## Post-Processor Hullform

During early stage ship design it can be useful to develop a rough hullform to better 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 relatively large amount of time, but for early stage designs a rough/approximate set of lines can be developed relatively quickly.

In general, there ae many ways to put together a rough set of lines, but here I have focused on three main options which mathematically relate a ship's hull shape to certain specific hull parameters.  These three options are;

• A method developed by RADM David W Taylor where a ship's Design Waterline and Sectional Area Curves are defined by separate 5th order polynomials for the fore and aft sections and below water section are defined by either a 4th order polynomial or a hyperbolic eqation based on its relative fullness.

• A method developed at the Netherlands Ship Model Basin by G. Kuiper and published in the March 1970 Issue of the Society of Naval Architects and Marine Engineers "Journal of Ship Production" where waterline curves are defined by separate 7th order polynomials for the fore and aft sections.  Here special curves are defined to relate parameters like LCF, 1/2 entrance angle, waerplane coefficient, etc vary over depth.

• A method developed by the Istanbul Technical University (ITU) where a ship's entire design waterline or sectional area curve is defined either a single by a 5th order polynomial for ships with no parallel midbody, or a single 7th order polynomial for ships with parallel midbody.

A Combined Method

For the purposes of the design tool I am developing, I have chosen to combine equations similar to tjose use in the ITU method for the sectional area curve and design waterline with RADM Taylor's equations for the shape of the underwater hull sections, as well as some additional information to define the hull's profile and above waterline shape.  I have set up an EXCEL spreadsheet to assist the user in doing these calcs.

Profile Defintion

The 1st step in this combined method is to define the ship's profile.  Starting with the ship's principle dimensions, the you can define a box of;

• Length = Lpp
• Height = Dm

Next the user needs to define;

• the bow slope
• the transom slope
• whether the ship has forward shear
• what type of shear it has
• where the shear starts
• the hieght of the shear at the bow

• whether the ship has aft shear
• what type of shear it has
• where the shear starts
• the hieght 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 trasnsom (derived from the Tt/Tx defined in the initial hullform 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 in showing what the ship will look like if it has a bow sonar dome.  [Currently, I have only one profile available, but I hope to add more later].

Sectional Area Curve Development

Next it is necessary to draw the ship's Sectional Area Curve.  For vessels of the type we are considering with their relatively low block coefficents, it is expected that they probably won't have much if any parallel midbody.  As such, I initially considered using the 5th order polynomial outlined in the OTU methodology.  However, 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.  I did this in part in order to give the user a little more control over the forbody shape as using the default 5th order polynomial sometimes gave sections that were a little too full to match up with the 1/2 entrance angles of the corresponding design waterlines typical of the type ships under investigation.

As such the selected polynomial is of the form;

y = a + bx + cx^2 + dx^3 + ex^4 + fx^5 + gx^6

And

y' = b + 2cx + 3dx^2 + 4ex^3 + 5fx^4 + 6gx^5 + 7hx^6

Where, @ x = 1

y' = b + 2c + 3d + 4e + 5f + 6g + 7h = slope of the curve @ the bow

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

• @ 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)
• S y dx = Cp
• S 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 shown below.

Design Waterline Development

Next it is necessary to draw the ship's Design Waterline.  As above, I initially considered using the 5th order polynomial outlined in the OTU methodology.  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, because the Design Waterline 1/2 Entrance Angle is one of the parameters needed for the Fung & Leibman 2 resistance estimation methodology that I am using.

This initially resulted in using a similar type 6th order polynomial for the Design Waterline as was used for the Sectional Area Curve.  However, eventually I found that it was sometimes difficult to control the shape of this curve for some desings.  As such, I decided to change one of the boundary conditions to allow the user to define the "1/2 Exit Angle" (or as it is often called the "1/2 Run Angle") similar to how the user inputs a "1/2 Entrance Angle".  In doing so the Longitudinal Center of Flotation has now been changed to be an "output" from the calculations rather than being a user input.

Additionally, I have also given the user the ability to enter a location for where the maximum section areas occurs rather than assuming that this value falls at the exact middle of the design waterline (0.50 of L).

An example of the Design Waterline Development is shown below.

Above Water Hullform Definition

Next, in order to help define the angle of flare of the sideshell along the length of the ship, it is necessary to better define the shape of the above water hullform.

To do this, I have chosen to set up a methodology where the user 1st defines the shape of the weatherdeck outline, and then the shape of an intermediate waterline halfway between the Design Waterline and the Weatherdeck.

For the intermediate upper waterline I chose to use the same type polynomial as for the Design Waterline, where the section of maximum area is assumed to be at the same location as for the Design Waterline.  For the Weatherdeck though, I decided to increase the order of the polynomial to allow the user to incorporate some degree of "parallel" length.  Here the fore and aft extents of the "parallel section" of the ship's deck edge is entered as a fraction of the ship's overall length, and due to the math used for the equation, if it is desired to draw a ship with little to no "parallel' shape the user should enter values for these entries close too each other, but not exactly the same, to avoide getting a math error.

For reference when laying out the weatherdeck I have added the outline of the Design Waterline for reference.  Here though, since the Weatherdeck will be longer, due to the bow and transom 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 Waterline and the Weatherdeck outline (appropriately scaled) for reference.

I believe that the revised formulas for the Design Waterline, Mid Upper Waterline and Weather Deck allow the user to better control the shape of a ship's upper hull, and in particular allows for expanded "flare" forward (as shown in the image a little further down this page).  By tweaking and adjusting the the coeffients and paramters of these upper waterlines 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 for Waterplane Coefficients, Longitudinal Centers of Flotation and the like for the upper waterlines based on the values used for the Design Waterline and Freeboard, etc.  Additionally I also want to better update the setup to allow for flare at the ship's midsection.

