I was reading something about someone who was doing this and I'm intrigued, but I have no idea what you actually do to optimize the engine to be better at lower altitudes and lower speeds?
As a summary, the engine generates thrust by changing the impulse (momentum) of the air. Please refer to this: NASA Genera Thrust Equation
So to optimize aircraft performance, the main options you can play with are that depend on your aircraft operation: V0: flight speed p0, r0: inlet pressure and density (function of altitude)
And the variables that depend on your engine design: Ve, pe, re: exhaust velocity pressure and density (depend on the thermal cycle and actual operating condition of the engine) A0, Ae: Inlet and nozzle areas (engine geometry, sizes derived from the design point)
Now what you need to see is that there is no single best option, aircraft design is an optimization problem, eg finding compromises. Flying lower and slower is actually an attractive option for greener aircraft, as it enables technologies that reduce drag, such as ultra high aspect ratio wings, laminar airfoils, boundary layer ingestion, electric propulsion and a few more.
Whether optimizing a turbofan for lower altitudes make sense can only be seen by comparing the wins against the losses, and it is hard to say absolute truths as it depends on many variables.
Now about the actual "how-to" optimize a gas turbine, essentially the key variable of the engine is the pressure ratio, PR, so how much it compresses the inlet air. For a given flight condition (thrust, speed and altitude) , there is an optimal pressure ratio. From that the number of compressor and turbine stages can be calculated (also if you need a multi-spool engine, do you use an intercooler, etc.), and also the geometrical dimensions of the engine. And this goes back to the comment I've made that it is a compromise. Building a very heavy, expensive, hard to maintain engine will not be a good choice as what you win by increased engine efficiency you will lose on the flight efficiency.
But suppose as an academic exercise you just want to alter an existing engine design to move its design point to a lower speed and altitude, then you need to change the design pressure ratio, and follow up with any internal changes (add/remove stages, change inlet, nozzle, etc.)
The real selling point for turbojets was the ability to fly "above" the weather at much faster speeds.
This was so attractive to airline companies that they literally stampeded right past turbo props to get to the fastest subsonic planform/engine combination available, the Boeing 707. Convair even tried to go one better, right to the edge of transsonic flight to say they were the fastest. Very inexpensive and plentiful fuel helped set the stage for this competition.
Today, with fuel costs weighing heavily on profit margins, there is the incentive to improve efficiency with slower turbo props, and indeed, routes up to 1000 miles would require only slightly longer airtime. 400 mph is still fairly rapid transportation.
Turboprops can also go low, but are much more efficient high as ground speed increases relative to airspeed with height. At 35,000 feet 250 mph indicated airspeed is around 590 mph groundspeed. 200 mph indicated gets you around 415 mph groundspeed.
Drag is much less at altitude for a given groundspeed. So low is out. Slow may be better, especially for freight.
EDITED TO RESPOND TO COMMENTS:
Look at the question: "lower and slower instead of higher and faster".
- Lower or higher altitude?
ANY AIRCRAFT benefits from flying higher as the same IAS gives a higher TAS or groundspeed. After climb fuel burn is won back, flying higher generates better miles per gallon fuel consumption. So going lower defeats the purpose of greater fuel efficiency.
- Faster or Slower IAS?
There is a significant gain in fuel economy by slowing down (IAS). However, schedules must be met, so how to slow down and be on time? The answer is to fly higher and slower (IAS), not lower and slower! As long as Mach limit is not broken, high and slow (IAS) is the way to go. It may be of interest to design a plane that flies the entire flight at Vy!
So you fly your jet at its most efficient IAS, which makes your TAS greater as you go higher. This is the best way to optimize it.