Helical Coordinates Research Articles

Stokes flow through a single‐screw extruder

AbstractStokes flow through a single‐screw extruder driven either by an axial pressure‐gradient, or by the rotation of the screw, or by a combination of both is analyzed. The geometry of the screw is chosen to resemble that encountered in the metering section of a real‐life industrial extruder. Working on the simplifying assumption of large helical pitch, a perturbation analysis is performed in non‐orthogonal helical coordinates, and contributions to the velocity and pressure fields are computed up to second‐order by finite element methods for unidirectional and two‐dimensional Stokes flow. Velocity fields are presented for different screw geometries, the axial flow rate is computed for pressure‐ and rotation‐driven flow, comparisons with simple models are made, and the effect of the pitch on the trajectories of passively convected particles is demonstrated. In the case of purely pressure‐driven flow, the flow rate decreases, whereas in the case of rotating flow the flow rate increases as the gap between the screw and the barrel is reduced. © 2006 American Institute of Chemical Engineers AIChE J, 2007

Stokes flow through a twisted tube

Pressure-driven flow through a tube with helical corrugations produced either by twisting a straight tube with arbitrary cross-section, or by embossing helical corrugations on a circular tube, or by inserting a helical fin inside a circular tube, is considered. The Stokes-flow problem is formulated in non-orthogonal curvilinear helical coordinates defined with respect to the helical pitch and azimuthal wavenumber, where the latter is determined by the tube cross-section rotational symmetry. In the first part of the paper, a perturbation analysis is carried out for a circular tube with small-amplitude sinusoidal corrugations, and the solutions of the first- and second-order perturbation problems are found by analytical methods. In the second part, an asymptotic analysis is performed for large-pitched helical corrugations and tubes with arbitrary cross-section, and the solutions of the zeroth-, first- and second-order problems are computed by finite-element methods for unidirectional and two-dimensional Stokes flow over the cross-sectional plane normal to the tube axis. The results illustrate the kinematic structure of the flow and demonstrate the dependence of the flow rate on the tube geometry.

Hybridization of the topological and dynamical phase in coiled optical fibres

We study the effect of hybridization of geometrical and dynamical phase for the fundamental modes of coiled weakly guiding fibres. A perturbation theory method of solving the vector wave equation written in the local nonorthogonal frame of helical coordinates is proposed. The phase corrections caused by the effect of the spin?orbit coupling on the geometric phase are found. It is demonstrated that in few-mode fibres such corrections can be comparable with post-adiabatic corrections to Berry's phase. The effect of hybridization of the topological phase and the dynamical phase in anisotropic coiled fibres is demonstrated. It is shown that in such fibres the topological phase can be considerably reduced by the dynamical phase.

Wplyw zmiany warunkow brzegowych na charakterystyke przeplywu slimakowego

Effect of change of boundary conditions on flow characteristic of viscous liquids in screw channel was analyzed theoretically. Two extreme variants of boundary conditions were assumed: rotating barrel and stationary screw (A) or rotating screw and stationary barrel (B). The analyses were performed for the flat, cylindrical or helical channel models using rectangular, cylindrical or helical coordinates, respectively (Fig. 1). In the case of Newtonian liquids it was shown that the solutions for any channel shape and corresponding to both variants of boundary conditions differ only in terms describing the solid body rotation. So both variants are equivalent i.e. the liquid displacement in relation to the screw surface, determining the output, is independent of the choice of boundary conditions. In case of the flow of non-Newtonian fluids in the flat channel (neglecting the real channel curvature), however, there are some differences in flow characteristics dependent on the boundary conditions' choice. For cylindrical or helical channels (taking into consideration the channel curvature) such differences do not appear.

Reversed field pinch current drive with oscillating helical fields

The use of oscillating helical magnetic fields to produce and sustain the toroidal and poloidal currents in a reversed field pinch (RFP) is investigated. A simple physical model that assumes fixed ions, massless electrons, and uniform density and resistivity is employed. Thermal effects are neglected in Ohm’s law and helical coordinates are introduced to reduce the number of coupled nonlinear equations that must be advanced in time. The results show that it is possible to produce RFP-like magnetic field profiles with pinch parameters close to the experimental values. The efficiencies obtained for moderate resistivity, and the observed scaling, indicate that this could be a very attractive method for high temperature plasmas.

