Integrated dynamics modeling for supercavitating vehicle systems

Transcription

1 SNAK, 015 Int. J. Nav. Arhit. Oean ng. (015) 7:346~363 pissn: , eissn: Integrated dynamis modeling or superavitating vehile systems Seonhong Kim and Nakwan Kim Department o Naval Arhiteture and Oean ngineering, Seoul National University, Seoul, Korea ASTRACT: We have perormed integrated dynamis modeling or a superavitating vehile. A 6-DOF equation o motion was onstruted by deining the ores and moments ating on the superavitating body surae that ontated water. The wetted area was obtained by alulating the avity size and axis. Cavity dynamis were determined to obtain the avity proile or alulating the wetted area. Subsequently, the ores and moments ating on eah wetted part-the avitator, ins, and vehile body-were obtained by physial modeling. The planing ore-the interation ore between the vehile transom and avity wall-was alulated using the apparent mass o the immersed vehile transom. We integrated eah model and onstruted an equation o motion or the superavitating system. We perormed numerial simulations using the integrated dynamis model to analyze the harateristis o the superavitating system and validate the modeling ompleteness. Our researh enables the design o high-quality ontrollers and optimal superavitating systems. KY WORDS: Superavitating vehile; Integrated dynamis modeling; Superavity; 6-DOF equation o motion; Planing ore; Cavity dynamis; Fritional drag ore; Cavity bubble; Cavity model; Open-loop numerial simulation. INTRODUCTION Objets moving in luid media suh as water are slowed down by luid ores that are olletively alled drag. Drag ores inrease quadradially with the objet speed, and hene, the thrust ore also inreases. Thereore, underwater vehiles have a veloity limit due to the limit o thrust ore. Numerous researhes have been onduted to redue the drag and inrease the speed o underwater vehiles. In the 1970s, Russian sientists proposed a radially dierent approah to solve this problem; they proposed redutions in the surae area o the body that is in ontat with water to eliminate one type o drag, the skin-rition drag. When an objet moves ast in water, an air bubble alled avity is ormed. The superavitating tehnology, proposed by the Russians, is based on the idea that skin-rition drag an be redued dramatially when a vehile is enompassed by large gas bubbles. A superavitating vehile hanges rom ully wetted ondition to superavitating ondition, and this auses unsteady hydrodynamial ores and moments beause the wetted area o the vehile body hanges. The wetted area o the vehile body is determined by the shape o the avity and the relative position between the avity and vehile. Thereore, the most important task is to alulate the avity size and axis. I avity modeling is suessully ompleted, the wetted part o the vehile an be determined and it beomes possible to alulate the ores and moments ating on the vehile body and ontrol surae. Many sientists have studied the dynamis and harateristis o the superavitating vehile, but their studies have been Corresponding author: Nakwan Kim, nwkim@snu.a.kr This is an Open-Aess artile distributed under the terms o the Creative Commons Attribution Non-Commerial Liense ( whih permits unrestrited non-ommerial use, distribution, and reprodution in any medium, provided the original work is properly ited.

