An investigation of the Zarnick non-linear mathematical model of planing craft motions

DeIft University of Technology Ship Hydrornechanics Laboratory Mekelweg 2

2628 CD DaIft The Netherlands PhoneOl5-786882

AN INVESTIGATION OF THE ZARNICK NON-LINEAR MATHEMATICAL MODEL OF PLANING CRAFT MOTIONS

by David Kring

22 February 1988. Reportno.: 786.

INTRODUCTION

The purpose of this report is to present an initial Investigation into an analytical method for determining planing craft motions. Ir. J.A. Keuning is considering a hypothesis concerning these motions. _{The severe bow} accelerations encountered on planing craft in waves has previously been attributed to the effects produced by slamming impact, but, as the work of Ing,. W. Beukelman' indicates, slamming pressures,, for section shapes

with more than a few degrees of deadrise, can _{not be cOnsidered the}

primary cause of these motions. Ir. Keuning. feels that the motions and

accelerations may be due predominantly to the non-linear effects of

bouyancy when the upperhull of the craft is plunged into waves. At the

transition speed between full, planing and full bouyancy conditions normally encountered by planing craft 'in' 'heavy, seas, the bouyancy may

well account for severe motions..

The scope: of this investigation is purposely. limited. This, pilot study is only meant to determine if further consideration of Ir. KeUning"s

hypothesis is warranted. This investigation consisted of analyzing a

planing craft model,, No. 219 tested at the Delft Hydromechanics Laboratory, through precious experimental data,, a non-linear method devised by Zarnick2 of DTNSRDC, and a version of the Zarnick method' modified by the author of this report. The model was compared for motions

in regular, head seas at only one foward spee& .A transition speed of

about

vi/T

_{= 2.5 was used as a fair example of. a planing craft in a sea.}

This speed corresponds to 22 knots for the model.

On this report, a general outline of Zarnick.'s method will be presented. More detailed 'understanding of his. method. can be. gained in, the.. DTNSRDC

report2 or in the discussion of Verkerk's3 .fifth .year work which dealt

extensively with the Zarnick method. Next,, ,'the modifications made to this method, and their supporting rationale in terms of this _{investigation will}

be given. Also, the model used as a sample and _{some specific details, of}

the experiment4, conducted in 1983, will be examined. Finally, the

results will 'be' discussed and sonie conclusions for this _{investigation}

GENERAL DESCRIPTION, OF ZARNICK METHOD

A method and accompanying FORTRAN 77 program to estimate the motions of planing craft in regular seas was developed by E. Zarnick of DTNSRDç in 1978. His method consits of a non-linear analytical approach _{to planing}

craft behaviour:. At successive time steps, the equations of motions, developed from modified low-aspect ratio and strip theories, are solved, and heave., _{pitch, and veitical accelerations at the bow and center of}

gravity are determined. Zarnick, in _{eveloping this method has made a} number of simplifying assumptions in order to deal with the _complexities

of non-linear motions. _{He has assumed wavelengths would be long in}

comparisson to the craft length (over

I

A/1 I) Also he assumed that

the. craft. would be- fully in- the planing region of speed and that the _free surface always. seperates at the chine.

The: general equations of motion are developed: from :Savitsky- with consideration given for the accelerations taken as- D'Alembert inertial forces. The hydrodynamic terms. are -developed from strip theory and Wagner's expanding plate theory for the added mass of -a wedge.

The two-dimensional hydrodynamic force considered at each position

depends upon the rate of change of momentum and drag components. A general formulation of the 2--D force, f, follows:

f =- -( _{(MaV) + CDC pbV2)}

where,

V velocity in -the plane of the cross-seOtion

M added -mass- of the sectiOn CD - -c-ross-flow drag coefficient

b _{instantaneous half-beam of section at free surface} p - density of fluid

The firs-t term, the ra-te of change of momentum, is highly dependent upon

the added mass for accuracy. Becaise the planing craft. in _{regular seas} doe-s not experience in encounter frequency -and the - frequency will be

relatively high Zarnick decided to use the Irequency-independant Wagner formulation of added mass rather than a method such as- Frank close-fit

or Lewis- transformation. For regular seas, this seems to be. a va-lid

In regular seas, even though some very low frequencies will be encountered,, the Wagner theory may be applicable because 'of the low density of occurrence of low frequency waves. At frequency ranges expected, _{Wagner's prediction varied from Frank's by less than two} percent.

