On transient hybrid Lattice Boltzmann–Navier-Stokes flow simulations

Highlights

• NS solvers can be used as efficient preconditioners to LB.

• Scaling number of LB time steps with domain size reduces compressibility artifacts.

• Two-way coupling yields acceptable results for Couette flows.

• Two-way coupling yields compressibility artifacts in Taylor–Green scenarios.

Abstract

We investigate one- and two-way coupled schemes combining Lattice Boltzmann (LB) and incompressible Navier-Stokes (NS) solvers. The one-way coupled simulation maps information from a coarse-grained NS system onto LB boundaries which allows for arbitrarily complex fluid flow boundary conditions on LB side. We find that this produces accurate velocity, pressure and stress predictions in Couette, Taylor–Green and Karman vortex scenarios. The two-way coupled simulation decomposes the computational domain into overlapping LB and NS domains. We point out that the weak compressibility of LB can have a major impact on the coupled system. Although very good agreement is found for Couette scenarios, this is not achieved to same extent in Taylor–Green flows.

Introduction

Both Lattice Boltzmann (LB) and incompressible Navier-Stokes (NS) solvers are well established methods to compute and, thus, predict fluid flow behavior. Both approaches come with advantages and disadvantages. On the one hand, LB methods were shown to provide accurate prediction of fluid flow over a wide range of temporal and spatial scales, including fluctuating hydrodynamics for nanoflow systems [1] and rarefied gas flows in the slip [2] and transition regime [3]. From the algorithmic perspective, they are rather simple and very local in their computations. This allows for highly efficient, massively parallel simulations as well as block-adaptive GPU variants, see amongst others [4], [5]. Incompressible NS solvers, on the other hand, typically come along with lower memory consumption, and often allow the use of coarser grids and rigorous higher-order discretizations such as discontinuous Galerkin approaches [6], [7] which are rather uncommon in the LB framework [8], [9]. For more detailed comparisons of LB and NS approaches, see amongst others [10], [11]. Hence, one or the other solver, or even a combination of both to exploit their features to highest extent may be favorable for a particular flow problem.

Various works on coupling LB solvers with respective finite difference or finite volume discretizations of the underlying, asymptotically approximated PDEs have been carried out. In the Navier-Stokes context, Latt et al. [12] have provided a coupling method for steady state channel flow scenarios. Yeshala [13] partly employs this approach to address flow around cylinders and other scenarios. Our group has recently developed and evaluated different Schwarz-like domain decomposition techniques for spatial steady-state LB–NS simulations [14].

In this contribution, we extend our considerations to transient coupling of LB and NS solvers for laminar flows. One- and two-way coupled systems are investigated in Couette, Taylor–Green and Karman vortex flows, the latter based on the 2D2 benchmark [15]. We show that one-way coupling, i.e. using a NS solver to impose arbitrary boundary/flow field conditions to LB, yields accurate results in all of these scenarios. This result has several, but most importantly two, implications. First, this allows to generate boundary conditions from a computationally cheap, coarse-grained flow model and plug them into the LB method to construct accurate predictions on finer scales at higher computational cost. Second, incompressible Navier-Stokes solvers may—based on these results—be used to represent an accurate initial flow field predictor. Another potential application may be found in parallel-in-time CFD simulations where LB is employed for the fine-scale simulation, cf. amongst others [16], and NS is used as coarse-scale predictor. This would further improve the (strong) scaling limits of this method.

Moreover, we consider two-way coupling where the domain is decomposed into separate LB and NS overlapping domains. We show that the weak compressibility can have a severe impact on the overall solution quality of the coupling, and briefly point out the importance of a valid choice of LB and NS time steps at the example of the Taylor–Green scenario.

We revise the theory for LB, NS, and their implementations in Section 2. The coupling methods—one- and two-way coupling—are discussed in Section 3. The studies of Couette, Taylor–Green and Karman vortex scenarios are presented in Section 4. We close with a short summary and outlook to future work in Section 5.