Section 8 Wind shifts

 

Those who race yachts inland and offshore have to race in a wind that swings in either direction and increases or decreases in speed in some not quite predictable way and that half of the fun lies in the challenge that this poses. I hear of skippers being able to predict and prepare for a “wind shift” and they get their warning of the arrival of a wind shift from, amongst other cues, the rudder.

 

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Fig 77 
It seems to me that this variation is the result of eddies in the wind and, if it is, there is something of interest here. Figures 76 a and 76 b show a windsock at the end of the long runway at Lasham Airfield in Hampshire England. The wind was blowing along the runway so there were no particular obstruction to make this a special case and the sock was swinging predictably for me to take these pictures.  have never seen any explanation of the motion of the windsock and yet it cannot be anything complex so what follows is my explanation.

 

I think that it is most unlikely that the wind can blow in blocks of slow-moving and fast moving air one bumping into the other. It has to be some commonly encountered flow pattern that can occur anywhere in a moving stream of fluid. The obvious thing is for the air to contain eddies of significant size ie when compared with a sailing boat. Then, what would be observed by the sailor, would depend on the course of the yacht through an eddy. Eddies will not be isolated but will form in streams. The eddy in figure 77 formed in a small river. It was one of a stream of eddies forming in the wake from the bridge pier. I think that it may be typical of eddies that form in the wind except that there is no free surface in the wind.

 

Sometimes, when moderate winds blow for long distances over the sea, the flow of air seems to be free from eddies which suggests that eddies do not form easily in an unobstructed flow. Eddies form when wind blows over the general contours of the land and, to a quite different scale, round isolated obstructions like buildings and trees. Thermals can punch through a steady wind to form thermal streets of clouds and create disturbance. There can be no doubt that the wind is disturbed when it flows over the land and the disturbance is most likely to be eddies like those in a river only, of course, on a much larger scale. Disturbances create angular momentum that persists and flowing fluid is very good at gathering angular momentum into organised, much larger, rotating systems, ie eddies.

 

Eddies may have axes in any orientation but I think that there is the possibility that those rotating about horizontal axes lose their energy in friction with the ground and die away relatively quickly whilst those that rotate about a more or less vertical axis are very persistent because they have no asymmetrical contact with the ground. I suspect that eddies with angled axes tend to become less vigorous than eddies with a vertical axis. This rotating flow is very common in nature and once formed is persistent because the flow pattern produces very little internal loss of energy.[1]

 

I do not know how deep upwards these eddies can be but I think that I can take it that the eddies are deep enough upwards to affect the whole rig of a yacht.

 

So I have a picture in my mind of a steady flow of air with eddies rotating in it and of yachts trying to sail with this air flowing over their sails. Sailors already have an image of their yachts sailing in an apparent wind that is the vector sum of the true wind speed and the speed of the yacht. When an eddy passes, the true wind is no longer in one direction but varying from time to time in both strength and direction. Sailors call it a wind shift. I think that there might be an advantage to be gained from investigating a model of the wind with it’s eddies.

 

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Fig 78
I need some simple way into this motion so let me look at the notional idea of just one ring of air rotating at constant tangential speed in a steady stream of otherwise undisturbed air with the centre of the ring moving with the flow. What would an observer on the ground experience as this ring passes him? On the face of it this seems to be a ridiculous idea but a vortex is made up of many such rings and it is worth thinking about a single ring because it establishes how to draw the vectors seen by the observer for all the other circles.

 

In figure 78 I have drawn a ring that is moving in an otherwise steady flow of air. The tangential speed is represented in magnitude and direction by the black arrows shown on the ring and labelled. I have drawn velocity diagrams for the air at points A, B, C and D and these give the resultant velocity of the flow at each point in green as a result of adding the red vector representing the mean velocity of the flow and the black vector.

 

Suppose that observer 1, who I represent by the eye, is stationary in a position so that points A and B pass him in sequence. As point A approaches the observer he will see the air approaching steadily at the velocity represented everywhere by the red arrow. When point A arrives, for an instant, the observer will see the true velocity of the air increase and come from the left as shown by the green arrow at A. As the flow continues to pass observer 1 it will be steady and un-diverted until point B reaches the observer when, for another instant, the air increases in speed again but comes from the right as shown again by a green arrow at B.

