Exact Free Surface Flows for Shallow Water Equations I: The Incompressible Case

A comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given. Several new families of exact solutions of the governing PDEs are found and are shown to embed the well-known self-similar or traveling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the traveling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied.     

Whitham-Broer-Kaup浅水波方程的Backlund变换和精确解

用一种新的方法并借助Mathematica,求出了Whitham-Broer-Kaup(简记WBK)方程的一种Backlund变换,并建立了WBK方程与热传导方程及Burgers方程的联系.利用这种关系得到了WBK方程的三组精确解,其中一组为孤波解.     

Regularized Equations for Numerical Simulation of Flows in the Two-Layer Shallow Water Approximation

Regularized equations describing hydrodynamic flows in the two-layer shallow water approximation are constructed. A conditionally stable finite-difference scheme based on the finitevolume method is proposed for the numerical solution of these equations. The scheme is tested using several well-known one-dimensional benchmark problems, including Riemann problems.     

齐次平衡法和浅水波中WBK方程的精确分析解

Based on the homogenous balance method and with the help of mathematica, the Backlund transformation and the transfer heat equation are derived. Analyzing the heat-transfer equation, the multiple soliton solutions and other exact analytical solution for Whitham-Broer-Kaup equations(WBK) are derived. These solutions contain Fan‘s, Xie‘s and Yan‘s results and other new types of analytical solutions, such as rational function solutions and periodic solutions. The method can also be applied to solve more nonlinear differential equations.     

Resonant Interactions between Vortical Flows and Water Waves. Part II: Shallow Water

Any weak, steady vortical flow is a solution, to leading order, of the inviscid fluid equations with a free surface, so long as this flow has horizontal streamlines coinciding with the undisturbed free surface. This work considers the propagation of long irrotational surface gravity waves when such a vortical flow is present. In particular, when the vortical flow and the irrotational surface waves are both periodic and have comparable length scales, resonant interactions can occur between the various components of the flow. The interaction is described by two coupled Korteweg-de Vries equations and a two-dimensional streamfunction equation.     

Exact Solutions of Compressible Flow Equations with Spherical Symmetry

In this paper, we construct spherically symmetric solutions of the equations of compressible flow, which are important in the theory of explosion waves in air, water, and other media. Following McVittie [ 1 ], we write a general solution form, in terms of velocity potential, as a product of a function of time and a function of a similarity variable. First, we find solutions to the equations of motion and continuity without reference to adiabatic or isentropic relation. These solutions are quite general and can be applied to nonadiabatic motions, such as the motions of interstellar gas clouds that lose energy by radiation. All the solutions found by McVittie [ 1 ] have linear velocity profile with respect to distance. We introduce a nonlinear form of the velocity function containing an arbitrary function of the similarity variable. Adiabatic conditions lead to a second-order ODE, which we discuss in some detail. We relate our work to the earlier investigations of Taylor [ 2 ], McVittie [ 1 ], and Keller [ 3 ].     

Exact Simple Waves on Shear Flows in a Compressible Barotropic Medium

Self-similar solutions describing simple waves on shear flows in a finite compressible barotropic atmosphere are found. These include the simple waves on shear flows in water as a special case. By making use of a number of transformations it becomes possible to write these solutions in an exact form. This form, though not explicit, is similar to the incomplete Beta function which seems to characterize this class of nonlinear physical problems.     

双参数假设与变形浅水波方程组的精确解

将双 参数假设加以推广,用直接积分的方法求得了菜浅水波方程的三族精确解。     

三重介质弹性渗流方程组的精确解

本文用分解的方法求得了有限封闭地层中三重介质弹性渗流的精确解.它不仅概括了已有的双重介质弹性渗流的主要结果,而且给出各重介质弹性渗流的基本特征.     

A Priori Estimates for the Compressible Euler Equations for a Liquid with Free Surface Boundary and the Incompressible Limit

In this paper, we prove a new type of energy estimate for the compressible Euler equations with free boundary, with a boundary part and an interior part. These can be thought of as a generalization of the energies in Christodoulou and Lindblad to the compressible case and do not require the fluid to be irrotational. In addition, we show that our estimates are in fact uniform in the sound speed k. As a consequence, we obtain convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Ebin to when you have a free boundary. In the incompressible case our energies reduce to those in Christodoulou and Lindblad, and our proof in particular gives a simplified proof of their estimates with improved error estimates. Since for an incompressible irrotational liquid with free surface there are small data global existence results, our result leaves open the possibility of long‐time existence also for slightly compressible liquids with a free surface.© 2017 Wiley Periodicals, Inc.     

The Boundary Value Problem for the Nonlinear Shallow Water Equations

We propose an approximate analytical solution of the boundary value problem (BVP) for the nonlinear shallow waters equations. Our work, based on the Carrier and Greenspan [ 1 ] hodograph transformation, focuses on the propagation of nonlinear nonbreaking waves over a uniformly plane beach. Available results are briefly discussed with specific emphasis on the comparison between the Initial Value Problem and the BVP; the latter more completely representing the physical phenomenon of wave propagation on a beach. The solution of the BVP is achieved through a perturbation approach solely using the assumption of small waves incoming at the seaward boundary of the domain. The most significant results, i.e., the shoreline position estimation, the actual wave height and velocity at the seaward boundary, the reflected wave height and velocity at the seaward boundary are given for three specific input waves and compared with available solutions.     

Large-time Behavior of Solutions to the Stokes Approximation Equations for Two Dimensional Compressible Flows

In this paper, we study the large time asymptotic behavior of solutions to both the Cauchy problem and the exterior problem of the Stokes approximation equations of two dimensional compressible flows.     

