Volume averaging for the analysis of turbulent spray flows

Spray flow calculations are usually based upon equations that have been developed by averaging droplet properties locally throughout the flow field. Presently, standard procedure for LES (large-eddy simulations) is to average these averaged equations once again to filter the short-length-scale fluctuations. In this paper, the theoretical foundations for the averaged spray equations are examined; then the volume-averaging process for LES and the volume-averaging process for two-phase flows are unified for the analysis of turbulent, two-phase flows. Comments are provided on the relationship between the averaging volume and the computational-cell volume. This paper provides generality to the weighting-function choice in the averaging process and precision to the definition of the volume over which the averaging is performed. New flux terms that result from the averaging process and appear in the governing averaged partial differential equations are identified and their modelling is discussed. Situations are identified where sufficient stratification of properties on the scale smaller than the averaging volume leads to the significance of these quantities. Evolution equations for averaged entropy and averaged vorticity are developed. The relationship amongst the curl of the average gas-phase velocity, the average of the gas-phase-velocity curl, and the rotation of the discrete droplets or particles is established. The needs and challenges for sub-grid modelling to account for small-vortex/droplet interactions are presented. Applications to spray combustion are discussed.     

An improved large eddy simulation of two-phase flows in a pump impeller

An improved large eddy simulation using a dynamic second-order sub-grid-scale (SGS) stress model has been developed to model the governing equations of dense turbulent particle-liquid two-phase flows in a rotating coordinate system, and continuity is conserved by a mass-weighted method to solve the filtered governing equations. In the current second-order SGS model, the SGS stress is a function of both the resolved strain-rate and rotation-rate tensors, and the model parameters are obtained from the dimensional consistency and the invariants of the strain-rate and the rotation-rate tensors. In the numerical calculation, the finite volume method is used to discretize the governing equations with a staggered grid system. The SIMPLEC algorithm is applied for the solution of the discretized governing equations. Body-fitted coordinates are used to simulate the two-phase flows in complex geometries. Finally the second-order dynamic SGS model is successfully applied to simulate the dense turbulent particle-liquid two-phase flows in a centrifugal impeller. The predicted pressure and velocity distributions are in good agreement with experimental results. The project supported by the National Natural Science Foundation of China (50779069 and 90510007), the Start-up Scientific Research Foundation of China Agricultural University (2006021) and the Beijing Natural Science Foundation (3071002).     

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.     

视密度加权的时平均统一二阶矩两相湍流模型

在气粒两相湍流的双流体模型中,颗粒相的视（表观）密度是有脉动的,在时平均的统一二阶矩（USM）模型中出现了和颗粒数密度或视密度脉动有关的项和方程,使模型方程比较复杂。实际上,用LDV或PDPA测量的流体（用小颗粒代表）和颗粒速度都是颗粒数加权平均的结果。因此,在视密度加权平均基础上推导两相湍流模型更为合理。通过推导和封闭了视密度加权平均的统一二阶矩模型（MUSM）方程组,改进了两相速度脉动关联的封闭,并引入了颗粒遇到的气体脉动速度及其输运方程。MUSM模型可以减少所用方程数,节省计算量。视密度加权平均的统一二阶矩两相湍流模型是一种对原有时间平均的统一二阶矩模型和改进和发展。     

NUMERICAL PREDICTION OF TWO-PHASE FLOW IN BUBBLE COLUMNS

A numerical model is described for the prediction of turbulent continuum equations for two-phase gas–liquid flows in bubble columns. The mathematical formulation is based on the solution of each phase. The two-phase model incorporates interfacial models of momentum transfer to account for the effects of virtual mass, lift, drag and pressure discontinuities at the gas–liquid interface. Turbulence is represented by means of a two-equation k–ϵ model modified to account for bubble-induced turbulence production. The numerical discretization is based on a staggered finite-volume approach, and the coupled equations are solved in a segregated manner using the IPSA method. The model is implemented generally in the multipurpose PHOENICS computer code, although the present appllications are restricted to two-dimensional flows. The model is applied to simulate two bubble column geometries and the predictions are compared with the measured circulation patterns and void fraction distributions.     

Description of local dilatancy and local rotation of granular assemblies by microstretch modeling

This study investigates the microstretch continuum modeling of granular assemblies while accounting for both the dilatant and rotational degrees of freedom of a macroelement. By introducing the solid volume fraction and the gyration radius of a granular system, the balance equations of the microstretch continuum are transformed into a new formulation of evolution equations comprising six variables: the solid volume fraction, the gyration radius, the velocity field, the averaged angular velocity, the rate of gyration radius, and the internal energy. The bulk microinertia density, the averaged angular velocity, and the microgyration tensor at a macroscopic point are obtained in terms of discrete physical quantities. The bulk part and the rotational part of the microgyration tensor are proposed as the two indices to measure the local dilatancy and local rotation of granular assemblies. It is demonstrated in the numerical simulation that the two indices can be used to identify the shear band evolution in a granular system under a biaxial compression.     

