Axisymmetric Media Research Articles

Abel inversion from central-projection background oriented schlieren observations for reconstruction of axisymmetric refractive media

The background oriented schlieren (BOS) technique provides quantitative measurements of integrated light deflection caused by gradients of refractive index in an optical medium. For an axisymmetric medium and parallel rays orthogonal to the axis of symmetry, Abel inversion enables fast, direct, and non-iterative reconstruction of the refractive index distribution. In this work, we relax the parallel ray hypothesis, demonstrating that Abel inversion can be effectively applied to observations obtained by central projection with a standard entocentric lens, with a simple correction for measured deflections. This correction derives from an original second-order approximation of ray deflection under the paraxial approximation. The accuracy of the method is demonstrated using a synthetic low-density jet and a cone of vision aperture of approximately 40°. Its practical relevance is further illustrated by reconstructing the temporal mean density fields of two experimental cases: a heated high-subsonic dual-stream jet and an under-expanded supersonic jet characterized by significant shock-induced discontinuities.

Development of Fringe Image Processing Algorithms for Tomograpic Reconstruction of Temperature Field in an Axisymmetric Medium

This work is part of the development of algorithms and digital tools for the tomographic reconstruction of the (3D) temperature field from images resulting from thermal metrology based on two optical techniques (Moire deflectometry and holographic interferometry). It is intended not only for digital scientists and image processing specialists but also for researchers and practitioners in instrumentation and measurement by non-invasive techniques in general and in thermal metrology by optical techniques in particular Tomography by optical techniques of axisymmetric transparent media, the main subject of application of this thesis, is a perfect example of an imaging system based on an elementary mathematical property (the Abel transform and its inverse in this case). This property comes up against in its practical application the hypersensitivity to noise and problems due to the intrinsic characteristics of this transform. However, the study of transparent media such as air in our case, which does not present any radiative, absorption or diffusion properties, makes the images resulting from optical techniques unusable only after a treatment which precedes the inverse calculation of the Abel transform. This treatment thus limiting the development of 3D imaging requires special attention. Furthermore, the tomographic reconstruction of the temperature field of such an environment will be done in two processing parts, the first of which will focus on the analysis of mud figures, while the second is dedicated to solving the inverse problem generated by the Abel transform.

Vectorial discrete ray tracing approach for solving radiative transfer in irregular variable index media via discrete transfer method

A novel vectorial discrete ray tracing approach is applied for tracking the curve paths of radiative beams through the variable index media with complex geometries. In this approach, the irregular solution domain is discretized by a boundary-fitted grid, where the radiative properties, including the refractive index, are considered to be constant for each control volume and control surface. Radiative beams travel along straight lines through each volume element until they hit the interfaces between adjacent volume elements, where the direction of rays changes according to the Snell's law of refraction. The radiation heat transfer in absorbing, emitting, and isotropically scattering media is solved by the discrete transfer method. The problem is solved for one-, two-, and three-dimensional media. Axisymmetric media are simulated by detaching a sector with specular sidewalls. Verification is done by comparing the results with available benchmark solutions in literature. For the cases where the data is not available, the accuracy is proved by satisfaction the energy conservation criterion.

Abel inversion method using Chebyshev wavelets for tomographic reconstruction of an axisymmetric medium.

We describe an Abel inversion method for tomographic reconstruction of an axisymmetric medium. The essence of this method is the approximation of data by Chebyshev wavelets. A numerical method and operational matrices are proposed. The accuracy of the method is tested in terms of the noise coefficient. The studied method is compared with the Legendre wavelets method. The results show that the proposed method is less sensitive to noise. We also demonstrate the efficiency of the developed method by obtaining good and expected results for the inversion of interferometric data.

