Mach Number Limit Research Articles

Modelling aerodynamic forces and torques of spheroid particles in compressible flows

In the present study, we conduct numerical simulations of compressible flows around spheroid particles, for the purpose of refining empirical formulas for drag force, lift force, and pitching torque acting on them. Through an analysis of approximately a thousand numerical simulation cases spanning a wide range of Mach numbers, Reynolds numbers and particle aspect ratios, we first identify the crucial parameters that are strongly correlated with the forces and torques via Spearman correlation analysis, based on which the empirical formulas for the drag force, lift force and pitching torque coefficients are refined. The novel formulas developed for compressible flows exhibit consistency with their incompressible counterparts at low Mach number limits and, moreover, yield accurate predictions with average relative errors of less than 5%. This underscores their robustness and reliability in predicting aerodynamic loads on spheroidal particles under various flow conditions.

Low Mach number limit of a diffuse interface model for two‐phase flows of compressible viscous fluids

AbstractIn this paper, we consider a singular limit problem for a diffuse interface model for two immiscible compressible viscous fluids. Via a relative entropy method, we obtain a convergence result for the low Mach number limit to a corresponding system for incompressible fluids in the case of well‐prepared initial data and same densities in the limit.

How general is the Jensen–Varadhan large deviation functional for 1D conservation laws?

Starting from a microscopic particle model whose hydrodynamic limit under hyperbolic space-time scaling is a 1D conservation law, we derive the large deviation rate function encoding the probability to observe a density profile which is a non entropic shock, and compare this large deviation rate function with the classical Jensen-Varadhan functional, valid for the totally asymmetric exclusion process and the weakly asymmetric exclusion process in the strong asymmetry limit. We find that these two functionals have no reason to coincide, and in this sense Jensen-Varadhan functional is not universal. However, they do coincide in a small Mach number limit, suggesting that universality is restored in this regime. We then compute the leading order correction to the Jensen-Varadhan functional.

An artificial compressibility approach to solve low Mach number flows in closed domains

An artificial compressibility approach is proposed to compute the solution of the compressible equations in the low Mach number limit, in closed domain with moving boundaries. The low Mach number stiffness is reduced by introducing an artificial sound speed, much lower than the physical one. This allows to avoid both the acoustic time step restriction and the loss of accuracy of classical compressible solvers, without solving a Poisson equation for the pressure or using the time-implicit discretization of the Turkel-type preconditioning technique. Moreover the proposed formulation involves the conservative variables plus the dynamic pressure, which facilitates the implementation of the approach in classical CFD codes for compressible flows. The numerical experiments presented show that the approach is both accurate and CPU efficient.

A curl preserving finite volume scheme by space velocity enrichment. Application to the low Mach number accuracy problem

In this article, we address the problem of accuracy of finite volume schemes in the low Mach number limit. It has been known for years that collocated finite volume schemes are naturally correctly behaving in this limit on triangular meshes [21,22,16], but fail in general on other types of mesh. We are first interested in the general problem of the conservation of vorticity for the wave system. By enriching the approximation space for vectors, we prove that the Hodge-Helmholtz context developed for triangular meshes in [16] can be recovered in the quadrangular mesh case. This leads to a numerical scheme for the wave system that naturally preserves the vorticity under mild assumption on the numerical flux. The new approximation space is then used with the barotropic Euler system. Numerical tests show that the new numerical scheme is accurate for both steady and acoustic problems at low Mach number.

Modeling of low Mach number unsteady turbulent pipe flows

AbstractUnder adiabatic conditions, and neglecting temperature variations due to entropy production, we present a set of Reynolds Averaged Navier–Stokes (RANS) equations for fluids of low compressibility, i.e., fluids in the liquid state. In the low Mach number limit, we specialize the RANS equations to the one-dimensional unsteady pipe flow, and we deduce the dimensionless number that plays a predominant role in the flow behavior. We reduce the system of equations to a linear damped wave equation, and use its analytical solution to investigate the propagation of large amplitude pressure waves in liquid-filled pipes (water hammer phenomenon). We test the model reliability by comparing the analytical solution of the proposed model against experimental data available in the literature.

