A simplified nonlinear analysis of a high-speed planning craft in calm water

A SIMPLIFIED NONLINEAR ANALYSIS OF A

HIGH-SPEED PLANING CRAFT

IN CALM WATER

ABSTRACT

A simplified theory for the steady behaviour ofa hgh-speed planing craft in calm water is presented. The hydrodynamic analysis is based on a 2 D nonlinear theory with effect of flow_{separation from chines. The theory is verified by comparing with}

a fully

nonlinear solution, an empirical formula and experimental droptests.

i INTRODUC1]ON

High-speed s'ender body theories

_{based on a 2-D}

approach are common to use in seakeeping analyses of

semi-displacement catamarans and monohulls (see for instance

Faltinsen and Zhao (1991 a,b)). 2D means that a o-dimensional Laplace equation with three-o-dimensional

free-surface conditions is solved in the cross-planes of the vessel.

A consequence is that the flow disturbance generated from one cross-plane of the ship will only propagate downstream. The solution is started at the bow of the ship and stepped downstream. When the Froude number is very high, gravity

can be neglected in the analysis.The problem has then similarities with the analysis of water entry

_{of a}

two-dimensional body.

Water entry of two-dimensional bodies

_{has been}

investigated and discussed by many authors,see for instance Wagner (1932), Faltinsen (1990), Zhao and Faltinsen (1993) and Zhao et al. (1996). Zhao and Faltinsen (1993) solved the

problem with the exact body boundary condition and free

surface conditions without gravity. The effect ofspray

generation is essential. No flow separation was accounted foç but this was included by Zhao et al. (1996) by satisfying a generalized Kutta condition at fixed separation points.

Zhao et al. (1996) presented also a simplified nonlinear

analysis without flow separation. The theorywas verified by

comparing with the fully nonlinear solution and validated

by comparisons with model tests. The simplified theory shows

the importance to satisfy the exact body boundary condition and include the local water elevation at the hull. Theexact

dynamic free-surface condition is less important.

The simplified nonlinear analysis of Zhao et al. (1996) will

be extended to include flow-separation from fixed separation

points. This is done by an iterative process. A shape of the separated free surface is assumed and the kinematic free-surface condition is satisfied on the assumed free free-surface. The shape of the separated free surface is iterated until the

difference between ,the pressure on the separated free

surface and atmospheric pressure is minimized. A Kutta condition is satisfied at the separation points. The solution is verified by comparing with the fully nonlinear numerical

solution of Zhao et al. (1996) for water entry of wedges

with knuckles. The method is also validated by comparing with experiments from drop tests of a wedge and a bow flare section with knuckles (Zhao et al . (1996)).

Figure 1: Coordinate system i = (x,y,z.) and symbols used in a prismatic planning boat analysis

The simplified theory is generalized to

a 2D approach

for the steady analysis of a high-speed planing craft in_calm

water. Flow separation from chines or spray rails is included.

The separation line has to be known. Rounded bilges are

therefore difficult to analyse.Stepped planing hulls and

transom flaps cannot be handled. There are otherwise no limitations on hull forms. The method is based on potential

flow theory and can be used to predict calm

water

resistance,sinkage and trim due to pressure effects. This includes the effect of spray and the local water elevation at the hull. The method is used in a parametric study of

prismatic planing hulls in calm water by varying the deadrise

angles,trim angles and length-beam ratio. The results are compared with an empirical formula by Savitsky(1964) for the lift and drag coefficients and the center of pressure.

4

The solution is numerically robust and fast to compute. The method can be generalized and applied in a time-domain analysis of dynamic stability and seakeeping performance

of a high-speed planing craft. This includes _nonlinear

hydrodynamic effects and water impact. A complete analysis

would require that viscous resistance, propulsion,_steering

and possible effects of cavitation and ventilation _are

incorporated. One relevant study for transverse stability ofa planing craft is to evaluate the roll moment as a function of

steady roll angle. The presented methodcan be used without any modifications. However only pressure loads on the hull due to potential flow are included.

