Numerical method for strongly variable-density flows at low Mach number: flame-sheet regularisation and a mass-flux immersed boundary method

Abstract

A low-Mach-number flow, in the laminar regime, has intrinsically two characteristic spatial scales for a given time scale, or two characteristic temporal scales for a given spatial scale, and these dual scales are very different due to the disparity between the flow and acoustic speed. Therefore low-Mach-number flows impose mathematical and computational challenges in their description. Standard numerical methods for compressible flows, which are typically designed for problems with a single dominant spatial and temporal scale, require alternative approaches such as preconditioning techniques or solvers tailored for low-Mach-number equations. The present work introduces a simplified fluid dynamics model for flows at low Mach number, based on the fractional time-step method. The proposed approach is suitable for handling strong temperature gradients and thermal diffusion, as encountered in combustion systems. To address discontinuities at the flame front in reacting-flow cases, due to the hypothesis of infinitely fast chemistry, a regularisation procedure is employed. Additionally, the immersed boundary method (IBM) is extended to handle mass flux across the boundary surface, enabling simulations of fuel ejection from an arbitrary burner geometry, using a convenient Cartesian grid. The numerical method utilises a predictor-corrector scheme for time integration on a collocated grid, with flux interpolation to prevent numerical pressure oscillations (``odd-even decoupling''). Relevant test cases are used to verify the methods and their implementations, demonstrating correctness and robustness.

Figures (13)

• Figure 1: A collocated bidimensional grid in Cartesian coordinates, with spacing $\Delta x_1$ and $\Delta x_2$ for $x_1$- and $x_2$-directions, respectively. Each variable is stored at the centre of grid cells (circles), while auxiliary fluxes are defined at cell interfaces of the corresponding direction (arrows), i.e, flux 1 ($F_1$) for the horizontal direction ($x_1$), and flux 2 ($F_2$) for the vertical direction ($x_2$).
• Figure 3: Streamlines of Taylor--Green vortices at $t \approx 1.3$. Light (dark) tones indicate low (high) local speeds.
• Figure 4: Axisymmetric representation of a Taylor--Couette flow.
• Figure 5: Comparison of penalised solutions accuracy (a) and order of convergence (b) for $\Rey = 1$. The behaviours under $\Rey = 10$ are practically indistinguishable from those presented, and have therefore been omitted for clarity.
• Figure 6: Scheme of enclosed square cavity of area $\hat{L}^2$, under thermally induced flow. The left wall is maintained at temperature $\hat{T}_H$, and the right wall, at $\hat{T}_C$, with $\hat{T}_H > \hat{T}_C$. Horizontal walls are considered adiabatic.