Possibilities and Limitations of Computer Simulations of Industrial Turbulent Dispersed Multiphase Flows

We give an overview on the usage of computer simulations in industrial turbulent dispersed multiphase flows. We present a few examples of industrial flows: bubble columns and bubbly pipe flows, stirred tanks, cyclones, and a fluid catalytic cracking unit. The fluid catalytic cracking unit is used to illustrate the complexity of the physical phenomena involved, and the possibilities and limitations of the different approaches used: Eulerian–Lagrangian (particle-tracking) and Eulerian–Eulerian (two-fluid). In the first approach, the continuous phase is solved using either RANS simulations (Reynolds-Averaged Navier–Stokes simulations) or DNS/LES (Direct Numerical Simulations/Large-Eddy Simulations), and the individual particles are tracked. In the second approach, the dispersed phase is averaged, leading to two sets equations, which are quite similar to the RANS equations of single-phase flows. The Eulerian–Eulerian approach is the most commonly used in industrial applications, however, it requires a significant amount of modelling. Eulerian–Lagrangian RANS can be simpler to use; in particular in situations involving complex boundary conditions, polydisperse flows and agglomeration/breakup. The key issue for the success of the simulations is to have good models for the complex physics involved. A major weakness is the lack of good models for: the turbulence modification promoted by the particles, the inter-particle interactions, and the near-wall effects. Eulerian–Lagrangian DNS/LES can play an important role as a research tool, in order to get a better physical understanding, and to improve the models used in the RANS simulations (either Eulerian–Eulerian or Eulerian–Lagrangian).     

Symmetry Reduced Dynamics of Charged Molecular Strands

The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are nonlocal when the screened Coulomb interactions, or Lennard–Jones potentials between pairs of charges, are included. The nonlocal dynamics is derived in the convective representation of continuum motion by using modified Euler–Poincaré and Hamilton–Pontryagin variational formulations that illuminate the various approaches within the framework of symmetry reduction of Hamilton’s principle for exact geometric rods. In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods. The motion equations in the convective representation are shown to arise by a classical Lagrangian reduction associated to the symmetry group of the system. This approach uses the process of affine Euler–Poincaré reduction initially developed for complex fluids. On the Hamiltonian side, the Poisson bracket of the molecular strand is obtained by reduction of the canonical symplectic structure on phase space. A change of variables allows a direct passage from this classical point of view to the covariant formulation in terms of Lagrange–Poincaré equations of field theory. In another revealing perspective, the convective representation of the nonlocal equations of molecular strand motion is transformed into quaternionic form.     

Well-Posedness of Surface Wave Equations Above a Viscoelastic Fluid

In this article, we prove the local well-posedness, for arbitrary initial data with certain regularity assumptions, of the equations of a Viscoelastic Fluid of Johnson–Segalman type in a domain with a free surface. Managing more general constitutive laws is also briefly depicted. The 2D geometry is defined by a solid fixed bottom and an upper free boundary submitted to surface tension. The proof relies on a Lagrangian formulation. First we solve two intermediate problems through a fixed point using mainly (Allain in Appl Math Optim 16:37–50, 1987) for the Navier–Stokes part. Then we solve the whole Lagrangian problem on [0, T ₀] for T ₀ small enough through a contraction mapping. Since the Lagrangian solution is regular enough and the change of coordinates invertible, we can come back to an Eulerian one.     

Using mesh-based methods to solve nonlinear problems of statics for thin shells

The paper outlines a numerical procedure for solving physically and geometrically nonlinear problems of statics for thin shells based on three mesh-based methods: finite-difference, variational difference, and finite-element methods. The methodological, algorithmic, and analytical aspects of implementing the Kirchhoff–Love hypotheses are analyzed. The algorithmic approach employs Lagrangian multipliers. The advantages and disadvantages of these methods are evaluated Translated from Prikladnaya Mekhanika, Vol. 45, No. 1, pp. 41–70, January 2009.     

