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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Complele Cancellation of Ship Waves in a Narrow Shallow Channel Xue-Nong Cheni, Som Deo Sharma2, Norbert Stuntz3 (i Forschungszentrum Karlsruhe, 2 Gerhard Mercator University Duisburg, 3 VBD-European Development Centre for Inland and Coastal Navigation Duisburg, Germany) ABSTRACT In this paper we report our recent experimental find- ing of the complete wave cancellation for a particular configuration of a ship in a narrow shallow channel. Chen & Sharma (1996,1997) predicted theoretically that a twin-oblique-soliton wave pattern can be fitted in the space between a ship and each of the two chan- nel sidewalls such that no waves exist either upstream or downstream and, therefore, the ship does not suf- fer any wave resistance. Recently, we carried out a fundamental model experiment in the VBD towing tank to validate this prediction. For designing the ex- periment an updated stationary shallow-water wave model of Boussinesq type was applied in the far- field, which allows only an approximate twin-soliton solution, while in the near-field an improved slender body theory was applied to obtain the required ship cross-sectional area and waterline inversely from the far-field solution. The experiment showed that in the exact design condition the ship's (reflected) bow and stern waves cancelled each other so completely that almost no waves were observed behind the ship and the measured total resistance was reduced so much that the estimated wave resistance became practically zero. In fact, an apparently slightly negative wave re- sistance provoked a scrutiny of the common viscous- resistance formulation. Additionally, numerical cal- culations of wave profile and wave resistance using an Euler solver were performed to supplement the theoretical and experimental results. INTRODUCTION By purely theoretical analysis, albeit inspired partly by previous experimental results, Chen & Sharma discovered and reported in (1996, 1997) that when a slender ship moves in a narrow shallow channel at a supercritical speed, its bow and stern waves can be made to cancel each other so completely by a proper choice of hull-channel geometry that there are no free waves behind the ship and so it experiences theoret- ically no wave resistance. This result was subject to the restriction that the theory accounts only for weak nonlinearity and ignores viscosity. The present paper reports results of our recent model experiments in the VBD Shallow Water Towing Tank that confirm the existence of a practically waveless hull-channel con- figuration in reality. The present model experiment was designed by means of an analytical theory, taking advantage of the improvements achieved since the original dis- covery in 1994-95, cf. Chen (19991. The predicted waveless state occurred exactly at the design speed to an extent never observed before. At supercritical speeds, i.e., V > /, where V is the ship speed, 9 the acceleration due to gravity and h the water depth, the ship wave pattern looks like the shock waves of a 2-D airfoil in supersonic flight. Both bow and stern waves extend aft obliquely along their characteristic lines. But normally the bow wave is a free surface elevation; the stern wave, a depression. By the nature of nonlinear shallow wa- ter waves, the elevation can form a permanent pure oblique soliton, provided the bow has a certain shape, but the depression can never form a permanent "neg- ative soliton". If the ship now moves symmetrically
