Low Mach Number Approximation Research Articles

Multiscale lattice Boltzmann schemes for low Mach number flows

A low Mach number approximation (LMNA) of the Navier-Stokes equations is widely used in numerical methods for the simulation of low-speed thermal and athermal flows. The advanced lattice Boltzmann approach (Bhatnagar-Gross-Krook) for the solution of the LMNA equations is discussed and its performance is compared with the performance of the commercial CFD code FLUENT 5.

Numerical analysis of the gravitational effect on the buoyancy-driven fluctuations in diffusion flames

Unsteady behavior of buoyant diffusion flames which were observed at various gravity fields was studiednumerically by a finite volume method using the low-Mach number approximation and the temperaturedependent physical properties such as viscosity and diffusivity. The numerical model was a time-dependent and axisymmetric flow including a single-step chemical reaction. The objective was to clarify a periodic instability mechanism of buoyant diffusion flames such as flickering flame. The parameters of the jet injection velocity, V f , and dimensionless gravity level, G , were varied to clarify the effects of shear force and buoyant force on the unsteady motion of the flame. The gravitational effect was evaluated by varying the gravity acceleration ranging from 0.5G to 5G. An oscillatory frequency increased with an increase in gravity level. Convective flow in the flame caused by the vortical roll-up motion was enhanced in a highgravity condition. To elucidate the mechanism of the instability motion, axial velocity of convection, oxygen mass fraction, vorticity, and the distribution of radial density gradient were derived from the calculated results. It was found that the convective velocity and the oxygen mass fraction played a very important role in flame instability, And it seemed that the origin of unsteady motion was the generation of the vorticity. The maximum value of the vorticity estimated in high-gravity field was greater than that of the normalgravity field. Furthermore, the unsteady behavior caused by the buoyancy was correlated to the radial density gradient.

Effect of wind velocity on flame spread in microgravity

A three-dimensional, time-dependent model is developed describing ignition and subsequent transition to flame spread over a thermally thin cellulosic sheet heated by external radiation in a microgravity environment. A low Mach number approximation to the Navier-Stokes equations with global reaction rate equations describing combustion in the gas phase and the condensed phase is numerically solved. The effects of a slow external wind (1–20 cm/s) on flame transition are studied in an atmosphere of 35% oxygen concentration. The ignition is initiated at the center part of the sample by generating a line-shape flame along the width of the sample. The calculated results are compared with data obtained in the 10 s drop tower. Numerical results exhibit flame quenching at a wind speed of 1.ccm/s, two localized flames propagating upstream along the sample edges at 1.5 cm/s, a single line-shape flame front at 5.0 cm/s, and three flames structure observed at 10.0 cm/s (consisting of a single line-shape flame propagating upstream and two localized flames propagating downstream along sample edges), followed by two line-shape flames (one propagating upstream and another propagating downstream) at 20.0 cm/s. These observations qualitatively compare with experimental data. Three-dimensional visualization of the observed flame complex, fuel concentration contours, oxygen and reaction rate isosurfaces, and convective and diffusive mass flux are used to obtain a detailed understanding of the controlling mechanism. Physical arguments based on the lateral diffusive flux of oxygen, fuel depletion, the oxygen shadow of the flame, and the heat release rate are constructed to explain the various observed flame shapes.

Numerical Simulation of Unsteady CH$_4$/Air Jet Diffusion Flame

The dynamic structures of unsteady CH/Air jet diffusion flame with a flame-vortex interaction were numerically investigated. A timed-dependent, axisymmetric computational model and a low mach number approximation were employed in the present calculation. A two-step global reaction mechanism which considers 6 species, was used to calculate the reaction rates. The predicted results including the gravitational effect show that the large outer vortices and the small inner vortices can be well simulated without any additional disturbances near nozzle tip. It was found that the temperature and species concentrations have deviated values even for the same mixture fraction in the flame-vortex interaction region. It was also shown that the flame surface is not deformed by the inner vortex in upstream region, while in downstream region, the flame surface is compressed or stretched by the outer vortex roll-up. The present unsteady jet flame configuration accompanying a flame-vortex interaction is expected to give good implications for the unsteady structures of turbulent flames.

