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Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model
Alexandria Engineering Journal (2018) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transitionsensitive turbulence model Mohamed M. Helal a,*, Tamer M. Ahmed b, Adel A. Banawan b, Mohamed A. Kotb c a
VSE Corporation, Alexandria, Egypt Dept. of Naval Architecture and Marine Engineering, Faculty of Engineering, Alexandria University, Alexandria, Egypt c Arab Academy for Science, Technology and Maritime Transport, Alexandria, Egypt b
Received 3 January 2018; revised 18 February 2018; accepted 12 March 2018
KEYWORDS CFD simulations; Marine propellers; Cavitation; Multi-phase flow; Turbulence models; Transition-sensitive models
Abstract One of the big challenges, yet to be addressed, in the numerical simulation of cavitating flow on marine propellers is; the existence of laminar and turbulence transition flows over the propeller’s blades. The majority of previous studies employed turbulence models that were only appropriate for fully turbulent flows. These models mostly caused high discrepancies between numerical predictions and experimental measurements especially at low rotational speeds where, Reynolds number decreases and laminar and transient flows exist. The present paper proposes a complete and detailed procedure for the CFD simulation of cavitating flow on marine propellers using the ‘K-Kl-x’ transition-sensitive model. Results are obtained using ‘ANSYS FLUENT 16’. The propeller under consideration is the ‘INSEAN E779A’ propeller model. The fully turbulent standard ‘k-e’ model is also adopted for comparison. Obtained results, based on both turbulence models, are validated by comparison with experimental data available in the literature. Predictions based on the ‘K-Kl-x’ transition-sensitive model are found to be in better agreement with experiments at lower rotational speeds i.e. at low Reynolds numbers. Ó 2018 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Cavitation is a widespread phenomenon that might exist in various marine applications. Prediction of cavitation, especially for marine propellers, is very important to avoid –or possibly eliminate– its adverse effects. Franc and Michel [1] * Corresponding author. E-mail address: [email protected] (M.M. Helal). Peer review under responsibility of Faculty of Engineering, Alexandria University.
defined cavitation as: ‘‘the evaporation of a liquid when its pressure decreases below its saturation pressure”. Applying this principle to marine propellers means that; when a propeller runs in water at high rotational speeds, the water’s local pressure decreases. This decrease in pressure is proportional to the local flow velocity squared. If the pressure of water drops below the vapor formation pressure corresponding to the flow’s temperature, bubbles of vapor (cavities) form. Subsequently, these cavities collapse with an explosive manner resulting in severe pitting to the blade surface. If the cavitation extent is sufficiently large, adverse effects will take place such
https://doi.org/10.1016/j.aej.2018.03.008 1110-0168 Ó 2018 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: M.M. Helal et al., Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.03.008
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Nomenclature cp D J k K Kl KQ KT L n p p1 pv Q r R Re T U V VA y+ y
pressure coefficient diameter (m) advance ratio turbulent kinetic energy per unit mass (m2/s2) turbulent kinetic energy (J) laminar kinetic energy (J) torque coefficient thrust coefficient length (m) rotational speed (rps) local pressure (Pa) free stream pressure (Pa) vapor pressure (Pa) torque (N m) radius (position) of any blade section (m) radius of the propeller (m) Reynolds number thrust force (N) incident velocity (m/s) total velocity (m/s) advance velocity (m/s) non-dimensional normal distance from the wall normal distance from the wall (m)
Greek symbols a vapor volume fraction e dissipation rate (W)
as; a decrease in produced thrust, an increase in required torque, damage to the propeller’s material (erosion), strong vibrations and, noise. Using photography–via– experiments to capture cavitation patterns and extents over marine propellers’ blades is a very complex task. Experiments are, in general, expensive to setup and time consuming to conduct. Numerical predictions of cavitating flows over marine propellers’ blades using CFD simulations have become a good alternative to experiments and subsequently, have become a trending research topic. Researchers nowadays may harvest the great advancement of computer performance that has made these simulations possible; refer to Blazek [2] as well as Versteeg and Malalasekera [3]. Previously, flow about propellers had been predicted using the lifting-line theory with, a vortex line representing the propeller’s blade and helicoidal vortices representing the propeller’s wake. Numerical models developed swiftly starting from the 1960s. Salvatore et al. [4] utilized the perturbation methods to demonstrate the lifting-line theory. The lifting surface model was developed later on. Dang [5] and Vaz [6] provided the possibility for boundary element methods (BEM), also referred to as the boundary integral or panel methods, to be considered for simulating the flow about twodimensional geometrically complex bodies. The application of BEM was first utilized for partially cavitating flows of two-dimensional foils by Uhlman [7] and then by Lee et al. [8]. Kinnas and Young [9] introduced the ‘PROPCAV’ method employing BEM for the numerical prediction of fully sub-
g l t q r swall x
efficiency viscosity (Pa s) kinematic fluid viscosity (m2 s1) density (kg/m3) cavitation number shear stress at the wall (N/m2) inverse turbulent time scale (s1)
Subscribts m L V Charact.
mixture liquid vapor characteristic
Abbreviations CFD computational fluid dynamics INSEAN Istituto nazionale per studi ed esperienze Di architettura navale (Research Institute active in the field of naval architecture and marine engineering within the frame of the National Research Council of Italy). PROPCAV propeller cavitation research BEM boundary element method RANS Reynolds-averaged Navier-Stokes equation MRF multiple reference frame
merged super-cavitating propellers along with experimental measurements. Viscous flow methods for predicting cavitating flows around two-dimensional foils started getting in use since the 1990s. Numerical cavitation methods based on Reynolds Averaged Navier-Stokes (RANS) models have been developed to deal with viscous flows and, are currently still an active topic of research. The first results obtained through RANS simulation of cavitating marine propellers were obtained by Maksoud [10] and watanabe et al. [11]. Salvatore et al. [12] carried out computational predictions using the ‘INSEAN’ propeller flow code. In addition, experiments were carried out to validate the predicted cavitation under uniform flow. Zhu and Fang [13] investigated the performance characteristics of propellers under cavitation using viscous multiphase flow methods based on Navier-Stokes and bubble dynamics equations. Pereira et al. [14] ran an experimental and theoretical study of a cavitating propeller in uniform inflow. Advanced imaging techniques were utilised to obtain quantitative data on the extent of the cavity. Pereira and Sequeira [15] developed a turbulent vorticity confinement strategy for RANS based industrial propeller flow simulations. The aim of the study was to improve the prediction of tip vortices. In general and from the presented brief literature, it is apparent that many researchers have significantly contributed to the prediction of propellers’ cavitating flow using CFD simulations to address various aspects of the problem. Among several challenges, not yet fully investigated is; the existence
Please cite this article in press as: M.M. Helal et al., Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.03.008
Numerical prediction of sheet cavitation on marine propellers of laminar and turbulence transition flows over the propellers’ blades especially, at low rotational speeds i.e. at low Reynolds numbers. It is necessary to take into account the transient flow problem by employing transition-sensitive models and, to avoid the assumption that the flow is fully turbulent over the whole propeller. In this paper, the detailed procedure of a CFD simulation is proposed for solving the threedimensional, viscous, cavitating flow of a marine propeller. Subsequently, the cavitation patterns, extents and, the propeller’s performance characteristics under cavitation are predicted. Additionally, the paper investigates the influence of the existence of transition flow on the predictions. This is achieved by using two different turbulence models, a transition-sensitive model and, a fully turbulent model, namely, the ‘K-Kl-x’ transition model [16,17] and the standard ‘k-e’ model [17], respectively. The CFD software ‘ANSYS FLUENT 160 is used to perform the simulations where, both turbulence models are implemented. The investigation is carried out for the ‘INSEAN E779A’ propeller model. Results from the two turbulence models are assessed against experimental data available in the literature [14,18]. 2. Theoretical analysis and numerical methods 2.1. Performance characteristics of marine propellers The performance of propellers is traditionally represented in terms of; the thrust coefficient KT, the torque coefficient KQ and the efficiency g, together with, their variation with the advance coefficient J. These parameters are introduced as follows: KT ¼
