Elsevier

Computers & Fluids

Volume 213, 15 December 2020, 104723
Computers & Fluids

A Lattice-Boltzmann-based perturbation method

https://doi.org/10.1016/j.compfluid.2020.104723Get rights and content

Highlights

  • A hybrid Lattice Boltzmann and potential flow formulation is derived.

  • The governing Navier-Stokes equations are recovered with this approach.

  • The method is verified and validated through a series of applications.

Abstract

In this work, we report on the development and initial validation of a new hybrid numerical model for the simulation of incompressible flow. A kinetic Lattice Boltzmann method (LBM) model using a reduced domain is nested within an inviscid flow field to provide increased simulation fidelity where desired, while leveraging the computational efficiency of inviscid solutions. We formulate a fully (or strongly) coupled approach, in which a Helmholtz decomposition is applied to the flow, separating the inviscid and viscous perturbation parts. The latter component is driven by the inviscid field through nonlinear inviscid-perturbation interaction terms that, in conventional Navier-Stokes solvers, would be expressed as volume forces. In the present work an equivalent LBM approach is presented where, as opposed to a body-force coupling, a strong coupling within the LBM collision operators is presented. The resulting hybrid LBM is applied to validation cases for a wave driven boundary layer and the flow past a cylinder.

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 IP 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 IP 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 IP 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.

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