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24th Symposium on Naval Hydrodynamics FuLuoka, JAPAN, 8-13 July 2002 A BEM Technique for the Modeling of Supercavitating and Surface-Piercing Propeller Flows Yin Lu Young Spyros A. Kinnas (The University of Texas at Austin, USA) ABSTRACT A three dimensional (3-D) boundary element method (BEM) is presented for the numerical mod- eling of supercavitating and surface-piercing pro- pellers. The method has been developed in the past for the prediction of unsteady sheet cavitation for conventional fully submerged propellers. To al- low for the treatment of supercavitating propellers, the method is extended to model the separated flow behind trailing edges with non-zero thickness. For surface-piercing propellers, the negative image method is used to account for the effect of the free surface. The current method is able to predict com- plex types of cavitation patterns on both sides of the blade surface with searched cavity detachments. The method is shown to converge quickly with grid size and time step size. The predicted cavity planforms and propeller loadings also compare well with exper- imental observations and measurements. Finally, a 2-D study to investigate the effects of jet sprays at the moment of blade entry is presented. INTRODUCTION Supercavitating propellers are often believed to be the most fuel efficient propulsive device for high speed vessels. The term supercavities refer to cavities that are longer than the blade. They tend to have smaller volume change and produce bubbles which collapse downstream of the blade trailing edge, which results in reduced noise and blade surface erosion. However, they are also difficult to model due to the unknown size of and the pressure in the separated region behind thick blade trailing edges, which are characteristic of supercavitating propellers. A surface-piercing (also called partially sub- merged ~ propeller is a special type of supercavitating propeller which operates at partially submerged con- ditions. Surface-piercing propellers are more efficient than submerged supercavitating propellers because: Reduction of appendage drag due to shafts, struts, propeller hub, etc.; 2. Larger propeller diameter since its size is not limited by the blade tip clearance from the hull or the maximum vessel draft; 3. Reduction of blade surface friction and erosion since cavitation is replaced by ventilation . In the past, the design of surface-piercing pro- pellers often involved trial-and-error procedure us- ing the measured performance of test models in free- surface tunnels or towing tanks. However, most of the trial-and-error approaches do not provide infor- mation about the dynamic blades loads nor the aver- age propeller forces (Olofsson 1996~. Model tests are extremely expensive, and are often hampered by scal- ing effects (Shell 1975) (Scherer 1977) and influenced by test techniques (Morgan 1966) (Suhrbier & Lecof- fre 1986~. Numerical methods, on the other hand, were not able to model the real phenomena. Difficul- ties in modeling surface-piercing propellers include: Insufficient understanding of the physical phe- nomena at the blade's entry to, and exit from, the free surface. Insufficient understanding of the dynamic loads accompanying a propeller piercing the water at high speed. · The modeling of very thick and very long venti- lated cavities, which are also interrupted by the free surface. · The modeling of water jets and associated change in the free surface elevations at the time of the blade's entry to, and exit from, the free surface. · The effect of blade vibrations due to the cyclic loading and unloading of the blades associated with the blade's entry to, and exit from, the free surface.
Supercavitating Propellers The development of numerical methods for the analysis and design of supercavitating propellers has been slow compared to conventional propellers. The main difficult arises from the unknown physics in the highly turbulent region behind the blunt trailing edge, which is characteristic of supercavitating pro- peller sections. The first theoretical design method was developed by (Tachnimdji & Morgan 1958), and followed by (Turin 1962, yenning & Haberman 1962, Cox 1968, Barr 1970, Yim & Higgins 1975, Yim 1976~. However, these methods were based on 2-D studies, and required many approximations and em- pirical corrections. Recently, more rigorous methods were developed by (Kamiirisa & Aoki 1994, Kikuchi et al 1994, Vorus & Mitchell 1994, Ukon et al 1995~. Nevertheless, these methods were still based on the optimization of 2-D cavitating blade sections to yield minimal drag for a given lift and cavitation number. A 3-D vortex-lattice method was developed by (Kudo & Ukon 1994) to predict the steady perfor- mance of supercavitating propellers. Their model assumed the pressure over the separated zone to be constant and equal to the vapor pressure. A variable length separated zone model using a similar vortex- lattice method was presented in (Kudo & Kinnas 1995~. It was found that the length of the separated zone had no effect on the results if all the blade sec- tions were covered by the supercavity. However, the length of the separated zone did have an effect on the pressure and cavity length near the blade trailing edge under fully wetted and partially cavitating con- ditions. Finally, (Kinnas et al 1999) further extended their method to search for midchord cavitation, and coupled it with an optimization method (Mishima & Kinnas 1997) for the design of supercavitating pro- pellers. However, all of the above mentioned lifting sur- face methods cannot capture accurately the flow de- tails at the blade leading and trailing edge due to the breakdown of linear cavity theory. In addition, the applicability of the thickness-loading coupling intro- duced by (Kinnas 1992) in the analysis of supercav- itating propellers is still under investigation. Surface-Piercing Propellers The first effort to model partially submerged propeller was carried out by (Oberembt 1968~. He used a lifting line approach to calculate the characteristics of partially submerged propellers. (Oberembt 1968) assumed that the propeller is lightly loaded such that no natural ventilation of the propeller and its vortex wake occur. A lifting-line approach which includes the ef- fect of propeller ventilation was developed by Furuya in (Furuya 1984, Furuya 1985~. He used linearized boundary conditions and applied the image method to account for free surface effects. He also assumed the face portion of the blades to be fully wetted and the back portion of the blades to be fully ventilated starting from the blade leading edge. The blades were reduced to a series of lifting lines, and method was combined with a 2-D water entry-and-exit theory developed by (Wang 1977, Wang 1979) to determine thrust and torque coefficients. Furuya compared the predicted mean thrust and torque coefficients with experimental measurements obtained by (Hadler & Hecker 1968~. In general, the predicted thrust co- efficients are within acceptable range compared to