T ; qn2 D4
KQ ¼
Q ; qn2 D5
g¼
J KT ; 2p KQ
J¼
VA nD
2.2. Cavitation calculations The onset of the process of cavitation is usually termed ‘cavitation inception’. The cavitation number r and the pressure coefficient cp may be defined as given below: p pv p p1 ; cp ¼ 1 r¼11 2 qðnDÞ qðnDÞ2 2 2 The traditional cavitation criterion based upon r and cp can be given as: If r 6 cp , cavitation occurs. 2.3. Multi-phase RANS method The cavitating flow is a mixture of two phases (vapor and liquid) that are simulated as one phase. Both phases share the same velocity and pressure fields. Assuming that the mixture is homogeneous, the multi-phase flow can be solved with conventional RANS equations after some modifications. The continuity equation of the mixture flow is given as: @ ! ðq Þ þ r ðqm V m Þ ¼ 0 @t m And the momentum equation can be expressed as:
ð1Þ
3 @ ! ! ! ! ðq V m Þ þ r ðq V m V m Þ ¼ rp þ r ½lm ðr V m @t ! þ rVT Þ þ q ! g þF m
m
ð2Þ
Eq. (2) represents the multi-phase RANS equation where, the left hand side represents changes in the mean momentum of a fluid element due to unsteadiness in the mean flow and is attributed to convection by the mean flow. These changes are balanced by the mean body force. As known from conventional RANS equations for single-phase flow and for the apparent stress to close, additional modeling is required. This has led to the development of many different turbulence models. A turbulence model is a tool for specifying the Reynolds stresses hence, closing the mean-flow equations. For multi-phase flows, the density constitution of each phase in a fluid cell is represented by a scalar volume fraction as follows: qm ¼ aqv þ ð1 aÞqL
ð3Þ
In a similar manner lm ¼ alv þ ð1 aÞlL
ð4Þ
Finally and to solve the system of the governing equations, turbulence closures are used. As clarified earlier, the transitionsensitive ‘K-Kl-x’ and the standard ‘k-e’ turbulence models are employed in this paper for this purpose. The ‘k-e’ model [17] deals with mechanisms that influence the turbulent kinetic energy. It is a two-equation model which means; it solves two additional partial differential equations for k and e in order to specify the Reynolds stress. The ‘k-e’ model is employed in the simulation presented here within as it converges relatively easily and, it has been reported suitable for simulating various turbulent flows. On the other hand, the ‘K-Kl-x’ transition model [16,17] is used to predict the boundary layer development and transition onset. This model is employed in the simulation presented here within as it addresses –effectively- the transition of the boundary layer from laminar to turbulent regime specially, in rotating flows. The model is a three-equation, eddy-viscosity type of model which includes transport equations for K, Kl and x. One unknown variable is yet to be identified in Eqs. (3) and (4), namely, the vapor volume fraction, a, which is obtained using the mass transfer (cavitation) model. 2.4. Mass transfer models Mass transfer models idealise the interphase mass transfer rate due to cavitation. Several mass transfer models have been proposed by researchers. In the current study, the ‘full cavitation model’ which was developed by Singhal et al. [19] and is available in ‘ANSYS FLUENT 160 , is employed. 3. Geometric modeling The propeller selected as a test case for this paper is the ‘INSEAN E779A’ propeller model which is a four bladed, fixed-pitch, right-handed propeller. Geometric modeling of this propeller is obtained using the ‘HYDROCOMP PROPCAD’ software as shown in Fig. 1. All geometrical data of this propeller is available in the literature [20] as summarized in Table 1.
Please cite this article in press as: M.M. Helal et al., Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.03.008
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M.M. Helal et al. 5. Grid generation
Fig. 1 The ‘INSEAN E779A’ propeller model developed in ‘PROPCAD’.
Table 1 model.
Geometric details of the ‘INSEAN E779A’ Propeller
E779a Propeller model Propeller Diameter Number of blades Pitch ratio Skew angle at blade tip Rake (nominal) Expanded area ratio Hub diameter (at prop. Ref. line) Hub length
227.27 mm 4 1.1 4°480 (positive) 4° 350 (forward) 0.689 45.53 mm 68.30 mm
4. Computational domain For the simulation under consideration, the computational domain is taken to be a cylinder as shown in Fig. 2. While the inlet is placed at a distance of 4D upstream from the propeller plane, the outlet is placed at a distance of 5D downstream from the propeller plane. In the radial direction, the domain radius is extended up to a distance of 2.5D from the axis of the hub.
Fig. 2
Design of the computational domain.
To generate an unstructured grid with hybrid cells, ‘ANSYS FLUENT 160 is used for grid generation. As concluded by Morgut and Nobile [21], this type of meshing is preferred since; hybrid meshes give higher accuracy for numerical prediction of propulsive performance. In addition, hybrid meshes need less effort in generation as compared to hexa-structured meshing. The mesh is generated taking into account that; the size of cells near the blade wall is small whereas, size increases towards the outer boundary. Fig. 3 shows size of the grid cells over the outer surface of the domain while Fig. 4 clarifies the size of hybrid cells inside the domain. After convergence, the total number of cells generated over the entire grid is 238,387. To obtain a suitable resolution of the viscous sublayer, the wall normal resolution over all surfaces of the blades is within a range of: y+ < 0.5 where, y+ is defined as: qffiffiffiffi sw y q yþ ¼ t 6. Boundary conditions The numerical simulation is performed over a range of advance coefficients and cavitation numbers equivalent to the range at which the propeller was experimentally tested [14,18]. Values assigned to the advance coefficient and cavitation number are as follows:
J = 0.9, r = 4.455 J = 0.83, r = 2.063 J = 0.77, r = 1.783 J = 0.71, r = 1.763
To obtain these values, various propeller rotational speeds of (1500 rpm, 1800 rpm, 2400 rpm and 3000 rpm) are used. The advance velocity values prescribed at the domain inlet are calculated and are as listed in Table 2. The free stream pressure p1 used in each run is calculated based on the corresponding value of r. In this study, it is assumed that Poutlet = p1. All boundary conditions imposed are shown in
Fig. 3 Grid over the domain surface developed in ‘ANSYSFLUENT’.
Please cite this article in press as: M.M. Helal et al., Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.03.008
Numerical prediction of sheet cavitation on marine propellers
5 For unsteady simulations, a second order implicit scheme is applied for time derivatives. The selected time step at each run corresponds to a rotation of 1°. The simulation is carried out by nine processors (Intel Ò Core(TM)2 Duo CPU E8400@ 3 GHZ). Table 3 shows details of the simulation settings. 8. Frame of reference
Fig. 4 Longitudinal section of the computational domain showing the size of cells.
The frame of reference of the ‘INSEAN E779A’ propeller model adopted in this study is shown in Fig. 6 where, the ‘xaxis’ points downstream, the ‘y-axis’ is directed upwards and, the ‘z-axis’ follows the right hand rule in the ‘x-y’ plane. Viewing form aft., the direction of rotation is clockwise as the propeller is a right-handed one. 9. Reynolds number distribution over the blades
Table 2
Flow parameters of each run.
n (rps)
VA (m/s)
J
r
25 30 40 50
5.1 5.65 6.99 8
0.9 0.83 0.77 0.71
4.455 2.063 1.783 1.763
Fig. 5. The continuum is chosen to be fluid with properties of water assigned to it. The propeller rotation is simulated using the Multiple Reference Frame (MRF) approach [17]. For the stationary region (the outer far field wall), the governing equations are solved in a fixed frame of reference while for all rotating regions (blade, hub and fluid zone), the governing equations are solved in a rotating frame of reference. A rotating frame of reference is assigned to the fluid at the selected absolute speeds given earlier in Table 2. A relative rotational speed of zero –with respect to adjacent cell zonesis prescribed to the walls of the propeller’s blade and hub. The far field boundary is treated as an inviscid wall to which, an absolute rotational speed of zero is prescribed. The four blades of the propeller are set at a regular angular interval of 90° hence, the modeling of one angular sector of 90° and one blade will suffice in solving the entire flow domain as shown previously in Fig. 5. The fact that other blades are present is taken into account by imposing periodic boundary conditions on the two sides of the blade. On these periodic boundaries, rotational periodicity is ensured.
