A Statistically Conditioned Averaging Formalism for Deriving Two-Phase Flow Equations†
01 Jan 1990-Particle & Particle Systems Characterization (WILEY‐VCH Verlag GmbH)-Vol. 7, Iss: 4, pp 191-202
TL;DR: In this paper, a statistical formalism overcoming some conceptual and practical difficulties arising in existing two-phase flow (2 PHF) formulations and modelling techniques is introduced. And basic theorems for the case of dispersed 2 PHF are presented.
Abstract: A statistical formalism overcoming some conceptual and practical difficulties arising in existing two-phase flow (2 PHF) formulations and modelling techniques is introduced. Basic theorems for the case of dispersed 2 PHF are presented. Phase interaction terms with a clear physical meaning enter the equations and this formalism provides some guidelines to avoid closure assumptions or to close those terms rationally. Continuous phase averaged continuity, momentum, turbulent kinetic energy and turbulence dissipation rate equations can be rigorously and systematically obtained with this methodology in a single step. These equations display a structure similar to that for single-phase flows. It is also assumed that the dispersed phase is well described by a "Boltzmann-type" equation and Eulerian "continuity", momentum and fluctuating kinetic energy equations for the dispersed phase are obtained. A k-e turbulence model for the continuous phase is used. A gradient transport model is adopted for the dispersed phase fluctuating fluxes of momentum and kinetic energy. Closure assumptions are proposed for the phase interaction terms. This model is then used to predict the behaviour of an axisymmetric turbulent jet of air laden with solid particles varying in sizes and concentrations. Numerical results compare reasonably well with available experimental data.
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TL;DR: In this article, the authors describe an extension and validation of the Euler/Lagrange approach for time-dependent calculations of the flow evolving in a bubble column. But they do not consider the effect of wake-generated turbulence by means of consistent Lagrangian-like terms.
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TL;DR: In this paper, the boundary conditions appropriate for use with averaged equations in the body of the water are obtained by integrating across the two-phase surface layer, where the mean pressure and the mean rate of strain have similarities to those for a compressible fluid.
Abstract: Strong turbulence at a water–air free surface can lead to splashing and a disconnected surface as in a breaking wave. Averaging to obtain boundary conditions for such flows first requires equations of motion for the two-phase region. These are derived using an integral method, then averaged conservation equations for mass and momentum are obtained along with an equation for the turbulent kinetic energy in which extra work terms appear. These extra terms include both the mean pressure and the mean rate of strain and have similarities to those for a compressible fluid. Boundary conditions appropriate for use with averaged equations in the body of the water are obtained by integrating across the two-phase surface layer.A number of ‘new’ terms arise for which closure expressions must be found for practical use. Our knowledge of the properties of strong turbulence at a free surface is insufficient to make such closures. However, preliminary discussions are given for two simplified cases in order to stimulate further experimental and theoretical studies.Much of the turbulence in a spilling breaker originates from its foot where turbulent water meets undisturbed water. A discussion of averaging at the foot of a breaker gives parameters that may serve to measure the ‘strength’ of a breaker.
156 citations
Cites methods from "A Statistically Conditioned Averagi..."
...After using (2.7) the equation for γ becomes ∂γ ∂t + ∂〈UiI〉 ∂xi = ∂γ ∂t + ∂(γUwi) ∂si = ∆ (4.8) where ∆ can be interpreted as a phase interaction term (see for instance Aliod & Dopazo 1990)....
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...The similarities between the methods used for single-phase turbulent flows and two-phase flows are even further stressed in Aliod & Dopazo (1990) who developed a statistically conditioned averaging formalism for deriving two-phase flow equations....
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TL;DR: In this article, the authors analyzed the underlying mechanisms in the continuum equations for non-colliding particles in dilute non-uniform two-phase flows and provided reasonable agreement in all available variables, including the particulate normal Reynolds stresses.
24 citations
29 Mar 2007
TL;DR: In this paper, the authors present a version of the CFI combustion model for application in evaporating fuel sprays, which includes the definition of a reaction progress variable representing the reduced chemistry yielding from CSP, a mixture fraction variable and an enthalpy variable.
Abstract: A gas turbine engine is an advanced apparatus for propulsion and power generation that has been developed over the last 60 years. The energy for this production
of propulsion and power in a gas turbine is generated by combustion.
It is feasible and relatively easy to solve the governing equations in combustion for one dimensional laminar hydrocarbon combustion with detailed chemistry. This has been done for several hydrocarbon fuels that are representative for liquid fuel combustion. The complex chemistry that is solved completely in a laminar flame is mostly modelled in simulations of turbulent combustion. Essential to this modelling is a correct understanding of the processes that govern the chemistry. Via the route of a numerical perturbation method, the CSP-method, this understanding can be developed. After analysis with CSP, the next step to a model describing turbulent combustion in gas turbines is taken using the CFI combustion model. This model comprises the definition of a reaction progress variable representing the reduced chemistry yielding from CSP, a mixture fraction variable and an enthalpy variable. The thesis presents a version of the CFI combustion model for application in evaporating fuel sprays.
11 citations
Cites background from "A Statistically Conditioned Averagi..."
...Several other authors have published on two-continua methods [113, 114]....
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TL;DR: In this paper, an in-house computational fluid dynamics code implementing a Euler-Lagrange approach is extended by incorporating the EulerEuler (two-fluid model) approach, to improve prediction capabilities of flow and thermal characteristics of turbulent evaporating sprays.