along the centerline of a narrow shallow channel of rectangular cross-section, its waves must be reflected by the vertical sidewalls. If the hull-channel geom- etry is adapted to the chosen depth Froude number as required by the theory, the bow wave after reflec- tion from the channel sidewall would hit the after- body and cancel the stern wave so that at the design speed the resultant wave in the ship wake would dis- appear totally, see Fig. 1. We have previously pro- posed the name "shallow channel superconductivity" for this phenomenon. BY ,~,,,,,,,,,~"""; ,~ ,........................................... \~N ~~ V .~ > 1 Nigh z~,~. / / ,,, ~~ ~~ z~ Figure 1: Schematic of ship wave pattern at super- critical speed in a narrow shallow channel. In this paper we present briefly the mathemat- ical model, its refined solution and a new ship de- sign based on it. The experimental results are shown in detail as measured resistance and wave profiles, illustrated by photographs. A comparison with the- oretical wave profiles is carried out. Some discus- sion of the friction line used to estimate the experi- mental wave resistance is included. Additionally, a fully nonlinear free-surface Euler numerical solution is presented. A video clip of the model experiment, documenting the waveless state, will be shown at the Symposium. MATHEMATICAL MODEL AND SOLUTION Here we describe the mathematical model and its so- lution only briefly and refer the reader for details to a parallel journal paper Chen, Sharma & Stuntz (20021. Unless otherwise stated, all variables are nondimensionalized by reference to water depth h, acceleration due to gravity 9, and water density p. The flow is governed by a 2-D steady shallow-water wave equation, (1U2)ixx + My + U~x~yy + 3Uix~xx where +2U~yi~y + 3 (a==== + ~==yy) = 0, (1) where So is a depth-averaged velocity potential and (nondimensional) ship speed U = V/ >/~ is identi- cal to the depth Froude number Fnh. The wave ele- vation ~ can be approximately expressed as (= Unix (2) In an improved slender body theory the bound- ary condition at the ship position becomes S9Y(X, +0) = ~ 2~1 + 5) { die t(U(P=)Sf (x)],, J (3) Sf (x) = So(x) + (s + ~X)b(x) + ((X)b(x), (4) s is sinkage and ~ is trim angle (positive bow down- ward). The no-flux boundary condition on the parallel vertical channel sidewalls simply reads, (x, iw/2) = 0. (5) It is difficult to find an exact twin-soliton solu- tion for Eq. (1) but one can obtain a good approxima- tion after Miles' weak-interaction solution (1977~. It is of the form ~ = F (A + feed (a)) + G (9 + fc~2 (I)) (6) with = x + cot °~1Yx1, 71 = x + cot ct2Y + X2, where F and G are single-soliton or solitary wave- train solutions, Al and ~2 are phase-shift functions due to the twin-soliton interaction, x1 and x2 are ini- tial phase constants, and °~1 and a2 are the angles be- tween the phase lines and the positive x-axis. It is un- derstood that the two component solitons are propa- gating in directions with equal positive x-component and possibly unequal but opposite y-components, implying that cot out and cot C>2 have different signs. The phase shift functions are found to be i = 2(U2~+ t2 ) 1) Gin), U¢3Cot2 ~~ ) Fig) 2(U2 + cot2 ~1 - 1) 2
The single-soliton solutions for F and G are of the same form, that is, F(~) = A1/kl tank, G(r') = A2/k2 tanh~k271), cottony = I + A ~~ -1, ki = 2U ' where Ai is the amplitude and ki the wave number of the soliton. To ensure exact symmetries about x = 0 and y = w/2, we set AT = A2 = A, kit = k2 = k, x~ = x2 = so, cot Ott = cot or and cot 0~2 = cot a, where cot a = >/(U2 - 1AU)/(1 + AU). The boundary condition on the channel sidewalls Eq. (5) is satisfied if the channel width is W = Pro/ cot or. Finally, fC in Eq. (6) is an improvement factor that is here assigned a value of 2.3 in the design condition. A theoretical solution corresponding to the following design is shown in Fig. 2. Figure 2: Theoretic solution of the ship wave pattern in the narrow channel. Here by virtue of symmetry only half of the free surface for w/2 ~ y > 0 is shown (height magnified ten times). DESIGN The design task consists of two steps. First, the cross- sectional area curve of the ship and the appropriate channel width are determined by theory for a chosen depth Froude number and a trial twin-soliton, requir- ing some iteration to achieve a prescribed displace- ment. Second, the detailed section shape, not pre- scribed by the theory, is selected to meet the practical demand of a fair and simple hull geometry. Also for simplicity, a "fixed" towing mode is assumed, i.e., no running sinkage and trim are permitted. This