Mach Number Limit Research Articles

Stabilized finite element method based on local preconditioning for unsteady compressible flows in deformable domains with emphasis on the low Mach number limit application

SUMMARYFlows with low Mach numbers represent a limit situation in the solution of compressible flows. The preconditioning of flow equations is one of the classical approaches proposed to capture the solution in the low Mach number limit. In this method, the time derivatives are premultiplied by a suitable preconditioning matrix in order to achieve a well‐conditioned system by means of the scaling of the system eigenvalues. Hence, the modified equations have only steady‐state solutions in common with the original system. For the application of these methods to unsteady problems, the dual time‐stepping technique has emerged, where the physical time derivative terms are treated as source and/or reactive terms. The use of a preconditioning matrix defined to compute steady‐state solutions may not be a good choice for unsteady problems, as showed by Vigneronit et al. (European Conference on Computational Fluid Dynamics, 2006). However, such matrices can be adapted to perform the computation of transient flows by means of the appropriate redefinition of some coefficients. The application of a ‘steady‐state’ preconditioning matrix to unsteady problems with an ALE (Arbitrary Lagrangian Eulerian) approach is presented. The equations are discretized in space using a stabilized Finite Element method and in time using finite differences. The preconditioning of the governing equations is not applied in the numerical scheme but is used, through the eigenvalues of the preconditioned system, to design appropriately the stabilization term. Several test cases are solved, including incompressible flows and the in‐cylinder flow in a motored opposed‐piston engine. Copyright © 2011 John Wiley & Sons, Ltd.

A note on incompressible limit for compressible Euler equations

This paper presents a simple justification of the classical low Mach number limit in critical Besov spaces for compressible Euler equations with prepared initial data. As the first step of this justification, we formulate a continuation principle for general hyperbolic singular limit problems in the framework of critical Besov spaces. With this principle, it is also shown that, for the Mach number sufficiently small, the smooth compressible flows exist on the (finite) time interval where the incompressible Euler equations have smooth solutions, and the definite convergence orders are obtained. Copyright © 2011 John Wiley & Sons, Ltd.

Rigorous derivation of the anelastic approximation to the Oberbeck–Boussinesq equations

It is shown that the anelastic Oberbeck-Boussinesq system is a small Mach, small Peclet and small Froude number limit of the complete Navier-Stokes-Fourier system for gases with large specific heat at constant volume. This result is obtained on an arbitrary large time interval. The proof allows an intrinsic view into the process of separation of fast oscillating acoustic waves, governed by a Lighthill-type equation, from the equations describing the slow fluid flows. This is a very useful information for numerical analysts. keywords: anelastic approximation

Dynamic steady-state crack propagation in quasi-crystals

The steady propagation of planar cracks in quasi-crystalline bodies with velocity lower than the one of bulk elastic macroscopic waves is under scrutiny. Closed-form solutions to the balance laws are provided. Unusual Mach number limits are determined. Numerical experiments describing peculiar aspects of the crack propagation in quasi-crystals are performed by varying parametrically the coupling coefficient between macroscopic deformation and substructural events. In this way, classes of quasi-crystals are then compared. Copyright © 2010 John Wiley & Sons, Ltd.

Influence of Strouhal number on pulsating methane–air coflow jet diffusion flames

Four periodically time-varying methane–air laminar coflow jet diffusion flames, each forced by pulsating the fuel jet's exit velocity U j sinusoidally with a different modulation frequency w j and with a 50% amplitude variation, have been computed. Combustion of methane has been modeled by using a chemical mechanism with 15 species and 42 reactions, and the solution of the unsteady Navier–Stokes equations has been obtained numerically by using a modified vorticity-velocity formulation in the limit of low Mach number. The effect of w j on temperature and chemistry has been studied in detail. Three different regimes are found depending on the flame's Strouhal number S = aw j /U j , with a denoting the fuel jet radius. For small Strouhal number (S = 0.1), the modulation introduces a perturbation that travels very far downstream, and certain variables oscillate at the frequency imposed by the fuel jet modulation. As the Strouhal number grows, the nondimensional frequency approaches the natural frequency of oscillation of the flickering flame (S ≃ 0.2). A coupling with the pulsation frequency enhances the effect of the imposed modulation and a vigorous pinch-off is observed for S = 0.25 and S = 0.5. Larger values of S confine the oscillation to the jet's near-exit region, and the effects of the pulsation are reduced to small wiggles in the temperature and concentration values. Temperature and species mass fractions change appreciably near the jet centerline, where variations of over 2 % for the temperature and 15 % and 40 % for the CO and OH mass fractions, respectively, are found. Transverse to the jet movement, however, the variations almost disappear at radial distances on the order of the fuel jet radius, indicating a fast damping of the oscillation in the spanwise direction.

