Basic Equations (3) – Oars and Blades

See this discussion on oars/sculls and levers. The rower pulls the scull at the handle. The scull rotates around the oarlock. The blade seems to be at a fixed point in the water, but in reality it slips slightly. The force on the blade F_{\rm blade} is given by:

F_{\rm blade} = F_{\rm handle} L_{\rm in}/L_{\rm out}

The sculls are at an angle \phi with respect to the axis perpendicular to the boat. The resulting blade velocity in the water has two components, a component v_b because of the boat velocity, and a component L_{\rm out} {\rm d}\phi/{\rm d} t due to the angular velocity of the scull.

The blade/water interaction results in drag and lift forces on the blade. The basic equations are explained nicely by Marinus van Holst, so I will just give a summary here. Basically, the harder the rower pulls, the more the blade slips. In light rowing, there is almost no slip, and the angular velocity of the scull is such that the blade remains at the same location in the water. When the rowers pull harder, this is result in a slip, such that the blade force equals F_{\rm blade} = F_{\rm handle} L_{\rm in}/L_{\rm out}.

The propulsive blade force is given by:

F_{\rm blade} = \sqrt{F_L^2 + F_D^2} \cos\phi

The lift force given by:

F_L = \frac{1}{2} C_L \rho A (u_l + u_p)^2

where A is the blade area, u_l, the velocity component perpendicular to the scull shaft, is given by u_l = \dot{\phi} L_{\rm out} - v_b \cos\phi and u_p, the component in the direction of the shaft, is given by u_p = v_b \sin\phi, and the “angle of attack ” is given by \alpha = \arctan(u_l/u_p).

The drag force is given by:

F_D = \frac{1}{2} C_D \rho A (u_l + u_p)^2

Similar to Atkinson and Van Holst, I am using the expression by Caplan and Gardner for the lift and drag coefficients C_L and C_D:

C_D = 2 C_{L,{\rm max}} \sin^2\alpha

C_L = C_{L,{\rm max}} \sin 2\alpha

For C_{L,{\rm max}}, I use a value of 1.0.

Finally, one must not forget to multiply the blade force with the number of blades for the boat, considering that scullers have two blades per rower. It took me a while to figure that out …

Note 2011/04/15: Removed \sin\alpha from the lift and drag force equations. See discussion below.


5 thoughts on “Basic Equations (3) – Oars and Blades

  1. Martijn

    In your equations for Fl and Fd you have the angle of attack. Isn’t the influence of the angle of attack already absorbed in the coefficients Cd and Cl?

    1. sanderroosendaal Post author

      Dear Martijn,

      I’ve been busy so I didn’t have too much time to look into this. However, the A \sin\alpha component represents the projected blade area in the direction of fluid flow.

      In ref. [1], Caplan and Gardner discuss the modelling of lift and drag components as a function of angle of attack. It is clear from their discussion, that they use the same definition of C_D and C_L is is used on this site. They find an additional dependence on angle of attack on top of the projected blade area dependence, which can be approximated well by the \sin^2 \alpha and \sin 2\alpha behaviour that I use in my model.

      So, yes and no to your question. Yes, the influence of the angle of attack as modelled by Caplan and Gardner are absorbed in the coefficients. No, the influence of the projected blade are is not absorbed in the coefficients.

      [1] N. Caplan and T. Gardner, “A mathematical model of the oar blade-water interaction in rowing”, Journal of Sports Science, July 2007; 25(9):1025-1034

  2. Martijn

    I agree that just stating ‘the projected area’ isn’t very clear. In some other article they define the area, A, more precisely.

    A is the projected area of the oar blade measured perpendicularly to the face of the blade

    Caplan, Nicholas and Gardner, Trevor N.(2007) ‘A fluid dynamic investigation of the Big Blade and Macon oar blade
    designs in rowing propulsion’, Journal of Sports Sciences, 25:6, 643 — 650

    1. sanderroosendaal Post author

      You are right, Martijn. I’ve been going back to the basics in order to understand the discrepancies in this post.

      Bill Atkinson pointed me to the same issue in a discussion on the comparison between models. I have now corrected it and will – in the course of the next weeks – go through my old simulations and gradually update the site.

      Thanks for pointing this out. I am sorry it took me such a long time to figure it out.

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