Finite Mach Number Research Articles

A dynamical system approach for solving the steady structure of high speed deflagrations

High-speed deflagrations have burning velocities much higher than laminar ones, and compressible effects become important. In the present paper, we study the structure of such high speed deflagrations in the thickened flame limit for a one-step Arrhenius rate law, whereby the transport coefficients are increased to give rise to buring velocities of finite Mach number. We study their steady structure and compare with the laminar low-Mach classical structure. The singular nature of both the fresh and burned gases conditions in the compressible regime, i.e., for large values of flame propagation Mach number, which are both saddle points, precludes the application of simple shooting methods to obtain their structure. The steps leading to the flame structure borrow techniques used to treat mathematical features commonly found in the study of dynamical systems (phase portrait of the flame structure equations system, eigenvalue decomposition of its linearized version), that can also be used to further comment on the nature of high-speed flames. The method proposed permits to determine the structure of deflagrations propagating up to speeds of aproximately 0.95 of the CJ-deflagration burning velocity, for a wide range of gas parameters commonly found in the field of numerical simulation of accelerating flames and their transitions to detonations. We comment on how the increasing role of compressibility modifies the structure of the laminar flame.

On the scattering of entropy waves at sudden area expansions

In this work, we investigate both numerically and theoretically the sound generated by entropy waves passing through sudden area expansions. This is a canonical configuration representing internal flows with flow separation and stagnation pressure losses. The numerical approach is based on a triple decomposition of the flow variables into a steady mean, a small-amplitude coherent part, and a stochastic turbulent part. The coherent part contains acoustic, vortical, and entropy waves. The mean flow is obtained as the solution of the Reynolds-Averaged Navier–Stokes (RANS) equations. The equations governing the coherent perturbations are linearised and solved in the frequency domain. To account for the effect of turbulence on the coherent perturbations, a frozen eddy viscosity model is employed. When entropy fluctuations pass through the area expansion, the generated entropy noise behaves as a low-pass filter. The numerical predictions of the noise at low frequencies are compared to the predictions of compact, quasi-one-dimensional, and isentropic theory and large discrepancies are observed. An alternative model for the generated entropy noise tailored for area expansions is then proposed. Such model is based on the conservation of mass, momentum, and energy written in integral form. The model assumes zero frequency and the one-dimensionality of the flow variables far upstream and downstream of the expansion. The predictions of this model agree well with the numerical simulations across a range of finite subsonic Mach numbers including low, intermediate, and high Mach numbers. The contributions of this work are both numerical and theoretical. Numerically, a triple decomposition adapted to high-Mach-number, compressible flows is introduced for the first time in the context of acoustic simulations. From a theoretical point of view, the quasi-steady model proposed here correctly captures the low-frequency entropy noise generated at sudden area expansions, including at high subsonic Mach numbers.

Density and tracer statistics in compressible turbulence: Phase transition to multifractality.

We study the statistics of fluid (gas) density and concentration of passive tracer particles (dust) in compressible turbulence. As Ma increases from small or moderate values, the density and the concentration in the inertial range go through a phase transition from a finite continuous smooth distribution to a singular multifractal spatial distribution. Multifractality is associated with scaling, which would not hold if the solenoidal and the potential components of the flow scaled differently, producing transport which is not self-similar. Thus, we propose that the transition occurs when the difference of the scaling exponents of the components, decreasing with Ma, becomes small. Under the smallness assumption, the particles' volumes obey a power-law evolution. That, by the use of conservation of the total volume of the flow, entails the volumes' shrinking to zero with probability 1 and formation of a singular distribution. We discuss various concepts of multifractality and propose a way to calculate fractal dimensions from numerical or experimental data. We derive a simple expression for the spectrum of fractal dimensions of isothermal turbulence and describe limitations of lognormality. The expression depends on a single parameter: the scaling exponent of the density spectrum. We demonstrate that the pair-correlation function of the tracer concentration has the Markov property. This implies applicability of the compressible version of the Kraichnan turbulence model. We use the model to derive an explicit expression for the tracer pair correlation that demonstrates their smooth transition to multifractality and confirms the transition's mechanism. The obtained fractal dimension explains previous numerical observations. Our results have potentially important implications for astrophysical problems such as star formation as well as technological applications such as supersonic combustion. As an example, we demonstrate the strong increase of planetesimal formation rate at the transition. We prove that finiteness of internal energy implies vanishing of the sum of Lyapunov exponents in the dissipation range. Our study leads to the question of whether the fluid density which is an active field that reacts back on the transporting flow and the passive concentration of tracers must coincide in the steady state. This is demonstrated to be crucial both theoretically and experimentally. The fields' coincidence is provable at small Mach numbers; however, at finite Mach numbers, the assumption of mixing is needed, which we demonstrate to be not self-evident because of the possibility of self-organization.

