Low Mach Number Approximation Research Articles

Low Mach number approximated linearized perturbed Euler equations-based unified computational flow-induced acoustic model for 2D planar/axisymmetric complex geometry

This article is on proposition of a unified 2D planar and axisymmetric differential formulation—for both fluid dynamics and acoustic governing equations. Further, using hydrodynamic splitting method for computational acoustics, a differential formulation is proposed by performing low Mach number approximation on Linearized Perturbed Euler Equations (LPEE). Finally, using the differential formulations, a unified numerical methodology is proposed for Computational Flow-induced Acoustics (CFiA)—involving complex geometry—on a body-fitted curvilinear grid. A finite volume and finite difference methods-based hybrid method is presented for algebraic formulation in CFiA. Further, solution methodology is presented with a semi-implicit pressure projection method for fluid flow, four step Runge-Kutta method along with sub time-stepping for acoustic perturbations, and a one-way explicitly coupled solution algorithm for the CFiA. A detailed validation study is presented separately for the present in-house computational-acoustic and CFiA solvers—by considering various test problems, involving 2D planar and axisymmetric complex geometry problems. Further, using an order of accuracy study, the present acoustic and CFiA solvers are demonstrated as IV-order and II-order accurate, respectively. Finally, using the CFiA solver, a superior performance is demonstrated for the proposed low Mach number approximated LPEE as compared to the traditional LPEE-based solver. For low Mach number CFiA, involving 2D planar and axisymmetric complex geometries, a novel method is presented here for computationally efficient CFiA development.

Decomposition of the mechanical stress tensor: from the compressible Navier–Stokes equation to a turbulent potential flow model

We frame the mechanical stress tensor decomposition in a general procedure which involves the Helmholtz–Hodge decomposition. We highlight the impact of the mechanical stress tensor decomposition on the Navier–Stokes equation, with emphasis on the dissipation function. For fluids with low compressibility, we draw some insights on the Reynolds Averaged Navier–Stokes equations, and on the Reynolds stress tensor decomposition. We derive a turbulent potential flow model, and investigate the transition from viscous potential flow to turbulent potential flow. Under low Mach number approximation, we apply the turbulent potential flow model to one-dimensional propagation of large amplitude pressure waves in liquid-filled pipe.

Large eddy simulation of fire-induced flows using Lattice-Boltzmann methods

Large-eddy simulations (LES) of the near-field region of large-scale fire plumes are performed for the first time with a pressure-based Lattice Boltzmann method (LBM) with low-Mach number approximation. Two scenarios are considered: the large-scale non-reactive helium plume and the 1 m methane pool fire, both investigated experimentally at Sandia. In the second scenario, a simplified modeling of the combustion and radiation processes is introduced involving a one-step irreversible reaction eddy-dissipation concept-based combustion model and a radiant fraction model, respectively. In both scenarios, a quantitative agreement is observed with the experimental data and model predictions are consistent with previously-published numerical studies. Our simulations demonstrate the computational efficiency of the proposed LBM solver to tackle fire-induced flows, suggesting that LBMs are a good alternative candidate for the modeling of fire-related problems.

DNS of buoyancy-driven flows using EDAC formulation solved by high-order method

Entropically Damped Artificial Compressibility (EDAC) equation-based solution method is investigated to simulate the incompressible Navier–Stokes equations for buoyancy-driven flows. The method introduces an evolution equation for pressure, which is used to close the system of equations. The resulting parabolic system removes the need to solve the traditional Poisson’s equation at each time step. The energy equation with the Boussinesq approximation and the EDAC system of equations with the low Mach number approximation is solved for the thermal convection problem. This system is discretized using a sixth-order compact difference in space and advanced in time using an explicit fourth-order Runge–Kutta scheme. To investigate the suitability of the EDAC model for buoyancy flows, two widely used benchmark problems, namely: thermal cavity and Rayleigh–Bénard problems, are simulated. The simulation results are compared against the literature data. An excellent agreement is obtained, showing the feasibility and accuracy of the EDAC method in simulating buoyancy-driven flows. The EDAC pressure equation derived from entropy balance, together with the energy equation, are shown in this paper to model thermally dominant flows accurately.

Numerical Modeling of Hydrogen Combustion: Approaches and Benchmarks

The paper is devoted to the analysis of two different approaches for the numerical simulation of gaseous combustion. The first one is based on a full system of Navier-Stokes equations describing the dynamics of the compressible reactive medium, while the second one utilizes low-Mach number approximation. The compressible model is realized by the traditional low-order numerical scheme and the contemporary CABARET method. The low-Mach approach is implemented on the base of a widely known FDS numerical scheme. The benefits and disadvantages of compressible and low-Mach approaches are discussed and demonstrated on a specially developed set of problem setups, applicable for validation and verification of the numerical methods for combustion analysis. In particular, the laminar flame velocity test, spherical bomb test, and multidimensional modeling of combustion development inside the rectangular closed vessel are performed via both techniques that allowed to determine the applicability limits of the low-Mach number approximation.