measured values. However, there were significant dis- crepancies with torque coefficients. Furuya (Furuya 1984, Furuya 1985) attributed the discrepancies to the effects of nonlinearity, absence of the blade and cavity thickness representation in the induced veloc- ity calculation, and uncertainties in interpreting the experimental data. He also stated that the applica- tion of lifting-line theory is limited due to the relative large induced velocities at low advance coefficients. An unsteady lifting surface method was em- ployed by (Wang et al 1990a) for the analysis of 3-D fully ventilated thin foils entering into initially calm water. The method was later extended by (Wang et al 1990b) and (Wang et al 1992) to predict the performance of fully ventilated partially submerged propellers with its shaft above the water surface. Similar to (Furuya 1984, Furuya 1985), the method assumed the flow to separate from both the lead- ing edge and trailing edge of the the blade, forming on the suction side a cavity that vents to the at- mosphere. Discrete line vortices and sources were placed on the face portion of the blade to simulate the effect of blade loading and cavity thickness, re- spectively. Line sources were also placed on the cav- ity surface behind the trailing edge of the blade to represent the cavity thickness in the wake. A heli- cal surface with constant radius and pitch were used to construct the trailing vortex sheets. The nega- tive image method was used to account for the effect of the free surface. The effect of the blade thickness was neglected in the computation. Comparisons were presented with both experimental measurements by (Hadler & Hecker 1968) and numerical predictions by (Furuya 1984, Furuya 1985~. The predictions were within reasonable agreement with experimental val- ues for a propeller with limited data range. However,
substantial discrepancies were observed for another propeller with both experimental values and numer- ical predictions by (Furuya 1984, Furuya 1985~. The 3-D vortex-lattice lifting surface method de- veloped by (Kudo & Ukon 1994) and (Kudo & Kin- nas 1995) for the analysis of supercavitating pro- pellers has also been extended for the analysis of surface-piercing propellers. However, the method performs all the calculations assuming the propeller is fully submerged, then multiplies the resulting forces with the propeller submergence ratio. As a result, only the mean forces can be predicted while the complicated phenomena of blade's entry to, and exit from, the water surface are completely ignored. A 2-D time-marching BEM was developed by (Savineau & Kinnas 1995) for the analysis of the flow field around a fully ventilated partially submerged hydrofoil. However, this method only accounts for the hydrofoil's entry to, but not exit from, the water surface. In addition, the negative image method was used so the effects of water jets and change in free surface elevation were ignored. Objectives The objective of this work is to extend an exist- ing 3-D potential based boundary element method to predict the performance of supercavitating and surface-piercing propellers. The method was first developed by (Kinnas & Fine 1991) for the nonlinear analysis of flow around partially and supercavitating 2-D hydrofoils. It was then modified to treat partially cavitating 3-D hy- drofoils (Fine & Kinnas 1993a). In (Kinnas & Fine 1992, Fine & Kinnas 1993b), the method was named PROPCAV (PROPeller CAVitation) for its added ability to analyze 3-D unsteady flow around cavi- tating propellers. Later, (Mueller & Kinnas 1999) modified the method to search for midchord cavita- tion on either the back or the face of propeller blades. Most recently, (Young & Kinnas 2001) extended the method to predict alternating or simultaneous face and back cavitation on conventional propeller blades subjected to non-axisymmetric inflow. The bound- ary element method inherently includes the effect of non-linear thickness-loading coupling by discretizing the blade surface instead of the mean camber sur- face. Thus, it requires more CPU time and memory than the lifting surface method. However, it offers a better prediction of the flow details at the propeller leading edge and tip than the lifting surface method. non-axisymmetric midchordpari~al wake effective inflow wake cavity on back side ~ An, ..,. yor:(myz)b - ~` ~ _ _..5 Ls-zxst ..-i~ 1 /d _ hub / {supercavity on face side l supercavity on back side Figure 1: Propeller subjected to a general inflow wake. The blade fixed (x, y, z) and ship fixed (xs~yS'zS) coordinate systems are shown. FORMULATION The general formulation for the prediction of unsteady sheet cavitation on conventional fully sub- merged propellers is presented in (Kinnas & Fine 1992, Fine October, 1992, Young & Kinnas 2001~. It is summarized in this section for the sake of com- pleteness. Consider a cavitating propeller subject to a gen- eral inflow wake, qw (AS ~ as ~ as ~ ~ as shown in Fig. 1. The inflow wake is expressed in terms of the absolute (ship fixed) system of coordinates (x,s,Ys,zs). The inflow velocity, -tin, with respect to the blade fixed coordinates (x, y, z), can be expressed as the sum of the inflow wake velocity, qw, and the propeller's an- gular velocity A, at a given location x: qirl (x, y, Z. t) = qw (X, r, (JBAt) + W X X (1) where r = x/~, cB = arctan~z/y), and x = (x, y, z). The inflow, qw, is assumed to be the effective wake, i.e. it includes the interaction between the vorticity in the inflow and the propeller (Choi 2000, Kinnas et al 2000~. The resulting flow is assumed to be incompressible, inviscid, and irrotational flow. Therefore, the time dependent perturbation poten- tial, ¢(x, y, z, t), can be expressed as follows: (X, y, Z' t) = qin(X'y, Z't) + VO(X'y, Z. t) (2) where ~ satisfies the Laplace's equation(V2<t) = 0) in the fluid domain. Note that in analyzing the flow around the propeller, the blade fixed coordinates sys- tem is used.
The perturbation potential, gb, at every point p on the combined wetted blade and cavity surface, SWB(t) U SC(t)' must satisfy Green's third identity: 2~¢ptt) = ~JrSWB(~)USC(~) [ q() ~nq(~) G(p;q)~3 q(~]dS Y ~ ~ ~ +I is Bird eq t) ~~,¢P¢;~) dS for p ~ (sweat) Ascot)) (3) where the subscript q corresponds to the variable point in the integration. G(p; q) = 1/R(p; q) is the Green's function with R(p; q) being the distance be- tween points p and q. nq is the unit vector normal to the integration surface, with the positive direction pointing into the fluid domain. /~<b is the potential jump across the wake surface, SW(t). As shown in Fig. 2, the symbols SWB (t)' SC(t), and SW(t) de- note the wetted blade and hub, blade sheet cavity, and wake surfaces, respectively. ~ WB Hi) / n V V) Sw~t) Figure 2: Top: Definition of the exact surface. Bot- tom: Definition of the approximated cavity surface. Equation 3 should be applied on the "exact" cavity surface Sc, as shown in the drawing at the top of Fig. 2. However, the cavity surface is not known and has to be determined as part of the solu- tion. In this work, an approximated cavity surface, shown