FAQ: How do Stream and Depth affect Rowing?

Prepared by Dr. A. Dudhia, Dept. Atmospheric Physics, Oxford University, an ARA
Silver Level coach and surprisingly slow sculler from St.Catherine's College BC

22nd July, 1995

  1. Introduction
  2. Viscosity
  3. Boat Resistance
  4. River Flow
  5. Upstream/Downstream Resistance
  6. Shallow Water Resistance
  7. Upstream/Downstream Times


1. Introduction

This Web FAQ page has been prompted by a discussion on rec.sport.rowing relating
to the why rowing upstream or downstream should feel different, and also the
difference between deep/shallow water. If anyone wants to query, argue or add to
any of this, feel free to email me. I'll start by saying I'm assuming that
rowing downstream feels 'heavier' than rowing upstream, which is my personal
impression although some would argue it's the other way
around - which I can't explain except as a purely psychological effect. Also,
I'm discounting the change in air resistance, i.e.when you row downstream you're
always moving faster relative to the air than rowing upstream, so relatively
speaking, downstream pieces are into more of a headwind than upstream pieces.
This is clearly correct, but I think, negligible - again speaking personally, I
think I can tell the difference between increased water resistance and increased
air resistance, and the upstream/downstream difference feels like water
resistance to me.



2. Viscosity

Consider two surfaces separated by fluid, a distance H apart, with the upper
surface moving at V and the lower surface fixed. The fluid adjacent to the upper
surface will be dragged along and also be moving at velocity V while the fluid
adjacent to the lower surface will be stationary, so that a constant velocity
gradient V/H (also known as `shear') is set up in the fluid.

<---- Vel V /_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/
               <---- v=V                 |      upper boundary
                <--- v=0.75V             |
                 <-- v=0.5V              H      shear layer
                  <- v=0.25V             |
            _______< v=0_________________|_____ lower boundary
      Vel 0 | | | | | | | | | | | | | | | | | |

                           Fig.(2.1)

As any physics textbook will tell you, the Resistance R (measured as forceper
unit area) caused by viscosity is given by:

(2.1)           R = e.dv/dz

where e is the coefficient of viscosity (measured in kg/m/s, assumed
constant), and the shear dv/dz = V/H in this case, so

(2.2)           R = e.V/H

This tells you that viscous drag (resistance) on the upper surface increases in
proportion to velocity. However, this is only really applies to situations where
the horizontal lengths are much larger than the separation H, so that the shear
layer is constant along the whole length. This is only true for boats rowing
over really shallow (inches) water. For most purposes a better model is
required.




3. Boat Resistance

As a boat moves through stationary water, the water in contact with the bows is
immediately accelerated to the boat speed V, but the shear layer can only grow
downwards at a fixed speed W (set by the mean free path of molecules).
So the lower (static) boundary of the shear layer slopes downwards from the
bows:


                          <- Boat, Vel=V
            ~~~~~~~~~~~\_____x____________/~~~~~~~~~~
                        \    |
                         \   |
                          \  h
               static ---->\ |   shear layer
               boundary     \| (V>0 this side)
           (V=0 this side)   \
                              \

                           Fig.(3.1)

Below point x along the hull, the boundary layer will have been growing for a
time t=x/V, so will have reached a depth h=W.t=W.x/V. So using Eq.(2.1), the
viscous drag at point x is given by:

(3.1)           R(x) = e.V^2/(W.x)

This is the origin of the V^2 law for boat resistance (cf. resistance
proportional to V for previous section).




4. River Flow

River flow is driven by the hydrostatic pressure gradient, which is
constantacross the whole cross section of the river. Were it not for
viscosityeffects, this would mean that the stream flowed at an equal speed at
allpoints within the cross-section since each point is driven with the
sameforce. However, due to viscosity, the flow is slower near the fixed
boundary(riverbed and banks) and faster near the free boundary (surface, since
theair offers relatively little resistance to flow), and the quickest flow
willbe the furthest from the fixed boundary, which means away from the sides
andwhere the river is deepest.

  _________________                        ________________
  ................|012222221111123333333210|...............
  ................\01111111000001222222210/................
  .................\0000000/---\011111110/.................
  ..................\-----/.....\0000000/..................
  ...............................\-----/...................

                         Fig.(4.1)

The diagram shows a cross-section of flow over an uneven river bed, thenumbers
represent flow speed. Flow is 0 next to fixed boundary, 1 for 1layer away, and
so on. As with most rivers, being wider than deep, the flowrate in most places
is determined by the depth rather than the distance fromthe sides. The surface
flow is therefore quickest (3) over the right handchannel and slowest tucked
right in to the sides, or over the central ridge.




5. Effect of River Flow on Resistance

Given that the stream has its own vertical shear due to flow, this will interact
with the vertical shear created by the passage of the boat throughthe water and
either increase/decrease the shear, and resistance, according to whether the
boat is moving downstream/upstream.