Self-consistent feedback mechanism for the sudden viscous dissipation of finite-Mach-number compressing turbulence.

Previous work [Davidovits and Fisch, Phys. Rev. Lett. 116, 105004 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.105004] demonstrated that the compression of a turbulent field can lead to a sudden viscous dissipation of turbulent kinetic energy (TKE), and that paper suggested this mechanism could potentially be used to design new fast-ignition schemes for inertial confinement fusion (ICF). We expand on previous work by accounting for finite Mach numbers, rather than relying on a zero-Mach-limit assumption as previously done. The finite-Mach-number formulation is necessary to capture a self-consistent feedback mechanism in which dissipated TKE increases the temperature of the system, which in turn modifies the viscosity and thus the TKE dissipation itself. Direct numerical simulations with a tenth-order accurate Padé scheme were carried out to analyze this self-consistent feedback loop for compressing turbulence. Results show that, for finite Mach numbers, the sudden viscous dissipation of TKE still occurs, for both the solenoidal and dilatational turbulent fields. As the domain is compressed, oscillations in dilatational TKE are encountered due to the highly oscillatory nature of the pressure dilatation. An analysis of the source terms for the internal energy shows that the mechanical-work term dominates the viscous turbulent dissipation. As a result, the effect of the suddenly dissipated TKE on temperature is minimal for the Mach numbers tested. Moreover, an analytical expression is derived that confirms the dissipated TKE does not significantly alter the temperature evolution for low Mach numbers, regardless of compression speed. The self-consistent feedback mechanism is thus quite weak for subsonic turbulence, which could limit its applicability for ICF.

Moderation of flame acceleration in obstructed cylindrical pipes due to gas compression

The role of gas compression in the process of extremely fast flame acceleration in obstructed cylindrical tubes is studied analytically and validated by computational simulations. This acceleration is associated with a powerful jet-flow produced by delayed burning in spaces between the obstacles. The acceleration mechanism is Reynolds-independent and conceptually laminar, with turbulence playing only a supplementary role. In this work, the incompressible formulation is extended to account for small but finite initial flame Mach number up to the first-order terms. As a result, it is shown that flame-generated compression mitigates the acceleration process noticeably. Being qualitatively the same as in two-dimensional channels, quantitatively, the effect is much stronger in cylindrical pipes, which reduces the validity domain of the incompressible cylindrical-axisymmetric formulation as compared to the incompressible two-dimensional theory. The latter is also tested when the theory is validated by the computational simulations.

Quantitative Pointwise Estimate of the Solution of the Linearized Boltzmann Equation

We study the quantitative pointwise behavior of the solutions of the linearized Boltzmann equation for hard potentials, Maxwellian molecules and soft potentials, with Grad's angular cutoff assumption. More precisely, for solutions inside the finite Mach number region, we obtain the pointwise fluid structure for hard potentials and Maxwellian molecules, and optimal time decay in the fluid part and sub-exponential time decay in the non-fluid part for soft potentials. For solutions outside the finite Mach number region, we obtain sub-exponential decay in the space variable. The singular wave estimate, regularization estimate and refined weighted energy estimate play important roles in this paper. Our results largely extend the classical results of Liu-Yu \cite{[LiuYu], [LiuYu2], [LiuYu1]} and Lee-Liu-Yu \cite% {[LeeLiuYu]} to hard and soft potentials by imposing suitable exponential velocity weight on the initial condition.

