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A multi-dimensional, compressible fluid flow solver, valid in both channel and porous media, is derived by volume-averaging the Navier-Stokes equations. By selecting an appropriate average density/velocity pair, a continuous, stab...
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A multi-dimensional, compressible fluid flow solver, valid in both channel and porous media, is derived by volume-averaging the Navier-Stokes equations. By selecting an appropriate average density/velocity pair, a continuous, stable solution is obtained for both pressure and velocity. The proposed model is validated by studying the pressure drop of two commonly used experimental setups to measure in-plane and through-plane permeability of fuel cell porous media. Numerical results show that the developed model is able to reproduce the experimentally measured pressure drop at varying flow rates. Further, it highlights that previously used methods of extracting permeability, which rely on the use of simplified one-dimensional models, are not appropriate when high flow rates are used to study the porous media. At high flow rates, channel-porous media interactions cannot be neglected and can result in incorrect permeability estimations. For example, at flow rates of 1 SLPM a discrepancy of 12% in pressure drop was observed when using previous permeability values instead of the values obtained in the article using the proposed 3D model. Given that at high flow rate one-dimensional models might not be appropriate, previous estimations of Forchheimer permeability might not be accurate. To illustrate the suitability of the numerical model to fuel cell applications, fluid flow bypass in serpentine and interdigitated fuel cell flow channels is also investigated.
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This paper introduces a high-order time stepping technique for solving the incompressible Navier-Stokes equations which, unlike coupled techniques, does not require solving a saddle point problem at each time step and, unlike proj...
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This paper introduces a high-order time stepping technique for solving the incompressible Navier-Stokes equations which, unlike coupled techniques, does not require solving a saddle point problem at each time step and, unlike projection methods, does not produce splitting errors and spurious boundary layers. The technique is a generalization of the artificial compressibility method; it is unconditionally stable (for the unsteady Stokes equations), can reach any order in time, and uncouples the velocity and the pressure. The condition number of the linear systems associated with the fully discrete vector-valued problems to be solved at each time step scales like O(tau h(-2)), where tau is the time step and h is the spatial grid size. No Poisson problem or other second-order elliptic problem has to be solved for the pressure corrections. Unlike projection methods, optimal convergence is observed numerically with Dirichlet and mixed Dirichlet/Neumann boundary conditions.
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In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. In contrast with such problems involving one smooth ...
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In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces. In contrast with such problems involving one smooth non-intersecting interface, that have been extensively studied, there are very few papers addressing elliptic interface problems with intersecting interfaces of coefficient jumps. It is well known that if the values of the permeability in the four subregions around a point of intersection of two such internal interfaces are all different, the solution has a point singularity that significantly affects the accuracy of the approximation in the vicinity of the intersection point. In the present paper we propose a fourth-order 9-point finite difference scheme on uniform Cartesian meshes for an elliptic problem whose coefficient is piecewise constant in four rectangular subdomains of the overall two-dimensional rectangular domain. Moreover, for the special case when the intersecting point of the two lines of coefficient jumps is a grid point, such a compact scheme, involving relatively simple formulas for computation of the stencil coefficients, can even reach sixth order of accuracy. Furthermore, we show that the resulting linear system for the special case has an M-matrix, and prove the theoretical sixth order convergence rate using the discrete maximum principle. Our numerical experiments demonstrate the sixth (for the special case) and at least fourth (for the general case) accuracy orders of the proposed schemes. In the general case, we derive a compact third-order finite difference scheme, also yielding a linear system with an M-matrix. In addition, using the discrete maximum principle, we prove the third order convergence rate of the scheme for the general elliptic cross-interface problem.(c) 2023 Elsevier B.V. All rights reserved.
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Let Gamma be a smooth curve inside a two-dimensional rectangular region Omega. In this paper, we consider the Poisson interface problem -del(2)u = f in Omega\Gamma with Dirichlet boundary condition such that f is smooth in Omega\G...
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Let Gamma be a smooth curve inside a two-dimensional rectangular region Omega. In this paper, we consider the Poisson interface problem -del(2)u = f in Omega\Gamma with Dirichlet boundary condition such that f is smooth in Omega\Gamma and the jump functions [u] and [del u.(n) over right arrow] across Gamma are smooth along Gamma. This Poisson interface problem includes the weak solution of -del(2)u = f + g delta(Gamma) in Omega as a special case. Because the source term f is possibly discontinuous across the interface curve Gamma and contains a delta function singularity along the curve Gamma, both the solution u of the Poisson interface problem and its flux del u.(n) over right arrow are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils and therefore, our proposed scheme can be easily implemented and is of interest to practitioners dealing with Poisson interface problems. Note that the curve Gamma splits Omega into two disjoint subregions Omega(+) and Omega(-)-. The coefficient matrix A in the resulting linear system Ax = b, following from the proposed scheme, is independent of any source term f, jump condition g delta(Gamma), interface curve Gamma and Dirichlet boundary conditions, while only b depends on these factors and is explicitly given, according to the configuration of the nine stencil points in Omega(+) or Omega(-). The constant coefficient matrix.. facilitates the parallel implementation of the algorithm in case of a large size matrix and only requires the update of the right hand side vector b for different Poisson interface problems. Due to the flexibility and explicitness of the proposed scheme, it can be generalized to obtain the highest order compact finite difference scheme for non-uniform grids as well. We prove the order 6 convergence for the proposed scheme using the discrete maximum principle. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources.
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In this paper we study the flux formulation of unsteady diffusion equations with highly heterogeneous permeability coefficients and their discretization. In the proposed approach first an equation governing the flux of the unknown...
