FIG 5 - uploaded by Chuandong Lin
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The sketch of the Maxwellian and actual distribution functions versus velocity v r and v θ , respectively. (a) the distribution functions at the leftmost line, (b) the distribution functions at the rightmost line. The long-dashed line is for distribution function f (v r ), the shot-dashed one is for distribution function f (v θ ), and the solid line is for f eq .
Source publication
We present a polar coordinate lattice Boltzmann model for the detonation
phenomena. In this model the fluid behavior is describe by a finite-difference
lattice Boltzmann(FDLB) model in polar coordinates, and the chemical reaction
is described by Cochran's rate function. The released chemical energy is
naturally coupled with the flow behavior. We in...
Context in source publication
Citations
... Discrete Boltzmann method (DBM) is a modeling and analysis method based on kinetic theory in statistical physics [73][74][75][76][77][78][79][80] . It provides various measures to detect, describe and exhibit the complex Thermodynamic Non-Equilibrium (TNE) behaviors, and has brought a series of new insights in several fields in recent years, such as multiphase flow [81][82][83][84][85] , rarefied gas flow 86,87 , combustion and detonation [88][89][90][91][92][93][94][95][96][97] , hydrodynamic instabilities [98][99][100][101][102][103][104][105] , etc. In contrast to conventional CFD methods, DBM is not restricted by the continuous and nearequilibrium assumptions, and can describe the cases of high Knudsen (Kn) number which contain strong TNE and noncontinuity effects. ...
Nonequilibrium kinetic effects are widespread in fluid systems and might have a significant impact on the inertial confinement fusion ignition process, and the entropy production rate is a key factor in accessing the compression process. In this work, we study the Richtmyer-Meshkov instability (RMI) and the reshock process by a two-fluid discrete Boltzmann method (DBM). Firstly, the DBM result for the perturbation amplitude evolution is in good agreement with that of experiment. Greatly different from the case of normal shocking on unperturbed plane interface between two uniform media, in the RMI case, the Thermodynamic Non-Equilibrium (TNE) quantities show complex but inspiring kinetic effects in the shocking process and behind the shock front. The kinetic effects are detected by two sets of TNE quantities. The first set are $\left |\Delta _{2}^{ {\rm{*}}}\right |$,$\left |\Delta _{3,1}^{ {\rm{*}}}\right |$, $\left | \Delta _{3}^{ {\rm{*}}}\right |$, and $\left |\Delta _{4,2}^{ {\rm{*}}}\right |$. All the four TNE measures abruptly increase in the shocking process. $\left |\Delta _{3,1}^{ {\rm{*}}}\right |$ and $\left |\Delta _{3}^{ {\rm{*}}}\right |$ show similar behaviors. They continue to increase in a much lower rate behind the shock front. $\left |\Delta _{2}^{ {\rm{*}}}\right |$ and $\left |\Delta _{4,2}^{ {\rm{*}}}\right |$ have different dimensions, but show similar behaviors. They quickly decrease to be very small behind the shock front. The second set of TNE quantities are ${\dot{S} _{NOMF}}$, ${\dot{S} _{NOEF}}$ and ${\dot{S} _{sum}}$. It is found that the mixing zone is the primary contribution region to the ${\dot{S} _{NOEF}}$, while the flow field region excluding mixing zone is the primary contribution region to the ${\dot{S} _{NOMF}}$. The light fluid has a higher entropy production rate than the heavy fluid.
... The DBM has been successfully applied to various flow problems and brought significant new insights into the complex flow systems. Typical applications include highspeed compressible flow [29][30][31] , phase separation 32-34 , nonequilibrium detonation [35][36][37][38][39][40][41][42][43] , hydrodynamic instability [44][45][46][47][48][49][50] , plasma shock wave 51 and non-equilibrium flow 25,26 . It has been found that, besides missing flow behaviours described by high-order moments, the short considering of TNE may significantly affect the flow field obtained by the molecular velocity distribution function when TNE is strong. ...
... Simulations are carried out under the meshes N x × N y = 1000 × 50. The time step is ∆t = 1 × 10 −3 and the relaxation time τ varies with Kn according to the relation in Eq. (36). Diffuse reflection boundary conditions are used on the upper and lower walls. ...
