1 1. INTRODUCTION 1.1 Membership Chairman: Prof. Fred Stern Iowa Institute of Hydraulic Research, UNITED STATES OF AMERICA Secretary: Dr. Hoyte C. Raven Maritime Research Institute Netherlands, NETHERLANDS Members: Dr. Ulderico Bulgarelli Instituto Nazionale per Studi ed Esperienze di Architettura Navale, ITALY Mr. Lars T. Gustafsson SSPA Maritime Consulting AB, SWEDEN Dr. Moustafa Abdel Maksoud Schiffbau-Versuchsanstalt Potsdam GmbH, GERMANY Prof. Luis Perez-Rojas Escuela Técnica Superior de Ingenieros Na- vales, SPAIN Prof. Toshio Suzuki Osaka University, JAPAN Prof. Lian-di Zhou China Ship Scientific Research Center, CHINA 1.2 Meetings The committee met 5 times: April 1997, Rome, Italy November 1997, Potsdam, Germany May 1998, Osaka, Japan August 1998, Iowa, USA November 1998, Madrid, Spain 1.3 Tasks and Report Structure Below we list the tasks given to the 22nd Resistance Committee (RC), and indicate how these have been carried out. • Review the state of the art, comment on the potential impact of new developments of the ITTC, and identify the need for research and development for resistance and flow. Monitor and follow the development of new experimental techniques and extrapolation methods. • Prepare an up-to-date bibliography of rele- vant technical papers and reports. • Monitor the development of CFD methods. State-of-the-art reviews are given regarding Resistance and Flow Physics (Section 2), Trends in Experimental Techniques (Section 3), and Trends in Computational Fluid Dynamics (CFD) (Section 4). Section 7 (Prognosis for Towing Tank Work) reviews trends in ship design and operation, and impacts of these and of the developments described in the preceding sections, on the future of our profession and the operation of towing tanks. The Resistance Commitee Final Report and Recommendations to the 22 nd ITTC 2 The reviews focus on the last three years, except for topics not covered in recent RC re- ports which cover a longer time period. The RC was unable to comprehensively re- view development of extrapolation methods, a task previously carried out by the Performance Committee. Some comments on extrapolation of viscous resistance are, however, included in Section 2. • Review the ITTC recommended procedures, benchmark data, and test cases for valida- tion and uncertainty analyses and update as required. Pass the information to the Qual- ity Systems group for publication in 1999. • Identify the requirements for new proce- dures, benchmark data, validation, uncer- tainty analyses and stimulate the necessary research for their preparation. • Review ASME and ITTC recommendations on quality assurance and uncertainty analyses. Derive procedures for imple- menting guidelines for typical ITTC ex- periments in the field of resistance and flow. Updated procedures for experimental un- certainty analysis methodology and guidelines for towing-tank experiments are described in Section 5, and illustrated with an example for a towing-tank resistance test based on collabora- tive work by most RC institutes. This effort demonstrates the procedures and provided some improvements. The resulting procedures are included in the Quality Manual (QM) and recommended for adoption by the 22 nd ITTC. Work on the development of procedures and methodology for verification and validation of CFD simulations with an example are sum- marized in Section 6. The resulting procedures are included in the QM and are recommended for interim adoption by the 22 nd ITTC. Available benchmark data for CFD valida- tion for resistance and propulsion is evaluated in Section 6.5. The resulting procedure is in- cluded in the QM and recommended for adop- tion by the 22 nd ITTC. The RC has cooperated closely with the Quality Systems Group in reviewing and edit- ing other procedures to be included in the QM based on previous RC recommendations. • Continue to encourage and monitor CFD validation including liaison with other or- ganizations such as ASME. All members were active in their respective regional professional organizations. In particu- lar, close liaison was maintained with European Thematic Network on CFD and ASME activiti- es concerning verification and validation of CFD simulations. 2. RESISTANCE AND FLOW PHYSICS The flow around a ship hull displays a large variety of physical phenomena, many of which are relevant for resistance and propulsive power. The RC restricts itself to physics of in- terest for a ship in steady motion in still water, including the effect of the propulsor on the hull flow. Response to a questionnaire distributed by the 21st RC indicated some flow phenomena that were expected to be relevant for future ship designs and would require attention from a physical point of view, viz. bow-wave breaking, bilge vortices, and separated flows. However, many more phenomena are of interest, such as: Reynolds number (R n ) effects and scaling, boundary layer and wake, stern flow, turbu- lence, vortex flow and separation, Froude num- ber (F n ) effects, wave breaking, bow flow, tran- som flow, propeller-hull interaction, wave- boundary layer and wake interaction, etc. This section will discuss several of these challenges, recent ideas, and developments. 2.1 Waves In the field of ship waves, there has been recent interest in the physics of breaking waves, the detailed phenomena at the ship bow, ship waves in restricted water, and wave-wash ef- fects. These topics are discussed below. Breaking Waves. Breaking wave phe- nomena are widely recognized to be of impor- tance in ship hydrodynamics. Breaking waves dissipate wave energy and affect resistance. Subsequent spray formation and air entrain- ment can be important for ship wake signatures. Under the safety point of view, breaking ocean waves interacting with ships are of main im- portance for ship capsizing. Although ship wave breaking phenomena are associated 3 mainly with bow flows, they may occur any- where in the ship wave pattern, and in particu- lar play an important role for transom sterns. Baba (1969) discussed a component of ship resistance generated by the breaking of waves, especially at the bow of full ships. Based on similarity with a shallow water hydraulic jump, he supposed this resistance component to fol- low F n law of similitude, like wave resistance. He also pointed out that this component can be caught by wake survey method and corresponds to the head losses found near the free-surface and outside the usual frictional wake belt. In recent experiments for a VLCC model, Van et al. (1998a,b) noted that low momentum fluid accumulated near the free-surface at the outside of the wake. Also in this line, Kanai et al. (1996) showed through numerical simula- tions using Reynolds-averaged Navier-Stokes (RANS) codes, that the energy deficit generat- ed near the bow is convected downstream and distributed near the free-surface. At the aft part of the ship, the loss due to wave breaking is mingled with the losses due to viscous effects, and apparently has an influence on the forma- tion of the wake. There have been several more fundamental studies of the physics of wave breaking. An excellent review of the current knowledge is given by Longuet-Higgins (1996). In particular, starting from the experimental observations of Duncan et al. (1994), the role of surface tension in characterizing the breaking wave field up to wavelengths of 2 meters is discussed. Although this is not necessarily of quantitative concern for full-scale ships, it has to be taken carefully into account when interpreting model-scale experiments. At full scale, viscous dissipation is of primary importance. Duncan (1983) made systematic measure- ments of breaking wave phenomena by towing a submerged hydrofoil in the laboratory. In his picture of the flow, a large part of the pressure drag on the hydrofoil, which is due to the presence of the free-surface, appears as mo- mentum loss in the turbulent surface wake. This corroborated Baba’s assumption. Like Baba, Dabiri & Gharib (1997) also suggested that wave breaking phenomena can be modeled as a hydraulic jump where F n is based on the thickness of the free-surface jet and on the velocity of the free-surface jet just prior to breaking. Also based on Duncan's experiments, Cointe & Tulin (1994) derived a physical and mathematical model for steady spilling break- ers. They suppose the breaker itself to be an essentially stagnant eddy, sitting on the forward face of the breaking wave. It is held in place by a balance between the weight of the breaking volume of fluid and the turbulent shear stresses acting on the streamline that separates the breaker and the underlying flow. Sadovnikov & Trincas (1998) consider that viscous processes should be taken into account, since energy dis- sipation plays a leading role in spilling breakers. They combine a model of a steady spilling breaker with a numerical technique based on fully non-linear potential theory, which implic- itly includes viscous effects. They demonstrate that the hydrostatic model by Cointe & Tulin (1994) is a particular case of their proposed model. In their opinion, the model should be improved on the basis of new experimental data concerning the shape of the spilling breaker and the density of aerated water inside. Obviously, the modeling of wave breaking phenomena is still incomplete. Also on the on- set or inception of wave breaking there seems to be incomplete understanding. There is exten- sive literature on breaking of ocean waves, but perhaps not all of their properties carry over to ship waves. It is known that the onset of a spilling breaker is connected with the near- surface vorticity as well as the dynamic char- acteristics of the near-surface shear layer. Dab- iri & Gharib (1996) pointed out that the vortic- ity injection due to the free-surface deceleration is dominant over the gravity-generated vorticity flux. Miller et al. (1998) found that the presence of free-surface drift layers reduces the maximum non-breaking wave height and that this wave height correlates with the surface- drift velocity. A somewhat separate problem occurs in calculations of ship wave patterns or ocean waves. If no modeling of breaking and the con- sequent dissipation is included, one at least wants to be able to continue the computation, and to approximately represent the effect of wave breaking on trailing wave amplitudes. This requires both a criterion for the occurrence of breaking, and a model for its dissipation. Recently, Subramani et al. (1998a,b) have pro- posed a curvature-based criterion using the fact 4 that when waves break, they attain a profile with a sharp crest of infinite curvature. Ales- sandrini & Delhommeau (1998) proposed a variation in order to take into account non- symmetrical free-surface flows. Bow Flow. There have been some recent detailed studies of local flow phenomena at the bow of ships. For sufficiently fine bows, a thin splash is created on the side of the hull, which rises and eventually falls, and oblique waves appear moving away on either side. A detailed numerical study of this phenomenon was made by Tulin & Wu (1996), using a non-linear two- dimensional plus time (2D+T) approach. This confirmed the strongly nonlinear nature of the near-bow flow. Explanations were presented for certain differences between the Kelvin pat- tern and a typical ship wave pattern, in which the crests of the divergent bow waves tend to be straight and have an inclination decreasing with increasing F n . The sheet of water rising along a fine ship bow, and the resulting spray formation, may affect the wave pattern and the electromagnetic scattering properties (radar signatures). There have been some recent detailed studies of these phenomena. Stern et al. (1996b) reinvestigated the bow flow of the Series 60 C B =0.6 ship model using both experiments and CFD. The data indicated a thin film and beads at the bow, which with distance from the bow underwent transition from steady laminar to unsteady tur- bulent flow and merged with the downstream bow wave with increasing wake width and de- creasing free-surface disturbances. The bead flow appears vortical and suggests significant viscous and surface tension effects. Dong et al. (1997a) studied, through parti- cle image velocimetry (PIV) measurements and free-surface visualisation, the flow around a ship model, focused on the flow within the liq- uid sheet forming around the bow. They de- monstrated that the formation of the bow wave and the thin liquid sheet on the body, upstream of the point at which the bow wave separates from the model, involved considerable vorticity production. Considerable energy loss occurred in the forward face of the wave, especially near the toe. Dai et al. (1996, 1998) made an experi- mental study of turbulent primary break-up of plane turbulent free jets, as a model of the sepa- rated portion of the bow sheet. Sarpkaya & Merrill (1998), through experimental investi- gation of the ligament and drop formation at the free-surface of liquid wall jets flowing over smooth and sand-roughened plates, suggested that smoother bow surfaces with suitable cur- vatures may help to alleviate the spray problem. In computations of bow flows it is often as- sumed that the free-surface is single-valued and air entrainment is not accounted for. To cir- cumvent this problem, Dommermuth et al. (1998) propose a LES formulation with a level- set approach, which permits the simulation of turbulent free-surface flows and the modeling of air entrainment. Restricted Water Effects. Restricted water affects not only the wave resistance but also the viscous resistance. The latter effect can arise even at low F n , when insignificant wave mak- ing occurs, due to the change in pressure and velocity field around the hull associated with proximity of the seabed. Not many studies on restricted water effects have been published in recent years. Chen & Sharma (1994) found by numerical computa- tion, and verified experimentally, that the wave resistance of a ship moving at supercritical speed in a channel can be reduced significantly by shifting its track from the channel centreline to a certain speed-dependent location near one of the channel sidewalls. Intuitively, since wave dispersion is weaker in shallow water, the inter- ference between waves arising from different origins becomes more effective than in deep water, especially at supercritical speeds. More recently, Chen & Sharma (1997) showed that the wave resistance acting on a slender body in a channel becomes zero for a suitable combi- nation of body speed, channel depth and width if the afterbody geometry is adapted to an arbi- trary forebody. Another significant aspect of waves gener- ated by fast ships in shallow water is their non- linearity. A ship moving in a shallow channel near the critical speed can shed solitary waves, which run ahead of the ship, travel a bit faster than it, and cause oscillations of the ship. Jiang & Sharma (1997) made numerical simulations of this. Jiang (1998) focuses his study on the numerical implementation of the Boussinesq equations. These equations combine the non- linear and dispersive effects of shallow water waves. 5 Wave Wash. The effects of waves generat- ed by ships, are of increasing concern. Prime reason is the continuing introduction of fast ferry services, supposed to lead to larger wave effects than conventional ships. In particular in Scandinavia, detailed studies of various aspects of fast ferry operation, including wash, have been carried out recently (Kofoed-Hansen, 1996; Forsman, 1996). The occurrence of detrimental effects of ship waves (e.g. damage to moored vessels or construction, danger to swimmers, coastal or bank erosion) depends on various aspects of the wave system and the local situation. This makes it impossible to indicate a single crite- rion governing the occurrence or absence of wash effects. The more serious wash problems predominantly had to do with coastal or bank erosion and danger to swimmers, caused by wave patterns generated in the open sea or in confined waters (Kofoed-Hansen, 1996; Ko- foed-Hansen & Mikkelson,1997). In restricted water, important factors are the critical speed effects and wave reflections. A prime factor in many wash problems is the am- plification of ship waves while they proceed from deep into shallow water. This amplifica- tion is determined by the ratio of wavelength to water depth. Therefore it may be much stronger for the long waves generated by a fast ferry, than for the shorter ones of a conventional ship running at perhaps half the speed. Conse- quently, fast ferry waves, while perhaps hardly visible in deep water, have occasionally been found to cause violent wave impacts on the coast and energetic plunging breakers on beaches; while waves from conventional ships, perhaps having a larger amplitude in deep wa- ter, undergo a much smaller amplification and may reach the coast as spilling breakers (Ko- foed-Hansen, 1996). Raven et al. (1998) show model test data for a ship wave pattern in a channel with a sloping bank, and in one case found a threefold increase of wave amplitude at the bank compared to a channel with a rectan- gular section. Besides, there may be effects of wave focusing due to a variable waterway cross section. 2.2 Viscous Flow In this subsection we address some of the physics connected with viscous flows; in par- ticular, turbulence, with emphasis on the effects of the free-surface on it, flow around append- ages, and stern and wake flows. Some remarks are made on scaling and the possible R n depen- dence of form factors. Turbulence. There are some measured data available of distributions of turbulence quanti- ties around ship hulls. One recent set is found in Suzuki et al. (1998a), who measured the tur- bulence in the flow around two ship models, by a triple-sensor hot wire in a wind tunnel. After studying the turbulent kinetic energy balance they found that a local equilibrium is not satis- fied in a stern flow field: production and dissi- pation of turbulence energy were not equal. An important but rarely addressed aspect for viscous ship flows is surface roughness effects. The principal effect of roughness is a change in the velocity and turbulence distribu- tion near the surface. Patel (1998) shows that recent applications of the k-ω turbulence model mimic the known effects of roughness rather well, and may be employed in complex flows, but there remains a need to make fresh ap- proaches to this old problem. An additional complication for ship flows is the modification of turbulence by the free- surface. This can be considered as one of the sources of error in CFD simulations compared to full scale data of surface-ship wakes (Hyman, 1998). There is quite little investigation in the literature on the effects of a free-surface on turbulence. A few of the recent numerical and experimental studies are mentioned by Sreed- har & Stern (1998a). These studies indicate that the inter-component transfer and the overall increase in kinetic energy near the free-surface is due to the anisotropic nature of the dissipa- tion tensor and an overall decrease in dissipa- tion rate near the free-surface. Sreedhar & Stern (1998b) indicate that the effect of the free- surface on turbulence is similar to that of the wall in some aspects, especially in the behavior of normal Reynolds stresses. The turbulent ed- dies are flattened near the surface, with the sur- face-normal fluctuations suppressed and the other two components of velocity fluctuations gaining energy. Unlike the wall region, there is negligible shear and turbulence production near the free-surface. This anisotropy of normal stresses is gener- ally known to be a cause of secondary flows. In this regard, Longo et al. (1998), in an experi- mental study for a surface-piercing flat plate, 6 report the thickening of the boundary layer and wake near the free-surface and the existence of two regions of high streamwise vorticity of opposite sign near the juncture region. Hyman (1998) indicates that the free- surface/turbulence interaction model does force anisotropy at the free-surface and leads to en- hanced spreading. At a more fundamental modeling level, it is known that coherent structures play an impor- tant role in wall turbulence (Robinson, 1991). A very important type of structure is low-speed streaks (Blackwelder & Kaplan, 1976). These