Modeling the dynamics of colliding particles in a turbulent shear flow

A kinetic model for the probability density function (PDF) of the particle velocity in a turbulent flow with account for particle collisions is presented. The model is tested by comparison with the results of a numerical experiment for a nonstationary homogeneous shear layer. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 105–112, July–August, 1998. The study received financial support from the International Science Foundation INTAS (project No. 94-4348) and the Russian Foundation for Basic Research (project No. 97-01-00398).     

Impact comminution of solids due to local kinetic energy of high shear strain rate: I. Continuum theory and turbulence analogy

The modeling of high velocity impact into brittle or quasibrittle solids is hampered by the unavailability of a constitutive model capturing the effects of material comminution into very fine particles. The present objective is to develop such a model, usable in finite element programs. The comminution at very high strain rates can dissipate a large portion of the kinetic energy of an impacting missile. The spatial derivative of the energy dissipated by comminution gives a force resisting the penetration, which is superposed on the nodal forces obtained from the static constitutive model in a finite element program. The present theory is inspired partly by Grady's model for expansive comminution due to explosion inside a hollow sphere, and partly by analogy with turbulence. In high velocity turbulent flow, the energy dissipation rate gets enhanced by the formation of micro-vortices (eddies) which dissipate energy by viscous shear stress. Similarly, here it is assumed that the energy dissipation at fast deformation of a confined solid gets enhanced by the release of kinetic energy of the motion associated with a high-rate shear strain of forming particles. For simplicity, the shape of these particles in the plane of maximum shear rate is considered to be regular hexagons. The particle sizes are assumed to be distributed according to the Schuhmann power law. The condition that the rate of release of the local kinetic energy must be equal to the interface fracture energy yields a relation between the particle size, the shear strain rate, the fracture energy and the mass density. As one experimental justification, the present theory agrees with Grady's empirical observation that, in impact events, the average particle size is proportional to the (−2/3) power of the shear strain rate. The main characteristic of the comminution process is a dimensionless number B_a (Eq. (37)) representing the ratio of the local kinetic energy of shear strain rate to the maximum possible strain energy that can be stored in the same volume of material. It is shown that the kinetic energy release is proportional to the (2/3)-power of the shear strain rate, and that the dynamic comminution creates an apparent material viscosity inversely proportional to the (1/3)-power of that rate. After comminution, the interface fracture energy takes the role of interface friction, and it is pointed out that if the friction depends on the slip rate the aforementioned exponents would change. The effect of dynamic comminution can simply be taken into account by introducing the apparent viscosity into the material constitutive model, which is what is implemented in the paper that follows.     

Large eddy simulation of bluff body wake in planar shear flow

Large eddy simulation of planar shear flow past a square cylinder has been investigated. Dynamic Smagorinsky model has been used to model subgrid scale stress. The shear parameter, K, namely the nondimensional streamwise velocity gradient in the lateral direction, is 0.0, 0.1 and 0.2. Reynolds number based on the centerline velocity is fixed at Re=21400. The time and span‐averaged velocity components, pressure coefficient, Reynolds stresses for uniform are in good agreement with the literature. In shear flow the calculated flow structure and mean velocity components are shown to be markedly different from those of the uniform flow. With increasing shear parameter, the cylinder wake is dominated by clockwise vortices. Both the velocity components in shear flow are compared with respective components in uniform flow. Comparison of normal and shear stresses between shear and no shear case have also been presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Flow birefringence and rheological measurements on shear induced micellar structures

Aqueous solutions of cationic surfactant systems with strongly binding counterions show the striking phenomenon of shear induced phase transitions. At low shear rates or angular frequencies, the solutions exhibit Newtonian flow. At high rates of shear, however, the rheological properties change dramatically. Above a well defined threshold value of the velocity gradient, a supermolecular structure can be formed from micellar aggregates. This shear induced structure (SIS) behaves like a gel and exhibits strong flow birefringence. The formation of the shear induced structure is very complicated and depends on the specific conditions of the surfactant system. In this paper we discuss new results which have been obtained from rheological measurements and from flow birefringence data. We examine the stability of the shear induced state as a function of temperature, surfactant concentration and salt concentration and we analyse the effect of solubilisation of alcohols and hydrocarbons. The results are interpreted in terms of a kinetic model which accounts for the observed behavior.Dedicated to the 60. birthday of Prof. H. Harnisch, Hoechst AGPartly presented at the 2nd Conference of European Rheologists, Prague, June 17–20, 1986     

