Zero Mach Number Research Articles

Zero Mach number limit of the compressible primitive equations: Ill-prepared initial data

In the work, we consider the zero Mach number limit of compressible primitive equations in the domain R2×2T or T2×2T. We identify the limit equations to be the primitive equations with the incompressible condition. The convergence behaviors are studied in both R2×2T and T2×2T, respectively. This paper takes into account the high oscillating acoustic waves and is an extension of our previous work in [29].

Representing the geometrical complexity of liners and boundaries in low-order modeling for thermoacoustic instabilities

This work introduces a novel method to represent the topological complexity of liners and boundaries in Low Order Models for thermoacoustic instabilities, under the assumption of zero-Mach number flow. In typical industrial combustion devices, the difficulty to model these elements is twofold: (1) they are characterized by complex-valued Rayleigh conductivities or acoustic impedances, and (2) they consist of large, curved panels whose geometries have first-order effect on the combustor thermoacoustic stability. To deal with the first point, the present approach makes use of a frame modal expansion recently introduced by Laurent et al. (2019) [41], which is a generalization of the classical rigid-wall Galerkin expansion, intended to deal with non-trivial boundary conditions. The core of this work lies in the second difficulty: complex-shaped liners and boundaries are modeled as two-dimensional manifolds, for which a specific set of curvilinear governing equations is derived. The inclusion of acoustic impedance or Rayleigh conductivity into these equations enforces the proper conservation equations at the frontiers of the adjacent volumes. Surface modal projections are then introduced to expand acoustic variables onto an orthogonal basis of modes solutions of a curvilinear Helmholtz eigenproblem. The resulting dynamical system is embedded into a state-space framework to build acoustic networks. A first non-reacting canonical test case, consisting of a multi-perforated liner in a cylindrical geometry is studied to assess the convergence and precision of the method. The ability of the approach to deal with realistic reacting cases is then illustrated by modeling the partially reflecting outlet of a multi-sector annular combustor typical of industrial gas turbines. This methodology enables the inclusion of liners and other boundaries of arbitrary geometrical complexity in modal projection-based thermoacoustic Low Order Models.

Zero Mach Number Limit of the Compressible Primitive Equations: Well-Prepared Initial Data

This work concerns the zero Mach number limit of the compressible primitive equations. The primitive equations with the incompressibility condition are identified as the limiting equations. The convergence with well-prepared initial data (i.e., initial data without acoustic oscillations) is rigorously justified, and the convergence rate is shown to be of order mathcal {O}(varepsilon ) , as varepsilon rightarrow 0^+ , where varepsilon represents the Mach number. As a byproduct, we construct a class of global solutions to the compressible primitive equations, which are close to the incompressible flows.

Zero Mach number limit to compressible quantum magnetohydrodynamic equations with vanishing viscosity coefficients

The connection between the compressible viscous quantum magnetohydrodynamic model with low Mach number and the ideal incompressible magnetohydrodynamic equations is studied in a periodic domain. More precisely, for well‐prepared initial data, we prove the convergence of classical solutions of the compressible viscous quantum magnetohydrodynamic model to the classical solutions of the incompressible ideal magnetohydrodynamic equations with a convergence rate when the Mach number, viscosity coefficient, and magnetic diffusion coefficient simultaneously tend to zero.

Volumetric displacement effects in Euler-Lagrange LES of particle-laden jet flows

Large-eddy simulations (LES) of a particle-laden jet under a range of volume loadings are performed using a modified point-particle Euler-Lagrange (EL) approach to evaluate the effect of the volume/mass displaced by the subgrid particles on the flow. The spatio-temporal variations in the volume fraction of the carrier phase are taken into account giving rise to a zero-Mach number, variable density formulation with source terms in both momentum and continuity equations. The influence of volume loading as well as particle relaxation time on these volumetric displacement effects are investigated by performing numerical tests at different inlet volume loadings (0.047%-37.6%) and Stokes numbers (0.038-11.6). It is shown that for volume loadings above 5%, the volumetric displacement effects tend to become important. For the cases studied, these effects increase the mean and r.m.s. velocities of the carrier phase due to the continuity source term; however, they decrease further downstream due to dispersion of particles and spreading of the jet. It is observed that reducing the particle Stokes number increases the volumetric displacement effects due to their quick response time to the changes in the background flow, lower dispersion and smaller reaction forces.

