Helical flow arising from the yielded annular flow of a Bingham fluid

The Bingham fluid model was developed to represent viscoplastic materials that change from rigid bodies at low stress to viscous fluids at high stress – a process termed yielding. Such a fluid model is used in the modeling of slurries, which occur frequently in food processing and other engineering applications.     

Alpha effect in helically forced turbulence

The effect on large-scale dynamics of small-scale helicity injection in three-dimensional resistive magnetohydrodynamic turbulence is investigated for a weak initial magnetic field. The magnetic configuration of α² dynamo prototype flows is such that, generally, the energy concentrates on one large-scale mode. However, we obtain that alpha effect is not limited to the dominant mode and that a non-local equation (in Fourier space) is more appropriate to describe it. It gives some insights into the non-local theory of Pouquet et al. [J. Fluid Mech. 77 (1976) 321] where the inverse cascade results from a competition between the helicity and Alfvén effects.     

Modeling fully developed laminar flow in a helical duct with rectangular cross section and finite pitch

The mass and momentum transport equations are written in an orthogonal coordinate system using Germano’s transformation to model a laminar flow in a helical duct with a rectangular cross section and finite pitch. The system of governing equations are discretized and solved by the finite-volume numerical method. The three dimensional domain is reduced to a two dimensional slab of cells, orthogonal to the main flow direction, enforcing the fully developed state for 2π/(τ · d_h) >> 1 where τ and d_h representing the duct’s centerline torsion and its hydraulic diameter. This approximation and the use of an orthogonal grid allow a great simplification on the numerical procedure. Comparisons of the numerical solution against experimental data are drawn to assess the accuracy of the approximation.     

Using coordinate transformation of Navier-Stokes equations to solve flow in multiple helical geometries

Recent research on small amplitude helical pipes for use as bypass grafts and arterio-venous shunts, suggests that mixing may help prevent occlusion by thrombosis. It is proposed here that joining together two helical geometries, of different helical radii, will enhance mixing, with only a small increase in pressure loss. To determine the velocity field, a coordinate transformation of the Navier-Stokes equations is used, which is then solved using a 2-D high-order mesh combined with a Fourier decomposition in the periodic direction. The results show that the velocity fields in each component geometry differ strongly from the corresponding solution for a single helical geometry. The results suggest that, although the mixing behaviour will be weaker than an idealised prediction indicates, it will be improved from that generated in a single helical geometry.     

Hermitian operator methods for reaction-diffusion equations

A variety of time-linearization, quasilinearization, operator-splitting, and implicit techniques which use compact or Hermitian operators has been developed for and applied to one-dimensional reaction-diffusion equations. Compact operators are compared with second-order accurate spatial approximations in order to assess the accuracy and efficiency of Hermitian techniques. It is shown that time-linearization, quasilinearization, and implicit techniques which use compact operators are less accurate than second-order accurate spatial discretizations if first-order approximations are employed to evaluate the time derivatives. This is attributed to first-order accurati temporal truncation errors. Compact operator techniques which use second-order temporal approximations are found to be more accurate and efficient than second-order accurate, in both space and time, algorithms. Quasilinearization methods are found to be more accurate than time-linearization schemes. However, quasilinearization techniques are less efficient because they require the inversion of block tridiagonal matrices at each iteration. Some improvements in accuracy can be obtained by using partial quasilinearization and linearizing each equation with respect to the variable whose equation is being solved. Operator-splitting methods which use compact differences to evaluate the diffusion operator were found to be less accurate than operator-splitting procedures employ second-order accurate spatial approximations. Comparisons among the methods presented in this paper are shown in terms of the L²-norm errors and computed wave speeds for a variety of time steps and grid spacings: The numerical efficiency is assessed in terms of the CPU time required to achieve the same accuracy.     

