The sailing properties of the Hanse cog in comparison with other cargo sailships
H. Brandt, K. Hochkirch , TU Berlin Institute of Naval Architecture, Marine and Ocean Engineering (ISM), TU Berlin, Salzufer 17-19, D-10587 Berlin, Germany
For a Hanse cog the stability was calculated, and wind-tunnel experiments were carried out to determine the sail-carrying capacity. Resistance and manoeuvering experiments yielded initial impressions of the travelling speeds that might be expected and proved the ship's steerability. With a full-scale replica of the Hanse cog, sailing performance tests were carried out under various weather conditions. The analyses showed that with a proper crew, in ballast or simulated barrel load condition, the cog demonstrates rough but safe seakeeping. At a broad reach to the wind, sailing performance was high compared to other cargo sailships. The Hanse cog's weaknesses appear when beating to windward.
Keywords: cog, sailing, performance, speed, manoeuvring, seakeeping.
A Hanse cog was discovered during dredging work in the port of Bremen in 1962. The cog's construction was dated to the year 1380 by means of dendrochronological analysis. Until this finding was made, virtually nothing was known about the shape and rigging of Hanse cogs. The sole source for information on their appearance had been illustrations on seal impressions, which provide only an approximate idea. Accounts of possible travelling speeds vary greatly, i.e. between 3.3 and 10 knots; data on weather conditions and wind directions are completely absent from reports.
The complete construction drawings, of the Hanse cog of 1380, have been published by Lahn (1992). On the basis of the various knowledge obtained, two reproductions of the cog have been built, Hoheisel (1988, 1993). For the present study the cog replica of Kiel is of particular interest as it corresponds largely to the original. This applies not only to the contours and rigging but, with a few minor exceptions, to the construction materials as well, Baykowski (1991, 1993). The first step in the study of the cog's sailing performance was to draw a line projection of the contours of the original hull, averaging out its asymmetries (Figs. 1 to 3). Since rudder, mast and yard were never found, there was no choice but to make assumptions based on seal depictions and new calculations. There is no definite proof as to whether the lengths of the mast and yard used on the Kiel reproduction of the cog correspond to the original. A full sheet of hydrostatic curves was computed to obtain information on the sail-carrying capacity. According to estimations of the centre of gravity and the necessary amount of ballast, stability calculations were carried out. With the projected square sail, Clausen (1988) undertook wind-tunnel experiments and determined the transverse, driving and wind-resistance forces of the sail and the hull above waterline. Finally, the towing experiments made by Postel (1989, 1992) with a model of the Hanse cog provided proof of her steerability and a prognosis for the hydrodynamic resistance and the transverse forces.
The estimation of the stability showed that the empty cog, weighing 51.8 t., required equipment weighing 12.9 t. and an additional 26.4 t of ballast in the form of fieldstones in her bilge. In ballast condition the Hanse cog thus has a total displacement of 91.1 t. Through heeling experiments the exact height of the centres of gravity in ballast and in load condition (displacement 127.8 t) were determined. These data allowed to establish the actual righting arms and with them the heeling stability (Fig. 4). Under these conditions sailing experiments sponsored by the German Society for the Advancement of Scientific Research were carried out with the Hanse cog of Kiel. The results of the sailing performance tests allow a comprehensive assessment of the sailing properties and seafaring capacities of the Hanse cog of 1380.
The sailing performance tests
The sailing performance tests served to establish polar plots for the sailing performance of the Hanse cog under various weather and wind conditions. They were also intended to allow analysis of her seaway and manoeuvering properties. In ballast condition the cog had a mean draught of 1.65 m at a stern trim of 0.5 m. To simulate a barrel cargo, the cog was loaded with 35 concrete ashlars weighing 1.05 t apiece, leading to a mean draught of 2.07 m and a stern trim of 0.38 m. The sailing experiments were carried out north of Sjaelland, off Warnemünde, in the Bay of Kiel, and east of Schleimünde.
