Duan Z. Zhang

The effects of mesoscale structures on the macroscopic momentum equations for two-phase flows

Abstract Mesoscale structures (bubbles, clusters and streamers) in two-phase flows, especially in gas–solid fluidized beds significantly affect macroscopic hydrodynamic behavior. For industrial-scale fluidized beds, it is typically impractical to simulate these structures directly due to the excessive resolution required. To model effects of mesoscale structures, the ensemble phase averaging method is extended to derive macroscopic averaged equations and their closures. It is found that added-mass and drag reduction effects due to mesoscale structures play essential roles in the macroscopic equations of motion. Unlike the classical added-mass force, which is proportional to the continuous fluid density, the mesoscale added-mass force is proportional to the mixture density. Thus for gas–solid systems wherein the classical added-mass force is almost always negligible, the mesoscale added-mass force is, in contrast, found to be quite important. Mesoscale drag reduction results from the fact that, in a particle rich region, there is significantly less relative velocity between particle and fluid phases than indicated by the macroscopic relative velocity. Possible effects of the new force terms in the macroscopic equations are examined from a one-dimensional simulation of a fluidized bed. Significant effects from the new terms on vertical pressure gradient and particle volume fraction distributions are observed.

Material point method enhanced by modified gradient of shape function

A numerical scheme of computing quantities involving gradients of shape functions is introduced for the material point method (MPM), so that the quantities are continuous as material points move across cell boundaries. The noise and instability caused by cell crossing of the material points are then eliminated. In this scheme, the formulas used to compute these quantities can be expressed in the same forms as in the original material point method, but with the gradient of the shape function modified. For one-dimensional cases, the gradient of the shape function used in the generalized interpolation material point (GIMP) method is a special case of the modified gradient if the characteristic function of a material point is introduced. The characteristic function of a material point is not otherwise needed in this scheme, therefore difficulties in tracking its evolution are avoided. Although the support of the modified gradient of a shape function is enlarged from the cell containing the material point to also include the immediate neighbor cells, all the non-local effects of a material point can be accounted for by two consecutive local operations. Therefore this scheme can be used in calculations with unstructured grids. This scheme is proved to satisfy mass and momentum conservations exactly. The error in energy conservation is shown to be second order on both spatial and temporal discretizations. Although the error in energy conservation is the same order as that in the original material point method, numerical examples show that this scheme has significantly better energy conservation properties than those of the original material point method.

Material point method applied to multiphase flows

The particle-in-cell method (PIC), especially the latest version of it, the material point method (MPM), has shown significant advantage over the pure Lagrangian method or the pure Eulerian method in numerical simulations of problems involving large deformations. It avoids the mesh distortion and tangling issues associated with Lagrangian methods and the advection errors associated with Eulerian methods. Its application to multiphase flows or multi-material deformations, however, encounters a numerical difficulty of satisfying continuity requirement due to the inconsistence of the interpolation schemes used for different phases. It is shown in Section 3 that current methods of enforcing this requirement either leads to erroneous results or can cause significant accumulation of errors. In the present paper, a different numerical method is introduced to ensure that the continuity requirement is satisfied with an error consistent with the discretization error and will not grow beyond that during the time advancement in the calculation. This method is independent of physical models. Its numerical implementation is quite similar to the common method used in Eulerian calculations of multiphase flows. Examples calculated using this method are presented.

Centrifugal Contactors: Separation of an Aqueous and an Organic Stream in the Rotor Zone (LA‐UR‐05‐7800)

Abstract A multi‐phase flow code is used to simulate the separation of an aqueous and an organic stream in the rotor zone of an annular centrifugal contactor. Different values for the mixture viscosity and for the initial volume fractions of the components are considered. A simple model for mass transfer of a species between phases is used. Geometrical effects are found to have significant influence on the separation of the two‐phase mixture.

Distribution coefficient algorithm for small mass nodes in material point method

When using the time explicit material point method to simulate interaction of materials accompanied by large deformations and fragmentation, one often encounters a numerical instability caused by small node mass, because acceleration on a mesh node is obtained by dividing the total force on the node by the mass of the node. When the material points are in the far sides of the cells containing the node, typically happening near material interfaces, the node mass can be very small leading to artificially large acceleration and then numerical instability. For the case of small material deformations, this instability is typically avoided by placing the material points away from cell boundaries. For cases with large deformations, with the exception of initial conditions, there is no control on locations of the material points. The instability caused by small mass nodes is often encountered. To avoid this instability tiny time steps are usually required in a numerical calculation. In this work, we present a numerical algorithm to treat this instability. We show that this algorithm satisfies mass and momentum conservation laws. The error in energy conservation is proportional to the second order of the time step, consistent with the explicit material point method. Numerical implementation of the algorithm is described. Numerical examples show effectiveness of the algorithm.

