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A boundary integral technique is developed to study the free surface flow of a steady, two‐dimensional, incompressible, irrotational and inviscid fluid flow which is produced by two submerged sinks (or sources) which are of equal strength, placed along a solid horizontal boundary with a stagnation point on the free surface in a two layer stratified fluid in the presence of gravity. A special form of the Riemann‐Hilbert problem, namely the Dirichlet boundary problem, is applied in the derivation of the governing non‐linear boundary integral‐differential equations which are solved for the fluid velocity on the free surface and this involves the use of an interpolative technique and an iterative process. Results have been obtained for the free surface flow for various values of the Froude number and sink locations on the solid horizontal boundary and we have also studied the largest value of the Froude number for which no convergent solutions are possible, namely the critical Froude number. We have found that the free surface profile is dependent on two parameters, namely the Froude number on the free surface and the non‐dimensional distance between the two sinks.

The purpose of this study is to examine theoretically the axisymmetric flow of a steady free-surface jet emerging from a tube for high inertia flow and moderate surface tension effect.

The method of matched asymptotic expansion is used to explore the rich dynamics near the exit where a stress singularity occurs. A boundary layer approach is also proposed to capture the flow further downstream where the free surface layer has grown significantly.

The jet is found to always contract near the tube exit. In contrast to existing numerical studies, the author explores the strength of upstream influence and the flow in the wall layer, resulting from jet contraction. This influence becomes particularly evident from the nonlinear pressure dependence on the upstream distance, as well as the pressure undershoot and overshoot at the exit for weak and strong gravity levels, respectively. The approach is validated against existing experimental and numerical data for the jet profile and centerline velocity where good agreement is obtained. Far from the exit, the author shows how the solution in the diffusive region can be matched to the inviscid far solution, providing the desired appropriate initial condition for the inviscid far flow solution. The location, at which the velocity becomes uniform across the jet, depends strongly on the gravity level and exhibits a non-monotonic behavior with respect to gravity and applied pressure gradient. The author finds that under weak gravity, surface tension has little influence on the final jet radius. The work is a crucial supplement to the existing numerical literature.

Given the presence of the stress singularity at the exit, the work constitutes a superior alternative to a computational approach where the singularity is typically and inaccurately smoothed over. In contrast, in the present study, the singularity is entirely circumvented. Moreover, the flow details are better elucidated, and the various scales involved in different regions are better identified.

The research is directed at development of an efficient and accurate technique for modelling incompressible free surface flows in which viscous effects may not be neglected. The paper describes the methodology, and gives illustrative results for simple geometries.

The pseudospectral matrix element method of discretisation is selected as the basis for the CFD technique adopted, because of its high spectral accuracy. It is implemented as a means of solving the Navier‐Stokes equations coupled with the modified compressibility method.

The viscous solver has been validated for the benchmark cases of uniform flow past a cylinder at a Reynolds number of 40, and 2D cavity flows. Results for sloshing of a viscous fluid in a tank have been successfully compared with those from a linearised analytical solution. Application of the method is illustrated by the results for the interaction of an impulsive wave with a surface piercing circular cylinder in a cylindrical tank.

The paper demonstrates the viability of the approach adopted. The limitation of small amplitude waves should be tackled in future work.

The results will have particular significance in the context of validating computations from more complex schemes applicable to arbitrary geometries.

The new methodology and results are of interest to the community of those developing numerical models of flow past marine structures.

The purpose of this paper is to extend the penalty concept to treat partial slip, free surface, contact and related boundary conditions in viscous flow simulation.

The penalty partial‐slip formulation is analysed and related to the classical Navier slip condition. The same penalty scheme also allows partial penetration through a boundary, hence the implementation of porous wall boundaries. The finite element method is used for investigating and interpreting penalty approaches to boundary conditions.

