Subject: Methods of Determining Ship Resistance
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Date: Thu, 28 Aug 2003 11:10:23 -0700

Methods of Determining Ship
Resistance

A Review of Empirical and
Numerical Methods to Determine Wave Resistance on Ship
Hull

ABSTRACT

The goal of this exercise was to
examine different methods that have been and are being used to calculate ship
resistance. The hydrodynamic drag of a body partially submerged in a liquid
determines the amount of energy required to get and keep the body in motion as
in travels from location to location. In the past scale tests have been used in
hopes of developing relationships of drag with the full scale, but suffer
(especially at high Froude Numbers) from scaling effects that introduce a great
deal of error into the model results. As computational fluid dynamics has
progressed, numerical models have become popular in determining drag
coefficients. These results, if accurate, represent a significant cost savings,
as both model and full-scale tests can be quite expensive. This paper explores
both the empirical and numerical approaches to see how they apply to
environmental fluid mechanics.

MOTIVATIONS AND OBJECTIVES

There are two main motivations
of this survey. The first is to gain a better understanding of hydrodynamics
through the study of a practical problem that served to form the basis of fluid
mechanics as it is traditionally regarded. The other motivation stems from the
desire to develop a numerical protocol to evaluate hulls of concrete racing
canoes. While many examples sited reference naval ships or other large sailing
vessels, the same types of methods could be used to evaluate any sort of object
in a flow field. If an accurate model can be developed that would save the
money, time and energy that it costs to construct scale models of concrete
canoes.

METHODS

Part 1- Empirical/ Model
Tests

In order to get a better
understanding of how ship resistance has reached its current state. In the
mid-late 1800, minimizing drag on a ship hull became a very important problem.
Since most nations’ economies depended on ships to move goods between each
other, fuel economy became important, as a ship business was limited to the
distance it could travel without refueling with coal. The inevitable problem
that arose was how to minimize drag with a very limited knowledge of the physics
of drag reduction.

William Froude was a pioneer in
the field of fluid mechanics and developed much of his legacy working on
reducing hull drag. Froude was convinced that trial and error with these huge
vessels represented an economically inefficient allocation of resources,
convinced that scale model testing was a valuable method of testing ships before
a prototype was created.

EFM relation #1-
Dimensional Analysis

Working with scale models
requires the understanding of how that results can be scaled up to the
prototype. Dimensionless numbers create a connection between parameters and
develop functional relationship to correlate information in the small-scale
tests.

For ship resistance, r,v,L,g,m
all serve to influence a
result, or:

Breaking the
terms down into their Mass/Length/Time components and applying the Buckingham Pi
Theorem, the three results become:

These values
are all significant. The first represents the equation for dynamic drag of a
fluid, while the second and third are the inverse of the Froude and Reynolds
numbers respectively.

EFM relation #2- Dynamic
Similitude

Part of the
problem with scale models is making sure all the scaling variables are accounted
for in the result. This requires geometric, kinematic and dynamic
similitude.  A well-constructed
experiment can ensure geometric and kinematic similitude, but problems arise
when trying to satisfy the third requirement. Evaluation of whether all the
dimensionless parameters in the model match those of the prototype proves
impossible because of the location of the velocity and length term in the two
equations. In most applications, one of the parameters can be ignored due to the
difference in magnitude, but in this case both wave and viscous forces
contribute to the total drag, so neither can be ignored.

Froude found a
way around this by breaking the resistance down, one component being a function
of the Froude number and one being a function of the Reynolds number. The
combination would yield a total coefficient of friction and introduce the idea
of incomplete dynamic similitude in which experiments are run based on known
values of the Froude number to determine total resistance. The Reynolds number
kernel could be determined using the idea of equivalent flat plate resistance,
and empirical method of determining surface resistance. The difference of the
Froude component and the Reynolds component yield the wave resistance or energy
loss as the body moves across the water surface.  The basic steps are as follows:

We know that drag force depends
only on Fr. Model speed can be estimated knowing:

Using the
Moody diagram, a Coefficient of Drag (CD) can be determined for both
the model and the prototype. Skin Drag on the model can now be calculated:

If total drag on the model is
measured in a flume, the difference between Dtotal and
Dskin is the wave resistance Dwave. Scale to
Prototype:

The friction drag can also be
obtained with prototype values exactly as the model was.

A follow-up technique (should
you have the luxury to have connections with the U.S. Navy) is an extrapolation
method discussed in Ferguson. He contends that a relationship can be developed
to link the sinkage and trim of a large ship to the sip resistance based on
model test that very the Froude number. The results were then extrapolated to
larger scale models and plotted against real data form those models.

