Discretizing two‐dimensional complex fractured fields for incompressible two‐phase flow

A method is introduced to discretize irregular and complex two‐dimensional fractured media. The geometry of the fractured media is first analysed by searching and treating the complex configurations. Based on that, the method generated a good mesh quality and allows for including finer grids. An incompressible two‐phase flow problem is solved to compare the developed method and a public method based on the approximation of a 1D fracture by the edges of a 2D finite element grid of the porous media. The comparison showed that the developed method (i) represents better the fractured domain by maintaining the geometric integrity of input surfaces and geologic data, (ii) provides, for sample and complex fractured domains, excellent and more accurate results, and (iii) is much less sensitive to the grid sizes. Furthermore, the method has to be more efficient than the other methods for transport problems and has to provide better predictable results; this is mainly based on point (ii) and because the method produces optimal triangular grids. Copyright © 2009 John Wiley & Sons, Ltd.     

Assessment of a high‐order MUSCL method for rotor flows

This work presents the implementation of a high‐order, finite‐volume scheme suitable for rotor flows. The formulation is based on the variable extrapolation MUSCL‐scheme, where high‐order spatial accuracy (up to fourth‐order) is achieved using correction terms obtained through successive differentiation. A variety of results are presented, including 2‐ and 3‐dimensional test cases. Results with the proposed scheme, showed better wake and higher resolution of vortical structures compared with the standard MUSCL, even when coarse meshes were employed. The method was also demonstrated for 3‐dimensional unsteady flows using overset and moving grids for the UH‐60A rotor in forward flight and the Enhanced Rotorcraft Innovative Concept Achievement tiltrotor in aeroplane mode. For medium grids, the present method adds reasonable CPU and memory overheads and offers good accuracy on relatively coarse grids.     

Finite‐volume‐type VOF method on dynamically adaptive quadtree grids

This paper describes a finite‐volume volume‐of‐fluid (VOF) method for simulating viscous free surface flows on dynamically adaptive quadtree grids. The scheme is computationally efficient in that it provides relatively fine grid resolution at the gas–liquid interface and coarse grid density in regions where flow variable gradients are small. Special interpolations are used to ensure volume flux conservation where differently sized neighbour cells occur. The numerical model is validated for advection of dyed fluid in unidirectional and rotating flows, and for two‐dimensional viscous sloshing in a rectangular tank. Copyright © 2004 John Wiley & Sons, Ltd.     

A dissipation‐free numerical method to solve one‐dimensional hyperbolic flow equations

In this paper, a numerical method to capture the shock wave propagation in 1‐dimensional fluid flow problems with 0 numerical dissipation is presented. Instead of using a traditional discrete grid, the new numerical method is built on a range‐discrete grid, which is obtained by a direct subdivision of values around the shock area. The range discrete grid consists of 2 types: continuous points and shock points. Numerical solution is achieved by tracking characteristics and shocks for the movements of continuous and shock points, respectively. Shocks can be generated or eliminated when triggering entropy conditions in a marking step. The method is conservative and total variation diminishing. We apply this new method to several examples, including solving Burgers equation for aerodynamics, Buckley‐Leverett equation for fractional flow in porous media, and the classical traffic flow. The solutions were verified against analytical solutions under simple conditions. Comparisons with several other traditional methods showed that the new method achieves a higher accuracy in capturing the shock while using much less grid number. The new method can serve as a fast tool to assess the shock wave propagation in various flow problems with good accuracy.     

Flow simulation on moving boundary‐fitted grids and application to fluid–structure interaction problems

We present a method for the parallel numerical simulation of transient three‐dimensional fluid–structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non‐overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time‐dependent domains. To this end, we present a technique to solve the incompressible Navier–Stokes equation in three‐dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time‐dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations. Here the grid velocity is treated in such a way that the so‐called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well‐known MAC‐method to a staggered mesh in moving boundary‐fitted coordinates which uses grid‐dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second‐order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid–structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid–structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical computation of three‐dimensional incompressible Navier–Stokes equations in primitive variable form by DQ method

In this paper, the global method of differential quadrature (DQ) is applied to solve three‐dimensional Navier–Stokes equations in primitive variable form on a non‐staggered grid. Two numerical approaches were proposed in this work, which are based on the pressure correction process with DQ discretization. The essence in these approaches is the requirement that the continuity equation must be satisfied on the boundary. Meanwhile, suitable boundary condition for pressure correction equation was recommended. Through a test problem of three‐dimensional driven cavity flow, the performance of two approaches was comparatively studied in terms of the accuracy. The numerical results were obtained for Reynolds numbers of 100, 200, 400 and 1000. The present results were compared well with available data in the literature. In this work, the grid‐dependence study was done, and the benchmark solutions for the velocity profiles along the vertical and horizontal centrelines were given. Copyright © 2003 John Wiley & Sons, Ltd.     

