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Development and application of a three dimensional numerical model for predicting pollutant and sediment transport using an Eulerian-Lagrangian marker particle technique

A computer coded Lagrangian marker particle in Eulerian finite difference cell solution to the three dimensional incompressible mass transport equation, Water Advective Particle in Cell Technique, WAPIC, was developed, verified against analytic solutions, and subsequently applied in the prediction of long term transport of a suspended sediment cloud resulting from an instantaneous dredge spoil release. Numerical results from WAPIC were verified against analytic solutions to the three dimensional incompressible mass transport equation for turbulent diffusion and advection of Gaussian dye releases in unbounded uniform and uniformly sheared uni-directional flow, and for steady-uniform plug channel flow. WAPIC was utilized to simulate an analytic solution for non-equilibrium sediment dropout from an initially vertically uniform particle distribution in one dimensional turbulent channel flow.

Pavish, D. L.↗

Numerical modeling of pollutant transport using a Lagrangian marker particle technique

A derivation and code were developed for the three-dimensional mass transport equation, using a particle-in-cell solution technique, to solve coastal zone waste discharge problems where particles are a major component of the waste. Improvements in the particle movement techniques are suggested and typical examples illustrated. Preliminary model comparisons with analytic solutions for an instantaneous point release in a uniform flow show good results in resolving the waste motion. The findings to date indicate that this computational model will provide a useful technique to study the motion of sediment, dredged spoils, and other particulate waste commonly deposited in coastal waters.

Spaulding, M.↗

Imposing a Lagrangian Particle Framework on an Eulerian Hydrodynamics Infrastructure in Flash

In many astrophysical simulations, both Eulerian and Lagrangian quantities are of interest. For example, in a galaxy cluster merger simulation, the intracluster gas can have Eulerian discretization, while dark matter can be modeled using particles. FLASH, a component-based scientific simulation code, superimposes a Lagrangian framework atop an adaptive mesh refinement Eulerian framework to enable such simulations. The discretization of the field variables is Eulerian, while the Lagrangian entities occur in many different forms including tracer particles, massive particles, charged particles in particle-in-cell mode, and Lagrangian markers to model fluid structure interactions. These widely varying roles for Lagrangian entities are possible because of the highly modular, flexible, and extensible architecture of the Lagrangian framework. In this paper, we describe the Lagrangian framework in FLASH in the context of two very different applications, Type Ia supernovae and galaxy cluster mergers, which use the Lagrangian entities in fundamentally different ways.

Dubey, A.↗

Numerical Simulation of the Instability of an Inviscid Liquid Jet in a Coflowing Compressible Airstream

The nonlinear interfacial instability of a liquid jet in a coflowing compressible airstream is studied numerically. A high-resolution scheme which has second-order accuracy in space and time is coupled with a Lagrangian marker particle algorithm to visualize the large-scale motion of the interfaces in compressible flow. A numerical algorithm based on an approximate equation of state of a compressible liquid is developed to allow this two-fluid system to be governed by the nonlinear unsteady Euler equations in conservative form. The initial growth of small disturbances given by the simulations agrees well with linear theory. The process of jet disruption in compressible flow is demonstrated to consist of the formation of liquid spikes, interweaving of the gas and liquid and stretching and detachment of the liquid main center core.

Li, H. -S.↗

Reduction of numerical diffusion in three-dimensional vortical flows using a coupled Eulerian/Lagrangian solution procedure

A new approach based on the coupling of an Eulerian and a Lagrangian solver, aimed at reducing the numerical diffusion errors of standard Eulerian time-marching finite-volume solvers, is presented. The approach is applied to the computation of the secondary flow in two bent pipes and the flow around a 3D wing. Using convective point markers the Lagrangian approach provides a correction of the basic Eulerian solution. The Eulerian flow in turn integrates in time the Lagrangian state-vector. A comparison of coarse and fine grid Eulerian solutions makes it possible to identify numerical diffusion. It is shown that the Eulerian/Lagrangian approach is an effective method for reducing numerical diffusion errors.

