Theo G. Theofanous

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.

The boiling crisis phenomenon. Part II: dryout dynamics and burnout

Abstract This is Part II of a two-part paper on the boiling crisis phenomenon. Here we report on burnout experiments conducted on fresh and aged heaters in pool boiling. Critical heat fluxes (CHFs) were found to vary from 50% to 140% of the hydrodynamic limit, previously thought to exist at well-wetting conditions. The burnout events were captured in action (for the first time), using high-speed, high-resolution infrared thermometry. Based on these observations and in conjunction with the levels of CHF reached, we are led to conclude that the phenomenon cannot be (macro)hydrodynamically limited, at least at normal pressure and gravity conditions. Based on infrared thermometry, and aided by X-ray radiography data on void fraction, the case for a scale separation phenomenon in high heat flux pool boiling is argued. This indicates that boiling crisis is controlled by the microhydrodynamics and rupture of an extended liquid microlayer, sitting and vaporizing autonomously on the heater surface. Further, the detailed dynamics of this microlayer, as revealed by our experiments, demonstrates that all previous thermally based models of boiling crisis are inappropriate.

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).

Aerobreakup in Rarefied Supersonic Gas Flows

We present new experimental results on the interfacial instabilities and breakup of Newtonian liquid drops suddenly exposed to rarefied, high-speed (Mach 3) air flows. The experimental approach allows for the first time detailed observation of interfacial phenomena and mixing throughout the breakup cycle over a wide range of Weber numbers. Key findings are that Rayleigh-Taylor instability alone is the active mechanism for freestream Weber numbers as low as 28 for low viscosity liquids and that stripping rather than piercing is the asymptotic regime as We→∞. This and other detailed visual evidence over 26<We<2,600 are uniquely suitable for testing Computational Fluid Dynamics (CFD) simulations on the way to basic understanding of aerobreakup over a broad range of conditions

How to solve compressible multifluid equations: a simple, robust and accurate method

Solving multifluid equations of compressible multiphase flows has proven to be extremely demanding because of some peculiar mathematical properties, such as nonhyperbolicity, nonconservative form, and stiffness due to disparity in fluid properties and flow scales occurring typically. In this paper, we first consider the mathematical issues concerning nonhyperbolicity and nonconservative form. Their effects on the stability and convergence of numerical solutions are the theme of our presentation; we shall present solutions for a range of problems selected to illuminate these numerical issues. To this end, we present a new numerical method that is simple to implement for a general class of fluids and yet is capable of robustly and accurately calculating phenomena involving material and shock discontinuities and interactions between them. Additionally, the paper is completed with a new information for ensuring hyperbolicity under an interfacial pressure representation.

Direct numerical simulation of interfacial instabilities: A consistent, conservative, all-speed, sharp-interface method

We present a conservative and consistent numerical method for solving the Navier-Stokes equations in flow domains that may be separated by any number of material interfaces, at arbitrarily-high density/viscosity ratios and acoustic-impedance mismatches, subjected to strong shock waves and flow speeds that can range from highly supersonic to near-zero Mach numbers. A principal aim is prediction of interfacial instabilities under superposition of multiple potentially-active modes (Rayleigh-Taylor, Kelvin-Helmholtz, Richtmyer-Meshkov) as found for example with shock-driven, immersed fluid bodies (locally oblique shocks)-accordingly we emphasize fidelity supported by physics-based validation, including experiments. Consistency is achieved by satisfying the jump discontinuities at the interface within a conservative 2nd-order scheme that is coupled, in a conservative manner, to the bulk-fluid motions. The jump conditions are embedded into a Riemann problem, solved exactly to provide the pressures and velocities along the interface, which is tracked by a level set function to accuracy of O(@Dx^5,@Dt^4). Subgrid representation of the interface is achieved by allowing curvature of its constituent interfacial elements to obtain O(@Dx^3) accuracy in cut-cell volume, with attendant benefits in calculating cell- geometric features and interface curvature (O(@Dx^3)). Overall the computation converges at near-theoretical O(@Dx^2). Spurious-currents are down to machine error and there is no time-step restriction due to surface tension. Our method is built upon a quadtree-like adaptive mesh refinement infrastructure. When necessary, this is supplemented by body-fitted grids to enhance resolution of the gas dynamics, including flow separation, shear layers, slip lines, and critical layers. Comprehensive comparisons with exact solutions for the linearized Rayleigh-Taylor and Kelvin-Helmholtz problems demonstrate excellent performance. Sample simulations of liquid drops subjected to shock waves demonstrate for the first time ab initio numerical prediction of the key interfacial features and phenomena found in recent experimental and theoretical studies of this class of problems [T.G. Theofanous, Aerobreakup of Newtonian and viscoelastic liquids, Ann. Rev. Fluid Mech. 43 (2011) 661-690.].

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.

Aerobreakup in disturbed subsonic and supersonic flow fields

This work concerns the breakup of millimetre-scale liquid droplets in gaseous flow fields that are disturbed from free-stream conditions by the presence of solid obstacles or other drops. A broad range of flow conditions is considered - from subsonic to supersonic, from highly rarefied to ambient pressures, and from fixed cylindrical obstacles to free liquid droplets (as obstacles). The liquid is water or tributyl phosphate, a water-like low-viscosity fluid of very low vapour pressure. We present data on deformation and breakup regimes, and, aided by numerical simulations, we discuss governing mechanisms and the time scaling of these events. Thereby a methodology is demonstrated for conveniently forecasting first-order behaviours in disturbed flow fields more generally. The highly resolved images lend themselves to testing/benchmarking numerical simulations of interfacial flows. These results, along with the experimental capability developed, constitute one of the key building blocks for our overall long-term aim towards predicting ultimate particle-size distributions from such intense aerodynamic interactions involving very large quantities of Newtonian and viscoelastic liquids.