Energy-saving and emission reduction are crucial since shipping activity due to the global maritime trade has increased exponentially. Several agreements have been engaged to optimize ship energy efficiency composed of ship design and shipping operation planning. However, most up-to-date studies focused on speed and route optimization. The interaction analysis between speed and route efficiency below varied environmental conditions is limited. To attain energy and cost efficiency, a study of cavitation on the propeller that considers the ocean environmental condition will be discussed in this work. Although researchers have previously observed cavitation phenomena, the predictability of simulations is not yet such that problems can be eliminated. Since the multiphase flow of water and vapor is sensitive to environmental conditions, it leads to varying observation accuracy. Thus, the current paper proposes a new performance indicator of the ship propeller under cavitation predicted by computational fluid dynamics (CFD). CFD-based simulation to observe the propeller cavitation was used to model the Zwart cavitation and Kunz cavitation models under two turbulence models of K−ε at different flow conditions and operating environments. Initial validation tests between experimental and numerical simulation show good agreement with a mean error of 4.7% in the Zwart model and 3.7% in the Kunz model, where the k−ε turbulence model provides an almost higher relative error. It is revealed from the result that the increase in temperature causes the rise in the cavitation problem. It is revealed from the result that the increase of temperature causes the increase in cavitation problem.
computational fluid dynamics
energy efficiency design index
general grid interface
global warming potential
International Maritime Organization
light fuel oil
liquefied natural gas
multiple reference frame
northern shipping route
potsdam propeller test case
Suez canal route
shear stress transport
Shipping is the most important mode of transportation for global commerce, accounting for about 90% of all global trade . However, worldwide shipping is subjected to marine accident risk [2,3,4,5], and it is capable of a noteworthy increment in greenhouse gas emissions, which share almost 3% carbon dioxide (CO2) and around 13% of total sulfur oxide emissions to the environment . This issue might cause changing climate designs, rising ocean levels, and diminishing biodiversity. Based on the International Maritime Organization (IMO)  study, the CO2 emissions from worldwide trade contain roughly 2.2% of the worldwide CO2 emissions. Because of the increasing volume of global commerce, by 2050 pollution could rise by 150–250% if no action is taken .
The physical impacts of climate change have become one of the foremost fundamental issues nowadays. As shipping accounts for expanding CO2 emissions, the Paris Agreement sets out a worldwide system to maintain a strategic distance from perilous climate change by restricting global warming to underneath 2°C and seeking after endeavors to constrain it to 1.5°C . In 2011, IMO accomplished new rules to regulate the energy efficiency for ships to diminish emissions delivered by ocean-going vessels . Based on those regulations, the vessels are committed to accomplishing a specific energy efficiency design index (EEDI) in order to extend energy efficacy and increase vessel cost-effectiveness.
Presently, there are generally three alternative ways to optimize energy efficiency. The first is from the point of view of a ship operation. The sailing speed and routes are optimized based on the relationship between fuel consumption and the vessel speed, related to the environmental conditions [10,11]. Then, the second is from the perspective of logistics. The economy of fleet navigation schedule and overall route planning is characterized to play down costs and outflow . Third, optimizing the design and arrangement of hull shape and propulsion system based is the last option . These three alternatives mean the industrial sector must consider long-term planning to adapt to new rules that comply with the EEDI to attain the shipping industry’s vitality investment fund and emission reduction goals. Therefore, this article introduces technical analysis according to the perspective of shipping efficiency parameters based on the first and the third alternative strategy. This article attempts to combine the philosophy of ship design with ship operating environment, in which propulsion system performance will be related to environmental conditions.
Cavitation, a multiphase phenomenon, founded during ship operation, should be investigated accurately to predict the propulsion performance . Cavitation can induce adverse effects such as performance loss, vibration, and noise . This problem cannot be eliminated, but the ship only wastes energy without generating the required power if it is not controlled. Early recognition of these negative effects is essential in shipping navigation passing through the Arctic Ocean. The shipping activities may provide more disadvantages in the Arctic environment due to fossil fuel burned emissions. Arctic’s viabilities have been the subject of numerous articles since the early decade [16,17,18,19]. So, it is crucial to account for the performance in propulsion design (e.g., by proper propeller design) and operations (control cavitation level) to prevent cavitation adverse effects and optimize fuel consumption and environmental conditions. With this issue, many engineering sectors are challenged to develop high-efficiency devices or energy alternatives. The designer is expected to solve these adverse effects to control the fuel cost of propeller efficiency and ensure compliance with the design requirement and green energy issue.
