The present work is concerned with the study of the hydrodynamic performance of an Olympic class “K-1” flat water racing Kayak. The evaluation of the hydrodynamic resistance of the vessel is of major importance since it is directly related to the human power required to sustain a specific speed. In this respect, experiments in calm water and regular waves were conducted at various speeds past the particular boat at the towing tank of the Laboratory for Ship and Marine Hydrodynamics (LSMH) of the National Technical University of Athens (NTUA). The calm water tests were performed in the range of speeds from 0.25 to 5m/s and useful conclusions were drawn concerning the influence of the wave formation on the non-dimensional resistance coefficients. Experiments in regular waves were carried out for two characteristic speeds and showed an increase of the hydrodynamic resistance of about 11%. Furthermore, systematic numerical tests using advanced computer codes developed at LSMHE have been performed in order to investigate whether Computational Fluid Dynamics (CFD) tools can be applied with confidence for predicting the calm water resistance of similar vessels. The scope of this part of the investigation is related to a rapid and cost-effective optimization of the shape of the boat. The computed results for the total resistance were in satisfactory agreement with the measurements, thus forming a basis for further investigation and deeper understanding of the athlete-boat interaction, especially for high performance and competition boats.
Under this study, every coach may form the way his athlete paddles, taking into consideration the hydrodynamic resistance during a canoe – kayak race with or without head waves. Additionally, this investigation is important for the canoe – kayak boat manufacturers since they can improve the boat shapes using existing CFD tools and taking into account the resistance increase due to waves.
**Key words:** racing-kayak, resistance, experiments, potential, RANS
The scope of the present work is to investigate the hydrodynamic behavior of an Olympic class K-1 Flat Water Racing Kayak boat at steady forward speed. In a first approximation, the complicated roll and yaw motion of the boat caused by the rower is simplified by regarding only the forward component including free heave and trim. The athlete is in any case replaced by a constant weight about his/her mean centre of gravity. The study includes both experimental and numerical tests. Basically, the aim of the experimental program was to measure the total resistance of the Kayak, covering a speed range of 0.25 to 5.15 m/s, at the towing tank of the Laboratory for Ship and Marine Hydrodynamics (LSMH) of the National Technical University of Athens (NTUA). The tests took place during the last week of January 2009. First, experiments were carried out in calm water at various speeds. Similar tests have also been performed by towing tanks past other types of vessels, e.g. (3). Next, the particular boat was tested at two characteristic speeds in low regular waves which were produced by the wave generator of the tank. These tests were made in order to assess the increase of the hydrodynamic resistance and the corresponding power which is required to sustain the particular speeds.
On the other hand, the dramatic development of Computational Fluid Dynamics (CFD) provides a valuable alternative for evaluating the hydrodynamic behavior of floating bodies. Many research groups have developed advanced computer codes which numerically solve the flow field around complicated geometries. So far, most of the applications are concerned with flows about ships and try to overcome the problem of extrapolating the towing tank measurements to full scale. However, this is not the case in the particular study because the real vessel is tested in the towing tank and, therefore, the experiments predict accurately its hydrodynamic behavior. The main reason for performing CFD tests is to evaluate the codes that have been developed at the LSMH in order to use them in a future optimization procedure regarding the shape of the boat. The application of reliable CFD tools requires substantially less cost than constructing various models and testing them in a towing tank, since the most favorable shapes can be detected numerically and then a limited number of experiments has to be carried out. In the present investigation two methods have been examined to calculate the boat resistance at steady forward speed; a non-linear potential flow solver as well as a Reynold’s Averaged Navier-Stokes (RANS) solver. Both of them are applied for the first time past the Kayak boat and useful conclusions are drawn.
#### Experimental Procedures
All the experiments were performed in the towing tank of the LMSH. The dimensions of the towing tank are 91 m (effective length), 4.56 m (width), and 3.00 m (depth). The towing tank is equipped with a running carriage that can achieve a maximum speed of 5.2 m/s. The tank is also equipped with a wave generating paddle (wave maker), located at one end of the flume. At the opposite end there is a properly shaped inclined shore, for the absorption of the waves. The wave making facilities can produce both harmonic and pseudorandom waves, in the frequency range from 0.3 to 1.4 Hz. The corresponding significant wave height can reach the level of 25 cm.
The hull provided by Pan-Hellenic Kayak and Canoe Trainers Association (PA.SY.P.K-C) was an Olympic class flat water racing Kayak, K-1 category, which refers to a single-seat boat, having the athlete paddling in a seated position. The weight category of the boat is M (medium), corresponding to an athlete’s weight in the range of 70 to 80 kg.
