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Twenty-Third Symposium on Naval Hydrodynamics (2001)
Naval Studies Board (NSB)

Page
191
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Page
191
Front Matter (R1-R19)
Modern Seakeeping Computations for Ships (1-45)
Forces, Moment and Wave Pattern for Naval Combatant in Regular Head Waves (46-65)
New Green-Function Method to Predict Wave-Induced Ship Motions and Loads (66-81)
Validation of Time-Domain Prediction of Motion, Sea Load, and Hull Pressure of a Frigate in Regular Waves (82-97)
Ship Motions and Loads in Large Waves (98-111)
Prediction of Vertical-Plane Wave Loading and Ship Responses in High Seas (112-125)
Basic Studies of Water on Deck (126-142)
Second Order Waves Generated by Ship Motions (143-156)
Prediction of Nonlinear Motions of High-Speed Vessels in Oblique Waves (157-170)
Optimizing Turbulence Generation for Controlling Pressure Recovery in Submarine Launchways (171-180)
Hull Design by CAD/CFD Simulation (181-190)
Steady-State Hydrodynamics of High-Speed Vessels with a Transom Stern (191-205)
Practical CFD Applications to Design of a Wave Cancellation Multihull Ship (206-222)
Simulation of Ship Maneuvers Using Recursive Neural Networks (223-242)
Flow- and Wave-Field Optimization of Surface Combatants Using CFD-Based Optimization Methods (243-261)
Marine Propulsor Noise Investigations in the Hydroacoustic Water Tunnel 'G.T.H.' (262-283)
Propulsor Design Using Clebsch Formulation (284-300)
Unsteady Flow Quantities on Two-Dimensional Foils: Experimental and Numerical Results (301-313)
Hydrofoil Turbulent Boundary Layer Separation at High Reynolds Numbers (314-329)
Pressure Fluctuation on Finite Flat Plate Above Wing in Sinusoidal Gust (330-341)
Control of the Turbulent Wake of an Appended Streamlined Body (342-354)
Investigation of Global and Local Flow Details by a Fully Three-Dimensional Seakeeping Method (355-367)
Prediction of Wave Pressure and Loads on Actual Ships by the Enhanced Unified Theory (368-384)
Frequency Domain Numerical and Experimental Investigation of Forward Speed Radiation by Ships (385-401)
International Collaboration on Benchmark CFD Validation Data for Surface Combatant DTMB Model 5415 (402-422)
Validation of High Reynolds Number, Unsteady Multi-Phase CFD Modeling for Naval Applications (423-440)
Free Surface Viscous Flow Computation Around A Transom Stern Ship by Chimera Overlapping Scheme (441-456)
Anti-Roll Tank Simulations With A Volume of Fluid (VOF) Based Navier-Stokes Solver (457-473)
Validation of Tab Assisted Control Surface Computation (474-484)
Experimental and Numerical Investigation of the Flow Around the Appendices of a Whitbread 60 Sailing Yacht (485-492)
Propeller Wake Analysis by Means of PIV (493-510)
Experimental and Numerical Investigation of the Unsteady Flow Around a Propeller (511-526)
Simulation of Incompressible Viscous Flow Around a Ducted Propeller Using a RANS Equation Solver (527-539)
On Submerged Stagnation Points and Bow Vortices Generation (540-552)
Numerical Prediction of Scale Effects in Ship Stern Flows with Eddy-Viscosity Turbulence Models (553-568)
The Experimental and Numerical Study of Flow Structure and Water Noise Caused by Roughness of a Body (569-578)
Large-Eddy Simulations of Turbulent Wake Flows (579-598)
Instability of Partial Cavitation: A Numerical/Experimental Approach (599-615)
An Unsteady Three-Dimensional Euler Solver Coupled with a Cavitating Propeller Analysis Method (616-638)
On the Flow Structure, Tip Leakage Cavitation Inception and Associated Noise (639-653)
An Experimental Investigation of Cavitation Inception and Development of Partial Sheet Cavaties on Two-Dimensional Hydrofoils (654-669)
Modeling 3D Unsteady Sheet Cavities Using a Coupled UnRANS-BEM code (670-686)
Ship Wake Detectability in the Ocean Turbulent Environment (687-703)
An Experimental and Computational Study of the Effects of Propulsion on the Free-Surface Flow Astern of Model 5415 (704-712)
Breaking Waves in the Ocean and Around Ships (713-745)
Numerical and Experimental Study of the Wave Breaking Generated by a Submerged Hydrofoil (746-761)
The Numerical Simulation of Ship Waves Using Cartesian Grid Methods (762-779)
Radiation Loads on a Cylinder Oscillating in Pycnocline (780-791)
Wave Resistance Computations - A Comparison of Different Approaches (792-804)
Computations of Nonlinear Turbulent Free Surface Flows Using the Parallel Uncle Code (805-819)
Submarine Maneuverability Assessment Using Computational Fluid Dynamic Tools (820-832)
Simulation of UUV Recovery Hydrodynamics (833-847)
Reynolds-Averaged Modeling of High-Froude-Number Free Surface Jets (848-862)
On Roll Hydrodynamics of Cylinders Fitted with Bilge Keels (863-880)
Combining Accuracy and Effciency with Robustness in Ship Stern Flow Computation (882-896)
An Unstructured Multielement Solution Algorithm for Complex Geometry Hydrodynamic Simulations (897-909)
Ship Stern Flow Calculations on Overlapping Composite Grids (910-926)
Study on the Prediction of Flow Characteristics Around a Ship Hull (927-940)
Analysis of Turbulence Free-Surface Flow Around Hulls in Shallow Water Channel by a Level-Set Method (941-956)
A Design Tool for High Speed Ferries Washes (957-967)
Flow Around Ships Sailing in Shallow Water - Experimental and Numerical Results (968-982)
Ship Stability Study in the Coastal Region: New Coastal Wave Model Coupled with a Dynamic Stability Model (983-992)
Waves and Forces Caused by Oscillation of a Floating Body Determined Through a Unified Nonlinear Shallow-Water Theory (993-1005)

