Study of a Cavitation Treatment in Kaplan Hydro-Turbine

Cavitation happens when the absolute static pressure of a liquid reaches or is lower than the vapor pressure (P_v) corresponding to the present temperature; the low pressure causes wide intermolecular gaps that initiate the cavity (i.e., vapor bubble). Once the flow absolute pressure rises over the vapor pressure, the formed cavitation form (bubble or the whole cluster) starts to vanish due to the generated microjets (100 m/s) and implosions of high-pressure waves (1 GPa) that propagate to surroundings [1].

In hydrosystems and turbomachinery, cavitation induces high vibrations and noise corresponding to the cavitation structure [2], erosion, and total failure for the system efficiency [3,4]. Expensive maintenance [5] or replacements are required. In turbines, cavitation limits the usage of the total available energy between the head and sink [6].

Introduced air into noncavitating is a technique used to avoid it happening at sections of low static pressure. Though it is not a common treatment, supplying air started to be explored for cavitating elimination in hydraulic systems. Zhi-Yong et al. [7] investigated experimentally and theoretically the cavitation control by aeration at the flow velocity range of V = 20–50 m/s. The experimental results show that aeration remarkably increases the pressure in the cavitation region. Meanwhile, the work presented a semicubical parabola relation between flow velocity and least air concentration to prevent cavitation erosion. Tomov et al. [8] compared aerated and nonaerated cavitation in a transparent horizontal venture nozzle using a high-speed camera. Aerated cavitation was done by injecting air bubbles for three different regimes: sheet cavitation, cloud cavitation, and super-cavitation. It was observed that the symmetrical cavitation structures were partially broken for sheet and cloud cavitation regimes, and they completely disappeared when the super-cavitation regime was reached. Lately, with a similar idea to the current work, Rivetti et al. [9] studied pressurized air injection in an axial hydroturbine (Kaplan). Air injection mitigates the erosive potential of tip leakage cavitation and decreases the vibration level of the structural components. A slight reduction in the turbine efficiency was the sole expense.

For this research, a numerical approach was investigated for a microscale Kaplan hydroturbine (3-in. diameter) to predict the cavitating flow behavior upon air redistribution over the blades and to minimize the cost and time of experimentation. The research outcomes guide the build-up of future experimental work outside the scope of this paper.

Working on a CFD model of a 3-in. (0.076 m) Kaplan turbine, cavitation was initially sought to set a baseline case and define a criterion related to the cavitation phenomena in the system. The upstream side is a 6-in. (0.15 m) diameter pipe with a uniform inlet velocity (V = 2 m/s). Then, the flow passes through a converging intake pipe confining the stator, and the average velocity increases to 10 m/s at the rotor section. Rotational speed (N) is preset at 1000 rpm for the rotational domain surrounding the rotor. The downstream of the rotor is designed to be a diffuser (i.e., diverging frustum), so the static pressure rises to an atmospheric exit. System configuration is illustrated in Fig. 1.

Such conditions were enough to drop the static pressure below the vapor pressure (P_v) and start the cavitation on the rotor blades (suction side) and the hub (intersection section with the blades).

In the advanced designs, the air is injected from 1-mm circular walled holes arranged in different patterns (1, 2, 4, and 12 circumferential ports), as seen in Figs. 2 and 3. Concerning the rotor blades, the holes are 1 mm (0.001 m) downstream of the leading edge so that the incoming jet can cover a major part of the blade surface. Though running pressure could be low enough for atmospheric pressure entrance, the air was pressurized at 5 x 10⁵ Pa to guarantee the stability of the injection at the startup when the water pressure was still high around the turbine. Another reason is that it acts as a cavitation precure by raising the pressure at the zones prone to vapor pressure.

The volume of fluid (VOF) is selected to be solved for the multiphase interactions since the expected forms of vapor and air are in continuous streams (i.e., large volumes), and interfaces are solved between these different phases [10]. At the mesh cell, all phases are defined by volume fractions (α), the cell volume percentage containing the specified phase. During marching, the fractions are updated by solving the corresponding transport equation. Finally, VOF computes the flow properties as a weighted average of the different phases. In the current simulation, the convection of diverse fluids is the second discretization.

Cavitation is considered an isothermal vaporization process. The best model to correlate the controlling forces (pressure, surface tension, and viscous) is defined by the Rayleigh–Plesset equation [11,12], which was later simplified by Sauer in 2000, and it omits the viscous and surface tension because of their diminished influence in most applications [13].

Additionally, the case is turbulent (Re = 3.4 × 10⁵). Thus, the unsteady random chaotic phenomena were simulated using large eddy simulation (LES), which filters the large-scale eddies to be solved in Navier–Stokes (N-S) equations from the small-scale ones which are shaped by the wall-adapting-local-eddy viscosity (WALE) Sub-Grid Scale (SGS) model. WALE is a recent model that exceeds the Smagorinsky SGS's performance in formulating the turbulent eddy viscosity (μ_t) and the accurate scaling near the wall without damping effects [14]. LES is more favorable than unsteady RANS in time-dependent numerical analysis. Some previous CFD work on turbomachinery showed the reliability of the LES [15,16].

