Jacek Pozorski

On the Lagrangian turbulent dispersion models based on the Langevin equation

Abstract A Lagrangian approach is used to describe turbulent two-phase particulate flows. Special attention focuses on a currently used model based on the Langevin equation with instantaneous relative velocity of fluid and particles. A detailed analysis of the model is performed and its drawbacks are discussed. Afterwards, a novel model for particle dispersion in homogeneous turbulence is proposed. It is built on physical arguments quite different from those underlying the previous model: rather than instantaneous relative velocities, averaged characteristics of the motion of solid–fluid particle pairs are considered. The new model accounts for both inertia and external force effects. It is validated by comparison with existing experimental data on particle dispersion in grid turbulence and with large-eddy simulations of homogeneous turbulence.

Analysis of the incompressibility constraint in the smoothed particle hydrodynamics method

SUMMARY Smoothed Particle Hydrodynamics (SPH) is a particle-based, fully Lagrangian method for fluid-flow simulations. In this work, fundamental concepts of the method are first briefly recalled. Then, we present a thorough comparison of two different incompressibility treatments in SPH: the weakly compressible approach, where a suitably chosen equation of state is used, and the truly incompressible method (in two basic variants), where the velocity field projection onto a divergence-free space is performed. A noteworthy aspect of the study is that in each incompressibility treatment, the same boundary conditions are used (and further developed) that allows a direct comparison to be made. Two-dimensional and three-dimensional validation cases are studied. Problems associated with the numerical setup are discussed, and an optimal choice of the computational parameters is proposed and verified. The efficiency issues of the incompressibility treatments are considered, and the speed-up features are highlighted. The results show that the present state-of-the-art truly incompressible methods (based on a velocity correction) suffer from density accumulation errors. To address this issue, an algorithm, based on a correction for both particle velocities and positions, is presented. The usefulness of this density correction is examined and demonstrated. Copyright

Probability density function computation of turbulent flows with a new near-wall model

The modeling and computation of near-wall turbulent flows is addressed with the probability density function (PDF) method for velocity and the turbulent frequency. Near-wall extensions are considered in detail and a new model for viscous transport is proposed. A method of elliptic relaxation for a blending function is applied to model the pressure–strain term. A numerical integration scheme is developed to deal with the near-wall singularity of coefficients that appears in the discrete formulation. The PDF equation is solved by a Monte Carlo method and the whole approach appears as a self-contained Lagrangian simulation using stochastic particles. For the sake of a numerical example, the fully developed channel flow case is computed; results are compared with the available direct numerical simulation data.

WALL-BOUNDARY CONDITIONS IN PROBABILITY DENSITY FUNCTION METHODS AND APPLICATION TO A TURBULENT CHANNEL FLOW

An application of a probability density function (PDF), or Lagrangian stochastic, approach to the case of high-Reynolds number wall-bounded turbulent flows is presented. The model simulates the instantaneous velocity and dissipation rate attached to a large number of particles and the wall-boundary conditions are formulated directly in terms of the particle properties. The present conditions aim at reproducing statistical results of the logarithmic region and are therefore in the spirit of wall functions. A new derivation of these boundary conditions and a discussion of the resulting behavior for different mean variables, such as the Reynolds stress components, is proposed. Thus, the present paper complements the work of Dreeben and Pope [Phys. Fluids 9, 2692 (1997)] who proposed similar wall-boundary particle conditions. Numerical implementation of these conditions in a standalone two-dimensional PDF code and a pressure-correction algorithm are detailed. Moments up to the fourth order are presented for a...

Derivation of a PDF model for turbulent flows based on principles from statistical physics

Classical ideas from statistical physics are used to derive a PDF model for turbulent flows. The model is built by adopting a Lagrangian point of view and by considering separately the statistical effects of the viscous and of the pressure gradient forces which act on a fluid particle. Closures are developed alternatively in terms of the pdf itself and of the trajectories of the stochastic process. The viscous force is shown to manifest itself as an anti-diffusion in phase space while modeling of the fluctuating part of the pressure gradient force is based on linear laws for non-equilibrium thermodynamics along Onsager’s regression-to-equilibrium hypothesis. The final expression is identical to a Langevin equation proposed by Pope which is thus seen to be obtained from underlying principles.