Conservative High-Order Finite-Difference Schemes for Low-Mach Number Flows

Three finite-difference algorithms are proposed to solve a low-Mach number approximation for the Navier–Stokes equations. These algorithms exhibit fourth-order spatial and second-order temporal accuracy. They are dissipation-free, and thus well suited for DNS and LES of turbulent flows. The key ingredient common to each of the methods presented is a Poisson equation with variable coefficient that is solved for the hydrodynamic pressure. This feature ensures that the velocity field is constrained correctly. It is shown that this approach is needed to avoid violation of the conservation of kinetic energy in the inviscid limit which would otherwise arise through the pressure term in the momentum equation. An existing set of finite-difference formulae for incompressible flow is generalized to handle arbitrary large density fluctuations with no violation of conservation through the non-linear convective terms. An algorithm which conserves mass, momentum, and kinetic energy fully is obtained when an approximate equation of state is used instead of the exact one. Results from a model problem are used to show both spatial and temporal convergence rates and several test cases are presented to illustrate the performance of the algorithms.

Turbulent transport in spatially developing reacting shear layers

Direct numerical simulation (DNS) of turbulent reacting shear layers with significant heat release and local Reynolds numbers up to 1000 based on the Favre-averaged vorticity thickness was performed. The combustion is simulated by a single-step, second-order reaction with an Arrhenius reaction rate. The transport equations are solved using a low Mach number approximation in which the effects of heat release are accounted for through variable density. Two-dimensional simulations illustrate how growth rate decreases with increasing heat release and how the shear layer is sensitive to reaction rate. A three-dimensional simulation is used to generate data to study in detail how the flame alters the turbulence using budgets of turbulent kinetic energy. The three-dimensional results reveal that turbulent transport and pressure transport have a unique role in the overall energy balance in that they tend to increase kinetic energy near the edges of the layer while decreasing it at the center. Opposite behavior is found for pressure dilatation. Dissiplation and convection everywhere act as sinks for turbulence. The DNS results are compared with the standard k-∈ model for production and turbulent transport. The Boussinnesq approximation with a typical eddy viscosity constant yields the proper profile for production but, in general, overpredicts the peak value. The evaluation also reveals that the dilatation dissipation is small compared with the solenoidal dissipation. The standard gradient-diffusion model for turbulent transport/molecular diffusion is satisfactory when the molecular diffusion is small. The similarity between pressuren transport and turbulent transport suggests that pressure transport could also be modeled using a gradient-diffusion expression.

A study of non-Boussinesq effect on transition of thermally induced flow in a square cavity

The low-Mach number approximation is considered to be less restrictive than the Boussinesq approximation. The former represents well the flow behavior in enclosures with relatively large temperature difference, and therefore permits a better understanding of thermally driven flows. However, as the flow becomes chaotic or weakly turbulent, it becomes uncertain whether the low-Mach number approximation can properly describe the transition behavior of the flow. In the present study, the behavior of transitional thermally driven flow in a two-dimensional (2-D) differentially heated square cavity filled with a gas has been numerically investigated in cases where the temperature difference increases. The exact governing equations, including the equation of state for ideal gas, are used to calculate the initial phase to transition, and the results are compared in detail with those obtained using the Boussinesq-approximated and low-Mach number equations. The applicability of these approximated governing equations has been discussed. It turned out that beyond a certain Rayleigh number, even the low-Mach number approximation may not always predict the transition behavior of the flow.

Direct Numerical Simulation of Reacting Mixing Layers. Effect of Heat Release and Density Difference of Mean Flow.