2 Int. J. Nav. Arhit. Oean ng. (015) 7:346~ limited to speii areas and their models do not enompass the entire superavitating system. In reent years, eorts have been made to study the dynamis o the superavitating vehile in various appliations. In these investigations, one researh ous has been the modeling o the system dynami under ully developed avitating onditions; ontrol problems have also been investigated. Dzielski and Kurdila (003) have researhed the modeling and ontrol problem in the early development phase. The longitudinal dynamis and ontrol problem have been studied by Vanek (007), Dzielski (011), and Fan et al. (011). Li et al. (014) and Hassouneh et al. (013) developed a vertial plane model inorporated with time-delay eet. Nguyen et al. (011) developed a dive-plane model with nonylindrial and nonsymmetri avity shape. Studies on superavitating low have also been onduted; early researhes into the partial avitation low o axisymmetri bodies using a steady potential-low boundary-element tehnique were perormed by Varghese et al. (005). Ahn et al. (01) studied the superavitating lows around a avitator or various two and three dimensional shapes. Park and Rhee (01) studied a high-speed super-avitating low around a two-dimensional symmetri wedge-shaped avitator using an unsteady Reynolds-averaged Navier-Stokes equations solver based on a ell-entered inite volume method. Reently, numerial approahes or a ventilated partial avity in the transition phase have been investigated. Zou et al. (010) established an empirial ormula or the gas-leakage rate o an unsteady ventilated superavity using the nonlinear least square method based on the mass balane equation; the results o numerial simulations were then ompared with the results rom experiments. Additionally, Xiang et al. (011) reported a numerial study on the drag redution mehanism reated by a ventilated partial avity. lbing et al. (008) studied the plate skin-rition drag redution aused by air bubbles; they tested the drag redution by injeting gas rom the line soure so that there are two distint drag-redution phenomena: bubble drag redution and air-layer drag redution. In this study, we have onstruted a 6-DOF equation o motion or a superavitating vehile system. To deine the ores and moments ating on the vehile body, the superavitating vehile system was divided into avitator, ins, and vehile body. ah part was modeled on the basis o previous researhes, and we integrated them to deine the dynamis o the superavitating system. In the modeling part, we deined all the terms inluded in the 6-DOF equation o motion. Ater the mathematial model was onstruted, we perormed open-loop numerial simulations. In this paper, all the terms are irst desribed and deined. Subsequently, we explain the alulations o the ores/moments and onstrut the mathematial model. Finally, by using the numerial simulation results, the physial harateristis o the superavitating vehile are analyzed and the modeling ompleteness is validated. The main ontribution o this study is integrated modeling o a 6-DOF superavitating vehile rom partially avitating ondition at low speed to superavitating ondition at high speed inluding transition phase. The model o a superavitating vehile onsists o every omponent o the dynamis suh as avity, avitator, body, ins, and planing, and aommodates oupled motion between the longitudinal and the lateral dynamis. The modeling o avity inludes avity dynamis with variation o the depth and the speed o the vehile, the eet o time-delay, and the deormation o avity axis due to the gravity eet whih generates asymmetri drag ores on rudders and reates in turn osillatory pithing motion. A desripttion o hydrostati and hydrodynami ore and moment applied on the wetted body is inluded in the modeling whih is pivotal in proper modeling o the vehile in the transition phase. In the ase o partial avity, the hydrostati and hydrodynamai ore and moment ating on the wetted body depend on how muh and whih diretion the vehile is in ontat with water. The diretion and the depth o immersion are determined by relative geometry o the vehile and the avity with the time delay eet, whih allows planing to arise in any radial diretion o the vehile. The integrated modeling o a superavitating vehile is expeted to be used as a test-bed or a design o ontroller or an optimization o onigurations o a superavitating vehile. MODLING OF SUPRCAVITATING VHICL 6-DOF equation o motion The avitator is loated at the ore body o the vehile, and a disk and our ins are loated at the at in the shape o a ross (+), as shown in Fig. 1. We seleted two oordinate systems, the earth-ixed oordinate system O XYZ and body-ixed oordinate system O XYZ, as shown in Fig. 1. The origin o the earth-ixed oordinate system was at sea level (zero) while the origin o the body-ixed oordinate system was loated at the enter o gravity.

3 348 Int. J. Nav. Arhit. Oean ng. (015) 7:346~363 Fig. 1 Superavitating vehile. The ores and moments ating on the vehile are generated by the our ins, avitator, gravity, thruster, wetted area o the body, and planing. The nonlinear equations o the superavitating vehile an be derived by using the linear momentum and angular momentum equations. u + qw vr M v ur pw + = F + F + F + F + F + F w + pv uq Thrust Cavitator Fins gravity wet Planing I xp + qr( I z I y) I q + qr( I I ) = M + M + M + M I zr + pq( I y I x) y x z Cavitator Fins wet Planing (1) Here, ( uvw,, ) are the linear veloities during the surge, sway, and heave motions, and (p, q, r) are the angular veloities during the roll, pith, and yaw motions. In this paper, it is assumed that the thrust ore is ating only in the X -diretion with magnitude T. FThrust 1 = T 0 0 () Cavity model The avity is a major omponent o the superavitating system. The behavior o the avity bubble around the vehile aets the ins and body immersion. The avitator ontinuously reates the avity while the vehile is moving. The avity axis, whih is integration o eah avity setion enter, is equal to the trajetory o the avitator, i there is no gravity eet and the avitator angle o attak is zero; in other words, the avity is axisymmetri. The plane that is perpendiular to the trajetory o the avitator is alled the avity setion and the avity ontour is obtained by integrating all the avity setions along the trajetory o the avitator. The avity hanges with time independent o the vehile dynamis. ah avity setion irst expands until it reahes its maximum radius and then starts to ontrat and disappear (Fig. ). Fig. Axisymmetri avity and avity setions. The important parameter that represents the avity harateristis is the avitation number σ :

