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Are any jet engines used in combat aircraft water cooled?

If they are not, and instead are air cooled, how can the engine ensure consistent cooling? Wouldn't the airflow passing through whatever radiator implementation it has be greatly affected by the airspeed? Consider low subsonic numbers with heavy maneuvering in a dogfight, vs. mach 2 sustained flight?

While I can imagine that variable air inlet for the radiator would change the airflow, I really have no idea how this would work.

I'm also aware that certain rocket engines, e.g. the F-1 allow the fuel itself to circle around the thrust chamber to cool it. Perhaps a similar system is used in jet engines.

Jet engines are cooled and temperature managed by the mass air flow itself which is used to surround and limit the boundaries of the flame, keeping it away from the burner can walls, and dilute the heat of the flame as it passes to the turbine (only a minority of the mass airflow's oxygen is actually burned - which is why afterburners work; there's lots of leftover oxygen to work with).

If you're talking about using a closed liquid cooling system with heat exchangers taking heat from the burner can and turbine somehow, well, I've never heard of a water jacketed burner can, and how you could use liquid in a closed system to cool a turbine, who knows.

Liquid cooling was used in the past as supplemental cooling to increase available power by spraying water directly into the mass flow as mentioned here, but in that case the water is a consumable.

The closest I could find for closed-cyle fluid cooling was a 1975 NASA study for supersonic aircraft, but this used no radiators, and wasn't for the engine. Instead, circulating refrigerant moved heat from the wing and fuselage to the fuel tank containing liquid hydrogen, in the hope that titanium skins could be replaced with cheaper aluminum. The conclusion was that this would help only at airspeeds much greater than Mach 3.

Even for piston-powered aircraft (and NASCAR automobiles for that matter), the drag due to airflow through a radiator is enormous. Some 1930's racing aircraft used radiators inset flush to the wings for this reason.

It may just be that the jet engine's mass airflow suffices, so there's no justification for the extra complexity and weight of a dedicated heat exchanger.

The thing most in need of cooling inside a jet engine is usually the high pressure turbine blades. The air entering the first stage of the turbine is in the neighborhood of 2500F (https://en.wikipedia.org/wiki/Turbine_blade). This is often higher than the melting temperature of the blades themselves. These blades are cooled using air bled off from the high pressure compressor, which is typically at a "cool" temperature of 1000F or so. Water would not be much help in cooling here, as it would instantly vaporize.