Slip flow is a common phenomenon in micro-/nano-electromechanical systems. It is well known that the mass and heat transfers in slip flow show many unique behaviors, such as the velocity slip and temperature jump near the wall. However, the kinetic understanding of slip flow is still an open problem. This paper first clarifies that the Thermodynamic Non-Equilibrium (TNE) flows can be roughly classified into two categories: near-wall TNE flows and TNE flows away from the wall. The origins of TNE in the two cases are significantly different. For the former, the TNE mainly results from the fluid–wall interaction; for the latter, the TNE is primarily due to the considerable (local) thermodynamic relaxation time. Therefore, the kinetic modeling methods for the two kinds of TNE flows are significantly different. Based on the Discrete Boltzmann Modeling (DBM) method, the non-equilibrium characteristics of mass and heat transfers in slip flow are demonstrated and investigated. The method is solidly verified by comparing with analytic solutions and experimental data. In pressure-driven flow, the DBM results are consistent with experimental data for the Knudsen number up to 0.5. It is verified that, in the slip flow regime, the linear constitutive relations with standard viscous or heat conduction coefficients are no longer applicable near the wall. For the Knudsen layer problem, it is interesting to find that a heat flux (viscous stress) component in the velocity (temperature) Knudsen layer approximates a hyperbolic sinusoidal distribution. The findings enrich the insights into the non-equilibrium characteristics of mass and heat transfers at micro-/nano-scales.
... Besides, on the microscopic level, the distribution function can be obtained by using the direct simulation Monte Carlo [9,10], or molecular dynamics [11,12]. As a kinetic mesoscopic methodology, the discrete Boltzmann method (DBM) is a special discretization of the Boltzmann equation in particle velocity space, and has been successfully developed to recover and probe the velocity distribution functions of nonequilibrium physical systems [8,[13][14][15][16][17]. ...
... In fact, the DBM is based on statistical physics and regarded as a variant of the traditional lattice Boltzmann method (LBM) [18][19][20][21][22]. Compare to standard LBMs, the DBM can address more issues, in particular to simulate the compressible fluid systems with significant nonequilibrium effects [13,17,[23][24][25][26][27]. At present, there are two means to recover the velocity distribution functions. ...
... At present, there are two means to recover the velocity distribution functions. One relies on the analysis of the detailed nonequilibrium physical quantities to obtain the main features of the velocity distribution function in a qualitative way [8,13,14,16]. The other is to recover the detailed velocity distribution function by means of macroscopic quantities and their spatio and temporal derivatives quantitatively, which can be derived by using the Chapman-Enskog expansion [15,17]. ...
How to accurately probe chemical reactive flows with essential thermodynamic nonequilibrium effects is an open issue. Via the Chapman-Enskog analysis, the local nonequilibrium particle velocity distribution function is derived from the gas kinetic theory. It is demonstrated theoretically and numerically that the distribution function depends on the physical quantities and derivatives, and is independent of the chemical reactions directly. Based on the simulation results of the discrete Boltzmann model, the departure between equilibrium and nonequilibrium distribution functions is obtained and analyzed around the detonation wave. Besides, it has been verified for the first time that the kinetic moments calculated by summations of the discrete distribution functions are close to those calculated by integrals of their original forms.
... In summary, this work investigates numerically the effect of small obstacle-induced perturbations on the re-initiation of a diffracting detonation wave in an unstable mixture of detonable gases. The computational model in this numerical investigation is based on the reactive Euler equations, which is simpler and of lower accuracy than Navier-Stokes or discrete kinetic models, such as the method of Lin et al. 47,48 The conclusions are reliable enough based on the current Euler equations, although the latter methods could provide more accurate predictions. The simulation results demonstrate that an incident detonation wave with fully developed cellular instabilities is prone to re-initiation in the unconfined space than a planar (laminar-like ZND) incident detonation. ...
