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.     

A high‐order discontinuous Galerkin method for all‐speed flows

In this article, we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of steady solutions of the compressible fully coupled Reynolds‐averaged Navier–Stokes and k ? ω turbulence model equations for solving all‐speed flows. The system of equations is iterated to steady state by means of an implicit scheme. The DG solution is extended to the incompressible limit by implementing a low Mach number preconditioning technique. A full preconditioning approach is adopted, which modifies both the unsteady terms of the governing equations and the dissipative term of the numerical flux function by means of a new preconditioner, on the basis of a modified version of Turkel's preconditioning matrix. At sonic speed the preconditioner reduces to the identity matrix thus recovering the non‐preconditioned DG discretization. An artificial viscosity term is added to the DG discretized equations to stabilize the solution in the presence of shocks when piecewise approximations of order of accuracy higher than 1 are used. Moreover, several rescaling techniques are implemented in order to overcome ill‐conditioning problems that, in addition to the low Mach number stiffness, can limit the performance of the flow solver. These approaches, through a proper manipulation of the governing equations, reduce unbalances between residuals as a result of the dependence on the size of elements in the computational mesh and because of the inherent differences between turbulent and mean‐flow variables, influencing both the evolution of the Courant Friedrichs Lewy (CFL) number and the inexact solution of the linear systems. The performance of the method is demonstrated by solving three turbulent aerodynamic test cases: the flat plate, the L1T2 high‐lift configuration and the RAE2822 airfoil (Case 9). The computations are performed at different Mach numbers using various degrees of polynomial approximations to analyze the influence of the proposed numerical strategies on the accuracy, efficiency and robustness of a high‐order DG solver at different flow regimes. Copyright © 2014 John Wiley & Sons, Ltd.     

A pressure‐based unstructured grid method for all‐speed flows

An all‐speed algorithm based on the SIMPLE pressure‐correction scheme and the ‘retarded‐density’ approach has been formulated and implemented within an unstructured grid, finite volume (FV) scheme for both incompressible and compressible flows, the latter involving interaction of shock waves. The collocated storage arrangement for all variables is adopted, and the checkerboard oscillations are eliminated by using a pressure‐weighted interpolation method, similar to that of Rhie and Chow [Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal 1983; 21 : 1525]. The solution accuracy is greatly enhanced when a higher‐order convection scheme combined with adaptive mesh refinement (AMR) are used. Copyright © 2000 John Wiley & Sons, Ltd.     

High‐order strand grid methods for low‐speed and incompressible flows

A novel high‐order finite volume scheme using flux correction methods in conjunction with structured finite differences is extended to low Mach and incompressible flows on strand grids. Flux correction achieves a high order by explicitly canceling low‐order truncation error terms across finite volume faces and is applied in unstructured layers of the strand grid. The layers are then coupled together using a source term containing summation‐by‐parts finite differences in the strand direction. A preconditioner is employed to extend the method to low speed and incompressible flows. We further extend the method to turbulent flows with the Spalart–Allmaras model. Laminar flow test cases indicate improvements in accuracy and convergence using the high‐order preconditioned method, while turbulent body‐of‐revolution flow results show improvements in only some cases, perhaps because of dominant errors arising from the turbulence model itself. Copyright © 2016 John Wiley & Sons, Ltd.     

Efficient high‐resolution relaxation schemes for hyperbolic systems of conservation laws

In this work, we present a total variation diminishing (TVD) scheme in the zero relaxation limit for nonlinear hyperbolic conservation law using flux limiters within the framework of a relaxation system that converts a nonlinear conservation law into a system of linear convection equations with nonlinear source terms. We construct a numerical flux for space discretization of the obtained relaxation system and modify the definition of the smoothness parameter depending on the direction of the flow so that the scheme obeys the physical property of hyperbolicity. The advantages of the proposed scheme are that it can give second‐order accuracy everywhere without introducing oscillations for 1‐D problems (at least with) smooth initial condition. Also, the proposed scheme is more efficient as it works for any non‐zero constant value of the flux limiter ? ? [0, 1], where other TVD schemes fail. The resulting scheme is shown to be TVD in the zero relaxation limit for 1‐D scalar equations. Bound for the limiter function is obtained. Numerical results support the theoretical results. Copyright © 2007 John Wiley & Sons, Ltd.     

