Comparison between Kelvin–Helmholtz instability experiments on OMEGA and simulation results using the CRASH code

The Center for Radiative Shock Hydrodynamics (CRASH) at the University of Michigan has developed a Eulerian radiation-hydrodynamics code with dynamic adaptive mesh refinement, CRASH, which can model high-energy-density laser-driven experiments. One of these experiments, performed previously on the OMEGA laser facility, was designed to produce and observe the Kelvin–Helmholtz instability. The target design included low-density carbonized-resorcinol-formaldehyde (CRF) foam layered on top of polyamide–imide plastic, with a sinusoidal perturbation on the interface and with the assembled materials encased in beryllium. The results of a series of CRASH simulations of these Kelvin–Helmholtz instability experiments are presented. These simulation results show good agreement both quantitatively and qualitatively with the experimental data.     

Theory on transverse migration of particles in a turbulent two-phase suspension flow due to turbulent diffusion—I

A new theoretical model has been developed to explain the behavior of transverse particle transport in turbulent flow of a dilute two-phase suspension due to turbulent diffusion. This model is based on the ability of a particle to respond to surrounding fluid motion and depends on particle size and density relative to the carrier fluid, the fractional variation in particle concentration in the transverse direction as well as the existing turbulence structure of the surrounding fluid. The model developed in this investigation has been formulated by dividing the transverse fluid velocity, as seen by a particular particle, into two superimposed components representing, respectively, the transverse turbulent fluid fluctuations and an apparent transverse local fluid drifting velocity due to the effect on the transverse oscillatory component of fluid motion by the transverse concentration distribution of particles. A subsequent paper will show that the theory (together with other new results on the concentration effects on particle drag and lift and fluid turbulence properties) can help to explain the phenomena measured previously.     

Multi-component Reynolds-averaged Navier–Stokes simulations of Richtmyer–Meshkov instability and mixing induced by reshock at different times

Turbulent mixing generated by shock-driven acceleration of a perturbed interface is simulated using a new multi-component Reynolds-averaged Navier–Stokes (RANS) model closed with a two-equation $K$ – $\epsilon $ model. The model is implemented in a hydrodynamics code using a third-order weighted essentially non-oscillatory finite-difference method for the advection terms and a second-order central difference method for the gradients in the source and diffusion terms. In the present reshocked Richtmyer–Meshkov instability and mixing study, an incident shock with Mach number $M\!a_{\mathrm{s}}=1.20$ is generated in air and progresses into a sulfur hexafluoride test section. The time evolution of the predicted mixing layer widths corresponding to six shock tube test section lengths are compared with experimental measurements and three-dimensional multi-mode numerical simulations. The mixing layer widths are also compared with the analytical self-similar power-law solution of the simplified model equations prior to reshock. A set of model coefficients and initial conditions specific to these six experiments is established, for which the widths before and after reshock agree very well with experimental and numerical simulation data. A second set of general coefficients that accommodates a broader range of incident shock Mach numbers, Atwood numbers, and test section lengths is also established by incorporating additional experimental data for $M\!a_{\mathrm{s}}=1.24$ , $1.50$ , and $1.98$ with $At=0.67$ and $M\!a_{\mathrm{s}}=1.45$ with $At=-0.67$ and previous RANS modeling. Terms in the budgets of the turbulent kinetic energy and dissipation rate equations are examined to evaluate the relative importance of turbulence production, dissipation and diffusion mechanisms during mixing. Convergence results for the mixing layer widths, mean fields, and turbulent fields under grid refinement are presented for each of the $M\!a_{\mathrm{s}}=1.20$ cases.     

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.     

Discrete particle modeling of granular Rayleigh–Taylor instability

The gravitational air–grain Rayleigh–Taylor (RT) flow instability in a Hele-Shaw cell was studied using a parallel three-dimensional discrete particle model (DPM). The onset of flow instability and the development of fingering flow structures were well captured by the model. Power spectra analysis of solid volume fraction field indicated the non-linear coarsening process of the fingering flow structures. The sensitivity of the flow patterns to the initial porosity, the Atwood number, and the ratio of particle size to the Hele-Shaw cell width was also demonstrated. The excellent agreement of DPM simulation results with the reported experimental observations proved the robustness and reliability of the numerical approach to model complex multiphase flows such as granular RT instability.     