Application of explicit Runge-Kutta-projection method to the low-Mach number approximation

Application of explicit Runge-Kutta-projection method to the low-Mach number approximation

A new hybrid lattice-Boltzmann method for thermal flow simulations in low-Mach number approximation

A new low-Mach algorithm for the thermal lattice Boltzmann method (LBM) is proposed aiming at reducing the computational cost of thermal flow simulations in the low Mach number limit. The well-known low Mach number approximation is adopted to accelerate the simulations by enlarging the time step through re-scaling the psuedoacoustic speed to the same order of the fluid motion velocity. This specific process is inspired by the similarity between the artificial compressibility method and the isothermal LBM and is further extended to its thermal counterpart. It must be emphasized that such low-Mach acceleration strategy is in a general form, thus can be easily applied to other compressible LB methods. The present method overcomes the drawback of the classical pressure gradient scaling method due to the pressure gradient changing. The new algorithm is validated by various well-documented academic test cases in laminar [one dimensional gravity column, 2D (two dimensional) rising thermal bubble, and 2D differentially heated square cavity] and turbulent [3D (three dimensional) Taylor–Green vortex and 3D heated cylinder] regimes. All the results show excellent agreement with the reference data and high computational efficiency.

Effect of molecular weight ratio on diffusion of light gases into air

We propose a numerical study on the effect of density of light gases on the behavior of a gas mixture (gas–air) in confined spaces and subjected to gradients of temperature and concentration. The diffusion of light gases requires the use of an appropriate model called low Mach number instead of the Boussinesq approximation which is not valid in such a situation. To solve the coupled momentum, energy, and concentration equations, an iterative methodology based on finite volume method using the low Mach number approximation was required. Representative results illustrating the effects of molecular weight ratio on the thermophysical and thermodynamic properties are reported. The average thermal and mass transfer rates on the active wall are presented and discussed. The results show that the hydrogen–air diffusion exhibits a particular behavior toward the compressibility of the binary mixtures. It is also found that the Boussinesq model is not sufficient to simulate diffusion of very light gases into air.

Development of unsteady natural convection in a square cavity under large temperature difference

To investigate how the nonuniform fluid density distribution caused by large temperature variations affects the development of unsteady natural convection, we perform a series of direct numerical simulations of two-dimensional compressible natural convection in an air-filled square cavity. The cavity has a hot wall on the left and a cold wall on the right, and two horizontal walls are adiabatic. The simulations are done using a kinetic approach based on a modeled Boltzmann equation, from which the fully compressible Navier–Stokes–Fourier equations are recovered. No Boussinesq approximation or low Mach number approximation is made. An extra source term is introduced to adjust the fluid Prandtl number. Simulations are performed for a range of Rayleigh numbers (107−109) with a fixed dimensionless temperature difference of ε=0.6 to determine the critical Rayleigh number and study the development of unsteady flow. To illustrate the instability mechanism, instantaneous fluctuation field, time trace of temperature, and velocity at selected monitoring points, the spectrum and other statistics are presented and discussed. As expected, significant differences are observed between the instability of compressible natural convection and the Boussinesq-type natural convection. With a large temperature difference, the transition to unsteady flow is asymmetric for the flows near the hot wall and cold wall. For the Rayleigh number range we studied, the cold wall region is dominated by low-frequency impact instability of the boundary thermal jet at the bottom corner. For the hot wall region, besides the upper corner impact instability, a boundary layer instability featuring high-frequency oscillations is observed.

A homogeneous relaxation low mach number model

In the present paper, we investigate a new homogeneous relaxation model describing the behaviour of a two-phase fluid flow in a low Mach number regime, which can be obtained as a low Mach number approximation of the well-known HRM. For this specific model, we derive an equation of state to describe the thermodynamics of the two-phase fluid. We prove some theoretical properties satisfied by the solutions of the model, and provide a well-balanced scheme. To go further, we investigate the instantaneous relaxation regime, and prove the formal convergence of this model towards the low Mach number approximation of the well-known HEM. An asymptotic-preserving scheme is introduced to allow numerical simulations of the coupling between spatial regions with different relaxation characteristic times.

