Numerical Analysis of Steady and Unsteady Sheet Cavitation on a Marine Propeller Using a Simple Surface Panel Method SQCM

Transcription

1 First International Symposium on Marine Propulsors smp 9, Trondheim, Noray, une 9 Numerical Analysis of Steady and Unsteady Sheet Caitation on a Marine Propeller Using a Simple Surface Panel Method SQCM Takashi Kanemaru, un Ando Graduate School of Engineering, Kyushu Uniersity, Fukuoka, apan Faculty of Engineering, Kyushu Uniersity, Fukuoka, apan ASTRACT This paper presents a calculation method for the steady and unsteady caitating propeller problem. The method is based on a simple surface panel method SQCM hich satisfies the Kutta condition easily. SQCM consists of Hess and Smith type source panels on the propeller or caity surface and discrete ortices on the camber surface according to Lan s QCM (Quasi-Continuous ortex lattice Method). The boundary conditions to determine these singularities are the constant pressure condition on the caity surface and the zero normal elocity condition on the propeller and caity surfaces. In this paper, e describe ho to apply SQCM to the calculation of nonlinear caitation on propellers and sho some calculated results about caitation patterns, caity shapes and caity olumes in steady and unsteady flo. Good agreements are obtained beteen the calculated results and published experimental data. Keyords SQCM(Source and QCM), Propeller Sheet Caitation, Free Streamline Theory INTRODUCTION Caity on the propeller blade produces the pressure fluctuation on the stern hull surface and also reduces the propeller performance. The recent large container ships gie us difficulties in design of propellers because they ill operate in heaily loaded condition. Therefore it is important to predict the caity pattern, olume, and propeller performance accurately. Panel methods, hich can represent the blade shape exactly, are idely used for the prediction of propeller performance. Moreoer Kinnas & Fine(99) and Kim & Lee(996) presented the caitation prediction method using panel method. In Kyushu Uniersity e deeloped a simple surface panel method SQCM and applied SQCM to the -D and 3-D sheet caitating hydrofoil problems. SQCM is also expected as an efficient numerical method for caitating propeller problem. In this paper e extend SQCM to the steady and unsteady caitating propeller problem and present the consistent calculation method to treat both the partial caitation and supercaitation. One of the features in our calculation is that the cross flo elocity near the tip of a propeller blade is taken into consideration at the constant pressure condition on the caity surface in order to get the reasonable results. We sho some calculated results for seeral kinds of propellers and compare them ith the published experimental data. In this paper, e treat the sheet caitation only. OUTLINE OF SQCM SQCM (Source and QCM) uses source distributions (Hess & Smith 964) on the propeller blade surface and discrete ortex distributions arranged on the mean camber surface according to QCM (Quasi-Continuous ortex lattice Method) (Lan 974). These singularities should satisfy the boundary condition that the normal elocity is zero on the propeller blade and the mean camber surfaces. Consider a K bladed propeller rotating ith a constant angular elocity (= n, n :number of propeller reolutions) in iniscid, irrotational and incompressible fluid. The space coordinate system O XYZ and the propeller coordinate system o xyz are introduced as Figure. The cylindrical coordinate system o xr is also introduced for conenience. Then the folloing relation transforms the cylindrical coordinate system o xr into the propeller coordinate system o xyz. x x, y r sin, z r cos () r y z, tan ( y / z) The propeller blade S is diided into M panels in the spanise direction. The face and back surfaces of the blade section are diided into N panels in the chordise direction, respectiely. Therefore the total number of source panels becomes ( M N) K and constant source m is distributed in each panel. The elocity ector V m due to the source distributions on the blade surfaces is expressed by using elocity potential as Vm m m ()