Green's function and pointwise estimate for a generalized Poisson‐Nernst‐Planck‐Navier‐Stokes model in dimension three

AbstractThe Cauchy's problem of the generalized Poisson‐Nernst‐Planck‐Navier‐Stokes model in dimension three is considered. First, after dividing the physical domain into two parts: finite Mach number region and outside finite Mach number region, we give pointwise estimates of the Green's function by using long‐wave short‐wave decomposition and weighted energy estimate method in each region separately. Then from Duhamel's principle and some estimates on the nonlinear interactions between different wave patterns, we give the pointwise estimates of the solution for the nonlinear problem –, which exhibit generalized Huygens' principle for mass density, macroscopic momentum, negative charge distribution and positive charge distribution. As a byproduct, we find that the decay rates in () for both negative charge distribution and positive charge distribution are faster than those for the mass density ρ and the macroscopic momentum. More importantly, we obtained both electrostatic potential and the difference between negative charge distribution and positive charge distribution have the exponential decay rate in ‐norm when .

Solving the non-isentropic Navier-Stokes equations in odd space dimensions: The Green function method

In this paper, we investigate the wave structure of the solution for the compressible Navier-Stokes equations in both one space dimension and three space dimensions. First, we give the pointwise estimate on the Green function for the linearized system by dividing the physical domain into two parts, which are the finite Mach number region and outside finite Mach number region. We use long-wave short-wave decomposition and a weighted energy estimate method in each region separately. The Green function provides a complete picture of the wave pattern, with explicit leading terms. Then with the help of the Duhamel principle, we give the pointwise estimate of the solution for the nonlinear problems (1.1) and (1.2). Specially, we find that the decay rate in Lp(R3) (2<p≤∞) for the energy ω̃ is faster than those for the density ρ̃ and the momentum m̃.

How vortices and shocks provide for a flux loop in two-dimensional compressible turbulence

Large-scale turbulence in fluid layers and other quasi-two-dimensional compressible systems consists of planar vortices and waves. Separately, wave turbulence usually produces a direct energy cascade, while solenoidal planar turbulence transports energy to large scales by an inverse cascade. Here, we consider turbulence at finite Mach numbers when the interaction between acoustic waves and vortices is substantial. We employ solenoidal pumping at intermediate scales and show how both direct and inverse energy cascades are formed starting from the pumping scale. We show that there is an inverse cascade of kinetic energy up to a scale $\ell$, where a typical velocity reaches the speed of sound; this creates shock waves, which provide for a compensating direct cascade. When the system size is less than $\ell$, the steady state contains a system-size pair of long-living condensate vortices connected by a system of shocks. Thus turbulence in fluid layers processes energy via a loop: Most energy first goes to large scales via vortices and is then transported by waves to small-scale dissipation.

Non-diffusive ignition of a gaseous reactive mixture following time-resolved, spatially distributed energy deposition

Systematic asymptotic methods are applied to the compressible conservation and state equations for a reactive gas, including transport terms, to develop a rational thermomechanical formulation for the ignition of a chemical reaction following time-resolved, spatially distributed thermal energy addition from an external source into a finite volume of gas. A multi-parameter asymptotic analysis is developed for a wide range of energy deposition levels relative to the initial internal energy in the volume when the heating timescale is short compared to the characteristic acoustic timescale of the volume. Below a quantitatively defined threshold for energy addition, a nearly constant volume heating process occurs, with a small but finite internal gas expansion Mach number. Very little added thermal energy is converted to kinetic energy. The gas expelled from the boundary of the hot, high-pressure spot is the source of mechanical disturbances (acoustic and shock waves) that propagate away into the neighbouring unheated gas. When the energy addition reaches the threshold value, the heating process is fully compressible with a substantial internal gas expansion Mach number, the source of blast waves propagating into the unheated environmental gas. This case corresponds to an extremely large non-dimensional hot-spot temperature and pressure. If the former is sufficiently large, a high activation energy chemical reaction is initiated on the short heating timescale. This phenomenon is in contrast to that for more modest levels of energy addition, where a thermal explosion occurs only after the familiar extended ignition delay period for a classical high activation reaction. Transport effects, modulated by an asymptotically small Knudsen number, are shown to be negligible unless a local gradient in temperature, concentration or velocity is exceptionally large.

