A Lattice-Boltzmann-based perturbation method
Introduction
Numerical models simulating the irrotational motion of an incompressible, inviscid fluid, based on potential flow theory, are computationally efficient and sufficiently accurate to simulate many engineering fluid problems, such as those involving free surface waves and wave-structure interactions (e.g., [19]). However, potential flow models cannot be used in applications where viscous effects are important, for instance, in the boundary layer near solid boundaries, or the ocean bottom, in the wake of bluff bodies, or to simulate surface wave breaking. Standard Computational Fluid Mechanics (CFD) Navier-Stokes (NS) solvers, such as based on a finite volume (e.g., [27]) or Lattice Boltzmann (LBM) method (e.g., [10], [15], [26], [28], [29], [34]), can model these as well as all types of flows, but are computationally costly. Additionally, for free surface flows, NS solvers often use a dissipative numerical scheme to capture the free surface boundary conditions, which is often too numerically dissipative to model wave propagation over long distances [7]. To more efficiently solve a broad class of hydrodynamics problems of interest to many engineering disciplines, in this work, we detail the development of a high-fidelity but low cost hybrid numerical model, that combines potential flow and NS models, and applies each model in the region where it is most efficient and accurate.
This hybrid model is based on a perturbation method, sometimes referred to in fluid mechanics as the Helmholtz decomposition, that was proposed in earlier work, but implemented using different numerical methods and applied to different problems than presented here [1], [18]. For instance, it was successfully used to model turbulent flows, using a finite volume method, and validated for turbulent channel and wave induced boundary layer flows [25] or for linear ship seakeeping [48]. Unlike one- or two-way coupled models applied over separate regions of the computational domain (e.g., [7], [23]), in this method, both the velocity and pressure fields are expressed as the sum of inviscid/irrotational (I) and viscous perturbation (P) components, each solved using different numerical models in separate but overlapping computational domains.
More specifically, here, the I fields are solved with a potential flow model typically over a larger size domain extending to the far-field, whereas the P fields are solved based on a modified (perturbed) NS equation, with a LBM model, in a smaller near-field domain in which viscous effects are deemed important based on the considered problem (this will be made more clear later). With this approach, the more computationally demanding perturbation LBM model, referred to as pLBM, is only applied to the smaller near-field domain where viscous/turbulent effects matter, with its solution forced by results of the potential flow model applied to the larger domain. Hence, this hybrid approach is much more computational efficient than applying a LBM (or similar NS) model to the entire domain, while ensuring that the complete NS solution is solved where the physics calls for it.
In engineering applications involving complex boundary conditions and/or boundary/structure geometry, the model solving potential flow equations over the entire computational domain must itself be an optimized generic numerical solver, such as based on the higher-order Boundary Element Method, and feature fully nonlinear free surface boundary conditions if applicable [25], [30]. Such cases, however, are not considered here, but left out for follow-up publications (see preliminary relevant work in [42], [43]). The present paper instead concentrates on detailing the development of a novel pLBM model and validating it on a series of simple, but representative, applications for which there are analytical solutions of the potential flow fields I that can be used in the hybrid model to force the pLBM solution.
In this work, the perturbation NS equations are solved by a LBM, rather than a finite volume solver as in earlier work. Some of the rationales for this are the data locality and kernel simplicity of the LBM, which allow for a very efficient parallel implementation of the model on a “General Purpose Graphical Processor Units” (GPGPU) [31], [50], [51]. While a single GPGPU still has a limited memory, a multi-GPGPU implementation of the LBM may achieve a higher computational efficiency, for a similar accuracy, than classical Navier-Stokes-based CFD solvers implemented on a massively parallel CPU cluster [2], [45], [55]. However, in the hybrid method context, for many engineering applications, the solution of the P-fields by the pLBM in a reduced-size computational domain can often be simulated using a single GPGPU [42], thus allowing complex simulations to be run on a standard desktop computer equipped with a relatively inexpensive GPGPU co-processor. When the potential flow part of the problem (I-fields) is also solved with a numerical model, e.g., BEM based, its solution may be calculated on the CPUs of the same computer, without affecting the LBM GPGPU calculation. If a traditional NS solver were to be used instead of the LBM, a significant number of CPUs would be required to run it at an accuracy equivalent to that of the LBM, leading to competing computational resources when combined with the potential flow solver [52].