Wagner predicts added mass as:

- kir/2 pb2

where,

correction for water pile-up

Because. theha'lf-beam of the-section 'is dependant. upon c'raft.position and .the submergence of the section,., the ..instantaneous.charige.in added mass

with time is:

K irpbb

a a

In order to predict the drag, term, Zarnick has assumed _{a Helmholtz-type}

flow. He estimates the cross-flow drag coefficient .by assuming the

V-section equal to a flat. plate with a Bobyleff flow coefficient':

CD,C - 1.0 cos

In determining the bouyant force acting on the 'section,. Zarnick releis on

the 'work of Shulford. In' s.teady-sta'te..planing,.condition':shulford. found the separation.along. the wetted' surface, 'chine,:and.stern, _{to reduce the} .bouyancy by a factor. oi..one-haiif'...-Zarn.ick..has _{not 'indicated. why he}

consideres this factor to be valid 'for unsteady motion in waves, but 'he uses it to determining bouyant 'force and moment. !fle force is determined

as:

ihere,

correction, factor'

A - instamtaneous submerged are of section

The strip wise integration of hydrodynarnic forces yields general

apgA

equations. of motion as follows:

-3-where,

Ma total added mass

Qa. = first .moment of adde mass about c.g. Ia .= second moment of added mass about c.,g

F' = components of hydrodynamic forces

D skin friction drag

T = thrust components = weight of craft

Using initial conditions of speed (assumed constant), position, and trim with a numerical ramp function for extering _{waves from a steady-state}

condition, _{the accelerations are calculated at each time step.. From} these results,. new positions are calculated and the. model is advanced a step.

In his strip theory, .Zarnick made:two. key assump.tions. _{One was the}

effect of seperation. on bouyancy mentioned.previously.. _{The.other was the}

effect of the.wave profile incontactwth. the planing _{draft. .At;planing}

speeds as he originally envisioned, there were two conditions of

submergence. for any section. Fist, for sections where the wave profile was below the chine, he assumed wave contact at the hull according _to the undisturbed wave profile with a correction added for wave-pile-up effects. As Froude, he assumed. the craft does _{not affect the. basic wave} profile. For this case, the wetted half-beam can be determined as:

b -, 6e cot = r/26cotfl

and, - r/2 cot 6

(M + Ma sin29) _{Xcg + (Ma sinOcosO) Zcg} (Qasin0)0

= Tx + Fx' - Dcos0

(Ma sinOcOsO)

cg + (M + Ma cos20) 'cg - (Q cos 0)0

= Tz= Fz' + D' sinO + W.

-(Qa s:in0)

cg - (Q, cos O).2cg + (I + Ia)0

where,

6e depth after pile-up correction 6 depth of submergence of the section

On the second case, where the wave profile is imposed over the chine, Zarnick assumes total seperation at the chine so that:

b b

max

and,

b=O

where, b - half-beam at chine

These key .. assumptions will be .. altered for the purposes of this

investigation.

MODIFIED ThOD

Planing craft in waves normally travel at speeds less than for full planing. The bow of the craft often plunges into waves causing water to come over the chine and contact the upper hull. Bouyancy plays _{a much}

greate.r part in the motions of planing craft than Zarnick seems to

indicate. This pilot study is meant to determine the relative importance

of bouyancy in the equations of motions: _{as well as begin an expansion of.}

this non-linear analytical. model into more useful. areas of performande.. Although this method may eventually be extensively modified to meet

various. criteria, _{the first modifications are. meant onlly to indicate a} suitable direction of study.

Observation of model tests and full-scale experience with planing craft indicates, three regions of immersion rather:than the two proposed by

Zarnick. When the wave profile is over the chine the f low does not always

seperate.. Depending upon speed, deadrise,. and wave profile, certain

portions of the upper hull become immersed while at other portions the flow still seperates at the chiie. In order to realistically model this flow a great deal, more study must be given to this problem. _{In order to} compare relative effects of added bouyancy to the hull along upper and lower surfaces, at this stage an assumption of no seperation _{at the chine} will be made. The bouyancy correction will be taken as one and the _wave profile will be used as the ftee-si.irace contact w.ith the hull, taking

It is believed that the correct results lie somewhere between _{these two} opposing assumptions of separation conditions. The modified method should. indicate the sensitivity of the buoyancy as well as the consIderation

warranted for Ir. Keuning's hypothesis.

The actual modifications to Zarnick's original method were actually quite

simple. The factors affected are _{the half-beams of the sections and}

bouyancy correction. The .bouyancy correction was set to one for comparison to the recommendation of Shuiford.

