Mach Number Limit Research Articles

Dynamical friction on hot bodies in opaque, gaseous media

We consider the gravitational force exerted on a point-like perturber of mass $M$ travelling within a uniform gaseous, opaque medium at constant velocity $V$. The perturber irradiates the surrounding gas with luminosity $L$. The diffusion of the heat released is modelled with a uniform thermal diffusivity $\chi$. Using linear perturbation theory, we show that the force exerted by the perturbed gas on the perturber differs from the force without radiation (or standard dynamical friction). Hot, underdense gas trails the mass, which gives rise to a new force component, the heating force, with direction $+V$, thus opposed to the standard dynamical friction. In the limit of low Mach numbers, the heating force has expression $F_\mathrm{heat}=\gamma(\gamma-1)GML/(2\chi c_s^2)$, $c_s$ being the sound speed and $\gamma$ the ratio of specific heats. In the limit of large Mach numbers, $F_\mathrm{heat}=(\gamma-1)GML/(\chi V^2)f(r_\mathrm{min}V/4\chi)$, where $f$ is a function that diverges logarithmically as $r_\mathrm{min}$ tends to zero. Remarkably, the force in the low Mach number limit does not depend on the velocity. The equilibrium speed, when it exists, is set by the cancellation of the standard dynamical friction and heating force. In the low Mach number limit, it scales with the luminosity to mass ratio of the perturber. Using the above results suggests that Mars- to Earth-sized planetary embryos heated by accretion in a gaseous protoplanetary disc should have eccentricities and inclinations that amount to a sizeable fraction of the disc's aspect ratio, for conditions thought to prevail at a few astronomical units.

Variable property-based lattice Boltzmann flux solver for thermal flows in the low Mach number limit

A variable property-based lattice Boltzmann flux solver (VPLBFS) is proposed in this paper for thermal flows with partial or total variation in fluid properties in the low Mach number limit. In the solver, one particle distribution function is introduced for pressure and momentum, and another for fluid temperature. The fluid properties are assumed to be functions of only the local fluid temperature. The macroscopic variables at cell centers are determined from the solution of macroscopic governing equations by the finite volume method. The fluxes at cell interfaces are evaluated by the local construction of the solution for the standard lattice Boltzmann equation. The additional terms due to fluid property variations and body forces are regarded as source terms and treated by the finite volume discretization. These features make its application on non-uniform grids with a fixed time interval more flexible in comparison with conventional lattice Boltzmann models. The VPLBFS is validated by several numerical examples, including fluid flows between two parallel plates, with one plate to be isothermally heated from a certain moment, the natural convection in a square cavity with a large temperature difference and the natural convection of supercritical carbon dioxide. Numerical results show the reliability of the VPLBFS for thermal flows with partial or total variation in fluid properties.

Low Mach number limit for the quantum hydrodynamics system

In this paper, we deal with the low Mach number limit for the system of quantum hydrodynamics, far from the vortex nucleation regime. More precisely, in the framework of a periodic domain and ill-prepared initial data we prove strong convergence of the solutions toward regular solutions of the incompressible Euler system. In particular, we will perform a detailed analysis of the time oscillations and of the relative entropy functional related to the system.

Study of a New Asymptotic Preserving Scheme for the Euler System in the Low Mach Number Limit

