With recent improvements in computer architecture, there are renewed interests in studying combustion instability in rocket engine via Computational Fluid Dynamics (CFD) simulations. Most existing numerical simulations are computationally expensive, with each simulation taking weeks to months to perform. The first objective of this dissertation is to develop a computationally inexpensive numerical solver. The solver, written in an axisymmetric formulation, is used to study longitudinal combustion instability in a single-injector rocket engine. For different stability domains, the results generated by the solver show good agreements with both experimental results and expensive 3D simulations. Compared to other axisymmetric solvers, the developed solver is at least an order of magnitude faster while using a more detailed chemical mechanism. Because of the canonical configuration of the rocket engine, fundamental understandings of the self-excited instability mechanisms under various stability domains are identified. Increasing the pressure oscillation leads to increase in mixing; making the flame inside the combustion chamber more compact. Pressure-Heat Release Rate coupling location plays a crucial role in driving the instability.

Subsequently, the flame dynamics under various oscillatory conditions are examined. Significant local extinctions and reignitions occur in the unstable cases compared to the stable cases. The complex chemical mechanism is proven to play an essential role in predicting the correct oscillation amplitude. Partially premixed flames are found in the combustion chamber.

In the last chapter, for an unstable oscillation, wall heat loss and chamber modifications was found to provide stabilizing effects. The stabilized chamber is then triggered using different types of disturbances with various durations. In some cases, a limit cycle behavior with much larger pressure oscillations than the initial conditions is observed. This triggered instability of the longitudinal mode in this dissertation is a novel contribution to the field.

The temporal evolution of three-dimensional instabilities on an incompressible planar liquid sheet segment is studied using direct numerical simulation (DNS), and the level-set and volume-of-fluid methods for the liquid-gas interface tracking. The purpose of this study is to reveal new crucial insights into the development of three-dimensional instabilities on liquid sheets, which result in formation of lobes, bridges and ligaments, and eventually break into droplets. Three atomization cascades are distinguished at early breakup, which are well categorized on a gas Weber number (We_g) versus liquid Reynolds number (Re_l) map. Each atomization domain has a distinct breakup mechanism with its own characteristic time and length scales in the cascade process. λ₂ method is used to identify the vortices near the liquid-gas interface and relate the vortex interactions to the surface dynamics at different stages of the jet breakup - namely, lobe formation, lobe perforation, ligament formation, stretching and tearing. Vortex dynamics explains the hairpin formation, and the interaction between the hairpins and the Kelvin-Helmholtz (KH) roller explains the perforation of the lobes at high We_g, the formation of corrugations on the lobe front edge at high Re_l, and the stretching of lobes into ligaments at low Re_l and low We_g. Streamwise vorticity generation - resulting in three-dimensional instabilities - is mainly caused by vortex stretching and baroclinic torque at high and low density ratios, respectively. Generation of streamwise vortices and their interaction with spanwise vortices produce the liquid structures seen at various flow conditions, understanding of which is crucial for controlling the droplet-size distribution and jet spread rate in primary atomization. Probability density functions (PDFs) of the local radius of curvature and the local cross-flow displacement of the liquid-gas interface are evaluated over wide ranges of Re_l, We_g, gas-to-liquid density ratio and viscosity ratio, and wavelength-to-sheet-thickness ratio. A novel analysis enables us to show the temporal cascade of liquid-structure length scales as well as quantify the spread rate of the liquid jet during primary atomization. The spray angle and the mean surface length scale are evaluated from the PDFs and are compared for wide ranges of dimensionless parameters. The validity and usefulness of the temporal analysis is established by a comparison with dynamics of a spatially developing liquid jet with slower coaxial gas flow. It is shown that the deformations in the upstream region of the jet cap follow a periodic behavior that can be well portrayed in a frame moving with the convective velocity of the liquid jet, as is implemented in our temporal model.