N3

N

Lflw

_{I w}

w

't,

ci) w C

Q) 0E

5LLJ

ri

Rong Zhao

_{MARINTEK, Norway}

Odd M Faltirisen

_{Norwegian University of Science and Technology}

Norway

A SIMPLIFiED NONLINEAR ANALYSIS OF A HIGH-SPEED PLANING CRAFT IN CALM WATER 2 THEORY

A planing boat with high forward speed U in calm water is considered. A right-handed coordinate system = (x,y,z)

fixed with respect to the vessel is used (see Figure 1). The forward speed of the vessel is constant. The surface z=O is the undisturbed water surface. The x-axis points in the

opposite direction of the forward speed of the ship.

The vessel is slender, so B(andD) is O(a). Here B,D and Lare

the beam, draught and length of the ship. a is a small U

parameter. The Froude number is O(e1), where g is the acceleration of gravity. Gravity may then be neglected.

(Ogilvie M 967]).

The fluid is assumed inviscid and incompressible. The flow is described by a velocity potential 1( .), which satisfies the three-dimensional Laplace equation.

The total velocity potential can be written as

=Ux.4-'Dj(.)

Equation (1) The order of magnitude of the derivatives of close to the

ship are

=O(E2)

Equation (2)

(and) = O(e)

Equation (3) The reason to choose

_{= O(e2) and not 0(1) is that}

the wetted surface changes rapidly near the bow. It follows

that the three-dimensional Laplace equation can be reduced

to a two-dimensional Laplace equation for CD1, namely a2 d2CD

2 2

Equation (4)

Based on a similar analysis as Wanger(1 932) did for the water

entry problem,it follows that the dynamic free-surface

condition can be written as

CD=O on Ç1(x) Equation (5)

Here Ç1(x) is the surface elevation at the intersection between

the body surface and the free surface for each side of the hull. This means that the dynamic free-surface conditon is satisfied on a horizontal line L. for each section at each side of the hull (see Figure 2). A coordinate system (y, z) which follows the keel of.a planing boat is used. lt is assumed n Figure 2 that all sections have the same geometry. However the method is applicable to a problem with changing cross-sections.The kinematic free-surface condition is used to

determine the surface elevation for each section. More details

about prediction of the intersection point will be given in the next section.

432 FAST '97 PAPERS

The body boundary condition can be written as

3N on SB

Equation (6)

where SB is the wetted body surface below z = Ç1(x) . Further

V, is the body velocity in the normal direction

j of the

body surface and 1Ç' is the two-dimensional normal vector. Positive directions of Ñ and j point into the fluid.

A

z,

(y,, z,')

(y,.,, z,'') PT

Figure 2: The coordinate system ly,zl,the real free surface 1, and

1,, and the horizontal lines L. and L,,, used in the numerical

simulation for sectioñs x, arid x,.,. Here one assumes that all sections have the same geometry.

The velocity potential CD, is found by using Green's second

identity for a number of sections. The velocity potential CD, at a point (y,z) in the fluid can be written as

2JrCD1(y, z) =

f[iogr

i(T1)

flaIor15()

aN

SB+SF+SC

Equation (7)

where r=[(y_q)2

+(z_)2J2

Sa horizontal

free surface which starts at the intersection between the body and free surface for each side of the hull, S, a control surface far away from the body. The contribution to the free surface integral from far away of the body can be rewritten. The velocity potential CD,for lyI>b(x), where 6(x) is large relative to the cross-dïmensions of the hull, can be expressed

as a vertical dipole (symmetrical part) and a multipole

(antisymmetrical part). CD, can be written as (for yl>b(x))

CD1 =C1(x)

2 2 +C2(x) 2 22

(y +z )

(y +z )

Equation (8)

where C,(x) and C2(x) are coefficients which can be obtained n the solution of Equation (7). C2(x) is zero for cross-sectional

shapes that are symmetric about the z-axis.In the numerical solution (inside yl>b(x)), the surface is divided into straight

line segments. The velocity potential and its derivatives are

assumed constant over each element. When (y,z) approaches

the mid-point of each element on andSFinEquation (7),

we haveNB+NF

equations, where N and N are the

number of elements on the body surface and the free surface.

The total number of unknowns is

NB + N+2.

