e.g. a parabola facing down gives max lift contribution & a backwards S on its side is in the middle & and a parabola facing up accelerating down draft.

This paper gives some sense of research newness, describing a 2016 NASA design that shaped the output exhaust of only the tail engine on a commercial jet. Computing the overall effect & control was described as challenging.

Edit: I am no expert, and I only took a quick glance at the paper…that said, it looked like the NASA design creates drag by curving the exhaust after its point of maximum acceleration. If that is true, it only adds energy efficiency in the sense of perhaps being a less costly way of steering up compared to wing flaps. It’s not equivalent to accelerating the exhaust in the desired direction.