This article deals with the discretization of the compressible Euler system for all Mach numbers regimes. For highly subsonic flows, since acoustic waves are very fast compared to the velocity of the fluid, the gas can be considered as incompressible. From the numerical point of view, when the Mach number tends to zero, the classical Godunov type schemes present two main drawbacks: they lose consistency and they suffer of severe numerical constraints for stability to be guaranteed since the time step must follow the acoustic waves speed. In this work, we propose and analyze a new unconditionally stable an consistent scheme for all Mach number flows, from compressible to incompressible regimes, stability being only related to the flow speed. A stability analysis and several one and two dimensional simulations confirm that the proposed method possesses the sought characteristics. 1. Introduction. Almost all fluids can be said to be compressible. However, there are many situations in which the changes in density are so small to be considered negligible. We refer to these situations saying that the fluid is in an incompressible regime. From the mathematical point of view, the difference between compressible and incompressible situations is that, in the second case, the equation for the conservation of mass is replaced by the constraint that the divergence of the velocity should be zero. This is due to the fact that when the Mach number tends to zero, the pressure waves can be considered to travel at infinite speed. From the theoretical point of view, researchers try to fill the gap between those two different descriptions by determining in which sense compressible equations tend to incompressible ones [2, 20, 21, 22, 33]. In this article we are interested in the numerical solution of the Euler system when used to describe fluid flows where the Mach number strongly varies. This causes the gas to pass from compressible to almost incompressible situations and consequently it causes most of the numerical methods build for solving compressible Euler equations to fail. In fact, when the Mach number tends to zero, it is well known that classical Godunov type schemes do not work anymore. Indeed, they lose consistency in the incompressible limit. This means that when close to the limit, the accuracy of theses schemes is not sufficient to describe the flow. Many efforts have been done in the recent past in order to correct this main drawback of Godunov schemes, for instance by using preconditioning methods [34] or by splitting and correcting the pressure on the collocated meshes [5], [9, 10], [12], [13, 14, 30], [15], [23, 24], [26, 27, 28], or instead by using staggered grids like in the famous MAC scheme, see for instance [3], [16], [17], [18], [19], [31]. Unfortunately, even if these approaches permit to bypass the consistency problem of Godunov methods, they all need to resolve the scale of the acoustic waves in the fluid in order to remain stable. This means that they suffer from a restrictive CFL (Courant-Frierichs-Levy) condition which is inversely proportional to the Mach number value. In this work, we derive a method which

Transition and Turbulence in a Wall-Bounded Channel Flow at High Mach Number

The turbulence in the viscous, compressible flow in a 3D wall-bounded channel, simulated using the direct simulation Monte Carlo (DSMC) method, has been used as a test bed for examining different aspects of transition and turbulence ([1, 2, 3]) at high Mach Ma = U_m / \\sqrt(gamma k_B T_w /m), and Reynolds numbers Re = (rho_m U_m H)/mu_w. Here, H is the channel half-width, U_m is the mean velocity, rho_m is the mean density, T_w is the wall temperature, m is the molecular mass, mu_w is the molecular viscosity based on the temperature at the isothermal wall, and k_B is the Boltzmann constant. The laminar-turbulent transition is accompanied by a discontinuous change in the friction factor even at high Mach number. The transition Reynolds number increases faster than linearly with Mach number, and the Knudsen number at transition (also proportional to the ratio of Mach and Reynolds numbers) passes through a maximum as the Mach number is increased. This maximum value is small, less than 0.009, indicating that transition is a continuum phenomenon even at high Mach numbers. The transition Reynolds number predicted by the linear stability analysis is significantly higher than that observed in the simulations, though its variation with Mach number is qualitatively similar. In a high Mach turbulent channel flow wall slip in the temperature and the velocities ([2]) are found to be significant. Interestingly, we find that the slip in the streamwise fluctuating velocity is higher than that in the mean velocity at high Mach number. We find that the amplitudes of the tangential fluctuating velocities increase between (0.2 - 0.4) power of the distance from the wall. In a compressible turbulent channel flow, we examine the result that the ratio of the mean free path and Kolmogorov scale increases proportional as (Ma/Re^{1/4}), and it increases asymptotically with Mach number in the high Mach number limit. The simulation show that the ratio (mean free path to Kolmogorov scale) does decrease as (Re^{-1/4}), but it does not increase linearly with Mach number. This is due to the decrease in the local Mach number within the channel, due to the increase in the temperature by viscous heating. We have also found that the smallest length scale for the velocity gradients is comparable to, or smaller than, the mean free path. Though this appears unusual, it should be noted that the smallest length for the gradients is the distance between molecules, and not the mean free path. In our simulations, the inter-molecular distance turns out to be much smaller than the mean free path. In fact, the inter-molecular distance is smaller than the cell size (we have 200 simulated molecules per cell), whereas the Kolmogorov scale and the mean free path are larger than the cell size. Even though the distance between molecules is the smallest length scale for gradients, the mean free path is the length scale for molecular transport, since the kinematic viscosity and thermal conductivity are proportional to the product of the mean free path and the fluctuating velocity. The present results suggest that the smallest scale for transport could be much larger than the smallest scale for gradients, thereby suggesting non-local transport in high Mach number turbulent flows at the smallest scales. A modification of the linear velocity profile in the viscous sub-layer near the wall, which takes into account temperature and density variations, is derived. The power law variation of the velocity and temperature is predicted under the assumption that the increase in temperature across the viscous sub-layer is larger than the wall temperature. It is found that the scaling laws do depend on the molecular model, through the dependence of viscosity and thermal conductivity on the temperature. The predicted power law, is found to be in good agreement with simulations, for two different molecular models, the hard-sphere and the variable hard-sphere.