Material point methods applied to one-dimensional shock waves and dual domain material point method with sub-points

Using a simple one-dimensional shock problem as an example, the present paper investigates numerical properties of the original material point method (MPM), the generalized interpolation material point (GIMP) method, the convected particle domain interpolation (CPDI) method, and the dual domain material point (DDMP) method.For a weak isothermal shock of ideal gas, the MPM cannot be used with accuracy. With a small number of particles per cell, GIMP and CPDI produce reasonable results. However, as the number of particles increases the methods fail to converge and produce pressure spikes. The DDMP method behaves in an opposite way. With a small number of particles per cell, DDMP results are unsatisfactory. As the number of particles increases, the DDMP results converge to correct solutions, but the large number of particles needed for convergence makes the method very expensive to use in these types of shock wave problems in two- or three-dimensional cases.The cause for producing the unsatisfactory DDMP results is identified. A simple improvement to the method is introduced by using sub-points. With this improvement, the DDMP method produces high quality numerical solutions with a very small number of particles.Although in the present paper, the numerical examples are one-dimensional, all derivations are for multidimensional problems. With the technique of approximately tracking particle domains of CPDI, the extension of this sub-point method to multidimensional problems is straightforward. This new method preserves the conservation properties of the DDMP method, which conserves mass and momentum exactly and conserves energy to the second order in both spatial and temporal discretizations.

Shock waves simulated using the dual domain material point method combined with molecular dynamics

In this work we combine the dual domain material point method with molecular dynamics in an attempt to create a multiscale numerical method to simulate materials undergoing large deformations with high strain rates. In these types of problems, the material is often in a thermodynamically nonequilibrium state, and conventional constitutive relations or equations of state are often not available. In this method, the closure quantities, such as stress, at each material point are calculated from a molecular dynamics simulation of a group of atoms surrounding the material point. Rather than restricting the multiscale simulation in a small spatial region, such as phase interfaces, or crack tips, this multiscale method can be used to consider nonequilibrium thermodynamic effects in a macroscopic domain.This method takes the advantage that the material points only communicate with mesh nodes, not among themselves; therefore molecular dynamics simulations for material points can be performed independently in parallel. The dual domain material point method is chosen for this multiscale method because it can be used in history dependent problems with large deformation without generating numerical noise as material points move across cells, and also because of its convergence and conservation properties.To demonstrate the feasibility and accuracy of this method, we compare the results of a shock wave propagation in a cerium crystal calculated using the direct molecular dynamics simulation with the results from this combined multiscale calculation.

A Separate-Phase Drag Model and a Surrogate Approximation for Simulation of the Steam-Assisted-Gravity-Drainage Process

General ensemble phase averaged equations for multiphase flows have been specialized for the simulation of the steam assisted gravity drainage (SAGD) process. In the average momentum equation, fluid-solid and fluid-fluid viscous interactions are represented by separate force terms. This equation has a form similar to that of Darcy’s law for multiphase flow but augmented by the fluid-fluid viscous forces. Models for these fluid-fluid interactions are suggested and implemented into the numerical code CartaBlanca. Numerical results indicate that the model captures the main features of the multiphase flow in the SAGD process, but the detailed features, such as plumes are missed. We find that viscous coupling among the fluid phases is important. Advection time scales for the different fluids differ by several orders of magnitude because of vast viscosity differences. Numerically resolving all of these time scales is time consuming. To address this problem, we introduce a steam surrogate approximation to increase the steam advection time scale, while keeping the mass and energy fluxes well approximated. This approximation leads to about a 40-fold speed-up in execution speed of the numerical calculations at the cost of a few percent error in the relevant quantities.

Combining dual domain material point method with molecular dynamics for thermodynamic nonequilibriums

Abstract The dual domain material point method (DDMP) combined with molecular dynamics (MD) is used to simulate a material undergoing a large deformation with a high strain rate. In the simulation, the continuum scale equation of motion is solved using DDMP, while the stresses required at the material points are obtained by performing MD simulations in small domains surrounding the material points following the entire history of the material deformation without reinitialization of the MD systems; therefore the history dependence, a common feature for thermodynamically nonequilibrium systems, can be tracked accurately. Two algorithms are introduced to avoid distortion of the MD domains and to ensure consistence in the energy density between the MD and macroscopic calculations. As an example, a hyper-velocity impact problem is simulated. The results of the combined DDMP–MD calculations are compared to a pure MD simulation. The method of the combined simulation is applicable to macroscopic systems, although the size of the simulation domain is only a few hundred nanometers in this example because of the limitation of the pure MD simulation.