Dam-Break Flows Over a Bottom Step

A single-layer shallow-water model is used to study the solvability of the problem of flows generated by dam break over a bed level discontinuity in the form of a step onto which water flows. Solutions in which the total flow energy is conserved on the step and solutions in which the energy is lost on the step are considered.     

Flows formed after the passage of a discontinuous wave over a bottom drop

The solvability of the problem of the flow formed after a discontinuous wave has passed over a bottom drop is studied within the framework of the first approximation of shallow water theory. Solutions in which the total energy of the flow is either conserved or lost at the drop are considered. Stable self-similar solutions of five qualitatively different types are derived and their domains of existence are determined in the dimensionless parameter plane.     

Flows Resulting from the Incidence of a Discontinuous Wave on a Bottom Step

The solvability of the problem of the flows resulting from the incidence of a discontinuous wave on a bottom step is studied using a single-layer shallow water model. Solutions in which the total energy of the flow is conserved at the step and those in which it is lost at the step are considered. Regions of double and triple hystereses in the obtained self-similar solutions are found. An analogy is drawn with the problem of single-layer flow over a bottom obstacle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 8–22, March–April, 2006.     

梯级溃坝洪水洪峰增强机制

我国在多条河流上修建了大量梯级水库, 梯级坝溃决诱发洪水大大超过单坝溃决洪水洪峰, 因此亟需加深对梯级坝溃决洪水洪峰增强机制的认识. 本文建立了梯级坝溃决洪水演进过程的一维浅水动力学模型, 发展了一套能捕捉激波、干湿边界和保平衡结构的数值求解方法, 通过大量算例, 系统研究了梯级坝溃决洪水演进过程的质量转化和能量转化机制. 研究结果表明, 梯级溃决中, 上游溃决诱发的洪水大大增大下游水库的质量和动量, 形成一个带动量的水塔, 同时在尾部残留一个动量较大的射流, 不断补充下游坝体溃决后水塔的质量和动量, 持续维持洪峰高度. 根据该射流-水塔机制, 建立了梯级坝溃决洪水演进过程对应的射流-水塔单坝溃决洪水过程等效模型, 该等效模型基本反映了梯级坝溃决诱发洪水的洪峰过程, 并成功预测了多个坝间距为百公里量级的梯级坝溃决洪水洪峰高程和流量, 可望为流域防洪和梯级坝设计提供理论依据.      

Depth above a Drop of the Channel Bottom after Free-Surface Discontinuity Decay

This paper reports experimental data on the depth above a bottom drop in a rectangular channel after removal of a shield that produces the initial difference in the free-surface level. It is shown that at a sufficiently large height of the drop, this depth is approximately 40 % smaller than that obtained in the first shallow-water approximation.     

On the discharge characteristic at the dam site after dam break

A review of the information available in the literature is given, and new experimental data on the depth and discharge at the dam site after a total and a partial dam break are presented. It is shown that in the case of a partial dam break with the formation of a rectangular breach, the specific discharge per unit width of the breach is higher than the specific discharge in the case of a total dam break with the same excess initial energy in the headwater. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 77–87, September–October, 2006.     

Gravity Waves due to Discontinuity Decay Over an Open-Channel Bottom Drop

Experimental data are given on wave shapes and propagation speeds and characteristic headwater and tailwater depths after removal of a shield producing an initial free-surface level drop and located above a bottom drop in a rectangular open channel. Check is performed of self-similar solutions of the problem obtained earlier using a hydraulic approximation. It has been established that in certain ranges of time, longitudinal coordinate, and problem parameters, these solutions are supported by experimental results.     

二维溃坝波的反射与绕射

本文推导了间断分解公式,建立了Godunov型格式,研究了二维溃坝波的传播、反射和绕射。计算结果符合物理实际,令人满意。方法是有效的,可以提供水工部门作为设计和防护的依据。     

Two‐dimensional dam break flow simulation

Numerical modelling of shallow water flow in two dimensions is presented in this work with the results obtained in dam break tests. Free surface flow in channels can be described mathematically by the shallow‐water system of equations. These equations have been discretized using an approach based on unstructured Delaunay triangles and applied to the simulation of two‐dimensional dam break flows. A cell centred finite volume method based on Roe's approximate Riemann solver across the edges of the cells is presented and the results are compared for first‐ and second‐order accuracy. Special treatment of the friction term has been adopted and will be described. The scheme is capable of handling complex flow domains as shown in the simulation corresponding to the test cases proposed, i.e. that of a dam break wave propagating into a 45° bend channel (UCL) and in a channel with a constriction (LNEC‐IST). Comparisons of experimental and numerical results are shown. Copyright © 2000 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.     

Spatial Stationary Long Waves in Shear Flows

The system of integrodifferential equations describing the spatial stationary freeboundary shear flows of an ideal fluid in the shallowwater approximation is considered. The generalized characteristics of the model are found and the hyperbolicity conditions are formulated. A new class of exact solutions of the governing equations is obtained which is characterized by a special dependence of the desired functions on the vertical coordinate. The system of equations describing this class of solutions in the hyperbolic case is reduced to Riemann invariants. New exact solutions of the equations of motion are found.     

