Robert Nourgaliev

The lattice Boltzmann equation method: theoretical interpretation, numerics and implications

Abstract During the last ten years the lattice Boltzmann equation (LBE) method has been developed as an alternative numerical approach in computational fluid dynamics (CFD). Originated from the discrete kinetic theory, the LBE method has emerged with the promise to become a superior modeling platform, both computationally and conceptually, compared to the existing arsenal of the continuum-based CFD methods. The LBE method has been applied for simulation of various kinds of fluid flows under different conditions. The number of papers on the LBE method and its applications continues to grow rapidly, especially in the direction of complex and multiphase media. The purpose of the present paper is to provide a comprehensive, self-contained and consistent tutorial on the LBE method, aiming to clarify misunderstandings and eliminate some confusion that seems to persist in the LBE-related CFD literature. The focus is placed on the fundamental principles of the LBE approach. An excursion into the history, physical background and details of the theory and numerical implementation is made. Special attention is paid to advantages and limitations of the method, and its perspectives to be a useful framework for description of complex flows and interfacial (and multiphase) phenomena. The computational performance of the LBE method is examined, comparing it to other CFD methods, which directly solve for the transport equations of the macroscopic variables.

Adaptive characteristics-based matching for compressible multifluid dynamics

This paper presents an evolutionary step in sharp capturing of shocked, high acoustic impedance mismatch (AIM) interfaces in an adaptive mesh refinement (AMR) environment. The central theme which guides the present development addresses the need to optimize between the algorithmic complexities in advanced front capturing and front tracking methods developed recently for high AIM interfaces with the simplicity requirements imposed by the AMR multi-level dynamic solutions implementation. The paper shows that we have achieved this objective by means of relaxing the strict conservative treatment of AMR prolongation/restriction operators in the interfacial region and by using a natural-neighbor-interpolation (NNI) algorithm to eliminate the need for ghost cell extrapolation into the other fluid in a characteristics-based matching (CBM) scheme. The later is based on a two-fluid Riemann solver, which brings the accuracy and robustness of front-tracking approach into the fast local level set front-capturing implementation of the CBM method. A broad set of test problems (including shocked multi-gaseous media, bubble collapse, underwater explosion and shock passing over a liquid drop suspended in a gaseous medium) was performed and the results demonstrate that the fundamental assumptions/approximations made in modifying the AMR prolongation/restriction operators and in using the NNI algorithm for interfacial treatment are acceptable from the accuracy point of view, while they enable an effective implementation and utility of the structured AMR technology for solving complex multiphase problems in a highly compressible setting.

High-fidelity interface tracking in compressible flows

The interface-capturing-fidelity issue of the level set method is addressed wholly within the Eulerian framework. Our aim is for a practical and efficient way to realize the expected benefits of grid resolution and high order schemes. Based on a combination of structured adaptive mesh refinement (SAMR), rather than quad/octrees, and on high-order spatial discretization, rather than the use of Lagrangian particles, our method is tailored to compressible flows, while it provides a potentially useful alternative to the particle level set (PLS) for incompressible flows. Interesting salient features of our method include (a) avoidance of limiting (in treating the Hamiltonian of the level set equation), (b) anchoring the level set in a manner that ensures no drift and no spurious oscillations of the zero level during PDE-reinitialization, and (c) a non-linear tagging procedure for defining the neighborhood of the interface subject to mesh refinement. Numerous computational results on a set of benchmark problems (strongly deforming, stretching and tearing interfaces) demonstrate that with this approach, implemented up to 11th order accuracy, the level set method becomes essentially free of mass conservation errors and also free of parasitic interfacial oscillations, while it is still highly efficient, and convenient for 3D parallel implementation. In addition, demonstration of performance in fully-coupled simulations is presented for multimode Rayleigh-Taylor instability (low-Mach number regime) and shock-induced, bubble-collapse (highly compressible regime).

Numerical prediction of interfacial instabilities: Sharp interface method (SIM)

We introduce a sharp interface method (SIM) for the direct numerical simulation of unstable fluid-fluid interfaces. The method is based on the level set approach and the structured adaptive mesh refinement technology, endowed with a corridor of irregular, cut-cell grids that resolve the interfacial region to third-order spatial accuracy. Key in that regard are avoidance of numerical mixing, and a least-squares interpolation method that is supported by irregular datasets distinctly on each side of the interface. Results on test problems show our method to be free of the spurious current problem of the continuous surface force method and to converge, on grid refinement, at near-theoretical rates. Simulations of unstable Rayleigh-Taylor and viscous Kelvin-Helmholtz flows are found to converge at near-theoretical rates to the exact results over a wide range of conditions. Further, we show predictions of neutral-stability maps of the viscous Kelvin-Helmholtz flows (Yih instability), as well as self-selection of the most unstable wave-number in multimode simulations of Rayleigh-Taylor instability. All these results were obtained with a simple seeding of random infinitesimal disturbances of interface-shape, as opposed to seeding by a complete eigenmode. For other than elementary flows the latter would normally not be available, and extremely difficult to obtain if at all. Sample comparisons with our code adapted to mimic typical diffuse interface treatments were not satisfactory for shear-dominated flows. On the other hand the sharp dynamics of our method would appear to be compatible and possibly advantageous to any interfacial flow algorithm in which the interface is represented as a discrete Heaviside function.