Numerical prediction of droplet dynamics in turbulent flow,using the level set method

In the present article, the droplet dynamics in turbulent flow is numerically predicted. The modelling is based on an interfacial marker-level set (IMLS) method, coupled with the Reynolds-averaged Navier–Stokes (RANS) equations to predict the dynamics of turbulent two-phase flow. The governing equations for time-dependent, two-dimensional and incompressible two-phase flow are described in both phases and solved separately using a control volume approach on structured cell-centred collocated grids. The topological changes of the interface are predicted by applying the level set approach. The kinematic and dynamic conditions on the interface separating the two phases are satisfied. The numerical method proposed is validated against a well-known computational fluid dynamics problem. Further, the deformation and breakup of a single droplet either suddenly moved in air or exposed to turbulent stream are numerically investigated. In general, the developed numerical method demonstrates remarkable capability in predicting the characteristics of complex turbulent two-phase flows.     

Properties and Averaged Equations for Flows of Bubbly Liquids

Averaged properties of bubbly liquids in the limit of large Reynolds and small Weber numbers are determined as functions of the volume fraction, mean relative velocity, and velocity variance of the bubbles using numerical simulations and a pair interaction theory. The results of simulations are combined with those obtained recently for sheared bubbly liquids [19] and the mixture momentum and continuity equations to propose a complete set of averaged equations and closure relations for the flows of bubbly liquids at large Reynolds and small Weber numbers.     

Investigation of the stability of a plane-channel suspension flow with account for finite particle volume fraction

Within the framework of the two-fluid approach, a variant of a heterogeneous-medium model which takes into account a finite volume fraction of the inclusions and a small but finite phase velocity slip is proposed. The interphase momentum exchange is described by the Stokes force with the Brinkman correction for the finite particle volume fraction. The suspension viscosity depends on the particle volume fraction in accordance with the Einstein formula. Within the framework of the model constructed, a formulation of the problem of linear stability of plane-parallel two-phase flows is proposed. As an example, the stability of a channel suspension flow is considered. The system of equations for small disturbances with the boundary conditions is reduced to an eigenvalue problem for a fourth-order ordinary differential equation. Using the orthogonalization method, the dependence of the critical Reynolds number on the governing nondimensional parameters of the problem is studied numerically. It is shown that taking a finite volume fraction of the inclusions into account significantly affects the laminar-turbulent transition limit.     

Evaluation de la fraction volumique locale de la phase solide dans un écoulement diphasique

Local particle volume fraction measurements in two-phase flows are rare. Generally, the concentration is supposed to be spatially homogeneous in the sedimentation flows fundamental experiments. This is far to be realistic in regimes different from Stokes' regime, particularly near the walls. This paper compares two methods of evaluation of the volume fraction in a three-dimensional two-phase flow.     

An efficient ψ-v scheme for two-dimensional laminar flow past bluff bodies on compact nonuniform grids

We recently proposed a second-order accurate ψ-v formulation of the steady-state Navier-Stokes (N-S) equations on compact Cartesian nonuniform grids. In the current work, we extend the ideas of the aforesaid formulation and propose a second-order spatially compact, implicit, stable ψ-v formulation for the unsteady incompressible N-S equations. Contrary to the existing ψ-v finite difference formulations which use grid transformation, the proposed scheme is developed for nonuniform Cartesian grids without transformation specifically designed for two-dimensional laminar flow past bluff bodies. It has been implemented on problems of internal flows inside curved regions as well as those involving fluid-embedded body interaction. However, the robustness of the scheme is highlighted by the accurate resolution of a host of complex flows past bluff bodies with different physical set-ups and boundary conditions. It was seen to handle problems involving both uniform and accelerated flows across a wide range of structures of varied shape, namely, a flat plate, a circular cylinder, inclined square cylinder, and a wedge in channel hinged to the wall. Apart from elegantly capturing all the details of the shedded vortex structures under different circumstances, the scheme was also able to handle both Dirichlet and Neumann boundary with equal ease. In all the cases, our results are found to be extremely close to the available numerical and experimental results.     

A thermodynamic model of turbulent motions in a granular material

This paper is devoted to a thermodynamic theory of granular materials subjected to slow frictional as well as rapid flows with strong collisional interactions. The microstructure of the material is taken into account by considering the solid volume fraction as a basic field. This variable is of a kinematic nature and enters the formulation via the balance law of the configurational momentum, including corresponding contributions to the energy balance, as originally proposed by Goodman and Cowin [1], but modified here. Complemented by constitutive equations, the emerging field equations are postulated to be adequate for motions, be they laminar or turbulent, if the resolved length scales are sufficiently small. On large length scales the sub-grid motion may be interpreted as fluctuations, which manifest themselves in correspondingly filtered equations as correlation products, like in the turbulence theory. We apply an ergodic (Reynolds) filter to these equations and thus deduce averaged equations for the mean motions. The averaged equations comprise balances of mass, linear and configurational momenta, energy, and turbulent kinetic energy as well as turbulent configurational kinetic energy. They are complemented by balance laws for two internal fields, the dissipation rates of the turbulent kinetic energy and of the turbulent configurational kinetic energy. We formulate closure relations for the averages of the laminar constitutive quantities and for the correlation terms by using the rules of material and turbulent objectivity, including equipresence. Many versions of the second law of thermodynamics are known in the literature. We follow the Müller-Liu theory and extend Müllers entropy principle to allow the satisfaction of the second law of thermodynamics for both laminar and turbulent motions. Its exploitation, performed in the spirit of the Müller-Liu theory, delivers restrictions on the dependent constitutive quantities (through the Liu equations) and a residual inequality, from which thermodynamic equilibrium properties are deduced. Finally, linear relationships are proposed for the nonequilibrium closure relations.Received: 21 March 2003, Accepted: 1 September 2003, Published online: 11 February 2004PACS: 05.70.Ln, 61.25.Hq, 61.30.-vCorrespondence to: I. Luca     

