Thank you for your efforts compiling your response. I had read the link that you provided but where he states:
Equations (1) and (4) imply that the tension is constant from the tip of the rod to the top of the loop and then decreases linearly (because m decreases linearly)
implies that it is a continuously tapering line ( the same as a whip) and not a DT as shown in the graph. Without knowing where the taper starts I could not see how the taper and velocity were related on the graph.
What I see as being consistent in both the level taper and DT line is the rapid accelerations at the tip as the loop unfolds. Goriely comments on this when he says:
This acceleration, sometimes refer to as the “Kucharski effect” is due to the rotation of the tip around the horizontal axis as the loop reaches the end [16]. As the tapering increases the maximum speed of the tip increases. At the maximal tapering, when the area
of the cross-section at the end of the tip is 1/20 the area at the handle, we get another doubling of the maximum tip speed. Again, no general relation between the tapering and the maximal tip speed has been found analytically, but a general trend can be observed. The maximal tip speed increases nonlinearly as a function of the tapering, with no obvious scaling.
I can see this in the bullwhip video and was trying to point Bernd to it but without prejudicing his view of the actual event. I have never heard of the Kucharski effect but suspect that it may be why Bernd can cause a sonic crack when he is casting.
implies that it is a continuously tapering line ( the same as a whip) and not a DT as shown in the graph.
Vince,
Not at all. His comment for a linear taper in the tension in the line would assume it was a level line.
The line in the top leg is being pulled by the force at the top of loop to accelerate it. Just behind the loop the mass that is being accelerated is the total mass of the top leg. Halfway down the line the mass that is being accelerated is only half of that value. At the end there is no mass left that is undergoing acceleration so the only tension is from the form drag force on the fly.
Since F=ma and all the line in the top leg has the same acceleration then the tension force in the top leg will go down by F(l)=a/(remaining mass(l)). For a level line that F(l) would be linear, for a tapered line it would not be exactly linear.
What I see as being consistent in both the level taper and DT line is the rapid accelerations at the tip as the loop unfolds.
This acceleration force for the rod leg has nothing to do with the loop unfolding. That is another can of worms that gets to be very complicated as Goriely showed in the whip wave paper.
I suspect that if Bernd still had a ball of fluff on the end of his leader the drag on the fluff would be high enough that it would not go supersonic so the sound he was hearing was not a true "crack."
After I few cracks in casting a line a line without a fly I generally see that the end of the leader is frayed like the cracker ends in a bull whip. I suspect it is those very small mass threads that are going supersonic, just as shown below for the wet towel crack.
Gordy
"Flyfishing: 200 years of tradition unencumbered by progress." Ralph Cutter
Not at all. His comment for a linear taper in the tension in the line would assume it was a level line.
In that case, I still cannot see what the mass distribution or taper is for the DT line.
This acceleration force for the rod leg has nothing to do with the loop unfolding. That is another can of worms that gets to be very complicated as Goriely showed in the whip wave paper.
Perhaps I did not make myself clear. Both your graphs and Lindegards show a rapid increase in velocity at the end of the graph, why is that?
Perhaps I did not make myself clear. Both your graphs and Lindegards show a rapid increase in velocity at the end of the graph, why is that?
Vince,
Did you understand Lingard's explanation?
Both lines are retarded in the early part of there trajectory because the aerodynamic drag outweighs the effects of the acceleration from the changing mass. A distinct acceleration phase is encountered later in the flight which, in the case of the level line, is sustained to the end. The double tapered line, however, attains a sharp velocity maximum and then decelerated finally as the effect of the taper is felt over the last few meters of the cast.
I don't think I could explain his reasoning any more clearly than I did back in post #42.
In that case, I still cannot see what the mass distribution or taper is for the DT line.
Lingard said the line parameters he used for his simulations were:
The line dimensions and mass distribution are those for a No.7 line, relative density 0.8, the first 9.14 m having a mass of 12.0 g.
As noted before the linear mass density at some point in the taper will be equal to the volume mass density (around 800 kg/m^3 for Lingard's 7wt line) times the cross section area of the line in m^2.
Thus the linear mass density will vary as the square of the taper diameter. I suspect that different line manufacturers use different taper functions in their lines, so you would need the diameter vs length values to come up with an exact rho_l variation in the tapered end of the line.
