Unsteady Flow Computations in Open Channel Hydraulics
Unsteady flow in complex systems of open channel courses (networks) can be simulated by a mathematical model consisting of the so called De Saint Venant equations. A derivation of the equations may be found by Stoker (1957) and Mahmood and Yevjevitch (197
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Unsteady Flow Computations in Open Channel Hydraulics J. Bouwknegt Heidemij Adviesbureau, Arnhem, Holland
INTRODUCTION Unsteady flow in complex systems of open channel courses (networks) can be simulated by a mathematical model consisting of the so called De Saint Venant equations. A derivation of the equations may be found by Stoker (1957) and Mahmood and Yevjevitch (1975). The numerical solutions of the De Saint Venant equations fall into one of the following categories: - explicit finite-difference method based on the characteristic equations - explicit finite-difference methods - implicit finite-difference methods - methods based on finite element schemes. Two of the categories are investigated in this paper: - The method of the characteristics. The two partial differential equations are transformed to 4 ordinary differential equations. A general technique for solving these 4 equations may be found by Lister. - Implicit finite difference methods based on the 4-point scheme proposed by A. Preissmann. The equation of motion is written in two different forms. This leads to two sets of difference equations. The advantages of both methods will be discussed. The basic approach to the three methods (characteristics, implicit difference method on two different equations of motion) will be discussed. For a hypothetical problem the three solutions are obtained and compared with the solutions of other authors. Finally the methods will be demonstrated on a very simple "network" of open channels including some construction works.
K. V. H. Smith et al. (eds.), Hydraulic Design in Water Resources Engineering: Land Drainage © Springer-Verlag Berlin Heidelberg 1986
354
BASIC EQUATIONS Unsteady flow in open channels can be described by the equations of continuity and motion. For the one-dimensional situation there are these equations: Continuity: IJh T--q=O IJt
Motion: IJQ .,. ( gA IJt
~).!!!.. .,. 2 £. !E. - gA A
IJx
A IJx
(1)
( So - Sf - S.., J
=O( 2A)
The equation of motion can also be written in the more compact form: Z ) IJQ - IJ (Q M -.,. at oX A .,. gA -oX - gA (S0 - S f - S10' J
=0
x = distance along channel axis. positive in downstream direction t = time A = cross sectional area of flow Q = discharge across a section h = waterdepth T = flow channel width q = lateral inflow per unit length g = acceleration due to gravity So= bottom slope Sf= friction slope Sw= surface slope, due to wind friction
(2B) em] [s] [mz] [m) 5-1]
em] em]
[m z S-1]
[ms-z] [-] [-] [-]
The space derivative of discharge of Equation 2B can be transformed as follows:
the cross-section is held constant for a branch so the last term vanishes. Substituting this results in Equation 2B gives equation 2A. A thorough treatment of these equations may be found by Cunge et al (1983). Equation 2A will be referenced throughout this paper by: equation of motion A; and equation 2B by: equation of motion B. The equations are solved for the two dependent variables Q and h as functions of the independent variables
355 x and t. A solution in clos
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