The planing craft was divided into two regions, _{wave profile above or} below, the chine. For sections with submergence less _{than the chine, the} program remained unaltered. For profile above, the chine, _{the deadrise}

angles,

2' of the upper hull were added to the program and the. half-beam becomes:

b - &:e_chine cot fi,, + (6e - 6 ) cot. chine

where,

6.e _{the effective depth to the chine, from the keel} chine

Added mass, a vital concern was satisfactorily determined _{to correspond} to Wagner's expanding plate theory for cases of immersion over the chine. The sigularity at the chine _{is simply ignored and the instantaneous}

change, b, is set equal to that for the lower hull. Comparison of Wagner with Frank close fit method yielded, less than _{two percent difference in}

added mass for a sample sectIon. Thisr analysis was..performed _{with the aid} of ir. Journe'é and: his Frank close fit routine.

EXPERIMENTAL RESULTS

in order to gain more than just a relative comparison of _{methods, and} experiment, performed on "Model No. 219 at the Delft Hydromechanics Laboratory in 1983, was used as a 'saiñple for analysis. The _{model had} nearly constant deadrise above and below the chine _{at e'ach station. It}

had been tested in regular wives of appropriate wavelengths _{(I L/A} 0.5 to 1.2) and: _speed .(V/1T1'

-

_{2.45) for the purposes of this}

investigation. The. model in these conditions was operating in a transition region between full planing and full displacement conditions..

Primarily, the experimental data was used to verify' the analytical methods and to examine Ir. Keuning's hypothesis.

-6-RESULTS

The. results of this investigation consits of a series of plots comparing the original Zarnick method, the modified version, and the. _experimental results.. The first figure shows heave amplitudes _{non-dimenslionalized by} wave elevation. The amplitudes shown are the highest magnitude of heave encountered at each wavelength. Non-dimensionalized p.itch amplitudes _are

seen in figure two, Figures three and four show the greatest

accelerations at bow and center of grav.ity respectively. The figures for

acceleration given, from the experiment had to be adjusted. in the lab. report the accelerations were incorrectly non-dimensionalized by dividing by acceleration due to gravity twice rather than once.

DISCUSSION

For heave., the predictions of the modified method came closer to

experiment in. longer waves while the Zarnick method predicts more. accurate values in shorter waves. _{The reason for this phenomenon is} unknown and will take more investigation to determine _{However, both}

predictions give fair correspondence to experimental results and seem to

bracket the experiment as expected. This indicates that there _{are greater}

bouyancy effects for this sample than Zarn.ick.predicted.

The relation of the predicted pitch motions ot the experiment for the modified version were vastly improved over the original method.

Correlation was within ten percent at all points between the mOdified and experiment results, while the original .method.varied.from _{experiment by}

up to 25 %. _{This seems. to indicate that there are bouyant effects and}

that the assumptions of this investigation are meaning ful.

The. accelerations gave the most interesting results in terms of this

study. in the cases of both bow and c.g. acceleration, the peaks

encountered by the experiment were accurately predicted by the modified

method but not the original method., The magnitude of the peaks, as

expected, was less than the modified version but much greater than the

original. 'This indicates' that the bo.iyant force is the predominant spring

force for this sample. The experiment appears as the modified result but with greater damping.

CONCLUSIONS

First, the relative difference between the original and modIfied Zarflick

methods indicates a greater need for attention to the bouyant spring force. The change to- bouyancy _{in terms of i'ave profile along the hull}

and bouyancy correction factors for seperation must be studied' more carefully. The accelerations strongly favor the hypothesis of ir.

Reuning. The agreement in' peak location 'of accelerations in even, one

sample make the bouyancy hypothesis very interesting and further study is

indeed warranted'.

As Zarnick 'originally intended, _{this. program.can' be very useful if the} user modifies, i,t _{for, his particular'.'needs. .The' program -is. a. simple} non-linear theoretical.model that allows for a' ;gr,eat' dea'l of extensIon.

By modifying this program individual..factors - can be -isolated, '..studied,

and modified. _{It will be possible to -incorporate transitional speed}

ranges, _{shorter wavelengths, and irregular seas if proper attention to.} details is applied.

REFERENCES

1] _{L B,eukelman, "Bottom impact Pressures Due to 'Forced Oscillation",}

international. Shipbuilding Progress,,. Marine Technology _Monthly,

Vol. 27,,, No. 309, May 1980.

[2'] Zarnick, Ernest E., _{"A. Non-linea'r,:.'Mathema'tica'l ModeL of.:Motions 'of}

a Planing Boat in Regular Waves"; DTNSRDS.-78/032, March1978.

Verkerk, Freek, "Pianerendé Schepen in Golven", 5e j'aars werk, Deift TU, Juni 1983.,.

Model No. 219, De'lft. Hydromechan'ics Laboratory, 19,83.

