Low Reynolds number hydrodynamic interaction of a solid particle with a planar wall

The slow viscous flow problem of an arbitrary solid particle in motion near a planar wall is recast into a boundary integral formulation. The present formulation employs the Green function appropriate to the planar wall problem and is developed in sufficient generality to allow calculations for arbitrary particles in any base flow which satisfies Stokes equations and no-slip on the wall. The resulting integral equations are easily discretized and solved for the particle surface tractions. Calculations are performed for axisymmetric motions of a variety of ellips?ids near the planar wall. Agreement with existing theory is excellent.     

The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions

Stokes and Couette flows produced by an oscillatory motion of a wall are analyzed under conditions where the no-slip assumption between the wall and the fluid is no longer valid. The motion of the wall is assumed to have a generic sinusoidal behavior. The exact solutions include both steady periodic and transient velocity profiles. It is found that slip conditions between the wall and the fluid produces lower amplitudes of oscillations in the flow near the oscillating wall than when no-slip assumption is utilized. Further, the relative velocity between the fluid layer at the wall and the speed of the wall is found to overshoot at a specific oscillating slip parameter or vibrational Reynolds number at certain times. In addition, it is found that wall slip reduces the transient velocity for Stokes flow while minimum transient effects for Couette flow is achieved only for large and small values of the wall slip coefficient and the gap thickness, respectively. The time needed to reach to steady periodic Stokes flow due to sine oscillations is greater than that for cosine oscillations with both wall slip and no-slip conditions.     

Turbulence in a skewed three‐dimensional wall‐bounded shear flow: effect of mean vorticity on structure modification

Mean‐flow three‐dimensionalities affect both the turbulence level and the coherent flow structures in wall‐bounded shear flows. A tailor‐made flow configuration was designed to enable a thorough investigation of moderately and severely skewed channel flows. A unidirectional shear‐driven plane Couette flow was skewed by means of an imposed spanwise pressure gradient. Three different cases with 8°, 34°and 52°skewing were simulated numerically and the results compared with data from a purely two‐dimensional plane Couette flow. The resulting three‐dimensional flow field became statistically stationary and homogeneous in the streamwise and spanwise directions while the mean velocity vector V and the mean vorticity vector Ω remained parallel with the walls. Mean flow profiles were presented together with all components of the Reynolds stress tensor. The mean shear rate in the core region gradually increased with increasing skewing whereas the velocity fluctuations were enhanced in the spanwise direction and reduced in the streamwise direction. The Reynolds shear stress is known to be closely related to the coherent flow structures in the near‐wall region. The instantaneous and ensemble‐averaged flow structures were turned by the skewed mean flow. We demonstrated for the medium‐skewed case that the coherent structures should be examined in a coordinate system aligned with V to enable a sound interpretation of 3D effects. The conventional symmetry between Case 1 and Case 2 vortices was broken and Case 1 vortices turned out to be stronger than Case 2. This observation is in conflict with the common understanding on the basis of the spanwise (secondary) mean shear rate. A refined model was proposed to interpret the structure modifications in three‐dimensional wall‐flows. What matters is the orientation of the mean vorticity vector Ω relative to the vortex vorticity vector ω _v, that is, the sign of Ω · ω _v. In the present situation, Ω · ω _v > 0 for the Case 1 vortices causing a strengthening relative to the Case 2 vortices. Copyright © 2011 John Wiley & Sons, Ltd.     

考虑振动非平衡的可压缩库埃特流动及其传热

新型近空间高超声速飞行器大多具有尖头薄翼的外形,驻点下游机身附近的强剪切流动及气动加热具有显著的非平衡特征.由于加热总量预估和实验测热数据辨识的需要,工程上越来越关注强剪切非平衡流动及气动加热预测问题.本文结合理论建模和直接模拟蒙特卡洛数值模拟,研究了振动非平衡条件下的可压缩库埃特流动的气动力/热问题.首先基于参考温度...     

Enhanced equilibrium distribution functions for simulating immiscible multiphase flows with variable density ratios in a class of lattice Boltzmann models

This research examines the behavior of a class of lattice Boltzmann (LB) models designed to simulate immiscible multiphase flows. There is some debate in the scientific literature as to whether or not the “color gradient” models, also known as the Rothman–Keller (RK) models, are able to simulate flow with density variation. In this paper, we show that it is possible, by modifying the original equilibrium distribution functions, to capture the discontinuity present in the analytical momentum profile of the two-layered Couette flow with variable density ratios. Investigations carried out earlier were not able to simulate such a flow correctly. Now, with the proposed approach, the new scheme is compatible with the analytical solution, and it is also possible to simulate the two-layered Couette flow with simultaneous density ratios of O(1000) and viscosity ratios of O(100). To test the model in a more complex flow situation, i.e. with non-zero surface tension and a curved interface, an unsteady simulation of an oscillating bubble with variable density ratio is undertaken. The numerical frequency of the bubble is compared to that of the analytical frequency. It is demonstrated that the proposed modification greatly increases the accuracy of the model compared to the original model, i.e. the error can be up to one order of magnitude lower with the proposed enhanced equilibrium distribution functions. The authors believe that this improvement can be made to other RK models as well, which will allow the range of validity of these models to be extended. This is, in fact, what the authors found for the method proposed in this article.     

