Initial Thesis Report: Surfboard Hydrodynamics
Figure 1: The other side – photograph of surfer from underneath a wave (Tan, E., 2008)
Rowan C. Beggs – French
Surfboards represent a unique type of watercraft, being propelled solely by the energy of the
breaking wave. The board is subject to a complex interaction of forces, as it rides the
interface between the water wave and the atmosphere. In order to assess whether current
theory accurately describes the forces acting on a board, calculations using the current
theory will be compared to what is observed in the experiments. The experiments will
include video recording the flow fields around a board in operation, which will help
qualitatively understand some of the flow phenomena impacting surfboard performance.
The next stage will examine the lift forces generated by the rails under a turning condition to
try and optimize their design. This initial report examines breaking wave and planing theory
before looking into the research which has been conducted into surfboard design. It will then
go on to look at how my work will build upon this.
Nomenclature
Ar
b
c
D
d
Fr
g
h
H
L
LB
l
m
Re
S
T
Vb
Vs
Vw
α
γ
= Wetted beam to length ratio
= Beam of planning craft [m]
= Celerity, wave speed [m.s-1]
= Drag force [N]
= Water depth [m]
= Froude number
= Acceleration due to gravity [ 9.81 m.s-2]
= Height of board on wave face [m]
= Height of wave, crest to trough [m]
= Lift Force [N]
= Lift due to buoyancy [N]
= Wetted length [m]
= mass [kg]
= Reynolds number
= Side force, perpendicular to velocity [N]
= Period [s]
= Break speed of wave, parallel to bottom contours [m.s-1]
= Speed of surfboard [m.s-1]
= Particle speed of water relative to wave crest [m.s-1]
= Free surface angle with horizontal [degrees]
= Angle of wave with bottom contour [degrees]
Δ
ρ
λ
τ
ψ
υ
μ
σ
= Load [N]
= Density [kg.m-3]
= Wavelength [m]
= Trim angle, board angle of attack relative to the water surface [degrees]
= Yaw angle, wetted area centerline relative to direction of velocity [degrees]
= Roll angle, board base relative to free surface [degrees]
= Dynamic Viscosity [N.s.m-2]
= Surface Tension [N.m-1]
I.
Introduction and background
A.
Introduction and Thesis Aims
Surfing over the past century has gone from being a novelty of the Pacific Islands to one of the most influential
and popular sports of today. The surfboard works by taking some of the vast energy which a breaking gravity wave
releases to propel the board and rider along the wave face. To this point there have been three research projects
conducted into the subject, utilizing planning hydrodynamics, model waves and CFD to analyze the complex
interplay of forces on a board. This project is aimed at anlysing the current knowledge of surfboard hydrodynamics
and then extending this through experimental work on a full size board. The bulk of the research will be based
around imagery of the flow fields of various surfboards riding different waves. If time permits, other work will be
conducted examining quantitatively the performance of rail design. The overall aim is to understand the forces
acting on a board and then use this to try and improve surfboard design. The project management documents are
enclosed at appendix 1.
B.
Basic water wave theory
In order to establish the forces acting on a surfboard, it is essential to understand the dynamics of a breaking
wave. Waves are generated by wind blowing over a region of ocean known as the fetch, they begin as small
capillary waves and over time they merge to form larger sinusoidal gravity waves. In capillary waves the mechanism
for the wave propagating is the surface tension of the water which dominates until waves reach a size of 2 cm
(Killen et al, 1976, P 468). Once the wave gets beyond this size gravitational forces begin to dominate the wave
motion. The wave will propagate at its phase velocity or celerity, c, which is given by :
Equation (1)
(This formula applies for both deep and shallow water waves)
Figure 2 is a graph showing c for various depths and periods. As can be seen the wave speed is constant above a
certain period. Generally for surfing waves in the vicinity of 1 m – 3m the wave period is 10 seconds, well into the
linear part of the graph.
In the open ocean where d>> λ the motion of individual water particles as the wave passes them is circular and
their velocity, Vw is much less than c. As the wave moves into shallow water its period, T remains constant, however
the wave will slow, and hence compress. The result of this is that our wave which was once sinusoidal has now
formed crests which are higher and sharper than in open water and troughs which are deeper and flatter (Kiss, A.