Numerical and Experimental Study of Rayleigh–Bénard–Kelvin Convection

We performed experimental and numerical studies of combined effects of thermal buoyancy and magnetization force applied on a cubical enclosure of a paramagnetic fluid heated from below and cooled from top. The temperature difference between the hot and cold wall was kept constant. After considering neutral situation (i.e. a pure natural convection case), magnetic fields of different intensity were imposed. The magnetization force produced significant changes in flow (transition from laminar to turbulent regimes), wall-heat transfer (enhancement) and turbulence (turbulence structures reorganization). The strong magnetic field and its gradients were generated by a superconducting magnet which can generate magnetic field up to 10 T and where gradients of the magnetic induction can reach up to 900 T²/m. A good agreement between experiments and numerical simulations was obtained in predicting the integral wall heat transfer over entire range of considered working parameters. Numerical simulations provided a detailed insights into changes of the local wall-heat transfer and long-term time averaged first and second moments for different strengths of the imposed magnetic induction.     

Spiral breakup in a RD system of cardiac excitation due to front–back interaction

We consider a two-variable partial differential equations model of cardiac excitation and study spiral wave instability in a one-parameter family of solutions. We investigate numerically the existence of periodic traveling wave solution and show the front and the back interaction far away from the bifurcation point in one dimension. In two dimensions, we show the emergence of a stable spiral pattern before the bifurcation point. The most complex spatiotemporal pattern is called ventricular fibrillation when the breakup of one spiral wave makes another wave and the medium becomes chaotic. We show spiral wave instability and periodic traveling wave instability in the same computational settings. It is found that the pattern of the front–back interaction in two dimensions is similar with that of in the one dimension.     

Rayleigh–Taylor instability of immiscible fluids in porous media

The time development of an interface separating two immiscible fluids of different densities in heterogeneous two-dimensional porous media is studied. The governing equations are simplified with the help of approximate Green’s functions which allow computation of the shape of the interface directly without resolving the fluid flow in the entire domain. The new formulation is amenable to numerical approximation, and the reduction in dimension leads to a significant gain in efficiency in the numerical simulation of the interfacial dynamics. Several test cases are investigated, and the numerical solutions are compared to known exact solutions and experimental data.     

The Richtmyer–Meshkov instability of a 'V' shaped air/helium interface subjected to a weak shock

The Richtmyer–Meshkov instability of a ‘V' shaped air/helium gaseous interface subjected to a weak shock wave is experimentally studied.A soap film technique is adopted to create a ‘V' shaped interface with accurate initial conditions.Five kinds of ‘V' shaped interfaces with different vertex angles are formed to highlight the effects of initial conditions on the flow characteristics.The results show that a spike is generated after the shock impact,and grows constantly with time.As the vertex angle increases,vortices generated on the interface become less noticeable,and the spike develops less pronouncedly.The linear growth rate of interface width after compression phase is estimated by a linear model and a revised linear model,and the latter is proven to be more effective for the interface with high initial amplitudes.The linear growth rate of interface width is,for the first time in a heavy/light interface configuration,found to be a non-monotonous function of the initial perturbation amplitude–wavelength ratio.     

Numerical study of Saffman–Taylor instability in immiscible nonlinear viscoelastic flows

In this paper, a numerical solution for Saffman–Taylor instability of immiscible nonlinear viscoelastic-Newtonian displacement in a Hele–Shaw cell is presented. Here, a nonlinear viscoelastic fluid pushes a Newtonian fluid and the volume of fluid method is applied to predict the formation of two phases. The Giesekus model is considered as the constitutive equation to describe the nonlinear viscoelastic behavior. The simulation is performed by a parallelized finite volume method (FVM) using second order in both the spatial and the temporal discretization. The effect of rheological properties and surface tension on the immiscible Saffman–Taylor instability are studied in detail. The destabilizing effect of shear-thinning behavior of nonlinear viscoelastic fluid on the instability is studied by changing the mobility factor of Giesekus model. Results indicate that the fluid elasticity and capillary number decrease the intensity of Saffman–Taylor instability.     