Cavitation may occur due to a low local static pressure combined with a high flow velocity, resulting in vapor pressure lower than the saturated water pressure of the liquid . Propeller blades are arranged by several hydrofoil sections where the hydrofoil’s upper surface has lower pressure than its bellow surface. This condition will remain to continue until the peak condition at a particular operation velocity. When the water pressure at the upper surface is lower than the saturated water pressure, liquid water will be transformed into the gaseous phase. This condition generates small vapor bubbles and then grows to a bigger diameter and then collects or coalesces to other macroscopic cavitation structures . These cavitation bubbles cause the drop of thrust and torque produced by the propeller. Furthermore, when the cavitation bubbles arrive at the blade trailing edge or behind the propeller, they will receive higher pressure. This condition makes bubbles rapidly collapse and produce a high frequency of shock waves. The microjet causes erosion damage on the propeller surface .
The numerical simulations of cavitation on the propeller have been previously investigated. Morgut and Nobile  have simulated cavitation on a model propeller with three cavitation models of Zwart, Kunz, and full cavitation model using ANSYS CFX. The cavitating flow of the E779A and potsdam propeller test case (PPTC) propeller models was predicted using these three calibrated models. The research has successfully generated the optimum cavitation coefficient of each mass transfer model according to the operational conditions. Moreover, Helal et al.  have performed a cavitation study numerically on a model propeller INSEAN E779A using ANSYS Fluent with the transition-sensitive model (K−Kl−ω) and fully turbulence model (K−ε). The research was purposed to understand the existence of laminar and turbulent flow regimes. The results showed that both experiment and simulation had good agreement with experiments. The accomplishment of the two selected turbulence models indicated that (K−Kl−ω) was suitable for low rotational speed. Besides, conditions that had more existence of transition flow were appropriate for K−ε. One of the challenges in investigating propeller cavitation is that the flow characteristic around the cavitating propeller contains multiscale flow physics, i.e., turbulence event, a phase change of the fluid, and unsteady loading. A comprehensive and detailed examination of the parameters influencing cavitation efficiency is needed for accurate performance prediction. With the case of different water properties, the result can be considered to enrich knowledge for the shipping industry in designing a propeller according to the needs and environmental conditions.
To address these issues, this work conducted a computational fluid dynamics (CFD)-based simulation to observe the propeller cavitation according to environmental conditions using ANSYS CFX. CEL expression language for CFX was used for modeling the Zwart cavitation model, which was known as the Zwart mass transfer model and the Kunz cavitation model [25,26]. The validation step was carried out with the Zwart and Kunz cavitation models and then was combined with two turbulence models of K−ε dan at three flow conditions and three operating environments. The structure of this article is constructed in sections as follows. Section 1 presents the introduction and objective of the study. A literature study on the Northern shipping route (NSR) commercial and environmental viability and previous cavitation studies related to temperature are reported in Section 2. Section 3 offers the mathematical model, and Section 4 reports the analysis of cavitation to its operating conditions. In the last, a few concluding remarks and summary are provided in Section 5.
2 Literature study
Exploration of the ocean is a big challenge for humans. There are regions blessed with tremendous and valuable common assets, e.g., renewable (wind, sun-based, wave) and non-renewable assets (oil, gas, minerals), which can advantage human life. It is attracting much interest in exploring and exploiting natural resources for human prosperity. The ocean must be managed equitably and sustainably because human activity can degrade and destroy the blue ecosystems.
2.1 Shipping in NSR and Suez canal route (SCR)
One of the extreme areas currently possible to conquer is the Arctic Ocean in the North Pole. Arctic shipping routes are the maritime route connected entirely of both east–west land and the Atlantic–Pacific Ocean. Located in the north polar of the Earth, it is characterized as a zone with a cold climate and secured by ice. This environment has been untouchable and nearly undisturbed by human activities for a long time. Recently, the route through parts of the Arctic is conceivable since the worldwide warming impact has made a more critical portion of the Arctic melting within the summer. Mainly it happens in March, as shown in Figure 1 .
Maritime container cargo contains an extraordinary potential request for the NSR shipping route, which is currently shipped through the SCR between East Asia and Northwest Europe (Figure 2). Vehicles are one of the foremost critical potential cargoes. Furthermore, other prospective cargoes, such as Murmansk gas condensate, Hammerfest liquefied natural gas (LNG), and Kirkenes iron ore, had already been commercially exported since 2009 to East Asia . As global warming has retreated from the Arctic seas, NSR trade in East Asia and Northwest Europe lately has started to gain momentum, leading to the direct result of a 40% reduction in sailing distances relative to the SCR. The NSR is expected to take 19.3 days to transport, which is 35.4% quicker than the SCR (30.4 days).