is not an inherent limitation imposed by the theory. But it does facilitate the design of the experiment by ex- cluding possible differences between the theoretical and real values of sinkage and trim in the "free" tow- ing mode. The trim, by the way, is always zero in theory for a fore-and-aft symmetric hull form in the waveless state. In principle, three nondimensional parameters, Go, A and U. can be freely chosen. The theory would then yield a nondimensional hull-channel configura- tion, i.e., a family of geosims with the property of being waveless at the design Froude number U. In practice, the physical model experiment has to be conducted on an object of given absolute size. The VBD towing tank available to us is 200 m long, 9.8 m wide and 1.3 m deep. The water depth is varied eas- ily and routinely by pumping water into another tank, while the width can be varied on demand by erect- ing a temporary intermediate wall with considerable yet justifiable effort. (It suffices to erect the interme- diate wall over the middle 80 m of the tank length where measurements are taken, leaving the run-in and roll-out stretches of the tank undivided.) Since it was obvious from previous designs that promising depth Froude numbers lay around 1.5 and the ratio of channel width to ship length around 0.5, we ar- bitrarily specified U = ok, along with h = 0.2 m to ensure a feasible towing speed, and w = 3.8 m to ensure a reasonable model length. With absolute size and two nondimensional parameters, namely U and with, now fixed, we were still free to manipu- late the pair A and JO to obtain any desired hull dis- placement within a certain range. We finally settled for A = 0.15 and x0 = 7.65636 as a compromise between extreme hull slenderness and unacceptable wave nonlinearity. The dynamic cross-sectional area curve Sf (x) can be obtained by integrating Eq. (3~: Aft) =U ~ ~y(1 + ()dx, y = +0. (7) However, in order to get the static cross- sectional area curve SO (x) from Eq. (4), we need to prescribe somehow the beam box) at the waterline. Partly anticipating the later determination of section shapes on practical grounds, we assume a uniform draft d and a uniform sectional area coefficient CM 3
over the entire hull length. It follows immediately that b(x) = So(x)/(CMd) (8) Substituting this into Eq. (4) we obtain easily So(x) = Sf (x) [1 + ((x, +O) ~ (9) Returning now to the prescription of section shapes, we ensure fair waterlines by declaring all sec- tions to be affine transforms and, hence, defined by a single nondimensional function y/(b(x)/2) = f (z/d), z =d, ...0. Further, we ensure mathematical simplicity by arbi- trarily choosing an exponential function f (z') = t1expel7.5(z' + 1)~/~1 - exp( - 7.5)], which can be integrated in closed form to yield a uni- forrn sectional-area coefficient To CM= J f(z')dz'. -1 Now, we can either freely choose uniform draft d and determine maximum beam b(0) to comply with Eq. (8) or vice versa. Our specific choice was a round draft d = 15 cm. One final detail remains to be explained. The theoretical sectional-area curve S(x) extends from minus infinity to plus infinity, implying an unrealis- tic ship of infinite length. Luckily, the curve decays exponentially so that an approximate practical hull form of finite length can be acquired by simple trun- cation of the bow and stern cusps. In absolute terms, we decided upon a round ship length of 6 m leaving the stem and stern as sharp edges of 2.7 mm thick- ness. , Item Symbol Value/Unit Length at waterline 6 m Beam at midship bm 0.3892 m Draft d 0.15 m Area of Sm 0.05063 m2 midship section Displacement V 0.1283 m3 Wetted surface area So, 2.437 m2 Block coefficient CB = V/l bm d 0.3663 Midship coefficient Crf = Sm/bmd 0.8672 Wetted Cw s = So / ~ 2.7776 surface coefficient Length/Depth I /h 30 Draft/Depth d / h 0.75 . Design speed V 1.9803 ms~ Design depth U = V/ ~ 1.414 Froude number Design water depth 0.2 m Design w 3.8 m . Width/Depth W / h 19 Table 1: Principal dimensions of the model hull and channel The principal dimensions of the final design are compiled in Table 1. The body plan is reproduced in Fig. 3; the cross-sectional area curve, in Fig. 4. 0.15 n ~ _ 0.05 N o - -0.05 -O.1 rrr : ~ i~ ~ ~ it. 1 _ - -0.15 0.2 -0.1 0.1 0.2 y [m] Figure 3: Body plan of the model hull with 21 uni- formly spaced sections 4