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Steady-State Hydrodynamics of High-Speed Vessels with a Hansom Stern Lawrence J. Doctors (The University of New South Wales, Australia) ATexancler H. Day (The University of Glasgow, Scol:lan(l) Abstract The inviscid linearized near-field solution for the flow past a vessel with a transom stern is de- veloped within the framework of classical thin-ship theory. However, the innovation in the current ap- proach is that the hollow in the water behind the stern is represented here by a virtual extension to the usual hull-centerplane source distribution. In the present work, the near-field solution to the flow using the thin-ship approximation must be computed. This idea clearly demands a consid- erable addition to the complexity of the numerical solution which contrasts with the traditional far- field method employed by Michell (1898~. The computer program functions by iterat- ing the geometry of the hollow until the criterion of atmospheric pressure on the surface of the hollow is zero. In addition, the iteration procedure includes adjusting the sinkage and trim of the vessel until it is in equilibrium. The latter component of the computation utilizes an integration of the resulting pressure distribution over the wetted surface of the vessel in an entirely consistent manner. Comparison of the theoretical results with a systematic series of twelve models shows excellent correlation with the towing-tank data. Indeed, the behavior of this approach appears to be much more robust than, for example, that of the Neumann- Kelvin problem and some fully nonlinear analyses. Introduction Literature Review Previous work on the subject of prediction of resistance of marine vehicles, such as monohulls and catamarans, has shown that the trends in the curve of total resistance with respect to speed can be predicted with excellent accuracy, using the tra- ditional Michell (1898) wave-resistance theory, to- gether with a suitable formulation for the compo- nent of frictional resistance. There have been fur- ther enhancements to this wave-resistance theory. These enhancements include the influences of finite depth and finite width of the canal by Lunde (1951) and Sretensky (1936~. A recent justification for this research, in which linearized free-surface conditions are em- ployed, is the very encouraging comparisons that were made by Doctors and Renilson (1993) for monohulls and catamarans with closed or pointed sterns and by Sahoo, Doctors, and Renilson ( 1999) for monohulls with open or transom sterns. One difficulty has been that nonlinear and viscous-wave effects are not included and, conse- quently, the correlation between theory and exper- iment has not been sufficiently good for the purpose of practical ship design. For this reason, a consid- erable effort has been invested in recent years in the development of fully nonlinear computer codes. The complete nonlinear kinematic body-boundary condition and the nonlinear free-surface kinematic and dynamic boundary conditions are satisfied in these programs. There has been excellent progress with such computer programs and they may even- tually be developed to the stage where they can be used for hull-form development. Currently, the execution time is too long for one to contemplate any realistic optimization of hull forms. Consequently, any type of optimiza- tion, such as the genetic-algorithm method of Doc- tors and Day (1995) and Day and Doctors (1997b) is not possible. This is because of the requirement to evaluate the resistance many thousands of times during the practical design process.

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:: R I Pi r :` . Mp Sp w i ,, REP op ~ Lxpxro~ i Y Hollow 1 .~-~1= 4_~ ~L;oU 1 ~ ~ Figure 1: Definition of the Problem (a) Geometry and Forces A further point is that more sophisticated computer codes, such as those briefly alluded to above, do not always lead to more accurate or reli- able predictions for sensitive quantities, such as re- sistance. This is because resistance can be affected markedly by minor inaccuracies in the computed pressure distribution over the surface of the hull. A revealing study of this troubling possibility was published by Sahoo, Doctors, and Renilson (1999~. It was demonstrated there that more reliable pre- dictions for the resistance were obtained from the consistent linearized approach, than from a mod- ern nonlinear code, for a set of fourteen modern high-speed vessels with transom sterns. Indeed, the linearized approach gave predictions which were within 5% for most of the test cases, while the errors from the competing nonlinear method were typically an order of magnitude greater. Further reductions in the errors inherent in the linearized theory can be obtained in a most practical manner by means of very easily esti- mated correction factors for the wave resistance and for the frictional resistance. Examples of this research were published by Doctors (1998a, 1998b, and l99Sc). Current Work These principles were advanced in the re- search presented by Doctors and Day (1997~. Firstly, transom-stern effects were included in the theory by accounting for the hollow in the water behind the vessel. This work was essentially a de- velopment of the ingenious and practical approach 2: Legend for panel types x Hull + Hollow Hull and hollow Null Figure 1: Definition of the Problem (b) Centerplane Paneling first presented by Molland, Wellicome, and Couser (1994) and Couser, Wellicome, and Molland (1998~. There, a simple virtual extension to the hull behind the transom was constructed in the com- puter program. The wave resistance for the vessel was deduced on the basis of an application of the Michell integral to the entire model and its exten- sion, together with an estimate of the hydrostatic resistance due to the existence of the dry transom. The extension was allowed to grow in length with increasing forward speed of the vessel in a physi- cally plausible manner as detailed by Doctors and Day (1997~. A more sophisticated enhancement of this work was presented recently by Doctors and Day (2000~. The local flow field was computed on the basis of the linearized theory. This permitted the squat (linkage and trim) to be determined; ex- cellent correlation with experimental data was ob- tained in this way. However, the shape of the hollow was still determined in a heuristic manner. It is the purpose of the current research effort to iterate the shape of this transom-stern hollow in order to improve the accuracy of the approach even further. Theory Definition of the Problem Figure lta) shows a typical arrangement for a vessel traveling at a constant speed U in calm

