Modified flux-difference splitting for simulating low Mach number flows, including combustion, on unstructured grids

Obtaining efficient and robust solutions to real reacting flow problems is a challenging task in numerical simulation due to severe nonlinearities in the conservation equations and the wide range of time scales of chemical reactions. However, there are many important industrial applications for such simulations, e.g., premixed and non-premixed combustion, catalytic combustion, or materials processing. A finite volume method on unstructured, non-staggered grids is presented for the solution of two-dimensional low Mach number reacting flows. The conservation law form of the flow equations are discretized in terms of primitive variables. An artificial compressibility method is employed so that low Mach number flows can be solved economically. The equations are marched in time using an implicit extrapolation method. Numerical results are obtained for three different applications: A reaction-diffusion controlled flow in the process chamber of a chemical downstream etch system, the simulation of a premixed ozone flame, and a methane diffusion flame.

Asymptotic and numerical analysis of an inviscid bounded vortex flow at low Mach number

Global, three‐dimensional, ideal MHD simulations of Earth's bow shock are reported for low Alfven Mach numbers MA and quasi‐perpendicular magnetic field orientations. The simulations use a hard, infinitely conducting magnetopause obstacle, with axisymmetric three‐dimensional location given by a scaled standard model, to directly address previous gasdynamic (GD) and field‐aligned MHD (FA‐MHD) work. Tests of the simulated shocks’ density jumps X for 1.4 ≲ MA ≲ 10 and the high MA shock location, and reproduction of the GD relation between magnetosheath thickness and X for quasi‐gasdynamic MHD runs with MA ≫ MS, confirm that the MHD code is working correctly. The MHD simulations show the standoff distance as increasing monotonically with decreasing MA. Significantly larger as are found at low MA than predicted by GD and phenomenological MHD models and FA‐MHD simulations, as required qualitatively by observations. The GD and FA‐MHD predictions err qualitatively, predicting either constant or decreasing as with decreasing MA. This qualitative difference between quasi‐perpendicular MHD and FA‐MHD simulations is direct evidence for as depending on the magnetic field orientation θ. The enhancement factor over the phenomenological MHD predictions at MA ∼ 2.4 agrees quantitatively with one observational estimate. A linear relationship is found between the magnetosheath thickness and X, modified both quantitatively and intrinsically by MHD effects from the GD result. The MHD and GD results agree in the high MA limit. An MHD theory is developed for as, restricted to sufficiently perpendicular θ and high sonic Mach numbers MS. It explains the simulation results with excellent accuracy. Observational and further simulation testing of this MHD theory, and of its predicted MA, θ, and MS effects, is desirable.

This work focuses on the study of the asymptotic preserving property of a semi-implicit method. The semi-implicit method, which is a pressure-based method, has been successfully used to simulate two-phase flows in numerous industrial applications. This method is used in our studies due to the fact that pressure-based methods generally perform well at low Mach numbers. The semi-implicit method is applied to the homogeneous equilibrium model (HEM) in this work to simulate two-phase flows. We show that the semi-implicit method is asymptotic preserving, i.e. the discretization for a compressible model tends to a consistent discretization for the related incompressible model at the low Mach number limit. Finally, test cases are performed to show that the numerical method is able to deal with low Mach number flows, as well as flows with a wide range of Mach numbers.

Flux-vector splitting for compressible low mach number flow

Combustion noise is becoming increasingly important as a major noise source in aeroengines and ground based gas turbines. This is partially because advances in design have reduced the other noise sources, and partially because next generation combustion modes burn more unsteadily, resulting in increased external noise from the combustion. This review reports recent progress made in understanding combustion noise by theoretical, numerical and experimental investigations. We first discuss the fundamentals of the sound emission from a combustion region. Then the noise of open turbulent flames is summarized. We subsequently address the effects of confinement on combustion noise. In this case not only is the sound generated by the combustion influenced by its transmission through the boundaries of the combustion chamber, there is also the possibility of a significant additional source, the so-called ‘indirect’ combustion noise. This involves hot spots (entropy fluctuations) or vorticity perturbations produced by temporal variations in combustion, which generate pressure waves (sound) as they accelerate through any restriction at the exit of the combustor. We describe the general characteristics of direct and indirect noise. To gain further insight into the physical phenomena of direct and indirect sound, we investigate a simple configuration consisting of a cylindrical or annular combustor with a low Mach number flow in which a flame zone burns unsteadily. Using a low Mach number approximation, algebraic exact solutions are developed so that the parameters controlling the generation of acoustic, entropic and vortical waves can be investigated. The validity of the low Mach number approximation is then verified by solving the linearized Euler equations numerically for a wide range of inlet Mach numbers, stagnation temperature ratios, frequency and mode number of heat release fluctuations. The effects of these parameters on the magnitude of the waves produced by the unsteady combustion are investigated. In particular the magnitude of the indirect and direct noise generated in a model combustor with a choked outlet is analyzed for a wide range of frequencies, inlet Mach numbers and stagnation temperature ratios. Finally, we summarize some of the unsolved questions that need to be the focus of future research.