4 Int. J. Nav. Arhit. Oean ng. (015) 7:346~ p p σ = (3) 0.5ρV p is the ambient pressure and p is the pressure inside the avity measured in Pa ; V is the vehile veloity. The avitation number σ is used to haraterize the potential o the low to avitate. When σ is low and the veloity o the vehile is ast or the avity pressure is high, large and wide avities tend to our. Numerous studies have investigated avity shape models; or example, Logvinovih (197) studied the avity radius and avity ontration rate or the disk type avitator in steady low. Two numeri onstants are deined to represent the avity model: k L = ( 3) 13 (4) Rn σ 4.5σ (40/17) 0.5 k = (1 (1 ) k1 ) (5) 1 + σ The ormula or the radius o the avity at a distane L rom the avitator is (0.8 (1 + σ ) ) = (6) n σ 0.5 R R k and the avity ontration rate R is (3/17) 1.9 R + σ σ = (0.8 ) V(1 ) k1 / ( k( 3)) (7) 17 σ 1+ σ σ The proposed ormulae or the avity shape are valid only i the ollowing inequality is satisied: 1.9 L > R n ( 3) (8) σ To represent the rontal part o the avity wherein the inequality is not satisied, the ollowing empirial ormula is usually used (Logvinovih, 197): R R n 1/3 3x = 1 +, when x< L Rn (9) The ormulae o Garabedian (1956) are useul or prediting the avity length L and the maximum avity radius R max : L 1 1 = CD ln (10) R σ σ n R R max n C σ D = (11)

5 350 Int. J. Nav. Arhit. Oean ng. (015) 7:346~363 C D is the avitator drag oeiient. The avity proile an then be estimated by May s ormula (May, 1975) as ollows: x L / R ( x) = Rmax 1 L / 1/.4 (1) The semi-empirial ormula o Savhenko (1998) also represents the avity radius o eah avity setion: CD x x R( x) = σ Rn( ) ln(1 / σ) R (13) ln(1 / σ) R n 1/ n Fig. 3 shows dierent avity shapes or various models, as determined aording to the ormulae proposed by Logvinovih (197), Savhenko (1998), and May (1975), or σ =0.01,0.0, and Fig. 3 Cavity proile aording to dierent avitation numbers. Calulation o avity axis The avitator ontinuously reates the avity while the vehile is moving. The avity model desribed in the previous setion is axisymmetri, and the avity axis is equal to the trajetory o the avitator. However, the avity axis is distorted by the eets o gravity and the angle o attak o the avitator. The eet o gravity on the avity axis is haraterized by a parameter alled the Froude number, whih is deined as the ratio o the low veloity to the gravitational wave veloity as ollows: V Frl = (14) gl Here, F rl is the Froude number with respet to the avity length L, and V is the low veloity. The avity axis is deormed in the upward diretion (loating up o avity tail) by gravity, and the eets are signiiant when the Froude number is relatively low; low Froude number implies low veloity. The axis deormation an be determined by the momentum theorem (Logvinovih, 197). The buoyany ore ρ gv must be equal to the vertial momentum:

6 Int. J. Nav. Arhit. Oean ng. (015) 7:346~ gq( x) h g ( x) = (15) VR ( x) π x g Qs () hg ( x) = ds V R () s (16) π 0 h is the upward deormation o the avity axis with respet to the distane rom the avitator along the avity axis, and Qx ( ) g is the avity volume rom 0 to x. More detailed modeling o the gravity eet was studied by Zou (013). In Savhenko s work (Savhenko, 1998), an approximation ormula was proposed: (1 +σ ) x hg ( x) = (17) 3Fr l The approximation ormula is valid in the ranges 0.05 σ 0.1 and.0 F 3.5. Fig. 4 shows the avity shape and avity axis deormation by gravity obtained using q. (16). The x and y axis represent the non-dimensionalized avity length and radius, respetively, or F rl = 10 and σ = Fig. 5 shows the omparison results o the avity axis deormation alulated by qs. (16) and (17) or F rl =.5 and σ = rl Fig. 4 Gravity eet on the avity entreline. Fig. 5 Comparison o avity entreline deormations. The angle o attak o the avitator also aets the avity axis. The deormation is derived rom Logvinovih s priniple that the momentum generated by the avitator must be equal and opposite to the momentum o the wake. L ds h ( x) = πρv (18) R x x 0 ( ) Here, h is the deormation o the avity axis along the avity axis, and L is the lit ore o the avitator, whih will be deined during the avitator modeling. Fig. 6 shows the avity axis deormation due to the angle o attak o the avitator when the angle o attak = 15 and σ =0.0.