N表面张力对近固壁二空化泡影响的数值研究

在忽略浮力下，用边界积分方法数值模拟了表面张力对固壁之上且靠近固壁的二轴对称空化泡生长和溃灭的影响，发现在下空泡最大等效半径为上空泡一半情形，若固壁对下空泡的Bjerknes力大于上空泡对下空泡的Bjerknes力，则表面张力的作用将使下空泡溃灭加速，使其向下的液体射流变强变宽；若固壁对下空泡的Bjerknes力小于上空泡对下空泡的Bjerknes力，则表面张力的作用将使下空泡溃灭变慢，使其向上射流变弱变细长；若这两个Bjerknes力近于相等，则表面张力将会对下空泡溃灭有重大作用，如改变下空泡射流的方向甚至形式(如由环状变向下或由向上变环状)，当上空泡等于或小于下空泡时，表面张力将不会对这两个空泡的行为产生显著影响，定性地分析了表面张力作用的机理。     

Turbulence in a three‐dimensional wall‐bounded shear flow

A new turbulent flow with distinct three‐dimensional characteristics has been designed in order to study the impact of mean‐flow skewing on the turbulent coherent vortices and Reynolds‐averaged statistics. The skewing of a unidirectional plane Couette flow was achieved by means of a spanwise pressure gradient. Direct numerical simulations of the statistically steady Couette–Poiseuille flow enabled in‐depth explorations of the turbulence field in the skewed flow. The imposition of a modest spanwise gradient turned the mean flow about 8° away from the original Couette flow direction and this turning angle remained nearly the same over the entire cross section. Nevertheless, a substantial non‐alignment between the turbulent shear stress angle and the mean velocity gradient angle was observed. The structure parameter turned out to slightly exceed that in the pure Couette flow, contrary to the observations made in some other three‐dimensional shear flows. Coherent flow structures, which are known to be associated with the Reynolds shear stress in near‐wall regions, were identified by the λ₂‐criterion. Instantaneous and ensemble‐averaged vortices resembled those found in the unidirectional Couette flow. In the skewed flow, however, the vortex structures were turned to align with the local mean‐flow direction. The conventional symmetry between Case 1 and Case 2 vortices was broken due to the mean‐flow three‐dimensionality. The turning of the coherent vortices and the accompanying symmetry‐breaking gave rise to secondary and tertiary turbulent shear stress components. By averaging the already ensemble‐averaged shear stresses associated with Case 1 and Case 2 vortices in the homogeneous directions, a direct link between the educed near‐wall structures and the Reynolds‐averaged turbulent stresses was established. These observations provide evidence in support of the hypothesis that the structural model proposed for two‐dimensional turbulent boundary layers remains valid also in flows with moderate mean three‐dimensionality. Copyright © 2009 John Wiley & Sons, Ltd.     

MOTION AND DEFORMATION OF VISCOUS DROP IN STOKES FLOW NEAR RIGID WALL

A boundary integral method was developed for simulating the motion and deformation of a viscous drop in an axisymmetric ambient Stokes flow near a rigid wall and for direct calculating the stress on the wall. Numerical experiments by the method were performed for different initial stand-off distances of the drop to the wall, viscosity ratios, combined surface tension and buoyancy parameters and ambient flow parameters. Numerical results show that due to the action of ambient flow and buoyancy the drop is compressed and stretched respectively in axial and radial directions when time goes. When the ambient flow action is weaker than that of the buoyancy the drop raises and bends upward and the stress on the wall induced by drop motion decreases when time advances. When the ambient flow action is stronger than that of the buoyancy the drop descends and becomes flatter and flatter as time goes. In this case when the initial stand-off distance is large the stress on the wall increases as the drop evolutes but when the stand-off distance is small the stress on the wall decreases as a result of combined effects of ambient flow, buoyancy and the stronger wall action to the flow. The action of the stress on the wall induced by drop motion is restricted in an area near the symmetric axis, which increases when the initial stand-off distance increases. When the initial stand-off distance increases the stress induced by drop motion decreases substantially. The surface tension effects resist the deformation and smooth the profile of the drop surfaces. The drop viscosity will reduce the deformation and migration of the drop.     