2 m 4 y S m( x, y, z, ds ( x x) ( y y) ( z z) Figure Coordinate systems of propeller Next the mean camber surface is diided into M segments in the spanise direction corresponding to the diision of the source panels and diided into N in the chordise direction. Here e introduce axis hose origin locates at the leading edge and is extended to the trailing ortex surface along the mean camber surface. The position of the bound ortex LP and control point on the mean camber surface are expressed by the folloing equations according to the QCM theory. LP Y V A T ( r ) L( r ) L( r ) cos N T ( r ) L( r ) r L( ) cos N r r r and are numbers in the spanise and chordise directions. L ( r ) and T ( r ) are the positions of the leading edge(l.e.) and trailing edge(t.e.), respectiely. And total ( M N ) K horse shoe ortices are located on the mean camber surface according to Equation (3) as illustrated in Figure. One set of ring ortices consists of one bound ortex, to free ortices and the first spanise shed ortex CH closest to the trailing edge in the trailing ake. In the case of the unsteady problem the shed ortex flos out from T.E.. Thus the ring ortex, hich represents the shed ortex, is located in the trailing ake. This ring ortex consists of to spanise shed ortex CH, EF and to streamise trailing ortex CE, HF. Here e define the induced elocity ector due to the -th ring ortex of unit strength hich starts from the -th strip and ones due to the -th ring ortex of unit strength (3) located in the trailing ake as k and k respectiely. These induced elocity ector are expressed as z o,o r Z S x, X k k k s k N F F s k k k s c c k k k k = induced elocity ector due to the bound ortex of unit strength on the mean camber surface F k = induced elocity ector due to the free ortex of unit strength on the mean camber surface s k = induced elocity ector due to the spanise shed ortex of unit strength in the trailing ake c k = induced elocity ector due to the streamise trailing ortex of unit strength in the trailing ake The induced elocity ector due to each line segment of ortex is calculated by the iot-saart la. If e define the segments of the ring ortex on the mean camber surface at time t L and the ring ortex in the trailing ake as k t L and k t L, the induced elocity ector due to the ortex model of the QCM theory is gien by the folloing equation. V c( r ) sin N N K M N k tl k k K ML ktl k k c ( r ) : chord length of section N, k tl k tl Equation (5) is used hen the control points are on the mean camber surface. When the control points are on the blade surface, the ring ortices are close to the control points especially for thin ing. In this case e treat the ring ortices on the mean camber surface and shed ortex nearest to T.E. as the ortex surfaces in order to aoid numerical error. We call this treatment Thin Wing Treatment (Maita et al. 996). Then e modify the ortex model of the QCM theory (Figure (a)) as follos: First, e close the ring ortex on the mean camber surface at T.E. and locate ring ortices ACH, HCDG and GDEF donstream from T.E. as illustrated in Figure (b). Next, e replace the -th ring ortex ith a set of ring ortices distribution along ( ~ ) on the mean camber surface as illustrated in the middle of Figure (b). Then the ring ortex ACH just donstream from T.E. and ring ortex HCDG are replaced ith a set of ring ortices distribution along in the trailing ake. The other ring ortices from are calculated as discrete ring ortices. (4) (5)

3 In this case, the induced elocity ectors due to the ortex systems on the mean camber surface and trailing ake are expressed as K M N k V k( tl) k( ) d k k K M N T ( r ) ACH k( tl) k ( ) d T ( r ) k K M T ( r ) HCDG k( tl) k ( ) d T ( r ) k K M GDEF k( tl) k k K ML k( tl ) k k ( ) k N F F ( k k ACH A C k ( ) k k ( HCDG k GDEF k k ( ) ( GD k F k DE k ( ) GF k F k ) ( )) A k AH HC ) k ( ) k ( ) HC CD HG GD ( ) k ( ) k ( ) k ( ) k FE k Here the first term in the right hand side of Equation (6) is diided according to Thin Wing Treatment, the second and third terms express the shed ortex just donstream from T.E. and the 4-the and 5-th terms express other ring ortices in the trailing ake. In this ay, the elocity ector V around a propeller in the propeller coordinate system is expressed as V V V (7) I V m V V I, V and m are inflo, induced elocity ectors due to ortex and source distributions, respectiely. The boundary conditions at the control points on the blade surfaces and mean camber surfaces are that there is no flo across the surfaces. Therefore the equation of the boundary condition is gien as follo: V n on Wing and Camber Surface (except T.E.) (8) V n V N at T.E. n is the normal ector on the blade surface and mean camber surfaces. V N is the normal