The Characteristics-Based Matching (CBM) Method for Compressible Flow With Moving Boundaries and Interfaces

Recently, Euterian methods for capturing interfaces in multi-fluid problems become increasingly popular While these methods can effectively handle significant deformations of interface, the treatment of the boundary conditions in certain classes of compressible flows are known to produce nonphysical oscillations due to the radical change in equation of state across the material interface. One promising recent development to overcome these problems is the Ghost Fluid Method (GFM). The present study initiates a new methodology for boundary condition capturing in multifluid compressible flows. The method, named Characteristics-Based Matching (CBM), capitalizes on recent developments of the level set method and related techniques, i.e., PDE-based re-initialization and extrapolation, and the Ghost Fluid Method (GFM). Specifically, the CBM utilizes the level set function to capture interface position and a GFM-like strategy to tag computational nodes. In difference to the GFM method, which employs a boundary condition capturing in primitive variables, the CBM method implements boundary conditions based on a characteristic decomposition in the direction normal to the boundary. In this way over-specification of boundary conditions is avoided and we believe so will be spurious oscillations. In this paper we treat (moving or stationary) fluid-solid interfaces and present numerical results for a select set of test cases. Extension to fluid-fluid interfaces will be presented in a subsequent paper.

On physics-based preconditioning of the Navier-Stokes equations

We develop a fully implicit scheme for the Navier-Stokes equations, in conservative form, for low to intermediate Mach number flows. Simulations in this range of flow regime produce stiff wave systems in which slow dynamical (advective) modes coexist with fast acoustic modes. Viscous and thermal diffusion effects in refined boundary layers can also produce stiffness. Implicit schemes allow one to step over the fast wave phenomena (or unresolved viscous time scales), while resolving advective time scales. In this study we employ the Jacobian-free Newton-Krylov (JFNK) method and develop a new physics-based preconditioner. To aid in overcoming numerical stiffness caused by the disparity between acoustic and advective modes, the governing equations are transformed into the primitive-variable form in a preconditioning step. The physics-based preconditioning incorporates traditional semi-implicit and physics-based splitting approaches without a loss of consistency between the original and preconditioned systems. The resulting algorithm is capable of solving low-speed natural circulation problems (M~10^-^4) with significant heat flux as well as intermediate speed (M~1) flows efficiently by following dynamical (advective) time scales of the problem.

A Reconstructed Discontinuous Galerkin Method for the Compressible Euler Equations on Arbitrary Grids

A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Euler equations on arbitrary grids. By taking advantage of handily available and yet invaluable information, namely the derivatives, in the context of the discontinuous Galerkin methods, a polynomial solution of one degree higher is reconstructed using a least-squares method. The stencils used in the reconstruction involve only the von Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The resulting RDG method can be regarded as an improvement of a recovery-based DG method, in the sense that it shares the same nice features, such as high accuracy and efficiency, and yet overcomes some of its shortcomings such as a lack of flexibility, compactness, and robustness. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results demonstrate that this RDG method is third-order accurate at a cost slightly higher than its underlying second-order DG method, at the same time providing a better performance than the third order DG method, in terms of both computing costs and storage requirements.

A Comparative Study of Different Reconstruction Schemes for a Reconstructed Discontinuous Galerkin Method on Arbitrary Grids

A comparative study of different reconstruction schemes for a reconstruction-based discontinuous Galerkin, termed RDG(P1P2) method is performed for compressible flow problems on arbitrary grids. The RDG method is designed to enhance the accuracy of the discontinuous Galerkin method by increasing the order of the underlying polynomial solution via a reconstruction scheme commonly used in the finite volume method. Both Green-Gauss and least-squares reconstruction methods and a least-squares recovery method are implemented to obtain a quadratic polynomial representation of the underlying discontinuous Galerkin linear polynomial solution on each cell. These three reconstruction/recovery methods are compared for a variety of compressible flow problems on arbitrary meshes to access their accuracy and robustness. The numerical results demonstrate that all three reconstruction methods can significantly improve the accuracy of the underlying second-order DG method, although the least-squares reconstruction method provides the best performance in terms of both accuracy and robustness.

Marker Redistancing/Level Set Method for High-Fidelity Implicit Interface Tracking

A hybrid of the front tracking (FT) and the level set (LS) methods is introduced, combining advantages and removing drawbacks of both methods. The kinematics of the interface is treated in a Lagrangian (FT) manner, by tracking markers placed at the interface. The markers are not connected—instead, the interface topology is resolved in an Eulerian (LS) framework, by wrapping a signed distance function around Lagrangian markers each time the markers move. For accuracy and efficiency, we have developed a high-order “anchoring” algorithm and an implicit PDE-based redistancing. We have demonstrated that the method is 3rd-order accurate in space, near the markers, and therefore 1st-order convergent in curvature; this is in contrast to traditional PDE-based reinitialization algorithms, which tend to slightly relocate the zero level set and can be shown to be nonconvergent in curvature. The implicit pseudo-time discretization of the redistancing equation is implemented within the Jacobian-free Newton-Krylov (JFNK) framework combined with ILU(k) preconditioning. Due to the LS localization, the bandwidth of the Jacobian matrix is nearly constant, and the ILU preconditioning scales as

Direct Numerical Simulation of Interfacial Flows: Implicit Sharp-Interface Method (I-SIM)

\sim N\log(\sqrt{N})