Zero Mach number limit of the compressible Hall-magnetohydrodynamic equations

In this paper, we study the low Mach number limit of the compressible Hall-magnetohydrodynamic equations. It is justified rigorously that, for the well-prepared initial data, the classical solutions of the compressible Hall-magnetohydrodynamic equations converge to that of the incompressible Hall-magnetohydrodynamic equations as the Mach number tends to zero.

The Low Mach Number Limit for a Barotropic Model of Radiative Flow

We aim at justifying rigorously the low Mach number asymptotics for a model of compressible fluid coupled to the radiation. For simplicity, we restrict ourselves to the barotropic situation and adopt the so-called $P1$-approximation to model the effects of radiation. We focus on small perturbations of stable constant equilibria. In the critical regularity framework, we establish estimates independent of the Mach number and convergence results to the incompressible Navier--Stokes equation, exactly as in the nonradiative case. Our results hold true in the whole space $\mathbb{R}^n$ as well as in a periodic box $\mathbb{T}^n$ with $n\geq2.$

Entropic lattice Boltzmann model for compressible flows.

We present a lattice Boltzmann model (LBM) that covers the entire range of fluid flows, from low Mach weakly compressible to transonic and supersonic flows. One of the most restrictive limitations of the lattice Boltzmann method, the low Mach number limit, is overcome here by three fundamental changes to the LBM scheme: use of an appropriately chosen multispeed lattice, accurate evaluation of the equilibrium, and the entropic relaxation for the collision. The range of applications is demonstrated through the simulation of a bow shock in front of an airfoil and the simulation of decaying compressible turbulence with shocklets.

Influence of combustion-preheating vitiation on operability of a hypersonic inlet

Vitiation of the test flow with combustion products is inherent in combustion wind tunnels, and its effect on experimental results needs to be clarified. In this study, the influence of air vitiation on the startability and performance of a hypersonic inlet is investigated through two-dimensional (2D) numerical simulation. The study examines the vitiation effects introduced by carbon dioxide and water vapor, on the basis of maintaining the static pressure, static temperature and Mach number of the incoming flow. The starting Mach number limits of the inlet are estimated, and it is found that both of these vitiation components lower the starting limit of the inlet. This suggests that the experimental results acquired by tests in combustion wind tunnels overestimate the startability of an inlet and, therefore, combustion-preheated facilities may not be completely trusted in this respect. Deviations in the inlet performance caused by the vitiation are also detected. These are nevertheless minor as long as the flow is at the same started or unstarted condition. A further analysis reveals that it is mainly the increase in the heat capacity, and the resulting weaker shock/compression waves and shock-wave/boundary-layer interactions that account for the aforementioned effects.

