Numerical Investigation of Compressibility Effects in Reacting Subsonic Flows

Citation

Beardsell, Guillaume (2021) Numerical Investigation of Compressibility Effects in Reacting Subsonic Flows. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/dtfx-gy14. https://resolver.caltech.edu/CaltechTHESIS:11102020-065104205

Abstract

Direct numerical simulations (DNS) of reacting flows are routinely performed either by solving the fully compressible Navier-Stokes equations or using the low Mach number approximation. The latter is obtained by performing a Mach number expansion of the Navier-Stokes equations for small Mach numbers. These two frameworks differ by their ability to capture compressibility effects, which can be broadly defined as phenomena that are not captured by the low Mach number approximation. These phenomena include acoustics, compressible turbulence, and shocks. In this thesis, we systematically isolate compressibility effects in subsonic flows by performing two sets of DNS: one using the fully compressible framework, and one using the low Mach number approximation. We are specifically interested in the interactions between turbulence, acoustics, and flames.

The addition of detailed chemistry in the compressible flow solver required the development of a novel time integration scheme. This scheme combines an iterative semi-implicit method for the integration of the species transport equations, and the classical Runge-Kutta method for the integration of the other flow quantities. It is found to perform well, yielding time steps limited by the acoustic CFL only. Furthermore, the computational cost per iteration of this hybrid scheme is low, being comparable to the one for the classical Runge-Kutta method.

After extensive validation, the first application is the investigation of flame-acoustics interactions in laminar premixed flames. The thermodynamic fluctuations that accompany the acoustic wave are shown to significantly impact the flame response. Using the Rayleigh criterion, the flame-acoustics system is found to be thermo-acoustically unstable for various fuels, flow conditions, and acoustic frequencies. As expected, the low Mach number approximation and the fully compressible framework are in good agreement at low frequencies, since the flame is very thin compared to the acoustic wavelength. The two frameworks differ for very large acoustic frequencies only. In the high frequency limit, the gain reaches a plateau using the low Mach number approximation, while it goes to zero using the fully compressible framework. This is related to the spatial variations in the acoustic pressure field, which are not present in the low Mach number approximation. However, for practically relevant acoustic frequencies, the low Mach number framework is found to yield accurate results.

Next, a numerical methodology to simulate compressible flows in geometries that lack a natural turbulence generation mechanism is presented. It is found that, unlike in incompressible flows, special care must be taken regarding the energy equation and the presence of standing acoustic modes. When using periodic boundary conditions, forcing the dilatational velocity field promotes the growth of unstable modes. This is explained by extracting the eigenvalues of the linearized forced Navier-Stokes equations. Based on these observations, it is found necessary to force the solenoidal velocity field only. This methodology is applied first to simulations of subsonic homogeneous non-reacting turbulence. We present simulations results for turbulent Mach numbers varying from 0.02 to 0.65. The Mach number dependence of various quantities, such as the dilatational to solenoidal kinetic energy ratio, is extracted. The Mach number scaling of all quantities of interest is found to be readily explained by the low Mach number expansion, specifically the zeroth and first order sets of equations, for turbulent Mach numbers up to 0.1.

Finally, the interaction between subsonic compressible turbulence and premixed flames is investigated. Compressibility effects are isolated by comparing results obtained with the low Mach number approximation and the fully compressible framework, at the same flow conditions. Compressibility effects on chemistry are found to be limited for turbulent Mach numbers at least up to 0.4, especially when contrasted with the large impact of the Karlovitz number. Compressibility effects give rise to significant thermodynamic fluctuations away from the flame front, but these remain small compared to the large fluctuations due to the presence of the turbulent flame brush. The low Mach number approximation thus remains a valid framework for the Mach numbers considered, when the primary goal is to characterize the impact of turbulence on the chemical processes at play.

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