Hydraulics of swirling flows along vortex drop shafts

Abstract

The vortex drop shaft is a benchmark structure in hydraulic engineering. It is often used in sewers and hydropower systems, given that a significant energy dissipation combined with a reduced space occupation is achieved. Conversely, the flow pattern establishing along the structure may lead to the occurrence of unstable phenomena as vibrations, abrasion and choking, particularly if the operational conditions are different from the standard design regime. It is advantageous to study the overall hydraulic efficiency of the structure, and particularly to derive the hydraulic conditions of the swirling flow along the vertical shaft. At this regard, the paper describes a physically based approach, based on the momentum conservation, to derive the rotational angle and the velocity profiles along the shaft. The method requires the calibration of an empirical parameter accounting for the increase of the wall friction stress due to the centrifugal force. The outcomes of the application of the proposed procedure are presented with reference to the operation of two physical models of supercritical vortex drop shafts, with a tangential or a spiral inlet at the shaft top. It is shown that the application of the empirically-modified momentum approach is necessary for accounting for the significant angular momentum imparted to the swirling flow by the spiral inlet and for modelling the rotational flow distribution along the vertical shaft accurately.

Energy Head Dissipation and Flow Pressures in Vortex Drop Shafts

“…(3) Equation ( 3) is derived under the simplifying assumption that the flow depth of the outflow from the shaft can be neglected. The flow thickness in the shaft decreases from the inlet device-to the shaft outlet cross-section to values of about 0.5-3.0% of D s [17,28], therefore this hypothesis does not lead to a significant error. If Equation ( 3) is applied, then it is possible to derive the energy efficiency η s limited to the flow along spiral inlet and the vertical shaft as…”

“…Otherwise, the energy head dissipation due to the flow passage along the spiral inlet and the vertical shaft only is herein computed as a function of the velocity V s of the flow outing from the shaft (at the horizontal section s-s as represented in Figure 1). The latter is a function of the axial and tangential velocity components of the swirling flow along the shaft and it can be obtained by following the computational procedure suggested by [17,28]. Consequently, the energy head H s at the shaft outflow is:…”

“…The computation of the flow velocity V s at the shaft outflow through the analytical procedure described by [27,28] allows also to calculate H s and, then, the shaft head loss coefficient K s :…”

“…V s is, therefore, smaller than in the condition where the quasi-uniform flow at the end of the vertical shaft is achieved, as instead observed by [8], and consequently K s increases. The computation of the flow velocity Vs at the shaft outflow through the analytical procedure described by [27,28] allows also to calculate Hs and, then, the shaft head loss coefficient Ks:…”