Post Processor HullForm

8/3/2019 Post Processor HullForm

1/23

Post-Processor Hull form

During early stage ship design it can be useful to develop a rough hull form tobetter estimate the ship's total enclosed volume and internal/external layouts.The development of a complete fully fair set of hull lines can take a relativelylarge amount of time, but for early stage designs a rough/approximate set oflines can be developed relatively quickly.In general, there are many ways to put together a rough set of lines, but here Ihave focused on three main options which mathematically relate a ship's hullshape to certain specific hull parameters. These three options are;

A method developed by RADM David W Taylor where a ship's DesignWaterline and Sectional Area Curves are defined by separate 5th orderpolynomials for the fore and aft sections and below water section aredefined by either a 4th order polynomial or a hyperbolic eqation based on

its relative fullness.

(More information on the Taylor methodologycan be found here) A method developed at the Netherlands Ship Model Basin by G. Kuiper

and published in the March 1970 Issue of the Society of Naval Architectsand Marine Engineers "Journal of Ship Production" where waterline curvesare defined by separate 7th order polynomials for the fore and aft

ti H i l d fi d t l t t lik LCF


2/23

sections Here special curves are defined to relate parameters like LCF

Length = Lpp Height = Dm

Next the user needs to define; the bow slope the transom slope whether the ship has forward shear

o what type of shear it haso where the shear startso the height of the shear at the bow

whether the ship has aft shearo what type of shear it haso where the shear startso the height of the shear at the bow

the main deck slope the location where the lower hull begins to sweep up towards the

transom the radius of the hull at the location where the lower hull begins to sweep

up towards the transom the bow forefoot radius (if any)

Combining this with the draft at the ship's transom (derived from the Tt/Txdefined in the initial hull form definition) you get a ship's profile as shown below.

Please note here that I've also allowed for adding in a Bow Sonar Profile to assist


3/23

As such the selected polynomial is of the form;

= + + + + + +

And

= + + + + + +

Where, @ x = 1 = + + + + + + = slope of the curve @ the bow

The other main constraints then are the same as for the 5th order curve, andinclude;

@ x = 0, y = the Transom Area Coefficient (At/Ax) @ x = 1, y = 0 (assuming the ship does not have a bulbous bow) @ x = 0.5, y = 1 (assuming that the midship section is equal to the

section of max area)

@ x = 0.5, y' = 0 (as above) y dx = Cp yx dx = Cp ( 0.5 - LCB / 100 )

For reference, I have included both the 5th order curve in the spreadsheet to

help give the user a reference point while manipulating the 6th order curve as


4/23

But, I eventually decided to revise this a little and use a 6th order polynomial

and an additional constraint based on the slope of the curve at the bow, becausethe Design Waterline 1/2 Entrance Angle is one of the parameters needed for theFung & Leibman 2 resistance estimation methodology that I am using.

As such, I have the same type 6th order polynomial for the Design Waterline asthe Sectional Area Curve.

However, as noted above for vessels of the type we are considering with their

relatively low block coefficients, although it is expected that they probably won'thave much if any parallel mid body, it is possible that they may have some flat ofside. As such, I am considering possibly updating the definition of the DesignWaterline Curve to a higher order polynomial to allow the user to better definethe extent of any possible flat of side on the ship.An example of the Design Waterline Development is shown below.


5/23

slope (rake). As such, I have scaled the design Waterline appropriately to give a

better sense of how the two waterlines relate to each other.Similarly, for the intermediate waterline I have added both the Design Waterlineand the Weather deck outline (appropriately scaled) for reference.By tweaking and adjusting the coefficients and parameters of these upperwaterlines the user can influence the shape and flare of the above water hullform. Eventually, as I review more information on existing designs, I hope to be

able to provide some guidance curves on what might be suitable values forWater plan Coefficients, Longitudinal Centers of Flotation and the like for theupper waterlines based on the values used for the Design Waterline andFreeboard, etc. Additionally I also want to better update the setup to allow forflare at the ship's midsection.

Below are samples of the Weather deck and Intermediate Deck Definition Panels.