1E, 0.8 0.6 - o us o.4 0.2 . n . . . . . . . . . . . . . . . . . . 0 10 20 n 30 40 Figure 4: Cross-sectional area curve from stern (sta- tion 0) to stem (station 40) MODEL EXPERIMENT The physical model experiment was carried out ac- cording to the foregoing theoretical design. The ship model was towed at a number of steady speeds en- compassing a wide range around the design value. Beside the design configuration (h = 0.2 m, w = 3.8 m) two off-design configurations were also tested, namely, full tank width (h = 0.2 m, w = 9.8 m) to approximate laterally unrestricted water as a reference, and a larger water depth (h = 0.3 m, w = 3.8 m) to study the effect of depth variation. All runs were executed in the so-called "fixed" mode to preclude any complications due to running trim or linkage. Results at Design Depth Fig. 5 shows our main result, namely, the curves of specific total resistance RT/(P9V) measured in the narrow channel of design width (solid line connect- ing dots) and in the undivided tank of full width (dashed line connecting circles), both at the same design depth. For the purpose of estimating wave resistance, which unfortunately cannot be measured directly, two curves of specific frictional resistance RF/(<P9V), one derived from the empirical plane friction line of Hughes (1952, 1954) and the other derived from the empirical ITTC 1957 ship-model correlation line, are also included. The former is claimed to be the true friction line for infinitely thin smooth plates in fully turbulent, two-dimensional flow, while the latter incorporates a constant viscous form factor of about 1.12, believed to be the mini- mum value for realistic hull forms. As theoretically predicted, the measured total resistance in the narrow channel drops dramatically at the exact design speed, whereas nothing conspic- uous happens in the wide tank. Quantitatively, at the design depth Froude number 1.414 the total spe- cific resistance in the narrow channel is 0.0102 com- pared to 0.0327 in the wide tank, i.e., 69 % less. At the same speed, the specific frictional resistance is 0.00994 after Hughes and 0.01116 after ITTC. Con- sequently, the residuary resistance (RR = RTRF) based on the Hughes line is reduced by 99 % and that based on the ITTC line is reduced by 104 %. Of course, we do not believe in negative wave resistance but will return to a discussion of this point in a later section. 0.05 0.04 JJ 0.03 sit 0.02 v ~ 0.01 7V~ I tar ~ . _, 0.6 0.8 1 1.2 1.4 1.6 1.8 2 depth Froude number Figure S: Measured specific total resistance in the design channel (solid line connecting dots) and in a much wider tank of same depth (dashed line connect- ing circles); the two smooth curves represent specific frictional resistance after Hughes (lower) and ITTC (upper). Longitudinal cuts through the wave pattern were acquired by taking time records of the free- surface elevation by means of stationary wave probes as the model passed by at constant speed. An array of six equidistant probes was installed along a line nor- mal to the tank centerplane. Measurements were car- ried out at several speeds both in the design configu- ration and the two off-design configurations. Five of the six transverse locations with respect to the model centerplane were identical in the narrow design chan- nel and the wide tank. is
~o:o~,- L: 6-~_ ~~ -0.2 ~ ·~-~` --' A I- rid -0.4 ~ -9 -1 0 1 _ ~) ~ . . . . 0.4 0.2 -f" -id O ~7 -0.2 -0.4 0.4 0.2 -0.2 -0.4 0.4 0.2 -0.2 -0.4 0.4 0.2 -0.2 -0.4 .'t ,%. 7 ~ 1 IN a\ " ,,_,_~% ~ `~- J \ At," -~ % '_ ail' -__ . , , , . . , . . . . . , . . . , -3 -2 -1 0 1 . . . . . , , . . . , . . , .~. . . , ~ ~ 1 - I I - Jl' ~ 1 . - . . , , . . , . , . . . . . , . 1 , . . . ~ -3 -2 -1 0 1 ~ ' ' ' ' ~ . . . . 0.41 0.2t i- O ~:~ Figure 6: Measured wave profiles at the design depth Froude number Fob = 1.414 (h = 0.2 m and V = 1.98 m/s) in the 9.8 m wide tank (dashed lines) and the 3.8 m narrow channel (solid lines); graphs from top to bottom are cuts at y = 0.3, 0.6, 0.9, 1.2, 1.5 and 1.8 m. Note: Probe at y = 1.5 m was missing in the wide tank. 6
