Influence of the nutation angle in gravitational settling of particles near vortices and in turbulence

The knowledge of the conditions in which particles denser than fluid settle is important in many areas of engineering, environmental sciences, meteorology, etc.For particle flows influenced by vortices, research mainly related to steady horizontal vortices has been undertaken. In this paper we determine the influence of the inclination of the vortex axis in the gravitational settling of particles.The results obtained, in relation to the trajectories, are qualitatively similar to previous ones for horizontal vortices. The main difference is this: in a horizontal vortex particles always remain in a plane perpendicular to the vortex axis and in an inclined vortex (angle θ) particles do not remain on that plane because there is a component v_tcosθ that takes them out.The average fall velocity 〈v_z〉 has an asymptote to the dimensionless terminal velocity v_t; this tendency is faster as the Stokes Number St increases and as v_t decreases. A fundamental result is the following: as θ decreases, v_t is reached faster because the component of the velocity u of the Rankine vortex over the Oz direction is small and because the v_t component that tries to keep the particles in a plane perpendicular to the vortex axis is small, so the vortex takes action over the particles for a small period of time.     

Heat and mass transfer characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing velocities

The mechanical and thermal characteristics of the self-similar boundary-layer flows induced by continuous surfaces stretched with rapidly decreasing power-law velocities U _w∝x ^m , m<?1 are considered. Comparing to the well studied cases of the increasing stretching velocities (m>0) several new features of basic significance have been found. Thus: (i) for m<?1 the boundary layer equations admit self-similar solutions only if a lateral suction is applied; (ii) the dimensionless suction velocity f _w<0 must be strong enough, i.e. f _w<f _w,max(m) where f _w,max(m) depends on m so that its absolute maximum max (f _w,max(m))=?2.279 is reached for m→?∞, while for m→?1, f _w,max(m)→?∞; (iii) the case {m→?∞, f _w,max(m)=?2.279} of the flow boundary value problem is isomorphic to the stretching problems with exponentially decreasing velocities U _w∝e ^ax with arbitrary a<0; (iv) for any fixed m<?1 and f _w<f _w,max(m) the flow problem admits a non-denumerable infinity of multiple solutions corresponding to the values of the dimensionless skin friction f ^″(0)≡s belonging to a finite interval s∈ [s _min(f _w,m), s _max(f _w,m)]; (v) the solution is only unique for f _w=f _w,max(m) where s=s _min(f _w,m)= s _max(f _w,m) holds; (vi) to every one of the multiple solutions of the flow problem there corresponds a unique solution of the heat transfer problem with a wall temperature distribution T _w?T _∞∝x ⁿ and a well defined and distinct value of the dimensionless wall temperature gradient ?^′(0), except for the cases n=(|m|?1)/2 where ?^′(0) has the same value ?^′(0)=Pr·f _w for the whole class of flow solutions with s∈[s _min(f _w,m), s _max(f _w,m)]; (vii) for f _w→?∞ one obtains the `asymptotic suction profiles' corresponding to s=s _min(f _w,m)?f _w and ?^′(0)?Pr·f _w in an explicit analytic form. The paper includes several examples which illustrate the dependence of the heat and fluid flows induced by surfaces stretching with rapidly decreasing velocities on the physical parameters f _w, m, n and Pr.     

On the similarity solutions of wave propagation for a general class of non-linear dissipative materials

Deductive similarity analysis is employed to study one-dimensional wave propagation in rate dependent materials whose constitutive laws are special cases of Maxwellian materials (σ_t = φ(ε, σ)ε_t + ψ(ε, σ), ε = strain, σ = stress). The general problem is shown not to have a similar solution although many special cases have the independent similar variable (x ? c)/(t ? d)^e. These cases are studied and tabulated. Analytic similar solutions are presented for several cases and a discussion of permissable boundary conditions is given.     

Similarity reductions and exact solutions of generalized Bretherton equation with time-dependent coefficients

In this paper, we apply Lie-group formalism to the generalized Bretherton equation with variable coefficients u _tt +α(t)u _xx +β(t)u _xxxx +δ(t)u ^m +θ(t)u ⁿ =0, to investigate the symmetries. We derive the infinitesimals and the admissible forms of the coefficients that admit the classical symmetry group. The ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.     