in the drawing at the bottom of Fig. 2, is used. The approximated cavity surface is comprised of the blade surface underneath the cavity on the blade, Scott, and the portion of the wake surface which is overlapped by the cavity, SCW(t) The justification for making this approximation, as well as a measure of its effect on the cavity solution can be found in (Kinnas & Fine 1993, Fine October, 1992~. Using the approximated cavity surface, Eqn. 3 may be decomposed into a summation of integrals over the blade surface, SO (_ SOB U SWB), and the portion of the wake surface which is overlapped by the cavity, sow. Boundary Conditions · Kinematic Boundary Condition on Wetted Blade and Hub Surfaces The kinematic boundary condition requires the flow to be tangent to the wetted blade and hub sur- face. Thus, the source strengths, - , are known in terms of the inflow velocity, qin in =qin n (4) · Dynamic Boundary Condition on Cavitating Sur- faces The dynamic boundary condition on the cavi- tating blade and wake surfaces requires the pressure everywhere on the cavity to be constant and equal to the vapor pressure, Pv By applying Bernoulli's equation, the total cavity velocity, qc, can be ex- pressed as follows: hi ~2 =n2D2`J +~q ~2+`,;2r2_29y 20<> (5) where In (PoPv)/~2n2D2) is the cavitation number; p is the fluid density and r is the distance from the axis of rotation. PO is the pressure far up- stream on the shaft axis; 9 is the acceleration of gravity and ys is the ship fixed coordinate, shown in Fig. 1. n = w/27r and D are the propeller rotational frequency and diameter, respectively. The total cavity velocity can also be expressed in terms of the local derivatives along the s (chordwise), v (spanwise), and n (normal) grid directions: q Vs [LS (S V) ~ + Vv [LV ~ )< + (Vn) Il (6) where s, v, and n denote the unit vectors along the non-orthogonal curvilinear coordinates s, v, and n, respectively. The total velocities on the local coordi- nates (Vs. Vv, An) are defined as follows: Vs ~3 + qin S; Vv ~ + qin V; Vn ~ + qin n (7)
Note that if s, v, and rz were located on the "ex- act" cavity surface, then the total normal velocity, On, would be zero. However, this is not the case since the cavity surface is approximated with the blade surface beneath the cavity and the wake sur- face overlapped by the cavity. Although Vn may not be exactly zero on the approximated cavity surface, it is small enough to be neglected in the dynamic boundary condition (Fine October, 1992~. Equations 5 and 6 can be integrated to form a quadratic equation in terms of the unknown chord- wise perturbation velocity A. By selecting the root which corresponds to the cavity velocity vectors that point downstream, the following expression can be derived: INS = --tin S + Vv COST + sin ¢ ~/~qc ~2 - Vv2 (8) where ~ is the angle between s and v directions, as shown in Fig. 2. Equation 8 can then integrated to form a Dirich- let type boundary condition for ¢: ,}: '[ --tin S + Vv COS ~ + sin ¢~/~qc ~2 _ VV2 ~ ds where s = sO corresponds to the cavity leading edge. The terms ~ and ~ inside A and Vv on the right- hand-side of Eqn. 9 are also unknowns and are deter- mined in an iterative manner. The value of otsO, v, t) is determined via cubic extrapolation of the unknown potentials on the wetted blade surface immediately upstream of the cavity. On the cavitating wake surface, the coordinate s is assumed to follow the streamline. It was found that the crossflow term (rev) in the cavitating wake region had a very small effect on the solution (Fine October, 1992, Fine & Kinnas 1993b). Thus, the to- tal cross flow velocity is assumed to be small, which renders the following expression for ~ on the cavitat- ing wake surface: as o+(s,v,t) = 0 (ST' V, t) + / {qin s + Iqcl} ds ST (10) where ST iS the s-direction curvilinear coordinate at the blade trailing edge. o+ and o- represent the potential on the upper and lower wake surface, re- spectively. The value of ¢+ (ST, v, t) can be obtained by applying Eqn. 9 at the blade trailing edge. · Kinematic Boundary Condition on Cavitating Sur- faces The kinematic boundary condition requires that the total velocity normal to the cavity surface to be zero: DO (nhas, v, t)) = (11) [ 63 + qua, y, z, t) Vie (nkits, v, t)) = 0 where n and h are the curvilinear coordinate and cavity thickness normal to the blade surface, respec- tively. Substituting Eqn. 6 into Eqn. 11 yields the fol- lowing partial differential equation for h on the blade (Kinnas & Fine 1992~: ,, As - cos¢Vv] + ,:' tVv - cos¢Vs] (12) sin2¢ (Vn8t ~ Assuming again that the spanwise crossbow ve- locity on the wake surface is small, the kinematic boundary condition reduces to the following equa- ¢(s, v, t) = ¢(sO, v, t)+ (9) tion for the cavity thickness (hw) in the wake: [ on on ~ ~~ = em As (13) Note that hw in Eqn. 13 is defined normal to the wake surface. In addition, the quantity hw at the blade trailing edge is determined by interpolating the upper and/or lower cavity surface over the blade and computing its normal offset from the wake sheet. The extent of the unsteady cavity is unknown and has to be determined as part of the solution. The cavity length at each radius r and time t is given by the function lfr, t). For a given cavitation num- ber, an, the cavity planform, lfr, t), must satisfy the following condition: ~ (I (r, t); r, cony _ h (I(r, t), r, t) = 0 (14) where ~ is the cavity height at the trailing edge of the cavity. Equation 14 requires that the cavity closes at its trailing edge. This requirement is the basis of an iterative solution method that is used to find the cavity planform (Kinnas & Fine 1993~. Solution Algorithm The unsteady cavity problem is solved by in- verting Eqn. 3 subject to conditions 4, 9, 10, and 14. An iterative pressure Kutta condition (Kinnas & Hsin 1992) is applied to ensure equality of pres- sures at both sides of the trailing edge everywhere
on the blade. The numerical implementation is de- scribed in detail in (Kinnas & Fine 1992~. In brief, for a given cavity planform, Green's formula is solved with respect to unknowns S) on the wetted blade and hub surfaces, and unknowns ~ on the cavity sur- face. The cavity heights on the blade and the wake are then computed by differentiating Eqns. 12 and 13 with a second order central finite difference method. Finally, the correct cavity planform for each time step is obtained iteratively using a Newton-Raphson technique which requires the cavity closure condition (Eqn. 14) to be satisfied. It should be noted that the split-panel method developed by (Kinnas & Fine 1993) is used to treat blade and wake panels that are intersected by the free surface. In addition, at each time step, the solution is only obtained for the key blade. The influence of each of the other blades is accounted for in a progressive manner by using the solution from an earlier time step when the key blade was in the position of that blade. · Cavity Detachment Search Criterion The cavity detachment location is determined via an iterative algorithm. First, the initial detach- ment lines at each time step (or blade angle) are obtained based on the fully wetted pressure distribu- tions. The search algorithm at each spanwise strip begins at the section leading edge and travels down- stream to the section trailing edge. The initial de- tachment location for each strip is given as the first point along the chordwise direction where the wetted pressure is less than or equal to the vapor pressure (i.e.,Cp > Any. The cavity detachment locations at each radial strip are then adjusted iteratively at every revolution until the following smooth detach- ment condition is satisfied: 1. The cavity has non-negative thickness at its leading edge, and 2. The pressure on the wetted portion of the blade upstream of the cavity should be greater than the vapor pressure. It can be shown that the above criterion is equiva- lent to the Villat-Brillouin condition (Brillouin 1911, Villat 1914). SUPERCAVITATING PROPELLERS Experimental evidence shows that the separated zone behind the thick blade trailing edge forms a closed cavity that separates from the practically ideal irrotational flow around a supercavitating blade sec- tion (Russel 1958~. Furthermore, the pressure within the separated zone (also called the base pressure) can be assumed to be uniform (Raibouchinsky 1926, Tulin 1953~. However, in high Reynolds number flows, the mean base pressure depends on the me- chanics of the wake turbulence (RoshLo 1955~. This implies that a turbulent dissipation model, such as the one used in (Vows & Chen 1987), is necessary to determine the mean base pressure and the extent of the separated zone. However, the use of such models in the prediction of unsteady 3-D cavitating propeller flows is not practical for engineering purposes. To simplify the physics, (Kudo & Ukon 1994) as- sumed the supercavitating blade section to be base ventilated (i.e. the mean base pressure equal to the vapor pressure), and solve the steady cavitating pro- peller problem using a 3-D vortex-lattice lifting sur- face method. Later, (Kudo & Kinnas 1995) modified the method to allow for a variable length separated zone model which determines the mean base pres- sure. However, the length of the separated zone is arbitrarily specified by the user, and has found to affect the pressure and cavity length near the blade trailing edge under fully wetted and partially cavitat- ing conditions. Furthermore, the method of (Kudo & Kinnas 1995) cannot be applied in unsteady cavi- tating analysis since the length of the separated zone changes with blade angle. In the present method, the assumption of (Kudo & Ukon 1994) is used for the analysis of supercav- itating propellers subjected to steady and unsteady inflow. The base pressure is assumed to be constant and equal to the vapor pressure, and the extent of the separated zone at each time step is determined iteratively like a cavity problem. The logic behind this assumption are: 1. The base pressure should equal to the vapor pressure in the case of supercavitation,. 2. The separated zone has to communicate with the supercavity in the span-wise direction in the case of mixed cavitation (i.e. one part of the blade is wetted or partially cavitating while an- other part is supercavitating). 3. Most supercavitating propellers operate in su- percavitating conditions. Hence, the present method solves for the sepa- rated zone like an additional cavitation bubble. How- ever, the "openness" at the blade trailing edge poses a problem for the panel method. Thus, a small clos- ing zone, shown in Fig. 3, is introduced. The precise geometry of the closing zone is not important, as long as it is inside the separated region and its trailing edge lies on the aligned wake sheet. ~ The method is
modified so that it treats the original blade and the closing zone as one solid body. Thus, the integral surface in Green's formula (Eqn. 3) must now in- cludes the blade and hub surfaces, the closing zones, and the wake surfaces. Moreover, Eqns. 9 and 12 should also be applied over the closing zone. The solution method is the same as that for conventional fully submerged propellers. However, additional care is needed to ensure the potential to be continuous between the wetted portions of the blade, the cav- ity surfaces, and the closing zones. Furthermore, an additional condition which requires the cavities to detach prior to the actual blade trailing edge is also needed. In the force calculation, the pressure acting on the thick blade trailing edge (shown in Fig. 4) must also be included. This is accomplished by mul- tiplying the separated region pressure acting normal to the blade trailing edge with the trailing edge area. Details of the numerical algorithm and numerical val- idations of the method are presented in (Young & Kinnas 2002, Young 2002~. Fully wetted G-~ ~ blade section _ ~ ~ r~ / """"my Amp., =T Partially cavitating cavity blade section (P= _ ~ Supercavitating can ty I ,\ = p\:) N\\ __ - closing zone wake - s~par<rtedlegion (P = Pa) / ;~ .~ closing zone / ~ wake ~! . ~ ~ ~- Sep<It~tC'~l ret'i()ll tP=P~) ---~~~~~~~~- - ~ ~~~~~ ~~ closing zone ~7""""' lo/ \~ G~ / wake blade section ~4'~" ~ Figure 3: Treatment of supercavitating blade sec- tions. Psuction ~ /W At/ ~ ~ Phase pressure Figure 4: Pressure integration over a blade section with non-zero trailing edge thickness. Validation with Experiments As depicted in Fig. 3, the present method is applicable to fully wetted, partially cavitating, and supercavitating conditions in steady and unsteady flows. Cavitation patterns on supercavitating pro- pellers that can be predicted by the present method are shown in Fig. 5. ., __ _ ~ ,_ _ ~ (c) (d) (b) ~_~ _ _R ~ ~ ~ I ~ ~ I l i ~ \ _ = ~ ~ ,1,,1~ ~ ~ _ 1 l 1 1 5111 ~ (f) x~ _ 1 0111 ~ _ _ ~ _ ~ ~ ~ ~ ~ ~ ~ _ _ ~ ~ ~ ~ ~ ~ _l ~ ~ ~ L __e ~ ! ~ (g) ==~~\ (h) _ (i) ~,,_. _ ~ __ I _ ~ _ ~ Figure 5: Cavitation patterns on supercavitating propellers that can be predicted by the present method. To validate the treatment of supercavitating propellers, the predicted force coefficients are com- pared with experimental measurements (Matsuda et al 1994) for a supercavitating propeller. The test geometry is M.P.No.345 (SRI), which is de- signed using SSPA charts under the following con- ditions: JA = V) = 1 10, TV = ~ = 0 40' and AT = ~ = 0.160. The discretized pro- peller geometry us shown in Fig. 6. Comparisons of the predicted and measured thrust (KT), torque (KQ = ~), and efficiency (71p = KT ~ ~ are shown in Fig. 7. It is worth noting that at JA = 1.3, there is substantial midchord detachment. Fig. 8 in- dicates that the detachment search condition is sat- isfied since the cavities have non-negative thickness and the pressures everywhere on the wetted blade surfaces are above the vapor pressure. As shown in Fig. 7, the numerical predictions compared well with experimental measurements. To further validate the method, the convergence of the predicted cavity planforms, as well as the thrust and torque coefficients, with number of panels are shown in Fig. 9 for JA = 1.3. AS depicted in Fig. 9, the predicted results converged quickly with number of panels.
