Low Mach Number Approximation Research Articles

Numerical Investigation of the Impact of Different Configurations and Aspect Ratios on Dense Gas Dispersion in Urban Street Canyons

The dispersion of chlorine gas in urban street canyons was numerically simulated using the fire dynamics simulator, a code developed by the National Institute of Standards and Technology of USA, which uses large eddy simulation coupled with the Smagorinsky sub-grid scale model. The unsteady flow fields were computed by solving the filtered incompressible Navier-Stokes equations under low Mach number approximation by the finite difference method. The studies analyzed the influence of different street canyon configurations and aspect ratios on the flow and chlorine gas dispersion. The geometric configuration and aspect ratio both affect the vortices and the local concentration distributions in street canyons.

Natural convection for large temperature gradients around a square solid body within a rectangular cavity

A numerical study of natural convection in cavity filled with air has been carried out under large temperature gradient. The flows under study are generated by a heated solid body located close to the bottom wall in a rectangular cavity with cold vertical walls and insulated horizontal walls. They have been investigated by direct simulations using a two-dimensional finite volume numerical code solving the time-dependent Navier–Stokes equations under the low Mach number approximation. This model permits to take into account large temperature variations unlike the classical Boussinesq model which is valid only for small temperature differences. We were particularly interested in the first transitions which occur when the Rayleigh number is increased for flows in cavities of aspect ratio A = 1, 2, 4. Starting from a steady state, the results obtained for A = 1 and A = 4 show that the first transition occurs through a supercritical Hopf bifurcation. The induced disturbances determined for weakly supercritical regimes indicate the existence of two instability types driven by different physical mechanisms: shear and buoyancy-driven instabilities, according to whether the flow develops in a square or in a tall cavity. For A = 2, the flow undergoes a pitchfork bifurcation leading to an asymmetric steady state which in turn becomes periodic via a supercritical Hopf bifurcation point. In both cases, the flow is found to be strongly deflected towards one vertical wall and instabilities are found to be of shear layers type.

Low Mach Number Modeling of Type I X‐Ray Burst Deflagrations

The low Mach number approximation (LMNA) is applied to two-dimensional hydrodynamic modeling of type I X-ray bursts on a rectangular patch on the surface of a nonrotating neutron star. Because such phenomena involve decidedly subsonic flows, the time step increase offered by the LMNA makes routine simulations of these deflagrations feasible in an environment where strong gravity produces significant stratification, while allowing for potentially significant lateral differences in temperature and density. The model is employed to simulate the heating, peak, and initial cooling stages in the deep envelope layers of a burst. During the deflagration, Benard-like cells naturally fill up a vertically expanding convective layer. The Mach number is always less than 0.15 throughout the simulation, thus justifying the low Mach number approximation. While the convective layer is superadiabatic on average, significant fluctuations in adiabaticity occur within it on subconvective timescales. Due to convective layer expansion, significant compositional mixing naturally occurs, but tracer particle penetration through the convective layer boundaries on convective timescales is temporary and spatially limited. Thus, mixing occurs on the relatively slow burst timescale through thermal expansion of the convective layer rather than from mass penetration of the convective layer boundary through particle convection. At the convective layer boundaries where mixing is less efficient, the actual temperature gradient more closely follows the Ledoux criterion.

Investigation of combustion noise using a LES/CAA hybrid approach

A hybrid approach combining large eddy simulation with computational aeroacoustics (LES/CAA) to describe combustion noise in the near field of a turbulent flame is presented. The approach is based on a LES of the reactive flow field using a low Mach number approximation and the solution of the wave equation with the appropriate combustion noise source term and inhomogeneous fluid properties. Two well established solvers are combined into a single tool for the description of the acoustics in the vicinity of the flame, exploiting the disparity of scales between the fluid flow and the acoustics. The complete approach is tested for a non-premixed turbulent jet flame and provides an insight into the combustion noise phenomenon. The LES is shown to yield good agreement with measurements for the velocity, the mixture fraction, as well as the temperature. The comparison of the acoustic results to an experimentally determined noise intensity spectrum allows to judge on the prediction quality of the presented LES/CAA hybrid approach.