There are

two unknowns which come from the representation of the far field solution, namely C1(x) and C2(x). Then we need two more equations. This is done by choosing two control points with yl -value >b(x) and satisfying Equation (7).

The initial conditions for t and Ç are I = O and Ç = O at the first section. The kinematic free-surface condition is used

to step the free-surface elevation Ç. The details will be

described later.

The pressure p on the body surface is calculated by Bernoulli's

equation, which can be written as

j

12

12

p-pa =-p(U----+((---) () ))

on S3

Equation (9)

Here Pa is atmospheric pressure. The pressure term

_O.5p()2

_{is higher order and has been neglected. The}

hydrostatic pressure is also negligible. Predicted negative

pressures near the spray roots are unphysical and set equal

to Pa (Zhao et al. )l 996}).The force components in the

longitudinal and vertical direction of the ship as well as the pitch moment can be written as

PJ(PPa)nidS

_i=

1,3,5

Equation (10)

Figure 3: Local polar coordinate system(r2, 92)and local Cartesian

system (y2,z2).lis the intersection point between the body and the

free surface.The local deadrise angle is a. The free surface condition is satisfied on the y2- axis in the numerical simulation,

Here =1, surge; =3,heave; i=5, pitch. n.is defined as

(fl1,n2,n3)=

Equation (11)

(n4,n5,n6)= rxni

Equation (12) where

_=xz+yjl-zk.

The formulated problem can be interpreted as a

two-dimensional time domain problem by the transformation

Ua-a

-Equation (13)

This transforms the steady problem to an unsteady problem

in an earth-fixed coordinate system.The ship cross-section will change with time in a fixed (y,z) plane in the earth-fixed

coordinate system. If on the other hand the problem of water entry of a two-dimensional body 'is solved, it can be changed

to a steady flow problem by x=Ut. This will be further

discussed durino the presentation of numerical results.

3 PREDICTION OF LOCAL FREE-SURFACE ELEVATION

We will first study a simple case of a ship with the same geometry for each section. The ship may have a constant

roll angle. Figure2 illustrates the free surfaceI and 1 at

two sections x, and x,,1.The free-surface conditions are satisfied on the horizontal linesL.and L11 at x and x,,,.

An integral equation following from Equation (7) is used at

each section x to find the velocity potential on the body

surface and the vertical velocities at L lt is assumed that the vertical velocity on L, for a given y-value is the same as

the vertical velocity on 1.

The solution can be stepped from x=x. to x=x.1 in the

following way. The procedure is first to decide the intersection

point (yi+l,z{')

for positive y-values at section x,,, and

then find what Ax = x.1 - x must be. By knowing Ax, the intersection point for negative y-values at x.1 is determined

afterwards. lt can be shown that Ax = x.1 - x. can be

approximated as

-yi)

dc Equation (14) where dx AP dc)m

2-P-UD(yj -y)

sr Equation (15) and AP

_{= "i+l}

FtI' Equation (16)

The coefficient D is from a local analysis. The relative vertical

velocity of a point between PT and PT.1 with y-coordinate is W. Close to the intersection between water and body surface, W can be written as

ir

R

2-r-sr

Equation (17)

ß and r2 are defined in Figure 3. D is found from the global solution and an average value is used from x to x,,,1. The details can be found in Zhao et al. (1996). If the body has a roll angle, D is determined as follows. An estimate of the horizontal line L.1 on the left hand side of the body (see

Figure 2) is based on the rate of change with x of the

intersection point from the previous section. D,,, is then again determ'ined as the average value at x, and x,,,1. Ax is found

by Equations (14) and (15).

Knowing Ax, thé intersection point on the left hand side of the body can be determined by a linear interpolation. This has been done by using similar equations, as used for the

D

A SIMPLIFIED NONLINEAR ANALYSIS

OF A HIGH-SPEED PLANING CRAFT IN CALM WATER

right hand side of the body, to calculate the Lsx that is needed

for points on the left hand free surface to move up to the body surface at section x,1. Then one can determine the y-coordinate of the intersection point_{by a linear interpolation.}

When the sections have different_{geometry, an iterative}

process is needed to find LiP and x.The _{reason is that Li?} depends on Lic. This can be done by_{guessing different}

tsP-values based on the value from a previous section and using

an interpolation method.