The Cardiocoil Stent-Artery Interaction

An analytical approach for the mechanical interaction of the self-expanding Cardiocoil stent with the stenosed artery is presented. The damage factor as the contact stress at the stent-artery interface is determined. The stent is considered as an elastic helical rod having a nonlinear pressure-displacement dependence, while the artery is modeled by an elastic cylindrical shell. An influence of a moderate relative thickness of the shell is estimated. The equations for both the stent and the artery are presented in the stent-associated helical coordinates. The computational efficiency of the model enabled to carry out a parametric study of the damage factor. Comparative examinations are conducted for the stents made of the helical rods with circular and rectangular cross sections. It was found, in particular, that, under same other conditions, the damage factor for the stent with a circular cross section may be two times larger than that for a rectangular one.

Reynolds‐stress balance equations in orthogonal helical coordinates and application

AbstractRecently, direct numerical simulations of turbulent flow in helically coiled pipes have been performed and published. Here, computational results are shown for low Reynolds number flow (Reτ = 230) through one helical pipe configuration (curvature κ R = 0.1, torsion τ R = 0.165). The results are shortly discussed and further information of other toroidal or helical pipe flow simulations are given. Besides, the Reynolds‐stress balance equations derived in orthogonal helical coordinates are presented to allow further insight into helical pipe flow. The Reynolds‐stress balance equations are of great importance for turbulence modelling. Compared to their formulation in cylindrical coordinates, further terms occur, describing centrifugal and other inertia force effects which result from pipe curvature and torsion. Except for toroidal flow, where torsion is zero and a symmetry plane exists, no possibility to simplify the equations has been found.

3‐D Seismic Data Discrete Smooth Interpolation Using the Conjugate Gradient Method

AbstractAimed at the problem of huge computation in the interpolation of three dimensional seismic data, we propose an interpolation method that applies the Laplacian algorithm to do smooth restraint. Using the idea of discrete smooth interpolation (DSI) studied by Mallet for reference, the precondition conjugated gradient method is introduced to compute the values straightly on the grids and avoid to find the function to fit the interpolating equation. In order to compute the converse of the Laplacian operator rapidly, the theory of helical spectral factorization method is applied. Under the helical coordinates, the matrix of the Laplacian algorithm has the Topelitz structure of which the converse can be got using spectral factorization quickly. Based on the two dimensional DSI, a flow of discrete smooth interpointion is presented to do DSI along the time slice by use of the conjugated gradient method combining with NMO. The good results have been obtained through the processing of synthetic data and real three‐dimensional data by using this method.

Influence of curvature and torsion on turbulent flow in helically coiled pipes

Fully-developed, statistically steady turbulent flow in straight, curved and helically coiled pipes is studied by means of direct numerical simulation for a Reynolds number of Reτ=230. The incompressible Navier–Stokes equations, written in orthogonal helical coordinates, are integrated numerically by means of a second order accurate finite volume method. It is shown that pipe curvature which induces a secondary flow has a strong effect on the flow quantities. Turbulence is significantly inhibited by streamline curvature and the flow almost relaminarizes for high values of the curvature parameter (κ=0.1). The torsion effect is weaker than the curvature effect. Nevertheless, it cannot be neglected. It influences the secondary flow induced by pure curvature and leads to an increase in fluctuating kinetic energy and dissipation rate.

Conformational Deformability of RNA: A Harmonic Mode Analysis

The harmonic mode analysis method was used to characterize the conformational deformability of regular Watson-Crick paired, mismatch- and bulge-containing RNA. Good agreement between atomic Debye-Waller factors derived from x-ray crystallography of a regular RNA oligonucleotide and calculated atomic fluctuations was obtained. Calculated helical coordinate fluctuations showed a small sequence dependence of up to ∼30–50%. A negative correlation between motions at a given base pair step and neighboring steps was found for most helical coordinates. Only very few calculated modes contribute significantly to global motions such as bending, twisting, and stretching of the RNA molecules. With respect to a local helical description of the RNA helix our calculations suggest that RNA bending is mostly due to a periodic change in the base pair step descriptors slide and roll. The presence of single guanine:uridine or guanine:adenine mismatches had little influence on the calculated RNA flexibility. In contrast, for tandem guanine:adenine base pairs the harmonic mode approach predicts a significantly reduced conformational flexibility in the case of a sheared arrangement and slightly enhanced flexibility for a face-to-face (imino proton) pairing relative to regular RNA. The presence of a single extra adenine bulge nucleotide stacked between flanking sequences resulted in an increased local atomic mobility around the bulge site (∼40%) and a slightly enhanced global bending flexibility. For an adenine bulge nucleotide in a looped-out conformation a strongly enhanced bulge nucleotide mobility but no increased bending flexibility compared to regular RNA was found.