Finite Mach number spherical shock wave, application to shock ignition

A converging and diverging spherical shock wave with a finite initial Mach number Ms0 is described by using a perturbative approach over a small parameter Ms−2. The zeroth order solution is the Guderley's self-similar solution. The first order correction to this solution accounts for the effects of the shock strength. Whereas it was constant in the Guderley's asymptotic solution, the amplification factor of the finite amplitude shock Λ(t)∝dUs/dRs now varies in time. The coefficients present in its series form are iteratively calculated so that the solution does not undergo any singular behavior apart from the position of the shock. The analytical form of the corrected solution in the vicinity of singular points provides a better physical understanding of the finite shock Mach number effects. The correction affects mainly the flow density and the pressure after the shock rebound. In application to the shock ignition scheme, it is shown that the ignition criterion is modified by more than 20% if the fuel pressure prior to the final shock is taken into account. A good agreement is obtained with hydrodynamic simulations using a Lagrangian code.

Linearized-Boltzmann-type-equation-based finite difference method for thermal incompressible flow

This study reports on further development of a finite difference method formulated on the basis of a linearized-Boltzmann-type-equation for thermal incompressible flows with external body force effect. In classical lattice Boltzmann methods, a pressure-density relation, and/or a finite Mach number, no matter how small, are required in the solution of the linearized Boltzmann-type equation, thus generating inherent compressibility error unavoidably. In the present approach, the pressure field is determined by a pressure-correction method to ensure incompressibility, thus the approach is valid for both liquid and incompressible gas flows. A variety of thermal laminar incompressible flows, such as Couette flow, falling thin liquid film flow, fluid flow through porous plates, and two- and three-dimensional natural convection flow are simulated. The results compared extremely well with analytical solutions and other known numerical simulations of the thermal incompressible flows investigated.

Linear analysis of the Richtmyer-Meshkov instability in shock-flame interactions

Shock-flame interactions enhance supersonic mixing and detonation formation. Therefore, their analysis is important to explosion safety, internal combustion engine performance, and supersonic combustor design. The fundamental process at the basis of the interaction is the Richtmyer-Meshkov instability supported by the density difference between burnt and fresh mixtures. In the present study we analyze the effect of reactivity on the Richtmyer-Meshkov instability with particular emphasis on combustion lengths that typify the scaling between perturbation growth and induction. The results of the present linear analysis study show that reactivity changes the perturbation growth rate by developing a pressure gradient at the flame surface. The baroclinic torque based on the density gradient across the flame acts to slow down the instability growth of high wave-number perturbations. A gasdynamic flame representation leads to the definition of a Peclet number representing the scaling between perturbation and thermal diffusion lengths within the flame. Peclet number effects on perturbation growth are observed to be marginal. The gasdynamic model also considers a finite flame Mach number that supports a separation between flame and contact discontinuity. Such a separation destabilizes the interface growth by augmenting the tangential shear.

Gas Separator with the Rarefied Gas Flow Induced by a Temperature Field

A gas separator that utilizes the thermal transpiration phenomena in a rarefied gas is proposed. The separator is devised to work without intermission and the device itself does not contain moving parts. One of its minimal implementation, which consists of an array of micro-channels with temperature gradient along the channel, is investigated numerically on the basis of the Boltzmann equation for binary gas mixture. It is shown that the separator generates a difference of molar concentration between its two exits when a gas flow with finite Mach number is supplied to the device.

An Eulerian method for the solution of the multi-species drift-kinetic equation

An Eulerian numerical discretization scheme for the solution of the first-order drift-kinetic equation is presented in detail. The approach is valid for multi-species plasmas, including complete impurity and electron physics, and thus ambipolarity is properly maintained. The code, NEO, provides a complete description of the second-order neoclassical transport fluxes and first-order flows, including the effects of strong (finite Mach number) toroidal rotation. Corrections to the weak rotation limit are demonstrated for multi-species plasmas over a wide range of collisionality.

Stability of Chapman–Jouguet detonations for a stiffened-gas model of condensed-phase explosives

The analysis of the linear stability of a planar Chapman–Jouguet detonation wave is reformulated for an arbitrary caloric (incomplete) equation of state in an attempt to better represent the stability properties of detonations in condensed-phase explosives. Calculations are performed on a ‘stiffened-gas’ equation of state which allows us to prescribe a finite detonation Mach number while simultaneously allowing for a detonation shock pressure that is substantially larger than the ambient pressure. We show that the effect of increasing the ambient sound speed in the material, for a given detonation speed, has a stabilizing effect on the detonation. We also show that the presence of the slow reaction stage, a feature of detonations in certain types of energetic materials, where the detonation structure is characterized by a fast reaction stage behind the detonation shock followed by a slow reaction stage, tends to have a destabilizing effect.