Recent developments in the computation of compressible low Mach number flows

The computation of low speed flows can usualy be performed by incompressible models or various Low Mach number approximations (Boussinesq, Anelastic, etc). However, there is a large number of flows where although the velocities are small, compressible effects cannot be ignored and the use of full compressible models is required. Unfortunately, standard finite element or finite volume method experiences various difficulties in the computation of these low Mach number flows. This talk will explain the origin of these difficulties and describe some techniques to circumvent them.

Numerical Methods for Chemically Reacting Fluid Flow Computation under Low-Mach Number Approximation

A mathematical model of environmental fluid is presented to describe fluid flow motions with large density variations. Moreover the associated numerical methods are discussed. The model of environmental fluid is formulated as an unsteady low-Mach number flow based on the compressible Navier-Stokes equations. For low-Mach number flows, the acoustic effects are assumed to be weak relative to the advection effects. Under this assumption, detailed acoustic effects can be removed from governing equations. The low-Mach number formulation thus enables numerical flow analysis with a projection methodology that uses high-order accurate upwind difference of the convection terms with a time step restricted solely by an advection Courant-Friedrichs-Lewy (CFL) condition. The algorithm presented here is based on an iterative implicit time evolution of second order accuracy and a high-accurate spatial discretization with TVD properties for unsteady low-Mach number flows. It is seen from the results on the verification for test cases of flows with a wide range of density variations that our numerical method is validated.

A lattice Boltzmann algorithm for calculation of the laminar jet diffusion flame

A new two-distribution lattice Boltzmann equation (LBE) algorithm is presented to solve the laminar diffusion flames within the context of Burke–Schumann flame sheet model. One distribution models the transport of the Schvab–Zeldovich coupling function, or the mixture fraction to combine the energy and species equations. The other distribution models the quasi-incompressible Navier–Stokes equations with the low Mach number approximation. In the quasi-incompressible flows, the thermodynamics quantities depend on the coupling function but not on the hydrodynamic pressure, and the fluid components are assumed to be compressible only in the mixing/reaction region. A systematic and consistent approach to deriving LBEs for the general advection–diffusion equation and the quasi-incompressible Navier–Stokes equations are also presented. The streaming step of the LBEs are discretized by the total variation diminishing (TVD) Lax–Wendroff scheme. Numerical simulations are carried out to reproduce the low frequency flame oscillation (or flame flicker) of buoyant jet diffusion flame. Comparison between the quasi-incompressible model and the incompressible model is presented and the role of non-solenoidal velocity is examined.

Modélisation et simulation numérique du changement de phase liquide–vapeur en cavité

A model for the simulation of boiling flow with phase change in a closed cavity is presented. A front-tracking method is used to deal with the liquid–vapor interface. The liquid phase is incompressible while the vapor phase is weakly compressible and obeys to the perfect gas law. This model can deal with large density ratio ( ρ l / ρ v ≃ 1000 ) flows while accounting for the saturation curve. Computations are performed on a 1D validation case, idealizing a pressure cooker. Results are compared with a low Mach number approximation. To cite this article: V. Daru et al., C. R. Mecanique 334 (2006).

Noise radiated by a non-isothermal, temporal mixing layer

The ability of Lighthill's analogy to predict the sound radiated by a transitional mixing layer is evaluated by means of direct numerical simulation (DNS). The specific case of low Mach number flows with density variations is investigated. In order to limit the global computational cost, the acoustic source information is based on numerical results where the sound waves have been removed. It is shown that the low Mach number approximation coupled with the acoustic analogy can lead to very accurate predictions for the radiated sound if the acoustic sources in Lighthill's equation are taken into account carefully. Results for the acoustic intensity deduced from a repeated use of the Lighthill's analogy over a wide range of Mach numbers allow us to discuss the adequacy of scaling laws proposed by previous authors (J. Sound Vib. 28(3), 563–585, 1973; 31(4), 391–397, 1973; 48(1), 95–111, 1976) for the prediction of noise from hot jets.