A hybrid building‐block and gridless method for compressible flows

A hybrid building‐block Cartesian grid and gridless method is presented to compute unsteady compressible flows for complex geometries. In this method, a Cartesian mesh based on a building‐block grid is used as a baseline mesh to cover the computational domain, while the boundary surfaces are represented using a set of gridless points. This hybrid method combines the efficiency of a Cartesian grid method and the flexibility of a gridless method for the complex geometries. The developed method is used to compute a number of test cases to validate the accuracy and efficiency of the method. The numerical results obtained indicate that the use of this hybrid method leads to a significant improvement in performance over its unstructured grid counterpart for the time‐accurate solution of the compressible Euler equations. An overall speed‐up factor from six to more than one order of magnitude and a saving in storage requirements up to one order of magnitude for all test cases in comparison with the unstructured grid method are demonstrated. Copyright © 2008 John Wiley & Sons, Ltd.     

Two‐dimensional modeling for stability analysis of two‐phase stratified flow

The effect of wavelength and relative velocity on the disturbed interface of two‐phase stratified regime is modeled and discussed. To analyze the stability, a small perturbation is imposed on the interface. Growth or decline of the disturbed wave, relative velocity, and surface tension with respect to time will be discussed numerically. Newly developed scheme applied to a two‐dimensional flow field and the governing Navier–Stokes equations in laminar regime are solved. Finite volume method together with non‐staggered curvilinear grid is a very effective approach to capture interface shape with time. Because of the interface shape, for any time advancement, a new grid is performed separately on each stratified field, liquid, and gas regime. The results are compared with the analytical characteristics method and one‐dimensional modeling. This comparison shows that solving the momentum equation including viscosity term leads to physically more realistic results. In addition, the newly developed method is capable of predicting two‐phase stratified flow behavior more precisely than one‐dimensional modeling. It was perceived that the surface tension has an inevitable role in dissipation of interface instability and convergence of the two‐phase flow model. Copyright © 2009 John Wiley & Sons, Ltd.     

Higher‐order upwind leapfrog methods for multi‐dimensional acoustic equations

A non‐dissipative and very accurate one‐dimensional upwind leapfrog method was successfully extended to higher‐order and multi‐dimensional acoustic equations. The governing equations in characteristic form and staggered grid were utilized to preserve the accuracy. Fourier analysis was performed to find the accurate scheme for acoustics and the resultant two‐dimensional methods were successfully applied to several classical test cases. Copyright © 2004 John Wiley & Sons, Ltd.     

Three‐dimensional instability of axisymmetric flows: solution of benchmark problems by a low‐order finite volume method

Several problems on three‐dimensional instability of axisymmetric steady flows driven by convection or rotation or both are studied by a second‐order finite volume method combined with the Fourier decomposition in the periodic azimuthal direction. The study is focused on the convergence of the critical parameters with mesh refinement. The calculations are done on the uniform and stretched grids with variation of the stretching. Converged results are reported for all the problems considered and are compared with the previously published data. Some of the calculated critical parameters are reported for the first time. The convergence studies show that the three‐dimensional instability of axisymmetric flows can be computed with a good accuracy only on fine enough grids having about 100 nodes in the shortest spatial direction. It is argued that a combination of fine uniform grids with the Richardson extrapolation can be a good replacement for a grid stretching. It is shown once more that the sparseness of the Jacobian matrices produced by the finite volume method allows one to enhance performance of the Newton and Arnoldi iteration procedures by combining them with a direct sparse linear solver instead of using the Krylov‐subspace‐based iteration methods. Copyright © 2006 John Wiley & Sons, Ltd.     

