Evolution of the free surface of a plane channel during filling with a viscous fluid

The behavior of the free surface of a viscous incompressible fluid is studied in the process of filling of a plane channel inclined to the gravity direction. The problem is numerically solved within the framework of the Navier-Stokes equations under the assumption that the fluid is Newtonian and capillary effects can be neglected. The numerical technique is based on the joint use of the SIMPLE and invariant methods. Two filling regimes are found to exist, namely, that with a steady shape of the free surface without the formation of internal cavities and that with cavity formation in the initial filling stage characterized by spouting behavior of the free boundary.     

Flow and heat transfer from a continuous surface in a parallel free stream of viscoelastic second-order fluid

Boundary layer solutions are presented to investigate the steady flow and heat transfer characteristics from a continuous flat surface moving in a parallel free stream of viscoelastic fluid. Numerical results are presented for the distribution of velocity and temperature profiles within the boundary layer. The effects of the viscoelastic parameter of the fluid on the shear stress at the wall and rate of heat transfer are studied. For the same Reynolds (based on the larger of the free stream and wall velocities) and Prandtl numbers and the same velocity difference |U _w >|, larger skin-friction and heat transfer coefficient result for U _w > than for U _w <.     

Influence of chemical reaction on heat and mass transfer of non-Newtonian fluid with yield stress by free convection from vertical surface in porous medium considering Soret effect

The effect of chemical reaction on free convection heat and mass transfer for a non-Newtonian power law fluid over a vertical flat plate embedded in a fluid-saturated porous medium has been studied in the presence of the yield stress and the Soret effect. The governing boundary layer equations and boundary conditions are cast into a dimen- sionless form by similarity transformations, and the resulting system of equations is solved by a finite difference method. The results are preSented and discussed for concentration profiles, as well as the Nusselt number and the Sherwood number for various values of the parameters, which govern the problem. The results obtained show that the flow field is influenced appreciably by the presence of the chemical reaction parameter γ the order of.the chemical reaction parameter m, the Soret number St, the buoyancy ratio N, the Lewis number Le, and the dimensionless rheological parameter Ω.     

Possible classification of the self-oscillatory spouting regimes of plane vertical submerged jets of a heavy fluid

New results of an experimental investigation of self-oscillatory regimes of plane vertical jet spouting from beneath the free surface of a heavy incompressible fluid are discussed. The experiments were performed on a setup with discharge over a weir. The range of dimensionless jet submergence values on which bifurcation change of spouting regime is observable is studied. It is established that on the Froude number and dimensionless jet submergence ranges considered in the study six characteristic spouting regimes differing in free surface shape and self-oscillation frequency can exist. It is shown that these regimes can be subdivided into three typical groups with respect to the dependence of the self-oscillation period on the jet flow rate. A dimensionless parameter that makes it possible to identify the boundaries of the bifurcation change in spouting regimes is obtained for each of these groups. For certain spouting regimes without the formation of free jets numerical calculations are carried out using the STAR-CD software package; the calculated results are in good agreement with experimental data.     

Formation of the free surface of a viscous fluid volume inside a rotating horizontal cylinder

Two-dimensional viscous flow with a free surface in a horizontal cylinder rotating at a constant speed is investigated numerically using the boundary element method. It is shown that in the initial stage of rotation of the cylinder four different variants of the behavior of the free surface can be realized in the stage of transition from horizontal to steady-state form.     

Simulation of interface and free surface flows in a viscous fluid using adapting quadtree grids

A new adaptive quadtree method for simulating laminar viscous fluid problems with free surfaces and interfaces is presented in this paper. The Navier–Stokes equations are solved with a SIMPLE‐type scheme coupled with the Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) (Numerical prediction of two fluid systems with sharp interfaces, Ph.D. Thesis, Imperial College of Science, Technology and Medicine, London, 1997) volume of fluid (VoF) method and PLIC reconstruction of the volume fraction field during refinement and derefinement processes. The method is demonstrated for interface advection cases in translating and shearing flow fields and found to provide high interface resolution at low computational cost. The new method is also applied to simulation of the collapse of a water column and the results are in excellent agreement with other published data. The quadtree grids adapt to follow the movement of the free surface, whilst maintaining a band of the smallest cells surrounding the surface. The calculation is made on uniform and adapting quadtree grids and the accuracy of the quadtree calculation is shown to be the same as that made on the equivalent uniform grid. Copyright © 2004 John Wiley & Sons, Ltd.     

Unsteady Self-Similar Flow of a Viscous Incompressible Fluid in a Plane Divergent Channel

The problem of two-dimensional unsteady flow of a viscous incompressible fluid in a sector-like domain is considered. Initially a strictly radial flow is imposed, which makes it possible to seek solutions within the class of self-similar flows. A numerical method based on mixed finite-difference and spectral spatial discretization is developed, making it possible to find the self-similar solution efficiently. The process of development and establishment of the steady Hamel-Jeffery and Moffatt flows is modeled mathematically.     

