| Autor(en) | Geier, Martin | Pasquali, Andrea |
|---|---|
| Titel | Fourth order Galilean invariance for the lattice Boltzmann method |
| Herausgeber | Computers & Fluids, Vol.166, pp. 139-151 |
| Erscheinungsjahr | 2018 |
| Abstract | Using the structure of a recursive asymptotic analysis we derive conditions on cumulants that guarantee a prescribed order of Galilean invariance for lattice Boltzmann models. We then apply these conditions to three different lattice Boltzmann models and obtain three models with fourth order accurate advection. One of the models uses 27 speeds on a body centered cubic lattice, one uses 33 speeds on an extended Cartesian lattice and one uses 27 speeds plus three finite differences on a Cartesian lattice. All models offer too few degrees of freedom to impose the conditions on the cumulants directly. However, the specific aliasing structure of these lattices permit fourth order accuracy for a model specific optimal reference temperature. Our theoretical derivations are confirmed by measuring the phase lag of traveling vortexes and shear waves. |
| Autor(en) | Kumar, Pradeep | Kutscher, Konstantin | Mößner, Michael | Radespiel, Rolf | Krafczyk, Manfred | Geier, Martin | Krafczyk, Manfred | Kutscher, Konstantin |
|---|---|
| Titel | Validation of a VRANS-model for Turbulent Flow Over a Porous Flat Plate by Cumulant Lattice Boltzmann DNS/LES and Experiments |
| Herausgeber | Journal of Porous Media |
| Erscheinungsjahr | 2018 |
| Abstract | The turbulent boundary layer over a flat plate with porous inlay is investigated using three different approaches. Experiments and large eddy simulations (LES) computations based on the cumulant lattice Boltzmann approach help to assess the performance of a newly developed module that enables a RANS-solver to compute turbulent flow over pore space. Although the experiments can provide reliable data about the flow field over the porous medium, the LES data completely resolve the porous medium and hence provide an insight into the phenomena inside the porous medium. It is shown that the RANS-solver can predict the effects of the porous medium considerably well. |
| Autor(en) | Kutscher, Konstantin | Geier, Martin | Kraczyk, Manfred |
|---|---|
| Titel | Multiscale simulation of turbulent flow interacting with porous media based on a massively parallel implementation of the cumulant lattice Boltzmann method |
| Herausgeber | Computers & Fluids, DOI: 10.1016/j.compfluid.2018.02.009 |
| Erscheinungsjahr | 2018 |
| Abstract | Flow noise during takeoff and landing of commercial aircrafts can be substantially reduced by the use of porous surface layers in suitable sections of wing profiles. On the other hand (passive) porosity and roughness of surfaces tend to have an adverse effect on the boundary layer and thus on the lift of wings. This results in the need to be able to predict the aerodynamic effects of porous segments of the surface by numerical methods for aerodynamic design of wings, taking into account porosity and roughness as a function of Reynolds number. The application of the RANS equations for this task requires additional modeling terms such as the permeability and the Kozeny–Carman parameter as well as the turbulent fluctuations of the velocity field to be determined by Direct Numerical Simulation (DNS) / Large Eddy Simulation (LES) simulations based on the lattice Boltzmann method which is the main focus of this contribution. For the simulation of the flow at the pore scale we use the cumulant lattice Boltzmann method. Due to the inherent requirements of resolving both the turbulence on the scale of an airfoil and the flow inside the pore scale resolved porous medium the resulting simulations require more than a billion grid nodes on a refined three-dimensional mesh leading to massively parallel simulations. We discuss modeling and computational aspects of our approach and present computational results including experimental validations. |
| Autor(en) | Uphoff, Sonja | Krafczyk, Manfred | Kutscher, Konstantin | Rurkowska, Katherina | Langer, Sabine | Lippitz, Nicolas | Faßmann, Benjamin |
|---|---|
| Titel | A HIERARCHICAL APPROACH TO DETERMINING ACOUSTIC ABSORPTION PROPERTIES OF POROUS MEDIA COMBINING PORE-RESOLVED AND MACROSCOPIC MODELS |
| Herausgeber | Journal of Porous Media, Volume 21, Begell House, 83-100, 2018 |
| Erscheinungsjahr | 2018 |