0.5 z~0.4 0.3 0.2 0.1 o ~ \~' -20 -10 0 ,... .... .... . / _ it" _ -20 -10 . . . 1 . . . . ~ 0 10 20 0 5 0.3 02 E: 1 ~ ~ o ~- 10 20 0.5 z~0.4 0 2 1 ~3 0 1 ~ O ~ ~ .,, .. .... ~ _ 1 . . . . . . . ~ . . . . -20 -10 0 10 20 Am'% ~~- 0.5 z~0.4 0.3 0.2 0.1 o 0.5 z~0.4 0.3 0.2 0.1 o 0.5 z~0.4 0.3 0.2 0.1 . . . . I . , . , . . . . : l : : _ , . . . of/ Elf/ at_ -20 -10 0 10 20 1 ' ' ' ' 1 1''''~ i... ~ .... ~ ~ 1'` I D A= ~ : . . . . 1 . . . . 1 . . . . 1 -20 -10 0 10 1 1 1 O - 2 0 - 1 0 Jim 20 . ~ it ~ 1 1 ~ 0 10 20 Figure 7: Comparison of theoretical (dashed lines), experimental (thick solid lines) and numerical (thin solid lines) wave profiles at design depth Froude number Fnh = 1.414 in the design narrow channel; graphs from top to bottom are cuts at y/h = 1.5, 3.0, 4.5, 6.0, 7.5 and 9.0. 7
The comparison of primary interest is, of course, that between the wide tank and the narrow channel at the design depth Froude number, as car- ried out in Fig. 6. Evidently, the strong free waves behind the model in the wide tank are almost totally absent in the narrow design channel, corroborating the dramatic drop in wave resistance. Quantitatively, the highest free-wave amplitude observed in the wide tank is 100 mm and in the narrow channel only 4 mm. We think it is fair to call it "complete" wave cancel- lation. The most direct test of the theory is to compare the theoretical and experimental wave patterns in the design condition. This has been carried out in Fig. 7 for six equidistant longitudinal cuts. There is striking agreement between theory and experiment except for a small phase shift, which may be due to bottom fric- tion or imperfect reflection from the sidewalls. Fig. 8 is a photographic attempt to convey a visual impres- sion of the wave pattern in the design condition of zero wave-resistance. Note the high wave crest be- side the model and the almost absolutely flat free- surface behind the model. = . ~ Figure 8: Twin photographs of the wave pattern in the design condition of zero wave-resistance, sepa- rately viewing the forebody (top) and the afterbody (bottom). Numerical Simulation The foregoing theoretical and experimental inves- tigation was supplemented by a numerical simula- tion using an Euler solver developed at the Merca- tor University in Duisburg as part of a larger project to compute fully nonlinear viscous free-surface flows around ships moving in restricted waters, see Bet et al. (1999~. Briefly stated, it is a computer code for solving the nonlinear Euler equations of steady three-dimensional flows of an incompressible invis- cid fluid on arbitrary grids employing a nodal finite- volume method, see Hanel et al. (20011. Major fea- tures of the algorithm are: (i) coupling of the mass and momentum equations by the method of artificial compressibility, (ii) integration in time by an explicit Runge-Kutta multi-stage time-stepping scheme, and (iii) free surface tracking by a level-set formulation on fixed grids. Fig. 7, already partly discussed, includes be- sides the theoretical and experimental wave profiles a third set designated as numerical. These were com- puted using the Euler solver. As expected, conver- gence problems arose in the transcritical speed range where the physical solution is known to be unsteady. Nevertheless, the numerically simulated wave pro- files at the design Froude number, almost adjacent to the transcritical range, are obviously in fair agree- ment with the theoretical and experimental profiles. The numerical wave resistance calculated by integrating the pressure on the hull surface in the Euler solution is compared in Fig. 9 to the resid- uary resistance obtained in the experiment by sub- tracting the frictional resistance after Huhges from the directly measured total resistance. The agree- ment is excellent except for a gap in the numerical results over the transcritical