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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Experimental and Numerical Investigation of the Cavitation Pattern on a Marine Propeller Francisco Pereira~, Francesco Salvatore~, Fabio Di Felice, Mauro Elefante2 Istituto NazionaTe per Studi ed Esperienze di Architettura NavaTe, 2 Centro Esperienze Idrodinamiche delIa Marina MiTitare Rome, Italy ABSTRACT An experimental investigation on a cavitating pro- peller in a uniform inflow is presented. Flow field investigations by advanced imaging techniques are used to extract quantitative information on both the cavity extension and thickness. A refined map of the propeller cavitating behavior is established. Measurements are compared to numerical results obtained using an inviscid flow boundary element method for the analysis of blade sheet partial and super-cavitation. The effect of the trailing wake vor- ticity on the prediction of the cavitation pattern is analyzed via a wake alignment technique. INTRODUCTION For two-dimensional and three-dimensional isolated hydrofoils, the cavitation literature offers a limited number of works where quantitative information has been successfully obtained from the direct observa- tion of the cavitation pattern (see e.g. Pereira et al., 1998~. Even more complicated is the measure- ment of the cavitation development on a marine pro- peller. The problem is not new, but is very little documented. Lehman (1966) seems to be the first to report such measurements, using multiple ortho- graphic views to build an estimate of the cavity vol- ume on a model propeller. Yamaguchi (1985) reports a stereo graphical technical to carry out the thickness of the cavity. Using a technique that has been much used in the case of isolated hydrofoils (Dupont and Avellan, 1991), Ukon et al. (1991 ) report measure- ments of the cavity thickness distribution. Full-scale measurements on propellers are rare because of the obvious difficulty: Tanibayashi et al. (1991) is, to the authors knowledge, the only work of this kind. In the present work, new developments are in- troduced into the observation and quantification of the cavitation pattern and are applied to a skewed four-blade model propeller in a uniform inflow. Us- in~ a novel cavitation analysis approach suitable for field applications, the area and the maximum thick- ness are estimated. A detailed chart of the cavitation figures is also determined and analyzed. A Wageningen modified type model propeller (INSEAN E779A) was selected for the present re- search project for two main reasons. First, this model propeller has been widely studied with the most advanced flow measurement and visualiza- tion techniques, such as laser Doppler velocimetry (Stella et al., 2000) and Particle Image Velocime- try (Di Felice et al., 20001. A large amount of data has been collected providing a thorough documen- tation on the non cavitating flow characteristics: the propeller geometry and LDV data are now available at http://crm.insean.it/E779A. The present work is intended to extend the existing database to cavitat- ing flow conditions. Second, the E779A model ge- ometric characteristics are very close to those used to design propellers for twin-screw ships. Further- more, the propeller performance and its cavitating
flow phenomenology are quite complex and provide a complete and challenging benchmark for the vali- dation of numerical codes. The unique data collected in the present work, covering a large number of flow conditions, provide a deep insight into different flow features such as sheet cavitation, tip and leading edge vortex cavi- tation, bubble cavitation and supercavitation. A rich cavitating flow database is being constructed, repre- senting a powerful tool to investigate in detail the range of applicability of new theoretical models and to assess the accuracy of predictions. As a mat- ter of fact, the validation of numerical predictions against experimental propeller flow data is not well addressed, even though the field is very active (Kim and Lee, 1997; Mueller and Kinnas, 1997; Dang, 2000~. Unfortunately, and because existing analyzes are usually based on experiments that cover a re- stricted set of flow conditions, only a limited num- ber of flow patterns are documented. Moreover, in- adequate data formats do not allow for a convenient exchange of information. State-of-the-art theoretical analysis of cavitating propellers is based on inviscid-flow models such as lifting surface methods and boundary element meth- ods. Viscous-how modeling by RANSE or LES ap- proaches are still limited to simple geometries as two- and three-dimensional hydrofoils in uniform flow. Propeller flow applications seem to be yet a long-term challenge. The theoretical formulation used in the present analysis is based on a well estab- lished boundary element approach that has been de- veloped by several authors over the past two decades (Lee, 1987; Kinnas and Fine, 1993~. The paper describes, in a first part, the experi- mental methodologies that are introduced to charac- terize the cavitation development on a rotating pro- peller. In a second part, a brief outline of the nu- merical approach is given. Finally, the results of the experimental investigation are used to assess the va- lidity of the numerical method. EXPERIMENTAL ANALYSIS Experimental Setup The experiments are carried out at the Italian Navy cavitation facility (C.E.I.M.M.) The tunnel, a closed type circuit, has a 0.6 m x 0.6 m x 2.6 m square test section. Optical access to the section is pos- sible through large Perspex windows. The nozzle contraction ratio is 5.96: 1 and the maximum wa- ter speed is 12 m sol. The maximum free stream turbulence intensity in the test section is 2%, and is reduced to 0.6% in the propeller blade inflow section atr/R=0.7,whereR=D/2=0.117misthepro- peller radius. The flow uniformity is within 1% for the axial component and 3% for the vertical one. The