A gaseous detonation wave that emerges from a channel into an unconfined space is known as detonation diffraction. If the dimension of the channel exit is below some critical value, the incident detonation fails to re-initiate (i.e., transmit into a self-sustained detonation propagating) in the unconfined area. In a previous study, Xu et al. [“The role of cellular instability on the critical tube diameter problem for unstable gaseous detonations,” Proc. Combust. Inst. 37(3), 3545–3533 (2019)] experimentally demonstrated that, for an unstable detonable mixture (i.e., stoichiometric acetylene–oxygen), a small obstacle near the channel exit promotes the re-initiation capability for cases with a sub-critical channel size. In the current study, two-dimensional numerical simulations were performed to reveal this obstacle-triggered re-initiation process in greater detail. Parametric studies were carried out to examine the influence of obstacle position on the re-initiation capability. The results show that a collision between a triple-point wave complex at the diffracting shock front and the obstacle is required for a successful re-initiation. If an obstacle is placed too close or too far away from the channel exit, the diffracting detonation cannot be re-initiated. Since shot-to-shot variation in the cellular wave structure of the incident detonation results in different triple-point trajectories, for an obstacle at a fixed position, the occurrence of re-initiation is of a stochastic nature. The findings of this study highlight that flow instability generated by a local perturbation is effective in enhancing the re-initiation capability of a diffracting cellular detonation wave in an unstable mixture.
... The LBM has been successfully applied to simulate a variety of complex flows, 18 including reactive flows. [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] Merits of the LBM include the algorithm simplicity and locality, which lead to excellent performance on parallel clusters. ...
... In recent years, the discrete Boltzmann method (DBM) has been constructed to model and simulate nonequilibrium systems, with various velocity and time-space discretization schemes. [31][32][33][34][35]39 The DBM and LBM can be viewed as two distinctive classes of discrete numerical methods based on the Boltzmann equation. Despite sharing the same origin and some similarities, the objectives, numerical implementation, and capabilities of the LBM and DBM are different. ...
... Since 2013, several DBMs for reactive flows have been proposed. [31][32][33][34][35]39 In 2013, Yan et al. proposed the first DBM for combustion. 31 Very recently, Lin et al. proposed a two-dimensional model for detonations and investigated the main features of the hydrodynamic and thermodynamic nonequilibrium effects. ...
Based on the kinetic theory, a three-dimensional multiple-relaxation-time discrete Boltzmann model (DBM) is proposed for nonequilibrium compressible reactive flows where both the Prandtl number and specific heat ratio are freely adjustable. There are 30 kinetic moments of the discrete distribution functions, and an efficient three-dimensional thirty-velocity model is utilized. Through the Chapman–Enskog analysis, the reactive Navier–Stokes equations can be recovered from the DBM. Unlike existing lattice Boltzmann models for reactive flows, the hydrodynamic and thermodynamic fields are fully coupled in the DBM to simulate combustion in subsonic, supersonic, and potentially hypersonic flows. In addition, both hydrodynamic and thermodynamic nonequilibrium effects can be obtained and quantified handily in the evolution of the discrete Boltzmann equation. Several well-known benchmarks are adopted to validate the model, including chemical reactions in the free falling process, thermal Couette flow, one-dimensional steady or unsteady detonation, and a three-dimensional spherical explosion in an enclosed cube. It is shown that the proposed DBM has the capability to simulate both subsonic and supersonic fluid flows with or without chemical reactions.
... The other type of kinetic models is to capture both hydrodynamic and thermodynamic nonequilibrium behaviours beyond the macroscopic models. The successful physical models include the unified gas kinetic scheme [24][25][26], the discrete unified gas kinetic scheme [27,28], the discrete Boltzmann method (DBM) [29][30][31][32][33][34][35][36][37][38][39], etc. These powerful models are suitable for continuum and rarefied systems with a wide range of Knudsen numbers, and capable to subsonic and supersonic flows with essential nonequilibrium effects. ...
... From a physical modeling perspective, a DBM is approximately equivalent to a continuous fluid model plus a coarse-grained model of other relevant thermodynamic nonequilibrium effects. At present, the DBM has been developed as a nonequilibrium flow simulation tool for various flow systems, including the fluid instability [29,32,38], multiphase flow [34], shock and detonation [30,31,33,36,37,39]. Roughly speaking, the DBMs can be classed into two categories: the single-relaxation-time DBM [29][30][31][32][33][34][35][36] and the multiple-relaxation-time DBM [37][38][39]. ...