Abstract: An “in-house” computational fluid dynamics code implementing a Euler-Lagrange approach is extended by incorporating the Euler-Euler (two-fluid model) approach, to improve prediction capabilities of flow and thermal characteristics of turbulent evaporating sprays. The performance of both approaches is assessed by comparing predictions with experimental data for a variety of evaporating-spray test cases. The applicability of the Euler-Lagrange and Euler-Euler approach is established in an isopropyl alcohol–air turbulent flow, in which characteristic droplet quantity predictions are in satisfactory overall agreement with measurements. The evaporating spray characteristics are then predicted under “stabilized cool flame” conditions and the computational results are compared to experimental data for a nonreactive case and a reactive case. In both cases, the velocity and thermal fields are successfully captured by both approaches. Overall, the article demonstrates an approach toward the development of performan...
11 citations
References
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TL;DR: In this paper, the forces on a small rigid sphere in a nonuniform flow are considered from first prinicples in order to resolve the errors in Tchen's equation and the subsequent modified versions that have since appeared.
Abstract: The forces on a small rigid sphere in a nonuniform flow are considered from first prinicples in order to resolve the errors in Tchen’s equation and the subsequent modified versions that have since appeared. Forces from the undisturbed flow and the disturbance flow created by the presence of the sphere are treated separately. Proper account is taken of the effect of spatial variations of the undisturbed flow on both forces. In particular the appropriate Faxen correction for unsteady Stokes flow is derived and included as part of the consistent approximation for the equation of motion.
3,130 citations
TL;DR: In this paper, the authors consider the properties of the bulk stress in a suspension of non-spherical particles, on which a couple (but no force) may be imposed by external means, immersed in a Newtonian fluid.
Abstract: The purpose of the paper is to consider in general terms the properties of the bulk stress in a suspension of non-spherical particles, on which a couple (but no force) may be imposed by external means, immersed in a Newtonian fluid. The stress is sought in terms of the instantaneous particle orientations, and the problem of determining these orientations from the history of the motion is not considered. The bulk stress and bulk velocity gradient in the suspension are defined as averages over an ensemble of realizations, these averages being equal to integrals over a suitably chosen volume of ambient fluid and particles together when the suspension is statistically homogeneous. Without restriction on the type of particle or the concentration or the Reynolds number of the motion, the contribution to the bulk stress due to the presence of the particles is expressed in terms of integrals involving the stress and velocity over the surfaces of particles together with volume integrals not involving the stress. The antisymmetric part of this bulk stress is equal to half the total couple imposed on the particles per unit volume of the suspension. When the Reynolds number of the relative motion near one particle is small, a suspension of couple-free particles of constant shape is quasi-Newtonian; i.e. the dependence of the bulk stress on bulk velocity gradient is linear. Two significant features of a suspension of non-spherical particles are (1) that this linear relation is not of the Newtonian form and (2) that the effect of exerting a couple on the particles is not confined to the generation of an antisymmetrical part of the bulk stress tensor. The role of surface tension at the particle boundaries is described.In the case of a dilute suspension the contributions to the bulk stress from the various particles are independent, and the contributions arising from the bulk rate of strain and from the imposed couple are independent for each particle. Each particle acts effectively as a force doublet (i.e. equal and opposite adjoining ‘Stokeslets’) whose tensor strength determines the disturbance flow far from the particle and whose symmetrical and antisymmetrical parts are designated as a stresslet and a couplet. The couplet strength is determined wholly by the externally imposed couple on the particle; but the stresslet strength depends both on the bulk rate of strain and, for a non-spherical particle, on the rate of rotation of the particle relative to the fluid resulting from the imposed couple. The general properties of the stress system in a dilute suspension are illustrated by the specific and complete results which may be obtained for rigid ellipsoidal particles by use of the work by Jeffery (1922).
1,428 citations
TL;DR: In this paper, a continuum mechanics approach to two-phase flow is reviewed and an averaging procedure is applied to the exact equations of motion, and the nature of the resulting equations is studied.
Abstract: : A continuum mechanics approach to two-phase flow is reviewed. An averaging procedure is discussed and applied to the exact equations of motion. Constitutive equations are supplied and discussed for the stresses, pressure differences and the interfacial force. The nature of the resulting equations is studied. (Author)
1,347 citations
TL;DR: In this paper, the axisymmetric turbulent incompressible and isothermal jet was investigated by use of linearized constant-temperature hot-wire anemometers and the quantities measured include mean velocity, turbulence stresses, intermittency, skewness and flatness factors, correlations, scales, low-frequency spectra and convection velocity.
Abstract: The axisymmetric turbulent incompressible and isothermal jet was investigated by use of linearized constant-temperature hot-wire anemometers. It was established that the jet was truly self-preserving some 70 diameters downstream of the nozzle and most of the measurements were made in excess of this distance. The quantities measured include mean velocity, turbulence stresses, intermittency, skewness and flatness factors, correlations, scales, low-frequency spectra and convection velocity. The r.m.s. values of the various velocity fluctuations differ from those measured previously as a result of lack of self-preservation and insufficient frequency range in the instrumentation of the previous investigations. It appears that Taylor's hypothesis is not applicable to this flow, but the use of convection velocity of the appropriate scale for the transformation from temporal to spatial quantities appears appropriate. The energy balance was calculated from the various measured quantities and the result is quite different from the recent measurements of Sami (1967), which were obtained twenty diameters downstream from the nozzle. In light of these measurements some previous hypotheses about the turbulent structure and the transport phenomena are discussed. Some of the quantities were obtained by two or more different methods, and their relative merits and accuracy are assessed.
1,287 citations