skewed four-blade model propeller has a uniform pitch-to-diameter ratio of 1.1 and a forward rake angle of 4°3". The blockage ratio in the test section is about 10%. A sketch of the propeller ge- ometry is given in Figure 1. _\ ___~___~__= __ : W~ W _ . . _i HI ~ ~ ~ ~ ~ Figure 1: E779A propeller geometry In the following sections, the advance ratio J. the cavitation number 60, the torque coefficient KQ and the thrust coefficient KT are defined respectively by vo/(nD), 60 = (PoPv)/(p vo/2), Q/(p n2 D5) and T/(p n2 D4), where po is the reference pressure measured at the propeller axis, Pv is the fluid vapor pressure, p is the fluid density, vo is the upstream velocity, and n is the propeller rotational speed (rps). The measurements described here have been per- formed across a large range of working conditions, 60 and J being the varying parameters, see Figure 2. The measurement configuration is pictured in Figure 3. The blade angular position is adjusted to allow a full view of the blade face. A CCD camera is dedicated to the measurement of the pattern area and is oriented at an angle with respect to the test section window. To minimize the aberrations intro- duced by the thick window and the water/glass/air interfaces, a glass tank in the form of a wedge and filled with water is clamped to the window in such a manner that the camera optical axis is normal to the wedge front window. The propeller shaft is equipped with a rotary encoder that supplies an electrical trig- ger signal to the image acquisition system. This lat-
o.9 0.85 0.8 0.75 0.65 + ~ 4h 4. ~ - I,_ - +*++4~+ +++ +~+ ++ + + + ~ 0.7 ++'+*+++ + ~11 1 1~ - 1- ~ + + + + + + + + + + 4. + + ~ + ~_, - + + ~ ~ ~ -4 ~ ~ + + + 4. . 0.6 0 2 4 6 Oo [ - ] 8 10 Figure 2: Hydrodynamic conditions ter controls the phase between the propeller and the image capture, and pilots a pulse delay generator ac- cordingly. This instrument sends the necessary trig- ger signals, with the appropriate time delays, to the camera and the illumination system. A single high intensity flash lamp, with a 10 Us light pulse, is used to observe the cavitation pattern. For every condition shown in Figure 2, 128 images are acquired. Simul- taneously, the flow parameters are recorded: So, J. Go, the cavitation number (Sn based on the propeller rotation speed, KT and KQ. Cavitation Extension Measurement We introduce a novel methodology to determine the cavity area, designed to be implementable in field situations where the experimental constraints do not always allow the use of standard techniques of image analysis. The common method in analyzing cavita- tion images consists in enhancing the contrast be- tween the cavitation pattern and the rest of the im- age. This is usually done by thresholding the image as used by Pham et al. (1998) for instance. How- ever, it is well known that this approach is very sen- sitive to variations of the illumination. The region of interest needs also to be easily identifiable from the background, either by running the experiment in such conditions that only the cavity scatters light or by removing a background image from the cavita- tion image. In a complex environment such as a ro- tating propeller, this approach is not robust enough: tip vortex, scattering of the blade, fluctuations of the illumination intensity, unwanted objects in the field of view, are among many other causes that make the analysis difficult or unreliable if done automatically. The approach used here, and depicted in Fig- ure 4, is based on the cross-correlation between a template image (Figure 4a) and the image under con- sideration (Figure 4b). The template image may be the blade viewed in non-cavitating conditions. The cross-correlation is a robust tool to make the compar- ison between images: a high correlation peak would indicate that the template and the image are simi- lar, while a slight difference would drop the correla- tion coefficient to low and distinct levels. However, in order to localize the transition regions between the cavitating and the non-cavitating situations, the cross-correlation is performed on small image re- gions, represented in red color in Figure 4a and Fig- ure 4b, with the resulting correlation image repre- sented in Figure 4c. The size of the correlation win- dows is set to 7 x 7 pixels in our case. The convo- lution operation is performed across the whole im- age to produce a correlation image, see Figure 4d, where the differences between the template and the image being analyzed are clearly and uniquely iden- tified as cavitation features: cavitating vortices from the blade tips, contour of the attached cavity. ~ _ ~~ ~ ~ _^A Figure 4: Image cross-correlation procedure: (a) tem- plate image; (b) cavitation pattern image; (c) local cross- correlation; (d) correlation image The correct quantitative evaluation of the cavita- tion area Ac is only possible if done in a known coor- dinate system. The registered image is a perspective representation of the three-dimensional blade sur- face. To report this image information into a plan view, where the area would be accurately measured,