Each test journey required fourteen persons to operate the cog and an escort vessel with a crew of three. The excursions had to be arranged plenty of time in advance; thus it was nearly impossible to take sudden changes in the weather conditions into account. Test journeys were made in winds measuring 0 to 7 on the Beaufort scale. For safety reasons, heeling angles of 12o to 15o were not exceeded: green sea on the unsealed, permeable deck would have endangered the ship. The outer planking and the bulwarks are tightly connected and not equipped with scuppers; all incoming water flows into the bilge and has to be pumped out to keep the ship from swamping and capsizing.
The sailing performance tests were carried out over periods of ten to fifteen minutes at constant sail settings and courses to the wind. The following readings were recorded and processed on-line:
The ship's motions, recorded with the aid of the horizontal gyroscope, are substitutional for a measurement of each respective seaway. For the calculation of the sailing performance diagram (polar plot), all of the values measured had to be converted to their "true" values. These include the true wind speed and wind direction in relation to the true course, differing from the ship's centreline by the amount of leeway, Fig. 5
The apparent wind angle (ßAK) to the true course is
where ßA denotes the direction of the wind relative to the centreline. The true ship speed can be determined by
and the leeway is therefore .
With the apparent wind speed (VA), the true wind speed (VT) can be calculated by .
The true wind angle (ßTK) to the true course is
and the velocity made good against or with the wind .
Despite the utilization of highly sensitive, calibrated measuring devices, under the natural conditions of the environment numerous factors influence the results. Although their essential causes are known for the most part, the consequences can hardly be eliminated. For this reason the careful analysis of the individual values and the integration of data recorded over relatively long measuring periods is imperative, Brandt (1992).
The wind speeds, measured at the top of the twenty-four-metre-high mast, remain uncorrected for use in the calculations. The somewhat lower value at the height of the centre of gravity of the sail would actually be more suitable for describing the ship's capacity to perform. That height depends on the sail area: with three bonnets and the mainsail it is at 9,65 m; when the wind is high and only one bonnet is in use, at 7.2 m. The wind distribution curves over the surface of the water, however, vary greatly. Not only are they a function of the wind speed, they also depend on the height of the seaway and on the weather conditions. Finally, the angle of inclination, i.e. the extent to which the wind direction deviates from the horizontal, is not considered.
Fig. 6 shows an example of plotted readings for the Hanse cog's polar plot. The measuring points shown have already been converted to their respective true wind speed and direction. Only the "quasi-steady" values were printed. The term "quasi-steady" indicates that during two measuring periods of approximately two to three seconds each, neither the ship's speed nor the leeway changed. This reduces the total number of measuring points to approximately one-third.
The smoothing method
Curves for each respective constant wind speed at all courses to the wind were smoothed as follows. The dependence of the ship's speed on the wind speed can be expressed as a polynomial statement for each constant angle of the wind:
A change in the ship's speed brought about by a change in the angle of the wind is replaced by a Fourier series whose coefficients are functions of the wind speed:
Selecting polynomials for p and q and regarding the cosine and sine terms separately, superposition gives this expression for the approximated ship's speed:
It is necessary to regard the sine and cosine terms separately to control the influence of asymmetry later on. For a completely symmetrical ship with symmetrical sail settings, the sine coefficients will be zero.
Np, Nps, Ns, Npc and Nc are constants characteristic of the model structure. These constants have to be selected such that a satisfactory approximation of the readings results.
Under the trivial condition that the ship's speed for VT = 0 disappears, for all *TK
the following applies to the coefficients:
The model (9) is thus simplified to (12) with altogether NK = Np + Nps * Ns + Npc * Nc coefficients.
This approximation model is linear in its coefficients and thus directly solvable. With the coefficients determined by a least square fit, the ship's speed can be established for all angles and wind speeds using Eq. (12). The sum of the squared errors is
Here (NN + 1) is the number of readings and VTi, ßTKi, VSi designate the readings of the ith measurement. The average error can be obtained using
The model structure according to Eq. (12) selected is stated in each graph in the form:
P: Np - Nps - Ns - Npc - Nc .