7 35 Int. J. Nav. Arhit. Oean ng. (015) 7:346~363 Fig. 6 Cavity axis deormation due to avitator angle o attak; angle o attak = 15, σ = 0.0. The avity shape and avity axis were alulated beause the immersion o the vehile, whih plays an important role in the hydrostati/dynami ores, is determined by the relative position between the avity and vehile. Fig. 7 shows the delayed avity setion. The vehile immersion an be alulated rom the loation o the avity enter at the in, X, and the avity avity radius. Let Xavitator () t be the present avitator position and τ be the time or the avity to reah the position o the ins. The position o the avitator at the time t t in the earth-ixed rame an be written as ollows: l Xavitator, ( t t) = X. g, ( t t) + R ( t t) 0 0 av (19) X and avitator, X are the avitator loation and enter o gravity in the inertial rame; g., l av is the distane between the enter o gravity and avitator; and R is the rotation matrix rom the body rame to the earth-ixed rame. The avity enter loation at in, X avity Fig. 7 Delayed avity setion., is on the trajetory o the avitator i there are no inluenes rom gravity and the angle o attak. The avity enter loation in qu. (19) an be hanged as ollows by inluding the eets o gravity and the angle o attak. X ( t) = X ( t t ) + gravity eet + A.O.A eet avity, avitator, 0 0 X ( t t) 0 h (, t t) h (, t t ) h (, t t ) = avitator, + + y, g z, (0)

8 Int. J. Nav. Arhit. Oean ng. (015) 7:346~ Here, X is the loation o the avity enter at the in in the inertial rame, and avity, h and h y,,y are the y- and z-axis deormations, respetively, due to the angle o attak o the avitator. As the avity may be vertially deormed due to the time delay o the avity generated at the avitator, the horizontal deormation o the avity axis an be addressed by the horizontally delayed avity setion. However, there is no gravity eet in the horizontal plane and the avity axis deormation by entriugal ore is assumed to be negligible beause the momentum o the luid around avity is not hanged by entriugal ore. Cavitator model The avitator is a undamental part o the vehile; it reates a avity bubble around the body and generates ores and moments by hanging the deletion angle to ontrol the vehile. The avitator shape was assumed to be a disk that has a rotating motion only about the y-axis. The ollowing relationships have been employed to estimate the drag and lit oeiients ating on the disk avitator (May, 1975) in the low axis: D ( sα, ) C ( sα, ) C (1 + s) os α (1) n D = x0 1/ρV A L ( sα, ) C ( sα, ) C (1 + s) osα sin α () n L = x0 1/ρV A a 1 = tan ( ) + u w w= w ql av δ (3) F = [ D osa L sin a, 0, D sina + L os a ] T C n n n n F = R F Cavitator C Cavitator (4) where C D and respetively; C L are the drag and lit oeiients, respetively; D n and F is the omponent o the avitator ore in the avitator rame; C Cavitator R is the rotation matrix rom the avitator rame to the body rame; L n are the magnitudes o the drag and lit ores, F is the omponent o the avitator ore in the body-ixed rame; C A is the disk area; V is the magnitude o the avitator veloity with its omponent, [ u ] T v w, at the avitator enter expressed in the body-ixed rame; and α is the angle o attak alulated rom the avitator deletion angle δ and heave veloity. From (1)-(4), the lit and drag oeiients o the avitator are untions o the angle o attak o the avitator and avitation number. The drag oeiient when the angle o attak was zero, C x0, was determined rom the experimental results o Kirshner et al. (00). The ritional drag ating on the avitator is negligible and the ore about the added mass was alulated rom Fan study (Fan et al., 001). Fin model The our ins are loated at the at o the vehile, and the loation o the in root rom the enter o gravity is deined as l. The horizontal ins are elevators while the vertial ins are rudders. The ores ating on the avitating ins are ompliated by the dierent low regimes (Kirshner et al., 00). At a low avitation number, whih is the superavitating state, the avity develops at the in base. At a moderate avitation number, a partial avity develops at the leading edge o the ins separate rom the base avity. It was assumed that the ins have a wedge ross setion and that the oeiients o the in ore and moment vary with the angle o attak o the in ( α ) and the immersion depth ( d ). The oeiients were determined by interpolating the data provided by Kirshner et al. (00). The ores and moments generated by the in are given in in oordinates as ollows:

9 354 Int. J. Nav. Arhit. Oean ng. (015) 7:346~363 Cx( α, d ) i i 1 F = ρv (, ) i S i Cy α d i i Cz( α i, d ) i (5) Cmx ( α, d ) i i 1 3 M = ρv (, ) i S i Cmy α d i i Cmz ( α, d ) i i (6) The terms ( Cx, Cy, C z) are the in ore and moment oeiients, ( mx, my, mz ) C C C, expressed in in oordinates, as shown in Fig. 8. The subsript i = 1,,3,4 reers to eah in; V is the veloity o eah in in in oordinates; and i S is the in span length. In this paper, the oordinate origins o the in are loated at the enter o the hydrodynami ore ating on the in (middle o the in immersion). Thereore, the oordinate origin o the in varies with its immersion depth and moments in in oordinate. The ores and moments in eah in oordinate are transormed to the body-ixed oordinate using the rotation matrix as ollows: Fig. 8 Fin oordinate system. Fig. 9 Angle o attak and deletion angle o in. F 4 = RF (7) ins i i i= 1 4 M = ( r F ( RF ) + RM ) (8) ins i i i i i i= 1 where Ri is the rotation matrix rom the in oordinate i to the body-ixed oordinate, whih is determined by the in deletion angle δ, and r i is the moment arm between the oordinate origin o eah in and the enter o gravity. Fig. 9 shows the relationships between the in oordinate and body-ixed oordinate, where δ and α are the deletion angle and angle o attak o the in, respetively. The angle o attak o the in an be alulated rom the deletion angle o the in and veloity in in oordinates as ollows: u u 0 r q V v v r 0 p = = + l w w q p 0 in (9) a a = artan(v / u ) + d, rudder rudder = artan(w / u ) + d, elevator elevator (30)

10 Int. J. Nav. Arhit. Oean ng. (015) 7:346~ where V is the veloity vetor at the oordinate origin o the in, expressed in body-ixed oordinates, and vetor rom the enter o gravity to the oordinate origin o the in. l is the position Immersion depth When a part o the in is immersed, two types o ores at on the in. The part o the in that is immersed outside the avity boundary generates ores and moments alulated using qs. (5) and (6), respetively. However, the non-immersed part o the in, whih is inside the avity boundary, annot generate as high ores and moment as the immersed part, beause the luid omposition is a mixture o water and air. For simpliity, it is assumed that the non-immersed part o the in does not generate ores and moments. As shown in Fig. 10, the in immersion depth is alulated by using the avity radius, whih was desribed previously in the avity model, and the loation o the avity axis ( X avity, ), whih an be alulated rom (0), at the point where the ins are loated. Fig. 10 Relative position o vehile and avity axis or alulating in immersion depth. The immersion o in is alulated by Vanek s method (Vanek, 008). In Fig. 10, d is immersion depth, y and z are oset values whih are position vetor rom avity enter to body enter in the inertial rame. The ins are numbered ollowing the positive diretion onvention starting with the starboard in at φ 1 = 0 and eah in is at 90 degrees inrement. Then, the in immersion depth ( d ) o eah in is as ollows. φi = ( i 1) π /, i = 1..4 (31) d = S + R y z R y + z S x i 0 x 1 Sat( x) = 0 i x < 0 1 i x > 0 d, i ( ) ( os( i) sin( i)) ( sin( i) os( i)) / = Sat( d ), i, i (3) Likewise, the immersion depth o the body an be alulated in the same manner. y + z R + R i y + z > R R h0 (, t t ) = 0 i y + z R R (33) The diretion o immersion relative to the body y-axis is deined as: 1 p tan ( z / y) φ = (34)

