Steering in a following current

There is a video of this article linked at the end. 

A few years ago we analyzed the grounding of the Ever Given in the Suez Canal just a few days after it happened. The results are presented in a 5-min video, followed by three support videos, one of which had to do with the effect of a following current on steerage, which was likely a factor in the event.  See discussion of tides and currents about 6 min into that video. There presented and discussed this diagram:

It shows what we mean when we say "steering is inhibited in a downbound vessel with following current."

When we turn left 30º relative to the water, we can see it on the  compass, but it is not apparent by looking at the water alone. Where we would see any change of course would be relative to, say, a rock ahead that we were turning away from. We will see that we have turned the heading away from the rock, but we will not be making good the full extent of our turn. The current causes us to lose some of the turn we made. In other words, the heading HDG (also called course through water, CTW) changes, but the course over ground (COG) changes less, which is the issue at hand. Conversely, when sailing into the current the maneuvers are enhanced relative to the ground.

Yesterday we got a call asking if this reckoning could be generalized into a formula which motivated some work on the topic.  The above diagram we got from qtVlm app using its simulator function plus a forces current function.

Here is the geometry

Vessel moving due west at 6 kts in a 3 kt following current. Then turns left 40º but only achieves a 27º turn in COG. 

Here is the trig and formula for the angle loss (beta) as a function of the turn (alpha) and the ratio of current to speed, r = C/S

We can also make a plot of the values:

For example, for r= 0.4, a 35º turn will lose 10º and you only make good 25º

Here are results as a table. Here speed over ground (SOG) is a ratio of the boat speed S.

Note in our figure above BA = SOG and we get SOG from:

Here are a couple examples from Excel:

Video presentation of the above discussion.