Instability inside a rotating gas cylinder subject to axial periodic strain

We study numerically the instability of a confined rotating gas flow subject to periodic strain along the axis of rotation, under the low Mach number approximation. An axisymmetric time-stepping spectral Galerkin-type code is used to investigate the viscous basic flow and its stability. Parametric resonance can lead to instability of this flow via the growth of inertial modes coupled by the oscillating strain. The marginal stability curve compares well with earlier experimental and (asymptotic) analytical results in the case of the axisymmetric inertial mode (1,1,0). The resulting flow is dominated by a time-oscillating toroidal vortex and differs very little from the theoretical mode. Two different nonlinear regimes are found, one with saturation to a constant modal amplitude, the other with weak periodic modulation. We also show evidence of the presence of an azimuthal circulation, apparently responsible for the observed modulation.

A 3D finite volume method for the prediction of a supercritical fluid buoyant flow in a differentially heated cavity

This paper describes a three-dimensional finite volume method for the prediction of supercritical fluid buoyant flows in heated enclosures. The space and the time accuracies of the method are checked on an exact analytical solution, and the solver is validated for several benchmark tests for natural convection inside a differentially heated cavity both in the Boussinesq and in the low Mach number approximations. Then, the influence of the linearization of the van der Waals equation of state on the global convergence is discussed. Comparisons, based on the Rayleigh number, between the supercritical fluid and the perfect gas flows in a side heated cavity, show many similarities in the thermal equilibria and considerable differences in the transient behaviours.

An adaptation of the low Mach number approximation for supercritical fluid buoyant flows

This Note describes an acoustic filtering of the equations governing the supercritical fluid buoyant flow driven by a weak heating. The resulting low Mach number approximation takes into account the compressibility of the fluid with respect to the hydrostatic pressure. Using the direct numerical simulation of a supercritical fluid flow in the Rayleigh–Bénard configuration, we show that the density stratification may be taken into account without further numerical effort and is fundamental for the prediction of the convective instability threshold induced by a weak heating. To cite this article: G. Accary et al., C. R. Mecanique 333 (2005).

A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio

A stable discretization of the lattice Boltzmann equation (LBE) for non-ideal gases is presented for simulation of incompressible two-phase flows having high density and viscosity ratios. The stiffness of the discretized forcing terms in LBE for non-ideal gases is known to trigger severe numerical instability and restrict practical application of the LBE method. Use of a proper pressure updating scheme is also crucial to the stability of the LBE method because of non-negligible pressure variation across the phase interface. To deal with these issues, we propose a stable discretization scheme, which assumes the low Mach number approximation, and utilizes the stress and potential forms of the surface tension force, the incompressible transformation, and the consistent discretization of the intermolecular forcing terms. The proposed stable discretization scheme is applied to simulate 1-D advection equation with a source term, a stationary droplet, droplet oscillation and droplet splashing and deposition on a thin film at a density ratio of 1000 with varying Reynolds numbers. The numerical solutions of stationary and oscillatory droplets agree well with analytic solutions including the Laplace’s law. The time history of the spread factor of the liquid sheet emitted after the droplet impact also follows the known spreading power law.

Rayleigh–Bénard and schwarzschild instability in a supercritical fluid

Due to the density stratification, the convection onset criterion in a supercritical fluid layer heated from below includes a stabilizing effect known as the Schwarzschild criterion, which requires an adaptation of the classical low Mach number approximation. We numerically solve the mathematical model describing the motion of a supercritical fluid in the classical Rayleigh–Bénard configuration and we bring to the fore two main events: (i) the reverse transition of an unstable layer to stability without any external intervention; (ii) the convection onset according to the Schwarzschild criterion. In addition, the 3D extension of the study is briefly described.

반응 및 비반응 제트 유동장에서 단일 와동의 전개 및 구조에 대한 수치모사

반응 및 비반응 유동장에서 단일와동의 전개 및 와(渦) 구조 검토를 위해 2 차원 직접 수치모사를 수행하였다. 수치기법으로는 낮은 마하수 근사법이 적용된 예측-교정자법이 적용되었으며, 연소모텔로는 2단계 총괄 반응식이 사용되었다. 반응 및 비반응 유동장에서 단일와동의 거동 비교를 통해, 와동의 전개특성 및 구조는 화학반응으로부터 생성된 열 뿐만 아니라 부력에 의해 생성된 외부 와동에 크게 영향을 받음을 확인하였다. 또한 반응장의 경우에 부피 팽창항, Baroclinic torque항 및 부력항에 의해 와동의 구조가 크게 변화될 수 있음을 알았다. A two-dimensional direct numerical simulation was performed to investigate the evolution and vortical structure of a single vortex in reacting and non-reacting jet flow fields. A predictor-corrector-type numerical scheme with a low Mach number approximation was used, and a two-step global reaction mechanism was adopted as the combustion model. Through the comparisons of single vortex behaviors in reacting and non-reacting jet flow fields, it was found that the evolution characteristics and vortical structure of the single vortex were significantly influenced by a outer vortex that was generated from the buoyance effect as well as the chemical heat release. Furthermore, it was also identified that the differences of the vortical structure in reacting and non-reacting jet flow fields were mainly attributed to the thermal expansion, Baroclinic torque and buoyance effect.