The reduced effect of CO2 pollution due to decreased distance is estimated between 13 and 35% for the NSR duty period of 105–225 days. This issue can entice the operators and shipowners based on the environmental benefits of shipping rather than the financial benefits . Lindstad et al.  have investigated the combination of the emission obtained with the specified global warming potential (GWP) factors for the last 20 years. Twenty years of assessment based on relevancy to identify fuel and technology relations are used, resulting in the best climate impact reduction. It is compatible with the 2050 target of maintaining temperature stability at 2°C above pre-industrial levels.
It can be found that the positive values shown in Table 1 indicate a warming impact, and negative values have a cooling effect. Figure 3 exemplifies comparison data of trades through SCR and trades through NSR. This study uses Panamax dry bulkers for both shipping trades. The fuel options are light fuel oil (LFO), marine gas oil, and LNG. Each fuel option is divided into low power and high power. The bar shows that NSR yields the lowest cost compared to 18 USD/ton versus 29 USD/ton for NSR and SCR, respectively. However, compared to CO2 equivalents (GWP20), trades through NSR have a more significant climate effect per ton transported than trades through SCR. The LFO used in NSR trades, for example, is 65 kg CO2-eq per ton transported, compared to 20 kg CO2-eq in SCR trades.
|Type of emission||CO2||BC||CH4||CO||N2O||NO x||SO2||OC|
|GWP20 world factor||1||1,200||85||5.40||264||−15.9||−141||−240|
|GWP20 world factor||1||6,200||85||5.40||264||−31.0||−47||−151|
|GWP100 world factor||1||345||30||1.80||265||−11.6||−38||−69|
|GWP100 world factor||1||1,700||30||1.80||265||−25.0||−13||−43|
2.2 Water characteristic in ocean
The water characteristics in the ocean are dependent on the temperature. The natural water phase is liquid, but it could turn to a solid phase and gaseous/vapor phase determined from heat energy transferred by sunshine. Water has low thermal conductivity than air. It needs about four times the amount of energy to increase the temperature by 1°C, but it also means water is more difficult to release heat. Figure 4 depicts the heat maps of the ocean surface temperature of the Earth. The water near the equator has yellow until the red represents the warm zone and gets cold to the poles. The temperature of the world ocean surface imposes on Earth’s climate and weather. The temperature of the ocean also varies according to the depth. There is a surface layer categorized up to 200 m deep, with the same temperature. Below the surface layer is a thermocline that reaches 1,000 m, colder from top to bottom. However, in this case, the study of ship operation is only on the surface. So, it is clear that the scope of ocean water in the following discussion means surface water of the ocean.
Besides temperature, the salinity of seawater is an exciting topic in oceanography. Seawater has different salinity levels influenced by temperature, pressure, conductivity, etc. The spectrum of practical salinity S p is in the range 2 ≤ S p ≤ 42 . The precise location of bulk material collection to prepare the standard seawater is not mentioned. The Standard Seawater Service issues guidance notes to ships collecting this bulk material, recommending that water be collected in deep water between longitudes 50 and 40°W during daylight hours. Millero et al.  created reference composition salinity S R (or reference salinity for short) to be the best approximation in the mass fraction absolute salinity S A of standard seawater. A seawater sample of reference composition whose practical salinity S P is 35 has a reference salinity S R of 35.16504 g kg−1 with absolute uncertainty is ±0.007 g kg−1. The difference between reference and practical salinities may be attributed to the early method of calculating salinity by evaporating water from seawater and weighing the remaining solid content. This method also evaporated certain volatile elements, accounting for the majority of the 0.16504 g kg−1 salinity discrepancy.
Veronis measurements  revealed that after a temperature drops to −40°C, newly formed ice had a salinity of 10.16 g kg−1. The ice salinity was as low as 5.64 g kg−1 after the temperature dropped to −16°C. Because of the salt solution, the salinity of ice reduces with age. It stays between the crystals during freezing since it is denser than ice and therefore leaches. Temperature readings have a minimum precision of 0.01°C. Ocean temperatures range from –1.9 to 30°C. The lower boundary is defined by the accumulation of ice. However, supercooled water with a temperature below the freezing point may be encountered. Water temperature in the surface layer of a closed sea can be higher than 30°C in extreme conditions.
