Mathematical models of particle crystallization in two-phase flows

The results of an analytic and numerical investigation of individual particle crystallization regimes are presented and it is shown that the equilibrium crystallization model [1, 2] is a limiting case of the nonequilibrium crystallization model [3]. An approximate method of solving the problem of nonuniform particle heating, which must be taken into account in determining the onset of particle crystallization, is described.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 77–84, November–December, 1989.     

Optimum nozzle configurations for two-phase flows

The structure of the optimum supersonic contour of an axisymmetric Laval nozzle is investigated within the framework of the nonequilibrium polydisperse two-phase flow model. In formulating the variational problem attention is focused on taking into account a restriction that makes it possible to construct an optimum contour with no or limited particle fall out. Tomsk. Translated from Izvesriya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 36–45, March-April, 1994.     

Lie symmetry solutions for two-phase mass flows

We apply Lie symmetry method to a set of non-linear partial differential equations, which describes a two-phase rapid gravity mass flow as a mixture of solid particles and viscous fluid down a slope (Pudasaini, J. Geophys. Res. 117 (2012) F03010, 28 pp [1]). In order to systematically explore the mathematical structure and underlying physics of the two-phase mixture flow, we generate several similarity forms in general form and construct self-similar solutions. Our analysis generalizes the results, obtained by applying the Lie symmetry method to relatively simple single-phase pressure-driven gravity mass flows, to the two-phase mass flows that include several dominant driving forces and strong phase-interactions. Analytical and numerical solutions are presented for the symmetry-reduced homogeneous and non-homogeneous systems of equations. Analytical and numerical results show that the new models presented here can adequately describe the dynamics of two-phase debris flows, and produce observable phenomena that are consistent with the physics of the flow. The solutions are strongly dependent on the choice of the symmetry-reduced model, as characterized by the group parameters, and the physical parameters of the flows. These solutions reveal strong non-linear and distinct dynamic evolutions, and phase-interactions between the solid and fluid phases, namely the phase-heights and phase-velocities.     

Equations for two-phase flows: A primer

The main goal of these notes is to give a review of the equations for two phase flow problems with an interface between the two phases in a self-contained way, and, in particular, to properly include surface tension into the interface balance equations.     

Averaged topological equations for dispersed two-phase flows

A general relationship between the volume fraction and the specific interfacial area for averaged dispersed two-phase flows is proposed. This relationship, expressed as a basic set of two scalar evolution equations and two vectorial non-uniformity state equations, is an analytical result obtained by a systematic approach using the derivatives of some generalized functions and a local volume-averaging technique. The proposed set of equations was expressed for measurable macroscopic parameters of the system and has the same generality as the averaged transport equations of two-phase flows. By combination of the basic set of equations, called the averaged topological equations (ATEs), second-order ATEs for the volume fraction were found. The second-order ATEs were expressed both by a Lagrangian formulation and by a Eulerian formulation. The importance and physical meaning of the ATEs developed in this study were clarified within the framework of the theory of kinematic waves.     

Calculation of two-phase swirling flows

The results of calculating the diffusion of a dispersed admixture in turbulent swirling jet flows using the model of momentum transfer in a turbulent gas—dispersion flow proposed by the authors are presented. These results are compared with experimental data and with calculations based on various mathematical models. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 71–78, January–February, 1994.     

Nonlinear systems synchronization for modeling two-phase microfluidics flows

The aim of this work is to identify a class of models that can represent the two-phase microfluidic flow in different experimental conditions. The identification procedure adopted is based on the nonlinear systems synchronization theory. The experimental time series were assumed as the asymptotic behavior of a generic state variable of an unknown Master system, and this information was used to drive a second Slave system, with a known model and undefined parameters. To reach the convergence between the time evolutions of the two systems, so the flow identification, an error was evaluated and optimized by tuning the parameters of the Slave system, through genetic algorithm. The Chua’s oscillator has been chosen as a Slave model, and an optimal parameters set of Chua’s system was identified for each of the 18 experiments. As proof of concept on approach validity, the changes in the parameters set in the different experimental conditions were discussed taking into account the results of the nonlinear time series analysis. The results confirm the possibility with a single model to identify a variety of flow regimes generated in two-phase microfluidic processes, independently of how the processes have been generated, no directed relations with the input flow rate used are in the model.     

