In this section we first want to explain the various reasons that make the direct numerical simulation of multiphase flow particularly costly from the computational point of view. Fol- lowing, we introduce the Adaptive Mesh Refinement technique as a tool to improve the ef- ficiency of simulations. Finally, we present the numerical framework analyzed in this work and the proposed numerical tests.
Mesh limitations in two-phase flow The Direct Numerical Simulation (DNS) of multi-
phase flow can result particularly challenging and expensive in terms of computational cost in the case of complex interfacial phenomena and high Reynolds numbers —a complete overview is given by Tryggvason et al. [1]. Indeed, in each part of the domain, different characteristic length requirements have to be ensured. In the first place, the mesh must be sufficiently fine to correctly model the Kolmogorov length scales that appear in turbulent flow. Second, the phenomena involving the instability and rupture of the interface must be correctly represented. That include the growth of waves and filaments, as well as the gen- eration of drops of varying size in primary and secondary atomization processes, with the smallest size equal or smaller than Kolmogorov length scales [2]. As a further requirement, the grid spacing at the interface has to be fine enough to properly evaluate interface prop- erties, as gradients and curvature. Due to these conditions, the same concept of DNS may be considered inappropriate for some kinds of turbulent multiphase flow, as the minimum mesh size that properly represent the interface phenomena cannot be defined apriori. In the practice, it is assumed that structures smaller than the minimum mesh size have no practical effect on the global behavior of the flow [3]. The engineering problems that better highlight these complex numerical issues and strong mesh limitations are constituted by atomizing two-phase flow, including injection sprays and coaxial jets, with important applications in the field of combustion injector and pharmaceutical sprays. Another notable example is the case of rising bubbles, where the leading forces are due to surface tension stresses that de- velop in proximity of the interface. Consequently, fine mesh resolutions are needed in the proximity of the interface to evaluate accurately these forces.
3.1. INTRODUCTION 63
Adoption of dynamic meshes An effective strategy to improve the feasibility of DNS in
multiphase flow consists in the adoption of the Adaptive Mesh Refinement (AMR) tech- nique. The tool performs a dynamic refinement of the mesh, depending on the local defini- tion requirement, with a global consistent reduction of the allocated computational resources. The robustness of AMR on 2D structured meshes was firstly demonstrated by Berger and Oliger [4] and by Berger and Colella [5]. More recently, AMR strategies have also ap- peared in unstructured hexahedral [6] and tetrahedral [7] meshes. The combined utiliza- tion of interface-capturing methods and AMR techniques was firstly introduced by Sussman et al. [8]. In Sussman [9], a parallelized algorithm to achieve mesh refinement in a cou- pled level-set/volume-of-fluid Navier-Stokes solver is described. More recently, Popinet et al. [10] combined the adaptive octree spatial discretization to Volume of Fluid (VOF) in- terface representation, by focusing on the correct balancing of surface tension forces and pressure gradients at the interface, while Zuzio and Estivalezes [11] coupled Level-Set and AMR, by applying particular procedures at the fine/coarse mesh interfaces to preserve the accuracy. Both authors were focusing on the simulation of laminar phenomena, as rising bubbles and Rayleigh-Taylor instabilities. Fuster et al. [3] also coupled octree AMR and VOF, and applied the method to atomization phenomena. One of the main issues connected to AMR discretizations, when dealing with incompressible problems, consists in the exten- sion of the divergence-free constraint to child cells. One of the most common strategies has been proposed by Balsara [12] and applied to two-phase simulations by [11]. An alternative scheme to deal with the incompressibility constraint can be found in Vanella et al. [13].
Current proposal In this work, we analyze a CLS-AMR framework that allows the sim-
ulation of turbulent multiphase flow in a wide range of situations, particularly, focusing on instability and atomization phenomena. In the presented method, the CLS scheme proposed and implemented by Balcàzar et al. [14] for interface-capturing of two-phase flows on 3D unstructured domains is adopted, due to its proven reliability on this class of phenomena —details above the scheme are reviewed in Chapter 1. It is coupled to an adaptive mesh refinement algorithm, whose implementation follows the work of Antepara et al. [15]. The algorithm carries out a quad/octree (2D/3D) hierarchical decomposition of the existing struc- tured mesh. In addition, a simple and low-cost reconstruction scheme for the face fluxes and cell-centered values has been introduced. Unlike similar approaches, this strategy is de- signed to exactly preserve mass, momentum and kinetic-energy in the sub-cells during the refinement/ coarsening procedures. This treatment allows to maintain the discrete conserva- tion properties of the adopted collocated scheme, demonstrated in [15, 16] for single-phase flows and analyzed in the context of multiphase flow by Schillaci et al. [17], thus, creating the basis for the correct resolution of turbulence. As demonstrated in the following sections, the analyzed scheme permits a considerable reduction of computational costs, both in terms of memory requirement and calculation time in comparison to fixed mesh schemes. At the same time, the method allows to obtain the required mesh size in each area of the domain,
without significantly affecting the stability and the accuracy of the simulation.
This Chapter is organized as follows. In Sec. 3.2, the general features of the AMR methods are explained. Then, in Sec. 3.3, the advantages led by the adoption of this strategy in contrast with static mesh schemes are highlighted by numerical tests. We consider the tests of a vortex flow and different set-up of rising bubbles, including the case of 3D merging bubbles in different configurations. In Sec. 3.4, the capability of the method to correctly simulate basic instability phenomena is shown, including the atomization of a 2-D coaxial jet.