11 356 Int. J. Nav. Arhit. Oean ng. (015) 7:346~363 Fores ating on wetted body For the partially avitating ase, the vehile body extends beyond the avity losure point. In this stage, hydrostati and hydrodynami ores are ating on the wetted area o the body inluding the added mass. In the ully developed avity stage (superavitating ondition), the avity extends the body dimensions, and thereore the ores are ating only on the ontrol surae, suh as ins and avitator. In this ase, the added mass is only ating on the avitator. The wetted area and volume o the body an be easily alulated by utilizing the avity proile. The hydrostati ores and moments are the buoyany ores and moments as alulated by (35) and (36), respetively. wet sinθ F = wet osθsinφ wet osθosφ (35) y b wet osθosφ zb wet osθsinφ M = z b wet sinθ x b wet osθosφ x b wet osθsinφ + y b wet sinθ (36) here, ( φθψ,, ) are the uler angles; wet is the magnitude o the buoyany and given by wet = ρ gv ; and wet X [ ] T buoy = xb yb zb is the position vetor rom the enter o gravity to the enter o buoyany. The hydrodynami ores ating on the wetted area are the pressure drag, ritional drag, and ores aused by the added mass. The hydrodynami ores are alulated in two diretions: X -axis and Y, vehile has the same setional shape or the hydrodynami ores as those ating along the Z -axis. This is beause a superavitating Y and Z -axes. Fig. 11 shows the hydrodynami ores ating on the X -axis. The pressure drag and ores that are aused by the added mass and at on the avitator are inluded in the avitator model. The normal pressure ontributions along the wetted body are assumed to have x- axis symmetry; urther, it is assumed that the pressure drag exists only at the avitator. The visous ontributions to the ritional drag oeiient ( C ) along the wetted portion o the body are alulated by using the Hughes line or the rition oeiient (Newman, 1977); thus, (37) expresses the ritional drag on the wetted F body. Fig. 11 Hydrodynami ores ating on X -axis. 1 DF = ρswetucf CF = (log Re ) 10 (37) D is the magnitude o the rition drag that ats in the negative diretion o the orward speed o the body; Re is the Reynolds F number; and ρ is the luid density. The ritional drag ore expressed in body-ixed oordinates an be written as ollows: F Frition DR DF (1 ) 100 = 0 0 (38)

12 Int. J. Nav. Arhit. Oean ng. (015) 7:346~ Drag redution ( DR ) is the perentage o the ritional drag redution. I the luid is a mixture o water and gas, the density o the luid an be attributed to the bubble and water. In this study, the rition drag is alulated using the water density, and the DR aused by the bubbles is addressed in the next setion. The hydrodynami ore ating on the Y Z plane ( F Morison ) is alulated by Morison s equations (Newman, 1977) as ollows (qs. (39) and (40)): FMorison = FI + FD (39) FC FC df = rc πr V ds + rc R V V ds (40) Morison m D F Morison is alulated by integrating df Morison, whih is the dierential hydrodynamis ore ating on the wetted part o the vehile body. ds is the dierential length along the X -axis; proportional to the square o the veloity, whih is the sum o the rition drag and pressure drag; F I is the ore proportional to the aeleration; C and mass and drag oeiients-1 and 1.3, respetively-or the ylindrial setion; and V and V F D is the ore C are the added m D are the veloity and aeleration, respetively, o the low in the Y Z plane relative to the body. The low veloity distribution relative to the body is determined by the body pith and yaw rate, and it is expressed in q. (41) and shown in Fig. 1 as the veloity distribution along the X -axis. u 0 r q ds dv = v r 0 p 0 + w q p 0 0 (41) Fig. 1 Veloity distribution along the X -axis in wetted body. The moment generated by the ore in q. (39) an be written as ollows: 0 M = l F l F Morison h. Morison, z h. Morison,y (4) where l h. is the distane rom the enter o gravity to the hydrodynami enter o F Morison along the X -axis; and F, and F are the Y Morison, z - and Z -axes omponents, respetively, o F Morison. The total ores and moments ating on the wetted body an be written as ollows: Morison y Fwet = F + FMorison + F (43) rition