Low Reynolds number effects in open-channel turbulent boundary layers

The present study reports measurements of a turbulent boundary layer in an open-channel flow using fiber-optic laser Doppler anemometry. The Reynolds numbers based on momentum thickness and depth of flow are in the range 750≤Re _θ ≤2,400 and 15,300≤Re _h ≤54,200, respectively. It is shown that an accurate estimate of the wall shear stress can be made by fitting a fifth-order polynomial to the near-wall data. The effect of Reynolds number on the mean turbulence intensity and triple correlation is examined using both conventional scaling laws and the recent scaling laws proposed by George and Castillo. The present results show that different scaling laws lead to different conclusions on low Reynolds number effects.     

Determination of particle exchange rates at over-flow weirs in horizontal fluidised beds by particle tracking velocimetry

Residence time distributions (RTDs) in horizontal fluidised beds have a huge effect on solid product properties and are influenced by the internal design of the apparatus, e.g. the separation into different compartments by weirs. Weirs can be passed in or against the overall solid transport direction, with the back-flow resulting in axial dispersion, which is a measure of the spread of the RTD. Therefore, the ratio of exchange rates at weirs under different fluidisation conditions provides information on axial dispersion. In this work, a methodology based on particle tracking velocimetry is presented to obtain information on the exchange rates of particles at weirs in horizontal fluidised beds. The internal recirculation is studied for over-flow weirs with respect to different fluidisation conditions, providing a first step towards determining the effects of weirs and fluidisation conditions on axial dispersion and RTDs in horizontal fluidised beds.     

Experimental Investigation of the Log-Law for an Adverse Pressure Gradient Turbulent Boundary Layer Flow at Re θ = 10000

We present an experimental investigation of a turbulent boundary layer flow at a significant adverse pressure gradient at Reynolds number Re _θ ?=?10000 using large field PIV. The testcase is designed to start from a zero pressure gradient flow at Re _θ ?=?8000 with a distinct log-law region following a slowly rising adverse pressure gradient. This allows to reveal a breakdown of the log-law under the effect of the adverse pressure gradient. The region described by the log-law is progressively reduced in terms of y ^?+? and then joins into a modified log-law which gives a good fit to the data up to at least y/δ ₉₉?≈?0.2. The scaling in the overlap region is demonstrated using the mean velocity slope diagnostic function, enabled due to the high quality of the PIV data. Locally, the velocity profile is measured down to the wall using long-range microscopic PIV with particle tracking velocimetry to determine the wall shear stress directly in the adverse pressure gradient region.     

Effect of Gaussian size distribution width on minimum fluidization velocity in tapered gas–solid fluidized beds

This paper investigated the effect of Gaussian distribution width, average particle diameter, particle loading, and the tapered angle on minimum fluidization velocity (U_mf) by conducting extensive experiments in tapered fluidized beds. Three powders with Gaussian size distribution and different distribution widths were used in the experiments. An increase in U_mf with increasing the average particle diameter, particle loading, and the tapered angle was observed. There was also a nonmonotonic behavior of U_mf as the Gaussian distribution width increased. An empirical correlation including dimensionless groups for predicting U_mf in tapered beds was developed in which the effect of distribution width was considered. The proposed correlation predictions were in good agreement with the experimental data, with a maximum deviation of 16.5% and average and standard deviations of, respectively, 6.4% and 7.4%. The proposed correlation was also compared with three earlier models, and their accuracy was discussed.     

Travel-time analysis of tracer tests in two geothermal fields

The most important results from a tracer test are whether or not tracer is detected at each observation well and the travel times to the wells that respond. A method developed by the authors for accurately calculating travel times for tracer movement in general flow fields enables the locations of major fractures in a reservoir to be deduced from the travel-time data. The procedure is applied here to data from Wairakei, New Zealand, and Palinpinon, Philippines.Notation H reservoir thickness, m - porosity, dimensionless - Q _c characteristic well volume flow rate, m³ s^-1 - R _c characteristic length, m - t _d time, s - t dimensionless time - t _td tracer travel time (without dispersion), s - t _t dimensionless tracer travel time - v _d background fluid speed, m s^-1 - v dimensionless background fluid speed - x _d Cartesian coordinate, m - x dimensionless Cartesian coordinate, m - y _d Cartesian coordinate, m - y dimensionless Cartesian coordinate     

A unified approach to closed-form solutions of moving heat-source problems

A closed-form model for the computation of temperature distribution in an infinitely extended isotropic body with a time-dependent moving-heat sources is discussed. The temperature solutions are presented for the sources of the forms: (i) ₀₁(t)=₀ exp(−λt), (ii) ₀₂(t) =₀(t/t ^*)exp(−λt), and ₀₃(t)=₀[1+a cos(ωt)], where λ and ω are real parameters and t ^* characterizes the limiting time. The reduced (or dimensionless) temperature solutions are presented in terms of the generalized representation of an incomplete gamma function Γ(α,x;b) and its decomposition C _Γ and S _Γ. The solutions are presented for moving, -point, -line, and -plane heat sources. It is also demonstrated that the present analysis covers the classical temperature solutions of a constant strength source under quasi-steady state situations. Received on 13 June 1997     

