Posted: October 9th, 2022

Boat Hull Dynamic Lift using Water Surface Impact

1. Introduction

Hydrofoil has been a popular way to lift a hull from the water to reduce hull drag hence improve the overall power and speed of the craft. However, in nature there are numerous ways for an animal to make use of a different ways to ‘run’ across water. The basilisk lizard is often recognized for their ability to run across water to evade predators, the size of their feet is relatively large comparable to their bodies, this has aided them to generate sufficient upward impulse to support their body weight. The movement of their tails provided some form of stability support preventing them to fall sideway whiles running (S. Tonia Hsieh and George V. Lauder, 2004). Many water birds have also been seen to use their wings for support and ran across water. The primary aim of this project is to investigate if a propulsion system could make use of basilisk lizard’s movement, and to prove that surface water impact would be sufficient to lift a hull from the water surface.

A simple paddle-wheel system was then decided to be used to mimic the behavior of the basilisk lizard movement and tested against common high-speed craft models for any advantage over them.

The idea of paddle propulsion has been around since ancient time, both Egyptians and Romans were aware of it. However, the first recorded use of this type of propulsion was in 1472 in ‘De Re Militari’ by R. Valturius. The use of paddle propulsion did not come in popular until a few decades later, it was the main form of propulsion for some time until screw propeller started to appear. However, It was not soon until paddle wheel become extinct from sea-going vessel, not because of their overall efficiency but due to its large change in draught and ease of being damaged (H. Volpich, I.C Bridge, 1955). Nowadays, paddlewheel vessel has become a specialized craft, primarily based on slow speed vessels in river. Although there are significant disadvantages on using the paddle wheel, the efficiency of the paddle wheel in shallow water application is significantly higher than waterjet or screw propulsion.

Chapter 2 would first look at some previous example of works done relating to paddlewheel and lizard, these papers would then be review for their usefulness to this project. A scaled preliminary design would be done in chapter 3 which made use of the size of the lizard then scaled accordingly to a model size for calculation. Chapter 4 would then develop a basis theory of force generated by the paddlewheel, the initial idea was taken from Alexander’s work in 1983. (Alexander K. V., 1983) Methodology would then be summarized in chapter 5 discussing design iteration and FEA to validate the strength of the structure. Motor power would also be estimate and the process of setting up CFD analysis would also be introduced. Chapter 6 would describe how experimental method would take place and variables that would affect the result. Chapter 7 would then look at the result of the CFD on the magnitude of lift force generated in it. Finally, the conclusion would be drawn in chapter 8.
2. Literature review

These literatures have provided supports and backgrounds knowledge for this project. Paddlewheel propulsion was only thought as a propulsive device only, limited paper addressed the lift force that paddlewheel generated, Alexander and Wray & Starret were the exception. They both addressed paddlewheel that ran in high RPS to provide. Although Alexander were the only one who investigate the lift force generated. One paper suggested a using a robot to mimic the basilisk lizard with four footpads which turned out to be unsuccessful.
Paddle wheels, Pt I: Preliminary model experiments. Paper no. 1193, Transations of the Institution of Shipbuilders and Engineers of Scotland
Volpich and Bridge, 1955-1957

These authors explored variables in a paddle wheel system which provided some measurement on efficiency and design of a paddle wheel. It was one of the first paper to test paddle wheel system in a tank test, providing some basis on any kind of further research. It also provided some preliminary setup to test the wheel, as it is different from a modern-day tank test. The paddle wheel must be able to spin whilst attached to a support as the hull will not be available by then. A figure of their set up was included in Figure 1. Their experiment setup was to measure thrust force as lift force was not in their interest as a form of propulsion back then.

Figure 1. The experiment set-up by Volpich and Bridge 1955

They tested on both fixed angle wheel and feathering wheel which the paddle angle changed during the rotation for an optimal angle of entry of paddle with the water surface. Their experiments have proven that feathering wheel will increase the efficiency of thrust produced, therefore any further tests were focused on feathering wheel. The star center of the feathering wheel is the center of the extra iron frame in a paddlewheel which can pivot the paddle slightly to optimize the angle of entry.

