Time evolution of water surface oscillations in surge tanks

Time evolution of water surface oscillations in surge tanks is of importance in pipeline and hydropower engineering, and is often solved graphically or numerically because the associated nonlinear second order ordinary differential equation, with a quadratic damping term, cannot be solved exactly. This research then innovates an accurate approximation based on the Lambert function, Padé approximant, and elliptic integral. Specifically, (i) the characteristic length depends on the pipe-tank geometry, the system resistance, and an unsteady factor; and the characteristic time depends on the pipe-tank geometry, the gravity, and the unsteady factor, but independent of the system resistance; (ii) the surge transit squared velocity is approximated by a Padé approximant of order [2/1], resulting in a simple approximation for the surge tank water surface oscillations in terms of an elliptic integral of the second kind; and (iii) the approximate solution accurately reproduces numerical and laboratory data thereby being applicable in practice.