Numerical investigation of the components of calm-water resistance of a surface-effect ship

Ocean Engineering 72 (2013) 375–385

Contents lists available at ScienceDirect

Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Numerical investigation of the components of calm-water resistance of a surface-effect ship Kevin J. Maki a,n, Riccardo Broglia b, Lawrence J. Doctors c, Andrea Di Mascio d a

University of Michigan, Ann Arbor, MI 48104, USA The Italian Ship Model Basin – CNR-INSEAN, 00128 Rome, Italy c The University of New South Wales, Sydney, NSW 2052, Australia d Istituto per le Applicazioni del Calcolo – IAC-CNR, 00185 Rome, Italy b

art ic l e i nf o

a b s t r a c t

Article history: Received 15 April 2013 Accepted 30 July 2013 Available online 15 August 2013

The elements of the calm-water resistance of an surface-effect ship are studied with two different numerical methods. A potential-flow-based method that satisfies linearized free-surface boundary conditions is used to predict the wave resistance of the sidehulls and air cushion. A RANS-based program that employs a single-phase level set method is used to simulate the flow around an SES of a nonlinear viscous fluid. Detailed comparison of the dynamic wetted surface, the free-surface elevation, and the wave, cushion, and frictional drag is made for a geometry that has experimental resistance data. It is shown that the linear free-surface boundary conditions of an inviscid fluid are accurate for prediction of wave drag. Disagreement is present between the two methods for the free-surface elevation behind the vessel, which might possibly be due to the transom-stern model that is used in the potential-flow method. The small difference between the numerically predicted resistance and the experimental measurement is attributed to the error in the seal and air drag models that are used in this study. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Surface-effect ship Calm-water resistance RANS

1. Introduction The Surface-Effect Ship (SES) is an attractive concept for applications that require a vessel to travel at high speeds. The vertical force that balances the weight of the craft is generated by a combination of buoyancy, hydrodynamic lift, and air-cushion support. For very high speeds, it is known that the wave resistance is small and the frictional drag dominates the total resistance of the vessel. The principal mechanism in which the SES shows an advantage over non-air-cushion-assisted vessels is the reduction in wetted surface that is achieved through reduced dependence of the vertical force on the action of buoyancy. Following this reasoning, the air-cushion vehicle (ACV) is a strong candidate for high-speed operation, although vessels that are fully supported by an air cushion have limited performance in medium to high sea states because it is difficult to maintain the air cushion. Thus, the SES shows a compromise between the displacement-type vessel and the ACV to deliver a ship that has reduced wetted surface and lower frictional drag but with desirable seakeeping properties. The SES has been used for high-speed naval vessels for many decades with an early example found in Ford (1964). The prediction of the power required to operate in calm water is a primary

n

Corresponding author. Tel.: +1 734 936 2804; fax: +1 734 936 8820. E-mail address: [email protected] (K.J. Maki).

0029-8018/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.07.022

characteristic that is used at the earliest stages of design to evaluate the SES concept and compare it to other candidates such as a planing boat, hydrofoil-assisted vessel, and/or multihull vessel. The total power required to propel an SES can be discussed as the sum of the power to generate the air cushion and that which is needed to overcome the aerodynamic and hydrodynamic forces on the hull. Indeed the two aspects are related, because the fan used to create the air cushion does alter the flow around the hull, but this interaction is assumed to be negligible with respect to the resistance of the vessel. The power required to generate the air cushion is generally significant with respect to the total power, although in this paper we focus solely on the fluid forces on the hull. 1.1. Previous work The numerical predictions in this paper include those based on the traditional potential-flow analysis employing a linearized dynamic condition and a linearized kinematic condition on the free surface. The method essentially allows one to represent the vessel by a distribution of singularities, which model the sidehulls (assumed to be thin) and the air cushion (assumed to be of low magnitude). The method has been described in detail by Doctors et al. (2005) and Doctors (2006). In these papers, the linearized approach allows one to compute the resistance using computations based only on the

376

K.J. Maki et al. / Ocean Engineering 72 (2013) 375–385

Nomenclature τ ΔF ΔL η r i n U μ ρa Aref B BC BS CD F f F RF fW; fF f W RW g L

viscous stress tensor displacement mass (full) displacement mass (light) free-surface elevation stress vector unit normal that is aligned with the free-stream velocity surface normal vector Reynolds-averaged velocity vector fluid molecular viscosity air mass density reference area for air-drag calculation waterline beam cushion beam waterline beam (sidehull) air-drag coefficient pffiffiffiffiffi Froude number F ¼ U= gL frictional resistance computed by linearized theory form factors for wave and frictional resistance wave resistance computed by linearized theory gravitational acceleration constant waterline length

downstream far-field wave system. Thus, the approach is very efficient in terms of computer resources. Comparison of these predictions with experimental data in the literature suggests very strongly that linear theory provides an excellent starting point for estimating the required propulsion power. More recent work is that of Doctors (2012), in which the identical basic theory is used. However, the advancement was to also compute the near-field wave system. This permits one to (a): study the details of the dynamic shape of the wetted surface, (b) predict the sinkage and trim, and (c) estimate more accurately the interaction between the cushion seals with the water. Fully nonlinear viscous methods have been used to study the calm-water resistance of air-cushion assisted vessels. The singlephase level-set method that is used in this paper was previously used to examine the components of the flow generated by a twodimensional pressure patch in Maki et al. (2012). Note that this reference also shows results from the same linearized theory that is employed in this paper. Additional examples of the numerical prediction with a RANS tool of the flow generated by an ACV are found in Bhushan et al. (2011). An interesting aspect of this paper is the numerical validation and verification study that is summarized, and the investigation of the side force that is generated during operation with a nonzero-yaw angle. The nonlinear method that is used in Bhushan et al. (2011) could be applied to a surface-effect ship, in principle. In our work, we extend what is presented in Bhushan et al. (2011) for the case of an SES and provide the details to properly account for the interaction between the hull and the air cushion. In the paper Donnelly and Neu (2011), a commercial CFD software that uses the volume-of-fluid method is evaluated to predict the drag on an SES. The commercial solver is a two-fluid implementation, and comparisons are made with physical model tests. In their simulations the seal geometry is modeled explicitly, but the seals are not allowed to deform. Also, the air-cushion is modeled using a momentum source technique that requires explicit discretization of the air plenum. In the current work, the

LC p pC RF Ra RC Rh RS RT S Sh Ss Sh;a Sh;c Sh;w Ss;a Ss;c Ss;w T U Ua x1 yþ

cushion length fluid pressure cushion pressure flat-plate friction predicted by friction line air drag cushion drag hydrodynamic drag seal drag total drag total surface of SES hull surface seal surface dry hull surface outside of the air cushion dry hull surface inside the air cushion wetted hull surface dry seal surface outside of the air cushion dry seal surface inside the air cushion wetted seal surface draft (20% hull-lift ratio) speed of the craft air speed cushion start station dimensionless distance to the surface using inner scaling

fully nonlinear method does not require discretization of the air plenum, and the seals are modeled in a flexible manner according to the approach described in Doctors and McKesson (2006). A unique numerical tool that simulates the operation of an SES in waves with detailed description of the lift-fan control characteristics is found in Connell et al. (2011). The simulation tool is called ACVSIM, and the hydrodynamic solution is found using the potential-flow based AEGIR program. Also, the paper Kring et al. (2011) demonstrates how AEGIR can predict wave forces on an SES while maneuvering. An interesting element of this paper is how the viscous code CFX is used to determine corrections that are used by AEGIR in the simulation of a maneuver in waves. 1.2. Current Work In this paper two numerical methods are used to predict the fluid force acting on an SES that is advancing at steady-forward speed in calm water. The first method exploits the irrotational nature of the flow around high-Reynolds number bodies such as ships, and computes the wave drag due to the sidehulls and air cushion through numerical evaluation of the integral equations governing a velocity potential. The second method uses a field discretization to find a numerical solution of the unsteady RANS equations. In the potential-flow based method the solution satisfies linearized free-surface boundary conditions on the calm-water plane. The RANS solution satisfies the fully nonlinear free-surface boundary conditions on the predicted free-surface elevation. This paper is organized into the following sections. First, a mathematical description of the fluid forces that act on an SES that is operating in calm-water is presented. This provides for the precise definition of each component of resistance, and allows for a clear connection to be made between the predictions gathered from each of the numerical tools and experiments. Then, an overview of each of the numerical methods is briefly presented. Next, the numerical predictions of each of the components of resistance are shown in the results section. This allows for evaluation of the different numerical methods, and elucidates the relative importance of each component of resistance. Finally,

K.J. Maki et al. / Ocean Engineering 72 (2013) 375–385

the conclusion section summarizes the utility of each of the numerical methods and provides suggestions for future research.

the integral over the hull and seal surfaces that are dry and exposed to the increased pressure in the cushion: Z RC ¼ r  i dS ð4Þ Sh;c þSs;c

2. Numerical methods In this section the exact mathematical expression for the drag on an SES is stated. This is useful to illustrate how the total drag can be thought of as a sum of forces on different components of the vessel. Also, the basic description serves as a common starting point from which the predictions from each numerical method can be examined. Specifically, it is necessary to consider the force on each component of the SES so that the quantities of cushion drag and hydrodynamic drag can be compared between different numerical and experimental methods. 2.1. Mathematical description of fluid force on an SES The drag on an SES can be determined by the integral of fluid stress over the surfaces of the vessel which are in contact with the air and water. We use a surface normal vector n that points into the fluid. The total force of the fluid on the surface S can be written as Z f ¼ ðpn þ τ  nÞ dS ð1Þ S

In this equation the fluid pressure is p, and the viscous stress tensor is defined as τ . Together they form the stress vector on the surface r ¼ pn þ τ  n. Evaluation of Eq. (1) requires knowledge of the fluid stress and position of the body with respect to the wave field generated by the vessel. It is noted that the seals are flexible and their equilibrium position depends on the speed of the vessel and pressure in the cushion. We break this integral into a series of expressions that correspond to different surfaces, starting by considering the hull and seals separately: S ¼ Sh þ Ss . Then, the surface of the hull that is in contact with fluid can be decomposed further as the portion in contact with water, that is in contact with air but not exposed to the increased pressure in the cushion, and finally the portion in contact with air and under influence of the cushion, i.e. Sh ¼ Sh;w þ Sh;a þ Sh;c . The surface of the seals is expressed in the same manner: Ss ¼ Ss;w þ Ss;a þ Ss;c . This decomposition of the surface of the SES results in six distinct elements. The streamwise component of the fluid-stress integral over the surface of the hull that is wetted is called the hydrodynamic drag: Z Rh ¼ r  i dS ð2Þ Sh;w

where i is the unit normal that is aligned with the free-stream velocity that is seen by the vessel. The drag due to the air flow around the hull is often neglected because of its small contribution to the force on the body. For an SES, the above-water portion of the hull is relatively large, and the air drag is not negligible with respect to the total drag. The air drag is defined as Z Ra ¼ r  i dS ð3Þ Sh;a þSs;a

It is important to note that this expression uses the external surface of the hull and seals. In this work, the air drag will be estimated by applying a drag coefficient that is appropriate for the shape of the above water portion of an SES. The air drag is small, although not negligible with respect to the accuracy of the numerical prediction of the total drag. The air cushion generates a wave system which is responsible for additional drag on the vessel. The cushion drag is defined as

377

In this work the air cushion is considered as a control volume that is bounded by the hull and seal surfaces, together with the freesurface, as well as any regions where air is passing into or out of the air cushion. During operation air will leak through gaps between the seals and the hull, and between the seals and the free surface. This leakage necessitates replenishment by the lift fans. The spatially varying velocity of the air in the cushion is due to the leakage and flow through the lift fans, and due to momentum exchange along the free-surface as the water passes at a velocity that is close to the free-stream value. Is it assumed that the dynamic pressure that is due to the velocity in the air and the viscous stress along the free surface is negligible. The spatially constant pressure in the cushion is labeled pC, and this is the increase in pressure over the atmospheric value. Also, several formulations for the total drag on an SES use a component labeled the momentum drag. We chose to omit this term because it requires accurate knowledge of the leakage of the air in order to assess its magnitude and direction. Indeed there are cases when a thrust is produced due to the large velocity through a small leakage area in the downstream direction. In this work the seals are not explicitly modeled in the hydrodynamic simulation. Instead, the pressure boundary condition on the free surface is used to account for the transition between the pressure outside of the air cushion to that on the free-surface inside the air cushion. The transition occurs over the wetted length of the seal. A common model for this behavior is found in Doctors and Sharma (1972): pðx; yÞ ¼ pC XðxÞYðyÞ

ð5Þ

  X ðxÞ ¼ 12 tanh½αx ðxÞtanh½αx ðxLC Þ g

ð6Þ

 YðyÞ ¼

1

for jyj oBC =2

0

otherwise

ð7Þ

The length and beam of the cushion are denoted by LC and BC, and the Cartesian coordinate system can be seen in Fig. 2. The smoothing parameter αx is chosen based on the longitudinal length of the seals, and αx is selected to be 0.15 m  1 at full scale for the case studied in this paper. This value is equal to that used in the paper by Doctors and McKesson (2006). The last portion of drag due to fluid stress acting on the SES is the integral over the wetted portion of the seals. The seal drag is defined as Z RS ¼ r  i dS ð8Þ Ss;w

The total drag on the SES is defined in terms of the components as RT ¼ Rh þ Ra þ RC þ RS

ð9Þ

In this paper both numerical methods will use the identical procedure for the prediction of the air drag and seal drag. The focus of the analysis in the paper is on the differences in the prediction of hydrodynamic and cushion resistance. The air drag is modeled as Ra ¼ 12ρa U 2a Aref C D

ð10Þ

where ρa is the mass density of air, Ua is the air speed, Aref is the reference area, and CD ¼0.826 is the drag coefficient. The reference area is the cross-sectional area of the vessel above the calmwater plane.

378

K.J. Maki et al. / Ocean Engineering 72 (2013) 375–385

2.2. Linearized theory

2.3. RANS method

The method based on the linearized theory is a product of many decades of research on high speed vessels, and in particular on vessels that use an air-cushion and multiple hulls to maximize performance. In its present form, the combination of the hydrodynamic and cushion drag is computed as the sum of three terms:

The second numerical tool is the finite-volume solver named χnavis (Di Mascio et al., 2007, 2009). This program calculates numerical solutions to the Reynolds-Averaged Navier–Stokes (RANS) equations. The free-surface position is calculated by using a single-phase level set method to satisfy the kinematic freesurface condition. The single-phase level set method only requires solution of the water phase. On the free surface, the pressure is specified by application of the dynamic viscous free-surface boundary condition. In this work the pressure is determined according to Eq. (5). The pressure variable in the field is determined by an artificial compressibility approach. The steady RANS equations are used in this work, although χnavis can solve the unsteady RANS equations. The fluid domain is discretized with overlapping block-structured grids. The program is written to take advantage of parallel computer architecture, and past experience has shown the parallel efficiency to be high over a wide range of problem type and size. The RANS method calculates the hydrodynamic drag by an integral over the wetted ship hull surface (Eq. (2)): Z Rh ¼ i  ½pn þ μð∇U þ ∇U > Þ  n dS ð12Þ

Rh þ RC ¼ f W RW þ RH þ f F RF

ð11Þ

The wave resistance, f W RW , is computed from a far-field integral of the wave elevation from the potential that is due to the sidehulls and air cushion. The potential satisfies linearized free-surface conditions on the calm-water plane, and a linearized bodyboundary condition on the center plane of each sidehull. Although the potential satisfies linearized conditions, the far-field wave integral accounts for the nonlinear interaction between the hull and the cushion-wave systems through the presence of the square of the wave-amplitude term in the integral. The frictional resistance is denoted f F RF . This component is modeled using an estimate of the wetted surface of the sidehulls S, together with the resistance predicted by a flat-plate friction line RF. The wetted surface that is used in the frictional resistance calculation is the atrest value. This is justified on the basis that it is consistent with linearized theory. A fortuitous aspect of the physics is that, as the speed changes, the vessel tends to sink and trim in sympathy with the changing local wave profile. Lastly, the symbols fW and fF in Eq. (11) are form factors that can be used to account for modeling errors on the wave and friction terms respectively. In Eq. (11) the cushion drag is computed through the contribution of the air cushion on the far-field wave system. The far-field calculation is equal to the near-field integral in Eq. (4). This is confirmed by considering the following connections. First, the near-field integral is equal to the integral of the cushion pressure times the wave slope over the region of nonzero cushion pressure in an inviscid fluid. This comes from conservation of momentum. Second, the assumption of a closed cushion of constant pressure relates the integral over the hull with that over the free surface (see Eq. (13)). The free-surface elevation is due to the interaction of both the sidehulls and the cushion. This means that the wave resistance of the cushion includes the influence of the sidehulls, just as the wave resistance of the sidehulls includes the influence of the air cushion. The transom stern of a sidehull is subject to ventilation . This is the process of unwetting of the transom, and in general the transom becomes less wetted as the speed increases. While the wave resistance is related to the near-field integral of dynamic pressure over the hull, the ventilation of the transom has a hydrostatic effect on the drag. In the case of a perfectly flat free surface and fully wetted transom, the integral of hydrostatic pressure has no contribution to the force in the streamwise direction. If the transom is not fully wetted, an imbalance in the x-component of the hydrostatic force arises and is called the transom-stern drag, RH. The subject of transom ventilation and its impact on the drag of vessels has been studied intensely in the past by many authors (for example see list of references in Maki, 2006). The formulation in Doctors (2003) is used in this paper. The linearized-theory method can be executed on a wide range of computational architectures, ranging from personal laptop computers to high-performance parallel computing platforms. For the results shown in this paper, the computer program is run as a serial process on a 2012 Apple MacBook Pro laptop with a 2.7 GHz Intel Core i7 processor. Each simulation requires approximately 0.1 s of CPU time. This time applies to a discretization of the sidehulls into 60 panels longitudinally and 20 panels vertically.

Sh;w

In this equation, p is the fluid pressure, μ is the molecular viscosity, and U is the Reynolds-averaged velocity vector. The one-equation Spalart–Allmaras turbulence model is used to calculate the eddy viscosity (which is nonzero outside of the viscous sublayer). The form of the model used in this paper requires that the computational grid resolves the viscous sublayer and the dimensionless near-wall spacing should satisfy the requirement of yþ o 1 everywhere on the hull surface. The cushion resistance is naturally expressed in terms of an integral over the ship hull and seals that are in contact with the increased pressure in the cushion (Eq. (4)). The accurate evaluation of this component of resistance requires specification of the hull and seal geometry. The hydrodynamic solver finds the nonlinear position of the free-surface, and it is preferred to express the cushion drag in terms of the hydrodynamic solution. If the mass of air in the cushion is in equilibrium, then the external force on the air cushion must be zero. This fact allows one to take the total integral of the surface force on the cushion and express it in terms of the portion of the bounding surface that is hull and seal, and the remainder that is free surface: Z Z F¼0¼ pn dS þ pn dS ð13Þ Sh;c þSs;c

Sfs;c

Note that the viscous stress along the free-surface boundary is ignored in this statement because it is assumed negligible compared to the pressure stress. Eq. (4) can be expressed as Z Z pnx dS ¼ pnx dS ð14Þ RC ¼ Sh;c þSs;c

Sfs;c

The free-surface elevation is denoted as η, and the normal cast in terms of the free-surface elevation allows for the cushion drag expression to be written (with no approximation) as Z RC ¼ pηx dx dy ð15Þ Sfs;c

The computational domain is discretized by means of an overlapped multiblock grid. The mesh around the model consists of about 4.7 million cells distributed in 71 patched and overlapped blocks. Due to the symmetry about the centerplane only one hull is discretized and symmetry boundary conditions are enforced on the plane y¼0. Overlapping grids capabilities are exploited to attain high quality meshes and for refinement purposes. An

K.J. Maki et al. / Ocean Engineering 72 (2013) 375–385

379

Fig. 1. Computational mesh, frontal and rear view (note: only the hydrodynamic surfaces are shown).

Table 1 Overlapping grid: resolution. Region

No. blocks

Volumes

Background Hull Outer and wake Inner Total

16 32 10 13 71

279,552 1,716,736 1,794,048 873,984 4,664,320

overview of the mesh around the catamaran is given in Fig. 1; a summary of the grid resolution is given in Table 1. The background grid, which covers the whole computational domain, is composed by 16 Cartesian blocks with a total of about 280,000 cells. The boundary conditions enforced on the blocks of the background grid are: inflow on the front face, outflow on the lateral, on the bottom and on the rear faces, symmetry on y¼0 and extrapolation at the top. The physical domain close to the hull surface is discretized by means of 20 body-fitted patched and overlapped blocks, with a total of about 1.7 million cells. Cells are clustered near the hull surface, so that the spacing normal to the wall is less than one wall unit (in the present simulations no wall functions have been used); moreover, no less than 30 cells are within the boundary layer. Cartesian blocks are used to increase the resolution around the water level both in the inner and wake regions (about 1.8 million cells) and in the outer regions (0.9 million cells). The numerical solutions were computed by means of a Full Multi Grid-Full Approximation Scheme (FMG-FAS) (see Favini et al., 1996), with four grid levels, each obtained from the next finer by removing every other grid point; the three finest grids (for which the refinement ratio is 2) have been used for verification and validation purposes. In the FMG-FAS approximation procedure, the solution is computed on the coarsest grid level first; then it is approximated on the next finer grid and the solution is iterated by exploiting all the coarser grid levels available with a V-Cycle. The process is repeated up to the finest grid level. On each level, the iterative procedure is conducted until the L2-norm of the residuals drops by four orders of magnitude, and the variation of the forces and moment is at most in the fourth significant digit. A verification procedure is conducted on for each simulation. The observed order of convergence varied between 1.9 and 2.7, which is near the theoretical value of 2. Indeed, although some aspects of the discretization such as the interpolation for the convective terms are third order, the time integration, the discretization of the viscous terms, and the surface integral for the evaluation of fluxes are of second order, therefore the overall accuracy of the scheme is second order. The Richardsonextrapolation-based error estimate yields a numerical uncertainty for the total drag based on grid convergence that is less than 4% for all cases. Iterative and other source of numerical uncertainty are much smaller.

Fig. 2. SES geometry.

The RANS simulations are run on a distributed-memory cluster. Each node has two quad core processors (INTEL Xeon X5462 at 2.8 GHz) with 8 GB of DDR3 RAM at 800 MHz. Nodes are linked by a standard GigaBit ethernet switch. Each simulation is run on a single node (i.e. 8 cores), and requires approximately 13 h of CPU time on the finest mesh. The attitude for the catamaran is fixed; in particular the sinkage was prescribed at the dynamic position taken from the measurements reported in Table 3, whereas the trim is kept fixed to zero. This allowed to perform steady simulations with a considerable reduction in computational expense.

2.4. Experimental data set The focus of this paper is on the different aspects of the flow that comprise the total drag on an SES that are predicted by two very different numerical methods. A great deal is learnt from the comparison of the two numerical results, but we also include comparison with experiment in order to assess errors that may be common to both the numerical tools. A data set based on experimental measurements of calm-water steady forward speed operation of an SES is used. The model was tested at two displacements, and a large range of speeds from Froude pffiffiffiffiffinumber of 0.18 to 0.92. The Froude number is defined as F ¼ U= gL, where U is the speed of the craft, L its length, and g the acceleration due to gravity. The data set has been the source of many other validation studies, including Doctors and McKesson (2006) and Maki et al. (2009). Error estimates are not available for the experimental data. The geometry can be seen as a wire-frame drawing in Fig. 2, and the principal particulars are shown in Table 2. The model was built to a scale factor of 1 to 17.5. The model test run conditions are summarized in Table 3. The model was free to sink and trim, and the cushion pressure varied depending on the speed and displacement.

380

K.J. Maki et al. / Ocean Engineering 72 (2013) 375–385

Table 2 SES particulars.

Table 3 Run conditions. Re

T/L

pC =ρgL

Full displacement 1 0.18 2 0.28 3 0.33 4 0.37 5 0.40 6 0.55 7 0.73 8 0.92

5.07  106 7.64  106 9.19  106 1.02  107 1.13  107 1.53  107 2.05  107 2.56  107

0.0338 0.0337 0.0364 0.0403 0.0364 0.0262 0.0187 0.0186

0.0118 0.0119 0.0108 0.0092 0.0108 0.0147 0.0173 0.0173

Light displacement 9 0.18 10 0.27 11 0.33 12 0.37 13 0.40 14 0.55 15 0.73 16 0.92

5.09  106 7.65  106 9.19  106 1.02  107 1.13  107 1.54  107 2.05  107 2.56  107

0.0307 0.0306 0.0303 0.0296 0.0204 0.0161 0.0153 0.0151

0.0106 0.0107 0.0108 0.0111 0.0143 0.0157 0.0160 0.0160

Item

Symbol

Value

Unit

Run

Displacement mass (light) Displacement mass (full) Waterline length (light) Draft (20% hull-lift ratio) Waterline beam Waterline beam (sidehull) Cushion length Cushion beam Cushion start station

ΔL ΔF L T B BS LC BC x1

1991 2239 80.14 1.374 24.5 3.546 72.00 17.50 2.00

t t m m m m m m m

3. Results In this section results from each of the computer programs are compared to the experimental measurements. The experimentally measured value of the cushion pressure is used as required by each numerical method.

Fr

3.1. Dynamic wetted surface A basic statement that expresses the motivating principle for selecting an SES is that it reduces drag at high speed by reducing the wetted surface through action of the air cushion. The first comparison that is made is the wetted surface that is predicted by the fully nonlinear method. Fig. 3 shows the dimensionless wetted surface S=L2 , for each case that was experimentally tested, as a function of Froude number. In this figure, three groups of symbols are present. The first, denoted ‘static no cushion’, is the hydrostatic wetted surface below the calm-water plane of the hulls, with the vessel in the dynamic equilibrium position (or the at-speed position of the model that was experimentally measured). Of course the data for this scenario have little meaning because the air cushion is necessary for the vessel to be situated in dynamic equilibrium, but the data do serve as a basis to observe how much the wetted surface is reduced when the air cushion is accounted for. The second group of data is denoted ‘static with cushion’. This is the hydrostatic wetted surface below the calm-water plane, with an adjustment for the air cushion. The reduction in wetted surface is estimated to be an area found by length of the cushion and the value of the pressure in the air cushion converted to units of length. It can be seen in Fig. 3 that the reduction in wetted surface is substantial due to the air cushion, at least according to the approximation that we have used. The final group of data represents the numerical predictions of the χnavis program. It is very interesting to see the strong agreement between the χnavis and the static with-cushion data, at least for speeds less than Froude number of 0.6. For the cases corresponding to speed greater than Froude number of 0.7, the nonlinear prediction shows an increase of wetted surface over the static prediction. Finally, we remark that some of the wave features predicted by χnavis may be unsteady. The predictions are from the steady solver, and perhaps wetted surface is truly an unsteady quantity when unsteady breaking waves are accounted for. 3.2. Free-surface elevation In this section we examine the predicted free-surface elevation. Fig. 4 shows the elevation on the centerline of the vessel from each method for the full displacement Runs 1–8. Note that the transom is at x=L ¼ 0, and bow at x=L ¼ 1. Run 1 has a speed of Froude number of 0.18. At this low speed, the waves that are generated by the hull and cushion are short and the wave resistance is expected to be small compared to the total

value. The presence of the air-cushion is clearly visible for this run, and linear theory predicts larger amplitude waves than those from the RANS method. While viscosity and surface tension may influence waves of this length, aspects that are omitted in the linear theory result, the RANS discretization is set for resolution of the more meaningful long waves associated with larger Froude numbers, and the poor resolution of the short waves may lead to an inaccurate prediction of the wave amplitude at this speed. Examination of the other seven runs in this figure shows very good agreement between the two numerical methods over the range of longitudinal coordinate that is in front of midships x=L 4 0:5. Aft this coordinate, the agreement deteriorates. Previous work which focused on the prediction of the free-surface for a two-dimensional pressure patch (Maki et al., 2012) shows near identical results from the same two numerical methods for a wide range of Froude number and value of cushion pressure. Thus it is unlikely that the disagreement seen in Fig. 4 is due to the presence of the air cushion. In order to investigate the difference in free-surface profile aft of midships in more detail, Run 6 is simulated with the cushion pressure set to zero. The body is restrained in the same position as when the cushion pressure has the experimentally measured value. In this way only the wave generated by the sidehull can be studied. Fig. 5 shows the free-surface elevation with the sidehulls and cushion on the left (the exact same data that are shown in Fig. 4), and that due to only the sidehulls on the right. The first difference that is seen when the air-cushion is absent is the reduction of the wave elevation in the vicinity of the hull 0 o x=L o1. Also, the agreement between the two methods is good in this range. Downstream, the agreement is poorer, in particular for the prediction of the large wave crest that is located near x=L  0:75. By comparing the plots on the left and the right it seems as though the sidehulls are mainly responsible for the large crest here. There are several possible explanations for the difference between the two methods, including: the influence of finitedepth effects and the position of the downstream boundary used in the RANS simulations, numerical errors in each method due to iteration and discretization, free-surface nonlinearity, viscosity and surface tension, and the transom-stern model. The linear theory simulates the geometry of the experimental channel, and the depth-based Froude number for the experiment is approximately 90% of the length-based value. Thus for the highest-speed runs the depth-based Froude number is about

K.J. Maki et al. / Ocean Engineering 72 (2013) 375–385

0.2

S /L2

S /L2

0.2

381

0.1 static - no cushion static - w/cushion RANS - w/cushion

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.0

1.5

1.0

1.5

1.0

1.5

1.0

1.5

F

F

Fig. 3. Wetted surface, left: full displacement, right: light displacement.

Run 1: F = 0.18

0.04

ζ/L

0.02

0

-0.02

-0.02 -0.5

ζ/L

0.5

1.0

1.5

-0.04 -1.0

0.02

0

0

-0.02

-0.02 -0.5

0.0

0.5

1.0

1.5

Run 5: F = 0.40

0.04

-0.04 -1.0

0.02

0

0

-0.02

-0.02 -0.5

0.0

0.5

1.0

1.5

Run 7: F = 0.73

0.04

-0.04 -1.0

0.02

0

0

-0.02

-0.02

-0.04 -1.0

-0.5

0.0

0.5

-0.5

1.0

1.5

-0.04 -1.0

0.5

0.0

0.5

Run 6: F = 0.55

-0.5

0.0

0.5

Run 8: F = 0.92

0.04

0.02

0.0

Run 4: F = 0.37

0.04

0.02

-0.04 -1.0

-0.5

0.04

0.02

-0.04 -1.0

ζ/L

0.0

Run 3: F = 0.33

0.04

ζ/L

0.02

0

-0.04 -1.0

Run 2: F = 0.28

0.04

linear theory RANS

-0.5

0.0

x/L

0.5 x/L

Fig. 4. Free-surface elevation on centerline for full displacement.

0.04 0.02 ζ/L

0.04

linear theory RANS

0

0

-0.02

-0.02

-0.04 -1.0

-0.5

0.0

0.5 x/L

pC = 0

0.02

1.0

1.5

-0.04 -1.0

-0.5

0.0

0.5

1.0

x/L

Fig. 5. Free-surface elevation on centerline for Run 6, left: sidehulls and cushion, right: sidehulls only.

1.5

382

K.J. Maki et al. / Ocean Engineering 72 (2013) 375–385

0.83, which is sufficiently high to have the bottom influence on the free-surface elevation. The bottom boundary of the RANS domain is one ship length. For Run 6, the depth-based Froude number is about 0.5, so for this speed the difference in the free-surface profile cannot be attributed to the differences in the position of the bottom boundary for each numerical method. Furthermore, numerical discretization has been carefully examined for each method. The finite spatial resolution of the RANS method could contribute to inaccurate solution of the smallest length scales of the wave, although the difference in the main features of the flow is probably a result of some other aspect of the two numerical methods. Free-surface nonlinearity could partially explain the differences between the two methods. The steepness of the waves computed by the linear theory defined using the ratio of the trough-to-peak height and the wave length is around 0.08, which is small compared to the approximate maximum value for a steady wave of 0.14. Also, free-surface nonlinearity may cause a shortening of the wave and a taller peak with shallower trough, but not to the extent to explain the difference seen in the free-surface elevation plots. On the other hand, the RANS simulation shows a very complicated free-surface solution behind the vessel, which has characteristics that indicate interaction between the transom wave and the wave system that originates near the bow. This can be seen in the contour plot shown in Fig. 6. This observation of the free-surface complexity aft the vessel is related to the description of the transom stern model that is treated in the following. The omission of viscosity and surface tension in the lineartheory prediction could contribute to the disagreement, although the Reynolds number used in the RANS prediction is relatively high for model scale, and it is unlikely that viscous damping is strong enough to influence the profile so close to the body. It would of course be important for the behavior of the wave in the far-field. Also, surface tension is important for the small scales of the free surface, but again it is unlikely that surface tension is responsible for the differences seen in the main features of the wave field. The next possible explanation for the disagreement in the predicted profile is related to the combination of viscosity and free-surface nonlinearity and is separately addressed in the linear theory by the transom-stern model. The model has been shown to accurately account for the resistance attributed to the transomstern geometry, but a close examination of the predicted freesurface elevation downstream of the transom for an SES has not been done before. Other researchers have looked at this issue closely for a transom stern catamaran, for example the paper by Lugni et al. (2004) provides analysis using experiments and several different numerical tools. The correlation between the experiment and the numerical results based on nonlinear potential flow is very similar to that between the linear-theory and the RANS simulations of the present paper. This suggests that the transom stern model that is necessary when employing inviscid fluid methods may need to be improved if one seeks accurate prediction of the free-surface elevation behind the vessel. 3.3. Hydrodynamic and cushion drag Fig. 7 shows the numerical prediction of the hydrodynamic and cushion drag, together with the experimentally measured total drag. The data are presented in terms of the resistance-to-weight ratio as a function of the Froude number (the symbol W denotes the weight of the vessel). Here it is seen that the data from the two numerical methods are strongly correlated. Also, it can be stated that for this vessel, the hydrodynamic and cushion components of the drag comprise a very large part of the total drag. This is true for the wide range of speeds and for each displacement.

Fig. 6. Free-surface elevation predicted by RANS method for Run 6.

3.4. Pressure drag In order to examine more closely how each tool computes the different components of resistance, we decompose the sum of the hydrodynamic and cushion drag into elements that can be expressed equivalently for each tool. We define the pressure drag for the linear tool as the sum of the wave and hydrostatic terms (Eq. (16)), and the pressure drag for the RANS tool is the integral over the hull and air cushion of only the pressure stress (Eq. (17)). In Eq. (16) the wave-resistance factor fW is set to the value of one: linear theory :

Rp ¼ f W RW þ RH Z

RANS :

ð16Þ

Z

Rp ¼  Sh;w

pnx dS þ

Sfs;c

pηx dx dy

|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

hull pressure

cushion pressure

ð17Þ

Fig. 8 shows the comparison of the specific pressure resistance as predicted by each numerical method. It is seen that the correlation between the two methods for all speeds and both displacements is very good, and particularly so at speeds less than a Froude number of 0.5. The linear method computes the wave drag through the farfield integration of the composite wave field due to the two sidehulls and the air cushion. The RANS method uses a nearfield evaluation of the hydrodynamic and cushion drag. Fig. 9 shows the total specific pressure resistance computed by χnavis, and the pressure resistance due to the hull and the cushion separately. For both values of displacement, the pressure drag is dominated by the cushion component for cases when the speed is greater than Froude number of 0.5. At lower speed, the full displacement data show that the hull pressure component is the largest portion of the total pressure drag, while for the light displacement, the hull and cushion components are of similar magnitude. 3.5. Frictional drag The last element of resistance is the frictional drag. This is defined in the linear method by an estimate of the wetted surface together with a friction line RF, seen in Eq. (18). In this work the ITTC-57 line is used and the form factor is fR ¼1.18. In the RANS method, the frictional drag is computed by an integral of the shear stress over the wetted portion of the hull, Equation (19). linear theory :

Rf ¼ f F RF

Z RANS :

Rf ¼

Sh;w

μe ð∇U þ ∇U > Þ  n dS

ð18Þ ð19Þ

Fig. 10 shows the prediction of the specific-frictional resistance from each numerical tool. The agreement between the two sets of data is very good, especially for the low-speed conditions of Froude number less than 0.4. At higher speeds, the small

K.J. Maki et al. / Ocean Engineering 72 (2013) 375–385

(Rh +RC ) / W

0.08

0.10

experimentRT / W linear theory RANS

0.08 (Rh +RC ) / W

0.10

0.06 0.04 0.02 0.00 0.0

383

0.06 0.04 0.02

0.2

0.4

0.6

0.8

0.00 0.0

1.0

0.2

0.4

F

0.6

0.8

1.0

F

Fig. 7. Numerical prediction of Rh þ RC , experimental data are the total-drag-to-weight ratio, left: full displacement, right: light displacement.

0.05

0.04

0.03

Rp / W

Rp / W

0.04

0.05

linear theory RANS

0.02

0.02 0.01

0.01 0.00 0.0

0.03

0.2

0.4

0.6

0.8

0.00 0.0

1.0

0.2

0.4

F

0.6

0.8

1.0

0.8

1.0

F

Fig. 8. Specific pressure resistance, left: full displacement, right: light displacement.

0.05

0.04

0.03

Rp / W

Rp / W

0.04

0.05

total hull cushion

0.02 0.01 0.00 0.0

0.03 0.02 0.01

0.2

0.4

0.6

0.8

0.00 0.0

1.0

0.2

0.4

F

0.6 F

Fig. 9. Specific pressure resistance due to hull and cushion from RANS solution, left: full displacement, right: light displacement.

linear theory RANS

0.03

0.02

Rf / W

Rf / W

0.03

0.01

0.00 0.0

0.02

0.01

0.2

0.4

0.6

0.8

1.0

0.00 0.0

0.2

0.4

0.6

0.8

1.0

F

F

Fig. 10. Specific friction resistance, left: full displacement, right: light displacement.

disagreement may be due to the difference in the predicted wetted surface between the linear and the nonlinear methods. Fig. 3 shows that the nonlinear wetted surface is greater than the linear estimate, and this can explain the differences in the frictional drag that is seen in the data at the higher speeds.

3.6. Total drag The hydrodynamic and cushion drags together form the largest portion of the total drag (see Fig. 7). Although the remaining two components are small, the air and seal drags are important for a

384

K.J. Maki et al. / Ocean Engineering 72 (2013) 375–385

0.10 0.08

0.08

0.06

R/W

R/W

0.10

experiment: total numerical : air numerical : seal

0.04 0.02

0.06 0.04 0.02

0.00 0.0

0.2

0.4

0.6

0.8

0.00 0.0

1.0

0.2

0.4

F

0.6

0.8

1.0

F

Fig. 11. Specific air and seal resistances, left: full displacement, right: light displacement.

0.10

0.08

0.06

RT / W

RT / W

0.08

0.10

experiment linear theory RANS

0.04 0.02 0.00 0.0

0.06 0.04 0.02

0.2

0.4

0.6

0.8

1.0

F

0.00 0.0

0.2

0.4

0.6

0.8

1.0

F

Fig. 12. Specific total resistance, left: full displacement, right: light displacement.

complete prediction of the total drag. The air drag is computed by using the frontal area of the hull structure including the portion between the sidehulls that is covered by the bow seal (Eq. (10)). The seal drag depends on the very complicated interaction between the flexible seal and the breaking bow wave. The seal drag has been the center of many focused research projects and papers in the literature, and the model that is used in this work is that which is presented in Doctors and McKesson (2006). Fig. 11 shows the specific air and seal resistances compared to the total resistance measured in the experiment. It is seen that the modeled seal drag has a larger relative contribution for Froude number less than 0.4. For speeds greater than 0.4 the seal drag is predicted to be very small. The air resistance is similarly a minor contribution to the total resistance, and increases with increasing speed according to the model. Fig. 12 shows the specific-total-resistance ratio predicted by each numerical tool and compared with the experimental measurement. The total resistance includes the seal and air, each of which is computed exactly the same for both numerical predictions. In this figure it is seen that the accuracy of the prediction is improved by including these two components (compare Fig. 12 with Fig. 7). Both methods tend to slightly under predict the total drag throughout the speed range and for both values of displacement.

4. Conclusions In this paper two different numerical methods are used to study the resistance developed by an SES that is advancing steadily through calm water. The first method solves for a velocity potential that satisfies linearized free-surface and body boundary conditions. The second solves the steady RANS equations and uses a single-phase level set method to track the nonlinear free surface. A test case is chosen of an SES which has high-Reynolds number experimental resistance data. Numerical results are shown for the

nonlinear wetted surface, the free-surface elevation, and the different components of resistance of an SES. In general the linear and RANS methods show strong agreement for the hydrodynamic and cushion drag. The computational requirements for each method are vastly different. The linearized program requires less than a second per speed, and the grid is generated automatically. The RANS method requires hundreds of CPU-hours for each simulation and the grid generation task takes days of effort. The linearized method certainly is attractive due to its significantly smaller computational effort, especially for design purposes. On the other hand, we must emphasize an important role of the RANS method. The fully nonlinear method has been used here as a high-fidelity source of information to validate the individual elements of the fluid solution of the linearized method. This has permitted us to understand more deeply the ability of the linearized method to accurately predict the total drag on an SES. The dynamic wetted surface was shown to deviate from the hydrostatically predicted value only for speeds greater than Froude number of 0.7. This is due to the large wave crest that the vessel generates, and the dynamic rise in elevation causes the increase in wetted surface on the insides of the sidehulls. The hydrodynamic and cushion drag is shown to be dominated by the cushion component for speed greater than Froude number of 0.5. The frictional drag computed with a friction line is very close to the RANS prediction, although nonlinear effects on the dynamic wetted surface may impose a restriction on the accuracy of the friction-line method in cases more extreme than those tested herein. The hydrodynamic and cushion resistance is studied in detail in this paper using two different numerical methods of varying approximation to boundary conditions of the mathematical equations governing fluid flow around the vessel. The disagreement between the two methods for the cushion and hydrodynamic drag, and for the frictional drag, is much less than the difference between either of the numerical predictions and the total drag

K.J. Maki et al. / Ocean Engineering 72 (2013) 375–385

that was experimentally measured. The air drag and seal drag can be the components responsible for the difference between the numerical and the experimental data. The seal drag in particular is recommended as the focus of future research. The complicated flow around the seal should be studied both as operating independent of the sidehulls and together as the integrated system.

Acknowledgments The authors would like to gratefully acknowledge the support of the US Office of Naval Research through the Grant titled “CalmWater Performance Prediction for the Design of Advanced Naval Vessels”, Award # N000141010303, under the direction of Kelly Cooper. The authors would also like to thank INSEAN for the support of the first author as a visiting professor. Finally, the experimental data that were presented in this paper were generously shared with the authors by JJMA (now Alion Science and Technology), and contributed greatly to the comparisons between the numerical results.

References Bhushan, S., Stern, F., Doctors, L.J., 2011. Verification and validation of urans wave resistance for air cushion vehicles, and comparison with linear theory. Journal of Ship Research 55 (December (4)), 1–19. Connell, B.S., Milewski, W.M., Goldman, B., Kring, D.C., 2011. Single and multi-body surface effect ship simulation for t-craft design evaluation. In: 11th International Conference on Fast Sea Transportation FAST2011, Honolulu, Hawaii. Di Mascio, A., Broglia, R., Muscari, R., 2007. On the application of the single-phase level set method to naval hydrodynamic flows. Computers & Fluids 36, 868–886.

385

Di Mascio, A., Broglia, R., Muscari, R., 2009. Prediction of hydrodynamic coefficients of ship hulls by high-order Godunov-type methods. Journal of Marine Science and Technology 14, 19–29. Doctors, L.J., 2003. Hydrodynamics of the flow behind a transom stern. In: TwentyNinth Israel Conference on Mechanical Engineering, Haifa, Israel, May. Doctors, L.J., 2006. A numerical study of the resistance of transom-stern monohulls. In: Fifth International Conference on High-Performance Marine Vehicles (HIPER'06), Launceston, Tasmania, November, pp. 1–14. Doctors, L.J., 2012. Near-field hydrodynamics of a surface-effect ship. Journal of Ship Research 56 (September (3)), 183–196. Doctors, L.J., McKesson, C., 2006. The resistance components of a surface effect ship. In: The Twenty-Sixth Symposium on Naval Hydrodynamics, Rome, Italy, September. Doctors, L.J., Sharma, S.D., 1972. The wave resistance of an air-cushion vehicle in steady and accelerated motion. Journal of Ship Research 16 (December (4)), 248–260. Doctors, L.J., Tregde, V., Jiang, C., McKesson, C., 2005. Optimization of a split-cushion surface effect ship. In: Eighth International Conference on Fast Sea Transportation (FAST '05), Saint Petersburg, Russia, June. Donnelly, D.J., Neu, W.L., 2011. Numerical simulation of flow about a surface-effect ship. In: 11th International Conference on Fast Sea Transportation FAST2011, Honolulu, Hawaii. Favini, B., Broglia, R., Di Mascio, A., 1996. Multi-grid acceleration of second order ENO schemes from low subsonic to high supersonic flows. International Journal for Numerical Methods in Fluids 23, 589–606. Ford, A., 1964. Captured air bubble over-water vehicle concept. Naval Engineers Journal 76 (April (2)), 223–230. Kring, D.C., Parish, M.K., Milewski, W.M., Connell, B.S., 2011. Simulation of maneuvering in waves for a high-speed surface effect ship. In: 11th International Conference on Fast Sea Transportation FAST2011, Honolulu, Hawaii. Lugni, C., Colagrossi, A., Landrini, M., Faltinsen, O., 2004. Experimental and numerical study of semi-displacement mono-hull and catamaran in calm water and incident waves. In: Twenty-Fifth Symposium on Naval Hydrodynamics, St. John's, Newfoundland and Labrador, Canada, August. Maki, K., 2006. Transom Stern Hydrodynamics. Ph.D. Thesis, University of Michigan. Maki, K., Broglia, R., Doctors, L.J., Di Mascio, A., 2012. Nonlinear wave resistance of a two-dimensional pressure patch moving on a free surface. Ocean Engineering 39, 62–71. Maki, K., Doctors, L., Broglia, R., McKesson, C., DiMascio, A., 2009. Calm-water resistance prediction of a surface-effect ship. In: Proceedings of International Conference on Air Cushion Vehicles & Surface Effect Craft, Royal Institution of Naval Architects, London, England, November 17–18, p. 12.