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Applied Ocean Research
j o u r n a l h o m e p a g e ; w w w . e l s e v i e r . c o m / l o c a t e / a p o r
O C E A N
R E S E A R C H
Review
Hydrodynamic analysis techniques for high-speed planing hulls
Reza Yousefi^ Rouzbeh Shafaghat^'', Mostafa Shalceri^
'Department of Mechanical Engineering, Babol Noshiravani University of Technology, Babol. Iran ''Department of Meclianical Engineering, University of California, Berkeley, CA 94720, USA
CrossMark
A R T I C L E I N F O
Article history:
Received 11 June 2012
Received i n revised f o r m 9 May 2013 Accepted 10 May 2013
Keywortls:
High-speed planing hull Analytical technique Experimental technique Numerical method Benchmark table A B S T R A C T A p l a n i n g h u l l is a m a r i n e vessel w h o s e w e i g h t is m o s t l y s u p p o r t e d b y h y d r o d y n a m i c p r e s s u r e s at h i g h -speed f o r w a r d m o t i o n . Its h i g h - s p e e d c h a r a c t e r has m a d e i t p o p u l a r a n d t h u s t h e i n t e r e s t f o r p l a n i n g h u l l s f o r m i l i t a r y , r e c r e a t i o n a l a n d r a c i n g a p p l i c a d o n s is p r o g r e s s i v e l y r i s i n g . The d e s i g n a n d a n a l y s i s p r o c e d u r e f o r h i g h - s p e e d p l a n i n g h u l l s , d u e t o t h e i r p e r f o r m a n c e a n d speed r e q u i r e m e n t s , is v e r y i m p o r t a n t . Access t o a fast, a c c u r a t e t e c h n i q u e f o r p r e d i c t i n g t h e m o t i o n o f these h u l l s p l a y s a s i g n i f i c a n t r o l e i n i m p r o v e m e n t i n t h i s f i e l d . O v e r t h e p a s t several decades, n u m e r o u s i n v e s d g a t i o n s have b e e n d o n e o n h y d r o d y n a m i c analysis o f h i g h - s p e e d p l a n i n g h u l l s . In t h i s s t u d y , t h e e x i s t i n g t e c h n i q u e s f o r analysis o f these h u l l s are r e v i e w e d . U n d e r s t a n d i n g t h e s t r e n g t h s a n d l i m i t a t i o n s o f t h e s e t e c h n i q u e s w i l l h e l p researchers a n d e n g i n e e r s select t h e m o s t a p p r o p r i a t e m e t h o d f o r o p t i m a l d e s i g n a n d analysis o f a h u l l . To p r e s e n t a c o m p r e h e n s i v e s t u d y o n t h e e x i s t i n g t e c h n i q u e s , t h e y are c l a s s i f i e d i n t o t w o m a j o r categories; a n a l y t i c a l - e x p e r i m e n t a l a n d n u m e r i c a l t e c h n i q u e s . T h e n u m e r i c a l t e c h n i q u e s are f u r t h e r d i v i d e d i n t o m e t h o d s f o r b o u n d a r y v a l u e p r o b l e m s a n d d o m a i n - d e p e n d e n t p r o b l e m s . Each t e c h n i q u e is a p p l i c a b l e o n l y f o r a l i m i t e d r a n g e o f cases. ® 2 0 1 3 Elsevier Ltd. A l l rights r e s e r v e d . 1. Introduction
Nowadays, high-speed vessels are used for military, recreational, racing, and transportation purposes. The number of high-speed hulls has significantly increased in recent years owing to their speed and performance. Accurate analysis of the hull behavior in motion plays an important role, mainly due to the significance of their optimum and reliable operation under a variety of sea conditions. The structural design of these hulls is also important as it has direct effects on their weight, cost and the load they can transport. The need for developing new concepts in the design of these hulls is becoming indispensable. Hydrodynamic forces on the high-speed hull, during its forward mo-tion, support most of its weight and thus lift a large portion of the hull out of water. Dynamic behavior of a hull in waves mutually alters its hydrodynamic performance. As the hull advances in water, both its underwater hydrodynamics and above-water aerodynamics affect its motion and thus contribute to the dynamic behavior of the hull.
For marine vessels that move in the displacement mode, the wave-making drag increases significantly with speed, requiring a higher level of power. Optimum design of the ship architecture can decrease the hydrodynamic drag at higher speeds. Planing hulls have charac-terisrics that distinguish them from other types of hulls. Understand-ing the characteristics of the hull is important in makUnderstand-ing an accurate prediction of the behavior of the planing hull under various operating
* Corresponding author. Tel.: + 9 8 912 375 7507.
E-mail adtlress: rshafaghatCfnit.ac.ir (R. Shafaghat).
0141-1187/$ - see front matter ® 2013 Elsevier Ltd. All rights reserved. http://dx,doi.org/10.1016/J.apor.2013.05.004
condirions. One of the greatest challenges in evaluating the perfor-mance of planing hulls is to obtain accurate and practical results from hydrodynamic analyses.
The emphasis of this study is to evaluate the existing hydrody-namic analysis techniques and determine the applicability of each method. A brief description of the high-speed planing hull is first pre-sented. Research studies in this field is then reviewed and classified. Finally, the applicability and limitations of each analysis method are presented in a benchmark table.
2. General specifications
In a planing vessel, hydrodynamic pressure distribution on the hull creates a lift that supports a significant portion of its weight. Hydrodynamic pressures also affect the stability of these hulls. In some cases, the hull speed exceeds 60 knots. At lower speeds, the hull displaces water to move forward. As the speed increases, a lift force is generated, which eventually supports the hull and moves it out of water. As the wetted surface area decreases, the hydrodynamic lift rises further.