APPLICATION OF CONVOLUTIONAL NEURAL NETWORKS FOR RESISTIVITY LOGS PROCESSING AND NON-ITERATIVE EXPRESS-INVERSION IN COMPLEX RESERVOIR ENVIRONMENTS

The work is devoted to the development of techniques and software for the quantitative interpretation of resistivity oil well logs. The article considers the results of applying the neural network approach to the processing of resistivity logging data measured at intervals composed of thin layers with contrasting electrical properties. The proposed algorithms combine the advantages of data interpretation based on a two-dimensional axisymmetric medium model and high performance, which allows them to be used at the primary processing stage, increasing the reliability of express interpretation.

Hydrodynamics of an inertial active droplet

Abstract

Nonlinear torsional wave propagation in cylindrical coordinates to assess biomechanical parameters

Abstract A formulation in cylindrical coordinates of the nonlinear torsional wave propagation on a hyperelastic material characterized by Hamilton's strain energy function is proposed. The objective of this formulation is to study and assess soft tissues, taking into account both geometrical and physical nonlinearity. Specifically, this work analyzes the propagation of torsional shear waves through an isotropic axisymmetric medium, so the only non-zero velocity component is associated with the angular coordinate. To transform the equations from Cartesian to cylindrical coordinates, the covariant and contravariant transformations are employed. A transverse torsional wave propagating through a quasi-incompressible hydrogel from the emitter to the receiver is considered. As the close form solution is not straightforward, a numerical simulation using the Finite Difference Time Domain method is performed. The results are obtained for a realistic range of wave frequencies and nonlinear parameters for medical applications.

Effects of inner soil on the vertical dynamic response of a pipe pile embedded in inhomogeneous soil

This paper is concerned with the effects of inner soil on the vertical dynamic response of a pipe pile embedded in a vertically multilayered and radially inhomogeneous soil. The soil is considered a three-dimensional axisymmetric medium, the radial inhomogeneity of which (due to the construction disturbance effect) is simulated by gradually varying the soil parameters in the radial direction. The pipe pile is treated as a vertical, hollow, viscoelastic, one-dimensional bar. The velocity admittance and reflected signal at the pile head are derived by solving the equilibrium equations for the pile and soil. They are then used to obtain an insight into the coupled effects of the inner soil and several related factors, such as the pile parameters, construction disturbance effect and pile defect characteristics, on the vertical dynamic response of the pipe pile concerned in pile integrity testing. A comparison with the measured data is also presented to verify the reliability of the proposed solution.

Computationally fast EM field propagation through axi-symmetric media using cylindrical harmonic decomposition.

We describe and provide a systematic procedure for computationally fast propagation of arbitrary vector electromagnetic (EM) fields through an axially symmetric medium. A cylindrical harmonic field propagator is chosen for this purpose and in most cases, this is the best and the obvious choice. Firstly, we describe the cylindrical harmonic decomposition technique in terms of both scalar and vector basis for a given input excitation field. Then we formulate a generalized discrete Fourier-Hankel transform to achieve efficient vector basis decomposition. We allow a slower, pre-computation step, that finds a representation of the axi-symmetric medium as a transfer matrix in a discrete, cylindrical-harmonic basis. We find this matrix from a series of axi-symmetric (2D) finite element simulations (also known as the 2.5D technique). This transfer matrix approach significantly reduces the computational load when the transverse size or range exceeds about 30 wavelengths. This matrix is independent of the input excitation field for a given space-bandwidth product and hence makes it reusable for different excitation fields. We numerically validate the above approaches for different axi-symmetric EM scattering media which include a hemispherical gradient-index Maxwell's fish-eye lens, a transformation optics designed spherical invisibility cloak, a thin aspheric lens, and a cylindrical perfect lens.

