1. INTRODUCTION

Kayaks and similar canoes have been in use for at least 2000 years, possibly as long as 4000 years (Noble et al. 1994). For the greater part of that time design improvements came via trial and error, by imitating the shapes of swimming animals, and through the inevitable progress made because humans can learn and pass that knowledge onto succeeding generations. Boat design became a science, irrevocably, about 100 years ago, primarily through the insights and investigations of William Froude (1872), and later through the mathematical labours of J. H. Michell (1898) and his successors. Thirty years ago, advances in computer technology enabled substance to be added to what had been, until then, largely ethereal. Today, computer-aided boat design is commonplace.

However, according to one kayak designer and builder, Winters (1996),

...canoes are rarely "designed"; they're more often adaptations or modifications of earlier shapes. Given the apparent success of this method, many would question the need for a more scientific approach. The value, of course, lies in the plodding nature of trial and error and the preponderance of failure over success. The designer, by applying hydrodynamic principles developed through experimentation (and, of course, trial and error), is able to improve the breed more rapidly while minimizing mistakes and risks. The process used is rarely inspirational (advertising hype not withstanding), and begins with a set of parameters for the proposed new canoe.

In this study, our set of design parameters is much smaller and simpler than that facing most kayak designers. We, for example, aren't concerned with stealth, whereas this is of paramount importance to native hunters of sea mammals which immediately dive below the surface when alarmed, Winter (1996). Nor are we too concerned with stability, which is of importance to those who design for whitewater and waves. Our simple aim is to minimise the upright, calm-water resistance of the submerged part of the kayak, while staying within the rules of international competition.

The International Canoe Federation (ICF) currently recognises three classes of kayaks, K1, K2 and K4 for one, two and four paddlers, respectively. The restrictions on the principal hull dimensions are given in Table 1.

The same constraints apply to both men's and women's events. Additionally, the section and longitudinal lines of the hulls are required to be convex.

Note that the minimum beam limits in the above table apply to the above-water portion of the hull. As there is no minimum beam limit for the underwater portion, (and because of a certain arbitrariness as regards I.C.F. kayak hull measurement), for the sake of the present report we have taken these limits as applying to the waterline.

At the 1996 Atlanta Olympics, kayak events were held over 500 metres and 1000 metres. Women's events were restricted to 500m.; male crews competed over both distances, except that there was no men's 500m. K4 event.

Table 2(a) shows the winning (Gold Medal) times recorded in Atlanta. Also shown are our estimates for the all up weights (i.e. paddlers + equipment) for each class, and the average speed over the 500 metre course.

Table 2(b) shows the same particulars as Table 2(a) but this time for 1000 metre events.

The (mean) volumetric Froude number, is defined by

where U is the (average) boat speed in metres per second, D is the total weight or displacement of the kayak (in tonnes), and g is gravitational acceleration, herein 9.81 ms^-2.

For the purposes of this study, we have included an additional displacement for each class, viz., 0.080, 0.155, and 0.300 tonnes for the K1, K2 and K4 classes, respectively. These are more representative of lightweight women's events and, although these events were not contested at the Atlanta Olympics, we have included them here to better gain an appreciation of how variations in displacement might affect performance.

1.1 Wave Resistance

We use Michell's integral (Michell 1898; see also Tuck 1987) to estimate the wave resistance of the kayaks. This requires evaluation of a triple integral, one integral in each of the length-wise and draft-wise co-ordinate directions, and one integral with respect to the angle theta of propagation of the boat-generated waves. The numerical method used here for evaluating these integrals is described fully in Tuck (1987). We use 33 stations, 33 waterlines, and 320 intervals for the integration with respect to theta.

1.2 Viscous Resistance

The viscous resistance R_v can be written as

R_v=1/2*rho*U²*S*C_v

where rho is the water density (herein 1000 kg/m³) and S the wetted surface area of the hull. When skin friction dominates, the drag coefficient C_v approximately equals C_f, where C_f is a skin friction coefficient which can be estimated using the ITTC 1957 ship correlation line (Proc. 8th ITTC).

