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Fluid Flow
Published in O.P. Gupta, Finite and Boundary Element Methods in Engineering, 2022
The general flow behaviour of fluids is highly non-linear and complex, involving turbulence, viscosity, volume change (compressibility) and so on. Analysis dealing with different cases of fluid flow has been carried out under a variety of simplifying assumptions applicable to specific cases. The fluid is taken to be inviscid and incompressible flowing under non-turbulent irrotational and steady conditions. This simplifies analysis but its applicability is thereby restricted. Complexity increases when these assumptions are dropped. Many attempts have been made to relax idealization through use of special factors and non-dimensional terms and fluid mechanics literature is replete with this heritage. Numerical techniques, especially the finite difference method, have been applied to complex cases but the situation quickly goes out of control with an increase in grid points required for more accurate results. Lately, FEM has been used with success and some hybrid methods have also been tried which make use of this method and special factors previously developed to arrive at approximate analytical solutions.
Simulation and Modeling of Chloride Transport in Cement-Based Materials
Published in Shi Caijun, Yuan Qiang, He Fuqiang, Hu Xiang, Transport and Interactions of Chlorides in Cement-Based Materials, 2019
Shi Caijun, Yuan Qiang, He Fuqiang, Hu Xiang
In 1996, Tang proposed this well-thought-out physical model in which many physical and chemical phenomena are taken into account. Finite difference method is used to solve the partial differential equations. However, in the model, chloride ion is treated as a neutral particle, and has no interaction with other species. It is well known that a chloride ion is a negative particle, and the pore solution of concrete is a concentrated solution with ions like K+, Na+, Ca2+, OH−, SO42−, etc. When the chloride transports in such a concentrated solution, the interactions between different species should be taken into account. It is worth mentioning that the ClinConc needs numerical iterations by means of computer. This may restrain the engineering applications of the ClinConc. Based on the ClinConc, Tang (2008) proposed a more engineer-friendly expression which doesn’t need numerical iterations.
FINITE-DIFFERENCE FORMULATION
Published in Wenquan Sui, Time-Domain Computer Analysis of Nonlinear Hybrid Systems, 2018
The finite-difference method is a straightforward and effective numerical technique for solving ordinary and partial differential equations. A continuous system, described by differential equations, is discretized and transformed into a set of linear algebraic equations, and the solution to a continuous system is expressed at discrete values with a set accuracy level.
Residence time distribution studies in air table
Published in International Journal of Coal Preparation and Utilization, 2022
Satyabrata Patro, Ganesh Chalavadi, Ranjeet Kumar Singh
The equation developed for residence time distribution of tracer on air table is a second-order parabolic partial differential equation and it determines the variation of concentration of tracer in space as well as time on air table. Eq. (10) requires one initial condition and two boundary conditions for solving. Finite difference method is used to solve the partial differential Eq. 10. Finite difference method uses Taylor series expansions to derive finite difference representations of partial differential equations. To approximate Eq. (10) by finite difference method, the closed domain of air table is divided by a set of lines parallel to the spatial and time axes to form a grid or a mesh as shown in Figure 3.The sets of lines are equally spaced for simplicity such that the distance between crossing points along the length and time is ∆x and ∆t respectively. The finite difference form of Eq. (10) using Forward Time Central Space (FTCS) is now given with the spatial second derivative evaluated from a combination of the derivatives at time steps j and j +1.
Heatline Analysis on Heat Transfer and Convective Flow of Nanofluids in an Inclined Enclosure
Published in Heat Transfer Engineering, 2018
As we know, the standard finite difference method has only the accuracy of second-order, and therefore could not solve the underlying problems effectively, unless a large number of mesh points are used. In order to obtain satisfactory numerical results with reasonable computational cost, a reasonable approach is to develop a higher-order compact finite difference method, which not only provides accurate numerical results and saves computational work, but also easily treat boundary conditions as pointed out in [37].