The role of the kinetic parameter in the stability of two-relaxation-time advection–diffusion lattice Boltzmann schemes

In general, explicit numerical schemes are only conditionally stable. A particularity of lattice Boltzmann multiple-relaxation-time (MRT) schemes is the presence of free (“kinetic”) relaxation parameters. They do not appear in the transport coefficients of the modelled second-order (macroscopic) equations but they have an impact on the effective accuracy and stability of the algorithm. The simplest uniform choice (the well known BGK/SRT model) is often inadequate, and therefore a compromise in the complexity of the model is sought. For this purpose, the von Neumann stability analysis is performed for the d1Q3 two-relaxation-time (TRT) advection–diffusion model. The extended optimal (EOTRT) model, which relates the two collision times such that the most stable scheme is set by a suitable choice of the equilibrium parameters, equal for any Peclet number, is then developed. This extends the very recently derived optimal subclass (OTRT) to larger combinations of “physical” and “kinetic” collision rates. Next, we provide the necessary and/or sufficient stability limits on the EOTRT subclass for a wide range of velocity sets, with and without numerical diffusion, and delineate the interesting choices of free equilibrium weights for the d2Q9 and d3Q15 models. The BGK/SRT model is without advanced advection properties; we prove (for minimal stencil schemes d1Q3, d2Q5 and d3Q7) that the non-negativity of the equilibrium distribution is necessary for its stability in the advection-dominated limit. Beyond the EOTRT and BGK/SRT subclasses of the TRT model, blind choices of the “ghost” collision number may result in quite unstable schemes, even for positive equilibrium. However, we find that the d1Q3 stability curves govern the advection properties of the multi-dimensional models and a fuller picture of the TRT stability properties begins to emerge.     

An evaluation of lattice Boltzmann schemes for porous medium flow simulation

We quantitatively evaluate the capability and accuracy of the lattice Boltzmann equation (LBE) for modeling flow through porous media. In particular, we conduct a comparative study of the LBE models with the multiple-relaxation-time (MRT) and the Bhatnagar–Gross–Krook (BGK) single-relaxation-time (SRT) collision operators. We also investigate several fluid–solid boundary conditions including: (1) the standard bounce-back (SBB) scheme, (2) the linearly interpolated bounce-back (LIBB) scheme, (3) the quadratically interpolated bounce-back (QIBB) scheme, and (4) the multi-reflection (MR) scheme. Three-dimensional flow through two porous media—a body-centered cubic (BCC) array of spheres and a random-sized sphere-pack—are examined in this study. For flow past a BCC array of spheres, we validate the linear LBE model by comparing its results with the nonlinear LBE model. We investigate systematically the viscosity-dependence of the computed permeability, the discretization error, and effects due to the choice of relaxation parameters with the MRT and BGK schemes. Our results show unequivocally that the MRT–LBE model is superior to the BGK–LBE model, and interpolation significantly improves the accuracy of the fluid–solid boundary conditions.     

A comparison study of multi-component Lattice Boltzmann models for flow in porous media applications

A comparison study of three different multi-component Lattice Boltzmann models is carried out to explore their capability of describing binary immiscible fluid systems. The Shan–Chen pseudo potential model, the Oxford free energy model and the colour gradient model are investigated using the multi-relaxation time scheme (MRT) algorithm to study the flow of binary immiscible fluids. We investigate Poiseuille flow of layered immiscible binary fluids and capillary fingering phenomena and evaluate the results against analytical solutions. In addition, we examine the capability of the various models to simulate fluids with significant viscosity and density contrast and suitable interface thickness. This is of great importance for large scale flow in porous media applications, where it is important to minimise the interfacial thickness from a computational point of view. We find that the Shan–Chen model can simulate high density ratios up to 800 for binary fluids with the same viscosity. Imposing a viscosity contrast will lead to highly diffusive interfaces in the Shan–Chen model and therefore this will affect significantly the numerical stability. The Free Energy model and the colour gradient model have similar capabilities on this point: they can simulate binary fluids having the same density but with significant viscosity contrast. This is of great importance to study the flow of water, supercritical CO₂ and oil in porous media, for CO₂ storage and Enhanced Oil Recovery (EOR) operations.     