Although the feathering wheel has been proven to have higher efficiency with a given power, the idea of employing a feathering wheel in this project was quickly given up as it would have complicated the wheel with an extra variable, however, it could be done as a further research into this subject in the future. Figure 2. shown an example of a feathering wheel.

Figure 2. An example of a feathering wheel
Running on water: Three-dimensional force generation by basilisk lizards
S. Tonia Hsieh and George V. Lauder 2004

In order to understand clearly on how the basilisk lizard moves, these authors conducted a series of test to generate data on the patterns of water flow induced by the lizard foot and tracked different location on the lizard to be tracked to develop a relationship between body parts of lizard. The size of the vortex ring was also measured by the system, the paddle chord and span were then decided using vortex ring generated by the lizard as the size of the lizard feet was not available anywhere.

Dynamic Modeling of a Basilisk Lizard Inspired Quadruped Robot Running on Water
Hyun Soo Park, Steven Floyd, and Metin Sitti

This paper modelled the motion of basilisk lizard using a quadruped robot. Their analysis mainly focused on stability issues as the center of gravity of the robot was situated far away from the water surface therefore creating a problem on the stability of roll motion. The main intake from this article was the dimension of the robot and a different kind of paddle used in their experiments. A circular foot with compliant flaps were deployed as an alternative footpad design. When the robot foot is pulling out of the water, the two flaps on the front and back of the foot collapse downward to reduce the area of the footpad normal footpad which could be useful to the project.

The robot did not manage to run across the water, as per the authors state the stability of the robot was an issue as there was a large roll moment and the force generated by the footpad was not great enough to overcome the weight of the body. The idea of producing a quadruped robot was quickly abandoned for this project, reasons are as follow:

Stability issue would be a problem as the robot would have to be balance using four singular points on the water surface, whilst the stability of a hull with paddlewheels would mainly be stabilize by the hull itself, introducing no extra variables to influence stability.
The cost of building robot would be significantly higher than a paddlewheel with near to no electronical part, except for the motor which drives the wheel.
This type of robot is impractical to build and employ in real world situation. Their work was mainly focused on proving lift force can be generated by viscous drag force.
As the author mentioned the robot did not manage to generate enough lift to overcome weight, the weight of the robot would be harder to decrease because of the essential component to drive it. However, the weight of a hull would be far easier to reduce as there won’t be as many components as the robot.

A Model Study of the Hydrodynamic Characteristics of a series of Paddle-Wheel Propulsive Devices for High-Speed Craft
Wray, G. A. & Starrett, J. A. (1970)

Wray and Starrett were the first few to investigate use of paddle wheel on a high-speed craft. Traditionally, paddle wheels are big and slow which are fitted in a steamer, however these authors had an idea to use a paddle wheel of small diameter with high rotational speed, to be employed on a planning hull patrol boat. Few points from their conclusions were proved to be important on the design of the paddle wheel.

The six-bladed wheel generates more thrust than the twelve-bladed wheel, it is also significantly more efficient than the twelve-blade wheel.
Thrust and efficiency increase with increasing immersion depth.
There was a break in the thrust curves at around 40% slip, that they have yet found a satisfactory explanation.

The use of high speed paddle wheel could be related to the this. The break in the thrust curves was to be identified as cavity intrusion (Alexander K. V., 1983), this will happen if a blade hit the air pocket generated by the previous blade, which will decrease the overall efficiency of the system and any increase in RPS will not benefit the craft speed.
The Lifting Paddlewheel: A non-buoyant wheel enabling a high speed wheeled amphibious craft to run on the water surface
K. V. Alexander (1983)

Alexander was the first and only person to propose the use of paddlewheel to lift a craft out of the water, theory was developed to describe the forces for simple flat-bladed paddle wheel. A total of 25 different type of wheels were tested in his thesis. His work was very relevant to this project as the result provided some basis information on dimensions and design of the paddle wheel. The main findings were listed below:

Alexander has produced an optimal 12 blades paddlewheel design with a wheel diameter of 152mm. Figure 3. shown his design in detail.
Lift force produced from the wheel was mostly dependent on the immersion and advance speed. As the immersion depth increases, the forces produced by the paddle would increase until it reaches the center line of the wheel.
In general, increases in wheel span and chord would increase the force produced by the wheel as more mass of water were actuates for each revolution.
The lifting paddlewheel craft has wake regime of a boat hull consist of displacement, semi-displacement and planing.