Decay of Almost Periodic Solutions¶of Conservation Laws

We consider the asymptotic behavior of solutions of systems of inviscid or viscous conservation laws in one or several space variables, which are almost periodic in the space variables in a generalized sense introduced by Stepanoff and Wiener, which extends the original one of H. Bohr. We prove that if u(x,t) is such a solution whose inclusion intervals at time t, with respect to ?>0, satisfy l _epsiv;(t)/t→0 as t→∞, and such that the scaling sequence u ^T (x,t)=u(T x,T t) is pre-compact as t→∞ in L _loc ¹(? ^{d +1} ₊, then u(x,t) decays to its mean value \(\), which is independent of t, as t→∞. The decay considered here is in L ¹ _loc of the variable ξ≡x/t, which implies, as we show, that \(\) as t→∞, where M _x denotes taking the mean value with respect to x. In many cases we show that, if the initial data are almost periodic in the generalized sense, then so also are the solutions. We also show, in these cases, how to reduce the condition on the growth of the inclusion intervals l _?(t) with t, as t→∞, for fixed ? > 0, to a condition on the growth of l _?(0) with ?, as ?→ 0, which amounts to imposing restrictions only on the initial data. We show with a simple example the existence of almost periodic (non-periodic) functions whose inclusion intervals satisfy any prescribed growth condition as ?→ 0. The applications given here include inviscid and viscous scalar conservation laws in several space variables, some inviscid systems in chromatography and isentropic gas dynamics, as well as many viscous 2 × 2 systems such as those of nonlinear elasticity and Eulerian isentropic gas dynamics, with artificial viscosity, among others. In the case of the inviscid scalar equations and chromatography systems, the class of initial data for which decay results are proved includes, in particular, the L ^∞ generalized limit periodic functions. Our procedures can be easily adapted to provide similar results for semilinear and kinetic relaxations of systems of conservation laws.     

Onset velocity of circulating fluidization and particle residence time distribution: A CFD–DEM study

Until now, the onset velocity of circulating fluidization in liquid–solid fluidized beds has been defined by the turning point of the time required to empty a bed of particles as a function of the superficial liquid velocity, and is reported to be only dependent on the liquid and particle properties. This study presents a new approach to calculate the onset velocity using CFD–DEM simulation of the particle residence time distribution (RTD). The onset velocity is identified from the intersection of the fitted lines of the particle mean residence time as a function of superficial liquid velocity. Our results are in reasonable agreement with experimental data. The simulation indicates that the onset velocity is influenced by the density and size of particles and weakly affected by riser height and diameter. A power-law function is proposed to correlate the mean particle residence time with the superficial liquid velocity. The collisional parameters have a minor effect on the mean residence time of particles and the onset velocity, but influence the particle RTD, showing some humps and trailing. The particle RTD is found to be related to the particle trajectories, which may indicate the complex flow structure and underlying mechanisms of the particle RTD.     

Finite element modeling of elasto-plastic contact between rough surfaces

This paper presents a finite element calculation of frictionless, non-adhesive, contact between a rigid plane and an elasto-plastic solid with a self-affine fractal surface. The calculations are conducted within an explicit dynamic Lagrangian framework. The elasto-plastic response of the material is described by a J₂ isotropic plasticity law. Parametric studies are used to establish general relations between contact properties and key material parameters. In all cases, the contact area A rises linearly with the applied load. The rate of increase grows as the yield stress σ_y decreases, scaling as a power of σ_y over the range typical of real materials. Results for A from different plasticity laws and surface morphologies can all be described by a simple scaling formula. Plasticity produces qualitative changes in the distributions of local pressures in the contact and of the size of connected contact regions. The probability of large local pressures is decreased, while large clusters become more likely. Loading-unloading cycles are considered and the total plastic work is found to be nearly constant over a wide range of yield stresses.     