Additive blending of local approximations into a globally-valid approximation with application to the dilogarithm

We show how local approximations, each accurate on a subinterval, can be blended together to form a global approximation which is accurate over the entire interval. The blending functions are smoothed approximations to a step function, constructed using the error function. The local approximations may be power series, asymptotic expansion, or other more exotic species. As an example, for the dilogarithm function, we construct a one-line analytic approximation which is accurate to one part in 700. This can be generalized to higher order merely by adding more terms in the local approximations. We also show the failure of the alternative strategy of subtracting singularities.     

A staggered-grid numerical algorithm for the extended Boussinesq equations

A second-order accurate numerical scheme is developed to solve Nwogu’s extended Boussinesq equations. A staggered-grid system is introduced with the first-order spatial derivatives being discretized by the fourth-order accurate finite-difference scheme. For the time derivatives, the fourth-order accurate Adams predictor–corrector method is used. The numerical method is validated against available analytical solutions, other numerical results of Navier–Stokes equations, and experimental data for both 1D and 2D nonlinear wave transformation problems. It is shown that the new algorithm has very good conservative characteristics for mass calculation. As a result, the model can provide accurate and stable results for long-term simulation. The model has proven to be a useful modeling tool for a wide range of water wave problems.     

A high order accurate numerical method for solving two‐dimensional dual‐phase‐lagging equation with temperature jump boundary condition in nanoheat conduction

Dual‐phase‐lagging (DPL) equation with temperature jump boundary condition (Robin's boundary condition) shows promising for analyzing nanoheat conduction. For solving it, development of higher‐order accurate and unconditionally stable (no restriction on the mesh ratio) numerical schemes is important. Because the grid size may be very small at nanoscale, using a higher‐order accurate scheme will allow us to choose a relative coarse grid and obtain a reasonable solution. For this purpose, recently we have presented a higher‐order accurate and unconditionally stable compact finite difference scheme for solving one‐dimensional DPL equation with temperature jump boundary condition. In this article, we extend our study to a two‐dimensional case and develop a fourth‐order accurate compact finite difference method in space coupled with the Crank–Nicolson method in time, where the Robin's boundary condition is approximated using a third‐order accurate compact method. The overall scheme is proved to be unconditionally stable and convergent with the convergence rate of fourth‐order in space and second‐order in time. Numerical errors and convergence rates of the solution are tested by two examples. Numerical results coincide with the theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1742–1768, 2015     

Asymptotic analysis and accurate approximate solutions for strongly nonlinear conservative symmetric oscillators

A new accurate iterative and asymptotic method is introduced to construct analytical approximate solutions to strongly nonlinear conservative symmetric oscillators. The method is based on applying a second-order expansion with the harmonic balance method and it excludes the requirement of solving a set of coupled nonlinear algebraic equations. Newton's iteration or the linearized model may be readily deduced by considering only the first-order terms in the model. According to this iterative approach, only Fourier series expansions of restoring force function, its first- and second-order derivatives for each iteration are required. It is concluded here that by only one single iteration, very brief and yet accurate analytical approximate solutions can be attained. Three physical examples are solved and accurate solutions are presented to illustrate the physics of the system and the effectiveness of the proposed asymptotic method.     

High-order accurate monotone compact running scheme for multidimensional hyperbolic equations

A high-order accurate monotone compact difference scheme proposed earlier by the author for one-dimensional nonstationary hyperbolic equations is extended to multidimensional equations. The resulting scheme is fourth-order accurate in space on a compact stencil and third-order accurate in time. Additionally, the scheme is conservative, absolutely stable, and efficient and can be solved using the running calculation method in space. By computing initial-boundary value problems for the linear advection equation and the nonlinear Hopf equation on refined meshes, it is shown that the orders of grid convergence of the multidimensional scheme are close to the corresponding orders of accuracy in independent variables. For the propagation of a two-dimensional rectangular pulse and the Hopf equation with a discontinuous solution, the multidimensional scheme is shown to inherit the monotonicity of its one-dimensional counterpart.