Coarse bifurcation studies of bubble flow microscopic simulations

The parametric behavior of regular periodic arrays of rising bubbles is investigated with the aid of 2-dimensional BGK Lattice-Boltzmann (LB) simulators. The Recursive Projection Method is implemented and coupled to the LB simulators, accelerating their convergence towards what we term coarse steady states. Efficient stability/bifurcation analysis is performed by computing the leading eigenvalues/eigenvectors of the coarse time stepper. Our approach constitutes the basis for system-level analysis of processes modeled through microscopic simulations. INTRODUCTION Multiphase flow chemical reactors are central to chemical engineering; they are vital in the production of a variety of chemicals where economy of scale is a driving factor. The hydrodynamic behavior of such gas-liquid and gas-liquid-solid reactors strongly affects their performance and scale-up. In particular, flow details at the level of individual bubbles affect the behavior of the overall system through phase interaction terms and effective stress tensors. It is therefore important both from a scientific and a technological point of view to assess the impact of such terms on the macroscopic hydrodynamics of bubbly flows [1,2]. In this work we use the LB method, based on kinetic theory, to simulate flow characteristics of individual bubbles [3,4]. Our objective is to expand the capabilities of the microscopic LB time-steppers, enabling them to perform system level tasks, such as stability analysis, continuation and bifurcation computations. Such tasks are currently inaccessible to microscopic simulators. Coarse (averaged) steady states of the system correspond to stationary profiles of moments of distributions from microscopic simulations. The parameter-dependent behavior of a system is traditionally analyzed by first obtaining a coarse model (e.g. a mean field PDE) and then performing bifurcation/stability analysis of this PDE. Here we propose an alternative methodology, building on the time-stepper-based approach to PDE bifurcation analysis. In particular, we adapt the Recursive Projection Method (RPM) of Shroff & Keller [5] to accelerate the convergence of the LB simulation to stable, and even to unstable coarse stationary (macroscopically steady) states. This implements coarse bifurcation analysis using the misroscopic evolution rules directly, and avoiding the intermediate step of construction of a coarse, mean-field PDE [6]. Such an approach is essential when a macroscopic description of the system is unavailable in closed form. Two-dimensional parametric studies of individual bubbles using the LB method are performed using periodic boundary conditions for the computational domain, thus corresponding to regular arrays of rising bubbles. Transitions to oscillatory patterns and formation of unstable wakes [2,7,8] are investigated. Our bifurcation parameter corresponds to variation of the average density difference between the gas and the liquid phase. The combination of LB and RPM allows the detection of coarse Hopf bifurcations and the corresponding coarse spectra (eigenvalues and eigenvectors). LATTICE BOLTZMANN SIMULATIONS Simulations of a single bubble rising in 2-D doubly periodic domains are performed employing a single species BGK-LB model [9] with a non-ideal equation of state. The governing evolution rule for this model, derived from the continuum BGK-Boltzmann distribution function for a single particle f(x,ξ,t) in position (x) and velocity (ξ ) phase space and time t, is given by the dimensionless explicit formula: Here τ is the relaxation time, related to the kinematic viscosity ν, ν=τ -1/2; fi’s are related to f(x,ξ,t) through: fi = (wi/w(ξi))f(x,ξ, t) where wi(ξ) are weights, w(ξi) weight functions: w(ξi)=(1/2π)exp(-ξiξi/2) (D=2, the dimension of the problem) and ξi’s the abscissas of the quadrature in velocity space as given in [10,11] for a square lattice in 2 dimensions: It has been shown [10] that a second order Hermite polynomial approximation equilibrium distribution f, is sufficient for isothermal flow calculations: where the local density n and the momentum nu (u being the local velocity vector) o node can be uniquely computed from successive moments of fi: The specific force a consists of the external specific force aext and that due to interactions, aint: a=aext + aint. The external force which causes the bubble to rise aext=g (1 / n ) being the average density of the mixture in the entire domain and g the acceleration due to gravity. The internal force according to [12,13 aint=G∑ψ(x+ξi)ψ(x)ξi, i=0,...8, where G denotes the Green’s function (here set to 2 the interaction potential, chosen according to [12]: ψ =1-exp(-n). The equation of sta system is [12]: P = nθ G/2ψ; P is the pressure and θ the dimensionless temperature 1 for an isothermal system). The LB simulations were performed in periodic domains consisting of 128x1 nodes. The area fraction of the bubble was ~5.8% and ν was set to 0.5. The dimensionless groups encountered in gas-liquid bubbly flows are the Reynolds (Re=ud/ν), the Morton number (Mo=gρ∆ρν/σ) and the Eotvos number (Eo=g∆ being the density of the liquid phase, d the effective diameter and u the slip veloc bubble, σ the interfacial tension and ∆ρ the density difference between the liquid an The parameter varied here was g, which, through the Mo and Eo numbers, corresp change in ∆ρ. At low values of g (g=0.00015 corresponding to Eo=3.0, Mo=0.0 Eo≈19994.3g and Mo≈12.6g) the bubble is almost spherical and rises vertically in the domain. In fig.1a three density snapshots are shown; dark grey color denotes low den light grey high density (liquid) and white the interface between the two phases. increases, a wake forms causing bubble shape deformation. Beyond a threshold valu wake starts shedding, and the shape of the rising bubble starts oscillating, clearly sug Hopf bifurcation in g. Fig.1b shows snapshots of an oscillating rising bubble for g=0.0 τ ) , ( ) , ( ) , ( ) 1 , ( t eq i f t f t i f t i i f i x x x ξ x − − = − + + ) 1 (