Spray penetration in a turbulent flow

Analytical expressions for mass concentration of liquid fuel in a spray are derived taking into account the effects of gas turbulence, and assuming that the influence of droplets on gas is small (intitial stage of spray development). Beyond a certain distance the spray is expected to be fully dispersed. This distance is identified with the maximum spray penetration. Then the influence of turbulence on the spray stopping distance is discussed and the rms spray penetration is computed from a trajectory (Lagrangian) approach. Finally, the problem of spray penetration is investigated in a homogeneous two-phase flow regime taking into account the dispersion of spray away from its axis. It is predicted that for realistic values of spray parameters the spray penetration at large distances from the nozzle is expected to be proportional to t2/3 (in the case when this dispersion is not taken into account this distance is proportional to t1/2). The t2/3 law is supported by experimental observations for a high pressure injector.

A New Stochastic Approach for the Simulation of Agglomeration between Colloidal Particles

This paper presents a stochastic approach for the simulation of particle agglomeration, which is addressed as a two-step process: first, particles are transported by the flow toward each other (collision step) and, second, short-ranged particle-particle interactions lead either to the formation of an agglomerate or prevent it (adhesion step). Particle collisions are treated in the framework of Lagrangian approaches where the motions of a large number of particles are explicitly tracked. The key idea to detect collisions is to account for the whole continuous relative trajectory of particle pairs within each time step and not only the initial and final relative distances between two possible colliding partners at the beginning and at the end of the time steps. The present paper is thus the continuation of a previous work (Mohaupt M., Minier, J.-P., Tanière, A. A new approach for the detection of particle interactions for large-inertia and colloidal particles in a turbulent flow, Int. J. Multiphase Flow, 2011, 37, 746-755) and is devoted to an extension of the approach to the treatment of particle agglomeration. For that purpose, the attachment step is modeled using the DLVO theory (Derjaguin and Landau, Verwey and Overbeek) which describes particle-particle interactions as the sum of van der Waals and electrostatic forces. The attachment step is coupled with the collision step using a common energy balance approach, where particles are assumed to agglomerate only if their relative kinetic energy is high enough to overcome the maximum repulsive interaction energy between particles. Numerical results obtained with this model are shown to compare well with available experimental data on agglomeration. These promising results assert the applicability of the present modeling approach over a whole range of particle sizes (even nanoscopic) and solution conditions (both attractive and repulsive cases).

Analysis of SGS Particle Dispersion Model in LES of Channel Flow

The wall-bounded two-phase turbulent flow with the dispersed heavy particles is modelled in the Eulerian-Lagrangian approach. The large-eddy simulation (LES) method with near-wall resolution is applied to compute the dynamics of the continuous phase (fluid). The particle tracking with account for drag and lift is used for the dispersed phase. A stochastic model for the residual fluid velocity along particle trajectories is used to account for the subgrid-scale (SGS) particle dispersion. Results for the fluid and particle velocity statistics are presented. In the case of particles undergoing deposition on the channel walls, the separation velocity curve, resulting from several variants of modelling, is presented and discussed.

Full velocity-scalar probability density function computation of heated channel flow with wall function approach

A joint velocity-scalar probability density function (PDF) method is presented to model and simulate turbulent flows with passive inert scalars (here temperature). The full PDF approach is applied for wall-bounded flows. In the present work, the boundary conditions are imposed in the logarithmic region and the modeling is therefore performed in the wall-function spirit. The PDF equation is solved by a Monte Carlo method and the whole approach appears as a Lagrangian simulation using stochastic particles. The purpose of the work is to analyze the behavior of classical PDF models in the near-wall region and to develop new particle boundary conditions for the velocity and scalars attached to each particle. First of all, the logarithmic region is described as an equilibrium zone and resulting analytical formulas for second-order temperature–velocity statistics 〈θ2〉, 〈uθ〉, 〈vθ〉 are derived. Boundary conditions for scalars are then developed and formulated in terms of instantaneous particle variables. These res...

Study on Langevin model parameters of velocity in turbulent shear flows

This paper deals with the stochastic equation used to predict the fluctuating velocity of a fluid particle in a nonhomogeneous turbulent flow, in the frame of probability density function (PDF) approaches. It is shown that a Langevin-type equation is appropriate provided its parameters (drift and diffusion matrices) are suitably specified. By following the approach proposed in the literature for homogeneous turbulent shear flows, these parameters have been identified using data from direct numerical simulations (DNS) of both channel and pipe flows. Using statistics extracted from the computation of the channel flow, it is shown that the drift matrix of the stochastic differential equation can reasonably be assumed to be diagonal but not spherical. This behavior of the drift coefficients is confirmed by the available results for a turbulent pipe flow at low Reynolds number. Concerning the diffusion matrix, it is found that this matrix is anisotropic for low Reynolds number flows, a property which has been ...