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water. The x,y,z coordinate system is also de- picted. The water is unbounded laterally (in the y direction) as well as having infinite depth. The components of the forces acting on the vessel are indicated. The vessel can either be self propelled or be towed. In the former case, the thrust from the pro- peller or the water jet acts along a defined line of action relative to the coordinate system attached to the vessel. Thus, the direction and position of the thrust line vary with the speed of the vessel. In the latter case, the vessel is towed at the specified speed from a particular point in the hull. Hence, the line of action of the thrust is longitudinal, but the line moves vertically in sympathy with the sinkage and trim. Discretization of the Hull Further details of the problem are provided in Figure lobe. Here, the centerplane paneling is seen to overlay the hull and the hollow in the water behind the transom stern. The panels or elements possess a flat facet and a rectangular base. They are employed, in particular, for the purpose of the numerical calculation of the pressure, or profile, re- sistance. These elements are chosen in order to approximate the centerplane area of the hull and the hollow as closely as possible. The longitudi- nal and vertical slope of the facets of the elements match the corresponding values of the hull surface in a root-mean-square sense. This type of panel is algebraically simpler than the "pyramids" or "tents" which were previ- ously employed by Day and Doctors (1997b) and Doctors and Day (1997a), for example. The use of flat facets implies a higher level of discontinuity on the hull surface. On the other hand, numerical convergence tests for wave resistance, based on the two types of panels, showed that a similar number of panels was required in either case; namely, 40 panels in the longitudinal direction and 8 panels in the vertical direction. Equations for the Potential We start in the traditional manner by uti- lizing the potential ¢, whose gradient gives the perturbation velocity. The potential satisfies the Laplace equation throughout the fluid domain: Axe + ~~y + fizz = 0 . (1) The linearized kinematic free-surface condition, namely ~z+U~ = 0 on z=0, (2) states that any water particles on the free surface remain there. Here, ~ is the elevation of the free surface measured upward from its undisturbed po- sition z = 0. In addition, the Bernoulli equation provides the linearized dynamic free-surface condi- tion Ups—9;—,ll; = 0 on z = 0, (3) in which 9 is the acceleration due to gravity. The Rayleigh (1883) artificial viscosity ,u, which is as- sumed to be vanishingly small and positive, has been introduced in this technique in order to im- pose the radiation condition, which states that waves must be propagated downstream. De Prima and Wu (1957) gave a clear description of this con- cept. Elimination of ~ from Equations (2) and (3) yields the linearized combined free-surface condi- tion: U2¢x2 + 9¢'z—,u~x = 0 on z = 0 . (4) Finally, the "bed" kinematic boundary condition states that As = 0 as z ~—no . (5) Potential-Flow Solution The solution for the flow past the vessel and its transom-stern hollow is obtained by using the equivalent centerplane-source distribution This distribution is assembled from panels, as noted ear- lier, while the panels are constructed from the el- ementary Kelvin point source. The potential due to a Kelvin point source of strength Q. obtained by Wehausen and Laitone (1960, p. 484, Equa- tion (13.36) ), is 47rT + her, + (P ~ (6) where we have defined the radial distances r = >/(x—x/~2 + (y _ y/~2 + (z _ z/~2 `7y r' = x,/(x—x/~2 + (y _ y/~2 + (z + z/~2 (8)

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Marl ~1 n Model, Figure 2: Sample Ship Models (a) Lego Ship Models 7 and 8 and the wave term is 7r oo ~ 47r2 | do | do (exp{k[z + zl + —7r 0 Model 12 it_ Figure 2: Sample Ship Models (b) Lego Ship Models 11 and 12 and the required arguments in the special function G4 are defined by x = Xi—xj + lax, + id—x') cost + iffy—y') sinew/ Z = pi—Zj + mi\z . /(ko—k cos2 ~—i,llcos0) . Here, the circular wave number is k = he sec2 0, the fundamental wave number is ho = g/U2, and the complex horizontal wave number is The first term in Equation (6) can be inte- grated for a constant-strength-source panel ok and a unit-constant-longitudinal-slope field panel in the so-called Galerkin manner. The result for the in- duced longitudinal gradient of the potential at the field panel is fx,1 = -4 ok ~ w' ~ wm G4(i' j,l,mj, I=—I m= - 1 (10) in which the weighting factor is given by the for- mula w' = ~ 2 ~ -1 —1 for 7 = 0 = 1 The four special functions needed for this analysis are interrelated by spatial integrations as follows: G~(x, z) = 1/~, G2(x,z) = ~G,(x,z)dz = sinh-i (z/~x~), (12) 1~ +iLy = 1cexpti8) . G3(x,z) = /G2~,z)(1:~ z sinh-i (x/~z~) + x sinh-i (z/~x~), G4~(x,z) = ~G3(x,z)dz 2 z2 sinh-i (x/~z~) + + xz sinh-i (z/~x~)— 2X~ . (13) (14) The second term in Equation (6) can be in- tegrated in a very similar manner. The third term can also be integrated with respect to the wavenumber it, as well as with re-

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specs to the spatial coordinates, to yield ~r/2 ~ ~ 27~2k2~i / cos36 ~ we ~ Wm. —7r/2 1= - 1 m= - 1 F4 (Z(i, j, 1, m)) do ~ (15) in which the argument of the function F4 is given by Z(i, j, 1, m) = k(zi + zj + mi\z) + + ik~(xi - Xj + If\x) . The special complex wave function F4 is closely related to the exponential integral and was defined by Doctors and Beck (1987). The zeroth wave function is co Fo(z) = f ~P( I)dk-7riexp(z) - 1/z o = exp(z) {E1 (z) - 21riH[Im(—z)] } - 1/z (16) and the complex exponential integral is 00 E1(z) = | P( )dt, z while the other four complex wave functions are Fl(Z) = /Fo(Z)dZ Modeling of the Hollow Fo(z) + 1/z, (17) F2 (z) = / F1 (z) dz F1 (z) + ln(—Z), (18) F3(z) = / F2(z) dz F2 (z) + z ln(—Z)—Z. (19) F4(z) = / F3(z) dz F3(z) + 2z2 ln(_z) _ 4 z2 . (20) The hollow is considered to be a virtual ex- tension to the hull of the vessel and is modeled by a continuation of the centerplane source distribution. The determination of the shape of the centerplane of this hollow represents a vital part of the hydro- dynamic problem. The length of the hollow is initially esti- mated by the method detailed by Doctors and Day (1997~. That is, the eq~bivalent radial position of a point in the region of the hollow behind the stern is taken to be r = ~/~. (21) We next state that the effective trajectory of a particle of water on the surface of the hollow is parabolic in nature and that it has parametric co- ordinates as follows: x = x~ran —Ut, (22) r = r ran—U — t _—gt2, (23 [ d~ ~ ~ 2=:~tran in which x~ran and r~ran are the coordinates of the equivalent springing point of the hollow on the transom girth. Equations (22) and (23) can be solved by setting r = 0 to yield the location of the vertex, or rooster tail, of the hollow ~ho~ Hence, the length of the hollow is evaluated as: Lho~ = XGran —Xho~ (24) In the current enhanced technique, the ini- tial estimated profile of the hollow is assumed to be defined by the parabola that springs longitudi- nally from the bottom of the transom and meets the vertex on the (un(listurbed) free surface. Forces and Moment on the Vessel One first assumes that the attitude of the vessel is the same as its static attitude, that is, with zero sinkage and trim. The required source strength is computed using the standard thin-ship result, namely, a. = - U—, ~x in which b is the local beam. The total gradi- ent of the potential ~x at the field panel i is next computed, by summing the contributions from the source panels presented in Equations (10) and (15~. From this, one can determine the pressure on the surface of the hull, as follows: p = p(U¢~ - gz), (25)