 

I have included a second observer 2 who is as far to the right of the path of the centre of the ring as observer 1 is to the left. As the ring passes he will also see the direction of flow change for two instants but he will also observe reductions in the true velocity of the air.

 

This all means that what a stationary observer sees depends on his position relative to the path of the centre of the ring and on the direction of rotation as well. This is consistent with the observed behaviour of the wind.

 

Quite obviously I am building up to considering not one ring but a whole rotating mass of air within an otherwise steady flow. So what we really want to know is what happens when a infinite set of rotating concentric rings like the one above in a steady flow of air pass a stationary observer. In other words we want to find out why the windsock that lets us become the stationary observer in figures 76a and 76b behaves as it does. We need some way of representing the rotating system that is likely to be a fair description of the real wind. I think that the only system that exists in nature is the free vortex. I have to describe such a vortex.

 

The free vortex

The free vortex is a well-known model for the rotation of water or any other fluid. It is based on the supposition that the fluid will settle down to flow in concentric circles so that the sum of the kinetic energy, the pressure energy and the potential energy throughout the pattern is constant. Then the tangential velocity at each radius is inversely proportional to the radius.

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Fig 79
Such a model has characteristics. In figure 79 I have plotted the variation of tangential velocity  with radius in red. As one would expect the velocity is trying to go to infinity at zero radius. This is not possible because the velocity gradient between adjacent circles becomes excessive and so does the loss of energy. I have plotted the velocity gradient in blue just to see how the shearing of the fluid increases with decreasing radius. The implication is that the middle of this model of the rotation of fluid cannot exist in reality because the shearing is too great to be sustainable. In the real system the vortex forms an eye that rotates like a cylinder in which the total energy is not uniform.[2] Then the velocity varies with radius as shown in figure 80.

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Fig 80

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Fig 81
Going back to figure 80 we can plot the pressure below some datum on the basis that the energy is constant throughout the model of the vortex. In figure 81 I have plotted the pressure below a datum of 10 units and it is the familiar free surface of a free vortex in water.

 

A vortex is, in principle at least, of infinite radius yet an eddy when it is shed from some obstruction such as a hill must have a finite size. There must be some sort of edge to the vortex where some other flow dominates and limits the diameter. This is evident in the eddying flow in figure 77 where the eddy is part of a more or less linear flow. At the edge of the eddy there must be fluid moving in a circle adjacent to liquid that is at rest relative to the centre of the eddy. That cannot persist. The eddy will cause some of the surrounding air to move with it and energy will be exchanged between the eddy and the surrounding fluid. It is a progressive process and leads ultimately to the decay of the eddy. So we must expect that the eddies in our wind will have a finite size and a core. If they have, the edge might well be detectable by a wily skipper.

 

All this gives a model for an eddy in an otherwise steady wind and it can be explored using a mathematics package to see whether it has recognisable features. In order to do so I have to put some ratios to the diameter of the “edge” of the vortex and the diameter of it’s core. I will start with 1 to 5.

 

The free vortex in a steady stream of air

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Fig 82
Figure 82 shows the basic diagram. The arc represents one streamline of the eddy. The vertical centre line lies along the direction of the undisturbed flow that contains the eddy. The line OO is across the flow and will be used as a reference line. The vertical line distance x from the centre is the track of the flow past the observer. As the eddy passes the observer the observer goes from a negative distance from OO to a positive distance. I shall call that distance y for computation. I want to see what happens to the true wind, that is, to the magnitude of its speed and its direction relative to the observer, in other words to extend figure78 to the whole set of circles representing a free vortex. The magnitude will give the gusting or weakening and the direction will show the veering or backing[3].

 

Obviously the true wind is the vectorial sum of the steady wind and the tangential velocity of the air flowing round the vortex. This is shown in figure 78. In order to find the speed and direction of the true wind I need to find first the radius of the streamline so that I can find the tangential speed for any point of the observer as the vortex passes.

