Mass flux of dispersed particles in turbulence: Representations and the influence of correlation structure in gravitational settling.

Different integral representations for the mass flux of inertial particles transported by turbulent gas flows have been proposed. These are discussed and analyzed. Each formulation provides its own insights into the underlying physical processes governing the resulting flux. However, none of the representations, as it stands, provides an explicit closed-form expression in terms of known statistical properties of the flow and parameters governing particle dynamics. We consider the representations in terms of their potential for reduction to closed-form models. To enable an analysis uncomplicated by the presence of many coupled interactions, we confine our attention to the classic test case of monodisperse particles in homogeneous, isotropic turbulent flows, and subject to a uniform gravitational field. The modification of the mean particle settling velocity resulting from their preferential sampling of fluid velocities is captured by the flux representations. A distribution-based symmetry analysis coupled with a correlation splitting technique is used to reduce and simplify the terms appearing in the flux integrals. This prompts a strategy for closure modeling of the resulting expressions in terms of correlations between the sampled fluid velocity and fluid strain-rate fields. Results from particle-trajectory-based simulations are presented to assess the potential of this closure strategy.

Stochastic transport of particles in straining flows.

Important features associated with the segregration of particles in turbulent flow are investigated by considering the statistical distribution (phase-space number density) of particles subject to the combined effects of straining flow and stochastic forcing. A Fokker-Planck model is used to obtain results for the phase-space distributions of particles that are entrained into straining flow fields. The analysis shows that, in marked contrast to the zero strain case, nonsingular steady-state distributions are generated, and also confirms that the diffusional effect resulting from stochastic forcing is sufficient to offset the otherwise singular distributions that would result from the indefinite accumulation of particles along stagnation lines. The influence of particle inertia (Stokes number) on the form of the resulting distributions is considered and several significant results are observed. The influence of strain rate on the attenuation of particle kinetic stresses is quantified and explained. The development of large third-order velocity moments is observed for Stokes numbers above a critical value. The mechanism underlying this phenomenon is seen to be a generic feature of particle transport in flows where vortex structures induce local counterflows of particles. The system therefore provides an ideal test for closure models for third-order moments of particle velocities, and here the standard Chapman-Enskog approximation is assessed.

A novel approach to the prediction of musculotendon paths.

One of the most important aspects of the modelling of musculoskeletal systems is the determination of muscle moment arms which are dependent upon the paths of the muscles. These paths are often required to wrap around passive structures that can be modelled as simple geometric shapes. A novel technique for the prediction of the paths of muscles modelled as strings when wrapping around smooth analytical surfaces is presented. The theory of geodesics is used to calculate the shortest path of the string on the surface and a smoothness constraint is used to determine the correct solutions for the string path between insertions. The application of the technique to tapered cylinders and ellipsoids is presented as an extension of previous work on right-circular cylinders and spheres. The technique is assessed with reference to a particular biomechanical scenario; string lengths and moment arms are calculated and compared with alternative approximate methods. This illustrates the potential of the technique to provide more accurate muscle moment arm predictions.

Algorithms for exact multi-object muscle wrapping and application to the deltoid muscle wrapping around the humerus.

Lines of action of muscle forces imply the function and performance of muscles acting around joints. It is not always possible to determine muscle force lines of action in vivo, and so computational techniques are often used to predict them. It is common to model a muscle as a taut elastic string that follows the shortest geodesic path between attachments over the wrapping geometry. A number of studies have been concerned with wrapping paths over single wrapping objects, and those that have considered more objects have applied the single-object solutions with iterative approaches to the search for a solution. This study presents a more efficient methodology for finding the exact solutions to a certain class of wrapping problems in which the path is constrained by multiple surfaces. It also introduces a more general wrapping technique based on the idea of energy minimization, which has been successfully validated against the exact solution. These methods are applied to the case of an element of the deltoid wrapping around the humerus modelled as a composite sphere-cylinder. Comparison of results with those obtained from approximated single-object solutions demonstrates the need to include correct multi-object wrapping algorithms in biomechanical models.

Is the kinetic equation for turbulent gas-particle flows ill posed?

This paper is about the kinetic equation for gas-particle flows, in particular its well-posedness and realizability and its relationship to the generalized Langevin model (GLM) probability density function (PDF) equation. Previous analyses, e.g. [J.-P. Minier and C. Profeta, Phys. Rev. E 92, 053020 (2015)PLEEE81539-375510.1103/PhysRevE.92.053020], have concluded that this kinetic equation is ill posed, that in particular it has the properties of a backward heat equation, and as a consequence, its solution will in the course of time exhibit finite-time singularities. We show that this conclusion is fundamentally flawed because it ignores the coupling between the phase space variables in the kinetic equation and the time and particle inertia dependence of the phase space diffusion tensor. This contributes an extra positive diffusion that always outweighs the negative diffusion associated with the dispersion along one of the principal axes of the phase space diffusion tensor. This is confirmed by a numerical evaluation of analytic solutions of these positive and negative contributions to the particle diffusion coefficient along this principal axis. We also examine other erroneous claims and assumptions made in previous studies that demonstrate the apparent superiority of the GLM PDF approach over the kinetic approach. In so doing, we have drawn attention to the limitations of the GLM approach, which these studies have ignored or not properly considered, to give a more balanced appraisal of the benefits of both PDF approaches.

Drift-free kinetic equations for turbulent dispersion.

The dispersion of passive scalars and inertial particles in a turbulent flow can be described in terms of probability density functions (PDFs) defining the statistical distribution of relevant scalar or particle variables. The construction of transport equations governing the evolution of such PDFs has been the subject of numerous studies, and various authors have presented formulations for this type of equation, usually referred to as a kinetic equation. In the literature it is often stated, and widely assumed, that these PDF kinetic equation formulations are equivalent. In this paper it is shown that this is not the case, and the significance of differences among the various forms is considered. In particular, consideration is given to which form of equation is most appropriate for modeling dispersion in inhomogeneous turbulence and most consistent with the underlying particle equation of motion. In this regard the PDF equations for inertial particles are considered in the limit of zero particle Stokes number and assessed against the fully mixed (zero-drift) condition for fluid points. A long-standing question regarding the validity of kinetic equations in the fluid-point limit is answered; it is demonstrated formally that one version of the kinetic equation (derived using the Furutsu-Novikov method) provides a model that satisfies this zero-drift condition exactly in both homogeneous and inhomogeneous systems. In contrast, other forms of the kinetic equation do not satisfy this limit or apply only in a limited regime.