Drag law for monodisperse gas–solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres

Gas–solid momentum transfer is a fundamental problem that is characterized by the dependence of normalized average fluid–particle force F on solid volume fraction ? and the Reynolds number based on the mean slip velocity Re_m. In this work we report particle-resolved direct numerical simulation (DNS) results of interphase momentum transfer in flow past fixed random assemblies of monodisperse spheres with finite fluid inertia using a continuum Navier–Stokes solver. This solver is based on a new formulation we refer to as the Particle-resolved Uncontaminated-fluid Reconcilable Immersed Boundary Method (PUReIBM). The principal advantage of this formulation is that the fluid stress at the particle surface is calculated directly from the flow solution (velocity and pressure fields), which when integrated over the surfaces of all particles yields the average fluid–particle force. We demonstrate that PUReIBM is a consistent numerical method to study gas–solid flow because it results in a force density on particle surfaces that is reconcilable with the averaged two-fluid theory. The numerical convergence and accuracy of PUReIBM are established through a comprehensive suite of validation tests. The normalized average fluid–particle force F is obtained as a function of solid volume fraction ? (0.1 ? ? ? 0.5) and mean flow Reynolds number Re_m (0.01 ? Re_m ? 300) for random assemblies of monodisperse spheres. These results extend previously reported results of  and  to a wider range of ?, Re_m, and are more accurate than those reported by Beetstra et al. (2007). Differences between the drag values obtained from PUReIBM and the drag correlation of Beetstra et al. (2007) are as high as 30% for Re_m in the range 100–300. We take advantage of PUReIBM’s ability to directly calculate the relative contributions of pressure and viscous stress to the total fluid–particle force, which is useful in developing drag correlations. Using a scaling argument, Hill et al. (2001b) proposed that the viscous contribution is independent of Re_m but the pressure contribution is linear in Re_m (for Re_m > 50). However, from PUReIBM simulations we find that the viscous contribution is not independent of the mean flow Reynolds number, although the pressure contribution does indeed vary linearly with Re_m in accord with the analysis of Hill et al. (2001b). An improved correlation for F in terms of ? and Re_m is proposed that corrects the existing correlations in Re_m range 100–300. Since this drag correlation has been inferred from simulations of fixed particle assemblies, it does not include the effect of mobility of the particles. However, the fixed-bed simulation approach is a good approximation for high Stokes number particles, which are encountered in most gas–solid flows. This improved drag correlation can be used in CFD simulations of fluidized beds that solve the average two-fluid equations where the accuracy of the drag law affects the prediction of overall flow behavior.     

Calculation of tracer travel times in fractured geothermal reservoirs

A new method is presented for calculating the time taken for tracer to move between wells in a fractured geothermal reservoir. The reservoir model considered is a two-dimensional confined layer, but many wells and a background regional flow can be included. Also, either a straight or dog-leg, finite length, high permeability fracture can be included. The fracture can alternatively be considered as a barrier to lateral flow. The flow field is represented by complex potentials which are used to accurately calculate the streamline locations and tracer travel times are evaluated by numerical integration along the streamlines. The methods developed are used to model the dispersion of tracer produced by large-scale differences in the flow paths along which the tracer travels from the release well to the observation well(s).Notation C _d concentration, kg m^-3 - C dimensionless concentration - C _obs dimensionless concentration at the observation well - f dimensionless distance between the injection and production wells - h _d fracture half length, m - h dimensionless fracture half length - H reservoir thickness, m - Ln(·) complex algorithm - M mass of tracer released, kg - n porosity, dimensionless - N _b number of streamlines calculated for blob release - N _f number of subdivisions for the high permeability fracture - N _w number of streamlines calculated for injection well release - ¢P _d complex potential, m² s^-1 - P dimensionless complex potential - Q _c characteristic well volume flow rate, m³ s^-1 - q _p production well volume flow rate, m³ s^-1 - R _c characteristic length, m - t _d time, s - t _b dimensionless response start time - t dimensionless time - t _td tracer travel time (without dispersion), s - t _t dimensionless tracer travel time - u average fluid velocity, ms^-1 - v _d background fluid speed, m s^-1 - v dimensionless background fluid speed - x _d Cartesian coordinate, m - x dimensionless Cartesian coordinate - y _d Cartesian coordinate, m - y dimensionless Cartesian coordinate - (·) Dirac delta distribution - _d velocity potential, m² s^-1 - dimensionless velocity potential - angle from the positive x axis to the direction of the background flow - _d stream function, m² s^-1 - dimensionless stream function - complex number - circle mapped to the fracture by the Joukowski transformation - region occupied by the blob - complex number - p production/observation well - r release well     

Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold

We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation u_t=D(u^m/m) = div (u^m-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for x ? \mathbb R^d{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold (\mathbb R^d,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard \mathbb R^d{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.     

