Exact Transport Equation Research Articles

Exact transverse macro dispersion coefficients for transport in heterogeneous porous media

We study transport through heterogeneous media. We derive the exact large scale transport equation. The macro dispersion coefficients are determined by additional partial differential equations. In the case of infinite Peclet numbers, we present explicit results for the transverse macro dispersion coefficients. In two spatial dimensions, we demonstrate that the transverse macro dispersion coefficient is zero. The result is not limited on lowest order perturbation theory approximations but is an exact result. However, the situation in three spatial dimensions is very different: The transverse macro dispersion coefficients are finite – a result which is confirmed by numerical simulations we performed.

Evolution of the volumetric interfacial area in two-phase mixtures

We investigate the time evolution of the density of interfaces in a two-phase mixture, with particular emphasis on the role of compressibility, dilatability and phase transitions. Two different and complementary routes are considered: a rather intuitive one based on exact results for dilute mixtures which are then interpolated to all concentrations, and a more systematic approach based on the statistical average of the exact transport equation for elementary pieces of interfaces. To cite this article: D. Lhuillier, C. R. Mecanique 332 (2004).

Velocity-scalar filtered density function for large eddy simulation of turbulent flows

A methodology termed the “velocity-scalar filtered density function” (VSFDF) is developed and implemented for large eddy simulation (LES) of turbulent flows. In this methodology, the effects of the unresolved subgrid scales (SGS) are taken into account by considering the joint probability density function (PDF) of the velocity and scalar fields. An exact transport equation is derived for the VSFDF in which the effects of the SGS convection and chemical reaction are closed. The unclosed terms in this equation are modeled in a fashion similar to that typically used in Reynolds-averaged simulation procedures. A system of stochastic differential equations (SDEs) which yields statistically equivalent results to the modeled VSFDF transport equation is constructed. These SDEs are solved numerically by a Lagrangian Monte Carlo procedure in which the Itô–Gikhman character of the SDEs is preserved. The consistency of the proposed SDEs and the convergence of the Monte Carlo solution are assessed by comparison with results obtained by a finite difference LES procedure in which the corresponding transport equations for the first two SGS moments are solved. The VSFDF results are compared with those obtained by the Smagorinsky model, and all the results are assessed via comparison with data obtained by direct numerical simulation of a temporally developing mixing layer involving transport of a passive scalar. It is shown that the values of both the SGS and the resolved components of all second order moments including the scalar fluxes are predicted well by VSFDF. The sensitivity of the calculations to the model’s (empirical) constants are assessed and it is shown that the magnitudes of these constants are in the same range as those employed in PDF methods.

Separation of Variables Applied to the Wave Equation in Laterally Inhomogeneous Media

—The new eigenfunction expansion formula based upon the method of separation of variables is derived. The method gives the exact transport equation and the generalized eikonal equation without the need of asymptotic series expansions. The generalized eikonal equation extends the classical eikonal equation to a rapidly varying medium. It governs the wave propagation with dispersive phase velocity depending upon the amplitude. Both equations are separated by making use of conventional phase-ray coordinates. The solution is expressed in terms of the elementary Sturm-Liouville eigenfunctions or modes for the vibrating string. The generalized eikonal equation is solved by employing the modified Hamiltonian formalism. This results in frequency-dependent phase-ray trajectories associated with the energy flux vector field.

Galilean invariance of velocity probability density function transport equation

Model equations for turbulent flows generally must have the same Galilean transformation properties as the exact transport equations. We examine the transport equation of one-point velocity joint probability density function (JPDF) under Galilean transformations. The JPDF and its transport equation as well as the unclosed terms in the equation are shown to be Galilean invariant. For constant density flows they are also invariant under extended Galilean transformations. It is also shown that conditional statistics of any Galilean invariant variables conditional on velocity are invariant. The present work provides a basis and justification for using such conditional statistics in studying turbulent flows.