Fully-implicit orthogonal reconstructed Discontinuous Galerkin method for fluid dynamics with phase change

A new reconstructed Discontinuous Galerkin (rDG) method, based on orthogonal basis/test functions, is developed for fluid flows on unstructured meshes. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton–Krylov based time integration. The method is designed for generic partial differential equations, including transient, hyperbolic, parabolic or elliptic operators, which are attributed to many multiphysics problems. We demonstrate the method's capabilities for solving compressible fluid–solid systems (in the low Mach number limit), with phase change (melting/solidification), as motivated by applications in Additive Manufacturing (AM). We focus on the method's accuracy (in both space and time), as well as robustness and solvability of the system of linear equations involved in the linearization steps of Newton-based methods. The performance of the developed method is investigated for highly-stiff problems with melting/solidification, emphasizing the advantages from tight coupling of mass, momentum and energy conservation equations, as well as orthogonality of basis functions, which leads to better conditioning of the underlying (approximate) Jacobian matrices, and rapid convergence of the Krylov-based linear solver.

The zero Mach number limit of the three-dimensional compressible viscous magnetohydrodynamic equations

This paper is concerned with the zeroMach number limit of the three-dimensional compressible viscous magnetohydrodynamic equations. More precisely, based on the local existence of the three-dimensional compressible viscous magnetohydrodynamic equations, first the convergence-stability principle is established. Then it is shown that, when the Mach number is sufficiently small, the periodic initial value problems of the equations have a unique smooth solution in the time interval, where the incompressible viscous magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number goes to zero. Moreover, the authors prove the convergence of smooth solutions of the equations towards those of the incompressible viscous magnetohydrodynamic equations with a sharp convergence rate.

A three dimensional lattice model for thermal compressible flow on standard lattices

A three-dimensional double distribution function thermal lattice Boltzmann model has been developed for simulation of thermal compressible flows in the low Mach number limit. Both the flow field and energy conservation equation are solved by LB approach. A higher order density distribution function on standard lattices is used to solve the flow field, while an energy distribution function is employed to compute the temperature field. The equation of state of thermal perfect gas is recovered by higher order Hermite polynomial expansions in Navier–Stokes–Fourier equations. The equilibrium distribution functions of D3Q15, D3Q19 and D3Q27 lattices are obtained from the Hermite expansion. They exhibit slight differences originating in differences in the discrete lattice symmetries. The correction terms in LB models for third order derivation are added using an external force in orthogonal polynomials form. Present models are successfully assessed considering several test cases, namely the thermal Couette flow, Rayleigh–Bénard convection, natural convection in square cavity and a spherical explosion in a 3D enclosed box. The numerical results are in good agreement with both analytical solution and results given by previous authors.

Uniform well-posedness and low Mach number limit to the compressible nematic liquid crystal flows in a bounded domain

We prove global-in-time and uniform-in-ϵ of the strong solutions to the 3D compressible nematic liquid crystal flows in a bounded domain, where ϵ is the Mach number. Consequently, we obtain the strong solution of compressible nematic liquid crystal model that converges to that of incompressible nematic liquid crystal model. This is the first result on the low Mach number limit for compressible nematic liquid crystal flows in 3D bounded domain. Our proof relies on the dedicated estimates on the solutions and the subtle use of the boundary conditions.

14-velocity and 18-velocity multiple-relaxation-time lattice Boltzmann models for three-dimensional incompressible flows

In this paper, 14-velocity and 18-velocity multiple-relaxation-time (MRT) lattice Boltzmann (LB) models are proposed for three-dimensional incompressible flows. These two models are constructed based on the incompressible LBGK model proposed by He et al. (2004) and the MRT LB model proposed by d’Humières et al. (2002), which have advantages in the computational efficiency and stability, respectively. Through the Chapman–Enskog analysis, the models can recover to three-dimensional incompressible Navier–Stokes equations in the low Mach number limit. To verify the present models, the steady Poiseuille flow, unsteady pulsatile flow and lid-driven cavity flow in three dimensions are simulated. The simulation results agree well with the analytical solutions or the existing numerical results. Moreover, it is found that the present models show higher accuracy than d’Humières et al. model and better stability than He et al. model.

