Front Tracking Scheme Research Articles

Global Well-Posedness of Shock Front Solutions to One-Dimensional Piston Problem for Combustion Euler Flows

This paper is devoted to the well-posedness theory of piston problem for compressible combustion Euler flows with physical ignition condition. A significant combustion phenomenon called detonation will occur provided that the reactant is compressed and ignited by a leading shock. Mathematically, the problem can be formulated as an initial-boundary value problem for hyperbolic balance laws with a large shock front as free boundary. In the present paper, we establish the global well-posedness of entropy solutions via wave front tracking scheme within the framework of space. The main difficulties here stem from the discontinuous source term without uniform dissipation structure and from the characteristic boundary associated with degenerate characteristic field. In dealing with the obstacles caused by ignition temperature, we develop a modified Glimm-type functional to control the oscillation growth of combustion waves, even if the exothermic source fails to uniformly decay. As to the characteristic boundary, the degeneracy of contact discontinuity is fully employed to get elegant stability estimates near the piston boundary. Meanwhile, we devise a weighted Lyapunov functional to balance the nonlinear effects arising from large shock, characteristic boundary, and exothermic reaction and then obtain the -stability of combustion wave solutions. Our results reveal that one-dimensional Zeldovich–von Neumann–Döring detonation waves supported by a forward piston are indeed nonlinearly stable under small perturbation in sense. This is the first work on well-posedness of inviscid reacting Euler fluids dominated by ignition temperature.

A machine learning strategy for computing interface curvature in Front-Tracking methods

In this work we have described the application of a machine learning strategy to compute the interface curvature in the context of a Front-Tracking framework. Based on angular information of normal and tangential vectors between marker points, the interface curvature is predicted using a neural network. The Front-Tracking-Machine-Learning method is validated using a sine wave and then applied in combination with a Marker-And-Cell method for solving a complex free surface flow. Our results indicate that it is feasible to employ machine learning concepts as an alternative approach for computing curvatures in Front-Tracking schemes.

Non-relativistic limits and stability of composite wave patterns to the relativistic Euler equations

We are concerned with the structural stability of composite wave patterns to the Cauchy problem for the relativistic full Euler equations consisting of a large 1-shock, a large 2-contact discontinuity, and a large 3-shock. When the initial data are bounded but possibly large total variations, approximate solutions have been constructed via the wave front tracking scheme, a weighted Glimm functional has been introduced and its monotonicity has been proved on the basis of the local wave interaction estimates, and then the global stability of these wave patterns has been established. Moreover, the non-relativistic limits of such solutions can be obtained as the light speed c → +∞.

Global Existence and Stability of Shock Front Solution to 1-D Piston Problem for Exothermically Reacting Euler Equations

In this paper, we are concerned with the global existence and stability of strong shock front for one-dimensional piston problem. We use the exothermically reacting Euler equations as a mathematical model to describe the gas motion. Under the assumptions that the total variations of initial data and the perturbation of piston velocity are sufficiently small, we establish the global existence and asymptotic behavior of entropy solutions including a strong shock front without restriction on the strength. A modified fractional wave front tracking scheme is developed and a modified Glimm-type functional is carefully designed.

Lyophilization model of mannitol water solution in a laboratory scale lyophilizer

The paper reports on the development of a numerical model for the simulation of a lyophilization process in a vial. Experimental analysis is presented of lyophilization dynamics inside a single vial in a laboratory scale lyophilizer. The problems of lyophilization modelling of a mannitol water solution are covered in detail. The effects of the small scale of the laboratory device with respect to a correct definition of boundary conditions for the numerical simulations are described, especially the effect of the comparatively high temperatures of the chamber walls. In the numerical model, a 1D vial approximation of the governing equations of heat and mass transport with moving front between the frozen and porous part of the cake is used and solved in a time stepping nonlinear iteration procedure. A water vapour diffusion model, implemented in the mass conservation equations, based on the Knudsen model of diffusivities, is applied and linked to the typical pore size of the porous cake. A front tracking scheme with moving computational grid is applied, and a dedicated sub-model of surface layer ice sublimation is introduced, based on the one-sided vapour diffusion model. The comparison of the numerical and the experimental results show that the developed numerical model is able to capture the transition points from primary to secondary drying very accurately, with accompanying accurate capturing of the temperature levels inside of the drying material.

Effect of Step Anisotropy on Crystal Growth Inhibition by Immobile Impurity Stoppers

Step pinning by immobile stoppers is the most important crystal growth inhibition mechanism. It was first studied by Cabrera and Vermilyea in 1958, who considered the macroscopic effect of a periodic array of pinning sites. However, their analysis (and others since) involved uncontrolled approximations and did not consider what happens when step anisotropy induces faceting. Here we revisit the motion of a step past a periodic array of pinning sites, simulating the evolution numerically using a semi-implicit front-tracking scheme for anisotropic surface energies and kinetic coefficients. We also provide exact formulas for the average step velocity when the anisotropy is such that the interface is fully faceted. We compare the average step velocities obtained numerically to the estimates derived in the isotropic setting by Cabrera & Vermilyea (1958) and Potapenko (1993), and to the exact results obtained in the fully faceted setting. Our results show that while the local geometry of the propagating step var...

