Engine with rotating detonation combustion system

A method for operating an Brayton cycle engine, the method comprising:

• flowing an oxidizer through a gas flowpath defined by a longitudinal wall extending in a lengthwise direction into a combustion section, wherein the flow of oxidizer includes a first portion and a second portion;
• actuating, via an actuator, an inner wall assembly to extend from the longitudinal wall along a depth of the gas flowpath perpendicular to a direction of the oxidizer flowing through the combustion section, the inner wall assembly including an upstream face and defining an adjustable rotating detonation combustion region at the upstream face in the gas flowpath adjacent to the longitudinal wall, the inner wall assembly being extendible into the gas flowpath to adjust a depth of the adjustable rotating detonation combustion region in correspondence with an adjustable depth of the upstream face;
• capturing, at the adjustable rotating combustion region, the first portion of the flow of oxidizer via the inner wall assembly being extended the depth into the gas flowpath by the actuator;
• flowing a first flow of fuel to the first portion of the flow of oxidizer captured at the adjustable rotating detonation combustion region via the inner wall assembly;
• producing, via a first mixture of the first flow of fuel and the first portion of the oxidizer at the adjustable rotating detonation combustion region, a rotating detonation wave of detonation gases;
• flowing at least a portion of the detonation gases downstream of the rotating detonation wave and mixing the detonation gases with the second portion of the flow of oxidizer;
• flowing a second flow of fuel to generate a mixture of the detonation gases and the second portion of the flow of oxidizer;
• and
• burning the mixture of the second flow of fuel, the detonation gases, and the second portion of the flow of oxidizer to produce thrust.