elocity at T.E. to satisfy the Kutta condition and expressed by the folloing equation. ( i) ( i) ( i) VN p / VI VN (9) And i is the number of iteration. The first term in the right hand side in Equation (9) means the corrector for the (i) alue of the preious step V N. V N is proportional to the pressure difference p expressed as (6) p ( ) ( V V ) () t Here and or V and V mean the perturbation potential or the magnitude of elocity at the control points on the upper and the loer surface, hich are closest to T.E.. The iteration is continued until p is small enough. means the acceleration factor and in this paper. is the density of the fluid. The Kutta condition of unsteady SQCM is gien by p () On the other hand, e consider that the first equation in Equation (8) includes the Kutta condition. ound Vortex Free Vortex (a) LP (b) Figure Arrangement of ortex system Spanise Shed Vortex The pressure p ( on the propeller blade is calculated by the unsteady ernoulli equation expressed as p( p V VI () t p = the static pressure in the undisturbed inflo = the perturbation potential in the propeller coordinate system A C D E H G F A H L ( r ) ( ) G T r F C D E Streamise Trailing Vortex I C DE AH GF I

4 Here the time deriatie term of in Equation () is obtained numerically by to points upstream difference scheme ith respect to time. The pressure of the propeller blade is expressed as the folloing pressure coefficient C pn in order to compare the calculated results ith experimental data. C pn p( p n D (3) D is the diameter of the propeller. The thrust T and the torque Q of the propeller are calculated by pressure integration. Denoting the X, Y and Z components of the normal ector on the blade surface by n X, n Y and n Z, respectiely, the thrust and the torque are expressed by T Q S S ( p( p ) n X ds ( p( p )( n Z n Y ) ds Y Z (4) Finally the adanced coefficient, the thrust and the torque coefficients, K Q are expressed as follo: VA T Q,, nd n D 4 K Q (5) n D 5 3 CALCULATION METHOD FOR CAVITATION In the present method e assume that the pressure on the caity surface is constant according to the free streamline theory. The difficulty in the analysis of the caitating flo is that not only zero normal elocity condition but also constant pressure condition must be satisfied on the caity surface. On the other hand the blade surface, hich is not coered ith caity, must satisfy the zero normal elocity condition only. Therefore e must sole the mixed boundary alue problem. We adopt the caitation number or n corresponding to those of published experimental data to be referred in this paper. and n are defined by p p (6) VA p p n (7) n D p = saturated apor pressure And e obtain the folloing relations. V T VI VA (8) t V T VI nn D (9) t The third term in the square root in Equation (8) and (9) is unknon. So this term makes the nonlinear caity flo theory complicate. Needless to say that this term is excluded in the case of steady problem. Here e describe the calculation procedure for caitation. (Step ) First of all e assume the initial caity length for each spanise section. Next e gie the caitation number or n to obtain the tangential elocity V T, and then determine the source and ortex distributions from the folloing equation. V T () s Here and are the total elocity potential on the control point of the -th and -th caity surface panels, respectiely. s means the distance beteen these to control points (See Figures 3 and 4). In the case of unsteady problem, the initial caity length and the initial alue of potential can be assumed using the conerged caity shape at the preious time step. V T means the modified tangential elocity by considering the cross flo component V Cr in the cylindrical coordinate system ( V C V Cx, VCr, VC, V C : the elocity ector on the caity surface) and is expressed by V T V T VCr () If e use V T in the boundary condition on the caity surface (Equation ()), the pressure condition is satisfied to-dimensionally for chordise section. ut using V T, e get the folloing equation. V Cx V V T VCx VCr VC () C s Equation () means that the tangential elocity condition (the constant pressure condition) is satisfied threedimensionally on the caity surface. D s Caity Surface E Figure 3 Model of a partially caitating section D s s c Camber Surface s c s E Camber Surface s E Caity Surface Figure 4 Model of a supercaitating section E

