A note on a plane creeping corner flow

Abstract

A simple method of similarity transformation is formulated to analyze a two‐dimensional creeping corner flow. By this peculiar transformation, governing equations for the plane velocity are reduced to a pair of ordinary differential equations. With a particular selection of appropriate boundary conditions, the field variables of velocity, pressure, vorticity, and stream function are obtained analytically. A special case with constant velocity at one boundary is explored. The salient characteristics of this example are used to compare with previous investigations. The present study shows that both approaches provide exactly the same solutions. A very interesting feature is that the velocity components in the coordinate system are independent of the radial direction.     

Superconducting State of a Disk with a Pentagonal/Hexagonal Trench/Barrier

We study the influence of a pentagonal (hexagonal) trench or barrier on the superconducting properties of a perforated disk. Effects associated to the pinning (anti-pinning) force of the central hole and the trench (barrier) and the interplay between the boundary conditions and the shape of the inner defects on the vortex configuration are studied for a thin disk. Also, we considered two cases for the value of the order parameter at the surface. The first one is ψ _s ≠0; this is the usual supercondutor/vacuum interface, for which the deGennes parameter is b→∞. The second one is ψ _s =0, which is the surface of the sample in a complete normal state, simulated by b=0. The vorticity, Gibbs free energy, magnetic induction, supercurrent density, magnetization and Cooper pairs density as a function of the external magnetic field are calculated. We show that in our sample new phenomena are possible due to the competing interactions of the boundary and the geometry of the sample and the added geometry of the nanoengineered trench and barrier.     

Analysis of three-dimensional natural convection of nanofluids by BEM

In this paper we analyse flow and heat transfer characteristics of nanofluids in natural convection flows in closed cavities. We consider two test cases: natural convection in a three-dimensional differentially heated cavity, and flow around a hotstrip located in two positions within a cavity. Simulations were performed for Rayleigh number values ranging from 10³ to 10⁶. Performance of three types of water based nanofluids was compared with pure water and air. Stable suspensions of Cu, Al₂O₃ and TiO₂ solid nanoparticles in water were considered for different volume fractions ranging up to 20%. We present and compare heat flux for all cases and analyse heat transfer enhancement attributed to nanofluids in terms of their enhanced thermal properties and changed flow characteristics. Results show that, using nanofluids, the largest heat transfer enhancements can be achieved in conduction dominated flows as well as that nanofluids increase the three-dimensional character of the flow field. We additionally examine the relationship between vorticity, vorticity flux and heat transfer for flow of nanofluids.The simulations were performed using a three-dimensional boundary element method based flow solver, which has been adapted for the simulation of nanofluids. The numerical algorithm is based on the combination of single domain and subdomain boundary element method, which are used to solve the velocity–vorticity formulation of Navier–Stokes equations. In the paper we present the adaptation of the solver for simulation of nanofluids. Additionally, we developed a dynamic solver accuracy algorithm, which was used to speed up the simulations.     

Boundary element analysis of wave-current interaction around a large structure

A numerical approach for wave-current interaction around a large structure is investigated, based on potential flow theory, linear waves and small current velocity approximation. The velocity potential in a wave-current coexisting field is separated into a steady current potential and an unsteady wave potential. The boundary element method was then employed to compute the unsteady wave potential with effects of both a uniform current and a large body taken into consideration. It is demonstrated that the steady current potential can be expressed as the sum of a uniform current and a steady disturbance due to the presence of the object. The variation of current velocity in the vicinity of the object is then calculated by using a surface vorticity boundary integral meethod. Boundary element analysis is also used for the numerical solutions of the surface vorticity method. Substituting both unsteady wave potential and current velocity into the first-order dynamic surface boundary condition, the water surface elevation around a large structure in a wave-current coexisting field can then be obtained. Comparisons of numerical predictions with experimental results ar also made; qualitative good agreements are obtained.     

Turbulent biomagnetic fluid flow in a rectangular channel under the action of a localized magnetic field

The fundamental problem of the turbulent flow of a biomagnetic fluid (blood) between two parallel plates under the action of a localized magnetic field is studied. The blood is considered to be an electrically conducting, incompressible and Newtonian fluid and its flow is steady, two-dimensional and turbulent. The turbulent flow is described by the Reynolds averaged Navier–Stokes (RANS) equations. For the numerical solution of the problem under consideration, which is described by a coupled and non-linear system of PDEs, with appropriate boundary conditions, the stream function–vorticity formulation is used. For the eddy-kinematic viscosity, the low Reynolds number k–ε turbulence model is adopted. The solution of the problem, for different values of the dimensionless parameter entering into it, is obtained by developing and applying an efficient numerical technique based on finite differences scheme. Results concerning the velocity and temperature field, skin friction and rate of heat transfer, indicate that the presence of the localized magnetic field, appreciable influences the turbulent flow field. A comparison is also made with the corresponding laminar flow, indicating that the influence of the magnetic field decreases in the presence of turbulence.     

