revision September 2007
1 Introduction
Maybe already twenty years ago the question was raised to
what extend lift or drag forces propelled the rowing boat. See e.g. the German publication of Volker Nolte: “Wie wird ein Ruderboot angetrieben? Theoretische Konzepte
bestimmen die Methodik der Rudertechnik”, Leistungssport #6, 1984. The subject was also discussed in
relation to swimming (Ross
Sanders and Edith Cowan ) and canoeing. The idea of Nolte was that the
modern rowing style was based on the use of
the lift force as the main propulsive force in rowing in the early phase of the pull. However, in
rowing the blade has fewer degrees of freedom than in swimming and canoeing.
About the only thing a rower can do is reaching further at the catch and that
is recommended by coaches who believe in the benefits of the lift force. Below
the nature of lift and drag will be explained, set in perspective and lift and
drag coefficients are presented.
2 Basics of lift and drag
When a
solid body is placed in a fluid flow and a nonsymmetrical situation occurs the
direction of the force on the body does not coincide with the direction of the
(undisturbed) flow. This principle makes flying possible. Discussion of lift
and drag starts usually with the introduction of an airfoil. (x is the
direction of the horizontal flow, z is vertical)
The airfoil
(e.g. the cross section of an airplane wing) is long in the direction
perpendicular to the plane of the drawing and the flow can be considered as two
dimensional. The airfoil is tilted with respect to the (undisturbed) flow
direction, defined by the angle of attack, AOA, α. the airfoil experiences
a force FR. Considering an airplane it is very useful to decompose
the force FR into components FL and FD
perpendicular and parallel to the flow direction. FL is the lift
force, it carries the plane, and by definition it does not do work. FD
is the drag force, the resistance to be balanced by the propulsion force
generated by the engines. The net power required is the product of drag force
times flow velocity. The lift and drag forces are expressed as:
with:
FL
and FD = lift and drag force
CL
and CD = lift and drag coefficient
ρ =
density of the fluid
A = projected area of the airfoil with e.g. 1m
length perpendicular to the plane of the drawing
u =
velocity of the undisturbed flow
Note that
the expression for FL and FD differ only in CL
and CD. The designer of an airplane tries to maximize CL
and to minimize CD. CL and CD are dependent on
the angle of attack. For an enormous number of airfoil profiles CL
and CD have been measured or calculated. Usually the CL
drops sharply and CD increases strongly at α = abt.150.
The force on the airfoil is the result of the integration of pressure around
the perimeter.
When not an
airfoil but a flat surface with zero thickness is placed in a flow a lift and
drag force can be distinguished as well.
z x
As the
force is the resultant of the pressure on the surface the direction of the
force cannot be different from perpendicular to the surface (shear forces neglected). This includes that
CD and CL cannot be independent of each other. Between
the two the next relation exists:
When a
curved surface with zero thickness is placed in a flow the force on every
surface element is perpendicular to that element but as the angle of attack
varies and also the pressure distribution not much can be said over the position
and the direction of the resulting force. See Fig 2.3. But when the curvature
is small as with a rowing blade, the situation cannot be very different from a
flat plate. Assume now that the forces are in the horizontal plane as is the
case with rowing. For an elaboration of the idea see section 5. CD
and CL as function of the angle of attack.
x
From the
explanation above follows:
the
distinction between lift and drag is not of a physical nature but it is a
functional one (carrying and resisting) or a geometrical one (perpendicular and
parallel to the flow direction) but the observation made before that the lift
force does not do work is of importance. In other words, the lift force does
not waste energy.
3 Kinematics of the blade
Fig 3.1
shows the velocities of the boat and the oar/scull.
O = thole pin
OA = shaft of the oar/scull, situation immediately after the catch
AB = blade
vb = boat velocity
pA = distance OA
pB = distance OB
The point A has the velocity of the boat vb and
the relative velocity of scull with respect to the 
. This results in the absolute velocity vA of the
point A with respect to the water. The velocity of the water with respect to A
is uA, uA = -vA.
The same holds for B.
From Fig 3.1 it becomes clear that points A and B have velocity components into the x-direction, they move into the same direction as the boat. At the same time they have a velocity component perpendicular to the blade. The discussed velocity components are shown in Fig 3.2.
The velocity vectors, relative magnitude and direction, are not to scale but I think (based on calculated guesses) that relative magnitude and resulting angle of attack are realistic for the situation immediately after the catch.
We arrived now at a situation as described in section 2. In the next section we shall continue our discussion on the description of the interaction between water and blade.