... At present, the DBM has been developed as a nonequilibrium flow simulation tool for various flow systems, including the fluid instability [29,32,38], multiphase flow [34], shock and detonation [30,31,33,36,37,39]. Roughly speaking, the DBMs can be classed into two categories: the single-relaxation-time DBM [29][30][31][32][33][34][35][36] and the multiple-relaxation-time DBM [37][38][39]. In the single-relaxation-time model, there is only one relaxation time that controls a thermodynamic nonequilibrium system approaching its equilibrium state. ...
A shock wave that is characterized by sharp physical gradients always draws the medium out of equilibrium. In this work, both hydrodynamic and thermodynamic nonequilibrium effects around the shock wave are investigated using a discrete Boltzmann model. Via Chapman–Enskog analysis, the local equilibrium and nonequilibrium velocity distribution functions in one-, two-, and three-dimensional velocity space are recovered across the shock wave. Besides, the absolute and relative deviation degrees are defined in order to describe the departure of the fluid system from the equilibrium state. The local and global nonequilibrium effects, nonorganized energy, and nonorganized energy flux are also investigated. Moreover, the impacts of the relaxation frequency, Mach number, thermal conductivity, viscosity, and the specific heat ratio on the nonequilibrium behaviours around shock waves are studied. This work is helpful for a deeper understanding of the fine structures of shock wave and nonequilibrium statistical mechanics.
... required number of high order moments are satisfied [33][34][35][36][37][38] . Different from traditional LBMs, DBM contains both equilibrium and nonequilibrium physical quantities that stem from the same discrete distribution function [33][34][35][36][37][38] . ...
... required number of high order moments are satisfied [33][34][35][36][37][38] . Different from traditional LBMs, DBM contains both equilibrium and nonequilibrium physical quantities that stem from the same discrete distribution function [33][34][35][36][37][38] . ...
... Over the past years, the versatile DBM has been effectively applied to thermal phase separation, fluid instabilities, reactive flows, etc [33][34][35][36][37][38] . The DBM for reactive flows was firstly presented by Yan et al. in 2013 35 . ...
We propose a multi-component discrete Boltzmann model (DBM) for premixed, nonpremixed, or partially premixed nonequilibrium reactive flows. This model is suitable for both subsonic and supersonic flows with or without chemical reaction and/or external force. A two-dimensional sixteen-velocity model is constructed for the DBM. In the hydrodynamic limit, the DBM recovers the modified Navier-Stokes equations for reacting species in a force field. Compared to standard lattice Boltzmann models, the DBM presents not only more accurate hydrodynamic quantities, but also detailed nonequilibrium effects that are essential yet long-neglected by traditional fluid dynamics. Apart from nonequilibrium terms (viscous stress and heat flux) in conventional models, specific hydrodynamic and thermodynamic nonequilibrium quantities (high order kinetic moments and their departure from equilibrium) are dynamically obtained from the DBM in a straightforward way. Due to its generality, the developed methodology is applicable to a wide range of phenomena across many energy technologies, emissions reduction, environmental protection, mining accident prevention, chemical and process industry.
... Recently, as a variant of the standard LBM, the discrete Boltzmann model (DBM) addresses the above issues [32][33][34][35][36]. In addition to recovering the reactive macroscopic equations, the DBM contains essential hydrodynamic and thermodynamic nonequilibrium information beyond the former. ...
... Roughly speaking, DBMs can be classified into two categories. One is the single-relaxation-time (SRT) DBM [32][33][34], which is based on the original Bhatnagar-Gross-Krook model. The relaxation speeds of various thermodynamic processes are simply taken the same, which results in some defects, including the fixed Prandtl number Pr = 1. ...
... The DBM has the merit of simplicity of the algorithm and coding due to its unified formulation. Different from previous models where the collision, force, and reaction terms are expressed in various forms [17,18,32,33,35,36], our DBM is more convenient to code as all terms are specified with the matrix inversion method in a uniform way. Furthermore, it is straightforward to extend this methodology to other physical mechanisms, such as the surface tension, electromagnetic action, etc. ...