The “avalanche effect” of an elasto-viscoplastic thixotropic material on an inclined plane

The so-called avalanche effect is one of the fingerprints of thixotropic materials. This self-reinforcing process where the decrease in viscosity, due to a rejuvenation process triggered by a stress field, induces a motion which in turn contributes to decrease the viscosity again, is well exemplified by the inclined plane problem. In this situation, the material in its fully-structured state is placed on an inclined plane with respect to the gravity force which is responsible for the beginning of the breakdown process. These thixotropic systems generally have a yield stress, a strength that must be overcome in order to induce rejuvenation. In addition, they exhibit elastic features, especially in the pre-yield state. In the present work we numerically solve the transient evolution of an elasto-viscoplastic thixotropic material subjected to the action of gravity on an inclined plane. In order to handle with the moving free-surface boundary condition encountered in the avalanche effect, we have used a combination of the Marker-And-Cell (MAC) method with the front-tracking scheme. This formulation was successfully employed for this kind of material in the recent paper of Oishi et al. (2016) [28]. In the present work, we have adapted our finite difference formulation to analyze the effects associated with an extended Herschel–Bulkley model in the simulation of a transient complex free surface flow. Concerning the parameters of the flow curve, it is shown that the dimensionless yield stress (plastic number) is the most significant one. However, for a fixed plastic number, different combinations of dimensionless consistency index and dimensionless Newtonian viscosity plateau can lead to a diversity of responses. The thixotropic equilibrium time had a significant impact on shifting the instant when the flow regime changes from an accelerating (when the front part of the material accelerates) to a retardation one (when this front part decelerates). Higher elasticity, as captured by the Weissenberg number, led to longer distances covered by the material.

Stability of rarefaction wave to the 1-D piston problem for exothermically reacting Euler equations

This paper is to devoted to the stability of the rarefaction wave for one dimensional piston problem of the exothermically reacting Euler equations. When the total variation of the initial data and the perturbation of the piston velocity are sufficiently small, we employ fractional wave front tracking scheme to establish the global existence and study the asymptotic behavior of entropy solutions as $$t\rightarrow +\infty $$ .

Global solutions for the ultra-relativistic Euler equations

A front tracking scheme for the ultra-relativistic Euler equations is introduced. This scheme is based on piecewise constant approximations to the front tracking Riemann solutions, where continuous rarefaction waves are approximated by finite collections of discontinuities, so-called non-entropy shocks. We study the interaction estimates of the generalized shocks (entropy and nonentropy shocks) of the ultra-relativistic Euler equations and the outcoming asymptotic Riemann solution. Moreover we use a new function to measure the strengths of the waves of the ultra-relativistic Euler equations. This function has the important implication that the strength is non increasing for the interactions of the generalized shocks. This enables us to define a new kind of total variation of a solution. The main application of this scheme, is proving the global existence of weak solutions for the ultra relativistic Euler equations.

Global weak solutions for a model of two-phase flow with a single interface

We consider a simple nonlinear hyperbolic system modeling the flow of an inviscid fluid. The model includes as state variable the mass density fraction of the vapor in the fluid, and then, phase transitions can be taken into consideration; moreover, phase interfaces are contact discontinuities for the system. We focus on the special case of initial data consisting of two different phases separated by an interface. We find explicit bounds on the (possibly large) initial data in order that weak entropic solutions exist for all times. The proof exploits a carefully tailored version of the front-tracking scheme.

Comparison and verification of enthalpy schemes for polythermal glaciers and ice sheets with a one-dimensional model

The enthalpy method for the thermodynamics of polythermal glaciers and ice sheets is tested and verified by a one-dimensional problem (parallel-sided slab). The enthalpy method alone does not include explicitly the transition conditions at the cold–temperate transition surface (CTS) that separates the upper cold from the lower temperate layer. However, these conditions are important for correctly determining the position of the CTS. For the numerical solution of the polythermal slab problem, we consider a two-layer front-tracking scheme as well as three different one-layer schemes (conventional one-layer scheme, one-layer melting CTS scheme, one-layer freezing CTS scheme). Computed steady-state temperature and water-content profiles are verified with exact solutions, and transient solutions computed by the one-layer schemes are compared with those of the two-layer scheme, considered to be a reliable reference. While the conventional one-layer scheme (that does not include the transition conditions at the CTS) can produce correct solutions for melting conditions at the CTS, it is more reliable to enforce the transition conditions explicitly. For freezing conditions, it is imperative to enforce them because the conventional one-layer scheme cannot handle the associated discontinuities. The suggested numerical schemes are suitable for implementation in three-dimensional glacier and ice-sheet models.

