A series of parametric studies were performed to determine the dependence of the vortex shedding frequency on the step height, mean flow speed, fuel type and equivalence ratio. These results were used in Rayleigh's criterion to determine regions of drift and damping of the oscillations. Finally, a velocity-sensitive volumetric source with a time delay is included as feedback to determine the linear stability characteristics of the system.
Chapter 1
Definition of Combustion Instability
In solid-propellant rockets, the response of heat release rate to pressure waves is not as clear, but unstable combustion will occur. In these combustors, unstable combustion can occur due to vortex separation at the separation point that forms the upstream end of the recirculation zone. The jet engine's intake shock will move due to oscillating pressures that occur during instability.
Previous Work
The postulated feedback does not include the natural acoustic modes of the combustion system and the average flow rate is important because entropy fluctuations are related to the flow. These investigations provide the background and a basis for discussing many aspects of the general field of combustion instabilities. It is hypothesized that the phase difference between the oscillating heat release rate and the oscillating pressure in the burner determines the stability of the combustion process.
Chapter 2
The Laboratory Combustor Facility
The mixture is enhanced by flow through a porous cone and several wire screens placed in the plenum section. A steel plate, with many small holes in it, can be placed in the plenum chamber together with steel wool upstream of the plate. A port is provided at the downstream end of the plenum section before the area contraction.
Instrumentation and Flow Visualization
The limitation of the fast Fourier transforms used for spectral analysis in this study is due to sampling rates in the digitization of the data. Most of the data shown in the following chapter represent cases where the finite-time averages were nearly identical; i.e. the signals represent stationary processes. A spark gap was used as a momentary light source for the purpose of freezing the motion of the fluid flow.
Reaction
The importance of the relationship between the mixing length scales and the thickness of the laminar flame zone was recognized by Damkohler [41]. First, mixing of the liquid element with the hot products must occur down to the molecular scale. Thus, the mechanical and diffusional mixing times of a fluid can be of the same order of magnitude as the chemical reaction time.
Chapter 3
Characteristics of the Combustion Instability
These eddies are much larger than those in the shear layer and are emitted at one of the frequencies present in the oscillating pressure signal. Second, the oscillation frequencies are generally close to the natural longitudinal acoustic modes of the system. The vortex shedding frequency is defined in the following section and always occurs at one of the dominant frequencies present in the pressure spectrum.
Vortex Shedding Frequencies
The frequency of the vortex shedding was usually very close to a frequency corresponding to a natural longitudinal acoustic mode of the system. Changes in the temperature or average flow velocity lead to shifts in the natural frequencies. Observing Figure 3.3, if the flame holder is replaced by the double step, the amplitude of the radiation intensity signal at the 231 Hz mode is greater than that of the 188 Hz mode at even very low speeds if the equivalence ratio is close to stoichiometric.
Map 3
Instability Frequencies
- Signals and Spectra: Two Examples
The magnitude of the largest peak in the spectrum of each signal was then taken. The acoustic model predicts the mode shape and the location of the modal nodes quite accurately. Phase changes at different locations in the system depend strongly on the reflection coefficients at the edges of the system.
However, the mode shapes and the locations of the nodes are predicted quite accurately by the model. A heat release rate distribution study was carried out for the two sample conditions using the radiation intensity data in the following manner. The magnitudes of both the average and oscillatory components of the radiation intensity signals at 0.5 inch increments along the combustion chamber were obtained.
This flow field results in the occurrence of the largest fluctuating radiation intensity near the end of the window. Both the vortex ejection and the largest fluctuating radiation intensities in most of the room occur in the 535 Hz mode. The peak in the fluctuating radiation intensity signal occurs approximately six inches downstream of the step.
In the first case, the control of the instability takes place near the area with the largest fluctuating radiation intensity.
EQUIVALENCE RATIO
Acoustic Equations with Sources
First, the normal acoustic modes of the system are specified as the eigenvalues of the acoustic system. In this case, the stability of the system depends on the sign of the imaginary part of the complex frequency. In this case, the effect of the mean flow Mach number can be retained and included in Equation 4.7.