It cannot be decided in advance which model structure will yield a satisfactory result, since the structure strongly depends on:
The sailing performance of the Hanse cog
Figs. 7 to 9 show three typical polar plots of the Hanse cog for various sail settings and load conditions. In this case the ship-speed curves for each constant wind speed are shown separately for port and starboard tilt since the graphs for the two sides differ considerably, especially in ballast condition. The causes lie not only in the existing asymmetry of the ship's hull, but can certainly also be ascribed to changes in the respective sail setting. The characteristic "apple form" of other polar plots, e.g. for large barks and full-rigged ships, is wholly absent for the cog. The ship achieves her highest speed towards her destination when sailing a direct course on a run.
An analysis of the polar plots reveals that when beating to windward the cog achieves only low VMG (velocity made good), and that the sailing performance values decrease as the wind increases and the sail area is reduced.
Can cogs beat against the wind? The polar plots show the maximally attainable angle of closeness to the true wind to be between 67o and 75o, depending on sail size and wind speed. These are course angles that can be sailed for at least short periods of time, resulting in a VMG value of approximately one knot. Over longer distances, however, VMG is only attained in very light winds in sheltered water, i.e. without seaway. Such were the results of measurements of speed and distance over ground with the aid of GPS devices. Taking two tacking manoeuvers into account, at wind speeds of Beaufort 2 and using the total sail area, the VMG amounted to 0.63 knots (Fig. 1010) in ballast condition. In load condition with two reefs in the sail and sailing at SSW 6 on a moderate sea in an area off Warnemünde, the cog was forced to turn 0.1 nm away from the wind (Fig. 11). In completely unloaded condition, sailing with one reef at ESE 5 to 6 in the Strander Bay, the loss of closeness to the wind was higher - 0.2 nm over distances of 1.8 and 2.4 nm. In such situations the leeway is quite considerable. It reaches 10o to 15o and can climb even higher when the sheet is hauled too close.
The cog's sailing performance at close hauls becomes worse the more the sail has to be reefed. For the most part this is due to the large lateral plane of the ship's hull above waterline. The proportion of the area of disturbance when maximum sail area is in use amounts to 40%, compared to nearly 60% with two reefs. The relationship of the driving force to the resistance and transverse forces decreases accordingly, as does the attainable angle of closeness to the wind. Thus Hanse cogs can hardly have beaten against the wind; they are suitable only for reaches.
The Hanse cog's average statistical speeds for all courses to the wind, calculated from the readings, were:
Sailing on a reach with a wind of Bft. 7, the ship attained a speed of 8 knots for short periods; this reading corresponds to a Froude number of 0.3.
An almost wholly independent criterion for the assessment of the Hanse cog's sailing performance is provided by the driving-force and resistance coefficients. It was possible to establish these coefficients on the basis of various previous experiments: The force components of the sail, including the ship's hull above waterline, had been established through wind-tunnel experiments. And the towing resistance had been determined accurately by tank experiments with the model and additional towing experiments with the full-size ship.
On the basis of the sail force coefficients in relation to the ship's longitudinal axis, determined in wind-tunnel experiments, the driving force coefficients in the direction of travel (CR) can be calculated for various wind speeds with the aid of the true leeway as a function of the apparent wind angle to the ship's course (ßAK) (Figs. 12, 15, 16). Here the calculation is based on the entire sail area, 192 m² (AS); it applies to the reefed sail as well. The driving force (FR) for various angles of attack are:
Since the driving force must correspond to the hydrodynamic resistance force when the ship's speed (VS) is constant, the following also applies:
Here WS is the wet outer surface of the ship and Ct the hydrodynamic resistance coefficient. The actual data for the ship's speed on various courses to the wind are derived from the polar plots. The hydrodynamic resistance coefficient of the cog can thus be calculated as a function of the Froude number and the respective wind direction (Fig. 13). At higher wind speeds the driving force coefficients from the wind-tunnel experiments had to be adjusted to the Reynolds numbers available for the Hanse cog. The actual hydrodynamic resistance was available from the towing experiments for use as a corrective.