I Figure 6: Discretized geometry of propeller SRI. 0.8 L 1°4 0.35 0.3 0.25 0.2 0.15 experiment 0.1 -am PROPCAV 0' ' ' I I, I, I,, 005 0.8 0.9 1 1.1 1 .2 1.3 1 .4 0.7 t 0.6 C 0.5 0.4 To 0.1 ~ ~ ~ ~ ~ G C /. 10*K - am. me. ~ ~ ~ o o 63 ~ 4E~-~WO ~Ceec 0.3 - 0.2 a JA l ( Figure 7: Comparison of the predicted and versus measured IT, KQ, and lip for different advance co- efficients. SURFACE-PIERCING PROPELLERS Since surface-piercing propellers are partially submerged, the computational boundary must also include the free surface. Hence, the perturbation po- tential, Op. at every point p on the combined wet- ted blade surface SWB(t)' ventilated cavity surface Scot (t)USc2(t) USC3(t), and free surface SF (t), must satisfy Green's third identity: 2~op~t)=J~is(~) [¢)q~t) anq(~) (P;q) ~nq(~)] (15) where S(t) _ Swatt) U Sc~(t) U SC2(t) U SC3(t) U SF (t) is the combined surfaced as defined in the blade section example shown on Fig. 10. nq is the unit vector normal to the integration surface, with the 'be,'\\ Figure 8: Predicted cavity shape and cavitating pres- sures for propeller SRI at JA = 1.3. 50x20 panels. Uniform inflow. I' 50X20: KT=0.1372, 1OKQ=O.4072 W-''''N'N' 70X30: K'O.1362, 1OKLQ=O.4048 \\' 70X20: KT=0.1366, KQ=0.4050 !_ 80XSO: KT=0.1353, 1OKQ=O.4031 Figure 9: Convergence of cavity shape and force coef- ficients with number of panels for JA = 1.3. Uniform inflow. positive direction pointing into the fluid domain. As in the case of fully submerged propellers, the "exact" ventilated cavity surfaces, Scat (`t) U SC2 (t) U SC3(t), are unknown and have to be determined as part of the solution. Thus, the ventilated cavity sur- faces are approximated with the blade surface under- neath the cavity, SC2(t) ~ SCatt), and the portion of the wake surface which is overlapped by the cavity, Scat (t) U Sc3(t) ~ Sow (t). The definition of SOB (t) and Smart) are shown in Fig. 10. Boundary Conditions · Dynamic Boundary Condition on the Free Surface and Ventilated Cavity Surfaces The dynamic boundary condition requires that the pressure everywhere on the free surface and on the ventilated cavity surface to be constant and equal to the atmospheric pressure, Palm. Redefining
If SF P = Patm P = Patm 1 1'1~ , /Sew "/ ¢+ known (P = Patm) ~ unknown D ~SWB | ~ unknown ~~ known (=qin n) Figure 10: Definition of "exact" and approximated flow boundaries around a surface-piercing blade sec- tion. an (PoPa~m)/~n2D2) as the ventilated cavi- tation number, the dynamic boundary condition re- duces to Eqns 9 and 10 on the on SoB(t) and Smart)' respectively. · Kinematic Boundary Condition on the Ventilated Cavity Surfaces The kinematic boundary condition on the ven- tilated cavity surfaces renders the same expression as Eqns. 12 and 13 on SoB(t) and SCw~t)' respec- tively. However, for partially submerged propellers, the cross-flow velocities are also assumed to be small on the blade surface (i.e. Vv ~ viscose on SCB(t)~. This reduces the 0oh term in Eqn. 12 to zero. The justification of this assumption can be found in (Fine October, 1992), where it is shown that the cross-flow term (evaluated iteratively) on the blade has a very small effect on the predicted supercavity on either a 3-D hydrofoil or a propeller blade. In addition, the 8oh term is difficult to evaluate due to the interrup- tion of the ventilated cavity by the free surface. · linearized Free Surface Boundary Condition on the Free Surface As a first step to model partially submerged pro- pellers in 3-D, the linearized free surface boundary condition is applied: 6~J~°((x, y, z, t) + g~33v (x, y, z, t) = 0 (~16) at Us =R + h (i.e. free surface) where h and R are the blade tip immersion and blade radius, respectively, as defined in Fig. 11. ys is the vertical ship-fixed coordinate, defined in Fig. 1. ' , . ...... .... ~ . . ~ ~ S~ : - split panels - - ~ x~S >4 z I J ' Lo . . .. Figure 11: Definition of ship-fixed (xs~yS'ZS) and blade-fixed (x,y,z) coordinate systems. Assuming that the infinite Froude number con- dition (i.e. Fr = rt2D/g ~ oo) applies, Eqn. 16 reduces to: ¢(x, y, z, t) = 0 at Us =R + h (17) The assumption that the Froude number grows with- out bounds is valid because partially submerged pro- pellers usually operate at very high speeds. Studies by (Shiba 1953, Brandt 1973, Olofsson 1996) have also shown that the effect of Froude number is neg- ligible for Fn<~ = >~ > 4 or Fr = ngD > 2 in the fully ventilated regime. The Negative Image Method Equation 17 implies that the negative image method can be used to account for the effect of the free surface. Consequently, only vertical motions are allowed on the free surface. This is accomplished by distributing sources and dipoles of equal strengths but with negative signs on the location of the mirror image with respect to the free surface. A schematic example of the negative image method on a blade section is shown in Fig. 12. Solution Algorithm For surface-piercing propellers, Green's formula is only solved for the total number of submerged pan- els on the key blade and the cavitating portion of the key wake. The values of the 0 and ~ are set equal to zero on the blade and wake panels that are above the free surface. Note that the current algorithm does not re-panel the blades and wakes at every time step, in order to maintain computation efficiency. As a result, there are some panels that are partially cut by the free surface. In the present algorithm, the
~~ 1 /1 ~ 1 1 1 ', .~ dipole (opposite normal) ~ 1 ~~ 1 / 1 sinkl,,',`~\~\ "in 1 1 1 I \ ,, ~ I let Figure 12: Schematic example of the negative image method on a partially submerged blade section. strengths of the singularities are also set equal to zero for the partially submerged panels. Neverthe- less, a method similar to the split-panel technique (Kinnas & Fine 1993) can be applied to account for the effects of these panels. The solution algorithm for partially submerged propellers is similar to that explained earlier for fully submerged supercavitating propellers. However, it- erations to determine the correct cavity lengths are no longer necessary since the ventilated cavities are assumed to vent to the atmosphere, as observed in experiments. · Ventilated Cavity Detachment Search Algorithm Depending on the flow conditions and the blade sec- tion geometry, the ventilated cavities may detach aft of the blade leading edge. Thus, the cavity detach- ment locations on the suction side of the blade are searched for iteratively at each time step until the smooth detachment condition is satisfied. In addi- tion, due to the interruption of the free surface, the following detachment conditions must also be satis- fied for partially submerged propellers: · The ventilated cavities must detach at or prior to the blade trailing edge; and · During the exit phase (i.e. when part of the Figure 14: Axial velocity distribution at the pro- blade is departing the free surface), the venti- pelter plane. Propeller model 841-B. h/D = 0.33. lated cavities must detach at or aft of the inter- Copied from (Olofsson 