Quadruple hot-wire probes in a simulated wall flow

A method for calibration and measurement with a fourwire probe is described. The method does not require any assumption about the response of the wires and it is not necessary to know the exact probe geometry. Measurements in a two-dimensional turbulent boundary layer showed large errors in the shear stress, although the probe had a wire arrangement practically insensitive to mean velocity gradients normal to the wall. The problem seems to be caused by the strong instantaneous spanwise velocity gradients, as suggested by the computed response of a simple probe model inserted into a numerical flow obtained by direct simulation.     

一类平衡大气边界层边界条件研究

基于标准k-ε湍流模型,首先利用湍流粘度方程和剪切应力在整个边界层内恒定的假设,推导出一类耗散率表达式,并根据常用的湍动能入口剖面方程以及平均风速剖面方程,计算获得相应的耗散率方程;然后在输运方程中添加自定义源项,通过已经确定的平均速度方程、湍动能方程、耗散率方程计算得到相应输运方程的自定义源项表达式,并进行空风洞数值模拟,从而得到了一类满足平衡大气边界层的来流边界条件．通过将这种边界条件与由湍流平衡条件得到的边界条件进行比较,表明本方法获得的边界条件更适用．并且,本方法无需考虑修正壁面函数和修正湍流模型常数,因而计算更为简单,可为平衡大气边界层的研究提供一种新的思路．     

Steady-shear viscosity and transient stress response for elasto-thixotropic fluids

A phenomenological model for dispersed systems which exhibit complex rheological behaviour such as shear and time-dependent viscosity, yield stress, and elasticity is proposed. The model extends the Quemeda model to describe the viscosity function with a structural parameter λ which varies according to different kinetic orders of particle aggregation and segregation. The transient stress response is obtained by solving an instantaneous Maxwell model with an assumed shear modulus functionG of the same form as the viscosity function η. Accuracy of the proposed model is verified experimentally with the results obtained for two oil (creosote)/water emulsions. The model that gives the best fit of experimental data appears to be the one with kinetic ordersn=m=2.     

The evolution equation of joint Pdf of turbulent velocity and dissipation

According to the hypothesis that the dissipation of turbulent kinetic energy satisfies log-normal distribution, a stochastic model of dissipation is provided and the Langevin model^[6] of velocity is modified. Then a joint Pdf equation of turbulent velocity and dissipation is derived. We solve numerically the joint Pdf equation using Monte Carlo method and obtain satisfactory results for decaying turbulence and homogeneous turbulent shear flow. The preliminary results show that the model is well working.     

Effect of inelastic collisions on the first-order slip coefficients for a diatomic gas with rotational degrees of freedom

When solving problems of inhomogeneous gas dynamics in the slip regime, it is necessary to know the boundary conditions for the velocity, temperature, heat fluxes, etc., that is, the boundary conditions for the gas macroparameters. In particular, such problems arise in developing the theory of thermophoresis of moderately large aerosol particles [1].The problem of monatomic and molecular (di- and polyatomic) gas slip along a boundary surface is considered in many publications (see, for example, [2–8]). The first-order effects include the isothermal and thermal gas slips characterized by the coefficients C_m and K_TS, respectively.In contrast to a monatomic gas, the molecules of diatomic and polyatomic gases have internal degrees of freedom, which considerably complicates the kinetic equation [9]. The lack of reliable models for the intermolecular interaction potential predetermines the need to construct model kinetic equations [10].In this study, for a diatomic gas whose molecules have rotational degrees of freedom, we propose a model kinetic equation obtained by developing the approach described in [6]. With the use of this model equation, the problem of diatomic gas slip along a plane surface is solved. As a result, for diatomic gases the coefficients C_m and K_TS, which depend on the thermophysical gas parameters and the intensity of inelastic collisions, are obtained.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 176–182. Original Russian Text Copyright © 2004 by Poddoskin.     

