A Time-Dependent Nodal-Integral Method for the Investigation of Bifurcation and Nonlinear Phenomena in Fluid Flow and Natural Convection

Wilson, Gary L.; Rydin, R.A.; Azmy, Yousry Y.

doi:10.13182/nse88-a23574

Cited by 25 publications

(7 citation statements)

References 2 publications

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“…Coarse-mesh nodal methods, including nodal integral methods (NIMs), have been developed over the last two decades to efficiently solve sets of linear and nonlinear PDEs [5][6][7][8][9][10][11][12]4]. Among the distinguishing features of some of the coarse mesh methods is the transverse integration process that-after the problem domain has been divided into nodes-reduces each of the PDE to an ODE by integrating over the node dimensions of all-but-one independent variables [6][7][8]10].…”

Section: Background and Motivationmentioning

confidence: 99%

An improved coarse-mesh nodal integral method for partial differential equations

Rizwan-uddin

1997

Numer. Methods Partial Differential Eq.

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An improved variation of the nodal integral method to solve partial differential equations has been developed and implemented. Rather than treating all of the nonlinear terms as the so-called pseudo-source terms (to be approximated), in this modified version of the nodal integral method, by approximating part of the nonlinear terms in terms of the discrete variable(s) that ultimately result at the end of the formulation process, some or all of the nonlinear terms are kept on the left-hand side in the transverse-integrated equations, which are to be solved analytically. Application of the method to solve the Burgers equation leads to exponential variation within the nodes and shows that the resulting scheme has inherent upwinding. Reconstruction of node interior solution -as a function of one independent variable, and averaged in all others -makes it possible to obtain rather accurate solutions even on a fine scale. Results of the numerical analysis and comparison with results of other methods reported in the literature show that the new method is comparable and sometimes better in accuracy than the currently used schemes. Extension to multidimensional problems is straightforward.

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Section: Background and Motivationmentioning

confidence: 99%

An improved coarse-mesh nodal integral method for partial differential equations

Rizwan-uddin

1997

Numer. Methods Partial Differential Eq.

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“…Because the derivation of the TDNIM equations has been described elsewhere [7], only a brief sketch of the derivation is included here. The static nodal integral method (SNIM) equations are obtained from the TDNIM equations by forcing the time-dependent faceaveraged values to take on the node-integral-average value, and then eliminating the nodeaverage value reducing the system of 11 dynamic equations (per node) to a system of eight static equations (per node) [9].…”

Section: Summary Solution Methodsmentioning

confidence: 99%

Multiple Equilibrium Solutions to the Bénard Problem at the Third Critical Rayleigh Number

Wilson

Rydin

1989

Numerical Heat Transfer, Part B: Fundamentals

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The exceptional accuracy and stability properties of static and dynamic nodal integral techniques have allowed them to be used to investigate the fine structure of the Benard natural convection problem as a function of aspect ratio for the untilled heated cavity. It was demonstrated that multiple unstable equilibrium solutions, corresponding to different values of net angular momentum, exist for the third critical Rayleigh number over the range of aspect ratios between approximately 1.1 and 2.5. These solutions were traced out by a path-dependent extrapolation process.

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“…NIM schemes have been developed and applied to solve fluid flow and heat transfer problems. In earlier implementations of this method, schemes were developed by clubbing the nonlinear convection terms of Navier-Stokes equations into inhomogeneous terms of ODEs generated from N-S equations [13][14][15][16]. This leads to an inefficient generation of shape functions for velocity components that captures only the diffusion process and not convection.…”

Section: Introductionmentioning

confidence: 99%

“…This leads to an inefficient generation of shape functions for velocity components that captures only the diffusion process and not convection. Moreover, use of the continuity equation to develop cell analytical solution leads to asymmetry in local solutions or shape functions of primitive variables (velocity components) in a cell [13][14][15][16]. The shortfalls of these schemes were addressed by the modified nodal integral method (MNIM) proposed by Rizwan-uddin [17].…”

Section: Introductionmentioning

confidence: 99%

Pressure Correction–Based Iterative Scheme for Navier-Stokes Equations using Nodal Integral Method

Kumar

Singh

Doshi

2012

Numerical Heat Transfer, Part B: Fundamentals

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A Time-Dependent Nodal-Integral Method for the Investigation of Bifurcation and Nonlinear Phenomena in Fluid Flow and Natural Convection

Cited by 25 publications

References 2 publications

An improved coarse-mesh nodal integral method for partial differential equations

An improved coarse-mesh nodal integral method for partial differential equations

Multiple Equilibrium Solutions to the Bénard Problem at the Third Critical Rayleigh Number

Pressure Correction–Based Iterative Scheme for Navier-Stokes Equations using Nodal Integral Method

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