5 We diide the caity surface into een spacing panels in the chordise direction. On the other hand, e diide the non-caity surface on the blade into cosine spacing panels. The caity detachment point D must be determined by the laminar separation point or the transition point for turbulence flo. ut this is so difficult that e ill treat the iscous effects in the future ork. Equation (8) is gien on the blade surface, hich is not coered ith the caity, and the mean camber surface. Here e consider the mean position beteen the back and face panels ith the caity surface to be the mean camber surface. Thus e can derie ( M ( N N )) K linear simultaneous equations hich determine m and. (Step ) When e calculate the elocity on the caity surface using (Step ), e can t generally satisfy the zero normal elocity condition on the caity surface. Therefore e need to arrange the source panels so as to coincide ith the flo direction using the folloing equation. h h V T V n (3) s t Here s ( s ) represents the distance from the caity detachment point along the caity surface (See Figures 3 and 4) and h means the caity thickness. Discretizing Equation (3), the caity height of the -th panel h ( is expressed by V h t t V n T ( ) h ( s t h ( (4) V T s t t means time step. In the case of steady problem, h is expressed by V n h h s V T (5) Then the opening of the caity end E is obtained by the integration of Equation (4) or (5) along s. (Step 3) We repeat (Step ) and (Step ) until the caity shape conerges. (Step 4) After the caity shape conerges, e compare E ith the target alue ET and adjust the caity length by using the folloing equation. ( n) ( n) ( ET E ) (6) Here n and are the iteration number for adjusting the caity length and a proper eight coefficient, respectiely. We adopt the small alue for ET as long as the caity shape conerges smoothly. As the result, the solutions by our method are similar to those of closed model. When is close to the chord length at the section, it is difficult to calculate the caity shape smoothly. Therefore e shift the section from the partially caitating model (Figure 3) to the supercaitating model (Figure 4) hen exceeds. c in Equation (6). (Step 5) We repeat described procedures until the caity length satisfies the folloing condition at all spanise sections. ( n) ( n). c (7) (Step 6) After the caity length conerges at all spanise section, e calculate the thrust and the torque by Equation (4). Here e integrate the pressure on the blade surface een on the caity area by using. 4 CALCULATED RESULTS 4. Steady Problem We select M.P.No.3 propeller and M.P.No.5 propeller for the calculations. oth of these propellers ere deeloped by Ship Research Institute in apan (SRI, present National Maritime Research Institute) and caitation tests ere conducted at SRI (Kadoi et al. 978). The principal particulars of these propellers are shon in Table. Only the pitch ratios are different beteen M.P.No.3 and M.P.No.5. Table Principal particulars of propellers NAME OF PROPELLER M.P.No.3 M.P.No.5 DIAMETER(m).5.5 NUMER OF LADE 6 6 PITCH RATIO AT.7R EXPANDED AREA RATIO.8.8 HU RATIO.8.8 RAKE ANGLE(DEG.) LADE SECTION SRI-a SRI-a Panel arrangements of propeller are shon in Figure 5. The propeller blade surface including caity surface are diided into 3 panels in the chordise direction (3 panels from leading edge to D, 5 panels from D to and panels from to c in the case of partial caitation, 3 panels from leading edge to D, 5 panels from D to c and panels from c to in the case of super caitation) and diided into panels in the spanise direction. Camber surface is diided into 3 segments in the chordise direction. Figure 5 shos the panel arrangements in the case of non-caitation at all sections. Here e ignore the hub surface. M.P.No.3 Figure 5 Panel arrangements for SRI-a propellers n M.P.No.5

6 Figures 6 and 7 sho the caitation patterns of M.P.No.3 and M.P.No.5 comparing ith the experimental data. The calculated results agree ell ith the experimental data. =5. r/r=.5 Caity End r/r=.5 =3. =.4 Figure 6 Caitation patterns (M.P.No.3, =.7) not considered in all calculated results. The calculated thrust and torque coefficients decrease abruptly at a lo adance coefficient as shon in experimental data. We emphasize that the calculated peak points of and K Q agree ith experimental data in most cases. In the case of lo caitation number, the agreements of the peak points are not so ell. One of the reasons is that the boundary beteen caity area and non-caity area can t be expressed by the present panel diision