Fundamental equations of turbulent two-phase flow

A new set of Reynolds equations for predicting turbulent two-phase flows has been developed by means of Reynolds averaging method on the unsteady laminar equations of two-phase flow. These equations involve average terms of products of turbulent fluctuations in some physical parameters in a large degree. The interaction forces between two phases, the pressures for dispersed phase, extra stresses except for pressure and the expressions for energy interchange between two phases have been discussed.     

EFFECT OF EMPIRICAL COEFFICIENTS ON SIMULATION IN TWO-SCALE SECOND-ORDER MOMENT PARTICLE-PHASE TURBULENCE MODEL

A two-scale second-order moment two-phase turbulence model accounting for inter-particle collision is developed, based on the concept of particle large-scale fluctuation due to turbulence and particle small-scale fluctuation due to collision. The proposed model is used to simulate gas-particle downer reactor flows. The computational results of both particle volume fraction and mean velocity are in agreement with the experimental results. After analyzing effects of empirical coefficient on prediction results, we can come to a conclusion that, inside the limit range of empirical coefficient, the predictions do not reveal a large sensitivity to the empirical coefficient in the downer reactor, but a relatively great change of the constants has important effect on the prediction.     

Modeling of Axisymmetric Two-phase Dilute Flows

This paper deals with the numerical treatment of Eulerian approach for dilute two-phase compressible flows (gas-particles mixtures) in axisymmetric configurations. For dilute flows, two classes of models depending on the dispersed phase volumetric fraction can be found. The volume occupied by the particles may be considered, that yields a model in which the gas phase and the dispersed phase equations are coupled through the void fraction and the source terms (Delhaye model). The void fraction effects can be neglected, that means the gas phase is a carrier phase for the particles (Ishii model). The mathematical nature of the two models is demonstrated from analysis of characteristic directions. For the Delhaye's model, a centered scheme is used to solve the system of partial differential equations, while an upwind TVD scheme is used for the Ishii's one. Then, it is shown that the problem of symmetry boundary conditions does not depend on the physical approach, as long as the flow remains dilute. However, a classical treatment for symmetry boundary conditions at the geometrical axis leads to large errors. A particular treatment for this boundary is presented: a new class of particles, described by a supplementary system of equations, is required.     

A combination of parabolized Navier–Stokes equations and level-set method for stratified two-phase internal flow

A novel methodology is proposed for the numerical computation of pressure-driven gravity-stratified flows along channels comprising two immiscible phases. The parabolized Navier–Stokes equations are combined with the level set approach, resulting into a downstream-marching problem in which the solution is computed at each cross-section based on upstream information only. A main difficulty in the implementation of the approach for internal flows is the conservation of the mass flow rates, which is addressed by extending to two-phase flows the method proposed by Patankar and Spalding (1972) and Raythby and Schneider (1979), and by adding an explicit forcing term in the equation for the advection of the level function. The combination of high-order finite differences and sparse storage and algebra used here allows a fully-coupled integration of the parabolized equations, as opposed to the more classical segregated approaches. This enables a very efficient calculation of the complete downstream-developing flow field.     

On basic equations of turbulent swirling gas-solid flows and their application in cyclones

The basic equations of turbulent gas-solid flows are derived by using the pseudo-fluid model of particle phase with a refined two-phase turbulence model. These equations are then applied to swirling gas-particle flows for analyzing the collection efficiency in cyclone separators.     

Finite element implementation of a generalised non-local rate-dependent crystallographic formulation for finite strains

This work describes the finite element implementation of a generalised strain gradient and rate-dependent crystallographic formulation for finite strains and general anisothermal conditions based on a multiplicative decomposition of the deformation gradient. The implementation involved the development of both a novel finite element formulation to determine the spatial slip rate gradients at each material point, and an implicit numerical integration scheme at the constitutive level to update the stresses and solution dependent variables. The time-integration procedure uses a Newton–Raphson scheme with a single level of iteration to solve the incremental non-linear equations associated with the non-local constitutive formulation. Closed-form solutions for the relevant fourth-order Jacobian tensors are given. The proposed numerical scheme is formulated in a general form and hence should be applicable to most existing crystallographic models. The crystallographic formulation is then used to investigate the effect of the morphology and volume fraction of the reinforcing phase of a two-phase single crystal on its macroscopic behaviour.