State-Of-The-Art Numerical Schemes Based on the Control Volume Method for Modeling Turbulent Flows and Dam-Break Waves

Abstract

Due to th ongoing progress in computational engineering and the emergence of new efficient algorithms, the sphere of applicability of numerical methods for hydraulic problems has makedly expanded. There are various algorithms and software packages modeling nonstationary open-channel flows, including turbulent flows, in one-dimensional (1D) and two-dimensional (2D) settings. The main requirements imposed on computation algorithms used, for instance, to calculate the dam-break wave parameters in preparing safety declarations for hydraulic structures are as follows: conservative approach, stability, ability to represent discontinuous flows, and calculation efficiency. The control volume method (CVM) makes it possible to develop 1D and 2D numerical schemes for systems of shallow water equations, which are conservative with respect to mass and pulse [1 – 3]. In a 2D case this method can relatively easily be adapted to irregular and triangular calculation grids. That is why lately foreign authors discussing numerical modeling of turbulent flows and discontinuous flows in the form of break waves, hydraulic jumps, or oblique waves pay special attention to various modifications of the control volume method [1 – 8]. The majority of studies are based on using explicit computational algorithms, which are more easily implemented for such problems than implicit algorithms and surpass the latter in efficiency. One must mention that publications of recent years display an active interest in schemes of the second (or higher) order of accuracy for coordinates and time [3 – 5, 7, 8]. The transition to such schemes makes it possible to obtain less “blurred” solution jumps in solving problems (for instance wave-fronts) [3, 7]. However, the need for changing over to such schemes in modeling results of experiments and the consequences of real hydrodynamic breakdowns is doubtful: calculations using schemes of the first order of accuracy in some cases appear quite acceptable [1, 9]. One of the calculation examples given below is interesting due to the fact that it corroborates the advantage of the second-order schemes, at least in numerical modeling of experiment results. A separate problem is numerical modeling of a flow spreading over a dry valley and a liquid flow in a multiply connected domain with the waterline changing with time. Such problems arise, for instance in modeling the consequences of hydrodynamic breakdowns at ash-and-slag disposal areas [10, 11]. In this case the most effective is the ripple-through carry method [12] adapted to “dry” bottom sites. One of the main criteria of applicability of developed algorithms to practical problems, for instance, to calculations of the consequences of a hydrodynamic failure, is testing the algorithm on some known analytical solutions or experimental and field data. A coordinated approach to the development of such tests and performing laboratory experiments has been proposed in the framework of the international project CADAM (1998 – 2000) [13]. Unfortunately, only the USA and some European countries participated in this project, whereas Russia and the CIS countries were not involved. However, some results of experiments carried out within the framework of the CADAM project have been published [10, 14] and later they have been used to test the numerical algorithms described below. In the present study the 1D version of the CVM algorithm is implemented on a coarse grid [10], which primarily simplifies representation of the discrete analog of the continuity equation and the impermeable boundary conditions. The method is based on the St. Venant system of one-dimensional equations that takes into account summands determined by turbulent viscosity of the liquid. This algorithm is a modification of the 1D CVM algorithm described in [10]; accordingly, we represent here only its distinctions from the previous version. First, instead of using iterations to account for the nonlinear summands of the St. Venant equations, the new 1D version of the algorithm uses time extrapolation of the check points on a free surface that make part of the channel cross-section area and the flow rates, which serve as factors in the nonlinear terms (convection, bottom friction). The absence of additional iterations has substantially increased the algorithm’s efficiency. Furthermore, the modified method, contrary to the method in [10] has the second order of accuracy for coordinates and time. The second order for the coordinate is obtained by introducing antidiffusion cor-