Cone Boundary Research Articles

Second mode unstable disturbance measurement of hypersonic boundary layer on cone by wavelet transform

Experimental investigation of hypersonic boundary layer instability on a cone is performed at Mach number 6 in a hypersonic wind tunnel. Time series signals of instantaneous fluctuating surface-thermal-flux are measured by Pt-thin-film thermocouple temperature sensors mounted at 28 stations on the cone surface in the streamwise direction to investigate the development of the unstable disturbance. Wavelet transform is employed as a mathematical tool to obtain the multi-scale characteristics of fluctuating surface-thermal-flux both in the temporal and spectrum space. The conditional sampling algorithm using wavelet coefficient as an index is put forward to extract the unstable disturbance waveform from the fluctuating surface-thermal-flux signals. The generic waveform for the second mode unstable disturbance is obtained by a phase-averaging technique. The development of the unstable disturbance in the streamwise direction is assessed both in the temporal and spectrum space. Our study shows that the local unstable disturbance detection method based on wavelet transformation offers an alternative powerful tool in studying the hypersonic unstable mode of laminar-turbulent transition. It is demonstrated that, at hypersonic speeds, the dominant flow instability is the second mode, which governs the course of laminar-turbulent transition of sharp cone boundary layer.

The hydrogen atom in a semi‐infinite space with an elliptical cone boundary

AbstractThe Schrödinger equation for the Hydrogen atom in spheroconal coordinates admits factorizable solutions, due to the fact that the Hamiltonian, the square of the angular momentum, and a linear combination of the squares of the Cartesian components of the angular momentum form a complete set of commuting operators. The linear combination defines the geometry of a family of elliptical cones, each one of which may serve as a confining boundary for the Hydrogen atom allowing the construction of its exact eigenfunctions expressed as the products of radial Laguerre polynomial functions and quasi‐periodic Lamé spheroconal harmonic functions in the elliptical cone coordinates. The boundary condition, requiring the vanishing of the wave function at the confining elliptical cone, introduces noninteger values of the angular momentum quantum label, as well as the breaking of the parity symmetry. Numerical results of the energy spectra, electric dipole moment, and pressure distribution of the Hydrogen atom are reported for different shapes and positions of the confining boundary. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Numerical Investigation of Wave Packets in a Mach 3.5 Cone Boundary Layer

Numerical Investigation of Wave Packets in a Mach 3.5 Cone Boundary Layer

Kinetics of first passage in a cone

We study statistics of first passage inside a cone in an arbitrary spatial dimension. The probability that a diffusing particle avoids the cone boundary decays algebraically with time. The decay exponent depends on two variables: the opening angle of the cone and the spatial dimension. In four dimensions, we find an explicit expression for the exponent, and in general, we obtain it as a root of a transcendental equation involving associated Legendre functions. At large dimensions, the decay exponent depends on a single scaling variable, while roots of the parabolic cylinder function specify the scaling function. Consequently, the exponent is of order one only if the cone surface is very close to a plane. We also perform asymptotic analysis for extremely thin and extremely wide cones.

Vortex-speed selection within the boundary-layer flow over a rotating sphere placed in an enforced axial flow

This paper furthers existing work into the instability mechanisms within the boundary-layer flow over a rotating sphere through the study of amplification rates within the convectively-unstable region. The onset of convective instability is associated with the experimentally observed onset of spiral vortices reported in the literature. Axial flow is found to stabilize the boundary layer by both delaying the onset of convective instability at all latitudes and also by significantly reducing the spatial amplification rates. We find that the type II (streamline curvature) mode becomes increasingly amplified with respect to the type I (crossflow) mode and is therefore likely to be selected in practice for sufficiently high axial flow rates. Furthermore, in experiments where special care is taken to remove all surface roughness, we predict that vortices will rotate at around 75% of the local surface speed. This is consistent with the experimental observations of Kobayashi & Arai who note a speed of around 76% under particular experimental conditions. These predictions are entirely consistent with related work on the rotating-disk and cone boundary layers.