Reynolds number varies along each blade with variations in the rotational speed and at every radius ratio r/R. Subsequently, and along the blade, the flow could change from laminar to transient to fully turbulent as Reynolds number values increase. Reynolds number Re is defined as: Re ¼
Ucharact Lcharact tfluid
For a propeller blade, Reynolds number may be estimated based on several characteristic dimensions, refer to literature [22] and [23]. In this study, it has been opted for a Reynolds number that is based on (Lcharct = r) and characteristic velocity (Ucharct = 2npr) at each radius ratio, i.e. at each r/R. The variation of Re with n for various blade sections (r/R = 0.3, 0.5, 0.7 and 1) is presented in Table 4 and Fig. 7. According to De Witt [24], the threshold for the turbulent region is at Re = 0.5 106. It can be noted from Table 4 and Fig. 7 that the flow becomes fully turbulent as the rotational speed increases and beyond a radius ratio of r/R = 0.5. From that, the standard ‘k-e’ turbulence model is expected to be more accurate for the higher rotational speeds at (r/R ˃ 0.5) since the flow is fully turbulent. On the other hand and for the range of radius ratios (r/R < 0.5), transient flow takes place and laminar flow exists specifically as the rotational speed decreases i.e. at low Re. Over this range, the ‘K-Kl-x’ transition model is expected to be well suited. 10. Results and discussion 10.1. Prediction of cavitation patterns and extents
7. Solution and solver settings The commercially available CFD software, ‘ANSYS FLUENT 160 is adopted to solve the three- dimensional viscous, unsteady, cavitating problem. The equations are solved as follows: The pressure and velocity components are solved using coupled technique. The second order ‘QUICK’ scheme is applied for convection terms in all transport equations.
In this section, predictions of the pressure distribution, based on the ‘k-kl-x’ transition-sensitive turbulence model, is presented. This distribution plays a significant role in predicting the cavitation inception. Later on, cavitation patterns -at the selected conditions in Table 2 will be shown based on both turbulence models. As mentioned earlier, the relevant experimental data of the cavitating flow for this propeller model can be found in the literature, Pereira et al. [14] and Salvatore et al. [18]. It covers the following (for various advance coefficients and cavitation numbers):
Please cite this article in press as: M.M. Helal et al., Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.03.008
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M.M. Helal et al.
Fig. 5
Table 3
Boundary Conditions: (a) 3D view. (b) Front view.
Solver settings.
Solver settings Pressure link Discretization scheme for convective fluxes and turbulence parameters Turbulence models
Near wall treatment Phases Solver Absolute Vapor pressure (KPa) CPU time/processor/run
SIMPLE QUICK 1. Standard (kЄ) 2. Transition (K-Kl-x) Standard wall functions 1. Water 2. Water vapor Unsteady 2.337 1–28 h (k-Є) 2–30.2 h (K-Klx)
Fig. 6
Frame of reference.
Please cite this article in press as: M.M. Helal et al., Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.03.008
Numerical prediction of sheet cavitation on marine propellers
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Table 4 Variation of Reynolds number (Re) with rotational speed (n) at various blade sections. r/R
n (rps)
Re 106
1
25 30 40 50 25 30 40 50 25 30 40 50 25 30 40 50
2 2.4 3.2 4 0.99 1.1 1.5 1.98 0.5 0.6 0.8 1 0.16 0.19 0.25 0.3
0.7
0.5
0.3
Fig. 8 Pressure distribution on the face and back sides of a blade section. r/R = 0.7, n = 50 rps and H = 135°.
Fig. 7 Variation of Reynolds number (Re) with rotational speed (n) at various blade sections.
Photographs of cavitation patterns. Measurements of cavitation extents. Measurements of performance characteristics. At the end of this section, predictions of the values of cavitation extents, based on both turbulence models, will be validated by comparison with corresponding experimental measurements. Figs. 8–10 show variations of the negative pressure coefficient (cp) with the position ratio (x/c) over the face and back sides of a blade section at (r/R = 0.7) and (n = 50rps) for (H = 135°), (H = 180°) and (H = 225°), respectively. By comparing the three figures it can be deduced that; the peak value of the negative pressure coefficient (cp) increases with a decrease in the hydrostatic pressure, i.e. a higher risk of cavitation. Subsequently, the probability of cavitation inception decreases as the blade rotates from H = 90° to H = 270° and then increases again as the blade rotates from H = 270°
Fig. 9 Pressure distribution on the face and back sides of a blade section. r/R = 0.7, n = 50 rps and H = 180°.
to H = 90°. Moreover, it is clearly observed that cavitation inception is more likely to occur over the back of the blade section (suction side) and close to the leading edge. An important issue that has been investigated in the past is; how to define the cavitation area [18], with two possible answers; either as an iso-surface of (cp = r) or based on the iso-surface of vapor volume fraction (a = const). This point is addressed in this study where, the cavitation area extent over the ‘E779A’ propeller blade is obtained based on
Please cite this article in press as: M.M. Helal et al., Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.03.008
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Fig. 10 Pressure distribution on the face and back sides of a blade section. r/R = 0.7, n = 50 rps and H = 225°.
both cavitation criteria; (a = const) and (cp = r) and, is assessed against photographs obtained experimentally to judge which of these criteria is more accurate in defining the cavitation area. Fig. 11 shows the predicted distribution of the pressure coefficient over the suction side of the propeller blade, based on the ‘K-Kl-x’ transition model, at (r = 1.76, J = 0.71, n = 50 rps and H = 90°). It is clear that large discrepancies exist between the numerically predicted and the experimentally obtained cavitation areas, at (cp = 1.76) [14,18]. In particular, the predicted and experimental cavitation areas don’t exhibit the same extent, shape or location on the blade surface.
M.M. Helal et al. This indicates that the cavitation criterion of (cp = r) is not reliable in defining cavitation. On the other hand, a variation of the vapor volume fraction a over the suction side of the propeller blade at the selected parameters is obtained based on both turbulence models. By comparing results predicted at various values of a with experiments, it is found that a value of (a = 0.5) led to the best cavitation predictions in terms of location, shape and extent. Subsequently, it is decided that cavitation predictions will be carried out based on the criterion of iso-suface of vapour volume fraction at (a = 0.5). Fig. 12 shows a comparison of predictions of cavitation patterns over the suction side of a blade, based on both turbulence models and at (a = 0.5), against corresponding patterns from photographs obtained experimentally. Comparisons reveal that the predicted cavitating regions, at the four operating conditions and based on both turbulence models, are in good agreement with those in the corresponding photographs [14,18]. Both predictions and photographs exhibit quite similar cavitation areas on the blade surface in terms of location, shape and extent. This demonstrates the correctness of the proposed simulation procedure as well as the validity of selection of the iso-surface of vapor volume fraction criterion and its assigned value. In addition, the cavitation regions are all found to be close to the leading edge of the suction side of the blade which emphasizes the previously drawn observations from Figs. 8– 10. An overall comparison of predictions and experiments shown in Fig. 12 reveals that; the ‘K-Kl-x’ transitionsensitive turbulence model is a better suited choice for high values of the advance ratio (J) while the ‘k-e’ fully turbulent standard model is a better suited choice for low values of (J). This is as expected since in this study, flow parameters were selected such that; a higher advance ratio means a lower rotational speed hence, less possibility for full turbulence to occur. Generally speaking and when compared to experiments, the extents of cavitation are over predicted -using both turbulence modelsand at all advance ratios.
Fig. 11 Cavitation area over the blade suction side at (J = 0.71, n = 50 rps, r = 1.76 and H = 90°): (a) Experimental photograph of the cavitation area (b) Predictions based on the ‘k-kl-x’ transition model and defined by (cp = r = 1.76).
Please cite this article in press as: M.M. Helal et al., Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.03.008
Numerical prediction of sheet cavitation on marine propellers
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Exp. J = 0.71, = 1.763
k- . J = 0.71, = 1.763
K-Kl- . J = 0.71, = 1.763
Exp. J = 0.77, = 1.783
k- . J = 0.77, = 1.783
K-Kl- .. J = 0.77, = 1.783
Exp. J = 0.83, = 2.063
k- . J = 0.83, = 2.063
K-Kl- .. J = 0.83, = 2.063
Exp. J = 0.9, = 4.455
k- . J = 0.9, = 4.455
K-Kl- .. J = 0.9, = 4.455
Fig. 12 Comparison of cavitation patterns, over the suction side of the blade, between experiments and predictions. Results are based on the ‘k-e’ and the ‘k-kl-x‘ turbulence models for the same operational conditions and at a = 0.5.
