Swimming for fun and (variable amounts of) profit

2014 May 14
by Daniel Lakeland

I've been working on my front-crawl swim stroke as an effort to improve my fitness. I really enjoy the challenge of getting better at the technique, and I also enjoy the challenge of not over-thinking things while I'm swimming. Of course, when I'm not in the water it's another story. So here's a blog post about dimensionless analysis and the forward crawl.

First off, let's be clear what we mean. Some people call this stroke "freestyle", some people the "forward crawl" others the "australian crawl", you can read about this stroke at the wikipedia page but almost everyone who can swim has used some variant of this stroke, so I'm going to assume you know what I'm talking about.

Many people of course are interested in how to swim quickly. The front crawl is generally considered the fastest stroke, which is why people use it during the "freestyle" event (where you really could swim whatever you wanted). Let's think about how we get velocity:

 v = s L r

In this equation, v is the velocity of the swimmer through the water, and sL is the "stroke length" (the distance traveled per stroke)  which we get by multiplying the swimmer's length L by a dimensionless factor s (this is already anticipating later analysis of lengths), whereas r is the stroke rate (the number of strokes per minute for example).

If we want to increase v we can either make sL bigger, or r bigger or both. Since we can't make the swimmer much longer without injury, to lengthen the stroke requires making s bigger which is part of working on technique.

Now, let's think about energy consumption. If we want to go fast in a long distance swim, we will need to make 1/t \int_0^t s(q)L r(q) dq large, which is the time averaged velocity, averaged over time t. To make this make sense we had better be going long enough that we take a lot of strokes and can consider the time between strokes as "infinitesimal" compared to the overall time. A good stroking rate is on the order of 1 stroke per second, so we're interested in averages over minutes or even tens of minutes.

Now, when trying to go fast, what will happen is we have some ability to output power P such that we can sustain it for the entire time t. And we will go as fast as that power budget will allow, in such a way as the power  from drag is equal to our sustained power output. So keeping drag low at a given velocity will let us go faster.

So let's try to model the power involved. The simplest model is a body averaged one:

P = v D

where D is the net drag force on your body.

To figure out what controls the drag D consider that it has dimensions of force, and using dimensional analysis we need to generate a force term from the following variables h the height of the bow wave d the depth of your body in the water, w the width of your body at the shoulders, L the length of your body, \rho the fluid density, \mu the dynamic viscosity, and v the velocity.

Our drag D will be made up of several components. We can use dimensional analysis to get the functional form of these components. We need combinations of the above variables which have dimensions of force for use in our power equation.

A good candidate force term is proportional to hw (\rho g h)=\rho g h^2 w. This is more or less related to the static pressure head generated by the bow wave in front of the swimmer. Another good candidate term is \rho dw v^2 which has dimensions of force and relates to the difficulty of pushing the fluid out of the way in front of you (without lifting it). Finally, we expect drag to be related to viscosity as well, we're going to find that \mu v L has dimensions of force, and is related to the drag along the length of the swimmer's body essentially caused by pulling the fluid along with us. Now we've identified several sources of drag. Let's write an equation for them:

 P = v(K \rho dwv^2 +K_2 \rho g h^2w + K_3 \mu v L)

In our overall problem, there are 2 equations and 9 variables (L,d,w,h,\rho,\mu,v,g,r) and 3 independent dimensions (length, time, mass) so we expect to have 6 independent dimensionless groups. One thing we can see though is that L,d,w,h are all lengths. Clearly we can make 3 dimensionless groups by doing h/L,w/L,d/L which makes sense since L is the largest dimension so all of these length ratios are less than 1.

