical modelling of linearized 3D circulation in shallow seas

A new method for solving the linearized equations of motion is presented in this paper, which is the implementation of an outstanding idea suggested by Welander: a transport approach to the convolution method. The present work focuses on the case of constant eddy viscosity and constant density but can be easily extended to the case of arbitrary but time-invariant eddy viscosity or density structure. As two of the three equations of motion are solved analytically and the main numerical ‘do-loop’ only updates the sea level and the transport, the method features succinctness and fast convergence. The method is tested in Heaps' basin and the results are compared with Heaps' results for the transient state and with analytical solutions for the steady state. The comparison yields satisfactory agreement. The computational advantage of the method compared with Heaps' spectral method and Jelesnianski's bottom stress method is analysed and illustrated with examples. Attention is also paid to the recent efforts made in the spectral method to accelerate the convergence of the velocity profile. This study suggests that an efficient way to accelerate the convergence is to extract both the windinduced surface Ekman spiral and the pressure-induced bottom Ekman spiral as a prespecified part of the profile. The present work also provides a direct way to find the eigenfunctions for arbitrary eddy viscosity profile. In addition, mode-trucated errors are analysed and tabulated as functions of mode number and the ratio of the Ekman depth to the water depth, which allows a determination of a proper mode number given an error tolerance.     

Large eddy simulation of vertical turbulent jets under JONSWAP waves

The effect of random waves on vertical plane turbulent jets is studied numerically and the mechanism behind the interaction of the jet and waves is analyzed. The large eddy simulation method is used and the σ-coordinate system is adopted. Turbulence is modeled by a dynamic coherent eddy model. The σ-coordinate transformation is introduced to map the irregular physical domain with a wavy free surface and an uneven bottom onto a regular computational domain. The fractional step method is used to solve the fil...     

Evaluation of the height of waves generated by an underwater landslide in a confined water reservoir

An equation was obtained for the motion of an underwater landslide over an uneven bottom slope under the action of gravity, buoyancy, friction, and water drag. Numerical modeling of the surface waves generated by landslide motion over an uneven bottom was performed using the nonlinear shallow water model. The effect of the landslide size, the friction coefficient, and other parameters on the maximum runup of landslide-generated waves on the shore with a parabolic shape of the bottom was studied.     

昆仑山隧道沟谷融区发育特征分析

利用多年冻土区昆仑山隧道2#冲沟帷幕注浆的机会,将注浆孔当作测试孔,测试每个孔的水位和孔温。通过测试孔水位、孔温和注浆量变化分析,判断2#冲沟融区发育特征,得出2#冲沟沟底的融化深度远大于通常的冻土上限;阳坡地温高于阴坡地温;在2#冲沟段,昆仑山隧道中心线左右至少7m范围内为融化区,融区深度在隧底2⁵m以下;阳坡地层的孔隙率高于阴坡孔隙率,沟底岩层与地表有较为畅通的地下水流通道,沟底地下水渗流通道明显优于两侧山坡,沟心纵断面位置附近存在至少一条未被冻结的地下水通道,且该通道的埋置深度至少可达17³0m,为昆仑山隧道渗漏水病害治理提供依据。     

Influence of turbulence intensity and free stream velocity oscillations on stagnation point heat transfer

A model is proposed for the momentum eddy diffusivity induced by free stream turbulence intensity and integral length scale. The eddy diffusivity model is applied to the stagnation point of a cylinder situated in a steady uniform crossflow in the presence of free stream turbulence. A numerical solution of the governing steady-state momentum and energy equations with the proposed eddy diffusivity model yielded results for the skin friction coefficient and the Nusselt number. Agreement between the numerical predictions of this work and experimental data is very good. The experimental data concerning the unsteady stagnation point heat transfer under the combined influence of free stream velocity oscillations and turbulence intensity have been successfully correlated by means of a new turbulence parameter     

THE MATHEMATICAL MODELLING OF NEAR COAST SHALLOW WATER CIRCULATION

This paper presents a finite element method to solve the shallow water circulation problem numerically. Considering the Coriolis effect, bottom friction and eddy viscosity, the continuity equation and momentum equation are integrated vertically. Using Galerkin weighted residual method, the weak variational formulation is derived for the finite element analysis. The split-time method is applied for the numerical integration instead of iteration for nonlinear terms. Moreover, an artificial smooth approach is proposed to suppress the short wavelength noise. In order to save Computer storage units, a densed storage scheme is set up, where all the zero elements in large scaled and sparse matrices are excluded.     