求解浅水波方程的熵相容格式

提出了一种求解浅水波方程组的熵相容格式．在熵稳定通量中添加特征速度差分绝对值的项来抵消解在跨过激波时所产生的熵增,从而实现熵相容．新的数值差分格式具有形式简单、计算效率高、无需添加任何的人工数值粘性的特点．数值算例充分说明了其显著的优点．利用新格式成功地模拟了不同类型溃坝问题的激波、稀疏波传播及溃坝两侧旋涡的形成,是求解浅水波方程组较为理想的方法.     

Regularized Equations for Numerical Simulation of Flows of Homogeneous Binary Mixtures of Viscous Compressible Gases

Computational Mathematics and Mathematical Physics - Regularized equations for binary mixtures of viscous compressible gases (in the absence of chemical reactions) are considered. Two new simpler...     

浅水流动与污染物扩散的高分辨率计算模型

将组合型TVD格式应用于守恒型的浅水方程和污染物扩散方程，建立了二者耦合求解的高分辨率有限体积计算模型。给出了溃坝水流、明渠突扩流和污染物输运计算的典型算例，并与实验数据或其它数值结果进行了比较，证实了该模型的有效性，表明它不但能处理有激波的非恒定流问题，也能较好地计算具有任意边界的一般的浅水流动和污染物扩散问题，为浅水流动和水环境模拟提供了精度高、稳定性好、普适性强的数值方法。     

Solutions to the Navier-Stokes Equations Describing Local Flows of Stratified and Compressible Fluids

A family of exact solutions of the Navier-Stokes equations is used to describe local flows of incompressible stratified and compressible fluids. For some of the flows, the coefficient of viscosity can depend on the temperature. An example of an incompressible stratified flow for which the analysis is applicable is the sheared swirling flow that is produced between two parallel plates that translate with different velocities and rotate with different angular velocities about different, but parallel, axes. The fluid may be stratified in the direction normal to the plates. These generalized von Karman flows are relevant to the study of strong local atmospheric disturbances, such as might be produced by the passage of a tornado. Also, when the coefficient of viscosity depends on the temperature, they can be used to analyze the flow of molten metals between surfaces that are in relative motion. An example of a compressible flow for which the analysis is applicable is that produced by a plane shock wave as it traverses a layer where the fluid is sheared in a direction normal to the shock.     

A Numerical Study of the Exact Evolution Equations for Surface Waves in Water of Finite Depth

We describe a pseudo-spectral numerical method to solve the systems of one-dimensional evolution equations for free surface waves in a homogeneous layer of an ideal fluid. We use the method to solve a system of one-dimensional integro-differential equations, first proposed by Ovsjannikov and later derived by Dyachenko, Zakharov, and Kuznetsov, to simulate the exact evolution of nonlinear free surface waves governed by the two-dimensional Euler equations. These equations are written in the transformed plane where the free surface is mapped onto a flat surface and do not require the common assumption that the waves have small amplitude used in deriving the weakly nonlinear Korteweg–de Vries and Boussinesq long-wave equations. We compare the solution of the exact reduced equations with these weakly nonlinear long-wave models and with the nonlinear long-wave equations of Su and Gardner that do not assume the waves have small amplitude. The Su and Gardner solutions are in remarkably close agreement with the exact Euler solutions for large amplitude solitary wave interactions while the interactions of low-amplitude solitary waves of all four models agree. The simulations demonstrate that our method is an efficient and accurate approach to integrate all of these equations and conserves the mass, momentum, and energy of the Euler equations over very long simulations.     

Unconditionally Stable Splitting Methods for the Shallow Water Equations

The front-tracking method for hyperbolic conservation laws is combined with operator splitting to study the shallow water equations. Furthermore, the method includes adaptive grid refinement in multidimensions and shock tracking in one dimension. The front-tracking method is unconditionally stable, but for practical computations feasible CFL numbers are moderately above unity (typically between 1 and 5). The method resolves shocks sharply and is highly efficient. The numerical technique is applied to four test cases, the first being an expanding bore with rotational symmetry. The second problem addresses the question of describing the time development of two constant water levels separated by a dam that breaks instantaneously. The third problem compares the front-tracking method with an explicit analytic solution of water waves rotating over a parabolic bottom profile. Finally, we study flow over an obstacle in one dimension.     

Fast Wave Averaging for the Equatorial Shallow Water Equations

The equatorial shallow water equations in a suitable limit are shown to reduce to zonal jets as the Froude number tends to zero. This is a theorem of a singular limit with a fast variable coefficient due to the vanishing of the Coriolis force at the equator. Although it is not possible to get uniform estimates in classical Sobolev spaces (other than L²) by differentiating the system, a new method exploiting the particular structure of the fast coefficient leads to uniform estimates in slightly different functional spaces. The computation of resonances shows that fast waves may interact with a strong external forcing, introduced to mimic the effects of moisture, to create zonal jets.     

浅水方程组的数值方法

构造了浅水方程组的二阶精度的TVD格式。格式由简单的TVD Runge-Kutta型时间离散和有坡度限制的空间对称离散格式组成。数值耗散项用局部棱柱化河道流的特征变量构造。格式的主要优点是能够计算天然河道中浅水方程组的弱解并且构造简单。格式能够求出天然河道或非平底部渠道中的精确静水解。给出了渠道溃坝问题数值解与解析解的比较,验证格式精度高。实际天然河道型梯级水库溃坝的数值实验表明格式稳定,适应性强。