CHME 7350
Transport Phenomena
Analysis of the Impact of Tidal Currents on Optimal Route Selection in Outrigger Canoe Racing
Andrew Luu
April 27, 2023
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Foreword
For my colleagues and instructor(s)
If you ever asked me how I would identify myself, it would be an aspiring engineer first, and
then as a paddler narrowly after. Paddling, and the entire community built around it, is of
immense personal significance to me, and I was extremely enthusiastic in applying what I
learned about vector math and continuum mechanics to a subject that I am deeply invested in.
Kula ‘Anela was in fact the first 6-man outrigger canoe race I ever competed in, and it is a
gratifying experience to revisit the same race through the lens of transport phenomena. I am also
well aware that I am far over the page limit for this paper, and I have no excuses besides my
avidity for this topic.
For my paddling friends and family
If you ever asked me how I would identify myself, it would obviously be as a paddler first, and
you can forget about the rest. Jokes aside, I have always believed in the effectiveness of applying
practical engineering principles to promote the understanding and adoption of effective strategies
for training or competing in any sport, especially when it comes to paddling. The paddling
disciplines of outrigger canoeing and especially dragon boat continue to grow in popularity,
competition, and technical knowledge, and we all participate in this dynamic ecosystem together.
The field is always evolving, and it is up to us to evolve with it, and if so inclined, to take part in
shaping the future of the sport.
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Introduction
Outrigger canoe racing is a watercraft racing sport with growing popularity around the
world. In most competitive races, canoes are crewed by one, two, or six paddlers through a given
racecourse that can vary in length from shorter sprints (200-1000 meters) to longer endurance
courses (5-35 miles).
For optimal race performance, paddlers must rely on a combination of their physical
ability and technical knowledge for how to best paddle under given water conditions. A key
aspect of this “technical knowledge” is the ability to intuit how water currents will develop
throughout the space of a race map and how they can affect boat travel velocity throughout the
racecourse. By applying the knowledge of how water current directions and magnitudes can vary
over the field, capable paddlers can select the most efficient route to take that would minimize
the total elapsed time to travel over region of space. As it currently stands, much of this paddling
intuition, while credible, remains vastly anecdotal in nature.
For this project term paper, I will model the water current velocity profile over a realistic
geographic body of water as a function of tide conditions by applying principles of continuum
mechanics. In this example, I will select the race map used in the Kula ‘Anela outrigger canoe
race, centered on the San Francisco Bay in California. Then, I will conduct a “route analysis” to
compare between different routes taken for the same racecourse to demonstrate the significance
of optimal route selection that considers the flow pattern of the region.
The overall objective of this project is to corroborate anecdotal race philosophy with a
semi-rigorous technical analysis by employing continuum modeling, finite element analysis, and
vector mathematics.
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Theory & Background
Susea Grant’s Kula ‘Anela Race
Kula ‘Anela (Hawaiian for Golden Angel) is a quintessential outrigger canoe race of the
NCOCA (Northern California Outrigger Canoe Association) annual race circuit that combines
long distances and challenging conditions for a demanding but rewarding race experience in the
San Francisco Bay Area. The race starts and ends at the northern tower of the Golden Gate
Bridge, and loops in a counterclockwise fashion around Angel Island for a roughly 10.5-mile
race. As the racecourse is positioned near the mouth of the San Francisco Bay, tidal flows are at
their highest relative to anywhere else in the bay, making this geographic region appropriate for
modeling dynamic surface water velocity profiles.
Figure 1. Kula Anela course map. The 10.5-mile “long course” is highlighted in orange.
Tidal Cycles and Tidal Flow Patterns
Tides are fluctuations in the water level of large bodies of water, such as the Pacific
Ocean, due to the gravitation pull of the moon as it orbits around the Earth. The tide cycle in the
San Francisco Bay follows a mixed semidiurnal pattern, in which there are two high tides and
two low tides of different size every lunar day.
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Figure 2. Mixed semidiurnal tide cycle, common along the western coastline of North America.
As the water level of the Pacific Ocean outside the mouth of the San Francisco Bay
fluctuates due to tide cycles, the difference in water heights drives the flow of water into or out
of the San Francisco Bay. As the tide rises from low to high tide, “flooding” activity is witnessed,
as water flows landwards into the bay, and as the tide falls from high tide to low tide, “ebbing”
activity proceeds, in which water recedes from the bay into the ocean.
