In this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory.An original Petrov-Galerkin formulation of ...In this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory.An original Petrov-Galerkin formulation of the Falkner-Skan equation is presented which is based on a judiciously chosen special basis function to capture the asymptotic behaviour of the unknown.A spectral method of remarkable simplicity is obtained for computing Falkner-Skan-Cooke boundary layer flows.The accuracy and efficiency of the Laguerre spectral approximation is illustrated by determining the linear stability of nonseparated and separated flows according to the Orr-Sommerfeld equation.The pentadiagonal matrices representing the derivative operators are explicitly provided in an Appendix to aid an immediate implementation of the spectral solution algorithms.展开更多
In the spectral solution of 3-D Poisson equations in cylindrical and spherical coordinates including the axis or the center,it is convenient to employ radial basis functions that depend on the Fourier wavenumber or on...In the spectral solution of 3-D Poisson equations in cylindrical and spherical coordinates including the axis or the center,it is convenient to employ radial basis functions that depend on the Fourier wavenumber or on the latitudinal mode.This idea has been adopted by Matsushima and Marcus and by Verkley for planar problems and pursued by the present authors for spherical ones.For the Dirichlet boundary value problem in both geometries,original bases have been introduced built upon Jacobi polynomials which lead to a purely diagonal representation of the radial second-order differential operator of all spectral modes.This note details the origin of such a diagonalization which extends to cylindrical and spherical regions the properties of the Legendre basis introduced by Jie Shen for Cartesian domains.Closed form expressions are derived for the diagonal elements of the stiffness matrices as well as for the elements of the tridiagonal mass matrices occurring in evolutionary problems.Furthermore,the bound on the condition number of the spectral matrices associated with the Helmholtz equation are determined,proving in a rigorous way one of the main advantages of the proposed radial bases.展开更多
A primitive variable spectral method for simulating incompressible viscous flows inside a finite cylinder is presented.One element of originality of the proposed method is that the radial discretization of the Fourier...A primitive variable spectral method for simulating incompressible viscous flows inside a finite cylinder is presented.One element of originality of the proposed method is that the radial discretization of the Fourier coefficients depends on the Fourier mode,its dimension decreasing with the increase of the azimuthal modal number.This principle was introduced independently by Matsushima and Marcus and by Verkley in polar coordinates and is adopted here for the first time to formulate a 3D cylindrical Galerkin projection method.A second element of originality is the use of a special basis of Jacobi polynomials introduced recently for the radial dependence in the solution of Dirichlet problems.In this basis the radial operators are represented by matrices of minimal sparsity-diagonal stiffness and tridiagonal mass-provided here in closed form for the first time,and lead to a Helmholtz operator characterized by a favorable condition number.Finally,a new method is presented for eliminating the singular behaviour of the solution originated by the rotation of the lid with respect to the cylindrical wall.Thanks to these elements,the resulting Navier-Stokes spectral solver guarantees the differentiability to any order of the solution in the entire computational domain and does not suffer from the time-step stability restriction occurring in spectral methods with a point clustering close to the axis.Several test examples are offered that demonstrate the spectral accuracy of the solution method under different representative conditions.展开更多
New direct spectral solvers for the 3D Helmholtz equation in a finite cylindrical region are presented.A purely variational(no collocation)formulation of the problem is adopted,based on Fourier series expansion of the...New direct spectral solvers for the 3D Helmholtz equation in a finite cylindrical region are presented.A purely variational(no collocation)formulation of the problem is adopted,based on Fourier series expansion of the angular dependence and Legendre polynomials for the axial dependence.A new Jacobi basis is proposed for the radial direction overcoming the main disadvantages of previously developed bases for the Dirichlet problem.Nonhomogeneous Dirichlet boundary conditions are enforced by a discrete lifting and the vector problem is solved by means of a classical uncoupling technique.In the considered formulation,boundary conditions on the axis of the cylindrical domain are never mentioned,by construction.The solution algorithms for the scalar equations are based on double diagonalization along the radial and axial directions.The spectral accuracy of the proposed algorithms is verified by numerical tests.展开更多
文摘In this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory.An original Petrov-Galerkin formulation of the Falkner-Skan equation is presented which is based on a judiciously chosen special basis function to capture the asymptotic behaviour of the unknown.A spectral method of remarkable simplicity is obtained for computing Falkner-Skan-Cooke boundary layer flows.The accuracy and efficiency of the Laguerre spectral approximation is illustrated by determining the linear stability of nonseparated and separated flows according to the Orr-Sommerfeld equation.The pentadiagonal matrices representing the derivative operators are explicitly provided in an Appendix to aid an immediate implementation of the spectral solution algorithms.
文摘In the spectral solution of 3-D Poisson equations in cylindrical and spherical coordinates including the axis or the center,it is convenient to employ radial basis functions that depend on the Fourier wavenumber or on the latitudinal mode.This idea has been adopted by Matsushima and Marcus and by Verkley for planar problems and pursued by the present authors for spherical ones.For the Dirichlet boundary value problem in both geometries,original bases have been introduced built upon Jacobi polynomials which lead to a purely diagonal representation of the radial second-order differential operator of all spectral modes.This note details the origin of such a diagonalization which extends to cylindrical and spherical regions the properties of the Legendre basis introduced by Jie Shen for Cartesian domains.Closed form expressions are derived for the diagonal elements of the stiffness matrices as well as for the elements of the tridiagonal mass matrices occurring in evolutionary problems.Furthermore,the bound on the condition number of the spectral matrices associated with the Helmholtz equation are determined,proving in a rigorous way one of the main advantages of the proposed radial bases.
文摘A primitive variable spectral method for simulating incompressible viscous flows inside a finite cylinder is presented.One element of originality of the proposed method is that the radial discretization of the Fourier coefficients depends on the Fourier mode,its dimension decreasing with the increase of the azimuthal modal number.This principle was introduced independently by Matsushima and Marcus and by Verkley in polar coordinates and is adopted here for the first time to formulate a 3D cylindrical Galerkin projection method.A second element of originality is the use of a special basis of Jacobi polynomials introduced recently for the radial dependence in the solution of Dirichlet problems.In this basis the radial operators are represented by matrices of minimal sparsity-diagonal stiffness and tridiagonal mass-provided here in closed form for the first time,and lead to a Helmholtz operator characterized by a favorable condition number.Finally,a new method is presented for eliminating the singular behaviour of the solution originated by the rotation of the lid with respect to the cylindrical wall.Thanks to these elements,the resulting Navier-Stokes spectral solver guarantees the differentiability to any order of the solution in the entire computational domain and does not suffer from the time-step stability restriction occurring in spectral methods with a point clustering close to the axis.Several test examples are offered that demonstrate the spectral accuracy of the solution method under different representative conditions.
文摘New direct spectral solvers for the 3D Helmholtz equation in a finite cylindrical region are presented.A purely variational(no collocation)formulation of the problem is adopted,based on Fourier series expansion of the angular dependence and Legendre polynomials for the axial dependence.A new Jacobi basis is proposed for the radial direction overcoming the main disadvantages of previously developed bases for the Dirichlet problem.Nonhomogeneous Dirichlet boundary conditions are enforced by a discrete lifting and the vector problem is solved by means of a classical uncoupling technique.In the considered formulation,boundary conditions on the axis of the cylindrical domain are never mentioned,by construction.The solution algorithms for the scalar equations are based on double diagonalization along the radial and axial directions.The spectral accuracy of the proposed algorithms is verified by numerical tests.
