A spectacular instability has been observed in low density round jets when the density ratio of jet fluid to ambient fluid falls below a threshold of approximately 0.6. This phenomenon has been observed in non-buoyant jets of helium in air, heated air jets and heated buoyant jets. The oscillation of the flow near the nozzle is extremely regular and periodic and consists of ring vortices. Even the smaller scale structures that appear downstream exhibit similar regularity. A theory for predicting the onset of this oscillation is based on finding regions of absolute instability from linear stability analysis of parallel flow. However, experiments suggest that the theory is at least incomplete and fortuitous as the oscillation is not a linear process. The present work is to observe and understand the process of regeneration of these oscillations by conducting numerical simulations. Here, two-dimensional, plane jets were simulated because they undergo a qualitatively similar process. A spatial and temporal picture of a heated jet has been obtained numerically. A perturbation expansion was used to obtain a system of conservation laws for compressible flows which is valid for low Mach numbers. The low Mach number approximation removes the high frequency acoustic waves from the flow field. This enables a larger time step to be taken without making the calculation unstable. To ensure that all the scales of motion are properly resolved, calculations were done at a low Reynolds number. The governing equations were discretized in space using second-order finite difference formulas on a staggered grid. Velocity fields were advanced using a second-order Adams-Bashforth explicit scheme and then corrected by solving for pressure such that continuity is satisfied at every time step. The Poisson problem for pressure requires the time derivative of the density which was approximated by a third-order backward difference formula. Gauss-Siedel iteration was used to find the pressure. Several numerical tests were conducted prior to simulations of variable density jets to check the stability and accuracy of the code. Two dimensional driven cavity flow calculations were done as a first test. Then a calculation of a forced, spatially developing, incompressible, plane mixing layer was done to check the time accuracy of the code. After obtaining satisfactory performance of the code for the different test cases, two-dimensional, variable density jets were simulated. Since the plane jet extends ad infinitum in the streamwise direction, a sufficiently large domain was used to capture all the relevant physics in the downstream regions of the jet. An advective boundary condition was imposed at the exit plane. Rigid, slipwall conditions were employed to prescribe lateral boundary conditions. A 2-D, incompressible plane jet was simulated first. The jet profile was approximated by two hyperbolic tangent shear layers. The most unstable mode of the inviscid shear layer for this profile, along with its first and second harmonics, was imposed on the velocity profile at the inlet plane. The amplitude of oscillation of the harmonics was chosen so as to provide sufficient energy in the perturbation to accelerate the growth of the layer. No explicit phase lag was introduced in the perturbation. The flow was allowed to develop long enough to wash out the effect of the initial condition. The results obtained for this case indicate that experimentally realized phenomena such as vortex pairing were captured in this simulation. Furthermore, to check the convective nature of instability of the incompressible jet, the forcing at the inlet plane was turned off. The disturbances were gradually convected downstream, out of the computational domain. Next, two-dimensional heated, non-buoyant jets were studied numerically. The effects of the ratio of jet density to ambient density S, the velocity ratio R, and jet width W, on the near field behavior of an initial laminar jet and the regeneration mechanism of the self-sustaining vortices were explored. The theory based on domain of absolute/convective instability identifies these three parameters. No initial perturbation was necessary to start roll-up of the shear layer. For certain choices, e.g., S= 0.75, R = 20, W =10.5, self-sustaining oscillations appeared spontaneously, and these cycles repeated for very long simulation intervals. Waviness on the jet shear layers grow and roll-up into vortices as in constant density shear layers. But unlike the incompressible plane jet, these vortices grow much larger and mixes more with the surrounding fluid. As these vortices evolve, packets of fluid break away as trailing legs similar to side jet expulsions observed in round jets and plumes. The growing vortices disturb the upstream shear layer. Consistently with linear theory, which predicts absolute instability for these parameters, these disturbances are able to grow and roll up. If these disturbances travelled faster than the downstream vortices, it would not be possible for the cycle to repeat. With sufficient shear between the co-flowing streams (R not too small), the entire regeneration process was found to begin from roughly the same streamwise location. Furthermore, it is the symmetric, varicose mode which occurs. At a slightly larger density ratio (S = 0.8, R = 10), self-sustaining oscillations appeared, but each new cycle began slightly farther downstream. It seems likely that these values are close to the boundary in parameter space between self-sustained oscillatory and convectively unstable behaviors. Jet width also influences the selection of these two behaviors. When jet width was reduced, W = 6, even for S = 0.75,R = 20, each new cycle began to shift downstream. For larger jet width (W = 12.3), self-sustaining oscillations occur but the response is now as an asymmetric sinuous mode after a short initial varicose mode. The detailed processes that have now been revealed in plane jets should serve as guidelines for the study of such processes in the technologically more important round jets.

This work represents the initial steps in a wider project that aims to map out the sensitive areas in fuel injectors and combustion chambers. Direct numerical simulation (DNS) using a Low-Mach-number formulation of the Navier–Stokes equations is used to calculate direct-linear and adjoint global modes for axisymmetric low-density jets and lifted jet diffusion flames. The adjoint global modes provide a map of the most sensitive locations to open-loop external forcing and heating. For the jet flows considered here, the most sensitive region is at the inlet of the domain. The sensitivity of the global-mode eigenvalues to force feedback and to heat and drag from a hot-wire is found using a general structural sensitivity framework. Force feedback can occur from a sensor-actuator in the flow or as a mechanism that drives global instability. For the lifted flames, the most sensitive areas lie between the inlet and flame base. In this region the jet is absolutely unstable, but the close proximity of the flame suppresses the global instability seen in the non-reacting case. The lifted flame is therefore particularly sensitive to outside disturbances in the non-reacting zone. The DNS results are compared to a local analysis. The most absolutely unstable region for all the flows considered is at the inlet, with the wavemaker slightly downstream of the inlet. For lifted flames, the region of largest sensitivity to force feedback is near to the location of the wavemaker, but for the non-reacting jet this region is downstream of the wavemaker and outside of the pocket of absolute instability near the inlet. Analysing the sensitivity of reacting and non-reacting variable-density shear flows using the low-Mach-number approximation has up until now not been done. By including reaction, a large forward step has been taken in applying these techniques to real fuel injectors.