| Abstract | Acoustic properties of porous media are very important for numerous industrial applications, the typical goal being to maximize broadband absorption to decrease the sound pressure level of the engineering system under consideration. Up to now acoustic absorption for porous media with complex inner geometry is determined experimentally, as acoustic simulations on the pore scale are computationally challenging due to the tedious geometric reconstruction of computer tomography (CT) data and the corresponding mesh generation as well as substantial computational requirements for the corresponding transient 3D solvers. The lattice Boltzmann method (LBM), which is an established computational approach to simulate pore-resolved porous media transport problems, has been used successfully for aeroacoustic setups and is utilized in this work to fill this gap. This paper presents a comparison of different experimental and numerical approaches to determine the acoustic absorption of different porous media. Experimental work with an impedance tube was carried out for comparison and CT scans were conducted to supply the detailed numerical simulation with geometry data of the porous samples. Results of LB simulations for the acoustic impedance of a microperforated plate and a felt are shown. Finally we demonstrate how microscopic parameters determined by a pore scale approach can be used to feed homogenized models to bridge the gap towards simulations of components where acoustic absorbers are applied to, e.g., wing flaps of airplanes. |
| Autor(en) | Geier, Martin | Pasquali, Andrea | Schönherr, Martin |
|---|---|
| Titel | Parametrization of the cumulant lattice Boltzmann method for fourth order accurate diffusion Part II: ap-plication to flow around a sphere at drag crisis |
| Herausgeber | Journal of Computational Physics, Vol. 348, No. 1, Elsevier, pp. 889-898, 2017 |
| Erscheinungsjahr | 2017 |
| Abstract | The optimized cumulant lattice Boltzmann method with fourth order accurate diffusion is used to simulate the flow around a sphere up to Reynolds number 106. The drag crisis is well captured by the method. We demonstrate with our results that the drag crisis corresponds to an almost discrete jump in the flow conditions. The intermediate values of drag in a small range of Reynolds numbers around the drag crisis observed in averaged data sets are found to originate from the flow switching between the high and the low drag conditions. Around the critical Reynolds number, the time spent in the low drag condition increases with the Reynolds number such that the average drag curve has a finite steepness. |
| Autor(en) | Geier, Martin | Schönherr, Martin |
|---|---|
| Titel | Esoteric Twist: An Efficient in-Place Streaming Algorithm for the Lattice Boltzmann Method on Massive-ly Parallel Hardware |
| Herausgeber | Computation, Vol. 5 No. 2, p. 19 |
| Erscheinungsjahr | 2017 |
| Abstract | We present and analyze the Esoteric Twist algorithm for the Lattice Boltzmann Method. Esoteric Twist is a thread safe in-place streaming method that combines streaming and collision and requires only a single data set. Compared to other in-place streaming techniques, Esoteric Twist minimizes the memory footprint and the memory traffic when indirect addressing is used. Esoteric Twist is particularly suitable for the implementation of the Lattice Boltzmann Method on Graphic Processing Units. |
| Autor(en) | Pasquali, Andrea | Geier, Martin | Krafczyk, Manfred |
|---|---|
| Titel | Near-wall treatment for the simulation of turbulent flow by the cumulant lattice Boltzmann method |
| Herausgeber | Computers & Mathematics with Applications, DOI: 10.1016/j.camwa.2017.11.022 |
| Erscheinungsjahr | 2017 |
| Abstract | We present a new wall function implementation for the cumulant lattice Boltzmann method that sets a partial slip velocity on the wall by computing a skin frictional coefficient. Our approach uses local information and is particularly appropriate for implementations on general purpose graphics processing units. The validation of the model has been conducted by performing numerical simulations of the turbulent channel flow test case with different grid resolutions and for different Reynolds numbers. The results showed encouraging agreement with Direct Numerical Simulation data for velocity profile, Reynolds shear and normal stresses. |
| Autor(en) | Yang, Xiaofan | Mehmani, Yashar | Perkins, William A. | Pasquali, Andrea | Schönherr, Martin | Kim, Kyungjoo | Perego, Mauro | Parks, Michael L. | Trask, Nathaniel | Balhoff, Matthew T. | Richmond, Marshall C. | Geier, Martin | Krafczyk, Manfred | Luo, Li-Shi | Tartakovsky, Alexandre M. | Scheibe, Timothy D. |
|---|---|
| Titel | Intercomparison of 3D pore-scale flow and solute transport simulation methods |
| Herausgeber | Advances in water resources, Vol. 95, pp. 176-189 |
| Erscheinungsjahr | 2016 |
