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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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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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Viscous shear flow near the bottom during the head-on collision of two solitary waves is investigated using a stream function-vorticity model. Fully nonlinear free-surface conditions are satisfied numerically at every time step. T...
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Viscous shear flow near the bottom during the head-on collision of two solitary waves is investigated using a stream function-vorticity model. Fully nonlinear free-surface conditions are satisfied numerically at every time step. The bottom boundary-layer flow region is locally refined to explore its physical characteristics through detailed vorticity transport in the shear layer beneath the two solitary waves during their collision. Both symmetric and asymmetric head-on collisions are examined and discussed. The wave deformations are compared with the results of other studies for validation, and the fluid mechanisms in the flow field are illustrated with streamline, equivorticity line, path line, and timeline plots to reveal the interaction processes. It is revealed that the wave interaction behaves nonlinearly not only at the free surface but also in the vorticity exchanges near the bottom.
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Correction methods for the steady semi-periodic motion of incompressible fluid are investigated. The idea is similar to the influence matrix to solve the lack of vor- ticity boundary conditions. For any given boundary condition of...
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Correction methods for the steady semi-periodic motion of incompressible fluid are investigated. The idea is similar to the influence matrix to solve the lack of vor- ticity boundary conditions. For any given boundary condition of the vorticity, the coupled vorticity-stream function formulation is solved. Then solve the governing equation with the correction boundary conditions to improve the solution.
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Natural convection in a rectangular cavity with a saturated porous layer on one of its vertical walls is studied numerically The governing equations using the Boussinesq approximation for the treatment of buoyancy term in the mome...
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Natural convection in a rectangular cavity with a saturated porous layer on one of its vertical walls is studied numerically The governing equations using the Boussinesq approximation for the treatment of buoyancy term in the momentum equation and the Darcy model are expressed using the vorticity-stream function approach These equations are discretized with the implicit finite-difference method. Thomas algorithm and Gauss-Seidel method are used to solve the resultant algebraic system equations. Results are presented in terms of streamlines, isotherms and isoconcentrations. The increase of the porous wall permeability leads to a more intensive buoyancy-driven flow through the porous wall and consequently an increase of heat and mass transfer occurs in the enclosure. Nusselt and Sherwood numbers are expressed as functions of dimensionless parameters as Darcy number, modified Grashof number and the geometric aspect ratio of the enclosure.
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In this paper we present a new method to solve the 2D generalized Stokes problem in terms of the stream function and the vorticity. Such problem results, for instance, from the discretization of the evolutionary Stokes system. The...
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In this paper we present a new method to solve the 2D generalized Stokes problem in terms of the stream function and the vorticity. Such problem results, for instance, from the discretization of the evolutionary Stokes system. The difficulty arising from the lack of the boundary conditions for the vorticity is overcome by means of a suitable technique for uncoupling both variables. In order to apply the above technique to the Navier-Stokes equations we linearize the advective term in the vorticity transport equation as described in the development of the paper. We illustrate the good performance of our approach by means of numerical results, obtained for benchmark driven cavity problem solved with classical piecewise linear finite element. (C) 2007 Elsevier B.V. All rights reserved.
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This investigation deals with some exact solutions of the equations governing the steady plane motions of an incompressible third grade fluid by using complex variables and complex functions. Some of the solutions admit, as partic...
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This investigation deals with some exact solutions of the equations governing the steady plane motions of an incompressible third grade fluid by using complex variables and complex functions. Some of the solutions admit, as particular cases, all the solutions of Moro et al. [1990, "Steady Flows of a Third Grade Fluid by Transformation Methods," ZAMM, 70(3), pp. 189-198].
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In this paper, we derive some exact solutions of the equations governing the steady plane motions of an incompressible second grade fluid. For this purpose, the vorticity and stream functions both are expressed in terms of complex...
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In this paper, we derive some exact solutions of the equations governing the steady plane motions of an incompressible second grade fluid. For this purpose, the vorticity and stream functions both are expressed in terms of complex variables and complex functions. The derived solutions represent the flows having streamlines as a family of ellipses, parabolas, concentric circles, and rectangular hyperbolas. Some physical features of the derived solutions are also illustrated by their contour plots.
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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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The main purpose of this work is to provide a Hilbertian functional framework for the analysis of the planar incompressible Navier-Stokes (NS) equations either in vorticity or in stream function formulation. The fluid is assumed t...
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The main purpose of this work is to provide a Hilbertian functional framework for the analysis of the planar incompressible Navier-Stokes (NS) equations either in vorticity or in stream function formulation. The fluid is assumed to occupy a bounded possibly multiply connected domain. The velocity field satisfies either homogeneous (no-slip boundary conditions) or prescribed Dirichlet boundary conditions. We prove that the analysis of the 2D Navier-Stokes equations can be carried out in terms of the so-called nonprimitive variables only (vorticity field and stream function) without resorting to the classical NS theory (stated in primitive variables, i.e. velocity and pressure fields). Both approaches (in primitive and nonprimitive variables) are shown to be equivalent for weak (Leray) and strong (Kato) solutions. Explicit, Bernoulli-like formulas are derived and allow recovering the pressure field from the vorticity fields or the stream function. In the last section, the functional framework described earlier leads to a simplified rephrasing of the vorticity dynamics, as introduced by Maekawa (Adv Differ Equ 18(1-2):101-146, 2013). At this level of regularity, the vorticity equation splits into a coupling between a parabolic and an elliptic equation corresponding respectively to the non-harmonic and harmonic parts of the vorticity equation. By exploiting this structure it is possible to prove new existence and uniqueness results, as well as the exponential decay of the palinstrophy (that is, loosely speaking, the H-1 norm of the vorticity) for large time.
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