Integrating 1D and 2D hydrodynamic,sediment transport model for dam-break flow using finite volume method

The purpose of this study is to set up a dynamically linked 1D and 2D hydrodynamic and sediment transport models for dam break flow.The 1D-2D coupling model solves the generalized shallow water equations,the non-equilibrium sediment transport and bed change equations in a coupled fashion using an explicit finite volume method.It considers interactions among transient flow,strong sediment transport and rapid bed change by including bed change and variable flow density in the flow continuity and momentum equations.An unstructured Quadtree rectangular grid with local refinement is used in the 2D model.The intercell flux is computed by the HLL approximate Riemann solver with shock captured capability for computing the dry-to-wet interface for all models.The effects of pressure and gravity are included in source term in this coupling model which can simplify the computation and eliminate numerical imbalance between source and flux terms.The developed model has been tested against experimental and real-life case of dam-break flow over fix bed and movable bed.The results are compared with analytical solution and measured data with good agreement.The simulation results demonstrate that the coupling model is capable of calculating the flow,erosion and deposition for dam break flows in complicated natural domains.     

Stability of an erodible bed in various shear flows

The 2D laminar quasi-steady asymptotically simplified and linearized flow with a simplified mass transport of sediments is solved over a slowly erodible bed in various laminar basic shear flow (steady, oscillating or decelerating). The simplified mass transport equation includes the two following phenomena: flux of erosion when the skin friction goes over a threshold value, and a non local effect coming either from an inertial effect or from a slope effect. It is shown that the bed is always unstable for small wave numbers. Examples of long time evolution in various shear régimes are presented, wave trains of ripples are created and merge into a unique bump. This coarsening process is such that the maximum wave length obeys a power law with time.     

Mathematical Development and Verification of a Finite Volume Model for Morphodynamic Flow Applications

The accuracy and efficiency of a class of finite volume methods are investigated for numerical solution of morphodynamic problems in one space dimension. The governing equations consist of two components, namely a hydraulic part described by the shallow water equations and a sediment part described by the Exner equation. Based on different formulations of the morphodynamic equations, we propose a family of three finite volume methods. The numerical fluxes are reconstructed using a modified Roe's scheme that incorporates, in its reconstruction, the sign of the Jacobian matrix in the morphodynamic system. A well-balanced discretization is used for the treatment of the source terms. The method is well-balanced, non-oscillatory and suitable for both slow and rapid interactions between hydraulic flow and sediment transport. The obtained results for several morphodynamic problems are considered to be representative, and might be helpful for a fair rating of finite volume solution schemes, particularly in long time computations.     

Unstructured finite volume discretisation of bed friction and convective flux in solute transport models linked to the shallow water equations

The finite volume discretisation of the shallow water equations has been the subject of many previous studies, most of which deal with a well-balanced conservative discretisation of the convective flux and bathymetry. However, the bed friction discretisation has not been so profusely analysed in previous works, while it may play a leading role in certain applications of shallow water models. In this paper we analyse the numerical discretisation of the bed friction term in the two-dimensional shallow water equations, and we propose a new unstructured upwind finite volume discretisation for this term. The new discretisation proposed improves the accuracy of the model in problems in which the bed friction is a relevant force in the momentum equation, and it guarantees a perfect balance between gravity and bed friction under uniform flow conditions. The relation between the numerical scheme used to solve the hydrodynamic equations and the scheme used to solve a scalar transport model linked to the shallow water equations, is also analysed in the paper. It is shown that the scheme used in the scalar transport model must take into consideration the scheme used to solve the hydrodynamic equations. The most important implication is that a well-balanced and conservative scheme for the scalar transport equation cannot be formulated just from the water depth and velocity fields, but has to consider also the way in which the hydrodynamic equations have been solved.     