贮腔类三维自由液面动力学问题数值研究

讨论了贮腔类三维自由液面动力学问题的数值研究,将任意的拉格朗日-欧拉运动学描述关系引入到系统的控制方程中,采用任意的拉格朗日-欧拉描述跟踪自由液面,推导了自由面上结点的法向矢量计算公式。采用Galerkin余量法推导了Navier-Stokes方程的空间离散有限元方程,采用三维自由液面上微分几何理论推导了表面张力计算公式。数值研究中考虑了接触角效应,最后进行了三维数值算例分析。     

Simulation of heat and mass transfer on peristaltic flow of hyperbolic tangent fluid in an asymmetric channel

In the present analysis, the influence of heat and mass transfer on the peristaltic flow of a hyperbolic tangent fluid in an asymmetric channel has been discussed. The highly nonlinear equations are simplified under lubrication approach. The perturbation and numerical solutions of the problem are not only discussed but the validity of the results is also being checked. The graphical results of the problem under discussion are also being brought under consideration to see the behavior of various physical parameters. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical solution of steady free‐surface flows by the adjoint optimal shape design method

Numerical solution of flows that are partially bounded by a freely moving boundary is of great importance in practical applications such as ship hydrodynamics. Free‐boundary problems can be reformulated into optimal shape design problems, which can in principle be solved efficiently by the adjoint method. In this work we investigate the suitability of the adjoint shape optimization method for solving steady free‐surface flows. The asymptotic convergence behaviour of the method is determined for free‐surface flows in 2D and 3D. It is shown that the convergence behaviour depends sensitively on the occurrence of critical modes. The convergence behaviour is moreover shown to be mesh‐width independent, provided that proper preconditioning is applied. Numerical results are presented for 2D flow over an obstacle in a channel. The observed convergence behaviour is indeed mesh‐width independent and conform the derived asymptotic estimates. Copyright © 2003 John Wiley & Sons, Ltd.     

Modeling jet flows caused by the incidence of a shock wave on a profiled free surface

The problem of the incidence of a shock wave with a front-pressure amplitude of about 30 GPa at the profiled free surface of an aluminum sample is studied. It is shown that in the case of large perturbations (amplitude 1 mm and wavelength 10 mm), jet flows occur on the free surface. The data obtained are described using a kinetic fracture model that takes into account the damage initiation and growth in the material due to tensile stress and shear strain. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 16–23, January–February, 2007.     

Transient free convection with mass transfer MHD micropolar fluid in a porous plate by the network method

The transient problem of coupled heat and mass transfer of a micropolar fluid in magneto‐hydrodynamic free convection from a vertical infinite porous plate with an exponentially decaying heat generating considering the viscous dissipation and ohmic heating effects is studied. Joule heating must be considered when the viscous dissipation and the Prandtl number are large. The non‐dimensional equations for the conservation of mass, momentum, energy and concentration are solved by means a numerical technique based on electric analogy (network simulation method). This method provides the numerical response of the system by running the network in circuit resolution software with the solution to both transient and steady‐state problems at the same time, and its programming does not require manipulation of the sophisticated mathematical software that is inherent in other numerical methods. The effects of the material parameters, viscous dissipation, internal generation and Joule heating on velocity, angular momentum and temperature fields across the boundary layer are investigated. In addition, the skin‐friction coefficient, couple stress coefficient, Nusselt number and Sherwood number are shown in tabular form. The numerical results for velocity and temperature distributions of micropolar fluids are compared with the corresponding flow problems for a Newtonian fluid. Copyright © 2007 John Wiley & Sons, Ltd.     

Unsteady three‐dimensional boundary layer flow due to a stretching surface in a micropolar fluid

In this paper, the unsteady three‐dimensional boundary layer flow due to a stretching surface in a viscous and incompressible micropolar fluid is considered. The partial differential equations governing the unsteady laminar boundary layer flow are solved numerically using an implicit finite‐difference scheme. The numerical solutions are obtained which are uniformly valid for all dimensionless time from initial unsteady‐state flow to final steady‐state flow in the whole spatial region. The equations for the initial unsteady‐state flow are also solved analytically. It is found that there is a smooth transition from the small‐time solution to the large‐time solution. The features of the flow for different values of the governing parameters are analyzed and discussed. The solutions of interest for the skin friction coefficient with various values of the stretching parameter c and material parameter K are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Transient free‐surface flow of a viscoelastic fluid in a narrow channel