probably come from the transition process and generate internal high-shear layer. Breakdown of the flow is possible when the shear reaches a certain threshold, leading to a turbulent burst, (Kim et al., 1971). It is now evident that wall turbulence is non-homogeneous and non- isotropic. The nearly periodic process of burst generation, the presence of low-speed streaks and the no-slip condition at the wall determine a complex flow that accurate models for cal- culations of averaged quantities in numerical codes are very difficult to make, even for sim- ple body shapes, (Speziale, 1994). Nevertheless, the use of turbulence models will be unavoid- able for many more years, for flows at high R n around bodies with complex geometry. Stern and Wake Flows. The viscous flow in the stern of a ship hull is characterized by a thick boundary layer, viscous-inviscid interac- tion, highly skewed flow, complex turbulent flow field, near wake and propeller action, and often a pair of longitudinal vortices. The latter are created where the flow passes over a region with large girthwise variations of the wall pres- sure field, causing near-wall flow convergence and open separation. The longitudinal vortices may pass through the propeller disk and cause a distortion of the inflow to the propeller. While the vortices in principle lead to an increased resistance, the flow distortion is often used to advantage for equalizing the wake field. V-shaped sterns generally induce weaker stern bilge vortices and lower viscous resistance; U- shaped sterns result in generation of stronger vortices but may thus make the propeller inflow more uniform. The designer must find the best compromise. This obviously requires that location and strength of the vortices (at full scale) can be predicted and influenced. This explains the large current interest in the computational pre- diction of longitudinal vortices in a ship’s wake. With standard turbulence models, current RANS solvers generally are unable to reliably predict the strength and location of the longitu- dinal vortices, the local axial velocity deficit in the vortex core, and the "hook shape" in the isolines for axial velocity. The longitudinal vortex strength at the propeller position is often underestimated by present codes; a near-wall non-isotropic turbulence model seems needed to do better (Deng & Visonneau, 1997). Appendages. A variety of appendages may be present on ship hulls, and these involve some particular physics. For high-speed ships to maintain stability and speed in a rough sea, some means of controlling the ship motions are necessary. Among various devices, fins have been recognized as very effective. Lee et al. (1998) found that the free-surface effect on the lift characteristics of fins attached to a body is significant when the submergence depth is less than three times the chord length. The domi- nant cause is the change in the flow incidence angle to the fins induced by the free-surface deformation caused by the strut. Masuko (1998) studied the effects of stern fins compu- tationally. Stern fins interrupt the downward flow of bilge vortices and the pressure above the fins increases. This pressure increase causes a reduction of resistance. It is known that adoption of stators located in front of the propeller can result in a reduc- tion of rotational energy losses, so that propul- sion efficiency can be increased. In addition, a non-axisymmetric upstream stator can alter the inflow to the propeller in such a way that un- steady forces and cavitation can be reduced. Recently Shen (1997), for an appended body of revolution, considered such a ”guide vane” composed of 4 radially placed foils. The flow at the junction of an appendage and a hull can be very complex. The hull boundary layer flow encounters an obstruction, often nearly perpendicular to the upstream flow. This results in both a non-recoverable loss of energy (drag) and a spatially varying flow field characterised by the so-called horseshoe vortex system shed around the obstacle. This structure is of interest since an excessive level of noise or vibration may have its origin in the genera- tion of additional turbulence in the core of the horseshoe vortex. Through calculations, Deng & Visonneau (1998) showed the strong turbu- 7 lence anisotropy associated with the develop- ment of this horseshoe vortex. More studies about appendages would be needed in order to understand fully the flow around them and their influence on the flow around the ship hull. Scale Effects and Form Factor. The flow around a model differs from the flow around the equivalent full-scale vessel due to the R n difference. In general, for increasing R n the flow features become more compact and com- pressed, and boundary layers and shear layers get thinner relative to the length of the vessel. However, other changes in the viscous flow may occur that are harder to foresee. While scale effects have different aspects, here we shall only address the scale effect on "form factor." The form factor tries to ap- proximate the influence of three dimensionality on viscous resistance, as a function of R n . Alt- hough in principle it may vary with R n as well as F n , in the ITTC (1987) the use of a R n - independent 1+K was recommended for routine work. In ITTC (1996a), the RC concluded that the variation of the form factor with R n depends on the distribution of resistance between friction and pressure. For a typical ship hull, the pres- sure component seemed to dominate and there- fore the form factor would increase with in- creasing R n . Nevertheless, the Powering Per- formance Committee at the same ITTC (1996b) concluded that the variation of form factor with scale might be within the level of uncertainty with which the form factor can be estimated. Grigson (1996) reanalysed the resistance tests of the Lucy Ashton and Victory geosims. Using the ITTC 1957 Correlation Line, he had got a marked scale effect, with a form factor larger at full-scale, in discord with what should be expected from the physics of the flow. The relative displacement thickness of the boundary layer decreases with scale, relatively the flow becomes less “full” as scale increases. Never- theless, any scale effect on the form factor is very small and if anything, the form factor is smaller at full scale than for the model when a fairly accurate friction law is used as the author provides. He is demanding a new Correlation Line. Bruzzone et al. (1997) presented the results from various geosim tests, highlighting the influence of R n and F n and of the hull form on the form factor but not with an unambiguous conclusion about scale effects. The values of the form factor sometimes increased and some- times decreased with scale. They propose to establish a minimum dimension of the model and for the large-scale models to make a cor- rection for blockage effects. Furthermore, it is necessary also to establish type, dimension, and position on the model of the turbulence stimu- lators. Garofallidis (1996) also pointed out that turbulence stimulation is of primary importance for the correlation, and raised questions about the applicability of the form-factor method for models 3 to 4 metres long. Kasahara & Masuda (1998), for different models of the ”DAIOH” ship, found that the measured form factor in- creases as R n increases for models longer than 7 m, but decreases up to this length. With the present developments in CFD, trends of the form factor in principle could be computed, particularly the R n -dependence. However, there are difficulties associated with this, such as doubts on the validity of turbu- lence models and the lack of validation data for very high R n . 2.3 Wave/Viscous Interaction Interaction between the wave making and viscous flow is conventionally disregarded in model test extrapolation and in computational prediction. Such interaction comes in many forms, such as effects of the inviscid pressure field upon the viscous flow, the effect of the modified (wavy) streamline pattern along the hull, the free-surface boundary conditions in the viscous-flow region, the effect of the free- surface on turbulence, and the different geometry of the wetted hull surface if waves are taken into account. The first phenomenon mentioned above, the effect via the pressure field, is easily under- stood from considering the inviscid pressure distribution on the hull. It is common to obser- ve that the pressure variations along a ship hull are drastically larger in the flow with free- surface, than in a double-body flow. This may, e.g., cause a much steeper pressure rise from the aft shoulder towards the stern, in viscous 8 flow possibly leading to local-flow separation near the waterline. This separation may occur in a certain F n range only, and may be supposed to lead to a rather drastic and sudden change of the viscous resistance and stern flow at a cer- tain speed; raising many questions on the validity of resistance extrapolation techniques. Choi & Stern (1993) showed that the free- surface boundary conditions in the viscous flow have an important influence in regions of large velocity gradients and wave slopes, including significant free-surface vorticity flux and com- plex momentum and vorticity transport in a layer close to the free-surface. Wave-Induced Separation. The separation solely due to free-surface wave-induced effects involves the complexities of free-surface de- formations, vorticity, and turbulence in addi- tion to the already formidable subject of three- dimensional (3D) boundary-layer separation. This phenomenon was first identified by Chow (1967) and has been studied by various authors. Recently, Zhang & Stern (1996) studied wave- induced separation for a surface-piercing NACA 0024 foil. The separation patterns were found to be F n -dependent. The free-surface is mainly a sink of vorticity. A part of the vortic- ity generated at the body surface fluxes up into the free-surface and the rest goes to the wake. Using video cameras above the surface Po- gozelski et al. (1997) observed that the reversed flow appeared only at F n exceeding 0.30. Connected to the complicated physics, the study of Zhang & Stern (1996) mentions sever- al difficulties in modeling that need to be ad- dressed in order to predict such flows. It is de- sired to evaluate the performance of turbulence models for prediction of free-surface induced separated flows, and the approximation used for the free-surface boundary conditions, and to obtain time-accurate unsteady solutions. Stern and Wake Flows. While in Section 2.2 stern and wake flows already have been considered from a viscous flow point of view, here the wave/viscous interaction is discussed. One aspect of this is the viscous effect on the stern wave system. Neglecting this, the stern waves are over predicted due to the neglect of the displacement effect of the hull boundary layer and due to the neglect of the viscous damping of the generated waves (Larsson, 1997b, Raven, 1998). This effect is considera- bly larger for bluff ships, where the stern waves sometimes almost disappear. For transom sterns, various flow regimes may occur. One is a flow with a clean free- surface separation from the edge of the transom, which is typical of higher speeds and lower transom immersions. In other conditions an area with highly turbulent and often unsteady flow behind the transom face may occur. In the transition from one regime to the other, wave breaking and wave/viscous interaction play a dominant role, but a precise model seems to be lacking and at present it is hard to predict which regime will occur in which speed range (Raven, 1998). It is noted that, since the same regimes occur just as well for transoms that are above the design waterline, a transom depth F n is not a useful parameter in this regard. Van et al. (1998b) made an experimental investigation on two models, a 3600TEU con- tainer ship and a 300K VLCC that can be con- sidered as two types of modern practical hull forms. For the container ship the wave eleva- tion near the stern was observed to be flatter due to the transom effects, although the tran- som stern of this ship (located above the design waterline) was not entirely cleared. In this case the so-called “dry transom” modelling is not entirely appropriate for accurate simulation. In the case of the VLCC, the designed waterline was located above the transom edge, and there was apparently no transom effect; but the local flow measurements revealed that the flow angle seems to have an abrupt change of direction due to the transom, and possibly flow reversal. 2.4 Propeller-Hull Interaction The presence of the propeller affects the flow both by inducing a swirling effect and by locally accelerating the flow. On the one hand, this helps to stabilize the boundary layer and prevents separation ahead of the propeller (Turnock and Molland, 1998), on the other hand the propeller may also induce separation, in particular on the hull above the propeller. Nevertheless, the major effect of the pro- peller on the flow upstream on the hull is the asymmetric acceleration of the flow, which leads to different pressure reductions on the port and starboard sides for a single screw ship. The cause is the interaction of the upward 9 component of the wake field and the direction of rotation of the propeller. Abdel-Maksoud et al. (1998a-c) point out the problems in the interaction between the flow around the stern of the ship and the flow induced by the propeller: the necessity to in- clude viscous forces and turbulence effects; the complexity of the geometry and the resulting effort to discretise the problem; the interaction between the rotating propeller and the station- ary ship; and the inherently unsteady nature of the problem. The results of their computations showed the strong influence of the propeller on the flow region, especially on the pressure field. 2.5 Conclusions Numerous studies have been conducted concerning ship resistance and flow physics. However, physical understanding of many de- tailed aspects of ship resistance remains in- complete since most studies are phenome- nological and fail to provide useful models. Limited study has been devoted to the impor- tant topic of scaling/extrapolation methods. More work is needed on wave breaking, turbu- lence, roughness effects, and viscous flow/free- surface interaction. 3. TRENDS IN EXPERIMENTAL TECH- NIQUES 3.1 Introduction Trends in experimental techniques over the last ten years of relevance to towing-tank testing are summarized. Techniques are consid- ered which are useful both for routine testing for design and evaluation as well as for more complex testing at model and full scale for physical understanding and CFD validation. Experimental techniques are divided into two categories: current and developing techniques. Current techniques refers to those already widely used, whereas developing techniques refers to those not widely used or from other fields which likely have near-term applicability to towing tanks. For many developing tech- niques, further developments are required, e.g., in extending a technique from physical to model or full scale testing. For each category, a sub division is made with regard to the scale of the experiment, i.e., physical-, model-, or full- scale testing. Physical-scale experiments are directed at providing data for documenting a particular physical phenomenon such as effects of pres- sure gradients on flow separation, effects of roughness on turbulent boundary layers and wakes, effects of turbulence and pressure gra- dients on corner vortices, etc. Such experiments are conducted using specialized/idealized ge- ometry, which may not resemble a ship's hull but may represent a local portion of it, e.g., appendage/hull juncture and flat plate with an imposed pressure gradient. Both time-mean and unsteady data are procured and used for physi- cal understanding, model development, and CFD validation. Model-scale experiments are directed at providing data for design and evaluation and for documenting particular physical phenomena such as boundary-layer and wake, vortex flow and separation, propeller-hull interaction, etc. Such experiments are conducted using scaled model ships, e.g., cargo/container, combatant, and tanker hull forms. The intention is to repli- cate full-scale conditions; however, lack of similitude and environmental conditions im- pose significant limitations. Both time-mean and unsteady data are procured and used for design and evaluation, physical understanding, model development, and CFD validation. Full-scale experiments are directed at providing data for sea trials, design and evaluation, and for documenting particular physical phenomena such as R n scale effects, turbulence, cavitation, etc. Such experiments are extremely difficult and subject to variable environmental conditions. Both time-mean and unsteady data are procured and used for sea trials, design and evaluation, physical under- standing, model development, and CFD vali- dation. These categories and divisions will change even in the near future as advancements are made, e.g., developing techniques for physical scale will be used for model scale and devel- oping techniques for model scale (e.g. flow measurements and flow observations) will be used for full scale. 3.2 Current Techniques 10 Physical Scale. Physical-scale tests are not usually conducted in towing tanks. Model Scale. Current techniques for model-scale tests were identified through a questionnaire distributed to RC and several Japanese ITTC members. The techniques are summarized in Table 1 and include measure- ment systems for forces (and moments), car- riage/model speed, water temperature, motion (sinkage and trim), flow visualization, surface pressure, nominal wake, wave profiles, and wave elevations. Additionally, in some cases, wind tunnel tests are done using double models. Current techniques are conveniently discussed with regard to routine and non-routine tests. Routine tests include forces, carriage/model speed, water temperature, sinkage and trim, flow visualization, wave profile, wave eleva- tions, and nominal wake at the propeller plane. For force/moment measurements, most towing tanks use load cells and only a few are still us- ing the counter-weight method. Measured forces are converted to digital format and aver- aged by computer programs (e.g., Longo & Stern, 1996). Model velocity is measured by two different methods. One is to measure ve- locity relative to the ground (carriage speed) using a speed circuit; the other is to measure the velocity relative to the water using a current meter. Conventional mercury and semi- conductive thermometers are used to measure water temperature. Sinkage and trim are meas- ured using potentiometers or ultra-sonic height meters. The measurements are taken at two points near the forward and after perpendicu- lars and converted to sinkage and trim. Flow visualization is performed using surface or depth tufts mounted to the hull surface. A tuft grid is also used to observe rotational flow at the propeller plane. Paint on the hull surface or dye injection is also used for flow visualization. Surface pressures are measured at select loca- tions using pressure taps and differential pres- sure transducers. Wave profiles along the hull are measured by photo or CCD camera. Servo- mechanism and finger wave probes are used to measure transverse wave elevations. Capaci- tance and resistance type wave probes are used to measure longitudinal wave elevations (cuts). Longitudinal wave cuts are used for deriving wave pattern resistance, CFD validation, etc. In the former case, the probe position is important. Each towing tank has its own standard for pro- be position and data acquisition time. Nominal wake is usually measured using Prandtl-tube or 5-hole pitot probe rakes and differential pres- sure transducers. Non-routine tests include the same variables as for routine testing, but with considerably larger mapping of the flow through dense data locations. Examples include surface pressure (Toda et al., 1990), wave elevations (Toda et al., 1992, Ikehata et al., 1998), and detailed mean velocity and pressure measurements using 5- hole pitot probes (Longo & Stern, 1996). Other non-routine tests are mean velocities and Rey- nolds stress measurements using double models in wind tunnels and triple hot wire sensor ane- mometer (Hyun & Patel, 1991a,b, Suzuki et al., 1997, 1998a-c). In these cases, six components of the Reynolds stress are measured, together with the three mean velocity components. The data are mainly used to validate CFD predic- tions. Full Scale. Routine tests for full-scale ships are measurement of propeller torque, propeller shaft revolution, and the velocity relative to ground and water. Sea conditions as wave height (observations) and relative wind velocity are also measured. Baba & Ikeda (1991) im- proved the Togino type torque meter for full- scale ships using a one line CCD sensor. Non-routine tests are thrust and towing force measurement, wake measurements by 5- hole Pitot tube or laser Doppler velocimetry (LDV) systems, cavitation observation, and measurement of pressure fluctuations around the propeller. Shaft thrust force was measured for the icebreaker SOYA (Uto & Narita, 1998) using direct measurement of propeller shaft compressive strain. Torque coupling effects caused by misalignment of the gauges were evaluated by Suzuki’s method (Suzuki et al., 1992). Towing force measurements were per- formed using a patrol boat (Hara et al., 1994) in the case of a rescue operation in a rough sea state. 