On shock wave propagation in a branched channel with particles

The problem of wave propagation in a dust–air mixture inside a branched channel has not been studied widely in literature, even though this topic has many important applications especially in process safety (dust explosions). In this paper, a shock wave interaction with a cloud of solid particles, and the further behaviour of both gas and particulate phases were studied using numerical techniques. The geometry mimicked a real channel where bends or branches are common. Two numerical approaches were used: Eulerian–Eulerian and Eulerian–Lagrangian. Using Eulerian-Lagrangian simulation, it was possible to include the effects of particle–particle and particle–wall collisions in a realistic and direct manner. Results are mainly shown as snap-shots of particle positions during the simulations and statistics for the particle displacement. The results show that collisions significantly influence the process of particle cloud formation. PACS47.40.Nm, 02.60.Cb, 47.55.kf     

On the dynamics of deployment of an orbital structure with elastic elements

The object to be studied is a spacecraft with a deployable pantograph structure as a solar-battery carrier. The objective of research is to design a mathematical model of this structure taking the elasticity of pantograph elements into account. The Lagrangian formalism is followed. To model the dynamic processes in the system, a software package has been developed, which can be adapted, if necessary, to study deployable structures of other types. The behavior of the structure during deployment, collapse, and redeployment under the action of various perturbations is modeled numerically. Plots illustrating the variation of characteristic variables are presented __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 136–144, August 2006.     

Forms of null Lagrangians in field theories of continuum mechanics

The divergence representation of a null Lagrangian that is regular in a star-shaped domain is used to obtain its general expression containing field gradients of order ≤ 1 in the case of spacetime of arbitrary dimension. It is shown that for a static three-component field in the three-dimensional space, a null Lagrangian can contain up to 15 independent elements in total. The general form of a null Lagrangian in the four-dimensional Minkowski spacetime is obtained (the number of physical field variables is assumed arbitrary). A complete theory of the null Lagrangian for the n-dimensional spacetime manifold (including the four-dimensional Minkowski spacetime as a special case) is given. Null Lagrangians are then used as a basis for solving an important variational problem of an integrating factor. This problem involves searching for factors that depend on the spacetime variables, field variables, and their gradients and, for a given system of partial differential equations, ensure the equality between the scalar product of a vector multiplier by the system vector and some divergence expression for arbitrary field variables and, hence, allow one to formulate a divergence conservation law on solutions to the system.     

Some aspects of solving interrelated problems of ecology and climate

Special features of interrelated problems of ecology and climate are analyzed. The technique proposed for solving this class of problems is demonstrated by an example of evaluating the atmospheric quality and monitoring and predicting the ecological consequences of man’s impact. An approach based on variational principles in combination with methods of splitting and decomposition is developed. The structure of algorithms implementing Eulerian and Lagrangian formulations of the problems is described. Examples of simulation scenarios for particular cases are given. Institute of Computational Mathematics and Mathematical Geophysics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 161–170, September–October, 2000.     

Simulation of bluff body stabilized flows with hybrid RANS and PDF method

The motivation of this study is to investigate the turbulence–chemistry interactions by using probability density function (PDF) method. A consistent hybrid Reynolds Averaged Navier–Stokes (RANS)/PDF method is used to simulate the turbulent non-reacting and reacting flows. The joint fluctuating velocity–frequency–composition PDF equation coupled with the Reynolds averaged density, momentum and energy equations are solved on unstructured meshes by the Lagrangian Monte Carlo (MC) method combined with the finite volume (FV) method. The simulation of the axisymmetric bluff body stabilized non-reacting flow fields is presented in this paper. The calculated length of the recirculation zone is in good agreement with the experimental data. Moreover, the significant change of the flow pattern with the increase of the jet-to-coflow momentum flux ratio is well predicted. In addition, comparisons are made between the joint PDF model and two different Reynolds stress models. The project supported by the National Natural Science Foundation of China (50506028), and Action Scheme for Invigorating Education Towards the twenty-first century.     

Comparison of near- and far-fault ground motion effect on the nonlinear response of dam–reservoir–foundation systems

In this paper, it is aimed to compare the near- and far-fault ground motion effects on the nonlinear dynamic response of dams including dam–reservoir–foundation interaction. Two different types of dams, which are concrete arch and concrete faced rockfill dams, are selected to investigate the near- and far-fault ground motion effects on the dam responses. The behavior of reservoir water is taken into account using Lagrangian approach. The Drucker–Prager material model is employed in nonlinear analyses. Near and far-fault strong ground motion records, which have approximately identical peak ground accelerations, of Loma Prieta (1989) earthquake are selected for the analyses. Displacements, maximum and minimum principal stresses are determined using the finite element method. The displacements and principal stresses obtained from the analyses of dams subjected to each fault effect are compared with each other. It is clearly seen that there is more seismic demand on displacements and stresses when the dam is subjected to near-fault ground motion.     