Felici, Helene M.↗

A coupled Eulerian/Lagrangian method for the solution of three-dimensional vortical flows

A coupled Eulerian/Lagrangian method is presented for the reduction of numerical diffusion observed in solutions of three-dimensional rotational flows using standard Eulerian finite-volume time-marching procedures. A Lagrangian particle tracking method using particle markers is added to the Eulerian time-marching procedure and provides a correction of the Eulerian solution. In turn, the Eulerian solutions is used to integrate the Lagrangian state-vector along the particles trajectories. The Lagrangian correction technique does not require any a-priori information on the structure or position of the vortical regions. While the Eulerian solution ensures the conservation of mass and sets the pressure field, the particle markers, used as 'accuracy boosters,' take advantage of the accurate convection description of the Lagrangian solution and enhance the vorticity and entropy capturing capabilities of standard Eulerian finite-volume methods. The combined solution procedures is tested in several applications. The convection of a Lamb vortex in a straight channel is used as an unsteady compressible flow preservation test case. The other test cases concern steady incompressible flow calculations and include the preservation of turbulent inlet velocity profile, the swirling flow in a pipe, and the constant stagnation pressure flow and secondary flow calculations in bends. The last application deals with the external flow past a wing with emphasis on the trailing vortex solution. The improvement due to the addition of the Lagrangian correction technique is measured by comparison with analytical solutions when available or with Eulerian solutions on finer grids. The use of the combined Eulerian/Lagrangian scheme results in substantially lower grid resolution requirements than the standard Eulerian scheme for a given solution accuracy.

Felici, Helene Marie↗

An Eulerian/Lagrangian coupling procedure for three-dimensional vortical flows

A coupled Eulerian/Lagrangian method is presented for the reduction of numerical diffusion observed in solutions of 3D vortical flows using standard Eulerian finite-volume time-marching procedures. A Lagrangian particle tracking method, added to the Eulerian time-marching procedure, provides a correction of the Eulerian solution. In turn, the Eulerian solution is used to integrate the Lagrangian state-vector along the particles trajectories. While the Eulerian solution ensures the conservation of mass and sets the pressure field, the particle markers describe accurately the convection properties and enhance the vorticity and entropy capturing capabilities of the Eulerian solver. The Eulerian/Lagrangian coupling strategies are discussed and the combined scheme is tested on a constant stagnation pressure flow in a 90 deg bend and on a swirling pipe flow. As the numerical diffusion is reduced when using the Lagrangian correction, a vorticity gradient augmentation is identified as a basic problem of this inviscid calculation.

Felici, Helene M.↗

A time-accurate finite volume method valid at all flow velocities

A finite volume method to solve the Navier-Stokes equations at all flow velocities (e.g., incompressible, subsonic, transonic, supersonic and hypersonic flows) is presented. The numerical method is based on a finite volume method that incorporates a pressure-staggered mesh and an incremental pressure equation for the conservation of mass. Comparison of three generally accepted time-advancing schemes, i.e., Simplified Marker-and-Cell (SMAC), Pressure-Implicit-Splitting of Operators (PISO), and Iterative-Time-Advancing (ITA) scheme, are made by solving a lid-driven polar cavity flow and self-sustained oscillatory flows over circular and square cylinders. Calculated results show that the ITA is the most stable numerically and yields the most accurate results. The SMAC is the most efficient computationally and is as stable as the ITA. It is shown that the PISO is the most weakly convergent and it exhibits an undesirable strong dependence on the time-step size. The degenerated numerical results obtained using the PISO are attributed to its second corrector step that cause the numerical results to deviate further from a divergence free velocity field. The accurate numerical results obtained using the ITA is attributed to its capability to resolve the nonlinearity of the Navier-Stokes equations. The present numerical method that incorporates the ITA is used to solve an unsteady transitional flow over an oscillating airfoil and a chemically reacting flow of hydrogen in a vitiated supersonic airstream. The turbulence fields in these flow cases are described using multiple-time-scale turbulence equations. For the unsteady transitional over an oscillating airfoil, the fluid flow is described using ensemble-averaged Navier-Stokes equations defined on the Lagrangian-Eulerian coordinates. It is shown that the numerical method successfully predicts the large dynamic stall vortex (DSV) and the trailing edge vortex (TEV) that are periodically generated by the oscillating airfoil. The calculated streaklines are in very good comparison with the experimentally obtained smoke picture. The calculated turbulent viscosity contours show that the transition from laminar to turbulent state and the relaminarization occur widely in space as well as in time. The ensemble-averaged velocity profiles are also in good agreement with the measured data and the good comparison indicates that the numerical method as well as the multipletime-scale turbulence equations successfully predict the unsteady transitional turbulence field. The chemical reactions for the hydrogen in the vitiated supersonic airstream are described using 9 chemical species and 48 reaction-steps. Consider that a fast chemistry can not be used to describe the fine details (such as the instability) of chemically reacting flows while a reduced chemical kinetics can not be used confidently due to the uncertainty contained in the reaction mechanisms. However, the use of a detailed finite rate chemistry may make it difficult to obtain a fully converged solution due to the coupling between the large number of flow, turbulence, and chemical equations. The numerical results obtained in the present study are in good agreement with the measured data. The good comparison is attributed to the numerical method that can yield strongly converged results for the reacting flow and to the use of the multiple-time-scale turbulence equations that can accurately describe the mixing of the fuel and the oxidant.