The interplay between inertia and elasticity is examined for transient free‐surface flow inside a narrow channel. The lubrication theory is extended for the flow of viscoelastic fluids of the Oldroyd‐B type (consisting of a Newtonian solvent and a polymeric solute). While the general formulation accounts for non‐linearities stemming from inertia effects in the momentum conservation equation, and the upper‐convected terms in the constitutive equation, only the front movement contributes to non‐linear coupling for a flow inside a straight channel. In this case, it is possible to implement a spectral representation in the depthwise direction for the velocity and stress. The evolution of the flow field is obtained locally, but the front movement is captured only in the mean sense. The influence of inertia, elasticity and viscosity ratio is examined for pressure‐induced flow. The front appears to progress monotonically with time. However, the velocity and stress exhibit typically a strong overshoot upon inception, accompanied by a plug‐flow behaviour in the channel core. The flow intensity eventually diminishes with time, tending asymptotically to Poiseuille conditions. For highly elastic liquids the front movement becomes oscillatory, experiencing strong deceleration periodically. A multiple‐scale solution is obtained for fluids with no inertia and small elasticity. Comparison with the exact (numerical) solution indicates a wide range of validity for the analytical result. Copyright © 2004 John Wiley & Sons, Ltd.     

On the accuracy of the desingularized boundary integral method in free surface flow problems

In this paper the numerical properties of the desingularized boundary integral formulation were studied within the framework of free surface potential problems. Several numerical experiments were carried out on simple test cases in order to investigate the effects on the accuracy of the distance between the singularity sheet and the free boundary. The optimum value of this distance was related to the mesh size by simple correlations. Once the desingularized boundary integral formulation had been so calibrated, it was implemented for the solution of two typical free surface flow problems: wave diffraction around a fixed obstacle and wave resistance of submerged bodies. Numerical results are discussed in comparison with experimental data; the computational efficiency and accuracy of desingularized algorithms are confirmed and specified. © 1997 John Wiley & Sons, Ltd.     

Solving viscoelastic free surface flows of a second‐order fluid using a marker‐and‐cell approach

This work is concerned with the numerical simulation of two‐dimensional viscoelastic free surface flows of a second‐order fluid. The governing equations are solved by a finite difference technique based on the marker‐and‐cell philosophy. A staggered grid is employed and marker particles are used to represent the fluid free surface. Full details for the approximation of the free surface stress conditions are given. The resultant code is validated and convergence is demonstrated. Numerical simulations of the extrudate swell and flow through a planar 4:1 contraction for various values of the Deborah number are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Effect of heat transfer on a third grade fluid in a flat channel

The heat transfer analysis on the laminar flow of an incompressible third grade fluid through a porous flat channel is examined. The lower plate is assumed to be at a higher temperature than the upper plate. Analytical solution for temperature distribution is obtained for various values of the controlling parameters and discussed. The obtained analytical solution is also compared with the numerical solution. The comparison shows the fact that the accuracy is remarkable. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical modelling of 3D stratified free surface flows: a case study of sediment dumping

A three‐dimensional numerical model has been developed to simulate stratified flows with free surfaces. The model is based on the Reynolds‐averaged Navier–Stokes (RANS) equations with variable fluid density. The equations are solved in a transformed σ‐coordinate system with the use of operator‐splitting method (Int. J. Numer. Meth. Fluids 2002; 38 :1045–1068). The numerical model is validated against the one‐dimensional diffusion problem and the two‐dimensional density‐gradient flow. Excellent agreements are obtained between numerical results and analytical solutions. The model is then used to study transport phenomena of dumped sediments into a water body, which has been modelled as a strongly stratified flow. For the two‐dimensional problem, the numerical results compare well with experimental data in terms of mean particle falling velocity and spreading rate of the sediment cloud for both coarse and medium‐size sediments. The model is also employed to study the dumping of sediments in a three‐dimensional environment with the presence of free surface. It is found that during the descending process an annulus‐like cloud is formed for fine sediments whereas a plate‐like cloud for medium‐size sediments. The model is proven to be a good tool to simulate strongly stratified free surface flows. Copyright © 2005 John Wiley & Sons, Ltd.     

Flow of branched polymer melts in a lubricated cross-slot channel: a combined computational and experimental study

Numerical simulations have been performed to evaluate the accuracy of the multimode Giesekus model in predicting the flow behavior of a rheologically well characterized low-density polyethylene melt in a lubricated cross-slot channel. Specifically, the fidelity of the numerical results is established by detailed comparison with flow-induced birefringence measurements in a new optical rheometer with lubricated side walls that allows the creation of ideal two-dimensional flow kinematics that lead to the elimination of end effects commonly encountered in flow birefringence measurements. Based on these comparisons, the ability of the multimode Giesekus model to capture the flow characteristics with reasonable accuracy in the experimentally available Wi range of 21 to 29 has been established. However, it should be noted that the model predictions are, at best, qualitative in the vicinity of the stagnation point. The discrepancy between numerically predicted and experimentally observed stresses in this region is mainly attributed to the inaccuracy of the experimental data that stem from the occurrence of multiple orders of retardation within the measurement volume. Overall, these studies have paved the way for the development of a hi-fidelity lubricated cross-slot channel rheometer.