| Abstract | Multiple numerical approaches have been developed to simulate porous media fluid flow and solute transport at the pore scale. These include 1) methods that explicitly model the three-dimensional geometry of pore spaces and 2) methods that conceptualize the pore space as a topologically consistent set of stylized pore bodies and pore throats. In previous work we validated a model of the first type, using computational fluid dynamics (CFD) codes employing a standard finite volume method (FVM), against magnetic resonance velocimetry (MRV) measurements of pore-scale velocities. Here we expand that validation to include additional models of the first type based on the lattice Boltzmann method (LBM) and smoothed particle hydrodynamics (SPH), as well as a model of the second type, a pore-network model (PNM). The PNM approach used in the current study was recently improved and demonstrated to accurately simulate solute transport in a two-dimensional experiment. While the PNM approach is computationally much less demanding than direct numerical simulation methods, the effect of conceptualizing complex three-dimensional pore geometries on solute transport in the manner of PNMs has not been fully determined. We apply all four approaches (FVM-based CFD, LBM, SPH and PNM) to simulate pore-scale velocity distributions and (for capable codes) nonreactive solute transport, and intercompare the model results. Comparisons are drawn both in terms of macroscopic variables (e.g., permeability, solute breakthrough curves) and microscopic variables (e.g., local velocities and concentrations). Generally good agreement was achieved among the various approaches, but some differences were observed depending on the model context. The intercomparison work was challenging because of variable capabilities of the codes, and inspired some code enhancements to allow consistent comparison of flow and transport simulations across the full suite of methods. This study provides support for confidence in a variety of pore-scale modeling methods and motivates further development and application of pore-scale simulation methods. |
| Autor(en) | Geier, Martin | Schönherr, Martin | Pasquali, Andrea | Krafczyk, Manfred |
|---|---|
| Titel | The cumulant lattice Boltzmann equation in three dimensions: Theory and validation |
| Herausgeber | Computers & Mathematics with Applications, ISSN 0898-1221, Vol. 70, Issue 4, pp. 507-547, 2015 |
| Erscheinungsjahr | 2015 |
| Abstract | We propose, analyze, and validate a lattice Boltzmann model with a cumulant collision operator. The new model is analytically and numerically shown to poses smaller errors than a moment based Multiple Relaxation Time lattice Boltzmann model. We demonstrate the usability of the cumulant lattice Boltzmann model by simulations of flow around a sphere for Reynolds numbers from 200 to 105. |
| Autor(en) | Krafczyk, Manfred | Kucher, Konstyantyn | Wang, Ying | Geier, Martin |
|---|---|
| Titel | DNS/LES Studies of Turbulent Flows Based on the Cumulant Lattice Boltzmann Approach |
| Herausgeber | High Performance Computing in Science and Engineering ‘14. Springer, pp. 519-531, Cham, 2015 |
| Erscheinungsjahr | 2015 |
| Abstract | In many industrial and environmental problems we encounter turbulent flows over porous surfaces which also penetrate the porous medium to different extents. Although there is a wealth of literature on macroscopic models of such phenomena which do not take the pore scale explicitly into account, these approaches typically require some additional transport coefficients to match experimentally obtained statistics for mass, momentum and energy transport across such interfaces. In this project we conduct Direct Navier-Stokes (DNS) and Large Eddy Simulation (LES) computations of turbulent flows which explicitly take into account specific pore scale geometries obtained from computer tomography imaging and do not use any explicit turbulence modeling. In this first part of the project we conducted validation studies for two canonical turbulent flows, i.e. flow around a plate and flow in a porous channel. Subsequently, we compare simulation results of turbulent flows over a porous sand and to experimental results and demonstrate the validity of our approach. Finally we discuss our approach to address evaporation processes on a pore scale which is based on a separation of time-scales. The newly developed cumulant Lattice Boltzmann scheme implemented as part of our research Code VirtualFluids shows a favorable behavior with respect to parallelization efficiency as well as to numerical stability and accuracy. |
| Autor(en) | Beck, Silja | Uphoff, Sonja | Langer, Sabine | Krafczyk, Manfred |
|---|---|
| Titel | Sensitivity of the slip rate coefficient in fluid flow – poroelastic coupling conditions |
| Herausgeber | Proceedings in Applied Mathematics and Mechanics, Volume 14, 699-700, Erlangen, 2014 |
| Erscheinungsjahr | 2014 |