11 Table 1. Current techniques for towing tank testing of resistance and flow. Type of test Equipment Comments Force Load cell + filter Counterweight Balance Measured forces are for example: resistance and trans- verse forces at FP and AP or at rudder; lift and drag forces measured on model advancing with drift angle; 6- component forces at constrained hull; unsteady meas- urements of resistance, pitching moment and heave and forces at fixed model conditions. The resistance force or added resistance in waves is bal- anced by weights. Velocity Current meter Wheel + pulse counter Velocity relative to the water typically measured at 0.5L –1.0L in front of model at half of mean model draft. Velocity of the carriage relative to the ground (rail). Temperature Mercury thermometer Resistive thermometer Quartz thermometer Temperature is typically measured as mean temperature over the measured distance at half of mean draft of model or at one/several fixed depths at fixed locations in the tank. Occasionally combined with artificial mixing of tank water. Motion (sink- age/trim) Potentiometer Ultrasonic distance me- ter Vertical displacements are measured in generic positions fore and aft of the model where after sinkage and trim are calculated. Flow visu- alization Wet paint Tufts on hull or grid Paint applied in stripes on hull, which becomes flow lines when model is towed or self-propelled through wa- ter. Tufts applied to hull (grid) with needles or with thin tape. Documented by video and observations. Hull pres- sure Pressure tap + DPT (DPT = Differentiated Pressure Transducer) Wake Prandtl tube rake + DPT 5-hole pitot probe rake + DPT Small propeller type velocimeter Normally measured in the propeller plane (typically at the intersection between r/R=0.7 and the generator line) for every 5-15 degrees, or with smaller intervals if neces- sary. Sometimes measured in a rectangular mesh. Wave eleva- tion at hull side Visual observations from photo or video Professional camera Spray paint Model is marked with stations and waterlines where after wave height along hull side is estimated from photos or video. Video and photographs of free-surface turbulence and wave breaking Wave profile obtained by moving the camera step by step from bow to stern. Wave eleva- tion at free- surface Resistance probe Capacitance probe Longitudinal cuts at different distances from hull side. Wave heights also measured at non-fixed locations e.g. stern wave measurements. 12 3.3 Developing Techniques Physical Scale. Many developing tech- niques have application to towing tanks, alt- hough as already noted in many cases further developments are required in extending a tech- nique from physical to model or full scale test- ing. The following discussions focus on several developing techniques, which hold promise for applications in towing tanks, i.e., PIV, LDV, deformation measurements, shear-stress meas- urements, pressure measurements, and com- bined experiments and CFD. PIV. 3D velocity measurements by image processing techniques have rapidly progressed in the last ten years; in particular the PIV tech- niques discussed below. In most common methods for measuring fluid flow velocities, the fluid is seeded with particles or markers, where after the flow field easily can be traced and imaged. In the absence of particles, flows have also been tagged with lines or grids using laser induced photochemical reactions or laser induced fluorescence. Barnhart et al. (1995), Slepicka & Cha (1997), and Fabry (1998) measured 3D velocity fields by using a holographic method. The 3D particle positions in the water channel are fro- zen in two holographic pictures with a small difference in time. The reproduced particle im- ages are detected as two-dimensional move- ments in the screen. In the case of 3D motions, the screen was moved to focus the particle im- ages. Adrian et al. (1997) and Gaydon et al. (1997) investigated the use of stereoscopic photographs and thus analysed the 3D particle motions. Kawakatsu et al. (1991) also used the stereoscopic photograph method but analysed it by a particle image correlation method. Koba- yashi et al. (1995) measured 3D positions and temperature simultaneously by the use of a mi- cro-capsulated liquid crystal particle. Kawasue & Ishimatsu (1996) introduced a very interest- ing method to measure 3D positions of particle images. They rotated the camera images and found that the diameters of circled images were related to the axial distance from the camera to particles. Raffel et al. (1995) applied a dual laser sheet technique to measure 3D velocity com- ponents, and succeeded to measure the 3D ve- locity fields around simple models in water channels. Post et al. (1994) developed a two- colour laser sheet method and measured the behaviour of high shear layer. Nishio (1995) proposed a statistical approach to measure the time-mean velocity. The frequency of image scattering was measured and a statistical ap- proach was applied. Okuno (1995) applied a spatio-temporal method to measure the velocity in a separated flow, and also applied this method to measure the movement of an oil film on a ship model surface. Okuno & Sakamoto (1990) applied Fourier transformation to the image picture and found the direction of mo- tion and the distance of the particles. Error analysis for PIV has been performed and compared with theoretical calculations. Wei et al. (1995) discussed the effects of vorti- ces and shear layers. Lourenco and Krothapalli (1995) discussed the accuracy of detecting the maximum auto-correlation point. Peysson & Guazzelli (1998) and Oschwald et al. (1995) analysed that if the light sheet plane is out of focus from the focal plane of the camera, the position in the image plane is not a linear func- tion of the position in the light-sheet plane. They also pointed out the systematic errors in PIV for a rotating mirror method. Thomas et al. (1993) studied the response of particles to a large velocity gradient field by measuring 3D particle velocities in a shock wave using PIV and 3-component LDV. LDV. Compton & Eaton (1996) succeeded to measure the viscous sublayer in the turbulent boundary layer by high resolution LDV using a small mirror in the flow. The measurement point closest to the wall surface was at about y + =5. The results showed good agreement with velocities and 6 components of Reynolds stresses measured by pressure probe or X-type hot wire anemometry. Deformation Measurements. Many papers focus on the correlation of two successive sca- lar images for the purpose of measuring imaged fluid motions. A methodology for direct meas- urement of velocity and velocity gradient field was developed by Tokumaru & Dimotakis (1995). They used a temporal spatial method and introduced the velocity and velocity gradi- ent as unknown parameters in an optimisation process. Su & Dahm (1996) propose a Scalar Imaging Velocimetry (SIV) technique for fully resolved four-dimensional (x, y, z, t) vector velocity field measurements in turbulent flows. SIV technique is one of the temporal spatial methods applicable to inner turbulent shear 13 flow. They succeed to measure the unsteady velocity field in a 3D volume. Laser excited fluorescence was studied by Hill & Klewicki (1996). They dealt with the LIPA (Laser Induced Photochemical Ane- mometry) and measured velocity and stream- wise vorticity distributions in the inner layer. This method entails the use of a laser and a light sensitive chemical. Two types of photo- chemical can be used: photochromic or phos- phorescent chemicals. The paper by Hill & Klewicki (1996) proves that this technique is a valuable measurement tool for understanding turbulence because of the high frequency re- sponse of the luminescent fluid. Gendrich & Koochesfahni (1996) present a spatial image correlation technique for estimating the dis- placement vector of the tagged regions with a much higher level of accuracy then had previ- ously been achieved. Shear-Stress Measurements. Various tech- niques for measuring shear stress on a wall have been developed, e.g. Preston tubes, float- ing elements, or laser based systems. The latter do not have probe disadvantages like mixed sensitivity, individual calibration, direct electri- cal contact, fragility of the sensors, etc. These advantages of laser systems are proven by the holographic fan fringe sensor, which Millerd et al. (1996) used to measure velocity gradients (shear stress and skin friction) inside the boundary layer. The dual-cylindrical wave (DCW) system, studied by Naqwi (1993), pro- duces an optical measuring volume by two in- terfering cylindrical waves from a laser. This system is a variation of LDV, and its applica- tion to shear-stress measurements in turbulent boundary layers and particle sizing are devel- oped in some studies. Shear stress near the wall is determined by the local velocity gradient immediately adja- cent to the wall. Nepomuceno & Lueptow (1997) measured it using a hot film wall shear- stress probe, mounted upstream of a hearing aid microphone for wall pressure measurements and a hot wire velocity probe. The shear stress probe was calibrated against a Preston probe. Preston probes cannot be used to measure wall friction in a ship model bow region due to the thin boundary layer. Ito & Oyanagi (1992) and Matsumura et al. (1995) proposed a non- contact measurement method using an oil film. The principle is that the oil dot or oil film on the model surface spreads at a rate proportional to the wall shear stress. The movements are detected by an image processing technique. Further investigations were proposed to find the precise relation between the oil dot velocity and the local skin friction. If successful the local skin friction of any type of body surface could easily be measured. Another idea, devel- oped by Wang (1993), was to select a material in which the fluid shear force can be transferred, but the movement of fluid is extremely re- stricted. A sintered metal using small spheres can meet these requirements. A sensor is there- after used to measure a pressure difference in- dicating the wall shear stress. Some other systems using laser light or floating elements have also been developed. Lubrication theory relates the local skin friction force to the thinning of an oil film placed on the test surface. Mateer & Monson (1996) de- veloped a laser interferometer skin-friction (LISF) technique. They measured the thickness of the oil film on a wing model by laser inter- ferometer and calculated the skin friction dis- tributions. They got good agreement with CFD results for both shear stress and pressure distri- bution. Liu & Sullivan (1998) used a lumines- cence intensity method. Luminescent molecules are dispersed in an oil film and the luminescent light is proportional to the thickness of the oil film. Three kinds of NACA wings were used in measurements, showing good agreement with CFD predictions. Micro-electro mechanical systems (MEMS) based sensors were applied to measure the shear stress on a two-dimensional airfoil (Na- gaoka et al. 1997). A micro-electro hot film sensor (0.2mm*0.2mm) on a small silicon wa- fer determined the separation region on the foil. The force acting on the large eddy break up (LEBU) devices was measured using a friction- al force balance to clarify the reduction of fric- tional force by LEBU (Lynn et al., 1995). Pressure Measurements. Pressure sensitive paint (PSP) seems to be a valuable technique for measuring surface-pressure distribution in wind tunnel models. PSP contains a component that is luminescent when excited by an appro- priate light source. The luminescence of the paint varies as a function of the partial pressure of oxygen, which is proportional to the static pressure of air at the coated surface. This en- ables measurement of essentially continuous property distributions. The limitation is only 14 the resolution of the imaging equipment. Mor- ris et al. (1993) applied this technique for a wing-body model and McLachlan et al. (1993) for a 2D NACA-0012 wing. The results were compared with pressure-tap measurements and showed good agreement. The brightness of a pressure-sensitive paint is a function of pres- sure, temperature, photo degradation, illumina- tion intensity, and coating thickness. Bell & McLachlan (1996) point out the importance of the model alignment and propose a projective equation of photo grammetry to relate model to image coordinates. Experiments performed by Woodmansee & Dutton (1998) confirm that the PSP are temperature sensitive, so a tempera- ture-correction data reduction method should be used to obtain quantitatively accurate sur- face-pressure measurements. Studies of wall-pressure fluctuation were motivated by an interest in hydro-acoustic properties of smooth surfaces with irregular geometry. Horne & Handler (1991) propose a methodology to cancel the contaminating noise in the measurement of turbulent wall pressure fluctuations using the signals from two flush mounted wall pressure transducers, directed transverse to the mean flow. Corrected results show good behaviour in the low frequency range. Instead of traditional plug-in probes, Nitsche et al. (1989) used miniature pressure transducers or piezoelectric foils. The piezoe- lectric effect of polarised plastic foils is used to register time-dependent pressure or shear loads. Löfdehl et al. (1994) use very small silicon based sensors manufactured using microelec- tronic technology, to measure wall pressure in turbulent flows. High frequency pressure fluc- tuations can be captured by the very small size of the transducers. Combined Experiments and CFD. Both CFD and flow field measurements have exten- sively been used to understand flow fields. Alt- hough CFD easily can estimate the entire flow field, it inevitably contains numerical errors. Meanwhile, the various field measurements usually require a lot of labour and also contain experimental errors. Although some experi- mental techniques can measure flow velocities at many points simultaneously, it is difficult to make a dense measurement for a large field. Therefore, some new techniques have been proposed to understand the whole flow field by combining various experimental techniques and CFD. Yamaguchi et al. (1996), Ohwaki et al. (1998), and Sugii et al. (1996a,b, 1997) pro- posed a technique to predict a complete flow field by combining PIV and CFD. In this tech- nique, velocities at points where no data have been measured are calculated by using locally obtained PIV data as a boundary condition. The PIV data are corrected simultaneously to reduce measurement errors using CFD results and fun- damental equations of fluid dynamics. In cor- recting the data, the cost function, which repre- sents the sum of the adjusted amount of ob- served data and the residual of the fundamental equations, is used; the whole flow field is ob- tained by minimizing the cost function. This technique has been applied to 3D non- isothermal flow fields and flow fields with ro- tation, shear, and expansion. Dong et al. (1997b) presented an approach to determine the pressure distribution by using measured velocity field data and RANS equa- tion. The approach was tried on an airfoil sec- tion at 8 degrees incidence. The velocity and Reynolds stress distribution around the foil were measured in a cavitation tunnel by LDV. The RANS equation, the Euler equation and the Bernoulli equation were employed separately to solve for the pressure while the velocities and Reynolds stresses were considered as known from the measured data. The results were com- pared with the pressure directly measured on the foil surface, showing good agreement. Ji et al. (1998) tried this approach to determine the pressure distribution on a body of revolution with tail fins. For validation, direct pressure measurements were carried out. Model Scale. Developing techniques at model scale include applications of some of the developing techniques discussed above by a few towing tanks as well as techniques for model-scale experiments from other fields. The following discussions focus on several devel- oping techniques, which hold promise for ap- plications in towing tanks, i.e., shear-stress measurements, velocity and turbulence meas- urements, wave-resistance measurements, and wave-pattern measurements. Shear-Stress Measurements. In order to re- duce turbulent frictional drag, several tech- niques can be used, all requiring the shear stress to be measured. Kato et al. (1990), Fujii 15 et al. (1991), Doi et al. (1991), Takahashi et al. (1997a,b), Watanabe et al. (1997), Larrarte & Kodama (1997), Sato et al. (1997), and Toku- naga et al. (1998) all succeeded to carry out shear stress measurements using similar float- ing element devices. Velocity and Turbulence Measurements. Turbulence measurements in the boundary layer on a ship model can be done using hot- wire or hot film anemometers. Mori & Hotta (1988) and Wu & Bose (1992) obtained veloc- ity profiles and boundary layer properties. In both cases a hot-film anemometer has been used, as it seems to be a practical, economical and accurate tool for towing-tank applications during ship model testing. Wu & Bose (1992) consider this equipment, which also can be used to measure high frequency components of the flow as well as the mean velocity compo- nents, as more accurate than Pitot tubes. A limitation is that it cannot be used in reversing flows. Kakugawa et al. (1989) applied one- dimensional LDV to measure the velocity field around a ship stern and compared the result with 5-hole Pitot tube data. The comparison showed good agreement. Eca et al. (1994) measured tip vortices in a cavitation tunnel by 3D LDV system. The results were compared with CFD results and good agreement was ob- tained. Hoekstra & Aalbers (1996) have made extensive wake measurements for 8 ship mod- els using 3D LDV. The two-colour backscatter system permitted simultaneous measurement of 3 velocity components in the towing tank, such that turbulence intensities and Reynolds stresses could be determined. Longo et al. (1998) performed measurements of solid/free- surface boundary layer and wake using a towed two-component LDV system. Traditionally, 5-holes Pitot probes have been used for measuring 3D aerodynamic flow fields. The development of 4-hole probes has brought the advantages of a smaller size, fewer measurements in calibration and application, and less instrumentation. Improvements of the 5-hole Pitot tube are made in the form of a 4- hole pyramid probe, as explained by Main et al. (1996), or a 7-hole probe by Payne et al. (1989). Zilliac (1993) analysed the performance of seven-hole pressure probes and found the maximum probe onset-flow angle is approxi- mately 70 degrees. Payne et al. (1989) com- pares its suitability and accuracy for delta wing vortex flow fields with LDV measurements. It was found that the seven-hole probe is reason- ably accurate for measurements at location be- fore and after vortex breakdown except near the vortex breakdown region. The major disad- vantage of this probe, however, is its inability to measure reversed flows, and its usage is limited in the breakdown region. Turbulence near the water surface has been measured by image processing (Peirson, 1997, Logory et al. 1996, Kumar & Banerjee, 1998). Shear stress or vorticity distributions close to the water surface are also obtained with this technique giving interesting results. Pogozelski et al. (1997) and Chang & Liu (1998) measured wave-breaking phenomena by PIV and found vortices behind the breaking regions. Kumar et al. (1998) measured upwelling in a channel flow. Hering et al. (1997a,b) measured drift current under the wind-wave interaction. Wave Resistance. Hirano et al. (1991) measured the pure wave-pattern resistance around a high-speed craft. They also quantified the spray drag by measuring the spray flux dis- tribution. Their approach was to take out the spray flux, which contaminated the wave pat- tern around the ship, by using a small bucket. Wave Pattern Measurements. Besides the conventional wave-cut measurements, wave patterns can be measured using image- processing techniques. Bonmarin et al. (1989) use a slit laser light sheet to measure wind- generated wave characteristics. Zhang & Cok (1994) colour-coded the wave slope by an opti- cal method, and Zhang (1996) integrated the slope to obtain the wave height. Oshima et al. (1994) measured the stern wave pattern of high-speed craft using laser sheet and CCD camera in a circulating water channel. They also measure the wave pattern of a high-speed container ship in a towing tank (Nisho et al., 1996). Suzuki & Sumino, (1993), Suzuki & He (1997) and Suzuki et al. (1994) measured the 2D wave pattern using projected light distribu- tions which are proportional to the free-surface curvature. They measured the wave-pattern resistance around the Series 60 (C B =0.6) model and compared it with experimental data. Full Scale. Use of developing techniques at full-scale tests is limited. Velocity Measurements. Kux (1990) and Tanibayashi (1990) performed wake measure- 16 ments at the propeller position by LDV. Komu- ra et al. (1991) performed 3D particle tracking velocimetry (PTV) and LDV measurements simultaneously. However, the comparison of data did not show a good agreement. Propeller Forces. Kamiirisa et al. (1991) and Uchida et al. (1989) measured propeller blade fluctuating stresses. Uchida et al. (1989) also measured the effects of the propeller blades crossing the free-surface. Ukon et al. (1990) and Ukon et al. (1991) measured the pressure distributions for a conventional and a highly skewed propeller. Sea Trials. Takezawa et al. (1994) per- formed sea trials of superconducting electro- magnetohydrodynamic propulsion ship ”Yamato 1.” They succeeded to propel a craft by superconducting electromagnetohydro- dynamic propulsive water jet pumps. 