High-Order Method with Moving Frames to Compute the Covariant Derivatives of Vectors on General 2D Curved Surfaces

The covariant derivative is a generalization of differentiating vectors.The Euclidean derivative is a special case of the covariant derivative in Euclidean space.The covariant derivative gathers broad attention,particularly when computing vector derivatives on curved surfaces and volumes in various applications.Covariant derivatives have been computed using the metric tensor from the analytically known curved axes.However,deriving the global axis for the domain has been mathematically and computationally challenging for an arbitrary two-dimensional(2D)surface.Consequently,computing the covariant derivative has been difficult or even impossible.A novel high-order numerical scheme is proposed for computing the covariant derivative on any 2D curved surface.A set of orthonormal vectors,known as moving frames,expand vectors to compute accurately covariant derivatives on 2D curved surfaces.The proposed scheme does not require the construction of curved axes for the metric tensor or the Christoffel symbols.The connectivity given by the Christoffel symbols is equivalently provided by the attitude matrix of orthonormal moving frames.Consequently,the proposed scheme can be extended to the general 2D curved surface.As an application,the Helmholtz‐Hodge decomposition is considered for a realistic atrium and a bunny.

Asymmetric and Moving-Frame Approaches to MHD Equations

The magnetohydrodynamic （MHD） equations of incompressible viscous fluids with finite electrical conductivity describe the motion of viscous electrically conducting fluids in a magnetic field. In this paper, we find eight families of solutions of these equations by Xu＇s asymmetric and moving frame methods. A family of singular solutions may reflect basic characteristics of vortices. The other solutions are globally analytic with respect to the spacial variables. Our solutions may help engineers to develop more effective algorithms to find physical numeric solutions to practical models.

Joint Invariants and Joint Differential Invariants of Some Transformation Groups

The theory of moving frames developed by Peter J Olver and M Fels has impor-tant applications to geometry, classical invariant theory. We will use this theory to classify joint invariants and joint differential invariants of some transformation groups.

A Violation of the Special Theory of Relativity Demonstrated Using the Correlation between Two Pair-Generated Photons

When Einstein developed the special theory of relativity (STR), he assumed the principle of relativity, i.e. that all inertial frames are equivalent. Einstein thought it was impossible to differentiate inertial frames into classically stationary frames where light propagates isotropically, and classically moving frames where light propagates anisotropically. However, the author has previously pointed out that classically moving frames have a velocity vector attached, and presented a thought experiment for determining the size of that velocity vector. The author has already shown a violation of the STR, but this paper presents a violation of the STR using different reasoning. More specifically, this paper searches for a coordinate system where light propagates anisotropically. This is done by using the correlation of two photons pair-generated from a photon pair generator. If the existence of such a coordinate system can be ascertained, it will constitute a violation of the STR.

非均匀船舶尾流中工作的常规和大侧斜船用螺旋桨的水动力和水声计算预报

Despite their high manufacturing cost and structural deficiencies especially in tip regions,highly skewed propellers are preferred in the marine industry,where underwater noise is a significant design criterion.However,hydrodynamic performances should also be considered before a decision to use these propellers is made.This study investigates the trade-off between hydrodynamic and hydroacoustic performances by comparing conventional and highly skewed Seiun Maru marine propellers for a noncavitating case.Many papers in the literature focus solely on hydroacoustic calculations for the open-water case.However,propulsive characteristics are significantly different when propeller-hull interactions take place.Changes in propulsion performance also reflect on the hydroacoustic performances of the propeller.In this study,propeller-hull interactions were considered to calculate the noise spectra.Rather than solving the full case,which is computationally demanding,an indirect approach was adopted;axial velocities from the nominal ship wake were introduced as the inlet condition of the numerical approach.A hybrid method based on the acoustic analogy was used in coupling computational fluid dynamics techniques with acoustic propagation methods,implementing the Ffowcs Williams-Hawkings(FW-H)equation.The hydrodynamic performances of both propellers were presented as a preliminary study.Propeller-hull interactions were included in calculations after observing good accordance between our results,experiments,and quasi-continuous method for the open-water case.With the use of the time-dependent flow field data of the propeller behind a nonuniform ship wake as an input,simulation results were used to solve the FW-H equation to extract acoustic pressure and sound pressure levels for several hydrophones located in the near field.Noise spectra results confirm that the highest values of the sound pressure levels are in the low-frequency range and the first harmonics calculated by the present method are in good accordance with the theoretical values.Results also show that a highly skewed propeller generates less noise even in noncavitating cases despite a small reduction in hydrodynamic efficiency.

Nongeneric Bifurcations Near a Nontransversal Heterodimensional Cycle

In this paper bifurcations of heterodimensional cycles with highly degenerate conditions are studied in three dimensional vector fields,where a nontransversal intersection between the two-dimensional manifolds of the saddle equilibria occurs.By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles,the authors construct a Poincar′e return map under the nongeneric conditions and further obtain the bifurcation equations.By means of the bifurcation equations,the authors show that different bifurcation surfaces exhibit variety and complexity of the bifurcation of degenerate heterodimensional cycles.Moreover,an example is given to show the existence of a nontransversal heterodimensional cycle with one orbit flip in three dimensional system.

New Algebraic Approaches to Classical Boundary Layer Problems

Classical non-steady boundary layer equations are fundamental nonlinear partial differential equations in the boundary layer theory of fluid dynamics. In this paper, we introduce various schemes with multiple parameter functions to solve these equations and obtain many families of new explicit exact solutions with multiple parameter functions. Moreover, symmetry transformations are used to simplify our arguments. The technique of moving frame is applied in the three-dimensional case in order to capture the rotational properties of the fluid. In particular, we obtain a family of solutions singular on any moving surface, which may be used to study turbulence. Many other solutions are analytic related to trigonometric and hyperbolic functions, which reflect various wave characteristics of the fluid. Our solutions may also help engineers to develop more effective algorithms to find physical numeric solutions to practical models.