Investigation of thermal large-eddy simulation approaches in a highly turbulent channel flow submitted to strong asymmetric heating

This study deals with thermal large-eddy simulation (T-LES) of anisothermal turbulent channel flow in the working conditions of solar receivers used in concentrated solar power towers. The flow is characterized by high-temperature levels and strong heat fluxes. The hot and cold friction Reynolds numbers of the simulations are, respectively, 630 and 970. The Navier–Stokes equations are solved under the low-Mach number approximation and the thermal dilatation is taken into account. The momentum convection and the density–velocity correlation subgrid terms are modeled. Functional, structural, and mixed subgrid-scale models are investigated. A tensorial version of the classical anisotropic minimum-dissipation (AMD) model is studied and produces good results. A Quick scheme and a second-order-centered scheme are tested for the discretization of the mass convection term. First, a global assessment of 22 large-eddy simulations is proposed, then six are selected for a careful analysis including profiles of mean quantities and fluctuation values as well as a comparison of instantaneous fields. Probability density functions of wall heat fluxes are plotted. The results point out that T-LESs performed with the Quick scheme tend to underestimate the wall heat flux whereas the second-order-centered scheme significantly improves its estimation. T-LESs tend to overestimate the peaks of velocity correlations. When regarding the dimensionless profiles of fluctuations, the tensorial AMD model provides better results than the other assessed models. For the heat flux estimation, the best agreement is found with the AMD model combined with the second-order-centered scheme.

Limit of the buoyancy ratio in Boussinesq approximation for double-diffusive convection in binary mixture

This paper deals with a mathematical and numerical investigation of double-diffusive natural convective heat and mass transfer in a cavity filled with Newtonian fluid with significant density and mass diffusivity changes. In such a situation, the assumption of the Boussinesq approximation is not justified, and an appropriate model based on a set of Low Mach Number equations is used. The active parts of two vertical walls of the cavity are maintained at fixed but different temperatures and concentrations, while the other two walls, as well as inactive areas of the sidewalls, are considered to be adiabatic and impermeable to mass transfer. The coupled momentum, energy, and solute transfer equations in binary mixtures of ideal gases are solved through a global iterative procedure based on the finite volume methods in the context of the low Mach number approximation. The study includes the effect of the buoyancy ratio N with the aim to find its application limit in the Boussinesq conditions. The results show that if we use the Boussinesq approximation to study double-diffusive convection, the value of parameter N must be between −6 and 27.

Optimizing the hypre solver for manycore and GPU architectures

The solution of large-scale combustion problems with codes such as Uintah on modern computer architectures requires the use of multithreading and GPUs to achieve performance. Uintah uses a low-Mach number approximation that requires iteratively solving a large system of linear equations. The Hypre iterative solver has solved such systems in a scalable way for Uintah, but the use of OpenMP with Hypre leads to at least 2× slowdown due to OpenMP overheads. The proposed solution uses the MPI Endpoints within Hypre, where each team of threads acts as a different MPI rank. This approach minimizes OpenMP synchronization overhead and performs as fast or (up to 1.44×) faster than Hypre's MPI-only version, and allows the rest of Uintah to be optimized using OpenMP. The profiling of the GPU version of Hypre shows the bottleneck to be the launch overhead of thousands of micro-kernels. The GPU performance was improved by fusing these micro-kernels and was further optimized by using Cuda-aware MPI, resulting in an overall speedup of 1.16—1.44× compared to the baseline GPU implementation.The above optimization strategies were published in the International Conference on Computational Science 2020 [1]. This work extends the previously published research by carrying out the second phase of communication-centered optimizations in Hypre to improve its scalability on large-scale supercomputers. This includes an efficient non-blocking inter-thread communication scheme, communication-reducing patch assignment, and expression of logical communication parallelism to a new version of the MPICH library that utilizes the underlying network parallelism [2]. The above optimizations avoid communication bottlenecks previously observed during strong scaling and improve performance by up to 2× on 256 nodes of Intel Knight's Landing processor.

Fully compressible simulations of the impact of acoustic waves on the dynamics of laminar premixed flames for engine-relevant conditions

Thermo-acoustic instabilities remain problematic in the design of propulsion systems such as gas turbine engines, rocket motors, and ramjets. They arise from the constructive interaction of heat release rate and acoustic pressure oscillations, and can result in increased noise and mechanical fatigue. In the present work, we are concerned with the flame response to the thermodynamic fluctuations that accompany an incident acoustic wave. The objective is to investigate the flame dynamics under engine-relevant conditions using high-fidelity numerical simulations and detailed chemical kinetics. The focus is placed on the combustion of hydrogen and n-heptane, as they are both of practical interest and behave very differently when subjected to acoustic waves. We extract the phase and gain of the unsteady heat release response, which are directly related to the Rayleigh criterion and thus the stability of the system. We highlight the differences between results obtained using the fully compressible Navier-Stokes equations and the low Mach number approximation. The two simulation frameworks agree very well for acoustic wavelengths much larger than the flame thickness. However, they differ significantly at high frequencies. The gain erroneously reaches a plateau under the low Mach number approximation, while it decays to zero using the fully compressible framework. This difference is attributed to the spatial variations in the acoustic pressure, which are not captured by the low Mach number approximation.