It appears the front taper on SA DT lines are quite short (5 feet), while for Triangle Taper lines use "a continuous forward taper in the head of the line, the first 27 to 40 feet depending on the application".
If I have done the conversions correctly, to get a rho_l of .012/9.14 (.00131 kg/m) in a level line with a volume mass density of 800 kg/m^3 the diameter would be 1.4 mm . You might want to check that to see if you get the same value.
Gordy
"Flyfishing: 200 years of tradition unencumbered by progress." Ralph Cutter
Thank you for that expansion. It is not the explanation which troubles me, it is the interpretation.
A distinct acceleration phase is encountered later in the flight which, in the case of the level line, is sustained to the end. The double tapered line, however, attains a sharp velocity maximum and then decelerated finally as the effect of the taper is felt over the last few meters of the cast.
Magnus also commented upon it:
Is it just me or is this chart odd? It reads like a fly attached to a level line never stops moving. And DT line cast fast (initial speed 30 m/s) and the same line cast slow (initial speed 10m/s) slow to a gentle stop at exactly the same displacement? Not in my experience - hit it hard and stop it abruptly and the tip of a tapered line kicks over - I don't see that in these histories.
For me the turn over off the DT lines in the graph do not match the real world.
Thus the linear mass density will vary as the square of the taper diameter. I suspect that different line manufacturers use different taper functions in their lines, so you would need the diameter vs length values to come up with an exact rho_l variation in the tapered end of the line.
It appears the front taper on SA DT lines are quite short (5 feet), while for Triangle Taper lines use "a continuous forward taper in the head of the line, the first 27 to 40 feet depending on the application".
This has been my problem, I could not work out the taper from Rh0_L alone. Consequently, I was not sure what the time histories were telling me. From your last post, it infers that he is using a 7wt floater with a 9m taper.
From your last post, it infers that he is using a 7wt floater with a 9m taper.
Vince,
I think Lingard used the same taper design for his DT line as was used by Dr. Spolek. That paper says:
Double Taper:
A commercially available line that is the most popular line for this type of cast. (20 meters). Each end of this line has a compound taper that consists of .61m (2 ft) of level line, followed by 3.05m (10 ft) of enlarging diameter line, followed by a central portion of level line.
So the variation in the diameter was over the last 3.6 meters of the fly end of the line.
For me the turn over of the DT lines in the graph do not match the real world.
I think he just assumed the end of the line followed the path of the line ahead of it as it went around the loop. I do not think he attempted to model the more complicated swing of the loop as it opens, nor did he have a leader on the the end of the line.
In my plots I stopped the ODE while there was .5 of level line left in the fly leg behind the loop since the book keeping for the looking at the effects of line taper and shortening line going around the loop were more than I wanted to get into.
So are you thinking the approach of Spolek, Lingard, Hendry, and Gatt-Bono are wrong, and that the acceleration of the loop that comes from the shortening mass in the traveling line is not related to the linear mass density of the line?
Does that mean you think the taper in a fly line will speed up the velocity of the fly at the end of the cast rather than to slow it down? Most people would think that since they have been exposed to the standard bull-whip analogy. You see the claim that:
The relationship between a whip and a fly line is obvious.
But most people who say that have never cracked a whip.
I am not sure what you are questioning in this analysis. It would be good for you to develop a program to look at this. I know I did not understand Dr. Spolek's results until I used Hendry's approach to come up with the ODE required to look at the problem. Once you play around with it, you soon realize how important the linear mass density of the line is in determining the velocity history of the fly as the loop propagates.
Gordy
"Flyfishing: 200 years of tradition unencumbered by progress." Ralph Cutter
So are you thinking the approach of Spolek, Lingard, Hendry, and Gatt-Bono are wrong, and that the acceleration of the loop that comes from the shortening mass in the traveling line is not related to the linear mass density of the line?
Does that mean you think the taper in a fly line will speed up the velocity of the fly at the end of the cast rather than to slow it down? Most people would think that since they have been exposed to the standard bull-whip analogy.
No I did not think that, back at your post 16, you said:
That is why the taper in a whip causes the whip wave to travel faster as it goes down the length of the whip, while the taper in a fly line will reduce the acceleration of the loop when the lower linear mass density of the line starts going around the loop.
This threw me becasue when I looked at the Lindgard graphs, the DT fly leg accelerated between 35 and 40m. This is why I became interested in the taper length.