Table 5 Relative error of the cavitation extent based on the ‘k-e’ and ‘k-kl-x’ turbulence models. J
0.9 0.83 0.77 0.71
r
4.455 2.063 1.783 1.763
Relative error (%) k-kl-x
k-e
+10 +11.6 +17.5 +24.7
+22.5 +20 +14.2 +9.4
Table 5 gives a detailed account of the relative error percentage of the predicted extent of cavitation i.e. predicted area of cavitation. This relative error may be calculated as:
R: Errorð%Þ ¼
CFD results Exp: results 100 Exp: results
10.2. Prediction of performance characteristics under cavitation conditions In this section, thrust and torque are obtained from numerical simulations, based on both turbulence models and, at various rotational speeds of the propeller as shown earlier in Table 2. Subsequently, the non-dimensional coefficients of performance (KT and KQ) are calculated for each value of J. A comparison of the numerically estimated coefficients, based on both turbulence models, against experimental measurements is carried out and plotted in Figs. 13 and 14 for
Please cite this article in press as: M.M. Helal et al., Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.03.008
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M.M. Helal et al. 0.28
Exp.data k-kl-ω k-ε
0.26 0.24 0.22 KT 0.2 0.18 0.16 0.14 0.12 0.1 0.7
0.75
0.8
0.85
0.9
0.95
J
Numerical/Experimental results of the thrust coefficient
Fig. 13 (KT).
0.48 0.46
k-klk-
0.44 0.42 0.4
Q
0.38 0.36 0.34 0.32 0.3 0.7
0.75
0.8
0.85
0.9
0.95
J
Fig. 14 (KQ).
Numerical/Experimental results of the torque coefficient
KT and KQ, respectively. Detailed values of the coefficients and of the relative error percentage are also shown in Table 6. It may be concluded from Figs. 13 and 14 and Table 6 that; the threshold for the occurrence of the effect of transient flow – for this propeller and associated flow conditions- is at an advance ratio of approx. J = 0.80. The agreement between predicted propeller characteristics, based on both turbulence models, and experiments is remarkably good for both, the KT and KQ coefficients. While –for both turbulence models- values of KT are slightly under-predicted at all advance ratios, values of KQ are slightly over-predicted; Table 6. At higher advance ratios (J), the ‘K-Kl-x’ transition-sensitive turbulence gives a slight advantage over the ‘k-e’ fully turbulent model when compared to experiments while at lower advance ratios, the advantage is
Table 6
reversed. This agrees well with conclusions drawn previously from Fig. 12 and Table 5. 11. Conclusions This paper proposes a complete and detailed CFD procedure for simulating the viscous, three-dimensional cavitating flow about marine propellers. Subsequently, the cavitation patterns, extents and the performance characteristics are predicted. This procedure is developed taking into account the possibility of existence of laminar and transient flow regimes over the propeller. The ‘K-Kl-x’ transition-sensitive turbulence model is selected for this purpose. The fully turbulent standard ‘k-e’ model is also employed for comparison. The investigation is carried out for the ‘INSEAN E779A’ propeller model using ‘ANSYS FLUENT 16’. The multiphase RANSE technique based on Multiple Reference Frame approach and full cavitation model is applied. The cavitation patterns and extents as well as the performance characteristics are predicted with results being validated by comparison with experimental data available in the literature. For the propeller under consideration, Reynolds number distribution against various blade radius ratios demonstrates the existence of laminar and transient flow regimes over the blade with decreasing rotational speeds. In this paper, the advance speed was varied -in a way- throughout all calculations to ensure an advance ratio that varies inversely with the rotational speed. One of the important issues investigated in this paper is evaluating the two commonly used criteria for defining the cavitation area. For the propeller under investigation and by comparison with available experiments, it is revealed that; the iso-surface of vapor volume fraction (a = const.) criterion is much more reliable in defining the cavitation area than the iso-surface of (cp = r) criterion. Subsequently, all predictions obtained in this paper are based on the former. Predictions of cavitation patterns, based on both turbulence models, are in good agreement with corresponding experiments. Both predictions and experiments exhibit similar cavitation areas on the blade surface in terms of location, shape and extent. Moreover, predictions have shown that cavitation inception is more likely to occur over the back of the blade section (suction side) and close to the leading edge. This agrees well with what is documented in the literature. As for the suitability of the two selected turbulence models, conclusions drawn agree with what is expected that is; while the ‘K-Kl-x’ transition-sensitive turbulence model is more suitable for low rotational speeds, the fully turbulent standard ‘k-e’ model is more suitable for high ones.
Numerical/Experimental results of the thrust (KT) and torque (KQ) coefficients.
J
r
R. error (%) for KT k-kl-x
k-e
k-kl-x
k-e
0.9 0.83 0.77 0.71
4.455 2.063 1.783 1.763
2.4 3.9 5.7 7.6
8.7 6.4 3.3 2.9
+1.5 +2.2 +2.8 +3.2
+3.8 +2.6 +1.9 +1.7
R. error (%) for KQ
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Numerical prediction of sheet cavitation on marine propellers
11
Obtained results of the cavitating performance characteristics (KT, KQ) of the propeller, based on both turbulence models, are compared with corresponding experimental results. Agreement between predictions and experiments is quite remarkable for all characteristics. Once again, while the ‘KKl-x’ transition-sensitive turbulence model gives an advantage over the ‘k-e’ fully turbulent model at low rotational speeds, the advantage is reversed at high rotational speeds. Overall, the proposed CFD simulation procedure based on the selected transition-sensitive turbulence model has proved to be an efficient and reliable tool in predicting the cavitation patterns and extents as well as the performance characteristics –under cavitation- of the marine propeller under consideration. This is particularly apparent at low Reynolds numbers i.e. low rotational speeds which correspond to high advance ratios in this case study. The CPU time/processor/run required for simulations, based on the transition sensitive turbulence model, is very close to that required for simulations based on the standard fully turbulent model. This concludes that the processing time may be eliminated as a factor when evaluating the feasibility of which turbulence model to opt for. The sole factor in deciding on the appropriate model should only be the possible existence of laminar and transient flow regimes over sections of the blade.
Symposium on Cavitation (CAV2001); 2001 June; California, USA, 2001. M. Maksoud, Numerical and experimental study of cavitation behaviour of a propeller, STG Sprechtag Kavitation; Hamburg, Germany, 2003. T. Watanabe, Kawamura, Takekoshi, Maeda, S.H. Rhee, Simulation of steady and unsteady cavitation on a marine propeller using a RANS – CFD Code, in: Fifth International Symposium on Cavitation (CAV 2003); 2003 November; Osaka, Japan, 2003. F. Salvatore, L. Greco, D. Calcagni, Computational analysis of marine propeller performance and cavitation by using an inviscid-flow BEM model, in: Second International Symposium on Marine Propulsors; 2011. September; Hamburg, Germany, 2011. Z. Zhu, S. Fang, Numerical investigation of cavitation performance of ship propellers, J. Hydrodynam. 24 (3) (2012) 347–353. F. Pereira, F. Di Felice, F. Salvatore, Measurement and modeling of propeller cavitation in uniform inflow, J. Fluids Eng. 126 (4) (2004) 671–679. F. Pereira, A. Sequeira, Propeller-flow predictions using turbulent vorticity-confinement, in: European Conference on Computational Fluid Dynamics (ECCOMAS CFD); Lisbon, Portugal, 2010. D.K. Walter, D. Cokljat, A three-equation eddy-viscosity model for Reynolds-averaged Navier-Stokes simulations of transitional flow, J. Fluids Eng. 130 (12) (2008) 121401. ANSYS Inc., ANSYS Fluent Theory guide, Release 15.0, 2013. F. Salvatore, H. Streckwall, T. Terwisga, Propeller cavitation modelling by CFD-results from the VIRTUE 2008, in: Proceedings of the 1st International Symposium on Cavitation (SMP ’09); 2009 June; Trondheim, Norway, 2009, 362–371. A.K. Singhal, M.M. Athavale, H. Li, Y. Jiang, Mathematical basis and validation of the full cavitation model, J. Fluids Eng. 124 (3) (2002) 617–624. F. Pereira, F. Di Felice, F. Salvatore, Numerical investigation of the cavitation pattern on a marine propeller: validation vs. experiments, in: Proceedings of the 23rd ITTC; Rome, Italy. Group discussion A.2, accuracy of CFD predictions, 2002. M. Morgut, E. Nobile, Comparison of hexa-structured and hybrid-unstructured meshing approaches for numerical prediction of the flow around marine propellers, in: First International Symposium on Marine Propulsors; 2009 June; Trondheim, Norway, 2009. Guilmineau, Deng, Leroyer, et al. Influence of the turbulence closures for the wake prediction of a marine propeller. In: Fourth International Symposium on Marine Propulsors (smp’15), Texas, USA, June 2015. Da-Qing, Validation of RANS Predictions of open water performance of highly skewed propeller with experiments, in: Conference of Global Chinese Scholars on Hydrodynamics, Shanghai, China, 2006. D.P. De Witt, F.P. Incropera, Fundamentals of Heat and Mass Transfer, third ed., Willy, New York, 1990.
