Effect of thermophoresis and other parameters on the particle deposition on a tilted surface

In this study a modified version of v2-f turbulence model (φ-α), is applied to simulate a non-isothermal air-flow. The φ-α model and a two-phase Eulerian approach complement each other to predict the rate of particle deposition on a tilted surface. The φ-α model can accurately calculate the normal fluctuations, which mainly represent the non-isotropic nature of turbulence regime near the wall. The Eulerian model was modified considering the most important mechanism in the particle deposition rate when compared to the experimental data. The model performance is examined by comparing the rate of particle deposition on a vertical surface with the experimental data in a turbulent channel flow available in the literature. The effects of lift force, turbophoretic force, thermophoreric force, electrostatic force, gravitational force and Brownian/turbulent diffusion were examined on the particle deposition rate. The results show that, using the φ-α model predicts the rate of deposition with reasonable accuracy. The results of modified particle model are in good agreement with the experimental data. This study highlights the paramount effect of thermophoretic force on the particle deposition rate and clearly shows that when the temperature difference exceeds a certain limit, the electrostatic force has insignificant effect on the particle deposition rate. Furthermore, it is indicated that even at small temperature differences, the effect of tilt angle on the particle deposition rate for intermediate-size particles is negligible.     

Analysis of unsteady gas–liquid flows in a rectangular tank: Comparison of Euler–Eulerian and Euler–Lagrangian simulations

We study the dynamics of gas–liquid flows experimentally and computationally in a rectangular bubble column where the gas source is introduced at the corner. The flow in this reactor is complex and inherently unsteady in nature. The two-dimensional liquid phase velocity field is calculated by an Eulerian approach solving the unsteady Reynolds Averaged Navier Stokes equations. The conservation equations are closed using a two parameter turbulence model. The two-way coupling was accounted for by adding source terms in the conservation equations of the continuous phase to take into account the interaction with the dispersed phase. Bubble tracking is achieved through a Lagrangian approach. Here the equations of motion are solved taking into account the drag, pressure, buoyancy and gravity forces. The time-averaged flows along with the variables which characterize turbulence are analyzed for a wide range of gas flow-rates using Euler–Lagrangian simulations. These simulation predictions are validated with Euler–Eulerian simulations where the gas-phase distribution is captured as a void fraction and PIV experiments. The motion of bubbles induces turbulence in the flow. The applicability of two parameter models for turbulence like the standard k–ε model on time-averaged flow properties is addressed. From the results of the time averaged velocity field, turbulence intensity, turbulent viscosity and gas hold-up profiles, it is concluded that the Euler–Lagrangian model is applicable at lower gas flow-rates. The Euler–Eulerian approach was found to be valid at lower as well as higher gas flow-rates.     

稳定分层二层流室内实验早期格栅湍流结构

采用室内实验混合箱和粒子图像测速技术,本文研究了稳定分层无平均剪切二层流(上层淡水、下层盐水)振动湍流结构。对实验录像进行粒子图像测速技术处理,获得垂向二维流场(垂直于格栅平面)瞬时速度和涡量,并用于计算:①时均速度和时均涡量;②均方根速度;③均匀程度和各向同性程度;④平均流强度;⑤时均泰勒的欧拉积分长度尺度;⑥时均湍动能和时均湍动能垂向通量;⑦水平和垂向速度的欧拉频谱。结果显示:(1)格栅方棒处时间平均速度方向垂直向上,而其两侧的时间平均涡量正负交替,表明格栅附近射流结构占据主要位置且存在反向涡旋对。(2)均方根速度随着距离格栅(水平面)高程的增大而减小,并且满足高程的-1.425(接近-3/2)幂次律,表明格栅湍流均方根速度的垂向变化较为剧烈。(3)靠近混合箱边壁处的均匀程度和各向同性程度都大于1,表明靠近混合箱边壁处存在各向异性湍流。(4)格栅反湍流强度的量级非0,但是,相对较小,表明平均流强度较低,故而,本实验结果仍可与无平均流的情况作对比。(5)时均泰勒的欧拉积分长度尺度随着距离格栅(水平面)高程的增大而线性增大,表明随着湍流向上发展,涡的平均尺度增大。(6)时均湍动能和时均湍动能垂向通量随着距离格栅(水平面)高程的增大而减小,而时均湍动能垂向通量为正值,表明远离格栅时均湍动能衰减,但是,始终向上传递。(7)水平和垂向速度的欧拉频谱随着距离格栅(水平面)高程的增大而减小,幂次律介于ω^-1和ω^-5/3(ω为频率)之间,表明水平和垂向的湍流脉动能量远离格栅均衰减,并受湍流的有限雷诺数效应的影响。     