Cavita~ patterned Water _ _ _ Control & acquisition 119~ Rotation pulse 1 ----------I - Camera ~ ~ Images controller Trigger signal Figure 3: Experimental setup one could perform a simple back projection using basic geometric optics, and this would correct for the variable magnification. However, this is only valid if the system is free of optical aberrations and requires an exact knowledge of the optical parame- ters. In fact, the complex lens system composed of the camera objective, the prism, the test section win- dow and the water medium introduce focusing aber- rations and optical distortions, which are the source of non-linear magnification. If the focusing aberra- tions can not be removed, image distortions can be compensated through calibration procedures (Soloff et al., 1997~. This operation, also very common in image processing techniques where it is referred to as warping transformation, is performed here. We use a second-order perspective transformation to ac- count for the non-linear effects. If x and y are the original image coordinates of a point, the warped im- age coordinates x' and y' are then defined by , a8x+aOy+aO+a3x2+aOy2+a X by x + bo Y + bo + ho x2 + bo y2 + bo xy , _ at x + ai y + a2 + a3 x2 + at y2 + aS by Y by x + by y + b2 + b3 x2 + b4 y2 + bS xy . . where at and by are the coefficients of the transform. . . al and by are unknowns that are determined by a Nelder-Mead least-square minimization method, us- ing reference points on the distorted image and the known corresponding points in the plan view. To perform this correspondence, a grid is printed on a blade, with lines drawn spanwise at various r/R, and radial lines regularly spaced chordwise, as shown on the top image in Figure 5. Instead of de- termining one unique transformation for the whole image, as commonly done, Equation (1) is calcu- lated for each polygon defined by the intersection of the radial lines and of the constant r/R lines. This approach refines the transformation and avoids the side-effects inherent to the use of non-linear polynomials in regions where calibration points are not available. Three sample distorted polygons are shown in the top image of Figure 5. Their correspon- dent warped image, determined using the locally cal- culated warping function expressed by Equation ( 1), is shown on the bottom image, which is the final undistorted image. For clarity, we show only four corresponding points for each polygon. <1~ The warping procedure is applied to the correla- tion image (Figure 4d), on which standard threshold techniques are utilized to track the cavitation pat- tern. The use of a threshold is now possible with- out the drawbacks outlined previously, for the corre- lation operation described above unequivocally en-
Figure 5: Warping procedure: original (top) and warped (bottom) images __ trances the features of interest. Morphological oper- ators may then be used to isolate the cavitation pat- tern from small and undesired features, such as trav- cling bubbles, tip vortices. This sequence of opera- lions results in a pattern image (Figure 6), where the cavitation region over the blade is clearly outlined. The undistorted and thresholded image is thus used to correctly estimate the area of the cavitation pattern, with a common coordinate system with the numerical grid used to perform the computations de- scribed in the next sections. The cavitation area Ac follows immediately and is expressed in percentage of the blade total area Ao starting at r/R = 0.3. We finally propose a rough estimate of the sheet cavitation thickness by simply measuring the thick- ness of the trailing section of the sheet or the size of the vortex structure immediately downstream the sheet. Figure 7 shows four cases. The thick- ness is measured on the average image and at the Figure 6: Cavitation extension location of minimum cross-section of the cavita- tion pattern. The measured dimension is then non- dimensionalized by the propeller diameter, for the purpose of comparison with the numerical solution of the thickness. 1__ In__ _ - _1 ma _: _ ~ 3 it_ - Figure 7: Cavitation thickness: a rough estimate THEORETICAL MODEL Governing equations Under the assumptions of inviscid and initially irro- tational fluid, the perturbation velocity field is irro- tational and hence it can be expressed in terms of a scalar potential, ¢. In a frame of reference (Oxyz) fixed to the propeller with the x-axis parallel to the
propeller axis, the unperturbed flow velocity reads Vet = VA + Q x x. (2) where VA is the incoming flow to the propeller, x = (x,y,z), and Q = (Q,0, 0) is the angular velocity of the propeller. Hence, the total velocity field is q = v, + Vo. `3y By incompressible flow assumptions, the poten- tial o satisfies the Laplace equation V20 = 0 in the unbounded fluid region Up surrounding the pro- peller, its trailing wake and the cavity. In the frame- work of potential flow modeling of bodies that are capable to generate lift or thrust, the wake denotes a zero thickness layer where the vorticity generated on the body is shed, and represents a discontinu- ity surface for the potential. The cavity denotes the fluid region where vaporization occurs. In the present approach the cavity is assumed to be a thin layer attached to the blade suction side and, if super- cavitation occurs, to the trailing wake surface. The Bernoulli equation gives the pressure p and, in the frame of reference fixed to the propeller, reads aO+ {q2+P+gzo= ~V2~+Pp°, (4) where t denotes time, q = I, Vet = v, g iS the gravity acceleration and z0 denotes depth. The Laplace equation for o is solved by impos- ing boundary conditions on &{p. On the cavitation- free portion of the propeller surface, namely the wet- ted body surface ~Y'WB, the impermeability condition yields q- n = 0, or, recalling Eq. (3) aa0 =_v,~n on3°WB' (5) where n is the outward unit normal to the surface. Across the wake surface 3°w the pressure is con- tinuous. By applying mass and momentum conser- vation laws under non cavitating conditions, yields ( an ) on amp, (6) where 3°ww is the cavitation-free wake portion, and denotes discontinuity across the two sides of the wake surface. Recalling Ap = 0 across the wake sur- face and combining the Bernoulli equation (4) and Eq. (6), one obtains that the potential discontinu- ity /\o is convected along wake streamlines, and the convection velocity is the averaged flow velocity on both sides of Law A further condition on o is required in order to assure that no finite pressure jump may exist at the body trailing edge (Kutta condition). Following Morino et al. (1975), this is equivalent to impose that the potential discontinuity at the wake trailing edge equals the difference between potentials on suction and pressure sides at the blade trailing edge A0 (XTE) = ATE OTE - (7) In