2008, p 14). As this happens, the path taken by each water particle changes from circular to elliptical and Vw
increases. This can be seen in figure 3, where streakline 3 is sinusoidal, showing the wave form in deep water, while
the wave in the shallower water is more sharply peaked.
The speed of the water particles will continue to increase as the wave propagates into shallower water, until the
wave speed c is reached. Generally this occurs as the ratio of the wave height to water depth reaches 0.78 (Kiss, A.,
2008, p 18). It is at this point that the wave „breaks‟ and the top of the wave falls forward overtaking the base. The
wave is steepest at the transition point, which is shown as streakline 1 in figure 3. The water particle velocity along
this streakline is equal to the celerity of the wave (Paine, M., 1976).
Figure 2: Celerity for various water depths and wave periods. Note: h is water depth in this plot (Kiss, A., 2008)
Figure 3: Diagram of a breaking wave, dashed line indicates critical depth contour causing breaking
Most waves will be angled to the bottom contours as they approach the beach, this is essential if a wave is to be
ridden. The angle that the wave peaks make with the bottom contour at the critical depth for the wave to break is
shown in figure 3 as γ, and the peeling speed is related to the celerity and γ by:
Equation (2)
Generally for surfing γ is between 30o and 60o (Pattiaratchi, 1999). If the angle is less then the wave will break to
fast for the surfer to keep up and this is said to be „closing out‟. Conversely if a wave makes too greater angle then
the wave will be very slow moving. By having the wave peeling it also transforms a 2 dimensional unsteady
problem of a wave breaking on a beach into a dynamically similar 3 dimensional steady state problem (Killen et al,
1976). If you were an observer watching from position x on the beach in figure 3, a wave normally incident on the
beach then it would rise and crash all at once, an unsteady event. If on the other hand the wave peels at a constant,
finite break velocity, then you will have a steady breaking wave, the break point moving at V b. The frame of
reference which is most useful for analyzing the motion of a surfboard on a wave is that of the crest at the point of
breaking, the peak of streakline 1 in figure 3. This point moves with a constant velocity of Vb parallel to the bottom
contours and c in the direction of propagation. For graphical description of angles and frames of reference please
refer to Appendix 2.
Along the breaking streakline, the speed Vw of the water particles with respect to an observer at rest on the beach
will be constant, at c. However, with respect to the peak of the wave, which is travelling horizontally toward the
beach also at c, the magnitude of the particle speed is given by the expression:
Equation (3)
Graphically this is represented in figure 3. This is an important relationship, as a surfer in equilibrium on a wave
moves at the same velocity as the breaking wave crest, our reference point. This means that the relationship derived
shows the water velocity relative to our board when the rider is at various wave heights, which is required in our
force balance.
The final factor of wave dynamics is the steepness and the manner in which a wave breaks. There is no
theoretical way to predict the shape of the wave surface, as it is dependant on the exact bathymetry of the break. So
while all waves will break with the same velocity in a given depth of water, their shape can be vastly different,
ranging from a crumbling wave with a gentle slope, through to a steep hollow barreling wave where the rider moves
along an almost vertical face (Paine, M., 1974). In order to analyse a wave we need to know the slope of the wave
face where a surfer is riding. In this thesis the analysis is based on measuring the geometry of a wave which is
photographed.
Figure 4: velocity profile for varying height above trough on breaking wave face
C.
Surfboard Hydrodynamics
A surfboard is designed to allow a rider to move freely along a wave and maneuver expressively to get the most
out of each ride. A surfboard slides along the water surface due to a combination of its buoyancy and planning lift,
whilst being held into the face of the wave by the side force generated by the fins and rail, the rounded edge of the
board. As with most designs a surfboard is a compromise between speed and maneuverability. For the sake of
maneuverability a board will be short with rounded rails and a high degree of rocker, or curvature from nose to tail.