On the yield functions in Lagrangian stress space of elastic–plastic materials with internal constraints

In the present paper it is shown that the elastic range in the second Piola–Kirchhoff stress space can be chosen in a hyperplane which is through the origin of Lagrangian stress space and perpendicular to the normal of the constraint manifold at the plastic configuration, if the determinate stress response of the elastic–plastic material with simple internal constraints with some condition is correctly chosen, otherwise, it is in general in a hypersurface and the normal flow rule by Il yushin’s postulate will have an indeterminate part. The choice of determinate stress response is probable because of its indeterminacy. Therefore the yield function should be a function of the second Piola–Kirchhoff stress lying in the hyperplane so that it is more simple and the back stress as the geometric center of the elastic range in general is inside the elastic range. Finally some examples are concerned. The project supported by the National Natural Science Foundation of China (10272055).     

Modeling non-equilibrium traffic dynamics in a Lagrangian framework

In this paper, we propose a formulation for modeling macroscopic traffic flow using a modified speed–density relationship. The flow model consists of a nonlinear hyperbolic system of conservation laws. The proposed modification distinguishes between acceleration and deceleration by assuming a different equilibrium velocity for a given traffic density based on whether a platoon of vehicles is accelerating or decelerating. We examine the appropriateness of this modification to two prominent traffic flow models in a Lagrangian reference frame, which we solve computationally. We show that a Lagrangian coordinate system is ideal for the incorporation of the proposed modification due to its ability to track the behavior of moving vehicles. We see that the modification is particularly well suited to “second order” models.     

Static analysis of two-dimensional elastic structures using global collocation

Based on previous findings concerning the numerical solution of one-dimensional elastodynamical problems [Provatidis in Arch Appl Mech 78(4):241–250, 2008] this paper extends the methodology to the static analysis of two-dimensional problems in quadrilateral domains. This target is achieved by replacing the Galerkin/Ritz procedure involved in Lagrangian (or Gordon–Coons) type finite elements by a global collocation scheme. In brief, the boundary conditions are fulfilled at all boundary nodes, while the governing equation is fulfilled at internal points. The theory is supported by four test cases concerning rectangular and curvilinear structures under plane-stress or plane-strain conditions, where the convergence rate is successfully compared with that of conventional bilinear finite elements with the same mesh density.     

A New Lagrangian Asymptotic Solution for Gravity–Capillary Waves in Water of Finite Depth

A third-order Lagrangian asymptotic solution is derived for gravity–capillary waves in water of finite depth. The explicit parametric solution gives the trajectory of a water particle and the wave kinematics for Lagrangian points above the mean water level, and in a water column. The water particle orbits and mass transport velocity as functions of the surface tension are obtained. Some remarkable trajectories may contain one or multiple sub-loops for steep waves and large surface tension. Overall, an increase in surface tension tends to increase the motions of surface particles including the relative horizontal distance travelled by a particle as well as the time-averaged drift velocity     

Evaluation of a Modified Reynolds Stress Model for Turbulent Dispersed Two-Phase Flows Including Two-Way Coupling

A modified Reynolds stress turbulence model for the pressure rate of strain can be derived for dispersed two-phase flows taking into account gas-particle interaction. The transport equations for the Reynolds stresses as well as the equation for the fluctuating pressure can be derived starting from the multiphase Navier–Stokes equations. The unknown pressure rate of strain correlation in the Reynolds stress equations is then modelled by considering the multiphase equation for the fluctuating pressure. This leads to a multiphase pressure rate of strain model. The extra particle interaction source terms occurring in the model for the pressure rate of strain can be constructed easily, with no noticeable extra computational cost. Eulerian–Lagrangian simulation results of a turbulent dispersed two-phase jet are presented to show the differences in results with and without the new two-way coupling terms.     

