Computing the force generated by a sail on a boat This is a follow up question to the question I asked yesterday: Problem understanding basic sail mechanics 
I am  focusing purely on the Newtonian aspect of motion and I am computing this as follows:
First I compute the relative wind ($v_r$) velocity by subtracting the ships velocity ($v_s$)from the true wind velocity ($v_t$)
$v_r = v_t - v_s$
Next I take the normal of the sail (in to the wind), $n_s$, and "bounce" the wind off it, computing the out velocity of the wind, $v_{out}$
$v_{out} = v_r - (2 * v_r \cdot n_s) * n_s $
Next I compute the change in wind velocity, $\Delta v$
$\Delta v = v_{out} - v_r $
This is linearly proportional to the force the sail exerts on the air, hence $-\Delta v$ is linearly proportional to the force exerted on the boat. 
Next, I compute the keel force that offsets the horizontal part of the sail force, allowing the boat to move against the wind. 
I take $n_k$ to be the normal of the keel going in the opposite direction to the sail force and perpendicular to the boat heading. I then compute the keel force, $F_k$ to be
$F_k = n_k * |-\Delta v|$
i.e. the magnitude of the sail force in the direction of the keel (perp to direction of boat facing).
To compute the resultant force, I of course sum these two forces.
Firstly, is all that correct. Clearly it is missing resistance from the water, but aside from that, do the calculations correctly reflect the answer given in my previous question.  
Secondly, what parametrises these forces and how. What I am seeing at the moment is, when sailing with the wind, the boat accelerates very quickly (up the the speed of the wind). However, when I try to sail near to the wind (is that the correct phrase? for when my boats forward and the direction of the wind are near and opposite) although I can see the apparent wind speed increasing, the force applied is still very small. Is this was you would expect to see? 
 A: Seems to me that you are trying to calculate the lift on an airfoil from first principles using only Newton's laws of motion. Chris Waltham did exactly that in his paper "Flight without Bernoulli". You might want to check it out.
http://users.df.uba.ar/sgil/physics_paper_doc/papers_phys/fluids/fly_no_bernoulli.pdf
Note that this is a non-standard way of doing the aerodynamic analysis.  His conclusion:

We have used a very simple physical model relying only on Newton’s
  second law to reproduce all the salient features of a rigorous fluid
  dynamical treatment of flight... The model has its limitations; we
  cannot calculate real  performance with it.

So, while your approach is reasonable, there's a reason why aerodynamic engineers take a different tack.
You also might want to check out Charles Eastlake's paper "An Aerodynamicist’s View of Lift, Bernoulli, and Newton" http://users.df.uba.ar/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf
His take is:

Measuring lift by measuring the increase in downward vertical velocity
  in the flow coming off the trailing edge of the airfoil is
  conceptually possible. This downward velocity is definitely there and
  is known as downwash.  I have never heard of anyone  actually
  measuring it with sufficient precision to calculate lift, not
  because it is physically unsound but because it is not a practical
  experiment.  It is not practical because the downwash is distributed
  over the entire flow field downstream of the trailing edge, and it
  would thus be very difficult to measure enough data points to
  integrate the distribution accurately.

Not to discourage you, but I think you may be heading towards a dead end.  Anyway, check out the papers above - I think you'll find them interesting.
A: 
Next I take the normal of the sail (in to the wind), n s  , and
  "bounce" the wind off it, computing the out velocity of the wind, v
  out

Nope. First you have to calculate the angle of the wind wrt the sail area. This depends both on the velocity vectors of the boat path and the angle of the sail wrt the boat. Then you compute the effective area of the sail to the wind, proportional to the cosine of the angle. The component of wind flow towards the sail is the cosine of the angle times the effective area.
All this assumes no aerodynamic behavior of the sail, which is not true. See https://www.youtube.com/watch?v=gNaEX6EGg7I for a start.
A: A sloop has two sails, the jib and the main.  There's a slot between the trailing edge of the jib and the leading edge of the main.  When sailing close-hauled into the wind, you try to get the air flowing off the leeward side of the jib (side away from the wind) to go through the slot and on to the windward side (side toward the wind) of the main, because the air accelerates out of the slot.  This creates a Bernoulli effect across the front of the main, and the lift caused by the lower pressure on the windward side of the main pulls the boat against the wind.
I think you need to add a force from the Bernoulli factor into your Newtonian model.  The Newtonian model will work for a boat sailing downwind, but there are more forces than you have taken into consideration, involved when sailing upwind.
A: There's a simple way to estimate the force generated by a sail when sailing into wind, using Newtonian mechanics based on the mass flow rate; using the equation: Force = ma = m/dt x dv.