On the other hand a board built for speed would be longer and finer, with razor sharp rails. Diagrams of the various
surfboard design features along with an explanation can be found at appendix 3. Years of trial and error have led to a
standard performance short board for waves between 1m – 3m. General features include a length around 6‟, with
hard or sharp rails on the rear third of the board, transitioning into rounded or softer rails further forward. It will
have a slight rocker and also light channels in the bottom known as concave, providing greater lift through directing
the flow in a similar way to winglets on an aircraft,
The approach which has been taken in the force analysis of a surfboard is through relating it to other watercraft,
which are better understood. At the most fundamental level there are two types of water craft namely displacement
and planing. A displacement craft relies upon hydrostatic forces (buoyancy) to get its lift while planing craft
generate hydrodynamic lift by having a flat hull at an angle to the water and going sufficiently fast to sit on the
water surface (Paine, M., 1974, p 13). Whether a watercraft is planing or displacement is dependant on its Froude
number, or speed to length ratio Fr:
Equation (4)
Where V is the craft speed and d is the craft‟s wetted length. Typically a planing craft will operate at
Fr > 1.5
-1
(Hornung et al, 1976). A surfboard operates at between 5 – 10 m.s (Carswell, D., et al, 2005) with a wetted length
ranging from 0.5 – 1 m . This gives us a range of 1.6 to 4.5 in Fr, which indicates surfboards operate at least in part
as a planing watercraft.
The research by Paine (1974) and Killen et al (1976) was based upon the planning craft theory developed in
NACA technical documents 1139 (Wagner. H, 1932) and 4187 (Savitsky, D, et al 1958). Paine used the Asymmetric
planing theory of NACA 4187 to achieve a theoretical force balance. This is dynamically similar to a surfboard,
which is essentially planing with a yaw, trim, and roll angle (please refer to appendix 2 for angle definitions).
This research provided a good foundation for further work into surfboard hydrodynamics. The asymmetric
planing in NACA 4187 is analogous to a surfboard, however these were conducted at Fr much higher than a
surfboard operates at. A surfboard also does not have the sharp edges of the plates tested, and the rail outline is
somewhat different to a rectangle.
The research of P. Killen et al, 1976 centered on the development of a stationary breaking wave, but also built on
the force balance Paine used by adding the theory developed by Wagner (1932).
Figure 5: Stationary oblique breaking wave developed for surfboard research (Killen, P. et al 1976)
The model wave seen above in Figure 5 had a height, H = 0.18 m, a free stream velocity, v = 2.24 m.s-1 and a
peeling angle, γ = 48o. The wave was able to have a board ride on it, suitably scaled and weighted with no external
forces acting. This allowed comparisons to be made between different standard surfboard design features such as
plan shape and bottom contours, and measure the differences.
Theoretically Killen‟s work took NACA 1139 (Wagner, H. 1932), and transformed the two dimensional
unsteady problem of a symmetric planning craft and converted it to a steady three dimensional problem by rolling
the plate sideways through a small angle. If an observer was to move with the same velocity as the plate then they
would see the flow as steady. This is essentially using the same transformation as was discussed earlier to transform
the unsteady two dimensional wave into a dynamically analogous three dimensional steady state problem. In figure
6 on the left is the unsteady problem of an infinitely wide plate, while on the right is a finite plate rolled through an
angle to create a steady state problem.
Figure 6: Conversion of unsteady 2D planning to analogous steady 3D planning (Killen et al, 1976)
This then allowed the lift generated by a surfboard to be calculated by the following equation:
Equation 5
This gave Killen et al results which were within 15% of the values experimentally obtained by Savitsky et al in
NACA 4187.
Hendrix (1969) discussed that surfboards like all hydrodynamic craft have four drag components:
Skin Friction – Due to viscosity of the fluid
Pressure drag – Due to the form or displacement of the craft in the fluid
Wave making drag – Generated by any object which moves on or close to a fluid surface
Splash drag – The drag created by the body changing the direction of the fluid flow at leading edge
In 1965 Sedov conducted an investigation into two dimensional hydrodynamics, and developed a relationship
between the wave making drag and splash drag for planing craft at various Froude numbers. Killen et al used this in
conjunction with the drag data from Savistky et al (1958) to predict the importance of the different drag components
on a full sized surfboard as:
Skin Friction – 22 %
Pressure – 58 %
Wave – 4 %
Splash – 16 %
Killen el at were very successful in advancing research into the field of surfboard hydrodynamics. The non
dimensional coefficients important to surfboard hydrodynamics however differ significantly between the models
tested and those expected on a real surfboard. Table 1 shows these differences and discusses their implication.
