1. A Boat on a River
Would a sailboat get to the other side of a river without a puff of wind? Let’s assume the water flows at a constant rate Vc relative to the shore. A sailing boat would usually go 30о - 150о to the true wind and may sail with speeds close to the true wind speed.
Solution: the water velocity Vc will give rise to a perceived wind ("apparent wind" is usually reserved to denote direction and strength of the wind corrected for the boat speed; therefore we would like to call the wind corrected for the current alone a “perceived wind”). Without any action, the sails start to luff. By trimming the sails, the boat starts to glide relative to the water - Vr. The absolute velocity relative to the ground Vabs = Vr + Vc, as presented by the diagram.
The lift created by the filled sails creates the driving force that moves the boat along the water. The perceived wind is the power supply that moves the boat. Note that the true wind velocity is zero and the source of perceived wind is actually the current or movement of water relative to the stationary air.
As shown by the diagram, the absolute velocity can vary in certain range depending on the boat point of sail. The boat is likely to cross the river faster if it’s reaching. Keep in mind that the boat can sail with speeds close to the perceived wind Vc if closehauled or reaching.
"Lift is the force that arises due to asymmetry of a gas flow and is perpendicular to the flow". Physical Encyclopedia, Volume 3, page 670, MOSKVA.1992.
The lift is the result of Bernoulli's law:
"With a steady flow of a fluid (gas) the pressure is greater in areas where the flow is slower, and conversely, lower where the flow is faster.".
The diagram below shows the trajectory of the airflow. The flow must be faster along the upper trajectory as the distance traversed is longer. Therefore the pressure on the lower surface of a sail is higher, which creates a lift perpendicular to the airflow. The angle of attack is the angle of the wind velocity relative to the plane of the airfoil. For the Bernoulli law to be valid and to avoid “drag”, or the force that drags the airfoil along the wind direction, the angle of attack should be relatively small to preserve laminar flow. So the sail looks like a wing of an airplane. The absolute magnitude of the lift and the optimal angle of attack depend on the sail/wing/airfoil shape, the condition of the surface, and the material they are made of.
2. Boat Velocity Polar Diagram
In the first part of the "A Boat on a River" we assumed that the boat speed is equal to the flow speed. In practice, one could construct a vector diagram and deduce the exact speeds of the boat from the wind speed and the boat course. The polar coordinate system that we use here consists of concentric circles centered at 0 and rays starting at center of coordinates. One of the beams corresponds to 0о and is used to measure the polar angle and is called a polar axis. A vector is fully specified by its length and polar angle.
A Boat Velocity Polar Diagram (BVPD) is presented in the following figure:
The polar axis coincides with the direction of the true wind. The right and left parts of the diagrams are totally symmetric. The left half corresponds to the boat sailing on the starboard tack, and the right to the port.
The curved lines are the boat velocity vectors depending on the wind and boat’s point of sail and can be determined experimentally with the use of modern navigational equipment (see, for example, www.oppedijk.com/zeilen/create-polar-diagram). Clearly, these curves vary greatly by the type and build of the boats. We marked different vectors by the characteristic points in the figure. The optimal vectors for sailing either upwind or downwind are determined by optimizing the corresponding projections.
If the boat speed equals the true wind speed then the apparent wind is always directed at half the angle of the true wind relative to the boat’s velocity vector (point of sail).
3. The Beating Triangle
Millions of people in large cities use subway. In some of them escalators move passengers up and down, who can stand still on the steps of the escalator and peruse the advertisement. Children experience fear and uncertainty when stepping on the escalator for the first time, but with time, as they become adults, they become bold and confident.
In some cases it happens that adults never earlier not made use of the escalator, still face this kind of problem, and it can be seen on the video.
But even experienced "users" escalators, may encounter difficulties, if the movement of the escalator stops suddenly, the passenger can fall because of the changing speed and the inertia unless he holds on to the handrails. Moreover, with time a strange thing happens: A person entering or coming off a still escalator experiences discomfort and uncertainty like the first time when he stepped on the escalator as a kid. This phenomenon is hopefully well known to all people: What people have learned in the childhood can be so engrained in their minds that they are confronted to learn the thing that comes naturally when you are a child.
