Limit Consistency of Lattice Boltzmann Equations

We establish the notion of limit consistency as a modular part in proving the consistency of lattice Boltzmann equations (LBE) with respect to a given partial differential equation (PDE) system. The incompressible Navier-Stokes equations (NSE) are used as paragon. Based upon the diffusion limit [L. Saint-Raymond (2003), doi: 10.1016/S0012-9593(03)00010-7] of the Bhatnagar-Gross-Krook (BGK) Boltzmann equation towards the NSE, we provide a successive discretization by nesting conventional Taylor expansions and finite differences. Elaborating the work in [M. J. Krause (2010), doi: 10.5445/IR/1000019768], we track the discretization state of the domain for the particle distribution functions and measure truncation errors at all levels within the derivation procedure. Via parametrizing equations and proving the limit consistency of the respective sequences, we retain the path towards the targeted PDE at each step of discretization, i.e. for the discrete velocity BGK Boltzmann equation and the space-time discretized LBE. As a direct result, we unfold the discretization technique of lattice Boltzmann methods as chaining finite differences and provide a generic top-down derivation of the numerical scheme which upholds the continuous limit

Fluid-Structure Interaction Simulation of a Coriolis Mass Flowmeter using a Lattice Boltzmann Method

In this paper we use a fluid-structure interaction (FSI) approach to simulate a Coriolis mass flowmeter (CMF). The fluid dynamics are calculated by the open source framework OpenLB, based on the lattice Boltzmann method (LBM). For the structural dynamics we employ the open source software Elmer, an implementation of the finite element method (FEM). A staggered coupling approach between the two software packages is presented. The finite element mesh is created by the mesh generator Gmsh to ensure a complete open source workflow. The Eigenmodes of the CMF, which are calculated by modal analysis are compared with measurement data. Using the estimated excitation frequency, a fully coupled, partitioned, FSI simulation is applied to simulate the phase shift of the investigated CMF design. The calculated phaseshift values are in good agreement to the measurement data and verify the suitability of the model to numerically describe the working principle of a CMF

Towards a hybrid parallelization of lattice Boltzmann methods

AbstractOngoing research towards the development of a hybrid parallelization concept for lattice Boltzmann methods is presented. It allows coping with platforms sharing both the properties of shared and distributed architectures. The proposed approach relies on spatial domain decomposition where each domain represents a basic block entity which is solved on a symmetric multi-processing (SMP) system. Emphasis is placed on the software design and the reworking needed to achieve good performance using OpenMP in that context. Those ideas are implemented in the C++ project OpenLB, which is also sketched in this article. The efficiency of the proposed approaches is tested on a 3D benchmark problem and compared with a purely MPI based approach

Simultaneous optimization of gait and design parameters for bipedal robots

A walking bipedal robot’s energy efficiency depends on its gait as well as its design, whereas design changes affect the optimal gaits. We propose a method to take these interdependencies into account via simultaneous optimization of gait as well as design parameters. The method is applied to a planar robot with hybrid zero dynamics control and a torsion spring between its thighs. Periodic gaits are simulated by means of the hybrid zero dynamics. The implementation of the simultaneous optimization of gait parameters and spring stiffness via sequential quadratic programming is presented. Subsequently, an error analysis is performed to gain good convergence and short computation times of the optimization. The evaluation of gradients is identified as crucial for the algorithm’s convergence and therefore performed via complex step derivative approximations. The resulting implementation exhibits good convergence behavior and is provided as supplement to this paper. At 2.3 m/s, the simultaneous optimization results in savings in energy expenditure of up to 55%. A consecutive optimization of first gait and then stiffness yields only 11%, demonstrating the advantage of the presented method

Lattice-Boltzmann coupled models for advection–diffusion flow on a wide range of Péclet numbers