3.4 Conclusions There is an increasing demand for more detailed model- and full-scale local-flow data, both for design and for CFD calibra- tion/validation. The advent of modern physical- scale LDV, PIV, surface shear stress and pres- sure distribution, and wave-elevation meas- urement systems (instrumentation, data acqui- sition and reduction) holds promise for meeting this demand. More work is needed on full-scale measurements, especially local flow. 4. TRENDS IN COMPUTATIONAL FLUID DYNAMICS The development of CFD for marine appli- cations has continued at an increased pace, as has its use in practical ship design. The fol- lowing sections summarise the trends over the last 3 years, of relevance to ship resistance and flow. Section 4.1 discusses application of CFD techniques in ship design practice with regard to status and needs. In Sections 4.2, 4.3 and 4.4, the present state of the art and recent develop- ments in inviscid and viscous flow methods and CFD-based optimisation are described. The discussion will be essentially limited to methods for steady flow calculations. 4.1 Practical Application of CFD; Status and Needs The actual application of CFD methods in ship design has always lagged behind the de- velopment of methods, which justifies a sepa- rate discussion. This survey partly updates the 21st ITTC Resistance Committee's discussion on "A Naval Architects View", and likewise is partly based on the committee's own perception, since the general status on applications is not easily retrieved in the literature. Inviscid Flow Calculation Methods. Most CFD applications in ship design today concern the wave pattern and inviscid flow around the hull. The methods used, discussed in Section 4.2 below, usually are panel methods imposing either linearised or nonlinear free-surface boundary conditions. The former are easier to deal with for less experienced users, as they do not iterate for the free-surface location. The nonlinear methods are significantly more accu- rate and complete, but all the same need not be appreciably more time-consuming or less ro- bust. Several shipyards have started collecting experience with these methods. Steps are being made to integrate these calculations into the design process (Tuxen et al., 1998, Kim et al., 1998c). This involves interfacing with CAD systems, automatic panel generation tools, postprocessing and visualisation programs. At towing tanks and institutes the use of these methods is more advanced and comprehensive. For some of them, inviscid flow calculations have become a standard component of any ship hull form design project, preceding model testing. Accreditation of these tools is a desired next step. While often the predicted wave resistance is the main result on which design variations are compared or optimised, there is increasing awareness that this is not the best way to ex- ploit these tools. In the first place, the resis- tance value gives no specific indication of which features of the design affect the wave making, how the design could be improved, or what aspects of the calculation are less reliable. Secondly, the predicted wave resistance may be the least accurate part of the results. In particu- lar linearised methods may give quite poor re- sistance predictions, due to their basic assump- 17 tions (Raven, 1990). Nonlinear codes at least provide positive and more consistent resistance estimates, but still suffer from numerical inac- curacy in the pressure integration over the hull and sensitivity to modelling details. Improve- ment may be obtained by applying wave pattern analysis to the calculated pattern (Nakos & Sclavounos, 1994, Raven & Prins, 1998). Even with sufficient numerical accuracy the wave resistance derived from an inviscid flow code still may be unreliable due to the neglect of viscous effects on wavemaking. While per- fectly justified over most of the hull, this ne- glect may lead to an appreciable overestimation of the stern wave system (and thereby of the wave resistance) for fuller hull forms, for cruis- er stern shapes, and in case of flow separation or dead-water areas aft of a transom (Raven, 1998). Corrections for viscous effects should improve this, but are hard to prescribe by sim- ple rules due to the sensitivity to the hull form. Consequently, wave resistance from invis- cid-flow codes will only be quantitatively accu- rate for rather slender vessels at higher speed, if predicted by a nonlinear method and with much care for numerical accuracy. For other applica- tions they may still be very useful for ranking design variations, but not for modifications that significantly affect the viscous effects on wave making, e.g. Janson & Larsson (1996). Pub- lished evidence that the predicted wave resis- tance is always good enough for ranking pur- poses, is rather limited. In any case it is prefer- ably always used in combination with a judge- ment on the predicted wave pattern. The predicted wave pattern is much more reliable and accurate. Predictions of the fore- body wave making and in particular the bow wave height at the hull are frequently used for assessing designs and ranking variations. Non- linear methods here give significantly more realistic and comprehensive predictions than linear ones. The wave profile along the remain- der of the hull is more easily predicted, and fore- and aft shoulder waves usually are accu- rate also for linearised methods. The use of predicted stern wave systems is much harder, and judgement still plays a sub- stantial role. In any case, linearised methods are not applicable to the common sterns with a transom above the still water level. Nonlinear methods perform much better in that regard but still suffer from the uncertainty whether the transom will be cleared or wetted by the flow, an intricate viscous effect not modelled. For a wetted transom, the stern wave system may be strongly overestimated. However, some suc- cesses have been shown for more slender ves- sels, for which the trends of the wave pattern and resistance with stern shape and transom height were well predicted (Raven & Valkhof, 1995, Raven, 1998). Experiments are desired for determining the dependencies and limits of inviscid methods for stern flows. Predicted wave patterns may be useful for estimating wash (Hughes, 1997, Raven et al., 1998), provided the wave evolution over large distances (and ideally other effects such as su- percritical flows or effects of bottom topogra- phy) can be accurately handled, which is a challenge. Viscous Flow Calculation Methods. For viscous flow calculation methods, RANS methods today are dominant. Besides dedicated codes for ship flows, also commercial general- purpose RANS solvers are being used. In most practical cases free-surface effects are disre- garded, but this may well change soon. Use of RANS solvers in ship design gener- ally is not yet a routine procedure but is largely limited to special applications or specific des- ign questions, mainly by towing tanks/institutes. The number of actual design calculations using viscous flow codes is limited but increasing. Also some large shipyards use viscous flow solvers on a more or less experimental basis. Reportedly, in the Far East the practical use of RANS codes at shipyards is more widespread (Larsson, 1997b). In principle, calculation of the viscous flow around the hull holds great promise for future applications: to support the extrapolation of model tests to full scale; to predict viscous re- sistance at model or full scale; to provide the effective wake field at full scale and permit an integrated optimisation of hull and propeller; and to predict the occurrence, extent and risk of separation at full scale, e.g. permitting to fix more precisely the limit of afterbody fullness. However, almost none of these examples has been realised so far and current possibilities are more limited, although quite helpful. A major restriction is that most current solvers have limited applicability for full scale. The very large velocity gradients at the wall 18 require large grid stretching and excessive cell aspect ratios, causing numerical problems for many codes. Wall functions alleviate this but are less accurate. Consequently, almost all practical use today is for model scale, and pub- lications on full scale viscous flow are rare. Eca & Hoekstra (1996) show accurate full-scale calculations without wall functions. Bull & Watson (1998) present scale effect studies for an appended submarine using some different turbulence models. Besides possible numerical problems, other issues are turbulence modelling for high R n , and the difficulty of carrying out experimental validation at full scale. Nevertheless, because of the scarceness of other information on scale effects, numerically accurate full-scale solu- tions already can be very useful and instructive. Regarding viscous resistance, substantial prediction errors are still found in the literature. In SRI (1994), results varied enormously be- tween methods, but about half of the predic- tions was within 10% of the data. Larsson et al. (1998) suppose that the prediction may be ac- curate enough for ranking design variations, provided much care is exercised in the genera- tion of the grid, particularly at the hull ends. Precise grid dependence studies are required. Bertram (1998) gives an example of a correct ranking of viscous resistance, although the magnitude of the (relatively small) differences was overestimated. Kasahara & Masuda (1998) apply a regression-analysis-based correction to their CFD-predictions, and thus predict form factors for a variety of ships to within 2 %. This suggests a correct ranking, but in absolute val- ue their CFD predictions for resistance often are 15 % in error. Therefore, as with inviscid methods the main benefit is not in predicting just resistance, but in providing comprehensive though qualitative flow field information and prediction of trends. As for the wake field in the propeller plane (with or without propeller effect), the general status is that its details cannot yet be reliably predicted. For slender ships good predictions may be obtained, for more critical cases usually the predicted wake contours are too smooth and show too little influence of longitudinal vor- tices. The circumferentially averaged wake is predicted better, and the wake fraction aver- aged over the propeller disk can be fairly good, according to limited information (e.g. Kasahara & Masuda, 1998). However, propeller design cannot fully rely on the wake field predictions now (Larsson et al., 1998). Two main causes of this are insufficient resolution of flow details, and deficiencies in turbulence modelling. Both may be significant in certain cases, but the cur- rent opinion seems to emphasise the latter cause. Predicted flow fields could also be very useful and practical for appendage alignment, design of energy saving devices and twin- gondola stern design. There is a large practical demand in this regard, since experimental tech- niques are subject to scale effects. However, calculations are hard due to the complicated geometries, and a very high accuracy of the result is often required. Larsson et al. (1998) state that useful predictions may be made for appendages that are not too close to the pro- peller plane, or for slender transom stern ves- sels. For other cases the flow directions may be expected to have a similar unreliability as the wake field. A most useful application for a class of cases is the prediction of occurrence and type of flow separation. Shortcomings in the turbu- lence modelling seem of less influence here, and calculations give much more information than is obtainable otherwise. Valkhof & Hoek- stra (1998) illustrate the practical benefit of such calculations. Summarising we believe that the very large potential of viscous flow calculations is not fully exploited yet in design; partly due to cur- rent limitations such as insufficient wake field predictions and problems for full scale; partly due to circumstances such as the required time for grid generation and geometry treatment, or the experience required for applying the methods successfully to a variety of cases. Therefore, attention is desired for: • improving wake and flow field predictions around the stern, by better representation of turbulence effects, e.g. using tuned eddy- viscosity models, Reynolds stress model (RSM) or even large eddy simulation (LES); • improving numerical accuracy; • improving the ease, speed and range of ap- plication, e.g. via multiblock, unstructured, or adaptive grids; • enhancing the applicability for full scale, and collecting full-scale validation data. 19 After this discussion of practical applications of CFD in ship design, we will now discuss recent advances in research and development. 4.2 Inviscid Flow Calculation Methods Introduction. Inviscid models for the steady flow around a ship hull predict the wave pattern and wave resistance, the velocity and pressure field; and lift effects (hydrofoils, sailing yachts). Usually the Laplace equation for the velocity potential is solved; in some cases, the Euler equations, but these mostly are intended as a step towards solving RANS equations with free-surface, and are discussed in that context later. For calculating the steady wave pattern one can either solve a transient problem until the steady state has been reached, or solve the steady problem directly. Unlike viscous flow computations, virtually all inviscid methods use the latter approach, which is successful and more efficient. This requires that a particular combination of kinematic and dynamic free- surface boundary condition is imposed, of which the Kelvin condition is one example. Steady Potential Flow Solution Methods. The exact inviscid free-surface boundary con- ditions are nonlinear and must be imposed on an unknown wave surface. Until around 1986 the problem was virtually always linearised, and research concentrated on devising suitable linearisations. Slow-ship linearised methods such as Dawson's were dominant, giving fairly realistic results at modest computational effort. While linearised codes are being used in in- dustry, during the last few years there is little development on these, as little further progress seems possible and linearisations are in most cases not needed anymore. Most publications now concern solution of the fully nonlinear free-surface potential flow problem. Taking into account the nonlinear effects improves the predictions much more than was expected before, and in other cases than was assumed. Bow wave height and di- verging bow wave system, severely underesti- mated by linearised methods, can now be quite well predicted. In Raven (1997) the differences between wave patterns found with linear and nonlinear methods are analysed and explained. Principal effects are due to imposing the free- surface boundary condition on the actual water surface instead of the still water plane, and due to the refraction of the ship's wave system by the velocity field around the hull. In addition, nonlinear methods include a more complete representation of several hull form features, dynamic trim and sinkage, and the flow off a (dry) transom stern. These methods seem mature now, and sev- eral development lines have converged to fairly similar solutions. Some references are Jensen (1988), Jensen et al. (1989), Raven (1993), Raven (1996), Kim et al. (1994), Janson (1997), and Hughes (1997). In Raven (1998) the main features of the leading methods are compared and capabilities and limitations of this flow model are outlined. The nonlinear problem is solved iteratively. Most methods start from an undisturbed free- surface and uniform flow, and obtain a con- verged result in O(10) iterations in practice. Each step solves a linearised problem that is fairly similar to that in Dawson's method. While the basic set-up is rather straightforward, care is needed for numerical details in order to come to a convergent and stable procedure. Within each iteration the Laplace equation for the potential, subject to hull and free- surface boundary conditions, is solved, usually by a Boundary Integral or Panel method, either in Green's identity or in source distribution form. Remarkably, the leading nonlinear methods now all use “raised singularities” or “desingularisation”, i.e. sources located at a distance above the wave surface. This uncon- ventional approach was first proposed by Xia (1986) and Jensen et al. (1986) for the wave resistance problem; and by Schultz et al. (1990) for unsteady free-surface problems. Raised sin- gularity methods owe their popularity in this application to some favourable properties. Us- ing the theoretical analysis method proposed by Sclavounos & Nakos (1988), Raven (1992, 1996) analyses the stability, numerical disper- sion and damping and concludes that raised- panel methods substantially reduce the numeri- cal dispersion and can eliminate point-to-point oscillations. Janson (1997) extends this analy- sis to higher-order raised panels and finds that these give no significant increase in accuracy compared to first order. The "radiation condition" that excludes any steady waves upstream of the disturbance, re- quires a particular treatment. There are two popular ways of imposing this. One is, using upwind difference schemes in the implementa- 20 tion of the free-surface boundary condition, a device introduced by Dawson (1977). This in- troduces some numerical damping, which, however, can be minimised by a suitable choice of the scheme, Raven (1998). The alternative way to satisfy the