Alexander’s results have reduced the number of variables this project has to investigate, and verified data that was scale from lizard to be similar with the results he produced.

Figure 3. The most promising wheel from Alexander’s thesis.
CFD Analysis of a High-Speed Paddlewheel
W.C. Bowen and K. V. Alexander (2010)

These authors conducted some CFD study on the preliminary work of Alexander back in 1983, their simulations were conducted in Ansys CFX v12.1. It involved of modelling transient analysis, which include solving VOF equation for two fluid (water and air) dispersing throughout a flow domain. Both 2D and 3D models were analyzed. The results were then compared against the work Alexander did in the previous thesis.

These results could be used to measure the accuracy of the experimental results produced by the current project. The 2D results did not reached a steady state force or repeating pattern, therefore Alexander selected a time interval and average out the force per unit spin, an example of the time interval selected was shown in Figure 4.

Figure 4 CFD Propulsive force for 5RPS wheel

The method of CFD could be apply to this project and helps to set the simulation up with small amount of time. They also did a test on how number of elements would affect the simulation outcome by modelling a flat blade dropping vertical downward to the water surface and observing the water splash generated by each case.
Summary

The idea of employing high speed paddlewheel was not tested as screw propellers often offered a more beneficial operating condition such as, paddlewheels were more likely to be damage and the overall operating equipment took up a lot of space on a vessel. However, it has an relatively high efficiency and it seemed like the limit of paddlewheel has yet to be achieved. Alexander had shown a reasonable progress on his idea of lifting paddlewheel, and Wray demonstrated that a six-bladed wheel was more efficient than a twelve-bladed wheel. It could be useful to produce a series of standard data for paddlewheel design which would improve the paddlewheel and explore alternative usage of it, for example as a small automatous craft working in shallow water.
3. Preliminary design

Information of the basilisk lizard were first collected and scaled to a craft size data. It was logical to keep the craft weight and size small to replicate it as a lizard model. In Table 1, it shown a scale factor of 5 from the lizard itself to the craft. This scale factor was chosen because it gave a reasonable size of a model craft size and it was convenient paddle size to compare against result from Alexander’s work. The weight of the craft could however be lighter to reduce the force required to generate.

As all lizards varied in weight, size and speed, an average value was taken to scale it from lizard size to craft size. The feet size of the lizard has proven hard to be found; therefore, the span and chord of the paddle wheel were estimated from the vortex ring size generated by the lizard feet of 0.03m2(S. Tonia Hsieh and George V. Lauder, 2004).
Scale Factor 5
Lizard Craft
Speed (m/s) 1.45 3.24
Weight (kg) 0.20 n.a
Force required (N) 0.98 n.a
Length(mm) 430 2150
Length without tail(mm) 116.10 580.5
Step period (s) 0.10 n.a.
Frequency (Hz) 5 5
Vortex Area (mm^2) 706.86 17671
Span (mm) n.a. 96
Chord (mm) n.a. 18

Table 1. Scaling calculation from lizard to craft size.

