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As far as I know from the working principle of jets engines, compressed air in the combustion chamber (or combustion canister) is mixed with fuel. The ignited mixture expands backwards to turn the turbine rotors and perpetuates the work cycle. Eventually, the left over hot gas creates the thrust.
Although heated gas expands in every direction in space, why does the combustion-gas only expand in one direction, towards the turbine?
It is actually not that simple to ensure the proper gas speed in a gas turbine. In the compressor you want to limit the flow speed over the compressor vanes to the high subsonic range, so the inlet has to decelerate the flow down to approx. Mach 0.4 - 0.5. Less would mean less throughput and consequently less thrust.
This speed, however, is far too high for ignition. The fuel needs some time to mix with the compressed air, and if flow speed is high, your combustion chamber becomes very long and the engine becomes heavier than necessary. Therefore, the cross section leading from the compressor to the combustion chamber is carefully widened to slow down the airflow without separation (see the section below named "diffusor"). Around the fuel injectors you will find the lowest gas speed in the whole engine. Now the combustion heats the gas up, and makes it expand. The highest pressure in the whole engine is right at the last compressor stage - from there on pressure only drops the farther you progress. This ensures that no backflow into the compressor is possible. However, when the compressor stalls (this is quite like a wing stalling - the compressor vanes are little wings and have the same limitations), it cannot maintain the high pressure and you get reverse flow. This is called a surge.
The graph below shows typical values of flow speed, temperature and pressure in a jet engine. Getting these right is the task of the engine designer.
Plot of engine flow parameters over the length of a turbojet (picture taken from this publication)
The rear part of the engine must block the flow of the expanding gas less than the forward part to make sure it continues to flow in the right direction. By keeping the cross section of the combustor constant, the engine designer ensures that the expanding gas will accelerate, converting thermal energy to kinetic energy, without losing its pressure (the small pressure drop in the combustor is caused by friction and the Rayleigh effect). Now the accelerated flow hits the turbine, and the pressure of the gas drops in each of its stages, which again makes sure that no backflow occurs. The turbine has to take as much energy from the flow as needed to run the compressor and the attached pumps and generators without blocking the flow too much.
The remaining pressure is again converted to speed in the nozzle. Now the gas is still much hotter than ambient air, and even though the flow at the end of the nozzle is still subsonic in modern airliner engines, the actual flow speed is much higher than the flight speed. The speed difference between flight speed and the exit speed of the gas in the nozzle is what produces thrust.
Fighter engines usually have supersonic flow at the end of the nozzle, which requires careful shaping and adjustment of the nozzle contour. Read all about it here.
The air in the compressor is both compressed and moving downstream towards the combustion section. The combustion does not create enough pressure to overcome all of that, and there is lower pressure as the air gets expanded through the turbine sections.
When the pressure in the compressor section drops too much, the combustion flames do expand in both directions. This is called a "compressor surge".
Disclaimer: I may have spent several hours on Wikipedia at one point trying to answer this question for myself!
Jet engines use the Brayton cycle, which is a "isobaric" process during combustion, meaning it keeps pressure constant during that phase. This is in contrast to the Otto cycle of a typical four-stroke piston engine, which is "isochoric" during combustion, meaning it keeps volume constant during that phase.
The Brayton cycle consists of 3 parts, of which combustion occurs in the middle
So how does this "constant pressure" thing work? Treat the combustion chamber as a sort of box for a moment. Whether there is combustion or not, there's going to be generally constant pressure inside the box. Air is being pushed into it by the compressor, with some velocity and pressure. If the turbine at the end of the chamber can "dispose" of air at a high enough velocity, it can keep the pressure at the far end of the chamber equal to that of the front end.
So how does this "constant pressure" thing actually keep the flame front from advancing forward? The trick is that the flame front is trying to advance forward, but the velocity of the air through the chamber matches the speed of the flame front, keeping it in a constant location in the chamber. This is a dynamic process, so we'll need some dynamics. They key detail is that the turbine and compressor are on one axle, so what happens to one affects the other.
Consider three cases which compare the airflow velocity to the fuel:
One takeaway from this pattern is why jet engines cannot change thrust quickly. If you rapidly add fuel by throttling up, you choke the engine, so you don't get much extra power until the turbine and compressor can spin up to equalize the engine.
I happened upon this question, and thought I could add some information, as I recognized the powerplant right away.
This is not another full answer, but only a response to @albin and @PeterKampf as to the engine model. I cannot yet add comments, so edit this as is necessary.
The engine picture you used in your answer is the power section of the Allison 501 series, which had several civil and military aircraft applications. The reduction gearbox and propeller was omitted from that picture (for whatever reason). That omission is confusing, please see my image below for the whole picture of what is happening in that powerplant. Usually an Aeroproducts or Hamilton-Standard propeller was fitted.
@PeterKampf, you were on the right track - there is little energy left over for jet thrust after the turbine has extracted it because most of the power needs to drive the RGB and propeller. Very typical of a turboprop (or turboshaft, for helicopters) design - residual jet thrust is of little use. In this case, given certain atmospheric conditions and at on-speed RPM (13,820), the turbine can extract roughly 10,000 horsepower from the gas stream. It takes ~6000 HP to drive the compressor, which leaves about 4000 shaft HP for the RGB and prop.
The Convair 580 uses 2 of these powerplants, The Lockheed C-130 Hercules uses 4.
If this were a turbojet design, the engine would likely have only 2 turbine stages instead of 4. This would extract only enough energy to drive the compressor (and thus self-sustaining the engine), with plenty of thrust left over for use.