6/23

Below Water Hullform DefinitionFor the underwater hullform definition I have chosen to use the samemethodology outlined in the Taylor Method. In doing so I have set up a lookuptable to define the appropriate value for the coefficient c, based on the sectionsare coefficient m and flare angle f, as shown below.


7/23

Table for Estimating Values of c

Table for Making Initial Estimate of Deadrise

m 0.75 0.74 0.72 0.70 0.68 0.66 0.64 0.62 0.60 0.58 0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 0.32 0.30

Deadris

e 18.95 20.35 23.83 28.63 35.55 45.95 61.84 84.11 90 90 90 90 90 90 90 90 90 90 90 88.57 83.65 73.56 58.93 37.09

f m

0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.8225 0.09248 0.06744 0.04750 0.03177 0.01933 0.00982 0.00294 -0.00076 -0.00174 -0.00030 0.00324 0.00854 0.0152920 0.23008 0.18256 0.14457 0.11385 0.08875 0.06802 0.05085 0.03678 0.02522 0.01586 0.00844 0.00832 0.0047915 0.43081 0.34752 0.28258 0.23100 0.18900 0.15446 0.12564 0.10159 0.08137 0.06428 0.04982 0.03763 0.0273310 0.72698 0.58434 0.47563 0.39148 0.32468 0.27063 0.22632 0.18904 0.15800 0.13170 0.10927 0.09005 0.073525 1.21249 0.94226 0.75383 0.61537 0.50914 0.42548 0.35846 0.30325 0.25748 0.21911 0.18650 0.15862 0.134610 2.10120 1.54374 1.18760 0.94554 0.77056 0.63863 0.53469 0.45349 0.38698 0.33124 0.28502 0.24570 0.21209

f m 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.92 0.93 0.94 0.9525 0.02316 0.03185 0.04107 0.05055 0.06002 0.06927 0.07811 0.08638 0.09396 0.10077 0.10680 0.11206 0.1166520 0.00246 0.00124 0.00091 0.00130 0.00224 0.00360 0.00528 0.00720 0.00935 0.01173 0.01443 0.01756 0.0213015 0.01871 0.01162 0.00594 0.00118 -0.00603 -0.00903 -0.01113 -0.01235 -0.01269 -0.01206 -0.01036 -0.00742 -0.0030110 0.05928 0.04700 0.03643 0.02736 0.01963 0.01311 0.00770 0.00341 0.00035 -0.00168 -0.00306 -0.00410 -0.004955 0.11376 0.09571 0.07997 0.06622 0.05419 0.04367 0.03448 0.02649 0.01957 0.01364 0.00863 0.00451 0.001340 0.18307 0.15788 0.13590 0.11673 0.09976 0.08484 0.07164 0.05999 0.04969 0.04055 0.03249 0.02539 0.01918


8/23

Body Plan Development


9/23

Bulkheads & Wateline DevelopmentAfter defining the shapes of the above waterline hull shapes and below waterlinesection shapes, it is a fairly easy task to interpolate other sections andwaterlines, and a section is incorporated into the spreadsheet to allow the userto do so.Hull Volume EstimateHaving defined the hullform section shapes it is also a fairly easy task to have thespreadsheet calculate the total enclosed volume of the hull. A plot showing hedistribution of this volume is shown below.


10/23


11/23

FB0 = 1.011827 * T - 0.0000209 * L ^ 2 + 0.027806 * LFinally, this paper also provides an equation for the Seakeeping Rank Estimatorthat was developed by Bales. Here;Rcap = 8.42 + 45.1 * Cwp(fwd) + 10.1 * Cwp(aft) - 378 * T / L + 1.27 * c / L - 23.5 *Cvp(fwd) - 15.9 * Cvp(aft)Where;

c = the Longitudinal Location of the hull cut-up aft of the FP T = Draft L = Lpp Cwp(fwd) = Waterplane Coefficient for the Fwd Section of the Hull Cwp(aft) = Waterplane Coefficient for the Aft Section of the Hull Cvp(fwd) = Vertical Prismatic Coefficient for the Fwd Section of the Hull Cvp(aft) = Vertical Prismatic Coefficient for the Aft Section of the Hull