There is a point at which the hydrodynamic lift balances the weight of the hull. Under this situation, the buoyancy forces decrease with the increase of hydrodynamic forces. In the displacement mode, to achieve a higher speed, a larger engine power is needed. However, as the vessel shifts to the planing mode, the hydrodynamic forces produced by the submerged portion of the vessel lift the hull toward the water surface and thus cause a faster morion (Fig. 1). The vessel in the planing mode has a higher efficiency and thus requires less power for the same forward speed. This is because the wetted surface area
106 Reza Youseflet al./Applied Ocean Research 42 (2013) 105-113
Fig. 1. A photograph of a high-speed boat in the planing mode.
E
Displacement Transition Planing
Mode Mode Mode
0;5 0.85
Pt = v/fgl
Fig. 2. Hull resistance as a function of speed, indicadng the three modes of motion. Graph reproduced f r o m [1 ].
and consequently the skin friction decrease.
Fig. 2 shows the hull resistance behavior in the three conventional modes of motion (displacement, semi-planing (transition), and plan-ing) as a function of the loaded hull Froude number (Fr = V / ^ , where Vis the forward velocity (m/s),gis the gravitational accelera-tion (m/s^), and L is the length of the hull at the waterline (m)). The Fr of the hull in the displacement mode is less than ~0.5. The hull speed beyond this falls within the "hump region" and is a transition between displacement and planing modes. Once Fr exceeds 0.85, it shifts to the planing mode. In comparison with other types of hulls, the performance of the planing hulls strongly depends on the loca-tion of its center of gravity. The hull experiences a different set of conditions in each of the aforementioned mode. For instance, the hull motion in the displacement mode is very similar to the motion of displacement hulls.
In the semi-planing mode, the hull shifts from displacement to planing mode and the Froude number stays less than -0.85. Similar to the displacement mode, the trim of the hull, wetted surface area, and drag all increase in this case. To achieve the planing mode, the hull has to overcome a so-called resistance barrier (Fr = ~0.5), which requires a relatively high power. The practical upper limit for the semi-displacement mode occurs when Fr reaches ~0.85. The hull will not shift to the planing mode if enough power is not supplied by the vessel engine. As the Froude number increases, the trim of the hull gradually decreases and tends to a constant value.
In the planing mode, the hydrodynamic lift and buoyancy forces support approximately 95% and 5% of the hull weight, respectively. In this mode of motion, the flow around the hull becomes two-phase and the solution of the governing equations using analytical techniques
SfiKiytail
Fig. 3. Schematic of a planing hull.
becomes impossible. Although experimental tests are the most reli-able way for modeling these flows, these techniques are very costiy and data are achievable only for a limited number of cases. The i n -herent limitations of analytical and experimental techniques have motivated the researchers to use computational fluid dynamics (CFD) methods in recent years.
3. Geometric characteristics
Planing hulls have a number of common geometric characterisrics which include the dead-rise angle, chine, and spray rail (Fig. 3).
The deadrise angle is defined as the angle between the bottom of the hull with the horizontal (Fig. 3). According to this definition, the hulls with a flat bottom have a deadrise angle of zero. Vessels with a deadrise angle of zero move constantly and comfortably in still water; however, in rough waters, they experience slamming. In the event of slamming, the passengers will be uncomfortable and, in extremely rough waters, slamming can hurt the passengers and cause severe damages to the hull and the equipment onboard. The hulls with a non-zero deadrise angle, on the other hand, break the water and move more smoothly in both waters and are lifted to the water surface while moving forward. Selecting a correct deadrise angle can help stabilize the hull and facilitate a smoother ride. It also reduces the wetted area and drag in the planing mode. To minimize slamming effects, hulls usually come in a V-shaped structure. It should be noted that hulls with a flat bottom have a much higher drag compared to their V-shaped counterparts. As the hull advances in water, the stern is the last secrion that comes out of water and is the first to touch the water on its way back. Therefore, larger sterns result in larger slamming effects. Slamming can be considerably reduced with increase in deadrise angle.
The intersection of the bottom and side of the hull forms a line that is called a chine. The chine becomes harder as the angle between the side and bottom surfaces increases. A chine can cause a smoother hull motion at higher speeds and in turbulent waters. It is also an important factor in keeping the hull more stable. In addition, a chine cuts the water and reduces the wetted area, which in turn, reduces the drag force and increases the speed.
Spray rails can improve the efficiency and performance of high-speed planing hulls. The shape, size, and location of the spray rails have important effects on their effectiveness. Spray rails redirect the upcoming water and confine i t through the bottom of the hull and thus cause a lift force. They improve the efficiency by reducing the wetted area. Spray rails also increase the longitudinal and transverse stability of the hull. Another advantage of spray rails is that they protect the side walls from incident water. Spray rails come in a variety of shapes; however, they all share a triangular cross-section. Spray rails have sharp edges in order to prevent stagnation points for the upcoming water.
Planing hulls have three main cross-sections: convex, concave, or flat (Fig. 4). Convex cross-sections are more popular in practice than the other two shapes. The convex hulls are very strong and, compared with other cross-sections require less material, causing the hull to become lighter. Another advantage of this shape is its reduced slamming loads [2].
107
Fig. 4. Common cross-sections used in liigh-speed planing hulls. Left: convex; middle: concave; and right: flat [2].
PlantfiEHLills Analysis TechniqiiGs I Analyticüir I Experlmenia! I Techniques
ZT
Numerical M e l h o d sSavitsky I Potential Flovj Technique i j Based Mettiods ^
Viscous How S a s e d Metiiods i BoundoryElemcnt Method ( B E M ) Finlto EICfTicnl Method (FEM) Finite Oiffcfcncc' Method (FDM) j Finite Volum Method IFVM)
Fig. 5. Hydrodynamic analysis tectiniques for plan ing hulls.