Figure 3. Visualization of water flow during “flood” and “ebb” activity. At the water level extrema (high or low
tide), there are no significant water currents due to tidal flow, which is referred to as “slack tide”.
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Velocity Profile Modeling with Continuum Mechanics
Continuum mechanics may be employed to model the water velocity profile around
Angel Island. To model our desired system, we must define the appropriate conservation
equations, region of analysis, and boundary conditions.
Conservation Equations
As we are interested in evaluating for velocity, the equation of continuity (conservation of
mass) and the equation of motion (conservation of momentum) are relevant to our analysis. In
this model, the equation is motion is simplified to a Newtonian fluid with constant density and
viscosity. This is a reasonable assumption for the scope of this project, as saltwater is
incompressible and does not vary significantly in viscosity over the supposed temperature range
of the region.
Equation of Continuity:
𝜕𝜌
+ (𝛻 ∙ 𝜌𝑣⃑) = 0
𝜕𝑡
Equation of Motion (Navier-Stokes):
𝜌
𝐷𝑣⃑
= −𝛻𝑃 + 𝜇𝛻 2 𝑣⃑ + 𝜌𝑔⃑
𝐷𝑡
Further simplifications are applied to the conservation equations. The equations are to be
solved in two-dimensional Cartesian coordinates and neglect body forces (gravity) in the x-y
plane. Additionally, each “snapshot” of the system can be placed under a quasi-steady-state
assumption, as it may be assumed that the timeframe for the velocity profile to develop is
significant shorter than the timeframe to proceed through different phases of a tide cycle. The
resulting partial differential equations are as follows:
Equation of Continuity:
𝜕
𝜕
(𝜌𝑣𝑥 ) +
(𝜌𝑣𝑦 ) = 0
𝜕𝑥
𝜕𝑦
Equation of Motion (vx-component):
𝜕𝑣𝑥
𝜕𝑣𝑥
𝜕𝑝
∂2 v𝑥 ∂2 v𝑥
𝜌(𝑣𝑥
+ 𝑣𝑦
)=−
+ 𝜇( 2 +
)
𝜕𝑥
𝜕𝑦
𝜕𝑥
∂x
∂y 2
Equation of Motion (vy-component):
𝜕𝑣𝑦
𝜕𝑣𝑦
∂2 v𝑦 ∂2 v𝑦
𝜕𝑝
𝜌(𝑣𝑥
+ 𝑣𝑦
)=−
+ 𝜇( 2 +
)
𝜕𝑥
𝜕𝑦
𝜕𝑦
∂x
∂y 2
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Region Selection and Boundary Conditions
The region of analysis is defined as the following 8x8 mile rectangle centered on Angel
Island. The following Dirichlet boundary conditions are imposed to simulate realistic flow
through the system:
•
Velocity: In the southwest corner, the velocity of tidal flow is fixed to a user-input value.
•
Pressure: Pressure sinks and sources are implemented along the northern, eastern, and
southeastern edges to simulate realistic flow through the rest of the bay outside of the
boundaries of the region. The values of the pressure sinks/sources are proportionally scaled
to the user-input velocity.
•
No-slip condition: Throughout the entire region, velocity magnitudes in both x and y
components are set equal to 0 along any physical edges.
Figure 4. Before-and-after of the defined region of analysis.
Green: Velocity is defined. Yellow: Pressure is defined. Orange: Velocity is equal to 0 (no-slip).
With the conservation equations, region, and boundary conditions fully defined, the
solution of this set of equations will provide the desired water velocity profile over the defined
region.
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Calculation of Boat Velocity and Total Travel Time
As a sport focused on racing, the total elapsed time to complete the course from start to
finish is of chief importance. The total elapsed time to complete a route is dependent on the path
length and the velocity of the boat as it proceeds along each part of the route.
𝐿
𝑡=∫
0
1
𝑣𝑛𝑒𝑡
𝑑𝑠
The net velocity of the boat is a function of paddling power (𝑣𝑝𝑎𝑑𝑑𝑙𝑖𝑛𝑔 ) from within the
boat and the tidal flow velocity profile (𝑣⃑𝑡𝑖𝑑𝑒 ). To simplify this calculation, paddling speed is
assumed to be held constant over all times and distances while paddling, which leaves only the
tidal velocity vector as a varying function of position. The effect that tidal flow has on boat speed
may be further resolved by separating it into tangential and orthogonal components.