Finite element method for solving geodetic boundary value problems

The goal of this paper is to present the finite element scheme for solving the Earth potential problems in 3D domains above the Earth surface. To that goal we formulate the boundary-value problem (BVP) consisting of the Laplace equation outside the Earth accompanied by the Neumann as well as the Dirichlet boundary conditions (BC). The 3D computational domain consists of the bottom boundary in the form of a spherical approximation or real triangulation of the Earth’s surface on which surface gravity disturbances are given. We introduce additional upper (spherical) and side (planar and conical) boundaries where the Dirichlet BC is given. Solution of such elliptic BVP is understood in a weak sense, it always exists and is unique and can be efficiently found by the finite element method (FEM). We briefly present derivation of FEM for such type of problems including main discretization ideas. This method leads to a solution of the sparse symmetric linear systems which give the Earth’s potential solution in every discrete node of the 3D computational domain. In this point our method differs from other numerical approaches, e.g. boundary element method (BEM) where the potential is sought on a hypersurface only. We apply and test FEM in various situations. First, we compare the FEM solution with the known exact solution in case of homogeneous sphere. Then, we solve the geodetic BVP in continental scale using the DNSC08 data. We compare the results with the EGM2008 geopotential model. Finally, we study the precision of our solution by the GPS/levelling test in Slovakia where we use terrestrial gravimetric measurements as input data. All tests show qualitative and quantitative agreement with the given solutions.

Direct generation of terahertz surface plasmon polaritons on a wire using electron bunches

We propose to generate terahertz surface plasmon polaritons (SPPs) on a metal wire by launching electron bunches onto a tapered end of the wire. To show the potential of this method, we solve Maxwell's equations for the appropriate boundary conditions. The metal wire tip is modeled by a perfectly conducting semi-infinite cone. It is shown that the SPPs can be recovered from the idealized fields by well-known perturbation techniques. The emitted radiation is strongly concentrated into a narrow solid angle near the cone boundary for cones with a small opening angle. We calculate that, using currently available technology, subpicosecond SPPs with peak electric fields of the order of MV/cm on a 1 mm diameter wire can be obtained.

Receptivity to free-stream disturbance waves for hypersonic flow over a blunt cone

A high-order shock-fitting finite difference scheme is studied and used to do direction numerical simulation (DNS) of hypersonic unsteady flow over a blunt cone with fast acoustic waves in the free stream, and the receptivity problem in the blunt cone hypersonic boundary layers is studied. The results show that the acoustic waves are the strongest disturbance in the blunt cone hypersonic boundary layers. The wave modes of disturbance in the blunt cone boundary layers are first, second, and third modes which are generated and propagated downstream along the wall. The results also show that as the frequency decreases, the amplitudes of wave modes of disturbance increase, but there is a critical value. When frequency is over the critial value, the amplitudes decrease. Because of the discontinuity of curvature along the blunt cone body, the maximum amplitudes as a function of frequencies are not monotone.

Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone

For an upstream supersonic flow past a straight-sided cone in $\R^3$ whose vertex angle is less than the critical angle, a transonic (supersonic-subsonic) shock-front attached to the cone vertex can be formed in the flow. In this paper we analyze the stability of transonic shock-fronts in three-dimensional steady potential flow past a perturbed cone. We establish that the self-similar transonic shock-front solution is conditionally stable in structure with respect to the conical perturbation of the cone boundary and the upstream flow in appropriate function spaces. In particular, it is proved that the slope of the shock-front tends asymptotically to the slope of the unperturbed self-similar shock-front downstream at infinity.

On the Structure of asymptotics of the solution of a second-order elliptic equation in a neighborhood of an angular point

The behavior of solutions of elliptic equations in neighborhoods of angular and conical boundary points has been well studied; the asymptotics of these solutions has been constructed. In the present paper, we propose a new approach to constructing asymptotic decompositions in a neighborhood of an angular boundary point, which allows us to describe the structure of these asymptotics in a relatively simple and illustrative way.