The coupling between continuum mechanics-based equations (or models), such as potential flow, and the kinetic-based LBM is less straightforward than the earlier implementation of the hybrid method based on a volume of fluid NS solver [25]. In particular, one must derive a pLBM equivalent to the nonlinear coupling terms that appear in the perturbation NS equations (see details below). To assess the ability of the LBM to simulate strongly nonlinear free surface flows, Janssen et al. [29], [30], [31] simulated the two-dimensional (2D) “weak coupling” wave breaking results reported in earlier work [7], [23], using a LBM in combination with a Volume Of Fluid (VOF) interface tracking method. In such cases, the LBM model was simply initialized with potential flow results for waves that had been propagated up to close to the breaking point in a potential flow BEM model [20], [21], [22]. Next, the same authors computed similar results with the hybrid method, in which the coupling terms were represented as LBM body force terms, using pre-computed I-fields to force the P field solution through these terms. This approach, while proven effective, required computing spatial derivatives of both the I and P-fields using finite difference approximations that yielded a compact but non-local LBM kernel. Additional analyses showed that this approach both caused higher truncation errors in the pLBM than in the original LBM collision operator and reduced the overall efficiency of the parallelized GPGPU solution. Therefore, Janssen [29] suggested instead to introduce the nonlinear coupling terms directly into the LBM equilibrium probability distribution functions (EPDFs), hence, to develop perturbation EPDFs or pEPDFs. The latter were incrementally developed, implemented, and validated as part of the development of a pLBM model component to a hybrid naval hydrodynamic solver, in which the potential flow solution, with fully nonlinear free surface boundary conditions (FNPF), was computed using a higher-order BEM model [42], [43], [44].
In this paper, we fully report on the rigorous development, and mathematical and numerical validation of the pEPDFs in an efficiently parallelized pLBM solver, implemented on a GPGPU as a component of a hybrid hydrodynamic solver. Janssen’s original approach [29] may be considered as a “top down” method, because the EPDFs were empirically derived based on the desired macroscopic quantities, i.e., the perturbation NS equations that they were meant to represent. Here, instead, a “bottom-up” derivation of the collision operators is carried out to derive the pEPDFs, through applying the Helmholtz decomposition directly to the PDFs. While we recover the same pEDPFs as in [29], we also provide a rigorous proof that these indeed solve the perturbation NS equations, through a Chapman-Enskog expansion. The appropriate initial conditions, boundary conditions, and grid refinement considerations for the pLBM are then discussed, and the method is validated for simple 2D and three-dimensional (3D) applications, for which accuracy and convergence properties are demonstrated. Simulations are run using the highly efficient, GPGPU-accelerated, Lattice Boltzmann solver ELBE ([32]; www.tuhh.de/elbe), developed at the Hamburg University of Technology, which features various LBM models, an on-device grid generator, higher-order boundary conditions, and the possibility of specifying overlapping nested grids.
Section snippets
Lattice Boltzmann method (LBM)
In part due to the method efficiency, models based on the LBM have become increasingly used for solving a variety of complex fluid dynamics and multi-fluid multi-physics problems (e.g., [3], [4], [5]). In contrast with classical CFD solvers that model the macroscopic NS equations on a continuum basis, the LBM simulates CFD problems on a mesoscopic scale, in which the fluid is represented by the PDFs of discrete particles moving on a fixed lattice. Macroscopic hydrodynamic quantities are
Perturbation lattice Boltzmann method (pLBM)
Before developing the pLBM and corresponding perturbation EPDFs (pEPDFs), we first derive the macroscopic perturbation NS equations that we seek to solve with the pLBM. As a canonical target application for the pLBM, we consider the viscous perturbation caused by a solid body boundary, occurring in a localized region of an otherwise inviscid, irrotational, and incompressible flow region. In this paper, we only consider direct NS simulation (DNS) at low Re value, which eliminates the additional
Applications
The accuracy and convergence properties of the novel pLBM approach are assessed in the following, on the basis of a series of simple but meaningful applications. Although the model is implemented in 3D, for the sake of validating the hybrid approach, only 2D applications are considered in this paper (albeit solved in 3D). More complex fully 3D cases are presented in [41], [42], [43], [44].
