Large eddy simulation of droplet dispersion for inhomogeneous turbulent wall flow

A large eddy simulation (LES) coupled with a Lagrangian stochastic model has been applied to the study of droplet dispersion in a turbulent boundary layer. Droplets are tracked in a Lagrangian way. The velocity of the fluid particle along the droplet trajectory is considered to have a large-scale part and a small-scale part given by a modified three-dimensional Langevin model using the filtered subgrid scale (SGS) statistics. An appropriate Lagrangian correlation timescale is considered in order to include the influences of gravity and inertia. Two-way coupling is also taken into account. The inter-droplet collision has been introduced as the main mechanism of secondary breakup. A stochastic model for breakup has been generalized for coalescence simulation, thereby two phenomena, coalescence and breakup are simulated in the framework of a single stochastic model. The parameters of this model, selectively for coalescence and for breakup, are computed dynamically by relating them to the local resolved properties of the dispersed phase compared to the main fluid. The model is validated by comparison with an agglomeration model and with experimental results on secondary breakup. The LES coupled with Lagrangian particle tracking and the model for droplet coalescence and breakup is applied to the study of the atmospheric dispersion of wet cooling tower plumes. The simulations are done for different droplet size distributions and volume fractions. We focused on the influence of these parameters on mean concentration, concentration variance and mass flux profiles.     

Modeling of binary droplet collisions for application to inter-impingement sprays

In this paper, we focused on modeling the collision phenomenon between two liquid droplets for application in spray simulations. It has been known that the existing O’Rourke collision model widely used in CFD codes is inaccurate in determining collision outcomes and droplet behavior. In addition, since the collision probability of the model follows a statistical approach involving computational cell geometry, the prediction results should be strongly dependent on the cell size. As a result, to more accurately calculate droplet collisions, the technique for predicting the droplet velocity and its direction after collision must be extended for use in spray modeling. Further, it is also necessary to consider all the possible collision outcomes, such as bouncing, stretching separation, reflexive separation and coalescence. Therefore, this paper describes the appropriateness of a composite concept for modeling collision outcomes and the implementation of deterministic collision algorithms into a multidimensional CFD code for the calculation of post-collisional droplet movements. Furthermore, the existing model does not consider the formation of satellite droplets. For this reason, our present modeling concept includes a fragmenting droplet collision model. Using the present model, we have validated the collision interactions between liquid droplets under high Weber number conditions by comparing our calculations with experimental results from a binary droplet collision. This paper also deals with the application of the model to inter-impingement sprays by analyzing the atomization characteristics, such as mean droplet size and velocity, spray tip penetrations and spray-shapes of the impinging spray using the suggested collision algorithms and then comparing the results with available experimental data.     

General solution for interaction of solitary waves including head-on collisions

A general solution of the Boussinesq equation is presented which solves the problem of interaction of any number of right-going and left-going solitary waves. The solution relies on the exact solution of Gardner, Greene, Kruskal, and Miura (1967), and has the same degree of accuracy as that solution, but has a wider scope of application. It is much simpler than, but as accurate as, Hirota's exact solution (1973) of the Boussinesq equation, to which the present solution is compared for the simplest case of two solitary waves in head-on collision.     

Diffuse interface model of surfactant adsorption onto flat and droplet interfaces

For applications where droplet breakup and surfactant adsorption are strongly coupled, a diffuse interface model is developed. The model is based on a free energy functional, partly adapted from the sharp interface model of [Diamant and Andelman 34(8):575–580, (1996)]. The model is implemented as a 2D Lattice Boltzmann scheme, similar to existing microemulsion models, which are coupled to hydrodynamics. Contrary to these microemulsion models, we can describe realistic adsorption isotherms, such as the Langmuir isotherm. From the free energy, functional analytical expressions of equilibrium properties are derived, which compare reasonably with numerical results. Interfacial tension lowering scales with the logarithm of the area fraction of the interface unloaded with a surfactant: . Furthermore, we show that adsorption kinetics are close to the classical relations of Ward and Tordai. Prelimary simulations of droplets in shear flow show promising results, with surfactants migrating to interfacial regions with highest curvature. We conclude that our diffuse interface model is very promising for apprehending the above-mentioned applications as membrane emulsification. Paper presented at the Annual European Rheology Conference (AERC) 2005, April 21–23, Grenoble, France.     

A model for droplet entrainment in heated annular flow

The ability to accurately predict droplet entrainment in annular two-phase flow is required to effectively calculate the interfacial mass, momentum, and energy transfer, which characterizes nuclear reactor safety, system design, analysis, and performance. Most annular flow entrainment models in the open literature are formulated in terms of dimensionless groups, which do not directly account for interfacial instabilities. However, many researchers agree that there is a clear presence of interfacial instability phenomena having a direct impact on droplet entrainment. The present study proposes a model for droplet entrainment, based on the underlying physics of droplet entrainment from upward co-current annular film flow that is characteristic to light water reactor safety analysis. The model is developed based on a force balance and stability analysis that can be implemented into a transient three-field (continuous liquid, droplet, and vapor) two-phase heat transfer and fluid flow systems analysis computer code.     