References [1] J.P. Franc, J.M. Michel, Fundamentals of Cavitation, Kluwer Academic Press, Dordrecht, 2004. [2] J. Blazek, Computational Fluid Dynamics: Principles and Applications, second ed., Elsevier, New York, 2005. [3] H.K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics: the Finite Volume Method, second ed., Pearson Education Limited, London, 2007. [4] F. Salvatore, C. Testa, S. Ianniello, F. Pereira, Theoretical modeling of unsteady cavitation and induced noise (INSEAN, Italian Ship Model Basin, Rome, Italy), in: Sixth International Symposium on Cavitation (CAV2006); Wageningen, Netherlands, 2006. [5] J. Dang, Numerical Simulation of Unsteady Partial Cavity Flows [doctoral dissertation], Delft University, Delft, Netherlands, 2001. [6] G. Vaz, Modeling of Sheet Cavitation on Hydrofoils and Marine Propellers Using Boundary Element Methods [doctoral dissertation], Universidad Te´cnica de Lisboa, Lisbon, Portugal, 2005. [7] J.S. Uhlman, The surface singularity method applied to partially cavitating hydrofoils, J. Ship Res. 31 (2) (1987) 107–124. [8] C.S. Lee, Kim, J.T. Lee, A potential based panel method for the analysis of two-dimensional super or partially cavitating hydrofoil, J. Ship Res. 36 (2) (1992). [9] S.A. Kinnas, Young, Numerical modelling of super cavitating and surface-piercing propeller flows, in: Fourth International
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[20]
[21]
[22]
[23]
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Please cite this article in press as: M.M. Helal et al., Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.03.008
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transitionsensitive turbulence model Mohamed M. Helal a,*, Tamer M. Ahmed b, Adel A. Banawan b, Mohamed A. Kotb c a
VSE Corporation, Alexandria, Egypt Dept. of Naval Architecture and Marine Engineering, Faculty of Engineering, Alexandria University, Alexandria, Egypt c Arab Academy for Science, Technology and Maritime Transport, Alexandria, Egypt b
Received 3 January 2018; revised 18 February 2018; accepted 12 March 2018
KEYWORDS CFD simulations; Marine propellers; Cavitation; Multi-phase flow; Turbulence models; Transition-sensitive models
Abstract One of the big challenges, yet to be addressed, in the numerical simulation of cavitating flow on marine propellers is; the existence of laminar and turbulence transition flows over the propeller’s blades. The majority of previous studies employed turbulence models that were only appropriate for fully turbulent flows. These models mostly caused high discrepancies between numerical predictions and experimental measurements especially at low rotational speeds where, Reynolds number decreases and laminar and transient flows exist. The present paper proposes a complete and detailed procedure for the CFD simulation of cavitating flow on marine propellers using the ‘K-Kl-x’ transition-sensitive model. Results are obtained using ‘ANSYS FLUENT 16’. The propeller under consideration is the ‘INSEAN E779A’ propeller model. The fully turbulent standard ‘k-e’ model is also adopted for comparison. Obtained results, based on both turbulence models, are validated by comparison with experimental data available in the literature. Predictions based on the ‘K-Kl-x’ transition-sensitive model are found to be in better agreement with experiments at lower rotational speeds i.e. at low Reynolds numbers. Ó 2018 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Cavitation is a widespread phenomenon that might exist in various marine applications. Prediction of cavitation, especially for marine propellers, is very important to avoid –or possibly eliminate– its adverse effects. Franc and Michel [1] * Corresponding author. E-mail address: [email protected] (M.M. Helal). Peer review under responsibility of Faculty of Engineering, Alexandria University.
defined cavitation as: ‘‘the evaporation of a liquid when its pressure decreases below its saturation pressure”. Applying this principle to marine propellers means that; when a propeller runs in water at high rotational speeds, the water’s local pressure decreases. This decrease in pressure is proportional to the local flow velocity squared. If the pressure of water drops below the vapor formation pressure corresponding to the flow’s temperature, bubbles of vapor (cavities) form. Subsequently, these cavities collapse with an explosive manner resulting in severe pitting to the blade surface. If the cavitation extent is sufficiently large, adverse effects will take place such
https://doi.org/10.1016/j.aej.2018.03.008 1110-0168 Ó 2018 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: M.M. Helal et al., Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.03.008
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Nomenclature cp D J k K Kl KQ KT L n p p1 pv Q r R Re T U V VA y+ y
pressure coefficient diameter (m) advance ratio turbulent kinetic energy per unit mass (m2/s2) turbulent kinetic energy (J) laminar kinetic energy (J) torque coefficient thrust coefficient length (m) rotational speed (rps) local pressure (Pa) free stream pressure (Pa) vapor pressure (Pa) torque (N m) radius (position) of any blade section (m) radius of the propeller (m) Reynolds number thrust force (N) incident velocity (m/s) total velocity (m/s) advance velocity (m/s) non-dimensional normal distance from the wall normal distance from the wall (m)
Greek symbols a vapor volume fraction e dissipation rate (W)
as; a decrease in produced thrust, an increase in required torque, damage to the propeller’s material (erosion), strong vibrations and, noise. Using photography–via– experiments to capture cavitation patterns and extents over marine propellers’ blades is a very complex task. Experiments are, in general, expensive to setup and time consuming to conduct. Numerical predictions of cavitating flows over marine propellers’ blades using CFD simulations have become a good alternative to experiments and subsequently, have become a trending research topic. Researchers nowadays may harvest the great advancement of computer performance that has made these simulations possible; refer to Blazek [2] as well as Versteeg and Malalasekera [3]. Previously, flow about propellers had been predicted using the lifting-line theory with, a vortex line representing the propeller’s blade and helicoidal vortices representing the propeller’s wake. Numerical models developed swiftly starting from the 1960s. Salvatore et al. [4] utilized the perturbation methods to demonstrate the lifting-line theory. The lifting surface model was developed later on. Dang [5] and Vaz [6] provided the possibility for boundary element methods (BEM), also referred to as the boundary integral or panel methods, to be considered for simulating the flow about twodimensional geometrically complex bodies. The application of BEM was first utilized for partially cavitating flows of two-dimensional foils by Uhlman [7] and then by Lee et al. [8]. Kinnas and Young [9] introduced the ‘PROPCAV’ method employing BEM for the numerical prediction of fully sub-
g l t q r swall x
efficiency viscosity (Pa s) kinematic fluid viscosity (m2 s1) density (kg/m3) cavitation number shear stress at the wall (N/m2) inverse turbulent time scale (s1)
Subscribts m L V Charact.
