From Reactive Boltzmann Equations to Reaction–Diffusion Systems

We consider the reactive Boltzmann equations for a mixture of different species of molecules, including a fixed background. We propose a scaling in which the collisions involving this background are predominant, while the inelastic (reactive) binary collisions are very rare. We show that, at the formal level, the solutions of the Boltzmann equations converge toward the solutions of a reaction-diffusion system. The coefficients of this system can be expressed in terms of the cross sections of the Boltzmann kernels. We discuss various possible physical settings (gases having internal energy, presence of a boundary, etc.), and present one rigorous mathematical proof in a simplified situation (for which the existence of strong solutions to the Boltzmann equation is known).     

Errors when assuming locally reacting boundary condition in the estimation of the surface acoustic impedance

Numerical simulations usually require boundary conditions in terms of surface acoustic impedance. The surface acoustic impedance depends on the porous material acoustic properties (e.g., characteristic impedance and wave number) and its thickness as well as the type of wave front impinging on its surface. The locally reactive behaviour hypothesis is often assumed to simplify the choice of proper boundary conditions assigning a constant acoustic impedance value on the porous material surface at a given frequency and for each angle of sound incidence. This hypothesis is also used in measurement procedures or for the estimation of the edge effects.     

Derivation of Slip boundary conditions for the Navier-Stokes system from the Boltzmann equation

Rarefied gas flow behavior is usually described by the Boltzmann equation, the Navier-Stokes system being valid when the gas is less rarefied. Slip boundary conditions for the Navier-Stokes equations are derived in a rigorous and systematic way from the boundary condition at the kinetic level (Boltzmann equation). These slip conditions are explicitly written in terms of asymptotic behavior of some linear half-space problems. The validity of this analysis is established in the simple case of the Couette flow, for which it is proved that the right boundary conditions are obtained.     

Localized partial traps in diffusion processes and random walks

Reaction-diffusion equations, in which the reaction is described by a sink term consisting of a sum of delta functions, are studied. It is shown that the Laplace transform of the reactive Green's function can be analytically expressed in terms of the Green's function for diffusion in the absence of reaction. Moreover, a simple relation between the Green's functions satisfying the radiation boundary condition and the reflecting boundary condition is obtained. Several applications are presented and the formalism is used to establish the relationship between the time-dependent geminate recombination yield and the bimolecular reaction rate for diffusion-influenced reactions. Finally, an analogous development for lattice random walks is presented.     

Classical versus quantum symmetries for Toda theories with a nontrivial boundary perturbation

In this paper we present a detailed study of the quantum conservation laws for Toda field theories defined on the half plane in the presence of a boundary perturbation. We show that total derivative terms added to the currents, while irrelevant at the classical level, become important at the quantum level and in general modify significantly the quantum boundary conservation. We consider the first nontrivial higher-spin currents for the simply laced a_n⁽¹⁾ Toda theories: we find that the spin-three current leads to a quantum conserved charge only if the boundary potential is appropriately redefined through a finite renormalization. Contrary to the expectation we demonstrate instead that at spin four the classical symmetry does not survive quantization and we suspect that this feature will persist at higher-spin levels. Finally, we examine the first nontrivial conservations at spin four for the d₃⁽²⁾ and c₂⁽¹⁾ nonsimply laced Toda theories. In these cases the addition of total derivative terms to the bulk currents is necessary but sufficient to ensure the existence of corresponding quantum exact conserved charges.     

Self-adjoint extensions and unitary operators on the boundary

We establish a bijection between the self-adjoint extensions of the Laplace operator on a bounded regular domain and the unitary operators on the boundary. Each unitary encodes a specific relation between the boundary value of the function and its normal derivative. This bijection sets up a characterization of all physically admissible dynamics of a nonrelativistic quantum particle confined in a cavity. Moreover, this correspondence is discussed also at the level of quadratic forms. Finally, the connection between this parametrization of the extensions and the classical one, in terms of boundary self-adjoint operators on closed subspaces, is shown.     

Effect of boundary layer losses on 2D detonation cellular structures

We evaluate the effect of boundary layer losses on two-dimensional H2/O2/Ar cellular detonations obtained in narrow channels. The experiments provide the details of the cellular structure and the detonation speed deficits from the ideal CJ speed. We model the effect of the boundary layer losses by incorporating the flow divergence in the third dimension due to the negative boundary layer displacement thickness, modeled using Mirels’ theory. The cellular structures obtained numerically with the resulting quasi-2D formulation of the reactive Euler equations with two-step chain-branching chemistry are found in excellent agreement with experiment, both in terms of cell dynamics and velocity deficits, provided the boundary layer constant of Mirels is modified by a factor of 2. A significant increase in the cell size is found with increasing velocity deficit. This is found to be very well captured by the induction zone increase in slower detonations due to the lower temperatures in the induction zone.     