The generalised penalty approach is verified by means of a novel variant of the circular‐Couette flow problem, having partial slip on one of the cylindrical boundaries, for which an analytic solution is derived. Further verificationis provided by consideration of viscous flow over a sphere with partial slip on the surface, and comparison of numerical and classical solutions. Numerical studies illustrate the versatility of the approach.

The penalty approach is applied to some different boundaries: partial slip and partial penetration with no/full slip/penetration as limiting cases; free surface; space‐ and time‐varying boundary conditions which allow progressive contact over time. Application is made to curved and inclined boundaries. Sensitivity of flow to penalty parameters is an avenue for continued research, as is application of the penalty approach for non‐Newtonian flows.

This is the first work to show the relation between penalty formulation of boundary conditions and physical boundary conditions. It provides a method that overcomes past difficulties in implementing partial slip on boundaries of general shape, and which handles progressive contact. It also provides useful benchmark problems for future studies.

This article is concerned with the numerical simulation of a reverse roller‐coating process, which involves the computation of Newtonian viscous incompressible flows with free‐surfaces. A numerical scheme is applied of a transient finite element form, a semi‐implicit Taylor‐Galerkin/pressurecorrection algorithm. For free‐surface prediction, we use kinematic boundary adjustment with a mesh‐stretching algorithm. In the present work, an alloy sheet (foil) passes over a large roller and then a smaller applicator roller, which provides the in‐feed. In combination, the applicator roller, the foil and the fluid form part of the underside coating mechanism. The aim of this study is to investigate fundamental aspects of the process, to ultimately address typical coating instabilities. These may take the form of chatter and starvation. A uniform coating thickness is the desired objective. A mathematical model is derived to describe the solvent coating applied to the underside of the sheet, assuming that the lacquer is a Newtonian fluid. In particular, the work has concentrated on the flow patterns that result and a parameter sensitivity analysis covering the appropriate operating windows of applied conditions. Effects of independent variation in roll‐speed and foil‐speed are investigated, to find that maxima in pressure, lift and drag arise at the nip and are influenced in a linear fashion.

The volume of fluid (VOF) method is a numerical technique to track the developing free surfaces of liquids in motion. This method can, for example, be applied to compute the liquid flow patterns in a rotating cone reactor. For this application a spherical coordinate system is most suited. The novel derivation of the extended VOF algorithms for this class of applications is presented here. Some practical limitations of this method, that are inherent in the geometry of the described system, are discussed.

This article focuses on the comparative study of annular wire‐coating flows with polymer melt materials. Different process designs are considered of pressure‐ and tube‐tooling, complementing earlier studies on individual designs. A novel mass‐balance free‐surface location technique is proposed. The polymeric materials are represented via shear‐thinning, differential viscoelastic constitutive models, taken of exponential Phan‐Thien/Tanner form. Simulations are conducted for these industrial problems through distributed parallel computation, using a semi‐implicit time‐stepping Taylor‐Galerkin/pressure‐correction algorithm. On typical field results and by comparing short‐against full‐die pressure‐tooling solutions, shear‐rates are observed to increase ten fold, while strain rates increase one hundred times. Tube‐tooling shear and extension‐rates are one quarter of those for pressure‐tooling. These findings across design options, have considerable bearing on the appropriateness of choice for the respective process involved. Parallel finite element results are generated on a homogeneous network of Intel‐chip workstations, running PVM (Parallel Vitual Machine) protocol over a Solaris operating system. Parallel timings yield practically ideal linear speed‐up over the set number of processors.

The purpose of this paper is to show how a surface tension model and an eddy viscosity based on the Smagorinsky sub‐grid scale model, which belongs to the Large‐Eddy Simulation (LES) theory for turbulent flow, have been introduced into ISPH (Incompressible smoothed particle hydrodynamics) method. In addition, a small modification in the source term of pressure Poisson equation has been introduced as a stabilizer for robust simulations. This stabilization generates a smoothed pressure distribution and keeps the total volume of fluid, and it is analogous to the recent modification in MPS.