Quasiequilibrium lattice Boltzmann models with tunable bulk viscosity for enhancing stability

Taking advantage of a closed-form generalized Maxwell distribution function [P. Asinari and I. V. Karlin, Phys. Rev. E 79, 036703 (2009)] and splitting the relaxation to the equilibrium in two steps, an entropic quasiequilibrium (EQE) kinetic model is proposed for the simulation of low Mach number flows, which enjoys both the H theorem and a free-tunable parameter for controlling the bulk viscosity in such a way as to enhance numerical stability in the incompressible flow limit. Moreover, the proposed model admits a simplification based on a proper expansion in the low Mach number limit (LQE model). The lattice Boltzmann implementation of both the EQE and LQE is as simple as that of the standard lattice Bhatnagar-Gross-Krook (LBGK) method, and practical details are reported. Extensive numerical testing with the lid driven cavity flow in two dimensions is presented in order to verify the enhancement of the stability region. The proposed models achieve the same accuracy as the LBGK method with much rougher meshes, leading to an effective computational speed-up of almost three times for EQE and of more than four times for the LQE. Three-dimensional extension of EQE and LQE is also discussed.

On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions

We study the low Mach number limit for the compressible Navier-Stokes system supplemented with Navier's boundary condition on an unbounded domain with compact boundary. Our main result asserts that the velocities converge pointwise to a solenoidal vector field - a weak solution of the incompressible Navier-Stokes system - while the fluid density becomes constant. The proof is based on a variant of local energy decay property for the underlying acoustic equation established by Kato.

Low Mach Number Limit for the Navier–Stokes System on Unbounded Domains Under Strong Stratification

We consider the low Mach number singular limit problem for the Navier–Stokes system describing the motion of a compressible fluid under strong stratification in an exterior domain. It is shown that the limit problem is represented by the so-called anelastic approximation. In particular, strong (pointwise) convergence of the acoustic components of the velocity is established by means of an abstract result of Tosio Kato.

Low Mach number asymptotics for reacting compressible fluid flows

We study the low Mach number limit for the fullNavier-Stokes-Fourier system describing the dynamics of chemicallyreacting fluids. The so-called reactive Boussinesq system isidentified as the asymptotic limit.

Finite Point Low Mach Number Algorithm for Unsteady Nonisothermal Flows Application to the Study of Reactive Flow Past a Thin Catalytic Wire

A numerical algorithm is presented for the solution of low Mach number unsteady nonisothermal flows at moderate Reynolds numbers (Re ∼ 100) with variable transport properties. The low Mach number limit is approached by using a relaxation method of the pseudo-compressibility type that has proved to be numerically robust and accurate enough to be used in industrial applications. Time integration is explicit and space integration is based on a finite point approach. Solver validation includes a sensitivity analysis and a comparison with previously published results concerning the flow around heated cylinders at different Re and temperature loadings. Then, the solver is applied to calculate the flow of a reacting mixture past a thin catalytic wire, a problem of interest for power MEMS design (micro catalytic reactors), aiming to determine the chemical-to-thermal energy conversion rate at the catalytic wire.

On the behavior of upwind schemes in the low Mach number limit. IV: P0 approximation on triangular and tetrahedral cells

Finite Volume upwind schemes for the Euler equations in the low Mach number regime face a problem of lack of convergence toward the solutions of the incompressible system. However, if applied to cell centered triangular grid, this problem disappears and convergence toward the incompressible solution is recovered. Extending the work of [3] that prove this result for regular triangular grid, we give here a general proof of this fact for arbitrary unstructured meshes. In addition, we also show that this result is equally valid for unstructured three dimensional tetrahedral meshes.