The meshing cell is an unstructured polyhedral, better than a structured mesh, for capturing physics like separation and wake region. After trying different mesh combinations, ranging from 0.3 × 10⁶ to 3.8 × 10⁶ cells, mesh size is selected fine to form 3.3 × 10⁶ cells with 8–12 prism layers maintaining the first layer y+ value less than 5. Mesh refinement is required for the solution's accuracy and independence and the optimal representation of the fluids' interfaces and propagation. Another reason for the small cell sizes is the LES recommendation of improved modeling when eddie volumes go to the universal Kolmogorov scale. A sectional plane of the meshed system is in Fig. 4, while the wall y+ of the turbine is depicted in Fig. 5.

Time marching in the case studied was important to capture the development of the vapor on the surfaces and its propagation with air within the liquid water. Temporal discretization was set to constant (10⁻⁵ s) with first-order solution accuracy, which was preferable over the second accuracy because of the faster performance and the stability granted by the computation process. The total computation time for a simulation is 2.5 days on 240 high-performance computing (HPC) cores.

Cavitation was first investigated on the conventional to check the initiation location and propagation of the vapor cloud. The vapor content is a function of location and time. Figure 6 shows the local and temporal variation in the distribution of VVF at the rotor. In Figs. 6(a)–6(e), cavitation behavior randomly fluctuates because of the vapor cloud's cyclic nature (formation, growth, separation, and collapse), and the relative position of the blade to the incoming flow from the stator. In Fig. 6(f), the cavitation modes are marked to illustrate the corresponding surface averaged VVF. The curves of VVF on blades and hub oscillate about the equilibrium mean with a similar pattern. Due to the inconsistencies of the first two rotations (t_cycle = 0.06 s), values at later times were averaged for the steady properties. The minimum pressure coefficient and cavitation number, calculated from the velocity and pressure entering the rotor, are found to be (|Cp_min| = 6.08, and σ = 6.04). Such status leads to the formation of vapor at four main locations on the rotor:

1. The blade leading edge and extending along the chord

2. The hub-blades intersection where the vapor spreads over part of the hub and the span of the blades

3. The blade tip where the tip vortex happens, and cavitation separates quickly

4. Part of the hub at the exit area of the blades, where the velocity increases because of the narrower path.

When air enters the system like a jet, it is affected by the three motion components: radially inward (i.e., penetrating the other fluids toward the turbine axis) due to the pressurization, axial flow with the liquid water, and a rotational with the cycle of the blade. Accordingly, the location of air varies with space and time. The air content is added to change the shape of the vapor cluster on the rotor. Figure 7 displays the air and vapor volume fraction on a midsectional plane after 0.36 s. The domination of air at one location (e.g., dashed circles) diminishes the existence of vapor at the same place, which indicates the merit of air injection. This observation leads to further analysis of the reduction in vapor content.

By scaling up the minimum bound of the absolute static pressure to the vapor saturation pressure (P_v = 2338 Pa), the air-filled zones have a rise in pressure to be above the cavitation possibility. Consequently, the vapor content at these areas is vanishing. A clear comparison is seen in Fig. 8, specifically at the numbered locations behind the blade (1 and 2) and the converging section of the hub (3). The minimum absolute static pressure at these locations is in the range of 17 kPa, which is 7.5 times over the cavitation limit.

The same concept applies to the suction side of the rotor blades. In Figure 9, pressure higher than P_v is attributed to the present air on the blade surface besides some nonaerated locations (1, 2, 3, respectively). In the meantime, the absolute pressure reaches/crosses the vapor limit at zones lacking air, and cavitation persists (4 and 5).

Time averaging the VVF on the rotor parts gives a unique number for each case. The surface and time-averaged VVF number are directly related to flow behavior and the interaction of air distribution over the rotor. VVF is used to compare the amount of vapor created on the blades and hub in each case (no-air, 1 port, 2 ports, 4 ports, and 12 ports). Table 1 lists VVF mean values around the blades and hub and records time-averaged P for each case of the five. On average, blade and hub VVF could be minimized by 18.7% and 10.9% using an air of Q_a/Q_w = 0.005%. A 4.8% average regained power is attributed to the same increase in Q_a.

Comparing the air injection cases together, the VVF reduction and power reclamation are plotted with Q_a/Q_w, as shown in Fig. 10. Although the trend is linear for Q_a, noticeable polynomial curves (second order) fit the positive outcomes. The positivity rise from one case to the following is illustrated in Table 2. The VVF benefit is more than the time of Q_a addition, which increases by placing more ports. Despite the mediocre start, the P benefit followed the same development at the larger values of Q_a.