Coefficient
Fr,
Model (orders
of magnitude)
1
Full scale (order
of magnitude)
10
Re,
105
106
W,
10-3
10-5
Implications
Relative importance of wave and splash drag is directly
proportional to Fr
Boundary layer laminar for model, transition for actual,
effecting viscous and pressure drag
Importance of capillary waves and gravity waves in the
wave drag varies with W
The most recent work which has been conducted into surfboard hydrodynamics is that of Brown et al, in their
CFD analysis of fin design. The work effectively placed the fins in a flume to determine the drag and lift forces
acting at speeds of 1 – 7 ms-1. It determined that for a standard three fin surfboard, that the side force which could be
generated by the fins was around 50 N ± 20 % at 7 ms-1 and 5o ψ, and that they would have a drag of 8 N ± 1 N.
Interestingly they found that the major drag component acting on the fins came once again from pressure. The
research treated the flow over the fins as 2 dimensional and did not look at the effect of fins only being partially
submerged, which as can be seen in figure 7 occurs during trim.
Figure 7: Photograph of board riding wave showing complex flow around fins (Blue Horizon, 2003)
The other interesting thing to note in this work is that it predicts that at the Froude numbers a surfboard operates
at that the wave drag would be considerably higher than the splash drag, in contradiction of the earlier work by
Killen et al.
II.
Thesis Methodology
A.
Scope of thesis
The scope of the thesis is to experimentally test the validity of the assumptions which constitute the current
understanding of surfboard hydrodynamics. The theoretical results attained by Paine and Killen et al in the 1970‟s
will compared this with what is observed through my testing. A large portion of the experimental work I am
undertaking is based around the video analysis of the flow fields that exist on a board as it rides a wave. This will
expand upon the work which Killen et al conducted, and will be an interesting comparison to what was seen on their
model wave. The next stage, time permitting of experimental work aims to quantitatively test the effectiveness of
different rail designs through towing them in a pool and measuring the lift forces. The intention is that once the
dominant forces are understood, that design features of craft working off similar principles will be taken and
examined for improving surfboard performance.
B.
Plan for conducting thesis
The plan for my thesis is evolving as more and more is understood about the complex subject. The plan as to
how this project will be achieved is as follows:
Stage 1: Literature Review
The most important part of any project is to understand what research has previously been covered and how it relates
to the project being undertaken. This stage of the project gives us our direction for the conduct of experiments in
order to further what is known.
Stage 2: Conduct Force Balance based on current theory
The stage which is underway at present is understanding the forces acting on a surfboard based upon the theory
which exists already. I am taking the work of Savitsky et al, and Killen and predicting the forces which are acting on
a surfboard when it is in the equilibrium condition on a breaking wave.
Stage 3: Begin Experimental work using underwater camera
This stage is concurrently being undertaken at present. An underwater video camera courtesy of Dr Michael Harrap
is being used for the task. The first part of this stage is designing a method of mounting the camera on the board in
such a way as to get clear images of the flow fields, and ensure that it does not affect them by being there. Two
designs have been trialed so far; please refer to work undertaken along with appendix 4. Once suitable mounts are
developed the experimental work will begin, video recording a range of surfboard types riding a variety of waves.
Stage 4: Analyse fluid flow and compare to current theory
After gathering the video footage of surfboards in action the videos will be analysed to understand the flow fields
which are being observed. A proof of concept on this stage has been conducted, and can also be found in appendix
4. By observing the water movement relative to the board the intention is to relate this to theoretical understanding
of how a surfboard works and determine if the current understanding describes the dynamics sufficiently. One point
of interest that may require further testing is the matter of gas bubbles in the surfboard wake. As can be seen in
figure 8 the complex wake of the board may contain bubbles, the nature of their formation is currently not
understood. It would be a worthwhile investigation to see how they get into the flow (see figure 8 below), whether
through mixing with the atmosphere at the free surface or through cavitation of the flow on the surfaces generating
sufficiently low pressures.