Turbulent Dissipation-Rate Equation for Compressible Flows

The turbulent dissipation rate in a compressible flow is modeled in terms of its solenoidal and dilatational parts. In the κ-e framework the solenoidal dissipation rate is computed using the modeled dissipation-rate equation for incompressible flows. This is based on the assumption that the solenoidal dissipation rate for compressible flows follows the same dynamics as the dissipation rate for incompressible flows. We test this assumption by comparing the exact transport equations of these two quantities, both analytically and using direct numerical simulation data of a Mach 4 boundary layer. The two equations are found to be equivalent except for an additional term caused by the variation of fluid viscosity in the compressible case. This forms the rigorous basis for using the incompressible modeled dissipation-rate equation in a compressible boundary-layer flow provided the extra term caused by variation of fluid viscosity is included. Finally, the implications of the dissipation-rate analysis on the κ-ω turbulence model are pointed out

Velocity filtered density function for large eddy simulation of turbulent flows

A methodology termed the “velocity filtered density function” (VFDF) is developed and implemented for large eddy simulation (LES) of turbulent flows. In this methodology, the effects of the unresolved subgrid scales (SGS) are taken into account by considering the joint probability density function of all of the components of the velocity vector. An exact transport equation is derived for the VFDF in which the effects of the SGS convection appear in closed form. The unclosed terms in this transport equation are modeled. A system of stochastic differential equations (SDEs) which yields statistically equivalent results to the modeled VFDF transport equation is constructed. These SDEs are solved numerically by a Lagrangian Monte Carlo procedure in which the Itô–Gikhman character of the SDEs is preserved. The consistency of the proposed SDEs and the convergence of the Monte Carlo solution are assessed by comparison with results obtained by an Eulerian LES procedure in which the corresponding transport equations for the first two SGS moments are solved. The VFDF results are compared with those obtained via several existing SGS closures. These results are also analyzed via a priori and a posteriori comparisons with results obtained by direct numerical simulation of an incompressible, three-dimensional, temporally developing mixing layer.

On the relation between the conditional moment closure and unsteady flamelets

We consider the relation between the conditional moment closure (CMC) and the unsteady flamelet model (FM). The CMC equations were originally constructed as global equations, while FM was derived asymptotically for a thin reaction zone. The recent tendency is to use FM-type equations as global equations. We investigate the possible consequences and suggest a new version of FM: coordinate-invariant FM (CIFM). Unlike FM, CIFM complies with conditional properties of the exact transport equations which are used effectively in CMC. We analyse the assumptions needed to obtain another global version of FM: representative interactive flamelets (RIF), from original FM and demonstrate that, in homogeneous turbulence, one of these assumptions is equivalent to the main CMC hypothesis.

Combustion and heat transfer monitoring in large utility boilers

As a result of the quick and vast development of instrumentation and software capabilities, the optimization and control of complex energy systems can presently take advantage of highly sophisticated engineering techniques, such as CFD calculations and correlation algorithms based on artificial intelligence concepts. However, the most advanced numerical prediction still relies on strong simplifications of the exact transport equations. Likewise, the output of a neural network, or any other refined data-processing device, is actually based in a long record of observed past responses. Therefore, the implementation of modern diagnosis tools generally requires a great amount of experimental data, in order to achieve an adequate validation of the method. Consequently, a sort of paradox results, since the validation data cannot be less accurate or complete than the predictions sought. To remedy this situation, there are several alternatives. In opposition to laboratory work or well-instrumented pilot plants, the information obtained in the full scale installation offers the advantages of realism and low cost. This paper presents the case-study of a large, pulverized-coal fired utility boiler, discussing both the evaluation of customary measurements and the adoption of supplementary instruments. The generic outcome is that it is possible to significantly improve the knowledge on combustion and heat transfer performance within a reasonable cost. Based on the experience and results, a general methodology is outlined to cope with this kind of analysis.

A radiative transport model for large-eddy fire simulations

Three-dimensional simulations of fires cannot be performed on present-day computers without devising simplifications to the governing equations. One such method is the large-eddy simulation (LES) approach for fires developed at NIST. This method results in computationally efficient fire simulations in which the buoyancy-generated motion of hot gases and smoke is driven by Lagrangian particles that carry the heat released by combustion. Complex geometries are represented by blocking cells interior to a rectangular domain. A P1 approximation to the radiation transport equation was developed to be consistent with the exact transport equation for scenarios based upon this model. An isolated fire plume above a semi-infinite solid with a constant absorption coefficient in each half-space was studied as an example. A direct elliptic solver required only a fraction of the total LES computational cost. Radiative fluxes and intensities from the numerical and exact solutions to the P1 approximation were in excellent agreement.

