Fourier Chebyshev Collocation Research Articles

Free vibration analysis of joined conical-cylindrical shells by matched Fourier-Chebyshev collocation method

A free vibration analysis of joined conical-cylindrical shells based on the first order shear deformation theory is developed. Deflections and rotations are represented by the expansions of Chebyshev polynomials and Fourier series. Equations of motion are collocated to yield the system of algebraic equations. Boundary conditions and compatibility conditions are considered as side constraints, and the set of algebraic equations is condensed so that the number of degrees of freedom matches the number of expansion coefficients. Numerical examples are provided for a joined conical-cylindrical shell, a complete conical shell attached to a cylindrical shell and a hermetic can.

A spectral approach to a constrained optimization problem for the Helmholtz equation in unbounded domains

We study some convergence issues for a recent approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains (Ciraolo et al. in J Comput Phys 246:78–95, 2013) where the index of refraction is not required to be constant at infinity. The approach is based on the minimization of an integral functional, which arises from an integral formulation of the radiation condition at infinity. In this paper, we implement a Fourier–Chebyshev collocation method to study some convergence properties of the numerical algorithm; in particular, we give numerical evidence of some convergence estimates available in the literature (Ciraolo in Helmholtz equation in unbounded domains: some convergence results for a constrained optimization problem, 2013) and study numerically the minimization problem at low and mid-high frequencies. Numerical examples in some relevant cases are also shown.

A Fourier–Chebyshev collocation method for the mass transport in a layer of power-law fluid mud

A Fourier–Chebyshev collocation spectral method is employed in this work to compute the Lagrangian drift or mass transport due to periodic surface pressure loading in a thin layer of non-Newtonian fluid mud, which is modeled as a power-law fluid. Because of the non-Newtonian rheology, these problems are nonlinear and must be solved numerically. On assuming that the solutions are of the same permanent waveform as the pressure loading, the governing equations are made time-independent by referring to a horizontal axis that moves at the same speed as the wave. The solutions are periodic in the horizontal direction, but are non-periodic in the vertical direction, and the computational domain is therefore discretized according to the Fourier–Chebyshev spectral collocation scheme. In this study, the spatial derivatives are computed with a differentiation matrix. In order to incorporate the boundary conditions, the matrix diagonalization technique is used to solve the matrix equation, and all the definite integrals in the vertical direction based on the collocation points are performed by the modified Clenshaw–Curtis quadrature rule. The developed method is applied to compute the first- and second-order motion of the mud. The comparison between the numerical results and the analytical solution in the Newtonian limit shows the good accuracy of the spectral method.

PARALLEL IMPLEMENTATION OF A FOURIER-CHEBYSHEV COLLOCATION METHOD FOR INCOMPRESSIBLE FLUID FLOW AND HEAT TRANSFER

A Fourier-Chebyshev collocation spectral method is parallelized to simulate the three-dimensional unsteady flow and heat transfer inside a cylindrical enclosure. Two solution approaches using different techniques for determining the pressure field and enforcing mass conservation are presented for shared memory applications using Cray directives and for distributed memory applications using MPI and SHMEM message passing libraries. Matrix diagonalization is employed for solving the pressure Poisson equation and Helmholtz equations for the velocity components and temperature. The parallelization approach is described and scaling results are presented for both platform types.

Direct numerical simulations of turbulent forced convection between counter-rotating disks

Abstract A Fourier–Chebyshev collocation spectral method is used to simulate the three dimensional unsteady flow inside a cylindrical annular enclosure comprised of two counter-rotating disks and sidewalls. Each disk is taken to be isothermal, with an imposed temperature difference between them, and the sidewalls are adiabatic. Buoyancy forces are neglected. A shallow annulus with aspect ratio, h / r o =0.06, and radius ratio, r i / r o =0.13, is considered (where 2 h is the distance between the disks and r i and r o are the inner and outer disk radii, respectively). One disk is rotating at the maximum speed and the other disk is rotating in the opposite direction. Results are obtained over a range of Re including the transition from laminar flow, for cases with both disks rotating at the same speed, Γ =−1.0, with one disk rotating more slowly than the other, Γ =−0.4, and with one stationary disk, Γ =0 (where Γ is the disk angular velocity ratio). The heat transfer rates between the disks are also obtained, showing the effects of transition to turbulence. The three-dimensional nature of the flows is characterized, and mean and RMS turbulence quantities are presented.

A Dynamical Pseudo-Spectral Domain Decomposition Technique: Application to Viscous Compressible Flows

A dynamical spectral domain decomposition method is presented. In each subdomain a transformation of coordinate is used. Both the locations of the interfaces and the parameters of the mappings are dynamically adapted by minimizing theHω2-norm of the calculated solution. We show on some functions that the total norm of the Chebyshev series depends on the location of the interfaces. Moreover, there exists a minimum that defines the best location of the interface. This defines a dynamical generation of Chebyshev collocation points. This numerical method is applied on partial differential equations and it is shown that both the overall accuracy and the matching at the interfaces are improved with respect to a fixed interface calculation. This algorithm is then used for the numerical solution of the time-dependent full Navier–Stokes equations. The solution technique consists in a Fourier–Chebyshev collocation method combined with a matching method. The computational domain is decomposed into subdomains in the vertical direction. In each subdomain a coordinate transform is used and the locations of the interfaces are dynamically determined. The elliptic problems coming from the viscous terms are solved by means of the Chebyshev acceleration method. Density is matched with an upwind procedure whereas the velocities, the temperature, and the concentration are handled with the influence matrix method. Numerical examples are carried out on the compressible Kelvin–Helmholtz and Rayleigh–Taylor flows.

A Fourier-Chebyshev collocation method for the shallow water equations including shoreline runup

A solution method for the shallow water equations governing wave motions in the nearshore environment is presented. Spatial derivatives contained in these equations are computed using spectral collocation methods. A high-order time integration scheme is used to compute the time evolution of the velocities and water surface elevation given initial conditions. The model domain extends from the shoreline to a desired distance offshore and is periodic in the longshore direction. Properly posed boundary conditions for the governing equations are discussed. A curvilinear moving boundary condition is incorporated at the shoreline to account for wave runup. An absorbing-generating boundary is constructed offshore. The boundary treatments are tested using analytical and numerical results. The developed method is applied to the prediction of neutral stability boundaries and equilibrium amplitudes of subharmonic edge waves. Numerical results are compared to weakly nonlinear theory and are found to reproduce the theory very well.

Nonlinear evolution of a second-mode wave in supersonic boundary layers

The nonlinear time evolution of a second-mode instability in a Mach 4.5 wall-bounded flow is computed by solving the full compressible, time-dependent Navier—Stokes equations. High accuracy is achieved by using a Fourier—Chebyshev collocation algorithm. Primarily inviscid in nature, second modes are characterized by high frequency and high growth rates compared to first modes. Time evolution of growth rate as a function of distance from the plate suggests this problem is amenable to the Stuart—Watson perturbation theory as generalized by Herbert.

Numerical experiments in supersonic boundary-layer stability

The three-dimensional (3-D) time-dependent compressible Navier–Stokes equations are numerically solved by a Fourier–Chebyshev collocation method to study the stability of supersonic flows over a flat plate. Several direct simulations carried out in this study suggest the existence of a secondary instability that might provide a route to transition. The interaction of the modes involved in the secondary instability is possibly amenable to a Floquet-type analysis. Pertinent differences between this instability and the analogous incompressible K-type secondary instability are pointed out. Some preliminary results of a 2-D direct simulation of the nonlinear evolution of a second mode perturbation wave are also discussed.