An assessment of the domain reduction method as an advanced boundary condition and some pitfalls in the use of conventional absorbing boundaries

AbstractThis paper assesses the performance of two commonly used absorbing boundaries in dynamic finite element analysis of geotechnical problems in conjunction with the domain reduction method (DRM). The DRM was originally developed by Bielak et al. (Bull. Seismol. Soc. Am. 2003; 93(2):817–824) to reduce the computational cost of seismological applications, whereas Yoshimura et al. (Bull. Seismol. Soc. Am. 2003; 93(2):825–840) showed that it can be effectively used as a boundary condition. In this study a practical methodology is proposed, which employs the cone boundary of Kellezi (Soil Dyn. Earthq. Eng. 2000; 19:533–547) on the outer boundary of the reduced (step II) model of the DRM. To verify the applicability of the proposed methodology, the results using both the cone boundary and the standard viscous boundary are compared with those using an extended mesh. Finally, results using the DRM as a boundary condition are compared with those using conventional boundary conditions. Some common pitfalls in the use of absorbing boundaries are highlighted and guidance for their correct use in engineering practice is given. Copyright © 2008 John Wiley & Sons, Ltd.

Angle Dependence of Nematic Liquid Crystal Dynamics with Conical Boundary Condition

In the manufacture of the thin film for liquid crystal devices, we can often find the situation in which the directors are equilibrated under the constraint that fixes only their polar angles on the film surface. We call this surface constraint the “conical boundary condition”. In this study, we investigate the dynamics of nematic liquid crystal under the conical boundary condition by a cell dynamical system (CDS) method. In particular, we focus on the anchoring and conical polar angle dependences of the annihilation time of topological defects and the time needed for equilibration. The angle dependence on the time characteristic of phase ordering is recognized and explained using the simplified scheme of director configuration under the hybrid condition of the conical and anchoring boundaries.

DNS of compressible turbulent boundary layer around a sharp cone

Direct numerical simulation of the turbulent boundary layer over a sharp cone with 20° cone angle (or 10° half-cone angle) is performed by using the mixed seventh-order up-wind biased finite difference scheme and sixth-order central difference scheme. The free stream Mach number is 0.7 and free stream unit Reynolds number is 250000/inch. The characteristics of transition and turbulence of the sharp cone boundary layer are compared with those of the flat plate boundary layer. Statistics of fully developed turbulent flow agree well with the experimental and theoretical data for the turbulent flat-plate boundary layer flow. The near wall streak-like structure is shown and the average space between streaks (normalized by the local wall unit) keeps approximately invariable at different streamwise locations. The turbulent energy equation in the cylindrical coordinate is given and turbulent energy budget is studied. The computed results show that the effect of circumferential curvature on turbulence characteristics is not obvious.

선형 안정성 이론을 이용한 압축성 축 대칭 원뿔 경계층의 천이지점 예측

본 연구에서는 축 대칭 원뿔 형상 위의 압축성 경계층의 천이 지점을 선형 안정성 이론과 -method를 이용하여 예측하였다. 축 대칭 좌표계에서의 압축성 유동 지배 방정식으로부터 압축성 원뿔 경계층의 선형 안정성 방정식을 얻었으며 안정성 방정식을 2차 정확도의 유한 차분법을 이용하여 계산하는 수치 프로그램을 개발하였다. 개발 된 코드로 원뿔 경계층의 안정성 특성 및 2차원 교란의 증폭률을 계산하고 실험결과와의 비교를 통해 검증을 수행하였다. 얻어진 교란의 증폭률을 활용하여 -method를 통해 천이지점 예측을 수행하였다. 풍동 시험 및 비행 시험 결과와의 비교를 통해 비행 조건에 있는 마하수 4와 8사이의 원뿔 경계층에 대한 본 연구의 천이지점의 예측 능력을 확인하였다. 또한 벽면 냉각이 경계층 내부 교란의 안정성 및 천이 지점에 미치는 영향을 분석하였다. In this study, the transition Reynolds number of compressible axi-symmetric sharp cone boundary layer is predicted by using a linear stability theory and the -method. The compressible linear stability equation for sharp cone boundary layer was derived from the governing equations on the body-intrinsic axi-symmetric coordinate system. The numerical analysis code for the stability equation was developed based on a second-order accurate finite-difference method. Stability characteristics and amplification rate of two-dimensional second mode disturbance for the sharp cone boundary layer were calculated from the analysis code and the numerical code was validated by comparing the results with experimental data. Transition prediction was performed by application of the -method with N=10. From comparison with wind tunnel experiments and flight tests data, capability of the transition prediction of this study is confirmed for the sharp cone boundary layers which have an edge Mach number between 4 and 8. In addition, effect of wall cooling on the stability of disturbance in the boundary layer and transition position is investigated.