Conditional Fluid-Particle Accelerations in Turbulence

Simple closures for average fluid-particle accelerations, conditional on fixed local fluid velocity, are considered in isotropic, homogeneous and stationary turbulence using exact probability density transport equations and are compared with direct numerical simulations (DNS). Such accelerations are common ingredients in Lagrangian stochastic models for fluid-particle trajectories in turbulence. One-particle accelerations are essentially trivial, so the focus is on two-particle relative accelerations, which are important in the relative dispersion process. The closure is simply a quadratic form in the velocity variable and this special form also defines the Eulerian velocity probability density function (pdf), and comparisons with DNS (for grids up to 5123) of both the acceleration closure and velocity pdf's are encouraging.

Analysis and modelling of anisotropies in the dissipation rate of turbulence

The modelling of anisotropies in the dissipation rate of turbulence is considered based on an analysis of the exact transport equation for the dissipation rate tensor. An algebraic model is systematically derived using integrity bases methods and tensor symmetry properties. The new model differs notably from all previously proposed models in that it depends nonlinearly on the mean velocity gradients. This gives rise to a transport equation for the scalar dissipation rate that is of the same general form as the commonly used model with one major exception: the coefficient of the production term is dependent on the invariants of both the rotational and irrotational strain rates. The relationship between the new model and other recently proposed models is examined in detail. Some basic tests and applications of the model are also provided along with a discussion of the implications for turbulence modelling.

Analytical theory of the destruction terms in dissipation rate transport equations

Modeled dissipation rate transport equations are often derived by invoking various hypotheses to close correlations in the corresponding exact equations. D. C. Leslie [Modern Developments in the Theory of Turbulence (Oxford University, Oxford, 1972)] suggested that these models might be derived instead from Kraichnan’s [J. Fluid Mech. 47 (1971)] wavenumber space integrals for inertial range transport power. This suggestion is applied to the destruction terms in the dissipation rate equations for incompressible turbulence, buoyant turbulence, rotating incompressible turbulence, and rotating buoyant turbulence. Model constants like Cε2 are expressed as integrals; convergence of these integrals implies the absence of Reynolds number dependence in the corresponding destruction term. The dependence of Cε2 on rotation rate emerges naturally; sensitization of the modeled dissipation rate equation to rotation is not required. A buoyancy related effect which is absent in the exact transport equation for temperature variance dissipation, but which sometimes improves computational predictions, also arises naturally. The time scale in the modeled transport equation depends on whether Bolgiano or Kolmogorov inertial range scaling applies. A simple extension of these methods leads to a preliminary dissipation rate equation for rotating buoyant turbulence.

A relaxation approximation for time-dependent second-order effects in shear turbulence

The Crow constraint on time-dependent turbulence models is generalized to effects of second order in the strain rate. For simple shear flow, this generalization requires agreement of the time-dependent normal stress deviators with the short-time rapid distortion limit and with long-time equilibrium data. A model consistent with these requirements is proposed based on a perturbative solution of the equations of the direct interaction approximation for shear flow. It requires a decomposition of the Reynolds stress into tensor components which satisfy coupled linear relaxation equations. This approach to time-dependent turbulence modeling based on analytical theories offers an alternative to the standard ones, in which various hypotheses are introduced to close correlations in the exact transport equation for second-order single-point moments.

Second-moment closure for modelling the near-wall turbulence in 3D mean flows

A second-moment closure for the near-wall turbulence is proposed. The limiting behaviour of this closure near a wall is consistent with that of the exact Reynolds-stress transport equations, and it converts asymptotically into a high-Reynolds-number closure remote from the wall. The closure is applied to a pressure-driven 3D transient channel flow. The predicted results are in fair agreement with the DNS data.