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where p is the density of the water and z is the elevation of the centroid of the field panel. Next, the three components of the general- ized forces on the vessel (pressure resistance, sink- age force, and bow-up moment) are then found from this pressure distribution: RP = - ||P~ dude, (26) s in which the generalized surface slope is given by the formula fib - Finally, the total resistance can be found by the simple summation RT = RP +RF +RA, (28) in which Rp is the pressure resistance computed from Equation (26), RF is the frictional resistance estimated from the 1957 International Towing Tank Committee (ITTC) formula, described by Lewis (1988, Section 3.5), and RA is the correlation resis- tance. Equilibrium of the Vessel At any stage of the iteration for the equilib- rium of the vessel, one can estimate the corrections to the sinkage and trim angle (positive bow down), with respect to the longitudinal center of flotation LCF: ~SI.CF = (Sp + W— Zprop)/(P9AW) ~ (~295) b) = ~—zpRp + (up—LCF)Sp—ME— —(xprop—LCF) Zprop + + ZpropXprop — —IF (RF + RA ) —Zstab Rstab + + (LCB—LCF)W]/(WGML) (30) The additional symbols introduced here are SO the sinkage pressure force, W the weight of the ves- sel, Zprop the vertical component of the propulsion fib ax fib LIZ z——x— for bow-up moment ax Liz for resistance force (equal to zero in the case of the vessel being towed), Aw the static waterplane area, zp the ver- tical lever arm for the pressure resistance (equal to zero), up the longitudinal lever arm for the pres- sure sinkage force (equal to—LCF), MP the bow- up pressure moment, xprop the longitudinal lever arm for the propulsion force, Zprop the vertical lever arm for the propulsion force, Xprop the longitudi- nal component of the propulsion force, IF the ver- tical arm for the frictional force (measured to the centroid of the wetted surface), Zs~ab the vertical lever arm to the stabilizers, Rub the resistance of the stabilizers (zero in the current work), LOB the longitudinal center of buoyancy, and GM the lon- gitudinal metacentric height. for sinkage force. (27) The sinkage at the coordinate origin x = 0 and the trim are s = sI,cF—~ LCF, (31) t = -Lid, (32) where L IS the length of the vessel. For simplicity, the hydrostatic stiffness coef- ficients were used for iterating the sinkage and trim of the vessel, as seen in Equations (29) and (30~. The use of the ideally consistent hydrodynamic stiffness coefficients would have posed a somewhat major computing challenge. Relative convergence of 1 x 1O-4 could be obtained within about eight iterations; once equilibrium is achieved, there is no error introduced by the simpler approach. Iteration of the Hollow The pressure at any point on the hollow is given by a formula of the type of Equation (254. If there are Thou panels that lie on the centerplane of the hollow, then there will be the same number of panel collocation points at which we desire the pressure to be zero (or atmospheric). The Kutta condition is applied by extending this requirement to include the last column of panels on the surface of the hull next to the transom. In addition, we must impose the closure con- ditions. Because the panels are of uniform rectan- gular shape, we just require the source strengths crj along a longitudinal line of panels (z is constant) to sum to zero. A set of overconstrained linear equations is then set up, in which the pressures are minimized

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in a least-square sense while the closure conditions are satisfied exactly. Next, the secant method is employed in which the length of the hollow is ad- justed until the average pressure on the surface of the hollow is minimized. The error in the average pressure on the sur- face of the hollow after the iteration process was typically one percent of the hydrostatic pressure at the keel of the vessel. In summary, it can be seen that the method- ology used in this work is analogous to that under- lying the solution of the problem of symmetric two- dimensional cavitating flow behind ~ body possess- ing a bluff stern, as presented by Newman (1980, Section 5.13, pp 208-215~. However, the following extensions have been introduced: 1. Three-dimensions; 2. Free-surface; 3. Equilibrium of the body; 4. Iteration of cavity shape. Simplistic Resistance The simpler and traditional approach to thin-ship resistance has been to utilize the Michell (1898) result for the resistance applied to the ves- sel, together with a suitable formulation for the shape of the hollow. Examples are the work of Day and Doctors (1997a), Day, Doctors, and Arm- strong (1997), Doctors and Day (1997), and Doc- tors (1999~. This approach leads to the following estimate of the total resistance: RT = RW + RH + RF + RA, (33) in which Rw is the wave resistance, computed as 7r/2 RW = PU2 / sec36(P'2 + Q'2)d6, (34) o where the complex Michell wave function is P'+iQ' = // ~ ' ~ exp~ik~x + hz~dxdz so - = ~ Pi' + iQ', (35) i=1 Item Length of bow section Waterline beam Draft Maximum-section coef. Symbol Lbow B T _ CM - Value 0.750 m 0.150 m 0.09375 ret 0.6667 Table 1: Lego Ship Models (Common Data) with SO being the centerplane area. The complex wave function for a panel with center (xi, Zi) is 4 1 1 Pi' + iQ' = k l sin(2kx/\x) sinh(2ki\z) '2 [A ] exptik~xi +hzi) . (36) Finally, RH is the so-called hydrostatic resis- tance, resulting from the imbalance of the hydro- static pressure owing to the transom being "dry": o RH =—P9 ~ b(xtran~z)zdz ~ (37) or —1 t ran in which Ttran is the draft at the transom. This simplistic approach to resistance pro- vides an identical result to that from the near-field approach described earlier in this paper, in the case of a vessel with no transom. Lego Ship Model Series This series of hulls was developed with the intention of studying the hydrodynamics of trarl- som sterns. Doctors (1998a) provided the details of the hull segments from which the ship models were assembled. There was a total of seven seg- ments. The bow and stern segments have parabolic waterplanes. The bow, stern, and parallel-middle- body segments all possess parabolic cross sections. Figure 2 shows views of four of the test models. Table 1 and Table 2 list the details of all twelve of these so-called Lego Ship Models. Results Numerical Convergence Tests The wave r esistance of a vessel without a transom stern, computed according to Equa- tions (34) to (36), is identical to that computed

