Analysis and reduction of spurious noise generated at grid refinement interfaces with the lattice Boltzmann method
Introduction
The lattice Boltzmann method (LBM) has emerged as a very efficient approach for computational fluid dynamics over the last two decades. Its high degree of versatility makes it applicable to a large variety of highly complex physical phenomena, such as turbulence [1], [2], multiphase flows [3], [4], porous media [5] or even hemodynamics [6], and it has increasingly interested both industrial and academic actors. Its main advantage are, inter alia, a very simple and weakly dissipative numerical scheme representing weakly compressible flows which makes the LBM suitable for aeroacoustic simulations. Furthermore, the well known collide and stream algorithm requires a Cartesian grid that allows for a seamless way to handle complex geometries [7] through automated octree meshes and immersed boundary conditions.
Current challenges faced by industrial companies require large scale problems to be simulated with a high degree of accuracy. This issue involves the use of non-uniform grids to reduce computational costs and focus the mesh refinements on regions of interest determined by the physical phenomena at stake. Unfortunately, such grid topologies are likely to generate spurious vorticity or acoustic disturbances that may pollute aeroacoustic simulations for which acoustic pressure fluctuations are much lower than the aerodynamic ones. In such situations, avoiding parasitic sources is of paramount importance. For now on, and considering the very enlightening bibliographic review of Gendre et al. [8], a few aeroacoustic studies in the presence of grid refinements can be found in the literature but these are limited to pure acoustic propagation problems [9], [10]. The first study with acoustic validation in the presence of vortices that cross a refinement interface has been performed by Gendre et al. using the standard BGK collision model with an increased kinematic viscosity for stability purposes. Despite the fact that vortices are brought to cross interfaces in many aeroacoustic applications such as turbulent jet noise [11], landing gear noise [12] or cavity noise [13], it is interesting to wonder why this a priori, simple case is almost never studied in the LBM literature.
Furthermore, the very same undesirable phenomenon occurs in the Navier-Stokes (NS) framework [14]. A first indication can be found in the PhD Thesis of Hasert [10], in which a turbulent flow crosses grid refinements. In this simulation, spurious vortices and pressure spots appear at grid interfaces, but also far from high fluctuation hydrodynamics areas. These spurious artifacts disappear, decreasing the order of the spatial interpolation scheme used at the interface to reconstruct missing data, which increases the dissipation in this region. Obviously, using a first-order interpolation scheme is not a reliable solution as the accuracy is decreased [15]. Another enlightening point can be found in [16], where a turbulent channel is simulated using an entropic lattice Boltzmann model [17]. Even if acoustic phenomena are not primarily investigated, the entropic stabilizer varies considerably close to the grid interface. Since the latter is related to interactions between stress variables and ghost ones [18], [19] which are closely linked with non-hydrodynamic modes (also referred to as ghost modes [20]) activity, a close relationship between grid refinement and these modes activity can be supposed.
Therefore, as these modes have an expected non-hydrodynamic contribution, they may be further investigated thanks to a von Neumann analysis [21]. This method is widely used to understand and predict the stability of a numerical scheme. It consists in evaluating the response of a system, described by a given set of equations, to linear perturbations. Sterling and Chen [22] were among the firsts to apply this method to the LBM scheme. Subsequently, this technique has allowed to exhibit the coexistence and the spectral properties of both hydrodynamic and non-hydrodynamic modes in a LBM scheme [20]. Moreover, in a recent study [23], an extended spectral analysis was proposed, consisting in an investigation of the eigenvector's macroscopic content of the linearized lattice Boltzmann scheme. This last reveals that non physical modes can have macroscopic contributions, and allows for their identification in term of acoustics or shear contribution.
This article aims at improving the analysis of spurious wave generation at grid interface, and proposing efficient solutions to damp them. Such an improvement could lead to the design of Lattice Boltzmann Methods well suited for aeroacoustic simulations on non-uniform grids. Until now, the origin of spurious phenomena has not been convincingly explained, which makes the resolution of this issue difficult. In this context, it is proposed (1) to understand and explain the spurious phenomena that occur at mesh refinement interface during the crossing of acoustics and vortices, by studying in particular the non-hydrodynamic modes involvement, and (2) to propose a collision model to avoid both spurious emissions and vorticity. It is worth emphasizing that the methodology introduced in the following is general and does not depend on the grid refinement algorithm.
The paper is organized as follows. First of all, the context of aeroacoustic simulations on non-uniform grids is introduced and the issues that may appear at grid interfaces are highlighted in Sec. 2. Next, key features of the lattice Boltzmann method are briefly summarized and three collision models are considered specifically for their interesting spectral properties regarding non-hydrodynamic modes in Sec. 3. Then, a grid refinement algorithm is described, along with a suggested modification to enable a proper velocity gradient computation which is required for the collision model studied in Sec. 4. Spectral analysis tools are subsequently presented in Sec. 5 to highlight the specific behavior of each collision model, especially with regard to the treatment of non-hydrodynamic modes. The effect of the resolution change on modes is then investigated in Sec. 6 by performing the spectral analysis of the projection of a fine mode onto a coarser resolution. It is important to notice that this projection is intrinsically tied to the use of different grid resolution, independently of the details of the grid transition algorithm. It is therefore a universal phenomenon, shared by all grid transition methods. Subsequently, in Sec. 7, sensors are developed so that non-physical modes can be identified in a systematic way during simulations. This step is necessary to make the link between spectral analysis observations, carried out in the very particular case of plane monochromatic waves in the linear approximation, and realistic LBM simulations. Afterwards, non-hydrodynamic effects on grid refinement will be meticulously studied in Sec. 8 for both pure acoustic propagation and shear flows on acoustic and transversal shear waves. Finally, a proposition for solving these spurious artifacts is proposed and validated on a convected vortex in Sec. 9 and on a highly turbulent flow around a cylinder in Sec. 10.