Simulation of rapidly varying flow using an efficient TVD–MacCormack scheme

An efficient numerical scheme is outlined for solving the SWEs (shallow water equations) in environmental flow; this scheme includes the addition of a five‐point symmetric total variation diminishing (TVD) term to the corrector step of the standard MacCormack scheme. The paper shows that the discretization of the conservative and non‐conservative forms of the SWEs leads to the same finite difference scheme when the source term is discretized in a certain way. The non‐conservative form is used in the solution outlined herein, since this formulation is simpler and more efficient. The time step is determined adaptively, based on the maximum instantaneous Courant number across the domain. The bed friction is included either explicitly or implicitly in the computational algorithm according to the local water depth. The wetting and drying process is simulated in a manner which complements the use of operator‐splitting and two‐stage numerical schemes. The numerical model was then applied to a hypothetical dam‐break scenario, an experimental dam‐break case and an extreme flooding event over the Toce River valley physical model. The predicted results are free of spurious oscillations for both sub‐ and super‐critical flows, and the predictions compare favourably with the experimental measurements. Copyright © 2006 John Wiley & Sons, Ltd.     

Topography and Dispersion Effects in Bottom Layer Dynamics

A hyperbolic model of a shallow water flow is considered with allowance for nonlinear and dispersion effects. The structure of traveling waves above a flat bottom is studied. Stability of small disturbances of a homogeneous flow and development of instability of a nonstationary flow above an inclined bottom are analyzed.     

Instability of the Motion of a Drop in Filtration Flows

The behavior of an inclusion (drop) of foreign fluid in a porous medium saturated with another fluid is considered. A steady-state regime of gravity-induced drop settlement is found and the instability of this regime is demonstrated. The horizontal motion of a liquid inclusion under the action of a stationary reservoir pressure gradient is also studied.     

HLLC scheme with novel wave‐speed estimators appropriate for two‐dimensional shallow‐water flow on erodible bed

This paper presents a first‐order HLLC (Harten‐Lax‐Van Leer with contact discontinuities) scheme to solve the Saint‐Venant shallow‐water equations, including morphological evolution of the bed by erosion and deposition of sediments. The Exner equation is used to model the morphological evolution of the bed, while a closure equation is needed to evaluate the rate of sediment transport. The system of Saint‐Venant–Exner equations is solved in a fully coupled way using a finite‐volume technique and a HLLC solver for the fluxes, with a novel wave‐speed estimator adapted to the Exner equation. Wave speeds are usually derived by computing the eigenvalues of the full system, which is highly time‐consuming when no analytical expression is available. In this paper, an eigenvalue analysis of the full system is conducted, leading to simple but still accurate wave‐speed estimators. The new numerical scheme is then tested in three different situations: (1) a circular dam‐break flow over movable bed, (2) an one‐dimensional bed aggradation problem simulated on a 2D unstructured mesh and (3) the case of a dam‐break flow in an erodible channel with a sudden enlargement, for which experimental measurements are available. Copyright © 2010 John Wiley & Sons, Ltd.     

A Godunov‐type fractional semi‐implicit method based on staggered grid for dam‐break modeling

A semi‐implicit finite volume model based upon staggered grid is presented for solving shallow water equation. The model employs a time‐splitting scheme that uses a predictor–corrector method for the advection term. The fluxes are calculated based on a Riemann solver in the prediction step and a downwind scheme in the correction step. A simple TVD scheme is employed for shock capturing purposes in which the Minmond limiter is used for flux functions. As a consequence of using staggered grid, an ADI method is adopted for solving the discretized equations for 2‐D problems. Several 1‐D and 2‐D flows have been modeled with satisfactory results when compared with analytical and experimental test cases. The model is also capable of simulating supercritical as well as subcritical flow. Copyright © 2010 John Wiley & Sons, Ltd.     

Nonlinear Capillary Oscillations of a Volumetrically Charged Dielectric Drop

For the axisymmetric capillary oscillations of a charged dielectric fluid drop an expression describing the shape of the generating surface of the drop as a function of time is obtained in the quadratic approximation in the amplitude of an arbitrary initial deformation of its spherical equilibrium shape. It is shown that in contrast to a perfectly conducting charged drop there is no displacement of the drop charge center during oscillation and, hence, such a drop cannot be a source of dipole electromagnetic radiation like a conducting drop in the quadratic approximation.     

Behavior of a Cylindrical Drop under Multi-Frequency Vibration

The behavior of an inviscid-fluid drop surrounded by a different fluid under the action of multi-frequency vibration is investigated. The second-order effects in the vibration amplitude are studied. A superharmonic resonance is registered. The stability of the forced oscillations with respect to small perturbations is studied. The condition of the onset of parametric resonance is found. An average drop shape is investigated. The two-frequency case is considered as a particular case of multi-frequency vibration.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 18–28.Original Russian Text Copyright © 2005 by Alabuzhev and Lyubimov.     

Explicit solution to the exact Riemann problem and application in nonlinear shallow‐water equations

The Riemann solver is the fundamental building block in the Godunov‐type formulation of many nonlinear fluid‐flow problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, this paper reports an explicit analytical solution to the exact Riemann problem. The present approach uses the homotopy analysis method to solve the nonlinear algebraic equations resulting from the Riemann problem. A deformation equation defines a continuous variation from an initial approximation to the exact solution through an embedding parameter. A Taylor series expansion of the exact solution about the embedding parameter provides a series solution in recursive form with the initial approximation as the zeroth‐order term. For the nonlinear shallow‐water equations, a sensitivity analysis shows fast convergence of the series solution and the first three terms provide highly accurate results. The proposed Riemann solver is implemented in an existing finite‐volume model with a Godunov‐type scheme. The model correctly describes the formation of shocks and rarefaction fans for both one and two‐dimensional dam‐break problems, thereby verifying the proposed Riemann solver for general implementation. Copyright © 2007 John Wiley & Sons, Ltd.