Traditional Lattice-Boltzmann modelling of advection–diffusion flow is affected by numerical instability if the advective term becomes dominant over the diffusive (i.e., high-Péclet flow). To overcome the problem, two 3D one-way coupled models are proposed. In a traditional model, a Lattice-Boltzmann Navier–Stokes solver is coupled to a Lattice-Boltzmann advection–diffusion model. In a novel model, the Lattice-Boltzmann Navier–Stokes solver is coupled to an explicit finite-difference algorithm for advection–diffusion. The finite-difference algorithm also includes a novel approach to mitigate the numerical diffusivity connected with the upwind differentiation scheme. The models are validated using two non-trivial benchmarks, which includes discontinuous initial conditions and the case Pe$_{g}$->$\infty$ for the first time, where Pe$_{g}$ is the grid Péclet number. The evaluation of Pe$_{g}$ alongside Pe is discussed. Accuracy, stability and the order of convergence are assessed for a wide range of Péclet numbers. Recommendations are then given as to which model to select depending on the value Pe$_{g}$ - in particular, it is shown that the coupled finite-difference/Lattice-Boltzmann provide stable solutions in the case Pe->$\infty$, Pe$_{g}$->$\infty

Lattice-Boltzmann LES modelling of a full-scale, biogas-mixed anaerobic digester

An Euler–Lagrange multicomponent, non-Newtonian Lattice-Boltzmann method is applied for the first time to model a full-scale gas-mixed anaerobic digester for wastewater treatment. Rheology is modelled through a power-law model and, for the first time in gas-mixed anaerobic digestion modelling, turbulence is modelled through a Smagorinsky Large Eddy Simulation model. The hydrodynamics of the digester is studied by analysing flow and viscosity patterns, and assessing the degree of mixing through the Uniformity Index method. Results show independence from the grid size and the number of Lagrangian substeps employed for the Lagrangian sub-grid simulation model. Flow patterns are shown to depend mildly on the choice of bubble size, but not the asymptotic degree of mixing. Numerical runs of the model are compared to previous results in the literature, from a second-ordered Finite-Volume Method approach, and demonstrate an improvement, compared to literature data, of 1000-fold computational efficiency, massive parallelizability and much finer attainable spatial resolution. Whilst previous research concluded that the application of LES to full-scale anaerobic digestion mixing is unfeasible because of high computational expense, the increase in computational efficiency demonstrated here, now makes LES a feasible option to industries and consultancies

Simulation of Dynamic Rearrangement Events in Wall-Flow Filters Applying Lattice Boltzmann Methods

Wall-flow filters are applied in the exhaust treatment of internal combustion engines for the removal of particulate matter (PM). Over time, the pressure drop inside those filters increases due to the continuously introduced solid material, which forms PM deposition layers on the filter substrate. This leads to the necessity of regenerating the filter. During such a regeneration process, fragments of the PM layers can potentially rearrange inside single filter channels. This may lead to the formation of specific deposition patterns, which affect a filter’s pressure drop, its loading capacity and the separation efficiency. The dynamic formation process can still not consistently be attributed to specific influence factors, and appropriate calculation models that enable a quantification of respective factors do not exist. In the present work, the dynamic rearrangement process during the regeneration of a wall-flow filter channel is investigated. As a direct sequel to the investigation of a static deposition layer in a previous work, the present one additionally investigates the dynamic behaviour following the detachment of individual layer fragments as well as the formation of channel plugs. The goal of this work is the extension of the resolved particle methodology used in the previous work via a discrete method to treat particle–particle and particle–wall interactions in order to evaluate the influence of the deposition layer topology, PM properties and operating conditions on dynamic rearrangement events. It can be shown that a simple mean density methodology represents a reproducible way of determining a channel plug’s extent and its average density, which agrees well with values reported in literature. The sensitivities of relevant influence factors are revealed and their impact on the rearrangement process is quantified. This work contributes to the formulation of predictions on the formation of specific deposition patterns, which impact engine performance, fuel consumption and service life of wall-flow filters