Finite difference method

Abstract: Homogeneous flow model is used to study the flow and heat transfer of carbon nanotubes (CNTs) along a flat plate subjected to Navier slip and uniform heat flux boundary conditions. This is the first paper on the flow and heat transfer of CNTs along a flat plate. Two types of CNTs, namely, single- and multi-wall CNTs are used with water, kerosene or engine oil as base fluids. The empirical correlations are used for the thermophysical properties of CNTs in terms of the solid volume fraction of CNTs. For the effective thermal conductivity of CNTs, Xue (Phys B Condens Matter 368:302–307, 2005) model has been used and the results are compared with the existing theoretical models. The governing partial differential equations and boundary conditions are converted into a set of nonlinear ordinary differential equations using suitable similarity transformations. These equations are solved numerically using a very efficient finite difference method with shooting scheme. The effects of the governing parameters on the dimensionless velocity, temperature, skin friction, and Nusselt numbers are investigated and presented in graphical and tabular forms. The numerical results of skin friction and Nusselt numbers are compared with the available data for special cases and are found in good agreement.

Abstract: In this paper, we consider a variable-order anomalous subdiffusion equation. A numerical scheme with first order temporal accuracy and fourth order spatial accuracy for the equation is proposed. The convergence, stability, and solvability of the numerical scheme are discussed via the technique of Fourier analysis. Another improved numerical scheme with second order temporal accuracy and fourth order spatial accuracy is also proposed. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Abstract: The theoretical basis and the numerical implementation of free-vortex filament methods are reviewed for application to the prediction and analysis of helicopter rotor wakes. The governing equations for the problem are described, with a discussion of finite difference approximations to these equations and various numerical solution techniques. Both relaxation and time-marching wake solution techniques are reviewed. It is emphasized how the careful consideration of stability and convergence (grid-independent behavior) are important to ensure a physically correct wake solution. The implementation of viscous diffusion and filament straining effects are also discussed. The need for boundary condition corrections to compensate for the inevitable wake truncation are described. Algorithms to accelerate the wake solution using velocity field interpolation are shown to reduce computational costs without a loss of accuracy. Several challenging examples of the application of free-vortex filament methods to helicopter rotor problems are shown, including multirotor configurations, flight near the ground, maneuvering flight conditions, and descending flight through the vortex ring state