Solenoidal Part Research Articles

Tomographic reconstruction of vector fields in variable background media

This paper considers the problem of determination of a planar vector field when its Doppler data are modified by the presence of an unknown scalar field. Such a problem occurs in the investigation of velocity distribution in a flow through media of variable sound speed. We show that the curl of the vector field can be stably recovered. In absorbing media the integrals are weighted to account for the attenuation along the path. When the boundary values of the vector field are known, its solenoidal part is determined from its curl. We use Bukhgeim's approach to the problem of tomography.

A primal formulation for the Helmholtz decomposition

In 1999, Jean-Paul Caltagirone and Jérôme Breil have developed in their paper [Caltagirone, J. Breil, Sur une méthode de projection vectorielle pour la résolution des équations de Navier–Stokes, C.R. Acad. Sci. Paris 327(Série II b) (1999) 1179–1184] a new method to compute a divergence-free velocity. They have used the grad(div) operator to extract the solenoidal part of a given vector field. In this contribution we explain how this method can be considered as a real Helmholtz decomposition and we present a stable approximation in the framework of spectral methods. Numerical results are presented to illustrate the efficiency of this approach.

Inversion of the x-ray transform for 3D symmetric tensor fields with sources on a curve

The x-ray transform I associates with a covariant symmetric tensor field f(x) of order , the line function If(l) by integration over the line l. In this paper, new reconstruction formulae for the solenoidal part of f are obtained. The case when If is known only for the family of lines intersecting a given curve in is considered as well. The formulae obtained contain only a finite number of differentiations, integrations, solving of linear algebraic equations and application the Laplacian powers. The formulae are obtained under the assumption that every plane intersecting the support of f intersects the curve m + 1 times. It is shown that the number of intersection points cannot be decreased. It is shown that the similar considerations are valid for two other admissible families of lines in : lines parallel to a given curve of directions and lines tangential to a given surface.

Error estimates for defect correction methods in Doppler tomography

The aim of Doppler tomography is the reconstruction of velocity fields from ultrasonic measurements. The linearized model of this inverse problem is the Doppler transform which integrates projections of the fields along the lines the ultrasonic beams are sent off. Unfortunately the Doppler transform has a non-trivial null space. As a consequence only the solenoidal part of the field can be recovered from the data. This is the reason to develop and investigate methods of defect corrections, which are to improve the reconstruction accuracy. In [J. Inv. Ill-Posed Prob., 12(6), 597–626, 2004] and [Inv. Prob., 21,75–91, 2005] two approaches for the defect correction are considered which rely on the Helmholtz decomposition of vector fields. They consist of the solution of corresponding boundary value problems. This article is concerned with a comparison of these two approaches. We state error estimates based on orthogonal decompositions of vector fields, which indicate the improvement in accuracy we may hope for.

A surface integral formulation of Maxwell equations for topologically complex conducting domains

A general and effective method is presented to numerically solve the electric field integral equation (EFIE) for topologically complex conducting domains by the finite element method. A new technique is proposed to decompose the surface current density into a solenoidal part and a nonsolenoidal remainder to avoid the low frequency breakdown. The surface current density field is approximated through div-conforming (facet) elements. The solenoidal part is represented through the null space of the discrete approximation D of the surface divergence operator in the subspace spanned by the facet elements, whereas the nonsolenoidal remainder is represented through its complement. The basis functions of the null space and its complement are evaluated, respectively, by the null and pseudo-inverse of the matrix D. The completeness of the null-pinv basis functions is studied. Unlike the loop-star and loop-tree basis functions, the null-pinv basis functions allow to deal with topologically complex conducting domains in a general and readily applicable way. A topological interpretation of the "null-pinv" decomposition is given and a general and simple method to evaluate the null and pseudo-inverse of D is proposed. The computational complexity of the proposed method is discussed.