Two-dimensional coupled mathematical modeling of fluvial processes with intense sediment transport and rapid bed evolution

Alluvial rivers may experience intense sediment transport and rapid bed evolution under a high flow regime, for which traditional decoupled mathematical river models based on simplified conservation equations are not applicable. A two-dimensional coupled mathematical model is presented, which is generally applicable to the fluvial processes with either intense or weak sediment transport. The governing equations of the model comprise the complete shallow water hydrodynamic equations closed with Manning roughness for boundary resistance and empirical relationships for sediment exchange with the erodible bed. The second-order Total-Variation-Diminishing version of the Weighted-Average-Flux method, along with the HLLC approximate Riemann Solver, is adapted to solve the governing equations, which can properly resolve shock waves and contact discontinuities. The model is applied to the pilot study of the flooding due to a sudden outburst of a real glacial-lake. Supported by the National Basic Research and Development Program of China (973 Program) (Grant No. 2007CB14106), the National Natural Science Foundation of China (Grant No. 50459001), and the Key Project of Chinese Academy of Sciences (Grant No. KZCX3-SW-357-02)     

Lattice Boltzmann Simulation of Free-Surface Temperature Dispersion in Shallow Water Flows

We develop a lattice Boltzmann method for modeling free-surface temperature dispersion in the shallow water flows. The governing equations are derived from the incompressible Navier-Stokes equations with assumptions of shallow water flows including bed frictions, eddy viscosity, wind shear stresses and Coriolis forces. The thermal effects are incorporated in the momentum equation by using a Boussinesq approximation. The dispersion of free-surface temperature is modelled by an advection-diffusion equation. Two distribution functions are used in the lattice Boltzmann method to recover the flow and temperature variables using the same lattice structure. Neither upwind discretization procedures nor Riemann problem solvers are needed in discretizing the shallow water equations. In addition, the source terms are straightforwardly included in the model without relying on well-balanced techniques to treat flux gradients and source terms. We validate the model for a class of problems with known analytical solutions and we also present numerical results for sea-surface temperature distribution in the Strait of Gibraltar.     

Scaling laws in aeolian sand transport

We report on wind tunnel measurements on saltating particles in a turbulent boundary layer and provide evidence that over an erodible bed the particle velocity in the saltation layer and the saltation length are almost invariant with the wind strength, whereas over a nonerodible bed these quantities vary significantly with the air friction speed. It results that the particle transport rate over an erodible bed does not exhibit a cubic dependence with the air friction speed, as predicted by Bagnold, but a quadratic one. This contrasts with saltation over a nonerodible bed where the cubic Bagnold scaling holds. Our findings emphasize the crucial role of the boundary conditions at the bed and may have important practical consequences for aeolian sand transport in a natural environment.     

A Riemann solver for unsteady computation of 2D shallow flows with variable density

A novel 2D numerical model for vertically homogeneous shallow flows with variable horizontal density is presented. Density varies according to the volumetric concentration of different components or species that can represent suspended material or dissolved solutes. The system of equations is formed by the 2D equations for mass and momentum of the mixture, supplemented by equations for the mass or volume fraction of the mixture constituents. A new formulation of the Roe-type scheme including density variation is defined to solve the system on two-dimensional meshes. By using an augmented Riemann solver, the numerical scheme is defined properly including the presence of source terms involving reaction. The numerical scheme is validated using analytical steady-state solutions of variable-density flows and exact solutions for the particular case of initial value Riemann problems with variable bed level and reaction terms. Also, a 2D case that includes interaction with obstacles illustrates the stability and robustness of the numerical scheme in presence of non-uniform bed topography and wetting/drying fronts. The obtained results point out that the new method is able to predict faithfully the overall behavior of the solution and of any type of waves.     

A Boundary Meshless Method for Solving Heat Transfer Problems Using the Fourier Transform

Fourier transform is applied to remove the time-dependent variable in the diffusion equation. Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation, which is solved by the method of fundamental solutions and the method of particular solutions. The particular solution of Helmholtz equation is available as shown in [4, 15]. The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm. Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response. Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.     

On the theory of acoustic solitons in a liquid with distributed gas bubbles

A close relation is established between numerical solutions to two systems of equations, viz., the two-level nonlinear wave dynamic model of a liquid with gas bubbles and the Korteweg-de Vries (KdV) equation. This model is used for deriving the KdV equation in the long-wave approximation for any dependent variable of the gas-liquid mixture. The KdV equations derived earlier using radically different approximations are particular cases of our equations.     