radiation condition is, shift- ing the free-surface collocation points forward over one panel length relative to the panels, a technique proposed by Jensen et al. (1986) and Jensen (1987). This permits to use analytical differentiation of panel inductions instead of a difference scheme, which is theoretically more accurate and free of numerical damping; alt- hough Janson (1997) found little advantage in practice. While the multiple iterations could require an order of magnitude more computational ef- fort than previous linearised methods, the dif- ference has largely disappeared due to better matrix solvers. The combined free-surface boundary condition together with a source-only formulation leads to a rather poorly conditioned system of equations, in the past solved by Gaussian elimination, requiring O (N 3 ) opera- tions for N panels. Today, several methods use an iterative matrix solver with proper precon- ditioning, reducing the effort to O (N 2 ) and saving much calculation time. The fastest methods now solve a fully nonlinear problem with e.g. 4000 panels in half an hour on a PC. There have been various proposals for further speedup. Soeding (1996) proposes a multigrid type approach for a panel method, and a panel clustering technique that combines the effect of several remote panels, reducing storage and computational time. More ad- vanced are "multipole acceleration" techniques (Korsmeyer et al., 1993) that approximate both far and local potential fields in spherical har- monics, permitting a reduction of the computa- tional effort for the entire method to O(N). In a simple example, for 5000 singularities the CPU time was reduced by a factor of 5 (Scorpio et al., 1996). An alternative, perhaps more easily applied to common steady wave pattern calcu- lation methods, is the "Precorrected FFT tech- nique" (Korsmeyer et al., 1996). Application of all these techniques to the problems considered here is just starting, and substantial further im- provement seems possible. However, the pri- mary benefit of these techniques will be for radiation/diffraction or wave propagation problems, where far larger panel numbers are required. Unsteady Potential Flow Solution Methods. Most unsteady potential flow methods are pri- marily meant for truly unsteady applications (seakeeping etc.), but some can be used to cal- culate the evolution of the wave pattern until a steady solution of the nonlinear problem has been found. All methods use the kinematic free-surface boundary condition to find an up- dated wave surface, and the dynamic condition to find a new potential at that surface. Rather few methods in this class permit a substantial forward speed. A noteworthy development is that of Beck et al. (1993), Scorpio et al. (1996), Subramani et al. (1998b), an unsteady “desin- gularised” point source method. For some steady wavemaking problems they get good agreement with the data (Ratcliffe, 1998). However, for steady applications this approach is less efficient, requiring two orders of mag- nitude more CPU time than a steady nonlinear panel code. Transom Sterns. There is continued interest in the flow off a transom stern, most research addressing the immersed transom sterns used for high-speed ships, rather than the transoms above the still waterline that are common for merchant vessels. Inviscid flow approximations only apply to a flow regime in which the tran- som is dry and the water surface detaches from the transom edge. There needs to be no diffi- culty in computing such a smooth transom flow as long as a potential flow solution exists. However, in a range of cases the transom will actually be wetted and the potential flow solu- tion is locally unrealistic, but the calculations do not indicate this (Raven, 1998). For dry-transom cases a particular treatment of the free-surface detachment from the tran- som edge is required. A variety of methods and physical models has been proposed, mostly for linearised methods. The fundamental inconsis- tency of a transom flow model with most line- arisation assumptions often has caused diffi- culties. Doctors & Day (1997) assume a shape of the hollow of the water surface aft of the transom, described by a few empirical parame- ters; and include this hollow as an extension of the hull in a Michell theory program. Telste & Reed (1993) model the flow off an immersed transom stern in a Neumann-Kelvin method, and propose a modified linearisation relative to a cylindrical surface extending aft from the transom edge. Wang et al. (1996) compare a Kelvin source and Rankine source method, and a method proposed by Tulin & Hsu (1986) 21 valid asymptotically for very high speed. For the former two methods the treatment of the transom flow is not discussed. Results shown, for the wave resistance of a single hull form, are inconclusive. In a nonlinear method the treatment of the flow off a transom stern is much more obvious. The free-surface boundary conditions are im- posed on the actual wave surface, and the mod- elling just needs to make sure that this wave surface has the proper behaviour at the transom edge. Raven (1993, 1996) argues that physi- cally no vorticity is involved, and points out the analogy with free-streamline theory. While the flow off the hull is tangential, the curvature of the streamlines at the transom edge may tend to infinity (although this seems to have little effect in practice). Comparison with dedicated ex- periments (Raven & Valkhof, 1995, Raven, 1998) shows that for a dry transom the correct trend of stern wave height and shape with tran- som immersion is predicted; but that some systematic deviations occur due to the neglect of viscous effects. Subramani et al. (1998b) apply a 2D version of their time-dependent nonlinear desingul- arised code to a semi-infinite body with tran- som. The calculations reproduce the two possi- ble flow regimes, one with a stagnation point at the transom face, another with a smooth flow off the transom edge; and agree very well with analytical results for the trailing wave steepness. Simplified Methods. There have been some recent proposals for simplified treatments, in- tended to give a clearer view of the physics or to permit solution of problems that cannot be dealt with by complete nonlinear codes. Noblesse et al. (1996) consider the Neu- mann-Kelvin problem, or the corresponding problem of radiation with forward speed in the frequency domain. They extend the Kochin theory to simplify the numerical evaluation of the velocity field induced by a distribution of Kelvin singularities on the hull surface. They derive an explicit form for the wavy velocity field away from the hull, in terms of a velocity distribution on a matching surface found from e.g. a nonlinear inviscid or viscous flow calcu- lation for the near field; thus enabling a promising composite approach. In the "2D+T" approximation for slender ships, the steady 3D velocity field is considered as a 2D field in cross-sectional planes, evolving in time. This approximation disregards trans- verse waves and upstream influences (via the pressure field). However, the 2D fully non- linear problems in crossplanes can be solved with very high resolution, and it is easier to include overturning wave crests. Tulin & Wu (1996) use this to study the origin and behav- iour of diverging bow waves, producing physi- cally plausible bow wave breaking. A compari- son with a fully nonlinear 3D result shows qualitative agreement for the wave pattern shape. 4.3 Viscous Flow Calculation Methods There has been much recent activity in de- velopment of viscous flow calculation methods for maritime applications. In virtually all cases this concerned solution methods of the RANS equations. This subsection reviews some of the main trends and achievements, addressing grid generation; numerical algorithms; free-surface treatment; turbulence; and unsteady flow cal- culations. Grid Generation and Domain Decomposi- tion. Grid generation often is the most time consuming part of a CFD computation, requir- ing substantial effort. To handle practical, often complicated geometries like ship hulls with appendages, the main difficulty is the genera- tion of a suitable body-fitted grid. CAD capa- bilities are required to construct the grid topol- ogy and to support the projection of grid nodes on the surface of the hull. In the last few years there is a tendency to use commercial codes for grid generation. The choice of the grid depends on the ap- plication and the numerical method used. The most common types of grids are described be- low, in order of increasing flexibility. Single-block Structured Grids. Structured grids are typically generated using either alge- braic functions or solution of partial differential equations which describe the transformation between a physical domain and a rectangular computational domain (Larsson et al., 1998, Kim et al., 1998a). In some cases 2D grids are generated in separate transverse planes, and corresponding grid nodes are simply connected in the longitudinal direction (Tzabiras, 1997a). Advantages of structured grids are the simplifi- cation of the programming, the reduction of the memory requirements due to the consecutive 22 numbering of grid lines, and the regular struc- ture of the matrix of algebraic equations that permits the use of a variety of efficient solvers (Ferziger & Peric, 1997). The most common single-block structured grid is an H-type grid. This has certain limita- tions, such as limited possibilities for accurate consideration of the stem and stern profiles, and may involve grid singularities in front of and behind the hull, which may affect the con- vergence of the numerical solution. For these reasons, some recent applications use an O-O or C-O topology, Tahara & Himeno (1998) and Kim et al. (1998b) presented a grid generation method based on solving a Poisson equation. The method was applied for a tanker with bul- bous bow and bulbous stern. The quality of the numerical grid in their case was considerably improved by using O-O grid topology. A single-block structured grid offers limited possibilities for local refinement. Concentration of points in a certain region, e.g. near the wall, may produce unnecessarily small spacing in other parts of the solution domain and lead to bad cell aspect ratios. Also, a sudden change of the ship form as in case of a transom stern or skeg may cause strong deformation of the grid. Multi-block Structured Grids. In many ap- plications, such as twin-screw ships, subma- rines, or sailing yachts, appendages have a strong influence on the flow. In such cases it is difficult or impossible to construct a single- block grid with satisfactory resolution. Multi- block grids then may provide a useful com- promise between the simplicity of single-block grids and the ability to handle complex geome- try that completely unstructured grids allow. They offer enough flexibility for most hull forms. The number of multi-block applications for practical ship forms is increasing rapidly in ship hydrodynamics literature during the last three years, and now they are applied nearly in every second computation of viscous flow for practical ship forms. Examples were presented by Bull & Watson (1998), Beddhu et al. (1998), Haussling et al. (1997), and Wilson et al. (1998). The first step of multi-block grid generation is the construction of the grid topology, i.e., subdividing the calculation domain into a rea- sonable number of blocks for the grid genera- tion. This is not at all straight forward; the quality of the topology influences the quality of the numerical grid directly (angle between grid lines, aspect ratio of control volumes, and the size ratio of two neighbouring volumes). Chap- pell & Bull (1998) discuss the difficulties of constructing a grid topology for a combatant with sonar dome, transom stern and append- ages. Bull (1996) investigates the effect of the topology of the numerical grid on numerical results around the “Suboff” fully appended geometry. The topology of the numerical grid can be parameterised with respect to the shape of the hull, e.g. for certain classes of bulbous bows or appendages. This has the advantage that the numerical grid can be automatically generated for similar forms. An example of application of a parameterised multi-block grid for a propeller is given in Abdel-Maksoud et al. (1998a). The simplest domain decomposition ap- proaches require regular connections between the grids in adjacent blocks; i.e. matching grid lines at interfaces. This makes it difficult to construct a good grid topology for hulls with different size of appendages. A more general technique allows non-matching sub-domain grids. This provides much flexibility, in par- ticular the possibility of using different grid resolution in different blocks (local refinement), and reduces the effort for the grid generation. Although the connectivity information between the cells faces at the sub-domains interfaces increases the memory requirements and com- puting time compared to a traditional multi- block technique, it is still less than for an un- structured grid. On the other hand, in most cases the spatial discretisation between the dif- ferent non-matching sub-domains is only first order, which affects the quality of the results. Also, for non-planar interfaces, grid gaps or overlapping can easily occur when different grid resolutions at both sides of the interface are applied. Non-matching multi-block grids are very powerful for complicated interaction problems between hull and appendages. Jiang & Remo- tigue (1998) applied it for the “Suboff” geome- try with full appendages, a submarine with ring-wing and a destroyer with rudders, pro- pellers and shaft supporting struts, and an ac- tuator disk for the propeller. Chappell & Bull (1998) presented a grid topology for a combat- ant with sonar dome and appendages. The ap- plication of non-matching multi-block tech- 23 nique for propeller blades and shaft, and in an investigation of propeller-hull interaction for a Series 60 C B = 0.6 model, is shown in Abdel- Maksoud et al. (1998b). Cura Hochbaum (1998) used the technique for computing the flow around a ship model in steady turn and in oblique motion. Composite or Overlapping Multi-block (Chimera) Grids. Overlapping grid techniques overcome the difficulty of matching the boundary surfaces between the different sub- domains and the necessity of applying complex grid topologies. In this case, each piece of the geometry can be treated as a complete separate component grid, which itself may consist of a multi-block grid (Lin et al., 1998). Also regions with large gradients such as boundary layers may be covered with separate grids embedded into one or more background grids (Larsson et al., 1998). A method is needed to interconnect the off- set grids, create proper hole regions and define interfaces between overlapping grids at which boundary conditions for one block are obtained by interpolating the solution from the other overlapped block (Lin et al., 1998). The disad- vantage of these grids is that conservation is not easily enforced at the interpolated irregular block boundaries (Ferziger & Peric, 1997). Masuko (1998) reported convergence difficul- ties when using staggered variable arrange- ments, which could be caused by the interpola- tion of the solution between the different com- ponent grids. The largest advantage of this technique is its applicability to, e.g., complex hull forms with moving appendages and for moving bod- ies. In this case, one or more blocks are cover- ing the body and moving with it, while a static grid covers the surroundings. Chen & Huang (1998) used this technique for investigating the unsteady induced flow by a full scale berthing vessel in a small harbour. Alessandrini & Del- hommeau (1998) used it for computing the viscous free-surface flow past a ship in drift and in yawing motion. The overlapping grid feature may be utilised in the calculation of the flow around a propeller operating in the wake behind the hull or the flow interaction between different ships moving with different velocities (Larsson, 1997a). Also it is powerful to inves- tigate the optimum location and size of certain appendages without significant modification of the numerical grid; Lin et al. (1998) presented results for a naval combatant with two different bow bulb configurations, and for a body of revolution with self adjusting control surfaces. Masuko (1998) investigated the flow around waterjet inlets, a stern with a fin in front of the propeller and a stern with a skeg. Korpus et al. (1998) applied the Chimera technique for a podded propulsor with strut and fin. Unstructured Grids. This is the most flexible type of grid, which can fit an arbitrary solution domain boundary. Unstructured grids are usually used with finite element methods and, increasingly, with finite volume methods. The elements or control volumes may have any basic shape but in practice tetrahedra or hexa- hedra are most often used. The aspect ratio can be easily controlled and the grid may be locally refined. Unstructured grids are very preferred for automation of grid generation. On the other hand, such grids require connectivity tables which identify the neighbours of each node; due to indirect addressing, larger memory re- quirement, and more complicated solvers for the linear equation systems, the computing time per iteration is usually longer on unstructured than on block-structured grids (Lilek et al., 1997). The application of unstructured grids in ship hydrodynamics is still relatively limited. Recent applications are wave resistance calcu- lation using a finite element solution of the Euler equations (Yang & Löhner, 1998), and in viscous flow computations, Löhner et al. (1998). Arabshahi et al. (1998) employed un- structured grid for viscous flow around the ap- pended Suboff configuration at different drift angles; Hino (1998) for free-surface viscous flow around a VLCC model with and without rudder. The difficulties of application of unstruc- tured grids for high R n boundary layers are dis- cussed in Larsson et al. (1998). To overcome these, Bull (1996) used a grid with prismatic cells surrounding the geometry and tetrahedral cells elsewhere. Prismatic cells have the ad- vantage that the grid lines may be orthogonal to the wall surface; this improves the accuracy of the computation in the boundary layer region. Algorithms for RANS Solvers. For solving numerically the equations of viscous incom- pressible flow for maritime applications, all common discretisation methods have been ap- plied, such as finite difference (Kim et al., 24 1998a, Hoekstra & Eca, 1998, Tahara & Himeno, 1998, Wilson et al., 1998), finite vol- ume (Masuko, 1998, El-Moktar & Muzaferija, 1998, Hino et al., 1998, Abdel-Maksoud et al., 1998a), and, more rarely, finite element (Löh- ner et al., 1998, Honzeaux & Codina, 1998). Also finite-analytic schemes have been used in many ship flow computations (Stern et al., 1996a, Tahara et al., 1998, Chen & Huang, 1998, Yabushita & Hiuata, 1998). It is worth mentioning the spectral and pseudo-spectral element methods that recently have been ap- plied, although not yet in ship hydrodynamics; see Quarteroni & Valli (1997). There still is much current research on techniques for CPU time reduction and parallel computing, space and time discretisation, ve- locity-pressure coupling and algebraic equation solvers. Below, just a few topics are discussed on which there is some recent development. Domain Decomposition. Domain decom- position (multiblock methods) may be an im- portant technique for CPU time reduction. Two types of domain decomposition exist. In the first one, called “non homogeneous domain decomposition approach” or “zonal approach”, different equations are solved in each zone, for example the RANS equations in boundary lay- ers and wakes and an inviscid model in the outer region, which is less CPU time intensive. In the second one, called ”homogeneous do- main decomposition approach” the same equa- tions are solved in all subdomains. This tech- nique facilitates not only the discretisation for complex geometries but also the use of parallel computing (Bull & Watson, 1998, Cowles & Martinelli, 1998). It also allows the use of grids with different spacing in each region (Paterson et al., 1996), useful in viscous free-surface