Numerical studies of the cold flow field in model combustion chambers

Numerical analyses of the cold turbulent flow in model combustion chambers weremade by usingκ.εturbulent model.The hybrid difference scheme and SNIP methodwere employed.Numerical solutions for retouchment length and velocity distributions wereobtained in the recirculating zone of the combustion chambers.The calculation results werein fairly good agreement with the reported experimental data.The work presented in thispaper was a basic part of the calculation model of sudden-enlarged combustion chambers.     

THE MESHLESS VIRTUAL BOUNDARY METHOD AND ITS APPLICATIONS TO 2D ELASTICITY PROBLEMS

A novel numerical method for eliminating the singular integral and boundary effect is processed. In the proposed method, the virtual boundaries corresponding to the numbers of the true boundary arguments are chosen to be as simple as possible. An indirect radial basis function network (IRBFN) constructed by functions resulting from the indeterminate integral is used to construct the approaching virtual source functions distributed along the virtual boundaries. By using the linear superposition method, the governing equations presented in the boundaries integral equations (BIE) can be established while the fundamental solutions to the problems are introduced. The singular value decomposition (SVD) method is used to solve the governing equations since an optimal solution in the least squares sense to the system equations is available. In addition, no elements are required, and the boundary conditions can be imposed easily because of the Kronecker delta function properties of the approaching functions. Three classical 2D elasticity problems have been examined to verify the performance of the method proposed. The results show that this method has faster convergence and higher accuracy than the conventional boundary type numerical methods.     

Compound wall treatment with low‐Re turbulence model

A generalized treatment for the wall boundary conditions relating to turbulent flows is developed that blends the integration to a solid wall with wall functions. The blending function ensures a smooth transition between the viscous and turbulent regions. An improved low Reynolds number k?ε model is coupled with the proposed compound wall treatment to determine the turbulence field. The eddy viscosity formulation maintains the positivity of normal Reynolds stresses and Schwarz' inequality for turbulent shear stresses. The model coefficients/functions preserve the anisotropic characteristics of turbulence. Computations with fine and coarse meshes of a few flow cases yield appreciably good agreement with the direct numerical simulation and experimental data. The method is recommended for computing the complex flows where computational grids cannot satisfy a priori the prerequisites of viscous/turbulence regions. Copyright © 2011 John Wiley & Sons, Ltd.     

A non‐linear dynamical system approach to finite amplitude Taylor‐Vortex flow of shear‐thinning fluids

The effect of shear thinning on the stability of the Taylor–Couette flow is explored for a Carreau–Bird fluid in the narrow‐gap limit. The Galerkin projection method is used to derive a low‐order dynamical system from the conservation of mass and momentum equations. In comparison with the Newtonian system, the present equations include additional non‐linear coupling in the velocity components through the viscosity. It is found that the critical Taylor number, corresponding to the loss of stability of the circular Couette flow, becomes lower as the shear‐thinning effect increases. That is, shear thinning tends to precipitate the onset of Taylor vortex flow, which coincides with the onset of a supercritical bifurcation. Comparison with existing measurements of the effect of shear thinning on the critical Taylor and wave numbers show good agreement. The Taylor vortex cellular structure loses its stability in turn, as the Taylor number reaches a critical value. At this point, an inverse Hopf bifurcation emerges. In contrast to Newtonian flow, the bifurcation diagrams exhibit a turning point that sharpens with shear‐thinning effect. Copyright © 2004 John Wiley & Sons, Ltd.     

Effect of confinement on droplet deformation in shear flow

The dynamics of a single droplet under shear flow between two parallel plates is investigated by using the immersed boundary method. The immersed boundary method is appropriate for simulating the drop-ambient fluid interface. We apply a volume-conserving method using the normal vector of the surface to prevent mass loss of the droplet. In addition, we present a surface remeshing algorithm to cope with the distortion of droplet interface points caused by the shear flow. This mesh quality improvement in conjunction with the volume-conserving algorithm is particularly essential and critical for long time evolutions. We study the effect of wall confinement on the droplet dynamics. Numerical simulations show good agreement with previous experimental results and theoretical models.     

Time-dependent MHD Couette flow of rotating fluid with Hall and ion-slip currents

The unsteady magnehydrodynamics (MHD) Couette flow of an electrically conducting fluid in a rotating system is investigated by taking the Hall and ion-slip currents into consideration.The derived fundamental equations on the assumption of a small magnetic Reynolds number are solved analytically with the well-known Laplace transform technique.The unified closed-form expressions are obtained for the velocity and the skin friction in the two different cases of the magnetic field being fixed to either the fluid or the moving plate.The effects of various parameters on the velocity and the skin friction are discussed by graphs.The results reveal that the primary and secondary velocities increase with the Hall current.An increase in the ion-slip parameter also leads to an increase in the primary velocity but a decrease in the secondary velocity.It is also shown that the combined effect of the rotation,Hall,and ion-slip parameters determines the contribution of the secondary motion in the fluid flow.     