Numerical difficulties with boundary element solutions of interior acoustic problems

Although boundary element methods have been applied to interior problems for many years, the numerical difficulties that can occur have not been thoroughly explored. Various authors have reported low-frequency breakdowns and artificial damping due to discretization errors. In this paper, it is shown through a simple example problem that the numerical difficulties depend on the solution formulation. When the boundary conditions are imposed directly, the solution suffers from artificial damping, which may potentially lead to erroneous predictions when boundary element methods are used to evaluate the performance of damping materials. This difficulty can be alleviated by first computing an impedance or admittance matrix, and then using its reactive component to derive the solution for the acoustic field. Numerical computations are used to demonstrate that this technique eliminates artificial damping, but does not correct errors in the reactive components of the impedance or admittance matrices, which then causes nonexistence and nonuniqueness difficulties at the interior resonance frequencies for hard-wall and pressure release boundary conditions, respectively. It is shown that the admittance formulation is better suited to boundary element computations for interior problems because the resonance frequencies for pressure release boundary conditions do not begin until the smallest dimension of the boundary surface is at least one half the acoustic wavelength. Aside from producing much more accurate predictions, the admittance matrix is also much easier to interpolate at low frequencies due to the absence of interior resonances. For the example problem considered, only the formulation using the reactive component of the admittance matrix produces accurate solutions as long as the surface element discretization satisfies the standard six-element-per-wavelength rule.     

The role of spontaneous waves in the deflagration-to-detonation transition in submillimetre channels

Flame acceleration and transition to detonation in submillimetre two-dimensional planar and three-dimensional square channels were simulated by solving the compressible reactive Navier–Stokes equations. A simplified chemical–diffusive model was used to describe the diffusive transport and chemical reaction of a highly reactive mixture, such as stoichiometric ethylene and oxygen in 2D and 3D channels. The walls of the channels were modelled as no-slip and adiabatic. The initial flame acceleration and precursor shock formation were consistent with earlier results. Viscous dissipation in the boundary layer heats the reactants, which have been compressed by the precursor shock. The strength of the precursor shock and the amount of viscous dissipation increase until the temperature of the boundary layer is high enough to ignite the reactants. This produces a spontaneous wave, which, in most of the cases considered, initiates the detonation. The spontaneous wave first forms where the flame attaches to the wall in the planar channels, and forms at the corner where two walls meet in the square channels. In a separate study, the boundary layer also ignited in a computation for a circular tube containing a mixture hydrogen and oxygen represented by a detailed chemical reaction mechanism. The formation of spontaneous waves to the extent studied appears to be robust, and is relatively insensitive to channel geometry, fuel and oxidiser mixture, and the level of detail in the chemical–diffusive models used.     

Wilson loops in $ \mathcal{N} = 2 $ superconformal Yang-Mills theory

We study the problem of imposing Dirichlet-like boundary conditions along a static spatial curve, in a planar Noncommutative Quantum Field Theory model.

After constructing interaction terms that impose the boundary conditions, we discuss their implementation at the level of an interacting theory, with a focus on their physical consequences, and the symmetries they preserve. We also derive the effect they have on certain observables, like the Casimir energies.

     

Computational modelling of multi-material energetic materials and systems

ABSTRACT

In this paper, we present a systematic roadmap for developing a robust and parallel multi-material reactive hydrodynamic solver that integrates historically stable algorithms with new and current modern methods to solve explosive system design problems. The Ghost Fluid Method and Riemann solvers were used to enforce appropriate interface boundary conditions. Improved performance in terms of computational work and convergence properties was achieved by modifying a local node sorting strategy that decouples ghost nodes, allowing us to set material boundary conditions via an explicit procedure, removing the need to solve a coupled system of equations numerically. The locality and explicit nature of the node sorting concept allows for greater levels of parallelism and lower computational cost when populating ghost nodes. Non-linear numerical issues endemic to the use of real Equations of State in hydro-codes were resolved by using more thermodynamically consistent forms allowing us to accurately resolve large density gradients associated with high energy detonation problems at material interfaces. Pre-computed volume tables were implemented adding to the robustness of the solver base.     

Three-dimensional boundary conditions for numerical simulations of reactive compressible flows with complex thermochemistry

The Navier–Stokes characteristic boundary conditions (NSCBC) is a very efficient numerical strategy to treat boundary conditions in fully compressible solvers. The present work is an extension of the 3D-NSCBC method proposed by Yoo et al. and Lodato et al. in order to account for multi-component reactive flows with detailed chemistry and complex transport. A new approach is proposed for the outflow boundary conditions which enables clean exit of non-normal flows, and the specific treatment of all kinds of edges and corners is carefully addressed. The proposed methodology is successfully validated on various challenging multi-component reactive flow configurations.     

On the theory of a boundary layer associated with wave motion on the free surface of a liquid

A modified theory of a boundary layer associated with a periodic capillary-gravitational motion on the free surface of an infinitely deep viscous liquid is proposed. The flow in the boundary layer is described in terms of a simplified (compared with the complete statement) model problem a solution to which correctly reflects the main features of an exact asymptotic solution: the rapid decay of the flow eddy part with depth of the liquid and insignificance of some terms appearing in the complete statement. The boundary layer thickness at which the discrepancy between the exact asymptotic solution and model solution is within a given margin is estimated.     