Minor alterations on the internal structure of the model were applied prior to the measurements, in order to accommodate the measuring equipment. This work was supervised by the personnel of PA.SY.P.K-C.
Both experimental and numerical tests were carried out with the boat having a displacement of Δ=86.8 kg (condition A). This is the sum of the bare hull weight with the added fixtures (11.8 kg) and the mean athlete’s weight, the last taken as 75 kg for the present study. The longitudinal position of the center of gravity (LCG) was chosen at the middle of the athlete’s seat. For the experiments, the rod of the resistance dynamometer was mounted on the hull at this location. The mounting was done using a heave rod – pitch bearing assembly, which allows for the vertical motions and trim angles (heave and pitch responses) of the boat.
The resistance measurements were performed for speeds in the range from 0.25 to 5.15 m/s, for the case of calm water and for two speeds (2.5 and 5.0 m/s) for the case of harmonic waves, (5). All the tests were performed in fresh water, at a temperature of 15 oC.
The boat resistance, the rise of the center of gravity (c.g.), the dynamic trim and the towing speed of the model were recorded during the runs on calm water. In this investigation, trim is defined as the signed rotation about the transverse axis passing through the c.g. and is considered positive when the bow of the kayak sinks. In addition, for the case of harmonic waves, the wave elevation was measured using wave probes.
#### Data Analysis
In order to investigate whether CFD tools can be applied with confidence to predict the calm water resistance of similar vessels under the scope of hull optimization, systematic numerical tests were carried out by applying the non-linear potential flow solver (7,8), as well as the RANS solver (6,8), both developed at LSMH.
The potential method is based on constant source quadrilateral panels that cover the wetted surface of the boat and the real free-surface (Figure 1). The latter is found by an iterative procedure which, after convergence, leads to the satisfaction of both the well known free surface conditions: the kinematic and the dynamic. The potential flow predicts the wave making component CW, whereas the total resistance coefficient CT is calculated by adding the corresponding 1957 International Towing Tank Conference (ITTC’57) value for the skin friction coefficient CF.
![Quadrilateral panels on the hull and water surface for the potential calculations.](/files/volume-13-number-4/1/figure-1.jpg “Quadrilateral panels on the hull and water surface for the potential calculations.”)
**Figure 1** Quadrilateral panels on the hull and water surface for the potential calculations.
Naturally, this procedure suffers from the potential flow drawbacks, i.e. the predicted wave pattern near and after the stern does not include any viscous effects. Besides, the so called form-resistance component including the skin friction alteration due to the shape of the hull and the viscous pressure component cannot be taken into account. These shortcomings disappear when the RANS equations are solved numerically. The latter, however, requires substantially higher computing power and time since a three-dimensional grid discretisation is required, Figure 2.
The employed method uses an H-O type numerical grid which is adjusted to the free-surface as the solution proceeds (6). To account for turbulence effects, the well known k-ε model with wall functions (2) is adopted.
![Numerical H-O type grid for the RANS calculations.](/files/volume-13-number-4/1/figure-2.jpg “Numerical H-O type grid for the RANS calculations.”)
**Figure 2** Numerical H-O type grid for the RANS calculations.
#### Calm water experiments
Calm water resistance tests were done for the speed range of 0.25 to 5.15 m/s. The experimental results concerning the calm water resistance, the CG rise, the dynamic trim and the towing speed of the kayak are presented in Table 1. The corresponding graphs for the resistance, dynamic trim and CG rise are presented in Figs. 3 to 5, respectively.
As observed in Fig. 4, the dynamic trim is negligible in the range of speeds 0-2.5 m/s while it increases rapidly after it, resulting in an increase of the draft at the stern and a raise of the bow. The CG –rise, Fig. 5, is always negative resulting in an increase of the mean vessel’s draft which presents a peak about the speed of 3.5 m/s. This behavior could be associated with the dynamic trim change and shows that the behavior of the boat is very sensitive with respect to the speed.
![Total Resistance](/files/volume-13-number-4/1/figure-3.gif “Total Resistance”)
**Figure 3** Total Resistance
![Dynamic Trim](/files/volume-13-number-4/1/figure-4.gif “Dynamic Trim”)
**Figure 4** Dynamic Trim
![C.G. Rise](/files/volume-13-number-4/1/figure-5.gif “C.G. Rise”)
**Figure 5** C.G. Rise
In order to study the usual Froude decomposition of the total resistance coefficient versus speed, the relation between the total resistance coefficient (CT) and the Froude number (Fn) is, firstly, depicted in Figure 6. These parameters are defined by the following relations:
where VS stands for the speed, g is the gravitational acceleration, L the waterline length, RT the total resistance, ρ the water density and WS the wetted surface.