Z-curve: A Computer Program Calculating DNA Helical Axis Coordinates for Three-dimensional Graphic Presentation of Curvature

Z-curve: A Computer Program Calculating DNA Helical Axis Coordinates for Three-dimensional Graphic Presentation of Curvature

A Three-Dimensional Analysis of Rotordynamic Forces on Whirling and Cavitating Helical Inducers

This paper investigates the linearized dynamics of three-dimensional bubbly cavitating flows in helical inducers. The purpose is to understand the impact of the bubble response on the radial and tangential rotordynamic forces exerted by the fluid on the rotor and stator stages of whirling turbomachines under cavitating conditions. The flow in the inducer annulus is modeled as a homogeneous inviscid mixture, containing vapor bubbles with a small amount of noncondensable gas. The effects of several contributions to the damping of the bubble dynamics are included in the model. The governing equations of the inducer flow are written in “body-fitted” orthonormal helical Lagrangian coordinates, linearized for small-amplitude perturbations about the mean flow, and solved by modal decomposition. The whirl excitation generates finite-speed propagation and resonance phenomena in the two-phase flow within the inducer. These, in turn, lead to a complex dependence of the lateral rotordynamic fluid forces on the excitation frequency, the void fraction, the average size of the cavitation bubbles, and the turbopump operating conditions (including, rotational speed, geometry, flow coefficient and cavitation number). Under cavitating conditions the dynamic response of the bubbles induces major deviations from the noncavitating flow solutions, especially when the noncondensable gas content of the bubbles is small and thermal effects on the bubble dynamics are negligible. Then, the quadratic dependence of rotordynamic fluid forces on the whirl speed, typical of cavitation-free operation, is replaced by a more complex behavior characterized by the presence of different regimes where, depending on the whirl frequency, the fluid forces have either a stabilizing or a destabilizing effect on the inducer motion. Results are presented to illustrate the influence of the relevant flow parameters.

Development of a helical coordinate system and its applications to analysis of polymer flow in screw extruders. Part II: A helical channel model for single screw extruders

AbstractBecause of its simplicity and originality, the parallel plates model (PPM) is often used to analyze flow in a single screw extruder. This model is developed in a rectangular coordinate frame and approximates the geometry of the screw as a hollow rectangle. As such, the geometrical features of the screw channel (curvature and helicity) are lost. In this paper, we present a helical channel model (HCM). It is developed in a helical coordinate frame that takes the real screw geometry into full consideration. A comparison is made between the HCM and PPM to better appreciate the effects of these inherent geometrical parameters of a screw channel on flow in terms of velocity profile, flow rate and shape factors. In order to obtain analytical solutions, a one‐dimensional flow of an isothermal, incompressible and Newtonian fluid is considered. It is concluded that the PPM is a special case of the HCM. In other words, the HCM reduces to the PPM when the channel depth‐to‐screw diameter ratio approaches zero.

Representing structural information of helical charge distributions in cylindrical coordinates

Structural information in the local electric field produced by helical charge distributions, such as dissolved DNA, is revealed in a straightforward manner employing cylindrical coordinates. Comparison of structure factors derived in terms of cylindrical and helical coordinates is made. A simple coordinate transformation serves to relate the Green function in cylindrical and helical coordinates. We also compare the electric field on the central axis of a single helix as calculated in both systems.

