## Combustion stabilization by forced oscillations in a duct.(English)Zbl 1087.80004

The author develops a model and mathematical technique in order to simulate an active control input-output mechanism. The developed model describes the interaction of an existing oscillation in the combustor with a control input and with a flame. Such a model and mathematical technique significantly differ from that used by previous authors for single oscillation/flame interactions. The model is used to demonstrate analytically the feasibility of stabilizing premixed combustion by forced oscillation. The author assumes that the oscillations at frequency $$\omega$$ generated by the loudspeaker are imposed on autonomous oscillations at frequency $$\omega_0$$ in a duct. The problem of the interaction of two oscillations can be solved by taking into account the following four facts:
(1) There is a flow in the combustor when the control input enters. Therefore, the corresponding velocity and pressure fields should be used as initial conditions.
(2) The result of interaction of the two different oscillations is complex, with unknown amplitude depending on time and location.
(3) The resulting oscillation should be found by the conjugation of two wave equation solutions obtained separately for each part of combustor (i.e., the fresh and burnt gases).
(4) While the wave equation is a second-order differential operator, there is only one boundary condition in each part of the combustor divided by the flame. Hence, two other boundary conditions are needed in the duct.
For such additional conditions, the author introduces two unknown functions in order to define the pressure and velocity amplitudes at the flame. The stability analysis is reduced to a system of two integro-differential equations that determine these unknown functions. Analysis shows that the stability domains of the time lag mainly depends on the flame location and the fresh/burnt gases temperature ratio. Numerical results are obtained for a centrally located flame and for the temperature ratio 1500K/300K. The developed mathematical technique may be applied to different combustion models, configurations, boundary and initial conditions.

### MSC:

 80A25 Combustion 93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) 93C73 Perturbations in control/observation systems 44A10 Laplace transform 45J05 Integro-ordinary differential equations
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