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In this paper we study the flux formulation of unsteady diffusion equations with highly heterogeneous permeability coefficients and their discretization. In the proposed approach first an equation governing the flux of the unknown scalar quantity is solved, and then the scalar is recovered from its flux. The problem for the flux is further discretized by splitting schemes that yield locally one-dimensional problems, and therefore, the resulting linear systems are tridiagonal if the spatial discretization uses Cartesian grids. A first and a formally second order time discretization splitting scheme have been implemented in both two and three dimensions, and we present results for a few model problems using a challenging benchmark dataset. (C) 2017 Elsevier B.V. All rights reserved.
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In this article, we present a direction splitting method, combined with a nonlinear iteration, for the compressible Navier-Stokes equations in spherical coordinates. The method is aimed at solving the equations on the sphere, and ...
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In this article, we present a direction splitting method, combined with a nonlinear iteration, for the compressible Navier-Stokes equations in spherical coordinates. The method is aimed at solving the equations on the sphere, and can be used for a regional geophysical simulations as well as simulations on the entire sphere. The aim of this work was to develop a method that would work efficiently in the limit of very small to vanishing Mach numbers, and we demonstrate here, using a numerical example, that the method shows good convergence and stability at Mach numbers in the range [10(-2), 10(-6)]. We also demonstrate the effect of some of the parameters of the model on the solution, on a common geophysical test case of a rising thermal bubble. The algorithm is particularly suitable for a massive parallel implementation, and we show below some results demonstrating its excellent weak scalability.
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In this paper we develop a high-order time stepping technique for the incompressible Navier-Stokes equations. The method is based on an artificial compressibility perturbation made high order by using a Taylor series technique. Th...
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In this paper we develop a high-order time stepping technique for the incompressible Navier-Stokes equations. The method is based on an artificial compressibility perturbation made high order by using a Taylor series technique. The method is suitable for time step control. It is unconditionally stable in the case of the unsteady Stokes equations and conditionally stable for the full Navier-Stokes equations. The numerical results presented in the paper suggest that the stability condition in the second case is of CFL type; i.e., the time step should be of the order of the ratio of the meshsize and the magnitude of the velocity. In principle, the technique can be developed to any order in time. We illustrate the idea by giving the third-order version of the methodology. We numerically illustrate the third-order convergence rate of the method on a manufactured solution. The scheme converges with time steps randomly chosen at each time level as the size of the average time step decreases. We also demonstrate the efficiency of a simple time step control on a realistic incompressible flow in 2D.
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This article introduces a second-order direction splitting method for solving the incompressible Navier-Stokes-Boussinesq system in a spherical shell region. The equations are solved on overset Yin-Yang grids, combined with spheri...
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This article introduces a second-order direction splitting method for solving the incompressible Navier-Stokes-Boussinesq system in a spherical shell region. The equations are solved on overset Yin-Yang grids, combined with spherical coordinate transforms. This approach allows to avoid the singularities at the poles and keeps the grid size relatively uniform. The downside is that the spherical shell is subdivided into two equally sized, overlapping subdomains that requires the use of Schwarz-type iterations. The temporal second-order accuracy is achieved via an artificial compressibility scheme with bootstrapping. The spatial discretization is based on second-order finite differences on the Marker-And-Cell stencil. The entire scheme is implemented in parallel using a domain decomposition iteration and a direction splitting approach for the local solves. The stability, accuracy, and weak scalability of the method is verified on a manufactured solution of the Navier-Stokes-Boussinesq system while its practicality is demonstrated on the natural convection problem in the gap between two concentric spheres.
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The stream function-vorticity formulation of the (Navier-)Stokes equations yields a coupled system of a parabolic equation for the vorticity and an elliptic equation for the stream function. The essential coupling between them occ...
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The stream function-vorticity formulation of the (Navier-)Stokes equations yields a coupled system of a parabolic equation for the vorticity and an elliptic equation for the stream function. The essential coupling between them occurs through the boundary conditions which in case of a Dirichlet boundary involve only the stream function. Therefore, the boundary condition for the vorticity must be derived from them and thus the vorticity equation must be coupled to the stream function equation via its boundary condition. In this paper we propose an unconditionally stable splitting scheme for the unsteady Stokes equations in a stream function vorticity formulation, that decouples the vorticity and stream function computations at each time step. The spatial discretization is based on a finite volume discretization on (generally) unstructured Delaunay grids and corresponding Voronoi finite volume cells. A generalization of the well-known Thom vorticity boundary condition is derived for such grids and the corresponding discrete problem is decoupled by a two-step splitting scheme which results in a decoupled discrete parabolic problem for the vorticity and an elliptic problem for the stream function. Furthermore, the scheme is extended to the unsteady Navier-Stokes equations. Finally, the stability and accuracy of the resulting schemes are demonstrated on numerical examples. (C) 2015 Elsevier B.V. All rights reserved.
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In this paper we discuss two methods for upscaling of highly heterogeneous data for parabolic problems in the context of a direction splitting time approximation. The first method is a direct application of the idea of Jenny et al...
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In this paper we discuss two methods for upscaling of highly heterogeneous data for parabolic problems in the context of a direction splitting time approximation. The first method is a direct application of the idea of Jenny et al. (2003) in the context of the direction splitting approach. The second method devises the approximation from the Schur complement corresponding to the interface unknowns of the coarse grid, by applying a proper L-2 projection operator to it. The spatial discretization employed in this paper is based on a MAC finite volume stencil but the same approach can be implemented within a proper finite element discretization. A key feature of the present approach is that it can extend to 3D problems with very little computational overhead. The properties of the resulting approximations are demonstrated numerically on some benchmark coefficient data available in the literature. (C) 2016 Elsevier Ltd. All rights reserved.
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