C_f = 0.075/(log₁₀ (R)-2)²

where R = UL/nu is the Reynolds number. Here L is the length of the waterline, and nu is the kinematic viscosity (herein, 1.054x10^-6m²s^-1).

1.3 Form Effects

Including a form factor specific to the hullform under consideration can often give better estimates of the viscous drag. This factor is difficult to estimate and may vary with speed because of (among other things) changes in trim and sinkage. In their examination of eight-oared rowing shells, Scragg and Nelson (1993) found a simple empirical formula for the form factor of these hulls. The viscous resistance coefficient is written as

C_v = (1+k) C_f

where

k = 0.0097(theta_entry + theta_exit)

Here, theta_entry and theta_exit are the half-angles (in degrees) of the bow and stern, respectively, at the waterplane.

Although Scragg and Nelson based their formula on empirical data for single, double, quad and eight-oared rowing shells, we suspect (almost as much as hope) that their formula will give reasonable estimates for the small, slender, convex hulls considered in the present study.

1.4 Shape Effects

A three parameter hullform family was used to investigate the effects of shape.

Let X(x) = 4x(x-1), with 0<=1.

Then the non-dimensional offsets, are given by

0 if X=0 or X^a₃<z,
Y(x,z; a₁,a₂,a₃) = 1/2FG^a₂ otherwise.

Here F = X^a₁, G = 1-z²X^-2a₃ and 0 <= x,z,a₁,a₂,a₃ <=1.

At any given length and draft, adjust the beam by uniform scaling of the non-dimensional offsets given by the above formula, so as to achieve the desired displacement.

In the above formulation, the first parameter, a₁, controls the shape of the waterlines, the second parameter, a₂, determines the cross-section shape, and the third parameter, a₃, controls the shape of the keel-line. If, for example, a₁=0.0, then the waterline shape is rectangular, if a₁=0.5 the waterline is elliptical and if a₁=1.0, the hull will have parabolic waterlines. And similarly for cross-sections and keel-lines.

Some other examples are shown in Table 3. For example, the PEP hull (a₁,a₂,a₃)=(1.0,0.5,1.0) has parabolic waterlines, elliptical cross-sections and a parabolic keel-line. The PER (a₁,a₂,a₃)=(1.0,0.5,0) hull is similar, but has a rectangular sideview.

Scragg and Nelson's form factor penalises hulls with large entrance and exit angles at the load waterline, and consequently there will be a strong push towards parabolic waterlines, i.e. toward a₁ = 1.0 in the formulation given above.

1.5 Some Effects Neglected

Wave-breaking and spray resistance are neglected. Wave-breaking for our fine, sharp-bowed hulls should be negligible. Spray resistance seems to be one of the reasons that form factors are difficult to calculate at high speeds.

We assume that there is no effect of dynamic vertical forces, which at low speeds account for sinkage and trim. These effects are currently being examined by the present authors. At high speeds, dynamic forces are upward and yield a lift rather than sinkage; hence planing, and we neglect that. Charlton, Denman and Millward (1987) have shown that it is theoretically possible for a top class, single sculler using a boat equipped with hydrofoils, to achieve foilborne motion. Such arrangements are of course prohibited by the ICF, but we make mention of them here to show that we cannot dismiss planing effects cursorily.

Shallow-water effects can be important in some applications, e.g. see Scragg and Nelson (1993) for eight-oared rowing shells. However, we retain the infinite-depth assumption here.

We do not make any allowance for changes in the centre of gravity or the consequent change in trim due to the continually changing position of the paddlers. The present analysis is done on a steady-flow basis, and hence relates to the average conditions during a race, neglecting speed variations due to racing conditions, as well as unsteady variations during an individual paddling stroke. Unsteady effects on rowing shell performance are currently being examined by the present authors.