Influence of inertia,topography and gravity on transient axisymmetric thin‐film flow

This study examines theoretically the development of early transients for axisymmetric flow of a thin film over a stationary cylindrical substrate of arbitrary shape. The fluid is assumed to emerge from an annular tube as it is driven by a pressure gradient maintained inside the annulus, and/or by gravity in the axial direction. The interplay between inertia, annulus aspect ratio, substrate topography and gravity is particularly emphasized. Initial conditions are found to have a drastic effect on the ensuing flow. The flow is governed by the thin‐film equations of the ‘boundary‐layer’ type, which are solved by expanding the flow field in terms of orthonormal modes in the radial direction. The formulation is validated upon comparison with the similarity solution of Watson (J. Fluid Mech 1964; 20 :481) leading to an excellent agreement when only 2–3 modes are included. The wave and flow structure are examined for high and low inertia. It is found that low‐inertia fluids tend to accumulate near the annulus exit, exhibiting a standing wave that grows with time. This behaviour clearly illustrates the difficulty faced with coating high‐viscosity fluids. The annulus aspect is found to be influential only when inertia is significant; there is less flow resistance for a film over a cylinder of smaller diameter. For high inertia, the free surface evolves similarly to two‐dimensional flow. The substrate topography is found to have a significant effect on transient behaviour, but this effect depends strongly on inertia. It is observed that the flow of a high‐inertia fluid over a step‐down exhibits the formation of a secondary wave that moves upstream of the primary wave. Gravity is found to help the film (coating) flow by halting or prohibiting the wave growth. The initial film profile and velocity distribution dictate whether the fluid will flow downstream or accumulate near the annulus exit. Copyright © 2004 John Wiley & Sons, Ltd.     

A hybrid spectral/finite‐difference to transient free‐surface flow inside thin symmetric cavities of longitudinally varying thickness

The lubrication theory is extended for transient free‐surface flow of a viscous fluid inside three‐dimensional symmetric thin cavities of thickness that varies in the flow direction. The problem is first formulated for a cavity of arbitrary shape. The moving domain is mapped onto a rectangular domain at each time step of the computation. The pressure, which in this case is governed by the modified Laplace's equation, is expanded in a Fourier series in the spanwise direction. The expansion coefficients are obtained using the finite‐difference method. Only a few modes are usually needed to secure convergence. The flow behaviour is strongly influenced by the cavity thickness. The flows inside a straight, contracting, expanding, and modulated cavities are examined. It is found that while the evolution of the front is always monotonic with time, that of the velocity at the front can be oscillatory if the degree of contraction of the cavity (whether modulated or not) is significant. The velocity of the contact point along the lateral walls is usually larger than that at the front, leading to the straightening of the front. However, for modulated cavities, the front may advance at a faster rate, leading to its own undulation. Copyright © 2005 John Wiley & Sons, Ltd.     

Computer‐algebra multiple‐timescale method for spatially periodic thin‐film viscous‐flow problems

A flexible, fully automated, computer‐algebra algorithm is developed for solving a class of non‐linear partial‐differential evolution equations arising frequently in the modeling of two‐dimensional transient free‐surface viscous thin‐film flows. The method, which is formulated for solving spatially periodic problems, is based upon an explicit multiple‐timescale asymptotic approximation of the thin‐film thickness. It admits the resolution of diverse physical phenomena by employing a finite geometric progression of increasingly slow timescales. The method is implemented on a challenging test problem comprising the evolution of an annular film of viscous liquid, with a free surface, adhering to the exterior of a horizontal rotating circular cylinder; as a model for numerous industrially motivated coating flows, this benchmark problem has been analyzed in diverse numerical and theoretical studies, against whose results those of the present method are compared. The explicit algebraic form of the solution admits a study of large‐time evolutionary dynamics that lies beyond the reach of considerably more expensive conventional numerical solvers, thereby shedding new light on the hitherto‐undiscovered explicit dependence of large‐time evolutionary fluid dynamics in terms of independent parameters describing gravitational and capillary effects. The results obtained from the new computer‐algebra procedure are demonstrated to be in good agreement with those obtained from a bespoke efficient numerical integration method that is spectrally accurate in space and 8th‐order (Runge–Kutta) in time. Newly discovered mechanisms describing the decay of free‐surface wave modes, from arbitrary initial conditions to the steady state, are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Transient two‐layer thin‐film flow

The transient two‐layer thin‐film planar flow is investigated theoretically in this study. The interplay among inertia, viscous and surface/interfacial tension is emphasized. It is found that the film and interface profiles, as well as the flow field, are strongly influenced by the viscosity ratio, velocity and film thickness ratios at inception, and the surface‐to‐interfacial tension ratio. The nonlinear stability of the steady state reveals the formation of a solitary wave after flow inception, which propagates in the form of a convective instability, with the steady state recovered only in the tail (upstream) region of the wave. In the presence of surface/interfacial tension, surface modulation appears, which grows in wavelength and amplitude with position. The flow is found to be particularly stable for higher viscosity of the lower film layer. Copyright © 2010 John Wiley & Sons, Ltd.     