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\l ~ R ~ · \ 1~ ~ ~ ' ~ 1 0.000 0.0000 0.7500 0.6666 2 0.000 0.1875 0.9375 0.7290 3 0.000 0.3750 1.1250 0.7499 4 0.000 0.5625 1.3125 0.7290 5 0.750 0.0000 1.5000 0.8332 6 0.750 0.1875 1.6875 0.8494 7 0.750 0.3750 1.8750 0.8499 8 0.750 0.5625 2.0625 0.8275 9 1.500 0.0000 2.2500 0.8888 10 1.500 0.1875 2.4375 0.8957 11 1.500 0.3750 2.6250 0.8928 12 1.500 0.5625 2.8125 0.8735 Table 2: Lego Ship Models (Variable Data) using the considerably more elaborate near-field so- lution. This assumes the same discretization of the hull centerplane and the use of the same mesh in the wave-angle integration. Thus, one has an inter- esting and powerful check of the computer coding. The four parts of Figure 3 show a test of convergence for Lego Ship Model 7 for four phys- ical parameters of interest. These parameters are the total resistance RT, the sinkage s, the trim t, and the hollow length Lam. These parameters have been rendered dimensionless using the weight of the ship W or its length L, as appropriate. It can be seen that using 40 panels longitudinally and 8 pan- els vertically is sufficient for the current purpose. In the same vein, one requires 32 points for half the range of the ~ integration in Equation (15) to obtain reasonable convergence of the results. . The annotation Holl—P,K indicates that in this case the pressure condition on the surface of the hollow and the Kutta condition on the transom were satisfied. Also, Free = Yes states that the model was free to sink and trim. The slenderness coefficient L/Vi/3 is also printed on the plots. Resistance Components Figure 4 depicts the theoretical computa- tions of the various resistance components referred to above, for four of the ship models. These show the hydrostatic resistance RH, the pressure resis- tance RP, the frictional resistance RF, the total resistance RT, and the total experimental resis- tance RT,E- In general, the correlation between the theory and the experiments is good at the higher speeds, which are of practical significance, that is, for a Froude number F of 0.6 or greater. The disagreement is greatest at the lower speeds, where the transom would in reality be partly wetted, thus reducing the drag. This phe- nomenon has been ignored in the current calcula- tions. An indication of the error involved at these low speeds is just the hydrostatic resistance; it can be observed that subtracting this quantity would bring the theoretical calculations into line with the experimental data. This process was, in fact, done by Doctors (1998d) in an approximate manner. Comparison with Experiments Figure 5 shows the specific resistance R/W for the four Lego Ship Models. The different meth- ods used are respectively: the simplified method of Doctors and Day (1997) (Holl = Simp), the simpli- fied method of Doctors and Day (2000) with the vessel also free to sink and trim (Free = Yes), the current method with the pressure and the Kutta conditions applied (Holl = P. K), and the current method with the pressure and Kutta conditions ap- plied and also the hollow length Lho~ iterated to minimize this pressure (Holl = P. K, L). It is noteworthy that the main factor of i~r~- portance is the need to permit the vessel to sink and trim in order to find its equilibrium position. The other various enhanced versions of the theory appear to offer little, if any, improved correlation with the experiments. We next examine Figure 6, which shows a comparison of the theoretical predictions and ex- periments for the dimensionless sinkage s/L for the four Lego Ship Models. The second, third, and fourth methods are considered here. The first method, of course, does not include sinkage and trim at all. Again, we observe that the simpli- fied method which permits determining the equi- libriurn of the vessel, namely that of Doctors and Day (2000), is as effective as the current more so- phisticated approaches. 1— —

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x 1ol2 - Curve No Nx No 10- ~ 32 20 8 32 40 4 8- 32 40 8 ~ a' 6- 2- O- D 1 0 0.2 0.4 D, 4- Series = Lego Model= 7 L = 1.875 m Holl = P,K Free = Yes 1 0.6 0.8 0 F x101° - Curve Ng Nx _ _ _ 16 40 8 - ~ 32 20 32 40 32 40 6- 2- O- _ No _ 8 8 4 8 Liz _~' ///~ // _ // // ,// Series = Lego Model = 7 L = 1.875 m Holl = P,K Free = YeF' Figure 3: Convergence Tests Figure 3: Convergence Tests (a) Resistance (b) Sinkage x 103-5 - Curve No _ 30—_ _ _ 16 40 ~ 32 20 25- ~ 32 40 32 40 20 - 15 - 5- 1C O- . _ . No 8 8 4 8 '/! 1 1 0 0.2 0.4 lay F Figure 3: Convergence Tests (c) Trim ,~ 1~ q c Series = Lego Model = 7 L = 1.875 m Holl = P,K Free = Yes 1 0.6 0.8 0 Similar sentiments can be expressed with re- gard to Figure 7, which presents the dimensionless trim t/L for the four Lego Ship Models. Again, the simplified method which includes the determi- nation of the equibrium of the vessel, is as accurate as the current methods. Finally, we present Figure 8, which shows the dimensionless hollow length Lho~/L for the four Lego Ship Models. It is curious to note that there is little significant difference between the length of the hollow computed in the current hydrodynamic manner and the length of the hollow resulting from the simplified geometric approach of Doctors and Day (2000), for example. At the very minimum, the general trend of the variation of the length of the hollow with the Fioude number is properly pre- dicted, but would seem to be rather too high. / o- Nx 40 20 40 40 / / 1 0.2 0.4 1 F Figure 3: Convergence Tests (d) Hollow Length Series= Lego Model = 7 L = 1.875 m Holl = P,K Free = Yes 1 0.6 0.8 It is also vital to point out the obvious fact, with regard to Figure 8, that the experimental data is at least as subject to debate as the theoretical calculations. The length of the hollow was deter- mined by means of a visual observation using a wire mesh as a reference frame. The accurate location of the rooster tail of the hollow Thou was not possi- ble because of various factors; it would even seem that the definition of this length itself poses a major difficulty. Concluding Remarks The present research has been instructive in showing that the precise length of the hollow be- hind the transom stern is not critical to predicting the resistance, sinkage and trim, of the vessel. In many ways, this is quite remarkable, because the