order to take into proper account crossbow ef- fects, Eq. (7) is coupled with a pressure-based itera- tive Kutta condition as proposed by Kerwin (19871. Boundary conditions on the cavity edge 3°c are imposed by assuming that the cavity is a fluid region where the pressure is constant and equal to the vapor pressure Pv By imposing p = pv on Arc, and by using the Bernoulli theorem (4), results q = [(nD) on +gZO) +v,2~ , (8) where CTn = (PoPv)/ 2 p~nD)2 denotes the cavita- tion number referred to the propeller rotational speed n (rps) and diameter D. Equation (8) is used to obtain a relationship between ~ on 3°c and On. By consid- ering on each blade surface a curvilinear coordinate system (s, u) with s in chordwise and u in spanwise directions, respectively, and s,u covariant base vec- tors, one has qS = qu cos ~ + ~ sin ~ ~ N/q2 _ q2 _ q2 (9) where qS = q s, qu = q u and qn = q n. Decom- posing qS by Eq. (3), and by integrating in chordwise direction from the cavity leading edge abscissa SCIE, yields As 0(5,U) = 0(SCLE,U) + J (qS V! s)ds. (10) S CLE Equations (8) to (10) combined provide a non lin- ear boundary condition for 0 on 3°c The present derivation differs from Kinnas and Fine (1992) in that quantity qn in Eq. (9) is not neglected in calcula- tions. In the case of supercavitating flows, a similar derivation leads to an expression for the velocity po- tential on the cavitating portion of the wake surface us O (s, u) = 0 (STE'U) + J (qs Vl S) ds, (1 1) S TE
where STE iS the blade trailing edge abscissa. The condition p = Pv and hence Eq. (8) are not valid in the aft portion of the cavity where pressure tends to wetted-flow conditions through complex two-phase flow phenomena. In the present work, a cavity-closure region is introduced in which pres- sure is forced to vary smoothly from p = Pv to wet- ted flow values behind the cavity trailing edge. The fraction ~ of the cavity length occupied by the clo- sure region is prescribed within a range that has no influence on the flowfield solution (0.1 < ~ < 0.31. An expression of the cavity thickness he is ob- tained by imposing a non-penetration condition on ~c. By combining the constant-pressure and the non-penetration conditions, follows that 3°c is a material surface. Denoting by 5~CB the cavi- tating portion Of 5~B' and by ~ the normal dis- tance to ~CB, the above condition corresponds to (~/3t + q v) (llhe)' or aahC +Vshc q = at +v, on 50CB~ (12) where Vs denotes the surface gradient on JAMB- In the case of supercavitating flow conditions, Kinnas and Fine (1992) show that a similar condition applies on the cavitating portion Ago of the wake surface. Equation (12) represents a partial differential equa- tion for he that may be solved once o, Do/On and O¢/Ot are known. Boundary integral formulation The Laplace equation for the velocity potential is solved by means of a boundary integral formulation. By assuming that the perturbation vanishes at infin- ity, the third Green identity yields at any point x ~ Dip E(`x>)o(<x) = JOB (an G - ~ an ) daffy) ,/- /~oaaGd~(y), (13) A where MOB = 3°WB U 3°C U ACE, G 1 / l l Y I I is the Green's function of the Laplace equation in an unbounded three-dimensional domain, E equals 0 inside, 1/2 on, and 1 outside amp and 0 = 0u _ 2~0TE if x ~ cow whereas o = o elsewhere. Equation (13) is solved using a boundary element approach. The boundary surface and the wake are divided into hyperboloidal quadrilateral elements. Equation (13) in discretized form is enforced at each surface element centroid on JIB. Flow quantities are assumed to be constant on each element. The numer- ical solution of Equation (13) determines 0 on JAMB and a¢/an on Tic once aO/an on ~°WB iS given by Eq. (5), Ao on the wake is given by Eq. (7), whereas 0 on JAMB iS given by Eq.~10), and ou on 3°cw is given by Eq.~111. Influence coefficients in the dis- cretized form of Eq. (13) are computed by using an- alytical integration (Morino et al., 1975~. Inviscid- flow hydrodynamic loads are computed integrating the pressure over the body surface. Viscosity effects are included in an approximate fashion by evaluat- ing the friction coefficient according to the Prandtl- Schlichting friction line formula. By observing that the cavity thickness is very small compared to the propeller dimensions, we as- sumed that in the solution of Eq. ( 13), 3°c may be re- placed by BOMB U Air- This simplification prevents additional computational effort required by surface re-gridding and influence coefficients re-computing during the cavity shape updating process. The issue of the wake is crucial to accurately cap- ture trailing vorticity effects that are responsible for load generation on the propeller and affect the cavi- tation pattern. In the present approach, the wake sur- face 3°w may be either prescribed (prescribed wake approach) or determined as a part of the flowfield solution (free wake approach). With the first ap- proach, Ad is an helical surface with a prescribed pitch distribution. At the trailing edge, a radially varying wake pitch pTE iS determined by the con- dition that the wake surface be tangent to the blade suction side. Far downstream the propeller, an aver- age pOO between the hydrodynamic pitch of the un- perturbed inflow and the blade pitch angle is used. In the near-wake region, the pitch is assumed to vary linearly between ATE and ho. In the free wake approach, the actual shape of 3°w is determined by imposing that wake points move according to the local flow velocity. A bound- ary integral representation of the perturbation veloc- ity field Vo is obtained by taking the gradient of both sides of Eq. (13~. Starting from an initial guess for Low, the computed velocity field is used to up- date the location of each wake node. Once 3°w has been modified, the potential field is re-computed by Eq. (131. The process is repeated until convergence of the aligned wake shape. Details of this technique are given in Giordani et al. (19991.