Compressible vapour flow in conduits and fractures

We consider the steady flow of a viscous compressible gas through an axisymmetric or two-dimensional porous medium whose properties in the direction of the flow are sufficiently slowly varying. The study is partly motivated by a number of different applications in the Earth sciences, including the release of magmatic volatiles from a magma chamber beneath an active volcano and the discharge of geothermal fluids. The results are also relevant to evaluating the consequences of an accidental release of carbon dioxide from a storage reservoir within the Earth, as might happen at a carbon capture and storage (CCS) site. We consider both slow, thermally equilibrated, flows and fast, adiabatic flows. Because the flow is compressible, it is the mass (and not the volume) flux which is conserved along the flow. We determine this constant mass flux and the velocity and pressure fields, both of which vary with position along the flow, as a function of all the physical parameters. We find that the resultant pressure gradient in the medium is largest at the far, low-pressure end of the conduit because the velocity is largest at that end due to the smallest density being associated with the smallest pressures. This means that the pressure in the permeable conduit is always larger than the linear pressure distribution which joins the given pressures at depth and at the surface, as would be the situation if the flow were incompressible. The detailed pressure distribution is shown to depend on the variation with depth of the quantity $\unicode[STIX]{x1D707}T/(ka^{2})$, where $\unicode[STIX]{x1D707}$ is the dynamic viscosity of the vapour, $T$ is the external temperature, $k$ the permeability and $\unicode[STIX]{x03C0}a^{2}$ the cross-sectional area of the conduit. The resultant mass flux is determined to be proportional to the mean along the flow of $\unicode[STIX]{x1D707}T/(ka^{2})$. We present two numerical illustrations of the results.

Laplace-Gauss and Helmholtz-Gauss paraxial modes in media with quadratic refraction index.

The scalar theory of paraxial wave propagation in an axisymmetric medium where the refraction index quadratically depends on transverse variables is addressed. Exact solutions of the corresponding parabolic equation are presented, generalizing the Laplace-Gauss and Helmholtz-Gauss modes earlier known for homogeneous media. Also, a generalization of a zero-order asymmetric Bessel-Gauss beam is given.

Filling Box Flows in an Axisymmetric Porous Medium

We present an analytical solution for buoyancy-driven “filling box” flows in axisymmetric porous media having closed bottom and side boundaries. The flow consists first and foremost of a descending, point source plume. When plume fluid reaches the (horizontal) bottom boundary, it begins to flow radially outward in the form of an axisymmetric gravity current. The leading edge of the gravity current advances with time as $$t^{1/2}$$ until it reaches the vertical sidewalls. At this point, the flow is characterized by a vertically ascending “first front” that steadily advects towards the plume source. We assume the plume to be in a Darcy regime, i.e. $$\text{ Re }\mathop {\sim }\limits ^{<}\mathcal {O}(10)$$ , with $$\text{ Pe }>\mathcal {O}(1)$$ , where Re and Pe are, respectively, the Reynolds and Peclet numbers, and derive a similarity solution for the plume by applying a boundary layer approximation. Formulas are therefore obtained for the vertical variation of the plume volume flux and area-averaged concentration. The former result shows important qualitative differences with the analogue equation derived in the limit $$\text{ Pe }<\mathcal {O}(1)$$ . In particular, the plume volume flux is now predicted to explicitly depend on the reservoir permeability, plume buoyancy flux and fluid viscosity. The gravity current problem is likewise solved using a self-similar solution, this time adapted from the work of Lyle et al. (J Fluid Mech 543:293–302, 2005) but connected to the outflow conditions of the plume. Finally, in solving for the motion of the first front, we apply a volume flux balance equation and therefore estimate the time scale required for the first front to advect from the bottom of the control volume to the source elevation. By synthesizing the above results, we can estimate the total volume of source fluid and mass of solute that can be injected into an axisymmetric reservoir without overflow. Predictions can also be made for the time-variable mean concentration of this contaminated fluid layer, which must obviously be less than the source concentration.