Influence of geometry on separation efficiency in a hydrocyclone

A numerical study of the gas–liquid–solid multiphase flow in hydrocyclones is presented. Three models of turbulence, the RNG k–ε model, the Reynolds stress model and Large eddy simulation with the volume of fluid model (VOF) multiphase model for simulating air core are compared in order to predict axial and tangential velocity distributions. This presentation is mainly aimed at identifying an optimal method, used to study effective parameters, based on which, eventually, effect of inlet flow rate variations and body dimension variations such as underflow diameter, overflow diameter and cone angle on the separation performance and pressure drop are investigated. The results are then used in the simulation of particle flow described by the stochastic Lagrangian model. The results suggest that the predicted size classifications are approximately similar to those of RSM and LES methods. Predictions using the RSM model are found in agreement with experimental results with a marginal error within the range of 4 to 8%. Proceeding model validation, parametric studies have been carried out concerning the influence of velocity inlet, particle size and body dimension such as underflow and overflow diameter and cone angle. The predictions demonstrate that the flow fields in the hydrocyclones with different sizes and lengths are different, which yields different performances.     

Complexification-averaging method via the averaged Lagrangian

In this paper, the complexification-averaging (CX-A) method for multi-DOF nonlinear vibratory systems is rederived in a new way based upon the averaged Lagrangian. The complex variables are introduced to represent the original displacements and velocities, and then the fast–slow decomposition of the complex variables is made. The time averaging of the Lagrangian over the fast variables is performed. Two different expressions for the kinetic energy are presented, and this results in two schemes for deriving the governing equations of the slow variables. For the scheme I, through the order analysis of the derivatives of the slow variables, it is shown that the second-order terms appeared in the averaged Lagrangian can be omitted, and thus a reduced averaged Lagrangian is obtained. Via the reduced averaged Lagrangian, the corresponding Lagrangian equations are derived. For the scheme II, through time averaging, the averaged Lagrangian is obtained, and then the corresponding equations for the slow variables can be obtained. Finally, two nonlinear vibratory systems with two-DOF and four-DOF, respectively, are given as examples to illustrate the new procedure for the CX-A method. The loci of nonlinear normal modes on the potential surface are studied in the first example, and the frequency-energy plot is investigated in the second example.     

Control of the spatial motions of a gantry manipulator with elastic links

A mathematical dynamic model is proposed for a controlled gantry robot with elastic compliance and inertia distributed along a two-link arm. The model includes a nonlinear system of hybrid differential equations. Kinematic and dynamic control problems for the robot are formulated. The dynamic characteristics of the robot are analyzed in comparison with an equivalent model of a robot manipulator with rigid links based on the Lagrangian formalism __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 121–128, February 2006.     

Numerical modelling of film cooling with and without mist injection

Two-phase CFD calculations, using a Lagrangian model and commercial code Fluent 6.2.16, were employed to calculate the gas and droplet flows and film cooling effectiveness with and without mist on a flat plate. Two different three dimensional geometries are generated and the effects of the geometrical shape, size of droplets, mist concentration in the coolant flow and temperature of mainstream flow for different blowing ratios are studied. A cylindrical and laterally diffused hole with a streamwise angle of 30° and spanwise angle of 0° are used. The diameter of film cooling (d) hole, and the hole length to diameter ratio (L/d) for both of geometries are 10 mm and 4, respectively. Also the blowing ratio ranges from 1.0 to 2.0, and the mainstream Reynolds number based on the mainstream velocity and hole diameter (Re _d) is 6,219. The results are shown for different droplets diameters (1–10 μm), concentrations (1–5%) and mainstream temperatures (350–500 K). The centreline effectiveness and distribution of effectiveness on the surface of cooling wall are presented.     

Lagrangian Covariance Analysis of β-Plane Turbulence

The effects of Rossby wave–turbulence interactions on particle dispersion are investigated in a Lagrangian analysis of the potential vorticity equation. The analysis produces several exact statistical results for fluid particle dispersion in barotropic turbulence on a β-plane. In the inviscid problem the first integral time scale of the meridional velocity is found to be zero, as might occur in pure wave processes, and the meridional particle dispersion is bounded. The second integral time scale, which determines the magnitude of the bound, is shown to depend explicitly on β, the enstrophy and the energy of the meridional velocity. Expressions relating the autocorrelation of the vorticity to the autocorrelation of the meridional velocity are derived and the Lagrangian integral time scale of the relative vorticity is diagnostically related to the meridional velocity correlation. The applicability of these predictions is verified in a series of numerical simulations. For a range of β values, the meridional extent of quasisteady alternating zonally averaged jets occurring in the numerical solutions scales with a length scale given by the the standard deviation of the meridional particle dispersion. Received 8 March 1999 and accepted 12 December 1999