1 Numerical Hydraulics Classification of the equations Wolfgang Kinzelbach with Marc Wolf and Cornel Beffa

2 General form of the equation systems of hydrodynamics u and v correspond for example to v and p,  and  for example to x and t The functions can be classified in different regions of the  - plane. Regions are bounded by lines, on which the derivatives of u and v are undetermined. These lines divide the plane into regions, which are kinematically not continuous. That means there are no solutions, which transgress these lines. (1)

3 Total differentials of u and v Together with equation system (1) the following matrix equation holds:

4 Lines of indeterminate deriatives The lines of indeterminate derivatives are obtained by setting the determinant of the system equal to zero: or: ab c

5 Lines of indeterminate deriatives: Characteristics Solution of the second order equation yields: D>0: 2 real characteristics through each point of the plane, equation system is called hyperbolic D=0: 1 real characteristic through each point of the plane, equation system is called parabolic D<0: characteristics imaginary, equation system is called elliptic.

6 Geometrical interpretation of the type of equation P   elliptic   parabolic P   hyperbolic P Region influenced from point P

7 Geometrical interpretation of the type of equation Hyperbolic: A system can develop from state P only to states between the two characteristic lines (e.g. pressure fronts of velocities c+v, -c+v) Elliptic: Development of the system from point P to any point in the plane is feasible (e.g. backflow) Parabolic: Only development to one side of P is feasible (e.g. only downstream in supercritical flow) Classification is of interest, as every type of equation requires different numerical solution methods.. The terminology is taken from the theory of the sections of the double cone, which satisfy the equation: