J Mar Sci Teclinol (2015) 20:278-291
D O I 10.1007/S00773-014-0282-1 CrossMark
O R I G I N A L A R T I C L E
Viscous and potential forces on an advancing surface-piercing flat
plate with a fixed drift angle
Babak Ommani • Odd M . Faltinsen
Received: 10 February 2014/Accepted: 24 August 2014/Published online: 16 September 2014 © J A S N A O E 2014
Abstract Transverse force and yaw moment acting on a surface-piercing fiat plate with yaw angle and forward speed are studied. Following the Froude hypothesis, the problem is decomposed into a tail-separated forward flow and a bottom-tip-separated cross-flow parts. The objective is to quantify the role of different components in the resulted force and moment acting on the plate. A 3D potential flow code using distribution of Rankine sources and dipoles is used to solve the linear potential flow problem. The free-suiface boundary condition is linearized based on the undisturbed flow velocity (Neumann-Kelvin linearization). The plate's trailing-edge flow separation is modeled using a dipole distribution behind the plate on a linearized vortex sheet. The force and moment due to the cross-flow separation from the plate's bottom tip are cal-culated by means of a slender body cross-flow method with a rigid surface boundary condition. Hence the free-surface effects are confined in the forward flow problem and neglected in the cross-fiow problem. Simulafions are carried out for different aspect ratios and drift angles. The plate's thickness is taken into account. Convergence and sensitivity studies are performed carefully. Results are compared with previous experimental and numerical studies. The importance of the second-order forces and validity of the linear assumptions are touched upon. The overall agreement of the results is acceptable. The relative importance of the two viscous and potential components
B. Ommani ( E l )
M A R I N T E K , Otto Nielsens vei 10, 7052 Trondheim, Norway e-mail: babak.ommani@marintek.sintef.no
O. M . Faltinsen
Department o f Marine Technology, N T N U , Centre f o r Autonomous Marine Operations and Systems ( A M O S ) , 7491 Trondheim, Norway
are studied. It is shown that the decomposed problem could follow the experimental results up to a relatively high Froude number.
Keywords Rankine Panel Method • Slender Body • Cross-Flow • 2D-^t
1 Introduction
The hydrodynamic forces acting on a ship with forward speed and drift angle are of interest in the ship maneuvering and dynamic stability calculations. Especially for high-speed vessels, which may suffer from dynamic instabilities such as calm water broaching, this type of calculations is important. In the recent years, there have been major the-oretical and computational advances in the methods for predicting the ship maneuverabiUty and dynamic stability. However, especially for high-speed semi-displacements, experimental techniques still play the major role.
To assess the vessel's dynamic stability, it is important to have an accurate hydrodynamic model which can pro-vide an estimation of the forces and moments acting on the vessel. However, due to complexities of the fiow and the ship geometry, developing such a model is not an easy task. Simplifications, such as assuming an inviscid-iiTOtational flow, linearization of the flow and the boundary conditions, and using simplified geometries were utilized in many past sttidies to overcome these problems.
The problem of a surface piercing, finite aspect ratio, fiat plate, with forward speed and drift angle has been studied experimentally and numerically before as a simplified alternative.
Chapman in [3] used a 2D+t method based on a slender body assumption with in potential flow theory to calculate
J Mar Sci Teclinol (2015) 20:278-291 279
the transverse force and yaw moment. He investigated the influence of linear, second-order and nonlinear free-surface boundary conditions and concluded that, while nonlinear free-surface boundary condition can influence free-surface elevation, it does not change side force and yaw moment significandy.
Maniar et al. [12] and Xii [19] solved the potential flow problem in three dimensions using the slender body assumption and Kelvin-Havelock Green function. They pointed out the incompatibility between the pressure Kutta condition and the linearized free-surface boundary condi-tion at the meeting point of the trailing edge and the free surface. They showed that the effects of this incompati-bility are local and do not change the global solution.
Landrini and Campana [11] used the double-body fine-arization to formulate the problem. They investigated the infiuence of the bottom-tip vortex on the side force and yaw moment. They showed that the bottom-tip separation (keel vortex shedding) plays an important role in the transverse force and yaw moment especially for lower draft-to-length ratios. They also demonstrated that, for their studied case, the linearized vortex sheets are sufficient to model the forces and there is no need for nonlinear vortex geometry. Zhu and Faltinsen [20] used a linear 3D Rankine panel method to solve the potential flow problem. They took the plate's thickness into account and neglected the tip vortex's effects.
In the present work, the problem was decomposed into a forward flow problem, considering the free-surface effects and trailing flow separation, and a cross-flow problem, considering the flow separation from the bottom-tip of the plate and neglecting the free-surface effects.
The method by Zhu and Faltinsen [20] was further developed for solving the forward flow problem. The transverse force and yaw moment acting on surface-piercing, finite aspect ratio fiat plate with forward speed and drift angle were studied. A potential irrotational flow was assumed and linearized using Neumann-Kelvin line-arization. The second Green identity was used by assuming a linear distribution of Rankine sources and dipoles over the boundaries to formulate the boundary integral equation system. The trailing-edge flow separation was considered by enforcing equality of pressure at the trailing edge. The trailing wake was modeled using a dipole singularity dis-tribution over a linearized vortex sheet. A fully 3D potential flow problem with both divergent and transverse wave systems and without any body slenderness assump-tion was solved.
In addition, the transverse force and yaw moment due to the cross-flow separation from the plate's bottom tip were estimated by means of a slender body cross-flow model. In analogy to the 2D-ft method, the steady 3D problem transformed into a transient 2D problem with rigid
surface condition. Doing so, the interactions between free-surface and cross-flow vortex, as well as, flow three-dimensionality were neglected. The objective was to quantify the importance of these interactions by evaluating the capability of this decomposed model in following the experimental results.
2 Theory 2.1 Formulation
A plate with finite aspect ratio and smafl drift angle was considered, advancing in negative X direction (Fig. 1). Two Cartesian coordinate systems OXYZ and Oxyz were defined. The XY plane of the Earth-fixed global coordinate system OXYZ corresponds to the undisturbed free surface. The local inertial coordinate system Oxyz moves with the plate's forward speed, with the Oxyz system being parallel to the OXYZ coordinate system, and the z-axis going through the plate's geometrical center. A third body-fitted coordinate system Ox'y'z' was obtained by rotating Oxyz around the z-axis with the drift angle a. Therefore, the obtained x' and axes are along and perpendicular to the plate's plane, respectively. The forces are presented in this coordinate system. The plate's dimensions and configura-tion are presented in Fig. 2. The plate's aspect ratio was defined as A = / / / L , where L is the plate's length and H is the mean draft.
The flow was assumed to be irrotational and the water to be incompressible. Therefore, a total velocity potential «ï exists which satisfies the Laplace equation (Eq. 1) in the water domain, together with the body, free-surface kine-matic, and dynamic boundary conditions, which are shown in Eqs. 2, 3 and 4, respectively.
V'(D = 0 a(D(x, 0 _
8/!
VB • n where x G S b 8/ 8A- dx dy dy dz on 8 0 1 -di + 2 8 0soy /80
9^7
. z = i:{x,y,t) (1) (2) ) (3) (4) where C is the free-surface elevation, is the body velocity and n is the normal vector to the body surface, which points inside the water domain. The radiation boundary condition which ensures solution uniqueness must also be satisfied. Infinite water depth was assumed in280 J Mar Sci Technol (2015) 20:278-291 F i g . 1 Schematic view o f the
problem and definition of the coordinate systems, a plate with d r i f t angle advancing in the negative X direclion, FS free surface. Body plate, VS vortex sheet, a drift angle
F i g . 2 Definition o f plate's dimensions, a plate with constant draft, b plate w i t h variable draft, y plates tapered angle
(b)
/ \
H :
\
the present work. Therefore, the bottom boundary condi-tion was simplified as shown in Eq. 5 below,
5cl)(x,0
0 as (5)
Only the steady motion was studied in the present work. Therefore, the problem was formulated independent of time.
Since the focus of the present work is on high Froude numbers of slender bodies, the Neumann-Kelvin lineari-zation is applied. In this approach, the undisturbed in-fiow potential is chosen as the base potential for the lineariza-tion, and the infiuence due to the presence of the body is assumed to be small. This is more consistent for high Froude numbers, comparing to the double-body lineariza-tion, which is more suitable for low Froude numbers and blunt ship forms (see, i.e., [15]). Then, the total velocity potential function may be described as shown in Eq. 6.
<l)(x) = to+(^(x) (6)
where IJ is the body's constant forward speed and (/) is the perturbation velocity potential, which must also satisfy the
Laplace equation. The expressions for linearized body boundary condition, combined free-surface boundary con-dition, and pressure are presented in Eqs. 7, 8 and 9, respectively. = - [ / i n 3;; a<^(x) , u'-^m^) Qz 8 8x2 = 0 on z = 0 p - p , = -pU^ - ^p(V(/) • V(/)) - pgz (V) (8) (9)
where pa is the atmospheric pressure. Equation 9 could be separated into a linear part, pi = -pU^ - pgz, and a nontinear part pj = - 5P(V(/) • Vcp). It must be noted that based on the linear assumptions made here, p2 must be close to zero, otherwise, the chosen free-suiface boundary condition is not consistent. This second-order pressure, which is calculated based on the first-order velocity potential, could indicate to what degree the linearization assumption is coiTect. The forces and moments acting on a body are obtained from pressure integration as follows.
J Mar Sci Technol (2015) 20:278-291 281 M -np ds - ( ( r - i - c ) X n)/7ds (10) ( H )
where r is the position vector of the points on the body surface and I'c is the position vector of the center. In this case, i-c points to the geometrical center of the plate in the free-surface plane. Substituting the pressure from Eq. 9 into Eq. 10, and keeping terms up to second order, it is possible to separate the first- and second-order contribution to the force as.
F(2) = y / :: - I I ;p(V./>-Vr/>) dl dz (12) d l d z + / n(-pgC^ ) d l wl (13)
Here, the integral over the plate is separated into two integrals, one from bottom up to the mean free surface, and the other from there to the actual wave elevation. More-over, 11'/ stands for the mean water-line curve. The second term in the second-order force was obtained by assuming small variations in the normal vector in z-direction close to the water line. It must be noted that this is not the complete second-order force, since the complete second-order force requires solution for the second-order velocity potential. Equation 13 is only the second-order force calculated from the first-order velocity potential. As second-order pressure, this second-order force indicates the validity of the linear assumptions and must be close to zero. The transverse force and yaw moment were non-dimensionalized as follows,
(14)
(15)
2.2 Trailing-edge separation
It is well known that due to fiuid viscosity the fiow cannot turn around a sharp corner and, therefore, leaves the trail-ing edge of a lifttrail-ing surface tangentially. This physical constraint, known as Kutta condition, could be ensured by enforcing the velocity potential to be continuous from the body into the fluid. As a result, the pressure is finite at the separation point. Since in the presence of a drift (yaw) angle the velocity potentials on the two sides of the body
are different, it requires the jump in the velocity potential to be continuous in the longitudinal direction. Morino and Kao [14] achieved this by introducing a flat surface of singularities into the fluid and relating the strength of sin-gularities to the circulation around tbe body. Although, in reality the wake surface is not flat, this type of linearization has been shown to be sufficiently accurate for small drift angles [14].
From the linear part of the Bernoulli equation Eq. 9, equality of the pressure at the trailing edge and along the vortex sheet leads to.
d{y-cp-)
= 0 (16)
This means that the velocity potential jump, generated at the trailing edge, does not change along the vortex sheet i n the x-direction. Therefore, the jump in the velocity poten-tial created at the body propagates with the flow downstream.
As pointed out by Maniar et al. [12] and Xtl [19], there is an inconsistency between the pressure Kutta condition and the linearized free-surface boundary condition at the meeting point of the trailing edge and the free surface. As observed in experiments before (for instance [1] and [12]), the free-surface elevations at the two sides of the plate are different due to the drift angle. This difference in the free-suiface elevations continues along the plate and into the fluid domain. In other words, while pressure is continuous at the trailing edge below the free surface, there is a dis-continuity in the free-surface elevation, and consequently the pressure, at the meeting point of the trailing edge and the free surface. This means that a sharp change in the free-surface elevation exists immediately after the plate, which collapses a short distance downstream. On the other hand, the free-surface elevation is assumed to be a single-valued function in the linear theory, which cannot account for such a sharp change.
To properly capture this nonlinear phenomenon, a nonlinear formulation seems to be necessary. However, as mentioned by Xii [19], and showed later on in this study, the eiTor due to this type of linearization is limited to a small area around the tail of the plate and has no significant infiuence on the induced forces. To get around this inconsistency, it is important to ease the Kutta condition and allow for pressure inequality at the meeting point of the trailing edge and the free surface.
2.3 Bottom tip separation
The separation from the bottom tip of the plate contributes to the transverse force and moment as pointed out by Chapman [3], and investigated later by Landrini and Campana [11]. They introduced a vortex sheet, starting
282 J Mar Sci Teclinol (2015) 20:278-291
from the plate's tip, to take into account the cross-flow tip separation effects. In the present method, a 2D-|-t-type approach based on the slender body assumption and using a zero Froude number free-surface boundary condition was used to account for this influence. There is obviously an inconsistency related to the potential flow method. The objective here is to decompose the problem into forward and flow parts and make an estimation of the cross-flow separation contribution to the transverse force and yaw moment.
Following the method presented by Faltinsen [4J, an Earth-fixed plane ( I I ) was defined nonnal to the plate as shown in Fig. 3a. The modon of the plate is viewed from this stationary plane. Due to the plate's forward speed and drift angle, the plate's section appears to move through the stadonary plane with velocity H] and slides side way with velocity vi (Fig. 3b). A right-handed Cartesian body-fixed coordinate, system Oxiym was defined with xo'rplane con-esponding to the mean free surface, xi along the plate center fine, and )']Zi-plane parallel to the Il-plane with Z] pointing upward. The origin of this coordinate system is attached to the intersection of the plate's leading edge and the free surface. From this frame of reference, there is an incident cross-flow velocity v i . Looking at the problem from the Earth-fixed plane, it transforms into a transient problem of a uniform cross-flow passing a 2D plate in presence of the free surface. Using a zero Froude number assumption, the free-surface boundary condition could be replaced by zero vertical velocity, which is the wall boundary condition (Fig. 4).
The drag force acting on a plate in infinite fiuid, or attached to a wall, is a well-studied fiuid dynamic prob-lem. However, most of the studies are concerned with the mean and oscillatory drag coefficient after the initial transient period. Fink and Soh [6] studied this problem numerically by a discrete vortex method and showed that the drag force coefficient starts from a large value (around 6) and then decreases rapidly during a short period in the beginning of the simulation, and approaches its mean value. However, due to numerical problems, it is difficult to calculate forces at the beginning of the simulation. Further studies by Faltinsen and Pettersen [5] using a vortex sheet model, and T0nnessen [18] with a finite element solution (FEM) of the laminar Navier-Stokes equation showed that due to these numerical problems, the obtained drag coefficient is unrealistically large at the few first time steps of the calculations. Similar studies were carried out by Koumoutsakos and Shiels [9] using a 2D adaptive numerical method based on the vortex methods in which they observed similar behavior at the start-up of the simulation.
The transient cross-flow drag coefficient (Co(.?)) could be presented as a function of a non-dimensional time
F i g . 3 Coordinate system fixed to the leading edge (.vo'iZi), a plate reaches the stationary plane, b plate is passing through the stationary plane
H
r r r r r r .
Q
<d
F i g . 4 Section o f the plate in cross-flow, leading-edge coordinate system (.vo'izi), v i cross-flow velocity, H plate's draft, d plate's thickness
defined as ^ = Vt/H, where V is the cross-flow velocity, / is time, and H is the characteristic dimension. In the present case, the cross-flow velocity would be vi, and the characteristic length was chosen to be the plate's draft. As demonstrated by Faltinsen [4], the sections of the plate downstream the flow, feel the vortex generated by the upper sections, depending on how far they are from the start point of the separation. In this way, time (r) could be related to the position of the section relative to the leading edge in Oxo'iZi coordinate system (see Fig. 3b). Using Xl = tilt and tana = V J / K I , we obtain
v ^ ^ n ^ x i ^ x i
H u\H H ^ ' Then the side force on the plate due to the cross-flow drag win be,
FcF = ^PV\iiiH^ CD{T)dT (18)
where L is the plate's length, a is the drift angle and p is the water density. Similarly, the yaw moment is,
, /<(£//ƒ) tana
yii]H' / Cr>ir)TdT (19) 2 Jo
McF=^FcF
Remembering A. — H/L, the non-dimensional force and moment are,
J Mar Sci Teciinol (2015) 20:278-291 283 FCF^ = A sin a cos a (L/H)ian2 CD{T)dT MrF = McF 1 tcF — A" cos a »(£//ƒ) lanj Co{T)rdT (20) (21
Therefore, the cross-flow side force and yaw moment are proportional to the area and moment of area under the curve ( C d ( T ) , T ) . It is interesting to point out that based on Eqs. 18 and 19 the integration interval is defined by tan a and H. Therefore, for a plate with constant H, the non-dimensional drag force and yaw moment (Eqs. 20, 21) are only a function of the drift angle and not the forward speed. It must be noted that this assumption does not account for the fiow three dimensionality and the free-suiface wave effects. It is expected that the non-dimen-sional cross-fiow drag force becomes Froude number dependent for high Froude numbers due to free-suiface effects. This type of dependency was neglected here by applying the rigid free-surface condition.
Looking at the results presented in [6, 9], and [18], a mean value of 5 was chosen for the drag coefficient at the beginning of the transition. A curve was fitted to the numerical values considering this start-up drag ( C b ( 0 ) — 5), which gave the best fit for all the experimental data considered here. A sensitivity study was canied out by changing the start-up value from 4 to 6. Although the results depended on the start-up value, the changes were not sig-nificant. Further studies using a full Navier-Stokes solver, considering the generation of small vortices at the plate's tip, and possible compressibility effects, would be neces-sary to obtain the behavior of the drag coefficient at the beginning of the flow more accurately.
It must be mentioned that the present method is funda-mentally different from the viscous cross-flow model usu-ally used in engineering analysis of, for instance, ship maneuvering. The latter method is based on 2D drag coef-ficients for steady flow and will, for a flat plate with constant draft, predict a constant force per unit length along the plate. Moreover, the intention here was not to solve the complete problem using 2D-|-t method, which ignores the flow three-dimensionality and transverse waves completely. The slender body cross-flow model used here to investigate the contribution to the transverse force and yaw moment from the cross-flow vortices by decomposing the problem. A fully 3D problem with proper free-surface boundary con-dition was solved for the forward-flow and trailing-edge separation, while for cross-flow separation the free-surface effects and flow three-dimensionality were neglected.
3 Numerical implementation
To solve 4) from the boundaiy value problem defined by Laplace equation and boundary conditions showed in Eqs. 7, 8, a collocation method was used based on the Green's second identity as follows.
C ( p ) 0 ( p ) Sf
+
Sv-S v + S v -8n-(q) (22)where C{p) is the solid angle at point p [13]. G refers to the Rankine source function, n is the surface normal vector pointing inside the water domain. SB and SF refer to the surface of the plate and free surface, respectively. Sv+ and Sv- are the two sides of the voitex sheet, while plus is the side with the normal vector n+ = (0,1,0) and minus is the side with n~ = ( 0 , - 1 , 0 ) .
There is a jump in (f> values on the two sides of the vortex sheet, while the normal velocity | ^ is continuous. Therefore, the integrals on the vortex sheet surfaces could be simplified as a dipole distribution on the positive side of the vortex sheet. The dipole strength on the sheet is equal to A(j) = (j)'^ — (l>~ at the trailing edge, which is, based on Eq. 16, constant along the .r-axis. Substituting the body and combined free-surface boundary conditions from Eqs. 7 and 8 into 22, gives
c{pmp) = / Hq)
Sl,QGip,q)
ds-l- ƒ G(p,q)Ui-n(q)ds
(23)
On the mean free surface, 8 0 / 9 « = -d(f>/dz, as a conse-quence of linearization. Letting the point p approaches q on a set of collocation points, a linear equation system is obtained, which could be solved using regular procedures such as LQ factorization.
284 J Mai- Sci Teclinol (2015) 20:278-291
F i g . 5 View of tlie numerical grid on plate, vortex sheet, and half o f the free-surface
3.1 Discretization
To solve Eq. 23, the boundaries were discretized into a finite number of rectangular panels. A linear distribution of unknowns on these panels was assumed. The size of the computational domain was chosen based on the guide lines suggested by Zhu and Falfinsen [20]. The dimensions and a sample of the computafional grid are shown in Fig. 5. The free-surface truncation boundaries' distances from the plate were determined by sensidvity studies to eliminate their infiuence on the calculated force and moment. The vortex sheet, which is bounded by the trailing edge and the free surface, was also limited to the extent of the free surface behind the plate. Although there were some local eiTors at the far downstream of the free surface, the effects of the finite vortex sheet on the forces acting on the plate were proven to be negligible by sensitivity studies.
Points were distributed on the boundaries by numerical grid-generation methods such as the hybrid curve point distribution algorithm (e.g., [17]), which provided the ability to control the grid-point density directiy. The method described in [16] was used to evaluate the singiüar integrals in Eq. 23.
A finite difference method was used to calculate the velocity potential detivatives. To do so, structured grids were generated for the surfaces.
The problem had no plane of symmetry, therefore, the velocity potential was solved everywhere on the bound-aries, except the vortex sheet.
As discussed by Bunnik [2] among others, it is important to choose a con-ect direction of differentiation to ensure stability and satisfy radiation condition in presence of high Froude numbers. Therefore, an upstream differentiation scheme was used for all derivatives on the free surface. The problem of saw-tooth instabilities, reported before in the literature (e.g., [15]), was treated by adjusting the damping of the finite difference scheme.
Since a 3D solver was used, the influence of the plate's thickness and drift angle was considered directly. In other words, the body boundary condition was satisfied on the actual position of the plate. To avoid numerical difficulties, the shape of the plate section was assumed to be a parabola with sharp corners at the leading and trailing edges.
3.2 Handling the Kutta condition
Satisfying the Kutta condition at the trailing edge was not straightforward. The goal was to enforce equality of the pressure, while afiowing a jump in the values of ( p . Unlike the method described in [14], by assuming linear panels instead of constant, collocation points exist at the trailing edge. Allowing for two different solutions of 4> at the same point makes the equation system singular. Therefore, the condition must be enforced with extra care. Here, the method proposed by Faltinsen and Pettersen [5] and further used by Kr-istiansen [10] was adopted. To avoid the sin-gularity, the value of (p extrapolated lineariy from the sides of the plate towards the trailing edge as follows,
Figure 6 shows one section of the plate and clarify the notations in Eqs. 24 and 25. In addition, equality of the pressure in its linear form states,
• 8 ^ ^ 9 ^ (26) Qx 6x
By not satisfying the body boundary condition at the points and f , the equation system is two equations short. Adding the two extrapolation Eqs. 24 and 25 and the pressure equality Eqs. 26, the number of equations exceeds the number of unknowns by one. To fix this, one more body boundary condition must be ignored, for example at (ƒ _ 1 ) - . To summarize, was calculated from Eq. 24, ^T from 26, and g^^r , from 25. Therefore, due to linear extrapolation it was important to have finer elements at the trailing edge. Yj is the strength of the dipole distribution on the vortex sheet, where F^- = 0+ - <p>~j • As Eq. 16 shows, this value does not change along the x-axis.
4 Results and discussion
Different experimental and numerical cases have been chosen for evaluating the validity of the proposed decom-position. First, the described method for enforcing Kutta
J Mar Sci Teclinol (2015) 20:278-291 285
condition was validated by solving the pressure coefficients for a 2D NACA profile. Then, plate A from [1] experiments was considered. Values for transverse force and yaw moment were calculated and compared with experiments and previous numerical results. Aspect ratios of 0.5 and 0.2 with drift angles of 4.5 and 9 degrees were considered. From experiments by Kashiwagi [8], the plate with aspect ratio 0.1 and drift angles of 4 and 8 degrees were consid-ered. The contributions of the forward-fiow and cross-flow to the transverse force and yaw moment were investigated.
X
( p j - 2 "t^j-l
, - -K X
( p j ^ T y - f l r y + 2
F i g . 6 Connection of the plate's tail (solid lines) to the vortex sheet
(dasiied line). Top view
4.1 Foil in infinite fluid
To validate the method for enforcing the Kutta condition, a simple case of a 2D symmetrical NACA0012 foil with an angle of attack in infinite fiuid, without the free surface, was investigated. A 3D NACA0012 model with length-to-cord ratio of 30 was modeled and the pressure distribution at the midsection was compared against the experimental results for a similar 2D foil from [7]. The results for two attack angles of 6 and 10 degrees are shown in Fig. 7.
Although the equality of pressure enforced for the linear pressure at the trailing edge. Fig. 7 shows that due to the thickness effects, the quadratic term in the pressure plays an important role, especiaUy at the leading-edge region.
4.2 Grid sensitivity study
Convergence study was carefully carried out for different cases to assess the effects of the grid density on the'final results for transverse force and yaw moment. The proper-ties of the four grid arrangements, which were used in the calculations, are summarized in Table 1. Besides the number of elements for the free surface and the plate, the attraction factors at the leading and trailing edges, as well as the water line and bottom tip line, are presented. These values are an indication of the grid points' density in the vicinity of each edge, and proportional to the inverse of the elements' span, i.e., a larger attraction factor means smaller elements, arcsinh was used as the control function in the Hybrid Curve Point Distribution Algorithm to generate the suitable point distributions along the boundaries using these attraction factors (please see section 9.2 in [17] for more details). 3 1 0.5 0 - 0 . 5 - 1 - 1 . 5 - 2 -2.5 3 2 1 0 - 1 - 2 - 3 - 4 - 5 - 6 (a)
o Gregory & O'Reilly (1996) Suction side Linear+Quad. Pressure side Linear+Quad.
0.2 0.4 0.6
( X - X , , ) / L
(b)
O Gregory & O 'Reiily (1996)
Suction side Linear+Quad. Pressure side Linear+Quad. Suction side Linear Pressure side Linear
0.2 0.4 0.6
( X - X , J / L
0.8
F i g . 7 Pressure coefficients on 2D NACA0012 section, a a = 6 ° , b a = 10°, Linear force f r o m linear pressure term, pi = —pUdijt/dx,
Quad, force f r o m second-order pressure tenn, p2 = —\p{V4> • V</>)
Figures 8 and 9 show the convergence study for trans-verse force and yaw moment on a plate with aspect ratio of 0.5 and drift angle of 4.5° using the grids described in Table 1. The values have an offset from the experiments but they show the convergence pattern. Besides the coarsest grid (Giid A), all the other three grids seem to follow the pattern of experimental values. The large vari-ation in the transverse force between Froude numbers of
0.4 and 0.7 is best captured by the finest grid (Grid D), while the results from the other two grids (Grid B and C) follow closely.
Similar convergence pattern was obtained for aspects ratios 0.2 and 0.1, although the relative difference between numerical and experimental results was increasing by decreasing the aspect ratio.
286 J Mar Sci Teciinol (2015) 20:278-291
4.3 Importance of nonlinearities
The importance of nonlinearities was investigated by cal-culating the second-order force and moment from the first-order velocity potendal. As discussed previously in Sect. 2.1, this is a part of the second-order force and it does not represent the complete value; which requires the solution of the second-order boundary value problem. However, still it could be of importance in evaluating the validity of the linearization and the importance of the nonlinear effects. Figures 10 and 11 show the values of the first- and second-order transverse force and yaw moment for aspect ratio 0.1 at drift angle of 4°.
Further investigations on aspect ratios of 0.5 and 0.2 showed that the difference between first- and second-order transverse force is negligible. However, this difference was more important for yaw moment, and increased for smaller aspect ratios. The second-order yaw moment showed smaller values than the first-order yaw moment. This was closer to the experiments for smaller aspect ratios which is an indication of the importance of the nonlinearities. On the other hand, the linear results for aspect ratios of 0.2 and 0.5 were closer to experiments than the results including the second-order effects. Therefore, the linear assumption suited better the larger aspects ratios.
T a b l e 1 Number of panels and grid properties f o r Plate A , A t t . F a c : Attraction Factor proportional to l/(Element span), W L : Water line, T L : T i p line
Grid Plate Surface Free Bow-Stern A t t . Fac, W L - T L Att. Fac. Total panels A 10 X 10 30 X 10 15 10 800 B 20 X 20 60 X 20 15 10 3,200 C 30 X 20 90 X 20 15 10 6,600 D 40 X 30 120 X 30 15 10 12,000 2.5 2 I 1 0,5 - Exp. Grid A > Grid B O Grid C + G r i d D 0.5 1.5
4.4 Slender body viscous cross-fiow
Looking at the results in Figs. 8, 9, 10, 11, it is clear that the values from the present potential fiow calculation are lower than experiments while they follow the same pattern. This difference is believed to be due to the cross-fiow separation effects from the bottom tip of the plate. As explained in Sect. 2,3, an attempt was made to consider these effects using a slender body viscous cross-flow model, with zero Froude number free-surface boundary condition. As shown in Eqs. 20 and 21, for a plate with a constant draft (zero tapered angle), the non-dimensional cross-fiow transverse force and yaw moment are only a function of the plate's drift angle and aspect ratio. There-fore, for a constant drift angle, the non-dimensional cross-flow force and moment will be constant for all Froude numbers. The values of the cross-flow transverse force and yaw moment, for different aspect ratios, are presented in Table 2. The relative importance of the cross-flow com-ponent in comparison to the total force is also listed in Table 2.
Figures 12 and 13 show the potential flow results for the first-order transverse force and yaw moment, with thick-ness ratio 0.01, plus the cross-flow coiTections. Here, the plate's aspect ratio is 0.5 and the attack angle is 4.5°. Since the viscous cross-flow moments were very small, they are not shown in the yaw moment plot (Fig. 13). The
F i g . 8 Convergence study for non-dimensional transverse force (Eq, 14), Exp. experimental results f r o m [ 1 ] , drift angle a = 4 , 5 ° , aspect ratio A = 0,5 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 - B - Exp, < Grid A > Grid B O Grid C + Grid D 0.5 1 1.5 2
F i g 9 Convergence study f o r non-diinensional yaw moment (Eq. 15), Exp. experimental results f r o m [ 1 ] , d r i f t angle a = 4 , 5 ° , ratio A = 0,5
aspect
calculated viscous cross-flow transverse force shifts the potential results towards the experimental values. Due to the zero Froude number free-surface boundary condition, which was used for calculating the cross-flow effects, a
J Mar Sci Technol (2015) 20:278-291 287 1 0.9 0.8 0.7 0.6 0.5 1 0.4 0.3 0.2 0.1 0 Exp. V Grid B (lineal) A Grid G (linear) O Grid B (linear+2ord) + Grid C (linear+2ord) >? p p 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Fn
Fig. 10 First- and second-order non-dimensional transverse force
(Eq, 14) for two different grids, Exp. experiinental results f r o m [8],
2ord second-order force f r o m first-order velocity potential, drift angle a = 4 . 0 ° , aspect ratio A = 0,1 0.25 0.2 Ö 0.15 0.1 0,05 - B - E x p , V Grid B (linear) A Grid C (linear) o Grid B (linear+2ord) + Grid C (linear+2ord) 0 I 0.15 0.2 0,25 0,3 0,35 0,4 F„ 0,45 0,5 0,55 0,6
Fig. 11 First- and second-order non-dimensional yaw moment (Eq.
15) f o r two different grids, Exp. experimental results f r o m [ 8 ] , 2ord second-order moment f r o m first-order velocity potential, drift angle
a = 4 . 0 ° , aspect rado A = 0.1
difference between experimental and computational results is expected at higher Froude numbers. However, this deviation starts at relatively higher Froude numbers than anticipated (around 1.0).
The potential flow results for the yaw moment are close to the experiments without the viscous cross-flow correc-tion. The viscous cross-flow coiTections to the yaw moment, as presented in Table 2, are negligible, except for aspect ratio 0.1. The non-dimensional yaw moment is much smaUer in this case than higher aspect ratios and the cross-flow correction, although small, is no longer negli-gible. This could be due to the increased importance of nonUnear effects.
Table 2 Values for viscous cross-flow correction on transverse force and yaw momenl, Fy = Fy/{O.SpU'LH), M-j = /{O.SpU^L^H), force and momenl are normalized by the drift angle in radians
Aspect ratio (A = H/L) 0,5 0,2 0,1 -Fyicr/a 0,39 0.37 0.34 Mz'CF/V. 0,002 0,006 0,008 Fy,cF/max{Fy,E,i,,) 0,16 0,29 0,51 A ? j , c f / m a x (M,,£v,,,) 0.003 0,014 0.051 2.5 1.5 1 0,5 - B- E x p ,
+ Present f^/lethod (linear) > Present Meltiod ( l i n e a r + C.F.)
0.5 1,5
Fig. 12 Comparison of non-dimensional transverse force (Eq. 14),
Exp. experimental results f r o m [1],' CF. viscous cross-fiow, d r i f t
angle a = 4 , 5 ° , aspect ratio A = 0,5
The obtained values for the aspect ratio 0.5 and drift angle of 4.5° are compared to other numerical results i n Figs. 14 and 15. A similar shift in the results was predicted by Landrini and Campana [11]. They solved the problem with and without bottom-tip voitex sheet. The difference between the two sets of their results is close to a constant value as well. The present results, without the cross-flow correction, are close to the Landrini and Campana [11] results without the tip vortex sheet, while the cross-flow coiTection shifts the present values towards their results with the tip vortex. There is a small difference between the Froude number of the peak force in the present and the results from [11]. This is believed to be due to their low Froude number free-suiface boundary condition. They used the double-body potendal as the base for linearizarion while the undisturbed inflow potential was used here which is believed to be more consistent at higher Froude numbers than the double-body potential.
Landrini and Campana [11] considered the vortex sheet and free-surface interactions. Although a simpler approach was used in the present calculation, their deviation from the experimental results is larger than the present calculations at higher Froude numbers. This could be also a result of
288 J Mar Sci Teclinol (2015) 20:278-291 1 0.9 0.8 0.7 Ö 0.6 0.5 1 ^ 0.4 0,3 0,2 0.1 0 - E x p .
Present Method (linear)
0.5 1.5
F i g . 13 Comparison of non-dimensional yaw moment (Eq. 15), Exp. experimental results f r o m [ 1 ] , d r i f t angle a = 4.5°, aspect ratio A = 0,5 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 - E x p .
• Landrini 8, Campana (1996) (tJo T.V.) • Landrini 8. Campana (1996) (Willi T,V,)
X u (1991) Cliapman (1976) Present fvlethod (linear)
0.5 1,5
F i g . 15 Comparison o f non-dimensional yaw moment (Eq, 15), Exp. Experimental results f r o m [1], T.V. tip vortex sheet, d r i f t angle a = 4 , 5 ° , aspect ratio A = 0.5 3 2,5 1,5 0.5 - B- E. X P ,
— - L,™drini & Camp.ina (1996) (No T,V,) Landrini & Campana (1996) (Wilh T . V . ) ,\u(1991)
- - - Chapman (1976) - n - Zhu. & Faltinsen (2007)
+ Present Method (linear) > Present Method (linear + C P . )
0.5 1,5
Fig. 14 Comparison o f non-dimensional transverse force (Eq, 14),
Exp. Experimental results f r o m [ 1 ] , CF. viscous cross-flow, T.V. tip
vortex sheet, d r i f t angle a = 4 , 5 ° , aspect ratio A = 0.5
their free-surface boundary condition which is not suitable for higher Froude numbers.
Maniar et al. [12] and Xii [19] solved this problem using a slender body assumption. The present results withoiU cross-flow correction are very close to their values for yaw moment. However, for the transverse force, the two curves are only close to each other away from the region of the maximum force while the position of the peak is predicted similarly in both calculations. The Rankine panel method results from Zhu and Faltinsen [20] are also presented for comparison.
Figures 16 and 17 show similar comparisons for aspect ratio 0.2 and drift angle 4.5°. Figures 18 and 19 also show comparisons with experimental results from [8] for a plate
with aspect ratio 0.1. The cross-flow model explains the difference between the potential-flow transverse force and the experiments for all aspect ratios. The yaw moment is predicted by the potential-flow model with reasonable accuracy for aspect ratios 0.5 and 0.2. The importance of nonlinearities is more pronounced in the yaw moment of aspect ratio 0.1, as shown-in Fig, 19, The results from calculations by Fandrini and Campana [11] using the double-body linearization and tip vortex sheet are closer to the experiments which could be due to their linearization method. It also indicates that, at this drift angle, the interaction between free-surface and tip-vortex may play' an important role.
Figures 20, 21, 22, 23 show the transverse force and yaw moment for plates with aspect ratio 0.5 and 0.2, with 9.0° drift angle. For the aspect ratio 0.5, the total transverse force deviation from the experimental results is larger and starts from lower Froude numbers in comparison to 4,5° drift angle. Moreover, the second-order forces turns out to be negligible. For aspect ratio 0.2, the effect of second-order forces starts to be important especially around the maximum transverse force. However, it is still negligible for Froude numbers approximately lower than 0.5 and higher than 1.5. On the other hand, the deviation from the experimental results, after the viscous cross-flow correc-tion, is much higher than for aspect ratio 0.5. The corrected results for aspect ratio 0.5 are closer to Landrini and Campana [11] results with a tip vortex, for aspect ratio 0.2 it shows a clear difference which does not exist at a 4,5° drift angle. This could indicate that the vortex generated due to the tip flow separation is stronger at higher drift angles and at smaller drafts its interaction with free-surface matters.
J Mar Sci Teclinol (2015) 20:278-291 289
-B-Exp.
- • - Landrini & Campana (1996) (No T.V.)
- X - Landrini & Campana (1996) (With T.V. + Present Mettiod (linear)
> Present Metiiod (linear + C P . )
0.5 1.5 1.2 ifeT 0.6 I 0.4 0.2 0 ' 0.15 - B- E x p . Kashiwagi (1983) (Theory) - o - Landrini & Campana (1996) (Wilh T.V.
O Present Method (linear+2ord)
+ Present Method (linear) > Present Melhod (linear + C F . )
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
F i g . 16 Comparison of non-dimensional transverse force (Eq. 14),
Exp. Experimental results f r o m [ 1 ] , CF. viscous cross-flow, T.V. tip
vortex sheet, drift angle a = 4 . 5 ° , aspect ratio A = 0.2
Fig. 18 Comparison of non-dimensional transverse force (Eq. 14),
Exp. Experimental results f r o m [ 8 ] , CF. viscous cross-flow, T.V. tip
vortex sheet, 2oid second-order force f r o m first-order velocity potential, drift angle a = 4 . 0 ° , aspect ratio A = 0.1
- B- E x p .
- • - Landrini & Campana (1996) (No T V . )
- X - Landrini & Campana (1996) (With T.V.) + Present Method (linear)
0.5 1.5
E„.
Fig. 17 Comparison of non-dimensional yaw moment (Eq. 15), Exp.
Experimental results f r o m [ I ] , T.V. tip voitex sheet, d r i f t angle
a = 4 . 5 ° , aspect ratio A = 0.2 0.25 0.2 0.15 0.1 0.05 - B- E x p . Kashiwagi (1983) (Theory.)
- X - Landrini & Campana (1996) (Wilh T.V.)
O Present Method (linear+2ord)
+ Present Method (linear)
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Fn
Fig. 19 Comparison of non-dimensional yaw moment (Eq. 15), Exp.
Experimental results f r o m [ 8 ] , T.V. tip vortex sheet, 2ord second-order moment f r o m first-second-order velocity potential, d r i f t angle a = 4 . 0 ° , aspect ratio A = 0.1
5 Conclusions
The transverse force and yaw moment acting on a surface-piercing flat plate were investigated by decomposing the problem into a linear potential forward-flow, and a viscous cross-flow problem. A 3D boundary integral equation was formulated based on the Green second identity using linear distribution of Rankine sources and dipoles over the plate and the free-surface boundaries to solve the potential for-ward-flow problem. Dipole singularities were introduced on a wake sheet behind the trailing edge of the plate together with a Kutta condition at the trailing edge. The Kutta con-dition was satisfied by enforcing continuity on the velocity
potential jump at the separation line. The shape of the sheet was finearized to a vertical flat surface and the dipoles' strengths were assumed to be invariant along the .v-axis.
The cross-flow separation from the plate's tip contrib-utes to the transverse force and yaw moment. A slender body cross-flow method using rigid free-surface boundary condition was used to solve the viscous cross-flow prob-lem. In analogy to 2D-M method, in this method the problem was transformed to a transient problem of a ver-tical plate attached to a wall in a start-up fiow. It was shown that the non-dimensional transverse force and yaw moment acting on the plate due to the tip separation with a
290 J Mar Sci Technol (2015) 20;278-291
1
0.5
-0.5 1 1.5 2 Fn
Fig. 20 Comparison o f non-dimensional transverse force (Eq. 14),
Exp. Experimental results f r o m [ 1 ] , CF. viscous cross-flow, T.V. tip
vortex sheet, drift angle a = 9.0°, aspect ratio A = 0.5
- Exp,
Landrini & Campana (1996) (With T.V.)
0 Present Melhod (linear+2ord)
+ Present Melhod (linear)
> Present Melhod (linear + CF.)
0.2
-0 -0.5 1 1.5 2
Fn
Fig. 22 Comparison of non-dimensional transverse force (Eq. 14),
Exp. Experimental results f r o m [1], lord second-order moment f r o m
first-order velocity potential, CF. viscous cross-flow, T.V. tip vortex sheet, d r i f t angle a = 9.0°, aspect ratio A = 0.2
1 0.9 0.8 0.7 h 0.6 0.5 0.4 0.3 0.2 0.1 - e - E x p .
- • - Landrini & Campana (1996) (Wilh T.V. + Present tylethod (linear)
ï-EiFTT^P V
F,
Fig. 21 Comparison o f non-dimensional yaw moment (Eq. 15), Exp.
Experimental results f r o m [ 1 ] , 7".V, tip vortex sheet, d r i f t angle
a = 9.0°, aspect ratio A = 0.5
rigid free-surface boundary condition depend on the drift angle and aspect ratio and not the Froude number. There-fore, it imposes only a constant shift on the curve of the non-dimensional transverse force against Froude number. This shift seemed to f i l l the gap between the potential flow results and the experiments coiTcctly, especially for Froude numbers approximately lower than 1.0. Deviations from the experimental values appeared by decreasing aspect ratio or increasing Froude number or drift angle. Neglect-ing the interactions between the cross-flow tip vortex and the free surface by imposing the rigid free-surface bound-ary condition is believed to be the reason. The influence of the tip separation on the yaw moment was shown to be negligible. 0.6 0.5 0.4 0.2 0.1 - f l - E x p .
Landrini & Campana (1996) (With T.V.) O Present tvlethod (linear+2ord) + Present Method (linear)
0,5 1.5
Fn
Fig. 23 Comparison o f non-dimensional yaw moment (Eq. 15), Exp.
Experimental results f r o m [ 1 ] , T.V. tip vortex sheet, 2onl second-order moment f r o m first-second-order velocity potential, d r i f t angle a = 9,0°, aspect rado A = 0.2
Although the second-order velocity potential was not studied in the present work, the contribution to the second-order force from the first-second-order velocity potential was cal-culated, and shown to have small infiuence, especially on the transverse force. However, its influence tends to grow rapidly by increasing the drift angle and decreasing the draft, especially for Froude numbers corresponding to the maximum transverse force. At larger drift angles, stronger interaction between the free surface and the cross-flow tip voitex is expected. As a consequence the cross-flow cor-rected results showed larger deviations from the experi-mental values especially at higher Froude numbers.
J Mai- Sci Tecl-iiiol (2015) 20:278-291 291
The results presented here suggest that the decomposi-tion of the forces into a potential forward-flow and a vis-cous cross-flow parts is valid in a pracdcal range of Froude numbers. Adopting a general method for calculating the transient drag coefficient, i.e., transient 2D Navier-Stokes solver, this method could be used to predict the transverse force and yaw moment acting on an advancing high-speed vessel with a drift angle; taking into account the com-plexities of the vessel's geometry.
Acknowledgments This work was supported by the Research
Council o f Norway through the Centres o f Excellence funding scheme A M O S , project number 223254 and CeSOS.
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