The driving force coefficients reconfirm that the cog's windward properties worsen when the sail is reefed. While at an apparent wind angle of 80o with maximum sail area (192 m²) driving force coefficients of 1.1 can be attained, these decrease with two bonnets (160 m²) to some 0.8 and with one bonnet (128 m²) to values of less than 0.65.
Marchaj (1964) investigated driving force coefficients for various sail forms (including square sails) and extension proportions (luff length to medium breadth); his findings correspond quite closely to the assessment of the Hanse cog with unreefed sail (Fig. 14). On a reach, however, Marchaj's maximum values are not attained. The driving force coefficient curves for the cog in reefed condition are even planer (Figs. 15 and 16). In Marchaj's experiments only the sail was investigated; the large surface of the hull above waterline was not taken into consideration. So the sailing performance of the cog when the sail is reefed is appreciably limited even on a reach.
The resistance coefficients increase considerably as the course to the wind decreases (Fig. 13). On a course of 90o to the true wind, for example, the coefficient increases to more than three times its value on a run. Once again Hanse cogs are suitable only for sailing on a reach.
The gradient of the resistance curves is so considerable at Froude numbers of larger than 0.2 that the maximum speed of nearly 8 knots (Froude number 0.3) established by the performance measurements can hardly be exceeded. All claims in the literature on Hanse cogs of even greater "average speeds" must therefore be classified as unrealistic.
The seakeeping of the Hanse cog
The metacentric heights over the centre of gravity - 0.72 m in ballast condition and 1.13 m in load condition - are sufficient, providing the sail is reefed in good time at wind speeds of Beaufort 5 to 6 and higher. The total sail area of 192 m² can be reduced by unlacing up to three bonnets, measuring 32 m² each. For the reefing process, however, the entire sail has to be struck for a time; because of the long, heavy yard this is quite a dangerous undertaking when the sea is rough. The extent of stability is exhausted at heeling angles of between 25o and 35o, depending on the ship's draught, for at those angles water is taken in. The fact that the incoming water runs into the bilge is in any case more advantageous for the stability of the ship than if the water stayed on deck for a longer period of time. A high, closed bulwark, on the other hand, was a vital necessity for the safety of the crew, who were required to spend their days and nights on deck without shelter.
Because of her compact shape and lack of a keel, the seakeeping of the Hanse cog is extremely rough. The roll and pitch periods measure between 6.5 and 6.8 seconds in ballast condition, depending on the load. When the ship is carrying a cargo of barrels her roll motions last only 5.4 seconds and the pitch 4.5 seconds. In comparison to other cargo sailships these periods are exceptionally short. In unfavourable seaways amplitudes of up to 10o were measured for both, pitch and roll angle, causing considerable physical strain for the crew, as we had the opportunity of experiencing ourselves.
The Hanse cog's steerability at sea was satisfactory, especially when one takes the low length-breadth ratio of the ship into account. On a ballast journey in seas of some 0.75 m high, rudder deflections of +6.5o were required to keep the cog on course. In loaded condition these increase to +15o, involving tiller deflections of more than one metre to port or starboard. When the wind is strong the four-metre-long tiller can be controlled only with the aid of a tackle requiring two crew members for its operation. When the seas are not too high and the sails managed correctly, the Hanse cog travels willingly through the wind. The manoeuverability of the cog is limited in the port area. Most likely, cogs lay in the roads even during loading and unloading. The ship would have had to be towed by rowboats to berth.