Cavitation phenomena cannot be separated from the evaporation and condensation process. This process depends on the temperature, pressure, water properties, and other thermodynamics properties in its specific environment. The water boiling point marks phase transformation from liquid to gaseous. Both freshwater and seawater have different boiling points because of their properties. The boiling point of water is determined by saturation pressure. In the case of this study, the saturation pressure of water is differentiated by freshwater and seawater. Figure 5 shows the difference between the boiling point and saturation point of both seawater and freshwater.
2.3 Previous cavitation research related to the temperature
Several researchers who studied the cavitation phenomena applied different techniques in approaching the cavitation phenomena. The effect of temperature on cavitation problems for various materials over a temperature range of 0–90°C has been studied by Plesset . The findings showed that the optimum damage rate occurred at temperatures ranging from 40 to 50°C. Furthermore, Al-Arabi  used experimental observation to investigate the impact of water temperature on the efficiency and cavitation inception of a centrifugal pump. The section test was created to monitor the ratio of flow rate, suction pressure, rotational speed, and fluid temperature. The findings revealed that increasing the water temperature sped up cavitation and increased the net positive suction head.
Furthermore, an experimental test was conducted by Dular and Coutier-Delgosha  to research the effect of thermodynamics on a single cavitation bubble’s growth and collapse. Their research focused on studying temperature changes in liquid environments. Experiments were conducted in a water-filled cylinder at room temperature and atmospheric pressure. The findings revealed that the growth of the bubble was caused by the extension of an initial air bubble. The experimental findings were compared to the “thermal delay” model’s predictions. Temperature fluctuations are due to latent heat exchanges during the vaporization and condensation processes in this approach.
Tanaka  observed a centrifugal pump’s thermodynamic cavitation efficiency experimentally in 2011. The findings found that cavitation performance with liquid nitrogen was superior to cavitation performance with cold water, and the expected temperature drop due to the thermodynamic effect decreased as the flow coefficient decreased. Furthermore, at the same flow coefficient, the predicted temperature drop attributable to the thermodynamic impact on the low cavitation condition was greater than on the high cavitation efficiency impeller. Kim and Song  conducted experimental research on the influence of temperature on the critical cavitation number in a turbopump inducer. According to the data, nondimensional thermal parameters increased the onset to a lower cavitation number and decreased cavitation rate. However, at 340 K and 5,000 rpm, the critical cavitation number and revolving cavitation onset were independent of the nondimensional thermal parameter for more than 0.540. Yusvika et al.  used CFD-based numerical simulation to simulate propeller cavitation based on water physical properties. The simulation effects of propeller output characteristics, propeller cavitation patterns, and fluid state were compared at observed temperatures ranging from 0 to 50°C. The cavitation phenomenon was modeled using the RANS system with several reference frames (MRF). The obtained results revealed a cavitation pattern occurring more vigorously as water temperature increased.
3 Mathematical model
3.1 Parameter of propeller performance
A parameter of propeller performances can be easily separated into open water and behind-hull properties, which have fundamental variations. This study observes the PPTC VP1304 propeller in open water by numerical simulation. The propeller thrust F (N) and torque are expressed in a series of nondimensional characteristics, which are as follows :
where and are the dimensionless coefficients of thrust and torque, is the diameter of the propeller, J is the advance coefficient, ( ) is the fluid density, is the open water efficiency, V A (m/s) is the freestream fluid velocity, and is the propeller rotational speed.
3.2 Calculation of cavitation event
Cavitation flow conditions were organized based on the cavitation number at the specified advance coefficient of testing. The pressure coefficient c p and cavitation number can be defined as follows :
where P v is the absolute vapor pressure, P is the local pressure, and is the pressure used for reference. Based on the theory of the cavitation inception criterion, cavitation will occur if ≤ c p.