Numerical analysis of thermal convections in a three-dimensional cavity: Influence of difference in approximation models on numerical solutions

In order to develop the practical approximation models suitable to flow fields at low Mach number with large temperature difference, the influence of difference in approximation models on numerical solutions was investigated by solving the natural convection in the 3-D enclosures with vertical sidewalls differentially heated and the heated bottom wall using 3 approximation models, that is Boussinesq approximation, low Mach Number approximation and approximation model proposed by Mlaouah. As results of the simulation, the effects of the differences in the three approximation models on the numerical solutions become clear.

An aeroacoustic hybrid approach for non‐isothermal flows at low Mach number

AbstractThis paper presents an aeroacoustic hybrid technique for the study of non‐isothermal flows at low Mach number. The flow dynamics and the acoustic production and propagation are computed separately. The fully compressible Navier–Stokes equations are modified through an expansion of the physical quantities using a low Mach number approximation. Compressibility effects are thus removed in the CFD while inhomogeneities of the flow related to heat transfer are preserved. One advantage is a reduction of the time step constraint. Another advantage is that the Mach number does not appear explicitly and a simple rescaling allows a study over a relatively wide band of subsonic Mach number flows with a single dynamic simulation.Compatible acoustic source terms for LEE based propagation have been defined and the procedure is implemented in the case of a temporal mixing layer. Compressible simulations for Mach numbers of 0.2, 0.3 and 0.4 are compared with the numerical results obtained using the proposed method. Very good agreement is obtained even at relatively high subsonic Mach number demonstrating the efficiency of the proposed technique. Copyright © 2004 John Wiley & Sons, Ltd.

低マッハ数近似式との比較によるBoussinesq近似式の予測精度の検討 : 圧縮性高浮力流れの数値解析に関する研究

In order to clarify the prediction accuracy of Boussinesq approximation, natural convection in a enclosure with heated and cooled vertical walls is analyzed by CFD technique based on Boussinesq approximation and low Mach number approximation. In the case of ΔT=300℃, the difference of the velocity profile between Boussinesq approximation and low Mach number approximation is great, and about 20% of the maximum velocity in the perpendicular direction. The cause which produces such a difference is mainly due to fact that the density of air in the continuity equation is constant in Boussinesq approximation.

A parallel adaptive projection method for low Mach number flows

AbstractWe describe an adaptive projection method for numerically simulating low Mach number flows. The projection method formulation enforces the velocity divergence constraint resulting from the low Mach number approximation. It is implemented on an adaptive hierarchy of logically rectangular grids, where each finer level is refined in space and in time. The adaptive algorithm has been shown in previous papers to be robust and second‐order accurate, and to satisfy the principles of conservation and free‐stream preservation as applicable. Here, the parallelization is described in some detail, and the methodology is demonstrated on two examples from premixed, low Mach number combustion. Published in 2002 by John Wiley & Sons, Ltd.

Benchmark solutions for the natural convective heat transfer problem in a square cavity

In this study, benchmark solutions are derived for the problem of two-dimensional laminar flow of air in a square cavity which is heated on the left, cooled on the right and insulated on the top and bottom boundaries. The temperature differences between the hot and the cold wall are large. No Boussinesq nor low-Mach number approximations of the Navier-Stokes equations are used. The ideal-gas law is used and viscosity is given by Sutherland's law. A constant local Prandtl number is assumed. An accurate and fast multigrid solver is used, which was developed in previous work. The solver is constructed so that its convergence behaviour is independent of grid size, grid aspect ratio, Mach number and Rayleigh number. Grid converged results with an accuracy of 4 up to 5 digits are obtained for a variety of Rayleigh numbers.