Eulerian simulation of sedimentation flows in vertical and inclined vessels

Sedimentation of particles in inclined and vertical vessels is numerically simulated using a finite volume method where the Eulerian multiphase model is applied. The particulate phase as well as the fluid phase is regarded as a continuum while the viscosity and solid stress of the particulate phase are modelled by the kinetic theory of granular flows. The numerical results show an interesting phenomenon of the emergence of two circulation vortices of the sedimentation flow in a vertical vessel but only one in the inclined vessel. Several sensitivity tests are simulated to understand the factors that influence the dual-vortex flow structure in vertical sedimentation. Results show that a larger fluid viscosity makes the two vortex centres much closer to each other and the boundary layer effect at lateral walls is the key factor to induce this phenomenon. In the fluid boundary layer particles settle down more rapidly and drag the local carrier fluid to flow downward near the lateral walls and thus form the dual-vortex flow pattern.     

Enumeration of totally positive Grassmann cells

Postnikov (Webs in totally positive Grassmann cells, in preparation) has given a combinatorially explicit cell decomposition of the totally nonnegative part of a Grassmannian, denoted Gr_k,n⁺, and showed that this set of cells is isomorphic as a graded poset to many other interesting graded posets. The main result of our work is an explicit generating function which enumerates the cells in Gr_k,n⁺ according to their dimension. As a corollary, we give a new proof that the Euler characteristic of Gr_k,n⁺ is 1. Additionally, we use our result to produce a new q-analog of the Eulerian numbers, which interpolates between the Eulerian numbers, the Narayana numbers, and the binomial coefficients.     

The reduction of graph families closed under contraction

Let be a family of graphs. Suppose there is a nontrivial graph H such that for any supergraph G of H, G is in if and only if the contraction G/H is in . Examples of such an : graphs with a spanning closed trail; graphs with at least k edge-disjoint spanning trees; and k-edge-connected graphs (k fixed). We give a reduction method using contractions to find when a given graph is in and to study its structure if it is not in . This reduction method generalizes known special cases.     

The traveling salesman problem in graphs with some excluded minors

Given a graph and a length function defined on its edge-set, the Traveling Salesman Problem can be described as the problem of finding a family of edges (an edge may be chosen several times) which forms a spanning Eulerian subgraph of minimum length. In this paper we characterize those graphs for which the convex hull of all solutions is given by the nonnegativity constraints and the classical cut constraints. This characterization is given in terms of excluded minors. A constructive characterization is also given which uses a small number of basic graphs.     

New results on the peak algebra

The peak algebra is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks. By exploiting the combinatorics of sparse subsets of [n−1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of . We discuss two peak analogs of the first Eulerian idempotent and construct a basis of semi-idempotent elements for the peak algebra. We use these bases to describe the Jacobson radical of and to characterize the elements of in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals of , j = 0,..., , such that is the linear span of sums of permutations with a common set of interior peaks and is the peak algebra. We extend the above results to , generalizing results of Schocker (the case j = 0). Aguiar supported in part by NSF grant DMS-0302423 Orellana supported in part by the Wilson Foundation     

The Eulerian distribution on involutions is indeed unimodal

Let In,k (respectively Jn,k) be the number of involutions (respectively fixed-point free involutions) of {1,…,n} with k descents. Motivated by Brenti's conjecture which states that the sequence In,0,In,1,…,In,n−1 is log-concave, we prove that the two sequences In,k and J2n,k are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers an,k such that

     

Polypseudologarithms revisited

Lee, in a series of papers, described a unified formulation of the statistical thermodynamics of ideal quantum gases in terms of the polylogarithm functions, . It is aimed here to investigate the functions , for s=0,−1,−2,…, which are, following Lee, referred to as the polypseudologarithms (or polypseudologs) of order n≡−s. Various known results regarding polypseudologs, mainly obtained in widely differing contexts and currently scattered throughout the literature, have been brought together along with many new results and insights and they all have been proved in a simple and unified manner. In addition, a new general explicit closed-form formula for these functions involving the Carlitz-Scoville higher tangent numbers has been established.