𝑣𝑛𝑒𝑡 = 𝑓(𝑣𝑝𝑎𝑑𝑑𝑙𝑖𝑛𝑔 , 𝑣𝑡𝑖𝑑𝑒,∥ , 𝑣𝑡𝑖𝑑𝑒,⊥ )
𝑣𝑝𝑎𝑑𝑑𝑙𝑖𝑛𝑔 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Tangential flow (𝑣𝑡𝑖𝑑𝑒,∥ ), in this scenario, is defined as the dot product between the “boat
travel direction” unit vector (𝑏⃑⃑) and the tidal flow vector (𝑣⃑𝑡𝑖𝑑𝑒 ) at any given point in the defined
region. This is a measure for the degree of agreement between the two vectors, in which a
positive value will add to the net velocity and a negative value will reduce the net velocity.
𝑣𝑡𝑖𝑑𝑒,∥ = 𝑏⃑⃑ ⋅ 𝑣⃑𝑡𝑖𝑑𝑒
Orthogonal flow (𝑣𝑡𝑖𝑑𝑒,⊥ ) is defined as the magnitude of the cross product between 𝑏⃑⃑ and
𝑣⃑𝑡𝑖𝑑𝑒 . This is a measure for sideways forces that act on the boat and requires “course correction”
to remain on track.
𝑣𝑡𝑖𝑑𝑒,⊥ = ‖𝑏⃑⃑ × 𝑣⃑𝑡𝑖𝑑𝑒 ‖
Now that tangential and orthogonal flow have been defined, the net velocity of the boat
as it proceeds in a given direction from a set point in space may be defined as the sum of the
“forward corrected velocity” (𝑣𝑓𝑜𝑟𝑤𝑎𝑟𝑑 ) and the tangential flow. The “forward corrected
velocity” is calculated from the degree of course correction required due to orthogonal flow.
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Figure 5. Calculation of “forward corrected velocity.”
Combining and substituting the above equations results in the following formula for the
net velocity of the boat at any given position and travel direction, as well as the total elapsed
time to complete the racecourse.
𝑣𝑛𝑒𝑡 = 𝑣𝑓𝑜𝑟𝑤𝑎𝑟𝑑 + 𝑣𝑡𝑖𝑑𝑒,∥
2
2
𝑣𝑛𝑒𝑡 = √(𝑣𝑝𝑎𝑑𝑑𝑙𝑖𝑛𝑔 ) − (𝑣𝑡𝑖𝑑𝑒,⊥ ) + 𝑣𝑡𝑖𝑑𝑒,∥
2
2
𝑣𝑛𝑒𝑡 = √(𝑣𝑝𝑎𝑑𝑑𝑙𝑖𝑛𝑔 ) − ‖𝑏⃑⃑ × 𝑣⃑𝑡𝑖𝑑𝑒 ‖ + (𝑏⃑⃑ ⋅ 𝑣⃑𝑡𝑖𝑑𝑒 )
𝐿
1
𝑡=∫
0
2
𝑑𝑠
2
√(𝑣𝑝𝑎𝑑𝑑𝑙𝑖𝑛𝑔 ) − ‖𝑏⃑⃑ × 𝑣⃑𝑡𝑖𝑑𝑒 ‖ + (𝑏⃑⃑ ⋅ 𝑣⃑𝑡𝑖𝑑𝑒 )
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Methods
Finite Element Analysis and Mesh Generation
The conservation equations, in the form of partial differential equations (PDEs), will be
solved numerically using finite element analysis, which is effective for complex regions and
across a large class of PDEs. As the PDEs (our conversation equations) and our boundary
conditions are already defined, we must next generate a mesh of the region of interest. This was
accomplished by converting elevation data of the geographic region into a contour plot, “slicing”
the plot at -30 feet below sea level, converting it to a boundary mesh, and then finally to an
element mesh region. This final step brings together all the necessary elements for the numerical
solution of the velocity profile over the defined region.
Figure 6a-d. Step-by-step conversion from contour plot to element mesh region.
Travel Path Discretization
The integration used to calculate the total time elapsed may be approximated as the sum
of a series of connected line segments that represent a discretized path function. This will
simplify the calculation process, as now the instantaneous net velocity may be calculated at each
point along a manually plotted route as opposed to integrating over an unwieldy parametric
function.
𝐿
𝑡=∫
0
1
𝑣𝑛𝑒𝑡
𝑑𝑠 ≈ ∑
𝑖
𝐿𝑖
𝑣𝑛𝑒𝑡,𝑖
𝑏⃑⃑𝑖 = [𝑥𝑖+1 − 𝑥𝑖 , 𝑦𝑖+1 − 𝑦𝑖 ]
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Results
Evolution of tidal flow velocity profile over partial tide cycle
Figure 7a-f. Overlaid contour and vector plots, from ebbing tide to flooding. Figure 8a-f. Stream plots.