4 FLOW SEPARATION FROM CHINES

When the flow reaches a chine, separation_{will occur. Zhao} et al. (1996) studied the problem of water entry with flow

separation by using a fully nonlinear analysis including a part

of the jet flow leaving a fixed separation point. Due to the thin jet flow, small time steps are needed in the numerical simulation. This implies that the method_{is time consuming.} A simple analysis of the flow separation problem will be described. The separated free surface _{from the chine is}

assumed to be a continuation of the _{body surface with}

pressure equal to atmospheric pressure. The horizontal lines L (see Figure 2) are determined in the same way as before as if the separated free surface is a part of the body surface.The

problem is how to find the shape of the fictitious "body

surface". Important information_{can be obtained by studying} the shape of the separated free surface near a separation point. Figure 4 shows a local polar coordinate system (r,9) and a local Cartesian coordinate _{system (s1,n1) at the} separation points. Using a Kutta condition_{at the separation} point S implies that the flow leaves tangentially and the velocities at point S are finite. In a small region near the

separation point, the behaviour of the velocities can be found

by a local analysis. A local solution of the velocity potential øloc, that satisfies the body boundary condition, can be written

øloc= A0 (x)+ U1(x)s1 + A, (x)r' cos(n9)

n>i

Equation (18)

By using a similar analysis as Zhao et al. (1996) and satisfying the dynamic free-surface condition, the solution in the vicinity

of the separatior point can be written as

øioc= A0 (x) + U1 (x)s' +A3(x)r2 cos( 9) + 0(r2)

Equation (19)

Equation (19) implies that the tangential velocity at the free

surface is _{U5(x).The free surface near the separation point}

can be found by Equation (19) and written as

n'

=C(x)(s')2 +0((s')2)

Equation (20)

This formula is valid for small _{However numerical tests} have shown that the formula_{can be used for quite large} 1. Further numerical results for the pressure on the "real" body surface are not sensitive to values of C(x). The calculation

procedure is an iterative process. A shape _{of the separated}

free surface s assumed based _{on Equation (20) and the}

kinematic free surface condition is satisfied _{on the assumed} free surface. The shape of the separated free surface is iterated

until the difference between the pressure on the separated

free surface and atmospheric pressure is minimised.

434 FAST '97 PAPERS

Figure 4:The local polar coordinate system _{(r,9) and the loca} Cartesian coordinate system (st,n1).Sis the separation point, SB the body surface, S the free surface.The _{s' coordinate coincides}

with the tangent of the body surface at the_{separation point S.}

5 VERIFICATION AND VALIDATION

The solution is verified by comparing with_{the fully nonlinear}

solution by Zhao et al. (1996). Figure 5 compares the time history of the vertical force during water entry of wedges with deadrise angles 20° and 30°. _{Generally speaking there}

are good agreement between the simplified and fully

nonlinear solutions both before and after flow separation from the knuckles. The force has a maximum value at the

time instant when flow separation of the spray roots

occurs.Results by the two-dimensional water entry theory of Vorus (1996) are also shown in Figure 5(a). When flow separation from the knuckles _{occurs initially,a singularity is} present in the Vorus (1996) method. This is clearly _{seen in}

Figure 5(a). Figure 5(a) demonstrates_{that the Vorus (1996)}

method predicts lower force than the two other _methods. The time history is also different and flow separation from the knuckles starts later according to Vorus (1996). Comparison of the predicted pressure distribution by the simplified and the fully nonlinear solution on the wedge with

deadrise angle 30° is shown in Figure 6. Good agreement is

documented.

n order to validate the theories, drop tests have been_carried

out at MARINTEK. Both a wedge section with_{30 degrees} deadrise angle and a typical ship bowsection were tested. The shape of the ship bow section is shown in Figure 7.

There is generally good agreement between the experiments

and the simplified and fully nonlinear theories. A detailed discussion and description of the drop tests_{can be found in} Zhao et al. (1996).

Figure 8 shows a comparison of numerical and experimental

values of slamming pressures on the bow flare section after

flow separation of the spray roots have occurred. The measured vertical velocity of the body during_{water entry is} presented in Figure 8(d). The fully nonlinear solution and

the simplified solution give nearly the _{same values. The}

experimental results give somewhat lower values than the

numerical results. Some of the differences may be attributed

to experimental errors (Zhao et al. (1996)).

The results of water entry of a two-dimensional body can be used n the analysis of steady flow past a planing craft. This will be demonstrated by comparing withan empirical

pressure coefficients for a prismatic and chine-wetted planing

hull. The formula is based on extensive experimental data. The lift coefficient can be written as

0.60

CL =C, - O.0065PCLO

Equation (21)

Here

CLO=F 1(0.5 pU2 B2 )

= 0.012112

and

CL FLß ¡(0.5 pU2B2 ) _{Further CLoCLßFLoFLß 'ß}

and B are lift coefficient for zero deadrise angle (/3), lift

coefficient, lift force for zero deadrise angle (/3), lift force, mean wetted length-beam ratio, trim angle of planing area (deg), angle of deadrise of planing surface (deg) and beam of planing surface. Equation (21) is valid for _{2 15}

and ,. 4. Figure 1 defines the hull and the angles ¡3 and 't.

The mean wetted length-beam ratio X is equal to

0.5(LK+Lc)IB(see

Figure 1).

The resistance componentD9due to the pressure force is in

this case simply

=FL tant

Equation (22)

The center of pressure C9 is expressed as

= = 0.75

Equation (23) whereI,,_{is the distance measured along the keel from the}

transom stern to the center of action of the hydrodynamic force.

Figure 9 shows comparisons between the empirical formula

and the numerical results for the lift coefficient and the center

of the pressure. Since there is a simple relationship (see

Equation (22)) betweenD and

Lß'results are not presented for

D .

There is reasonabe agreement between the theory and tAe empirical formula. The results are either presented as a function of dearise angle /3 or as function of X. The detailed theoretical force distribution along the ship when_/3

= 200 and /3 = 30° can approximately be obtained from

Figure 5 by setting V_{= U'r and x=Ut. This means that the}

horizontal axis in Figure 5 becomes tx/B. The maximum force

value is at x = LK - L (see definition in Figure 1), ie. when flow separation from the chines starts. The theoretical force

A SIMPLIFIED NONLINEAR ANALYSIS OF A HIGH-SPEED PLANTNG CRAFT IN CALM WATER

Figure 5: Vertical forces on symmetric wedges during water entry with constant V. _{, fully nonlinear solution; - - - simplified solution; - - -,} solution of Vorus (1996). p is mass density of the fluid, B ¡s breadth of the wedge, t is time variable, a) deadrise angle a = 20°, b) deadrise angle z = 30°.

Figure 6:The pressure (p - p) distribution during the water entry

of a wedge with deadrise angle 30° and knuckles.Calculated by the fully nonlinear solution and the simplified theory. V is constant drop velocity, p is atmospheric pressure, p is mass density of the fluid and B is breath of wedge. y is horizontal coordinate on the body surface. t0 is the time ïnstance when the spray roots of the jets reach the separation points.The results of the simplified solution are given by marks and the results of the fully nonlinear solution are given by the lines. ,t - t0; ,t =

1.1t0;---;t=1.3t0,---,t=2.0t0.

distribution up to x = LK - L is the same for all X. The

difference in theoretical values for different values of X is simply

a matter of how large maximum'cc/B(or Vt/B in Figure 5) is.

Even if the local force aft of the chine position where flow separation starts, are smaller then ahead of the separation

point, they are not negligible. The X dependence of the results demonstrate that. This means that the effect of flow separation

on the hydrodynamic loads are important. The theory

overpredicts CL8 and underpredicts C relative to the empirical

formula. The differences between teory and the empirical

formula does not vary much with X. A possible reason to the difference is that three-dimensional flow effect are of some importance from x=0 to x=LK

_-

_Lc.This is likely since

_(L

-L0)/B varies from 0.63 to 2.66 in the numerical results

presented in Figure 9. The values of (LK

_-

_{Lc)/B imply} a rapid change of the flow in the longitudinal direction

from x=0 to x=L - L0. The general trend is that

8'- 4- 2-A p -o _0-r O o

0--0--L

tpV° *

_{0 ...}

m.

-0.5 0.0 0.5 B

A SIMPLIFIED

NONLINEAR ANALYSIS OF A HIGH-SPEED

PLANING CRAFT IN CALM WATER

F,

Dummy

Section Measuring_Section Dummy

Section

100mm

Pressure gauges

Figure 7: The geometry of

the bow flare section_{and the locations of the}

pressure gauges used in the drop test experiments.

4 o 0.0 0.2 0.4 _{0.6 o.s} 1.0

z-;

z0

z-;

2.4 2.2 1.8 1.6 5.4 20 -0.0 Vit) 0000 0.2 0.4 0.020 _0.040 0.6 _0.8 0.060 _0.080 1.0

(d)

Z-Z,1 Z0

Figure 8 : The pressure

distribution (p_{- p) at the three time instances}

during water entry of a bow fiare section with

knuckles (see Figure 7).V(t) is the measured drop velocity, p is atmospheric

pressure pis mass density of the fluid, t is time_{variable, t=0 corresponds}

to that the keel touches_{the water} surface, z is vertical coordinate

on the body surface

, z is vertical coordinate of the keel and

z0 is the draft of the body_{. o, experimental results;}

--, fully nonlinear solution; _{- - - - ,simplified solution,}

t =0.06s is the time instant when

the flow seperation

occurs. (a) t=0.06 s _{(b) t=O.07 s (C)}

t0.08 s ; (d) experimental_{drop velocity}

436 FAST '97 PAPERS

'V

_'V

.P3

t.P4

E Cv:, y L 400 mm

_j.

_¡

400 mm

_J

o

Figure 9: Comparison between the empirical formula by Savitsky (1964) ( _{) and the numerical results (A) for the lift coefficient}

and the center of pressure. ß is the angle of deadrise of planing surface(deg),C is the center of pressure _{Tis the trim angle of planing area (deg) and X ¡s the mean}

wetted lengthbeam ratio, a) lift coefficient foro 40 and X = 4, _{) lift coefficient for t = 6° and X = 4, c) centre of pressure fort = 4° and X = 4, d) centre}

of pressure for t = 6° and IL_{= 4, e) lift coefficient for t = 4° and 3 = 20°, f) centre of pressure for t = 4° and 3 = 20°.} (LK - Lc)/B decreases with deadrise angle ß for fixed value

of trim angle -r. f ß is fixed, then_{(LK - L0)/B decreases with} increasing value of -r. If the three-dimensional flow effects

are mainly in the bow region, it can explain why the

differences betweentheory and the empirical formula does

not vary much with X. Lai's (1994) three-dimensional

numerical results are also an indication of thepresence of

three-dimensional effects.

A rational approach for three-dimensional correction_factors

to our theory should be derived. The three-dimensional

near-A SIMPLIFIED NONLINEnear-AR near-ANnear-ALYSIS OF near-A_{HIGH-SPEED PLANING CRAFT IN CALM WATER}

bow solution presented by Fontaine and Faltinsen (1997) could be a starting point for such an analysis. The analog near-bow

solution in the planing problem could be constructed from the solution of steady flow with a small angle of attack past an infinitely long delta-wing. The near-bow flow solution_and the 2-D solution should then be matched.

A conventional slender body theory with no effect on the local water elevation gives _{very different results for the}

hydrodynamic forces and moments. The maximum local force

A 0.12- 0.10- 0.08- 0.06- 0.040.02 -0.05 o CL A 10 20 A = = A 40 20° 40 e) A 1.0 0.80.8 0.4 - 0.2- 0.0-0 p 10 20 30 o = 4° 200 40

f)

¿ 0. A A 0 10 20 30 40 o 10 20 30 40 A d) 1.0 c) t = 40 1.0-t = 6° 0.8 _{IL =4 0.8} -A A A A A A 0.6 0.8 0.4 _0.4 0.2 _0.2 -00 ₀₀ A 0.14- CL _a) 0.25 CL b) 0.12-0.20 = 6° 0.10- _{t = 40} 0. A 0.15 0.08 A 0.06 0.10 0.04 oo-a - 0.05 0.05 0.05