Comparative study of lattice‐Boltzmann and finite volume methods for the simulation of laminar flow through a 4:1 planar contraction

In the present paper, a comparative study of numerical solutions for Newtonian fluids based on the lattice‐Boltzmann method (LBM) and the classical finite volume method (FVM) is presented for the laminar flow through a 4:1 planar contraction at a Reynolds number of value one, Re=1. In this study, the stress field for LBM is directly obtained from the distribution function. The calculations of the stress based on the FVM‐data use the evaluations of velocity gradients with finite differences. The stress field for both LBM and FVM is expressed in the present study in terms of the shear stress and the first normal stress difference. The lateral and axial profiles of the velocity, the shear stress and the first normal stress difference for both methods are investigated. It is shown that the LBM results for the velocity and the stresses are in excellent agreement with the FVM results. Copyright © 2004 John Wiley & Sons, Ltd.     

Flow field investigation in rotating rib-roughened channel by means of particle image velocimetry

The turbulent velocity field over the rib-roughened wall of an orthogonally rotating channel is investigated by means of two-dimensional particle image velocimetry (PIV). The flow direction is outward, with a bulk Reynolds number of 1.5 × 10⁴ and a rotation number ranging from 0.3 to 0.38. The measurements are obtained along the wall-normal/streamwise plane at mid-span. The PIV system rotates with the channel, allowing to measure directly the relative flow velocity with high spatial resolution. Coriolis forces affect the stability of the boundary layer and free shear layer. Due to the different levels of shear layer entrainment, the reattachment point is moved downstream (upstream) under stabilizing (destabilizing) rotation, with respect to the stationary case. Further increase in rotation number pushes further the reattachment point in stabilizing rotation, but does not change the recirculation length in destabilizing rotation. Turbulent activity is inhibited along the leading wall, both in the boundary layer and in the separated shear layer; the opposite is true along the trailing wall. Coriolis forces affect indirectly the production of turbulent kinetic energy via the Reynolds shear stresses and the mean shear. Two-point correlation is used to characterize the coherent motion of the separated shear layer. Destabilizing rotation is found to promote large-scale coherent motions and accordingly leads to larger integral length scales; on the other hand, the spanwise vortices created in the separating shear layer downstream of the rib are less organized and tend to be disrupted by the three-dimensional turbulence promoted by the rotation. The latter observation is consistent with the distributions of span-wise vortices detected in instantaneous flow realizations.     

A dynamic two-equation Sub Grid Scale model

In this paper we demonstrate that the transport equation of the generalised subgrid scale (SGS) turbulent stress tensor is form-invariant but not frame-indifferent under Euclidean transformations of the frame. A new closure equation between the generalized SGS turbulent stress tensor and the resolved kinematic quantities is proposed. The closure equation at the basis of the proposed model (Two-Equation Model, TEM): a) respects the principle of the turbulence frame indifference [1]; b) takes into account both the anisotropy of the turbulence velocity scales and turbulence length scales; c) removes any balance assumption between the production and dissipation of SGS turbulent kinetic energy; d) assumes scale similarity in the definition of the second-order tensor representing the turbulent velocity scales. In the proposed model: a) the closure coefficient C which appears in the constitutive equation is uniquely determined without using Germanos dynamic procedure [2]; b) the generalized SGS turbulent stress tensor is related exclusively to the generalized SGS turbulent kinetic energy (which is calculated by means of its balance equation) and the modified Leonard tensor; c) the viscous dissipation of the generalized SGS turbulent kinetic energy is calculated by solving the balance equation. The proposed model is tested for a turbulent channel flow at Reynolds numbers (based on friction velocity and channel half-width) ranging from 180 to 2340.Received: 11 February 2004, Accepted: 20 August 2004, Published online: 22 February 2005PACS: 02.60.Cb, 47.27.Eq, 47.11. + j Correspondence to: F. Gallerano     

稠密气固两相流各向异性颗粒相矩方法

基于气体分子动力学和颗粒动理学方法,考虑颗粒速度脉动各向异性,建立颗粒相二阶矩模型.应用初等输运理论,对三阶关联项进行模化和封闭.考虑颗粒与壁面之间的能量传递和交换,建立颗粒相边界条件模型.采用Koch等计算方法模拟气固脉动速度关联矩.考虑气体-颗粒间相互作用,建立稠密气体-颗粒流动模型.数值模拟提升管内气固两相流动特性,模拟结果表明提升管内颗粒相湍流脉动具有明显的各向异性.预测颗粒速度、浓度和颗粒脉动速度二阶矩与Tartan等实测结果相吻合.模拟结果表明轴向颗粒速度脉动强度约为平均颗粒相脉动强度的1.5倍,轴向颗粒脉动能大约是径向颗粒脉动能3.0倍.     