A high‐order element based adaptive mesh refinement strategy for three‐dimensional unstructured grid

Adaptive mesh refinement (AMR) shows attractive properties in automatically refining the flow region of interest, and with AMR, better prediction can be obtained with much less labor work and cost compared to manually remeshing or the global mesh refinement. Cartesian AMR is well established; however, AMR on hybrid unstructured mesh, which is heavily used in the high‐Reynolds number flow simulation, is less matured and existing methods may result in degraded mesh quality, which mostly happens in the boundary layer or near the sharp geometric features. User intervention or additional constraints, such as freezing all boundary layer elements or refining the whole boundary layer, are required to assist the refinement process. In this work, a novel AMR strategy is developed to handle existing difficulties. In the new method, high‐order unstructured elements are first generated based on the baseline mesh; then the refinement is conducted in the parametric space; at last, the mesh suitable for the solver is output. Generating refined elements in the parametric space with high‐order elements is the key of this method and this helps to guarantee both the accuracy and robustness. With the current method, 3‐dimensional hybrid unstructured mesh of huge size and complex geometry can be automatically refined, without user intervention nor additional constraints. With test cases including the 2‐dimensional airfoil and 3‐dimensional full aircraft, the current AMR method proves to be accurate, simple, and robust.     

A high‐resolution wetting and drying algorithm for free‐surface hydrodynamics

A new wetting and drying algorithm for numerical modeling free‐surface flows is proposed and analyzed. A well structured, mildly nonlinear system for the discrete water surface elevation is derived from the governing differential equations by requiring a correct mass balance in wet areas as well as in the region of transition from wet to dry and from dry to wet. Existence and uniqueness of the numerical solution, along with a convergence analysis of an iterative scheme for the mildly nonlinear system, is provided. The present algorithm is devised to use high‐resolution bathymetric data at subgrid level. The resulting model is quite efficient, does not require a threshold value for minimal water depth, does not produce un‐physical negative water depths and generates accurate results with relatively coarse mesh and large time step size. These features are illustrated on a severe test‐case with known analytical solution. Copyright © 2008 John Wiley & Sons, Ltd.     

A sharp‐interface immersed boundary framework for simulations of high‐speed inviscid compressible flows

A new finite‐volume flow solver based on the hybrid Cartesian immersed boundary (IB) framework is developed for the solution of high‐speed inviscid compressible flows. The IB method adopts a sharp‐interface approach, wherein the boundary conditions are enforced on the body geometry itself. A key component of the present solver is a novel reconstruction approach, in conjunction with inverse distance weighting, to compute the solutions in the vicinity of the solid‐fluid interface. We show that proposed reconstruction leads to second‐order spatial accuracy while also ensuring that the discrete conservation errors diminish linearly with grid refinement. Investigations of supersonic and hypersonic inviscid flows over different geometries are carried out for an extensive validation of the proposed flow solver. Studies on cylinder lift‐off and shape optimisation in supersonic flows further demonstrate the efficacy of the flow solver for computations with moving and shape‐changing geometries. These studies conclusively highlight the capability of the proposed IB methodology as a promising alternative for robust and accurate computations of compressible fluid flows on nonconformal Cartesian meshes.     

