Category Archives: rowing physics

Comparisons … and questions

It has been silent on this blog for too long. It’s been a busy time for me, both at home and at work.

And I had another excuse. Bill Atkinson asked me to add my model to the comparison between Van Holst and Atkinson. It turned out the numbers didn’t compare too well. So I went through my code again and again to find discrepancies. Unfortunately, I didn’t find any. This evening, I went through everything again in a final effort to understand the difference between my model and the two others.

I give up. It doesn’t make sense to keep looking myself, to stubbornly restrict my thinking power to only my (limited) brain. Are we in the age of open source and web 2.0 and on-line collaboration or not? It’s time to show the unfinished work and let it be criticized in the open. So, here we go.

inputs value unit remarks
Mass of rower 90 kg
Mass of shell 14 kg
Peak blade force 368 N Trapezium profile (function of oar angle)
Oar outboard 2 m
Catch angle -1.18 rad
Release angle 0.68 rad
Stroke length 1.37 m
Rating 29.4 stroke/min
Blade area surface 0.142 m^2 sum of 2 scull blades
Hull resistance factor (varies) ITTC approximation
Blade lift/drag coeff (varies) simplified flat plate
results value unit
average shell velocity 4.75 m/s
minimum shell velocity 3.3 m/s
shell velocity at release 5.25 m/s
recovery time 1.3 s
Total power 547 W

The chart in figure 1 gives the power distribution. All values are in Watt, averaged over the stroke. The numbers in bold are direct outputs of my model. The other numbers are calculated. So, blade losses are calculated directly from the blade force and blade slip. Drag losses are calculated directly from the drag force and hull velocity. The power going into “momentum” is calculated directly from the acceleration of the rower’s mass, and the power at the oarlock is calculated from the propulsive force and boat velocity.

Figure 1: Power Distribution (updated)

For comparison, here’s the same chart for Marinus van Holst’s model (data taken from Atkinson’s site):

Figure 2: Power distribution according to Van Holst

Figure 3 shows the same chart for Atkinson’s model:

Figure 3: Power distribution according to Atkinson

Atkinson calls one of the cells “Footboard”. In my model, I have access to the energy “stored” in kinetic energy (rower and shell – as in excess of the average system velocity), but I am not able to calculate directly the power exercised “at the footboard”. This is the reason for the question mark behind “footboard” in figure 1. I have estimated the power dissipated in the rower’s body using Van Holst’s explanation, see figure 2 and the explanation just below it here.

Clearly, the numbers are different. I am especially concerned by my inability to achieve the same value for blade losses, where Atkinson and Van Holst agree. Time to consult the community …

Well, that didn’t hurt, didn’t it? Not yet.

Update 15 April 2011

Bill Atkinson pointed me to the same issue as Martijn has done a long time ago. In my equation for drag and lift, I multiplied the blade area A with the sine of the angle of attach \sin\alpha, which was wrong. I’ve corrected for this and updated the numbers in figure 1.


Drag revisited (2)

In this post, I postulated that skin drag amounts to about 80% of drag and simply divided the skin drag by 0.8 to obtain the total drag.

Bill Atkinson pointed me to this work by Tuck and Lazauskas. First of all, form drag is estimated there by an equation from a paper by Scragg:

C_f = k C_{\rm skin}


k = 0.0097(\theta_{\rm entry} + \theta_{\rm exit})

where \theta_{\rm entry} and \theta_{\rm exit} are the half angles (in degree) of the bow and stern, respectively. Tuck and Lazauskas do a complete modeling of wave drag as well, and come up with something that we could almost call a phase diagram for rowing shells. Looking at the figure 2b here, the wave drag is around 6% of total drag for shells optimized for racing speed, a Froude number around 2 in the figure. Of course, wave drag varies quite strongly with the shell’s velocity. However, the shell velocity varies during the stroke, and it seems a good approximation to use an average value for wave drag which is only a minor portion of the total drag. Doing the integrals to calculate the instantaneous value would slow down the simulation software too much, for a relatively small improvement in accuracy.

As far as the bow and stern angles are concerned, I collected data from boat manufacturers sites about the boat length, beam and draft as a function of the displacement.The results are depicted in figure 1. The solid line is a linear fit, showing that all boats are basically of (more or less) the same shape. Assuming a parabolic boat shape, the bow half angle would be 8 degrees, with an elliptic shape around 6.3 degrees. This would give a value of k between 0.12 and 0.15. So, the total drag (wave drag, skin drag and form drag) would be about 1.2 to 1.22 times the skin drag. Or, in other words, skin drag accounts for about 82% of the total drag.