problems where we have to discretise flow features with very different characteristic lengths. Domain decomposition poses requirements to the solution algorithm. One approach is to carry out the iteration in each block and to up- date the values at the interfaces before going to the next iteration, (Cura Hochbaum, 1998a, Tzabiras, 1997a). This is effective if the num- ber of blocks is small. For each separate block the organisation of the data in the code and the structure of the matrix of the algebraic equation system is similar as in a single-block applica- tion. The other approach, more often used in newly developed codes, considers all control volumes simultaneously, independent of to which block they belong. This results in an irregular structure of the matrices of linear equations, and an efficient solver for these is required. The advantages are good convergence characteristics also for the case when the pres- sure is crucial, and the fact that the number of blocks does not influence the computing time directly. Velocity Pressure Coupling. For setting up efficient algorithms for solving the incom- pressible RANS or Navier-Stokes equations one should take into account their particular structure in which no time derivative of the pressure appears. This can be done in different ways, the most popular being “pressure Pois- son” or “pressure based” methods, and the arti- ficial compressibility approach. In both meth- ods, the coupling of the pressure to the velocity field is achieved indirectly through iteration or time stepping. The first class is based on projection schemes (Miyata, 1996, Alessandrini & Del- hommeau, 1996, Ramamurti & Löhner, 1996, Löhner et al., 1997). A velocity field is predict- ed in a first step. The conservation of mass is enforced in a second step by solving a Poisson equation, the right-hand side of which depends on the predicted velocity field, which gives a new pressure. Finally, the velocity field is up- dated with this new pressure. Hence these methods are closely related to the family of algorithms called semi-implicit method for pressure link equations (SIMPLE) method. A further development is the method called pres- sure-implicit with splitting of operators (PISO) scheme, used in Hamasaki et al. (1996) and Abdel-Maksoud et al. (1998b). The second class, artificial compressibility schemes, is used by, e.g., Cowles & Martinelli (1998), Hino (1997), and Bet et al. (1998). The infinite speed of sound of the incompressible medium is reduced to a finite number by ad- ding a time derivative of the pressure to the divergence equation. At steady state the time derivative vanishes, yielding the proper incom- pressible solution. This approach enables the use of techniques developed for compressible flow simulation, such as limiters (Cowles & Martinelli, 1998) and upwind differencing of higher order (Mompean, 1998). Besides the two basic methods mentioned, there are also solution techniques in which the 25 continuity equation is fully coupled with the momentum equations and satisfied at every instant in the solution process (Hoekstra & Eca, 1998). Time Discretisation. In many cases the un- steady RANS equations are solved also for steady flow applications, by using explicit, semi-implicit or implicit time stepping. Most computations use second order time stepping schemes (Boukir et al., 1997, Rodes et al., 1998), although in general schemes are used from first order explicit or implicit (Schumann, 1998a,b, Brakkee et al., 1998) to multistage Runge-Kutta (Cowles & Martinelli, 1998), sometimes with coefficients optimised for computational performance (Martinelli & Cowles, 1998). Also second order semi- implicit fractional step and Adams-Moulton multistep methods have been used (Mayer et al., 1998, Codina et al., 1998). Space Discretisation. In general, the vis- cous terms are discretised by a central scheme of order from second to fourth (Sung et al., 1996, Cura Hochbaum, 1998, Nirata, 1996), while the convective terms are discretised tak- ing into account their hyperbolic behaviour, that is by various upwind schemes (Tan, 1996, Bull & Watson, 1998, Alessandrini & Del- hommeau, 1996, Nirata, 1996). The Total Variation Diminishing (TVD) and Essentially Non Oscillatory (ENO) schemes should also be mentioned in this re- gard; see Shu (1998) and the references listed therein. The basic idea of these schemes was to prevent the occurrence of oscillations in the solution near shocks in compressible flow. In incompressible flow they may be useful for stability and accuracy in high-gradient regions. In TVD schemes, the idea is to reduce the total variation of the conserved quantity by limiting the flux of the quantity through the control volume interfaces (Godlewski & Raviart, 1996). The simplest idea is to choose a high order flux (e.g. the Lax-Wendroff flux) that works well in smooth regions, and a low order flux (typically some monotone first order method) that be- haves well near discontinuities and steep gradi- ents; the scheme combines these two into a single flux expression. One such scheme is known as the slope limiter: Monotonic Up- stream Centred Schemes for Conservation Laws (MUSCL), applied in ship hydrodynam- ics by Gatiganti et al. (1998) and Hino (1998). Algebraic Equation Solvers and Accelera- tion Techniques. The numerical solution of the incompressible Navier-Stokes equations almost ever requires solving large systems of linear algebraic equations. Common discretisation methods give linear systems with sparse matri- ces. Hence, iterative solvers are the most effec- tive, but their convergence depends on the structure of the matrices themselves. Different approaches are used to improve the convergen- ce properties, such as multigrid techniques (Spyropoulos et al., 1998) and Krylov subspace methods (Tefy & Leyland, 1998, Oosterlee et al., 1998), although also Cholesky and ap- proximate factorisation are used (Mompean, 1998, Haussling et al., 1997). The combination of Krylov subspace projection methods and good preconditioning is state of the art. The generalised minimum residual method (GMRES) is often the preferred method for the solution of large, sparse, and possibly non- symmetric systems such as arising in CFD (Spyropoulos et al., 1998, Dwyer & Grear, 1998). Vuik (1996), Brakkee (1996) and Tan (1996) present a recent application of GMRES as an accelerator in domain decomposition methods. For this approach, new techniques with inaccurate subdomain solvers are under investigation, Brakkee et al. (1998). One of the more widely used techniques for accelerating the solution of the discretised equations is multigrid, e.g. Cowles & Marti- nelli (1998). Coarser grids are used to cancel the error components whose spatial frequency is too low for the finer grid. Large CPU time- savings may result. In the application of the standard multigrid methods for the solution of the Navier-Stokes equations in complicated domains, two prob- lems may arise. First, coarsening is not possible to the full extent since the geometry and flow must be resolved by the coarsest grid used. Second, there may be a stability restriction for convection-diffusion problems, especially for higher R n . As it is shown in Griebel et al. (1998) the use of algebraic multigrid (AMG), which is another type of multilevel method, overcomes this. This does not make use of any geometrical information of the grid. Free-Surface Treatment. Until recently, RANS solvers were usually applied to double- body flows, i.e. with symmetry conditions at 26 the still water surface. The last few years have seen much development on methods to solve the coupled problem of ship wave making and viscous flow. These in principle take into ac- count interactions between both effects, re- moving some assumptions underlying all pre- vious CFD and towing tank work. In the "RANS/FS" problem, a no-slip con- dition is imposed on the wetted part of the hull surface, and a set of free-surface boundary con- ditions (FSBC) at the actual position of the water surface. These are: a kinematic condition that there be no flow across the wave surface; and 3 dynamic conditions stating that the nor- mal stress at the free-surface balances the am- bient pressure and surface tension and that tan- gential stress components vanish. See Choi & Stern (1993) and Delhommeau et al. (1996) for a discussion. The kinematic condition can be stated in Eulerian or in Lagrangian form, de- pendent on whether the free-surface is single- valued or not. Surface tension and free-surface boundary layers usually being unimportant, most methods simplify the dynamic FSBC, requiring the pressure to be atmospheric and the normal derivatives of tangential velocities to vanish. Dynamic trim and sinkage of the hull ought to be taken into account but are usually neglected, except in Orihara & Miyata (1997), Akimoto & Miyata (1998). Lacking other in- formation, for turbulence models generally the same conditions as for symmetry planes are used. The infinite flow domain is truncated, and absorbing boundary conditions are needed to avoid wave reflections and possible instabilities. Downstream usually the same conditions (lon- gitudinal derivatives zero) are used. This may cause problems if the outlet is too close to the hull, Takai & Zhu (1994). Similar problems may occur if the lateral outer boundary is too close to the hull, unless the method is matched to some outer solution. For the free-surface treatment, two main approaches can be distinguished, denoted as "free-surface fitting" and "free-surface captur- ing", in analogy with the treatment of shock waves in aerodynamics. Free-Surface Fitting. In the applications considered means that (at least) one boundary of the domain is the (guessed) wave surface. On this boundary the FSBC are imposed. The free-surface thus is a sharp interface, the mo- tion of which is followed. Obviously the mesh needs to be adapted in the course of the solu- tion process to conform to the changing free- surface location. This grid adaptation may be either general or simplified, e.g. grid points sliding along predefined lines (spines). Sack- inger et al. (1996), Kim et al. (1998a) and Beddhu et al. (1998) first define a background grid that determines the paths along which grid points may slide. Löhner et al. (1998) only up- date the mesh every 100-250 time steps, and take into account the free-surface change at intermediate steps through the pressure boundary condition only. Free-surface fitting techniques are poten- tially quite accurate, and require rather little change to the RANS solver itself. But they are less suitable for large free-surface distortions or topology changes, such as for overturning waves or when the grid has to be moved along walls of a complicated shape. Unstructured and multi-block grids could be a solution in such cases. It appears that the majority of recent methods for steady flow around the hull now use free-surface fitting techniques; but free- surface capturing methods are coming up. Free-Surface Capturing. The alternative approach, means solving the RANS equation on a predetermined grid, which is not fitted to the wave surface, extends also into the air re- gion, and therefore usually is not adapted dur- ing the calculation process. As the free-surface does not coincide with a domain boundary, its position needs to be resolved on the grid. Typical techniques in this category are: • Marker-and-Cell method (MAC): Massless particles (markers) are initially introduced into the water near the free-surface and followed during the calculation. The sche- me can compute complex phenomena, e.g. Park & Miyata (1994) computed breaking bow waves for a tanker model obtaining good agreement with experiments. However, the computing effort is large. • Volume-of-Fluid method (VOF): A trans- port equation for the volume fraction of the water (1 for full, 0 for empty cells) is solved, in addition to the RANS equations. From the distribution of these volume fractions the free-surface shape can be reconstructed. The method is more efficient than MAC 27 and is well suited to changing flow domain topologies and wild free-surface motion such as occur in sloshing problems or breaking of waves. However, the free- surface contour is not sharply defined and it is not easy to keep accuracy. To obtain an accurate free-surface with a reasonable number of cells, various special techniques for the free-surface reconstruction and VOF transport algorithm have been developed (Lafaurie et al., 1997, Muzaferija & Peric, 1998, Azcueta et al., 1998). Some im- provements are possible by choosing a par- ticular basis for the free-surface description, such as in the spine-flux method (Mashayek & Ashgriz, 1995). Lowry et al. (1997) study the accuracy of a VOF method for wave propagation problems, and confirm that at least for larger-scale problems the accuracy is of much concern. Improvements pro- posed are an adaptive grid refinement strat- egy, and the use of the PLIC surface recon- struction algorithm. Schumann (1998b) ap- plies a VOF method in a finite-volume Euler equation solver. With sufficient grid density a fair wave pattern prediction is obtained for a Series 60. He also computed a steady breaking bow wave for a tanker, obtaining some agreement with the experi- mental hull wave profile. In De Jouette et al. (1996), a VOF approach is applied to a submerged foil and a surface-piercing strut. Because of the artificial compressibility method used, the solution evolves in “pseudo-time”. The admissible time step size is limited due to the explicit update of the VOF function. Similar to VOF is the “density function technique” used in (Kanai et al., 1996). The calculation includes the air domain above the free-surface, and a convection equation is used for the density. The latter poses large requirements on the discretisation, and the resulting smearing of the density jump covers some 25 cells in the example shown. Nevertheless, a good rep- resentation of regular waves is achieved. • Level set technique: In this interface cap- turing technique (Sethian, 1996) a scalar “level set function” is defined everywhere in the domain (i.e. also above the free- surface). Initially its value is equal to the distance to the free-surface, which therefore is defined by the zero subset of this function. The level set function is convected as a pas- sive scalar, such that the interface remains defined by its zero value. Some “smearing” of the interface results from the algorithm to locate the free-surface. On the other hand, if the RANS solver is applied to the air region as well, it has to handle a large jump of the density, which may require additional smearing to maintain stability. While the accuracy of the free-surface reconstruction is better than with the VOF method, the latter feature may partly spoil this. Vogt (1997, 1998) finds a smearing of the density jump over 4 cells to be sufficient; a larger “interface thickness” causes a larger nu- merical dispersion of free-surface waves in his method. The same references indicate very high resolution requirements, 300 cells per wavelength being needed for acceptable numerical dispersion in the case studied. Bet et al. (1998) use a level set treatment of the free-surface in an artificial compressi- bility method. Their method differs from those mentioned above by not solving any flow in the air domain, but using a simple extrapolation of flow quantities towards the free-surface. Good predictions of hull wave profiles for standard test cases are obtained. In summary, free-surface capturing tech- niques are often of low order and may lack ac- curacy when surface tension or viscous boundary layer phenomena are dominant. Maintaining a sufficiently sharp interface is a point of concern. The performance of various methods for cases with large deformations is compared in Rider & Kothe (1995). On the other hand, main advantages of free-surface capturing are robustness, relative simplicity and the ability to handle complex geometry and wave breaking. However, while such computa- tions may correctly indicate the inception and occurrence of wave breaking, they may not be expected to give useful predictions of the trail- ing wave system without any special modelling of the physics playing a role inside a breaker, Kanai et al. (1996). Solution Approach. A complication, in particular for surface-fitting methods, is that the FSBC are to be imposed on an initially un- known wave surface. This requires either itera- tion or time stepping. One of the rare examples of the former approach is (Tzabiras, 1997b). This is a solution method for 2D steady free- surface flows, consisting of a steady RANS solver and a quasi-time-dependent free-surface update based on the kinematic boundary condi- tion. Compared with a fully time-dependent 28 formulation, this is found to converge signifi- cantly faster to the steady solution. Virtually all methods opt for a fully time- dependent solution. The problem is thus con- sidered as transient. The hull is accelerated to the desired speed and the time integration is continued until a steady state has been obtained. The time-dependent solution approach is more obvious than the steady one: The kinematic boundary condition, which explicitly contains the free-surface motion, is commonly used to update the free-surface position at each time step, while the dynamic conditions are imposed in the RANS solution. Usually the free-surface update is uncoupled from the solution of the flow equations. This may result in a stability limit for the time step, which is possibly ineffi- cient for steady flow applications. Alternatively, coupled solution methods are therefore being proposed. Such coupling is possible via a cou- pled time-dependent mapping of the flow do- main or via an additional iteration at each time step (Muzaferija et al., 1996, Wilson et al., 1998). Alternatively, Alessandrini & Delhom- meau (1996) update the wave elevation from the normal component of the dynamic condi- tion, and solve a coupled system for the wave height change and the velocity field. They thus obtain a significant improvement in conver- gence rate, but this might rather be connected with details of the numerical scheme used. Mayer et al. (1998) study the effect of grid density on the wave length and energy loss for a free-surface Euler solver. They emphasise that numerical conservation of energy, and a proper transfer between potential and kinetic energy in the discretised system, is most critical in RANS and Euler solvers for free-surface problems. For accurate modelling of the energy transfer properties they find it necessary to sol- ve an additional Poisson equation for the pres- sure, instead of just using the pressure correc- tion algorithm. They conclude that at compara- ble resolution the numerical damping of an Euler solver is larger than that of a potential flow method. With 50 points per wavelength and 50 time steps per period, for a standing wave case they find a numerical energy loss of 0.5% per wave period. Also Kang (1997) con- cludes that 50 cells per wavelength are needed in his method for good accuracy. For e.g. a tanker at F n = 0.15, this guideline will lead to 350 cells over the length of the hull. The transverse spacing even must be smaller to resolve diverging waves. Larsson et al. (1998) estimate that some 10 6 grid points on the free- surface would then be needed for adequate re- solution of the wave pattern. None of the publi- cations on 3D cases so far satisfies this re- quirement. Besides the large number of cells needed, also a large number of time steps is often required due to a slow and oscillatory approach of the solution to steady state. Calcu- lation times mentioned vary over orders of magnitude, but generally are quite substantial even with the much lower number of grid points now common. Results Achieved. As there is much devel- opment on this topic, an assessment of achievements is of just temporary validity. Nevertheless, it is of interest to distinguish some common features. What one would hope to achieve with these RANS/FS methods is an improved prediction of (in particular) the stern wave system com- pared to inviscid codes; and an improved vis- cous flow field compared to double-body RANS calculations. So far these benefits have not yet materialised, although progress is being made. Most published RANS/FS calculations for ships show a good prediction of the steady wave profile along the hull, comparable to that of nonlinear inviscid calculations; as is expect- ed since they impose nonlinear free-surface boundary conditions and have ample resolution in this area. Improvements in the stern wave- making predictions by including the viscous effects are often hard to distinguish. In some cases a direct comparison is shown between an Euler and a RANS solution. In Cowles & Mar- tinelli (1998) both give good hull wave profiles, Euler sometimes being slightly better. Löhner et al. (1998) obtain perfect agreement with the experimental wave profile of a submerged foil using Euler equations, but less perfect using RANS. Ratcliffe (1998) compares predictions by some inviscid and viscous codes with data for Series 60 C B =0.6 and DTMB model 5415. While in particular for the latter the wave pro- files of the RANS/FS codes are far better, the potential flow predictions included are much worse than state of the art, and the differences probably are due to numerics rather than mod- elling. 