A high‐order backward forward sweep interpolating algorithm for semi‐Lagrangian method

Conventional semi‐Lagrangian methods often suffer from poor accuracy and imbalance problems of advected properties because of low‐order interpolation schemes used and/or inability to reduce both dissipation and dispersion errors even with high‐order schemes. In the current work, we propose a fourth‐order semi‐Lagrangian method to solve the advection terms at a computing cost of third‐order interpolation scheme by applying backward and forward interpolations in an alternating sweep manner. The method was demonstrated for solving 1‐D and 2‐D advection problems, and 2‐D and 3‐D lid‐driven cavity flows with a multi‐level V‐cycle multigrid solver. It shows that the proposed method can reduce both dissipation and dispersion errors in all regions, especially near sharp gradients, at a same accuracy as but less computing cost than the typical fourth‐order interpolation because of fewer grids used. The proposed method is also shown able to achieve more accurate results on coarser grids than conventional linear and other high‐order interpolation schemes in the literature. Copyright © 2017 John Wiley & Sons, Ltd.     

A robust multi‐grid pressure‐based algorithm for multi‐fluid flow at all speeds

This paper reports on the implementation and testing, within a full non‐linear multi‐grid environment, of a new pressure‐based algorithm for the prediction of multi‐fluid flow at all speeds. The algorithm is part of the mass conservation‐based algorithms (MCBA) group in which the pressure correction equation is derived from overall mass conservation. The performance of the new method is assessed by solving a series of two‐dimensional two‐fluid flow test problems varying from turbulent low Mach number to supersonic flows, and from very low to high fluid density ratios. Solutions are generated for several grid sizes using the single grid (SG), the prolongation grid (PG), and the full non‐linear multi‐grid (FMG) methods. The main outcomes of this study are: (i) a clear demonstration of the ability of the FMG method to tackle the added non‐linearity of multi‐fluid flows, which is manifested through the performance jump observed when using the non‐linear multi‐grid approach as compared to the SG and PG methods; (ii) the extension of the FMG method to predict turbulent multi‐fluid flows at all speeds. The convergence history plots and CPU‐times presented indicate that the FMG method is far more efficient than the PG method and accelerates the convergence rate over the SG method, for the problems solved and the grids used, by a factor reaching a value as high as 15. Copyright © 2003 John Wiley & Sons, Ltd.     

A stabilized finite element formulation for high‐speed inviscid compressible flows using finite calculus

This paper aims at the development of a new stabilization formulation based on the finite calculus (FIC) scheme for solving the Euler equations using the Galerkin FEM on unstructured triangular grids. The FIC method is based on expressing the balance of fluxes in a space–time domain of finite size. It is used to prevent the creation of instabilities typically present in numerical solutions due to the high convective terms and sharp gradients. Two stabilization terms, respectively called streamline term and transverse term, are added via the FIC formulation to the original conservative equations in the space–time domain. An explicit fourth‐order Runge–Kutta scheme is implemented to advance the solution in time. The presented numerical test examples for inviscid flows prove the ability of the proposed stabilization technique for providing appropriate solutions especially near shock waves. Although the derived methodology delivers precise results with a nearly coarse mesh, a mesh refinement technique is coupled to the solution process for obtaining a suitable mesh particularly in the high‐gradient zones. Copyright © 2014 John Wiley & Sons, Ltd.     

A two grid algorithm based on perturbation and homotopy methods

In this paper, we propose a new class of bi-grid algorithm to solve large scale linear algebraic equations. This method is based on homotopy, perturbation technique and Padé approximants. To cite this article: R. El Mokhtari et al., C. R. Mecanique 330 (2002) 825–830.     