Hall effects on the unsteady hydromagnetic flows of an Oldroyd-B fluid

The influence of Hall currents and rotation on the oscillatory flows of an infinite plate is investigated. Exact solutions for the two problems are obtained.The fluid considered is a homogeneous Oldroyd-B. During the mathematical analysis it is found that governing differential equation for steady flow in an Oldroyd-B fluid is identical to that of viscous fluid. Further, it is observed that in absence of the strength of transverse magnetic field (B₀) the solution in resonance case does not satisfy the boundary condition at infinity. Physical significance of mathematical results is also discussed.     

Complete decomposition of axisymmetric Stokes flow

As Stokes has shown, axisymmetric, incompressible, viscous creeping flow can be studied through the use of a stream function Ψ which belongs to the kernel of the fourth-order differential operator E⁴, where E²Ψ measures the vorticity of the flow. In fact, irrotational flows are described by stream functions that belong to the kerE², while rotational flows are described by stream functions that do not belong to the kerE². It is shown that a decomposition, of the form Ψ=Ψ₁+r²Ψ₂, for any stream function Ψ is possible, where Ψ₁ and Ψ₂ belong to kerE² and r is the radial spherical variable. Consequently, a stream function that describes a rotational flow can always be divided by a stream function that describes an irrotational flow in a way that renders the ratio always equal to the square of the Euclidean distance. If no singularities are observed on the axis of symmetry then the above decomposition is unique.     

Vorticity evolution mechanism in near‐field of a transverse jet

Abstract

Flow structure and vorticity evolution processes in the near field of an elevated jet in a crossflow are experimentally studied in a wind tunnel. The instantaneous and time‐averaged flow field characteristics are observed and measured by using a flow visualization technique and a high‐speed Particle Image Velocimeter (PIV). Time histories of the instantaneous velocity of the vortical flows in the shear‐layer are recorded by a hot‐wire anemometer and a high‐speed data acquisition system in order to analyze the frequency characteristics of the traveling coherent structure in the shear‐layer. Experiments are performed between two different jet‐to‐crossflow momentum flux ratios R = 0.08 and 0.56, which are selected from two regimes with different kinds of flow patterns at a fixed crossflow Reynolds number 2051. The behaviors and mechanisms of the vortical flow structure and the vorticity evolution mechanisms appear to be distinct in different flow regimes. By analyzing the pictures of the smoke flow visualization and the instantaneous vorticity contour maps, two kinds of vorticity evolution mechanisms, “shear‐induced vortices” and “swing‐induced vortices”, can be identified in the shear‐layer evolving from the jet exit. The time‐averaged velocity field and vorticity properties are also discussed in this paper.     

Sound Produced When a Vortex Pair is Swallowed by a Jet

An analysis is made of the sound generated during the high-Reynolds-number convection of a vortex pair in a jet of water exhausting from a large vessel through a slit aperture. The equations of motion are linearized about the classical free-streamline solution describing steady flow through the aperture. It is assumed that the vortex pair is swept through the aperture into the jet by the steady mean flow, with no account taken of the nonlinear influence on the motion of ‘images’ in the boundaries. Additional vorticity is shed from the edges of the aperture in order that the flow should remain smooth and continuous (the Kutta condition). This vorticity is convected away within a sheet of ‘bound’ vorticity on the mean free streamlines of the jet. A strong peak in the bound vorticity is established when the vortex pair enters the aperture. Both the incident and the shed vorticity generate sound, but their respective contributions to the acoustic pressure are of opposite phase. The dominant radiation in the water above the aperture is produced as the vortex enters the jet, and has the form of a pressure pulse of width ~h/M and monopole strength , where h is the width of the aperture, ρ _o the density of the water, v a typical flow velocity, and M is the jet Mach number.     

The Use of the RANS/ILES Method to Study the Influence of Coflow Wind on the Flow in a Hot,Nonisobaric, Supersonic Airdrome Jet during Its Interaction with the Jet Blast Deflector

The influence of the coflow wind on the flow in a hot, nonisobaric, supersonic airdrome jet from a biconical nozzle and its interaction with a jet blast deflector (JBD) are studied by the RANS/ILES method. The conditions at the external boundary of the computational domain are formulated for the problem of jet interaction with the JBD. All calculations were performed at the Joint Supercomputer Center of the Russian Academy of Sciences with a MVS-10P supercomputer. The features of method parallelization for the supercomputer with modern architecture are described. The total temperature of the jet at the nozzle output is T₀ = 1050 K and π_с = 4. The wind velocity ranges from 0 to 20 m/s. Two JBD positions are examined: at distances of 5 and 15D_e of the nozzle cross section. The computation grids consist of (6.33–8.53) × 10⁶ cells. Fields of the flow parameters and of their turbulent pulsations near the jet are obtained. The dimensions of the “safety zone” for people and machinery is determined by the temperature, pressure pulsations, and velocity near the airdrome surface. The influence of wind velocity on the size and shape of the safety zone are revealed. The distributions of pressure and temperature and their pulsations over JBD altitude are presented as a function of JBD position and wind velocity.     

