## Mathematical Modelling of Natural Phenomena

### Numerical Modeling of the Stream Dynamics for River Channels with Complex Spatial Configuration

Institute for Water and Environmental Problems of SB RAS (Novosibirsk Branch) 630090 Novosibirsk, Morskoy pr., 2, Russia

Abstract

Mathematical modeling provides a particularly important tool for studying the stream runoff formation processes, and its role is enhanced in the case of a sparse, obsolete monitoring network characteristic of most regions of Siberia. When analyzing spatio-temporal regularities of the water and sediment runoff in river systems, serious problems are caused by lack of the basic hydrological model capable of handling real-time data of hydrological measurements. Calculations of unsteady flows in stream channels draw heavily on one-dimensional numerical models which are relatively easy to use and yield reliable results. Numerical investigations into the hydraulic regime of natural streams involve specific difficulties caused by the presence of nonlinear frictional forces in a turbulent flow, a variability in the channel geometry, the braiding of flows, the presence of floodplain depressions, riffles, etc. For especially complex stretches of rivers, a onedimensional approximation no longer fits the reality sufficiently adequately, so that the planar flow structure must be taken into account. For this purpose Saint Vennant's plane system of equations was used as the basis in order to develop further the numerical model due to this author which is intended for calculating the flow field, flow rates, levels, and impurity concentrations in natural water bodies of an arbitrary configuration or in a part of them. Fundamental laws of fluid mechanics are used as the basis for the model. Spatial modeling of flows in complex regions necessitates reliable, consistent methods providing acceptable accuracy. As far as hydrological problems are concerned, the control volume method that allows the use of curvilinear grids was found to be the most powerful tool for obtaining the initial finite-difference relations. This paper offers a number of examples illustrating the capabilities of the planar model for streams which is intended to resolve real problems arising at the design, construction and operation stages of engineering structures in river channels and floodplains.

(Online publication October 2 2009)

Key Words:

• numerical modelling;
• hydrodynamics;
• planar flows;
• complex channel systems

Mathematics Subject Classification:

• 86A05

Correspondence: