# On the Vaneless Space Vortex Structures in a Kaplan Turbine Model Operating at Speed No Load
## 1 Introduction  
The increase in the installation of new intermittent renewable energy sources, such as solar and wind power, must be complemented with other energy sources, such as hydropower, able to stabilize the grid, storage energy and supply peak load. Nevertheless, these conditions force turbines to be operated for longer periods of time in off-design and transient conditions [1; 2; 3; 4]. For instance, a hydraulic turbine under a grid stabilization mode works 4% of the time at speed no load (SNL) and 24% at deep part load (DPL), while it only works 1% at SNL and 0% at DPL under a base load mode [5]. Deleterious hydrodynamic phenomena, such as vaneless space vortex (VSV) structures and stochastic flow phenomena, occur at SNL and DPL, which may cause high-pressure fluctuations, damage the runner blades [3] and impose more regular refurbishment requirements [6; 7]. At SNL, turbines rotate at nominal speed without generating power while the discharge is not zero. The hydraulic energy is dissipated through the generation of large-scale unsteady vortex structures that break down into smaller vortices until the energy dissipation occurs at the smallest scales [8]. The flow at SNL is primarily stochastic and provokes high dynamic stresses on runner blades of Francis turbines [5; 9; 10; 11; 12; 13; 14; 15]. Most investigations focused on determining the relative fatigue damage inflicted on runner blades when operated at SNL or DPL compared to that induced at best efficiency point (BEP). SNL and DPL operation have been concluded to generate most fatigue damage due to stochastic flow phenomena [5; 8; 10; 11; 12; 13; 14; 15]. Other researchers have found that in addition to stochastic excitations, specific Francis turbinesalso present periodic excitations induced by coherent vortex structures [16; 17]. [PERSON] et al. [18] simulated several operating points of a Francis turbine from full load (FL) to DPL and identified up to three distinct runner channel vortices occurring simultaneously at low discharges. In the same study, the results of pressure measurements on the blades and on the band of a reduced-scale Francis turbine runner showed that the channel vortices caused significant induced pressure fluctuations. [PERSON] et al. [19] investigated the development and the flow characteristics associated with the formation of inter blade vortices in a Francis runner at DPL numerically. It was concluded that the appearance of a backflow region in the vicinity of the hub near the trailing edge was closely linked with the inter-blade vortex development. This finding was also proved experimentally by the same group of researches in later studies [20]. In axial turbines, SNL operation also presents stochastic flow phenomena and coherent vortex structures, which are usually originated in the vaneless space and cause low-frequency pressure fluctuations and significant vibrations [21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. [PERSON] et al. [21] investigated numerically the origin of strong pressure fluctuations and vibrations in an axial turbine operated at SNL. They observed vortices and a distorted flow in the runner and concluded that the origin of these instabilities was a combined action of the centrifugal force and the low flow rate. [PERSON] et al. [22] investigated numerically the unsteady flow characteristics and flow instabilities in the SNL condition of a bulb turbine. A rotating vortex array was found below the runner which was composed by two vortices self-rotating and precessing at frequencies lower than the runner rotating frequency. It was concluded that the vortex array was the major source of pressure and torque fluctuations at SNL. [PERSON] et al. [23; 24] measured strains and pressure on a runner blade of a Kaplan turbine prototype at SNL and detected low-frequency pulsations. Additionally, torsional strains on the shaft were found to have 12-fold larger peak-to-peak values at SNL than those at BEP. [PERSON] et al.