Solution procedure The proposed methodology for the analysis of cav- itating propeller flows is based on the assumption that the flowfield perturbation induced by the cav- ity has a negligible impact on the location of the wake surface. The validity of this assumption is discussed elsewhere (Arndt et al., 1991; Kinnas and Pyo, 1999~. Thus, the wake alignment procedure is performed under non-cavitating flow conditions. A wake geometry of prescribed type is used as the ini- tial guess. Once the actual shape of the trailing wake is determined, the cavitation model is turned on and an initial guess ~OCB°) for the cavity shape is imposed. In the present analysis, leading edge cavity detach- ment is prescribed by experimental observations. By solving Eq. (13), O¢/an on ~OCB°) is obtained and an estimate in(°) of the cavity thickness is eval- uated by Eq. (12~. A free cavity length approach is used here. The cavity trailing edge is determined by imposing he = 0 at each surface strip in chordwise direction. If this condition is not verified, it is as- sumed that the guessed planform is too small and the cavity shape is extrapolated. Thus, an updated cavity planform CAB) is obtained and the procedure is re- peated until the difference between cavity volumes computed at two successive iterations is less than a prescribed value Marc = 1 x 10-5 in the present analysis). Convergence after more than 12 iterations is infrequent and usually occurs only for those flow conditions that are outside the proper range of appli- cability of the present cavitating flow methodology. As a fundamental step to assess the accuracy of the present methodology to predict non-cavitating and cavitating flow features, the effects of surface discretization on the numerical results have been investigated and are briefly addressed here. A family of grids characterized by a constant ratio between the number of blade elements in chord- wise direction MB' in spanwise direction NB' and the number of wake elements in streamwise direc- tion per turn, MW is used. In the present anal- ysis, MB/NB = 4, MB/MOO = 4/5, with MB = {24, 36, 48, 60, 72, 84, 96) is considered. Hub sur- face is discretized by a number of elements ranging from 528 (MB = 24) to 1248 (MB = 96~. Figure 8 shows the effect of grid refinement on the calculated propeller thrust and torque in the case of advance coefficient J = VA /nD = 0.71 in non-cavitatina and cavitating flow conditions at On = 1 .5 1 5. 1 0.300 0.290 0.280 0.270 0.260 0.250 0.240 0.230 non-cavitating flow model 0 cavitating flow model 0 ~3 n Awn - - - , 0.03 0.04 0.05 0.06 1/MB 0.00 0.01 0.02 0.050 0.048 0.046 by 0.044 0.042 0.040 non-cavitating flow model 0 cavitating flow model o jV ~ EGO A a A ' A 0.038 0.00 0.01 0.02 0.03 0.04 0.05 0.06 1/MB Figure 8: Effect of discretization on calculated propeller thrust (top) and torque (bottom) coefficients. Non cavitat- ing flow at J = 0.71, and cavitating flow at J = 0.71 and On = 1.515 Smooth convergence 1S observed under non- cavitating flow conditions, whereas oscillations are present if the cavitating flow model is included. Figure 9 illustrate the effects of grid refinement on the calculated cavity area AC and on the cavity volume Vc (AT denotes the total blade area). Results are reported in the following table: MB 36 48 60 72 84 96 C .305 .265 .225 .183 .176 .172 AT . V 3 .103 .383 .318 .276 .242 .230 .221 Using MB > 72, discretization errors are less than 10% of corresponding extrapolated values for MB ~ °° All the numerical investigations described in the next section have been obtained by using the MB = 72 surface grid. In such case, the computa- tional time required by a non-optimized code to per-
form one step of the cavitating flow solution scheme is 53 sees. on a 800MHz PC. °8r 0.7 ~ \ /M~ _ 36 ~ / . / , .. ~ ~ ~ / ME; = GO / P~S {2 / MB=~4 / MB = 96 Figure 9: Effect of discretization on the calculated cavity planform (J = 0.71,c,,' = 1.515) 0.6 ~ 0.2 t 0.1 O _ \ 0 0.2 0.4 0.6 J KT (Exp.) [a lO*KQ (Exp.) KT (Num.) 10AKQ (Num.) to Hi\ ~E' C EN 0.8 1 1.2 Figure 10: Open water characteristics of model propeller E779A. 0.50 0.48 FLOW FIELD INVESTIGATIONS 0.46 First, the non cavitating flow features of the model propeller E779A are considered. The experimen- tal data have been taken from previous works per- formed at the CEIMM facility. Figure 10 shows the open water characteristics. In the range of J considered in the present investigations (0.65 < J < 0.88), the numerical predictions of the thrust coeffi- cient are in good agreement with the measurements, whereas discrepancies between computed and mea- sured torque coefficient are observed for J < 0.8. This may be motivated by the approximate approach used to compute the viscous flow contributions to the hydrodynamic loads that, in the considered range of J. are relevant for torque and almost negligible for thrust. Figure 11 shows the location of the tip vortex for two values of the advance coefficient. The vor- tex spatial location has been determined by means of the vorticity field measured using the PIV tech- nique. The