Free surface Stokes flows obstructed by multiple obstacles

Gravity‐driven Stokes flow down an inclined plane over and around multiple obstacles is considered. The flow problem is formulated in terms of a boundary integral equation and solved using the boundary element method. A Hermitian radial basis function (RBF) is used for the interpolation of the free surface, generation of the unit normal and curvature, and to prescribe the far‐field conditions. For flow over an obstacle, hemispheres are taken. For flow around an obstacle, circular cylinders are modelled and the contact angle condition on the obstacle/free surface intersection specified using the RBF formulation. Explicit profiles are produced for flow over and around two obstacles placed in various locations relative to one another. Interaction due to two obstacles is given by comparisons made with the profiles for flow over and around individual obstacles. In general, when the obstacles are separated by a sufficiently large distance the flow profiles are identical to a single obstacle analysis. For flow over and around two obstacles in‐line with the incident flow, effects of the governing parameters are examined, with variations in plane inclination angle, Bond number, obstacle size, and in the case of obstacles intersecting the free surface, static contact angle is considered. Finally flows over and around three obstacles are modelled. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical solutions of peristaltic flow of Williamson fluid with radially varying MHD in an endoscope

In this article we have studied the peristaltic motion of an incompressible Williamson fluid with constant and radially varying magnetohydrodynamics (MHD) in an endoscope. Using the low Reynolds and long wavelength assumptions, the equations of Williamson fluid model in simplified form are solved using (i) the HAM method and (ii) the Shooting method. The comparisons of both solutions have been found a very good agreement. Graphical results have been presented for various emerging parameters. © British Crown Copyright 2010/MOD. Reproduced with permission. Published by John Wiley & Sons, Ltd.     

A hybrid FVM–LBM method for single and multi‐fluid compressible flow problems

The lattice Boltzmann method (LBM) has established itself as an alternative approach to solve the fluid flow equations. In this work we combine LBM with the conventional finite volume method (FVM), and propose a non‐iterative hybrid method for the simulation of compressible flows. LBM is used to calculate the inter‐cell face fluxes and FVM is used to calculate the node parameters. The hybrid method is benchmarked for several one‐dimensional and two‐dimensional test cases. The results obtained by the hybrid method show a steeper and more accurate shock profile as compared with the results obtained by the widely used Godunov scheme or by a representative flux vector splitting scheme. Additional features of the proposed scheme are that it can be implemented on a non‐uniform grid, study of multi‐fluid problems is possible, and it is easily extendable to multi‐dimensions. These features have been demonstrated in this work. The proposed method is therefore robust and can possibly be applied to a variety of compressible flow situations. Copyright © 2009 John Wiley & Sons, Ltd.     

Hydraulic analysis of free-surface flows with solidification

Summary The paper is concerned with a one-dimensional analysis of plane open-channel flow with continuous solidification. The process is of relevance for recent developments in the casting of steel and other metals. The bottom of the channel consists of a rotating casting roll and a horizontal cooling table, where the solidified material is withdrawn with given velocity. The study is restricted to the region downstream of the top of the casting roll. Surface tension is neglected. In the main part of the analysis inviscid fluid flow is considered since the Reynolds number is very large in the applications. It is found that the steady-state solutions are nonunique in a certain parameter range. In addition to a continuous solution, there are two solutions including hydraulic jumps, with one hydraulic jump being located on the casting roll, the other one on the cooling table. Regarding the stability of the non unique solutions, the evolution of disturbances is investigated numerically as an initial-value problem. It is concluded that the hydraulic jump on the cooling table is unstable, while the other discontinuous solution as well as the continuous solution are stable for sufficiently small disturbances. Which stable solution is attained in the steady state, depends on the history of the process. Friction at the liquid/solid interface is taken into account in the last part of the analysis. A constant friction coefficient is assumed. It is found that the history of the process determines the steady-state solution if, and only if, the friction coefficient is sufficiently small. For larger values of the friction coefficient, the steady-state solution is unique and independent of the history of the transient process. Furthermore, for sufficiently large friction coefficients, stable hydraulic jumps are found, in contrast to the inviscid case, also on the cooling table. Received 19 March 1999; accepted for publication 3 May 1999     

The Use of DNS to Define Stress Producing Events for Turbulent Flow over a Smooth Wall

Over the past 15 years direct numerical simulations (DNS) of turbulent flow and particle image velocimetry (PIV) have provided the opportunity to obtain information about a turbulent velocity field simultaneously at a large number of locations. This paper gives a personal viewpoint of how these techniques are providing new insights about the Reynolds stress producing structures in turbulence generated by flow over a smooth boundary. This revised version was published online in August 2006 with corrections to the Cover Date.     

Investigation of droplet size distribution for liquid-liquid emulsions in Taylor-Couette flows

In this paper, we investigate oil-in-water emulsions in a Taylor-Couette flow. A high-speed camera was employed to record the formation of those emulsions, and image processing was used to obtain the diameter of the droplets. No surfactants were added in order to study the pure effect of the fluid dynamical forces on the droplets. The results for three different oil-in-water emulsions show that the Sauter mean diameter considerably depends on the local shear rate and the material properties and that the droplet size distribution follows a log-normal distribution. We, therefore, propose to express the Sauter mean diameter normalized by Prandtl mixing length in terms of a correlation, which is based on the Kolmogorov turbulence theory. This correlation subsequently depends on the local shear rate and the material properties such as viscosity, density, and interfacial tension. The predictions of the correlation show fairly good agreement with the experimental measurement the Sauter mean diameter. Finally, comparing the predictions of the correlation to the data presented by Eskin et al. [Chem. Eng. Sci. 161 36–47; 2017] shows excellent agreement in the case, where the droplets are larger than the Kolmogorov length scale.     

High‐order finite difference schemes for incompressible flows

This paper presents a new high‐order approach to the numerical solution of the incompressible Stokes and Navier–Stokes equations. The class of schemes developed is based upon a velocity–pressure–pressure gradient formulation, which allows: (i) high‐order finite difference stencils to be applied on non‐staggered grids; (ii) high‐order pressure gradient approximations to be made using standard Padé schemes, and (iii) a variety of boundary conditions to be incorporated in a natural manner. Results are presented in detail for a selection of two‐dimensional steady‐state test problems, using the fourth‐order scheme to demonstrate the accuracy and the robustness of the proposed methods. Furthermore, extensions to higher orders and time‐dependent problems are illustrated, whereas the extension to three‐dimensional problems is also discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

A boundary condition for arbitrary shaped inlets in lattice–Boltzmann simulations

We introduce a mass‐flux‐based inlet boundary condition for the lattice‐Boltzmann method. The proposed boundary condition requires minimal amount of boundary data, it produces a steady‐state velocity field which is accurate close to the inlet even for arbitrary inlet geometries, and yet it is simple to implement. We demonstrate its capability for both simple and complex inlet geometries by numerical experiments. For simple inlet geometries, we show that the boundary condition provides very accurate inlet velocities when Re?1. Even with moderate Reynolds number, the inlet velocities are accurate for practical purposes. Furthermore, the potential of our boundary condition to produce inlet velocities which convincingly adapt to complex inlet geometries is highlighted with two specific examples. Copyright © 2009 John Wiley & Sons, Ltd.