Hyperbolicity study of models for turbulent two-phase flows obtained from the variational principle

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

Nonlinear algebraic Reynolds stress model for two-phase turbulent flows laden with small heavy particles

The purpose of this article is to propose a generic methodology to build consistent Lagrangian models for polydisperse turbulent two-phase flows where the main issue is to devise a stochastic model for the velocity of the fluid seen by discrete particles. By consistent, it is meant that such models should meet the requirements set forth by Minier et al. (Phys. Fluids, 26, 113303, 2014) and, in the limit of vanishing particle inertia, retrieve the state-of-the-art stochastic models referred to as generalized Langevin models (GLMs) used for the simulation of turbulent single-phase flows. The methodology is generic in the sense that the resulting stochastic models for polydisperse two-phase flows are not limited to one particular fluid model but allows extending any GLM formulation to the two-phase flow situation. This is obtained by introducing a specific operator, which represents how statistical characteristics of fluid particles are transformed, or mapped, to the ones pertaining to the velocity of the fluid seen. In practice, this operator can be worked out separately from first principles or by resorting to some physical inputs. Once it is expressed, the present methodology shows how to extend a GLM for fluid particles to obtain a two-phase GLM formulation in a consistent manner. This is helpful to decouple physics-based developments used to obtain such an operator from the construction of practical stochastic models while ensuring that they remain consistent with fluid descriptions.

A two-fluid model of turbulent two-phase flow for simulating turbulent stratified flows

Computational fluid dynamics (CFD) models of a HighRiseTM hog building (HRHB) were developed to simulateair velocity and ammonia distribution within the building under minimum ventilation conditions. Because bothlaminar/transient and turbulent flow conditions exist in the building, two different flow simulations were used. Air velocityprofiles from both turbulent and laminar flow models indicated that some air moves from the lower to the upper level, whichaffects the distribution of ammonia in the pig space. The simulation results were compared to air velocities and ammoniaconcentrations measured within an experimental HRHB. The turbulent flow model more closely matched measured ammoniavalues at locations in the HRHB than the laminar flow model. Using the turbulent flow model, ammonia concentration in thepig space would be below 25 ppm, and the NH3 emission factor for the HRHB during winter (low ventilation) conditions wouldbe 4.6 kg pig1 yr1. Although limited by the representation of real building geometry and processes, the twodimensional CFDmodels allowed rapid simulation of airflow and species gradients in the HRHB.

Recent experiments have shown that the high Reynolds number turbulent flow of superfluid helium is similar to classical turbulence. To understand this evidence we have developed an idealized model of normal fluid turbulence which is based on vorticity tubes and we have studied numerically the behavior of superfluid quantized vortex lines in this model of turbulent normal flow. We have found that the vortex lines form ordered superfluid vortex bundles in regions of high normal fluid vorticity. A vortex wave instability and mutual friction are responsible for generating a high density of vortex lines such that the resulting macroscopic superfluid vorticity and the driving normal fluid vorticity patterns match. The results are discussed from the point of view of the idea, put forward to explain experiments, that in the isothermal, turbulent flow of He II a high density of vortex lines locks the two fluid components together and the resulting turbulent flow is that of a classical Navier–Stokes fluid.

An improved approach is presented for the hybrid Eulerian-Lagrangian modeling of turbulent two-phase flows. The hybrid model consists of a nonlinear k–ε model for the fluid flow and an efficient Lagrangian trajectory model for the particulate flow. The improved approach avoids an empirical correlation required to determine the dispersion width for the existing Stochastic-Probabilistic Efficiency Enhanced Dispersion (SPEED) model. The improved SPEED model is validated using experimental data for a poly-dispersed water spray interacting with a turbulent annular air jet behind a bluff-body. Numerical results for the number-mean and Sauter-mean droplet diameters, as well as mean and fluctuating droplet velocities are compared with the experimental data and with the predictions of other dispersion models. It is demonstrated that higher computational efficiency and smoother profiles of Sauter-mean diameter can be obtained with the improved stochastic-probabilistic model than with the eddy-interaction model.

The thermodynamically consistent, rate dependent model for turbulent two-phase flows was used to study the special case of a simple shear. The variations of the fluctuation kinetic energy with the solid volume fraction was evaluated. The kinetic model for rapid flows of granular materials, which includes frictional losses, was used for studying gravity flow down an inclined chute. The velocity profiles were obtained and the results were compared with the data of Johnson et al. The effect of diameter to height ratio was also studied. Further progress has been made in developing a computational model for rapid granular and two-phase flows in complex geometries. The discrete element scheme was used and the unsteady developing granular flow down a chute was analyzed. The results are compared with the experimental data of Savage. The numerical procedure for analyzing two-phase flows was further developed. The special case of duct flow in a gravitational field is analyzed. Further progress has been made in the construction of the experimental monolayer simple shear flow device. Assembling the device was completed.