A multiple-relaxation-time discrete Boltzmann model (DBM) is developed for compressible thermal reactive flows. A unified Boltzmann equation set is solved for hydrodynamic and thermodynamic quantities as well as higher order kinetic moments. The collision, reaction, and force terms are uniformly calculated with a matrix inversion method, which is physically accurate, numerically efficient, and convenient for coding. Via the Chapman-Enskog analysis, the DBM is demonstrated to recover reactive Navier-Stokes (NS) equations in the hydrodynamic limit. Both specific heat ratio and Prandtl number are adjustable. Moreover, it provides quantification of hydrodynamic and thermodynamic nonequilibrium effects beyond the NS equations. The capability of the DBM is demonstrated through simulations of chemical reactions in the free falling process, sound wave, thermal Couette flow, and steady and unsteady detonation cases. Moreover, nonequilibrium effects on the predicted physical quantities in unsteady combustion are quantified via the DBM. It is demonstrated that nonequilibrium effects suppress detonation instability and dissipate small oscillations of fluid flows.
... In fact, as a mesoscopic method, the DBM has the capability of modeling and simulating nonequilibrium systems with relatively high efficiency [19][20][21][22][23][24][25] . The idea of investigating nonequilibrium behavior with the DBM was explored by Xu et al. [19] . ...
... Furthermore, Yan et al. proposed the first DBM for combustion and detonation in 2013 [20] . Later, Lin et al. presented a polar-coordinate DBM for explosion or implosion, and investigated the main features of the velocity distribution function by analyzing nonequilibrium manifestations [21] . Zhang et al. then demonstrated that nonequilibrium characteristics are related to the entropy increasing rate [26] . ...
Thanks to its mesoscopic nature, the recently developed discrete Boltzmann method (DBM) has the capability of providing deeper insight into nonequilibrium reactive flows accurately and efficiently. In this work, we employ the DBM to investigate the hydrodynamic and thermodynamic nonequilibrium (HTNE) effects around the detonation wave. The individual HTNE manifestations of the chemical reactant and product are probed, and the main features of their velocity distributions are analyzed. Both global and local HTNE effects of the chemical reactant and product increase approximately as a power of the chemical heat release that promotes the chemical reaction rate and sharpens the detonation front. With increasing relaxation time, the global HTNE effects of the chemical reactant and product are enhanced by power laws, while their local HTNE effects show changing trends. The physical gradients are smoothed and the nonequilibrium area is enlarged as the relaxation time increases. Finally, to estimate the relative height of detonation peak, we define the peak height as H(q)=(qmax−qs)/(qvon−qs), where qmaxis the maximum of q around a detonation wave, qsis the CJ solution and qvonis the ZND solution at the von-Neumann-peak. With increasing relaxation time, the peak height decreases, because the nonequilibrium effects attenuate and widen the detonation wave. The peak height is an exponential function of the relaxation time.
... Different from traditional LBMs, the DBM employs only one set of discrete distribution function to describe various physical quantities, including the density, temperature, velocity, and other high order kinetic moments, which is in line with the Boltzmann equation. Since 2013, several Single-Relaxation-Time DBMs have been formulated for exothermic reactive flows [38][39][40] . Yet, the Prandtl number in those proposed model is fixed at Pr = 1 . ...
An efficient, accurate and robust multiple-relaxation-time (MRT) discrete Boltzmann method (DBM) is proposed for compressible exothermic reactive flows, with both specific heat ratio and Prandtl number being flexible. The chemical reaction is coupled with the flow field naturally and the external force is also incorporated. An efficient discrete velocity model which has sixteen discrete velocities (and kinetic moments) is introduced into the DBM. With both hydrodynamic and thermodynamic nonequilibrium effects under consideration, the DBM provides more detailed and accurate information than traditional Navier–Stokes equations. This method is suitable for fluid flows ranging from subsonic, to supersonic and hypersonic ranges. It is validated by various benchmarks.