Figure 8: Bubble stream clearly visible off right hand fin tip (Smith, N., 2006)
Stage 5: Design and conduct experiment to test surfboard rail effectiveness
If time permits then I would like to extend the work to better understand another common situation in surfing
and that is a steady turn. Often if a wave peel speed is low then a surfer will outrun the wave, requiring him to turn
back to get back to the part of the wave just breaking. In this situation the surfboard rail is generating the centripetal
force allowing the turn to be sustained. The more lift that the rail generates, the deeper the rail will cut into the
water, allowing a greater force to change the board direction. The effectiveness of surfboard rail design could be
tested by designing a test which tows a full size or model rail through water such as a swimming pool, and
measuring the lift generated through the depth the rail sits in the water whilst being towed.
Stage 6: Identify future design direction based on results.
Hopefully after the investigation of the two common dynamic conditions of a surfboard I will be in a position to
recommend areas of further development in their design. I would really like to have the time at the conclusion of my
experiments to review the design of other water craft relying on the same forces as a surfboard and then optimize
surfboard design using these.
B.
Work undertaken
So far I have mainly focused my work on understanding the broad literature which is applicable to this topic. As
can be seen above, there is considerable background which is required to be processed before any meaningful work
in this area can be conducted.
The next stage I have begun is designing different mounting systems for the underwater camera on my
surfboards so that experiments could begin as soon as possible. The designs thus far can be found in appendix 3,
along with some of the footage which has been captured. All the designs so far mount the camera to an Aluminium
plate which is secured by Velcro to the nose of each board. It then looks down along the rail showing the flow of
water around it as the board traverses a wave. Technical drawings along with photographs are included at
Appendix 4.
The first test of this design in the water showed that the design could capture the footage as desired. However, the
camera in the first instance was too low, and hence there was a lot of interference of the splashed water off the nose
of the board obscuring the flow which we want to analyse. Also the angle was too flat, which mean we looked at too
much of the board and not enough of the flow itself. Hence a second design was made which was 100mm higher
allowing better observation of the flow with less interference and a better angle. At the time of writing this design
was yet to be tested in the water. A third design is also in the process of being developed which will allow a view of
the flow patterns on the bottom of the board.
In conjunction with this I am conducting a force analysis based upon the work of Paine and Killen et al. So far I
have done the force analysis treating the board as a simple planning craft in asymmetric motion, not accounting for
the fin forces or the rail design which will necessarily change the force balance. For the full calculations and
diagrams, please refer to appendix 5. The angles were approximated by measuring the geometry of figure 9, and
deducing α, h/H, υ, ψ. The results of this are as follows:
For W = 883 N, α = 40o, and H = 2 m:
L = 750 N
D = 300 N
S = 450 N
From the planing theory alone of Savitsky et al, the forces calculated are, assuming a trim angle of 6o (measured):
L = 690 N
D = 150 N
S = 170 N
This result is quite reassuring showing that this theory at least partially describes the motion of a surfboard. The lift
is very close to that required for the force balance, which is a good sign, as hydrodynamic lift at this speed
dominates. The drag and side force are considerably lower but this is where I would predict the Fr difference would
have the greatest bearing. In addition to that, the fact a surfboard does not have a razor sharp edge, and has the fins
generating the majority of the side force.
II.
Conclusions
The subject of surfboard hydrodynamics is vast in the areas it draws upon. Thus far it has proved to be
fascinating looking at the research already conducted and the conclusions drawn. The literature has shown that
surfboard motion is dynamically similar to asymmetric planing however there are some differences this thesis aims
to investigate through underwater photography. The thesis is now in a strong position to undertake this work, as the
necessary background has been collected and understood. It will be fascinating to see what the nature of the bubbles
is in the flow off the rails and fins. From the initial iteration of the force balance that asymmetric planing provides a
good basis, however the fins and rail design substantially change the side and drag forces which could be expected
on a full scale board.
Figure 9: Understanding the complex flow around a board will hopefully lead to better performance
(Onarati, M., 2008)
References:
1.
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3.
4.
5.
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