This equation is set up to estimate the relative seakeeping capability of a vesselbased on the above parameters, with Rcap ranging from 0 to 10. The higher theresultant Rcap, the better the relative seakeeping capabilities of the hull isestimated to be.I have included these equations into the spreadsheet to give guidance on

h th d i h d t f b d b d th d ti


12/23

RADM DW Taylor's Mathematical LinesAs long ago as the early 20th Century Rear Admiral David W. Taylor (USN) proposed a methodology where a ship's underwaterhullform could be defined through a set of mathematical equations. Specifically, RADM Taylor proposed that a ship's sectional area

curve could be broken into segments, including the fore section, the aft section, and a mid section (if the ship has any parallelmidbody). In this scheme the fore and aft sections of the curve are approximated by fifth order polynomials, of the form:

= + + + + +If you non-dimensionalize the curves, where both x and y range from 0 to 1, such that;

y = the ordinate of the sectional area curve at any atation along its length (as expressed as a fraction of the maximum) x = the distance along the curve. For a ship with no parallel middle body, and an equal length fore and aft section, x = 1

would be equivalent to 1/2 the ship's Lpp t = the slope of the curve at its origin (numerically equal to the intercept of the tangent of the origin on the maximum

ordinate (eg x = 1), see figure below


13/23

Then, if the ship does not have a bulbous bow

@ x = 0, y = 0 @ x = 1, y = 1 @ x =1, y' = 0 (where y' = the 1st derivative of the curve)

= + + + +


14/23

Therefore

+ + + = @ x = 1, y" = (where y" = the 2nd derivative of the curve)

= + + +

Therefore+ + + =

the area under the curve (as defined by y dx) = Cpfwhere,

= +

+

+

+

= + +

+

+

Then, you end up with 5 equations and 5 unknowns. As such, it is fairly easy to solve for the coefficients, as shownbelow. Specifically; a = 60 Cpf - 6t -30 -1/2 b-180 Cpf + 12t +100 +21


15/23

c = 180 Cpf - 10t - 105 - 5/2 1 d = -60 Cpf + 3t +36 + 1

The aft section of the sectional area curve can be handled similarly, as can the fore and aft ends of the design waterline. For theaft end of the Sectional Area curve, you just need to substitute Cpa for Cpfand use an appropriate value of t to give you the shapethat you wish for this section. Similarly the value of1 should be the negative of the value of1 from the fore section to ensurecontinuity. Typically in many/most cases 1 can often be assumed to be 0 for both segments. For the Design Waterline, youshould substitute either Cwl(fwd) or Cwl(aft) for Cpf and either the 1/2 entrance angle or the 1/2 exit angle for t,depending on which end of the waterline that you are interested in.For the underwater hull sections RADM Taylor found that a single equation did not appear to be suitable for all possible sections so;

for sections with a sectional area less than about 0.75 he proposed a 4th order parabola = + + +

for sections with a sectional area greater than about 0.667 he proposed a hyperbolic equation

= + ( + )

for sections in between 0.667 and 0.75, it was not that either equation gave reasonably similar shapes, as such herecommended using a sectional area of about 0.72 as the dividing point to determine which curve to use

By doing similar mathematical manipulations as above (bearing in mind that the coefficients a, b, c, and d here are not the same asthose above) you can solve for the coefficients in terms of other known values.Specifically, if;

y = the half-breadth at any waterline as a fraction of the value at the design waterline (ie y = 1 @ x = 1) x = the distance above the keel as a fraction of draft (ie x = 1 @ the design waterline)


16/23

m = the local sectional area of the section in question = the reciprocal of the dead rise angle for the section f= the flare angle for the section

Then, for the 4th order parabola

a = 3/2 f - 9/2 + 30 m - 12 b = -4 f + 6 - 60 m + 28 c = 5/2 f - 5/2 + 30 m + 15

And, for the hyperbola a = f - c ( 1 - f ) b = ( 1 - f ) * ( 1 + c )^2 d = ( 1 - f ) * ( 1 + c ) ^2 * c

It is harder to solve for c but since y dx = mThen,

= + . +

Or substituting a, b, &d from above

=

+ ( ) ( + )

(

+

) . +

Using a spreadsheet like MS Excel it becomes fairly easy to set up an interpolation table, where for various values of m and fyoucan estimate the appropriate value ofc to use.


