Numerical solution of the Navier-Stokes equations
01 Jan 1968-Mathematics of Computation (American Mathematical Society (AMS))-Vol. 22, Iss: 104, pp 745-762
TL;DR: In this paper, a finite-difference method for solving the time-dependent Navier-Stokes equations for an incompressible fluid is introduced, which is equally applicable to problems in two and three space dimensions.
Abstract: A finite-difference method for solving the time-dependent Navier- Stokes equations for an incompressible fluid is introduced. This method uses the primitive variables, i.e. the velocities and the pressure, and is equally applicable to problems in two and three space dimensions. Test problems are solved, and an ap- plication to a three-dimensional convection problem is presented.
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TL;DR: In this article, a general, numerical, marching procedure is presented for the calculation of the transport processes in three-dimensional flows characterised by the presence of one coordinate in which physical influences are exerted in only one direction.
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TL;DR: In this article, a new finite element formulation for convection dominated flows is developed, based on the streamline upwind concept, which provides an accurate multidimensional generalization of optimal one-dimensional upwind schemes.
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TL;DR: This paper is concerned with the mathematical structure of the immersed boundary (IB) method, which is intended for the computer simulation of fluid–structure interaction, especially in biological fluid dynamics.
Abstract: This paper is concerned with the mathematical structure of the immersed boundary (IB) method, which is intended for the computer simulation of fluid–structure interaction, especially in biological fluid dynamics. The IB formulation of such problems, derived here from the principle of least action, involves both Eulerian and Lagrangian variables, linked by the Dirac delta function. Spatial discretization of the IB equations is based on a fixed Cartesian mesh for the Eulerian variables, and a moving curvilinear mesh for the Lagrangian variables. The two types of variables are linked by interaction equations that involve a smoothed approximation to the Dirac delta function. Eulerian/Lagrangian identities govern the transfer of data from one mesh to the other. Temporal discretization is by a second-order Runge–Kutta method. Current and future research directions are pointed out, and applications of the IB method are briefly discussed. Introduction The immersed boundary (IB) method was introduced to study flow patterns around heart valves and has evolved into a generally useful method for problems of fluid–structure interaction. The IB method is both a mathematical formulation and a numerical scheme. The mathematical formulation employs a mixture of Eulerian and Lagrangian variables. These are related by interaction equations in which the Dirac delta function plays a prominent role. In the numerical scheme motivated by the IB formulation, the Eulerian variables are defined on a fixed Cartesian mesh, and the Lagrangian variables are defined on a curvilinear mesh that moves freely through the fixed Cartesian mesh without being constrained to adapt to it in any way at all.
4,164 citations
Cites methods from "Numerical solution of the Navier-St..."
...From Chorin, I learned fluid mechanics, and also his projection method (Chorin 1968, Chorin 1969) for incompressible flow, upon which foundation the immersed boundary method was built....
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...From Chorin, I learned fluid mechanics, and also his projection method ( Chorin 1968, Chorin 1969) for incompressible flow, upon which foundation the immersed boundary method was built....
[...]
TL;DR: In this article, a non-iterative method for handling the coupling of the implicitly discretised time-dependent fluid flow equations is described, based on the use of pressure and velocity as dependent variables and is hence applicable to both the compressible and incompressible versions of the transport equations.
4,019 citations
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01 Jan 196111,419 citations
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30 Nov 1961TL;DR: In this article, the authors propose Matrix Methods for Parabolic Partial Differential Equations (PPDE) and estimate of Acceleration Parameters, and derive the solution of Elliptic Difference Equations.
Abstract: Matrix Properties and Concepts.- Nonnegative Matrices.- Basic Iterative Methods and Comparison Theorems.- Successive Overrelaxation Iterative Methods.- Semi-Iterative Methods.- Derivation and Solution of Elliptic Difference Equations.- Alternating-Direction Implicit Iterative Methods.- Matrix Methods for Parabolic Partial Differential Equations.- Estimation of Acceleration Parameters.
5,317 citations
TL;DR: In this article, a numerical method for solving incompressible viscous flow problems is introduced, which uses the velocities and the pressure as variables and is equally applicable to problems in two and three space dimensions.
2,797 citations
TL;DR: In this article, the authors considered the Navier-Stokes equation for 3-dimensional flows and deduced the existence theorems for 3D flows through a Hilbert space approach, making use of the theory of fractional powers of operators.
Abstract: : The initial value problem for the nonstationary Navier-Stokes equation is considered. New results were obtained with the aid of various methods from modern functional analysis. The existence of unique and global (in time) solutions for 2-dimensional flows as well as the existence of unique and local (in time) solutions for 3-dimensional flows was established. In most of these works the word solution was interpreted in a generalized sense. The purpose is to deduce such existence theorems for 3-dimensional flows through a Hilbert space approach, making use of the theory of fractional powers of operators and the theory of semi-groups of operators. (Author)
1,255 citations