method. Also, e hae to determine the caity detachment point theoretically in order to simulate the caity accurately near the root., K Q.5 M.P. No.3,σ=6. K Q Non-Ca, K Q.5 M.P. No.3,σ=3. K Q Non-Ca r/r=.5 Caity End r/r=.5 =3.9 =.9 =.4 Figure 7 Caitation patterns (M.P.No.5, =.) Figure 8 shos the comparisons of the pressure distributions on the blade and caity surfaces to inestigate the effects of the cross flo consideration. The results ith cross flo sho the good agreement ith negatie caitation number hich is the target pressure coefficient on the caity surface. On the other hand, the calculated results ithout cross flo don t satisfy the constant pressure condition sufficiently. -C pn 5-5 M.P.No.5 =. σ=.9 With cross flo r/r=.7 r/r=.8 r/r=.9 n Chordise Position, x/c -C pn Chordise Position, x/c Figure 8 Caity thickness ith and ithout cross flo Figures 9 and sho the thrust coefficient and the torque coefficient K Q of M.P.No.3 and M.P.No.5 comparing ith the experimental data (Kadoi et al. 978). In these figures, experimental data and calculated results ithout caitation are also shon. The iscous effects are 5-5 M.P.No.5 =. σ=.9 r/r=.7 r/r=.8 r/r=.9 Without cross flo, K Q, K Q, K Q M.P. No.3,σ=. K Q Non-Ca, K Q Figure 9 Thrust and torque coefficients (M.P.No.3) M.P. No.5,σ=..5.5 M.P. No.5,σ=6. K Q K Q Non-Ca Non-Ca, K Q, K Q Figure Thrust and torque coefficients (M.P.No.5) 4. Unsteady problem We select conentional propeller () and highly skeed propeller (HSP) of Seiun-Maru-I in order to compare the experimental data at SRI (Kudo et al. 989). The experimental data ere obtained by a highly adanced M.P. No.3,σ=.8 Non-Ca K Q M.P. No.5,σ=3. K Q.5.5 M.P. No.5,σ=.8 K Q Non-Ca Non-Ca

7 method of measuring the three dimensional caity shape using a laser beam coupled ith a CCD (Charge Coupled Deice) camera and image processor. Table shos the principal particulars and Figure shos the panel arrangements of these propellers. The time step t is gien by the folloing equation. t / 36. / n (8) is chosen as =.5 degrees. Calculations are performed using the ake distribution (see Figure ) gien by the reference (Kudo et al. 989). Panel diision method and other treatments are the same in the steady case described in 4.. Table Principal particulars of propellers (Seiun-Maru-I) NAME OF PROPELLER HSP DIAMETER (m).95. NUMER OF LADE 5 5 PITCH TARIO AT.7R EXPANDED AREA RATIO.65.7 HU RATIO.97 RAKE ANGLE (DEG.) LADE SECTION MAU Modified SRI- V T (9) s The differences of calculated results ith and ithout cross flo are small ith regard to. In HSP, the calculated caity shapes considering cross flo effect agree ell ith the experimental data. Figures 7 and 8 sho the ariations of caity area and caity olume ith regard to both of and HSP comparing ith experimental data. In these figures, e also sho the calculated results ithout cross flo. Calculated results ith cross flo can express the characteristic of ariations and the differences beteen and HSP qualitatiely. Caity End r/r=.3 Caity End 35. deg. deg. deg. deg 3. deg Figure 3 Caitation patterns (, =.7, n =3.6) Figure Panel arrangements for Seiun-Maru-I propellers =. HSP Caity End. deg r/r=.3 Caity End. deg 3. deg 4. deg 5. deg 7 9 Propeller Disk 8 Figure Axial ake elocity distribution Figures 3 and 4 sho the calculated caitation patterns of and HSP. The caity end lines by experimental sketch are added in these figures. We can see the good agreements beteen calculated results and experimental data in both and HSP. Figures 5 and 6 sho the calculated caity shape of and HSP comparing ith experimental data. In the figures of r / R =.95, e added the calculated results ithout cross flo component. In the calculation ithout cross flo component, folloing equation is used instead of Equation (). Figure 4 Caitation patterns (HSP, =., n =.99) 5 CONCLUSIONS In this paper e presented a calculation method for steady and unsteady caitating propeller using a simple surface panel method SQCM. We considered the cross flo elocity in the boundary condition on the caity surface. Comparison beteen the calculated results and experimental data led the folloing conclusions: () The present method can predict the thrust and the torque in extensie adance coefficients and caitation numbers. In unsteady case, the present method can express the ariations of the caity area and caity olume. () The cross flo effect on the caity surface gies reasonable caity shapes and ariations of caity area and olume especially in HSP.

