For car engines, the cylinder volume is often associated with the engine power, which suggests scaling of the power as $L^3$ where L is the linear size. Consider a system consisting of a motor and its energy supply, e.g., an internal combustion engine with a fuel tank, or an electric motor with a battery, or a steam turbine with a nuclear reactor etc. Now we scale the system geometrically, up or down by some factor, using same materials, and assume that in the scaled system nothing breaks or burns through, so it can function. What should we expect for the scaling of the output mechanical power? Would it be close to $L^3$? Can an example be constructed where the scaling is very different from $L^3$?
[Physics] scaling of motor power
electricityheat-enginethermodynamics
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What you are talking about is called a combined cycle engine. They are commonplace in stationary power generation, i.e. utility-scale electricity generation. There has even been some talk of combined cycle engines in cars.
As pointed out in the answer by dmckee, the reason this hasn't been widely applied in cars is that no one has demonstrated an economically competitive combined-cycle car. I promise you, if such a thing can pay for itself in gas savings then it will eventually be built and sold, unless some better technology makes it irrelevant.
In general there are many reasonable ideas that are physically permissible but economically or technically difficult or nonviable. You are effectively suggesting to add a steam engine to a car, which is quite a difficult proposal. I'd suggest that a hybrid gas-electric car is more economical than what you suggest, and even they have had a hard time catching on. In electric power generation it matters much less that the combined cycle engine has a larger sunk cost than a normal engine, is heavier, etc., so the economic balance works out.
Bringing the question back to physics, no matter what you use for heat scavenging, your engine including all of its "subengines" cannot exceed the Carnot efficiency corresponding to the largest temperature difference in the engine. Adding additional heat engines will help to approach the Carnot limit. In order to beat Carnot, you can't use heat as an intermediate step between chemical energy (fuel) and mechanical work.
The key is the combustion of fuel in the combustor. This adds energy to the flow so there is plenty available for the turbine to drive the compressor.
Depending on flight speed, the intake does already a fair amount of compression by decelerating the flow to Mach 0.4 - 0.5. More would mean supersonic speeds at the compressor blades, and the intake ensures a steady supply of air at just the right speed.
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 in the diagram 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). 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 is needed to run the compressor and the engine accessories (mostly pumps and generators) without blocking the flow too much. Without the heating, the speed of the gas would drop to zero in the turbine, but the heated and accelerated gas has plenty of energy to run the turbine and exit it at close to ambient pressure, but with much more speed than the flight speed, so a net thrust is generated.
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 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.
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This is really a comment not an answer, but it got a bit long for the comment box.
The problem with your question is that the power of the engine is related to the amount of fuel/air mixture that burns with each cycle, and the frequency (i.e. engine speed). There is clearly some connection between the amount of fuel/air mixture burned per cycle and the size of the cylinder it's burning in, but the efficiency of cylinder filling is affected by so many other factors that we rarely see any but the most basic correlation.
For example, most family cars come in a range of types from those aimed at grannies to those aimed at excessively enthusiastic teenagers and over this range you'll find that engines of the same size can produce very different powers. Typically this is because the valve timing and lift differs between the different engines and hence the cylinder filling efficiency varies. You might wonder why car manufacturers don't always aim at maximum filling efficiency - it's because if you maximise filling efficiency you tend to find you get a high efficiency at only a narrow range of engine speeds, which is great for the racetrack but not much use driving to supermarket.
If you take some example engine cylinder of capacity $C$ then the fuel/air mixture that fills it has to flow in through the inlet valves, and because air has a non-zero viscosity this takes time. If you now double the cylinder size (i.e. raise its volume to $8C$) but don't change the valve size the air/fuel flow rate wouldn't be adequate to fill the cylinder in the time available between cycles. So you double the valve size as well, but of course this only increases the valve area, $A$, by a factor of four to $4A$, and the ratio of valve area ($\approx$ flow rate) to cylinder volume has fallen by a factor of two. This means power won't scale with $L^3$. In practice engine manufacturers combine increased cylinder volume with changes to the valve design, and not simply size. Even so it's hard to maintain efficiency over a large range of cylinder sizes, which is why large engines tend to have more cylinders rather than the same number of bigger cylinders.