A Plethora of Non-Bending Surfaces of Revolution: Classifications and Explicit Parameterizations

As the title itself suggests here we are presenting extremely reach two/three parametric families of non-bending rotational surfaces in the three dimensional Euclidean space and provide the necessary details about their natural classifications and explicit parameterizations. Following the changes of the relevant parameters it is possible to trace out the ``evolution'' of these surfaces and even visualize them through their topological transformations. Many, and more deeper questions about their metrical properties, mechanical applications, etc. are left for future explorations.

The notes on rotational surfaces in Galilean space

In this study, we provide a brief description of rotational surfaces in 4-dimensional (4D) Galilean space using a curve and matrices in [Formula: see text]. That is, we provide different types of rotational matrices, which are the subgroups of [Formula: see text] by rotating a selected axis in [Formula: see text]. Hence, we choose two parameter matrices groups of rotations and we give the matrices of rotation corresponding to the appropriate subgroup in Galilean 4-space and we generate rotated surfaces.

On the Formulation of Nonreflecting Boundary Conditions for Turbomachinery Configurations: Part I — Theory and Implementation

With unsteady flow simulations of industrial turbomachinery configurations becoming more and more affordable there is a growing need for accurate inlet and outlet boundary conditions as numerical reflections alone can lead to incorrect trends in engine efficiency, noise and aeroelastic analysis parameters. This is the first of two papers on the formulation of unsteady boundary conditions which have been implemented for both time-domain and frequency-domain solvers. Giles’ original idea for steady solvers to formulate the boundary condition in terms of characteristics generalizes to frequency-domain solvers. The boundary condition drives the value of the incoming characteristics to ideal values that are computed using the modal decomposition of linearized 2D Euler flows. The present paper explains how to generalize 2D nonreflecting boundary conditions to real 3D annular domains by applying them in certain conical rotational surfaces. For a flow with zero radial component and an annular boundary that is perpendicular to the machine axis, these surfaces are the cylindrical streamsurfaces. For more general flows and geometries, however, there is no natural choice for the rotational surfaces. In this paper, two choices are discussed: the surfaces that are generated by the boundary normals and those that are defined by the circumferentially averaged meridional velocity. The impact of the boundary condition on the stability of the harmonic-balance solver is analyzed by studying the pseudo-time evolution of certain energy integrals. For a model problem which consists of a small disturbance of an inviscid flow, the increase or decrease of this energy integral is shown to be directly related to the normal characteristic variables along the boundary. This shows that the actual boundary condition should be formulated as a control problem for the normal characteristics. Moreover, the application of the harmonic balance solver to a simple duct configuration with prescribed disturbances demonstrates that using the characteristics based on the meridional velocity may prevent the solver from converging. In contrast, the 2D theory can be formulated in a different surface without impairing the robustness of the overall approach. These findings are illustrated by a simple test case. The impact of the choice of the rotational surface for the 2D theory is studied for various duct segments and a low-pressure turbine configuration in the second paper. There it is shown that applying the 2D theory to the meridional-velocity surfaces may be advantageous in that it leads to more accurate results.

On the Formulation of Nonreflecting Boundary Conditions for Turbomachinery Configurations: Part II — Application and Analysis

The flow in turbomachinery components is complex due to the relative motion of rotating and non-rotating elements. A proper design and prediction of physical phenomena requires reliable CFD tools. One important aspect is the incorporation of sophisticated algorithms at the boundaries of the computational domain. For inviscid, one-dimensional and two-dimensional Euler-flows there exist analytical solutions for the formulation of a boundary condition. Realistic applications, however, are viscous and consist of a complex three-dimensional character. Nevertheless, the analytical 2D nonreflecting boundary conditions are commonly used in CFD codes for their high computational efficiency and numerical robustness. The application becomes more challenging when the boundaries are close to geometrical features such as blades and vanes. In practical applications, the position of the boundaries is dictated by geometrical constraints and hence the proximity to the blading cannot always be avoided. The interaction of rotating and non-rotating geometrical features in a turbomachine produces complex flow patterns that propagate in the form of acoustic, vorticity and entropy waves. A boundary condition must be implemented in such a way that waves can propagate undisturbed out of the computational domain. Any reflection may unphysically affect the solution within the computational domain which is especially harmful to sensitive values such as unsteady aeroelastic quantities. But also steady-state computations may suffer from errors produced by reflective boundary conditions. The following paper is the second of two papers on the formulation of unsteady boundary conditions based on a two-dimensional analytical approach. The first part of this paper [6] explains how to extend 2D nonreflecting boundary conditions to real 3D annular domains by applying them in certain conical rotational surfaces. Two different formulations are discussed referring to the orientation of said rotational surfaces. In the first case the surfaces are oriented perpendicular to the boundary panel. In the second case the surfaces are aligned with the circumferentially averaged meridional flow velocity. In the present paper a thorough analysis of the two different approaches will be given. Both formulations of the boundary algorithm are validated on the basis of several elementary model flows. The behavior is analyzed for various unsteady wave patterns of different propagation directions with respect to the boundary. It will be shown that the alignment of the rotational surfaces with the meridional flow has a beneficial effect on the reflective behavior for the majority of the investigated flow conditions. The boundary conditions are then tested on realistic turbomachinery components in order to analyze their applicability on complex flows.

On parametrizations of loxodromes on time-like rotational surfaces in Minkowski space-time

We investigate the parametrizations of loxodromes on the time-like rotational surfaces by using suitable Lorentzian angles in Minkowski space-time. Also, some examples and corresponding graphs are given by using Mathematica.