Such a result is more related to the arrangement of the ports, which allows more chances for the air to contact the cavitating parts while rotating. The last result highlights a very important step in allocating the ports to allow for the good distribution of air inside the system. Targeting the blade cavitation area can effectively and efficiently use a certain amount of air.

Additionally, an experiment was conducted in the lab for a closed system where water flows from a head tank to the turbine, discharges to a sink tank, and then pumps the water again from the sink tank to the head one. The turbine stator and rotor blades are 3-D printed of NGen, sanded, and painted to have a smooth surface, which captures flow so the generated video and pictures can reveal the flow behavior around the turbine under different running conditions. The camera was intended to be in a straight plane, facing the clear turbine housing with no inclination. The aperture and focal point were always adjusted to the perfect view. The images were recorded, analyzed, and saved by photronfastcamviewer (PFV) software. 1000 W lights were placed nearby to supply the turbine with a good amount of light. The images were taken for a streamwise turbine at 2000 rpm and an inlet flow velocity of 0.45 m/s in the 15 cm pipe. Figure 11(a) shows the instantaneous cavitation formation on two turbine blades after entering the steady-state phase. After the comparison of different time frames, the blade walls (dashed lines) and cavitation boundaries (solid lines) were tracked and defined in each frame, as illustrated in Fig. 11(b). With such a definition, matching the CFD scene was easier. The cavitation starts from the blade's leading edge and extends upwards to touch the pressure side of the succeeding blade, then declines while going downstream to join a merging zone after the blades. The same phenomenon can be interpreted from the time-averaged cavitation cloud (white formation on the brown rotor) seen in Fig. 11(c).

As a quantitative validation, an evaluation of a motor's power spinning the rotor under the flow and cavitation conditions is conducted. For the case of (V = 0.45 m/s N = 2000 rpm), the power drawn in the experimental setup watts (55 ± 5% watts), while the CFD resulted in 70 Watts from the torque calculation on the blades. The CFD result did not change due to any further mesh refinement, and it was accepted as the closest to be offered by the numerical models used (i.e., VOF, LES, Unsteady Rotational Domain).

A second case increased the rotational speed to N= 3000 rpm, and consequently, the inlet velocity increased to V= 0.56 m/s in the 0.15 m pipe. The motor power was found to be 197 and 220 Watts in the experiment and CFD, respectively. It is worth mentioning that the CFD simulations could predict the power type (i.e., motor consumption) by generating a negative sign to the torque calculated.

By studying the effect of air on a cavitating flow in turbomachinery, a time-dependent CFD case was made for a 3-in. Kaplan hydroturbine undergoing cavitation. The case is developed as jet-in-cross-flow by radial injection of pressurized air at the rotor blades. Once initial trials showed significant cavitation, different patterns of injection (1, 2, 4, and 12 ports) were suggested to check the treatment effectiveness and understand the impact of the airflow rate. Unifying the steady-state vapor content over the rotor components, a surface averaging of the vapor volume fraction (VVF) followed by a time averaging was made for each case. VVF values with the mechanical output power (P) were compared for the proposed cases. Air admission into the system increases the static pressure over the vapor pressure, leading to fewer chances of cavitation in areas of high air presence. Adding more air volumes through multiple jets (Q_a/Q_w = up to 40%) could reach a VVF reduction of 55% with 12.4% power regain compared to a pure cavitation situation. Understanding the effect of air increase, both VVF reduction and power rise follow second polynomial correlations with the increased air volume. While the VVF reduction is directly proportional to the Q_a, the power increase seems to have an inflection to a constant or even a decline with Q_a because of the adverse nature of excessive air (i.e., entrapment) in turbomachinery. The importance of work falls under the safety of pumps, turbines, and even hydrofoils of hulls. It is worth mentioning that air could be used at almost atmospheric conditions, and any pressurization will add up to the cost of the treatment besides the installation. Future analysis is planned for the optimal position of the ports on the rotor parts and the effectiveness of the injection pressurization.

• The U.S. Department of Energy (DE-FOA-0001006; Funder ID: 10.13039/100000015).

The authors attest that all data for this study are included in the paper.

CP =

pressure coefficient

CP_min =

minimum pressure coefficient

CFD =

computational fluid dynamics

D =

diameter

LES =

large eddy simulation

N =

rotational speed

N-S =

Navier–Stokes

P =

mechanical power

P_v =

vapor pressure

Q_a =

air flow rate

Q_w =

water flow rate

RANS =

Reynolds‐averaged Navier-Stokes

Re =

Reynolds number

SGS =

sub-grid scale

t =

time

VOF =

volume of fluid

VVF =

vapor volume fraction

WALE =

wall-adapting local eddy viscosity

y+ =

non-dimensional wall distance

Greek Symbols

$αl$ =

phase volume fraction

μ_t =

turbulent eddy dynamic viscosity

σ =

cavitation number

$∀$ =

volume