3.3 Multiphase Reynolds-averaged Navier–Stokes (RANS) method
Cavitation flow analysis is gas-liquid two-phase mixture that combine water & water vapor into a single fluid mixed-phase state. The RANS equation can solve the multiphase flow if the combination of the two liquids is homogeneous. To approach turbulent flow, the RANS equation assumes that the fluid is incompressible and employs the eddy viscous model . The following is the governing equation:
This equation represents the multiphase RANS equations. The volume continuity and momentum equation for the liquid–vapor mixture is shown on the left, and on the right, the volume fraction for the liquid phase. and are liquid and vapor density, respectively. represents the average pressure, indicates the average velocity, is the additional momentum, and indicates the water volume fraction. Then, is the vapor volume fraction:
The multiphase flow is constructed from the mixing of liquid–vapor density as well as its dynamics viscosity. Each phase’s density is expressed by a scalar volume fraction, which is as follows:
where represents the mixture density, is the interphase transfer rate due to cavitation, and is the dynamic viscosity of the mixture. Furthermore, the eddy viscous turbulent model was used to close the governing equation model and treat the two liquid–vapor phases as one phase. The standard fully k−ε turbulence model and shear stress transport (SST) turbulence models were used for each simulation in the test case. Meanwhile, for the observation part, only the SST turbulence model was used because it considers the validation in the test case, which has superior accuracy with the SST turbulence model to k−ε.
3.4 Bubble cavity inception process and mass transfer model
The first stage of cavitation begins with the formation of bubbles. It assumes that the cavitation bubble is a homogeneous phase and analyzes only one bubble during the flows, despite it being impossible in practice. The bubbles grow larger following its pressure at the local area. The Rayleigh–Plesset equation is used to model bubble growth. It represents the increasing diameter of bubbles in a fluid as a time function. The equation measures growth in a single bubble when assuming no thermodynamic flow or temperature decrease from bubble initiation to collapse. For test and observation scenarios, the equation has been used as a mass transfer model in CFX, and it is as follows :
where represents the bubble radius, indicates the pressure of the bubble at the fluid reference temperature, and is the pressure around the bubble. and reflect the fluid density and surface tension coefficient of water vapor and fluid, respectively. When the second derivative is removed, the equation is reduced to:
The rate of bubble volume growth is described as follows:
while the rate of change in mass of the bubble can be described as follows:
If the number of bubbles per unit volume is defined by , then the equation for the bubble vapor volume fraction is
The total mass transfer rate per volume is then set as follows:
As the vaporization process occurs, the bubble volume fraction increases with increasing bubble diameter, and the nucleation density decreases. As a result, the volume fraction of water vapor produced during vaporization is a feature of . The rate of vaporization and condensation bubble formation can be described as follows:
where represents the nucleation site of the vapor volume fraction, and reflect the empirical calibration coefficient of evaporation and condensation, respectively, and represents the vapor volume fraction. The following model parameters have been shown to be effective for a wide range of fluids: . This model of cavitation flow is known as the Zwart mass transfer model or the Zwart cavitation model.
3.5 Kunz cavitation model
This mass transfer model parameter is set up in two ways to build and destroy of liquid. The liquid-to-vapor transition is measured as a function of how much pressure is below the vapor pressure. Otherwise, the conversion of vapor to liquid is dependent on a third-order polynomial function of volume fraction, C. The specific mass transfer rate is denoted by the formula m = (m + ) + (m −).
where = L/U denotes the mean flow time scale, (m/s) represents the freestream velocity, L is the characteristic length scale, and C dest and C prod are two empirical coefficients, respectively, where C dest = 100 and C prod = 100 are assumed.
3.6 Thermodynamic parameter
The thermodynamic parameter is known as the change in water temperature and is used to calculate the behavior of bubble dynamics. Thermodynamic behavior ∑ can be denoted as :
where L denotes the latent heat of vaporization, represents the density, the specific heat at constant pressure is indicated by , and is the temperature of the liquid.
4 Numerical modeling
4.1 The cavitation tests conducted in the Schiffbau-Versuchsanstalt (SVA) potsdam
In this work, the cavitation experiments were conducted using the cavitation tunnel K15A (Kempf and Remmers) of the SVA Potsdam . A dimension with a length of 2593.5 mm and a cross-section of 850 × 850 mm was used for cavitation observation tests. Figure 6 depicts the cavitation tunnel in the potsdam cavitation test.
Throughout measurement, the propeller was mounted according to the configuration shown in Figure 6, with a 12° inclination of the propeller toward the inflow direction or the X-axis. In this case, the propeller in an inhomogeneous environment is powered by the cavitation tunnel’s homogeneous inflow. Tangential velocities work on one side of the propeller against and on the other side with the direction of blade rotation. Figure 7 shows the cavitation test section.
For the simulation, this study has slight differences compared to the testing model. The simplification technique is applied, which only involves a propeller without a hub and shaft to achieve optimal element mesh due to resource limitation of computer capabilities. This method was valid because the hub behind the propeller would not disrupt the inflow. This technique was also used in simulation techniques by Morgut et al. , as shown in Figure 8.