The zero Mach number limit of the three-dimensional compressible viscous magnetohydrodynamic equations

This paper is concerned with the zeroMach number limit of the three-dimensional compressible viscous magnetohydrodynamic equations. More precisely, based on the local existence of the three-dimensional compressible viscous magnetohydrodynamic equations, first the convergence-stability principle is established. Then it is shown that, when the Mach number is sufficiently small, the periodic initial value problems of the equations have a unique smooth solution in the time interval, where the incompressible viscous magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number goes to zero. Moreover, the authors prove the convergence of smooth solutions of the equations towards those of the incompressible viscous magnetohydrodynamic equations with a sharp convergence rate.

Analysis of an inviscid zero-Mach number system in endpoint Besov spaces for finite-energy initial data

The present paper is the continuation of work [18], devoted to the study of an inviscid zero-Mach number system in the framework of endpoint Besov spaces of type B∞,rs(Rd), r∈[1,∞], d≥2, which can be embedded in the Lipschitz class C0,1. In particular, the largest case B∞,11 and the case of Hölder spaces C1,α are taken into account.The local in time well-posedness of this system is proved, under an additional finite-energy hypothesis on the initial data. The key to get this result is new a priori estimates for parabolic equations with variable coefficients in endpoint spaces B∞,rs(Rd), which are of independent interest.In the special case of space dimension d=2, we are able to give a lower bound for the lifespan, such that the solutions tend to be globally defined when the initial inhomogeneity is small. There, we will show refined a priori estimates in endpoint Besov spaces for transport equations.

The well-posedness issue for an inviscid zero-Mach number system in general Besov spaces

The present paper is devoted to the study of a zero-Mach number system with heat conduction but no viscosity. We work in the framework of general non-homogeneous Besov spaces Bp,rs(Rd), with p∈[2,4] and for any d⩾2, which can be embedded into the class of globally Lipschitz functions. We prove a lo cal in time well-posedness result in these classes and we are also able to show a continuation criterion and a lower bound for the lifespan of the solutions. The proof of the results relies on Littlewood–Paley decomposition and paradifferential calculus, and on refined commutator estimates in Chemin–Lerner spaces.

Using Boundary Conditions to Account for Mean Flow Effects in a Zero Mach Number Acoustic Solver

The present study is devoted to the modeling of mean flow effects while computing thermoacoustic modes under the zero Mach number assumption. It is first recalled that the acoustic impedance modeling of a compressor or a turbine must be prescribed under an energetical form instead of the classical acoustic variables. Then we demonstrate the feasibility to take into account the coupling between acoustic and entropy waves in a zero Mach number framework to capture a family of low frequency entropic modes. The proposed approach relies on a new delayed entropy coupled boundary condition (DECBC) and proves able to capture a family of low frequency entropic mode even though no mean flow term is included in the fluctuating pressure equation.

A simple analytical model to study and control azimuthal instabilities in annular combustion chambers

This study describes a simple analytical method to compute the azimuthal modes appearing in annular combustion chambers and help analyzing experimental, acoustic and large eddy simulation (LES) data obtained in these combustion chambers. It is based on a one-dimensional zero Mach number formulation where N burners are connected to a single annular chamber. A manipulation of the corresponding acoustic equations in this configuration leads to a simple dispersion relation which can be solved by hand when the interaction indices of the flame transfer function are small and numerically when they are not. This simple tool is applied to multiple cases: (1) a single burner connected to an annular chamber (N=1), (2) two burners connected to the chamber (N=2), and (3) four burners (N=4). In this case, the tool also allows to study passive control methods where two different types of burners are mixed to control the azimuthal mode. Finally, a complete helicopter chamber (N=15) is studied. For all cases, the analytical results are compared to the predictions of a full three-dimensional Helmholtz solver and a very good agreement is found. These results show that building very simple analytical tools to study azimuthal modes in annular chambers is an interesting path to control them.

Extension of the pressure correction method to zero-Mach number compressible flows

In the present paper, the classical pressure correction method was extended into low Mach number compressible flow regime by integrating equation of state into SIMPLE algorithm. The self-developed code based on this algorithm was applied to predicting the lid-driven cavity flow and shock tube problems, and the results showed good agreement with benchmark solutions and the Mach number can reach the magnitude of as low as 10−5. The attenuation of sound waves in viscous medium was then simulated. The results agree well with the analytical solutions given by theoretical acoustics. This demonstrated that the present method could also be implemented in acoustics field simulation, which is crucial for thermoacoustic simulation.