The Soave-Redlich-Kwong cubic equation of state (EoS) was used in real-gas investigations for a mixture of species. An algorithm for a direct solution using a modified version of the Cardano’s method is described in order to solve the cubic EoS more efficiently compared to previous methods. Isentropic, choked flow through a convergent-divergent nozzle was investigated using classical mixing rules and compared to flow with the ideal-gas law. Two novel methods were described to solve for the downstream conditions of a normal shock and the results compared to ideal-gas solutions. A computer script was written to solve the cubic EoS. A wide range of stagnation pressure and stagnation temperatures were investigated to analyze the effects on the real-gas solutions. Upon tabulation of results, it was concluded that most solutions with lower stagnation temperatures and higher stagnation pressures are furthest in value to ideal-gas solutions. However, the range of agreement is dependent upon the species present as well as the concentration of those species in a gas.

A numerical study of the temporal and spatial instabilities appearing on the interface of an annular liquid jet emerging from an orifice and flowing into a high pressure gas medium has been performed using Direct Numerical Simulation. The purpose of this study is to gain a better insight into the dominant mechanisms in the atomization of annular liquid jets during the start-up portion of the injection. The effects on the growth rate and wavelength of the emerging Kelvin-Helmholtz and Rayleigh-Taylor instabilities of various flow parameters have been investigated: the Reynolds and Weber numbers; fluids properties like gas-to-liquid density and viscosity ratios; and geometrical parameters involved in the problem such as thickness-to-diameter ratio of the liquid sheet. The Reynolds numbers used in this study are in the range from 3,000 to 30,000, and the Weber numbers are in the range of 6,000 up to 150,000. The convergence rate and length of the liquid jet has been also computed and compared for different cases. A characteristic convergence time has been proposed based on the obtained results. Use has been made of an unsteady axisymmetric code with a finite-volume solver of the Navier-Stokes equations for liquid streams and adjacent gas and a level-set method for the liquid/gas interface tracking.

Two significant velocity reversals were detected on the axis of symmetry for all flow Reynolds numbers; the one closer to the nozzle exit being attributed to the recirculation zone, and the one farther downstream corresponding to the annular jet collapse on the centerline. The effects of different flow parameters on the location of these velocity reversals are studied. The results indicate that the convergence length and time increase significantly with the gas density and liquid viscosity and decrease with the liquid sheet thickness, while the effects of the gas viscosity and the surface tension are not so considerable.

The range of unstable Kelvin-Helmholtz and Rayleigh-Taylor wavelengths have been also studied. The statistical data obtained from the numerical results show that, the average normalized wavelength of the KH instabilities decreases with the Reynolds and Weber numbers and the sheet thickness, and increases with the gas-to-liquid density ratio, and is independent of the viscosity ratio. The wavelength of the KH instabilities were observed to increase in time, except for the very thin liquid sheet, where the average KH wavelength oscillates between two values, indicating occurrence of different sheet breakup cycles. The sheet breakup times and lengths were reported up to the second sheet breakup, and it is shown that the later sheet breakups happen closer to the nozzle exit plane. The RT wavelengths tend to decrease during the start-up period of injection.

This thesis presents a numerical study of the heat and mass transfer occurring when a cool liquid is suddenly introduced to a hotter gas at supercritical pressures. Different binary mixtures of heavy hydrocarbons and light gases are considered and fluid properties are obtained using a real-gas equation of state and various high-pressure models. Liquid-gas interface dynamics and mass and thermal diffusion for different pressures are analyzed. Then, a comparison with expected growth rates of the Kelvin-Helmholtz (KH) instability is provided to learn whether a phase equilibrium is well established before hydrodynamic instabilities can become important.

Two phases may still appear at supercritical pressures because mixture critical properties differ considerably from pure species critical properties. The diffusion time scales in both phases are comparable to the KH instability transient (i.e., 20-100 microseconds). That is, diffusion layers of 10 microns thickness in the liquid and 30 microns in the gas are observed for the oxygen/n-decane mixture, suggesting that variations of fluid properties around the liquid-gas interface may affect breakup mechanisms. Furthermore, condensation by increasing pressure can be observed, even at supercritical pressures, while the First and Second Law of thermodynamics are obeyed.