Figure 1 Shell length vs beam (both in m) for Vespoli and Empacher shells

Regarding the scaling equation used for different boat types, it is clear that if all boats are basically scaled versions of the same shape, then beam, draft and length should be proportional to the 1/3 power of displacement. This is clearly shown in figures 2 and 3.


Figure 2: Beam as a function of displacement for different makes and types of shell.


Figure 3: Shell length vs displacement for different makes and types of rowing shells

Finally, figure 4 shows the wetted area as quoted on various manufacturer’s websites as a function of the displacement. The solid line shows the \propto D^{2/3} scaling that I use in my model.

Figure 4: Wetted area as a function of displacement for rowing shells of various manufacturers

So, in conclusion, the mystery isn’t a mystery any more. The lower value quoted by Slasias and Tullis is just the skin drag portion calculated from a 5m² wetted area. In reality, the wetted area of a four is slightly higher, and one has to add 20% for wave drag and form drag. Looking at the scatter of the data in the various graphs, I would also say that the accuracy of my model is good within about 10%, not better. This is not a disaster, as the relative results predicted by the model are still valid. Finally, boat design is found to be extremely important, as a 5% reduction of skin drag can means a mean velocity increase of 2.6%. This amounts to more than 2 seconds per 500m for an eight.

Drag revisited

A discussion on the Concept 2 UK Forum reminded me of the unsolved mystery of the amount of drag experienced by a rowing shell. So I went back to my old textbook on Fluid Mechanics (Robert A. Granger, “Fluid Mechanics”, Dover 1995). Assuming fully turbulent boundary layer flow, the drag coefficient for skin drag is a function of the shell’s velocity or Reynold’s number. Granger recommends the following equation:

C_{D_f} = \frac{0.455}{(\log R_l)^{2.58}}

for Reynold’s numbers R_l larger than 10^7. The  International Towing Tank Conference (ITTC) recommends a slightly different equation:

C_{D_f} = \frac{0.075}{(\log R_l - 2)^2}.

For a single with a length of 8.2m and a velocity of 8.42 m/s (1:43 per 500m), the Reynold’s number amounts to 3.3\times 10^7 for water of 10 degrees C. This gives C_{D_f} = 2.459 \times 10^{-3} with the ITTC equation. Assuming the wetted area for a single with a rower weighing 80kg to be 2.25 m² as claimed by Cambridge Racing Shells to be the value for a Fluidesign single, I obtain \alpha_{\rm skin} = 2.77 N s²/m². Using Granger’s equation, I get 2.81 N s²/m². For a four with a wetted surface of 6 m², I get 6.7 N s²/m² using the ITTC equation.

Skin drag is not the only form of drag. There is also form drag and wave drag. It is said that skin drag accounts for about 80% of the drag (Ana Dudhia). So, dividing my number by 0.8 I get a value of \alpha amounting to 3.51 for the single scull at race speed. Pretty close to the value I have used so far. The difference is a slight dependence on the shell’s velocity for the value of \alpha. In figure 1, the drag force using the ITTC equation is compared with the simplified method of taking constant drag coefficient, for a single scull with a rower weighting 80kg (wetted surface of 2.25 m²). The differences are small but at maximum velocity they can be of the order of 10%.

Figure 1: Drag Force vs velocity for a single scull with a rower of 80kg. Green line: Constant drag coefficient approximation. Blue line: ITTC 1957 drag coefficient

The division by 0.8 to account for other forms of drag seems a bit arbitrary. Interestingly, when I try to validate the model on the Lucerne 2010 (no wind) boat speed and ergometer scores of Ondřej Synek, the 2010 world champion in the single, I get a consistent story. Synek’s Lucerne achievement was a power of 586W on the 2k (calculated using the ITTC equation value for drag). Assuming 25W is consumed in moving a 98kg heavy athlete up and down the Concept 2 slide, the remaining ergometer power of 561W corresponds to a 2000m time of 5:42 on the ergometer. Synek is known to have rowed 5:41.8 (Czech National record).