A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

AbstractA coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 John Wiley & Sons, Ltd.

A fourth-order auxiliary variable projection method for zero-Mach number gas dynamics

A fourth-order numerical method for the zero-Mach-number limit of the equations for compressible flow is presented. The method is formed by discretizing a new auxiliary variable formulation of the conservation equations, which is a variable density analog to the impulse or gauge formulation of the incompressible Euler equations. An auxiliary variable projection method is applied to this formulation, and accuracy is achieved by combining a fourth-order finite-volume spatial discretization with a fourth-order temporal scheme based on spectral deferred corrections. Numerical results are included which demonstrate fourth-order spatial and temporal accuracy for non-trivial flows in simple geometries.

A note on the zero Mach number limit of compressible Euler equations

This note presents a short and elementary justification of the classical zero Mach number limit for isentropic compressible Euler equations with prepared initial data. We also show the existence of smooth compressible flows, with the Mach number sufficiently small, on the (finite) time interval where the incompressible Euler equations have smooth solutions.

Zero Mach number limit for compressible flows with periodic boundary conditions

We address the question of convergence to the incompressible Navier-Stokes equations for slightly compressible viscous flows with ill-prepared initial data and periodic boundary conditions (the case of the whole space has been studied in an earlier paper by the author). The functional setting is very close to the one used by H. Fujita and T. Kato for incompressible flows. For arbitrarily large initial data, we show that the compressible flow with small Mach number exists as long as the incompressible one does. In particular, it exists globally if the corresponding incompressible solution exists for all time. We further state a convergence result for the slightly compressible solution filtered by the group of acoustics.

A Semi-implicit Numerical Scheme for Reacting Flow: I. Stiff Chemistry

An additive semi-implicit projection scheme for the simulation of unsteady combustion in two dimensions is constructed. The scheme relies on a zero-Mach number formulation of the compressible conservation equations with detailed chemistry. The governing equations are discretized in space using second-order differences and integrated in time using a semi-implicit approach. Time integration of the evolution equations for species mass fraction, thermodynamic pressure, and density is performed using a semi-implicit, nonsplit scheme that combines a second-order predictor–corrector treatment of convection and diffusion terms, and a stiff integrator for the reaction source terms. Meanwhile, the momentum equations are integrated using a second-order projection scheme. The projection scheme is based on a predictor–corrector approach that couples the evolution of the velocity and density fields in order to stabilize computations of reacting flows with large density variations. A pressure Poisson equation is inverted following both the predictor and corrector steps using a fast solver. The advantages of the stiff integration of reaction source terms are analyzed by comparing the performance of the scheme to that of a predictor–corrector scheme in which reaction and diffusion are integrated in a similar nonstiff fashion. The comparison in based on both one-dimensional (1D) unsteady tests of a premixed methane–air flame, and unsteady two-dimensional tests of the same flame interacting with a counterrotating vortex pair. In both cases, the GRImech1.2 reaction mechanism with 32 species and 177 elementary reactions is used. Computed results show that the stiff reaction scheme enables selection of larger time steps and thus leads to substantial improvement in the performance of the computations. For the present reaction mechanism and flame conditions, speedup factors of about 10 are achieved in the 1D tests and about five in two dimensions. Possible extensions of the present scheme to further improve efficiency are also discussed.

The Derivation and Numerical Solution of the Equations for Zero Mach Number Combustion

Abstract We present a limiting system of equations to describe combustion processes at low Mach number in either confined or unbounded regions and numerically solve these equations for the case of a flame propagating in a closed vessel. This system allows for large heat release, substantial temperature and density variations, and substantial interaction with the hydrodynamic flow field, including the effects of turbulence. This limiting system is much simpler than the complete system of equations of compressible reacting gas flow since the detailed effects of acoustic waves have been removed. Using a combination of random vortex techniques and flame propagation algorithms specially designed for turbulent combustion, we describe a numerical method to solve these zero Mach number equations. We use this method to analyze the competing effects of viscosity, exothermicity, boundary conditions and pressure on the rate of combustion for a flame propagating in a swirling flow inside a square.