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R— Xlol2 - Curve Comp L/V1/3 = 7.615 1 x10-2 Curve Comp L/V1/3 = 8.187 10 ~ ~ Free — Y.s 10 Free — Yes R 6—Series = Lego to ,, 6- Series = Lego GIG Model = 7 /O ° ~ ~ Model = 8 / ° ~ ~ -8-° ° -I '< oO 2 ~ _ _— _ _ _ D_ - - - - = o 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 F i Figure 4: Resistance Components Figure 4: Resistance Compc (a) Lego Ship Model 7 (b) Lego Ship Model 8 x 101° - Curve Comp o o o o T,E ] ~ F 6— T Series = Lego 4—Model = 11 L = 2.625 m 2- O- L/V1/3 = 9.375 Hall = P,K ° Free = Yes 0 / 0 0.' /c 2.625 m ~° / 0 1 0.2 0.3 0.4 F Figure 4: Resistance Components (c) Lego Ship Model 11 10 - xlo-2 A— _ 6- ' V<4- . it' ________ 2- 1 1 0.5 0.6 flow of the water in the region of the hollow must surely affect the pressure distribution on the hull itself. There should be a resulting influence on the behavior of the vessel. It now appears that the principal features of the hollow can indeed be mod- eled in the simplified manner, allowing one to find (within engineering accuracy) the essential hydro- dynamic performance factors of interest. Other models for the profile of the hollow have been considered. For example, it is thought that an S-shaped profile might also be worthy of study. Here, one end of the S corresponds to the springing point from the bottom of the transom and the other end corresponds to the rooster tail. Both ends of the S would be horizontal in this alternative model. O- Curve Comp o 0 0 0 T,E ~ _ H _______ P _ F T Series = Lego Model = 12 L =2.813 m L/V1/3—9889 Holl = P,K Free = Yes o ~ 0-~ 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 F Figure 4: Resistance Components (d) Lego Ship Model 12 It is necessary to emphasize that, at least with regard to resistance, the current approach still uses a traditional estimator for the frictional com- ponent. That is, ideally, one might consider utiliz- ing a form factor to correct the frictional resistance. Alternatively, and preferably, a more sophis- ticated approach could be considered. That is, the influence of the actual geometry of the hull on the frictional resistance would be computed. Such an approach would take into account the non-zero pressure gradient on the surface of the hull and would be consistent with the current thin-ship the- ory. Finally, it would be feasible to incorporate the influence of the deformation of the free-surface on the actual submerged volume of the vessel. This

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x 1 ol 2 - Curve Holl Free Comp ~ x 1 o-2 Curve Holl Free Comp o o o o Yes T,E o o o o Yes T,E 10 - _ Simp No T ° ~ 10 - _ _ _ Simp No T . Simp Yes T ~ , - ~ Simp Yes T At,/ ,, ~ ~ 8 1 _ | P,K,L | Ye9 | T | ~ ' | ~ 1~ | P,~ ~ Yes | T | i, -a l ty6- igloo''' 6- / '',o,,- ~_ ~~ 9~- 4- /~o' _,~Oo ° Series = Lego ,~,O'o Series = Legs o° Model = 7 2— :'' o ° Model = l ,~ L = 1.875 m ~ L = 2.064 ~, O ¢b L/V1/3 = 7.615 L/V1/3 = 8.187 O- 1 1 ' 1 O- 1 1 1 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 F ~ Figure 5: Comparison of Resistance Figure 5: Comparison of Resist (a) Lego Ship Model 7 (b) Lego Ship Model 8 10 - xlo-2 8- Curve | Holl | Free | Comp o o o o Yes T. E __ __ _ __ Simp No T __ _____ Simp Yes T _ _ _ _ _ P,K Yes T _ P,K,L Yes T I: me' —''o ~ Series = Lego to ° Model = 11 O ° L = 2.625 m a O ° L/Vi/3 = 9376 4- O- o 0 i' ,' 0 0.1 0.3 F 0.4 0.5 0.6 Figure 5: Comparison of Resistance (c) Lego Ship Model 11 could be effected approximately by means of a local vertical distortion of the sectional geometry. The resulting influence may well be of similar signifi- cance to that of permitting the vessel to find its equilibrium. Acknowlecigments The authors would like to thank the Direc- torate of Naval Platform Systems Engineering, De- partment of Defence, for its support through Con- tract 9627MZ. They also gratefully acknowledge the assistance of the Australian Research Coun- cil (ARC) Large Grant Scheme (via Grant Num- ber A89917293~. The support of this work by The University of New South Wales is also greatly ap- preciated. 0.7 10 xlO_2 - Curve 0 0 0 0 6- 4- 2- Holl Simp Simp P,K _ P,K,L Free | Comp | Yes T. E No T Yes T Yes T Yes T o of art' Series = Lego to Model = 12 ~Oo°° L =2.813 m O ° ° L/V1/3 = 9.889 T I I T I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 F Figure 5: Comparison of Resistance (d) Lego Ship Model 12 References COUSER, P.R., WELLICOME, J.F., AND MOL- LAND, A.F.: "An Improved Method for the Theoretical Prediction of the Wave Resistance of Transom-Stern Hulls Using a Slender Body Approach", International Shipbuilding Progress, Vol. 45, No. 444, pp 331-349 (December 1998) DAY, A.H. AND DOCTORS, L.J.: "Design of Fast Ships for Minimal Resistance and Motions", Proc. Sixth International Marine Design Confer- ence /(IMDC '97J, Newcastle upon Tyne, England, pp 569-583 (June 1997) DAY, A.H. AND DOCTORS, L.J.: "Resistance Op- timization of Displacement Vessels on the Basis of Principal Parameters", J. Ship Research, Vol. 41, No. 4, pp 249-259 (December 1997)

