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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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A stability condition is provided for a class of vorticity boundary formulas used withthe second order finite-difference numerical scheme for the vorticity-stream function formulation ofthe unsteady incompressible Navier-Stokes eq...
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A stability condition is provided for a class of vorticity boundary formulas used withthe second order finite-difference numerical scheme for the vorticity-stream function formulation ofthe unsteady incompressible Navier-Stokes equations. These local vorticity boundary formulas arederived using the no-slip boundary condition for the velocity. A new form of these long-stencilformulas is needed to classify the stability property, in which local terms are controlled by globalquantities via discrete elliptic regularity for the stream functions.
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Computation of viscous fluid flow is an area of research where many authors have tried to present different numerical methods for solution of the Navier-Stokes equations. Each of these methods has its own advantages and weaknesses...
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Computation of viscous fluid flow is an area of research where many authors have tried to present different numerical methods for solution of the Navier-Stokes equations. Each of these methods has its own advantages and weaknesses. In the meantime, many researchers have attempted to develop coupled numerical algorithms in order to save storage for computational purposes and to save computational time. In this paper, a new coupled method is presented for the first time by combining FDM and DRBEM for solving the stream function-vorticity formulation of the Navier-Stokes equations. The vorticity transport equation is analyzed using a finite difference technique while the stream function Poisson's equation is solved using a dual reciprocity boundary element method. Finally, the robustness and accuracy of the coupled FDM-DRBEM model is proved using the benchmark problem of the flow in a driven square cavity.
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We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity form. The classical finite element method of degree one usually used does not allow the vorticity on the boundary of the domain to...
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We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity form. The classical finite element method of degree one usually used does not allow the vorticity on the boundary of the domain to be computed satisfactorily when the meshes are unstructured and does not converge optimally. To better approach the vorticity along the boundary, we propose that harmonic functions obtained by integral representation be used. Numerical results are very satisfactory, and we prove that this new numerical scheme leads to an optimal convergence rate of order 1 for the natural norm of the vorticity and, under higher regularity assumptions, from 3/2 to 2 for the quadratic norm of the vorticity.
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Two-dimensional oscillatory Stokes flow in the region with a semi-infinite plate parallel to an infinite plane wall is examined numerically by using the vorticity and the stream function formulation. Two kinds of flows are conside...
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Two-dimensional oscillatory Stokes flow in the region with a semi-infinite plate parallel to an infinite plane wall is examined numerically by using the vorticity and the stream function formulation. Two kinds of flows are considered: one due to the oscillatory uniform pressure gradient far inside the channel formed by the planes and the other the oscillatory slide of the infinite plane. A vorticity extrapolation scheme at the wall resolving the Stokes layer structure is proposed. The stress distributions on the plate and the wall and the instantaneous stream lines are presented. In Appendices A and B, the asymptotic behaviors for both flows at a distant from the mouth are discussed.
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We work on a vorticity, velocity and pressure formulation of the bidimensional Stokes problem for incompressible fluids. In previous papers, the authors have developed a natural implementation of this scheme. We have then observed...
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We work on a vorticity, velocity and pressure formulation of the bidimensional Stokes problem for incompressible fluids. In previous papers, the authors have developed a natural implementation of this scheme. We have then observed that, in case of unstructured meshes with Dirichlet boundary conditions on the velocity, the convergence is not optimal. In this paper, we propose to add "bubble" velocity functions with compact support along the boundary to improve convergence. We then prove a convergence theorem and illustrate by numerical results better behaviour of the scheme in general cases.
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We consider the finite element solution of the stream function-vorticity formulation for a large-scale ocean circulation model. First, we study existence and uniqueness of solution for the continuous and discrete problems. Under a...
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We consider the finite element solution of the stream function-vorticity formulation for a large-scale ocean circulation model. First, we study existence and uniqueness of solution for the continuous and discrete problems. Under appropriate regularity assumptions we prove that the stream function can be computed with an error of order h in H~1-seminorm. Second, we introduce and analyze an h-adaptive mesh refinement strategy to reduce the spurious oscillations and poor resolution which arise when convective terms are dominant. We propose an a posteriori anisotropic error indicator based on the recovery of the Hessian from the finite element solution, which allows us to obtain well adapted meshes. The numerical experiments show an optimal order of convergence of the adaptive scheme. Furthermore, this strategy is efficient to eliminate the oscillations around the boundary layer.
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We consider the bidimensional Stokes problem for incompressible fluids and recall the vorticity, velocity and pressure variational formulation, which was previously proposed by one of the authors, and allows very general boundary ...
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We consider the bidimensional Stokes problem for incompressible fluids and recall the vorticity, velocity and pressure variational formulation, which was previously proposed by one of the authors, and allows very general boundary conditions. We develop a natural implementation of this numerical method and we describe in this paper the numerical results we obtain. Moreover, we prove that the low degree numerical scheme we use is stable for Dirichlet boundary conditions on the vorticity. Numerical results are in accordance with the theoretical ones. In the general case of unstructured meshes, a stability problem is present for Dirichlet boundary conditions on the velocity, exactly as in the stream function-vorticity formulation. Finally, we show on some examples that we observe numerical convergence for regular meshes or embedded ones for Dirichlet boundary conditions on the velocity.
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A two-dimensional stochastic solver for the incompressible Navier-Stokes equations is developed. The vorticity-stream function formulation is considered. The polynomial chaos expansion was integrated with an unstructured node-cent...
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A two-dimensional stochastic solver for the incompressible Navier-Stokes equations is developed. The vorticity-stream function formulation is considered. The polynomial chaos expansion was integrated with an unstructured node-centered finite-volume solver. A second-order upwind scheme is used in the convection term for numerical stability and higher-order discretization. The resulting sparse linear system is solved efficiently by a direct parallel solver.The mean and variance simulations of the cavity flow are done for random variation of the viscosity and the lid velocity. The solver was tested and compared with the Monte- Carlo simulations and with previous research works. The developed solver is proved to be efficient in simulating the stochastic two-dimensional incompressible flows.
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We deal in this work with the Stokes equations set in a three-dimensional axisymmetric bounded domain. The boundary conditions that we consider are given on the normal component of the velocity and the tangential components of the...
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We deal in this work with the Stokes equations set in a three-dimensional axisymmetric bounded domain. The boundary conditions that we consider are given on the normal component of the velocity and the tangential components of the vorticity. Under assumptions on the data of the problem, the three-dimensional problem is reduced to a two-dimensional one. We write a stream function-vorticity formulation for this problem with two scalar unknowns. For the discretization, we use a domain decomposition method: the spectral element method which is well-adapted here. We prove the well-posedness of the obtained formulations and we derive error estimates between the exact solution and the discrete one.
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