The escalator is used to change the altitude in the earth gravitational field. A person, when in a hurry, can accelerate the change by moving up or down the ladder relative to the escalator. Sailboats use wind and, as in the gravitational field, can change the height relative to the wind velocity. A boat is gaining "altitude" when it is moving upwind and losing "altitude" when it is sailing downwind. The escalator is similar to the current: the wind will generally move a boat down the wind, but the current and crew managing the boat will move it “higher” up the wind.
The same “escalator” phenomenon manifests itself in sailing when a boat is beating in the presence of currents.
No sailboat can sail directly into the wind. A typical angle to the true wind for optimal "altitude" gathering is about 45о. Modern fast racing boats are capable of beating at an 30о angle. In order to achieve objectives along the coastline or the upwind mark the boat should tack. The resulting path can be portrayed as zigzags. In the sailboat races the boat’s first leg is usually upwind and after the start of the race the boats diverge to the left and to the right of the start area. The "the beating triangle" (Fig. 1) shows how far apart the boats can diverge in the opposite directions.
Fig. 1
"The beating triangle" is an isosceles triangle, which sides are fetchlayn (leylayn). The base of the triangle is the windward isoline. The vertex of the triangle is located at the upwind mark or at the location of the target. The symmetry axis of the triangle coincides with the direction of the wind. The triangle rotates about its peak at an angle α as the wind shifts. The windward isolines are similar to the steps of the still escalator. The steps should be overcome by applying the sailing skills.
The points along the start line are usually equivalent and the boat’s advantage does not depend on the position at the start line if the start line is positioned exactly perpendicular to the wind. One can also rearrange the start line along the windward isolines ("altitude") within the area of "the beating triangle" while preserving the distance travelled over the water. If one of the start marks is positioned "above" with respect to the second buoys, then the total the travelled distance will depend on the position at the start. A wind shift leads to the same result as the triangle shifts as a whole. The Figures 2-4 illustrate how one should start to take advantage of the wind shift. One conclusion is transparent: you need to maneuver within the "the beating triangle" if you do not know exactly the direction and time of the wind shift. Going beyond the fetchline (out of a triangle) leads to an unnecessary increase in the total distance.
Figure 2 - Figure 3 - Figure 4
Animated Figure
4. A Current and a Wind
The title of the fourth part of this article is just a permutation of words in the overall title since these two are often parts of the same phenomena that move a sailing boat. This is particularly true when the sail’s lift is the driving force as opposed to just a pressure created by the wind (as if running). For the high-performance sailing boats these are points of sail from close-hauled to the reaching.
The velocities of current and wind are traditionally measured relative to the Stationary Frame of Reference (SFR). Let’s assume we measure the speeds with a GPS or relative to a fixed point on a shore or a lake seabed.
When sailing, a boat is placed on the boundary of air and water. The relative speed of air relative to the water moves the boat. Since two substances are involved, the keel (centerboard) also forms a “sail” under the water and it’s lift is practically identical to the sail’s lift. One of the major differences though is the density of air and water. The water density is about a thousand times higher than that of the air. The magnitude of lift is proportional to the kinetic energy of the fluid per unit volume and the wing area.
| ∼ | ρV2 |
| 2 |
(where ρ is the density of the medium and V is the speed of the medium relative to the wing/sail/keel/centerboard, and is much higher for water). Thus, the keel/centerboard area is usually much smaller.
Let’s assume that a boat is experiencing a current and is far from the shore or navigational marks so that it is hard to detect the current (we are expanding on part 1 "A Boat on a River."). The boat crew sets the sails in accordance with the perceived wind, which is the sum of the true wind (the crew might only guess about it) and the wind induced by the current (see “A Boat on a River”). The moving water is the coordinate system for the boat helmsman and the crew. The captain might throw a floating object overboard and measure crew boat’s speed relative to the floating objects which are drifting along with the water, but he can only guess the absolute speed (or measure it with navigational tools like GPS).
For the boat in this moving coordinate system that has relative speed equal to the flow velocity - Vc, everything might look like the system is stationary (no flow). The perceived wind velocity is not the true wind velocity but can be calculated by using a polar diagram for vector addition (if one knows exactly the corresponding absolute speeds of the wind and the current at the boat’s location).