mixture liquid vapor characteristic
Abbreviations CFD computational fluid dynamics INSEAN Istituto nazionale per studi ed esperienze Di architettura navale (Research Institute active in the field of naval architecture and marine engineering within the frame of the National Research Council of Italy). PROPCAV propeller cavitation research BEM boundary element method RANS Reynolds-averaged Navier-Stokes equation MRF multiple reference frame
merged super-cavitating propellers along with experimental measurements. Viscous flow methods for predicting cavitating flows around two-dimensional foils started getting in use since the 1990s. Numerical cavitation methods based on Reynolds Averaged Navier-Stokes (RANS) models have been developed to deal with viscous flows and, are currently still an active topic of research. The first results obtained through RANS simulation of cavitating marine propellers were obtained by Maksoud [10] and watanabe et al. [11]. Salvatore et al. [12] carried out computational predictions using the ‘INSEAN’ propeller flow code. In addition, experiments were carried out to validate the predicted cavitation under uniform flow. Zhu and Fang [13] investigated the performance characteristics of propellers under cavitation using viscous multiphase flow methods based on Navier-Stokes and bubble dynamics equations. Pereira et al. [14] ran an experimental and theoretical study of a cavitating propeller in uniform inflow. Advanced imaging techniques were utilised to obtain quantitative data on the extent of the cavity. Pereira and Sequeira [15] developed a turbulent vorticity confinement strategy for RANS based industrial propeller flow simulations. The aim of the study was to improve the prediction of tip vortices. In general and from the presented brief literature, it is apparent that many researchers have significantly contributed to the prediction of propellers’ cavitating flow using CFD simulations to address various aspects of the problem. Among several challenges, not yet fully investigated is; the existence
Please cite this article in press as: M.M. Helal et al., Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.03.008
Numerical prediction of sheet cavitation on marine propellers of laminar and turbulence transition flows over the propellers’ blades especially, at low rotational speeds i.e. at low Reynolds numbers. It is necessary to take into account the transient flow problem by employing transition-sensitive models and, to avoid the assumption that the flow is fully turbulent over the whole propeller. In this paper, the detailed procedure of a CFD simulation is proposed for solving the threedimensional, viscous, cavitating flow of a marine propeller. Subsequently, the cavitation patterns, extents and, the propeller’s performance characteristics under cavitation are predicted. Additionally, the paper investigates the influence of the existence of transition flow on the predictions. This is achieved by using two different turbulence models, a transition-sensitive model and, a fully turbulent model, namely, the ‘K-Kl-x’ transition model [16,17] and the standard ‘k-e’ model [17], respectively. The CFD software ‘ANSYS FLUENT 160 is used to perform the simulations where, both turbulence models are implemented. The investigation is carried out for the ‘INSEAN E779A’ propeller model. Results from the two turbulence models are assessed against experimental data available in the literature [14,18]. 2. Theoretical analysis and numerical methods 2.1. Performance characteristics of marine propellers The performance of propellers is traditionally represented in terms of; the thrust coefficient KT, the torque coefficient KQ and the efficiency g, together with, their variation with the advance coefficient J. These parameters are introduced as follows: KT ¼
T ; qn2 D4
KQ ¼
Q ; qn2 D5
g¼
J KT ; 2p KQ
J¼
VA nD
2.2. Cavitation calculations The onset of the process of cavitation is usually termed ‘cavitation inception’. The cavitation number r and the pressure coefficient cp may be defined as given below: p pv p p1 ; cp ¼ 1 r¼11 2 qðnDÞ qðnDÞ2 2 2 The traditional cavitation criterion based upon r and cp can be given as: If r 6 cp , cavitation occurs. 2.3. Multi-phase RANS method The cavitating flow is a mixture of two phases (vapor and liquid) that are simulated as one phase. Both phases share the same velocity and pressure fields. Assuming that the mixture is homogeneous, the multi-phase flow can be solved with conventional RANS equations after some modifications. The continuity equation of the mixture flow is given as: @ ! ðq Þ þ r ðqm V m Þ ¼ 0 @t m And the momentum equation can be expressed as:
ð1Þ
3 @ ! ! ! ! ðq V m Þ þ r ðq V m V m Þ ¼ rp þ r ½lm ðr V m @t ! þ rVT Þ þ q ! g þF m
m
ð2Þ
Eq. (2) represents the multi-phase RANS equation where, the left hand side represents changes in the mean momentum of a fluid element due to unsteadiness in the mean flow and is attributed to convection by the mean flow. These changes are balanced by the mean body force. As known from conventional RANS equations for single-phase flow and for the apparent stress to close, additional modeling is required. This has led to the development of many different turbulence models. A turbulence model is a tool for specifying the Reynolds stresses hence, closing the mean-flow equations. For multi-phase flows, the density constitution of each phase in a fluid cell is represented by a scalar volume fraction as follows: qm ¼ aqv þ ð1 aÞqL
ð3Þ
In a similar manner lm ¼ alv þ ð1 aÞlL
ð4Þ
Finally and to solve the system of the governing equations, turbulence closures are used. As clarified earlier, the transitionsensitive ‘K-Kl-x’ and the standard ‘k-e’ turbulence models are employed in this paper for this purpose. The ‘k-e’ model [17] deals with mechanisms that influence the turbulent kinetic energy. It is a two-equation model which means; it solves two additional partial differential equations for k and e in order to specify the Reynolds stress. The ‘k-e’ model is employed in the simulation presented here within as it converges relatively easily and, it has been reported suitable for simulating various turbulent flows. On the other hand, the ‘K-Kl-x’ transition model [16,17] is used to predict the boundary layer development and transition onset. This model is employed in the simulation presented here within as it addresses –effectively- the transition of the boundary layer from laminar to turbulent regime specially, in rotating flows. The model is a three-equation, eddy-viscosity type of model which includes transport equations for K, Kl and x. One unknown variable is yet to be identified in Eqs. (3) and (4), namely, the vapor volume fraction, a, which is obtained using the mass transfer (cavitation) model. 2.4. Mass transfer models Mass transfer models idealise the interphase mass transfer rate due to cavitation. Several mass transfer models have been proposed by researchers. In the current study, the ‘full cavitation model’ which was developed by Singhal et al. [19] and is available in ‘ANSYS FLUENT 160 , is employed. 3. Geometric modeling The propeller selected as a test case for this paper is the ‘INSEAN E779A’ propeller model which is a four bladed, fixed-pitch, right-handed propeller. Geometric modeling of this propeller is obtained using the ‘HYDROCOMP PROPCAD’ software as shown in Fig. 1. All geometrical data of this propeller is available in the literature [20] as summarized in Table 1.
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M.M. Helal et al. 5. Grid generation
Fig. 1 The ‘INSEAN E779A’ propeller model developed in ‘PROPCAD’.
Table 1 model.
Geometric details of the ‘INSEAN E779A’ Propeller
E779a Propeller model Propeller Diameter Number of blades Pitch ratio Skew angle at blade tip Rake (nominal) Expanded area ratio Hub diameter (at prop. Ref. line) Hub length
227.27 mm 4 1.1 4°480 (positive) 4° 350 (forward) 0.689 45.53 mm 68.30 mm
4. Computational domain For the simulation under consideration, the computational domain is taken to be a cylinder as shown in Fig. 2. While the inlet is placed at a distance of 4D upstream from the propeller plane, the outlet is placed at a distance of 5D downstream from the propeller plane. In the radial direction, the domain radius is extended up to a distance of 2.5D from the axis of the hub.
Fig. 2
Design of the computational domain.
To generate an unstructured grid with hybrid cells, ‘ANSYS FLUENT 160 is used for grid generation. As concluded by Morgut and Nobile [21], this type of meshing is preferred since; hybrid meshes give higher accuracy for numerical prediction of propulsive performance. In addition, hybrid meshes need less effort in generation as compared to hexa-structured meshing. The mesh is generated taking into account that; the size of cells near the blade wall is small whereas, size increases towards the outer boundary. Fig. 3 shows size of the grid cells over the outer surface of the domain while Fig. 4 clarifies the size of hybrid cells inside the domain. After convergence, the total number of cells generated over the entire grid is 238,387. To obtain a suitable resolution of the viscous sublayer, the wall normal resolution over all surfaces of the blades is within a range of: y+ < 0.5 where, y+ is defined as: qffiffiffiffi sw y q yþ ¼ t 6. Boundary conditions The numerical simulation is performed over a range of advance coefficients and cavitation numbers equivalent to the range at which the propeller was experimentally tested [14,18]. Values assigned to the advance coefficient and cavitation number are as follows:
J = 0.9, r = 4.455 J = 0.83, r = 2.063 J = 0.77, r = 1.783 J = 0.71, r = 1.763
To obtain these values, various propeller rotational speeds of (1500 rpm, 1800 rpm, 2400 rpm and 3000 rpm) are used. The advance velocity values prescribed at the domain inlet are calculated and are as listed in Table 2. The free stream pressure p1 used in each run is calculated based on the corresponding value of r. In this study, it is assumed that Poutlet = p1. All boundary conditions imposed are shown in
Fig. 3 Grid over the domain surface developed in ‘ANSYSFLUENT’.