4. Analysis tectiniques for planing liulls
The design of a planing hull requires tools to allow for estimating the dimensions, propelling power, hydrodynamic drag, and predicting the hull modon behavior. Due to the complexity of the behavior of these hulls, hydrodynamics of the flow around them has not been fully understood. Hydrodynamic analysis of these hulls is much more complicated than displacement hulls due to the spray drag, wave-making drag, free-surface simulation, and the two-phase nature of the flow. The best and most accurate results are from experiments that have been conducted over the past several decades. Numerical techniques can be used to determine the flow unknowns and predict the hydrodynamic behavior of these hulls. Numerical methods can be instrumental at different stages of the planing vessel design specially in selecting the optimum hull form. In general, there are a variety of theorerical and numerical techniques for hydrodynamic analysis of planing hulls, which can be classified as in Fig. 5.
The main objective of this study is to describe the numerical tech-niques that are more popular among researchers. FVM and BEM are the dominant methods for viscous and potential flows, respectively. However, FDM and FEM are among the least popular methods for hydrodynamic analysis of the flow around planing hulls. It should be emphasized that the FVM was originally developed in the form of FDM [3]. Among the methods presented in Fig. 5, FEM and FDM have limited applications in hydrodynamic analysis of planing hulls. Therefore, in the following sections, among the methods based on viscous flow, only FVM will be described.
4.1. Analytical-experimental teclmiques
There are a limited number of techniques for analysis of planing hulls. Savitsky's design and analysis method is among the most pop-ular techniques. In 1964, Savitsky [4] conducted a series of tests on various wedge-like hulls and obtained semi-empirical correlations for estimating lift and drag forces. He obtained equations through a regression procedure and suggested a method for estimating the drag of high-speed planing hulls. To use his equations, the hull is assumed to be in the planing mode, which in turn, requires that the pressure applied to the bottom of the hull support the entire weight and thus neglect buoyancy forces.
4.2. Boundary element mettiod (BEM)
Fluid motion is described by the conrinuity equarion in conjunc-tion with the Navier-Stokes equaconjunc-tions. These equaconjunc-tions are based on the conservation of mass and momentum, respectively. In gen-eral, solution of Navier-Stokes equations is both complicated and time-consuming. In practice, to simplify the governing equations, a number of assumptions are made. Simplification of governing equa-tions limits their applicability. In inviscid flows, in most applicaequa-tions, the potential theory is used, which assumes negligible variations of properties in the computational domain.
BEM uses the properties of the Green's second identity to solve a set of differential equations. In this technique, the flow field is not separated from the boundary; the equations are solved only on the boundary. This method reduces one dimension of the problem, lead-ing to fewer unknowns. As a result, less memory and time are needed. In BEM, the integral over the entire fluid region can be related to the integral over the boundary, which results in easier meshing and higher computational speed.
Due to the semi-analytical nature of this method and the use of integrals, the function for the Laplace equation is exact. Discretization of boundaries can be a source of errors in BEM. Using this technique, partial velocity potentials are first calculated, followed by the total potentials. Therefore, by calculating the partial potentials, velocity is obtained by differentiation. Knowing the velocities, pressures and forces are computed. Although BEM can considerably reduce the anal-ysis time and provide reasonable solutions compared to the viscous flow based techniques, they lead to significant errors for problems in which viscous effects are not negligible and/or wave breaking occurs.
4.3. Finite volume mettiod (FVM)
In FVM, integrals over the control volume are discretized in the computational domain. Navier-Stokes equations (3 equations) and continuity for an incompressible flow have a total number of 4 un-knowns (3 velocities and 1 pressure). These equations can be solved simultaneously or iteratively. In the simultaneous method, a set of equations are solved for the four unknowns. This method is costiy and requires a relatively high memory and computational time. Due to the high volume of computations in this method, the computer speed plays an important role [3].
FVM has the following algorithm:
1. Integration of the governing equations over the control volume. 2. Discretization, which includes replacing approximations for
inte-gral terms and converting the inteinte-gral equations to a set of alge-braic equations.
3. Selection of a method to solve the set of equations.
The first step, i.e. integration over the control volume, distin-guishes FVM from other CFD methods.
The problem modeling can be done in three steps: calculating the velocity and pressure distributions, modeling the free surface, and simulation of the hull motion. To obtain the velocity and pressure
108 Reza Yousefiet al./Applied Ocean Researcli 42 (2013) 105-113
distributions, Navier-Stol<es equations are solved. Normal stresses (pressure) and tangential stresses (due to viscous forces) are then calculated from pressure and velocity distnbutions. Forces and mo-ments are obtained from normal and tangential stresses. Furthermore, linear and angular displacements are calculated using conservation of linear and angular momentum equations.
In FVM, solution of the governing equations is significantiy influ-enced by the type and quality of the mesh, type of flow and turbulence model (if used), and solution algorithm of the velocity and pressure fields. The first step in numerical simulation of the flow around a high-speed planning hull is to create an appropriate mesh. There are a variety of meshes, each of which having advantages and drawbacks depending on the specific problem. Based on the geometry of the problem, a structured or unstructured mesh can be used. Structured meshes are much simpler and require less information for discretiza-tion and computadiscretiza-tion of the mesh. However, these meshes encounter difficulty when they are used for complex geometries. On the other hand, unstructured meshes are more efficient but require high capac-ity storage for the mesh data. To resolve this, multi-block meshes are used. The quality of the mesh is improved by increasing the resolu-tion of the mesh in regions with a high gradient of flow parameters. Multi-block meshes improve the accuracy of the solution [5].