The surface tension force in free surface flow is evaluated without a direct modeling of surrounding air for decreasing computational costs. The proposed model was validated by calculating the surface tension force in the free surface interface for a cubic‐droplet under null‐gravity and the milk crown problem with different resolution models. Finally, effects of the eddy viscosity have been discussed with a fluid‐fluid interaction simulation.

From the numerical tests, the surface tension model can handle free surface tension problems including high curvature without special treatments. The eddy viscosity has clear effects in adjusting the splashes and reduces the deformation of free surface in the interaction. Finally, the proposed stabilization appeared in the source term of pressure Poisson equation has an important role in the simulation to keep the total volume of fluid.

An incompressible smoothed particle hydrodynamics is developed to simulate milk crown problem using a surface tension model and the eddy viscosity.

The paper targets on providing new experimental data for validation of the well-established mathematical models within the framework of the lattice Boltzmann method (LBM), which are applied to problems of casting processes in complex mould cavities.

An experimental campaign aiming at the free-surface flow within a system of narrow channels is designed and executed under well-controlled laboratory conditions. An in-house lattice Boltzmann solver is implemented. Its algorithm is described in detail and its performance is tested thoroughly using both the newly recorded experimental data and well-known analytical benchmark tests.

The benchmark tests prove the ability of the implemented algorithm to provide a reliable solution when the surface tension effects become dominant. The convergence of the implemented method is assessed. The two new experimentally studied problems are resolved well by simulations using a coarse computational grid.

A detailed set of original experimental data for validation of computational schemes for simulations of free-surface gravity-driven flow within a system of narrow channels is presented.

The jet impingement usually accompanying large interface movement is studied by the in-house solver MLParticle-SJTU based on the modified moving particle semi-implicit (MPS) method, which can provide more accurate pressure fields and deformed interface shape. The comparisons of the pressure distribution and the shape of free surface between the presented numerical results and the analytical solution are investigated. The paper aims to discuss these issues.

To avoid the instability in traditional MPS, a modified MPS method is employed, which include mixed source term for Poisson pressure equation (PPE), kernel function without singularity, momentum conservative gradient model and highly precise free surface detection approach. Detailed analysis on improved schemes in the modified MPS is carried out. In particular, three kinds of source term in PPE are considered, including: particle number density (PND) method, mixed source term method and divergence-free method. Two typical kernel functions containing original kernel function with singularity and modified kernel function without singularity are analyzed. Three kinds of pressure gradient are considered: original pressure gradient (OPG), conservative pressure gradient (CPG) and modified pressure gradient (MPG). In addition, particle convergence is performed by running the simulation with various spatial resolutions. Finally, the comparison of the pressure fields by the modified MPS and by SPH is presented.

The modified MPS method can provide a reliable pressure distribution and the shape of the free surface compared to the analytical solution in a steady state after the water jet impinging on the wall. Specifically, mixed source term in PPE can give a reasonable profile of the shape of free surface and pressure distribution, while PND method adopted in the traditional MPS is not stable in simulation, and divergence-free method cannot produce rational pressure field near the wall. Two kernel functions show similar pressure field, however, the kernel function without singularity is preferred in this case to predict the profile of free surface and pressure on the wall. The shape of free surface by CPG and MPG is agreement with the analytical solution, while a great discrepancy can be observed by OPG. The pressure peak by MPG is closer to the analytical solution than that by CPG, while the pressure distribution on the right hand side of the pressure peak by latter is better match with the analytical solution than that by former. Besides, fine spatial resolution is necessary to achieve a good agreement with analytical results. In addition, the pressure field by the modified MPS is also quite similar to that by SPH, and this can further validate the reliable of current modified MPS.

The present modified MPS appears to be a stable and reliable tool to deal with the impinging jet flow problems involving large interface movement. Mixed source term in PPE is superior to PND adopted in the traditional MPS and divergence-free method. The kernel function without singularity is preferred to improve the computational accuracy in this case. CPG is a good choice to obtain the shape of free surface and the pressure distribution by jet impingement.