Chebyshev expansion methods for the solution of the extended graetz problem

Ku, Hwar-Ching; Hatziavramidis, Dimitri

doi:10.1016/0021-9991(84)90109-8

Cited by 47 publications

(24 citation statements)

References 12 publications

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“…An example of the azimuthal velocity profiles obtained by solving equations (1)(2)(3)(4)(5) is shown in Figure 1. We refer to this solution as Type 1 to distinguish it from another solution termed as Type 2 that will be discussed later.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…As shown by Ku and Hatziavramidis [4,5] these values uniquely define the coefficients in the linear combinations of Chebyshev polynomials used in the spectral approximation of the solution. Therefore, the function values at any point within a computational domain can be found since the explicit expressions for Chebyshev polynomials are readily available.…”

Section: C50mentioning

confidence: 96%

“…Thus the spectral accuracy of the method is ensured. At the same time as discussed by Ku and Hatziavramidis [4,5] the collocation formulation enables one to reduce the integration and differentiation procedures to a simple matrixvector multiplication. Namely, introducing the vector…”

Section: C50mentioning

confidence: 97%

“…The governing equations are solved using the multi-dimensional version of the Chebyshev pseudospectral collocation method initially proposed by Ku and Hatziavramidis [4,5] and implemented by Suslov and Paolucci [7,8]. In this method the solution is represented by a linear combination of Chebyshev polynomials of degree ranging from 0 to the pre-set values N r and N z that define the total number of Chebyshev modes used for approximating the solutions in the radial and depth directions, respectively.…”

Section: C50mentioning

confidence: 99%

“…, and the standard N r × N r matricesĜ N r ×N r 1 and W N r ×N r are defined by Ku and Hatziavramidis [4,5]. Differentiation and integration in z is performed using the similar N z ×N z matrices.…”

Section: C50mentioning

confidence: 99%

See 4 more Smart Citations

Numerical modelling of axisymmetric electromagnetically driven flows in thin layers

Suslov

Cuevas

2017

ANZIAMJ

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We present the results of the numerical modelling of deceptively simple steady axisymmetric electromagnetically driven flows in thin disk-like layers of a weakly conducting electrolyte. The fluid motion is caused by an azimuthally acting Lorentz force appearing when a radial current flows in the electrolyte layer placed on top of a magnet with a vertical polarisation. The small layer thickness and the circumferential direction of the driving force suggest that the flow in such a system should be essentially uni-directional. However, it was found that not only is the flow fully three-dimensional, but multiple solutions can exist for the same set of governing parameters.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: C50mentioning

confidence: 96%

Section: C50mentioning

confidence: 97%

Section: C50mentioning

confidence: 99%

Section: C50mentioning

confidence: 99%

See 3 more Smart Citations

Numerical modelling of axisymmetric electromagnetically driven flows in thin layers

Suslov

Cuevas

2017

ANZIAMJ

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A spectral method to solve the equations of linear elasticity for the transient response of a tube subjected to impact

Kjellmert

1999

Int. J. Numer. Meth. Engng.

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SUMMARYThe transient response of a tube subjected to impact is described through Fourier-Galerkin and Chebyshev collocation multidomain discretizations of the equations of linear elasticity. The trapezoidal rule is used for time integration. For each Fourier mode the spatial collocation derivative operators are represented by matrices, and the subdomains are patched by natural and essential conditions. At each time level the resulting governing matrix equation is reduced by two consecutive block Gaussian eliminations, so that an equation for the complex Fourier coe cients at the subdomain corners has to be solved. Back-substitution gives the coe cients at all other collocation points. An inverse discrete Fourier transform generates, at optional time levels, the three components of the displacement ÿeld. Through this method the long-term evolution of the ÿeld may be calculated, provided the impact time is long enough. The time history as represented by computed contour plots has been compared with photos produced by holographic interferometry. The agreements are satisfactory.

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Multidomain collocation techniques for a viscous compressible flow

Kjellmert

1991

Numerical Methods in Fluids

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SUMMARYThis paper presents a viscous compressible flow problem to which an equilibrium solution, in terms of density and velocity, can be given implicitly by elementary functions. The corresponding initial boundary value problem is solved by time discretization by the Crank-Nicolson method, Newton linearization and space discretization using multidomain Chebyshev collocation techniques. The physical interval is covered by subintervals of equal length. Each subinterval utilizes the same number of collocation points and each interface consists of one or two points. Six ways of patching are tested. All of them yield solutions with spectral accuracy for a few time steps, but only three are stable in the long run. Details of the density evolution are illustrated.

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Chebyshev expansion methods for the solution of the extended graetz problem

Cited by 47 publications

References 12 publications

Numerical modelling of axisymmetric electromagnetically driven flows in thin layers

Numerical modelling of axisymmetric electromagnetically driven flows in thin layers

A spectral method to solve the equations of linear elasticity for the transient response of a tube subjected to impact

Multidomain collocation techniques for a viscous compressible flow

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