A SIMPLIFIED NONLINEAR ANALYSIS OF A _{HIGH-SPEED PLANING CRAFT N CALM WATER}

per unit length is lower and happen aft of x

_{= Li L and at}

the intersection between the calm water surface and the

chine lines. Further the local two-dimensional force_{is nearly} zero aft of where the local maximum force occurs. Applying Wagner's theory gives similar results as ours for very small deadrise angles and ahead of where maximum local force

occurs. t cannot be applied where flow separation from

chines occurs. By using Figure 5(a) together with_{a change} of variables demonstrates that the Vorus (1 996) method results in lower Uertical forces and different values of Cthen our numerical method. Lai (1994) has presented a three-dimensional numerical method where the wettedarea was

predicted by the Vorus (1996) method. Good agreement with Savitsky's formula was demonstrated.

CONCLUSIONS

A simplified theory for the steady behaviour of_{a high-speed}

planing craft is presented. The theory is verified by_comparing

with a fully nonlinear solution for the water-entry problem and validated by experimental drop tests.Good agreement is demonstrated. The theory is compared with the_empirical formula by Savitsky (1964) for the lift and drag forceand

the center of pressure of a planing craft. The_{agreement is}

satisfactory. The differences are explained by

three-dimensional flow effects in the bow region.

REFERENCES

Falt:nsen, 0M. 'Sea loads on ships and offshore structures," Cambridge University Press, 1990.

Faltinsen, 0M. and Zhao,R., "Numerical predictions of ship motions at high forward speed," Phil. Trails. R. Soc. Lond.A,

Vol.334, pp.241-252, 1991a.

Faltinsen, 0M. and Zhao, R., "Flow prediction around high-speed ships in waves," Mathematical Approaches in Hydrodynamics, Editor: T.Miloh, Soc. nd. Appl. Math. (SIAM), Philadelphia, 1991b. Fontaìne, E., Faltinsen, O., "Steady flow near a wedge shaped bow," Twelfth International Workshop on WaterWaves

andFloating Bodies, Carry-le-Rouet, France, 1997.

Lai, C., "Three-dimensional planing hydrodynamics based on a vortex lattice method," Ph.D thesis, Dept. 0f Naval Architecture and Marine Engineering, University of Michigan, 1994.

Ogilvie, T.F., "Nonlinear high-Froude-number free-surface problems," Journal of Engineering Mathematics, Vol 1,

pp.215-235, 1967.

.Savitsky, D., "Hydrodynamics design of planing hulls," Marine Technology, Vol. 1, No.1, 1964.

Vorus, W.S., "A flat cylinder theory for vessel impact and steady planing resistance," J. Ship Research,Vol.40, No.2, pp.89-106,

1996.

Wagne H.," Überstoss und Gleitverg änge ari der Oberflache von

Flüssigkeiten,"Zeitschr. f. Angew. Math, und Mech., Vol.12, No.4,

pp.193-235, 1932.

Zhao, R., Faltinsen, 0M., "Water entry of two-dimensional

bodies," J.Fluid Mech., VoI.246,pp.593-612, 1993.

Zhao, R., Faltinsen, 0M., Aarsnes, J,/. "Water_{entry of arbitrary} two-dimensional sections with and without flow separation,"

Proceedings, Twenty-first Symposium on Naval Hydrodynamics, Trondheim, Norway, 1996.

438 FAST '97 PAPERS

RONG ZHAO

Sivilingenier, Norwegian Institute of Technology, Norway, 1984 Dr.ing (PhD), Norwegian Institute of Technology, Norway, 1990 Dctechn,. Norwegian Institute of Technology, Norway, 1994

if

Visiting scientist: Department of Ocean Engineering,

MITi September 1992- August 1993

Presently: Senior Research Engineer (since 1989).,,

Norwegian Marine Technology Research Institute. Trondheim, Norway

Between 1985 and 1989, Rong Zhao was Research

assistant at the Norwegian Institute of Technology,