Let's rewrite our power equation in dimensionless form. We could normalize the power equation by one of the types of drags, since we're interested in going fast, consider that the term multiplied by v^2 may dominate and normalize by that. Let's assume that over the range of speeds of interest to swimmers the dimensionless multiplier of this drag term is near constant and therefore the drag is equal to K \rho d w v^3 for some dimensionless constant K as shown above. Dividing through we get:

 \frac{P}{(K\rho d w v^3)} = 1 + K_w \frac{g h^2}{dv^2} + K_l \frac{\mu L}{d\rho vw}

In this equation, we've identified 3 important dimensionless groups g h^2 / (dv^2), \mu/\rho v d and w/L, there is also another important dimensionless group, v/Lr, this basically measures our velocity as a fraction of one body length per stroke interval, and is equal to s the dimensionless stroke length. Finally, there is h/L,d/L which are the natural measures of bow height, and depth in the water (we are essentially measuring all lengths as a fraction of the swimmer's full length). That is the full 6 dimensionless groups promised. If we write r/L = r', d/L = d', v/Lrs=v'=1 and h/L = h' we can rewrite our equations as (note, maxima computer algebra system is helpful in avoiding mistakes):

 v' = 1


P' = 1 + K_w \frac{g}{r^2L}\frac{h'^2}{d's^2} + K_L\frac{\mu}{\rho r L^2}\frac{1}{d'sw'}

In other words, if we measure velocity in stroke lengths per stroke time, distance in units of your body length, and power in units of piston drag (the term proportional to v^2) we get things into a very simple form. Clearly we can learn a lot from this. It becomes obvious why tall swimmers are desirable, their wave drag decreases like 1/L and lateral drag like 1/L^2. Plugging the following approximate order of magnitude constants in g=9.81,L=2,r=1.25,\mu=.001,\rho=1000,w'=1/3 we get order of magnitude estimates of how important our different drags are:

 P' = 1 + 3.1 K_w h'^2 / (d' s^2) + 6\times 10^{-7} K_L/(d's)

Now, we can't estimate K_w or K_L without data, but the fact that the typical size of the dimensionless group in front of the third term is orders of magnitude smaller than 1, suggests that lateral drag is really not very important in the real world (though we haven't accounted for kicking here, it's quite possible that excessive kick churn would enter into essentially lateral drag). To support this conclusion, consider that the Mythbusters tested out swimming in various viscosities and found that for non-elite swimmers the viscosity had essentially no important effect on speed over multiple orders of magnitude of viscosity.

Clearly though, with a dimensionless premultiplication factor of 3 or so, wave drag is important. One reason to keep your head low in the water (h' small) and your stroke length long (s \sim O(1)) is that small stroke lengths and high head positions will severely increase your wave drag. This is why swimmers try to stay under the water after a push off, so that waves aren't sapping their momentum.

So, to swim efficiently, we need small total power, which means a long stroke (large s), low in the water (small h'), and we need to keep our body stretched out and lying flat in the water to maximize L and minimize w' (which produces a small dimensional piston drag dwv^2\rho). Keep the stroke rate high, but keep it smooth so as to minimize other kinds of wave drag (choppiness) which is a kind of drag not included in this model. Since piston drag is proportional to depth in the water, and wave drag is proportional to height of the wave above the water squared, there is an optimum depth that involves not being too deep, but also not creating too big of a wave. It's probably a good idea to try to stay deeper in the water, with head down and see how well that works for you. Obviously if it interferes with breathing, or alters the constant in front of the piston drag due to strange turbulence around your head, it won't help. There is some optimum depth tradeoff for each swimmer.

What about the "glide"?? this is something that some people advocate, because it lengthens the stroke, and people have found that long strokes are efficient. However, it decreases the stroke rate as well. As long as rs increases, everything is good. So in other words, if you are "over paddling" with a short stroke length and a high stroke rate, slowing down your strokes and making them longer makes sense. But if you are "over-gliding" with a long stroke length (s) but a slow stroke rate (r) you should instead speed-up your stroking so that the overall combination rs increases. Note how the relative wave drag is inversely proportional to (rs)^2, which is related to the velocity. When we slow down, piston drag decreases, so wave drag becomes a bigger relative factor. Ideally, we're maximizing v, which means that we're maximizing piston drag, and wave drag becomes relatively less important. Eventually we come to a steady state where the drag force and velocity are high enough that we can't go faster without exceeding our power budget and therefore getting tired out too soon. Since piston drag is related to v^2 which is what we're trying to maximize, we can't get rid of it entirely, but we can try to minimize the additional drag caused by waves.


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