Low Mach number limit on perforated domains for the evolutionary Navier–Stokes–Fourier system

We consider the Navier–Stokes–Fourier system describing the motion of a compressible, viscous and heat-conducting fluid on a domain perforated by tiny holes. First, we identify a class of dissipative solutions to the Oberbeck–Boussinesq approximation as a low Mach number limit of the primitive system. Secondly, by proving the weak–strong uniqueness principle, we obtain strong convergence to the target system on the lifespan of the strong solution.

One-dimensional mechanism of gaseous deflagration-to-detonation transition

A one-dimensional mechanism of deflagration to detonation transition is identified and investigated by an asymptotic analysis in the double limit of large activation energy and small Mach number of the laminar flame velocity. The unsteady analysis concerns the self-accelerating tip of an elongated flame in a smooth walled tube. The flame on the tip, considered as plane and orthogonal to the tube axis, is pushed from behind by the longitudinal flow resulting from the cumulative effect of the radial flows of burned gas issued from the lateral flame of the finger-like front (called backflow in the following). The analysis of the one-dimensional dynamics is performed by coupling the flame structure with the downstream-running compression waves propagating in the external flows. A critical elongation is identified from which the slightest increase in elongation leads to a pressure runaway producing the flame blow-off. The dynamics of the inner structure of the laminar flame on the tip which is accelerated by the self-induced backflow is characterized by a finite-time singularity of the reacting flow in the form of a dynamical saddle-node bifurcation.

Electron beam driven ion-acoustic solitary waves in plasmas with two kappa-distributed electrons

Formation and the basic features of arbitrary amplitude ion-acoustic solitary waves (IASWs) in a plasma consisting of warm positive ions, two kappa-distributed electrons and an electron beam are investigated by using the Sagdeev pseudopotential approach. It is shown that the soliton existence domain (Mach number limits) sensitively depends on temperature of ions, spectral index of cool electrons and concentration of hot electron species while spectral index of hot electrons, hot-to-cool electron temperature ratio and also concentration of electron beam do not considerably affect this domain. It is also found that temperature of electron beam only affect the existence domain of rarefactive solitons. Furthermore, it is shown that considered plasma medium supports the coexistence of positive and negative IASWs. Moreover, effect of different plasma parameters such as hot-to-cool electron density ratio, ion-to-cool electron temperature ratio, beam-to-ion density ratio, hot-to-cool electron temperature ratio and superthermality index of electron species on the basic features of positive and negative IASWs is investigated numerically. Finally, the effect of plasma parameters on the parametric regime of coexistence of compressive and rarefactive IASWs is studied and, for example, effect of temperature of positive ions and number density of hot electrons on polarity of IASWs is numerically investigated.

On a fully-implicit VMS-stabilized FE formulation for low Mach number compressible resistive MHD with application to MCF

This study presents the development and evaluation of a fully-implicit variational multiscale (VMS) stabilized unstructured finite element (FE) formulation for compressible magnetohydrodynamics (MHD) model, at low Mach number regime. The model describes the dynamics of a compressible conducting fluid in the low Mach number limit in the presence of electromagnetic fields and can be used to study aspects of astrophysical phenomena, important science and technology applications, and basic plasma physics phenomena. The specific applications that motivate this study are macroscopic simulations of the longer time-scale stability and disruptions of magnetic confinement fusion (MCF) devices, specifically the ITER tokamak. The discussion considers the development of the VMS FE representation, the structure of the stabilizing terms that deal with significant convective flows, the stabilization of the nearly incompressible response of the fluid flow, and the stabilization of the constraint that enforces the solenoidal involution on the magnetic field. The nonlinear discretized system is solved with scalable preconditioned Newton–Krylov iterative methods, which employs a multiphysics block preconditioning method based on approximate block factorizations and Schur complements. The study presents an evaluation of the VMS method on a 2D cartesian tearing mode instability, and illustrates the scalability of the solvers on MCF relevant problems. A set of results are also presented for longer time-scale stability and disruptions for the ITER tokamak. These include a vertical displacement event (VDE), and a (1,1) internal kink mode. The formulation is demonstrated to be scalable and also reasonably robust with respect to the Lundquist number scaling.