Section snippets
Aeroacoustic context
This section aims at introducing specific concerns that may appear in the aeroacoustic framework on non-uniform grids. When dealing with non-uniform grids in standard LBM, two main categories of algorithms exist in the literature: on one hand, the so-called “cell-vertex” formulation [24], [25], [15] and on the other hand, the one referred to as “cell-centered” [26], [27]. These algorithms are highly dissimilar at the first glance, as in the cell-centered case, fine and coarse nodes are never
The lattice Boltzmann method with the BGK collision operator
The lattice Boltzmann method describes the time and space evolution of the discrete particle distribution functions , which can be viewed as the probability density of finding fictive particles at position x, at time t and advected at discrete velocities . With no body-force term, it can be expressed as
The collision operator can be approximated by the single-relaxation time Bhatnagar-Gross-Krook (BGK) model [30]
Grid refinement algorithm
Before introducing a particular grid refinement algorithm, it is worth mentioning that the concepts presented in the following are independent of the grid refinement algorithm and have been validated for both cell-vertex and cell-centered algorithms in two and three dimensions. Since the aim of the article is not a comparison of grid coupling algorithms, the one from Lagrava et al. [15] is chosen as it is one of the most popular and one of the simplest in term of implementation.
This algorithm
Von Neumann analyses of LBM schemes
The von Neumann analysis [21] is a very powerful tool to investigate the behavior of numerical schemes as the LBM, in terms of stability and accuracy properties. This method consists of evaluating the response of a system, which is described by a given set of equations, to small disturbances. It can exhibit the coexistence of physical and non-physical modes [20] in a computation.
The standard von Neumann analysis principles can be found in [21]. Sterling and Chen were among the first to apply
Energy transfer induced by a change of resolution
The aim of this section is to study the effect of a resolution change on the LBM modes, regardless the grid coupling algorithm. Since the spectral properties of the LBM schemes strongly depend on the dimensionless wavenumber vector k, then on the mesh resolution, it is interesting to wonder how a given mode may be affected by a resolution change. To address this question, it is proposed here to study the passage matrix P between modes with a wavenumber and those with a wavenumber .
Introduction of sensors to locate non-hydrodynamic modes
This section aims at proposing different kinds of sensors in order to detect the presence of non-physical modes in a simulation. The objective is to make the link between modes exhibited by the von Neumann analysis and phenomena observed during simulations.
Currently in the literature, the entropic lattice Boltzmann models are based on a similar attempt to systematically identify non-hydrodynamic content [17]. More precisely, it is proposed to decompose the populations into three parts
Non-hydrodynamic modes effects on grid refinement algorithms
The aim of this section is to highlight some issues of grid refinement algorithms on very simple cases. First of all, a convected shear wave will be introduced to characterize the effect of the spuriousS () modes. Then an upstream acoustic wave is studied to look at the influence of the spuriousAc () modes.
Improvement of fluid modeling for grid refinement algorithms
The previous section has highlighted the undesirable effects of non-hydrodynamic modes at grid refinement interfaces. Since these lasts can exchange energy with physical modes, and as their amplitude is inverted at each iteration, they are very difficult to handle properly with classical grid refinement algorithms. In light of this, any lattice Boltzmann scheme that can effectively attenuate non-hydrodynamic modes seems to be a good candidate to avoid these exchanges.
To this end, many ways
Validation on a high Reynolds number turbulent flow around a cylinder
In this section, a validation is carried out on a three-dimensional high Reynolds turbulent flow around a circular cylinder. The flow physics is not examined here, since it depends mainly on parietal modeling, which is not the subject of this paper. The objective is to simulate a low-viscosity turbulent flow across refinement interfaces, minimizing parasitic vorticity and spurious noise.
This test case is purely qualitative and intends to highlight specific problems that may occur when adding
Conclusion
This paper has investigated the transfer of energy between non-hydrodynamic and physical modes occurring at a grid refinement interface. More precisely, by clearly sorting the modes by their carried macroscopic information, referred to above as shear or acoustic modes, and by systematically identifying them in a simulation thanks to adequate newly proposed sensors, it has been shown that the energy of a non-hydrodynamic mode can be redistributed on every mode carrying a quantity of the same
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.
Acknowledgements
The authors would like to gratefully acknowledge Felix Gendre for the fruitful discussions on grid refinement algorithms. Acknowledgments are also expressed to Airbus Operations and ANRT/CIFRE for the financial support.
References (64)
Toward advanced subgrid models for lattice-Boltzmann-based large-eddy simulation: theoretical formulations
Comput. Math. Appl.
(2010)- et al.
Numerical analysis of the pressure drop in porous media flow with lattice Boltzmann (BGK) automata
Comput. Phys. Commun.
(2000) - et al.
Advances in multi-domain lattice Boltzmann grid refinement
J. Comput. Phys.
(2012) - et al.
Stability analysis of lattice Boltzmann methods
J. Comput. Phys.
(1996) - et al.
Grid refinement for lattice-BGK models
J. Comput. Phys.
(1998) - et al.
Grid refinement in lattice Boltzmann methods based on volumetric formulation
Physica A
(2006) - et al.
Lattice Boltzmann method with regularized pre-collision distribution functions
Math. Comput. Simul.
(2006) - et al.
Direct and large-eddy simulation of turbulent flows on composite multi-resolution grids by the lattice Boltzmann method
J. Comput. Phys.
(2014) - et al.
Lattice Boltzmann method with selective viscosity filter
J. Comput. Phys.
(2009) - et al.
The cumulant lattice Boltzmann equation in three dimensions: theory and validation
Comput. Math. Appl.
(2015)