A New Transport Equation for Single-Time Green′s Functions in an Arbitrary Quantum System. General Formalism

Using new renormalization methods for nonequilibium real-time Green′s functions a set of exact self-consistent non-linear transport equations is derived for all one-particle distribution functions occurring in an arbitrary inhomogeneous multi-component quantum many-body system. In order to develop the theory for a general initial state even if the particle number is not conserved, a new generalization of Wick′s theorem is presented. The transport equation is given as a systematic perturbation expansion in the coupling parameter around a generalized Boltzmann and Enskog equation, whereby the non-locality of scattering processes, arbitrarily strong driving forces, collisional broadening effects, and energy renormalizations due to Hartree-Fock potentials are incorporated. Grand canonical, as well as canonical, ensembles can be treated, which makes the theory applicable to macroscopic, as well as mesoscopic, systems. The theory is also well suited for the calculation of transport coefficients in linear or non-linear response theory and can describe quantum interference effects like localization phenomena in disordered systems. Furthermore, it is shown that the diagonal singularity property of van Hove is equivalent to (quasi-)momentum conservation and the validity of the Zwanzig inversion method of classical kinetic theory is proven for an arbitrary quantum many-body system. A generalized Kadanoff-Baym ansatz is derived for systems without particle number conservation.

General Transport Theory for Weak Inhomogeneities and Quantum Solids in High External Fields

The exact transport equation obtained in a recent work on new renormalization methods for single-time Green′s functions is applied to weakly inhomogeneous quantum systems. The initial density matrix and the external fields are assumed to vary slowly over microscopic length scales like the de Broglie wavelength λ F of the particles, the range λ V , of the interaction and the lattice constant a. In addition, the Hamiltonian without the external fields has to fulfil (quasi-)momentum conservation. The final result is given by a set of local and nonlinear integral equations for the Weyl transforms of all one-particle distribution functions occurring in the system. As an application a quantum solid of Bloch electrons, phonons, and impurities in arbitrarily time-dependent and weakly inhomogeneous electric and magnetic fields is studied. The fields can be moderately high in the sense that r L ⪢ max {λ F , λ V , a} and eE max{λ F , λ V , a} ⪡ ϵ F . Here r L denotes the radius of the Landau orbits, E is the electric field, and ϵ F is the fermi energy of the electrons. In Born approximation, a set of generalized Boltzmann equations is obtained which include interband transition terms from scattering and Zeener tunneling as well as collisional broadening and intracollisional field effects.

Experimental analysis of flamelet models for premixed turbulent combustion

Flamelet models are now widely used to predict turbulent premixed combustion because they allow theseparation of chemical features from the description of the turbulent flow field. Some of them introduce a flame surface density (flame surface per unit volume) modelled either by an algebraic closure (Bray-Moss-Libby model) or from a transport equation. This equation may be exactly written but needs closure assumptions. In the present work, a two-dimensional turbulent premixed propane-air V flame is studied. Velocity profiles are obtained from laser-Doppler velocimetry. Flame front visualizations are achieved by tomography with high-speed cinematography using a copper vapor laser. Images are then processed to extract flame front characteristics (flame surface density, vector normal to the flame front, curvature). The Bray-Moss-Libby model is found to have a good trend but needs a precise closure of the flame wrinkling length scale. Then some terms of the exact transport equation for flame surface density are examined. A simple model is proposed to close the strain rate term acting on the flame surface and due to the mean flow. Curvature and propagation terms are found to act as a source term on the fresh gases side and as a consumption term on the burnt one. This fact points out a lack in the generally used closures of the flame surface density equation where the propagation term is neglected and the curvature term modelled as a destruction one.

TLM modelling of diffusion, drift and recombination of charge carriers in semiconductors

AbstractThe paper provides an extension to the one‐dimensional TLM diffusion model. The extended diffusion node presented here models the exact transport equation with diffusion drift and recombination of charge carriers in semiconductors. A general algorithm for providing a numerical solution to the transport phenomena is also presented here. The analytical solution for infinitely long semiconductors is compared with the TLM numerical results.

Vorticity form of turbulence transport equations

The exact transport equations describing the turbulent kinetic energy, Reynolds stresses, and the dissipation rate of the turbulent kinetic energy are written in their vorticity form. The limiting behavior of the transport equations describing the irrotational fluctuations at the free-stream edges of shear flows is directly deduced from the exact transport equations. Additional relations of pure kinematic origin are also indicated. It is suggested that the formulation developed may be useful in improving the turbulence transport models.