8 (3) The iscous effect on the caity detachment point should be considered in order to improe the accuracy of calculation. z μ /c μ z μ /c μ z μ /c μ deg. r/r=.8 (/ cross flo). deg.... deg. 3. deg deg.. Chordise position x μ /c μ 35. deg.. deg. r/r=.95 (/ cross flo) (/o cross flo). deg.... deg. 3. deg..5.5 Chordise position x μ /c μ Figure 5 Caity shapes (, =.7, n =3.6)..... deg.. deg. HSP r/r=.95 (/ cross flo) (/o cross flo) 4. deg deg. 5. deg..5.5 Chordise position x μ /c μ Figure 6 Caity shapes (HSP, =., n =.99) ) / cross flo ) /o cross flo 5 ) ) HSP ) ) 5 Caity Area (mm ) Caity Volume (mm 3 ) ) / cross flo ) /o cross flo HSP 3 ) ) ) ) Angular Position (deg) Angular Position (deg) Figure 7 Variations of caity area and olume ( : =.7, n =3.6, HSP : =., n =.99) REFERENCES Ando,., Maita, S. and Nakatake, K. (998). A Ne Surface Panel Method to Predict Steady and Unsteady Characteristics of Marine Propeller. Proceedings of st Synposium on Naal Hydrodynamics, Washington D.C., pp Ando,., Matsumoto, D., Maita, S., Ohashi, K. and Nakatake, K. (999). Calculation of Three- Dimensional Steady Caitation by a Simple Surface Pnanel Method. ournal of the Society of Naal Architects of apan, Vol.86, pp.7-7. Ando,., Kanemaru, T., Ohashi, K. and Nakatake, K. (). Calculation of Three-Dimensional Unsteady Sheet Caitation by a Simple Surface Pnanel Method. ournal of the Society of Naal Architects of apan, Vol.88, pp.9-3. Fine, N.E. (99). Nonlinear Analysis of Caitating Propellers in Nonuniform Flo. Ph.D. Thesis, M.I.T., Cambridge, Mass. Hess,.L. and Smith, A.M.O. (964). Calculation of Nonlifting Potential Flo about Arbitrary Three Dimensional odies. ournal of Ship Research, Vol. 8, No., pp.-44. Kadoi, H., Kokubo, Y., Koyama, K. and Okamoto, M. (978). Systematic Tests on the SRI-a-Propeller. Report of the SRI, Vol.5, No.. Kanemaru, T., Ando,. (8). Numerical Analysis of Steady Sheet Caitation on a Marine Propeller Using a Simple Surface Panlel Method SQCM. Proceeding of 3 rd PAAMES and AMEC8, pp Kim, Y.-G., Lee, C.-S. and Suh,.-C. (996). Prediction of Unsteady Performance of Marine Propellers ith Caitation Using Surface-Panel Method. Proceedings of st Synposium on Naal Hydrodynamics, Trondheim, pp Kinnas, S.A. and Fine, N.E. (99). A Numerical Nonlinear oundary Element Method for the Analysis of Unsteady Propeller Sheet Caitatyon. Proceedings of 9th Synposium on Naal Hydrodynamics, Seoul, pp Kinnas, S.A. and Fine, N.E. (993). A Numerical Nonlinear Analysis of the Flo around -D and 3-D Partially Caitating Hydrofoils. ournal of Fluid Mechanics, Vol. 54, pp.5-8. Kudo, T., Ukon. Y., Kurobe, U. and Tanibayashi, H. (989). Measurement of Shape of Caity on a Model Propeller lade. ournal of the Society of Naal Architects of apan, Vol.66, pp Lan, C.E. (974). A Quasi-Vortex-Lattice Method in Thin Wing Theory. ournal of Aircraft, Vol., No. 9, pp.58-57