1996~. section between the blade section and the free surface. It should be noted that the ventilated cavities on the pressure side of the blade are always assumed to detach from the blade trailing edge. It is pos- sible to also search for cavity detachment locations on the pressure side. However, such occurrence is unlikely due to the high-speed operation of partially submerged propellers. Validation with Experiments In order to validate the partially submerged pro- peller formulation of the method, numerical predic- tions for propeller model 841-B are compared with experimental measurements collected by (Olofsson 1996~. A photograph of the partially submerged pro- peller and the corresponding BEM model are shown in Fig. 13. The velocity distribution at the propeller plane is shown in Fig. 14. The experiments were conducted at the free-surface cavitation tunnel at KaMeWa of Sweden. Details of the experiments are given in (Olofsson 1996~. am_ ~ - ~ ~ _ _~ ~ ~ ez) Figure 13: Photograph of propeller model 841-B shown in (Olofsson 1996), with corresponding BEM model on the right. o on 0.4 ..................... 0.6 _ 0.E flat plate ~ free surface It.,.,,,_,,,,,,,_ _ v 1 , . ~ ................................................... ` velocity distribution in propeller plane 'I, l . ~ . a. - 0.25 0.5 V' x 0.75 1 The current method assumes the blade to be
rigid, the cavities to be fully ventilated, and the Ffoude number to be infinite. Thus, the following combination of test conditions were selected to mini- mize the effect of Fioude number, cavitation number, and blade vibration: shaft yaw angle: ~ = 0° shaft inclination angle: by= 0° blade tip immersion: h/D = 0.33 advance coefficient: Froze number: cavitation number: JA = Vie = 1.01.2 F'l D = N/~ = 6. O ~V= ~=0.25 Note that PO is the pressure far upstream on the shaft axis. Comparisons of the observed and predicted ven- tilated cavity patterns are shown in Figs. 15 to 17. Comparisons of the measured and predicted individ- ual blade force and moment coefficients are shown in Fig. 18. The solid lines and the symbols in Fig. 18 represent the load coefficients predicted by the present method and measured in experiments, re- spectively. (,KFX, KEY, KFZ, KMX, KMY, KMZ ~ are the six components of the individual blade force and moment coefficients defined in the coordinate system shown in Fig. 1. ~ = oO7 gO°, ~ soo Figure 15: Comparison of the observed (top) and predicted (bottom) ventilated cavity patterns for JA = 1.2. Propeller model 841-B. 4 Blades. h/D = 0.33. 60x20 panels. /\d = 6°. As shown in Figs. 15 to 18, the predicted venti- lated cavity patterns and blade forces agree well with experiments for JS = 1.2. In addition, the method also converged quickly with time step size and grid size, as shown in Figs. 19 and 20. However, there is more discrepancy between the predicted and mea- sured individual blade forces for JS = 1.0, which is shown in Fig. 21. if_ ~ = 30°, 120°, 210° Figure 16: Comparison of the observed (top) and predicted (bottom) ventilated cavity patterns for JA = 1.2. Propeller model 841-B. 4 Blades. h/D = 0.33. 60x20 panels. /\d = 6°. 9=60° 150° 240° Figure 17: Comparison of the observed (top) and predicted (bottom) ventilated cavity patterns for JA = 12 Propeller model 841-B. 4 Blades. h/D = 0.33. 60x20 panels. i\§ = 6°. The authors believe that the discrepancies are attributed to: . . Inadequate simulation of the blade entry phe- nomena. At the instant of impact, a very strong jet is developed near the blade leading edge, which results into very high slamming forces. In the current formulation, the presence of the jet cannot be captured due to the application of the negative image method. In other words, a non- linear free surface model should be applied to capture the development of the jet, so that the added hydrodynamic force can be directly eval- uated. Inability of the current method to capture the increase in free surface elevation. The overall free surface rises due to the cavity displacement
O.O5 1 -0.04 .0.03 = -0.02 Fox (P) ·1811~1~1~1~1~1 an (P) ~ Pox A) O KPZ (E) 0. 0 1 . ... .. . O. tic 0.01 0.02 1 1 0.015 0.01 0.005 -0.00 0.0025 0.0025 0.005 ~ -0.025 -0.02 ~~~~~ ~~~~--~ ~ KF / ~ C) -0.015 blade angle 0.01 360 it, -0.01 ~ -0.005 KM,, (P) ·I.I.I.I.I.I.I KMZ (P) '` KM,. (E) 0 KMZ (E) _ .K /- I 90 1 80 blade angle 270 .02 0.01 _ ~60 .I-~ ; KMy (P) KMY (E) . . ~ . . . . . . . . . . ~ . ~0 ~ ~ 0 005 0 90 180 270 360 blade angle Figure 18: Comparison of predicted (P) and mea- sured (E) blade forces for JA = 1.2. Propeller model 841-B. 4 Blades. h/D = 0.33. 60x20 pan- els. /\§ = 6°. . effect (Olofsson 1996~. As a result, the actual immersion of the propeller increases, which in turn adds to the hydrodynamic blade load. This effect can be observed in the experimental data shown in (Olofsson 1996) for low JA values. Inability of the current method to model the ef- fect of blade vibrations. Blade vibration is a resonance phenomenon which affect the blade shapes and loadings. The effect of which is evi- dent via the "humps" (amplified fluctuations su- perimposed on the basic load) observed in the experimental data shown in Figure 21. It was also observed during experiments that the fre- quencies of these fluctuations modulate between the blade's fundamental frequency in air and in water (Olofsson 1996~. However, the current model assumes rigid body motion. Thus, the effects of blade vibrations cannot be captured. 2-D STUDY OF FREE SURFACE EFFECTS In order to quantify the added hydrodynamic forces associated with jet sprays generated at the AD = 9O AD = 6° AD = 3° o 0.005` ) 90 180 to blade angle (degrees) ~ , , 1 _ 360 Figure 19: Convergence of thrust (KT) and torque (KQ) coefficients (per blade) with time step size. Propeller model 841-B. JA = 1.2. 70x30 panels. 6 propeller revolutions. blade entry and exit phase, a systematic 2-D study has been initiated. The objective of the 2-D study is to find a simplified approach to quantify the added hydrodynamic forces due to slamming and change in free surface elevations. The progression of the pro- posed 2-D study is shown in Fig. 22. Previous Work The problem of a 2-D rigid wedge entering the water was first studied by (don Karman 1929) and (Wagner 1932~. Both assumed that the velocity field around the wetted part of the body can be approx- imated with the flow field around an infinitely long flat plate. The model in (don Karman 1929) as- sumed that the free surface is flat, while the model in (Wagner 1932) accounted for the deformation of the free surface. However, the similarity method of (Wagner 1932) reduced the unsteady problem to a steady one. Since then, the slamming problem on a 2-D body has been extensively studied by (Makie 1969, Cox 1971, Yim 1974~. In particular, (Yim 1974) applied a linearized theory to study the wa- ter entry and exit of a thin foil and a symmetric wedge with ventilation. Later, (Wang 1977, Wang 1979) also applied linear theory to study the verti- cal and oblique entry of a fully ventilated foil into a horizontal layer of water with arbitrary thickness. The method of (Wang 1977, Wang 1979) was later extended by (F'uruya 1984, Furnya 1985) for the per- formance prediction of surface-piercing propellers. More recently, the 2-D wedge entry problem was thoroughly investigated by (Zhao & Faltinsen 1993,