29 Far-field waves (at e.g. > 0.2 L off the hull) are consistently underpredicted by RANS/FS codes and lack detail; at least for somewhat shorter waves (lower F n , or diverging waves). At the 1994 workshop, the wave pattern pre- dictions were just poor Mori and Hinatsu (1994). Progress has been made, and it may be assumed that this shortcoming will be over- come by better spatial resolution. An improvement in viscous flow predic- tions, as a result of taking into account the free- surface, has rarely been demonstrated. In Rat- cliffe (1998), quantitative differences between predicted and measured wake contours are still significant, even for these rather slender vessels. An application of substantial practical im- portance is the prediction of the viscous flow off a transom stern; in particular for partially wetted transoms for which inviscid flow codes fail. This is a quite difficult problem in view of the physical sensitivity of this flow, and due to difficulties in the modelling and grid generation. Haussling et al. (1997) show good predictions for a dry-transom case, indicating that viscous effects are responsible for a forward shift of the rooster tail compared to inviscid predictions. In Ratcliffe (1998), for DTMB 5415 with partially wetted transom one of the two RANS codes predicts the stern wave system fairly well. Wil- son et al. (1998) for the same case predict a too low stern wave system, while Cowles & Marti- nelli (1998) miss the wetted transom flow alto- gether and predict a dry-transom flow. Obvi- ously, improvement in this area is desired. Turbulence Modeling. For predicting the complex flow around a ship stern, turbulence modelling is critical. In the 1994 CFD Work- shop (SRI, 1994), most computations for the HSVA tanker failed to catch the detailed fea- tures, in particular the "hook shape" in axial velocity contours. Many other turbulence mod- els have been tested since, and recent advances for application to ship hydrodynamics are re- viewed below. For other fields of application, Hanjalic (1994) reviewed conventional two- equation eddy viscosity models and Reynolds stress models, and concluded that the latter will be used more in the future. Marvin & Huang (1996) reviewed turbulence models in aerody- namic applications. Zero-Equation Models (Algebraic Models). The zero-equation models such as Baldwin- Lomax and Cebeci-Smith are still widely used in ship flows. They need little computing time and give reasonable eddy viscosity values, alt- hough often with an unrealistic distribution. Wilson et al. (1998) used the Baldwin-Lomax model for unsteady ship flows, McDonald & Whitfield (1996) for the flow around Suboff with rotating propeller. Ishikawa (1994) made some ad-hoc corrections to the Baldwin-Lomax model to prevent too large values of eddy vis- cosity. Tahara & Himeno (1996) made another modification to include anisotropy of turbu- lence and effects of pressure gradient; interest- ing results with this modification are shown in Kodama (1998). One-Equation Models. The one-equation models to take into account the history effects on the flow, so called non-equilibrium models, such as Johnson-King, Baldwin-Barth, and Spalart-Allmaras, have been used in aerody- namics for unsteady problems. Wernert et al. (1996) showed the performance of the Johnson- King model for dynamic stall of an airfoil. The- se models have not been used widely in ship hydrodynamics. Hoekstra & Eca (1998) found that for the HSVA tanker the Baldwin-Barth model did not give better results than two- equation or algebraic models. Hsiao & Pauley (1998) used the Baldwin-Barth model to com- pute the steady state tip vortex flow over a fi- nite span hydrofoil. Two-Equation Models. Different versions of the k-ε model have been used in ship hydro- dynamics. The low R n k-ε model solves the k and ε transport equations in the entire flow field including the near wall region. The two- layer k-ε model solves only the k transport equation in the near wall region where the ε value is obtained from an algebraic equation. Alessandrini & Delhommeau (1996) used the former model for the flow around Series 60 with free-surface, and presented the high regu- larity of turbulent viscosity that comes from the k-ε model. Unfortunately, no experimental tur- bulence data are available for comparison. Hoekstra & Eca (1998) compared both k-ε model versions with algebraic and one- equation turbulence models for the flow around the HSVA tanker. The results of two-equation model showed better agreement with the ex- perimental data than other models but still did not catch the details of the flow. Recently, the k-ω model has been used very often in ship hydrodynamics. This model has good stability properties and accurate predic- 30 tion of the logarithmic layer for pressure gradi- ent flows, but Wilcox’s k-ω model (1988) has a strong dependency on free-stream turbulence. Menter (1993) suggested two new versions (BSL, SST) that combine the advantage of k-ω near the surface with the superior characteris- tics of the k-ε model near the boundary layer edge. These new models do not have a strong sensitivity to the free-stream values. Deng & Visonneau (1996) tested the Wilcox k-ω model, the k-ω BSL model of Menter, and a Reynolds Stress Model for the flow around HSVA tanker. They found that the original k-ω model offered the best compromise for this particular class of flows. Watson & Bull (1998) used the k-ω model for full scale without using wall func- tions, and found a large change in the stern flow field with increasing R n , in particular for the Menter model; but this result was to be considered preliminary. Hino (1998) used Menter’s k-ω SST model in an unstructured free-surface RANS code, and obtained reason- able agreement with the data for a VLCC. Flow around a hull in oblique motion was simulated with the k-ω model by Alessandrini & Del- hommeau (1998) and Cura Hochbaum (1998). The results of Cura Hochbaum show good agreement with experiments. The k-ω model seemed to be able to predict complex flow phe- nomena like 3D separation and vortex shedding around the Series 60 in oblique motion. Nonlinear Eddy Viscosity Models (Alge- braic Stress Models). The above eddy viscosity based linear models cannot predict turbulence anisotropy. Considering the increased compu- tational complexity of the Reynolds Stress Model, it is desirable to develop simpler mod- els that account for turbulence anisotropy. Thus, several groups are now looking into nonlinear eddy viscosity schemes. Gatski & Speziale (1993) proposed a quadratic constitutive rela- tion, Craft et al. (1993) one involving terms up to third order. Sreedhar & Stern (1998a) tested the model of Myong & Kassagi (1990) for a free-surface piercing flat plate. Sofialidis & Prinos (1996) simulated the effect of wall suc- tion on the structure of fully developed pipe flow using the linear and nonlinear k-ε model or k-ω low R n models, where the nonlinear model used Craft’s cubic relation. The comput- ed results for the turbulent shear stress were in close agreement with experiments and espe- cially the k-ω model predicted the distribution of the turbulent kinetic energy better. Sotiro- poulos & Ventikos (1998) used the two non- linear variants of the k-ω model based on Gat- ski and Speziale’s and Craft’s constitutive re- lations for flow through a 90-deg rectangular duct; the cubic nonlinear k-ω closure was the only model that successfully reproduced most of the experimental features of mean flow and turbulence. Svennberg et al. (1998) also tested quadratic and cubic algebraic stress models based on the k-ε model for two test cases; a vortex in free flow with different axial velociti- es in the vortex core, and a vortex pair em- bedded in a turbulent boundary layer on a flat plate. The cubic model gave approximately the same results as the Reynolds stress model in the first case and more accurate results than the simple models in the second case. Reynolds Stress Models. At the 1994 CFD Workshop, two research teams used RSM (Sotiropoulos & Patel, 1994, Chen et al., 1994). Their results showed the superiority over iso- tropic eddy viscosity models and well captured the hook shape in axial velocity contours. As opposed to this, for the same case Deng & Vi- sonneau (1996) got better results with the k-ω model than with RSM, which yielded a too intense secondary flow in the near wake and gave robustness problems. Svennberg et al. (1998) showed the superiority of RSM for vor- tex flow. Reynolds Stress Models often provide the best results, as they take into account the an- isotropy of turbulence. This is found to result in prediction of more flow details. However, they are numerically less stable than two equation models and therefore make it much harder to get a converged solution. In addition, solving the seven additional transport equations re- quires more computational effort. This makes the use of RSM not yet practical now; but with the rapid development of efficient numerical methods and faster computers, this requirement may not be too restrictive for steady state simulations. In summary, the k-ω model has become more popular in both zero and non-zero Froude number cases, and might currently be the best two-equation model. Reynolds Stress Model- ling may provide a major step forward, but is considered by many to be too impractical at the moment. Nonlinear eddy viscosity models are promising but have not been tried for ship flows so far. Near-future goals with large de- mands for turbulence modelling are flows around practical hull forms including surface roughness, propeller, free-surface and high R n . 31 Unsteady Flows. The further development of RANS methods and increase of computer resources recently allowed the investigation of unsteady viscous flow problems. Below we briefly review some computations of genuinely unsteady flows in ship hydrodynamics (i.e. not methods using time marching to find a steady solution). Some recent 3D applications are: manoeuvring of a submarine with rotating pro- peller (McDonald & Whitfield, 1996); ship in head waves (Rhee & Stern, 1998, Wilson et al., 1998); slamming and sloshing in a tank (Muzaferija et al., 1998, Azcueta, 1998); inter- action between propeller and hull (Abdel- Maksoud et al., 1998b, McDonald & Whitfield, 1996); flows induced by a berthing ship (Chen & Huang, 1998). As many RANS solvers for steady problems already follow a transient approach, application to unsteady problems may be a relatively small step. Time-accuracy may pose some additional demands. E.g. in artificial compressibility algo- rithms, a pseudo-time is used for letting the solution settle to an incompressible steady limit. Time-accurate calculations thus require solu- tion for a number of pseudo-time steps at each true time level. This type of algorithm was used by Makino & Kodama (1997) for the flow around two full hull forms in oblique or steady turning motion; and by Davoudzadeh et al. (1997), who coupled RANS equation and equations of motion to simulate submarine ma- noeuvres, including crashback. Just qualitative analysis of what happens physically was made. The time discretisation technique is essen- tial for computing unsteady flows. Different classes of methods can be used, e.g. two-level methods (explicit or forward Euler, implicit or backward Euler, midpoint rule, trapezoidal rule etc.), multi-level method (second order forward, second order backward, etc.), predictor–cor- rector method, Runge-Kutta methods. In most ship hydrodynamics applications, two-level methods were used. The backward Euler sche- me was applied by Chen & Huang (1998), Ab- del-Maksoud et al. (1998b), Arabshahi et al. (1998). Linear or quadratic backward Euler schemes can be used in the method of Muzaferija et al. (1998) and Azcueta et al. (1998). Increasing the order of accuracy of the temporal discretisation has similar conse- quences as the spatial one, a better numerical accuracy but a reduced numerical stability. To overcome this problem, different temporal dis- cretisation schemes can be applied for the con- vection, diffusion and source terms, e.g. Wilson et al. (1998). For unsteady problems, the accuracy is also affected by the time step size, the level of con- vergence at each time step and the time needed to achieve a solution that is independent of the initial conditions. In general, for smaller time steps more details of the unsteady behaviour are captured, e.g. for the vertical force in a wa- ter entry problem, Muzaferija et al. (1998). Lilek et al. (1997) found that the lift coefficient on a cylinder depended considerably on the time step size if an implicit Euler scheme was used, in comparison with a three-level scheme. Rhee & Stern (1998) studied the effect of the convergence level and time step and found that the influence was restricted to the near hull region. For ship-propeller interaction, Abdel- Maksoud et al. (1998a) found that about two propeller revolutions were necessary to get a periodic behaviour of the propeller-induced forces on the hull. The application of uncertainty analysis is important especially for unsteady flow compu- tations, to quantify the influence of the parameters used in the computation. 4.4 CFD-Based Optimisation There is renewed interest in methods for CFD-based automatic hull form optimisation. As such methods search for optimal values of design parameters, based on repeated applica- tion of a CFD tool, useful results are only ob- tained if the underlying CFD code gives results that are at least accurate in a comparative sense; and if the limitations of the code are sufficiently taken into account. However, with the increasing accuracy and scope of CFD methods, optimisation tools are gaining im- portance. Design methods based on the concept of Di- rect Numerical Optimisation may be formed by coupling hydrodynamic analysis methods with minimisation schemes. The user specifies the design requirements in terms of an objective function and (geometrical) constraints. The flow around an initial design and a design per- turbed by a small change of a single design parameter are then computed. The derivative of the objective function with respect to this parameter is then calculated by a finite differ- 32 ence approximation. Doing this for all design parameters in turn yields the gradient of the objective function with respect to the parame- ters. This is then used by the optimisation algo- rithm to derive an improved hull form, and the process starts all over again. Janson & Larsson (1996) used this method to compute ship forms of minimum wave + viscous resistance. While the optimisation process worked well, the final design was un- successful due to restrictions of the flow code. Chou et al. (1998) apply a similar optimisation technique, but prescribe a desired hull pressure distribution and apply the optimisation method to automatically modify a basic hull form to match it. The advantage of this optimisation ap- proach is its black-box character, which re- quires no extra mathematical analysis to obtain the flow sensitivity information. However, at each step of the optimisation cycle the flow code must be run as many times as the number of design parameters. Therefore, efficient and accurate methods to compute the gradient of the objective function become an important area of research. Various optimisation and sen- sitivity analysis methods are therefore used. Papanikolaou et al. (1996) optimised the seakeeping and wave resistance of catamarans by applying the so-called Reduced Gradient method and local form optimisation by La- grange method. Hamasaki et al. (1996) use Dawson's method to predict the wave resistance, and a RANS solver for the viscous resistance and wake. The variations to the original hull forms are expressed by B-spline functions, whose coefficients are used as design variables and optimised using the nonlinear program- ming approach. Tahara et al. (1998) extend and modify this method, using successive quadratic programming (SQP), the convergence of which is faster than successive linear programming (SLP). The aft part of the hull is modified by a 6-parameter function to minimise the viscous resistance. Huan & Huang (1998) present a method for shape optimisation for minimum wave resis- tance, for potential flow with non-linear free- surface boundary conditions. The sensitivities to a perturbation of the hull shape are directly derived by solving a separate set of equations. Alternatively, an explicit form of the sensitivi- ties is obtained through introduction of a set of adjoint equations. Regardless of the number of design parameters, the gradient of the objective function with respect to the hull shape can be obtained from one solution to the original flow problem and one solution to the adjoint equa- tions, followed by a simple integration over the hull for each design variable. This replaces the separate evaluation of all flow sensitivities in the formulation, making it more efficient if there are many design parameters. Huan and Huang applied the proposed methods in an in- verse design computation of a 2D hydrofoil underneath a non-linear free-surface to match a prescribed pressure distribution. Hino et al. (1998) presented a hydrodynamic shape opti- misation system for 3D ship hulls by the com- bination of a RANS solver, the adjoint equa- tions method for the sensitivity analysis, and the SQP procedure. The system is applied for total drag minimisation for a simple ship form. Hirayama et al. (1998) propose a method in which, based on a computed free-wave spec- trum for a basic design a modification of the hull is derived that should partly cancel the wave spectrum according to thin-ship theory. A similar approach has in the past been proposed by Sharma & Naegle (1970), but based on ex- perimental wave spectra. Some successful ap- plications are shown. In general it seems that these CFD-based optimisation methods have not yet made their way to actual routine application, although al- ready they might be helpful to give indications for possible design changes. 4.5 Conclusions Wave pattern prediction based on inviscid- flow panel codes is well developed and rou- tinely utilised in ship design. Further progress in this field mainly requires incorporation of viscous effects on the stern wave making. Viscous-flow RANS methods are increas- ingly being used in practical ship design. Ap- plicability is being extended by use of commer- cial grid-generation codes, multi-block and overlapping-grid methods, or unstructured-grid codes. Efficiency is enhanced due to fast matrix solvers, multi-grid methods, and high- performance parallel computing. Recent work includes free-surface algorithms and turbulence modelling. Limited study has been devoted to full-scale simulations. 33 Improvements are desired in turbulence models and numerical methods for accurate prediction of thick boundary layer and wake, full-scale flow and wave pattern using RANS codes. Additionally, pronounced user variabil- ity and restrictions in applicability and ease of use should be removed. 