A comparison of the Hanse cog of 1380 with other cargo sailships
The comparison of the Hanse cog with other cargo sailships is considerably limited in regard to the reliability of its results. To a great extent the existing data on the performance of both older freight sailers and those of the generations following the Hanse cog is inadequate for this purpose. There are, however, a few exceptions: Through Braemer's (1991) study of a sailing vessel of the Mediterranean region - the "Galateia" built in the fourth century B.C. - we have reliable performance diagrams of model experiments (Fig. 17). Wagner (1967) ran extensive wind-tunnel tests on a four-masted bark and calculated its efficiency. Experiments for the model of a modern freight sailer ("Indosail Version 50/3") should also be taken into account for such comparisons (Fig. 17). Projects dealing with extremely large modern cargo sailships are less suitable; only the polar plots pertaining to the six-masted Prölss sailship are comparable to the cog data, Wagner (1967). There have been publications of various comparative studies on the efficiency of freight sailers, almost all of which are based on the data gathered by Prager in his study of 1905.
Prager (1905) researched the travelling speed of sailing ships, classifying wooden barks and full-rigged ships into one group and iron barks and four-masted ships into another. His work also includes the analysis of the performance data of a five-masted bark and a five-masted full-rigged ship. He divided the directions of the wind into four sectors: "windward" (0-6 points), "broadside" (6-9 points), "quarter wind or broad reach" (9-15 points) and "before the wind" (15-17 points); one point corresponding to 11.25 . Prager based his calculations on wind speeds entered in the deck logs, i.e. on estimate by the respective ship's master. Thus it is to be expected that on smaller ships the wind speeds entered were subjectively higher. Light winds interrupted by lulls were not analyzed, nor were heavy sea and storms. The ships' speeds are calculated on the basis of the distances between locations entered in the logs and the average is taken for each respective category of ship, wind speed and wind-direction sector. Only data pertaining to ships carrying ballast or loaded with lightweight general cargo was included.
For the comparison of the Prager data with the Hanse cog, the square-riggers listed in Table 1 were chosen. Correspondingly, in the case of the cog the readings for draft with ballast were used, as were the measurement results at 1800 when sailing "before the wind" and the corresponding data at 800 (7 points) for the "windward" sector since smaller angles to the true wind were seldom attained by the cog. The comparison with a somewhat larger wind angle appears justified, however, as the data according to Prager is likely to be based on the apparent wind direction, i.e. the direction to the ship, the difference from the true wind direction being at least 10o.
While the waterline length of the Hanse cog measures a mere 17.4 m, the wooden bark ships are an average of 49.66 m, the four-masted ships 88.81 m and the "Preußen" 122.53 m long. In Figs. 19 to 22 the results were plotted as a function of the Froude number, producing entirely different results. Here the cog attains better values than the other ships, especially when sailing with a stern wind, and when sailing "windward" her performance is only slightly worse.
Smaller ships generally achieve better results, at least at low-to-medium wind speeds. Thus a simplest possible coefficient was sought which would also take displacement and sail area into account. The driving force and resistance coefficients as determined for the cog (see Equations 16 and 17) can unfortunately not be included in the comparison, since the necessary bases for the calculation are not available.
Let us now define the "coefficient for sailing vessels". As a function of the true wind speed VT and the true course to the wind ßTK, the performance achieved by a sailship can be represented by the coefficient CVs, independent of ship size. * designates the displacement and AS the sail area when sailing windward.
The Froude number (Fn) as well as the relationship between displacement and sail area are independent of a scale ratio, a rule which also applies to CVs. The coefficient for sailship efficiency is meant to represent the relationship between utility - a first approximation of the displacement as transport cargo - and the expenditure - the sail area. The larger the CVs value, the more efficient the ship. A large sail area, like a small amount of displacement, leads to a small CVs value.
The coefficient formulated above is based on the assumption that the ships tested have been constructed for the purpose of freight transport. For this reason, the comparison with either a heavily ballasted sailboat or one equipped with a fin keel is hardly possible. In such cases the stability and the draft would also have to be taken into account in order to better comprehend the windward performance characteristics.