Rheology of SiO2/(acrylic polymer/epoxy) suspensions. II. Nonlinear stress relaxation

Large deformation, nonlinear stress relaxation modulus G(t, γ) was examined for the SiO₂ suspensions in a blend of acrylic polymer (AP) and epoxy (EP) with various SiO₂ volume fractions (?) at various temperatures (T). The AP/EP contained 70 vol.% of EP. At ??≤?30 vol.%, the SiO₂/(AP/EP) suspensions behaved as a viscoelastic liquid, and the time-strain separability, G(t, γ)?=?G(t)h(γ), was applicable at long time. The h(γ) of the suspensions was more strongly dependent on γ than that of the matrix (AP/EP). At ??=?35 vol.% and T?=?100°C, and ??≥?40 vol.%, the time-strain separability was not applicable. The suspensions exhibited a critical gel behavior at ??=?35 vol.% and T?=?100°C characterized with a power law relationship between G(t) and t; G(t)?∝?t ^???n . The relaxation exponent n was estimated to be about 0.45, which was in good agreement with the result of linear dynamic viscoelasticity reported previously. G(t, γ) also could be approximately expressed by the relation $G(t,\gamma) \propto t^{-n^{\prime}}$ at ??=?40 vol.%. The exponent n ^′ increased with increasing γ. This nonlinear stress relaxation behavior is attributable to strain-induced disruption of the network structure formed by the SiO₂ particles therein.     

Incorporating engineering intuition for parameter estimation in thermal sciences

This paper proposes a new method of incorporating priors based on engineering intuition for solving inverse problems. The thesis of this paper is that if an asymptote can be found to a problem in applied sciences or engineering, estimation of parameters can be first done for this asymptotic variant, which in principle should be simpler, since one or more parameters of the original problem may vanish for the asymptotic variant. Even so, by solving the inverse problem associated with the asymptotic variant, estimates of key parameters of the full problem can be obtained. This information can then be quantitatively incorporated as priors in the estimation of parameters for the full version of the problem which we call as prior generation through asymptotic variant. The goal is to see if this methodology will significantly reduce the uncertainties in the resulting estimates. To demonstrate this methodology, the classic problem of unsteady heat transfer from a one dimensional fin is chosen. The inverse problem is posed as the simultaneous estimation of the temperature dependent transfer coefficient (h _θ ) and the thermal diffusivity (α) of the fin material, given experimentally measured temperature–time histories at various locations along the fin. The asymptotic variant θ _(x,t) is the steady state problem where the influence of thermal diffusivity vanishes. Using surrogate temperature data generated from assumed values of h _θ , first the asymptotic variant of the problem is solved using the Markov Chain Monte Carlo method in a Bayesian framework to generate an estimate of h _θ . The estimate of h _θ is then used as an informative prior for solving the inverse problem of determining h _θ and α from θ _(x,t), and the effect of prior is quantitatively assessed by performing estimation with and without the prior. Finally, for purposes of validation, in-house experiments have been done where θ _(x,t) is generated using liquid crystal thermography and these data are used to estimate h _θ and α. A comparison of experimentally measured temperatures with those that are simulated by using estimated values of (h _θ , α) to solve the governing equation to the problem is also done.     

Effects of boundary slippage on thin-film lubrication between two nonparallel plane plates

Hydrodynamic lubrications between two planeplates with an intersection angle θ have been investigated using the boundary slippage theory,and relations are obtainedbetween dimensionless pressures and coordinate x,betweenbearing capacity,friction force,friction coefficient and dimensionless slipping size factor.The results show that bearing capacity of two plane plates without boundary slippagesignificantly increases with increasing intersection angle θwhen 0 θ 1,whereas decreases with increasing intersection angle θ when θ 1.The results also show thatnegative pressure occurs in fluid entrance region and bearingcapacity decreases,and friction force and friction coefficientincrease with the increase of dimensionless slipping size factor.     

Conservation laws of multidimensional diffusion--convection equations

All possible linearly independent local conservation laws for n-dimensional diffusion–convection equations u _t=(A(u)) _ii +(B ⁱ(u)) _i were constructed using the direct method and the composite variational principle. Application of the method of classification of conservation laws with respect to the group of point transformations [R.O.~Popovych, N.M. Ivanova, J. Math. Phys. 46, 2005, 043502 (math-ph/0407008)] allows us to formulate the result in explicit closed form. Action of the symmetry groups on the conservation laws of diffusion equations is investigated and generating sets of conservation laws are constructed.