Sensitivy Analysis for Waterhammer Problems in Pipes

Abstract

The differential method was applied to the sensitivity analysis for waterhammer problems in hydraulic networks. Starting from the classical waterhammer equations in a single-phase liquid with friction (the direct problem) the state vector comprising the piezometric head and the
velocity was defined.
Applying the differential method the adjoint operator, the adjoint equations with the general form of their boundary conditions, and the general form of the bilinear concomitant were calculated. The discretized adjoint equations and the corresponding boundary conditions
were programmed and solved by using the so called method of characteristics.
As an example, a constant-level tank connected through a pipe to a valve discharging to atmosphere was considered. The bilinear
concomitant was calculated for this particular case. The corresponding sensitity coefficients due to the variation of different parameters by using both the differential method and the response surface generated by the computer code WHAT, solver of the direct problem, were also
calculated. The results obtained with these methods show excellent agreement.