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10— X10-3 03 X10-9 6- Curve Holl 0 0 0 0 8 - Simp P,K P,K,L 6- 4- 2- Data Exp Theory Theory Theory //' ~ ,-~o/ , W~-0 ~ O~ O- 1 1 0 0.2 0.4 /Io /,/m F Figure 6: Comparison of Sinkage (a) Lego Ship Model 7 Curve Holl 0 0 0 0 __ _ _ __ ___ Simp 5 - ~ P,K P,K,L 4- 3 - 2- O- Data | Exp Theory Theory _ Theory K;°° oo c, 0 0 - 0 0.1 0.2 0.3 F 75 Series = Lego Model = 11 L = 2.625 m L/vl/3 = 9375 Free = Yes 1 1 1 0.4 0.5 0.6 Figure 6: Comparison of Sinkage (c) Lego Ship Model 11 DAY, A.H., DOCTORS, L.J., AND ARMSTRONG, N.A.: "Concept Evaluation for Large Very-High- Speed Vessels", Proc. Fourth International Confer- ence on Fast Sea Transportation (FAST '97), Syd- r~ey, Australia, Vol. 1, pp 65-75 (July 1997) DE PRIMA, C.R. AND WU, T.Y.: "On the Theory of Surface Waves in Water Generated by Moving Disturbances", California Institute of Technology, Engineering Division, Pasadena, California, Report 21-23, 40+i pp (May 1957) DOCTORS, L.J.: "Modifications to the Michell Integral for Improved Prediction of Ship Resis- tance", Proc. Twenty-Seventh Israel Conference on Mechanical Engineering, Technion, Haifa, Is- rael, pp 502-506 (May 1998) D OCTORS, L. J .: "Improvement of the Correla- tion of Michell's Integral for Displacement Ves- 10 x 10-3 Curve Holl 0 0 0 0 8 - Simp P,K P,K,L 6- 4- Series = Lego Model = 7 L = 1.875 m L/V1/3 = 7.615 / Free = Yes 1 O- 1 0.6 0.8 0 0.2 — — o o x10-3 Curve Holl a,, 5- 0 0 0 0 o 0 ~ Simp P,K 4— P,K,L 3- - xlO-3 2- n- Data Exp Theory Theory ,Y to Series = Lego Model = 8 L = 2.063 m L/V1/3 = 8.187 Free = Yes 1 0.4 0.6 0.8 F Figure 6: Comparison of Sinkage (b) Lego Ship Model 8 Data | Exp Theory Theory _ Theory too O o ° OO0 ~ o,'~ o,? 0 d5~ Series = Lego I) Model = 12 L =2.813 m L/V1/3 = 9 889 Free = Yes 1 1 1 1 1 1 1 0.7 0 0.1 0.2 0.3 0.4 F 0.5 0.6 0.7 Figure 6: Comparison of Sinkage (d) Lego Ship Model 12 eels", Proc. Third Biennial Engineering Mathe- matics and Applications Conference (EMAC '98J, Adelaide, South Australia, pp 183-187 (July 1998) DOCTORS, L.J.: "Intelligent Regression of Resis- tance Data for Hydrodynamics in Ship Design", Proc. Twenty-Second Symposium on Naval Hydro- dynamics, Washington, DC, pp 33-48, Discussion: 49 (August 1998) DOCTORS, L.J.: "An Improved Theoretical Model for the Resistance of a Vessel with a Transom Stern", Proc. Thirteenth Australasian Fluid Me- chanics Conference (13 AFMCJ, Monash Univer- sity, Melbourne, Victoria, Vol. 1, pp 271-274 (De- cember 1998) DOCTORS, L.J.: "On the Great Trimaran- Catamaran Debate", Proc. Fifth Interna- tional Conference on Fast Sea Transportation

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8 4 1 —~ 1 30 - X10-8 25 - 20 - 1' 10 - o Hell Simp P,K P,K,L Data Exp Theory Theory Theory By, Boo // o my/ ° ,,~c~=~==<' ~ ~ ~ tare °~ _ _—-~~ ~ X1 ,~o ° ~ oo 0 o Series = Lego Model = 7 L = 1.875 m L/V1/3 = 7.615 Free = Yes 0.2 0.4 0.6 ~ Figure 7: Comparison of Itin. (a) Lego Ship Model 7 Holl Simp P,K P,K,L _ Data Exp Theory Theory Theory lo i, ~0°° Series = Lego ~ /' o Model = 11 o 0 0 oooOo° °OoO L =2.625 m L/V1/3 = 9.375 Free = Yes 1 1 1 1 1- 1 0.1 0.2 0.3 0.4 0.5 0.6 F Figure 7: Comparison of Trim (c) Lego Ship Model 11 (FAST '99g, Seattle, Washington, pp 283-296 (August-September 1999) DOCTORS, L.J. AND BECK, R.F.: "Convergence Properties of the Neumann-Kelvin Problem for a Submerged Body", J. Ship Research, Vol. 31, No. 4, pp 227-234 (December 1987) DOCTORS, L.J. AND DAY, A.H.: "Hydrodynami- cally Optimal Hull Forms for River Ferries", Proc. International Symposium on High-Speed Vessels for Transport and Defence, Royal Institution of Naval Architects, London, England, pp 5-1-5-15 (Novem- ber 1995) DOCTORS, L.J. AND DAY, A.H.: "Resistance Pre- diction for Transom-Stern Vessels", Proc. Fourth International Conference on Fast Sea Transporta- tion (FAST '97J, Sydney, Australia, Vol. 2, pp 743- 750 (July 1997) 35 - O ~ 30 - 25 - 20 - 15 - 10 - 5— Curve 0 0 0 c Holl Simp P,K . P,K,L Data Exp Theory Theory ,^~ Theory /~~ O ° ° ,/ o o ,y 0 ,y 0 lo° ~ 0 lo ~Oo 0 Series = Lego Model = 8 L = 2.063 m L/Vt/3 = 8.187 =_ I Free = Yes 0.8 0 0.2 0.4 0.6 F Figure 7: Comparison of Trim (b) Lego Ship Model 8 Gus xlo-3 12 - 0.7 0 0.8 8 4— O— -4- 1 0.1 . Holl Simp P,K . P,K,L . . Data Exp Theory Theory Theory 0 ~ 0 ~ 0 0 / o / 0 /O Series = Lego ~' 0 Model = 12 o o o OoO=O° OoO L =2.81~3 m L/V1/3 = 9 889 Free = Yes 1 1 1 1 0.2 0.3 0.4 0.5 0.6 0.7 F Figure 7: Comparison of Trim (d) Lego Ship Model 12 DOCTORS, L.J. AND DAY, A.H.: "The Squat of a Vessel with a Transom Stern", Proc. Fif- teenth Inter national Workshop on Water Waves and Floating Bodies (15 I WWWFB), Caesarea, Is- rael, pp 40-43 1:February-March 2000) DOCTORS, L.J. AND RENILSON, M.R.: "The In- fluence of Demihull Separation and River Banks on the Resistance of a Catamaran", Proc. Second In- ternational Corlfererlce on Fast Sea Transportation (FAST '93J, Yokohama, Japan, Vol. 2, pp 1231- 1244 (December 1993) LEWIS, E.V. (ED.~: Principles of Naval Architec- ture: Volume II. Resistance, Propulsion and Vibra- tion, Society of Naval Architects and Marine Engi- neers, Jersey City, New Jersey, 327+vi pp (1988) LUNDE, J.K.: "On the Linearized Theory of Wave Resistance for Displacement Ships in Steady and

