What curves do photographers use when modifying the contrast of their photographic images?¹
The vast majority don't do their own curves programmatically, which seems to be what you are seeking.
Instead, most start with a "canned" default curve, such as your example of y = a*x + b, applied by their chosen raw processing application, and then either ...
I would suggest trying a large, diffuse flash (softbox perhaps, or firing a flash into a diffuser) lighting the water either from the front or from behind. Then back down the ambient light exposure to darken the background. That will give you a lot more contrast between the water and the background, and also let you control the way you light the water.