Progess of discrete Boltzmann modeling and simulation of combustion system

Progess of discrete Boltzmann modeling and simulation of combustion system

A new front tracking scheme for the ultra-relativistic Euler equations

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure p, the spatial part u∈R3 of the dimensionless four-velocity and the particle density n. Two schemes for these equations are presented in one space dimension, namely a front tracking and a cone-grid scheme. A new front tracking technique for the ultra-relativistic Euler equations is introduced, which gives weak solutions. The front tracking method is based on piecewise constant approximations to Riemann solutions, called front tracking Riemann solutions, where continuous rarefaction waves are approximated by finite collections of discontinuities, so-called non-entropy shocks. This method can be used for analytical as well as for numerical purposes. A new unconditionally stable cone-grid scheme is also derived in this paper, which is based on the Riemann solution for the ultra-relativistic Euler equations. Both schemes are compared by two numerical examples, where explicit solutions are known.

Numerical methods with controlled dissipation for small-scale dependent shocks

We provide a ‘user guide’ to the literature of the past twenty years concerning the modelling and approximation of discontinuous solutions to nonlinear hyperbolic systems that admitsmall-scale dependentshock waves. We cover several classes of problems and solutions: nonclassical undercompressive shocks, hyperbolic systems in nonconservative form, and boundary layer problems. We review the relevant models arising in continuum physics and describe the numerical methods that have been proposed to capture small-scale dependent solutions. In agreement with general well-posedness theory, small-scale dependent solutions are characterized by akinetic relation, a family of paths, or anadmissible boundary set. We provide a review of numerical methods (front-tracking schemes, finite difference schemes, finite volume schemes), which, at the discrete level, reproduce the effect of the physically meaningful dissipation mechanisms of interest in the applications. An essential role is played by theequivalent equationassociated with discrete schemes, which is found to be relevant even for solutions containing shock waves.

Simulating the kinematics of completely faceted surfaces

We fully generalize a previously-developed computational geometry tool [1] to perform large-scale simulations of arbitrary 2+1D faceted surfaces z=h(x,y). The method is an explicit front-tracking scheme that uses a compact, three-component facet/edge/junction storage mode. Because it naturally mirrors the intrinsic surface structure, this scheme allows both rapid simulation of large ensembles, and easy extraction of geometrical statistics. To do so, it must overcome the barrier of detecting and resolving a wide variety of topological changes that occur during surface evolution. However, while the variety of topological events is larger than in the case of 2D cellular networks, it is still limited, and our main result is a comprehensive algorithm performing these changes in the code.

Surfactant effects on drop formation in co-flowing fluid streams

We conduct a numerical study to investigate the effect of surfactants on the drop formation process in a co-flowing system using a hybrid volume-of-fluid (VOF) method combined with a front-tracking scheme. The drop and bulk phases are treated as incompressible Newtonian fluids, and the surfactants are modeled using a Langmuir adsorption framework. We consider the effect of soluble surfactants in the adsorption–desorption limit on the drop formation process. A drop in the co-flowing geometry typically breaks up at the primary neck, close to the primary drop, in the absence of surfactants. When surfactants are present, they accumulate in the neck region resulting in Marangoni stresses that slow down the neck thinning rate. This results in longer breakup times with larger drop volumes. At high surfactant coverages, the primary neck formation slows down enough and breakup occurs at the secondary neck, close to the remnant drop. Increasing the outer co-flowing flow weakens the retarding effect of the high surfactant coverage leading to breakup again at the primary neck. The adsorption–desorption kinetics also affects the neck breakup position, and the primary drop volume and breakup time depend non-linearly on the Biot number. For any given Biot number, a critical fractional coverage exists beyond which the drop fails to neck. The presence of a confining wall may lower the value of the critical equilibrium fractional coverage required for the drop to enter the no-necking regime.