Combustion at elevated pressure brings advantages in terms of efficiency and emissions. In many different devices a liquid reactant is injected into a high pressure chamber where the reaction occurs. Common applications are diesel, jet, and rocket engines. Frequently, the thermodynamic conditions of the ambient fluid in the combustion chamber are supercritical for both fuel and oxidizer. However, the injected fluid is usually at subcritical temperature. Consequently, two phases exist until the temperature of the liquid rises to its critical value by means of heat transfer. This process is known as transcritical vaporization. Important and challenging problems associated with this scenario have been studied with increasing interest during the past years. The main aspects of study include transcritical single-droplet and spray vaporization, mixing, combustion, phase equilibrium, and transport.

Despite an extensive and growing list of publications, there are still unanswered questions. For example, the presence of sharp gradients in density or composition are not well understood under conditions that one would expect to be supercritical. The possible existence of a heat of vaporization and surface tension in the supercritical regime has been highlighted by some researchers. Understanding the physics governing these phenomena is crucial to control combustion chambers, as well as to provide an opportunity for improving their design.

Other relevant applications for transcritical combustion are the burning of gas hydrates and direct water injection. It is well known that large quantities of methane are stored in form of gas hydrates in the ocean at a depth of the order of 1 km. The possibility of burning these materials locally to extract energy has been suggested. The conditions in which combustion would occur with so much water in the environment and at high pressure are also not well understood.

A one-dimensional counterflow diffusion flame model is used in this dissertation together with real-fluid thermodynamics and laws of phase equilibrium. The presented solutions provide insights that are relevant to the scenarios posed above.

The injection of liquid fuel at supercritical pressures is a necessary but overlooked topic in combustion. Under the premise that the liquid will swiftly transition to a supercritical state, the role of two-phase dynamics is typically ignored. Nevertheless, a transcritical domain exists for which a sharp interface prevails before substantial heating occurs. This transcrit- ical scenario may persist, for example, during the early deformation of liquid hydrocarbon fuels injected into high-pressure combustion chambers. Local thermodynamic phase equi- librium (LTE) enhances the dissolution of the oxidizing species into the liquid phase, and vaporization or condensation may occur simultaneously in several regions along with the in- terface. Following species and thermal mixing, fluid properties vary greatly, with liquid- and gas-mixture properties being more comparable near the interface. Small surface instabilities emerge early due to the combination of low, variable surface-tension force and gas-like liquid viscosities.

The mixing process, interface thermodynamics, and early deformation of a cool liquid n- decane jet submerged into a hotter moving gas initially composed of pure oxygen are studied under different ambient pressures and gas velocities. For this purpose, a two-phase, low- Mach-number flow solver for variable-density fluids is proposed. Since the jet flow evolution corresponds to transitional turbulence, no turbulence models are considered. The interface is captured using a split Volume-of-Fluid method, generalized for a non-divergence-free liq- uid velocity. Both phases exchange mass across the interface. Numerical challenges emerge compared to incompressible two-phase algorithms, which are successfully addressed for the first time in this type of flow. The importance of transcritical mixing effects for increasing pressures is demonstrated. At first, local deformation features emerge, which differ signifi- cantly from earlier incompressible works. Then, instead of the traditional atomization into droplets, the minimal surface-tension force is responsible for the folding and layering of liquid sheets. Thus, rather than spray formation, surface-area increase at transcritical conditions is primarily the result of gas-like deformations under shear. Moreover, the interface may be easily perturbed in hotter regions submerged in the faster oxidizer stream under specific triggers. The net mass exchange limits the liquid-phase vaporization to the small liquid structures at high pressures. Then, the dynamical vortex identification method is used to identify the interaction between vortical structures and the liquid surface, explaining various deformation mechanisms. Details on the evolution of vortex structures are provided, showing that layering traps and weakens the vortices.