Turbulence with Large Thermal and Compositional Density Variations

Density variations in fluid flows can arise due to acoustic or thermal fluctuations, compositional changes during mixing of fluids with different molar masses, or phase inhomogeneities. In particular, thermal and compositional (with miscible fluids) density variations have many similarities, such as in how the flow interacts with a shock wave. Two limiting cases have been of particular interest: ( a) the single-fluid non-Oberbeck–Boussinesq low–Mach number approximation for flows with temperature variations, which describes vertical convection, and ( b) the incompressible limit of mixing between miscible fluids with different molar masses, which describes the Rayleigh–Taylor instability. The equations describing these cases are remarkably similar, with some differences in the molecular transport terms. In all cases, strong inertial effects lead to significant asymmetries of mixing, turbulence, and the shape of mixing layers. In addition, density variations require special attention in turbulence models to avoid viscous contamination of the large scales.

Peculiarities of mathematical modeling of combustion of hydrogen-air mixtures

Paper presents a comprehensive study of contemporary opportunities in the numerical analysis of transient combustion regimes on the example of the topical problem of hydrogen combustion. The elementary processes determining the development of combustion under different conditions are considered. In view of this analysis, it is concluded that the most appropriate for numerical modeling of classic deflagration inside confined space is the low-dissipation numerical schemes of low (second) order of approximation. For near-critical conditions where distinct mechanisms determine the combustion development one is able to utilize the simplifications such as low Mach number approximation.

Density gradient calculation in a class of multiphase lattice Boltzmann models.

The multiphase lattice Boltzmann (LB) models based on pairwise interactions show great potential for simulating multiphase flows due to the conceptual and computational simplicity. Although the dynamics of multiphase flows are reproduced by the pairwise interaction force, the gradient of density (or effective density, i.e., pseudopotential) is implicitly involved in these models via the specialized forcing scheme or the consistent scheme for ɛ^{3}-order term. This work focuses on the calculation of density gradient in this class of multiphase LB models. Theoretical analyses are first carried out to reveal the involvement and calculation of density gradient. On the basis of a low Mach number approximation, an improved scheme is then proposed to calculate the density gradient for the recent LB model with self-tuning equation of state. Analytical and numerical calculations show that the improved scheme is more accurate and can help to reduce the numerical error when the reduced temperature is relatively low.

Hybrid Lattice Boltzmann-finite difference model for low mach number combustion simulation

A hybrid Lattice Boltzmann(LB)-finite difference(FD) numerical scheme for the simulation of reacting flows at low Mach numbers is presented. The FD solver is used to model the energy and species fields while the LB model computes the flow field. The proposed LB solver is a modified version of the classical iso-thermal weakly compressible LB scheme with the hydrodynamic pressure as its zeroth-order moment instead of density, recovering the well-known low Mach number approximation for thermo-compressible flows. The proposed numerical approach is used to model a variety of test-cases ranging from 1-D to 3-D configurations, involving premixed and non-premixed flames. In all cases results obtained by this solver are shown to agree very well with reference data.

Penetration of the laminar flames of natural gas–oxygen mixtures through conical obstacles

It has been experimentally found that a flame of dilute natural gas–oxygen mixtures does not penetrate through the central opening of a confuser, but it penetrates only through the central opening of a diffuser, even if there are additional openings on the cone elements. The numerical modeling performed using compressible dimensionless reactive Navier–Stokes equations in a low Mach number approximation made it possible to qualitatively interpret the results.

The Rayleigh–Bénard convection in near-critical fluids: Influences of the specific heat ratio

This article studies the natural convection in near-critical fluids under the Rayleigh–Bénard configuration. The governing equations are non-dimensionalized by using modified scaling factors and five dimensionless parameters are determined, out of which the specific heat ratio γ is investigated thoroughly in terms of its influences on the convection. The governing equations are solved by the finite volume method along with the low Mach number approximation technique. It is revealed that the strength of the piston effect increases with γ. Since the piston effect is more efficient than the convection in terms of heat transfer, as γ is reduced, the relaxation of temperature field is weakened, while the development of convection is enhanced. By comparing the temperature and velocity fields in different cases and analyzing the heat transfer characteristic, we conclude that the final quasi-steady state is almost unaffected.