The inflow velocity in the numerical calculations can be determined with the advance coefficient. The pressure level in the numerical calculations can be resolved with the cavitation number. All evaluations shall be conducted with the propeller being in the zero-degree position. The origin of the geometry corresponds to the propeller plane position (x = 0 m). The calculations must be carried out according to the thrust identity (KT values are provided, non-cavitating), with the inflow velocity free to alter. The revolution rate is kept constant as it is used to define the cavitation number.
4.2 Propeller model and test case scenario
With the aim of testing and validation, SVA Potsdam developed a PPTC model VP1304 propeller model. The propeller consists of five blades with a controllable pitch propeller and a right-handed direction of rotation, as shown in Figure 9. The experimental data and geometries are available for viewing on the company website . The VP1304 model was used in previous research for numerical simulation and validation results. Instead of experimental analysis, numerical simulation of the propeller output under cavitating flows using CFD-based software has been a good choice.
The simulation began with the simulation of the cavitating flow for validation purposes under existing conditions and then continued with the observational scenario. Fluid properties of water and vapor are set according to the water condition in the experimental testing tank. The physical properties used in numerical simulations are assumed that water is freshwater and seawater. The saturation vapor pressure is a representation of the reference temperature. As shown in Table 2, this section consists of three operating cases.
|Parameters||Symbol||Unit||Case 1||Case 2||Case 3|
Table 2 displays the operational parameter, i.e., the observation scenario as a case to be studied. The simulation uses , , and as fixed parameters, while other parameters are temperature dependent and are the most important to observe. Water temperature varies at 1, 10, and 20°C. The current cavitating flow for the test case is subjected to fluid properties at 22, 21.9, and 23.2°C. New water material at specified water temperature is set to define water and vapor’s thermodynamic and transport properties. Furthermore, Table 3 offers a comprehensive description of the fluid’s properties. Using the CalcSteam online calculator, all fluids are measured based on the water temperature. In Table 3, all parameters with different temperatures are shown, with the new outlet boundary conditions as the static pressure choice determined using the equation of cavitation number.
denotes fluid and vapor pressure at reference temperatures, respectively; is the water density; represents the vapor density; and are dynamic viscosity of water and vapor, respectively; , , , and are thermodynamic parameters consisting of enthalpy and entropy of liquid and vapor, respectively; and denote the isobaric heat capacity of fluids; and and represent thermal conductivity of water and vapor.
4.3 Model validation
For validation purposes, the design of the computational domain used for the simulation test under consideration is a cylinder, as shown in Figure 10. The diameter D of the VP1304 propeller is 250 mm. The width and length of the rotating domain are 270 and 300 mm, respectively. The test section has an area of 850 mm × 850 mm perpendicular to the flow direction. The fixed domain is extended to 2593.5 mm overall. The simulations were run in 3D and assumed a steady-state condition. Because of the open water conditions, the propeller moved forward in a homogeneous uniform flow. The multiple reference frame (MRF) method was used to simulate propeller rotation, with the rotational speed remaining constant at . The domain is divided into two parts: a rotating part and a fixed part. The MRF method analyzes the rotating domain in relation to other domains, while the general grid interface (GGI) method approximates a continuous surface .
The single rotational speed of the water was applied to obtain the value of advance coefficient and cavitation number . The advance coefficient was used to measure the inlet velocity. The turbulent pressure boundary on the inlet was set at 1%. The pressure outlet for each run was determined using the cavitation number equation. The pressure outlet was with a static pressure option for all simulations. The free slip boundary condition was added to the domain’s outer boundary, while the propeller or solid surface was set with a no-slip boundary condition. The domain motion on the rotating domain was set to rotate with angular velocity. Furthermore, the stationary motion was applied. First, SST with automated wall treatment was used to influence the turbulent transition flow. In this work, the standard k−ε is applied in the next step for the full turbulence model with a scalable wall function. The Rayleigh–Plesset equation was used to describe cavitation bubble growth. Table 4 shows additional configuration parameters.