4 Blade-flow interaction
Consider the situation in Fig 4.1. A curved surface, the blade, in a flow. The figure is a combination of elements from Fig 2.3 and Fig 3.1.
Important
differences with the situations described in section 2 exist. The system is not
two-dimensional, the object is very close to the air-water interface, the
velocity field is not homogeneous and the situation changes rapidly with time.
Another complication is that at the whole convex side of the blade the water is
in turbulence. All these complications will be neglected in the following
discussion and the blade is considered flat as stated before.
In Fig 4.2
the force on the blade due to the flow according to Fig 4.1 has been drawn. As
demonstrated in section 2, the resulting force FR is perpendicular
to the blade. (It is a weak argument but I think that when I am rowing I “feel”
a force perpendicular to the blade). Of course FR can be decomposed
in a lift and a draft force FL and FD.
(Also a
lift force in the vertical direction exists. This is the force that prevents
the blade from diving to deep in the water and facilitates the extraction of
the blade at the end of the pull. It does not contribute to the propulsion) )
Otherwise than
in the case of a wing of an airplane the direction of the lift and drag force
has no functional meaning. They can be decomposed again in a transverse and a
longitudinal direction as has been done by the Dreissigacker brothers (FISA Coaches
Conference, Seville 2000). But this is a rather indirect approach. The same
result is obtained by decomposing FR directly in its transverse and
longitudinal components. See Fig 4.3.
In this
situation the force on the blade is mainly a lift force that does not waste
energy but unfortunately the longitudinal component that propels the boat is
very small. In swimming or canoeing the “blade” has more degrees of freedom.
The blade can turned in such a way that the longitudinal component is greater.
The only possibility to influence the direction of the lift force is to change
the position of the blade (with respect to the shaft).
Turning the
blade with respect to the shaft as in Fig 4.4 might be attractive.
The force
on the blade becomes more effective. However, the angle of attack becomes
smaller and that is a drawback. This can be compensated by a bigger rotational
speed of the oar. The effect will be positive only in the first phase of the
pull and negative in the last phase but the net effect could be positive.
Turning the
blade is this way has been proposed before by M.N. Brearley ("Modeling the
rowing stroke in racing shells", The Mathematical Gazette, Nov 1998) but
was never(?) seen during a regatta.
5 CD
and CL as function of the angle of attack.
Atkinson presents on his website values
for CD and CL as a function of the angle of attack as
taken from Hoerner's "Fluid Dynamic Drag" for a flat plate. These
experimental values obey approximately the tangent relation presented in
section 2. CD and CL both increase with the AOA until
about 42 degrees, then they fall sharply in magnitude and CD grows
again and CL goes to zero for AOA going to 90 degrees. The sharp
decrease of the coefficients is a well known phenomenon observed in experiments
with a fixed position of the plate or airfoil. The experiments are carried out
in a stationary situation. I think it is not likely to occur in the rowing situation.
No sudden decrease of the force on the blade is (ever?) observed during the
pull through because flowing patterns belonging to this decrease have no time
to develop. Based on the consideration above the following rules are presented.
For CD:
if
(α < π/4) then
CD = m * α * tan
α else CD = c
and for CL:
if (α < π/4) then CL = m * α else CD
= c / tan α
where,
α = angle of attack, AOA in radians
c = maximum value for CD and CL
m = c / (π/4)
Adapting this rules simplifies the selection of the coefficients to the selection of c.
c = 1.2 brings us rather close to Hoerner's values.
Fig 5.1 is the graphical representation of the rules.
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6 New values for CD an CL Recently new experiments were carried out to determine the hydrodynamic coefficients for rowing blades. See: Journal of Sports Sciences, April 2007; 25(6): 643-650 Nicholas Caplan & Trevor N. Gardner, "A fluid dynamic investigation of the Big Blade
and Macon oar blade designs in rowing propulsion" The following expression for CD and CL are derived from the results in this paper but are the interpretation of this author:
These expressions fulfil the requirement Caplan and Gardner found values |
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CD CL Compare this graph with Fig 5.1 |
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The differences with Fig 5.1 are striking. How they can be explained is a difficult question. Caplan and Gardner do not refer to previous results. See also Atkinson who discusses in more detail the results of Caplan and Gardner. It is clear that these more smooth functions for the coefficients are an advantage for the simulation because of the iterations it contains. Therefore the simulations with the complete model will be (partly) rerun with these new values for CD and CL. It is hoped that some hydrodynamic scientist will comment on the validity of the coefficients in the future. |