Simulation of viscous liquid flow under the action of a pressure drop with channel filling

The modeling of the hydrodynamic process, realized during liquid drain from the blender with the simultaneous filling of the vertically fixed mold, was carried out. The mathematical problem is formulated in a creeping flow of the Newtonian liquid approximation. On the basis of the boundary element method, a numerical algorithm for solving the problem in a planar formulation is developed. Parametric studies of the main characteristics of the process as functions of control parameters are performed. The parametric investigations of the process, including the stages of the container emptying, jet formation, liquid spreading on the horizontal surface, and filling after contact of the side walls of the free surface front, were carried out.     

Modeling of the Viscous Flow with a Free Surface inside a Rotating Horizontal Cylinder

The flow of a high-viscosity liquid that partially fills a horizontal cylinder rotating at constant velocity is considered. A problem is mathematically formulated in the creeping-flow approximation and is numerically solved by the boundary-element method. Parametric studies of the kinematic and dynamic characteristics of the process as functions of control parameters are performed. Two flow patterns are revealed, and critical parameter values separating these patterns are found. Curves for the specific flow power as a function of the control parameters are obtained; these curves characterize the liquid dispersion intensity in the flow studied.__________Translated from Teoreticheskie Osnovy Khimicheskoi Tekhnologii, Vol. 39, No. 3, 2005, pp. 303–309.Original Russian Text Copyright © 2005 by Shrager, Shtokolova, Yakutenok, Milekhin, Merkulov, Banzula, Karyazov, Glushkov.     

数值模拟中低Reynolds数下简单剪切流流过单液滴的稳态流场(英文)

This work presents a numerical investigation on steady internal, external and surface flows of a liquid sphere im-mersed in a simple shear flow at low and intermediate Reynolds numbers. The control volume formulation is adopted to solve the governing equations of two-phase flow in a 3-D spherical coordinate system. Numerical re-sults show that the streamlines for Re=0 are closed Jeffery orbits on the surface of the liquid sphere, and also closed curves outside and inside the liquid sphere. However, the streamlines have intricate and non-closed struc-tures for Re≠0. The flow structure is dependent on the values of Reynolds number and interior-to-exterior vis-cosity ratio.     

Numerical simulation of steady flow past a liquid sphere immersed in simple shear flow at low and moderate Re

This work presents a numerical investigation on steady internal, external and surface flows of a liquid sphere im-mersed in a simple shear flow at low and intermediate Reynolds numbers. The control vol...     

Destabilization of leidenfrost boiling by a sudden rise of ambient pressure

The contact of two liquids, one at a temperature significantly above the boiling point of the other, can lead to fast vapor formation on an explosive time scale causing mechanical damage. Such an event, generally termed a “vapor explosion”, has been recognized as the cause of accidents in a wide variety of smelting and liquefied natural gas (LNG) industries. The increase of awareness of the problem, especially in hypothetical accidents in liquid metal fast breeder reactors (LMFBR), has resulted in a broad research effort to investigate the sequence of events and the necessary conditions leading to a vapor explosion.A key step in the explosion scenario has been identified as the destabilization of film boiling, resulting in liquid-liquid contact over a large area. It is found experimentally that, when the ambient pressure is suddenly increased around an evaporating volatile drop suspended above a hot liquid surface, the vapor film may or may not collapse. A theory is developed here in which instability corresponds to resonance between the oscillations in the creeping-flow vapor film thickness and the capillary waves on the underside of the drop surface.Two separate estimates are made of the critical vapor film thickness at which instability occurs, one by a creeping flow model of the film thickness oscillation, and the other by a Taylor-Helmholtz instability model. The two models are in agreement with each other and with the data.     