Degree of freedom of the hull, motion amplitude and relative mo-rion between the hull and fluid play an important role in selecting a suitable mesh. Cartesian, overset, and body-attached meshes are mainly used by researchers in this fleld. In some cases, remeshing is also used to improve the simulation results. Cartesian meshes are fixed and the effect of the motion of the structure is applied to the discretized equations or the shape of the cells on the boundaries. However, in overset method, a number of overiapped meshes are used to discretize the computational domain. In this technique, one simple mesh covers the entire domain, and for each moving section or complex geometry, a separate mesh is used.
In remeshing, an unstructured mesh containing the boundaries is first created. Linear and angular displacements are then applied to the structure. Given the initial computational domain, which is kept fixed throughout the solution, and displaced boundaries of the structure, a new mesh on the entire domain is created. In the attached mesh technique, by computing the linear and angular displacements, the structure is first displaced, and then the undeformed mesh is displaced accordingly. This method is appropriate in the simulation of 6 degree-of-freedom small amplitude motion [6].
The flow regime around a high-speed hull is normally turbulent. In turbulent flow, transport quantities such as momentum and energy fluctuate at a high frequency. Simulations of these fluctuations are time consuming and numerically expensive. Instead of direct sim-ulation, the governing equations can be averaged over time (RANS equations) to reduce the time and expense. The averaged equations have additional unknowns, which can be obtained using turbulence models.
In the analysis of turbulent flows, it is very important to select an appropriate model. In hydrodynamic analysis of high-speed hulls, k-s and k-ü) have mostly been used by researchers, k-e model is popu-lar due to its accuracy both for simple and complex flows including recirculation, streamline curvature and swiri flows. The k-e model is further divided into three types: standard, RNG (re-normalizarion), and realizable. While the standard model is used in high Re flows, the RNG theory uses a differential equation to account for viscous effects, which become important in low Re flows. However, effec-tive use of this model depends on appropriate behavior of the flow near the wall. The RNG model has a significant improvement over the standard model specially where the streamlines are highly curved and thus vortices and circulation exist. In flows with reduced veloc-ity and separation due to reversed pressure gradient, RNG performs better than standard k-s. k-s models lead to more accurate results in regions w i t h higher Re. However, near the wall, where the Re is
relatively small, it runs into troubles and results in less accurate es-timation of the flow parameters. Therefore, k-w model can be used to predict turbulent variables near the rigid wall using finer mesh elements.
In control volume method, there are two methods to solve the Navier-Stokes and continuity equations: simultaneous and iterative methods. In simultaneous method, all flow variables (velocities and pressure) are first discretized to obtain a system of algebraic equa-tions. These linear equations are then simultaneously solved. This method is very expensive and requires powerful computers. In this method, the velocity field is fist solved and the pressure is then com-puted. On the other hand, in iterative method, the velocity and pres-sure terms are discretized in two ways: estimation-correction and partial step. Both of these discretization methods are usually used in hydrodynamics problems and are relatively accurate. There are dif-ferent methods for discretization of the velocity and pressure terms. Among them, 1 st and 2nd order up-wind are mostly used in this field. In comparison to the 1st order up-wind discretization, the 2nd order up-wind is less stable but more accurate.
On the convergence of the solution, it should be noted that in addi-tion to the residuals plots, another criterion should be monitored. For instance, in problems in which hydrodynamic forces are important, variations of drag and lift with time can be plotted to monitor conver-gence. Or, if the maneuverability of the hull is important, variations of heave and pitch of the hull are usually monitored, as in [7]. A bench-mark table, summarizing major velocity-pressure coupling schemes, meshes, and turbulence models used in numerical investigations on planning hulls, is presented in Section 6.
5. Review of research on high-speed planing hulls
A great deal of theoretical and experimental research has been done on high-speed planing hulls since eariy 20th century and a num-ber of techniques have been developed for hydrodynamic analysis of these hulls. Eariier research on hydrodynamics of these vessels was based on analytical methods. Due to the natural limitations of these techniques, they were limited to two-dimensional (2D) studies with the exception of the work by Wagner [8], Mauro [9], and Tulin [10], which was based on three-dimensional (3D) flow. Doctors [11] per-haps conducted the first comprehensive 3D study on planing hulls with no limitations on the Froude number. Most of the research then was based on BEM. Because BEM reduces one dimension of the flow field, computations were quite fast and since the computers were slow, it was a great advantage. However, an important factor, i.e. viscosity, is neglected in this technique. This method was based on potential theory and thus the results were not reliable for viscous flows. With the advances of computer technology in the later decades of the 20th century, interest in FVM, which is capable of providing details about the flow field, gained momentum and has constantiy improved. FVM is now used in CFD codes or commercial software. It should be noted that BEM is still being used. Depending on what information about the flow is needed and the time constrains, either of BEM or FVM is used. In the following sections, 2D and 3D ana-lytic studies based on potential theory will first be presented. This is followed by studies based on viscous flow. Finally, experimental investigations w i l l be briefly discussed.
5.1. Potential flow
One of the eariier studies on high-speed hulls is by Von Karman [12], which was based on conservation of momentum for analysis of 2D hulls. In 1932, Lamb [13] investigated a 2D planing problem. He used an integral equation to obtain the pressure distribution as a function of slope for a 2D planing plate with a small aspect ratio. Later in 1951, Mauro [9] solved the 2D planing problem based on Lamb's method and by a Fourier series expansion to obtain an un-known pressure distribution on a plane. He used the slope of the
109
Table 1
List of software programs and their ma j or capabilities and characteristics.
Capabilities Software
Capabilities
Fluent ANSYS-CFX CFDShip-lOWA OpenFOAIVl Star-CD Tdyn
Computational Excellent Excellent Good Good Excellent Good
stability
Scripting capability Good Good — Excellent
-
-Access to source No No No Yes No No
code
Availability of Easy Easy Very difficult Easy Difficult Difficult
software
Table 2
List of major numerical investigations performed distribution, t r i m diagram and drag curve. It also
after year 2000 and summary o f their results. The x symbol next to each research represents whether the study obtains the pressure shows whether the study simulates the free-surface and investigates stability, wave pattern and maneuverability.
Pressure T r i m Drag Free-surface Wave
Date Author(s) Hull Software/analysis method distribution diagram curve simulation stability pattern Maneuverabilitj Finite volume method (FVM)
2000 Subramani et a l [49]
Planing Series 60
CFD-Ship-lowa: RANS, VOF, dynamic mesh
X X X
2001 Caponnetto 137]
Planing COMET: HRIC interpolation for VOF
x X X X
2002 Thornhill 140]
Planing FLUENT: RANS, VOF x X X
2003 Caponnetto etaL 139]
Planing COMET: dynamic mesh, RANS,
k-e
X X X X X
2005 Senocakand laccarino [47]
DTMB 5415 FLUENT: VOF, RANS, k-e X X X
2007 Özdemir [3] Planing FLUENT: RANS, k - E , k-m X X X
2007 Subramanian etaL [48] Planing w / tunnel FLUENT: RANS, k-s, single-phase X X 2007 Javanmardi etaL [7]
Trimaran NUMELS: dynamic mesh, ClCSAMforVOF
X X X X X X
2008 Fultz [42] Pentamaran FLUENT: k-e, PLlCforVOF X X X X
2009 Jahanbakhsh et at [45]
Catamaran NUMELS: dynamic mesh, VOF X X X
2009 S e i f e t a l . [44] Planing NUMELS: ClCSAMforVOF X X X X
2009 Panahietal. 143]
Catamaran planing and wedge
NUMELS: dynamic mesh, k-e. ClCSAMforVOF
X X X X
2011 Pranzitelli et a L [ l ]
Semi-planing FLUENT and SHIPFLOW: RANS, VOF, panel method, two-phase
X X X
2011 Brizzolara and Serra 141]
Planing wedge Star-CD: RANS, VOF X X X X
Boundary element method (BEM) 2002 Savanderet
a l [ 2 5 ]
Planing plate Potential perturbation, vortex distribudon X X X 2006 Kihara [34] Planing pyramid 2D -F T, domain decomposition in spray region X 2008 Chassemi and Yu-min 127] Series 62 planing
BEM for pressure drag, BL for frictional drag and empirical method for spray drag
X X X 2010 Ghassemi and Kohansal [28] Wedge/ flat plate, variable deadrise
Coupled BEM and boundary layer theory X X X X 2010 Sun and Faltinsen 132] Planing 2D + T X X X X
Finite element method (FEM) 2000 Yangetal.
129]
Planing, Series 60
FEM, dynamic mesh X X
2005 Xie etaL [30] Planing FEM X
plane as a boundary condition in liis solution. His method was valid for plates moving at typical Froude numbers. Cumberbatch [14] was able to solve the 2D planing problem for high Froude numbers. In his method, the integral equation was expanded, with a power series of inverse Froude number, and solved using an iterative technique. He showed that the right combination of a geometric shape and flat surface could remove the singularity of the leading edge. His results showed a considerable reduction in drag force. Clement and Blount
[15] evaluated the existing techniques for predicting the motion of high-speed hulls and compared them with experimental results.
Doctors [11] presented a finite pressure element method for solv-ing the flow around a 2D plate. In his problem, an inviscid flow ran over the free-surface. His idea was similar to Mauro's; however, a different pressure distribution was used. A linear potential flow was used in his work. His method allowed for a variety of pressure distri-butions for the planing plate.
110 Reza Yousefi et al./Applietl Ocean Research 42 (2013) 105-113
The Strip theory, which is based on potential flow, is used to deter-mine the drag and lift coefficients for planing hulls. In this technique, the hull is divided into thin strips and each strip is then analyzed. This method assumes a two-dimensional flow around the hull. Strip theory was first used in Salvensen's study [16] and has been improved over the years. Frandoli et al. [17] showed that the strip theory provides relatively good results even at higher Froude numbers.
Another method for analysis of high-speed hulls is 2,5D or 2D -i- T. This technique assumes a fixed imaginary plane which intersects w i t h the hull as it moves through. In this technique, the problem reduces to a plane moving the water surface. This technique was first used by Tulin [10] and then Zhao et al. [18| extended the method for analysis of high-speed hulls in calm water.
Three-dimensional methods are much more complicated than 2D techniques. In 3D problems, the wetted surface area is much larger and calculation of the integral equation is more sophisticated. In the eariier decades of the 20th century, due to the limitations on com-putational hardware, the 3D problem was tackled by only a limited number of investigators. These studies were limited in hull speed and aspect ratio. Earlier attempts for modeling 3D-planing problem were made by Wagner [8]. He modeled the hydrodynamics of the planing problem by a slender body moving in water. Furthermore, Tulin [10] used a vortex distribution to determine the flow around the hull and the plunging jet impact on the free-surface. Wagner's water entry and Tulin's jet models laid the foundations for future studies.
Mauro [19] presented'an integral equation for the 3D-planing problem, which related the slope of the surface to the velocity po-tential terms using a function of unknowns. His integral equation was similar to the integral equation for the vortex distribution over a thin planar foil. He then solved the integral equation by both a large and a small aspect ratio approximation. He expanded the integral equation with respect to the aspect ratio and only the first term was taken into account, neglecting higher orders. Therefore, the solution was an approximation. For smaller aspect ratios, his method required a high Froude number. Mauro's method was incapable to be used for rectangular planing plates. Wang and Risipin [8] also solved the 3D steady state potential flow around a planning rectangular plate with a moderate aspect ratio and a large Froude number. They ob-tained the pressure distriburion on the hull in the form of a series and compared it experiments. Their results were in good agreement w i t h experiments.
Doctors [20] perhaps conducted the first comprehensive 3D study on planing hulls w i t h no limitations on the Froude number and as-pect ratio. He used an integral equation that related the pressure distribution to the velocity potential, which was earlier obtained by Wehausen and Laitone [21 ]. The flow around the hull was modeled by finite pressure elements. The pressure was allowed to vary with the position of each element and the overall distribution was continuous. The double integral equation was transformed to a line integral using a special function. One of his problems was that the wetted area was unknown a priori and thus it was part of the solution. Through an iter-ative procedure, the wetted area was adjusted in such a way to satisfy the trailing-edge Kutta condition until it reached a constant value. The pressure distribution was found to be oscillatory. The pressure oscillation was attributed to the inaccuracy of the pressure elements. Because the pressure elements were not uniform on the free-surface, the pressure distribution was not correctly modeled on the surface.
Wellicome and Jahangeer [221 studied the 3D-planing problem based on the pressure distribution on the wetted area using rectan-gular elements of constant pressure. Tong [23] also used these integral equations for pressure to study the planing plate problem. In his study, the elements were constant and matched the leading edge profile of the wetted surface area. The shape of the wetted surface was known a priori and the draft was determined at the transom.
In addition, Cheng and Wellicome [24| used pressure strips in the
transverse direction to study hydrodynamics of planing hulls. Pres-sure along each strip was assigned a sinusoidal series, and thus each term was represented by a mathematical formulation. The wetted area was determined when the draft and the transom profile were as-sumed unknown. In his case, there was no limit on the Froude number and the aspect ratio. One drawback of this technique was that variable transverse pressure strips produced a continuous pressure distribu-tion only in the transverse direcdistribu-tion; however, in the longitudinal direction, the pressure distribution was discontinuous.
Savander et al. [25] applied the boundary value problem to a plan-ing plate and obtained relationships between potential perturbation and vortex distribution. They calculated the hydrodynamic pressure, lift and drag forces for the planning plate at different speeds. Ghas-semi et al. [26-28] have developed a computer code, based on BEM in conjunction with boundary layer, for hydrodynamic analysis of plan-ing and non-planplan-ing hulls. One of the drawbacks of this code is that it does not take into account a two-phase flow model. It addition, the code cannot be used for complex geometries and high Froude number cases. They also used this code to study the wave pattern and pressure coefficients. Furthermore, Ghassemi and Yu-Min [27], and Ghassemi and Ghiasi [26] developed a hybrid technique to determine the hy-drodynamic forces for steady state flow around a planing hull. In all of these studies, good agreement between BEM and experimental results was reported.
Yang et al. [29] used an FEM to simulate the flow around a planing hull. In their simulation, the draft of the hull was determined through an iteration procedure and then the heave and pitch were deter-mined using the balance of normal forces and moments. They applied a moving mesh near the hull. Their iterations were continued until it converged to a dynamic balance. Tests were also done for the Wigley and Series 60 for an extensive range of Froude numbers. When the trim and draft were fixed, this technique revealed considerable differ-ence with experiments compared to the variable trim and draft case (two degrees of freedom). Xie et al. [30] also investigated 3D-planing hulls using FEM. They used the potential theory and determined the free-surface by adopting a coordinate system normal to the hull and by assuming a zero pressure condition. In this technique, each ele-ment on the planing plate was set to a constant power pressure field. The pressure distribution obtained with this technique was in good agreement w i t h the results of Tong [231, Cheng and Wellicome [24], and Wang and Risipin [31 ]. In this study, unlike previous studies, the pressure was not oscillatory. The oscillarion is believed to be related to the constant pressure distribution and induced coefficient for pres-sure elements, which was not used previously.
Sun and Faltinsen [32] investigated the performance of a planing hull with unsteady flow assumption for the incident waves using the BEM and 2D + T techniques. They also studied the resulting waves due to the heave and pitch motions. The results were compared w i t h the experiments of Fridsma [33]. In addirion, Kihara [34] used a 2D -i-T technique along with BEM to investigate nonlinear free-surface flow including the spray. Their idea was based on domain decomposition in the spray region using boundary elements.
The aforementioned theorerical studies were all based on potenrial theory and the free- surface fluctuations were assumed to be small. In reality, the flow around a planing hull is a nonlinear free-surface phenomenon. A number of researchers have studied the nonlinear planing hull problem using a variety of techniques. These techniques will be described in the following section.
5.2. Viscous flow
With the advancement of computer technology, researchers have started to widely use the FDM for solving the 3D, nonlinear problem for displacement hulls. Hino et al. [35] utilized the FDM to study the hydrodynamics of two simple prismatic geometries. They used the Euler equation along with a nonlinear free-surface condition and a
Marker-and-Cell scheme. The wave at the stern was accurately ap-proximated; however, the pressure at the bow was not in agreement. Richard et al. [36] used the RANS model to predict the planing lift in 2D flat plates. They calculated the lift force for the plate planing at the free-surface and with a slip wall approximation for the free-surface. They found that the approximated free-surface produced better re-sults than those from the calculated free-surface when compared against the empirical correlarions. They suggested enhancements in the VOF method before these calculations could be used for design of planing hulls.
In the same year, Caponnetto [37] used a two-phase, FVM to find the pressure distribution on the planning hull. They urilized a uni-form mesh in their analysis with the COMET software (zero degree-of-freedom). For each speed, three trim angles and three drafts were used. For other cases, the results were interpolated for a given lift and LCP relative to the equilibrium state of the hull. Balance is achieved when the lift equals the weight of the hull and the center of pressure coincides w i t h the center of gravity. They compared the results using the Savitsky empirical correlarions [4,38]. Caponnetto etal. [39] later extended this work using CFD to solve the 3D, high-speed hull prob-lem. In the new technique, they employed a two degree-of-freedom moving mesh in the simulation. The two degrees of freedom included balances of moments and normal forces to obtain the trim and draft, respectively. The results were found to be quite accurate; however, the computational cost was relatively high.
Pranzitelli et al. [ 1 ] used the panel method along with FVM to sim-ulate the free-surface, two-phase flow around a semi-displacement hull advancing steadily in calm water. A volume of fluid (VOF) method was used to predict the free-surface profile and the drag. Using the FLOWTECH SHIPFLOW software, they employed the panel method to calculate the waves generated around the hull. In addition, Javan-mardi et al. [7] used their NUMELS code to study the effects of the three configurations of the Trimaran hull with a two-phase, viscous flow model using FVM. They presented the hull maneuverability, sta-bility, trimangle, and the drag. Furthermore, Thornhill etal. [40] used FVM and the VOF method to investigate the steady, ti/vo-phase flow around the hull. They assumed a 3D flow in calm water and used an unstructured mesh. To find the equilibrium state for the hull, balances of normal forces and moments were used to determine the draft and trim angle, respectively. The results included diagrams for trim, drag, and pressure as a function of speed and for three cases of zero, one and two degrees of freedom. Their results were found to be in good agreement with experiments. Finally, Brizzolara et al. [41 ] simulated the free-surface flow around a wedge-like hull using FVM and the VOF method. Lift, drag, and the trim angle were obtained and com-pared against experiments and the results from Savitsky empirical correlarions.
Fultz [42] has recently investigated the flow around a Pentamaran hull using FVM for two fluid conditions: single phase, and two-phase (using the VOF). Panahi et al. [431 have also used the FVM and VOF and the partial step method for velocity and pressure coupling to simulate hydrodynamics of two different hulls. The first hull was a two-dimensional wedge with a two degrees of freedom. They then analyzed the motion of a planing Catamaran hull. In these studies, they obtained the drag and trim angle curves and compared their findings with other existing numerical results and found relatively good agreement. They used the NUMELS-NUMERIC code, which was developed and optimized by Seif et al. [44]. This code allows for hydrodynamic analysis and maneuverability of high-speed planing hulls. One limitation of this code is that suffers solution divergence at higher Froude numbers. They have recently published several studies including three-dimensional simulation of the nonlinear motion of a high-speed hull, simulation of the motion of a Catamaran hull using a moving mesh [45] and numerical simulation of a high-speed planing hull.
5.3. Analysis using commercial software
With the development of computational techniques in recent years, calculation of drag forces for a floating body using commer-cial software has become reliable. MARNET-CFD research team in Barcelona investigated the effects of incident water on transverse sections of a planing hull, using both an experimental technique and a numerical method with the FLUENT software. In the ANSYS users' conference in 2008, Godderidge [46] presented a modeling of free-surface flow using the CFX software. His results included the pressure and trim diagrams as a function of speed and in the presence of a free-surface pattern.
Senocak and laccarino [47| urilized the FLUENT software to in-vestigate the free-surface flow around the DTMB5415 ship model. They adopted the VOF method to track the free-surface and the k-s turbulence model in their simulations. A structured mesh was used in their study. They found that the quality of the mesh considerably affected their computations. Özdemir [3] used the FLUENT software to compute the lift and drag for a high-speed hull. He obtained the velocity and pressure fields and compared the accuracy of k-s and
k-w turbulence models.
Subramanian et al. [48] used the FVM based CFD-Ship-Iowa code to simulate a Series 60 hull. VOF method was used to trace the free-surface. With a moving mesh, the position of the hull changed with time, which in turn, caused the trim and draft of the hull to change toward a balance state. This procedure was continued until balance was achieved. The results agreed with experiments very well. The trim diagram and drag of the hull for a variety of speeds and free-surface shapes were among their results. Subramani et al. [49] also used FVM for a single-phase, RANS model to investigate the effect of a tunnel on the pressure drag and lift. They used the FLUENT software and compared the results with Savitsky experiments.
It should be mentioned that there are a number of computer soft-ware that have the capability to simulate the flow around a hull, tak-ing into account maneuverability, wave and free-surface effects, and esrimate the hull drag. ANSYS-FLUENT ANSYS-CFX, CFD Ship-Iowa, Open FOAM, Ship Flow, Tdyn, and Star-CD are among the available software. To date, the first four software programs are the most popu-lar ones having excellent computational stability. ANSYS-FLUENT and ANSYS-CFX are easily available and have programming capabilities, though they are not open source. Open FOAM is readily available, open source and allows for user programming. Although CFDSHIP-Iowa is a powerful and professional program for simulation of the flow around ships; i t is not easily available and has no user programming capabil-ities. In addition, i t is not open source. It should be noted that Open FOAM has not yet gained popularity and FLUENT and CFX are still the dominant software for simulation of the flow in this field. Table 1 summarizes the characteristics of a number of popular software packages.
5.4. Experimental studies
Although the main focus of the current study is on computational techniques, a brief description of some important experimental inves-tigations is presented. Experimental measurements made by Sottorf [50] were among the early experimental studies on planing hulls. He conducted a series of systematic model tests for planing hulls and measured the hydrodynamic drag, center of pressure, and pressure distriburion for a planning flat plate. In 1934, he extended his analysis to include the V-shaped models [51 ]. His work was followed by Shoe-maker [52], Sambraus [53], Sedov [54], and Locke [55]. These efforts led to a large volume of experimental data on hydrodynamic charac-teristics of planing plates under special conditions. Shoemaker [52] reported data for hydrodynamic drag, center of pressure, wetted line, and draft for a variety of trim angles, loading, and forward speeds. Clement and Blount [15] further performed extensive model tests on Series 62. These experimental data on planing hulls became a good
112 Reza Yousefi et al./Applied Ocean Research 42 (2013) 105-113
T a b l e s
List of major experimental investigations on planning hulls and a summary of results.
Date Author(s) Hull Major results
1929 Sottorf150) Planing flat plate Drag, center of pressure, pressure
distribution
1934 Sottorf[511 V-shaped model Drag and center of pressure
1934 Shoemaker [521 Sottorrs V-shaped model Drag, center of pressure, wetted length and d r a f t
1938 Sambraus [53] Extended SottorPs V-shaped model Drag, center of pressure, wetted length 1939 Sedov [54] Extended SottorPs V-shaped model Drag, center of pressure, wetted length 1948 Locke[55] Extended Sottorfs V-shaped model Drag, center of pressure, wetted length
1963 Clement and Blount [15] Series 62 model Database f o r future studies
1964 Savitsky [4] W/edge-shaped planing hull Semi-empirical correlations for drag, lift, center of pressure
1969 Fridsma [33,56] Wedge-shaped Drag and center of pressure i n
1971 planing hull regular and irregular waves
1976 Savitsky and Brown [38] Extended method to include effects of Drag, lift, and center of pressure waves on drag
1953 Kapryan and Boyd [59] Pnsmatic planing hull, variety o f Lift and pressure distribution deadrise angles
2002 Katayama et al. [57] Wedge-shaped planing hull Drag and l i f t coefficients at various forward speeds
2005 Bowles and Denny [58] Planing hull Predicted turbulent water surface at the
2007 Savitsky |60] Prismatic planing hull bow Wetted surface characteristics and spray
database for future studies.
In 1964, Savitsky [4] performed experimental tests and provided empirical correlations for calculation of lift, drag, and center of pres-sure for wedge-shaped high-speed planing hulls. His results were reported for a vanety of speeds, deadrise angles, and loading. Based on Fridsma's tests [33,56] on high-speed planing wedges in regular and irregular waves, Savitsky and Brown [38| revisited and modified their original method to include the effects of waves on accelera-rion and wave drag. The main advantage of this method is that i t is simple and provides relatively accurate results for a number of hulls with a regular shape. However, his method has some drawbacks. This technique cannot be used for analysis of hulls with variable deadrise angles in the longitudinal and transverse directions. In addition, this method is semi-static and is unable to predict transient behavior. It provides the total force through a series of correlations and cannot be used to calculate the force at a point or on a particular panel.
Katayama et al. [57] also performed model tests to find hydrody-namic drag for wedge-shaped high-speed planing hulls at different speeds and obtained the lift and drag coefficients. In addition, Bowles and Denny [58] found a model for prediction of turbulent water sur-face at the bow of the planing hull.
Most of the previously mentioned investigations were focused on force and moment measurements. Their main objective was to provide experimental correlations for prediction of hydrodynamic drag; however, there has been littie experimental research on pres-sure distribution meapres-surement. Kapryan and Boyd [ 59| meapres-sured the pressure distribution on planing hulls. They performed a series of ex-periments on prismatic hulls w i t h a variety of deadrise angles. The lift force variations were obtained by integrating the pressure distribu-rion. Savitsky et al. [60] has also recently studied the wetted surface characteristics and spray to determine the spray drag.
6. Benchmark table
To help future researchers and engineers select the most appro-priate design and analysis techniques for high-speed planing hulls, the most recent research studies are summarized in Table 2. The main purpose for this table is to compare the analysis techniques and list their capabilities and limitations. Research studies on the hy-drodynamic simularions of planing hulls published after 2000 have been reported herein. In addition to listing the analysis technique for
each study, important results including pressure distribution, tiim diagram, drag curve, free-surface shape and effects, stability, wave pattern, and maneuverability of the hull are presented. This bench-mark table also contains information about the range of applicability of each technique. Furthermore, given the importance of experimen-tal techniques in the analysis of the performance of high-speed hulls, a brief summary of experimental research is presented in Table 3.
7. Conclusions
Estimation of hydrodynamic forces is the most important compo-nent in the analysis of high-speed planing hulls. Planing hulls create complex free-surface flows. These flows include nonlinear phenom-ena such as plungingjets and irregular waves. To date, model testing is the best way to predict the hydrodynamic performance of hulls. With the advancement in the computer hardware and software, numerical techniques have become effective tools for hydrodynamic analysis. The most important advantage of numerical methods is that they do not suffer limitations that are normally encountered in model test-ing such as the size of the hull, environmental conditions, analysis and interpretation of results for prototype hulls. They also eliminate the cost for construction of laboratory models. Numerical techniques allow for hydrodynamic modeling of real size hulls, investigation of design components in eariy phases, and obtaining detailed informa-tion, which are otherwise impossible to obtain with experiments.
Computational fluid dynamics can be used in applications where analytical solution of the governing equations is not possible, using two methods: potential theory for inviscid flow and viscous flow. FDM and FEM are nowadays rarely used in hydrodynamic analysis of high-speed planing hulls, although FEM is mainly urilized for struc-tural analysis of marine vessels. Potenrial flow-based methods are applicable to steady state inviscid flows, in which the viscous effects are negligible. Computational time for this method w i t h an advanced computer is on the order of minutes and thus is relatively fast.
Based on previous investigations, BEM is an appropriate technique to obtain wave patterns; however, it is not recommended for simu-lation of viscous flows with complex free-surface profiles. In these flows, the viscous effects cannot be neglected and thus a comprehen-sive analysis of the flow is required. FVM is a more appropriate and accurate technique for modeling turbulent, free-surface flows. In re-cent decades, turbulent models have been improved and solution of
113
free-surface flows has become possible. In conclusion, FVM is capable of solving complex, free-surface flows such as breaking waves. It can also be used to predict maneuverability, seakeeping capability, and the equilibrium state of the hulls with complex geometries.
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