An implicit pseudo‐spectral multi‐domain method for the simulation of incompressible flows

A pseudo‐spectral method for the solution of incompressible flow problems based on an iterative solver involving an implicit treatment of linearized convective terms is presented. The method allows the treatment of moderately complex geometries by means of a multi‐domain approach and it is able to cope with non‐constant fluid properties and non‐orthogonal problem domains. In addition, the fully implicit scheme yields improved stability properties as opposed to semi‐implicit schemes commonly employed. Key components of the method are a Chebyshev collocation discretization, a special pressure–correction scheme, and a restarted GMRES method with a preconditioner derived from a fast direct solver. The performance of the proposed method is investigated by considering several numerical examples of different complexity, and also includes comparisons to alternative solution approaches based on finite‐volume discretizations. Copyright © 2003 John Wiley & Sons, Ltd.     

A Legendre‐spectral element method for flow and heat transfer about an accelerating droplet

The problem of flow and heat transfer associated with a spherical droplet accelerated from rest under gravitational force is studied using a Legendre‐spectral element method in conjunction with a mixed time integration procedure to advance the solution in time. An influence matrix technique that exploits the superposition principle is adapted to resolve the lack of vorticity boundary conditions and to decouple the equations from the interfacial couplings. The computed flow and temperature fields, the drag coefficient, the Nusselt number, and the interfacial velocity and vorticity are presented for a drop moving vertically in a quiescent gas of infinite extent to illustrate the evolution of the flow and temperature fields. Comparison of the predicted drag coefficient and the Nusselt number against previous numerical and experimental results indicate good agreement. Copyright © 2000 John Wiley & Sons, Ltd.     

Computations of a laminar backward‐facing step flow at Re=800 with a spectral domain decomposition method

The two‐dimensional laminar incompressible flow over a backward‐facing step is computed using a spectral domain decomposition approach. A minimum number of subdomains (two) is used; high resolution being achieved by increasing the order of the basis Chebyshev polynomial. Results for the case of a Reynolds number of 800 are presented and compared in detail with benchmark computations. Stable accurate steady flow solutions were obtained using substantially fewer nodes than in previously reported simulations. In addition, the problem of outflow boundary conditions was examined on a shortened domain. Because of their more global nature, spectral methods are particularly sensitive to imposed boundary conditions, which may be exploited in examining the effect of artificial (non‐physical) outflow boundary conditions. Two widely used set of conditions were tested: pseudo stress‐free conditions and zero normal gradient conditions. Contrary to previous results using the finite volume approach, the latter is found to yield a qualitatively erroneous yet stable flow‐field. Copyright © 1999 John Wiley & Sons, Ltd.     

Large‐eddy simulation with complex 2‐D geometries using a parallel finite‐element/spectral algorithm

A parallel stabilized finite‐element/spectral formulation is presented for incompressible large‐eddy simulation with complex 2‐D geometries. A unique discretization scheme is developed consisting of a streamline‐upwind Petrov–Galerkin/Pressure‐Stabilized Petrov–Galerkin (SUPG/PSPG) finite‐element discretization in the 2‐D plane with a collocated spectral/pseudospectral discretization in the out‐of‐plane direction. This formulation provides an efficient approach for solving 3‐D flows over arbitrary 2‐D geometries. Utilizing this discretization and through explicit temporal treatment of the non‐linear terms, the system of equations for each Fourier mode is decoupled within each time step. A novel parallelization approach is then taken, where the computational work is partitioned in Fourier space. A validation of the algorithm is presented via comparison of results for flow past a circular cylinder with published values for Re=195, 300, and 3900. Copyright © 2003 John Wiley & Sons, Ltd.     

An algebraic factorisation scheme for spectral element solution of incompressible flow and scalar transport

A spectral element algorithm for solution of the unsteady incompressible Navier–Stokes and scalar (species/heat) transport equations is developed using the algebraic factorisation scheme. The new algorithm utilises Nth order Gauss–Lobatto–Legendre points for velocity and the scalar, while (N-2)th order Gauss–Legendre points are used for pressure. As a result, the algorithm does not require inter-element continuity for pressure and pressure boundary conditions on solid surfaces. Implementations of the algorithm are performed for conforming and non-conforming grids. The latter is accomplished using both the point-wise matching and integral projection methods, and applied for grids with both polynomial and geometric non-conformities. Code validation cases include the unsteady scalar convection equation, and Kovasznay flow in two- and three-dimensional domains. Using cases with analytical solutions, the algorithm is shown to achieve spectral accuracy in space and second-order accuracy in time. The results for the Boussinesq approximation for buoyancy-driven flows, and the species mixing in a continuous flow micro-mixer are also included as examples of applications that require long-time integration of the scalar transport equations.     