Simulations of high‐aspect‐ratio jets

There are many practical situations when jets are emanating from non‐axis‐symmetric apertures, yet numerical simulations of such three‐dimensional jets are scarce and most of them have failed to reproduce some of the unique flow features. Examples of this type of jets are gas leaks from flanges. These can be treated as jets issuing from high aspect ratio rectangular orifices. The present work consists of a series of large eddy simulations typifying such jets using different inflow boundary conditions. Good agreement with available experiments was observed provided appropriate boundary conditions were present. A discrete method for formulating turbulence data with a known energy spectrum for an inflow condition is outlined and evaluated with three other inflow conditions–a steady uniform profile, a steady parabolic profile and pseudo‐random noise. The implementation of the new inlet condition results in a more realistic centreline velocity decay where the division between the end of the potential core region and the start of the characteristic decay region is clearly visible. Large velocity oscillations are also observed in the final quarter of the domain (15–20 slot widths downstream). Similar oscillations have been observed in real jets. Off‐centre mean velocity peaks are present along the major axis 10 slot widths downstream of the exit in all the simulations. The peaks are approximately 3% of the centreline velocity. The presence of the off‐centre peaks are proved to be independent of jet inflow boundary conditions and an explanation for the mechanism causing the off‐centre peaks is given. Copyright © 2002 John Wiley & Sons, Ltd.     

Semi‐Lagrangian advection on a spherical geodesic grid

A simple and efficient numerical method for solving the advection equation on the spherical surface is presented. To overcome the well‐known ‘pole problem’ related to the polar singularity of spherical coordinates, the space discretization is performed on a geodesic grid derived by a uniform triangulation of the sphere; the time discretization uses a semi‐Lagrangian approach. These two choices, efficiently combined in a substepping procedure, allow us to easily determine the departure points of the characteristic lines, avoiding any computationally expensive tree‐search. Moreover, suitable interpolation procedures on such geodesic grid are presented and compared. The performance of the method in terms of accuracy and efficiency is assessed on two standard test cases: solid‐body rotation and a deformation flow. Copyright © 2007 John Wiley & Sons, Ltd.     

Non‐oscillatory relaxation methods for the shallow‐water equations in one and two space dimensions

In this paper, a new family of high‐order relaxation methods is constructed. These methods combine general higher‐order reconstruction for spatial discretization and higher order implicit‐explicit schemes or TVD Runge–Kutta schemes for time integration of relaxing systems. The new methods retain all the attractive features of classical relaxation schemes such as neither Riemann solvers nor characteristic decomposition are needed. Numerical experiments with the shallow‐water equations in both one and two space dimensions on flat and non‐flat topography demonstrate the high resolution and the ability of our relaxation schemes to better resolve the solution in the presence of shocks and dry areas without using either Riemann solvers or front tracking techniques. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulation of high‐Reynolds number flow around circular cylinders by a three‐step FEM–BEM model

An innovative computational model, developed to simulate high‐Reynolds number flow past circular cylinders in two‐dimensional incompressible viscous flows in external flow fields is described in this paper. The model, based on transient Navier–Stokes equations, can solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the projection method. The pressure is assumed to be zero at infinite boundary and the external flow field is simulated using a direct boundary element method (BEM) by solving a pressure Poisson equation. A three‐step finite element method (FEM) is used to solve the momentum equations of the flow. The present model is applied to simulate high‐Reynolds number flow past a single circular cylinder and flow past two cylinders in which one acts as a control cylinder. The simulation results are compared with experimental data and other numerical models and are found to be feasible and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

Computations of strongly swirling flows with second‐moment closures

The present study is concerned with simulating turbulent, strongly swirling flows by eddy viscosity model and Reynolds stress transport model variants adopting linear and quadratic form of the pressure–strain models. Flows with different inlet swirl numbers, 2.25 and 0.85, were investigated. Detailed comparisons of the predicted results and measurements were presented to assess the merits of model variants. For the swirl number 2.25 case, due to the inherent capability of the Reynolds stress models to capture the strong swirl and turbulence interaction, both the linear and quadratic form of the pressure–strain models predict the flow adequately. In strong contrast, the k–ϵ model predicts an excessively diffusive flow fields. For the swirl number 0.85 case, both the k–ϵ and Reynolds stress model with linear pressure–strain process, show an excessive diffusive transport of the flow fields. The quadratic pressure–strain model, on the other hand, mimics the correct flow development with the recirculating region being correctly predicted. Copyright © 1999 John Wiley & Sons, Ltd.