Kim, S.-W.↗

Detailed Studies on the Structure and Dynamics of Reacting Dusty Flows at Normal and Microgravity

Dusty reacting flows are of particular interest for a wide range of applications. Inert particles can alter the flammability and extinction limits of a combustible mixture. Reacting particles can release substantial amount of heat and can be used either for power generation or propulsion. Accumulation of combustible particles in air can result in explosions which, for example, can occur in grain elevators, during lumber milling and in mine galleries. Furthermore, inert particles are used as flow velocity markers in reacting flows, and their velocity is measured by non-intrusive laser diagnostic techniques. Despite their importance, dusty reacting flows have been less studied and understood compared to gas phase as well as sprays. The addition of solid particles in a flowing gas stream can lead to strong couplings between the two phases, which can be of dynamic, thermal, and chemical nature. The dynamic coupling between the two phases is caused by the inertia that causes the phases to move with different velocities. Furthermore, gravitational, thermophoretic, photophoretic, electrophoretic, diffusiophoretic, centrifugal, and magnetic forces can be exerted on the particles. In general, magnetic, electrophoretic, centrifugal, photophoretic, and diffusiophoretic can be neglected. On the other hand, thermophoretic forces, caused by steep temperature gradients, can be important. The gravitational forces are almost always present and can affect the dynamic response of large particles. Understanding and quantifying the chemical coupling between two phases is a challenging task. However, all reacting particles begin this process as inert particles, and they must be heated before they participate in the combustion process. Thus, one must first understand the interactions of inert particles in a combustion environment. The in-detail understanding of the dynamics and structure of dusty flows can be only advanced by considering simple flow geometries such as the opposed-jet, stagnation-type. In such configurations the imposed strain rate is well characterized, and the in-depth understanding of the details of the physico-chemical processes can be systematically obtained. A number of computational and experimental studies on spray and particle flows have been conducted in stagnation-type configurations. Numerically, the need for a hybrid Eulerian-Lagrangian approach has been identified by Continillo and Sirignano, and the use of such approach has allowed for the prediction of the phenomenon of droplet flow reversal. Gomez and Rosner have conducted a detailed study on the particle response in the opposed-jet configuration, and the particle thermophoretic diffusivities were determined experimentally. Sung, Law and co-workers have conducted numerical studies on the effect of strain rate and temperature gradients on the dynamics of inert particles, as a way of understanding potential errors in experimental LDV data that may arise from thermophoretic forces. This investigation is a combined experimental and numerical study on the details of reacting dusty flows. The specific tasks are: (1) Experimental determination of laminar flame speeds, and extinction strain rates of dusty flows at normal- and micro-gravity as functions of the particle type, particle initial diameter, particle initial number density, and gas phase chemical composition; (2) Detailed numerical simulation of the experiments. Results are compared with experiments and the adequacy of theoretical models is assessed; and (3) Provision of enhanced insight into the thermo-chemical coupling between the two phases.

Andac, M. Gurhan↗