Numerical simulation of moving free surface problems in polymer processing using volume‐of‐fluid method

We have developed a numerical algorithm based on 2D/3D finite element method for solving non‐Newtonian fluid flow with the moving free surface encountered in polymer processing. The power law model is considered as a rheological constitutive equation. The standard Galerkin finite element formulation/penalty formulation are applied to discrctize the governing equations, the volume‐of‐fluid (VOF) scheme is used to track the moving free surface, and the donor‐acceptor model introduced by Hirt and Nichols is modified and implemented on FEM. We applied the numerical scheme to simulate fountain flow and viscous buckling problems. For fountain flow, the numerical prediction of this study is in good agreement with the experimental results of other investigators. For viscous buckling, both 2D and 3D numerical simulations show that the shear thinning effect retards buckling. As this algorithm is very effective in treating moving free surface problems and requires less memory than previous algorithms, it may help solve challenging problems in polymer processing such as transient visroelastic flow simulations with moving free surfaces.     

Shear-thinning film on a porous substrate: Stability analysis of a one-sided model

The temporal stability of a Carreau fluid flowing down an inclined porous substrate is considered. A reduced model is derived under the assumption of small permeability which decouples the flow in the liquid layer from the filtration flow in the porous medium and incorporates the effect of the porous medium by means of an effective slip condition at the liquid–solid interface. The slip coefficient in the effective slip condition is a function of the structure, permeability of the porous medium and the rheology of the fluid saturating the porous medium. The effects of shear-thinning rheology and permeability of the substrate on the stability of the film flow system are investigated. This problem gives rise to a generalized eigenvalue formulation which is solved through two approaches. The problem is solved analytically for long waves in the limiting cases of weakly and strongly non-Newtonian behaviors (power-law limit). A numerical investigation is carried out in the general case. The results are shown to agree well for the weakly non-Newtonian limit. Further, the power-law model and the Carreau model agree on a wide range of shear-thinning parameter values for a thin film over a rigid substrate. However, when considering a porous medium, this trend is not observed. The Carreau model gives valid results for the entire range of shear-thinning parameter values for a film over a rigid/porous substrate. The novelty of the present investigation lies in the inclusion of both the effects of bottom permeability and shear-thinning rheology. Both permeability and shear-thinning rheology have a destabilizing effect on the film flow system. The numerical results indicate the correlation between the effects due to shear-thinning properties and permeability. An energy balance analysis performed on the perturbation fields shows that destabilization induced by both shear-thinning and permeability is linked to the viscous shear work rate on the free surface.     

Characteristic time scales for predicting the scalar flux at a free surface in turbulent open‐channel flows

The purpose of this study is to predict the turbulent scalar flux at a free surface subject to a fully developed turbulent flow based on a hydrodynamic analysis of turbulence in the region close to the free surface. The effect of the Reynolds number on turbulent scalar transfer mechanisms is extensively examined. A direct numerical simulation technique is applied to achieve the purpose. The surface‐renewal approximation is used to correlate the free‐surface hydrodynamics and scalar transport at the free surface. Two types of characteristic time scales have been examined for predicting turbulent scalar flux. One is the time scale derived from the characteristic length and velocity scale at the free surface. The other is the reciprocal of the root‐mean‐square surface divergence. The results of this study show that scalar transport at the free surface can be predicted successfully using these time scales based on the concept of the surface‐renewal approximation. © 2012 American Institute of Chemical Engineers AIChE J, 2012     

Reflection of a shock wave in a dusty cloud

The problem of shock-wave passage along a cloud of particles adjacent to a solid surface is studied numerically and analytically. The wave pattern of the flow near the shock wave reflected from this surface is analyzed within the framework of the equilibrium approximation of mechanics of heterogeneous media. The conditions of the transition from regular to irregular reflection from the substrate of the refracted shock wave inside the cloud are obtained analytically. The results of numerical simulations of a nonequilibrium flow in the two-velocity two-temperature approximation are compared with data obtained in the equilibrium approximation. Nonequilibrium and equilibrium flows are found to become more similar as the particle size decreases. __________ Translated from Fizika Goreniya i Vzryva, Vol. 43, No. 1, pp. 121–131, January–February, 2007.     