A Lagrangian boundary element approach to transient three‐dimensional free surface flow in thin cavities

The lubrication theory is extended for transient free‐surface flow of a viscous fluid inside a three‐dimensional thin cavity. The problem is closely related to the filling stage during the injection molding process. The pressure, which in this case is governed by the Laplace's equation, is determined using the boundary element method. A fully Lagrangian approach is implemented for the tracking of the evolving free surface. The domain of computation is the projection of the physical domain onto the (x, y) plane. This approach is valid for simple and complex cavities as illustrated for the cases of a flat plate and a curved plate. It is found that the flow behavior is strongly influenced by the shape of the initial fluid domain, the shape of the cavity, and inlet flow pressure. Copyright © 2001 John Wiley & Sons, Ltd.     

Stability of high‐speed two‐layer film casting of Newtonian fluids

The steady‐state flow and its linear stability are investigated for the isothermal two‐layer film casting process. Newtonian fluids are considered in this study. The continuity of traction is ensured at the interface, and the axial velocity is assumed to be uniform across each film layer separately. The effects of inertia, gravity, fluid parameters and processing conditions on the steady‐state flow and its stability are studied. The results indicate that the fluid properties and the processing conditions have significant influence on the flow. The flow stability is strongly dependent on the layer layout with respect to the take‐up rolling process. The frequency of the (unstable) disturbance is insensitive to flow and processing parameters. Copyright © 2006 John Wiley & Sons, Ltd.     

Multi‐material incompressible flow simulation using the moment‐of‐fluid method

This paper compares the numerical performance of the moment‐of‐fluid (MOF) interface reconstruction technique with Youngs, LVIRA, power diagram (PD), and Swartz interface reconstruction techniques in the context of a volume‐of‐fluid (VOF) based finite element projection method for the numerical simulation of variable‐density incompressible viscous flows. In pure advection tests with multiple materials MOF shows dramatic improvements in accuracy compared with the other methods. In incompressible flows where density differences determine the flow evolution, all the methods perform similarly for two material flows on structured grids. On unstructured grids, the second‐order MOF, LVIRA, and Swartz methods perform similarly and show improvement over the first‐order Youngs' and PD methods. For flow simulations with more than two materials, MOF shows increased accuracy in interface positions on coarse meshes. In most cases, the convergence and accuracy of the computed flow solution was not strongly affected by interface reconstruction method. Published in 2009 by John Wiley & Sons, Ltd.     

A high‐order triangular discontinuous Galerkin oceanic shallow water model

A high‐order triangular discontinuous Galerkin (DG) method is applied to the two‐dimensional oceanic shallow water equations. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high‐order Lagrange polynomials on the triangle using nodal sets up to 15th order. Both the area and boundary integrals are evaluated using order 2N Gauss cubature rules. The use of exact integration for the area integrals leads naturally to a full mass matrix; however, by using straight‐edged triangles we eliminate the mass matrix completely from the discrete equations. Besides obviating the need for a mass matrix, triangular elements offer other obvious advantages in the construction of oceanic shallow water models, specifically the ability to use unstructured grids in order to better represent the continental coastlines for use in tsunami modeling. In this paper, we focus primarily on testing the discrete spatial operators by using six test cases—three of which have analytic solutions. The three tests having analytic solutions show that the high‐order triangular DG method exhibits exponential convergence. Furthermore, comparisons with a spectral element model show that the DG model is superior for all polynomial orders and test cases considered. Copyright © 2007 John Wiley & Sons, Ltd.     

Three‐dimensional effects in dynamically loaded journal bearings

The effects of non‐Newtonian lubricants on the dynamics of a 3D journal bearing are investigated using a moving spectral element method. Comparisons are made with the findings reported for the 2D case. The variation of L/D, the ratio of the length of the bearing to its diameter, is shown to have a significant effect on the stability properties of the journal. Copyright © 1999 John Wiley & Sons, Ltd.     

