European Office of Aerospace Research and Development (EOARD)

This work is concerned with the implementation and testing, within a structured collocated finite-volume framework, of seven segregated algorithms for the prediction of multi-phase flow at all speeds. These algorithms belong to the Geometric Conservation Based Algorithms (GCBA) group in which the pressure correction equation is derived from the constraint equation on volume fractions (i.e. sum of volume fractions equals 1). The pressure correction schemes in these algorithms are based on SIMPLE, SIMPLEC, SIMPLEX, SIMPLEM, SIMPLEST, PISO, and PRIME. Solving a variety of oneand two-dimensional laminar and turbulent two-phase flow problems in the subsonic, transonic, and supersonic regimes and comparing results with published numerical and/or experimental data assess the performance and accuracy of these algorithms. The SG method is used to solve for the one-dimensional test problems and the effects of grid size on convergence characteristics are analyzed. On the other hand, solutions for the two-dimensional problems are generated for several grid systems using the single grid method (SG), the prolongation grid method (PG), and the full non-linear multi-grid method (FMG) and their effects on convergence behavior are studied. The main outcomes of this study are the clear demonstrations of: (i) the capability of all GCBA algorithms to deal with multifluid flow situations; (ii) the ability of the FMG method to tackle the added non-linearity of multi-fluid flows; (iii) and the capacity of the GCBA algorithms to predict multi-fluid flow at all speeds. The Geometric Conservation Based Algorithms for Multi-Fluid Flow at All Speeds 3 Nomenclature AP (k ) ,.. coefficients in the discretized equation for φ (k ) . BP (k ) source term in the discretized equation for φ (k ) . B ) body force per unit volume of fluid/phase k. Cρ (k ) coefficient equals to 1/ RT (k ) . DP [φ (k ) ] the D operator. HP[φ (k ) ] the H operator. HPP[φ (k ] the HP operator working on φ (φ=u,v, or w). HPP[u (k ) ] the vector form of the HP operator. I ) inter-phase momentum transfer. D k f ) ( J diffusion flux of ) (k φ across cell face ‘f’. J f (k)C convection flux of φ (k ) across cell face ‘f’. ) (k M& mass source per unit volume. P pressure. ) k ( t ) k ( Pr , Pr laminar and turbulent Prandtl number for fluid/phase k. ) (k q& heat generated per unit volume of fluid/phase k. Q ) general source term of fluid/phase k. r (k ) volume fraction of fluid/phase k. R ) gas constant for fluid/phase k. f S surface vector. t time. T (k ) temperature of fluid/phase k. U f (k ) interface flux velocity v f .S f ( ) of fluid/phase k. u velocity vector of fluid/phase k. u,v,.. velocity components of fluid/phase k. x, y Cartesian coordinates. b , a the maximum of a and b. The Geometric Conservation Based Algorithms for Multi-Fluid Flow at All Speeds 4 Greek Symbols ρ ) density of fluid/phase k. Γ (k ) diffusion coefficient of fluid/phase k. Φ (k ) dissipation term in energy equation of fluid/phase k. φ (k ) general scalar quantity associated with fluid/phase k. κ f space vector equal to ( ) f f f S ˆ ˆ d n γ − [ ] ) (k P φ ∆ the ∆ operator. ) ( ) ( , k t k μ μ laminar and turbulent viscosity of fluid/phase k. Ω cell volume. β (k ) thermal expansion coefficient for phase/fluid k. δt time step. Subscripts e, w, . refers to the east, west, ... face of a control volume. E,W,.. refers to the East, West, ... neighbors of the main grid point. f refers to control volume face f. P refers to the P grid point. Superscripts C refers to convection contribution. D refers to diffusion contribution. (k) refers to fluid/phase k. * ) (k refers to updated value at the current iteration. (k ) o refers to values of fluid/phase k from the previous iteration. (k ′ ) refers to correction field of phase/fluid k. old refers to values from the previous time step. Introduction The last two decades have witnessed a substantial transformation in the CFD industry; from a research means confined to research laboratories, CFD has emerged as an every day engineering tool for a wide range of industries (Aeronautics, Automobile, HVAC, etc...). This increasing dependence on CFD is due to a multitude of factors that have rendered practical the simulation of complex problems. Some of these factors are directly related to the maturity of several numerical aspects at the core of CFD. These include: multi-grid acceleration techniques [1-4] with enhanced equation solvers [5,6] that have decreased the computational cost of tackling large problems, better discretization techniques, unstructured grids [7-12], bounded high resolution schemes [13-18], as well as improved pressure-velocity (and density) coupling algorithms for fluid flow at all speeds [19-27]. Other factors, independent of the CFD industry, have to do with the exponential increase in processor power and decrease