Anisotropy measurements in the boundary layer over a flat plate with suction

Laser Doppler velocity measurements are carried out in a turbulent boundary layer subjected to concentrated wall suction (through a porous strip). The measurements are taken over a longitudinal distance of 9× the incoming boundary layer thickness ahead of the suction strip. The mean and rms velocity profiles are affected substantially by suction. Two-point measurements show that the streamwise and wall-normal autocorrelations of the streamwise velocity are reduced by suction. It is found that suction alters the redistribution of the turbulent kinetic energy k between its components. Relative to the no-suction case, the longitudinal Reynolds stress contributes more to k than the other two normal Reynolds stresses; in the outer region, its contribution is reduced which suggests structural changes in the boundary layer. This is observed in the anisotropy of the Reynolds stresses, which depart from the non-disturbed boundary layer. With suction, the anisotropy level in the near-wall region appears to be stronger than that of the undisturbed layer. It is argued that the mean shear induced by suction on the flow is responsible for the alteration of the anisotropy. The variation of the anisotropy of the layer will make the development of a turbulence model quite difficult for the flow behind suction. In that respect, a turbulence model will need to reproduce well the effects of suction on the boundary layer, if the model is to capture the effect of suction on the anisotropy of the Reynolds stresses.     

Hybrid RANS/LES of flow and heat transfer in round impinging jets

Fluid flow and convective heat transfer predictions are presented of round impinging jets for several combinations of nozzle-plate distances H/D = 2, 6 and 13.5 (where D is the nozzle diameter) and Reynolds numbers Re = 5000, 23,000 and 70,000 with the newest version of the k-ω model of Wilcox (2008) and three hybrid RANS/LES models. In the RANS mode of the hybrid RANS/LES models, the k-ω model is recovered. Three formulations are considered to activate the LES mode. The first model is similar to the hybrid models of Davidson and Peng (2003) and Kok et al. (2004). The turbulent length scale is replaced by the grid size in the destruction term of the k-equation and in the definition of the RANS eddy viscosity. As grid size, a maximum measure of the hexahedral grid cell is used. The second model has the same k-equation, but the eddy viscosity is the minimum of the k-ω eddy viscosity and the Smagorinsky eddy viscosity, following a proposal by Batten et al. (2004). The Smagorinsky eddy viscosity is formed with the cube root of the cell volume. The third model has, again, the same k-equation, but has an eddy viscosity which is an intermediate between the eddy viscosities of the first and second models. This is reached by using the cube root of the cell volume in the eddy viscosity formula of the first model.The simulation results are compared with experimental data for the high Reynolds number cases Re = 23,000 and Re = 70,000 and LES data for the low-Reynolds number case Re = 5000. The Reynolds numbers are defined with the nozzle diameter and the bulk velocity at nozzle outlet. At low nozzle-plate distance (the impingement plate is in the core of the jet), turbulent kinetic energy is overpredicted by RANS in the stagnation flow region. This leads to overprediction of the heat transfer rate along the impingement plate in the impact zone. At high nozzle-plate distance (the impingement plate is in the mixed-out region of the jet), the turbulence mixing is underpredicted by RANS in the shear layer of the jet which gives a too high length of the jet core. This also results in overprediction of the heat transfer rate in the impingement zone caused by too big temperature gradients at impingement.All hybrid RANS/LES models are able to correct the heat transfer overprediction of the RANS model. For good predictions at low nozzle-plate distance, it is necessary to sufficiently resolve the formation and development of the near-wall vortices in the jet impingement region. At high nozzle-plate distance, the essence is to capture the evolution and breakup of the flow unsteadiness in the shear layer of the jet, so that accurate mean and fluctuating velocity profiles are obtained in the impingement region. Although the models have a quite different theoretical justification and generate a quite different eddy viscosity in some flow regions, their overall results are very comparable. The reason is that in zones that are crucial for the results, the models behave similarly.     