A NUMERICAL STUDY ON MIXING IN THE KENICS STATIC MIXER

Mixing in the Kenics Static Mixer is analyzed in this paper. The velocity field is calculated using helical coordinates. The twist angle per element, s, and the aspect ratio of a mixing element, L/R, are taken as parameters. Poincare sections show that uniform mixtures cannot be produced for small s or small L/R, since the corresponding element is either too flat or too short to produce effective mixing. Furthermore, an alternation of mixing windows where uniform mixing takes place and island regions in the parameter space is found using bifurcation analysis. The rate of mixing for various combination of parameters is calculated, based on the evaluation of uniformity using the Kolmogorov test. An optimal design of the element is suggested, which enables producing quality mixtures with minimum pressure drop. A good agreement between simulation of some mixing processes and existing experiments is observed.

Waves in a cylindrical shell stiffened by a helical rib

In vacuo elastic wave propagation characteristics in an infinite cylindrical shell stiffened by an infinite helical rib are investigated analytically. Equations of motion of the helical rib and of the cylindrical shell in helical coordinates are developed. They are combined into a single set of equations using appropriate boundary conditions. Dispersion curves of the elastic waves propagating in the shell are obtained for different pitch angles. Comparisons with wave propagation characteristics in an unstiffened shell are discussed.

Computational study of three-dimensional magnetohydrodynamic equilibria in toroidal helical systems

Computational methods based on the time-dependent relaxation technique for obtaining finite beta stellarator equilibria with no net current are described. The computation grid is Eulerian corresponding to space-fixed non-orthogonal helical coordinates. The poloidal cross section, which is composed of rectangular grids, rotates with the same pitch as that of external helical conductors along the toroidal direction. By this choice, it is possible to handle complicated field topologies and to economize the computer memory. Also, numerical convergence is greatly improved compared with that in cylindrical coordinates. The developed code is applied to a typical low shear l=2 stellarator (Wendelstein VII-A). Furthermore, breaking up of magnetic surfaces due to finite β effect in a low aspect ratio l=2 torsatron is shown.

Three-dimensional Boundary Layer Flow on Rotating Blades

In order to determine the flow characteristics of the laminar and the turbulent boundary layers on rotating blades of an axial-flow turbomachine, a theoretical analysis was performed taking the chord-wise pressure gradient into account with the aid of the rotating helical coordinates. Comparison of the calculated results of rotating blades with those of two-dimensional stationary ones showed that the secondary flow induced over the rotating blades has strong effect of suppressing the boundary layer growth, which results in that the transition and the separation of the boundary layer are much delayed. This effect increases with an increase of attack angle. To prove the validity of the theory, measurements were performed with an axial blower equipment. The measured results showed that the present analysis is well applicable to the estimation of the flow characteristics of the boundary layers on relatively thin and small-cambered rotating blades to the chord length.

Kinematic dynamo action with helical symmetry in an unbounded fluid conductor: Part 1. Formulation and survey of low order cases

Abstract Kinematic dynamo action in an unbounded homogeneous, fluid conductor is studied from an exact, analytical viewpoint, by exploiting the mathematical tractibility of “helical symmetry,” uncovered by Lortz [1968a). The principal motivation is to elucidate how helical dynamos operate, to clarify their energetics, to extend Lortz's work where possible and to place helical dynamos within the more general context of dynamo theory. Lortz's novel helical coordinate system is first motivated and derived rationally. The velocity and magnetic fields are then projected onto helical coordinates and represented in terms of helical defining scalar functions dependent on only two helical coordinates. The unsteady dynamo equations, in helical variables, are next derived, and their invariance properties deduced. Attention is then focused on the steady state. Low order finite Fourier expansions for the dependence on the primary helical coordinate are introduced. Three different possible low order dynamo classes are ...

Elliptically polarized modes in gyrotropic waveguides. II. An alternative treatment of the longitudinally magnetized case

For pt.I see ibid., vol.9, p.1957, (1976). The gyrotropic waveguide, magnetized along the direction of propagation, is analysed by a conventional method which does not involve any initial assumptions regarding the state of polarization of waveguide modes. The problem is simplified by the use of complex helical coordinates in which the gyrotropic tensor assumes the diagonal form. The waveguide modes are shown to be elliptically polarized and dielectric boundary conditions are applied leading to a waveguiding condition which is outwardly similar to that applicable to isotropic dielectric guides but more complicated in content.