good agreement between the measure- ments and the numerical results confirms the valid- ity of the present wake alignment technique in de- termining both the trailing wake contraction rate and the pitch. ~ Due to the low blockage ratio of the facility ( 10%), no tunnel correction has been considered in the numerical calculations. or\ J=0.748 (Exp.) O I \N J=0.748 (Num.) - 1 ~ ~ J=0.880 (Exp.) I ~ rum) 0.44 0.42 0.40 _ . -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x/D Figure 11: Tip vortex location in non cavitating flow con- ditions for two advance conditions. Using the measurement chart shown in Figure 2 and analyzing the individual cavitation images, the E779A model propeller can be described in terms of distinct cavitation patterns. Figure 12 represents a map of the cavitating behavior of the propeller, also known as a bucket diagram, where the main figures of cavitation have been identified. The pro- peller presents four main patterns: bubble cavita- tion/traveling cavitation, supercavitation, sheet cav- itation, leading edge (LE) and tip vortex cavita- tion. The bubble cavitation is present at low val- ues of the cavitation number (fin and is coexisting in most of the cases with supercavitation. This situa- tion can be observed for J = 0.65,on = 0.528 and J = 0.88,on = 0.387: bubble cavitation occurs in
· Bubble and supercavitation ~ Partial and tip vortex cavitation · Supercavitation ~ Leading edge vortex cavitation 0.9 0.S 1 3.(' 2.5 2.0 1.5 c)" 1.0 0.5 -0.5 . . ~ 1~, tic 0.2 0.4 0.6 0.8 1.0 Yl~ 1 ~ ~ ~ ~ cavitation flow model i. I_ convergence limit , ti-~- - · ~ 0.7 0.6 \ \ ~ ~1 I; · !~. r`~e ·~. · ~ o .; · · \ _ ~ ·) · ~ ~ \ . · ·\ , / , 1 \ , 1 \ 1 1 0 / 1\ 2 \ 3 J = 0.65, an = 0.528 I r/R=0.4 I ..R ~.~j I r/R=0.8 ~ Bubb~ec<.~v a, . ~.. ^\ J=0.88,(5n=0.387 r/R=0.4 !','~...~~ r/R=0.8 Bu 1: bte GHAN.'. Or) - ...~....... 6n [ ] 4.0 3.0 2.0 1.0 0.0 . ~ Cav. riR=O.9 Wet. r/R=O.9 -1 .0 0.0 0.2 0.4 0.6 0.8 1.0 x/C J = 0~77, <in = 2.082 . . ;s.u r/R=0.4 4.O 2.5 . !/R--'0;8 = 05 1~ ~ ]~\~\. -1 .0 , . . . -1.0 -1 .0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x/C x/C x/C _ _ ~ _ _:'. ~~ ~ __ Figure 12: E779A model propeller cavitation chart; Cp pressure coefficient distributions on the blade suction side, at characteristic conditions, and associated images O. ~ , ·.w 3.0 2.0 1.0 0.0 . ~ ~ ................ Cav. rfR=O.9 Wet. r/R=O.9 - -1 .0 0.0 0.2 0.4 0.6 0.8 1.0 xlC the inner part of the blade where the Cp distribution ical predictions using a non-cavitating flow model tends to flatten. This is consistent with the numer- (we recall that the present theoretical approach does
not take into account bubble cavitation). At higher on, the Cp presents a steep gradient at the leading edge, responsible for a sharp start of the leading edge cavitation. This attached sheet cavitation mixes with the vortex structure downstream and, for the highest values of (fin, transforms into the so-called leading edge cavitating vortex, for it starts far from the tip and along the leading edge, as visible for J = 0~77~on = 2,082 and (Sn = 3.268. The pro- peller also presents a form of streak cavitation, see the case J = 0.88,6n = 0.387. This pattern is char- acterized by two or three vapor streaks detaching at precise locations along the leading edge. This sit- uation is thought to be caused by irregularities of the model blade geometry (i.e. local roughness) that, combined with the flat Cp distribution, become the place of early and very localized cavitation incep- tion. A curve dentin = Thin fJ) is also plotted in Fig- ure 12 to identify the locus of (5n values such that, if on < anon, the predictions by the present numerical cavitation flow model are not reliable. In such cases, the iterative cavitating flow procedure usually fails to converge. Images of the cavitation pattern for 9 selected flow conditions are shown in Figures l 3 to l S. These images have been obtained using the warp procedure described earlier. The calculated cavity areas by us- ing both a free wake model and a prescribed wake model are also shown for comparison. It is apparent that the predicted extension of the cavitating area is affected by the shape of the trailing wake used in the calculations. In particular, if a cor- rect location of the trailing vorticity is obtained by means of the wake alignment technique (free wake model), numerical results (red plots) are in closer agreement with experiments as compared to those obtained by using a prescribed wake model (blue curves). A quantitative comparison between the observed and the calculated cavity extensions is made possi- ble using the area measurement technique described in this work. Results are shown in Figure l 6, where the data referred to four values of the advance coef- ficient are shown. The area is expressed as the ra- tio between the measured cavitation area Ac and the blade planform area Ao for r/R > 0.3. The experi- mental data are represented with the corresponding standard deviation. The cavitation area fluctuations are in general very small, in the range of l to 3%, except for the cases at the lowest values of the cav- 6n = 1.00 :7 On = 1.51 On = 2.02 Figure 13: Planform view of the cavitating blade and pre- dicted cavitation area at J=0.71: prescribed wake model (_), free wake model (_)