          <- A: vel=-V+U               B: vel=+V+U ->
  ~~~~~~~~\____________/~~~~~~~~~~~~~~~\____________/~~~~~
                  surface layer, stream=+U ->
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
                  lower layer, stream=+a.U ->
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

                           Fig.(5.1)

Boats A and B are both moving at speed V relative to the water, except that A is
moving upstream and B is moving downstream. The surface water is movingat
velocity U (relative to banks) and, for simplicity, we'll just considerthe boat
shear layer extending down into the next layer, moving at velocitya.U where a is
some number between 0 and 1.

     (A)             (B)       (A)                (B)
Vel=-V+U     +U      V+U      -V+U       +U       V+U
      :       :       :         :         :        :
      +-------+-------+-->      +----+----+--------+--> Velocity
       \      :D     /           |   :    :D     _/
        \     :e    /            |   :    :e   _/
         \    :p   /              |  :    :p _/
          \   :t  /               |  :    :t/
           \  :h /                 | :   _:h
            \ : /                  | : _/ :
             \:/                    |:/   :
              |                      |    :
             +U                    +a.U

                       Fig.(5.2)

The left diagram shows the shear set up by A and B in water moving at constant
velocity +U throughout its depth. Since both gradients are the same, both A and
B perceive the same drag. The right diagram shows the effect of stream shear,
where the lower boundary is moving at velocity a.U. Boat A, moving upstream, now
experiences less overall shear therefore decreased resistance, while Boat B,
moving downstream, experiences more overall shear, therefore increased
resistance.

In general, the faster the flow, or the shallower the water, the greater the
shear so the greater the difference in resistance.

Back to Contents
--------------------------------------------------------------------

6. Shallow Water Resistance

The difference between sections 1 and 2 was that in the former case the shear
layer had fixed depth whereas in the latter it grew continuosly as the boat
passed over.

In shallow water, the shear layer may touch the bottom, in which case it
obviously ceases to grow and Eq.(2.2) applies instead of Eq.(3.1). At first
sight this might seem like a good thing, since in Eq.(2.2) the drag only
increases linearly with velocity rather than the square of the velocity from
Eq.(3.1). However you have to remember that bottom effects are felt at low
velocities rather than high velocities (since the shear layer has more time to
grow downwards at low speeds) and the point where the two become equal is
where the shear layer separates from the bottom.

  R ^                   x
  e |                   x     +
  s |                  x  +
  i |                 *
  s |              + x
  t |           +   x
  a |        +     x           [ + = linear regime ]
  n |     +      x             [ x = quadratic regime ]
  c |  +     x                 [ * = separation of shear layer]
  e o------------------------->
       Velocity
                     Fig.(6.1)

So at low velocities the shallow water resistance is linear (shown by + in the
diagram) and greater than would be expected from the quadratic regime (shown by
x) with an unbounded shear layer. Changing the depth of the water has the effect
of reducing the slope of the linear regime (differentiating Eq.(2.2)):

(6.1)           dR/dV = e/H^2

so that the transition (shown by *) from linear to quadratic occurs at lower
velocities.

So how shallow does the water have to be before you notice the bottom? You can
get an idea of the depth of the shear layer by observing the extent of the
sideways turbulence at the stern of the boat (the shear layer probably grows
downwards at much the same rate as outwards), i.e. around 1 metre. Any deeper
and you shouldn't notice the bottom at all. The minimum depth for Olympic
Regatta courses is, I believe, 2 metres, just to be on the safe side.

Of course, as the speed tends to zero, the shear layer will extend to infinity
so the actual depth of the water will always be `noticed' eventually, but
usually at such low resistances that it will be impossible to distinguish
between the linear and quadratic regimes.

Note that moving water will also have flowed-induced shear (section 5), an
entirely separate effect. In that case (i.e. rivers) the total depth will always
be significant.


7. Upstream/Downstream times

It is a common misconception that if you row, say, 2000m upstream, and 2000m
downstream, measured against some fixed points on the bank, your average time is
the same as if you rowed 2000m in still water (I'm ignoring any change in speed
due to fatigue or the effects discussed in the previous sections). For low
speeds it is a reasonable approximation, but the average in stream will always
be slower than your still water time. Why? Because...

Suppose your intrinsic speed through the water is V, the stream speed is U,and
you're rowing a distance L measured along the bank.

(7.1) Still water Time, Ts = L/V

(7.2)    Upstream Time, Tu = L/(V-U)
(7.3)  Downstream Time, Td = L/(V+U)

(7.4)     Average Time, Ta = (Tu+Td)/2 = L.V/(V^2 - U^2)

As you'd expect, as stream speed U tends to zero, then the average of your
upstream+downstream times tends towards your still water time, but for any non
zero stream, Ta is longer than Ts (because V^2-U^2 is always less than V^2).

How much different? Take V=5 m/s (corresponding to Ts=6:40 for 2000m in still
water). Rowing in a (slowish) stream of U=10 cm/s, you'd row 2000m upstream in
Tu=408.2s, and downstream in Td=392.2s, giving an average of Ta=400.2s, ie only
0.2s slower, not important. But in a quicker stream of U=1m/s, you'd get
Ta=416.7s, ie 16.7s out, or you'd think you were about 5 lengths slower than you
really are. Note that although the stream speed increased by a factor 10, the
error increased by a factor 100 (depends on U^2).