| Abstract | To model flow‐induced structural vibrations, an interface to couple fluid flow and poroelastic material in a finite element formulation has been developed. One parameter of this interface condition is the slip rate coefficient, resulting from the so‐called Beavers‐Joseph‐Saffman condition. This condition states that the jump in tangential velocity at a fluid flow – porous interface is proportional to the shear stress. Up to now no a priori determination of this parameter exists, and the known parameter range has been deducted from measurements, i. e., in our case from the results of the pore‐resolving simulations. When modeling realistic problems assuming incompressible fluids, vectorial flow velocity and scalar pressure interact with the poroelastic material. As the slip rate coefficient only influences the tangential contributions, its overall influence is not clear. In this work, the sensitivity of the slip rate coefficient regarding the interface's coupling conditions is evaluated. |
| Autor(en) | Beck, Silja | Uphoff, Sonja | Langer, Sabine | Krafczyk, Manfred |
|---|---|
| Titel | Sensitivity of the slip rate coefficient in fluid flow – poroelastic coupling conditions |
| Herausgeber | Proceedings in Applied Mathematics and Mechanics, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics, Erlangen |
| Erscheinungsjahr | 2014 |
| Abstract | To model flow-induced structural vibrations, an interface to couple fluid flow and poroelastic material in a finite element formulation has been developed. One parameter of this interface condition is the slip rate coefficient, resulting from the so- called Beavers-Joseph-Saffman condition. This condition states that the jump in tangential velocity at a fluid flow – porous interface is proportional to the shear stress. Up to now no a priori determination of this parameter exists, and the known parameter range has been deducted from measurements, i. e., in our case from the results of the pore-resolving simulations. When modeling realistic problems assuming incompressible fluids, vectorial flow velocity and scalar pressure interact with the poroelastic material. As the slip rate coefficient only influences the tangential contributions, its overall influence is not clear. In this work, the sensitivity of the slip rate coefficient regarding the interface’s coupling conditions is evaluated. |
| Autor(en) | Liu, Geng | Geier, Martin | Liu, Zhenyu | Krafczyk, Manfred | Chen, Tao |
|---|---|
| Titel | Discrete adjoint sensitivity analysis for fluid flow topology optimization based on the generalized lattice Boltzmann method |
| Herausgeber | Computers and Mathematics with Applications, Vol. 68, Issue 10, p. 1374 |
| Erscheinungsjahr | 2014 |
| Abstract | A discrete adjoint sensitivity analysis for fluid flow topology optimization based on the lattice Boltzmann method (LBM) with multiple-relaxation-times (MRT) is developed. The lattice Boltzmann fluid solver is supplemented by a porosity model using a Darcy force. The continuous transition from fluid to solid facilitates a gradient based optimization process of the design topology of fluidic channels. The adjoint LBM equation, which is used to compute the gradient of the optimization objective with respect to the design variables, is derived in moment space and found to be as simple as the original LBM. The moment based spatial momentum derivatives used to express the discrete objective functional (cost function) have the advantage that the local stress tensor is a local quantity avoiding the numerical computation of gradients of the discrete velocity field. This is particularly useful if dissipation is a design criterion as demonstrated in this paper. The method is validated by a detailed comparison with results obtained by Borrvall et al. for Stokes flow. While their approach is only valid for Stokes flow (i.e. very low Reynolds numbers) our approach in its present form can in principle be applied for flows of different Reynolds numbers just like the Navier–Stokes equation based approaches. This point is demonstrated with a bending pipe example for various Reynolds numbers. |
| Autor(en) | Uphoff, Sonja | Krafczyk, Manfred | Rurkowska, Katherina | Langer, Sabine | Lippitz, Nicolas | Faßmann, Benjamin |
|---|---|
| Titel | A hierarchical approach to determine acoustic absorption properties of porous media combining pore-resolved and macroscopic models |
| Herausgeber | Journal of Porous Media |
| Erscheinungsjahr | 2014 |