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Numerical prediction of sheet cavitation on marine propellers
5 For unsteady simulations, a second order implicit scheme is applied for time derivatives. The selected time step at each run corresponds to a rotation of 1°. The simulation is carried out by nine processors (Intel Ò Core(TM)2 Duo CPU E8400@ 3 GHZ). Table 3 shows details of the simulation settings. 8. Frame of reference
Fig. 4 Longitudinal section of the computational domain showing the size of cells.
The frame of reference of the ‘INSEAN E779A’ propeller model adopted in this study is shown in Fig. 6 where, the ‘xaxis’ points downstream, the ‘y-axis’ is directed upwards and, the ‘z-axis’ follows the right hand rule in the ‘x-y’ plane. Viewing form aft., the direction of rotation is clockwise as the propeller is a right-handed one. 9. Reynolds number distribution over the blades
Table 2
Flow parameters of each run.
n (rps)
VA (m/s)
J
r
25 30 40 50
5.1 5.65 6.99 8
0.9 0.83 0.77 0.71
4.455 2.063 1.783 1.763
Fig. 5. The continuum is chosen to be fluid with properties of water assigned to it. The propeller rotation is simulated using the Multiple Reference Frame (MRF) approach [17]. For the stationary region (the outer far field wall), the governing equations are solved in a fixed frame of reference while for all rotating regions (blade, hub and fluid zone), the governing equations are solved in a rotating frame of reference. A rotating frame of reference is assigned to the fluid at the selected absolute speeds given earlier in Table 2. A relative rotational speed of zero –with respect to adjacent cell zonesis prescribed to the walls of the propeller’s blade and hub. The far field boundary is treated as an inviscid wall to which, an absolute rotational speed of zero is prescribed. The four blades of the propeller are set at a regular angular interval of 90° hence, the modeling of one angular sector of 90° and one blade will suffice in solving the entire flow domain as shown previously in Fig. 5. The fact that other blades are present is taken into account by imposing periodic boundary conditions on the two sides of the blade. On these periodic boundaries, rotational periodicity is ensured.
Reynolds number varies along each blade with variations in the rotational speed and at every radius ratio r/R. Subsequently, and along the blade, the flow could change from laminar to transient to fully turbulent as Reynolds number values increase. Reynolds number Re is defined as: Re ¼
Ucharact Lcharact tfluid
For a propeller blade, Reynolds number may be estimated based on several characteristic dimensions, refer to literature [22] and [23]. In this study, it has been opted for a Reynolds number that is based on (Lcharct = r) and characteristic velocity (Ucharct = 2npr) at each radius ratio, i.e. at each r/R. The variation of Re with n for various blade sections (r/R = 0.3, 0.5, 0.7 and 1) is presented in Table 4 and Fig. 7. According to De Witt [24], the threshold for the turbulent region is at Re = 0.5 106. It can be noted from Table 4 and Fig. 7 that the flow becomes fully turbulent as the rotational speed increases and beyond a radius ratio of r/R = 0.5. From that, the standard ‘k-e’ turbulence model is expected to be more accurate for the higher rotational speeds at (r/R ˃ 0.5) since the flow is fully turbulent. On the other hand and for the range of radius ratios (r/R < 0.5), transient flow takes place and laminar flow exists specifically as the rotational speed decreases i.e. at low Re. Over this range, the ‘K-Kl-x’ transition model is expected to be well suited. 10. Results and discussion 10.1. Prediction of cavitation patterns and extents
7. Solution and solver settings The commercially available CFD software, ‘ANSYS FLUENT 160 is adopted to solve the three- dimensional viscous, unsteady, cavitating problem. The equations are solved as follows: The pressure and velocity components are solved using coupled technique. The second order ‘QUICK’ scheme is applied for convection terms in all transport equations.
In this section, predictions of the pressure distribution, based on the ‘k-kl-x’ transition-sensitive turbulence model, is presented. This distribution plays a significant role in predicting the cavitation inception. Later on, cavitation patterns -at the selected conditions in Table 2 will be shown based on both turbulence models. As mentioned earlier, the relevant experimental data of the cavitating flow for this propeller model can be found in the literature, Pereira et al. [14] and Salvatore et al. [18]. It covers the following (for various advance coefficients and cavitation numbers):
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Fig. 5
Table 3
Boundary Conditions: (a) 3D view. (b) Front view.
Solver settings.
Solver settings Pressure link Discretization scheme for convective fluxes and turbulence parameters Turbulence models
Near wall treatment Phases Solver Absolute Vapor pressure (KPa) CPU time/processor/run
SIMPLE QUICK 1. Standard (kЄ) 2. Transition (K-Kl-x) Standard wall functions 1. Water 2. Water vapor Unsteady 2.337 1–28 h (k-Є) 2–30.2 h (K-Klx)
Fig. 6
Frame of reference.
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Numerical prediction of sheet cavitation on marine propellers
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Table 4 Variation of Reynolds number (Re) with rotational speed (n) at various blade sections. r/R
n (rps)
Re 106
1
25 30 40 50 25 30 40 50 25 30 40 50 25 30 40 50
2 2.4 3.2 4 0.99 1.1 1.5 1.98 0.5 0.6 0.8 1 0.16 0.19 0.25 0.3
0.7
0.5
0.3
Fig. 8 Pressure distribution on the face and back sides of a blade section. r/R = 0.7, n = 50 rps and H = 135°.
Fig. 7 Variation of Reynolds number (Re) with rotational speed (n) at various blade sections.
Photographs of cavitation patterns. Measurements of cavitation extents. Measurements of performance characteristics. At the end of this section, predictions of the values of cavitation extents, based on both turbulence models, will be validated by comparison with corresponding experimental measurements. Figs. 8–10 show variations of the negative pressure coefficient (cp) with the position ratio (x/c) over the face and back sides of a blade section at (r/R = 0.7) and (n = 50rps) for (H = 135°), (H = 180°) and (H = 225°), respectively. By comparing the three figures it can be deduced that; the peak value of the negative pressure coefficient (cp) increases with a decrease in the hydrostatic pressure, i.e. a higher risk of cavitation. Subsequently, the probability of cavitation inception decreases as the blade rotates from H = 90° to H = 270° and then increases again as the blade rotates from H = 270°
Fig. 9 Pressure distribution on the face and back sides of a blade section. r/R = 0.7, n = 50 rps and H = 180°.
to H = 90°. Moreover, it is clearly observed that cavitation inception is more likely to occur over the back of the blade section (suction side) and close to the leading edge. An important issue that has been investigated in the past is; how to define the cavitation area [18], with two possible answers; either as an iso-surface of (cp = r) or based on the iso-surface of vapor volume fraction (a = const). This point is addressed in this study where, the cavitation area extent over the ‘E779A’ propeller blade is obtained based on
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Fig. 10 Pressure distribution on the face and back sides of a blade section. r/R = 0.7, n = 50 rps and H = 225°.
both cavitation criteria; (a = const) and (cp = r) and, is assessed against photographs obtained experimentally to judge which of these criteria is more accurate in defining the cavitation area. Fig. 11 shows the predicted distribution of the pressure coefficient over the suction side of the propeller blade, based on the ‘K-Kl-x’ transition model, at (r = 1.76, J = 0.71, n = 50 rps and H = 90°). It is clear that large discrepancies exist between the numerically predicted and the experimentally obtained cavitation areas, at (cp = 1.76) [14,18]. In particular, the predicted and experimental cavitation areas don’t exhibit the same extent, shape or location on the blade surface.