speed range for reasons just mentioned. Fig. 10 shows a perspective view of the starboard-half of the wave pattern in the design con- dition as numerically simulated by the Euler solver. The upper right yellow surface is the model running from left to right. The lower left yellow surface is the channel bottom with isobars plotted on. The blue band in the middle is the free water surface with the channel sidewall at the left edge. More clearly than in the photograph of the physical experiment (Fig. 8), we can see here the theoretical semi-rhombic wave pattern (Fig. 1 and Fig. 2) comprising two oblique wave crests alongside the model merging into a sin- 8
ale crest of almost double the height at the channel sidewall. This is a purely local wave bound to the model. No free waves can be seen behind the model. In short, the computationally extremely effi- cient 2D shallow-water theory seems to be at least as close to physical reality as the more general but also computationally far more demanding 3D Euler simulation. 0.04 0.03 U2 ,, Ul 0.02 3 i, 0. 01 au U2 . ~£ . . . , ~ 0 5 0.75 1 1.25 1.5 depth Froude number 1.75 2 Figure 9: Comparison of measured residuary resis- tance from model experiment (solid line connect- ing dots) with calculated wave resistance from Euler simulation (circles) over a wide speed range in the narrow design channel. Figure 10: Perspective view of the numerically sim- where ulated wave pattern in the design condition (height magnified four times). and Discussion of Viscous Resistance The only reasonably practical and reliable method available to date for obtaining wave resistance from a model experiment is to resort to a modified Froude hypothesis, i.e., to subtract from the directly mea- sured total resistance the best possible empirical es- timate of the viscous resistance, thereby ignoring any wave-viscous interaction. It is current standard practice in tankery to estimate viscous resistance of streamlined hull forms by applying a viscous form- factor to some agreed friction line, usually the 1957 ITTC ship-model correlation line. Our hull is extremely thin (bm/1 = 0.0649) and slender (V/13 = 0.000594), entailing a form-factor only slightly above unity. On the other hand it has also an extremely small aspect ratio (2d/1 = 0.05) which would tend to induce edge effects and increase the form factor. In any case, the purely viscous form- factor cannot lie below unity. However, we observe in Fig. 5 that in order to avoid the physical impossi- bility of a negative wave resistance in the design con- dition, an implausible viscous form-factor less than unity would be required. We think that this paradox can be explained by taking account of wave-viscous interaction. Our de- sign condition is characterized by exclusively posi- tive wave elevation alongside the model (see Fig. 7), which is most unusual, specially over the afterbody. This induces two opposite effects: (i) the increased wetted surface would tend to increase the frictional resistance and, hence, increase the apparent form fac- tor, but (ii) the absolute forward motion of the water under the wave crests would tend to decrease the rel- ative velocity of water past the hull and, hence, de- crease the apparent form-factor. The following rough calculation shows that the latter effect predominates. By definition, the frictional resistance ignoring the wave effects is RFO = 2PV SWCF(RU). (10) By analogy, the frictional resistance including the above wave effects would be RFW = 2 P(VU) 2 ~5W + /\ SW ) OF (Rn ), ( 1 1 ) 1 {~/2 u = - / ups, 0)dx, I J-~/2 rt/2 /\Sw= / ((x,0)dx, -~/2 The ratio of frictional resistance accounting for wave interaction to the frictional resistance ignoring wave interaction becomes RFW ~ _uN2l' ASWN\ RFO = ~1 vJ <1+ sw ) 9