Stochastic analysis of steady seepage underneath a water-retaining wall through highly anisotropic porous media

Steady seepage is determined by a head drop upstream/downstream of a water-retaining wall. Due to its erratic variations, hydraulic log-conductivity$Y=\ln K$is modelled as a stationary random space function (RSF). We deal with a highly anisotropic porous formation, i.e. an axisymmetric medium where the horizontal correlation integral scale of$Y$is much larger than the vertical one. The goal of computing the resulting flow field within a stochastic framework is complicated by non-uniformity of the mean flow. Simple (closed-form) expressions for the correlation functions of the flow variables as well as the mean head are derived. We use these results to quantify the impact of spatial variability of$Y$upon the probability that the exit volumetric flow rate downstream of the wall is greater than that obtained by regarding the formation as homogeneous (with constant hydraulic conductivity). In particular, we show that the spatial variability of$Y$may lead to predictions (and consequently to design choices) which significantly differ from those achieved by regarding the porous formation as homogeneous.

Three-Dimensional Scattering and Inverse Scattering from Objects With Simultaneous Permittivity and Permeability Contrasts

There has been increasing efforts in enhanced biomedical and geophysical imaging in the recent years exploring the magnetic contrast agent. The handling of both dielectric and magnetic contrasts adds on difficulties to the forward and inverse scattering problems. In this paper, the 3-D scattering and the inverse scattering from objects having simultaneous electric and magnetic contrasts are presented, where the permittivity, conductivity, and permeability of the objects can all be different from the background. To cope with the challenging high computation cost in the 3-D scattering problem, we formulate the combined field volume integral equations and extend the stabilized biconjugate gradient method and fast Fourier transform (BCGS-FFT) method to compute the electromagnetic field incorporating both the electric and magnetic contrasts. The variational Born iterative method for electrical contrast inversion in axisymmetric media is generalized to 3-D and to the simultaneous reconstruction of objects with electric and magnetic contrasts. The BCGS-FFT method provides the predicted scattered field from the 3-D heterogeneous objects and the Frechet derivatives in the inverse scattering problem. The efficient forward solver also dramatically reduces the computation time of the inverse problem. Numerical results are presented to validate the forward solver and to demonstrate the effectiveness of the inverse scattering method.

Seismic wave propagation in fully anisotropic axisymmetric media

We present a numerical method to compute 3-D elastic waves in fully anisotropic axisymmetric media. This method is based on a decomposition of the wave equation into a series of uncoupled 2-D equations for which the dependence of the wavefield on the azimuth can be solved analytically. Four independent equations up to quadrupole order appear as solutions for moment-tensor sources located on the symmetry axis while single forces can be accommodated by two separate solutions up to dipole order. This decomposition gives rise to an efficient solution of the 3-D wave equation in a 2-D axisymmetric medium. First, we prove the validity of the decomposition of the wavefield in the presence of general anisotropy. Then we use it to derive the reduced 2-D equations of motions and discretize them using the spectral element method. Finally, we benchmark the numerical implementation for global wave propagation at 1 Hz and consider inner core anisotropy as an application for high-frequency wave propagation in anisotropic media at frequencies up to 2 Hz.

AxiSEM: broadband 3-D seismic wavefields in axisymmetric media

Abstract. We present a methodology to compute 3-D global seismic wavefields for realistic earthquake sources in visco-elastic anisotropic media, covering applications across the observable seismic frequency band with moderate computational resources. This is accommodated by mandating axisymmetric background models that allow for a multipole expansion such that only a 2-D computational domain is needed, whereas the azimuthal third dimension is computed analytically on the fly. This dimensional collapse opens doors for storing space–time wavefields on disk that can be used to compute Fréchet sensitivity kernels for waveform tomography. We use the corresponding publicly available AxiSEM (www.axisem.info) open-source spectral-element code, demonstrate its excellent scalability on supercomputers, a diverse range of applications ranging from normal modes to small-scale lowermost mantle structures, tomographic models, and comparison with observed data, and discuss further avenues to pursue with this methodology.