远场模型方程的同伦近似对称约化

以同伦近似对称法为理论依据研究了远场模型方程, 通过归纳各阶相似约化解和各阶相似约化方程的通式构造相应的同伦级数解. 各阶相似约化方程均为线性变系数常微分方程, 并且可以从零阶开始依次求解. 同伦模型中的辅助参数影响同伦级数解的收敛性. 关键词： 同伦近似对称法 远场模型方程 同伦级数解     

Contaminant Flow and Transport Simulation in Cracked Porous Media Using Locally Conservative Schemes

The purpose of this paper is to analyze some features of contaminant flow passing through cracked porous medium, such as the influence of fracture network on the advection and diffusion of contaminant species, the impact of adsorption on the overall transport of contaminant wastes. In order to precisely describe the whole process, we firstly build the mathematical model to simulate this problem numerically. Taking into consideration of the characteristics of contaminant flow, we employ two partial differential equations to formulate the whole problem. One is flow equation; the other is reactive transport equation. The first equation is used to describe the total flow of contaminant wastes, which is based on Darcy law. The second one will characterize the adsorption, diffusion and convection behavior of contaminant species, which describes most features of contaminant flow we are interested in. After the construction of numerical model, we apply locally conservative and compatible algorithms to solve this mathematical model. Specifically, we apply Mixed Finite Element (MFE) method to the flow equation and Discontinuous Galerkin (DG) method for the transport equation. MFE has a good convergence rate and numerical accuracy for Darcy velocity. DG is more flexible and can be used to deal with irregular meshes, as well as little numerical diffusion. With these two numerical means, we investigate the sensitivity analysis of different features of contaminant flow in our model, such as diffusion, permeability and fracture density. In particular, we study $K_d$ values which represent the distribution of contaminant wastes between the solid and liquid phases. We also make comparisons of two different schemes and discuss the advantages of both methods.     

非驻定过滤燃烧的均相模拟

基于一维层流反应流模型,构建了新的准稳态均相模型,并对堆积床内充分发展后的低速过滤贫燃过程进行数值模拟.将计算结果与传统双相模型进行比较,分析弥散效应和化学反应机理等对计算结果的影响,并开展输运项分析;将均相模型的数值结果与准稳态和瞬态的理论结果进行比较,验证理论方法.     

Dynamic aspect of solid solution cathodes by method of integral transform

A mathematical model is presented for a thin film, spherical and cylindrical particles electrodes under galvanostatic discharge. The model available to simulate the electrochemical behavior is discussed considering not only their electrochemical representation (transport phenomena), but also the mathematical techniques, i.e. Integral transform that have been used for solving the equations. We examine the non-homogeneous material balance equation in the rectangular, spherical and cylindrical coordinate system; determine the elementary solutions, the norms and the eigenvalues of the problems for galvanostatic boundary conditions and systematically tabulate the resulting expressions. Expressions are developed for plane, cylindrical and spherical particles giving the relation between battery load and the amount of cathode material utilized. The particle shape and a single parameter Q is used to describe cathode performance.     

两相湍流色噪声扩维法PDF模型及数值模拟

由颗粒运动的朗之万方程出发,对流体脉动速度采用扩维方法,得到两个不同层次的PDF输运方程.通过对颗粒运动方程求解和高斯分布假设,解决PDF方程的封闭问题,获得颗粒二阶矩模型,然后将颗粒应力方程简化成代数方程,建立代数应力模型.将对流扩散方程的有限分析法运用到求解两相流模型中,对壁面两相射流进行数值模拟,对比分析数值结果与实验结果.     

An Eulerian method for multi-component problems in non-linear elasticity with sliding interfaces