The objective of this paper is to clarify the effect of heat release and density difference in mean flow on the development of mixing layers. To achieve this purpose, we have performed a direct numerical simulation of a two-dimensional chemically reacting mixing layer by means of a spectral method. Low Mach number approximation was used to take into account the effect of density change. From these results, we have obtained the following conclusions. (1) Heat release causes baloclinic torque in the outside region of the coherent structure. Because of this baloclinic torque, the growth of the mixing layer and chemical reaction are suppressed. (2) The density difference of mean flow suppresses the growth of the mixing layer and chemical reaction because of the baloclinic torque and expansion. (3) The growth of the mixing layer and chemical reaction are further suppressed by the coexistence of heat release and density difference.

DIRECT SPECTRAL METHODS FOR THE LOW MACH NUMBER EQUATIONS

The low Mach number approximation of the Navier—Stokes equations is of similar nature to the equations for incompressible flow. A major difference, however, is the appearance of a space‐ and time‐varying density that introduces a supplementary non‐linearity. In order to solve these equations with spectral space discretization, an iterative solution method has been constructed and successfully applied in former work to two‐dimensional natural convection and isobaric combustion with one direction of periodicity. For the extension to other geometries efficiency is an important point, and it is therefore desirable to devise a direct method which would have, in the best case, the same stability properties as the iterative method. The present paper discusses in a systematic way different approaches to this aim. It turns out that direct methods avoiding the diffusive time step limit are possible, indeed. Although we focus for discussion and numerical investigation on natural convection flows, the results carry over for other problems such as variable viscosity flows, isobaric combustion, or non‐homogeneous flows.

Vortex simulation of a three-dimensional reacting shear layer with infinite-rate kinetics

The three-dimensional transport element method is extended to solve the conservation equations for reacting flow. The numerical scheme belongs to an adaptive, Lagrangian class of field methods in which computational effort is concentrated in zones of finite vorticity and chemical reaction. We use the low Mach number approximation of combustion and restrict our attention to the case of diffusion flames with no heat release. A single-step, second-order, infinite-rate kinetics chemical reaction model is employed. The scheme is applied to study the effect of flow-induced instabilities on the reaction field in a temporal shear layer

Combustion of a finite quatity of gas released in the atmosphere

Low Mach number and large activation energy asymptotic approximations are used to derive a model for the thermal behaviour of a finite planar, cylindrically or spherically symmetric fuel region released in an unconfined oxidant environment. A single one-step exothermic reaction is taken to occur. The induction stage has been studied in a previous paper [1] where it was shown that ignition occurs provided the fuel region exceeds a critical size. Below this size diffusive dilution starves the reaction too quickly for ignition to be possible. In this paper combustion of the system following ignition is studied. The analysis is equally valid whether ignition arises through self-heating or whether it is brought about by some other means such as a spark. Deflagration flames are formed which quickly consume the locally lean species at any point. A criterion is identified for any diffusion flame to exist after the passage of deflagration flames. A non-linear moving boundary formulation is developed to describe the progress and eventual collapse of such diffusion flames as the remaining fuel is consumed. The linearised version of these equations gives an upper limit for the time in which all combustion ceases.

Ignition and criticality of a finite quantity of gas released in the atmosphere

Low Mach number and large activation energy asymptotic approximations are used to establish a model for the thermal behaviour of a finite planar, cylindrically or spherically symmetric fuel region released in an unbounded oxidant environment. A single one-step exothermic reaction is taken to occur and solutions are obtained for the induction stage. In all cases thermal runaway, taking on a characteristic asymptotic form, is found to develop for large enough quantities of fuel; except near the critical size, thermal runaway develops more quickly than it does for the release of an infinite amount of fuel. Starvation of the reaction through diffusive dilution of the fuel is seen as the mechanism preventing thermal runaway for sub-critical sizes. A simple analysis illustrates the process and indicates that the rate of dilution (even for very small quantities) may not always be fast enough to prevent thermal runaway. In fact it is found numerically that the planar symmetric case with a first-order reaction does not show signs of this form of criticality. In all other cases, with integral reaction orders, criticality through starvation of the reaction does exist.