13 358 Int. J. Nav. Arhit. Oean ng. (015) 7:346~363 Mwet = M + M (44) Morison Planing ore model The planing o the body on the avity generates ores and moments. The planing ore is the interation ore between the vehile transom and avity wall. The planing ore model has been investigated by Logivinovih (1980) and Vasin and Paryshev (001). In this paper, Paryshev s model is employed sine it was shown to it the experimental data better (Dzielski, 006). Fig. 13 shows the planing o a body on the avity wall. The oordinate ζ is at a distane along the X -axis rom the transom. α is the planing angle and h 0 is the immersion depth when ζ = 0 ; it is the maximum immersion depth. The p apparent mass per unit length an be alulated by using q. (45). Fig. 13 Planing o body on avity wall. * mp ( h) = pρ R 1 ( + h) (45) here, is a gap between the body and avity radius: length, Dzielski (011) expressed the magnitude o planing ore as ollows: = R R. h is the immersion depth. Using the apparent mass per unit v M F = M w + w (46) * * p T T h0 here, w T is the transom veloity, whih is equal to the immersion veloity o the body into the avity wall, and apparent mass expressed as ollows: * M is the h0 / tana * p * M = m( hd ) ζ 0 h0 / tana p * m ( h 0 0 ζ tan ap ) dζ (47) = h0 = pρ R ( + h ) tana 0 p The planing moment an be obtained by alulating the enter o pressure o the planing ore, x p x p h0( + h0) = ( + h ) tana 0 p (48) The diretion o planing ore and moment an be desribed using the diretion o immersion o body φ in q. (34). In p this planing model, several onditions are assumed-the onstant immersion veloity, instantaneous avity ormation, the steady planning, and a positive small value o the gap. The shape o the avity surae where the immersion ours is assumed to be ylindrial. For the planing with nonylindrial avity, see the Nguyen s study (Nguyen, 011).

14 Int. J. Nav. Arhit. Oean ng. (015) 7:346~ NUMRICAL SIMULATIONS Numerial simulations o the integrated model were perormed to analyze the harateristis o the system and validate the modeling. The vehile parameters are shown in Table 1; it is based on the benhmark high-speed superavitating vehile model used by Dzielski and Kurdila (003). The body has a uniorm density ρ b, and the ratio o body density to water density is m = ρb / ρ. The vehile mass M, the moment o inertia relative to the avitator-ixed rame, I, and the loation o the enter yy o gravity relative to the avitator, x, an be expressed as ollows: g 7 = ( ) (49) 9 M mρπ RL 11 I ( m ) RL ( m ) RL yy = ρπ + ρπ (50) xg 17 = L (51) 8 Table 1 System parameters or vehile model Parameters Desription Value and units g Gravitational aeleration 9.81 m/ s m Density ratio ( ρb / ρ ) R n Cavitator radius m R Vehile radius m S Fin span length 0.1 m L Vehile length 1.8 m C x0 Drag oeiient 0.8 Senario 1 : vertial open-loop simulation Fig. 14 shows the open-loop time response o the integrated model. The simulation onditions are shown in Table. The variables shown in the igure are vertial plane variables beause the lateral plane variables are onstant and equal to zero. Here, X and Z positions, respetively, in the earth-ixed rame and u and w are the veloities, respetively. θ is the pith angle and q is the angular veloity in pith. In initial stage, espeially beore 1 seond, the pith angle o the vehile negatively inreases beause the ins (elevators in this ase) and the avitator generate negative pith moment. The orward speed is always positive due to the thrust ore, and the heave veloity is also always positive due to the gravity ore. Thereore, the angle o attak o the avitator and the Z -axis ore are positive aording to qs. (3) and (4), respetively, and the avitator generates negative pith moment. Similarly, the ins generate negative lit ore and moments. There is unstable regime around 1 seond. In that regime, the avity is developed beyond the in root and in (rudder in this ase) immersion depth is hanged (not zero). Asymmetry o immersion depth, whih is aused by gravity eet on the avity axis, makes asymmetri drag ore and pith moment. The unstable pith moment is disappear when the avity size is derease as depth inreases.

15 360 Int. J. Nav. Arhit. Oean ng. (015) 7:346~363 Fig. 15 shows the avity radius and length aording to the avitation number. In the initial stages, the avitation number is relatively small and the avity is developed. The growth o the avity redues the wetted area o the body as well as the ritional drag. Thereore, the orward speed u rapidly inreases in the initial state. The drag oeiient o the avitator has a minimum value when the avitation number is zero and inreases with avitation number (q. (1)). The maximum drag oeiient, C is 1.17 in this paper, whih is idential to the disk drag oeiient in water. Hene, the orward speed onverges D,max to a onstant value (approximately 90 m/s), and the drag oeiient reahes its maximum value. Fig.14 Vertial open-loop simulation results o the integrated model. Table Conditions or vertial open loop simulation. State Desription Value T Magnitude o thrust ore 9000 N z 0 Initial depth 1 m u 0 Initial orward veloity 10 m/s δ Cavitator deletion angle 0 deg. δ e levator deletion angle 0 deg. δ r Rudder deletion angle 0 deg. Fig. 15 Cavity proile versus avitation number.