Regularity of ghosts in tensor tomography

We study on a compact Riemannian manifold with boundary the ray transform I which integrates symmetric tensor fields over geodesics. A tensor field is said to be a nontrivial ghost if it is in the kernel of I and is L2-orthogonal to all potential fields. We prove that a nontrivial ghost is smooth in the case of a simple metric. This implies that the wave front set of the solenoidal part of a field f can be recovered from the ray transform If. We give an explicit procedure for recovering the wave front set.

Evolution of a passive scalar spectrum in the flow of random waves

We consider a passive pollutant advected by the flow due to linear random waves with finite attenuation. We derive the equation that governs the evolution of the pair correlation function of pollutant concentration and show that it coincides with the equation for the case of a short-correlated velocity. Due to a finite wave attenuation, nontrivial evolution (particularly, the growth of inhomogeneities) appears already in the second order in wave amplitudes. We show that random potential waves lead to the growth of concentration inhomogeneities. We identify two stationary solutions for the spectral density of concentration, equipartition, and flux state. Which one is established depends on the relation between mean square velocity gradients due to potential and solenoidal parts of the flow, respectively. We also analyze transient regimes and show how periodic component in the concentration distribution appears and disappears.

An approach to direct reconstruction of a solenoidal part in vector and tensor tomography problems

An approach to direct reconstruction of a solenoidal part in vector and tensor tomography problems

An approach to direct reconstruction of a solenoidal part in vector and tensor tomography problems

The problem of reconstruction of vector and 2-tensor fields, given in refracting medium, is considered. The approach of direct reconstruction of a solenoidal part is presented. Least squares method is exploited for a reconstruction of solenoidal parts of original vector and 2-tensor fields by usage of certain solenoidal bases of polynomial type constructed for this purpose. The approach is justified theoretically by consideration of polynomial decompositions of polynomial vector and 2-tensor fields. The problems of existence, uniqueness and convergence of obtained approximation to the solenoidal part of original field are solved. The results of numerical simulation are presented and discussed.

Singular value decomposition for the 2D fan-beam Radon transform of tensor fields

In this article we study the fan-beam Radon transform Dm of symmet- rical solenoidal 2D tensor fields of arbitrary rank m in a unit disc D as the operator, acting from the object space L2(D ; Sm) to the data space L2((0, 2π) × (0, 2π)). The orthogonal polynomial basis s (±m) n,k of solenoidal tensor fields on the disc D was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator Dm was obtained. The inversion formula for the fan-beam tensor transform Dm follows from this decomposition. Thus obtained inversion formula can be used as a tomographic filter for splitting a known tensor field into potential and solenoidal parts. Numerical results are presented.

Singular value decomposition for the 2D fan-beam Radon transform of tensor fields

In this article we study the fan-beam Radon transform of symmetrical solenoidal 2D tensor fields of arbitrary rank m in a unit disc as the operator, acting from the object space L 2 (; S m ) to the data space L 2 ([0, 2π) × [0, 2π)). The orthogonal polynomial basis of solenoidal tensor fields on the disc was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator was obtained. The inversion formula for the fan-beam tensor transform follows from this decomposition. Thus obtained inversion formula can be used as a tomographic filter for splitting a known tensor field into potential and solenoidal parts. Numerical results are presented.

WAVES AND SHEAR FLOWS

The exact analytical solution of the extended Rayleigh equation for the case of periodic compressible shear flow is found. The dispersion relation of the problem is the Hill determinant. It is found that the sound waves in shear flow have a dispersion and its velocity field contains a solenoidal part. Besides the sound waves, new wave modes, namely phonon, waveguide and vortex wave modes, are revealed. The phonon mode is similar to phonons in the crystal lattice but they are not connected to heat transfer. The vortex mode is a singular solenoidal mode. The vortex modes are negative-energy waves and it is possible that they possess dissipative instability. The absolute phonon-vortex instability appears for a Mach number Ma≥0.4. The interplay of waves and solar granulation is considered.