The response amplitude as a function of location in the system can be determined for a given frequency and forced amplitude. The amplitude and phase of the oscillating pressure in the drop plane are shown in Figure 4.2 as a function of the forced unit frequency. The solid line represents the amplitude of the response, and the dotted line represents the phase difference between the response immediately downstream of the step and the forcing.
The frequencies of the peaks generally correspond to the lengths of the separate chambers used in the model. These phase differences have important implications for system stability, and further discussion is included in the next section. The constant F is the proportionality constant between the velocity and amplitude of the volume source.
In the first figure it can be seen that near the natural frequencies of the system the forcing amplitude required for instability is very small.
This is substituted into equation 4.7, neglecting mass sources, and equation 4.11 is multiplied by this expansion. 7 is then multiplied by 1pn and both equations are then subtracted and integrated over the length of the chamber. Using orthogonality relations 4.13 and 4.14, Green's theorem, and boundary conditions 4.10 and 4.12, construct a set of ordinary differential equations for the modal coefficients such that 4.IG), where the first term on the right-hand side represents the drive with heat addition, and the second term represents the drive with non-ideal boundary conditions.
The potential for energy addition to these small pressure perturbations by heat release is much greater than other energy addition mechanisms. In Appendix A it is shown that if the fluctuating heat release rate is in phase with the pressure, energy is added to small perturbations. This can also be shown by solving equation 4 .16 for a particular form of heat release rate.
This solution method provides valuable insight into the time evolution of pressure fluctuations associated with common comb acoustic modes. However, when the combustion of the reactants causes a rather sudden expansion of the gases, a phase change of the rate fluctuations is expected as modeled. This results in changes in the phase difference between p' and u' at different locations in the system and for different oscillation frequencies, as shown in Figure 4.4.
Representing these phase changes at normal states of the system may require many terms in the expansion.
Nonlinear Development of Combustion Instability
Path l represents the case in which oscillations at a fixed frequency near a natural frequency of the system increase and reach a finite limiting amplitude. This pathway would result if the interaction index, F, between the velocity oscillations and the magnitude of the oscillatory heat release rate were a sensitive function of the oscillation amplitude. If the amplitude of the fluctuating heat release rate becomes saturated as the rate fluctuations increase in magnitude, Path l is the result and the endpoint is a stable equilibrium point.
From equation B.10 it is observed that for a fixed oscillation frequency to occur as the amplitude of the pressure oscillations varies, the interaction index, Qn must vary in a particular way with the time delay parameter. Thus, some saturation of the interaction index may occur and contribute to the nonlinear behavior of the system. This model predicts that for the higher frequency modes of the system to be excited, the fast firing required for feedback would require low arnpli-.
The limit cycle behavior observed occurs when driving and damping of the acoustic field are of equal magnitude. Hencricks determined that as the amplitude of the velocity fluctuation is increased, the time from vortex shedding to vortex impact on the lower wall below the staircase is reduced. Thus, as growth in the amplitude of the velocity fluctuation occurs, T decreases and Wr increases for constant F and a.
For amplification of higher frequency modes in the system, this model predicts larger amplitudes, resulting in improved mixing to provide the shorter time delays required for feedback.
Chapter 5
Decreasing the step height also results in the excitation of higher frequency modes. The results indicate that regions of strong attenuation usually occur immediately upstream of the location where the amplitude of the radiation intensity peaks. Similarly, regions of strong attenuation of the acoustic energy usually occur just downstream of this location.
Modeling the feedback mechanism to determine the phase difference between the pressure and heat release oscillations traditionally uses a pressure-sensitive time delay to predict the linear stability characteristics of a given system. A linear stability model based on a velocity-sensitive mass source is used for the geometry of the laboratory cornbustor. The solution method requires that certain values of the time delay and the interaction index determine the amplification rate and frequency of the oscillation.
Large changes in the phase difference between the pressure oscillation and the velocity oscillation occur near the natural frequencies of the system. Finally, possible nonlinear growth mechanisms are described in the context of a rate-sensitive mass source, linear stability theory. These results indicate a non-linear evolution involving an increase in the oscillation frequency and a decrease in the time delay parameter as the oscillation amplitude increases and reaches a final ~ threshold amplitude.
A., “A study of the radiation from laminar and turbulent open propane-air flames as a function of flame area, l~quiv.