In the calculation of the total resistance coefficient, the wetted surface used was the one calculated by means of the potential method. The variation total resistance coefficient vs. Fn, presented in Fig.6, shows that it is influenced strongly by the wave formation. The main hump is located in the region of Fn 0.4÷0.45, i.e. it is moved to the left with respect to the predicted one by the linear wave theory (about 0.5) (4). However, the prismatic hump is missing while a “hollow” appears about Fn=0.3 which is moved to the right with respect to the predicted one by the linear wave theory (about 0.24), while the higher values at the low Fn show a dominant effect of skin friction.
![Total resistance coefficient.](/files/volume-13-number-4/1/figure-6.gif “Total resistance coefficient.”)
**Figure 6** Total resistance coefficient.
According to the standard Froude approach, the total resistance coefficient can be decomposed into the friction (CF) and the residual (CR) components as:
The friction coefficient (CF) can be calculated by the ITTC’57 formula as:
represents the corresponding Reynolds number, L is the immersed waterline length and ν the kinematic viscosity.
Furthermore, the residual resistance may be regarded as equal to the so-called wave-making resistance CW, i.e. CR ≈ CW. The three coefficients with respect to the Froude number are presented in Table 2. The negative or very low values of CR at the lower Froude numbers show that the skin friction formula rather over-predicts CF and, therefore, an extended laminar region may cover the front part of the vessel. It should be noted here that no turbulence stimulators were applied since the real hull was tested. The slender form of this hull should result in a thin boundary layer region over the major part of the wetted surface, thus permitting the existence of a laminar zone especially at low speeds, which in any case is favorable because it leads to a reduction of the total resistance.
The residual resistance coefficient, plotted vs. Fn in Fig. 7, shows similar trends with Fig. 6 and influences accordingly the total coefficient. CR is comparable to CF after Fn=0.3, but in any case is lower than that, implying that skin friction plays an important role for the total resistance. This trend is due to the very slender form of the particular boat which was designed to produce low waves, as far as possible.
![Wave, Pressure and Residual resistance coefficients.](/files/volume-13-number-4/1/figure-7.gif “Wave, Pressure and Residual resistance coefficients.”)
**Figure 7** Wave, Pressure and Residual resistance coefficients.
#### Potential results
In order to validate the use of the non-linear potential solver (7) for the examined type of vessel, systematic numerical tests were conducted for the same speed range as the experiments.
The solver has been developed at the LSMH and solves the wave problem by covering the hull and the free-surface with quadrilateral panels. The hull geometry is represented by the conformal mapping approach which exhibits the advantage of a fast and effective reconstruction of panels as the free-surface changes. A special feature of the code is the calculation of the free-surface by combining an integral with a differential method. The total number of panels used was 12,000 while the trim angle as well as the dynamic rise of the c.g., were calculated numerically. The potential results of the examined cases are shown in Table 3. Essentially the method predicts only the wave resistance component CW, while CF is derived under the ITTC’57 skin friction approximation. The predicted CW is compared to the measured one in Fig. 7. Evidently it exhibits the same variations, but it is lower than the experimental in the whole range of Fn. This is an expected behavior according to the aforementioned shortcomings. The potential theory predicts higher waves at the stern region, resulting in increased pressures underneath the stern that in turn lead to a reduction of the total wave resistance. However, the total resistance coefficient appears closer to the experimental in Fig. 6 where the skin friction was added. This is reflected also to the calculation of the total resistance (which is the meaningful quantity) in Fig. 3, where the calculated results are in satisfactory agreement with the measurements up to the speed of 3.5 m/s (~7%) while deviations increase at higher speeds.
#### RANS results
In order to explore the possibility of obtaining better results at high speeds with RANS computations, three test cases were examined, corresponding to the speeds of 3, 4 and 5m/s. The relevant code has also been developed at the LSMH and, unlike other methods, uses the concept of orthogonal curvilinear co-ordinates to solve the viscous flow equations. This feature is beneficial for obtaining effectively converged solutions. The free-surface is calculated iteratively by applying a surface-tracking method that has been developed for the first time in (6).
