Direct numerical simulations of gas–solid–liquid interactions in dilute fluids

The dynamic interactions between gas bubbles, rigid particles and liquid can lead to profound nonlinearities in the aggregate behavior of a multiphase fluid. Predicting the nonlinear dynamics of the multiphase mixture hence requires understanding how the phases interact at the scale of individual interfaces, but these interactions are notoriously difficult to resolve in models. The goal of this paper is to develop and validate a computational method capable of capturing the complex flow interactions between gas bubbles and rigid particles immersed in a Newtonian liquid. We focus on multiphase systems that are dilute enough for the solid and gas components to move through and be moved by the ambient liquid. We use level sets with a topology-preserving advection scheme to track the gas interfaces. To include the motion of the rigid particles, we couple distributed Lagrange multipliers to an immersed-boundary method. The high viscosity contrast between the liquid and the gas requires both time splitting and approximate factorization to efficiently solve the governing equations consisting of the conservation of mass, momentum and energy. To resolve interactions between interfaces that vary drastically in size, we refine our mesh adaptively in the vicinity of the boundary.     

Numerical simulations of the process of multiple shock–flame interactions

Based on a weighted essentially nonoscillatory scheme, the multiple interactions of a flame interface with an incident shock wave and its reshock waves are numerically simulated by solving the compressible reactive Navier–Stokes equations with a single-step Arrhenius chemical reaction. The two-dimensional sinusoidally perturbed premixed flames with different initial perturbed amplitudes are used to investigate the effect of the initial perturbation on the flame evolutions. The results show that the development of the flame interface is directly affected by the initial perturbed amplitudes before the passages of reshock waves, and the perturbation development is mainly controlled by the Richtmyer–Meshkov instability(RMI). After the successive impacts of multiple reshock waves, the chemical reaction accelerates the consumption of reactants and leads to a gradual disappearance of the initial perturbed information. The perturbation developments in frozen flows with the same initial interface as those in reactive flows are also demonstrated.Comparisons of results between the reactive and frozen flows show that a chemical reaction changes the perturbation pattern of the flame interface by decreasing the density gradient,thereby weakening the baroclinic torque in the flame mixing region, and therefore plays a dominant role after the passage of reshock waves.     

Spatiotemporal dynamics of a predator–prey model

Spatial component of ecological interactions has been identified as an important factor in how ecological communities are shaped. In this paper, we consider a Holling?CTanner model with spatial diffusion. Choosing appropriate parameter values in parameter spaces, we obtain rich patterns, including spotted, black-eye, and labyrinthine patterns. The numerical results show that predator?Cprey system can exhibit complicated behavior.     

Solution and asymptotic analysis of a boundary value problem in the spring–mass model of running

We consider the classic spring–mass model of running which is built upon an inverted elastic pendulum. In a natural way, there arises an interesting boundary value problem for the governing system of two nonlinear ordinary differential equations. It requires us to choose the stiffness to ascertain that after a complete step, the spring returns to its equilibrium position. Motivated by numerical calculations and real data, we conduct a rigorous asymptotic analysis in terms of the Poicaré–Lindstedt series. The perturbation expansion is furnished by an interplay of two time scales what has an significant impact on the order of convergence. Further, we use these asymptotic estimates to prove that there exists a unique solution to the aforementioned boundary value problem and provide an approximation to the sought stiffness. Our results rigorously explain several observations made by other researchers concerning the dependence of stiffness on the initial angle of the stride and its velocity. The theory is illustrated with a number of numerical calculations.

     

Hidden dynamics in a fractional-order memristive Hindmarsh–Rose model

Nonlinear Dynamics - By showing that there is no any memristor in the integer-order Hindmarsh–Rose (H–R) model, the slow ion channel is remodeled by a fractional-order memristor, where...     

Rich dynamics in a China Telecom–cable television model

The policy and schedule of the integration of the three networks, i.e., Internet, telecom, and cable television (catv), were formally released by the Chinese government. However, under this circumstance, the dynamical behaviors between the telecom and catv industry have not been studied. As a result, a discrete-time China Telecom–cable television model by qualitative analysis and numerical simulation is investigated. It is verified that there are phenomena of the transcritical bifurcation, flip bifurcation types, and chaos. The results obtained show that such a model can have rich dynamical behavior and the market behavior cannot be well predicted due to the existence of chaos. It may suggest that the telecom and the catv industry should operate their traditional business more efficiently and strengthen the development and propagation of novel businesses in their own industry.     

Nonlinear dynamics of a delayed Leslie predator–prey model

The present paper is concerned with a delayed Leslie predator–prey model. The conditions of boundedness of the solutions of the system, existence, and stability of the equilibrium of the system are investigated. Meanwhile, we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the delay crosses through a sequence of critical values. The extensive simulations carried out show that the bifurcations arise around the positive equilibrium.     

Toward numerical simulations of fluid–structure interactions for investigation of obstructive sleep apnea

Obstructive sleep apnea (OSA) is a medical condition characterized by repetitive partial or complete occlusion of the airway during sleep. The soft tissues in the airway of OSA patients are prone to collapse under the low-pressure loads incurred during breathing. This paper describes efforts toward the development of a numerical tool for simulation of air–tissue interactions in the upper airway of patients with sleep apnea. A procedure by which patient-specific airway geometries are segmented and processed from dental cone-beam CT scans into signed distance fields is presented. A sharp-interface embedded boundary method based on the signed distance field is used on Cartesian grids for resolving the airflow in the airway geometries. For simulation of structure mechanics with large expected displacements, a cut-cell finite element method with nonlinear Green strains is used. The fluid and structure solvers are strongly coupled with a partitioned iterative algorithm. Preliminary results are shown for flow simulation inside the three-dimensional rigid upper airway of patients with obstructive sleep apnea. Two validation cases for the fluid–structure coupling problem are also presented.     