Timing of traffic lights and phase separation in two-dimensional traffic flow

In this paper, we study the effects of traffic light period in two-dimensional Biham-Middleton-Levine (BML) traffic flow model. It is found that a phase separation phenomenon, in which the system separates into coexistence of free flow and jam, could be observed in intermediate vehicle density range when traffic light period T?4. We have explained the reason of occurrence of phase separation and investigated its behavior in different traffic light period.     

Diffusion of fluid confined to nanotube with rectangular cross section

A dynamical model is proposed to study self-diffusion coefficient by confining the fluid in rectangular nanotube. The theoretical model is based on the consideration that the confinement affects the movement at atomic level. The model predicts that the diffusion parallel to walls of channel is different from that of diffusion perpendicular to the walls. Near the walls the dynamics of fluid has been found to slow down to an extent that below a certain value of ratio of width to the diameter of particle, the molecules behave as if these belong to solid. The results are contrasted with the result obtained from the model based on similar considerations for a fluid confined only in one direction. It is found that tendency of freezing near the wall increases due to confinement from second direction. Empirical relation which governs the behavior of diffusion coefficient as function of distance from the confining walls has also been proposed. The effect of confinement is more pronounced for denser fluids than for dilute fluid.     

Multirange multi-relaxation time Shan–Chen model with extended equilibrium

The work presents simulations with the multirange Shan–Chen model developed by Sbragaglia et al. (2007) [18], which improved the Shan–Chen model for the proper surface tension term. Also, by introducing the matrix collision operator and extended equilibrium density distribution function, the density ratio is increased from 100 to 160. The Multi-Relaxation Time (MRT) method attracted the attention of researchers due to several advantages, such as better stability, simulations with Prandtl number different from unity, and possibilities to improve the accuracy of the scheme compared with BGK Single Time Relaxation model. Our recent results have shown that the combination of MRT methods with multiphase flow models can improve the achievable gas–liquid density ratio.     

Effect of temperature gradient orientation on the characteristics of mixed convection flow in a lid-driven square cavity

The effect of temperature gradient orientation on the fluid flow and heat transfer in a lid-driven differentially heated square cavity is investigated numerically. The transport equations are solved using the high-order compact scheme. Four cases are considered depending on the direction of temperature gradient imposed. The differentially heated top and bottom walls result in gravitationally stable and unstable temperature gradients. While the differentially heated left and right side walls lead to assisting and opposing buoyancy effects. The governing parameters are Pr = 0.7 and Ri = 0.1, 1, and 10. It is found that both Richardson number and direction of temperature gradient affect the flow patterns, heat transport processes, and heat transfer rates in the cavity. Computed average Nusselt number indicates that the heat transfer rate increases with decreasing Ri regardless the orientation of temperature gradient imposed. And the assisting buoyancy flows have best performance on heat transport over the other three cases.     

Analysis of Lattice-Boltzmann methods for internal flows

The applicability of several Lattice-Boltzmann methods to wall-bounded turbulent flows is investigated. The various methods consist of the standard Bhatnagar–Gross–Krook (BGK) method with 19 (BGK19) and 27 (BGK27) discrete velocities, the multiple-relaxation-time (MRT) model with 19 discrete velocities and the cascaded Lattice-Boltzmann method (CLB). Based on the findings of turbulent channel flow it can be concluded that stability considerations, predicting the superiority of the advanced moment based schemes like the CLB and MRT method not necessarily hold for wall-bounded turbulent flows. Moreover, in some flow problems the simple BGK method with 19 discrete velocities delivers reasonable and stable results, where the other methods yield unphysical solutions.     

Effective scheduling of local interactive processes and parallel processes in a non-dedicated cluster environment

This paper presents a performance evaluation of the interactions between local sequential processes running on behalf of interactive applications and parallel processes running as part of an overall parallel application on a non-dedicated cluster environment. To control the interactions between the two types of processes we propose to constrain the scheduling of local interactive processes (IPs) by a measure of the maximum response time (MRT) expected by the workstation (WS) user. The measure is assumed obtained through empirical studies. We propose a mathematical model of the scheduling problem based on the usage of the MRT measure. In addition, we propose a scheduling scheme that within the MRT cycle computes the time quanta needed to satisfy the requirements of both local IPs and the parallel task process present in the system. A colored Petri net (CPN) is used to model the scheduling scheme. Simulations of the CPN model and numerical results have shown the effectiveness of the proposed scheduling scheme in allowing the parallel task to ensure a minimum speedup even in heavy-loaded situations and to maximize the speedup adaptively depending on load conditions. In addition the simulation results revealed the sensitivity of the interactions to factors such as interactive job demand, parallel job demand, and arrival rate of interactive jobs among other. Finally, simulation and analytical results have been found to agree nicely thus confirming the correctness of the proposed analytical model.     