| Abstract | Acoustic properties of porous media are very important for numerous industrial applications, the typical goal being to maximize broad band absorption to decrease the sound pressure level of the engineering system under consideration. Up to now acoustic absorption for porous media with complex inner geometry is determined experimentally, as acoustic simulations on the pore scale are computationally challenging due to the tedious geometric reconstruction of computer tomography (CT) data and the corresponding mesh generation as well as substantial computational requirements for the corresponding transient 3D-solvers. The Lattice Boltzmann method (LBM), which is an established computational approach to simulate pore resolved porous media transport problems, has been used successfully for aero-acoustic setups and is utilized in this work to fill this gap. This paper presents a comparison of different experimental and numerical approaches to determine the acoustic absorption of different porous media. Experimental work with an impedance tube was carried out for comparison and CT scans were conducted to supply the detailed numerical simulation with geometry-data of the porous samples. Results of LB simulations for the acoustic impedance of a micro-perforated plate and a felt are shown. Finally we demonstrate how microscopic parameters determined by a pore scale approach can be used to feed homogenized models to bridge the gap towards simulations of components where acoustic absorbers are applied to e.g. wing flaps of airplanes. |
| Autor(en) | Geller, Sebastian | Uphoff, Sonja | Krafczyk, Manfred |
|---|---|
| Titel | Turbulent jet computations based on MRT and Cascaded Lattice Boltzmann models |
| Herausgeber | Computers and Mathematics with Applications, Vol. 65, Iss. 12, 2013, pp. 1956-1966 |
| Erscheinungsjahr | 2013 |
| Abstract | In this contribution a numerical study of a turbulent jet flow is presented. The simulation results of two different variants of the Lattice Boltzmann method (LBM) are compared. The first is the well-established D3Q19 MRT model extended by a Smagorinsky Large Eddy Simulation (LES) model. The second is the D3Q27 Factorized Cascaded Lattice Boltzmann (FCLB) model without any additional explicit turbulence model. For this model no studies of turbulent flow with high resolution on nonuniform grids existed so far. The underlying computational procedure uses a time nested refinement technique and a grid with more than a billion DOF. The simulations were conducted with the parallel multi physics solver VirtualFluids. It is shown that both models are feasible for the present flow case, but the FCLB outperforms the traditional approach in some aspects. |
| Autor(en) | Krafczyk, Manfred | Uphoff, Sonja | Schönherr, Martin | Geier, Martin | Kucher, Kostyantyn | Stiebler, Maik |
|---|---|
| Titel | HPC CFD Simulations Based on Kinetic Methods Using Multi- and Many-Core Systems |
| Herausgeber | Developments in Parallel, Distributed, Grid and Cloud Computing for Engineering, Chapter 6, 2013, pp. 125-149 |
| Erscheinungsjahr | 2013 |
| Abstract | In recent years it has become obvious that modern computer hardware essentially will rely on multi- and many-core architectures to increase computational performance. Yet, only very few computational frameworks are ready to meet this challenge from an algorithmic and software engineering point of view. This chapter gives an overview about the recent work undertaken by our group in the context of high-performancecomputing (HPC) simulations in the area of computational fluid dynamics (CFD) in civil engineering. We will address different aspects of the simulation pipeline ranging from the efficiency of explicit numerical kernels capable of exploiting modern central processing units (CPU) as well as general purpose graphics processing units (GPGPU) architectures as well as domain-decomposition aspects for local time stepping schemes on Eulerian grids and the importance of hardware aware data structures. Our basic modeling approach for CFD is based on various lattice-Boltzmann models (LBM) tailored for turbulent flows. After a short introduction into the basic methods we will discuss several computational examples of complex flow simulations with up to five billion degrees of freedom. These examples demonstrate the feasibility of LBM for complex flow problems and the efficiency of GPGPU based simulations which allow simulations with more than a billion degrees of freedom (DOF) on desktop systems. Thus kinetic techniques for transport simulation may play a more prominent role in computational engineering due to their intrinsic suitability for present and future multi- and many-core architectures. |
| Autor(en) | Uphoff, Sonja |
|---|---|
| Titel | Development and Validation of turbulence models for Lattice Boltzmann schemes |
| Herausgeber | Dissertation TU-Braunschweig, 2013 |
| Erscheinungsjahr | 2013 |