A capillary force model for interactions between two spheres

Based on the method of energy principle, an analytical approach for computing the capillary force for sphere/sphere geometry is presented in this paper. In modeling the capillary force, we consider spheres with both equal and non-equal radii, for both symmetric and asymmetric configurations at liquid/solid interfaces. We use numerical analysis to investigate the validity and efficiency of the derived model. The effect of various parameters including humidity, distance between two spheres, radii of spheres and contact angles on the meniscus force are investigated. Finally the results obtained from the model are compared with experimental measurements, and the accuracy and precision of the presented approach is verified.     

A numerical model for the nonlinear interaction of elastic waves with cracks

Microstructures such as cracks and microfractures play a significant role in the nonlinear interactions of elastic waves, but the precise mechanism of why and how this works is less clear. Here we simulate wave propagation to understand these mechanisms, complementing existing theoretical and experimental works.We implement two models, one of homogeneous nonlinear elasticity and one of perturbations to cracks, and then use these models to improve our understanding of the relative importance of cracks and intrinsic nonlinearity. We find, by modeling the perturbations in the speed of a low-amplitude P-wave caused by the propagation of a large-amplitude S-wave that the nonlinear interactions of P- and S-waves with cracks are significant when the particle motion is aligned with the normal to the crack face, resulting in a larger magnitude crack dilation. This improves our understanding of the relationship between microstructure orientations and nonlinear wave interactions to allow for better characterization of fractures for analyzing processes including earthquake response, reservoir properties, and non-destructive testing.     

A new fully non‐hydrostatic 3D free surface flow model for water wave motions

A new fully non‐hydrostatic model is presented by simulating three‐dimensional free surface flow on a vertical boundary‐fitted coordinate system. A projection method, known as pressure correction technique, is employed to solve the incompressible Euler equations. A new grid arrangement is proposed under a horizontal Cartesian grid framework and vertical boundary‐fitted coordinate system. The resulting model is relatively simple. Moreover, the discretized Poisson equation for pressure correction is symmetric and positive definite, and thus it can be solved effectively by the preconditioned conjugate gradient method. Several test cases of surface wave motion are used to demonstrate the capabilities and numerical stability of the model. Comparisons between numerical results and analytical or experimental data are presented. It is shown that the proposed model could accurately and effectively resolve the motion of short waves with only two layers, where wave shoaling, nonlinearity, dispersion, refraction, and diffraction phenomena occur. Copyright © 2010 John Wiley & Sons, Ltd.     

Modeling droplet dispersion and interphase turbulent kinetic energy transfer using a new dual-timescale Langevin model

Dispersion of spray droplets and the modulation of turbulence in the ambient gas by the dispersing droplets are two coupled phenomena that are closely linked to the evolution of global spray characteristics, such as the spreading rate of the spray and the spray cone angle. Direct numerical simulations (DNS) of turbulent gas flows laden with sub-Kolmogorov size particles, in the absence of gravity, report that dispersion statistics and turbulent kinetic energy (TKE) evolve on different timescales. Furthermore, each timescale behaves differently with Stokes number, a non-dimensional flow parameter (defined in this context as the ratio of the particle response time to the Kolmogorov timescale of turbulence) that characterizes how quickly a particle responds to turbulent fluctuations in the carrier or gas phase. A new dual-timescale Langevin model (DLM) composed of two coupled Langevin equations for the fluctuating velocities, one for each phase, is proposed. This model possesses a unique feature that the implied TKE and velocity autocorrelation in each phase evolve on different timescales. Consequently, this model has the capability of simultaneously predicting the disparate Stokes number trends in the evolution of dispersion statistics, such as velocity autocorrelations, and TKE in each phase. Predictions of dispersion statistics and TKE from the new model show good agreement with published DNS of non-evaporating and evaporating droplet-laden turbulent flow.     

Design and mechanical characteristics analysis of a new viscous damper for piping system

In this paper, the design of a new viscous damper is presented and its mechanical characteristics are investigated experimentally. The motion equation of a system consisting of a drop machine and the damper is set up. By numerically simulating this equation, the curve of the damper cavity generatrix is obtained on the assumption that the resisting force is constant. Then the new damper with big capacity and high-energy dissipation rate is designed. Drop tests using this damper and a Pro225-damper bought in the market are performed, respectively. On one hand, the experimental resisting forces of the new damper approximate constants, which illustrates that the simulation is viable. On the other hand, some advantages of the new damper over the Pro225-damper are found.     

A predictive model for impact response of viscoelastic polymers in drop tests

We show for the first time that a classical Hookean viscoelastic constitutive law for rubbery materials can predict the impact forces and deflections measured with a commercial drop tester when a mass, or tup with a flat impacting surface is dropped onto a flat pad of commercial impact-absorbing rubber. The viscoelastic properties of the rubber, namely the relaxation times and strengths, are obtained by a standard rheological linear-oscillatory test, and the equation of momentum transfer is then solved, using these measured parameters, assuming a uniaxial deflection of the pad during the impact. Good agreement between measured and predicted forces and deflections is obtained for a series of various drop heights, tup masses, impact areas, and pad thicknesses, as long as the deflection of the pad relative to its thickness is small or modest (<50% or so), and as long as the area of the pad is less than or equal to that of the tup. When the pad area is greater than the tup, forces are higher than predicted, unless an empirical factor is introduced to account for the nonuniaxial stretching of the ring of material that extends outside of the impact area. These results imply that the impact-absorbing properties of a rubbery polymeric material can be assessed by simply examining the material's linear viscoelastic spectrum.     