Detailed chemistry flamelet modeling of mixed-mode combustion in spark-assisted HCCI engines

A detailed chemistry mixed-mode flamelet model for the prediction of combustion in spark-assisted homogeneous charge compression ignition (HCCI) engines is presented in this paper. The complex phenomena of spark-channel processes (turbulent corrugation, multiple restrikes) and of early flame kernel propagation (localized flame kernel formation, non-spherical early flame shapes), both induced by spray-guided spark-ignition combustion initiation, are captured by the recently introduced SparkCIMM ignition model. In this paper, laminar burning velocities and flame thicknesses are calculated along the mean turbulent flame front, using tabulated detailed chemistry flamelet calculations to appropriately consider locally rich, highly diluted, and auto-igniting stratified mixtures. Flame extinction criterions are formulated and incorporated into an extended G-equation flame front tracking scheme. Auto-ignition processes in the unburnt mixture, controlled by the flame-induced pressure and temperature increase, are captured by a recently developed multi-zone flamelet model that accounts for detailed chemistry and effects of scalar mixing on turbulent combustion. The laminar burning velocity is shown to increase significantly as the flame propagates into the chemically reacting mixture within the first stage of auto-ignition. After the initiation of the thermal runaway, however, flame extinction occurs rapidly. These interactions of mixed-mode combustion processes are captured by the presented flamelet model. It is developed based on a time and length scale analysis, revealing a scale separation between the ignition delay of diffusion combustion and the flame time of flame propagation. The analysis of the mixture preparation process, along with the simulation of turbulent flame front propagation and its extinction, demonstrates that the prediction of combustion in spark-assisted HCCI engines requires the on hand comprehensive mixed-mode combustion model. A comparison of simulation results from this new model with data from experiments and combustion models of reduced physical complexity proofs its qualification.

On high-order shock-fitting and front-tracking schemes for numerical simulation of shock–disturbance interactions

High-order methods that can resolve interactions of flow-disturbances with shock waves are critical for reliable numerical simulation of shock wave and turbulence interaction. Such problems are not well understood due to the limitations of numerical methods. Most of the popular shock-capturing methods are only first-order accurate at the shock and may incur spurious numerical oscillations near the shock. Shock-fitting algorithms have been proposed as an alternative which can achieve uniform high-order accuracy and can avoid possible spurious oscillations incurred in shock-capturing methods by treating shocks as sharp interfaces. We explore two ways for shock-fitting: conventional moving grid set-up and a new fixed grid set-up with front tracking. In the conventional shock-fitting method, a moving grid is fitted to the shock whereas in the newly developed fixed grid set-up the shock front is tracked using Lagrangian points and is free to move across the underlying fixed grid. Different implementations of shock-fitting methods have been published in the literature. However, uniform high-order accuracy of various shock-fitting methods has not been systematically established. In this paper, we carry out a rigorous grid-convergence analysis on different variations of shock-fitting methods with both moving and fixed grids. These shock-fitting methods consist of different combinations of numerical methods for computing flow away from the shock and those for computing the shock movement. Specifically, we consider fifth-order upwind finite-difference scheme and shock-capturing WENO schemes with conventional shock-fitting and show that a fifth-order convergence is indeed achieved for a canonical one-dimensional shock-entropy wave interaction problem. We also show that the method of finding shock velocity from one characteristic relation and Rankine–Hugoniot jump condition performs better than the other methods of computing shock velocities. A high-order front-tracking implementation of shock-fitting is also presented in this paper and nominal rate of convergence is shown. The front-tracking results are validated by comparing to results from the conventional shock-fitting method and a linear-interaction analysis for a two-dimensional shock disturbance interaction problem.

Thermocapillary migration of nondeformable drops

In this paper, we present a numerical study on the thermocapillary migration of drops. The Navier–Stokes equations coupled with the energy conservation equation are solved by the finite-difference front-tracking scheme. The axisymmetric model is adopted in our simulations, and the drops are assumed to be perfectly spherical and nondeformable. The benchmark simulation starts from the classical initial condition with a uniform temperature gradient. The detailed discussions and physical explanations of migration phenomena are presented for the different values of (1) the Marangoni numbers and Reynolds numbers of continuous phases and drops and (2) the ratios of drop densities and specific heats to those of continuous phases. It is found that fairly large Marangoni numbers may lead to fluctuations in drop velocities at the beginning part of simulations. Finally, we also discuss the influence of initial conditions on the thermocapillary migrations.

Nonlinear Hyperbolic Systems: Nondegenerate Flux, Inner Speed Variation, and Graph Solutions

We study the Cauchy problem for general nonlinear strictly hyperbolic systems of partial differential equations in one space variable. First, we re-visit the construction of the solution to the Riemann problem and introduce the notion of a nondegenerate (ND) system. This is the optimal condition guaranteeing, as we show it, that the Riemann problem can be solved with finitely many waves only; we establish that the ND condition is generic in the sense of Baire (for the Whitney topology), so that any system can be approached by a ND system. Second, we introduce the concept of inner speed variation and we derive new interaction estimates on wave speeds. Third, we design a wave front tracking scheme and establish its strong convergence to the entropy solution of the Cauchy problem; this provides a new existence proof as well as an approximation algorithm. As an application, we investigate the time regularity of the graph solutions (X,U) introduced by LeFloch, and propose a geometric version of our scheme; in turn, the spatial component X of a graph solution can be chosen to be continuous in both time and space, while its component U is continuous in space and has bounded variation in time.