The thermodynamically consistent, rate dependent model for turbulent two-phase flows was used to study the special case of a simple shear. The variations of the fluctuation kinetic energy with the solid volume fraction was evaluated. The kinetic model for rapid flows of granular materials, which includes frictional losses, was used for studying gravity flow down an inclined chute. The velocity profiles were obtained and the results were compared with the data of Johnson et al. The effect of diameter to height ratio was also studied. Further progress has been made in developing a computational model for rapid granular and two-phase flows in complex geometries. The discrete element scheme was used and the unsteady developing granular flow down a chute was analyzed. The results are compared with the experimental data of Savage. The numerical procedure for analyzing two-phase flows was further developed. The special case of duct flow in a gravitational field is analyzed. Further progress has been made in the construction of the experimental monolayer simple shear flow device. Assembling the device was completed.

An extended quasi two-phase mass flow model

This paper deals with the statistical modeling of turbulent two-phase flows consisting of particles or droplets immersed in a gas. The problem of gaseous turbulence alone being very complex, we concentrate here on the simpler case of an a priori given forced isotropic homogeneous turbulence acting on the particles, whose mean square velocity and integral Lagrangian time-scale are given constants. Our main objective is to derive a hierarchy of reduced models from the joint particle-gas pdf (probability density function). The latter equation may therefore be regarded as a master equation for our problem. The reduced models describe the dispersion of a cloud of particles observed at di®erent time scales compared to the dynamic response time of the particles and the characteristic time scale of the turbulence along their trajectories. These derivations rely on very classical Chapman-Enskog expansions. We recover in particular the result of Tchen [C. M. Tchen, /i>Mean value and correlation problems connected with the motion of small particles suspended in a turbulent fluid, PhD thesis, Delft, The Hague, Martinus Nijhoff, 1947] stating that the diffusion rate is the same for small or large particles in homogeneous turbulence, under the assumption that the lagrangian statistical properties along their paths are the same. Moreover, our approach allows us to prove that the long-time limit of the joint particle-gas distribution function is a bi-maxwellian distribution, whatever the size of the particles. This is consistent with some usual assumptions made in the literature for the derivation of particle collision models [J. Lavieville, E. Deutsch and O. Simonin, Large eddy simulation of interactions between colliding particles and a homogeneous isotropic turbulence field, Gas-Solid Flows, ASME, 228, 347-358, 1995], [Leonid I. Zaichik, Olivier Simonin and Vladimir M. Alipchenkov, Two statistical models for predicting collision rates of inertia particles in homogeneous isotropic turbulence, Phys. Fluids, 15(10), 2995-3005, 2003].

An improved stochastic separated flow model is proposed to obtain reasonable statistical characteristics of a two-phase flow. Effects of the history of a particle and its current trajectory position on the mean-square fluctuating velocity of the dispersed phase are continuously considered in this model. Comparing with the conventional model, results using the improved model are more reasonable and can also be obtained more easily. Furthermore, the improved model requires less computational particles for simulating dispersed-phase turbulence at the beginning of the stochastic trajectory. In this paper, an application in turbulent two-phase flow of planar mixing layer is carried out. Numerical results including velocity, mean-square fluctuating velocity, particle number density and pdf of fluctuation velocity of dispersed phase are shown to compare well with experimental data.

Low Mach number preconditioning techniques for Roe-type and HLLC-type methods for a two-phase compressible flow model

Statistical descriptions of polydisperse turbulent two-phase flows

The USM-ϑ model of Bingham fluid for dense two-phase turbulent flow was developed, which combines the second-order moment model for two-phase turbulence with the particle kinetic theory for the inter-particle collision. In this model, phases interaction and the extra term of Bingham fluid yield stress are taken into account. An algorithm for USM-ϑ model in dense two-phase flow was proposed, in which the influence of particle volume fraction is accounted for. This model was used to simulate turbulent flow of Bingham fluid single-phase and dense liquid-particle two-phase in pipe. It is shown USM-ϑ model has better prediction result than the five-equation model, in which the particle-particle collision is modeled by the particle kinetic theory, while the turbulence of both phase is simulated by the two-equation turbulence model. The USM-ϑ model was then used to simulate the dense two-phase turbulent up flow of Bingham fluid with particles. With the increasing of the yield stress, the velocities of Bingham and particle decrease near the pipe centre. Comparing the two-phase flow of Bingham-particle with that of liquid-particle, it is found the source term of yield stress has significant effect on flow.