Numerical simulation of free surface flows on a fish bypass

A computational fluid dynamics (CFD) model is developed to better understand the complex flow field inside a free surface fish bypass constructed at Rocky Reach Dam. This facility consists of two identical parallel channels, with fish screens on the side walls of each channel, and a pump station recirculating 96% of incoming flows into the forebay. The model is based on the Reynolds-averaged Navier-Stokes (RANS) equations, with a standard κ-ε turbulence model. The volume of fluid (VOF) method is used to predict free surface elevations. A proportional controller is implemented in the model to achieve a target flow rate at the pump exits. The pressure drop in the fish screens is modeled using porous media. Quantitative validation and visualization of the flow field characteristics indicate that CFD modeling may be a useful tool for fish passage design.     

Two secret sharing schemes based on Boolean operations

Traditional secret sharing schemes involve complex computation. A visual secret sharing (VSS) scheme decodes the secret without computation, but each shadow is m times as big as the original. Probabilistic VSS solved the computation complexity and space complexity problems at once. In this paper we propose a probabilistic (2,n) scheme for binary images and a deterministic (n,n) scheme for grayscale images. Both use simple Boolean operations and both have no pixel expansion. The (2,n) scheme provides a better contrast and significantly smaller recognized areas than other methods. The (n,n) scheme gives an exact reconstruction.     

Numerical analysis of confined turbulent flow

This work outlines a second order accurate, coupled, conservative new numerical scheme for solving a two dimensional incompressible turbulent flow filed. Mean vorticity, ω, and mean stream function, ψ, are used as the mean flow dependent variables. The turbulent kinetic energy $\text{k}$ and the turbulent energy decay rate, ?, are used to define the turbulent state. In the present computational scheme two systems of equations and variables are considered: the mean flow system, ψ-ω, and the turbulent state system, $\text{k-?}$. Every system is solved implicity in a coupled double loop manner, and all the flow equations are solved iteratively in the global sense. Since the turbulence boundary conditions have a non-regular variation near a solid wall, they are coupled to the equations implicitly in both systems. In this way the numerical instabilities due to the irregular form of the equations near the solid walls are suppressed. The rate of convergence of the new numerical scheme of the coupled systems ψ-ω and $\text{k-?}$ is twice that realized when solving these equations separately. The necessary conditions for convergence of the numerical equations are investigated as well as the rate of convergence features. The detailed stability conditions are derived. As an example, the axisymmetric mixing of two confined jets with an internal heat source is considered with this numerical scheme.     

Numerical prediction of confined swirling jets

The axisymmetric flow of a swirling viscous, incompressible fluid jet inside confining cylindrical boundaries has been numerically investigated using the well-known implicit finite-difference scheme. For a swirling jet confined by cylindrical tube, the vortex-breakdown or the formation of an axisymmetric isolated eddy occurs at high values of swirl ratios at a moderate flow rate or Reynolds number. With the introduction of artificial adverse pressure gradients, such as one studied in the case of a step-up cylindrical tube, the vortex-breakdown occurs at a relatively lower swirl ratio at a given flow rate. For a swirling jet discharging in a coaxial non-rotating surrounding stream enclosed by a cylindrical tube, the vortex-breakdown and its structure depend on various parameters such as the flow rate of the jet, surrounding stream velocity, the swirl of the jet and on the radius of the enclosing cylindrical tube. In general increasing Reynolds numbers, swirl ratio, decreasing surrounding stream velocity and increasing size of the cylindrical tube enhance the occurrence and size of the vortex-breakdown.     