Multiple synchronization transitions due to periodic coupling strength in delayed Newman–Watts networks of chaotic bursting neurons

In this paper, we numerically study the effect of time-periodic coupling strength on the synchronization of firing activity in delayed Newman–Watts networks of chaotic bursting neurons. We first examine how the firing synchronization transitions induced by time delay under fixed coupling strength changes in the presence of time-periodic coupling strength, and then focus on how time-periodic coupling strength induces synchronization transitions in the networks. It is found that time delay can induce more synchronization transitions in the presence of time-periodic coupling strength compared to fixed coupling strength. As the frequency of time-periodic coupling strength is varied, the firing exhibits multiple synchronization transitions between spiking antiphase synchronization and in-phase synchronization of various firing behaviors including bursting, spiking, and both bursting and spiking, depending on the values of time delay. These results show that time-periodic coupling strength can increase the synchronization transitions by time delay and can induce multiple synchronization transitions of various firing behaviors in the neuronal networks. This means that time-periodic coupling strength plays an important role in the information processing and transmission in neural systems.     

Particle image velocimetry study of the shock-induced single mode Richtmyer–Meshkov instability

Full field particle image velocimetry (PIV) measurements are obtained for the first time in Richtmyer–Meshkov instability shock tube experiments. The experiments are carried out in a vertical shock tube in which the light gas (air) and the heavy gas (SF₆) flow from opposite ends of the shock tube driven section and exit through narrow slots at the interface location. A sinusoidal perturbation is given to the interface by oscillating the shock tube in the horizontal direction. Richtmyer–Meshkov instability is then produced by the interaction with a weak shock wave (M _s  = 1.21). PIV measurements are obtained by seeding the flow with 0.30 μm polystyrene Latex spheres which are illuminated using a double-pulsed Nd:YAG laser. PIV measurements indicate the vorticity to be distributed in a sheet-like distribution on the interface immediately after shock interaction and that this distribution quickly rolls up into compact vortices. The integration of the vorticity distribution over one half wave length shows the circulation to increase with time in qualitative agreement with the numerical study of Peng et al. (Phys. Fluids, 15, 3730–3744, 2003).     

Turing instability and pattern formation of neural networks with reaction–diffusion terms

In this paper, a model for a network of neurons with reaction–diffusion is investigated. By analyzing the linear stability of the system, Hopf bifurcation and Turing unstable conditions are obtained. Based on this, standard multiple-scale analysis is used for deriving the amplitude equations of the model for the excited modes in the Turing bifurcation. Moreover, the stability of different patterns is also determined. The obtained results enrich the dynamics of neurons’ network system.     

Viscous effects on the non-classical Rayleigh–Taylor instability of spherical material interfaces

The viscous and conductivity effects on the instability of a rapidly expanding material interface produced by a spherical shock tube are investigated through the employment of a high-order WENO scheme. The instability is influenced by various mechanisms, which include (a) classical Rayleigh–Taylor (RT) effects, (b) Bell–Plesset or geometry/curvature effects, (c) the effects of impulsively accelerating the interface, (d) compressibility effects, (e) finite thickness effects, and (f) viscous effects. Henceforth, the present instability studied is more appropriately referred to as non-classical RT instability to distinguish it from classical RT instability. The linear regime is examined and the development of the viscous three-dimensional perturbations is obtained by solving a one-dimensional system of partial differential equations. Numerical simulations are performed to illustrate the viscous effects on the growth of the disturbances for various conditions. The inviscid analysis does not show the existence of a maximum amplification rate. The present viscous analysis, however, shows that the growth rate increases with increasing the wave number, but there exists a peak wavenumber beyond which the growth rate decreases with increasing the wave number due to viscous effects.