Solution of magnetohydrodynamic flow problems using the boundary element method

A boundary element solution is implemented for magnetohydrodynamic (MHD) flow problem in ducts with several geometrical cross-section with insulating walls when a uniform magnetic field is imposed perpendicular to the flow direction. The coupled velocity and induced magnetic field equations are first transformed into uncoupled inhomogeneous convection–diffusion type equations. After introducing particular solutions, only the homogeneous equations are solved by using boundary element method (BEM). The fundamental solutions of the uncoupled equations themselves (convection–diffusion type) involve the Hartmann number (M) through exponential and modified Bessel functions. Thus, it is possible to obtain results for large values of M (M≤300) using only the simplest constant boundary elements. It is found that as the Hartmann number increases, boundary layer formation starts near the walls and there is a flattening tendency for both the velocity and the induced magnetic field. Also, velocity becomes uniform at the center of the duct. These are the well-known behaviours of MHD flow. The velocity and the induced magnetic field contours are graphically visualized for several values of M and for different geometries of the duct cross-section.     

Separation and velocity dependence of the drag force applied to a rigid ovoid of Rankine nosed projectile penetrating an elastic–perfectly-plastic target

Yarin et al. have developed an analytical solution for normal penetration of a rigid projectile of the shape of an ovoid of Rankine into an incompressible elastic–perfectly-plastic target. Here, the closed form expressions from this solution are used to analyze separation and velocity dependence of the drag force applied to the projectile. It is shown that for penetration velocities V below a critical value V_s, the target material maintains full contact with the projectile's surface, the drag force is due solely to the resistance of plastic flow in the target and it is independent of the velocity V. For V equal to V_s the separation point jumps from the projectile's tail to a point closer to its tip. For increasing values of V the separation point moves even further towards the projectile's tip causing a strong dependence of the drag force on velocity. In contrast, results of the cavity expansion model applied to this problem indicate that separation cannot occur and that the drag force depends on velocity for all values of V. These results emphasize the importance of accurately modeling the flow field around the projectile.     

Boundary element method solution of magnetohydrodynamic flow in a rectangular duct with conducting walls parallel to applied magnetic field

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct with one conducting and one insulating pair of opposite walls under an external magnetic field parallel to the conducting walls, is investigated. The MHD equations are coupled in terms of velocity and magnetic field and cannot be decoupled with conducting wall boundary conditions since then boundary conditions are coupled and involve an unknown function. The boundary element method (BEM) is applied here by using a fundamental solution which enables to treat the MHD equations in coupled form with the most general form of wall conductivities. Also, with this fundamental solution it is possible to obtain BEM solution for values of Hartmann number (M) up to 300 which was not available before. The equivelocity and induced magnetic field contours which show the well-known characteristics of MHD duct flow are presented for several values of M.     

Unsteady mixed-convection boundary layer flow along a symmetric wedge with variable surface temperature

Laminar two-dimensional unsteady mixed-convection boundary-layer flow of a viscous incompressible fluid past a sharp wedge has been studied. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equations is reduced to local non-similarity boundary layer equations, which are solved analytically for small time. Perturbation solutions are also obtained for small and large dimensionless time, τ. Solutions of the governing equations for all time are obtained employing the implicit finite difference method. Here we have focused our attention on the evolution of skin-friction coefficient (C_f) and local Nusselt number (Nu) (heat transfer rate), fluid velocity and fluid temperature with the effects of different governing parameters such as different time, τ, the exponent, m (=0.2, 0.4, 0.6, 0.8, 1.0), mixed convection parameter, λ (= 0.0, 0.5, 1.0) for fluids having Prandtl number, Pr = 0.1, 0.7, and 7.0.     

Concerning closed-streamline flows with discontinuous boundary conditions

High-Reynolds-number (Re) flow containing closed streamlines (Prandtl-Batchelor flows), within a region enclosed by a smooth boundary at which the boundary conditions are discontinuous, is considered. In spite of the need for local analysis to account fully for flow at points of discontinuity, asymptotic analysis for Re 1 indicates that the resulting mathematical problem for determining the uniform vorticity ₀) in these situations, requiring the solution of periodic boundary-layer equations, is in essence the same as that for a flow with continuous boundary data. Extensions are proposed to earlier work [3] to enable ₀ to be computed numerically; these require coordinate transformations for the boundary-layer variables at singularities, as well as a two-zone numerical integration scheme. The ideas are demonstrated numerically for the classical circular sleeve.     