M.M. Helal et al. This indicates that the cavitation criterion of (cp = r) is not reliable in defining cavitation. On the other hand, a variation of the vapor volume fraction a over the suction side of the propeller blade at the selected parameters is obtained based on both turbulence models. By comparing results predicted at various values of a with experiments, it is found that a value of (a = 0.5) led to the best cavitation predictions in terms of location, shape and extent. Subsequently, it is decided that cavitation predictions will be carried out based on the criterion of iso-suface of vapour volume fraction at (a = 0.5). Fig. 12 shows a comparison of predictions of cavitation patterns over the suction side of a blade, based on both turbulence models and at (a = 0.5), against corresponding patterns from photographs obtained experimentally. Comparisons reveal that the predicted cavitating regions, at the four operating conditions and based on both turbulence models, are in good agreement with those in the corresponding photographs [14,18]. Both predictions and photographs exhibit quite similar cavitation areas on the blade surface in terms of location, shape and extent. This demonstrates the correctness of the proposed simulation procedure as well as the validity of selection of the iso-surface of vapor volume fraction criterion and its assigned value. In addition, the cavitation regions are all found to be close to the leading edge of the suction side of the blade which emphasizes the previously drawn observations from Figs. 8– 10. An overall comparison of predictions and experiments shown in Fig. 12 reveals that; the ‘K-Kl-x’ transitionsensitive turbulence model is a better suited choice for high values of the advance ratio (J) while the ‘k-e’ fully turbulent standard model is a better suited choice for low values of (J). This is as expected since in this study, flow parameters were selected such that; a higher advance ratio means a lower rotational speed hence, less possibility for full turbulence to occur. Generally speaking and when compared to experiments, the extents of cavitation are over predicted -using both turbulence modelsand at all advance ratios.
Fig. 11 Cavitation area over the blade suction side at (J = 0.71, n = 50 rps, r = 1.76 and H = 90°): (a) Experimental photograph of the cavitation area (b) Predictions based on the ‘k-kl-x’ transition model and defined by (cp = r = 1.76).
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Numerical prediction of sheet cavitation on marine propellers
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Exp. J = 0.71, = 1.763
k- . J = 0.71, = 1.763
K-Kl- . J = 0.71, = 1.763
Exp. J = 0.77, = 1.783
k- . J = 0.77, = 1.783
K-Kl- .. J = 0.77, = 1.783
Exp. J = 0.83, = 2.063
k- . J = 0.83, = 2.063
K-Kl- .. J = 0.83, = 2.063
Exp. J = 0.9, = 4.455
k- . J = 0.9, = 4.455
K-Kl- .. J = 0.9, = 4.455
Fig. 12 Comparison of cavitation patterns, over the suction side of the blade, between experiments and predictions. Results are based on the ‘k-e’ and the ‘k-kl-x‘ turbulence models for the same operational conditions and at a = 0.5.
Table 5 Relative error of the cavitation extent based on the ‘k-e’ and ‘k-kl-x’ turbulence models. J
0.9 0.83 0.77 0.71
r
4.455 2.063 1.783 1.763
Relative error (%) k-kl-x
k-e
+10 +11.6 +17.5 +24.7
+22.5 +20 +14.2 +9.4
Table 5 gives a detailed account of the relative error percentage of the predicted extent of cavitation i.e. predicted area of cavitation. This relative error may be calculated as:
R: Errorð%Þ ¼
CFD results Exp: results 100 Exp: results
10.2. Prediction of performance characteristics under cavitation conditions In this section, thrust and torque are obtained from numerical simulations, based on both turbulence models and, at various rotational speeds of the propeller as shown earlier in Table 2. Subsequently, the non-dimensional coefficients of performance (KT and KQ) are calculated for each value of J. A comparison of the numerically estimated coefficients, based on both turbulence models, against experimental measurements is carried out and plotted in Figs. 13 and 14 for
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M.M. Helal et al. 0.28
Exp.data k-kl-ω k-ε
0.26 0.24 0.22 KT 0.2 0.18 0.16 0.14 0.12 0.1 0.7
0.75
0.8
0.85
0.9
0.95
J
Numerical/Experimental results of the thrust coefficient
Fig. 13 (KT).
0.48 0.46
k-klk-
0.44 0.42 0.4
Q
0.38 0.36 0.34 0.32 0.3 0.7
0.75
0.8
0.85
0.9
0.95
J
Fig. 14 (KQ).
Numerical/Experimental results of the torque coefficient
KT and KQ, respectively. Detailed values of the coefficients and of the relative error percentage are also shown in Table 6. It may be concluded from Figs. 13 and 14 and Table 6 that; the threshold for the occurrence of the effect of transient flow – for this propeller and associated flow conditions- is at an advance ratio of approx. J = 0.80. The agreement between predicted propeller characteristics, based on both turbulence models, and experiments is remarkably good for both, the KT and KQ coefficients. While –for both turbulence models- values of KT are slightly under-predicted at all advance ratios, values of KQ are slightly over-predicted; Table 6. At higher advance ratios (J), the ‘K-Kl-x’ transition-sensitive turbulence gives a slight advantage over the ‘k-e’ fully turbulent model when compared to experiments while at lower advance ratios, the advantage is
Table 6
reversed. This agrees well with conclusions drawn previously from Fig. 12 and Table 5. 11. Conclusions This paper proposes a complete and detailed CFD procedure for simulating the viscous, three-dimensional cavitating flow about marine propellers. Subsequently, the cavitation patterns, extents and the performance characteristics are predicted. This procedure is developed taking into account the possibility of existence of laminar and transient flow regimes over the propeller. The ‘K-Kl-x’ transition-sensitive turbulence model is selected for this purpose. The fully turbulent standard ‘k-e’ model is also employed for comparison. The investigation is carried out for the ‘INSEAN E779A’ propeller model using ‘ANSYS FLUENT 16’. The multiphase RANSE technique based on Multiple Reference Frame approach and full cavitation model is applied. The cavitation patterns and extents as well as the performance characteristics are predicted with results being validated by comparison with experimental data available in the literature. For the propeller under consideration, Reynolds number distribution against various blade radius ratios demonstrates the existence of laminar and transient flow regimes over the blade with decreasing rotational speeds. In this paper, the advance speed was varied -in a way- throughout all calculations to ensure an advance ratio that varies inversely with the rotational speed. One of the important issues investigated in this paper is evaluating the two commonly used criteria for defining the cavitation area. For the propeller under investigation and by comparison with available experiments, it is revealed that; the iso-surface of vapor volume fraction (a = const.) criterion is much more reliable in defining the cavitation area than the iso-surface of (cp = r) criterion. Subsequently, all predictions obtained in this paper are based on the former. Predictions of cavitation patterns, based on both turbulence models, are in good agreement with corresponding experiments. Both predictions and experiments exhibit similar cavitation areas on the blade surface in terms of location, shape and extent. Moreover, predictions have shown that cavitation inception is more likely to occur over the back of the blade section (suction side) and close to the leading edge. This agrees well with what is documented in the literature. As for the suitability of the two selected turbulence models, conclusions drawn agree with what is expected that is; while the ‘K-Kl-x’ transition-sensitive turbulence model is more suitable for low rotational speeds, the fully turbulent standard ‘k-e’ model is more suitable for high ones.
Numerical/Experimental results of the thrust (KT) and torque (KQ) coefficients.
J
r
R. error (%) for KT k-kl-x
k-e
k-kl-x
k-e
0.9 0.83 0.77 0.71
4.455 2.063 1.783 1.763
2.4 3.9 5.7 7.6
8.7 6.4 3.3 2.9
+1.5 +2.2 +2.8 +3.2
+3.8 +2.6 +1.9 +1.7
R. error (%) for KQ
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Numerical prediction of sheet cavitation on marine propellers
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Obtained results of the cavitating performance characteristics (KT, KQ) of the propeller, based on both turbulence models, are compared with corresponding experimental results. Agreement between predictions and experiments is quite remarkable for all characteristics. Once again, while the ‘KKl-x’ transition-sensitive turbulence model gives an advantage over the ‘k-e’ fully turbulent model at low rotational speeds, the advantage is reversed at high rotational speeds. Overall, the proposed CFD simulation procedure based on the selected transition-sensitive turbulence model has proved to be an efficient and reliable tool in predicting the cavitation patterns and extents as well as the performance characteristics –under cavitation- of the marine propeller under consideration. This is particularly apparent at low Reynolds numbers i.e. low rotational speeds which correspond to high advance ratios in this case study. The CPU time/processor/run required for simulations, based on the transition sensitive turbulence model, is very close to that required for simulations based on the standard fully turbulent model. This concludes that the processing time may be eliminated as a factor when evaluating the feasibility of which turbulence model to opt for. The sole factor in deciding on the appropriate model should only be the possible existence of laminar and transient flow regimes over sections of the blade.
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Please cite this article in press as: M.M. Helal et al., Numerical prediction of sheet cavitation on marine propellers using CFD simulation with transition-sensitive turbulence model, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.03.008