Titan's hydrodynamically escaping atmosphere: Escape rates and the structure of the exobase region

In Strobel [Strobel, D.F., 2008. Icarus, 193, 588–594] a mass loss rate from Titan's upper atmosphere, ∼ 4.5 × 10 28 amu s − 1 , was calculated for a single constituent, N 2 atmosphere by hydrodynamic escape as a high density, slow outward expansion driven principally by solar UV heating due to CH 4 absorption. It was estimated, but not proven, that the hydrodynamic mass loss is essentially CH 4 and H 2 escape. Here the individual conservation of momentum equations for the three major components of the upper atmosphere (N 2, CH 4, H 2) are solved in the low Mach number limit and compared with Cassini Ion Neutral Mass Spectrometer (INMS) measurements to demonstrate that light gases (CH 4, H 2) preferentially escape over the heavy gas (N 2). The lightest gas (H 2) escapes with a flux 99% of its limiting flux, whereas CH 4 is restricted to ⩾75% of its limiting flux because there is insufficient solar power to support escape at the limiting rate. The respective calculated H 2 and CH 4 escape rates are 9.2 × 10 27 and 1.7 × 10 27 s − 1 , for a total of ∼ 4.6 × 10 28 amu s − 1 . From the calculated densities, mean free paths of N 2, CH 4, H 2, and macroscopic length scales, an extended region above the classic exobase is inferred where frequent collisions are still occurring and thermal heat conduction can deliver power to lift the escaping gas out of the gravitational potential well. In this region rapid acceleration of CH 4 outflow occurs. With the thermal structure of Titan's thermosphere inferred from INMS data by Müller–Wodarg et al. [Müller-Wodarg, I.C.F., Yelle, R.V., Cui, J., Waite Jr., J.H., 2008. J. Geophys. Res. 113, doi:10.1029/2007JE003033. E10005], in combination with calculated temperature profiles that include sputter induced plasma heating at the exobase, it is concluded that on average that the integrated, globally average, orbit-averaged, plasma heating rate during the Cassini epoch does not exceed ∼ 5 × 10 8 eV cm − 2 s − 1 ( ∼ 0.0008 erg cm − 2 s − 1 ).

Low Mach number limit for the non-isentropic Navier–Stokes equations

We study the low Mach number limit of the local in time solutions to the compressible Navier–Stokes equations with zero heat conductivity coefficient as the Mach number tends to zero. A uniform existence result for the one-dimensional initial–boundary value problem is proved provided that the initial data are “well-prepared” in the sense that the temporal derivatives up to order two are bounded initially.

The Perturbation and ADAE Index of a Degenerated Hyperbolic System

We consider a linear version of the zero Mach-number limit of the Euler equations of compressible fluid flow. This system turns out to be a coupled hyperbolic/parabolic equation with coupled, time-dependent boundary conditions. Using the theory of abstract differential-algebraic equations it is shown that the frozen coefficient system has ADAE index 1. Moreover, the much stronger result is proven that the system has time-perturbation index one and space-perturbation index two even in the case of time-dependent boundary conditions. The results are stated in terms of the original physical variables. The estimates agree well with numerical experiments.

Modeled Boltzmann Equation and Its Application to Shock-Capturing Simulation

A modified equilibrium distribution function for the Bhatnagar-Gross-Krook-type modeled Boltzmann equation has recently been proposed. The function was deduced using acoustics scaling to normalize the equation and allowed a correct recovery of similarly normalized Euler equations. It is a combination of a Maxwellian distribution plus three other terms that are moments of particle velocity. The lattice counterpart of the modified equilibrium distribution function also led to an exact recovery of the Euler equations; therefore, there is no Mach number limitation in the entire approach. This lattice counterpart was able to replicate aeroacoustics problems involving vorticity-acoustic and entropy-acoustic interactions correctly, and the simulations were carried out using a finite difference lattice Boltzmann method employing only a two-dimensional, nine-velocity lattice. Thus formulated, the numerical scheme has no arbitrary constants and all calculations were carried out using one single relaxation time and a set of constants derived from the analysis. This paper investigates the validity and extent of the formulation to capture shocks and resolve contact discontinuity and expansion waves in one- and two-dimensional Riemann problems. The simulations are carried out using the same two-dimensional, nine-velocity lattice, and identical set of constants and relaxation time; they are compared with theoretical results and those obtained by solving the Euler equations directly using Harten's first-order numerical scheme. Good agreement is obtained for all test cases. However, the modified equilibrium distribution function is not suitable for shock structure simulation; for that, an exact recovery of the Navier-Stokes equations is required.