-0.025 -0.02 C) -0.01 5 cur ~0.005 -0.01 o 0.005 ~f ~ I I ~ I I I I , , I C) 90 180 270 360 blade angle (degrees) Figure 20: Convergence of thrust (KT) and torque (KQ) coefficients (per blade) with panel discretiza- tion. Propeller model 841-B. JA = 1.2. /\§ = 6°. 6 propeller revolutions. Zhao & Faltinsen 1998~. They applied a boundary element method with constant source and dipole dis- tributions. The exact nonlinear free surface bound- ary condition was used. A special model was used to treat the thin jet that develops at the intersection between the free surface and the body. The method was verified by comparisons with similarity solutions by (Dobrovol'skaya 1969) and asymptotic analysis by (Wagner 19321. Similar methods were also developed by (tin & Ho 1994, Falch 1994, Fontaine & Cointe 19971. However, the focus of the previous investiga- tions was on slamming loads on ship hulls with small dead rise angles and no ventilation. Formulation The first step for the proposed 2-D analysis in- volves analyzing the flow around a rigid hydrofoil entering the water at an arbitrary angle of attack. The formulation is similar that presented in (Zhao & Faltinsen 1993), which was derived for the vertical water entry of a symmetric wedge without ventila- tion. Consider a rigid, 2-D hydrofoil entering into initially calm water of an unbounded domain at a constant velocity V and angle attack or, as shown in Fig. 23. For incompressible, inviscid, and ir- rotational flow, the perturbation potential ¢, de- fined with respect to the undisturbed free surface coordinates (x, y) shown in Fig. 23, at any time t satisfies Laplace's equation in the fluid domain: (V24(x, y, t) = 0~. Thus, the perturbation potential on the bound- -o~o5 -0.04 50X10 -0.03 60X20 70X30 ~ 0.02 -0.01 O. _ 0.02 n n1 ~ ~0.01 _ 0.005 O' ...... .. ....... -0.005 n 90 180 270 360 blade angle .os- 0.04 o 03 ~ 0.02--~= 0 01 ~ _~~~~ 1:' ~~_ {~ ~ 0.0 1 ~ lo blade angle . fox (P) ..... A ~ .. .. .. . ~I.I.I.I.I.I.' KFZ(P) ^^ ~ ~ K1:Y ~ Kp7 (E' KM,` (P) I _ ~I~I~I~I~I~I~I KMZ (P) K VIX (E) KMZ (E) ~~~ KMX -0.02 0.01 F~ , 0.0025 O al~lalelelalai KMy (P) ~ KMY (E) ~~~- ~~ ~~~ ~~ ~ K ; . ~ . it. . ''. - - ' ' ''-'- - -- - -- 1-- ' - - -' ---- - - -- - - -- - . - - . ~~ ~ -0.0025 -0 005 . . , . . 1 . . . . . I O 90 180 270 360 blade angle Figure 21: Comparison of predicted (P) and mea- sured (E) blade forces for JA = 1.2. Propeller model 841-B. 4 Blades. h/D = 0.33. 60x20 pan- els. /\§ = 6° ary, S(t), of the computation domain, is represented by Green's third identity: o~x,y,t) = Jr [_ r1,t)~G(,;~71'~' (18) + ,94~;'71'~) G((, rid, tic dS((, r1, t) where G = inn, r = :/(x(12 + (Y _ 7112 and S(t) = SWB (t) U SF (t) U SOO (defined in Fig. 23~. Notice that SF(t) includes the free surface and the ventilated surface as a whole. n is the unit vector normal to the integration surface, which points into the fluid domain. It should be noted that for this problem, the perturbation potential (~) is the same as the total potential (~) since the system is defined with respect to the undisturbed free surface coordi- nates (x,y). · Kinematic Boundary Condition on SF: The kine- matic free surface condition requires fluid particles on the free surface and ventilated surface to remain
(1) (2) A. (3) , (4) n (~ Scc ~ 1 so ~ 1 Figure 22: Planned progression of the 2-D nonlin- ear study for the water entry and exit problem of a surface-piercing hydrofoil. on the surface: 09 + 04 00 = 0° (19) at ~xbx by DO Dt where 71(x, t) iS the vertical coordinate of the fluid which can be written as: particle, as shown in Fig. 23. · Dynamic Boundary Condition on SF: On the exact free surface and ventilated surface, where the pressure should be constant and equal to the at- mospheric pressure: bt + 2 [( 0~ ) + ( i9y ) ] + 90 = 0 (20) · Combined Kinematic and Dynamic Boundary Con- dition on SF: Equations 19 and 20 can be combined to form ((x, t) and 71(X, t) are the horizontal and vertical co- a system of three equations using the definition of ordinates of the fluid particle, which at t = 0 was Ox ~ (5) it------ / r ~ v24=o on SF: _ Dt ax it\_ ' ]~ So D71 Dt By Dt 2 [(0X) + (0Y) ] 99 // Figure 23: Definition of coordinate system and con- trol surface for the water entry problem of a 2-D rigid hydrofoil. substantial derivative, Dt = aft + V) · V: D: So = Dt fix DO So _ = Dt by 52 + (0o j2~ - (21> DF G (22) F = ~ 71 G = ~ 1 l 2 [(~¢,)2 + (~,)2~ J (23)
located on the undisturbed free surface at (x, t = 0), as shown in Fig. 23. · Kinematic Boundary Condition on SWB: The kinematic boundary condition requires the following condition to be satisfied on the wetted body surface: (<Vo- V) n= 0 (~24) · Kinematic Boundary Condition or SOO: The kinematic boundary condition on the infi- nite boundary requires zero normal velocity across the boundary: ,,; = 0 (25) · Initial Boundary Condition on SF: The initial boundary condition on the free sur- face are set as follows: o~x,y,O)=0 ~ 71(X, 0) = 0 ~ at t = 0 (~26) ((x, 0) = x ) Solution Algorithm AMP At each time step, Green's formula (Eqn. 18) is fly i+ ~ ~+~ solved with respect to unknowns o on SWB and SOO, and unknowns ~ on SF. The control surface, S. is discretized into a number of straight segments. In order to avoid singularities at intersection points, ~ and ~ for each panel are approximated with constant and linear strength distributions, respectively. The values of ~ are assigned at the panel mid-points, and the values of o are computed at the panel end points. The generation of the jet panel follows the same method presented in (Zhao & Faltinsen 1993~. In the time integration, the new particle posi- tion and new values of o on the free surface and ven- tilated surface are calculated by time stepping the fluid particle at the mid-points of the old elements. As explained in (Zhao & Faltinsen 1993), the details of the new element procedure is very important in order to conserve fluid mass, particularly in regions of high curvature. In the current scheme, a cubic spline method is used to: (1) calculate the position and potential of particles at panel mid-points based on the existing values at panel endpoints, and (2) cal- culate the position and potential at panel endpoints based on the updated positions of panel mid-points. It should be noted that the cubic spline method is only employed for regions of high curvature, i.e. re- gions near the jet and ventilated surface. The time stepping procedure used to solve Eqn. 22 is explained below. · Predictor step for time t + 1: 1. Compute A, it, and o at panel mid-points on SF at time t + 1 using a second-order explicit Adams-Bashforth predictor method: FiP+ 2 ,~+i = Fi+ 2 't + 2 t3Gi+ 2 'tGi+ 2 't- i] (27) 2. Refine the panel distribution in highly curved _ _ _ regions. Calculate Fi+~ i;, Gi+~ i;, and Gi+1 ~_~ at new panel midpoints. 3. Apply Eqns. 24 and 25 to determine the known values of - A+ t+~ on the wetted body bound- ary and infinite boundary, respectively. 4. Solve Green's formula, Eqn. 18, to obtain jiPt+i and - a.+ ~+~ everywhere. 5. Calculate velocities at panel mid-points: ~ = [ As=+us] By i+ 2 ,[+l IS 0~ i+ 2't+i [prissy + d~rlrtY] ~ (28) where s = (Sx'Sy) and n = (ninny) are the tangential and normal unit vectors, respectively. Note that values of fib are known at the panel endpoints and values of ~ are known at the panel mid-points. Thus, ~ is calculated as fol- lows: UP = A+,+ - ~i,~+i (29) IS i+ 2 a+ ~ SP+ ~ t+ ~SP t+ ~ where so+ i,+ is the arclength of panel endpoint i + 1 at time t + 1 for the predictor step. 6. Calculate GP+! !+, on SF. · Corrector Step for Time t + 1: I. Compute A, 71, and ~ at panel mid-points on SF at time t+ 1 using a third-order implicit Adams- Moulton corrector method: Fi+ 2 ,~+l = Fi+ 2 't + ~2 t5G~P+~ i+, +8Gi+ 2,~ - Gi+ 2't-~]