Cures for the shock instability: Development of a shock-stable Roe scheme

A preconditioning scheme has been implemented into a three-dimensional viscous computational fluid dynamics code for turbomachine blade rows. The preconditioning allows the code, originally developed for simulating compressible flow fields, to be applied to nearly-incompressible, low Mach number flows. A brief description is given of the compressible Navier-Stokes equations for a rotating coordinate system, along with the preconditioning method employed. Details about the conservative formulation of artificial dissipation are provided, and different artificial dissipation schemes are discussed and compared. The preconditioned code was applied to a well-documented case involving the NASA large low-speed centrifugal compressor for which detailed experimental data are available for comparison. Performance and flow field data are compared for the near-design operating point of the compressor, with generally good agreement between computation and experiment. Further, significant differences between computational results for the different numerical implementations, revealing different levels of solution accuracy, are discussed.

In this paper, radiated sound from low Mach number turbulent boundary layers is studied using Lighthill’s analogy. The focus is to investigate the behavior of wall pressure fluctuations on a flexible wall boundary. Comparisons are made between the results from rigid and flexible walls. At low frequencies, the flexible wall results have shown much higher levels of power spectra. At high frequencies, the spectra of the flexible wall sound radiation and those of the rigid wall sound radiation merge. These trends are in agreement with other experimental and theoretical results in literature. This theoretical study of the two-dimensional turbulent boundary layer can clarify the effects of a flexible wall on radiated sound in low Mach number flows.

An approximate Riemann solver for real gas parabolized Navier–Stokes equations

In this study, the sound generation in a constricted pipe at low Mach number (M=0.003,Re=1170 at the inlet) flow is investigated using a hybrid modeling approach of computational aero-acoustics (CAA).Large Eddy Simulation (LES) model is used simulate the flow and accurately capture the flow instabilities associated with the constriction.The Acoustic Perturbation Equation (APE) method is used to model the acoustic field where the source terms are derived from the incompressible flow simulation results.The acoustic spectra from the APE simulation results showed good agreement (within 9%) with measured values on the pipe wall downstream of the constriction.The strongest APE acoustic sources were found to occur in the jet core breakdown region that contained the highest pressure fluctuations in the flow field.Proper Orthogonal Decomposition (POD) was used to study the Coherent flow structures in the jet core breakdown region in order to further understand the characteristics of acoustic sources.The strongest structure corresponds to the peak frequency of the measured sound pressure spectra.The study results improve our understanding of the sound generation mechanisms in biomedical applications involving the airways such as phonation and partial airway obstruction.

A high order finite difference scheme is introduced to deal with thermo-hydraulic quench simulation in superconducting magnets, specifically in the case of cable-in-conduit conductors (CICC). Liquid helium flow in CICC is governed by the unsteady convection-diffusion conservation equations, in the low Mach number and high Reynolds number flow region. In the past several years many methods have been considered for quench simulation in superconducting magnets, for example, finite element, finite volume, and finite difference methods. One of the main challenges is the accurate simulation of flow during a superconductor quench, given the large gradients in pressure and temperature that arise during conductor normalization. These simulations are key to proper magnet protection design. The Dispersion Relation Preserving (DRP) scheme is applied to the quench simulation problem. The computational results are compared to those obtained using a well-proven commercial quench simulation code (Gandalf), as well as experimental data. Preliminary results are promising.

A new two-dimensional formula to describe aeolian tones radiated from rigid bodies in a uniform flow at low Mach numbers is proposed as an improved approximation of Curle’s dipole solution. This modified Curle’s dipole is composed mainly of two simple terms; one depends on time and the other does not, which represent the acoustic propagation and the hydrodynamic mean effect, respectively. The acoustic term includes the Doppler effect by regarding the sound speed to be directional in the source-fixed frame. The formula is verified in comparison with the results by direct numerical simulations (DNS) of the two-dimensional compressible Navier–Stokes equations for sounds from a circular cylinder in low Mach number flows. The results show that the modified Curle’s dipole approximates well the DNS results not only for the fluctuation pressure but also for the mean pressure in the far field. The mathematical basis of the formula is also presented in relation to the exact dipole term of the Ffowcs Williams–Hawkings equation.

Preconditioning methods for compressible flow CFD codes: Revisited

ABSTRACTAs a type of shock-capturing scheme, the traditional Roe scheme fails in large eddy simulation (LES) because it cannot reproduce important turbulent characteristics, such as the famous k−5/3 spectral law, as a consequence of the large numerical dissipation. In this work, the Roe scheme is divided into five parts, namely, ξ, δUp, δpp, δUu, and δpu, which denote basic upwind dissipation, pressure difference-driven modification of interface fluxes, pressure difference-driven modification of pressure, velocity difference-driven modification of interface fluxes, and velocity difference-driven modification of pressure, respectively. Then, the role of each part in the LES of homogeneous decaying turbulence with a low Mach number is investigated. Results show that the parts δUu, δpp, and δUp have little effect on LES. Such minimal effect is integral to computational stability, especially for δUp. The large numerical dissipation is due to ξ and δpu, each of which features a larger dissipation than the sub-grid scale model. On the basis of these conditions, an improved all-speed Roe scheme for LES is proposed. This scheme can provide satisfactory LES results even for coarse grid resolutions with usually adopted second-order reconstructions for the finite volume method.