|Inlet domain||Normal speed|
|Inlet heat transfer||Static temperature|
|Turbulent intensity||Low intensity|
|Outlet pressure||Static pressure|
|Fluid definition||Volume fraction||Water (1) and vapor (0)|
|Rotating domain||Rotational speed||20 rps|
|Heat transfer||Homogenous||Total energy|
|Turbulent||SST||Automatic wall function|
|Nucleation||Mean diameter||2 × 10−6|
4.4 Meshing strategy
The chosen meshing method is an unstructured grid and hybrid mesh, which provides high precision. The rotating and fixed domain parts of the domain were segregated in this analysis. The mesh was created with CFX’s solver mesh (Figure 11). The rotating domain was initialized automatically, and the fixed domain was defined using the tetrahedron method. The CFX solver will join the rotating and fixed interfaces meshing method using a simplified grid interface (GGI). The mesh sizing with the body of influence selection to approach more precision for the critical segment was used to reduce the number of mesh elements and maximize the mesh density. Then, 11 layers with inflation on the propeller surface to obtain the viscous effect as the turbulent boundary layer were placed. For all propeller surfaces, all meshes have the same value of approximately 10. The following equation can be used to calculate the value :
where denotes the kinematic viscosity of the fluid, represents the shear stress at the wall, denotes the density of the water, and represents the normal distance from the wall.
4.5 Solution and solver setting
The CFD platform ANSYS CFX 19 was used for simulation, which offers solutions for three-dimensional viscous scenarios. The propeller cavitation simulation was performed under the assumption of steady-state conditions. The “upwind” advection scheme was added to the turbulent numerical high-resolution transport equation. The dynamic control model used the velocity–pressure coupling and multiphase control models.
4.6 Verification and validation tests
Verification and validation tests are conducted to investigate the effects of fluid properties and temperature effects on the cavitation process. Following that, the effects of propeller output at three operating conditions: , are compared. The output of thrust and torque coefficient is calculated in this section using CFX results by comparing two turbulence models of k−ε and SST. Following that, in each case, all nondimensional coefficients of output ( are determined. The Zwart cavitation and Kunz Cavitation models’ validation results under various operating conditions are shown in Tables 5 and 6, respectively.
|Experiment||Simulation||Relative error (%)|
|Experiment||Simulation||Relative error (%)|
Furthermore, the validation stage was extended by performing a mesh dependence analysis to achieve optimum meshing. Table 7 displays the meshing information for the rotating and fixed domains depending on the number of elements. The optimum meshing value was calculated by taking simulation time and data error precision into account. It can be found that the longer the running time, the more the total of mesh elements. The result of value was determined using Case 3 at the existing temperature, and the error value was determined by comparing the simulation results to the experimental data collected in SVA potsdam’s cavitation tunnel .
|Mesh case||Element in rotating domain||Element in a fixed domain||Error (%)|
There are inconsistencies in the results of simulation data in experiments of both cavitation models with variable errors in Tables 5 and 6, assuming steady state and heat transfer during the process are set with total energy. The comparison of numerical results shows that the predicted cavitation at the three operating conditions by both turbulence models is in good agreement with the corresponding experimental results. For the coefficients, the agreement between predicted results based on both numerical and experiments is good. The relative error value differs between k−ε and SST turbulence models, with the k−ε turbulence model providing an almost higher relative error, while SST produces more variable results. Based on the simulation results in Tables 5 and 6, the SST turbulence model is superior to the k−ε turbulence model for modeling cavitation. As a result, the simulation in the second stage was entirely based on SST turbulence models. Figures 12–15 provide a comparative graphic of values based on observations and simulations of various turbulence models.
5 Observation of simulation results
The influence of simulation results on propeller performance under different operational conditions is discussed in this section. Tables 8–10 compare thrust and torque coefficients and the relative error under three different case scenarios. The torque and thrust coefficients under three temperature values are compared to the value calculated in the existing temperature. To illustrate in more detail, Figures 16 and 17 depict a schematic comparison between the advance coefficient (J) and the performance coefficient of and . Based on the result in Tables 8–10, it is understood that at the temperature of 1°C, the torque and thrust coefficient values have the highest difference compared to the data in normal temperature conditions; this occurs in all cases (Cases 1–3). It can be seen in the deviation percentage in all evaluated cases that the higher the temperature, the higher the deviation occurs. With increasing temperature, it can be realized that the propeller performances experience a decreasing trend. It means that the rising cavitation pattern is defined by decreased propeller output as temperature rises. The gap in ambient temperature significantly impacts propeller performance under cavitation conditions. Moreover, comparing the performance of all evaluated cases, the cavitation that occurs in case 1 is the most aggressive.
|Temperature (°C)||(%) to at 22°C||(%) to at 22°C|
|Existing temperature (22.0)||0.383||—||0.097||—|
|Temperature (°C)||(%) to at 21.9°C||(%) to at 21.9°C|
|Existing temperature (21.9)||0.209||—||0.070||—|
|Temperature (°C)||(%) to at 23.2°C||(%) to at 23.2°C|
|Existing temperature (23.2)||0.144||—||0.058||—|
The findings are consistent with Al-Arabi , who investigated the cavitation in the centrifugal pumps at various water temperatures. It can be found that the decrease in the pump head is proportional to the rise in temperature. Moreover, Dular and Petkovšek  investigated liquid nitrogen and water cavitation at temperatures ranging from 20 to 90°C. It is found that the erosion rate was found to be proportional to the rise in water temperature in tests. Moreover, Plesset  conducted cavitation studies with water temperature changes ranging from 0 to 90°C. It is found from the result that the increase in temperature causes the rise in the cavitation problem.