| Abstract | Computational fluid mechanics has become a standard approach in many branches of engineering. Simulation of flow on the building- and infrastructure scale, however, remains very challenging and is mostly restricted to basic research at the present stage. In particular, accurate, three-dimensional, time-resolved simulation such as Large Eddy Simulation is still rarely used despite its potential. On the other hand, it is reasonable to expect a growing influence of these methods as computers become more powerful and numerical methods evolve. In the present work the Lattice Boltzmann method is chosen as a starting point to analyze simulations of flow around buildings. This approach appears to be particularly apt for such applications due to its very good scalability with respect to parallel computing. Different variants of the Lattice Boltzmann method, namely the Lattice Bhatnagar-Gross-Krook (LBGK) method, the Multiple Relaxation Time (MRT) method, and variants of the Cascaded Lattice Boltzmann (CLB) method have been implemented and compared on the basis of standard benchmarks. Furthermore, several turbulence models, such as the Smagorinsky model, the wall adapting local eddy-viscosity model, and Vreman’s model have been investigated. One focus was on the applicability of the Lattice Boltzmann method to turbulent flows, considering also the interdependence between the numerical method and the LES model. Particular attention was paid to the ability of these models to correctly reproduce turbulent shear flows. Some typical infrastructure elements have been studied and compared to windtunnel data. The simulations were carried out on a PC cluster and on graphics processing chips (GPGPUs). Overall, the Lattice Boltzmann method has yielded good results for turbulent flow simulations, which is documented in several benchmarks. In particular, the results for the FCLB model show for the first time for a reasonably complex benchmark, that the model performs well for turbulent flows, for which an explanation is attempted. |
| Autor(en) | Uphoff, Sonja | Kucher, Kostyantyn | Krafczyk, Manfred |
|---|---|
| Titel | Simulation of Turbulent Boundary Layer Flow over Porous Media with a Lattice Boltzmann Model |
| Herausgeber | 5th GACM Colloquium on Computational Mechanics, Hamburg, 2013 |
| Erscheinungsjahr | 2013 |
| Abstract | Simulation of Turbulent Boundary Layer Flow over Porous Media with a Lattice Boltzmann Model |
| Autor(en) | Ahrenholz, Benjamin | Niessner, Jennifer | Helmig, Rainer | Krafczyk, Manfred |
|---|---|
| Titel | Pore-scale determination of parameters for macroscale modeling of evaporation processes in porous media |
| Herausgeber | Water Resour. Res., 47, W07543, doi:10.1029/2010WR009519, 2011 |
| Erscheinungsjahr | 2011 |
| Abstract | Evaporation is an important process in many natural and technical systems, such as the unsaturated zone of the subsurface or microchannel evaporators. For the understanding and prediction of the involved processes, numerical simulations of multiphase flow and transport processes are an important tool. In order to achieve an accurate, physically based description of kinetic interphase mass and heat transfer occurring during evaporation, the numerical model has to account for the interfacial areas between phases. A recently developed model for two-phase flow in porous media is able to account for the involved processes by using interfacial areas explicitly as parameters in the model. The crucial issue, however, is the determination of the relationships between specific interfacial areas, capillary pressure, and saturation in this paper, we present a multiphase lattice Boltzmann model, which allows us to determine these relationships. On the basis of the scanned geometry of a natural porous medium, the relationships between specific interfacial areas, capillary pressure, and saturation are determined. To the best of our knowledge, this is the first time that fluid-solid specific interfacial area relationships have been obtained from 19 pore-scale data. Using these functions, we present the results of macroscale simulations of an evaporator device and of drying in a porous medium. |
| Autor(en) | Schönherr, Martin | Kucher, Kostyantyn | Geier, Martin | Stiebler, Maik | Freudiger, Sören | Krafczyk, Manfred |
|---|---|
| Titel | Multi-thread implementations of the lattice Boltzmann method on non-uniform grids for CPUs and GPUs |
| Herausgeber | Computers and Mathematics with Applications, Vol. 61, 12, 2011, pp. 3730-3743 |
| Erscheinungsjahr | 2011 |
| Abstract | Two multi-thread based parallel implementations of the lattice Boltzmann method for nonuniform grids on different hardware platforms are compared in this paper: a multi-core CPU implementation and an implementation on General Purpose Graphics Processing Units (GPGPU). Both codes employ second order accurate compact interpolation at the interfaces, coupling grids of different resolutions. Since the compact interpolation technique is both simple and accurate, it produces almost no computational overhead as compared to the lattice Boltzmann method for uniform grids in terms of node updates per second. To the best of our knowledge, the current paper presents the first study on multi-core parallelization of the lattice Boltzmann method with inhomogeneous grid spacing and nested time stepping for both CPUs and GPUs. |