Dynamic fluid–structure interaction of an elastic capsule in a viscous shear flow at moderate Reynolds number

The unsteady behavior of a 2-D circular elastic capsule was investigated in three viscous shear flows. An immersed boundary method (IBM) has been used to solve the dynamic fluid-structure interaction of the capsule. Computations were carried out in finite parameter ranges where the Reynolds number is Re=1-40 and the capillary number is Ca=0.0005-0.05, which is the ratio of the external viscous shear stress to the resistant elastic tensions of the membrane. For the simple shear flow, the effect of inertia on the transient behavior of the capsule was studied. For the pulsatile shear flow, two values of the peak fluid strain, T_f=1 and 5, were considered for the quasi-steady capsule mechanics. The capsule shows a cyclic structural response that includes subharmonics as the Reynolds number is elevated to 10 and 40. The capsule dynamic response includes a phase lag, which is a function of the capillary number, the Reynolds number, and the peak fluid strain. Finally, the capsule flowing in the Couette flow shows lateral migration due to the transient lift force, which is higher for lower Ca and higher Re. When capsules with diverse elasticity are dispersed along the velocity gradient, the capsule with a hard membrane experienced greater lift than the one with a soft membrane.     

A numerical method for the evaluation of hydrodynamic forces of translating bodies under a free surface

This paper presents a numerical method to evaluate the hydrodynamic forces of translating bodies under a free surface. Both steady and unsteady problems are considered. Analytical and numerical studies are carried out based on the Havelock wave‐source function and the integral equation method. Two main problems arising inherently in the proposed solution method are overcome in order to facilitate the numerical implementation. The first lies in evaluating the Havelock function, which involves integrals with highly oscillatory kernels. Particular integration contours leading to non‐oscillatory integrands are derived a priori so that the integrals can be evaluated efficiently. The second problem lies in evaluating singular kernels in the boundary integral equation. The corresponding non‐singular formulation is derived using some theorems of potential theory, including the Gauss flux theorem and the property related to the equipotential body. The subsequent formulation is amenable to the solution by directly using the standard quadrature formulas without taking another special treatment. This paper also attempts to enhance the computational efficiency by presenting an interpolation method used to evaluate matrix elements, which are ascribed to a discretization procedure. In addition to the steady case, numerical examples consist of cases involving a submerged prolate spheroid, which is originally idle and then suddenly moves with a constant speed and a constant acceleration. Also systematically studied is the variation of hydrodynamic forces acting on the spheroid for various Froude numbers and submergence depths. Copyright © 2000 John Wiley & Sons, Ltd.     

Direct numerical simulation of flow in channel with time-dependent wall geometry

A numerical scheme is developed to extend the scope of the spectral method without solving the covariant and contravariant forms of the Navier-Stokes equations in the curvilinear coordinates. The primitive variables are represented by the Fourier series and the Chebyshev polynomials in the computational space. The time advancement is accomplished by a high-order time-splitting method, and a corresponding high-order pressure condition at the wall is introduced to reduce the splitting error. Compared with the previous pseudo-spectral scheme, in which the Navier-Stokes equations are solved in the covariant and contravariant forms, the present scheme reduces the computational cost and, at the same time, keeps the spectral accuracy. The scheme is tested in the simulations of the turbulent flow in a channel with a static streamwise wavy wall and the turbulent flow over a flexible wall undergoing the streamwise traveling wave motion. The turbulent flow over an oscillating dimple is studied with the present numerical scheme, and the periodic generation of the vortical structures is analyzed.     

The void-size effect on plastic flow localization in the Gurson model

Recent studies have shown that the size of microvoids has a significant effect on the void growth rate. The purpose of this paper is to explore whether the void size effect can influence the plastic flow localization in ductile materials. We have used the extended Gurson‘s dilatational plasticity theory, which accounts for the void size effect, to study the plastic flow localization in porous solids with long cylindrical voids. The localization model of Rice is adopted, in which the material inside the band may display a different response from that outside the band at the incipient plastic flow localization. The present study shows that it has little effect on the shear band angle.     

The Calculation of Semi-circular Corrugated Tube——An Application of the General Solution of Ring Shell

In this paper, the deformation and stress distribution in semi-circular corrugated tube under axial force are calculated by means of the general solutions of circular ring shell given in previous paper[1].