 

Then it is a matter of trigonometry to find the speed and direction of the true wind.

 

Four graphs are needed.

 

(I have drawn an eye to represent the observer of the approach and passing of the vortex in the wind. In order to maintain this perspective I will draw graphs in which the distance from OO is represented by a vertical distance starting at some negative value and going to an equal positive value.)

 

I shall have to let y start at some negative value and increase to an equal positive value. I plan to set this at 10 arbitrary units of length. So y will change from –100 to +100. For every value of y there will be a radius that can be calculated.

                                                  sets the range of y and         

                                          calculates the radius. Mathcad is reading the letter r as “a function of” but the “r” identifies the thing calculated in the programme ie the radius.

 

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Fig 83
The graph for x = 5 is shown in figure 83. The graph can be used to calculate the tangential velocity of the air moving along the streamline to give figure 84

 

 

 

 

 

 

 

 

 

 

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Fig 84
Given the radius the tangential velocity can be calculated for each value of streamline in terms of its radius. In order to do this the eddy must be given some speed/radius relationship. I found that I could have a maximum speed of about 10% greater than the mean if I put speed = “strength”/radius and let the strength be 6. The strength for the eddy can be changed at will.

                          calculates the tangential speed of air at radius r. It is a scalar quantity. The choice of 6 for the strength suits a notional undisturbed speed of the wind of 10 knots.

 

The choice of x =0.5 ensures that the core of the vortex misses the observer. As a result the tangential speed rises as shown as the observer approaches OO and then decays again as the observer recedes from OO.

 

Now to evaluate the vector diagram for any point on the track. This involves the combination of the undisturbed wind speed, which I plan to give a value of 10 knots, and the velocity of the air at the point. (Note that I am now interested in the vector representing the air in speed and direction). In order to do this I shall use the cosine rule and for that I need angle q.

 

                   calculates this angle in radians. The outcome is shown in figure 85.

 

It is clearly what one would expect.

 

Now I need a step to find the magnitude of the true wind, that is of the combination of the eddy and the undisturbed flow. (I do not know what else to do with this because sailors imagine the wind to be steady, call it the true wind, and combine it with the speed of the boat to give the apparent wind. Now, I am saying that the true wind veers and gusts.)

 

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Fig 85
                  gives my true wind in knots. It is an application of the cos rule.

This is shown in figure 56 where the speed is in knots. For the path that I have chosen the speed of the true wind increases slowly at first and then, over about 25 units of distance increases very rapidly (ie it gusts) and then falls back to its original value in a mirror process.

 

Had the observer been an equal distance to the left of the centre of the vortex the graph would have shown a weakening of the wind.

 

Now I want to know the angle between the true wind and the direction of the undisturbed flow.

 

Text Box:  Fig 86                                             will give that angle using the cos rule again. This needs some manipulation to eliminate unwanted graphs and leave just one graph.

 

The graph is shown in figure 86 and will be recognisable by all off shore sailors and it is pertinent to note that, as the whole stream, complete with vortex, is moving at constant speed, the vertical axis is also time to some scale.

 

 This graph suggests to me that the basic idea that a “wind-shift” is caused by a free vortex in a steady stream of air is sound. In effect this paper sets out the mechanics of the wind shift.

 

However I have drawn graphs for only one value of x, one strength of the vortex, and, for the case of the core being to the left of the observer. There are lots of other combinations.

 

This is too much for a web site but the basis is there for anyone to explore.

 

 

 



[1] I want to look at the plumes from tall chimneys. In some photographs they look like the combination of a horizontal vortex and steady flow.

[2] Whenever there is a major hurricane we see photographs of the storm taken by satellites. The “eye” at the centre is a very obvious feature. In passing there is a physical problem to think about there. Those storms are often in excess of 1,000 miles in diameter yet they are only 6 miles thick at most.  What keeps them together?

[3] When the true wind appears to swing clockwise it is said to veer and when it swings anti-clockwise to back.