Uniform asymptotic approximations of Green's functions in a long rod

AbstractAsymptotic approximations for Green's function for the operator −Δ in a long rod are derived. These approximations are uniformly valid over the whole domain including the end regions of the rod. Connections are established between the asymptotic approximations in a long rod and the asymptotics in thin domains. Overview of asymptotic approximations of Green's kernels in a domain with a small hole and domains with singularly perturbed smooth or conical boundaries is also given. Copyright © 2008 John Wiley & Sons, Ltd.

Mechanism of transition in a hypersonic sharp cone boundary layer with zero angle of attack

Firstly, the steady laminar flow field of a hypersonic sharp cone boundary layer with zero angle of attack was computed. Then, two groups of finite amplitude T-S wave disturbances were introduced at the entrance of the computational field, and the spatial mode transition process was studied by direct numerical simulation (DNS) method. The mechanism of the transition process was analyzed. It was found that the change of the stability characteristics of the mean flow profile was the key issue. Furthermore, the characteristics of evolution for the disturbances of different modes in the hypersonic sharp cone boundary layer were discussed.

Operators with numerical range in a conic domain

Given a convex domain C with conic boundary, a linear operator A with numerical range contained in , and a rational function bounded on , we are interested in estimating the norm of r(A) in terms of the supremum of r in . In particular, we show that ellipses, hyperbolas and parabolas are K-spectral sets with K = 2 + 2= p 3 3:16. 2000 Mathematical subject classications : 47A12 ; 47A25

Numerical investigation of evolution of disturbances in supersonic sharp cone boundary layers

The spatial evolution of 2-D disturbances in supersonic sharp cone boundary layers was investigated by direct numerical simulation (DNS) in high order compact difference scheme. The results suggested that, although the normal velocity in the sharp cone boundary layer was not small, the evolution of amplitude and phase for small amplitude disturbances would be well in accordance with the results obtained by the linear stability theory (LST) which supposes the flow was parallel. The evolution of some finite amplitude disturbances was also investigated, and the characteristic of the evolution was shown. Shocklets were also found when the amplitude of disturbances increased over some value.

Generalizations of McShane's identity to hyperbolic cone-surfaces

We generalize McShane's identity for the length series of simple closed geodesics on a cusped hyperbolic surface to a general identity for hyperbolic cone-surfaces (with all cone angles ≥ π), possibly with cusps and/or geodesic boundary. The general identity is obtained by studying gaps formed by simple-normal geodesics emanating from a distinguished cone point, cusp or boundary geodesic. In particular, by applying the generalized identity to the quotient orbifolds of a hyperbolic one-cone/one-hole torus by its elliptic involution and of a hyperbolic closed genus two surface by its hyperelliptic involution, we obtain general Weierstrass identities for the one-cone/one-hole torus, and an identity for the genus two surface, which are also obtained by McShane using different methods. We also give an interpretation of the general identity in terms of complex lengths of the cone points, cusps and geodesic boundary components.

Multicellular convection in rotating annuli

The onset of convection in a rotating cylindrical annulus with sloping conical boundaries is studied in the case where this slope increases with the radius. The critical modes assume the form of drifting spiralling columns attached to the inner cylindrical wall at moderate and large Prandtl numbers, but they become attached to the outer wall at low Prandtl numbers. These latter 'equatorially attached' modes are multicellular at intermediate rotation rates. Through a perturbation analysis which is validated by a numerical code, we show that all equatorially attached modes are quasi-inertial modes and analyse the physical mechanisms leading to multicells. This is done for both stress-free and no-slip boundary conditions. At finite amplitudes the convection generates a Reynolds stress which leads to the development of a mean zonal flow, and a geometrical analysis of the mechanisms leading to this zonal flow is presented. The influence of Ekman friction on the zonal flow is also studied.