As Bill Atkinson rightly pointed out, the scaling equation I used to calculate for different boat types, depends heavily on the wetted shape. I took the \propto m^{2/3} intuitively without giving it much thought, except that Anu Dudhia uses it here. The numbers given for wetted area on the CRS site seem to confirm this. Also, my validation with the 2010 Lucerne results gives confidence, but, admittedly, I have not given it enough thought from the basic physics side. To be continued …

Stroke Profile III – realistic force profiles

In the two previous posts, I have looked at hypothetical force profiles. Here I present the simulation results on more realistic ones.

Figure 1: Force profiles used to simulate more realistic rowing styles

Figure 1 shows the profiles used in the simulation. As you can see, all are defined by two points (x1,F1) and (x2,F2). The first point defines the handle position at maximum leg force. From that point on, the force can stay on a plateau or go downhill immediately. This is defined by the second point (x2, F2).  The force curves look more or less like what can be observed on a Concept2 ergometer screen. One could interpret the differences as differences in style, and perhaps also body proportions. For example, T3 could be a rower with shorter legs and a longer torso compared to T2. T5 is equal to “trapezium” in my two previous posts.

Figure 2: Mean boat velocity vs rower power for different force curves

Figure 2 shows the mean boat velocity achieved for a rower in the single, using standard rigging (catch angle 63 degrees). Each data point represents a different stroke rate, increasing from 25 spm to 35 spm in steps of one stroke per minute. Again, the differences are small. The most efficient strokes seem to be T2 and T5. T3 seems to be the most inefficient. It is interesting to look at the difference between T2 and T3. It seems like keeping a strong stroke as long as possible pays off. This is in agreement with common coaching practices.

Figure 3: Mean boat velocity vs rower power, smaller catch angle

Figure 3 shows the same plot as figure 2, with a changed rigging. The rower has moved the footboard towards the bow and reduced the catch angle to 43 degrees.  Still, the rowing style “T2” is the most efficient, although the differences are reduced. In conclusion, a flatter force profile leads to more efficient rowing, independent of the catch angle, although smaller catch angles are more forgiving.

The differences between T2 and T3 are that the short-legged rower T3 has to push harder in the beginning of the stroke to reach the same velocity. He therefore loses more energy due to blade slip. The more balanced stroke of T2 wins.

Stroke force profile II – a detailed look

In the previous post, I wondered about the differences in rower power for the different stroke profiles. Here’s a comparison between an ideal “flat” profile and the “trapezium” profile from the previous post.

Figure 1: Energy consumed during one stroke of "trapezium" style rowing

The red line in figure 1 is the energy used at the oarlock to accelerate the boat. The black line is energy dissipated at the blade (creating puddles). This rower is doing 35 spm and rowing 1:57 per 500m.

Figure 2: Energy dissipation for one stroke, "flat" styleFigure 2 shows the same plot for the “flat” stroke profile. The mean boat speed achieved is almost the same, 1:57 per 500m at 35 spm. However, the energy consumed is higher. The “trapezium” sculler is rowing at 507W, while the “flat” sculler is rowing at 531W. So, where’s the difference?

Here’s a table with some data:

So, indeed the robot rowing a “flat” stroke force profile wastes more energy at the blade. However, he also puts more energy into the propulsion of the boat … at the same average speed. How is that possible? Is he more inefficient? First, I looked at the maximum and minimum boat speeds, and it seemed that the “trapezium” rower had a more unbalanced stroke, with larger speed differences during the stroke. Also, the mean square deviation from the mean velocity (a very very mean paramater) is higher. However, the mean v^3 which is proportional to the energy lost in drag is higher for the “flat” rower. Especially, during the recovery, the “flat” rower dissipates more energy at the hull through drag than the “trapezium” rower.

Let’s not forget that we used some extreme force profiles, which are never seen in real rowing. In reality, force profiles for different rowing styles will be closer to each other. I have a suspicion that physiology plays a larger role here than physics. It will be the movement which is most efficient and/or most powerful for the body which will make the difference. Another source of inaccuracy might be that my model does not take into account a pitching movement of the boat (around the lateral axis) which might be a source of additional energy dissipation.