34 5. UNCERTAINTY ANALYSIS FOR EXPERIMENTAL FLUID DYNAMICS 5.1 Introduction Reporting of experimental uncertainties continues to be a problem for the ITTC and related disciplines such as aerospace and mechanical engineering. Problems include both implementation procedures (e.g., simple re- peatability tests are often done in lieu of careful estimates for bias and precision limits) and documentation and presentation of results. Clearly experimental uncertainty estimates are imperative for risk assessments in design both when using data directly or in calibrating and/or validating simulation methods. Within the ITTC several discussions have been devoted to the subject of uncertainty ana- lysis started already in 1987. The Panel on Validation Procedures published in the 19 th Proceedings (ITTC, 1990) [Vol. 1, Section II.3.2] their work ”Guidelines for Uncertainty Analysis of Measurements” based on the ANSI/ASME (1985) standard together with examples for various ITTC related tests such as resistance and manoeuvring tests and speed and power trials. As recommended by the Panel of Validation Procedures the guidelines were ap- proved by the full conference. Since then large efforts have been made to improve the existing methodologies. Recently, the American Institute of Aero- nautics and Astronautics (AIAA) in conjunc- tion with Working Group 15 of the Advisory Group for Aerospace Research and Develop- ment (AGARD) Fluid Dynamics Panel has put forth a standard for assessment of wind tunnel data uncertainty (AIAA, 1995). This standard was developed in order to provide a rational and practical framework for quantifying and reporting uncertainty in wind tunnel test data. The quantitative assessment method was to be compatible with existing methodologies within the technical community. Uncertainties that are difficult to quantify were to be identified and guidelines were to be given on how to report these uncertainties. Additional considerations included: integration of uncertainty analyses into all phases of testing; simplified analysis while focusing on primary error sources; incor- poration of recent technical contributions such as correlated bias errors and methods for small sample sizes; and complete professional analy- sis and documentation of uncertainty for each test. The uncertainty assessment methodology has application to a wide variety of scientific and engineering measurements, including tow- ing tank experiments. The AIAA (1995) stan- dard is based on Coleman & Steele (1999), which is an update to the ANSI/ASME (1985) standard, and the most current drafts of inter- national guidelines and standards (ISO, 1992, 1993a,b). This makes the AIAA (1995) stan- dard the most recent update of the uncertainty analysis methodology previously adopted and currently used by the ITTC. It can also be worth noting that some ITTC members have already implemented the AIAA (1995) standard (e.g., Forgach, 1992). With this background, the 22 nd ITTC RC recommends that the AIAA uncertainty as- sessment methodology [i.e., Chapter 2 of AIAA (1995) standard] be adopted as the ITTC stan- dard for towing tank experiments. To insure proper application, the methodology is repro- duced verbatim as procedure 4.9-03-01- 01, ”Uncertainty Analysis in EFD (Experi- mental Fluid Dynamics), Uncertainty Assess- ment Methodology,” in the QM with minor modifications for terminology and figure, table, and equation numbering and by royalty free licence from the AIAA. To aid in application of the methodology for towing tank experiments, QM procedure 4.9-03-01-02 ”Uncertainty Analysis in EFD, Guidelines for Resistance Towing Tank Tests,” provide guidelines for towing tank experiments. The guideline para- phrases AIAA (1995), but adapted for towing tank experiments. The guideline includes a philosophy for testing and recommendations for application/integration of uncertainty as- sessment methodology into the test process and documentation of results as well as recommen- dations for management. In addition to above, QM procedure 4.9-03-02-02 ”Uncertainty Analysis in EFD, Example for Resistance Test,” provides an example for a towing tank resistance test. In the example, the uncertainty for the total resistance coefficient C T for a model scale resistance test is established. The Committee recommends also the guidelines and example to be adopted. In Section 5.2 through 5.4 extracts, summa- ries and discussions from the respective QM procedures are given. Based on work by the Committee, uncertainties from 7 facilities are quoted and compared in Section 5.5. Lastly, conclusions are given in Section 5.6. 35 5.2 Uncertainty Assessment Methodology The methodology for estimating the uncer- tainties in measurements and in the experi- mental results calculated from them must be structured to combine statistical and engineer- ing concepts. This must be done in a manner that can be systematically applied to each step in the data uncertainty assessment determina- tion. In the methodology discussed below, the 95% confidence large-sample uncertainty as- sessment approach is used as recommended by the AIAA (1995) for the vast majority of engi- neering tests. Overview. The word accuracy is generally used to indicate the closeness of the agreement between an experimentally determined value of a quantity and its true value. Error is the differ- ence between the experimentally determined value and the truth. Accuracy is said to increase as error approaches zero. The true values of standard measurement quantities (e.g., mass, length, time, volts, etc.) generally only reside in national standards laboratories. Only in rare instances is the true value of a quantity known. Thus, one is forced to estimate error, and that estimate is called an uncertainty, U. In general, the uncertainty of a quantity is a function of the value of that quantity. However, it is common practice to quote the same value of uncertainty for a range of values of the quantity, e.g., per- cent of full scale of an instrument. In this methodology all estimates are assumed made at a 95-percent confidence level, meaning that the true value of the quantity is expected to be within the ±U interval about the experimentally determined value 95 times out of 100. As shown in Figure 1, errors can be consid- ered to be composed of two components: a pre- cision (random) component and a bias (system- atic) component. An error is classified as preci- sion if it contributes to the scatter of the data; otherwise, it is a bias error. It is assumed that corrections have been made for all systematic errors whose values are known. The remaining bias errors are thus equally as likely to be posi- tive as negative. A general representation of the data reduc- tion equation is r = r (X 1 , X 2 , ..., X J ) (1) where r is the experimental result determined from J measured individual variables X i . Each of the measured variables contains bias and precision errors. As shown in Figure 2, the er- rors in the measured variables propagate through the data reduction equation, thereby generating the bias and precision errors in the experimental result. -0.005 P i MAGNITUDE OF X X i true X i P i µ β F R E Q U E N C Y O F O C C U R A N C E Figure 1. 95-percent confidence precision limit interval (P i ) around a single reading of a variable X i . The bias limit is denoted β and represents the difference between the true value true i X and the biased mean value µ for many repetitions under the same condition using the same equipment. The uncertainty assessment methodology can be used for calculating the uncertainty for different measurement procedures such as single and multiple tests. To estimate the preci- sion limit for a single test an end-to-end ap- proach can be used with measurements taken over an appropriate time interval including all factors causing variability. If this is not the case the precision limits can be determined at an elementary level. For towing-tank tests, the end-to-end approach is difficult to use for a single run as effects of model misalignment, trim, heel, residual current or waves, tempera- ture variability, etc. vary over a longer time interval than the testing time and/or are not changed within the same set up. Therefore the best method is to use the theory for multiple tests including several runs over an appropriate time interval with the model removed and rein- stalled a few times. With this information, the precision limits for the average result or for a single test (one run) of a set of multiple tests can be determined. In the daily commercial work, such repeat tests are not possible and the experimenter must estimate a value for the pre- cision limit using the best information available 36 at this time, for example previously made in- vestigations for models of similar geometry. Figure 2. Propagation of errors into an experi- mental result. Single Tests. In single tests, r is determined from a single set of measurements (X 1 , X 2 , …, X J ) at a given test condition. The uncertainty in r is the root-sum-square (RSS) of the bias and precision limits. 2 r 2 r 2 r P B U + = (2) The bias limit of the result is given by − = + = = θ θ + θ = 1 J 1 i J 1 i k ik k i 2 i J 1 i 2 i 2 r B 2 B B (3) where θ i are the sensitivity coefficients i i X r ∂ ∂ = θ (4) B i are the bias limits in X i , and B ik are the correlated bias limits in X i and X k α α = α = ) B ( ) B ( B k i L 1 ik (5) where L is the number of correlated bias error sources that are common for measurement of variables X i and X k . Assuming no correlated precision errors the precision limit is estimated from the scatter in the measured values by r r S K P = (6) where K is the coverage factor and equals 2 for a 95% confidence interval and large sample size (N≥10) and S r is the standard deviation of the sample of N readings of the result r. The value of S r is determined from N readings over an appropriate time interval, i.e., includes all factors causing variability in the result. Alter- natively P r can be estimated by the RSS of the precision limits for the measurements of the individual variables = θ = J 1 i 2 i 2 i 2 r P P (7) where θ i are the sensitivity coefficient defined by equation (4) and P i =K S i are the precision limits in X i (where K and S i are defined simi- larly as S r in equation (6)). Often the time in- terval is insufficient and P i ’s or P r must be es- timated based on previous readings taken over an appropriate time interval. For towing-tank applications, N is equal to the number of sam- ples over the run. Even a very long run can not include all factors causing variability and there- fore the methodology for multiple tests are rec- ommended. Multiple Tests. In multiple tests, an aver- age result r is determined from M sets of measurements (X 1 , X 2 , …, X J ) k at the same given test condition = = M 1 k k r M 1 r (8) If the M sets of measurements is taken over an appropriate time interval, the precision limit of a single result of the M measurements is r r S K P = (9) where K=2 as above for large sample sizes (M≥10) and S r is the standard deviation of the sample of M results ( ) 2 1 M 1 k 2 k r r r 1 M 1 S ù ê ë é − − = = (10) 37 where M is the number of repeat tests per- formed for the same test point. The uncertainty of a single result (one run) of the sample of M tests is obtained from equation (2) with equa- tion (3) and (9) and can be written as ( ) 2 r 2 r 2 r S 2 B U + = (11) The precision limit for the average result is given by M S K P r r = (12) The uncertainty for the average result of M tests is then 2 r 2 r 2 r M S 2 B U ö ç ç è æ + = (13) where B r is given by equation (3). The use of K=2 assumes a large sample size and Gaussian error distribution. A large sample size implies that the test is taken over a time that is comparable to the period of all factors, which causes variability in r. Ideally (N, M)≥10, but this is often not the case. For (N, M)3 may be required. Consider the situation for 3 solutions corresponding to fine 1 k S ˆ , medium 2 k S ˆ , and coarse 3 k S ˆ values for the k th input parameter. Solution changes ε=for medium-fine and coarse-medium solutions and their ratio R k are defined by 1 2 k k k 21 S ˆ S ˆ − = ε 2 3 k k k 32 S ˆ S ˆ − = ε (32) k k 32 21 k R ε ε = Three convergence conditions are possible: (i) Converging condition: 0 < k R < 1 (ii) Oscillatory condition: k R < 0 (33) (iii) Diverging condition: k R > 1 For the converging condition (i), the solu- tions exhibit monotonic convergence and gen- eralised Richardson extrapolation is used to estimate k U or ∗ δ k =and C k U . Richardson ex- trapolation is generalised for J input parameters and accounting for effects of higher-order terms. Power-series expansions about S C for each in- put parameter using m solutions are used to obtain estimates for the ∗ δ k ’s in equation (31). ∗ δ I must be accurately estimated or be negligi- ble for each solution. The effects of higher- order terms are important for practical applica- tion of CFD; however, the number of terms (n) that can be determined depends on the number of solutions (i.e., for m=3, n=1; m=5, n=2, etc.) and m>3 is undesirable from a resources point of view. Correction factors are proposed to account for the effects of higher-order terms based on only m=3 solutions. The methods proposed are tentative and need further testing. Hopefully through practical application the present or alternative strategies for estimating effects of higher-order terms based on limited number of solutions will prove satisfactory. Figure 9. Definition of comparison error. For the oscillatory condition (ii), the solu- tions exhibit oscillations, which may be erro- neously identified as condition (i) or (iii). Methods for estimating uncertainties k U for the oscillatory condition (ii) require more than m=3 solutions and are based on the upper and lower bounds of the solution oscillation. For the diverging condition (iii), the solu- tions exhibit divergence and errors and uncer- tainties can not be estimated. The preparation and verification steps must be reconsidered. Improvements in iterative convergence, parameter specification (e.g., grid quality), and/or CFD code may be required to achieve converging or oscillatory conditions. Validation. Validation is defined as a proc- ess for assessing modelling uncertainty SM U by using benchmark experimental data and, when conditions permit, estimating the sign and magnitude of the modelling error SM δ itself. Thus, the errors and uncertainties in the ex- perimental data must be considered (see Sec- tion 5). The validation comparison is shown in S + U s E U D U x r X D S 45 figure 9. The experimentally determined r- value of the ( ) i i r , X data point is D and simu- lated r-value is S. The benchmark experimental data with er- ror δ D is used for the truth in equation (19) (i.e., T=D-δ D ) to define the comparison error ) ( S D E SN SPD SMA D S D δ + δ + δ − δ = δ − δ = − = (34) with δ SM decomposed into the sum of δ SPD , error from the use of previous data such as fluid properties, and δ SMA , error from model- ling assumptions. Thus E is the resultant of all the errors associated both with the experimental data and with the simulation. For the approach in which no estimate * SN δ of the sign and mag- nitude of SN δ is made, all of these errors are estimated with uncertainties. If i i r , X , and S share no common error sources, then the uncertainty E U in the com- parison error can be expressed as 2 S 2 D 2 S 2 2 D 2 2 E U U U S E U D E U + = ö ç è æ ∂ ∂ + ÷ ö ç è æ ∂ ∂ = (35) or 2 SN 2 SPD 2 SMA 2 D 2 E U U U U U + + + = (36) Ideally, one would postulate that if the ab- solute value of E is less than its uncertainty E U , then validation is achieved (i.e., E is “zero” considering the resolution imposed by the “noise level” E U ). In reality, there is no known approach that gives an estimate of SMA U , so E U cannot be estimated. That leaves a more stringent validation test as the practical alterna- tive. If the validation uncertainty V U is defined as the combination of all uncertainties that we know how to estimate (i.e., all but SMA U ), then 2 SN 2 SPD 2 D 2 SMA 2 E 2 V U U U U U U + + = − = (37) If |E| is less than the validation uncertainty V U , the combination of all the errors in D and S is smaller than the estimated validation un- certainty and validation has been achieved at the V U level. V U is the key metric in the vali- dation process. V U is the validation “noise level” imposed by the uncertainties inherent in the data, the numerical solution, and the previ- ous experimental data used in the simulation model. It can be argued that one cannot dis- criminate once |E| is less than this; that is, as long as |E| is less than this, one cannot evaluate the effectiveness of proposed model “im- provements.” If the corrected approach of equations (21)- (24) is used, then the equations equivalent to equations (34) and (37) are ) ( S D E SN SPD SMA D C C ε + δ + δ − δ = − = (38) for the corrected comparison error and 2 N S 2 SPD 2 D 2 SMA 2 E 2 V C C C U U U U U U + + = − = (39) for the corrected validation uncertainty. Note that S C and E C can be either larger or smaller than their counterparts S and E, but C E U and C V U should be smaller than E U and V U , re- spectively, since N S C U should be smaller than SN U . Additional discussion is provided in the QM, including methods for estimating uncer- tainties in the data due to measurement of inde- pendent variables and in the simulation due to use of previous data. Also, discussion is provided for validation of a single CFD code, 46 validation for the comparison of multiple codes and/or models, validation for the prediction of trends, and use of corrected vs. uncorrected simulation results. 6.2 Example for RANS CFD Code Example results of verification and valida- tion are presented for a single CFD code and for specified objectives, geometry, conditions, and available benchmark information. A RANS CFD code developed for computational ship hydrodynamics was used (Paterson et al., 1998, Wilson et al., 1998). The RANS equations are solved using higher-order upwind finite differ- ences, PISO, k-ω turbulence model, and exact and approximate treatments, respectively, of the kinematic and dynamic free-surface boundary conditions. The objectives are to de- monstrate the usefulness of the proposed verifi- cation and validation procedures and method- ology and establish the levels of verification and validation of the simulation results for an established benchmark for ship hydrodynamics CFD validation. Geometry, Conditions, and Benchmark Data. The geometry is the Series 60 C B =0.6 cargo/container ship (see section 6.3). The con- ditions for the calculations are F n = 0.316, R n = 4.3x10 6 , and zero sinkage and trim. These are the same conditions as the experiments, except the resistance and sinkage and trim tests, as explained next. The variable selected for verifi- cation and validation is resistance C T (integral variable). The benchmark data is provided by Toda et al. (1992). The uncertainty estimates were recently confirmed/updated following the Section 5 procedures (Longo and Stern, 1998). The resistance is known to be larger for free vs. fixed models. Data for the Series 60 indi- cates about an 8% increase in C T for the free vs. fixed condition over a range of F n including F n =0.316 (Ogiwara and Kajatani, 1994). The Toda et al. (1992) resistance values were cali- brated (i.e., reduced by 8%) for effects of sink- age and trim for the present comparisons. Verification and Validation of Integral Variable: Resistance. Verification was per- formed with consideration to iterative and grid convergence studies, i.e., G I SN δ + δ = δ and 2 G 2 I 2 SN U U U + = . The studies were conducted for m=3 grids with fixed grid refinement ratio 2 r G = in each coordinate direction. The sizes of the coarse, medium, and fine grids are 101x26x16 = 42,016, 144x36x22 = 114,048, and 201x51x31 = 317,781. The grids were gen- erated using the commercial code GRIDGEN (Pointwise, Inc.). The grids are body-fitted, structured, single block with an H-type topolo- gy and grid clustering near the bow and stern in the ξ-direction, at the hull in the η-direction, and near the free surface in the ζ-direction. Near-wall spacing is determined by turbulence modeling considerations where the first point from the no-slip surface is placed at a normal- ized wall distance of y + =(0.7, 1.0, 1.4). The iteration errors and uncertainties were negligible in comparison to the grid errors and uncertainties for all three solutions i.e. δ I