Nonlinear dynamics of a planar beam–spring system: analytical and numerical approaches

The multiple timescales method is applied to the exact partial differential equations of the planar motion of a hinged–simply supported beam with a linear axial spring of arbitrary stiffness. The forced-damped and free oscillations of the system around frequencies corresponding to nth natural bending mode are examined thoroughly and compared with numerical simulations as well as with already published results obtained by Lindstedt–Poincaré method. A special numerical technique using explicit finite element method to draw the frequency–response curves is appositely developed. The well-known jump phenomena between resonant and non-resonant branches, as well as superharmonic resonances, have been detected numerically.     

On the nonlinear dynamics of an ecoepidemic reaction–diffusion model

A reaction–diffusion ecoepidemic model of predator–prey type with a transmissible disease spreading among the predator species only is considered. The longtime behavior of solutions is analyzed and, in particular, absorbing sets in the phase space are determined. Conditions guaranteeing the non existence of non-constant equilibria have been found. Linear and non-linear stability conditions for biologically meaningful equilibria are determined.     

Pattern dynamics of a spatial predator–prey model with noise

A spatial predator–prey model with colored noise is investigated in this paper. We find that the number of the spotted pattern is increased as the noise intensity is increased. When the noise intensity and temporal correlation are in appropriate levels, the model exhibits phase transition from spotted to stripe pattern. Moreover, we show the number of the spotted and stripe pattern, with respect to both noise intensity and temporal correlation. These studies raise important questions on the role of noise in the pattern formation of the populations, which may well explain some data obtained in the ecosystems.     

Critical oscillations of mass–spring systems due to nonsmooth friction

Critical values of the parameters governing the dynamics of simple systems appear when Coulomb friction is not regularized. We explore such systems using a method based on the fact that under constant or analytical data the trajectory exists, is unique and is also sufficiently regular. In fact these properties justify elementary analytical computations on successive time intervals where the condition used to connect the solution from one interval to the other is due to the regularity. Although the systems are simple the dynamics turn out to be quite complex and thus furnish an interesting benchmark for contact dynamics numerical codes. Among other possible applications we choose to present here how to use a mass–spring chain with Coulomb friction to slow down in a progressive and regular manner an oncoming mass with a given initial velocity.     

Finite element simulations of dynamics of multivariant martensitic phase transitions based on Ginzburg–Landau theory

A finite element approach is suggested for the modeling of multivariant stress-induced martensitic phase transitions (PTs) in elastic materials at the nanoscale for the 2-D and 3-D cases, for quasi-static and dynamic formulations. The approach is based on the phase-field theory, which includes the Ginzburg–Landau equations with an advanced thermodynamic potential that captures the main features of macroscopic stress–strain curves. The model consists of a coupled system of the Ginzburg–Landau equations and the static or dynamic elasticity equations, and it describes evolution of distributions of austenite and different martensitic variants in terms of corresponding order parameters. The suggested explicit finite element algorithm allows decoupling of the Ginzburg–Landau and elasticity equations for small time increments. Based on the developed phase-field approach, the simulation of the microstructure evolution for cubic-tetragonal martensitic PT in a NiAl alloy is presented for quasi-statics (i.e., without inertial forces) and dynamic formulations in the 2-D and 3-D cases. The numerical results show the significant influence of inertial effects on microstructure evolution in single- and polycrystalline samples, even for the traditional problem of relaxation of initial perturbations to stationary microstructure.     

Eulerian–Lagrangian 3-D simulations of unsteady two-phase gas–liquid flow in a rectangular column by considering bubble interactions

This work discusses the development of a three-dimensional Eulerian–Lagrangian CFD model for a gas–liquid flow in a rectangular column. The model resolves the time-dependent, three-dimensional motion of small gas bubbles in a liquid to simulate the dynamic characteristics of the oscillating bubble plume. Our model incorporates drag, gravity, buoyancy, lift, pressure gradient and virtual mass forces acting on a bubble rising in a liquid, and accounts for two-way momentum coupling between the phases. We use MUSIG model that provides a framework in which the population balance method together with the break up and coalescence models can be incorporated into three-dimensional CFD calculations. We use turbulent flow to describe liquid flow field. The standard κ–ε of turbulence is selected for calculating the properties of turbulent flow. The effect of aspect ratio of the column on the flow pattern, liquid velocity and gas hold-up profiles is discussed.     

Pattern dynamics in a toxin-producing phytoplankton–zooplankton model with additional food

In this paper, we investigate the pattern dynamics in a spatial plankton model including phytoplankton which can release toxins, and zooplankton provided with additional food. The combined effects of toxin liberation and additional food on the stability of the system are explored. It is found that intermediate amounts of additional food and toxin liberation promote the stable coexistence of phytoplankton and zooplankton. A large quantity of additional food can generate Turing patterns more easily, whereas excessive toxin liberation leads to the extinction of zooplankton. Moreover, we obtain the conditions for the occurrence of Turing instability by linear stability analysis. Near the critical value of Turing instability, a multiple-scale method is applied to derive the amplitude equations based on which we can consider the selection and stability of different patterns. The corresponding theoretical results are illustrated by numerical simulations. Furthermore, we show the transitions of pattern formations due to varying the amounts of additional food and toxin liberation, which provides us with particular insight regarding the control of plankton distribution.