Force profile – surprising results

Figure 1: Stroke Force profiles

In the previous post, I looked at the influence of different recovery styles. Now, it’s time to look at the influence of the force profile during the stroke. I varied the force profile of the applied force during the stroke, keeping stroke length constant, and scaling the maximum force such that the average handle force per stroke remains constant. Figure 1 shows the different force profiles that I tried. The system used for the simulations was a single scull with a 80kg rower and fairly standard rigging. Catch angle -63 degrees, finish angle 43 degrees (zero being sculls perpendicular to the boat’s long axis).

Figure 1 shows the various stroke profiles that I tested. I used extreme, unrealistic profiles on purpose. We’re still in validation mode so consider the results more as a test of the simulation than an exercise using realistic stroke profiles. No sculler will be able to pull a force profile like “strong finish” in figure 1. Once we have confidence in the simulations and a feeling for the error margins, we can look at realistically achievable force profile variations.

Figure 2 shows the results. As said, I kept the average handle force per stroke constant and increased the stroke frequency from 25spm to 35spm for the five different force profiles. The duration of the stroke is a function of the force profile and of the boat velocity at the catch, so the recovery speed, or ‘the rhythm’ will be different for the different rowing styles, even when rowing at the same stroke frequency.

Figure 2: Mean velocity vs power for different force profiles (stroke frequency varying between 25spm and 35spm)

Interestingly, all force profiles except “strong catch” seem to fall around a common Power-velocity curve, although the stroke frequency needed to achieve a certain power varies. Apparantly, some force profiles are “heavy” strokes where the rower achieves a higher average power with a lower stroke frequency. It’s a bit puzzling to me to see such large differences in power at the same stroke frequency for the different force profiles. My working assumption right now is that the constant average handle force in figure 1 is really a “stroke length average” and not a time average, but I am not 100% sure that this is the right explanation. I need to dive into my model to find the explanation.

The “strong catch” curve is the outsider. Higher end speeds are achieved, at relatively low frequency, however at the cost of much higher power. So, is this a “heavy, inefficient” stroke? The equations for drag and lift are symmetric around the 0 degree oar angle point. However, the scull arc in my example is not symmetric around 0. So, I performed an additional set of simulations where I moved the foot board towards the bow in such a way that the catch angle is now only -43 degrees and the finish angle is 63 degrees. The results can be seen in figure 3.

Figure 3: Velocity vs power. Same as figure 2, but with a lower catch angle

Indeed, the “strong finish” stroke has now become the most inefficient one. Looking at Van Holst, figure 4.3 here, the cause of this is that the rowing is pulling hard in the part of the stroke where the blade efficiency is low. We’ll look at this conclusion again in a subsequent post on catch angle dependence.

Note to self: In another subsequent post, I need to look deeper into the power differences for the different stroke force profiles.

Managing the recovery

Boat velocity vs rower power for different types of recovery

The figure shows the boat velocity vs power chart for different types of recovery. Each data point represents a calculation at different ratings between 25 spm and 35 spm. Keeping the force constant and varying the rating, the higher powers are obtained by racing the slides more. I modeled 4 types of recovery:

  1. Constant handle speed
  2. Sinus function handle speed
  3. Triangle handle speed (linear acceleration and deceleration of handles)
  4. “Realistic” handle speed (roughly modeled from rowing videos)

Note that I model the velocity profile of the handle. Of course, the body center of mass responds differently, the center of mass being slower than the handle at the start of the recovery where only the arms move, and faster during the second half of the recovery.

The figure shows that – within the accuracy of the model and at constant rower power – the average boat speed is independent of the recovery style. Knowing how coaches like to stress the importance of the recovery, this seems a counterintuitive result. However, my results are in agreement with Van Holst, who concludes quite strongly: “It seems useless to coach on a very pronounced mode of seat motion.”

Actually, coaches are right to pay attention to the return, but not for energy efficiency reasons. It is the quality of the return that determines the accuracy of the catch. This effect is strongest in sweep rowing: If the crew does not manage the recovery phase in perfect sync, the boat balance will be disturbed. Apart from the balance, there are more subtle effects. I believe that any disturbance resulting from imperfect crew synchronization or abrupt body movements during the return compromises the quality of the catch, which will have an effect on the stroke quality.

You can try this with your crew by training them to start from the release position. Another way of illustrating the effect is by training start sequences. The first stroke is usually done with less power and more attention to balance than the second one, because a bad first stroke can ruin the first return, which in turn ruins the first catch (of the second stroke), and so forth.