17/23

The figure below shows a set of section shapes developed using this methodology, for a section area ratio (m) ranging from 0.475to 0.95, with flare angle (f) of zero.

An important factor to bare in mind here, is that when using this method for drawing up a ship's rough lines, you will have totransform some data for use. Specifically, using the polynomial equations for the Design Waterline and Sectional Area Curvecan give you values for 1/2 breadth as a fraction of the ship's maximum overall 1/2 breadth, and sectional area for the station as a

fraction of the ship's maximum sectional area. To draw the actual section shape you should then transform the sectional are valuethat you get from the equations into full scale sectional area, and then figure out what the local sectional area value would be forthe max beam and draft at that station.


18/23

The method outlined above can be very useful, but because it treats the fore and aft sections of the ship separately, it relies oninputs of hull form coefficients such as Cpf, Cpa, Cwl (fwd), Cwl(aft), LCB(fwd), LCB(aft), LCF(fwd), and LCF(aft), etc thatare not commonly available for most ships.


19/23

Under Development


20/23

The Istanbul Technical University MethodHere is another method, that I believe is from the Istanbul Technical University, in which a ship's entire waterline or sectional areacurve can be treated as a single 5th order polynomial for ship's with no parallel midbody or as a 7th order polynomial for ships with

parallel midbody.Specifically, for waterlines and sectional area curves without parallel midbody;

= + + + + + + Where x = 0 is the transom and x = 1 is the bowHere, for the sectional area curve, the boundary conditions are;

@ x = 0, y = the Transom Area Coefficient (At/Ax) @ x = 1, y = 0 (assuming the ship does not have a bulbous bow) @ x = 0.5, y = 1 (assuming that the midship section is equal to the section of max area) @ x = 0.5, y' = 0 (as above) y dx = Cp yx dx = Cp ( 0.5 - LCB / 100 )

For the Design Waterline, these boundary conditions are; @ x = 0, y = the Transom Beam Coefficient (Bt/Bx) @ x = 1, y = 0 (assuming the ship does not have a bulbous bow) @ x = 0.5, y = 1 (assuming that the midship section is equal to the section of max beam) @ x = 0.5, y' = 0 (as above) y dx = Cwl yx dx = Cwl ( 0.5 - LCF / 100 )


21/23

For ships with parallel midbody

= + + + + + + +

Here, for the sectional area curve, two additional points are defined as

pf = the forward extent of the parallel midbodypa = the aft extent of the parallel midbody

The boundary conditions are; @ x = 0, y = the Transom Area Coefficient (At/Ax) @ x = 1, y = 0 (assuming the ship does not have a bulbous bow) @ x = pf, y = 1 (assuming that throughout the extent of the parallel midbody sectional area = maximum area) @ x = pa, y = 1 (as above) @ x = pf, y' = 0 (as above) @ x = pa, y' = 0 (as above) y dx = Cp yx dx = Cp ( 0.5 - LCB / 100 )

For the Design Waterline, these boundary conditions are; @ x = 0, y = the Transom Beam Coefficient (Bt/Bx) @ x = 1, y = 0 (assuming the ship does not have a bulbous bow) @ x = pf, y = 1 (assuming that throughout the extent of the parallel midbody sectional area = max beam) @ x = pa, y = 1 (as above) @ x = pf, y' = 0 (as above) @ x = pa, y' = 0 (as above) y dx = Cwl yx dx = Cwl ( 0.5 - LCF / 100 )


22/23

Using the Matrix ("Array") capabilities of a Spreadsheet Program like Excel it is fairly easy to solve for the coefficients in the originalpolynomials.

A sample of the 5th order and 7th order polynomials is shown below.


23/23

Post Processor HullForm

Documents

Transcript of Post Processor HullForm