The cavitation number is known to affect the reference pressure. The formation of cavitation bubbles becomes very destructive when the cavitation number is high. This theory is founded in the previous studies indicating that serve roles as the primary physical properties that induce cavitation . If cavitation is characterized as a decrease in the underwater vapor pressure of the environment, then the ratio of water vapor pressure becomes inversely proportional. The lower the ratio, the more violent the cavitation bubble’s inception.
Figure 18 depicts the vapor volume fraction pattern at a ratio of 0.4. The red area denotes the low-pressure contour indicating cavitation bubbles are formed. It can be found in all cases that the lower pressure contours of the propeller rise with increasing temperature. In cavitation with variable temperature, vapor density plays an important part. Water viscosity has a minor effect on the change in Reynolds number, but it significantly impacts the temperature distribution . To illustrate the cavitation phenomenon in more detail, Figure 19 compares pressure contours on the suction side with a condition similar to Figure 18. From the comparison of pressure contour in Figure 19, it can be found that cavitation initiation occurs in the blue area, which represents low-pressure zones. In contrast, the red color represents areas with a higher vapor fraction. It can be summarized that the pressure contour on the suction side is related to cavitation initiation. According to the concept of cavitation, cavitation bubbles will form in environments where the local pressure is smaller than the saturated pressure of water.
6 Concluding remarks
This work investigated the complete and detailed CFD procedures to discuss the cavitation phenomenon of the three-dimensional propeller under various operating conditions. To examine the cavitation phenomena on the (PPTC1) VP1304 propeller model, multiphase problems were formulated using the RANS method with the MRF approach in ANSYS CFX 19 software. The initial study is first conducted for validation purposes. In this case, the numerical result of propeller performances is compared with the data of experimental test using Zwart and Kunz cavitation models with K− dan turbulence models under various operating conditions. The prediction is that the SST turbulence model has a reasonably good agreement that produces a more reliable result than the K– turbulence model.
Based on the simulation result, the following conclusions are summarized:
Cavitation increased similarly with the temperature increase, while cavitation inhibited the formation of cavitation bubbles at low temperatures. The comparative data show that the most severe cavitation problem exists in Case 1.
Dependent on the various physical properties of water such as the water vapor pressure (P v), vapor density ( ), latent heat (L), and surface tension (σ), all played essential roles in the rate of cavitation formation.
The Rayleigh–Plesset equation can be used to model cavitation and provide a general overview of the results of various physical properties of water.
Based on these investigations, it is considered that this research might be turned into a topic for future work. One alternative that might be pursued is numerical research using compressible cavitation models with a mass transfer that does not occur instantly. To get relevant to the actual phenomenon, an unstable simulation should be executed. Also, an experimental cavitation test is another option that needs to be performed in future work.
This work was supported by the RKAT PTNBH Universitas Sebelas Maret – Year 2022, under Research Scheme of “Penelitian Unggulan Terapan” (PUT-UNS), with Research Grant/Contract No. 254/UN27.22/PT.01.03/2022. The support is gratefully acknowledged by the authors.
Author contributions: Conceptualization: M.Y., A.R.P., and A.F.; methodology: M.Y., A.R.P., S.H., and I.Y.; software: M.Y., A.R.P., T.T., and I.Y.; validation: A.R.P., S.H., I.Y., T.M., and F.B.L.; formal analysis: M.Y., A.R.P., T.T., and A.F.; investigation: M.Y., A.R.P., I.Y., T.M., and F.B.L.; resources: A.R.P., S.H., and I.Y.; data curation: M.Y. and A.R.P.; writing – original draft preparation: M.Y. and A.R.P.; writing – review and editing: M.Y., A.R.P., T.T., and A.F.; visualization: A.R.P. and T.T.; supervision: A.R.P.; project administration: A.R.P.; funding acquisition: A.R.P.
Conflict of interest: Authors state no conflict of interest.
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