水平集方法数值模拟单液滴的生成过程(英文)

液滴生成的动力学过程和机理,对溶剂萃取、喷墨打印机设计、飞行器保护等有重要意义。虽然已有很多相关的实验、理论和计算研究,但对涉及拓扑变形界面的该问题研究仍富有挑战性,常用的有限元等模拟方法还有待改进。1988年提出的水平集(Level set)方法,可以比较简方便地模拟多相变形界面问题,已成功用于多相流、结晶、浇铸、模式识别等过程的计算。作者在对水平集方法进行改进并用于相间传质模拟的基础上,本文数值模拟了单个液滴在毛细管口的生长和脱离过程。假设流体均为不可压缩液体或气体作层流流动,采用二维轴对称欧拉坐标和交错网格,利用控制容积法和SIMPLE算法离散求解藕合水平集函数的运动方程组。水平集函数的发展方程和重新初始化方程的空间与时间离散格式,分别采用5阶WENO(Weighted essentially nonoscillatory)和3阶TVD(Total variation diminishing)Runge-Kutta格式。初步计算了不同参数的单液滴在气体或另一不互溶液体中的生成过程,液滴的拓扑形状、两相流场和液滴大小与实验或文献报道的模拟结果基本吻合,为今后拓展模拟三维空间的不稳定运动液滴或气泡生成过程的流动和传递奠定了基础。     

Fiber mat deformation in liquid composite molding. II: Modeling

Mathematical models are proposed to simulate the flow induced fiber mat deformation during liquid composite molding. The fiber bed is treated as an elastic beam and the load acting on the bed causes its deformation. The lubrication approximation is used to simplify the resin flow equation in the fiber free region, while Darcy's law is used to calculate the pressure and velocity fields in the deformed fiber bed. The governing equations are solved using the control volume/finite element method. The numerical results show reasonable agreement with the experimental results from Part I.     

Motions of a fluid drop in linear and quadratic flows

A theoretical analysis is presented of the flow field near a spherical fluid drop immersed in an incompressible Newtonian fluid which, at large distances from the drop, is undergoing an undisturbed flow. The undisturbed flows considered here are relevant to studies of drop motions near a phase boundary, and to some aspects of the coalescence of liquid drops. Exacl solutions in closed form have been found using the harmonic function expansion in spherical coordinates. Calculation of the hydrodynamic force on the drop leads to a generalization of Faxen’s law to afluid particle in anarbitrary undisturbed creeping-flow. The solutions are then expressed in terms of she fundamental singularity solutions for Stokes flow in anticipation of future analysis of the drop coalescence. In addition, the deformed shapes are determined for a fluid drop freely suspended in an axisymmetric Poiseuillian flow.     

Laminar natural convection over a slender vertical frustum of a cone with variable surface temperature

The problem of laminar, natural convection flow over a slender frustum of a cone with variable surface temperature is treated in this paper. The governing differential equations are solved by a combination of quasilinearization and finite-difference methods. As an illustration of the formulation, numerical solutions are obtained for a range of Prandtl numbers (0.1 to 100) and the transverse curvature parameter for the case of power-law surface temperature distribution. The formulation presented in this paper provides a way to analyze the heat transfer over a slender frustum of a cone where both the transverse curvature and the surface temperature variation have to be considered simultaneously.     

Numerical solution of viscous flow past a solid sphere with the control volume formulation

The control volume formulation with the QUICK finite difference scheme is used to solveincompressible liquid flow past a solid sphere in terms of stream function and vorticity.Several tech-nical points are addressed on improving the accuracy and efficiency of numerical simulation of similarproblems of fluid flow.In particular,the importance of suitable specification of the distortion func-tion to enforcing the far field boundarv conditions is emphasized.     

Limits of Existence of a Two-Dimensional Steady-State Structure of a Liquid Film in Combustion-Wave Propagation