On the 2D isentropic Euler system with unbounded initial vorticity

This paper is devoted to the study of the low Mach number limit for the 2D isentropic Euler system associated to ill-prepared initial data with slow blow up rate on log⁡ε−1. We prove in particular the strong convergence to the solution of the incompressible Euler system when the vorticity belongs to some weighted BMO spaces allowing unbounded functions. The proof is based on the extension of the result of [6] to a compressible transport model.

Numerical simulation of flow dynamics in the periodic regime in an idealized thermoacoustic engine

Flow dynamics in the stack and heat exchangers of a standing wave thermoacoustic engine is studied using two-dimensional direct numerical simulations. The numerical approach is based on asymptotic coupling in the low Mach number limit of a nonlinear dynamic model in the active cell with linear acoustics in the resonator. Computed results of the initial amplification and of the periodic regime eventually reached are shown. For the former, results show the existence of a critical temperature for which the system becomes unstable so that the engine starts, which is strongly dependent upon the load. Analysis of the results in the periodic regime shows the importance of vortex dynamics and the role of vortex shedding at sharp heat exchanger corners.

Singular limit of a dispersive Navier–Stokes system with an entropy structure

We prove a low Mach number limit for a dispersive fluid system [3] which contains third-order corrections to the compressible Navier–Stokes. We show that the classical solutions to this system in the whole space ℝn converge to classical solutions to ghost-effect systems [7]. Our analysis follows the framework in [4], which is built on the methodology developed by Métivier and Schochet [6] and Alazard [1] for systems up to the second order. The key new ingredient is the application of the entropy structure of the dispersive fluid system. This structure enables us to treat cases not covered in [4] and to simplify the analysis in [4].

Pfirsch–Schlüter neoclassical heavy impurity transport in a rotating plasma

In this work, we extend previous analytic theories for the neoclassical transport of a trace heavy impurity in a rotating plasma in the Pfirsch–Schlüter regime. The complete diffusive and convective components of the ambipolar particle flux are derived. The solution is valid for arbitrary impurity charge and impurity Mach number and for general geometry. Inclusion of finite main-ion temperature gradient effects is shown in the small ion Mach number limit. A simple interpolation formula is derived for the case of high impurity charge and circular geometry. While an enhancement of the diffusion coefficient is found for order one impurity Mach number, a reduction due to the rotation-driven poloidal asymmetry in the density occurs for very large Mach number.

Godunov-type schemes with an inertia term for unsteady full Mach number range flow calculations

An inertia term is introduced in the AUSM+-up scheme. The resulting scheme, called AUSM-IT (IT for Inertia Term), is designed as an extension of the AUSM+-up scheme allowing for full Mach number range calculations of unsteady flows including acoustic features. In line with the continuous asymptotic analysis, the AUSM-IT scheme satisfies the conservation of the discrete linear acoustic energy at first order in the low Mach number limit. Its capability to properly handle low Mach number unsteady flows, that may include acoustic waves or discontinuities, is numerically illustrated. The approach for building the AUSM-IT scheme from the AUSM+-up scheme is applicable to any other Godunov-type scheme.

Existence of strong solutions to the steady Navier–Stokes equations for a compressible heat-conductive fluid with large forces

We prove that there exists a strong solution to the Dirichlet boundary value problem for the steady Navier–Stokes equations of a compressible heat-conductive fluid with large external forces in a bounded domain Ω⊂Rd (d=2,3), provided that the Mach number is appropriately small. At the same time, the low Mach number limit is rigorously verified. The basic idea in the proof is to split the equations into two parts, one of which is similar to the steady incompressible Navier–Stokes equations with large forces, while another part corresponds to the steady compressible heat-conductive Navier–Stokes equations with small forces. The existence is then established by dealing with these two parts separately, establishing uniform in the Mach number a priori estimates and exploiting the known results on the steady incompressible Navier–Stokes equations.