2. Refine the panel distribution in highly curved regions. Calculate Fi+1 t+i and Gi+~ ~ at new panel midpoints. Apply Eqns. 24 and 25 to determine the known values of Pi+ i- i;+ L on the wetted body bound- ary and infinite boundary, respectively. 4. Solve Green's formula, Eqn. 18, to obtain Hi,+ and ~ i+ ~ t+~ everywhere. 5. Calculate velocities, ~i+1 i+ and ~i+'- +' at panel mid-points. Calculate Gi+ 1 ~+~ on the free surface. ~ 6 · Pressure and Impact Force Calculation The pressure at the wetted body surface is calculated at panel mid-points of each time t via Bernoulli's Equation: ;' 0.4 ~loo.3 V=0.2 and ( P )i+2,t let i+ 2 ,t 99i+:,t (31) 2 [(do) (0Y ) ~ it - where ~i+1 t iS calculated as follows: TV _ DO ~ ~ `9o: 2 ~ `9o) 2] 0t i+2,! Dt i+2,t L<5XJ ~ YJ ~ i+2,{ (32) Dt in 2 ,t 2/\~ (hi+ 2 ,t+ihi+ 2 ,t-~ ) (33) Once the pressure has been computed, the im- pact force can be calculated by integrating the pres- sure over the wetted area on the solid body. Preliminary Results · Vertical Entry of a Symmetric Wedge In order to validate the method, the predictions for the 2-D wedge entry problem are first compared with those presented in (Zhao & Faltinsen 19934. Note that the formulation for the water entry prob- lem of 2-D symmetric wedge is the same as that ex- plained in above with the following exceptions: · The wedge is symmetric with respect to the y- axis. · There is no ventilated cavity surface. Thus, SF only includes the free surface. For surface-piercing hydrofoil/propeller applica- tions, the dead-rise angle is often very high (i.e. a < 10°~. Thus, the case of or = 9° (highest dead- rise angle presented in (Zhao & Faltinsen 1993~) is selected for validation studies. The predicted free surface elevation and pressure distribution on the body are shown in Fig. 24. The current method com- pares very well with predictions by (Zhao & Faltinsen 1993), which is also shown in Fig. 24. n -1 L 10C = 9°1 . . . . . O 0.5 1 XIV: present method I_ (Zhao & Faltinsen 1993) ~. 1 -0.5 0 ylvt Figure 24: Predicted free surface elevation and pres- sure distribution during water entry of a 2-D wedge. 00 = 9°. · Oblique Entry of a Flat Plate To further validate the method, numerical pre- dictions for the oblique entry of a flat plate are also presented. The predicted pressure distributions for or = 5° and 8° are shown in Figs. 25 and 26, re- spectively. Also shown in Figs. 25 and 26 are the results obtained using the method of (Savineau & Kinnas 1995), which applied the linearized free sur- face boundary conditions. As expected, the current method predicted higher forces and increased wetted area compared to (Savineau & Kinnas 1995~. CONCLUSIONS A 3-D boundary element method has been ex- tended to predict the performance of supercavitating and surface-piercing propellers. The current method is able to predict complex types of cavity patterns on
6 r 5 4 5,- 3 2 present method (Savineau, 96) do, la = sol - 0 N\ ~ present method (Savineau, 96) 1 -0.8 -0.6 -0.4 -0.2 0 0.2 ylvt 1 -0.8 -0.6 -0.4 -0.2 0 ylvt Figure 25: Pressure distribution on the wetted body Figure 26: Pressure distribution on the wetted body surface predicted by the present method and by the surface predicted by the present method and by the method of (Savineau 1993~. Flat plate. or = 5°. both sides of the blade surface for conventional and supercavitating propellers in steady and unsteady flow. The method is also able to simulate the un- steady separated region behind blade sections with non-zero trailing edge thickness. For surface-piercing propellers, the negative image method is used in the 3-D model to account for the effect of the free surface. The method is also able to search for detachment locations of ventilated cavities. In general, the pre- dicted cavity planforms and propeller loadings com- pare well with experimental measurements and ob- servations. The method also appeared to converge quickly with grid size and time step size. A 2-D study using the exact free surface boundary conditions has been initiated to quantify the added hydrodynamic forces associated with jet sprays during the entry phase. An overview of the formulation and preliminary results for the 2-D study was presented. Current efforts include the following: · Determine the effect of prescribed pressure on the separated zone behind non-zero trailing edge sections in fully wetted and partially cavitating conditions. Complete the 2-D nonlinear study of surface- piercing hydrofoils, and find a simplified ap- proach to incorporate the results into the 3-D model. Couple the hydrodynamics with a structural analysis model to include the effect of blade vi- bration. Validate the results with experimental measure- ments, and perform convergence studies with time and space discretizations. . method of (Savineau 1993~. Flat plate. of = 8°. ACKNOWLEDGMENT Support for this research was provided by Phase III of the "Consortium on Cavitation Performance of High Speed Propulsors" with the following members: AB Volvo Penta, American Bureau of Shipping, El Pardo Model Basin, Hyundai Maritime Research In- stitute, John Crane-Lips Norway AS, Kamewa AB, Michigan Wheel Corporation, Naval Surface Water Center Carderock Division, Ulstein Propeller AS, and VA Tech Escher Wyss GMBH. REFERENCES BARR, R. A. 1970 Supercavitating and superventi- lated propellers. Transactions of SNA ME, 78, PP. 417-450. BRANDT, H. 1973 Modellversuche mit schiffspro- pellern an der wasseroberflache. Schiff and Hafen, 25, 5, PP. 415-422. BRILLOUIN, M. 1911 Les surfaces de glissement de Helmholtz et la resistance des fluides. Annales de Chimie and de Physique, vol. 23, pp. 145-230. CHOT, J. 2000 Vortical inflow - propeller interac- tion using unsteady three-dimensional euler solver. Doctoral dissertation, Department of Civil Engi- neering, The University of Texas at Austin. Au- gust. Cox, B. 1971 Hydrofoil theory for vertical water entry. Doctoral dissertation, Department of Naval Architecture, MIT. May.
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