Naturally, the "efficiency coefficient" for freight sailing vessels also allows only an approximate criterion for comparison since it leaves many influences, particularly those resulting from the sea and weather conditions, out of consideration.
For the comparison of the coefficient, data for the five-masted full-rigged ship "Preußen" was selected from the Prager material, as this was the only case for which the data concerning volume of displacement and sail area was reliable (Figs. 23 to 26). In these graphs the values of the cog, a much smaller vessel, have been raised. As already evident from the comparative curves discussed above, the cog exhibits her particular strength on reaches and runs. At 80o to the true wind, however, the Hanse cog also performs better or at least as well, providing one takes into consideration that the speed of the "Preußen" was averaged in one category for all angles to the apparent wind of less than 6 points (68o), possibly causing a certain reduction of the performance data.
For further comparison, a varied spectrum of sailing ships was investigated for which data regarding the sailing performance, volume of displacement and sail area were relatively dependable. Table II lists the most significant data on the vessels and their rigging types. An important aspect here was that values pertaining to smaller ships could be included, permitting a better estimation of the above-mentioned degree of influence of the ship's size. In addition to the cog in ballast and in barrel-load condition (corresponding to lengths between perpendiculars of 17.4 and 18.2 m), data was compared from the model of the "Galateia", equipped with a square sail and exhibiting a waterline length of 6.71 m, and the model of the "Indosail Version 50/3", a three- masted schooner with a length of 5.88 m at water-surface level. All remaining square riggers included in the studies of Wagner (1967) and Walter (1993) are barks and full-rigged ships, appreciably larger than the cog, with waterline lengths of between 76.5 and 96 m. In Walter's material the wind directions are unfortunately not defined; it is assumed, however, that his results primarily reflect reaches. Finally, the data pertaining to three sailing yachts, all somewhat smaller than the Hanse cog, was recorded "non-competitively." One, a vessel with sloop rigging, sails without a spinnaker even on a broad reach (length between perpendiculars 8.0 m); the other, slightly larger vessel is a ketch (length between perpendiculars 13.56 m) whose spinnaker was taken into account for the comparison. Both are relatively heavy yachts, i.e. not intended for racing. For the sake of completeness let it be mentioned that, for the sloop vessel, data derived from sailing performance measurements was employed while the ketch data and the data of the IMS racing yacht "Mumm 36" was taken from a Velocity Prediction Program.
The results were plotted on the basis of wind speed, in relation to the non-dimensional coefficient CVs for the courses 80o and 180o to the wind (Fig. 27 to 28). For the "run" course the maximum leeward speeds were selected from the polar plots since even at high wind speeds the polar-plot curves of the multi-mast ships characteristically exhibit a pronounced "apple form" in contrast to other sailing vessels, because at an attack angle of 180o the sails cover each other. As mentioned above, certain data regarding the "close haul" courses of the barks and full-rigged ships are missing from Walter's material.
Surprisingly, in comparison to Wagner's four-masted bark and the "Galateia" the quite bulky Hanse cog of 1380 achieves relatively good values even on a close haul (Fig. 27). This claim must be qualified, however, by pointing out that the four-masted bark and most of the other ships included in the comparison could sail closer to the wind and were thus superior to the cog regarding VMG. The beating properties of the freight vessels are to be discussed at a later date. The Indosail project yielded test results which are particularly difficult to interpret; the values are even higher than the coefficients of the yachts. In comparison with the Hanse cog we are dealing here with a relatively narrow ship exhibiting an L/B ratio of 4.17 and an L/V1/3 (length/displacement) ratio of 4.4 while the cog's L/B measures 2.4 and the longitudinal coefficient according- ly 3.6.
The cog performs outstandingly on a run (Fig. 28). If one leaves the evaluation of the Indosail model out of account, the cog is superior to all other square riggers on this course, and regarding the degree of efficiency she and the yachts are equally good. Only the "Galateia", also a much narrower vessel (L/V1/3 = 4.2) exhibits a somewhat better coefficient.
