Numerical simulation of two-phase flow for sediment transport in the inner-surf and swash zones

A two-dimensional two-phase flow framework for fluid–sediment flow simulation in the surf and swash zones was described. Propagation, breaking, uprush and backwash of waves on sloping beaches were studied numerically with an emphasis on fluid hydrodynamics and sediment transport characteristics. The model includes interactive fluid–solid forces and intergranular stresses in the moving sediment layer. In the Euler–Euler approach adopted, two phases were defined using the Navier–Stokes equations with interphase coupling for momentum conservation. The k–ε$k''–''\epsilon$ closure model and volume of fluid approach were used to describe the turbulence and tracking of the free surface, respectively. Numerical simulations explored incident wave conditions, specifically spilling and plunging breakers, on both dissipative and intermediate beaches. It was found that the spatial variation of sediment concentration in the swash zone is asymmetric, while the temporal behavior is characterized by maximum sediment concentrations at the start and end of the swash cycle. The numerical results also indicated that the maximum turbulent kinetic energy and sediment flux occurs near the wave-breaking point. These predictions are in general agreement with previous observations, while the model describes the fluid and sediment phase characteristics in much more detail than existing measurements. With direct quantifications of velocity, turbulent kinetic energy, sediment concentration and flux, the model provides a useful approach to improve mechanistic understanding of hydrodynamic and sediment transport in the nearshore zone.     

Hydrodynamics of coupled flow above and below a sediment–water interface with triangular bedforms

The hydrodynamics of a system where there is a coupled flow above and below a sediment–water interface (SWI) are not completely understood. We numerically simulate mean two-dimensional, unidirectional, steady, viscous flow in these systems using a sequentially coupled formulation. Simulations were conducted to determine fundamental relationships between bedform geometry, Reynolds number for the water-column flow (Re), interfacial exchange zone depth (d_z) in the sediments, and flux through the SWI (q_int); the latter two parameters play a significant role in biogeochemical and aquatic-life processes across the SWI. d_z and Re are functionally related through an asymptotic growth-curve model while q_int and Re follow a power function. These relationships are dynamically explained by the manner in which pressure gradients along the SWI develop due to current–bedform interactions at different Res and by Darcy’s Law. We found that the coupling between water column and exchange zone flow is controlled by the behavior of the water-column eddy. The eddy detaches at or near the point of minimum pressure along the interface, and reattaches near the point of maximum pressure. These two critical points determine the pressure gradient along the bed surface that controls the exchange zone flow field. Moreover, the reattachment point corresponds to flow divides within the sediments. Lastly, pore-water velocities drop with depth below the SWI, and are larger below the bedform crests than below the troughs.     

AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER——FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.     

交叉裂隙水力学开度的计算及非线性水力特性研究

基于人工交叉裂隙模型,通过室内透水试验,得到了不同进口与出口组合时的流量和压力变化的关系曲线。同时结合串联裂隙水力学开度的计算公式,利用最小二乘法求解超静定方程组,计算交叉裂隙中每个裂隙单元的水力学开度。通过拓扑得到裂隙交叉点的几何尺寸,并求解Navier-Stokes方程,研究裂隙交叉对其水力特性的影响。结果表明,通过室内试验和串联裂隙水力学开度计算公式相结合的方法,可以准确计算每个裂隙单元的水力学开度。当雷诺数Re较小时,交叉点内部的流体成稳态层流流动;当雷诺数较大（比如Re≥100）时,可以观测到明显的漩涡,说明流体的惯性力远大于黏性力,经典的立方定律不再适用。对于1个进口2个出口的情况,出口的流量分配比率与雷诺数成二次函数关系,随着雷诺数的增大,流量分配比率的非线性越来越明显,其最大分配比率变化超过15%。出口的水力学开度e与初始水力学开度（即力学开度E0）的比值e/E0和雷诺数Re也具有二次函数关系。当Re10时,e/E0呈现出较强的非线性;利用该关系式可以得到修正的立方定律,从而进一步求解交叉裂隙的水力特性问题。     

流体-结构相互作用下用微分求积法求弹性板上的动水压力场

采用微分求积法(DQ)在流体-结构相互作用(FSI)框架下求解原变量形式的Navier-Stokes(N-S)方程和板的弯曲振动方程,获取作用在弹性板上的动水压力场.分析对象是4块矩形板组成的箱型流道,其中两块为弹性薄板,另外两块为刚性板.为比较FSI对动水压力场的影响,数值结果给出了1.5m×0.5m×0.2m流道、来流雷诺数分别为000和10000、相应DQ网格为25×25×25和29×29×29考虑FSI与否的压力分布.数值计算结果表明,所建立的考虑FSI的DQ法数值求解原变量形式N-S方程和弹性板振动方程,获取弹性板上的动水压力场是可行的,FSI效应对动水压力场的影响是明显的.