On the numerical approximation of a degenerated hyperbolic system

The heat exchanger in a heat pump system may be conveniently described by a degenerated hyperbolic system, namely the zero Mach-number limit of the Euler equations. This leads to a mixed hyperbolic/parabolic system with coupled time-dependent boundary conditions. We propose a method-of-lines discretisation by using an upwinding scheme. We derive stability estimates for the linearisation with frozen coefficients. The resulting differential–algebraic equation has a perturbation index of 2 and a weak instability with respect to the space step size. The latter property is validated experimentally even for the nonlinear system. In contrast, the perturbation index did not exceed one in the numerical experiments.

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

The inviscid low Mach number compressible flow developing from a plane incompressible vortex with constant density in a bounded domain is studied. A reference solution for this model flow is obtained by two-time scale asymptotic development in the zero Mach number limit. The solution can be decomposed into variations with a slow convective and a fast acoustic time scale. A selection of numerical schemes widely used to solve unsteady compressible low Mach number flows has been implemented to simulate this flow for two Mach number values, M = 0.1 and M = 0.01. The ability of the schemes to predict low Mach number flows and their behavior with the value of the Mach number have been analyzed.

Heating of the fuel mixture due to viscous stress ahead of accelerating flames in deflagration-to-detonation transition

The role of viscous stress in heating of the fuel mixture in deflagration-to-detonation transition in tubes is studied both analytically and numerically. The analytical theory is developed in the limit of low Mach number; it determines temperature distribution ahead of an accelerating flame with maximum achieved at the walls. The heating effects of viscous stress and the compression wave become comparable at sufficiently high values of the Mach number. In the case of relatively large Mach number, viscous heating is investigated by direct numerical simulations. The simulations were performed on the basis of compressible Navier–Stokes gas-dynamic equations taking into account chemical kinetics. In agreement with the theory, viscous stress makes heating and explosion of the fuel mixture preferential at the walls. The explosion develops in an essentially multi-dimensional way, with fast spontaneous reaction spreading along the walls and pushing inclined shocks. Eventually, the combination of explosive reaction and shocks evolves into detonation.

Layered incompressible fluid flow equations in the limit of low Mach number and strong stratification

We investigate the low Mach number regimes of the complete Navier–Stokes–Fourier system which describes an evolution of a viscous, heat conducting gas with a heat source term interpreted as a radiation cooling in models of atmospheric flows. This investigation is performed in the context of weak solutions on an arbitrary large time interval and for the ill-prepared initial data. The physically expected limiting equations as the Mach and Froude numbers tend to zero at the same rate are layered incompressible flow equations with strong stratification. We give a rigorous mathematical proof of this result.

An All-Speed Roe-type scheme and its asymptotic analysis of low Mach number behaviour

A new scheme, All-Speed-Roe scheme, was proposed for all speed flows. Compared with traditional preconditioned Roe scheme, All-Speed-Roe scheme changes non-linear eigenvalues in the numerical dissipation terms of Roe-type schemes. With an asymptotic analysis, the low Mach number behaviour of the scheme is studied theoretically in two ways. In one way, All-Speed-Roe scheme is regarded as finite magnification of Low-Speed-Roe scheme in the low Mach number limit. In the other way, a general form of All-Speed-Roe scheme is analyzed. Both ways demonstrate that All-Speed-Roe scheme has the same low Mach number behaviour as the original governing equation in the continuous case, which includes three important features: pressure variation scales with the square of the Mach number, the zero order velocity field is subject to a divergence constraint, and the second order pressure satisfies a Poisson-type equation in the case of constant-entropy. The analysis also leads to an unexpected conclusion that the velocity filed computed by traditional preconditioned Roe scheme does not satisfy the divergence constraint as the Mach number vanishes. Moreover, the analysis explains the reason of checkerboard decoupling and shows that momentum interpolation method provides a similar mechanism as traditional preconditioned Roe scheme inherently possesses to suppress checkerboard decoupling. In the end, general rulers for modifying non-linear eigenvalues are obtained. Finally, several numerical experiments are provided to support the theoretical analysis. All-Speed-Roe scheme has a sound foundation and is expected to be widely studied and applied to all speed flow calculations.