In any case the grid size had 2.65 million grid points. To reduce the computation cost as well as the uncertainties related with the longitudinal position of the center of gravity, the trim angle of the vessel was taken from the experiments while it was assumed free to heave. The results acquired via the RANS solver are shown in Table 4. First, it is important to notice that the calculated skin friction coefficient CF is in very good agreement with the empirical ITTC’57 formula in Table 2, which justifies the relevant assumption when the potential method is adopted. The calculated values of the total resistance coefficients are presented in Table 4. Evidently, the total resistance is predicted with satisfactory agreement with respect to the experimental values for the examined speeds. The larger deviation at the highest speed may be a result of the extended wave breaking which was observed during the experiments in this case, which cannot be simulated numerically. The deviations percent of the calculated vs. the experimental total resistance is depicted in Table 5 for both methods, where the superiority of the RANS approach is obvious at high speeds.
The calculated wave patterns about the boat by the RANS computations are plotted in Figs. 8 to 10 for the speeds of 3.0 m/s, 4.0 m/s and 5.0 m/s, respectively. The full lines represent wave crests while the dashed lines correspond to wave troughs. These plots show a regular formation which is similar to the real one observed during the experiments.
![Water surface elevation contour, RANS solver, VS =2.995 m/s.](/files/volume-13-number-4/1/figure-8.jpg “Water surface elevation contour, RANS solver, VS =2.995 m/s.”)
**Figure 8** Water surface elevation contour, RANS solver, VS =2.995 m/s.
(Full lines: wave crests, dashed lines: wave troughs)
![Water surface elevation contour, RANS solver, VS =3.989 m/s.](/files/volume-13-number-4/1/figure-9.jpg “Water surface elevation contour, RANS solver, VS =3.989 m/s.”)
**Figure 9** Water surface elevation contour, RANS solver, VS =3.989 m/s.
(Full lines: wave crests, dashed lines: wave troughs)
![Water surface elevation contour, RANS solver, VS=5.153 m/s.](/files/volume-13-number-4/1/figure-10.jpg “Water surface elevation contour, RANS solver, VS=5.153 m/s.”)
**Figure 10** Water surface elevation contour, RANS solver, VS=5.153 m/s.
(Full lines: wave crests, dashed lines: wave troughs)
#### Experimental tests in regular waves
The tests in regular waves were done at the speed of 2.5 m/s for wave frequencies of 0.3 Hz, 0.5 Hz, 0.7 Hz, and 0.9 Hz and at the speed of 5.0 m/s for wave frequencies of 0.3 Hz and 0.5 Hz (5).
During the tests, the following responses were measured:
– C.G. rise
– Added resistance
– Wave Height
The experimental results for these tests are presented in Table 6. Based on the recorded time histories of the boat responses, the Response Amplitude Operators (RAOs) in heave (at the CG position) and in pitch motion were calculated and presented also in this Table, together with the measured values of wave amplitude and mean added resistance.
The non-dimensional RAO values were calculated using the following formulae:
– RAOHEAVE = ξ0 / ζ0
– RAOPITCH = θ / (k ξ0)
– ξ0 : heave response amplitude
– ζ0 : wave amplitude
– θ : pitch amplitude [rad]
– k : wave number (k=2π/λ)
– λ : wave length
The most important result is the resistance increase presented in the last column of Table 6. It can be concluded that the added resistance is negligible for wave lengths much larger than the boat length (low frequency range, examined frequency 0.3 Hz) and can reach values from 7 to 12% for faster waves (examined frequencies 0.5, 0.7, and 0.9 Hz) and for both wave heights. This resistance increase reflects directly on the power required by the athlete.
The measured total resistance coefficient shows a minimum about the vessel speed of 1.5m/s and a maximum at 3.0 m/s. These values appear as a result of the interactions of the generated wave systems about the boat. In addition, the Froude decomposition of the total resistance coefficient demonstrates that skin friction is higher than the residuary component at all speeds, while at low speeds the appearance of laminar flow regions about the bow is rather possible. Wave breaking was also observed at speeds above 3.5 m/s.
The performance of the boat subjected to low amplitude heading harmonic waves was also investigated. The main conclusion is that short waves (high frequencies) may increase the boat resistance and, therefore, the required human power by almost 10%.
The applications of the employed CFD approaches have shown that the computation of the total resistance by applying a non-linear potential flow code in conjunction with the ITTC’57 skin friction formula is in good agreement with the measured one for speeds up to 3.5 m/s. Above this level, viscous effects are dominant and RANS methods have to be employed to obtain accurate results. However, in the usual range of speeds of the particular vessel, the potential approach may produce reliable results and, therefore, can be involved in optimization procedures concerning the hull geometry.