A finite volume scheme for two-phase compressible flows

In gas-particle two-phase flows, when the concentration of the disperesed phase is low, certain assumptions may be made which simplify considerably the equations one has to solve. The gas and particle flows are then linked only via the interaction terms. One may therefore uncouple the full system of equations into two subsystems: one for the gas phase, whose homogeneous part reduces to the Euler equations; and a second system for the particle motion, whose homogeneous part is a degenerate hyperbolic system. The equations governing the gas phase flow may be solved using a high-resolution scheme, while the equations describing the motion of the dispersed phase are treated by a donor-cell method using the solution of a particular Riemann problem. Coupling is then achieved via the right-hand-side terms. To illustrate the capabilities of the proposed method, results are presented for a case specially chosen from among the most difficult to handle, since it involves certain geometrical difficulties, the treatment of regions in which particles are absent and the capturing of particle fronts.     

Large eddy simulation of homogeneous shear flows with several subgrid‐scale models

In this article, large eddy simulation is used to simulate homogeneous shear flows. The spatial discretization is accomplished by the spectral collocation method and a third‐order Runge–Kutta method is used to integrate the time‐dependent terms. For the estimation of the subgrid‐scale stress tensor, the Smagorinsky model, the dynamic model, the scale‐similarity model and the mixed model are used. Their predicting performance for homogeneous shear flow is compared accordingly. The initial Reynolds number varies from 33 to 99 and the initial shear number is 2. Evolution of the turbulent kinetic energy, the growth rate, the anisotropy component and the subgrid‐scale dissipation rate is presented. In addition, the performance of several filters is examined. Copyright © 2005 John Wiley & Sons, Ltd.     

A probabilistic method for field solutions in two-phase flows

Solid-particle motion and related transport phenomena in two-phase flow are fluctuating processes in space and time. A deterministic method can describe only partially the intrinsic physics of these processes. In this paper, the fluctuations of the flow parameters are modelled by considering the spatial correlations, and a probabilistic computational method for two-phase flow is presented. The probabilistic governing equations have been discretized in space using a finite volume method, and then solved by applying the Neumann expansion method. This last method is time efficient, and its convergence can be guaranteed even for large fluctuations. A liquid-solid particle mixture flow in a circular pipe is taken as an example. Computational results illustrate the merit of the probabilistic approach for the prediction of two-phase flow phenomena.     

The fast procedure for predicting transient subcooled two-phase flows

This paper describes a method of predicting transient, two-phase flows in channels, and presents predictions for several problems. The model is based on a Lagrangian drift-flux formulation of the equations of mass and energy in which the liquid phase can be subcooled. The advantage of the present model over previous models lies in the solution technique, which yields accurate solutions very inexpensively and without problems related to stability. In the technique used, analytical solutions to the differential equations that are valid over limited time and space intervals are used to construct the global solution. The example problems include subcooled boiling, flow reversals and blowdown transients.     

Finite-particle-size effects in disperse two-phase flows

Three aspects of the finite radius of spherical particles in disperse two-phase flows are described. The first one is the relation between the exact volume fraction and the widely used approximation nv (n is the particle number density and v is the particle volume). The approximation affects the behavior of the effective equations at short wavelengths with possible consequences on stability and hyperbolicity. Secondly, the dilute theory of inviscid suspensions is corrected retaining the next leading order in the particle size and an application of this result to the linear problem is described. Thirdly, it is shown how several important properties of suspensions such as effective thermal conductivity and viscosity depend on the subtle effect of translation of the average fields over distances of the order of the particle size.This work has been supported by DOE and NSF under Grants DE-FG02-89ER14043 and CBT-8918144, respectively.     

