Quadrature based moment methods

Quadrature-based moment methods (QBMM) are a class of computational fluid dynamics (CFD) methods for solving Kinetic theory and is optimal for simulating phases such as rarefied gases or dispersed phases of a multiphase flow. The smallest 'particle' entities which are tracked may be molecules of a single phase or granular 'particles' such as aerosols, droplets, bubbles, precipitates, powders, dust, soot, etc. Moments of the Boltzmann equation are solved to predict the phase behavior as a continuous (Eulerian) medium, and is applicable for arbitrary Knudsen number ( K n ) {displaystyle (Kn)} and arbitrary Stokes number ( S t ) {displaystyle (St)} . Source terms for collision models such as Bhatnagar-Gross-Krook (BGK) and models for evaporation, coalescence, breakage, and aggregation are also available. By retaining a quadrature approximation of a probability density function (PDF), a set of abscissas and weights retain the physical solution and allow for the construction of moments that generate a set of partial differential equations (PDE's). QBMM has shown promising preliminary results for modeling granular gases or dispersed phases within carrier fluids and offers an alternative to Lagrangian methods such as Discrete Particle Simulation (DPS). The Lattice Boltzmann Method (LBM) shares some strong similarities in concept, but it relies on fixed abscissas whereas quadrature-based methods are more adaptive. Additionally, the Navier–Stokes equations(N-S) can be derived from the moment method approach. Quadrature-based moment methods (QBMM) are a class of computational fluid dynamics (CFD) methods for solving Kinetic theory and is optimal for simulating phases such as rarefied gases or dispersed phases of a multiphase flow. The smallest 'particle' entities which are tracked may be molecules of a single phase or granular 'particles' such as aerosols, droplets, bubbles, precipitates, powders, dust, soot, etc. Moments of the Boltzmann equation are solved to predict the phase behavior as a continuous (Eulerian) medium, and is applicable for arbitrary Knudsen number ( K n ) {displaystyle (Kn)} and arbitrary Stokes number ( S t ) {displaystyle (St)} . Source terms for collision models such as Bhatnagar-Gross-Krook (BGK) and models for evaporation, coalescence, breakage, and aggregation are also available. By retaining a quadrature approximation of a probability density function (PDF), a set of abscissas and weights retain the physical solution and allow for the construction of moments that generate a set of partial differential equations (PDE's). QBMM has shown promising preliminary results for modeling granular gases or dispersed phases within carrier fluids and offers an alternative to Lagrangian methods such as Discrete Particle Simulation (DPS). The Lattice Boltzmann Method (LBM) shares some strong similarities in concept, but it relies on fixed abscissas whereas quadrature-based methods are more adaptive. Additionally, the Navier–Stokes equations(N-S) can be derived from the moment method approach. QBMM is a relatively new simulation technique for granular systems and has attracted interest from researchers in computational physics, chemistry, and engineering. QBMM is similar to traditional CFD methods, which solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy) numerically, but QBMM accomplishes this by modeling the fluid as consisting of fictive particles, or nodes, that constitute a discretized PDF. A node consists of an abscissa/weight pair and the weight defines the probability of finding a particle that has the value of its abscissa. This quadrature approximation may also be adaptive, meaning that the number of nodes can increase/decrease to accommodate appropriately complex/simple PDF's. Due to its statistical nature, QBMM has several advantages over other conventional Lagrangian methods, especially in dealing with complex boundaries, incorporating microscopic interactions (such as collisions), parallelization of the algorithm, and computational costs being largely independent of particle population. The numerical methods for solving the system of partial differential equations can be interpreted as the propagation (with a flux term) and interactions (source terms) of fictitious particle probabilities in an Eulerian framework. QBMM is a family of methods encompassing a variety of models, some of which are designed specifically to handle PDF's of passive variables, and others more complex, capable of multidimensional PDF's of active variables (such as velocity). Note that the full representation of the PDF is f ( t , x ; ξ ) {displaystyle f(t,mathbf {x} ;mathbf {xi } )} , where the parameters t {displaystyle t} and x {displaystyle mathbf {x} } represent the external coordinates of time and space respectively, while the internal coordinate vector, ξ {displaystyle mathbf {xi } } , may contain any additional desired degrees of freedom to represent the particles, e.g., temperature ( T ) {displaystyle (T)} , diameter ( L p ) {displaystyle (L_{p})} , velocity ( v ) {displaystyle (mathbf {v} )} , angular velocity, etc. The applicability of these methods depends upon which particle parameters are important (velocity, diameter, temperature, etc.), and importantly upon two values of the phase: ( K n ) {displaystyle (Kn)} and ( S t ) {displaystyle (St)} . For example, a monokinetic fluid will have a single velocity vector at each point in space, v ( t , x ) {displaystyle mathbf {v} (t,mathbf {x} )} ; therefore, its corresponding PDF, f ( v ) {displaystyle f(mathbf {v} )} , is a Dirac Delta function at every point in space. Similarly, a monodisperse phase has a constant diameter for all particles so that f ( L p ) {displaystyle f(L_{p})} is also a Delta function at every point in space. In those cases a PDF is superfluous and can instead be modeled by just tracking a single value corresponding to the abscissa of the Delta function, and the Navier-Stokes equations may be far more optimal to implement. One of the earliest applications of QBMM was the Quadrature Method of Moments (QMOM) by McGraw in 1997. This method was used mainly for aerosol sprays and droplets by tracking their diameters through phenomenon such as breakage, coalescence, evaporation, etc.