A local domain‐free discretization method to simulate three‐dimensional compressible inviscid flows

In this paper, the domain‐free discretization method (DFD) is extended to simulate the three‐dimensional compressible inviscid flows governed by Euler equations. The discretization strategy of DFD is that the discrete form of governing equations at an interior point may involve some points outside the solution domain. The functional values at the exterior‐dependent points are updated at each time step by extrapolation along the wall normal direction in conjunction with the wall boundary conditions and the simplified momentum equation in the vicinity of the wall. Spatial discretization is achieved with the help of the finite element Galerkin approximation. The concept of ‘osculating plane’ is adopted, with which the local DFD can be easily implemented for the three‐dimensional case. Geometry‐adaptive tetrahedral mesh is employed for three‐dimensional calculations. Finally, we validate the DFD method for three‐dimensional compressible inviscid flow simulations by computing transonic flows over the ONERA M6 wing. Comparison with the reference experimental data and numerical results on boundary‐conforming grid was displayed and the results show that the present DFD results compare very well with the reference data. Copyright © 2009 John Wiley & Sons, Ltd.     

A Cartesian grid technique based on one‐dimensional integrated radial basis function networks for natural convection in concentric annuli

This paper reports a radial basis function (RBF)‐based Cartesian grid technique for the simulation of two‐dimensional buoyancy‐driven flow in concentric annuli. The continuity and momentum equations are represented in the equivalent stream function formulation that reduces the number of equations from three to one, but involves higher‐order derivatives. The present technique uses a Cartesian grid to discretize the problem domain. Along a grid line, one‐dimensional integrated RBF networks (1D‐IRBFNs) are employed to represent the field variables. The capability of 1D‐IRBFNs to handle unstructured points with accuracy is exploited to describe non‐rectangular boundaries in a Cartesian grid, while the method's ability to avoid the reduction of convergence rate caused by differentiation is instrumental in improving the quality of the approximation of higher‐order derivatives. The method is applied to simulate thermally driven flows in annuli between two circular cylinders and between an outer square cylinder and an inner circular cylinder. High Rayleigh number solutions are achieved and they are in good agreement with previously published numerical data. Copyright © 2007 John Wiley & Sons, Ltd.     

A collocated grid,projection method for time‐accurate calculation of low‐Mach number variable density flows in general curvilinear coordinates

A time‐accurate algorithm is proposed for low‐Mach number, variable density flows on curvilinear grids. Spatial discretization is performed on collocated grid that offers computational simplicity in curvilinear coordinates. The flux interpolation technique is used to avoid the pressure odd–even decoupling of the collocated grid arrangement. To increase the stability of the method, a two‐step predictor–corrector time integration scheme is employed. At each step, the projection method is used to calculate the hydrodynamic pressure and to satisfy the continuity equation. The robustness and accuracy of the method is illustrated with a series of numerical experiments including thermally driven cavity, polar cavity, three‐dimensional cavity, and direct numerical simulation of non‐isothermal turbulent channel flow. Copyright © 2012 John Wiley & Sons, Ltd.     

Two‐dimensional compact finite difference immersed boundary method

We present a compact finite differences method for the calculation of two‐dimensional viscous flows in biological fluid dynamics applications. This is achieved by using body‐forces that allow for the imposition of boundary conditions in an immersed moving boundary that does not coincide with the computational grid. The unsteady, incompressible Navier–Stokes equations are solved in a Cartesian staggered grid with fourth‐order Runge–Kutta temporal discretization and fourth‐order compact schemes for spatial discretization, used to achieve highly accurate calculations. Special attention is given to the interpolation schemes on the boundary of the immersed body. The accuracy of the immersed boundary solver is verified through grid convergence studies. Validation of the method is done by comparison with reference experimental results. In order to demonstrate the application of the method, 2D small insect hovering flight is calculated and compared with available experimental and computational results. Copyright © 2009 John Wiley & Sons, Ltd.     

Non‐intrusive reduced‐order modeling for multiphase porous media flows using Smolyak sparse grids

In this article, we describe a non‐intrusive reduction method for porous media multiphase flows using Smolyak sparse grids. This is the first attempt at applying such an non‐intrusive reduced‐order modelling (NIROM) based on Smolyak sparse grids to porous media multiphase flows. The advantage of this NIROM for porous media multiphase flows resides in that its non‐intrusiveness, which means it does not require modifications to the source code of full model. Another novelty is that it uses Smolyak sparse grids to construct a set of hypersurfaces representing the reduced‐porous media multiphase problem. This NIROM is implemented under the framework of an unstructured mesh control volume finite element multiphase model. Numerical examples show that the NIROM accuracy relative to the high‐fidelity model is maintained, whilst the computational cost is reduced by several orders of magnitude. Copyright © 2016 John Wiley & Sons, Ltd.     