Benchmark computations based on lattice-Boltzmann, finite element and finite volume methods for laminar flows

The goal of this article is to contribute to the discussion of the efficiency of lattice-Boltzmann (LB) methods as CFD solvers. After a short review of the basic model and extensions, we compare the accuracy and computational efficiency of two research simulation codes based on the LB and the finite-element method (FEM) for two-dimensional incompressible laminar flow problems with complex geometries. We also study the influence of the Mach number on the solution, since LB methods are weakly compressible by nature, by comparing compressible and incompressible results obtained from the LB code and the commercial code CFX. Our results indicate, that for the quantities studied (lift, drag, pressure drop) our LB prototype is competitive for incompressible transient problems, but asymptotically slower for steady-state Stokes flow because the asymptotic algorithmic complexity of the classical LB-method is not optimal compared to the multigrid solvers incorporated in the FEM and CFX code. For the weakly compressible case, the LB approach has a significant wall clock time advantage as compared to CFX. In addition, we demonstrate that the influence of the finite Mach number in LB simulations of incompressible flow is easily underestimated.

The influence of the Mach number on the stability of radiative shocks

We study the stability properties of hydrodynamic shocks with finite Mach numbers. The linear analysis supplements previous analyses which took the strong shock limit. We derive the linearised equations for a general specific heat ratio as well as temperature and density power-law cooling functions, corresponding to a range of conditions relevant to interstellar atomic and molecular cooling processes. Boundary conditions corresponding to a return to the upstream temperature ($R$ = 1) and to a cold wall ($R$ = 0) are investigated. We find that for Mach number $M > 5$, the strong shock overstability limits are not significantly modified. For $M < 3$, however, shocks are considerably more stable for most cases. In general, as the shock weakens, the critical values of the temperature power-law index (below which shocks are overstable) are reduced for the overtones more than for the fundamental, which signifies a change in basic behaviour. In the $R$ = 0 scenario, however, we find that the overstability regime and growth rate of the fundamental mode are increased when cooling is under local thermodynamic equilibrium. We provide a possible explanation for the results in terms of a stabilising influence provided downstream but a destabilising effect associated with the shock front. We conclude that the regime of overstability for interstellar atomic shocks is well represented by the strong shock limit unless the upstream gas is hot. Although molecular shocks can be overstable to overtones, the magnetic field provides a significant stabilising influence.

Simulation of two-dimensional decaying turbulence using the “incompressible” extensions of the lattice Boltzmann method

Incompressible extensions of the lattice Boltzmann method [Zou Q, Hou S, Chen S, Doolen GD. J Stat Phys 1995;81:35, He X, Luo LS. J Stat Phys 1997;88:927, Fang HP, Wan RZ, Lin ZF. Phys Rev E 2002;66:036314] are studied simulating two-dimensional decaying turbulence in a double-periodic domain. The simulation results are compared with reference calculations obtained by a pseudo-spectral code. It is shown that the standard method can perform better than the incompressible extensions even in the case of finite Mach number flows as long as density variations (pressure effects) are not significant in the domain. The importance of initializing not only the initial values of macroscopic variables but also the non-equilibrium part of the distribution functions is emphasized, and deviations arising from underspecified initial conditions are shown.

Sound generation by a two-dimensional circular cylinder in a uniform flow

The sound generated by a circular cylinder in a flow at low Mach numbers is investigated by direct solution of the two-dimensional unsteady compressible Navier–Stokes equations. Results show that sound pressure waves are generated primarily by vortex shedding from the cylinder surface into its wake. When a vortex is shed from one side of the cylinder, a negative pressure pulse is generated from that side whereas a positive pressure pulse is generated from the other side; alternate vortex shedding from the upper and lower sides of the cylinder produces negative and positive pulses alternately and thus produces sound pressure waves on both sides. The dipolar nature of the generated sound is confirmed; lift dipole dominates the sound field. The Doppler effect is shown to play an important role at finite Mach numbers. The direct solutions are also compared with the solutions obtained by Curle's acoustic analogy. The results show that Curle's solution describes well not only the generation mechanism of the sound but also the propagation process if we take the Doppler effect into consideration.