Modeling Volumetric Coupling of the Dispersed Phase using the Eulerian-Lagrangian Approach

The Eulerian-Lagrangian approach is commonly used in modeling two-phase flows wherein liquid droplets, solid particles, or bubbles are dispersed in a continuum fluid of a different phase. Typically, the motion of the dispersed phase is modeled by assuming spherical, point-particles with models for added mass effects, drag, and lift forces. The effect of the dispersed phase on the fluid flow is modeled using reaction forces in the fluid momentum equation. Such an approach is valid for dilute regions of the dispersed phase. For dense regions, however, the point-particle approach does not capture the interactions between the fluid and the dispersed phase accurately. In this work, the fluid volume displaced by the dispersed phase is taken into account to model the dense regions. The motion of the dispersed phase results in local, spatio-temporal variations of the volume fraction fields. The resultant divergence in the fluid velocity acts as a source or sink displacing the flow due to dispersed phase and is termed as volumetric coupling. The size of the dispersed phase is assumed smaller than the grid resolution and for the continuum phase. The variable–density, low–Mach number equations based on mixture theory are solved using a co–located, finite volume scheme. The interphase momentum exchange due to drag forces is treated implicitly to provide robustness in the dense regions. The volumetric coupling approach is first validated with analytical studies for flow induced by oscillating bubbles and gravitational settling of particles. Simulations of Rayleigh-Taylor instability, particle-laden jet impingement on a flat plate, and particle-laden jet in a cross are performed to test the robustness of the scheme. ∗Corresponding Author: [email protected] Introduction Majority of spray systems in propulsion applications involve complex geometries and highly unsteady, turbulent flows near the injector. The numerical models for spray calculations should be able to accurately represent droplet deformation, breakup, collision/coalescence, and dispersion due to turbulence. In the traditional approach for spray computation, the Eulerian equations for gaseous phase are solved along with a Lagrangian model for particle transport with two-way coupling of mass, momentum, and energy exchange between the two phases [1]. Typically simulations of spray systems use DNS, LES or RANS for the carrier phase whereas the motion of the dispersed phase is modeled. The ‘point-particle’ (PP) assumption is commonly employed where forces on the dispersed phase are computed through model coefficients. The effect of the particles on the carrier phase is represented by a force applied at the centroid of the particle. The disperse phase equations are typically solved in a Lagrangian frame by tracking a few set of computational particles or parcels [2] with models for droplet breakup, collision/coalescence, evaporation, dispersion, and deformation. Fully resolved simulations involving comprehensive modeling of interfacial dynamics are being developed [3, 4], however, are computationally expensive. Several simulations of particle-laden flows have been performed with the carrier fluid simulated using direct numerical simulation ([5, 6],[7],[8]), largeeddy simulation ([9, 10, 11, 12]), or Reynoldsaveraged Navier Stokes equations [13], where the dispersed phase is assumed subgrid (so dp < LK , the Kolmogorov length scale, for DNS whereas dp < ∆, the grid size, in LES or RANS). However, modeling the dispersed phase using point-particle approach does not always provide the correct results. For moderate loadings and wall-bounded flows [11] have shown that the point-particle approximation fails to predict the turbulence modulation compared to experimental values. In addition, if the particle size is comparable to the Kolmogorov scale (for DNS) or the grid size (for LES/RANS), simple drag/lift laws typically employed in PP do not capture the unsteady wake effects commonly observed in full DNS studies ([14, 15]). These effects become even more pronounced in dense particulate regions. In many practical applications, the local particle size and concentrations may vary substantially. In liquid atomization process, e.g., the droplet sizes may 1In this paper, particle may mean solid particle, liquid droplets, or bubbles depending upon the case being studied. range from 1 mm to 1 μm with dense regions near the injector nozzle. The point-particle assumption is invalid under these conditions. In the present work, we extend the point-particle approach by accounting for the volumetric displacements of the carried phase due to the motion of particles or droplets. The disperse phase also affects the carrier phase through mass, momentum, and energy coupling. The combined effect is termed as ‘volumetric coupling’. This approach is based on the the original formulation by Duckowicz [1] and later modified by Joseph & Lundgren [16]. The approach is derived based on mixture theory that account for the droplet (or particle) volume fraction in a given computational cell. This effect is important in dense spray regimes, however, are typically ignored in the context of LES or DNS simulations [17, 12]. A similar formulation has been applied to bubbly flows at low bubble concentrations (up to 0.02) to investigate the effect of bubbles on drag reduction in turbulent flows [8, 18]. Several studies on laminar dense granular flows [19, 20, 21] also use this approach. Recently, Apte etal. [22] have shown the effect of volumetric displacements on the carrier fluid in dense particle-laden flows. They compared the solutions for the carrier phase and the particle dispersion obtained from the point-particle assumption and accounting for volumetric displacements to show large differences. If the volume displaced by the disperse phase is taken into account, thte velocity field is no longer divergence free in the regions of variations in volume fractions. This has a direct effect on the pressure Poisson equation, altering the pressure field through a local source term. These effects may become important in dense regions of spray system. However, computing dense spray systems by accounting for volume displacements due to droplet motion could be numerically challenging. The temporal and spatial variations in fluid volume fractions could be locally large and make the computation numerically unstable. This is specifically true if the interphase coupling of mass, momentum, and energy is treated explicitly. In the present work, we focus on non-reacting flows and only momentum exchange between the two-phases is considered. A numerical approach based on co-located grid finite-volume method is developed with part of the momentum exchange terms treated implicitly. The approach is similar to the fractional step algorithms for particlein cell methods on staggered grids [19, 21, 20]. Implementation in co-located finite-volume formulation is discussed and is applicable to unstructured grids. Governing Equations The formulation described below consists of the Eulerian fluid and Lagrangian particle equations, and accounts for the displacement of the fluid by the particles as well as the momentum exchange between them ([16]). An Eulerian-Lagrangian framework is used to solve the coupled two-phase flow equations. The disperse phase equations are solved in a Lagrangian frame with models for for drag, buoyancy, and inter-particle collision forces. Continuum-phase equations In the present formulation, both continuity and momentum equations account for the local concentration of particles in the continuum phase. The fluid mass for unit volume satisfies a continuity equation, ∂ ∂t (ρfΘf ) +5 · (ρfΘfuf ) = 0 (1) where ρf , Θf , and uf are density, concentration, and velocity of the fluid phase respectively. Local spatio-temporal variations of particle concentration, generate a non-divergence free velocity field in the flow. The non-zero velocity divergence can be shown by rearranging the equation 1. ∇ · uf = − 1 Θf DΘf Dt (2) where D Dt is the material derivative with respect to fluid velocity. Fluid concentration is calculated as Θf = 1−Θp, where Θp is particle concentration. Lagrangian quantities, such as particle concentration, are interpolated to the Eulerian control volumes effectively, using the following interpolation function,