Separation control by tangential blowing inside the bubble

 Experiments have been carried out investigating the effectiveness of steady tangential blowing (inside the separation bubble) to control an axisymmetric separated flow at low speeds. Turbulent boundary separation was induced on a contoured afterbody and the separated shear layer reattached on a narrow cylindrical sting. Measurements made consisted of model surface pressures, mean velocity, turbulent shear stress and kinetic energy profiles using a 2-component LDV system. The results explicitly demonstrate that blowing downstream of the separation location, but within the bubble, can be an effective means of separation control, considering both wall and wake flow reversals. Received: 16 October 1998/Accepted: 27 September 1999     

A k–$${\varepsilon}$$ turbulence closure model of an isothermal dry granular dense matter

The turbulent flow characteristics of an isothermal dry granular dense matter with incompressible grains are investigated by the proposed first-order k–\({\varepsilon}\) turbulence closure model. Reynolds-filter process is applied to obtain the balance equations of the mean fields with two kinematic equations describing the time evolutions of the turbulent kinetic energy and dissipation. The first and second laws of thermodynamics are used to derive the equilibrium closure relations satisfying turbulence realizability conditions, with the dynamic responses postulated by a quasi-linear theory. The established closure model is applied to analyses of a gravity-driven stationary flow down an inclined moving plane. While the mean velocity decreases monotonically from its value on the moving plane toward the free surface, the mean porosity increases exponentially; the turbulent kinetic energy and dissipation evolve, respectively, from their minimum and maximum values on the plane toward their maximum and minimum values on the free surface. The evaluated mean velocity and porosity correspond to the experimental outcomes, while the turbulent dissipation distribution demonstrates a similarity to that of Newtonian fluids in turbulent shear flows. When compared to the zero-order model, the turbulent eddy evolution tends to enhance the transfer of the turbulent kinetic energy and plane shearing across the flow layer, resulting in more intensive turbulent fluctuation in the upper part of the flow. Solid boundary as energy source and sink of the turbulent kinetic energy becomes more apparent in the established first-order model.     

可压缩均匀剪切湍流的二阶矩模拟

基于对可压缩湍流中脉动压力场和脉动速度场特征的理论分析以及ＤＮＳ结果,建立了可均匀剪切湍流中压力－变形率关联的压缩性修正模式,应用这个模式,加上Ｓａｒｋａｒ等建立的脉动体胀率项（ｄｉｌａｔａｔｉｏｎａｌ ｔｅｒｍｓ）的模式,预测可压缩均匀剪切湍流随时间的发展,所得雷诺应力各是性张量的平衡值与Ｂｌａｉｓｄｅｌｌ等的ＤＮＳ数据非常一致。这个模式准确地预测出均匀剪切湍流中压缩性导致的雷诺应力结构的“流向     

Control of diffuser jet flow: turbulent kinetic energy and jet spreading enhancements assisted by a non-thermal plasma discharge

An axisymmetric air jet exhausting from a 22-degree-angle diffuser is investigated experimentally by particle image velocimetry (PIV) and stereo-PIV measurements. Two opposite dielectric barrier discharge (DBD) actuators are placed along the lips of the diffuser in order to force the mixing by a co-flow actuation. The electrohydrodynamic forces generated by both actuators modify and excite the turbulent shear layer at the diffuser jet exit. Primary air jet velocities from 10 to 40 m/s are studied (Reynolds numbers ranging from 3.2 to 12.8 × 10⁴), and baseline and forced flows are compared by analysing streamwise and cross-stream PIV fields. The mixing enhancement in the near field region is characterized by the potential core length, the centreline turbulent kinetic energy (TKE), the integrated value of the TKE over various slices along the jet, the turbulent Reynolds stresses and the vorticity fields. The time-averaged fields demonstrate that an effective increase in mixing is achieved by a forced flow reattachment along the wall of the diffuser at 10 m/s, whereas mixing enhancement is realized by excitation of the coherent structures for a primary velocity of 20 and 30 m/s. The actuation introduces two pairs of contra-rotating vortices above each actuator. These structures entrain the higher speed core fluid toward the ambient air. Unsteady actuations over Strouhal numbers ranging from 0.08 to 1 are also studied. The results suggest that the excitation at a Strouhal number around 0.3 is more effective to enhance the turbulence kinetic energy in the near-field region for primary jet velocity up to 30 m/s.