The question remains as to the cog's seakeeping in head winds. The test results substantiate the assertion that, although she managed to make progress against the wind for short periods of time at every wind speed investigated, she could not repeat this achievement over longer distances in stronger winds when the sail was reefed, particularly when she had to tack occasionally. Thus at winds of up to 3 Beaufort, VMG values of 0.16 to 0.63 were achieved over greater distances.
The comparison with other freight sailing vessels yields similarly negative results for the Hanse cog (Figs. 29 and 30). In regard to absolute VMG values, the "six-masted windship" by Prölss with her 150-metre length is far superior to all of the other smaller freight vessels. The 96-m-long four-masted bark tested by Wagner only reaches half the VMG of the Prölss while the cog hardly manages to make any headway at all against the wind.
The data plotted in relation to the coefficient CVs modify these results (Fig. 30). The ship's speed has been substituted here by VMG. Fig. 30 makes particularly evident that even Wagner's four-masted bark hardly exhibits better sailing performance than the cog on a beat. The "Galateia", on the other hand, is at an advantage in a head wind. The windward sailing properties of the modern yachts with their deep-draft keels and greater stability are particularly excellent. The evaluation of the Indosail model is to be reserved for analysis at a later time.
To summarize, on a "close haul" the Hanse cog of 1380 was nearly equal in performance to later square riggers, while her relative sailing performance can be assessed even higher on reaches and runs. Remarkably, even the large full-rigged ships hardly attained Froude numbers exceeding 0.2 on average. So their sailing speed was within a range in which the wave-resistance factor played only a minor role and thus the quality of the hull's outer surface below waterline was of great significance for reducing frictional resistance.
Even much narrower ships such as the "Galateia" of pre-Christian times exhibit similar performance values in a beam wind and only slightly better values on a run.
The cog's weaknesses manifest themselves in head winds. Here she is inferior to many sailing vessels of the past and present; only the large square riggers have comparable properties. Headway against the wind can be made only in light winds. This handicap plays a larger role for the cog than for the large square riggers since the latter could make good use of reaches in the trade wind zones while the cog's territory was limited primarily to the North and Baltic Seas. Delays were therefore unavoidable; the cog and her crew had to wait for favourable winds.
Table 1: Square rigged vessels
Table 2: Ships for comparison
Figure 1: Reconstruction of the Hanse cog of 1380 built in Kiel
Figure 2: Hanse Cog Lines
Figure 3: Hanse Cog sail
Figure 4: Righting Arms
Figure 5: Definitions
Figure 6-9: Polar plots
Figure 10: Way over ground while beating
Figure 11: Way over ground while beating
Figure 12: Aerodynamic driving force coefficient for 192mē
Figure 13: Hydrodynamic resistance coefficients
Figure 14: Driving force coefficient by Marchaj for different aspect ratios (AR) compared with Hanse cog's unreefed sail including the effect of the hull above waterline
Figure 15: Hanse cog aerodynamic driving force coefficients for 160mē mainsail and 2 bonnets
Figure 16: Hanse cog aerodynamic driving force coefficients for 128mē mainsail and 1 bonnet
Figure 17: Model "Galateia" Scale 1:2
Figure 18: Model "Indosail"
Figure 19: Hanse cog compared with square rigged vessel; beating
Figure 20: Hanse cog compared with square rigged vessel; beam wind
Figure 21: Hanse cog compared with square rigged vessel; reaching
Figure 22: Hanse cog compared with square rigged vessel; run
Figure 23: Hanse cog compared with square rigged vessel; beating
Figure 24: Hanse cog compared with square rigged vessel; beam wind
Figure 25: Hanse cog compared with square rigged vessel; reaching
Figure 26: Hanse cog compared with square rigged vessel; run
Figure 27: Comparison of different ships; beating
Figure 28: Comparison of different ships; run
Figure 29: Comparison of different ships; Velocity made good
Figure 30: Comparison of different ships; Velocity made good