I. ~ - ~ - ~ ~ - - ~ a - - ~ ~ ~ - ~O ~ - ~ ~ ~~ ~ - _ ~ it_ ~1 - · - - ~ ~ ~ - ~ - _ ~ ~~ _I 1 ~ 1 ~~ On = 1.19 On = 1.38 _ e _ __ _ _, 1 . a_ - _ ·_n_ mu__ rem_ I~ _ l ~! d ~' ~~a _~' ___ ~ ;_ ___ _ ; ~~ _ [~ ~~ On = 1.78 On = 2.06 1 1 1 ~ _! 1 em_ 1 1 ~ 1 __ 1 1; · I,. 1 ~ ' 1 i ~ ~' 1 __ 1 1 ~_' I row I I .~' _ 1 _ 1 l~ 1 1 l~ 1 1_~ 1 1 1_D" 1 · ~ ~ 1 1 . _ ~~ 1 _ ~ 1 1 _1 _ 1 ._''' ~3~ 1 1 _' .. *'_ 1 1 1 On = 2.38 1 1 On = 2.75 Figure 14: Planform view of the cavitating blade and pre- dicted cavitation area at J=0.77: prescribed wake model (_), free wake model (_) Figure 15: Planform view of the cavitating blade and pre- dicted cavitation area at J=0.83: prescribed wake model (_), free wake model (_)
tation number (fin, where the occurrence of bubble cavitation (see, for instance, the plot at J = 0.88) or of streak cavitation (J = 0.77) is the source of im- portant changes of the pattern aspect. The numerical results are in quite good agreement with the exper- imental data, in particular at high (Sn The results tend to diverge for lower values of (fin. A higher discrepancy is observed at J = 0.71, yet the differ- ence is within only 15% from the experimental data, and can be explained by the limited resolution of the computational grid (see the table above). 50% 40% ct a' c'' 30% o co - ct Cal a) - 20% 10% 0% .... . ~ i, ...... ... .. .... .... _ ._ · J=0.710 + 4 E-4 * J=0.769 + 6 E-4 · J=0.830 + 2 E-4 · J=0.879 + 3 E-4 J=0.71, free wake model - ~ `` J=0.77, free wake model ~.~ \ J=0.83, free wake model - J=0.88, free wake model :...~. ~ ...:~... 0 1 2 On [~] 3 4 Figure 16: Effect of parameters J and On on the cavity area Ac. Comparison between measurements and numeri- cal results (free wake model) A general feature of the present numerical results is that the predicted cavities are overestimated com- pared to the measurements. Large discrepancies oc- cur at low (fin where attached cavities are thick and strong cavitating vortices are present at the blade tip, and at low J values, where the propeller blades are heavily loaded. The characteristic of the present numerical invis- cid flow model to overpredict the extension of the cavitating flow region may be further explained by existing results that highlight the effect of viscous how modeling. Salvatore and Esposito (2001) show that the inclusion of viscous flow effects determines a reduction of both cavity extension and thickness as compared to inviscid-flow calculations. Specifically, for the case of a three-dimensional hydrofoil in uni- form inflow (60 = 1.148, Re = v0c/v = 9 x lO5), viscous flow cavity area is up to 15% smaller than the inviscid flow one. The calculated cavity volume Vc and maximum thickness he are displayed in Figure 17. Results ob- tained using both the prescribed and free-wake mod- eling are compared. As observed in the case of the cavity area predictions, the extension of the cavitat- ing flow region is generally overestimated when us- ing a trailing wake that is not correctly aligned with the flow. This trend is confirmed by both the cavity volume and the maximum thickness predictions. The procedure described above to provide an es- timate of the cavity maximum thickness on the ba- sis of the measurement of the trailing vortex thick- ness has been used to compare against the numerical predictions. Though only approximate, the results shown in Figure 18 are found to be fairly represen- tative of the true situation. In particular, the levels and trends are compatible with the numerical results. Based on the good agreement found on the cavity area between the numerical and the experimental re- sults, see Figure 16, we can confidently consider the trailing vortex (or the downstream cavity) thickness as a fair estimate of the true cavity thickness, in a first order approximation. The cavitation literature reports a number of experiments where the cavity length and the cav- ity thickness have been measured, though es- sentially on two-dimensional hydrofoils and in quasi three-dimensional foils: Dupont and Avel- lan (l991),Pereira (1997), Laberteaux and Ceccio (1998~. These works show that the cavity maximum thickness is found to vary in a closely linear trend with the cavity length, at least in the steady situ- ations (i.e., free of cloud cavitation), which is our case here as per the low fluctuations observed and reported in Figure 16. Figure 19 displays the maxi- mum thickness, which we recall is only a rough es- timate of the cavity true maximum thickness, versus a length Ic taken as the square root of the attached cavitation area Ac; hence, the values of IC are only indicative. The numerical results fall on a straight line, whereas the experimental data follow the same linear trend, with the additional dispersion due to the coarse thickness evaluation. This result seems to in- dicate that steady leading edge cavitation on a com- plex geometry presents the same macroscopic fea- tures as on simple geometries. Yet, such a result needs to be confirmed by an accurate measurement