Incompressible limit for compressible viscoelastic flows with large velocity

Abstract We are concerned with the incompressible limit of global-in-time strong solutions with arbitrary large initial velocity for the three-dimensional compressible viscoelastic equations. The incompressibility is achieved by the large value of the volume viscosity, which is different from the low Mach number limit. To obtain the uniform estimates, we establish the estimates for the potential part and the divergence-free part of the velocity. As the volume viscosity goes to infinity, the dispersion associated with the pressure waves tends to disappear, but the large volume viscosity provides a strong dissipation on the potential part of the velocity forcing the flow to be almost incompressible.

Lattice Boltzmann solidification modeling of forced convection internal flows applied to Gen-IV nuclear reactor coolants

Generation-IV nuclear reactors involve novel coolants such as molten salts and lead. One of the issues that must be assessed with these coolants is the risk of solidification during reactor operation owing to their high melting temperature. However, the numerical simulation of solidification problems is computationally demanding. In this work, we assess the applicability of Lattice Boltzmann Method (LBM) to nuclear coolants solidification problems under laminar internal forced flow conditions. The double distribution Total Enthalpy-Partially Saturated Method is implemented in the open-source code OpenLB and compared against more standard Finite-Volume Computational Fluid Dynamics (FV-CFD) results, which uses the enthalpy-porosity Eulerian Multiphase method, in commercial code Star-CCM+. The test cases comprise the solidification of two fluids with significantly different Prandtl numbers, molten salt and lead, in a 3D pipe geometry in laminar flow conditions with an initial Reynolds number of 100. The solid interface advances from the cooled wall towards the bulk of the pipe reducing the area of flow passage as the transient evolves. Results of LBM velocity fields, solid fraction profiles, temperature fields, and pressure fields are in good agreement with FV-CFD results for both coolants. Furthermore, LBM stability and accuracy limits are independently explored for molten salts and lead using Two-Relaxation Time (TRT) and Bhatnagar-Gross-Krook (BGK) collision operators. Simulation time between FV-CFD and LBM is also contrasted for both coolants. In this work, LBM solvers use explicit time schemes with a regular grid, whereas FV-CFD solutions are able to include implicit time schemes and near-wall mesh refinement. For molten salt flows (high Prandtl number) FV-CFD allows for considerably larger timesteps than LBM, which is limited by the maximum lattice Mach number limitation and stability considerations. For lead flows (low Prandtl number), the implicit scheme in FV-CFD is limited in terms of accuracy for large timesteps due to the limiting Fourier-number. On the other hand, LBM does not present notable stability issues for lead and the lattice Mach number limit is not as relevant as for the molten salt case for this Reynolds number, allowing for faster simulation times than FV-CFD.

Singular limit of planar magnetohydrodynamic equations with vacuum free boundary

In this paper, we study the low Mach and Froude number limit of global strong solutions to the fluid‐vacuum free boundary problem of the planar magnetohydrodynamic (MHD) equations with degenerate viscosity coefficients and well‐prepared initial data. Only the initial energy at the basic level is required to be small. The main difficulties are the degeneracy of the system yielded by vanishing of the density close to the free boundary, the singular terms due to the large coefficients inversely proportional to the Mach number and the Froude number, and the strong coupling of the magnetic field and the velocity. The uniform estimates of solutions with respect to the Mach number and the Froude number are established, which is the first result on the MHD equations with vacuum free boundary. At the same time, we obtain the convergence rates for the strong solutions global in time with well‐prepared initial data as the Mach number and the Froude number vanish.