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- o ~- 20 - xlo-2 16 - 12- 8- ~- . Curve Holl 0 0 0 0 Simp P,K P,K,L Series = Lego Model = 7 L = 1.875 m . Data . Exp Theory Theory . Theory L/V1/9 - 7.615 / Free = Yes / ,~ooooooocoo<~oo ° ° . . . 0 0.2 0.4 0.6 0.8 Figure 8: Comparison of Hollow Length (a) Lego Ship Model 7 16 - xlo-2 12 - 8— 4- O— A 16 - xlo-2 12 - _ 8- 4- Curve Holl 0 0 0 0 Simp ~ Pa P,K,L Series = Lego Model = 8 L = 2.063 m Data Exp Theory Theory Theory / L/V1/3 = 8.187 Free = You / 00000000000000 0 0 ° . . . 0 0.2 0.4 0.6 0.8 F Figure 8: Comparison of Hollow Length (b) Lego Ship Model 8 _ Curve Holl Data L/V l /3 = 9.875 x 1 ol 2 - Curve Holl Data L /V 1/3 = 9.889 o o o o Exp Free = Yes o o o o Exp Free = Yes ~ Simp Theory / 10 - ~ Simp Theory Off ~ _ ~ P,~ Theory / ~ ~ P,K Theory I P,K,L Theory /: ~ 8- P,K,L Theory / Series = Lego / ~ ~ Series = Lego a' _ Model = l 1 / ~ 6—Model = 12 / / 4- / /OOOooooOOooooooo ° ° ° 1 24 / ooooooooooooo Do 0 0 ° 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 F F Figure 8: Comparison of Hollow Length (c) Lego Ship Model 11 Accelerated Motion", Trans, Society of Naval Ar- chitects and Marine Engineers, Vol. 59, pp 25-76, Discussion: 76-85 (December 1951) MICHELL, J.H.: "The Wave Resistance of a Ship", Philosophical Magazine, London, Series 5, Vol. 45, pp 106-123 (1898) MOLLAND, A.F., WELLICOME, J.F., AND COUSER, P.R.: "Resistance Experiments on a Sys- tematic Series of High Speed Displacement Cata- maran Forms: Variation of Length-Displacement Ratio and Breadth-Draught Ratio", University of Southampton, Department of Ship Science, Re- port 71, 82+i pp (March 1994) NEWMAN, J .N.: Marine Hydrodynamics, The MIT Press, Cambridge, Massachusetts, 402+xiii pp (1980) Figure 8: Comparison of Hollow Length (d) Lego Ship Model 12 RAYI,EIGH, LORD: "The Form of Standing Waves on the Surface of Running Water", Proc. London Math. Society, Vol. 15, pp 69-78 (December 1883) SAHOO, P.K., DOCTORS, L.J., AND RENILSON, M. R.: "Theoretical and Experimental Investi- gation of Resistance of High-Speed Round-Bilge Hull Forms", Proc. Fifth International Conference on Fast Sea Transportation (FAST '99j, Seattle, Washington, pp 803-814 (August-September 1999) SRETENSKY, L.N.: "On the Wave-Making Resis- tance of a Ship Moving along in a Canal", Philo- sophical Magazine, Series 7, Supplement, Vol. 22, No. 150, pp 1005-1013 (November 1936) WEHAUSEN, J.V. AND LAITONE, E.V.: "Surface Waves", Encyclopedia of Physics: Fluid Dynam- ics III, Ed. by S. Flugge, Springer-Verlag, Berlin, Vol. 9, pp 445-814 (1960)

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DISCUSSION L. Rahe a Indian Inshtute of Technology, India It was a nice presentahon. It is not a queshon but more of a comment. It seer s that the primary reason or monvahon for using such a simp ined model (i.e. ccuterplaue source di.vribuuon I is because it is expected or presumed that the wave drag will be much smaller than She f ichona pa t. I propose Chat the model cou d be improved upon by using a desingu anzed panel method where She sources cou d be placed lid e inside the bounds y which wou d sell preserve She imp ined ca cu anon aspect and may work better in She sense of di using She prearm Fr on. AUTHOR'S REPLY I appreciate your comment. Since we were getting good resu h in this case, so we continued with the model. However, we may improve upon it in subsequent work. Tha k you very much for your qllo.r on.

Representative terms from entire chapter:

theory theory theory