Be-04 6e-04 2e-04 Oe+OO awns _ _ 2e-02 1~ n . Oe+OO l Free Wake, J=0.71 Free Wake, J=0.77 Free Wake, J=0.83 Prescr. Wake, J=0.71 - Prescr. Wake, J=0.77 - O Prescr. Wake. J=0.83 ~ O _ 0.5 1 1.5 2 2.5 3 3.5 4 Free Wake, J=0.71 ~ Free Wake, J=0.77 0 Free Wake, J=0.83 Prescr. Wake, J=0.71 - Prescr. Wake, J=0.77 o '< Prescr. Wake,J=0.83 ~ 0.5 1 1.5 2 2.5 3 3.5 4 On Figure 17: Effect of parameters J and on on the computed cavity volume Vc (top) and cavity maximum thickness he (bottom). Comparison between free wake model and pre- scribed wake model. of the three-dimensional shape of the cavity. The propeller thrust and torque coefficients for three values of the advance coefficient are shown in Figure 20. A common feature is that both KT and KQ increase as the cavitation number is increased. This trend is more evident at relatively low J and is associated with extensive cavitation on the blades. Numerical calculations are also given for compari- son. For flow conditions characterized by moderate to light cavitation, i.e. Gn > 1.5 at J = 0.71, 0.77, and 2%~ 1% 0% _ o · J=0.710+4 E-4 ~ J=0.769 + 6 E-4 · J=0.830 + 2 E-4 · J=0.879 + 3 E-4 J=0.71, free wake model J=0.77, free wake model J=0.83, free wake model - J=0.88, free wake model 3 4 Figure 18: Effect of parameters J and 6,' on the cavity maximum thickness ho. Comparison between measure- ments and numerical results (free wake model) 4% q0/^ ~ _ . =~2% 1 oak O · . · Experiment · Free wake model Linear regression · e: ON .~. _ :.~. .~e ~ 0% 2% 1 4% 6% 8% Ic Figure 19: Behavior of the cavity maximum thickness he with the cavity length Ic. Comparison between measure- ments and numerical results (free wake model) On > 1.0 at J = 0.83, a similar degree of accuracy as the one observed in Figure 10 for non cavitating flow conditions is obtained. At lower (fin values, both KT and KQ are underpredicted. This may be related to the overestimated cavitation patterns at low (fin, as discussed above.
0.5 ° 0.3 y 0.2 0.1 o 0.6 0.5 0.4 0.3 0.2 0.1 o 0.5 0.4 0.3 0.2 0.1 o KT (Num ) KT (EXP ) K`: (Num.) VOODOO O O O O O KQ (EXP.) Oo~ O 0.5 1 1.5 2 2.5 o o <,n 3 3.5 4 4.5 K (Num ) AT (EXP ) O - Kit (Num.) Q (EXP) O c~aooo(3o 0 0 0 0 0 c' ~ ' - _ 000~ O 0.5 1 1.5 2 2.5 3 3.5 4 4.5 On KT (Num ) KT (EXP.) K`, (Num.) O C O O O ~ O ~ G O G KO (EXP ) 3 ~ - , , , O 0.5 1 1.5 2 2.5 3 3.5 4 4.5 On Figure 20: Effect of cavitation number Cyn on propeller thrust and torque coefficients at J = 0.71 (top), J = 0.77 (center) and J = 0.83 (bottom) CONCLUDING REMARKS An experimental investigation on a cavitating pro- peller in a uniform inflow has been presented. The cavitation figures have been analyzed through new and robust analysis methods, which have been found to provide information otherwise difficult to assess. The propeller cavitating characteristics have been quantified in terms of cavitation extension. A mea- sure of the cavity thickness was established, in a first order approximation, from the measure of the dimensions of the downstream section of the sheet cavitation. A complete mapping of the propeller hy- drodynamic characteristics has been established. Measurements are used to assess an inviscid flow boundary element method for the analysis of blade partial sheet and super-cavitation. The methodology includes an accurate evaluation of the trailing wake vorticity path by means of a wake alignment tech- n~que. Comparison with the experimental measure- ments shows that the present theoretical methodol- ogy is able to accurately predict the sheet cavitation area across a wide range of advance ratios and of cavitation number values. In addition, the predicted cavity thickness is found to be in satisfactory agree- ment with the experimental data. The importance of an accurate evaluation of the trailing wake vorticity is highlighted and the range of applicability of the present prediction model is also clearly identified. The present work constitutes a first step of a joint experimental and theoretical research project, with the following objectives: i. the improvement of the cavity thickness measurement; ii. the definition of a volume measurement technique; iii. the extension of the cavitating flow model to unsteady propeller cavitation; iv. the inclusion in the theoretical model of viscous effects; v. the modeling of the tip vortex. A database accessible to the public domain will be regularly updated with the latest results and de- velopments: http://crm.insean.it/E779A ACKNOWLEDGMENTS The authors are grateful to the CEIMM personnel. The work was supported by the Italian Ministero dei Trasporti e delta Navigazione in the frame of the IN- SEAN 2000-2002 Research Program. REFERENCES Arndt, R. E. A., Arakeri, V. H., and Higuchi, H., "Some observations of tip-vortex cavitation," Journal of Fluids Mechanics, vol. 229, (1991), pp. 269-289. Dang, J., Numerical Simulation of Unsteady Partial Cav-
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