Multiple vortex formation in steady laminar axisymmetric impinging flow

Steady confined laminar axisymmetric impinging flow of a Newtonian fluid is relevant in many situations, an important application being heat and mass transfer from a solid surface to an impinging jet. This paper focuses on the evolution of the structure of the radial flow field in the channel region beyond the impingement zone. We employ an upwind scheme with an established numerical technique to solve the stream function and vorticity equations for a range of Reynolds numbers Re and geometrical aspect ratios e. Our results show the progressive complexity in the radial flow due to multiple points of flow separation and reattachment, and we provide a detailed demarcation of the Re-e plane based on flow separation behavior. In addition to the primary and secondary vortices anchored on the confining and impinging surfaces, respectively, we describe the formation and properties of a tertiary vortex which is wholly enclosed within the primary vortex. At a fixed Reynolds number, the tertiary vortex is observed only for a specific range of the aspect ratio, and we catalog its birth, growth and demise as the aspect ratio is varied. The range of aspect ratios over which the tertiary vortex exists is seen to increase with the Reynolds number. These results show that the fine structure of the radial flow at high Reynolds number continues to be dependent on the aspect ratio in a complex manner. At a given aspect ratio, the sizes of the vortices increases with Reynolds number, scaling as ∼Re^1/3, and for sufficiently large Re, the length of the tertiary vortex can exceed that of the secondary vortex. The primary and secondary vortex lengths satisfy an asymptotic relationship independent of Re and e, the numerically computed value of α being ∼2. Similarly, the locations of these vortices bear simple linear relationships independent of Re and e. Furthermore, despite the complex fine structure of the flow field, macroscopic flow properties such as vortex circulation and excess pressure loss continue to exhibit relatively simple dependence on Re and e, in accordance with previous results at much lower Reynolds numbers. Finally, some comments are made regarding the possibility of additional cascaded or isolated vortices occurring at even higher Reynolds numbers and aspect ratios.     

Free surface flow simulations on GPGPUs using the LBM

In this paper, we present the implementation of a volume-of-fluid-(VOF)-based algorithm for the simulation of free-surface flow problems on general purpose graphical processing units (GPGPUs). For the solution of the flow field and the additional advection equation for the VOF fill level, the lattice Boltzmann method on the basis of an MRT collision operator is used. A Smagorinsky LES model serves to capture the small-scale turbulent structures of the flow. We show that despite the additional non-local operations near the phase interface, we end up with an algorithm with good overall performance, which is suitable for the simulation of demanding real-world engineering applications.     

Control of numerical effects of dispersion and dissipation in numerical schemes for efficient shock-capturing through an optimal Courant number

Composite schemes consist of several steps of a dispersive scheme followed by one step of a dissipative scheme [Liska Richard, Wendroff Burton. Composite schemes for conservation laws. SIAM J Numer Anal 1998;35(6):2250-71]. The latter [Liska Richard, Wendroff Burton. 2D shallow water equations by composite schemes. Int J Numer Meth Fluids 1999;30:461-79] acts as a filter reducing oscillations in regions of discontinuity. Liska and Wendroff have derived the composite Lax-Wendroff/Lax-Friedrichs (LWLF) [Liska Richard, Wendroff Burton, 1998] scheme which blends the Lax-Wendroff (LW) scheme with the 2-step Lax-Friedrichs (LF) scheme. The formulation of the 2-step Lax-Friedrichs scheme [Liska Richard, Wendroff Burton, 1998] is different from that of the classic Lax-Friedrichs scheme and has been devised by Liska [Liska Richard, Wendroff Burton, 1998]. In this work, we propose to replace LW scheme by MacCormack (MC) scheme since the latter is less dispersive. We obtain a new composite scheme in 1-D and in 2-D by blending the MacCormack scheme with the 2-step Lax-Friedrichs scheme which we term as the composite MacCormack/Lax-Friedrichs (MCLF) scheme. This is followed by analytical work on the effective amplification factor (EAF) and the relative phase error (RPE) for both families of schemes in 1-D and 2-D: LWLFn and MCLFn, consisting of (n − 1) steps of the dispersive scheme (LW or MC) and 1 step of the dissipative LF scheme. We introduce a new concept, baptised as Curbing of Dispersion by Dissipation for Efficient Shock-capturing, CDDES in which a cfl number is computed whereby dissipation curbs dispersion. This cfl number is termed as optimal in this work. We conduct a comparative study based on numerical experiments in 2-D namely: contact-discontinuity problem [Ould Kaber SM. A legendre pseudospectral viscosity method. J Comput Phys 1996;128:165-80], rotating hill problem [Ould Kaber SM, 1996] and the deformative flow of Smolarkiewicz [Dabdub Donald, Seinfeld John H. Numerical advective schemes used in air quality models-sequential and parallel implementation. Atmos Environ 1994;28(20):3369-85, Ghods A, Sobouti F, Arkani-Hamed J. An improved second order method for solution of pure advection problems. Int J Numer Meth Fluids 2000;32:959-77, Nguyen K, Dabdub D. Two-level time-marching scheme using splines for solving the advection equation. Atmos Environ 2001;35:1627-37] to show that the MacCormack/Lax-Friedrichs (MCLF) scheme is more efficient than LWLF scheme to capture shocks in regions of discontinuity. We also show that better results are obtained at optimal cfl numbers for some variants of LWLFn and MCLFn schemes, with n = 2, 3, 4 and 5.     