Low Mach number limit of the global solution to the compressible Navier–Stokes system for large data in the critical Besov space

In this paper, we consider the compressible Navier–Stokes system around the constant equilibrium states and prove the existence of a unique global solution for arbitrarily large initial data in the scaling critical Besov space provided that the Mach number is sufficiently small, and the incompressible part of the initial velocity generates the global solution of the incompressible Navier–Stokes equation. Moreover, we consider the low Mach number limit and show that the compressible solution converges to the solution of the incompressible Navier–Stokes equation in some space time norms.

Measure-Valued Low Mach Number Limits of Ideal Fluids

As a framework for handling low Mach number limits we consider a notion of measure-valued solution of the incompressible Euler system which explicitly formulates the usually suppressed dependence on the pressure variable. We clarify in which sense such a measure-valued solution is generated as a low Mach limit and state sufficient conditions for this convergence. For the special case of pressure-free solutions we are able to give such sufficient conditions only on the incompressible solution itself. As a necessary condition for low Mach limits we obtain a Jensen-type inequality. The same necessary condition actually holds true for limits of weak solutions or vanishing viscosity limits. We illustrate that this Jensen inequality is not trivially fulfilled. Since measure-valued solutions in the classical sense are always generated by weak solutions, this shows that our solution concept contains more information and that low Mach limits lead to a partial selection criterion for incompressible solutions.

Low Mach number limit of the compressible Navier–Stokes–Cattaneo equations with general initial data

In this paper, we investigate the low Mach number limit of the compressible Navier–Stokes–Cattaneo equations in the whole space R3. In particular, the large variations of density, temperature and heat flux are taken into account. By introducing an appropriate time-weighted normed space, the uniform estimates of the solutions are established for general initial data. Then, the convergence of the solutions is proved by combining the uniform estimates and the local energy decay of acoustic waves. Finally, we derive a new limit equations which reveal the effect of heat flux on the field of velocity.

Incompressible limit for the free surface Navier-Stokes system

We establish uniform regularity estimates with respect to the Mach number for the three-dimensional free surface compressible Navier-Stokes system in the case of slightly well-prepared initial data in the sense that the acoustic components like the divergence of the velocity field are of size $$\sqrt{\varepsilon }$$ , $$\varepsilon $$ being the Mach number. These estimates allow us to justify the convergence towards the free surface incompressible Navier-Stokes system in the low Mach number limit. One of the main difficulties is the control of the regularity of the surface in presence of boundary layers with fast oscillations.

All-Speed Numerical Methods for the Euler Equations via a Sequential Explicit Time Integration

This paper presents a new strategy to deal with the excessive diffusion that standard finite volume methods for compressible Euler equations display in the limit of low Mach number. The strategy can be understood as using centered discretizations for the acoustic part of the Euler equations and stabilizing them with a leap-frog-type (“sequential explicit”) time integration, a fully explicit method. This time integration takes inspiration from time-explicit staggered grid numerical methods. In this way, advantages of staggered methods carry over to collocated methods. The paper provides a number of new collocated schemes for linear acoustic/Maxwell equations that are inspired by the Yee scheme. They are then extended to an all-speed method for the full Euler equations on Cartesian grids. By taking the opposite view and taking inspiration from collocated methods, the paper also suggests a new way of staggering the variables which increases the stability as compared to the traditional Yee scheme.

Incompressible limit for a fluid mixture

In this paper we discuss the incompressible limit for multicomponent fluids in the isothermal ideal case. Both a direct limit-passage in the equation of state and the low Mach-number limit in rescaled PDEs are investigated. Using the relative energy inequality, we obtain convergence results for the densities and the velocity-field under the condition that the incompressible model possesses a sufficiently smooth solution, which is granted at least for a short time. Moreover, in comparison to single-component flows, uniform estimates and the convergence of the pressure are needed in the multicomponent case because the incompressible velocity field is not divergence-free. We show that certain constellations of the mobility tensor allow to control gradients of the entropic variables and yield the convergence of the pressure in L1.

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].