This ratio can be interpreted as a Froude-number de- pendent frictional "form factor", although it is obvi- ously not a true viscous form-factor. The latter would have to be determined in the absence of waves, e.g., by towing a deeply submerged double model. The numerical value of this apparent form- factor in the design condition is easily calculated from the known theoretical flow and is indeed found to be less than unity, namely 0.947. This explains the observed paradox and does not violate any physi- cally motivated expectation. On the contrary, our de- sign displays an unexpected fringe benefit, namely, it not only eliminates wave resistance but also reduces frictional resistance. Depth Variation With a view to possible application of this design in actual practice, it is of interest to examine how sensi- tive the state of zero wave-resistance is to unwanted variation of external parameters. It is already appar- ent from Fig. 5 that the resistance minimum is rather sharp and sensitive to speed. The practical conse- quence is that passively stable steady motion is only possible at speeds somewhat higher than the point of minimum resistance. As already stated, beside the design depth h = 0.2 m we also tested one off-design water-depth h = 0.3 m in the same narrow channel. Fig. 11 shows the measured specific total resistance at both depths along with the corresponding specific frictional re- sistance curves after the ITTC line. We note that the dramatic drop in total resistance almost down to the ITTC line still occurs at the 50 To higher off-design depth but at a reduced depth Froude number corre- sponding, however, to a 13 % higher absolute veloc- ity. Measured wave cuts at the speed of minimum resistance in the off-design depth are reproduced in Fig. 12. As may be expected, the free waves behind the model are quite weak but not completely elimi- nated as in the design condition. We conclude that a near-optimum state of extremely low wave resistance can be maintained over a range of water depths if the ship has sufficient power reserve to adapt its speed. 0.07 0.06 0.05 0.04 0.03 0.02 0.01 . _ _ .3 m 1 ~ _ 7 ~ _ _ ~ 1~;~ =0.2 IT/ ' _ _ _~ ~ f; ~ V _~ ~~ ~ 0.6 0.8 1 1.2 1.4 1.6 1.8 2 depth Froude number Figure 11: Measured specific total resistance at off- design water-depth h = 0.3 m and design water- depth h = 0.2 m along with the corresponding spe- cific frictional resistance curves after the ITTC line. CONCLUSIONS The purely theoretical prediction of a state of no trail- ing waves and hence zero wave-resistance for a ship of an ingenious mathematical hull form moving at a chosen supercritical design speed in a rectangu- lar channel of appropriate depth and width has been verified by numerical simulation using a nonlinear 3D Euler solver and validated by physical model ex- periments conducted in a specially erected narrow shallow-water channel. ACKNOWLEDGMENT We thank the management and staff of VBD- European Development Centre for Inland and Coastal Navigation at Duisburg, Germany, for their valuable support in conducting the model experi- ments. 10
0.4 urn 0 20 -0.2 -0.4 0.4 z,7l 0 . 20 -0.2 -0.4 0.4 At 0 . 20 -0.2 -0.4 0.4 z/77v 0 · 20 -0.2 -0.4 0.4 0.20 -0.2 -0.4 0.4 z/qrl 0 . 20 -0.2 -0.4 . ~ .. .... .... . .... , - . ~~ A_ ~ : , .... .... _ -4 -3 -2 -1 . . . . 0 1 _ 1 ~ F ~/ . -1 0 1 ... .... .... .... _ -4 -3 -2 -1 0 1 .... .... .... ..·· . _ J _ . ., .... -1 0 1 . . . . . . . . . . . . . ~~ . . . . . -1 0 1 Figure 12: Measured wave profiles at the off-design depth h = 0.3 m (in the narrow channel of 3.8 m width) at "optimum" depth Froude number Fnh = 1.30 (V = 2.23 mist; graphs from top to bottom are cuts at y = 0.3, 0.6,0.9, 1.2, l.Sandl.8m. 11
REFERENCES [1] Bet, F., Stuntz, N., Hanel, D., and Sharma, S. D., "Numerical Simula- tion of Ship Flow in Restricted Water," 7th International Conference on Numerical Ship Hydrodynamics, Nantes, France, 1999. [2] Chen, X.-N. and Sharma, S. D., "Non- linear theory of asymmetric motion of a slender ship in a shallow channel," Proc. 20th Symp. on Naval Hydrodynamics, Santa Barbara, California, 1994, pp. 386-407. [3] Chen, X.-N. and Sharma, S. D., "On ships at supercritical speeds," Proc. 21st Symp. on Naval Hydrodynamics, Trondheim, Norway, 1996, pp.715-726. [41 Chen, X.-N. and Sharma, S. D., "Zero wave re- sistance for ships moving in shallow channels at supercritical speeds," J. Fluid Mechanics, Vol. 335, 1997, pp. 305-321. [51 Chen, X.