To assess the basic convergence and accuracy properties of the proposed pLBM, we first consider two
Conclusions
In this paper, we provided both a rigorous derivation of the fundamentals and a numerical validation through a series of applications, of the perturbation LBM (pLBM) approach. We showed that a “bottom up” approach in which the Helmholtz decomposition is applied to the PDFs indeed recovers the perturbation collision operators that were originally derived following a “top down” approach by Janssen [29]. We rigorously proved that the perturbation NS equations are recovered using the modified
CRediT authorship contribution statement
Christopher M. O’Reilly: Methodology, Software, Validation, Writing - original draft, Writing - review & editing. Christian F. Janßen: Methodology, Software, Validation, Writing - original draft, Writing - review & editing. Stephan T. Grilli: Conceptualization, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
C. O’Reilly, and S.T. Grilli gratefully acknowledge support for this work from grants N000141310687 and N000141612970 of the Office of Naval Research (PM Kelly Cooper). As well as the support of XSEDE grant ENG170010 for GPU computational resources.
References (55)
- et al.
Efficient GPGPU implementation of a lattice Boltzmann model for multiphase flows with high density ratios
Comput Fluids
(2014) - et al.
An efficient lattice Boltzmann multiphase model for 3d flows with large density ratios at high Reynolds numbers
Comput Math Appl
(2014) - et al.
Parametrization of the cumulant lattice Boltzmann method for fourth order accurate diffusion Part I: derivation and validation
J Comput Phys
(2017) - et al.
Parametrization of the cumulant lattice Boltzmann method for fourth order accurate diffusion Part II: application to flow around a sphere at drag crisis
J Comput Phys
(2017) - et al.
Benchmark computations based on lattice-Boltzmann, finite element and finite volume methods for laminar flows
Comput Fluids
(2006) - et al.
Validation of the GPU-accelerated CFD solver ELBE for free surface flow problems in civil and environmental engineering
Computation
(2015) Hydrodynamic limit of lattice Boltzmann equations
(2007)- et al.
Solution of viscous flows in a hybrid naval hydrodynamic scheme based on an efficient lattice Boltzmann method
Proc. 13th intl. conf. on fast sea transportation (FAST 2015; Washington D.C., September 1-4, 2015)
(2015) - et al.
Multiple-GPU acceleration of the boussinesq type wave model FUNWAVE-TVD
J Adv Model Earth Syst
(2020) Thèse d’habilitation en vue de diriger les recherches
(2007)
The actuator line model in lattice Boltzmann frameworks: numerical sensitivity and computational performance
J Phys Conf Ser
The simulation of turbulent particle-laden channel flow by the lattice Boltzmann method
Int J Numer Methods Fluids
A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems
Phys Rev
Numerical analysis of the internal kinematics and dynamics of three-dimensional breaking waves on slopes.
Int J Offshore Polar Eng
Momentum transfer of a lattice-Boltzmann fluid with boundaries
Phys Fluids
Water wave mechanics for engineers and scientists (vol. 2)
Multiple relaxation-time lattice Boltzmann models in three-dimensions
R Soc London Philos TransSer A
A parallelization concept for a multi-physics lattice Boltzmann solver based on hierarchical grids
Prog Comput Fluid Dyn
Lattice gas hydrodynamics in two and three dimensions
J Complex Syst
Multi-reflection boundary conditions for lattice Boltzmann models
Phys Rev E
Fully nonlinear potential flow simulations of wave shoaling over slopes: spilling breaker model and integral wave properties
Water Waves
On the development and application of hybrid numerical models in nonlinear free surface hydrodynamics
Keynote lecture in proc. 8th intl. conf. on hydrodynamics (Nantes, France, September 2008)
Progress in fully nonlinear potential flow modeling of 3D extreme ocean waves
Chapter 3 in Advances in numerical simulation of nonlinear Water Waves (Vol. 11 in Series in Advances in Coastal and Ocean Engineering)
Numerical generation and absorption of fully nonlinear periodic waves
Journal of Engineering Mechanics
Breaking criterion and characteristics for solitary waves on slopes
J Waterway Port Coastal Ocean Eng
Numerical modeling of wave breaking induced by fixed or moving boundaries
Comput Mech
Computation of shoaling and breaking waves in nearshore areas by the coupling of BEM and VOF methods
Proc. 9th offshore and polar engng. conf. (ISOPE99, Brest, France, May 1999), Vol. III
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