Generalized thermoelastic solutions for the axisymmetric plane strain problem

Thermal disturbance propagates in the elastic medium with a finite speed during the transient heat transfer, which leads to the thermal-mechanical behavior of materials being significantly different than that of conventional heat transfer. The axisymmetric plane strain problem is investigated in this paper, the generalized thermoelastic solutions for different generalized thermoelasticity are derived by means of the Laplace transform and its limit theorem and the asymptotic formulas of Bessel functions. A case of an infinite axisymmetric medium with a cylindrical hole under thermal shock is analyzed. It is pointed out that each of physical field has a phased distribution when the propagation of thermal disturbance with a finite speed, and both the temperature and stresses have jumps at the location of thermal elastic wavefront and thermal wavefront. The L-S and G-N theories predict similar patterns for the distributions of displacement, temperature and stresses in the transient thermal shock problem, but for G-L theory a Dirac-delta-type result for the stresses is predicted and the displacement is discontinuous at both the wavefronts.

Study on a Type of Low-Leakage Brush Seal Porous Media Model

Computation Fluid Dynamics (CFD) has been employed to calculate the pressure, flow distributions and leakage of brush seal by using Reynolds-Averaged-Navier-Stokes (RANS) method and two-dimensional axisymmetric anisotropic porous medium model. The leakage in brush seal with radial clearance has a marked increase compared with contact brush seal. The leakages of brush seal with different radial clearances have been investigated comparing contact brush seal. A type ofretaining ring structure has been employedto reduce the leakage on radial clearance condition. Also the disturbance effect of retaining ring on bristle pack has been studied.

Pitfalls in the finite element modeling of the buckling of sandwich shells of revolution

The objective of this work is to answer the following question: Is a multilayer shell model always appropriate for the prediction of the buckling of a thin structure? In order to do that, we chose to analyze the buckling of a three-layer cylindrical tube in uniform axial compression. We compared three approaches for the calculation of the theoretical buckling load: a fully analytical formula, a Fourier series buckling model of a multilayer axisymmetric shell and a calculation based on a mechanical mesh of the axisymmetric medium of the middle layer. This analysis shows that two buckling modes are in competition: a global beam mode and a local skin buckling mode (wrinkling) which develops when the stiffness of the middle layer is quite small. The critical loads can be estimated analytically for both types of modes, leading to a critical length under which the cylinder fails by skin buckling; otherwise, it fails in beam mode. Then the problem was solved by finite elements using two models: first, a multilayer axisymmetric shell model, then a model with two types of elements in each section (an 8-node axisymmetric element for the middle layer, and axisymmetric shell elements for the two skins). We showed that both models can predict the beam mode, but only the latter was capable of predicting the skin buckling mode. Similar results are proposed for the prediction of the buckling of a three-layer cone under internal pressure. We also compared the formulations for geometrically nonlinear problems with nonlinear materials and showed that in such problems the nonlinearities cause even more pronounced differences. This clearly raises the question of the choice of the type of finite element for the prediction of buckling in a multilayer structure in which some layers have only a small relative stiffness. The analytical solution and the analysis of the nonlinear response in the case of the cone confirm that the poor prediction of the skin buckling mode is due to a failure to account for the flexibility perpendicular to the mean surface.

A simple Abel inversion method of interferometric data for temperature measurement in axisymmetric medium

In this work, we describe a simple method of Abel inversion for temperature measurement in a natural convection axisymmetric flow. The essence of the method is that the measured lateral fringe shift profile is fitted with a polynomial with only even powers and then Abel inverse integral is evaluated analytically. This technique is compared with recent existing methods to test the accuracy and error propagation using a simulated interferogram of natural convection flow below a downward-facing heated horizontal disk in air. For this comparison, lateral fringe profiles are simulated using temperature fields computed by solving Navier Stokes and energy equations. Through random-number generation, noise profile is artificially added to the simulated noise-free lateral fringe shift profile. The results showed that the proposed technique for Abel inversion leads to accurate temperature profiles when the lateral fringe shift profile is fitted with even-power polynomials having degrees ranging from 20 to 30.