Below are samples of the Weatherdeck and Internediate Deck Defintion Panels.

Below Water Hullform Definition

For the underwater hullform definition I have chosen to use the same methodology outlined in the Taylor Method.  In doing so I have set up a lookup table to deine the appropriate value for the coefficient c, based on the sections are coefficient m and flare angle f, as shown below.

For sections with an area coefficent m of less than 0.72 I have also setup a simple lookup to recommend a value of recipricald eadrise angle l to use, as shown below.  In general these values will help the user make a first initial estimate, but this may need to be revised for each individual ship.  To assist in better refining these values for the finer hull sections I have set up the EXCEL Spreadsheet to plot out the calculated deadrise of the other sections to give the user a feel for the range of values and any trends through them to assist in estimating alternate values to use.

An example of this section of the spreadsheet is shown below.

Table for Estimating Values of c

 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 0.82 25 0.09248 0.06744 0.04750 0.03177 0.01933 0.00982 0.00294 -0.00076 -0.00174 -0.00030 0.00324 0.00854 0.01529 20 0.23008 0.18256 0.14457 0.11385 0.08875 0.06802 0.05085 0.03678 0.02522 0.01586 0.00844 0.00832 0.00479 15 0.43081 0.34752 0.28258 0.23100 0.18900 0.15446 0.12564 0.10159 0.08137 0.06428 0.04982 0.03763 0.02733 10 0.72698 0.58434 0.47563 0.39148 0.32468 0.27063 0.22632 0.18904 0.15800 0.13170 0.10927 0.09005 0.07352 5 1.21249 0.94226 0.75383 0.61537 0.50914 0.42548 0.35846 0.30325 0.25748 0.21911 0.18650 0.15862 0.13461 0 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 0.95 25 0.02316 0.03185 0.04107 0.05055 0.06002 0.06927 0.07811 0.08638 0.09396 0.10077 0.10680 0.11206 0.11665 20 0.00246 0.00124 0.00091 0.00130 0.00224 0.00360 0.00528 0.00720 0.00935 0.01173 0.01443 0.01756 0.02130 15 0.01871 0.01162 0.00594 0.00118 -0.00603 -0.00903 -0.01113 -0.01235 -0.01269 -0.01206 -0.01036 -0.00742 -0.00301 10 0.05928 0.04700 0.03643 0.02736 0.01963 0.01311 0.00770 0.00341 0.00035 -0.00168 -0.00306 -0.00410 -0.00495 5 0.11376 0.09571 0.07997 0.06622 0.05419 0.04367 0.03448 0.02649 0.01957 0.01364 0.00863 0.00451 0.00134 0 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

Table for Making Initial Estimate of Deadrise

 m 0.75 0.74 0.72 0.7 0.68 0.66 0.64 0.62 0.6 0.58 0.56 0.54 0.52 0.5 0.48 0.46 0.44 0.42 0.4 0.38 0.36 0.34 0.32 0.3 Deadrise 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

Body Plan Development

After defining the shapes of the above waterline hull shapes and below waterline section shapes, it is a fairly easy task to interpolate other sections and waterlines, and a section is incorporated into the spreadsheet to allow the user to do so.  I have updated the methodology of drawing these Bulkheads & Waterlines based on some of the work incorporated into the Aircraft Carrier Design Worksheet that I have put together and the resultant output is shown below.

Hull Volume Estimate

Having defined the hullform section shapes it is also a fairly easy task to have the spreadsheet calculate the total enclosed volume of the hull.  A plot showing he distribution of this volume is shown below.

Initial Seakeeping Assessment

Here I have added a section for Freeboard checks and other preliminary seakeeping assessments.  Currently I have the spreadsheet plot the vessel's flare @ station 3, freeboard and length against recommended trend data from from DDS 079-2, as shown below.

There also is a paper that was published by the American Society of Naval Engineers, entitled "Methods for Designing Hull Forms with Reduced Motions and Dry Decks" by David A. Walden & Peter Grundmann that provide some additional recommendations on Freeboard.  These include:

For L/T < 27.5

• FB0 = 0.1 * V ^(1/3) * [ 5.4 - ( L - 120 ) / 40 * 0.5 ]
• FB1.5 = 0.1 * V ^(1/3) * [ 5.3 - ( L - 120 ) / 40 * 0.19 ]
• FB6 = 0.28 * V ^(1/3)

For L/T => 27.5

• FB0 = 0.1 * V ^(1/3) * [ 5.15 - ( L - 120 ) / 40 * 0.45 ]
• FB1.5 = 0.1 * V ^(1/3) * [ 5.1 - ( L - 120 ) / 40 * 0.09 ]
• FB6 = 0.1 * V ^(1/3) * [ 2.6 - ( L - 120 ) / 40 * 0.16 ]

This paper also provides some guidance on Freeboard from other sources.  Specifically, the paper gives the following recommendation that is attributed to Bales:

• FB0=10.5 - 0.045 * ( L - 150 ) - 0.00002 * ( L - 150 ) ^ 2 - 0.20 * [ ( L / T ) - 27.5 ]

The paper also gives this equation, which is attributed to Gale:

• FB0 = 1.011827 * T - 0.0000209 * L ^ 2 + 0.027806 * L

Finally, this paper also provides an equation for the Seakeeping Rank Estimator that 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 vessel based on the above parameters, with Rcap ranging from 0 to 10.  The higher the resultant Rcap, the better the relative seakeeping capabilities of the hull is estimated to be.

I have included these equations into the spreadsheet to give guidance on whether a design has adequate freeboard based on these recommendations.

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