40Cr轴面激光熔覆Ni60数值模拟

目的 针对激光熔覆过程中熔池内部复杂的传热和对流现象,分析激光功率和扫描速度对熔池内部温度场、流场演变和分布的影响.方法 采用双椭球热源模型,建立了40Cr轴面基体激光熔覆Ni60粉末过程的三维温度场流场数值模型,并进行试验验证.结果 熔覆过程形成了近似椭球体的熔池,最高温度位于移动光斑中心偏后方,达到了2080.4 ...     

Non-unique solution for combined-convection assisting flow over vertical flat plate

Non-unique solutions of flow and temperature field are reported here for the first time for non-similar flows given by the laminar boundary layer equations for combined-convection flow past a vertical flat plate. The solution of the boundary layer equation for natural convection constitutes the self-similar solution whose perturbation with respect to the small parameter (ε), which is inversely proportional to the square root of the Richardson number (G _x ), provides the non-similar solution. Solutions obtained by the shooting method indicate two sets for the self-similar solution (ε = 0) — one of them showing positive velocity everywhere inside the shear layer (well-known oft-reported physical result). The other self-similar solution shows that recirculation in the outer part of the shear layer may not be physical — as it has not been experimentally demonstrated so far. In contrast, the perturbative part of the non-similar solution (ε ⊋ 0) is seen to be either convergent or divergent depending upon the choice of integration domain of the shear layer equations — bringing forth the question of the validity of such perturbation procedures and possible stability of the basic solution itself.     

Simulation of 3D flow in porous media by boundary element method

In the present paper problem of natural convection in a cubic porous cavity is studied numerically, using an algorithm based on a combination of single domain and subdomain boundary element method (BEM). The modified Navier–Stokes equations (Brinkman-extended Darcy formulation with inertial term included) were adopted to model fluid flow in porous media, coupled with the energy equation using the Boussinesq approximation. The governing equations are transformed by the velocity–vorticity variables formulation which separates the computation scheme into kinematic and kinetic parts. The kinematics equation, vorticity transport equation and energy equation are solved by the subdomain BEM, while the boundary vorticity values, needed as a boundary conditions for the vorticity transport equation, are calculated by single domain BEM solution of the kinematics equation. Computations are performed for steady state cases, for a range of Darcy numbers from 10^?6 to 10^?1, and porous Rayleigh numbers ranging from 50 to 1000. The heat flux through the cavity and the flow fields are analyzed for different cases of governing parameters and compared to the results in some published studies.     

Minimum spouting velocity of binary mixture with non-spherical particles

The minimum spouting velocity, U_ms, defined for stable external spouting, is found also crucial to ensure good mixing when multi-component particles are involved in spouted beds. In this study, experiments were performed in a cylindrical-conical spouted bed to study the influences of diverse factors on the U_ms of a binary system with the cylindrical particles and the spherical bed material. The results showed that the changes in U_ms with the particle properties (particle shape, size, and density) and operating conditions were closely related to the blending ratio of the mixture. When the volume fraction of the non-spherical particles was relatively small (less than 40% to 50%), U_ms mainly depended on the properties of the bed material. It was considered acceptable to estimate U_ms by assuming that the system only consisted of the spherical bed material. Otherwise, the cylindrical particle shape has a significant influence on the flow dynamics and U_ms. For such spouting systems, an equivalent diameter of the bed material was proposed to reflect the shape effects of non-spherical particles, whereby U_ms would be independent of the blending ratio. Consequently, a novel empirical correlation is proposed to quantitatively predict the U_ms of binary mixtures.     

Isothermal slip flow over curved surfaces

It has long been recognised that the no-slip-boundary condition employed in the Navier-Stokes equations can only be applied when the Knudsen number, Kn?10⁻³. If the Knudsen number is increased beyond this value, rarefaction effects start to influence the flow and the molecular collision frequency per unit area becomes too small to maintain the no-slip-boundary condition. Unfortunately, Maxwell's famous slip equation describing the velocity discontinuity at the wall is often misapplied when analysing flows over curved or rotating boundaries. In the present study, a generalised version of Maxwell's slip equation is used to investigate low Knudsen number isothermal flow over walls with substantial curvature. The generalised slip equation is written in terms of the tangential shear stress to overcome the limitations of the conventional slip-boundary treatment. The study considers a number of fundamental, but challenging, rarefied flow problems and demonstrates that Maxwell's conventional slip equation is unable to capture important flow phenomena over curved or rotating surfaces.