-8-APPDIX I:: MODIFICATIONS TO FORTRA1 77 PROGRAM CODE OF ZARNICK METHOD

The program SBSLKEUN.CNTL (ODWH) in the user-id SBSLLEX _{on the tso ystem}

was used as the basic Zarnick method. Two versions of this code were

copied for purposes of modification. The first was;

ODWDAVE

;which contains the modification of the Zarnickmethod _{to be compared}

with his original version.. The second method was left essentially

unchanged and results from this were coSidered as the Zarnick predictions. It s named:

ODWDAVE 2

Both programs required modification .in,the :execution:scommands.in, _order

to receive output.. _{This task was performed with the aid of ing. De} Zwaan. Also, the data for model 219 _{was checked and correlated wi:th the} experiment for both programs. Data for model 219 is contained in, _ODWH, but this was found to contain many differences from the real model data

and required checking.

The specif ire FORTRAN code modifications to, ODWDAVE, are relatively

few and are as follows:

the addition of

BETAZ (120), CDZ(120),, TAZ(120)

to the common blocks of all routines 'at

COMMON/CONST/

Various correction to the dock block according to the model No. 219

including

DATA BETAZ/- , ,

-for inclusion of the upper büi

also, according to this modified _{heory, the bouyancy correction factor,} ABM, was set equal to 1.0

in subroutine input, at lines 30 through 32, the following was added:

BETAZ(I) BETAZ(I)* C. 017453292519 CDZ(I) = COS (BETAZ(I))

TAZ( I) TAN( BETAZ (I)>

in order to account for the new data

in subroutine DAIJX,. the. half beam and added mass terms are determined. These were altered to conform to the modified theory, as Follows:

at lines 32 to 34 for the sections where the chine is immersed,; the

code is now:

32 70 B(I) TEST(I)*1.jTA(I).*yH + (D(I)-TEST(i:))*.P/H

+ (D(I) -TEST(I)*.(.1 /TAZ(I) )*P/H 33 Bi(.I) - B:(I)

34 NA( I) - I(*P}1ALF*B( I) *B (I)

this replaces the previous code where

B(I) TEST(I)

BI(I)= 0

MA(I) = *PLF*B(I)*B(I.)

where TEST(I,) maximum beam at chine

The .bouyancy correction is accountedfor.:in line 70 of subroutine FUNCT in the term DS(I).

The rate of change in added mass is accountedfor.through the _{term Bi(I)}

in line Si of FUNCT.

At slight error exists in this method,, which will not adversely affect.

the results of the pilot study.

For the immersed chine the Statement at line Si of, FUNCT should _read. D1(I) VEL*BL(I)*(.X(Z) X(ç) + (cXG*E(i) + S XC + N(I))-DRDT)/TAZ(I,) For the nonimmersed chine the line rernans as befOre.

APPE1DIX Ii: PROGRAM LISTING

A complete record of the modified FORTRAN 77 code and an example of output is included.

Comparisons - using Model 219 results

Zar Zarnick, original method of prediction (assumes complete

separation at chine)

Mod Modified Zarnick prediction (assume full bouyancy effects and no seperatiori at chine)

Exp Experimental results from Model 219

- 12

-'Heave Amplitude (.Za/ea) Pitch amplitude ($a/(211c/A))

I

L/A'

Zar. Mod. Exp. 'Zar.. Mod. Exp.

0.5 1.56 1.29 1.34 1.20 0.83 0.97 0.6 1.41 1.30 1.43 1.19 0.88 0.91 0.65 0.98 1.18

l27

0.85 0.86 0.87 0.7 0.66 0.95 1.04 0.60 0.69 0.79 0.75 0.38 0.80 0.79 0.37 0.56 0.66 0.8 0.21 0.64 0.51 0.21 0.41 0.45 0.9 0.09 0.34 0.08 0.09 0f24 0.16 1.0. 0.04 0.16 0.09 0.03 .0.12 0;'06 1.1 0.06 .0.05 0.11 0.03 r 0.05 0.02 1.2 0.05 - 0.09 0.02 - 0.01 Magnitude of Acceleration

atBow

a.lwl g a

atCG

Zar. Mod. Exp. Zar. Mod Exp:.

I

L/A' 0.5 13.5 11.8 13.7 10.5 8,8 10.3 0.6 28.3 25.0 ., 24,5 16.4 16.6 16.2 0.65 34.1 38.7 35.3 l7.2 19.4 17.7 0.7 22.9 6,9.1 44.1

]l9

' .15.6 17.2

O75

20.7 ' 79.9 52.0 8.06 20.4 17.7 0.8 15.2 . 68.7 49.5 5.7 18.9 13..2 0.9 10.4 ' 56.2 32.4 2.8 13.7 "' 9.,3 1.0 9.8 .24.8 ' 28.9 ' 2.5 13.0 ,, 7.8 1.1 10.9 24.4 ''28.0 3.5 1.8 7.4

Model 219

V 22 knots (3.582 mis) Stact-Os Ramp /s

Z0=2..5 mm _{Run: 0 to 8 s}

3.7

a

Unmodified Method of Zarnick (complete separation at chine)

A(m) Heave Pitch Bow Accel. 2 (in/s ) CC Accel. 2 (rn/s ) .1 L7A'

a/2i/A) max

mm max mm

0.5 8.48 1.29 0.834 5.44 - 4.24 4.09 -3.77 0.6 5.8,9 ,1.30 0.882 11.6 -11.5 7.67 -5.27 0.65 5.02 . 1.18 0.859 12.9 -.17.9 8.97 -8.32 0.7 .4.33 0.947 0.69.2 12.1 -32.0 6.10 -7.23 0.75 3.77 0.803 0.556 12.6

370

7.99 -9.44 3.31 0.643 0.413 13.0 3.i.8 _{7.15 -8.76} 2.62 0.342 0.239 12.3 -26.0 6.36 -5.70 1.0

212

0.162 O.115 . I1..,5 - 8..90 6.02 -4.61 1.1 1.75 0.051 0.054 11.3 - 7.421 36'2 -3.15 1.2 1.47 - - . -0.85 2.93 0.482 0.352 13.0 -30.3 . 7.5.8 -8.53 0.95 2.35 0.245 0.167 11.7 20.7 6..77 -5.63

Modified Method of Zarnick (assuming no separation)

-14

A(m) Hea;e Pitch

Bow Accel. (m/s2) CC Accel.. i

(W2)

I

L/." max mm max 0.5 8.48 1.56 :1.20 6.25 - 5.36' 4.87 -4.24 .0.6 5.89 1.41 H i.i9 10.9' -13.1 7.58 -'6.5,5 0 65 5 02 0 975 0 851 11 2 -15 8 7 94 -5 89 0.7 4.33 0.664 0.598 H 10.6 - 8.30 5.52 -4.84 0.75 3.77 0.37.7 0.371 _H 8.90 - 9.57 3.72 -3.73 '0.8. 3.31 0.214 0.209 .7.04 - 6.50 2.65 -2.4:6 O:.9 .2.62 0.090 0.086 4.82.: - 4.43 1.30 -0.9.88 1.0 2.12 '0.043 0.031 4.55 .: - 4.19 1.20 -1.07 1.1, 1.75 ' ' '0.:061 0.025 H 3.46 - 503 .1.62 ' -1.25 1.2 1.47 0.046 : 0.019 . '' 4.68 - 5.67 1.65 -1.45 0.55

7.1

1.57 1:29 . 8.38 - 7.27' 6.83. -5.53 0.775' 3:.35 '. 0.280 : 0.25,5 7.18 - 5.541 1.75 -3.04

$0.8

MODEL 219 at 22 knots

15 -Zarnick, unmodified

- - - modified

-. - - - experiment 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

8.48

5.8:9 4.33 3.31 2.62 2.12 1.7,5 1.47 - A(m)

Fig. 1. Heave amplitudes for model 219 in long-crested,

regular waves.

1.4 1.2 1 ,O .0 8 0.. 6

0.4-0.2

MODEL 219 at 22 knots

- 16 -Zarnick, unmodified

modified

experiment A(m)

Fig. 2. Pitch amplitudes for model 219 in long-crested,

regular waves.

0.5 0.6 0.7 0.8 0.9 1.0 :1.1 1.2

MODEL 219 at 22 knots

I

/

/

/

I

I

I

Zarnick, unmodified

-modified - experiment 17 -A(m.)

Fig. 3. Magnitude of acceleration. at 10% aft bow in

long-crested, regular waves.

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

8.48

_5.89

_4.33

_3.31

_{2.62 2.12}

_1.75

_1.47 80 70 60 50 40 30 '. 20 10 0 0

0 C) :20 10

MODEL 219 at 22 knots

Zarnick, unmodified

- - modified

-! - experiment 0

Fig. 4. Magnitude of acceleration at c.g., 60% aft ç

bow in long-crested, regular waves.

18 -0.5 0.6 0.7 0.8 0.9 1.0 1.1 8.48 5.89

4.33

3.31

2.62

2.12

1.75

A(rn) 40 30