Resource allocation decisions under various demands and cost requirements in an unreliable flow network

This paper considers resource allocation decisions in an unreliable multi-source multi-sink flow network, which applies to many real-world systems such as electric and power systems, telecommunications, and transportation systems. Due to uncertainties of components in such an unreliable flow network, transmitting resources successfully and economically through the unreliable flow network is of concern to resource allocation decisions at resource-supplying (source) nodes. We study the resource allocation decisions in an unreliable flow network for a range of demand configurations constrained by demand-dependent and demand-independent cost considerations under the reliability optimization objective. Solutions to these problems can be obtained by computing the resource allocation for each demand configuration independently. In contrast, we pursue an updating scheme that eludes time-consuming enumeration of flow patterns, which is necessary in independent computation of resource allocations for different demand configurations. We show that updating is attainable under both demand-independent and demand-dependent cost constraints when demand incurs an incremental change, and demonstrate the proposed updating scheme with numerical examples.     

A new 3D parallel SPH scheme for free surface flows

We propose a new robust and accurate SPH scheme, able to track correctly complex three-dimensional non-hydrostatic free surface flows and, even more important, also able to compute an accurate and little oscillatory pressure field. It uses the explicit third order TVD Runge-Kutta scheme in time, following Shu and Osher [Shu C-W, Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J Comput Phys 1988;89:439-71], together with the new key idea of introducing a monotone upwind flux for the density equation, thus removing any artificial viscosity term. For the discretization of the velocity equation, the non-diffusive central flux has been used. A new flexible approach to impose the boundary conditions at solid walls is also proposed. It can handle any moving rigid body with arbitrarily irregular geometry. It does neither produce oscillations in the fluid pressure in proximity of the interfaces, nor does it have a restrictive impact on the stability condition of the explicit time stepping method, unlike the repellent boundary forces of Monaghan [Monaghan JJ. Simulating free surface flows with SPH. J Comput Phys 1994;110:399-406]. To asses the accuracy of the new SPH scheme, a 3D mesh-convergence study is performed for the strongly deforming free surface in a 3D dam-break and impact-wave test problem providing very good results.Moreover, the parallelization of the new 3D SPH scheme has been carried out using the message passing interface (MPI) standard, together with a dynamic load balancing strategy to improve the computational efficiency of the scheme. Thus, simulations involving millions of particles can be run on modern massively parallel supercomputers, obtaining a very good performance, as confirmed by a speed-up analysis. The 3D applications consist of environmental flow problems, such as dam-break flows and impact flows against a wall. The numerical solutions obtained with our new 3D SPH code have been compared with either experimental results or with other numerical reference solutions, obtaining in all cases a very satisfactory agreement.     

An adaptive scheme for fault-tolerant scheduling of soft real-time tasks in multiprocessor systems

The scheduling of real-time tasks with primary-backup-based fault-tolerant requirements has been an important problem for several years. Most of the known scheduling schemes are non-adaptive in nature meaning that they do not adapt to the dynamics of faults and task's parameters in the system. In this paper, we propose an adaptive fault-tolerant scheduling scheme that has a mechanism to control the overlap interval between the primary and backup versions of tasks such that the overall performance of the system is improved. The overlap interval is determined based on the observed fault rate and task's soft laxity. We also propose a new performance index, called SR index, that integrates schedulability (S) and reliability (R) into a single metric. To evaluate the proposed scheme, we have conducted analytical and simulation studies under different fault and deadline scenarios, and found that the proposed adaptive scheme adapts to system dynamics and offers better SR index than that of the non-adaptive schemes.     

Image Compression by Layered Quantum Neural Networks

We have proposed the qubit neuron model as a new scheme in non-standard computing. Identification problems have been investigated on neural networks constructed by this qubit neuron model, and we have found high processing abilities of them. In this paper, we evaluate the performance of the quantum neural network of large size in image compression problems to estimate the utility for the practical applications comparing with the conventional network consists of formal neuron model.