Linear and non‐linear simulations of feedback control in plane Poiseuille flow

This paper examines the performance of optimal linear quadratic state and output feedback controllers in stabilizing two‐dimensional perturbations in a plane Poiseuille flow. The synthesis of the controllers is based on a linearized model of the flow using a new set of interpolating polynomials in the wall‐normal direction, which automatically satisfy the homogeneous Dirichlet and Neumann boundary conditions at the walls and eliminate spurious eigenvalues. The controllers are implemented into a non‐linear Navier–Stokes solver, which is modified to compute the evolution of the flow perturbations. Two cases are examined, one with small initial disturbances that do not violate the linearity assumptions and the other with much larger disturbances that trigger the non‐linear convection terms. For the smallest disturbances, the solver accurately reproduced the results of the linear simulations of open‐ and closed‐loop systems. The simulations for the larger disturbances without control showed a rapid initial growth but the flow soon reached a saturated state in agreement with previous findings in the literature. The large initial growth is a consequence of the non‐normal nature of the system dynamics. The state feedback and output feedback controllers were able to reduce significantly the perturbation energy. For the larger disturbances, the energy calculated from the state variables is well below the energy evaluated by direct integration of the velocity field. This is probably due to the non‐linear terms transferring energy to harmonics of the considered wavenumber, which are not sensed by the linear controller. Copyright © 2008 John Wiley & Sons, Ltd.     

A second‐order time accurate finite element method for quasi‐incompressible viscous flows

A finite element method for quasi‐incompressible viscous flows is presented. An equation for pressure is derived from a second‐order time accurate Taylor–Galerkin procedure that combines the mass and the momentum conservation laws. At each time step, once the pressure has been determined, the velocity field is computed solving discretized equations obtained from another second‐order time accurate scheme and a least‐squares minimization of spatial momentum residuals. The terms that stabilize the finite element method (controlling wiggles and circumventing the Babuska–Brezzi condition) arise naturally from the process, rather than being introduced a priori in the variational formulation. A comparison between the present second‐order accurate method and our previous first‐order accurate formulation is shown. The method is also demonstrated in the computation of the leaky‐lid driven cavity flow and in the simulation of a crossflow past a circular cylinder. In both cases, good agreement with previously published experimental and computational results has been obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

A stabilized Nitsche‐type extended embedding mesh approach for 3D low‐ and high‐Reynolds‐number flows

This paper presents a stabilized extended finite element method (XFEM) based fluid formulation to embed arbitrary fluid patches into a fixed background fluid mesh. The new approach is highly beneficial when it comes to computational grid generation for complex domains, as it allows locally increased resolutions independent from size and structure of the background mesh. Motivating applications for such a domain decomposition technique are complex fluid‐structure interaction problems, where an additional boundary layer mesh is used to accurately capture the flow around the structure. The objective of this work is to provide an accurate and robust XFEM‐based coupling for low‐ as well as high‐Reynolds‐number flows. Our formulation is built from the following essential ingredients: Coupling conditions on the embedded interface are imposed weakly using Nitsche's method supported by extra terms to guarantee mass conservation and to control the convective mass transport across the interface for transient viscous‐dominated and convection‐dominated flows. Residual‐based fluid stabilizations in the interior of the fluid subdomains and accompanying face‐oriented fluid and ghost‐penalty stabilizations in the interface zone stabilize the formulation in the entire fluid domain. A detailed numerical study of our stabilized embedded fluid formulation, including an investigation of variants of Nitsche's method for viscous flows, shows optimal error convergence for viscous‐dominated and convection‐dominated flow problems independent of the interface position. Challenging two‐dimensional and three‐dimensional numerical examples highlight the robustness of our approach in all flow regimes: benchmark computations for laminar flow around a cylinder, a turbulent driven cavity flow at Re = 10000 and the flow interacting with a three‐dimensional flexible wall. Copyright © 2016 John Wiley & Sons, Ltd.     

Modelling of two‐dimensional laminar flow using finite element method

In this paper, a Galerkin weighted residual finite element numerical solution method, with velocity material time derivative discretisation, is applied to solve for a classical fluid mechanics system of partial differential equations modelling two‐dimensional stationary incompressible Newtonian fluid flow. Classical examples of driven cavity laminar flow and laminar flow past a cylinder are presented. Numerical results are compared with data found in the literature. Copyright © 1999 John Wiley & Sons, Ltd.     

On a boundary condition for pressure‐driven laminar flow of incompressible fluids

We prove in Theorem 1 a new relationship between the stress, pressure, velocity, and mean curvature for embedded surfaces in incompressible viscous flows. This is then used to define a corresponding modified pressure boundary condition for flow of Newtonian and generalized Newtonian fluids. These results agree with an intuitive notion of the flow physics but apparently have not previously been shown rigorously. We describe some of the implementation issues for inflow and outflow boundaries in this context and give details for a penalty treatment of the associated tangential velocity constraint. This is then implemented and applied in high‐resolution 3D benchmark calculations for a representative generalized viscosity model. Copyright © 2007 John Wiley & Sons, Ltd.