As the craft’s length and speed was determined, it was then logical to find suitable hull parameters for the craft. Table 2shown ratios such as L/B and B/T, to calculate the craft basic dimension. These ratios were taken assuming the hull type was a high-speed planing hull, these ratios would differ from cargo ship but closer to passenger ship. (Professors R A Shenoi and R V Pomeroy, Mrs. G A Keane and Dr. D J Taunton, 2017)
L(m) 0.581
B(m) 0.073
T(m) 0.015
D(m) 0.048
L/B 8
B/T 5
B/D 1.5
Fn 1.358681
∇ (m^3) 0.611303

Table 2. Derived Dimension of the proposed model craft

The buoyancy of the craft in an equilibrium case could therefore be derived by simply multiplying the water density and gravity into it. In fresh water, ρ = 1000kgm-3, therefore the derived buoyancy would be 6kN. However, this buoyancy would only be required where the craft is stationary. When the paddlewheel is operating, it would generate lift force and it would match ∇ therefore the craft should start rising from the surface and ‘walks’ on the water surface. And when the transfer of momentum becomes larger, this force would take over the buoyancy where it becomes negligible as the hull of the craft would be out of the water. The hull drag would then reduce to aerodynamic drag rather than hydrodynamic drag. To demonstrate the difference of the hull in water and in air, the drag equation was employed and displayed below in equation 1.

To help visualize this, Figure 6 was plotted below.

Figure 6. Visualization of vertical force acting on the water

The angle θ depended on the immersed depth, the deeper the wheel was immersed, the larger θ would be. A simple trigonometry equation can be made below to calculate the angle θ.

sin⁡θ= r-immersed depthr

( 10 )

Table 4 was made to estimate the lift force generated by the blade. Immersed area and radius would follow what it was in the estimation for thrust.

Table 4. Lift Force estimation with varied RPS and Immersion Depth
RPS Lift Force (N)
5 2.728 3.312 3.604 3.750 3.824
6 3.273 3.975 4.325 4.501 4.588
7 3.819 4.637 5.046 5.251 5.353
8 4.364 5.299 5.767 6.001 6.118
9 4.910 5.962 6.488 6.751 6.882
10 5.455 6.624 7.209 7.501 7.647
1.425 2.85 5.7 11.4 22.8 Immersion depth (mm)
4.3 Summary

Lift force and thrust force estimation were done in this chapter. The thrust force magnitude mainly depends on immersed area and RPS value, while the lift force would also depend on immersion depth on top of those included in thrust force. Higher immersion depth would provide a larger lift force in Table 4. However, the estimation done in Table 4 did not account for the downward Vv­which would drag the wheel downward instead of the upward force it needed. Summarizing through Table 3 and Table 4, it seems logical that a higher RPS would increase the magnitude of force generated, however, these values also excluded the effect of cavity intrusion which was discussed in Alexander’s and Wray & Starrett work. A wheel can only have a limited maximum RPS, in higher RPS the blade would hit the air bubble generated by a previous blade, which decrease efficiency. This would be dealt with more in chapter 6.
5. Methodology
5.1 Wheel Design Iterations

There were three different paddle wheels made to test the optimal setup of it, as different angles, blade shape, and number of blade would impact the size of both thrust and lift force of the wheel. The surface area of the paddlewheel was to keep as a near constant for the ease to compare their performance against each other. The first design of the wheel was taken from Alexander’s thesis proposed with the dimension listed shown in Fig.5 (Alexander K. V., 1983) which was similar in size from the scaling of lizard to craft size listed in Table.1. Another two wheels were designed with different angle or shape of blade to verify if that design was the optimal and were included in the Appendix.

The paddlewheel was first designed in Solidworks such that it can be 3D printed for accuracy and quality (Solidwork 2016). The 3D printed model should have the strength to withstand bending stress cause by the rotation of blade across the water surface. As ABS polymer lack the strength to resist stress cracking, structures were added in to strengthen it. Figure 7 shown the wheel in Solidworks and Figure 9. shown the same wheel with the top removed for the ease of observation. The span of each paddle was supported by 7 points which were connected to the side wall and to the center wheel which would be connect to the shaft from the motor. The design was overbuilt such that any stress cracking can be avoided during the testing phase in the towing tank. The weight of the wheel also needed to be kept to a minimum as the rotation of the wheel cause a dynamic problem, keeping the weight down would increase the natural frequency of the system to reduce resonance during testing. As the natural frequency depends on the stiffness, k, and the mass, m following the equation below. As the equation suggested, the stiffer the object and the lighter the mass is, the higher the natural frequency would be.