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Using Mathematica’s finite element analysis function, the velocity profile around Angel
Island was successfully calculated across a partial tide cycle, starting from ebbing tide,
progressing through low tide (“slack”), and ending with flooding tide. Viewing both the vector
and stream plots, one can visualize the progression of tidal currents over time.
Sample Route Analysis
With the successful generation of a velocity field across the defined region, route
comparisons are made viable. Three sample paths were manually plotted: a direct route with the
minimum total travel distance, a “flood favored” route designed to take advantage of flooding
conditions, and an “ebb favored” route designed to take advantage of ebbing conditions.
Figure 9. Overlaid routes over the racecourse.
Figure 10a-c. Individual paths for each route plotted against the velocity plot of flooding conditions. At each point
is an arrowhead representing the “boat travel direction” unit vector 𝑏⃑⃑𝑖 , and the length between points is Li.
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As described in the “Theory” and “Methods” section, the relative total elapsed times were
calculated for each route as they progressed through changing tidal flow patterns as displayed in
Figures 7 and 8.
Direct
Route
Flood
Favored
Ebb
Favored
Distance
A
Moderate
Ebb
B
Weak
Ebb
C
Slack
Tide
D
Weak
Flood
E
Moderate
Flood
F
Strong
Flood
1.06
1.77
1.43
1.31
1.27
1.29
1.42
1.16
1.83
1.64
1.41
1.32
1.30
1.36
1.19
1.61
1.51
1.50
1.65
1.78
1.96
Relative Total Times
2.2
2
1.8
Direct Route
1.6
Flood Favored
1.4
Ebb Favored
1.2
1
Moderate Ebb
Weak Ebb
Slack Tide
Weak Flood
Moderate
Flood
Strong Flood
Table 1 and Figure 11. Comparison of relative time elapsed for each route taken depending on tidal conditions.
Discussion
By employing the appropriate conservation equations and boundary conditions over a
defined element mesh region, we can visualize how water currents may evolve throughout a
region over the course of a tide cycle. One can also see how “boundary layers” develop along the
shorelines due to the no-slip condition, which roughly mimics how drag from the seafloor along
shallow regions will reduce flow through the area. With enough experience, seasoned paddlers
can intuit these same velocity profiles as they paddle through the same waters.
Additionally, using the derived equations for total time elapsed to take each route
depending on tidal conditions, we can demonstrate how tidal conditions can impact route
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selection. At the extremes of the tide cycle, when flooding or ebbing are at their highest, the
routes that are designed for each type of flow pattern (flooding or ebbing) result in the fastest
total elapsed time. Under conditions with less significant tidal currents, the more direct route
results in the fastest time as expected.
Limitations
It is important to note that a plethora of idealizations and simplifications were employed
in the development and analysis of this system. For example, in simulating the water velocity
profile in the San Francisco Bay, laminar flow over a two-dimensional, quasi-steady-state system
with static spatial boundaries was assumed, when in actuality none of the assumptions hold true.
Having static spatial boundaries is a particularly egregious limitation to this system, as tides are
well-understood to push forward or pull back the water line of shores, resulting in moving
boundaries. Additionally, other external weather conditions such as wind direction and
magnitude have been neglected—under the right conditions, wind currents can even predominate
over tidal currents in terms of pushing the canoe in any given direction. Finally, the human
element of paddling must not be ignored. For example, the paddling power output throughout the
length of a racecourse inevitably fluctuates, often intentionally—paddlers will frequently
partition their energy to sprint through regions with unfavorable conditions, and then conserve
their energy on more favorable stretches of the racecourse. Other technical skills such as drafting
wake or surfing waves are part of a paddler’s toolkit but are not taken into consideration here.
Ultimately, it is important to recognize that this simulation is designed to elucidate the
significance of optimal route selection in accordance with water velocity profiles, with all other
considerations for realism being secondary.
Conclusion
Through the application of continuum mechanics and vector math, I was able to simulate
the flow field of a geographic body of water subject to tidal flows. Using the vector field data, I
could then calculate and compare the velocity and total time elapsed of different routes
optimized for varying tidal conditions. The results of this analysis support colloquial racing
philosophy to always “choose the best line!”