16 Int. J. Nav. Arhit. Oean ng. (015) 7:346~ Senario : lateral open-loop simulation In lateral open loop simulation, o whih onditions are same as in Table exept the rudder deletion angle ( δ r ). The rudder deletion angle is hanged rom 0 to 0.65 deg. at 1 seond Fig. 16 Lateral open-loop simulation results, position and uler angle. here, X, Y and Z are positions, respetively, in the earth-ixed rame and φ, θ and ψ are uler angle. u, v and w are the veloities, respetively, in the body-ixed rame and p, q and r are the angular veloities. The rudder ore and moment are generated due to the rudder deletion angle ( δ r ) ater 1 seond. As seen in Fig. 17, the variables assoiated with lateral dynamis suh as v, p, r are zero beore 1 seond. The rudder generates ore in Y -axis and yaw moment. The roll angle also inreases beause the longitudinal and lateral dynamis are oupled. Roll angle ( φ ) is bigger than yaw angle ( ψ ) eause X -axis moment o inertia ( I xx ) is muh smaller than the Z -axis moment o inertia ( I zz ). Fig. 18 shows the avity axis oset rom X -axis alulated by q. (0) and the vehile position. Fig. 19 shows the diretion o immersion alulated by q. (34) and magnitude o planing ore/moment. eore 1 seond, there is only vertial distane ( Z ) between avity axis and vehile enterline due to the gravity eet. The in and body immersion depth are alulated by the oset values by qs. (3)-(33). The vertial oset ( Z ) has relatively large value in initial phase, where the gravity eet is signiiant when the Froude number is relatively low; low Froude number implies low veloity. In that phase, the diretion o immersion is ± π /. Fig. 17 Lateral open-loop simulation results, linear and angular veloities.

17 36 Int. J. Nav. Arhit. Oean ng. (015) 7:346~363 Fig. 18 Cavity axis oset rom X -axis. Fig. 19 Diretion o Immersion and Planing ore/moment. In both ases o simulation osillatory responses are present around one seond. The osillatory response has nothing to do with the deletion o rudders, but it is due to the longitudinal asymmetry o ores exerted on the upper rudder and the lower rudder. eause o the eet o gravity, the avity is deormed astern upward. Larger drag ore exerted on the lower rudder than on the upper rudder generates negative (nose-down) pithing moment, whih in turn immerses the upper rudder and produes more drag ore than the lower rudder, and engenders positive pithing moment. This phenomenon would repeat and may destabilize the system as long as the avity ould maintain its size and length. However, the simulation results in Figs show that the osillatory response dies out in a short period o time as the depth o the vehile gets deeper. The avitation number whih is a untion o the depth and speed o a vehile inreases and the size o the avity is redued. The sensitive response o the vehile is quikly suppressed as the avity ontrats. The avity is redued to be partial avity that does not reah the loation o the rudders, and the asymmetri drag ore on the rudders disappears. CONCLUSIONS In this study, we perormed integrated dynamis modeling o a superavitating vehile. A 6-DOF equation o motion was onstruted by deining the ores and moments ating on the superavitating body. ah part o the vehile was modeled by reerring to previous researhes, and we integrated the dierent models to obtain the dynamis o a superavitating system. The avity modeling onsisted o alulations involving the size o the avity and avity axis. The wetted area o eah part o the vehile was deined based on the avity model. Subsequently, by using the integrated dynamis model, we onduted numerial simulations to analyze the harateristis o the superavitating system and validate the modeling ompleteness. The simulation results demonstrated that the vehile is unstable when the in immersion depth is asymmetri and dynamis o horizontal & vertial is oupled. Depth ontrol is required to maintain the superavitating ondition, and moreover, the light envelope should be determined. Our researh results an be used or understanding and employing superavitating vehile systems. Further, based on this researh, high-quality ontrollers and optimal superavitating systems an be designed, and the speiiations o omponents suh as ventilation systems, avitators, or ins an be eiiently hosen. ACKNOWLDGMNT This researh was supported by asi Siene Researh Program through the National Researh Foundation o Korea (NRF) unded by the Ministry o duation, Siene and Tehnology (NRF-01R1A1A008633). RFRNCS Ahn,.K., Lee, T.K., Kim, H.T. and Lee, C.S., 01. xperimental investigation o superavitating lows. International Journal o Naval Arhiteture and Oean ngineering, 4(), pp

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