Acoustic attenuation of hybrid silencers

The acoustic attenuation of a single-pass, perforated concentric silencer filled with continuous strand fibers is investigated first theoretically and experimentally. The study is then extended to a specific type of hybrid silencer that consists of two single-pass perforated filling chambers combined with a Helmholtz resonator. One-dimensional analytical and three-dimensional boundary element methods (BEM) are employed for the predictions of the acoustic attenuation in the absence of mean flow. To account for the wave propagation in absorbing fiber, the complex-valued characteristic impedance and wave number are measured. The perforation impedance facing the fiber is also presented in terms of complex-valued characteristic impedance and wave number. The effects of outer chamber diameter and the fiber density are examined. Comparisons of predictions with the experiments illustrate the need for multi-dimensional analysis at higher frequencies, while the one-dimensional treatment provides a reasonable accuracy at lower frequencies, as expected. The study also shows a significant improvement in the acoustic attenuation of the silencer due to fiber absorption. Multi-dimensional BEM predictions of a hybrid silencer demonstrate that a reactive component such as a Helmholtz resonator can improve transmission loss at low frequencies and a higher duct porosity may be effective at higher frequencies.     

The virial,boundary kinetic-energy density and electron density,in one dimension and in spherical symmetry

The virial is expressed in terms of the actual forces appearing in Schrödinger's equation and a term involving the kinetic-energy density at the boundary, and the derivatives of the electron density on the boundary.     

An experimental study of the acoustic impedance characteristics of human hair

Previous analytical and empirical studies of the human auditory system have shown that the cues used for localization are modified by the inclusion of nonrigid scattering surfaces (clothing, hair etc). This paper presents an investigation into the acoustic impedance properties of human hair. The legitimacy of a locally reactive surface assumption is investigated, and an appropriate boundary condition is formulated to account for the physiological composition of a human head with hair. This utilizes an equivalent impedance parameter to allow the scattering boundary to be defined at a reference plane coincident with the inner rigid surface of the head. Experimental examination of a representative synthetic hair material at oblique incidence is used to show that a locally reactive surface assumption is legitimate. Additional experimental analysis of a simple scattering problem illustrates that the equivalent impedance must be used in favor of the traditional surface impedance to yield physically correct pressure magnitudes. The equivalent acoustic impedance properties of a representative range of human hair samples are discussed, including trends with sample thickness, fiber diameter, bulk density, and mass.     

On the Riemann problem for a system of balance laws modelling a reactive gas mixture

This paper deals with the Riemann problem for a gas mixture undergoing reversible and irreversible bimolecular reactions governed by a suitable closure at Euler level of the Boltzmann equation, which results to be a set of balance laws. The aim of the paper is to investigate questions like compatibility with entropy principle and dissipative character of the system in order to better understand the effects of the source terms on the structure of the solution of the Riemann problem and its relation with the corresponding solution in absence of chemical reactions, which is here deduced explicitly. Numerical simulations show the space–time evolution of the solution profiles for the proposed reactive system.     

Characteristic boundary conditions for simulations of compressible reacting flows with multi-dimensional,viscous and reaction effects

A generalized formulation of the characteristic boundary conditions for compressible reacting flows is proposed. The new and improved approach resolves a number of lingering issues of spurious solution behaviour encountered in turbulent reacting flow simulations in the past. This is accomplished (a) by accounting for all the relevant terms in the determination of the characteristic wave amplitudes and (b) by accommodating a relaxation treatment for the transverse gradient terms with the relaxation coefficient properly determined by the low Mach number asymptotic expansion. The new boundary conditions are applied to a comprehensive set of test problems including: vortex-convection; turbulent inflow; ignition front propagation; non-reacting and reacting Poiseuille flows; and counterflow cases. It is demonstrated that the improved boundary conditions perform consistently superior to existing approaches, and result in robust and accurate solutions with minimal acoustic wave interactions at the boundary in hostile turbulent combustion simulation conditions.     

Improved algorithm for solving nonlinear parabolized stability equations

Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations(NPSE) approach has been widely used to study the stability and transition mechanisms. However,it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation(DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers.     

Spectral Action for Torsion with and without Boundaries

We derive a commutative spectral triple and study the spectral action for a rather general geometric setting which includes the (skew-symmetric) torsion and the chiral bag conditions on the boundary. The spectral action splits into bulk and boundary parts. In the bulk, we clarify certain issues of the previous calculations, show that many terms in fact cancel out, and demonstrate that this cancellation is a result of the chiral symmetry of spectral action. On the boundary, we calculate several leading terms in the expansion of spectral action in four dimensions for vanishing chiral parameter θ of the boundary conditions, and show that θ = 0 is a critical point of the action in any dimension and at all orders of the expansion.