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Wave drag: some theory

Wave drag: some theory

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When swimming near the surface, the pressure field around the swimmer sets up a wave system similar to what occurs for ships (Figure 1). Both the wave-length (l) and the wave amplitude increase with increasing swimming speed.

Figure 1: Wave formation: the positive pressure at bow and stern create ‘positive’ waves, while negative pressure point induce ‘negative’ waves around the ship. Dependent on the velocity of the ship, the created waves will have a certain wavelength (l) (after: Carlton 1994).

The wave system that is created by the ship, or for that matter by the swimmer, will travel with the same speed as of the ship on the surface. The crest-to-crest distance of the wave system (l depends on this speed (v) according to:

(where g is the gravitational acceleration = 9.8 m/s^2)

Hence, the crest-to-crest distance will increase when the speed increases. This is illustrated in the left panel of Figure 2, where the wave system surrounding the ship will lengthen when the ship is sailing at higher speeds. At a certain speed the wavelength will equal the “water-line length” of the ship. At that velocity the ship is trapped in a self-created hollow between crests of the bow wave and the stern wave. More effort will lead to a higher wave amplitude leading to a deeper hollow and thus further attempts to increase speed will be extremely costly as most energy is used to “climb out of the hollow”.

Figure 2: Left panel: increase of speed of the ship (v) leads to a longer wave length l. Finally the wave length equals waterline length and the ship has reached hull speed. Right panel: interference of bow and stern wave depends on wavelength and thus on ship speed. In the top panel the bow wave is in synchrony with the stern wave inducing reinforcement of wave formation; in the bottom panel the bow wave cancels the stern wave (after Carlton 1994).

The dependence of wavelength on speed induces another effect given the fixed waterline length of the ship. Especially the bow wave can interfere with the stern wave inducing either reinforcement of the stern wave or cancellation of the bow wave. This is illustrated in Figure 2 (right panel) and Figure 3 (left panel). Apart from the effect of speed this interference has an additional effect on wave amplitude and thus on wave drag illustrated in the right panel of Figure 3. Wave formation in swimming the front crawl seems to some extent similar to wave formation in ships. As can be seen in Figure 4 a clear ‘bow’ and ‘stern’ wave are formed when swimming at higher speeds.

Figure 3: Left panel: interference of bow and stern wave depends on wavelength and thus on ship speed and leads to reinforcement of cancellation of the stern wave. Right panel: both speed and interference will influence the amplitude of the created waves and therefore the wave making resistance (adapted from Carlton, 1994).

Figure 4: Wave length (L) of wave system created by the swimmer.

Similar to ships, Figure 6 suggests that at a certain speed swimmers can be trapped between bow and stern wave and hence that also for swimming a "hull speed", occurs (Vogel, 1994; Aigeldinger & Fish, 1995; Fish & Baudinette, 1999). For ships the hull speed (vh) is a function of the square root of the waterline length of the hull or body (Prange & Schmidt-Nielsen, 1970), assuming l equal to the waterline length recasting equation 3 into:

vh


where lw is the waterline length along the longitudinal axis of the body (in m). With an arbitrary height of 2 m, a hull speed of 1.77 m•s-1 is found. Since real maximum swim speed is about 2 m/s this suggests that 1) humans seem to be able to swim faster than the hull speed and 2) wave making resistance matters at competitive swimming speed. Apparently, the “non-stationarity” of the hull of the swimmer (technique) has an effect on wave drag. A re-evaluation of Figure 2 seems to support this suggestion. In Figure 5 the passive drag recordings seem to exhibit the ‘humps and hollows’ indicative of wave interference whereas the active drag curve seems smooth suggesting that the arm action and body roll prevents wave interference to have an additional drag effect. In line with these suggestions it has been observed that proficient swimmers create waves of lower amplitude than less skilled swimmers (Takamoto, Ohmichi & Miyashita, 1985, see Figure 6).


Figure 5: Active drag and passive drag (in two positions; head low = head in the water, high = head out of the water) presented dependent on velocity for one subject. The passive drag determinations seem to exhibit humps and hollows due to wave interference.



Figure 6: Wave height dependent on swimming speed and swimming technique (adapted from Takamoto, Ohmichi & Miyashita, 1985).

The results of recent experiments (Toussaint, Stralen & Stevens, 2002) show that wave drag cannot be neglected when contemplating improvement of competitive swimming speed. It remains to be determined whether wave drag and thus total drag may be diminished by improving the swimming technique as was suggested previously (Counsilman, 1968; Bober & Czabanski, 1975; Maglischo, 1982; Toussaint & Beek, 1992). It could be theorized that the forward stretched arm increases the length of the ‘hull’, with consequent reduction of the wave-making resistance. Also, the gliding arm could reduce the pressure above it and in front of the head, thereby reducing the amplitude of the bow-wave, i.e. similar to function of the cone shaped nose (bulbous bow, Figure 7) below the water-line in large ships (Larsen, Yancher & Bear, 1981). In effect, measurement of wave amplitudes of swimmers revealed that the amplitude of the bow wave was smaller than halve the amplitude of the stern wave.

Figure 7: Bulbous bow induces a flow field in the region where normally the pressure builds up responsible for bow wave formation. The reduced pressure leads to a reduced amplitude bow wave (adapted from Carlton, 1994).

Another approach to reduce total drag is to evade wave drag by swimming substantially below the water surface after start-dive and turns. This approach is in line with the observation that some excellent swimmers showed outstanding results in competition by covering up to 50% of the competitive distance under water using the butterfly kick only. This suggests that there may be a performance advantage when the swimmer ‘dives under’ the wave-making-resistance at the short competitive distances where a high swimming speed can be developed (Toussaint, Hollander, Berg & Vorontsov, 2000).

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