Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.

A finite difference technique for simulating unsteady viscoelastic free surface flows

Abstract: This work is concerned with the development of a numerical method capable of simulating viscoelastic free surface flow of an Oldroyd-B fluid. The basic equations governing the flow of an Oldroyd-B fluid are considered. A novel formulation is developed for the computation of the non-Newtonian extra-stress components on rigid boundaries. The full free surface stress conditions are employed. The resulting governing equations are solved by a finite difference method on a staggered grid, influenced by the ideas of the marker-and-cell (MAC) method. Numerical results demonstrating the capabilities of this new technique are presented for a number of problems involving unsteady free surface flows.

Globally optimal finite-difference schemes based on least squares

Abstract: Spatial finite-difference (FD) coefficients are usually determined by the Taylor-series expansion (TE) or optimization methods. The former can provide high accuracy on a smaller wavenumber or frequency zone, and the latter can give moderate accuracy on a larger zone. Present optimization methods applied to calculate FD coefficients are generally gradient-like or global optimization-like algorithms, and thus iterations are involved. They are more computationally expensive, and sometimes the global solution may not be found. I examined second-order spatial derivatives and computed the optimized spatial FD coefficients over the given wavenumber range using the least-squares (LS) method. The results indicated that the FD accuracy increased with increasing operator length and decreasing wavenumber range. Therefore, for the given error and operator length, globally optimal spatial FD coefficients can be easily obtained. Some optimal FD coefficients were given. I developed schemes to obtain optimized LS-...

Velocity-scalar filtered density function for large eddy simulation of turbulent flows

Abstract: A methodology termed the “velocity-scalar filtered density function” (VSFDF) is developed and implemented for large eddy simulation (LES) of turbulent flows. In this methodology, the effects of the unresolved subgrid scales (SGS) are taken into account by considering the joint probability density function (PDF) of the velocity and scalar fields. An exact transport equation is derived for the VSFDF in which the effects of the SGS convection and chemical reaction are closed. The unclosed terms in this equation are modeled in a fashion similar to that typically used in Reynolds-averaged simulation procedures. A system of stochastic differential equations (SDEs) which yields statistically equivalent results to the modeled VSFDF transport equation is constructed. These SDEs are solved numerically by a Lagrangian Monte Carlo procedure in which the Ito–Gikhman character of the SDEs is preserved. The consistency of the proposed SDEs and the convergence of the Monte Carlo solution are assessed by comparison with results obtained by a finite difference LES procedure in which the corresponding transport equations for the first two SGS moments are solved. The VSFDF results are compared with those obtained by the Smagorinsky model, and all the results are assessed via comparison with data obtained by direct numerical simulation of a temporally developing mixing layer involving transport of a passive scalar. It is shown that the values of both the SGS and the resolved components of all second order moments including the scalar fluxes are predicted well by VSFDF. The sensitivity of the calculations to the model’s (empirical) constants are assessed and it is shown that the magnitudes of these constants are in the same range as those employed in PDF methods.

Dynamic fission instabilities in rapidly rotating n=3/2 polytropes: a comparison of results from finite-difference and smoothed particle hydrodynamics codes

Abstract: The effectiveness of three different hydrodynamics models is evaluated for the analysis of the effects of fission instabilities in rapidly rotating, equilibrium flows. The instabilities arise in nonaxisymmetric Kelvin modes as rotational energy in the flow increases, which may occur in the formation of close binary stars and planets when the fluid proto-object contracts quasi-isostatically. Two finite-difference, donor-cell methods and a smoothed particle hydrodynamics (SPH) code are examined, using a polytropic index of 3/2 and ratios of total rotational kinetic energy to gravitational energy of 0.33 and 0.38. The models show that dynamic bar instabilities with the 3/2 polytropic index do not yield detached binaries and multiple systems. Ejected mass and angular momentum form two trailing spiral arms that become a disk or ring around the central remnant. The SPH code yields the same data as the finite difference codes but with less computational effort and without acceptable fluid constraints in low density regions. Methods for improving both types of codes are discussed. 68 references.