ω2=km

( 11 )

Figure 7. Designed wheel for ABS 3D print Figure 8 Designed wheel for ABS 3D printed with the top removed

This design was then disregarded due to the unavailability of the ABS machine installed in the University. A simplified version of the original wheel was made as metal print was offered as an option to manufacture the wheel, these structures would no longer be useful as the strength of (what type of metal) are higher than that for ABS polymer. Figure 9 shown the simplified wheel designed in Solidworks to be metal printed. Majority of the supports were removed to reduce weight and cost of the production.

Figure 9 Simplified wheel removing supporting structures to minimize weight.

The wheel was to be finalized here, however once entering the production stage of the wheel, it was realized that supporting structure were needed as 3D printed parts are built layer by layer, a previous layer was required for any structure to be built upon (Cain, 2018). Figure 10 shown the layer needed for the penultimate design. The supporting structure would then have to be removed, the removing process would likely to damage the model leading to an imperfection which contradict the choice of using 3D printing. Therefore, another new model was made for the ease of production and post processing which is shown in Figure 11. The new model would be easier to print and the size were identical to the previous wheel shown in Figure 9with one significant different, the outer wall was added in to support the paddle as a finite element analysis shown the region was prone to a high stress, this would be discussed more in depth in the next chapter.

Figure 10. The supporting structure needed for the model to be 3D printed

Figure 11. The final design of the paddlewheel
5.2 Finite Element Analysis (FEA)

Finite element analysis is a numerical method to solve engineering problems, the object was to be mesh into smaller parts and equations were used to model these small parts and assembled back into the system to model the problem. (Harish, 2018) Series of FEA were done to ensure the strength of these models were sufficient to withstand the torque and to locate any excessive deflection, these tests were all done in Solidworks Simulation. These included: static test for stress, strain and displacement. Although these tests would not be perfectly accurate because of the constantly changing force opposing the motion in the fluid, it would be enough to show the structures would not crack under the load condition. These tests would show any stress over the tensile strength of the material and marked them with a darker color. A comparison between a wheel with support and without was also tested to demonstrate the difference of stress of the models. This is shown in Figure 12.

Figure 12. Stress test for wheel with outer ring support and without.

In Figure 12, each of the immersed blade were subjected to a constant normal force of 12N which should be high enough to simulate the blade in action as suggested in table 3 and 4. 12N was chosen based on the resultant of both force by square root of the sum of the squared lift force and thrust force. In reality, the force would be distributed through the structure as most of the momentum exchange would happened instantaneously when the blade hit the water surface, and no single blade should have to handle the 12N all together.

Two materials were available in the 3D printer, aluminum and stainless steel. Whilst aluminum is a light weight metal which benefit the experiment, stainless steel has higher overall strength, therefore stainless steel was chosen instead of aluminum because of the torque and force the wheel would experience during the testing phase. Their properties were listed in Table 5for ease of comparison.

Table 5. Properties of Stainless Steel 316L and Aluminum
Stainless Steel Aluminum
Yield Strength, N/mm2 374 ± 5 211 ± 4
Tensile Strength, N/mm2 650 ± 5 329 ± 4
Elongation, % 65 ± 4 9 ± 1
Young’s Modulus, N/mm2 200 x 103

103

5.3 Motor Power Estimation

Motor size was determined by the needed rpm for paddle wheel which is between 5-10rpm. Motors that are oversized and underloaded will overhead and lose efficiency, therefore a suitable size must be determined.

To determine the size of the motor, the torque must be calculated. Using the following relationship:

τ=F r

( 12 )

where r =radius, F = force

The force produced by the blade was unknown to start with, therefore this was assumed by taking the result from Alexander’s tank test. (Alexander K. V., 1983) And the corresponding torque was calculated and therefore the required power could be determined by multiplying the angular velocity (rad/s) with torque. Table 6 listed the power required for different RPS and force setting.

Table 6. Required power with different RPS and force

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