Self-excited oscillations of a confined flame, burning in
the wake of a bluff-body flame-holder, are considered. These
oscillations occur due to interaction between unsteady
combustion and acoustic waves. According to linear theory, flow disturbances
grow
exponentially with time. A theory for nonlinear oscillations
is developed, exploiting the
fact that the main nonlinearity is in the heat release rate,
which essentially ‘saturates’.
The amplitudes of the pressure fluctuations are sufficiently
small that the acoustic waves remain linear. The time evolution of the
oscillations
is determined by numerical
integration and inclusion of nonlinear effects is found to lead to
limit cycles of finite
amplitude. The predicted limit cycles are compared with results from experiments
and from linear theory. The amplitudes and spectra of the limit-cycle oscillations
are in reasonable agreement with experiment. Linear theory is found to
predict the
frequency and mode shape of the nonlinear oscillations remarkably well.
Moreover,
we find that, for this type of nonlinearity, describing function
analysis enables a good
estimate of the limit-cycle amplitude to be obtained from linear theory.Active control has been successfully applied to eliminate these oscillations.
We
demonstrate the same effect by adding a feedback control system to our
nonlinear
model. This theory is used to explain why any linear controller capable
of stabilizing the linear flow disturbances is also able to stabilize
finite-amplitude oscillations in the nonlinear limit cycles.
Finite amplitude electron-acoustic waves in the electron diffusion region