The current investigation has been based on the fruitful collaboration of three research groups, i.e. the Laboratory for Ship and Marine hydrodynamics of NTUA, the Pan-Hellenic Canoe – Kayak Trainers Association, and the Department of Physical Education and Sport Science of the University of Athens. The groups combined their efforts for the first time, and the data acquired can form a basis for further investigation and deeper understanding of the athlete-boat interaction, especially for high performance and high competitive boats, like the case at hand. The research will be continued toward the hull optimization of the boat as well as the experimental study of the effect of the yaw and roll motions by designing the proper experimental apparatus. The numerical tools will be further developed to simulate these motions as well as to take into account the unsteady influence of waves.
The systematic numerical experiments have shown that both potential and RANS methods can be applied in order to calculate the calm water resistance of a flat water racing kayak. The potential solver provided results in good qualitative agreement with the experiments and, therefore, can be involved in optimization procedures concerning the hull geometry. The RANS solver gave very accurate predictions for the total resistance and therefore can be used with confidence for predicting the resistance of vessels of similar geometry.
### Applications in Sport
In the last several years we have seen a tremendous rise in new technologies (construction materials, e.g. carbon fiber) (1) which in their way affect the increasing improvement of results in canoe – kayak. The main factor for the accomplishment of better times in canoeing is the hydrodynamic resistance of the boat’s hull. With this study, every coach may develop the way his athlete paddles, taking into consideration the hydrodynamic resistance which is observed depending on the waves appearing during a canoe – kayak race.
Additionally, this study is very important for the canoe – kayak boat manufacturers, since they can achieve the making of more improved boat hulls, taking into account the hydrodynamic resistance appearing under different types of waves.
The authors wish to thank the personnel of LSMH and particularly Mr. I. Trachanas who has carried out the measurements in the Towing Tank as well as Mr. D. Triperinas, Ms. D. Damala and Mr. G Katsaounis for designing the experiments and interpreting the results.
The authors would also like to thank Lloyd’s Register Educational Trust (LRET), since Mr. Polyzos’ Phd studies are supported by LRET.
The Lloyd’s Register Educational Trust (LRET) is an independent charity working to achieve advances in transportation, science, engineering and technology education, training and research worldwide for the benefit of all.
#### Table 1
Experimental results for the calm water resistance tests, condition: Δ=86.8 Κp.
|Speed||Froude Number||Total Resistance (Rr)||Dynamic Trim (+) by bow, (-) by stern||C.G. Rise|
#### Table 2
Experimental results for the calm water resistance tests.
|Speed||Froude Number||Total Resistance (Rr)||Total Resistance Coefficient (CF)||Frictional Resistance Coefficient (CT)||Residual Resistance Coefficient|
#### Table 3
Numerical results for the calm water resistance tests, potential method.
|Speed||Froude Number||Dynamic Trim (+) by bow, (-) by stern||C.G. Rise||Wave Resistance Coefficient (CW)||Frictional Resistance Coefficient (CF) (ITTC’57)||Total Resistance Coefficient (CT)||Total Resistance (RT)|
#### Table 4
Numerical results for the calm water resistance tests, RANS method.
|Speed||Froude Number||Pressure Resistance Coefficient (CP)||Frictional Resistance Roefficient (CF)||Total Resistance Coefficient (CT)||Total Resistance (RT)|
#### Table 5
Experimental results for the calm water resistance tests.
|Speed||Froude Number||Deviation in Total Resistance δRT (%)|
#### Table 6
Experimental results for the tests in regular waves.
|Speed||Wave Frequency||Wave Amplitude||RAO Heave||RAO Pitch||Added Resistance||Resistance Increase|
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### Corresponding Author
Mr. Stylianos Polyzos
Laboratory for Ship and Marine Hydrodynamics
9 Heroon Polytechniou str. NTUA Campus, Zografos 15773, Greece
### Author Bios
George Tzabiras is a Professor and Head of the Laboratory for Ship and Marine Hydrodynamics at the National Technical University of Athens (NTUA).
Stylianos Polyzos and Konstantina Sfakianaki are Phd Candidates at the Laboratory for Ship and Marine Hydrodynamics.
Athanasios D. Villiotis and Konstantinos Chrisikopoulos are members of the Pan-Hellenic Canoe – Kayak Trainers Association
Vassilios Diafas and Sokratis Kaloupsis are Professors at the University of Athens, Department of Physical Education and Sport Science, Faculty of water sports