Linear aerospike engines are an old idea that seem so full of promise. Why are they not widely used today by the likes of Boeing, SpaceX, etc.?
Aerospikes are notoriously difficult to cool efficiently.
With a bell nozzle, you have a minor part of rapidly expanding(+cooling) exhaust touching the broad, actively cooled nozzle - that means little conductive heat transfer, lower temperature gradient, lots of area for coolant plumbing on the outside (or within) the bell, and outer area radiating a lot of heat out (or passing it to air while in atmosphere) besides coolant drawing it away.
In aerospike the pressure (and temperature) of the gas remains very high all along the spike surface, and the sharp tip leaves very little room for cooling systems. You have a lot of extra-hot, very dense gas in contact with the narrow spike that must pass all the coolant and dissipate the heat somehow, not to melt.
That means short test runs of aerospike engines, proving all the benefits, are perfectly viable - but the experimental rigs are shut down before they could overheat and suffer critical damage. They just couldn't run continuously for as long as a typical rocket needs to lift payload to orbit. Work on efficient, fault-proof cooling of the engines is ongoing, but it's not nearly as easy as in case of bell nozzles - a major challenge that seriously throws a wrench into broad adoption of this engine type.
Money. Most engine designs we're now using are an evolution of the space race era in one form or another, from the times when financing research in rocket engine / nozzle design wasn't such an issue. Linear spike fundamentally changes rocket design, for one engine support structure, and would as such require a revolution in engineering if someone expects it on the cheap. Steady evolution simply isn't good enough, the changes required are too big.
So either rocket engine research & development would get a huge injection of money from somewhere, or we reach some engineering breakthrough that doesn't require so much of it, like maybe more reliable engineering simulators (one fine example, but not enough). Short of that, designing, building and testing cycles are simply too expensive. Engineering side of it is challenging, but no engineer will ever tell you it's impossible.
The answer is simple
lack of proven reliability
lack of proven flight experience
lack of performance validation..
- more surface area exposed to hot gas . So it must be cooled which is difficult ..
Having worked at Rocketdyne for many years my understanding was that although they wanted to put it on the Space Shuttle, the politics of a single source caused NASA to write a proposal that required usual rocket nozzles.
At that point there was no funding to finish full scale development and flight testing.
Arcaspace have been bulding a hydrogen peroxide test engine they hope to fire later in 2018, with the stated aim of flying a RP1/H2O2 eSSTO. https://newatlas.com/arc-aerospike-linear-engine-complete/51431/
Firefly Aerospace will use an (annular) aerospike engine with their Alpha rocket. They have a planned launch in 2018.
This answer is supplemental to the other excellent answers here.
This Curious Droid video also confirms it's just a question of time and money. The first few words "History is full of decisions..." Due to the details of development history, bell shaped nozzles made it first, and people stuck with it.
This may be slightly reminiscent to the current non-use of Thorium based nuclear power, though that is also related to the need for nuclear weapons development.