Part 2 Numerical Models

EFM
relation #3- Governing Equations and Boundary Conditions

Even the best experiment still
takes time and money to construct and run. As computers become more powerful and
the knowledge base of computational fluid dynamics continues to grow, scientist
and mathematicians have begun to search for new ways to evaluate hulls with
computer modeling. A code that could determine the optimal hull design for
applied conditions would be an invaluable resource for designers as it would
eliminate any need for model testing.

There is a reason that this
perfect code does not yet exist. The complicated mathematics coupled with
various model assumptions has made reliable solutions more elusive than would be
expected. The following will show some of the techniques used to solve this
complicated problem using the basic governing equations of fluid mechanics.

One example (Sorti et al.) shows
how conservation of momentum can be used in the calculation of wave resistance.
Using a Laplacian governing equation, flow is assumed to be irrotational and
incompressible and that Bernoulli’s equation applies at the free surface waves
created in the control volume are assumed to pass through with the “absorbing
boundary” that eliminates the viscous effects and allowing energy to leave the
system and validating the momentum flux assumption.

(From
Sorti et al.)

If these conditions are met, a
simple momentum balance at the plane ahead of the ship and behind the ship will
produce a net decrease in flux equal to the wave-making resistance. Using the
Bernoulli assumption, the description for momentum of a vertical plane of water
before hitting the hull can be described as:

Once the water passes the body a
perturbation technique to linearized to:

A finite element analysis uses this
information to evaluate drag coefficients at different Froude numbers and
outputs results like:

Idelson et al. also uses a
finite element approach, but develops the solution a step further by using for
Euler and Navier-Stokes Flow as a free surface boundary. The code predicts a
velocity field and uses that information in a mass conservation to calculate a
pressure field (using a Poisson Equation). The pressures obtained are then
updated in the program and a new iteration is calculated. Like the previous
technique, an equation is included to incorporate a radiation condition so as to
eliminate any reflected energy that might occur in the artificial flow field.
Some results of the analysis look like:

Meshes for 1. the Euler Equation, the Navier-Stokes

Results of wave pattern behind a sailboat
(Idelson).

DISCUSSION

The methods that have been
described in this brief survey have all provided an application to all the note
that were take in CEE 618 this year. The process of solving problems by first
identifying variables, choosing governing equations and making simplifying
assumptions, setting boundary conditions and the applying them to the to the
governing equations to solve an eventual outcome, can bee seen in each reference
listed.

As for the practicality of this
author developing a code that will be helpful for evaluating a canoe hull, it
may take a bit more research and training. Hopefully the reader has gained some
appreciation of what goes into the process and how complicated it can actually
become.

REFERENCES

Bessho, M.
(1994). On a Consistent Linearized Theory of the Wave-Making Resistance

of Ships. Journal of Ship
Research, Vol.38, No.2, pp. 83-96.

Chen, C.Y. and Noblesse, F. (1983) Comparison Between
Theoretical Predictions of

Wave Resistance and Experimental
Data for the Wigley Hull. Journal of Ship Research, Vol. 27, No.4, pp.
215-228

Ferguson, A.M. (1984) Extrapolation Method from Ship
Resistance Based on the

Variation of Sinkage and Trim with
Froude Number. Naval Architect, January 1984, pp. 17-32

Gillmer, T.C. and Johnson, B (1982) Introduction to Naval
Architecture. Naval Institute

Press, Annapolis, Maryland

Idelsohn, S.R. , Onate, E. and Sacco, C. (1999) Finite
Element Solution of Free-Surface

Ship-Wave Problems,
International Journal for Numerical Methods in Engineering, No. 45, pp.
503-528

Kostyukov, A.A.
(1968) Theory of Ship Waves and Wave Resistance, E.C.I., Iowa
City,

Iowa

Kundu, P.J. (1990) Fluid Mechanics, Academic Press,
San Diego, California

Larsson, L. and Baba, E. (1996) Ship Resistance and Flow
Computations. Advances in

Marine Hydrodynamics, Vol.
5, pp 1-75, Computational Mechanics Publications, Southampton, Boston

Roberson, J.A. and Crowe, C.T. (1997) Engineering Fluid
Mechanics,  John Wiley &

Sons, Inc., New York, New York

Storti, M. , D'elia, J. and Idelsohn, S. (1998) Computing
Ship Wave Resistance from

Wave Amplitude with Non-local
Absorbing Boundary Condition. Communications in Numerical
Method

of Engineering, no. 14, pp.
997-1012.