This paper is devoted to developing a multi-material numerical scheme for non-linear elastic solids, with emphasis on the inclusion of interfacial boundary conditions. In particular for colliding solid objects it is desirable to allow large deformations and relative slide, whilst employing fixed grids and maintaining sharp interfaces. Existing schemes utilising interface tracking methods such as volume-of-fluid typically introduce erroneous transport of tangential momentum across material boundaries. Aside from combatting these difficulties one can also make improvements in a numerical scheme for multiple compressible solids by utilising governing models that facilitate application of high-order shock capturing methods developed for hydrodynamics. A numerical scheme that simultaneously allows for sliding boundaries and utilises such high-order shock capturing methods has not yet been demonstrated. A scheme is proposed here that directly addresses these challenges by extending a ghost cell method for gas-dynamics to solid mechanics, by using a first-order model for elastic materials in conservative form. Interface interactions are captured using the solution of a multi-material Riemann problem which is derived in detail. Several different boundary conditions are considered including solid/solid and solid/vacuum contact problems. Interfaces are tracked using level-set functions. The underlying single material numerical method includes a characteristic based Riemann solver and high-order WENO reconstruction. Numerical solutions of example multi-material problems are provided in comparison to exact solutions for the one-dimensional augmented system, and for a two-dimensional friction experiment.     

Approximate homotopy symmetry method: Homotopy series solutions to the sixth-order Boussinesq equation

An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving...     

The Riemann Problem for the one-dimensional,free-surface Shallow Water Equations with a bed step: Theoretical analysis and numerical simulations

In this paper, the solution of the Riemann Problem for the one-dimensional, free-surface Shallow Water Equations over a bed step is analyzed both from a theoretical and a numerical point of view. Particular attention has been paid to the wave that is generated at the location of the bed discontinuity. Starting from the classical Shallow Water Equations, considering the bed level as an additional variable, and adding to the system an equation imposing its time invariance, we show that this wave is a contact wave, across which one of the Riemann invariants, namely the energy, is not constant. This is due to the fact that the relevant problem is nonconservative. We demonstrate that, in this type of system, Riemann Invariants do not generally hold in contact waves. Furthermore, we show that in this case the equations that link the flow variables across the contact wave are the Generalized Rankine–Hugoniot relations and we obtain these for the specific problem. From the numerical point of view, we present an accurate and efficient solver for the step Riemann Problem to be used in a finite-volume Godunov-type framework. Through a two-step predictor–corrector procedure, the solver is able to provide solutions with any desired accuracy. The predictor step uses a well-balanced Generalized Roe solver while the corrector step solves the exact nonlinear system of equations that consitutes the problem by means of an iterative procedure that starts from the predictor solution. In order to show the effectiveness and the accuracy of the proposed approach, we consider several step Riemann Problems and compare the exact solutions with the numerical results obtained by using a standard Roe approach far from the step and the novel two-step algorithm for the fluxes over the step, achieving good results.     

Variable viscosity effects on the flow of MHD hybrid nanofluid containing dust particles over a needle with Hall current—a Xue model exploration

This study scrutinizes the flow of engine oil-based suspended carbon nanotubes magneto-hydrodynamics (MHD) hybrid nanofluid with dust particles over a thin moving needle following the Xue model. The analysis also incorporates the effects of variable viscosity with Hall current. For heat transfer analysis, the effects of the Cattaneo-Christov theory and heat generation/absorption with thermal slip are integrated into the temperature equation. The Tiwari-Das nanofluid model is used to develop the envisioned mathematical model. Using similarity transformation, the governing equations for the flow are translated into ordinary differential equations. The bvp4c method based on Runge-Kutta is used, along with a shooting approach. Graphs are used to examine and depict the consequences of significant parameters on involved profiles. The results revealed that the temperature of the fluid and boundary layer thickness is diminished as the solid volume fraction is raised. Also, with an enhancement in the variable viscosity parameter, the velocity distribution becomes more pronounced. The results are substantiated by assessing them with an available study.     

ADER schemes for the shallow water equations in channel with irregular bottom elevation

This paper deals with the construction of high-order ADER numerical schemes for solving the one-dimensional shallow water equations with variable bed elevation. The non-linear version of the schemes is based on ENO reconstructions. The governing equations are expressed in terms of total water height, instead of total water depth, and discharge. The ENO polynomial interpolation procedure is also applied to represent the variable bottom elevation. ADER schemes of up to fifth order of accuracy in space and time for the advection and source terms are implemented and systematically assessed, with particular attention to their convergence rates. Non-oscillatory results are obtained for discontinuous solutions both for the steady and unsteady cases. The resulting schemes can be applied to solve realistic problems characterized by non-uniform bottom geometries.