A two-dimensional steady-state structure of a film flow of a combustible liquid on a heat-conducting substrate in the case of combustion-wave propagation is considered in the hydrodynamic formulation. The physical mechanism of formation of this structure is analyzed. It is shown that the thermocapillary effect plays an important role. The conclusion is justified that the existence of a two-dimensional regime is possible only for rather low values of the temperature gradient on the film surface. A critical condition is obtained, which determines the transition to a three-dimensional regime. This condition implies that the flow velocity is equal to the velocity induced by the thermocapillary force. If the temperature gradient is higher than a certain critical value, a zone with a reverse flow should appear, in accordance with the two-dimensional model. It is assumed that such a regime cannot exist due to its instability to three-dimensional perturbations. Experiments with a liquid film flowing down due to gravity in the presence of an immovable heat source (without the combustion wave) support the conclusion of the transition to a three-dimensional regular flow structure if the temperature gradient is rather high. The first part of the paper deals with simulation of the film structure for the critical condition satisfied. The second part deals with generalization of the problem to the case of a movable heat source moving with a constant velocity. This formulation of the problem includes the situation with combustion-wave propagation. The mathematical formulation of this problem allows us to assume that the existence of a two-dimensional steady-state regime in this case is limited by the same critical condition. If the temperature gradient on the film surface is higher than the critical value, the two-dimensional steady-state solution does not exist. This concept justified in the present work offers a generic explanation of phenomena observed in liquid films in the presence of local heat sources of various natures.     

Finite-amplitude surface waves on a thin film flow subject to a unipolar-charge injection

The interaction of an electric field with a dielectric liquid film is investigated as it drains under gravity down an inclined plane electrode emitting uniform positive ions into the liquid region. By applying long-wave approximation to the governing equations, the evolution equation for the free surface is derived up to the first order of a thin film parameter ξ To investigate the space charge effect on the development of a finite-amplitude surface wave, a neutral stability condition is obtained as a critical Reynolds number through a linear stability analysis, and the amplitude and velocity of a periodic disturbance are also calculated within a supercritically stable flow region. The presence of a unipolar space charge in the fluid makes a steady surface wave take on even higher amplitude and faster wave speed compared with the case of no space charge.     

Calculation of the parameters of an ensemble of particles in a well-stirred fluidized-bed reactor

Consideration is made of the problem of determining the parameters (the mass, the particle size distribution, and the entrainment, settling, and drain flows) of a fixed fluidized bed of reacting particles in a continuous reactor in the approximation that the solid phase is well stirred. A closed set of equations for solving this problem is derived. Special cases are studied, namely, beds of constant-size particles or particles contracting or growing during processing in a reactor with or without overflow devices. Formulas are obtained for the fluidized-bed parameters at different particle size distribution functions in the feed flow and also different size dependence of the particle contraction or growth velocity.     

Analysis of the Centrifugal Filtration Process in a Rotor with a Conic Filter Surface

The centrifugal filtration process in a centrifuge with a conic filter surface has been analyzed under the assumption of a one-dimensional flow. The problem of calculating the process parameters, such as the speed, the filtrate production capacity, and the liquid pressure, has been solved. The solution is illustrated based on an example of calculations for these parameters and presented in the form of their plots. The obtained plots reflect the specific process features and confirm the validity of the one-dimensional problem formulation.     

Coherent wave structures on falling fluid films flowing down a flexible wall

The dynamics of a thin fluid film flowing down a flexible vertical wall at moderate flow rates is studied in order to identify the dominant wave structures that will be observed in experiments. An asymptotic reduction using boundary-layer theory, and the von Kármán–Polhausen approximation, leads to coupled partial differential equations governing the nonlinear dynamics of the flow rate, and the gas–liquid and liquid–solid interfaces; closure is provided by a semi-parabolic fluid velocity profile. Fluid inertia, capillarity and viscous retardation effects are incorporated as are wall damping and tension. The validity of our approach is demonstrated using direct comparisons with predictions from the Orr–Sommerfeld equations. Nonlinear steady-travelling waves are identified from a nonlinear eigenvalue problem illustrating a multiplicity of solutions from which the dominating (attracting) solutions can be identified. Subsequent time-dependent numerical simulations of the fully nonlinear partial differential equations demonstrate the selection of these dominant solutions, and, as such, they then constitute a point of direct comparison with physical experiments.     

模拟变形液滴和气泡运动的改进水平集算法

An improved level set approach for computing the incompressible two-phase flow with significantly deformed free interface in presented.The control volume formulation with the semi-implicit method for pressure-linked equations consistent(SIMPLEC)algorithm incroporated is used to solve the governing equations on a staggered grid.Several improvements concerning the convergence and numerical stability.The motion of a bubble or drop in a liquid with large density ratio,viscosity ratio and surface tension in numerically simulated.The computational results are in good agreement with the reported experimental data.