in microprocessor cost, whereby multiprocessors systems with large memory can now be set up at a fraction the cost of the super-computers of a decade ago. Challenges still abound in relation to increasing the robustness of numerical techniques, improving the models used (e.g. turbulence), and extending the currently used algorithms [28-34] for the simulation of multi-phase flows at all speeds [35]. In this last area a number of algorithms have been recently reviewed and new ones proposed [36]. The basic difficulty in the simulation of multi-phase flows [36] stems from the increased algorithmic complexity that need to be addressed when dealing with multiple sets of continuity and momentum equations that are inter-coupled (interchange momentum by inter-phase mass and momentum transfer, etc.) both spatially and across fluids. Despite these complexities, successful segregated incompressible pressure-based solution algorithms have been devised. The IPSA variants devised by the Spalding Group at Imperial College [37-39] and the set of algorithms devised by the Los Alamos Scientific Laboratory (LASL) group [40-42] are examples of The Geometric Conservation Based Algorithms for Multi-Fluid Flow at All Speeds 6 incompressible multiphase algorithms. When dealing with all-speed flows, pressurevelocity-density coupling has to be accounted for. Pressure-based algorithms have been extended successfully [19-27] to account for this additional coupling. Recently, Darwish et al. [36] extended the applicability of the available segregated singlefluid flow algorithms [35] to predict multi-fluid flow at all speeds. In their work, it was shown that the pressure correction equation can be derived either by using the geometric conservation equation or the overall mass conservation equation. Depending on which equation is used, the segregated pressure-based multi-fluid flow algorithms were classified respectively as either the Geometric Conservation Based family of Algorithms (GCBA) or the Mass Conservation Based family of Algorithms (MCBA). Moukalled et al. [43-46] implemented and tested the MCBA family and proved its capability to predict multi-fluid flow at all speeds. On the other hand, the GCBA family has not yet been implemented nor tested. The objective of the present work is to implement and test the GCBA family within a structured finite-volume framework with the convection terms along the control volume faces evaluated using a High Resolution (HR) scheme applied within the context of the Normalized Variable and Space Formulation methodology (NVSF) [15]. To reduce the overall computational cost, the convergence rate is accelerated through the use of a non-linear full multi-grid method. The discretization scheme is second-order accurate in space and first order accurate in time. In what follows, the governing equations are first introduced, followed by a brief description of the discretization procedure. Then the GCBA algorithms are presented, their capabilities to predict multi-fluid flow phenomena at all speeds demonstrated, and their performance characteristics (in terms of convergence history and speed) assessed. For that purpose, a total of twelve laminar and turbulent incompressible and compressible problems encompassing The Geometric Conservation Based Algorithms for Multi-Fluid Flow at All Speeds 7 dilute and dense gas-solid, and bubbly flows in the subsonic, transonic and supersonic regimes are solved. In addition, the performance of these algorithms is evaluated using (i) a single grid approach (SG), (ii) a prolongation only approach (PG) whereby the solution moves in one direction starting on the coarse grid and ending on the finest grid with the solution obtained on level n used as initial guess for the solution on level (n+1), and (iii) finally a Full Multi-Grid (FMG) approach with a W cycle. The Governing Equations In multi-phase flow the various fluids/phases coexist with different concentrations at different locations in the flow domain and move with unequal velocities. Thus, the equations governing multi-phase flows are the conservation laws of mass, momentum, and energy for each individual fluid. For turbulent multi-phase flow situations, an additional set of equations may be needed depending on the turbulence model used. These equations should be supplemented by a set of auxiliary relations. The various conservation equations needed are: ( ) ( ) ) k ( ) k ( k ) k ( ) k ( ) k ( ) k ( M r r t r & = ρ ∇ + ∂ ρ ∂ ) ( u . (1) ∂ r ρ u ( ) ∂t + ∇. r ρ u (k u ) ( )= ∇. r ) μ (k) + μ t(k) ( )∇u (k) [ ]+ r k) −∇P + B ( )+ I M ( k ) (2) ( ) ( ) ( ) ( ) ) k ( P ) k ( E ) k ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k ( P ) k ( ) k ( ) k ( t ) k ( t ) k ( ) k ( ) k ( k k ) k ( ) k ( k ) k ( ) k ( c I q . P P . t P T c r T Pr Pr r T r t T r +       + Φ +     ∇ − ∇ + ∂ ∂ β +         ∇       μ + μ ∇ = ρ ∇ + ∂ ρ ∂ & u u . u . ) ( ) ( ) (