By L. Quartapelle (auth.)
ISBN-10: 3034885792
ISBN-13: 9783034885799
ISBN-10: 3034896891
ISBN-13: 9783034896894
This ebook provides varied formulations of the equations governing incompressible viscous flows, within the shape wanted for constructing numerical resolution methods. The stipulations required to meet the no-slip boundary stipulations within the numerous formulations are mentioned intimately. instead of focussing on a specific spatial discretization approach, the textual content presents a unitary view of numerous equipment at the moment in use for the numerical resolution of incompressible Navier-Stokes equations, utilizing both finite alterations, finite components or spectral approximations. for every formula, a whole assertion of the mathematical challenge is supplied, comprising some of the boundary, in all probability imperative, and preliminary stipulations, compatible for any theoretical and/or computational improvement of the governing equations. The textual content is acceptable for classes in fluid mechanics and computational fluid dynamics. It covers that a part of the subject material facing the equations for incompressible viscous flows and their selection through numerical equipment. a considerable element of the e-book includes new effects and unpublished material.
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Additional resources for Numerical Solution of the Incompressible Navier-Stokes Equations
Sample text
Here, ~ = In(rla), (r, e) are the polar coordinates and a is the radius of the cylinder. 13) i = 1,2, where A is the Galerkin projection of the nonlinear term -J((, 'l/J) on the k-th function of the basis whereas ak and bk are the expansion coefficients of the boundary data a and b; it has been assumed that 6 ~ ~ ~ 6. 14) where e±k~ = 7]k(~) are two linearly independent solutions of the harmonic modal equation (d2lde - k2)7]k(~) = O. 13) for the expansion coefficients and to resort to Green identity for the ordinary differential operator d2 I de, which reads where 'l/J(~) and ¢(~) are arbitrary functions.
10. Numerical schemes: local discretizations 41 methods which belong to this class, we describe the method first introduced in finite difference methods to circumvent the difficulty caused by the absence of boundary conditions for the vorticity (Woods 1954). 1 Boundary vorticity formula methods Such a method consists in defining the boundary values of vorticity in terms of the stream function by means of some approximate formula-hence the name of vorticity boundary formula method. The various formula are derived by expanding the stream function in a Taylor series along the inward normal to the boundary, n = -n, 'ljJl = 'ljJls 2 8 2'IjJ I 0ri2 S I h + h 8'IjJ Ori s + 2 3 3 h 8 'IjJ I + "6 0ri3 S + ...
Therefore it is (-'\1~N)t = _'\1 2, where '\1 2 denotes the Laplace operator with no boundary condition. Now, N( - '\1 2 ) is the linear space of functions 'f/ that satisfy the equation - '\1 2 'f/ = 0 (without boundary conditions). Thus ( must be orthogonal to the space of the harmonic functions, namely, ( -1 {'f/ I - '\1 2 'f/ = O}. Let us now assume that ( is to be found as the solution of the elliptic equation (-'\1 2 + 'Y)( = j, with 'Y ~ 0 fixed, and let us introduce the linear space Z-y defined as follows: Z-y = {z I (- '\1 2 + 'Y) z = g, for all g}.
Numerical Solution of the Incompressible Navier-Stokes Equations by L. Quartapelle (auth.)
by Michael
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