RCM‐TVD hybrid scheme for hyperbolic conservation laws

We describe a hybrid method for the solution of hyperbolic conservation laws. A third‐order total variation diminishing (TVD) finite difference scheme is conjugated with a random choice method (RCM) in a grid‐based adaptive way. An efficient multi‐resolution technique is used to detect the high gradient regions of the numerical solution in order to capture the shock with RCM while the smooth regions are computed with the more efficient TVD scheme. The hybrid scheme captures correctly the discontinuities of the solution and saves CPU time. Numerical experiments with one‐ and two‐dimensional problems are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimized sixth‐order monotonicity‐preserving scheme by nonlinear spectral analysis

In this paper, sixth‐order monotonicity‐preserving optimized scheme (OMP6) for the numerical solution of conservation laws is developed on the basis of the dispersion and dissipation optimization and monotonicity‐preserving technique. The nonlinear spectral analysis method is developed and is used for the purpose of minimizing the dispersion errors and controlling the dissipation errors. The new scheme (OMP6) is simple in expression and is easy for use in CFD codes. The suitability and accuracy of this new scheme have been tested through a set of one‐dimensional, two‐dimensional, and three‐dimensional tests, including the one‐dimensional Shu–Osher problem, the two‐dimensional double Mach reflection, and the Rayleigh–Taylor instability problem, and the three‐dimensional direct numerical simulation of decaying compressible isotropic turbulence. All numerical tests show that the new scheme has robust shock capturing capability and high resolution for the small‐scale waves due to fewer numerical dispersion and dissipation errors. Moreover, the new scheme has higher computational efficiency than the well‐used WENO schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

A three‐dimensional hydrodynamic model for shallow waters using unstructured Cartesian grids

This paper presents a three‐dimensional unstructured Cartesian grid model for simulating shallow water hydrodynamics in lakes, rivers, estuaries, and coastal waters. It is a flux‐based finite difference model that uses a cut‐cell approach to fit the bottom topography and shorelines and, at the same time, has the flexibility of discretizing complex geometries with Cartesian grids that can be arbitrarily downsized in the two horizontal directions simultaneously. Because of the use of Cartesian grids, the grid generation is very simple and does not suffer the grid generation headache often seen in many other unstructured models, as the unstructured Cartesian grid model does not have any requirements on the orthogonality of the grids. The newly developed unstructured Cartesian grid model was validated against analytical solutions for a three‐dimensional seiching case in a rectangular basin, before it was compared with another three‐dimensional model named LESS3D for circulations and salinity transport processes in an idealized embayment that is driven by tides and freshwater inflows. Model tests show that the numerical procedure used in the unstructured Cartesian grid model is robust. Similar to other unstructured models, a variable grid size has resulted in a smaller number of grids required for a reasonable model simulation, which in turn reduces the CPU time used in the model run. Copyright © 2010 John Wiley & Sons, Ltd.     

A high‐order alternating direction implicit method for the unsteady convection‐dominated diffusion problem

A high‐order alternating direction implicit (ADI) method for solving the unsteady convection‐dominated diffusion equation is developed. The fourth‐order Padé scheme is used for the discretization of the convection terms, while the second‐order Padé scheme is used for the diffusion terms. The Crank–Nicolson scheme and ADI factorization are applied for time integration. After ADI factorization, the two‐dimensional problem becomes a sequence of one‐dimensional problems. The solution procedure consists of multiple use of a one‐dimensional tridiagonal matrix algorithm that produces a computationally cost‐effective solver. Von Neumann stability analysis is performed to show that the method is unconditionally stable. An unsteady two‐dimensional problem concerning convection‐dominated propagation of a Gaussian pulse is studied to test its numerical accuracy and compare it to other high‐order ADI methods. The results show that the overall numerical accuracy can reach third or fourth order for the convection‐dominated diffusion equation depending on the magnitude of diffusivity, while the computational cost is much lower than other high‐order numerical methods. Copyright © 2011 John Wiley & Sons, Ltd.