Stability analysis of the Rijke tube with a Greenʹs function approach

An analysis is made of the stability of a Rijke tube. The tube is open at both ends and contains an acoustically compact flame holder that “blocks” the acoustic motions and across which there is a jump in the tube cross-sectional area. Oscillations are described in terms of an acoustic Greenʹs function obtained in analytic form. The blocked motion near the flame holder can be regarded as incompressible; on either side of the flame holder full acoustic wave propagation is assumed. Velocity potentials of the incompressible and acoustic regions are matched by requiring continuity of pressure and volume flow. A linear heat release model is introduced that relates heat transfer from the flame to the acoustic field and provides the acoustic feedback necessary to maintain the oscillations. The oscillations can then be described in terms of the eigenmodes of an integral equation derived using the Greenʹs function. Growth rates predicted from this equation are expressed in terms of properties of the heat release model.