Continuous thermodynamics for droplet vaporization: Comparison between Gamma-PDF model and QMoM

The Continuous Thermodynamics Model (CTM) (Cotterman et al., 1985) is a suitable method to reduce computational cost of multi-component vaporization models. The droplet composition is described by a probability density function (PDF) rather than tens of components in the classical Discrete Component Model (DCM). In the first CTM method developed for this application, the PDF was assumed to be a Γ-function (Hallett, 2000), but some problems had appeared in the case of vapor condensation at the droplet surface (Harstadt et al., 2003). The method put forward in this article, the Quadrature Method of Moments (QMoM), enables one to avoid any assumption on the PDF mathematical form. Following Lage who has developed this method for phase equilibria (Lage, 2007), this article widens the scope of QMoM to the modelling of multi-component droplet vaporization. The different CTM approaches are presented in the first part and the results obtained for a vapor condensation test case are then compared and analysed to illustrate improvements made by QMoM. To cite this article: C. Laurent et al., C. R. Mecanique 337 (2009).     

A predictive model for pressure fluctuations on sloping channels using support vector machine

In this study, support vector machine (SVM) is proposed as a new predictive model for pressure fluctuations beneath free jump occurring on sloping channels. The proposed model reproduces the pressure fluctuation intensity C′_p in terms of normalized flow and channel section characteristics. The model variables were derived from dimensional analysis. The proposed model is calibrated and validated by using a wide range of experimental data. The SVM predicted C′_p with a correlation of coefficient (CC) of 0.989 and a root mean square error (RMSE) of 0.004. Also, linear and nonlinear regression analyses are applied on the same experimental data set, and the SVM model is compared to the equations obtained from these regression analyses. CC, RMSE and average absolute deviation (D_i) are used in the evaluation of performance of each model. The SVM model predicted the measured pressure fluctuations better than conventional regression equations. The results of this study reveal that the proposed SVM model can be effectively used in predicting the pressure fluctuation beneath free jump. Copyright © 2010 John Wiley & Sons, Ltd.     

A simple and explicit algebraic expression for the Rayleigh wave velocity

In this note a factorization technique based on the theory of the Riemann problem is used to derive a compact algebraic formula for the velocity of Rayleigh waves. Unlike previous results based on rationalization and Cardan’s solution of a cubic, the present formulation leads to a formula for the velocity which is a continuous function of the Poission’s ratio and yet is simple enough to be of practical interest. The new formula also enables us to express the complex roots associated with the Rayleigh wave equation as simple functions of the Rayleigh wave velocity.     

A new microstructure-based constitutive model for human blood

A new constitutive equation for whole human blood is derived using ideas drawn from temporary polymer network theory to model the aggregation and disaggregation of erythrocytes in normal human blood at different shear rates. Each erythrocyte is represented by a dumbbell. The use of a linear spring law in the dumbbells leads to a multi-mode generalized Maxwell equation for the elastic stress and both the relaxation times and viscosities are functions of a time-dependent structure variable. An approximate constitutive equation is derived by choosing a single mode corresponding to the cell aggregate size where the largest number of cells are to be found. This size is identified in the case of steady flows. The model exhibits shear-thinning, viscoelasticity and thixotropy and these are clearly related to the microstructural properties of the fluid. Agreement with the experimental data of Bureau et al. [M. Bureau, J.C. Healy, D. Bourgoin, M. Joly, Rheological hysteresis of blood at low shear rate, Biorheology 17 (1980) 191–203] in the case of a simple triangular step shear rate flow is convincing.     

A new matrix dissipation model for central scheme

In this paper, a simple and efficient improvement to the famous Swanson–Turkel matrix dissipation model for the central scheme is proposed. In the new matrix dissipation model, the accuracy is improved by eliminating the second‐difference dissipation added to the characteristic fields representing the vorticity waves. This strategy is proposed based on analyzing the flow‐physics about shock‐vortex interaction using the Rankine–Hugoniot jump condition. In this paper, the behavior of central scheme for rotational flow is also theoretically and numerically analyzed. Results show a newfound problem of the original scalar and matrix dissipation models, in which for rotational flow excessive second‐difference dissipation is added due to the pressure‐based shock sensor. With current new matrix dissipation model improved accuracy is obtained at minimal cost overhead, especially, in the highly vortical region where the second‐difference dissipation is reduced. At the same time, it preserves the excellent shock capturing capability and convergence speed of original method. Numerical properties of this new matrix dissipation model are validated with a series of numerical experiments and results comparison with original model verifies improved performance of current method. Copyright © 2013 John Wiley & Sons, Ltd.