Influence of flow path modification on oscillation of cavity shear layer

This study investigates the influence on the oscillating characteristics of a cavity shear layer by introducing either a sloped bottom or a flow path modifier at the bottom of the cavity. All the experiments are performed in a recirculating water channel. The laser Doppler velocimetry system and the laser sheet technique are employed to perform the quantitative velocity measurements and the qualitative flow visualization, respectively. The Reynolds number, based on the momentum thickness at the upstream edge of the cavity, is kept at about Re _{θ 0}=194 ± 3.4. It is found that, in addition to the feedback effect, the upstream moving part of the recirculating flow inside the cavity also plays an important role in changing the oscillating characteristics of the unstable shear layer. As the bottom of the cavity is either negatively or positively sloped, the oscillating characteristics of the cavity shear layer are modified to different extents. Significant reduction of the oscillating amplitude within the cavity is found while the bottom slope increases up to d/L=± 2/5. As the bottom slope further increases up to d/L=± 1/2, the self-excited oscillation is completely suppressed. In addition, the ability to suppress the self-excited oscillation by the negative bottom slopes is superior to that in the case of a positive bottom slope. Depending upon the fence locations, the upstream moving part of the recirculating flow will perturb the unstable shear layer at different x/L locations, leading to different oscillating amplitudes. The ability to promote the enlarged oscillating amplitude of the unstable shear layer is better for a fence inclined at a positive angle than for one at a negative angle. Received: 31 May 2000/Accepted: 11 January 2001     

On the accuracy of finite difference and modal methods for computing tidal and wind wave current profiles

This paper deals with the comparative accuracy of using finite difference grids or a modal representation through the vertical in modelling tidally or wind wave induced current profiles. A point model is used in the vertical, with a no-slip condition at the sea bed. In the finite difference approach the high-shear bottom layer is resolved using either a regular grid on a logarithmic or log-linear transformed co-ordinate or an irregular grid, varying in such a manner as to retain second-order accuracy. The accuracy of these various grid schemes is considered in detail. The relative merits of using either the Crank-Nicolson or Dufort-Frankel time integration methods are considered; in the case of a fine grid in a high-viscosity region, some numerical problems are found with the Dufort-Frankel method. An alternative approach to using a finite difference grid in the vertical, namely a modal (spectral) method, is described. The form of the modes is such that they can accurately resolve the high-shear bottom boundary layer. Calculations show that the thickness of the bottom boundary layer in relation to the total water depth is important in determining the choice of grid transform and rates of convergence of solutions using finite difference or modal methods. However, for the majority of problems the modal solution is numerically attractive owing to its computational efficiency and the ease with which solution algorithms based upon it can be coded in vectorizable form suitable for the new generation of vector computers. The influence of viscosity profile, its time variation and water depth upon tidally induced or wave induced currents is considered. Calculations suggest that near-bed measurements of tidal flow in shallow water together with associated modelling would enable appropriate formulations of eddy viscosity to be determined. Similar measurements, though using a laboratory flume, would be appropriate for wind wave problems.     

Experimental Study of Sediment Flow Discharge in New System of Bottom Intakes with Porous Media

Bottom rack intake is one of the most popular structures for diverting water in steep rivers. The problems of corrosion, deformity, and clogging of the bottom racks in long term inspire a new system of bottom intake in which a filled trench of porous media replaces the bottom racks. Diversion of a specified amount of water through the porous media, when the void space in the granular material is filled with fine sediments in comparison to the bottom racks, requires much larger structure. For the proposed water intake, lower cost of construction and maintenance and higher compatibility with the river morphology are considered as major advantages. This research deals with an experimental model with two-story channel, the lower one is used to convey diverted water through the porous media and the upper one is used to carry the remained flow to downstream. Measurements of the diverted discharge were performed for different rates of flow, grain, and sediment size distributions as well as surface slopes of intake. Results show that despite clogging of the suspended sediments in porous media, using appropriate grain size with a surface slope of the porous media increases the discharge coefficient of system. In comparison to the clear water flow, the diverted discharge reduction induced by clogging is not significant and the efficiency of system in long term with no operation effort is confirmed. An empirical formulation has been proposed based on the concept of Darcy’s law for surface infiltration. The equation is found to be dependent on size ratio, Reynolds number, and hydraulic gradient. The theoretical predictions in comparison with the experimental results have shown a good consistency.     

局部与整体水压法在水下饱和边坡稳定分析中的应用

针对以考虑土条侧向条间力来提高瑞典条分法的计算精度方法, 作者基于物理学思想, 提出了局部与整体水压法以进行水下饱和边坡稳定分析, 并采用该法对瑞典条分法中一对相互矛盾的"真假李逵" 式进行了辨析. 指出李鬼式精度低的原因在于直接提取土条顶面和底面水压力沿着竖向的分量来建立平衡, 而忽略了其沿着水平方向分力的影响, 并证明了局部与整体水压法在提高有水边坡计算精度和便捷度时具有的普适性.     

Introducing the theory of successful settling in order to evaluate and optimize the sedimentation tanks

A rectangular settling tank in full scale is investigated using the Fluent software to increase its efficiency. First, the pure water is simulated in the absence of particles. Then particles are injected into the flow field and tracked by means of the discrete phase model. Three methods are presented to optimize the settling tank: (1) adding a baffle which is mounted in the bottom and is extended up to the near of free surface, (2) adding a baffle which is mounted in the free surface and is extended up to the near of the tank bottom and (3) install a bi-directional baffle which is mounted in the free surface and is extended up to the near of the bottom. These three suggestions are checked using the short-circuiting phenomenon and the successful settling theory. The successful settling theory states that a particle can be accounted as a “trapped” particle if (1) settled particle stays static and stable in the tank bottom and (2) settled particle doesn’t return to purified sewage. Although the second method provided higher efficiency, the third method was selected as the most appropriate method in order to optimize the settling tank.