-N., "Hydrodynamics of Wave- Making in Shallow Water," Doctoral Disser- tation, University of Stuttgart, Shaker Verlag, 1999. [6] Chen, X.-N., Sharma, S. D., and Stuntz, N., "Zero wave resistance for ships moving in shal- low channels at supercritical speeds. Part 2. Im- proved theory and model experiment," submit- ted to J. Fluid Mechanics, 2002. [7] Hanel, D., Dervieux, A., Cloth, O., Fournier, L., Lanteri, S., and Vilsmeier, R., "De- velopment of Navier-Stokes ~ ' Hybrid Grids," In: E.H. al. (Ed.~: Numerical Flow Simulation: Notes on Numerical Fluid Mechanics, Vol. 75, Springer Verlag, 2001, pp. 49-66. [81 Hughes, G., "Frictional resistance of smooth plane surface in turbulent flow," Trans. INA, Vol. 94, 1952. Solvers on Hirschel et [91 Hughes, G. "Friction and form resistance in tur- bulent flow and a proposed formulation for use in model and ship correlation," Trans. INA, Vol. 96, 1954. t10] Miles, J. W., "Obliquely interacting solitary waves," J. Fluid Mechanics, Vol. 79, 1977, pp.157-169. 12
DISCUSSION Hang S. Choi Seoul National University, Korea When we look at the related waves at the sidewall, it looks like the so-called Mach reflection. Have you perhaps examined this phenomenon during your experiments? AUTHORS' REPLY Yes, indeed, there exists a well-known fluid- dynamic analogy between ship waves at supercritical speed in shallow water and shock waves of an aircraft at supersonic speed. Our experiments can well be used to illustrate Mach waves and Mach reflection. DISCUSSION Turgut Sarpkaya Naval Postgraduate School, USA Our experience has shown that the type of the model described in your paper is very sensitive to yaw instability and, as a single ship, makes its use nearly impossible. Have you carried out a yaw-stability analysis for the single model in confined as well as unconfined environment? AUTHORS' REPLY We thank Prof. Sarpkaya for his corroborative evidence (oral remarks). It is true for almost any ship model in a shallow channel of rectangular cross-section that if ship length and speed are in the right proportion to channel depth and width, the bow waves reflected from channel sidewalls would significantly cancel the stern waves and, hence, result in a corresponding reduction of wave resistance. However, to our knowledge a complete elimination of wave resistance has never been recorded previously. The question of stability raised by Prof. Sarpkaya is indeed very important and goes far beyond just yaw-stability. It is well known that various ship types are not passively yaw-rate stable and that no ship can be passively directionally stable. Nonetheless, almost every ship can be actively stabilized on course by means of a rudder and a controller (misleadingly called autopilot) enabling it to safely navigate channels and cross the oceans. This would be all the easier for a slender ship like ours. What is really crucial in our design is the stability of forward motion. The fact that the curve of total resistance does not monotonically rise with increasing speed but, instead, dramatically falls just before reaching the design speed implies a regime of unstable equilibrium of forward motion. Assuming a traditional propeller characteristic, a slight transient increase in resistance due to an external disturbance would throw the ship back to a low subcritical speed. In fact, an enormous power reserve would be required in the first place to boost the ship from rest over the huge transcritical resistance hump on to its favorable supercritical design speed. This is a problem our design has in common with various high speed craft, such as planing hulls, hydrofoil boats, hovercraft and wigs. A less critical but also practically relevant issue is the weak transverse static stability of our design due to a relatively low mean beam-to-draft ratio. Hence, the present monohull can be only a stepping stone to progress. A truly practical design would have to be some kind of a multi- hull exploiting the principle of mutual cancellation of non-dispersive bow and stern waves at supercritical speed.