LDA signal discrimination in two-phase flows

The results of an investigation into the application of LDA for measurement of two-phase flows are presented. It is determined that an important aspect of such measurement is the discrimination between the signals originating from different phases in the flow. It is shown that in the regions where the number density of the discrete phase is high, the error could become considerable. Existing methods of discrimination are investigated and shown to be inadequate. A novel method of signal discrimination is presented and is shown to be capable of complete removal of this error. A two-component LDA system is modified to incorporate the discriminator. Experimental data are presented in support of the method which are in good agreement with the calculated results.     

Improved THINC/SW scheme for computing incompressible two-phase flows

In this paper, we propose an improved tangent of hyperbola for interface capturing with slope weighting (THINC/SW) scheme for computing incompressible two-phase flows with surface tension on fixed Eulerian grids. The new scheme possesses the following major new properties in comparison with the original THINC/SW scheme: (i) providing a simple and accurate approach for generating a smooth level set (LS) field from the discontinuous volume-of-fluid field, (ii) determining the interface slope from the cogenerated LS field, and (iii) evaluating the surface tension force by the cogenerated LS field. We verified the proposed scheme with the widely used advection benchmark tests and multifluid simulations. Numerical results reveal that the new scheme can not only improve the solution quality of interface capturing but also increase the accuracy of curvature estimates and suppress the parasitic currents.     

Numerical investigations of the XFEM for solving two-phase incompressible flows

This paper discusses the application of the extended finite element method (XFEM) to solve two-phase incompressible flows. The Navier–Stokes equations are discretised using the Taylor–Hood finite element. To capture the different discontinuities across the interface, kink or jump enrichments are used for the velocity and/or pressure fields. However, these enrichments may lead to an inappropriate combination of interpolations. Different polynomial enrichment orders and different enrichment functions are investigated; only the stable combination will be used afterward.

In cases with a surface tension force, the accuracy mainly relies on the precise computation of the normal and curvature. A novel method for computing normal vectors to the interface is proposed. This method employs successive mesh refinements inside the cut elements. Comparisons with analytical and numerical solutions demonstrate that the method is effective. Moreover, the mesh refinement improves the sub-integration in the XFEM and allows for a precise re-initialisation procedure.     

On the surface equations in two-phase flows and reacting single-phase flows

This paper summarizes the mathematical surface equations which are useful in two-phase flows and single-phase reacting flows. The connection between the interfacial area concentration transport equation for two-phase flows and the flame surface density transport equation for turbulent reacting flows is established. Several analytical examples are given to clarify the physical significance of the different quantities involved in the different transport equations. An introduction to the mathematical treatment of anisotropic interfaces is also given. This theory is illustrated on two different numerical examples: a single inclusion in a simple shear and a single inclusion in an uni-axial elongation.     

Improved subgrid scale model for dense turbulent solid-liquid two-phase flows

The dense solid-phase governing equations for two-phase flows are obtained by using the kinetic theory of gas molecules. Assuming that the solid-phase velocity distributions obey the Maxwell equations, the collision term for particles under dense two-phase flow conditions is also derived. In comparison with the governing equations of a dilute two-phase flow, the solid-particle‘s governing equations are developed for a dense turbulent solid-liquid flow by adopting some relevant terms from the dilute two-phase governing equations. Based on Cauchy-Helmholtz theorem and Smagorinsky model, a second-order dynamic sub-grid-scale (SGS) model, in which the sub-grid-scale stress is a function of both the strain-rate tensor and the rotation-rate tensor, is proposed to model the two-phase governing equations by applying dimension analyses. Applying the SIMPLEC algorithm and staggering grid system to the two-phase discretized governing equations and employing the slip boundary conditions on the walls, the velocity and pressure fields, and the volumetric concentration are calculated. The simulation results are in a fairly good agreement with experimental data in two operating cases in a conduit with a rectangular cross-section and these comparisons imply that these models are practical.