Fig. Determination of the Perceived Wind
What does the observer see in a stationary reference frame SFR? The race judges on an anchor Race Committee Boat can measure the true wind and the current since they are anchored. If the speed and trajectories of boats vary greatly experienced observers can notice the discrepancy between the true wind and the direction of the perceived wind pointed out by the sails.
One can draw velocity (vector) in the polar diagram of vector addition in the Mobile and Stationary Frames of Reference (MFR) or SFR. The actual speed of the wind and currents are functions of time and are subject to statistical fluctuations.
We orient BVPD in the direction of the perceived wind to compute the relative velocity - Vr.
Fig. BVPD rotated in the direction of Perceived Wind
Any movement can be represented as the sum of two orthogonal motions, so we represent the speed (courses) yacht for currents in the direction perpendicular to the true wind and parallel to the true wind.
Fig. Designation of Vectors
Current: weak - strong.
Fig. True wind direction and parallel to the flow
Current: weak - strong.
Fig. The direction of the true wind and currents are orthogonal
The analogy with the behavior of the passengers on an escalator in motion is as follows. When approaching a mark an inexperienced helmsman steers as if the boat moves along the green vector. The helmsman discovers that boat often skips past the sign since in the absolute system relative to the mark the boat moves along the red vector. Only a very experienced helmsman, watching the unexpected tacks of the other boats, assess the situation quickly and steer the boat the mark accounting for the current.
The effect of current can be enormous in the calm weather. Diagram below shows the flows in the region of Mauritania. Without sails, boat can just circle in one place around a spot. However, one can guess the “perceived” wind by just reversing the current vectors, which can be used to get out of this infinite circle.
Figure - Map of flows in the region of Mauritania
Figure taken from the site - Fish-Scouting Department.
5. The Prize for Winning
The document ISAF «Race Management Manual» Part 2 contains drawing.
This picture reminded me of the final race of the "Golden Cup", specifically the start of it. In this race I finished first after all and received a beautiful trophy.
It was in France in 1966. I describe a similar situation in my article "A Wind and a Current». The current, wind and starting line in the final race had the same pattern. That day I arrived to the start line in advance, sailed up to the outside starting mark and noticed the current along the starting line. The detection of the flow vector was helped by the fixed mark on the start line. Most of the other participants were far from the start area and the mark and could not see the current. The events developed as follows in this article. Let’s assume the wind to current speed ratio to be 3 to 1. The polar diagram helps to find the “perceived” wind.
Let’s rotate BVPD in accordance with the direction of the perceived wind to determine the relative boat speed and define the racing close-hauled speed. Let’s assume, as in the first article, that the racing speed close-hauled is equal to the perceived wind. The results of the computation are shown in Fig.
The figure shows:
1. Isosceles triangle in the presence the current of orthogonal to the true wind is converted into oblique triangle.
2. Yacht 1 has advantage with this relative position the starting line and the first mark of the distance.
3. The windward isolines are not applicable and to assess the relative position for boats to left of the beating triangle. In first approximation, the distances isolines to the mark can be represented by arcs of circles of radius R centered at the location of the mark.
4. The position of the external mark should be chosen as to ensure equal chance to be first at the upwind mark: if the start line is located within the beating triangle, then (move pin down wind) - a mark to pull down wind (the starting line is parallel to the windward isolines, if the same starting line out the beating triangle, then (move pin up wind) - mark up to the wind in accordance with the distance isoline to the upwind mark.
In that distant time I was lucky. More than a hundred participants the World Championship gathered at external mark since the perceived wind gave the impression of it to be "the windward mark." Only by the boat in the immediate vicinity of the mark could have detected the current.
I had had already decided to start at the anchored Race Committee Boat and revealed this plan only two minutes before the start signal and sailed along the starting line to the anchored Race Committee Boat. About a minute later another boat repeated my maneuver and was second at the finish as the result.
In the final figure yacht 1 is fetching the mark and secured first place at the finish. Boats that sailed to the left side course areas on the starboard tack from the start moved away from the mark 1 in fact.
(It is recommended that you read "Wind and Current," to understand the full derivation and picture - First part - 1. A Boat on a River ).
Victor Kozlov,
translation - Alex Kozlov,
design - Timofey Soloveychik