Turbulent Dissipation-Rate Equation for Compressible Flows

The turbulent dissipation rate in a compressible flow is modeled in terms of its solenoidal and dilatational parts. In the κ-e framework the solenoidal dissipation rate is computed using the modeled dissipation-rate equation for incompressible flows. This is based on the assumption that the solenoidal dissipation rate for compressible flows follows the same dynamics as the dissipation rate for incompressible flows. We test this assumption by comparing the exact transport equations of these two quantities, both analytically and using direct numerical simulation data of a Mach 4 boundary layer. The two equations are found to be equivalent except for an additional term caused by the variation of fluid viscosity in the compressible case. This forms the rigorous basis for using the incompressible modeled dissipation-rate equation in a compressible boundary-layer flow provided the extra term caused by variation of fluid viscosity is included. Finally, the implications of the dissipation-rate analysis on the κ-ω turbulence model are pointed out

The effects of heat release on the energy exchange in reacting turbulent shear flow

The energy exchange between the kinetic and internal energies in non-premixed reacting compressible homogeneous turbulent shear flow is studied via data generated by direct numerical simulations (DNS). The chemical reaction is modelled by a one- step exothermic irreversible reaction with Arrhenius-type reaction rate. The results show that the heat release has a damping effect on the turbulent kinetic energy for the cases with variable transport properties. The growth rate of the turbulent kinetic energy is primarily in uenced by the reaction through temperature-induced changes in the solenoidal dissipation and modifications in the explicit dilatational terms (pressure–dilatation and dilatational dissipation). The production term in the scaled kinetic energy equation, which is proportional to the Reynolds shear stress anisotropy, is less affected by the heat release. However, the dilatational part of the production term increases during the time when the reaction is important. Additionally, the pressure–dilatation correlation, unlike the non-reacting case, transfers energy in the reacting cases, on the average, from the internal to the kinetic energy. Consequently, the dilatational part of the kinetic energy is enhanced by the reaction. On the contrary, the solenoidal part of the kinetic energy decreases in the reacting cases mainly due to an enhanced viscous dissipation. Similarly to the non-reacting case, it is found that the direct coupling between the solenoidal and dilatational parts of the kinetic energy is small. The structure of the flow with regard to the normal Reynolds stresses is affected by the heat of reaction. Compared to the non-reacting case, the kinetic energy in the direction of the mean velocity decreases during the time when the reaction is important, while it increases in the direction of the shear. This increase is due to the amplification of the dilatational kinetic energy in the x2-direction by the reaction. Moreover, the dilatational effects occur primarily in the direction of the shear. These effects are amplified if the heat release is increased or the reaction occurs at later times. The non-reacting models tested for the explicit dilatational terms are not supported by the DNS data for the reacting cases, although it appears that some of the assumptions employed in these models hold also in the presence of heat of reaction.

AN INTEGRAL GEOMETRY PROBLEM IN A NONCONVEX DOMAIN

We consider the problem of recovering the solenoidal part of a symmetric tensor field f on a compact Riemannian manifold (M,g) with boundary from the integrals of f over all geodesics joining boundary points. All previous results on the problem are obtained under the assumption that the boundary ∂M is convex. This assumption is related to the fact that the family of maximal geodesics has the structure of a smooth manifold if ∂M is convex and there is no geodesic of infinite length in M. This implies that the ray transform of a smooth field is a smooth function and so we may use analytic techniques. Instead of convexity of ∂M we assume that ∂M is a smooth domain in a larger Riemannian manifold with convex boundary and the problem under consideration admits a stability estimate. We then prove uniqueness of a solution to the problem for

An efficient mollifier method for three-dimensional vectortomography: convergence analysis and implementation

We consider the problem of three-dimensional vector tomography, that means thereconstruction of vector fields and their curl from lineintegrals over certain components of the field. It is well knownthat only the solenoidal part of the field can be recovered fromthese data. In this paper the method of approximateinverse is modified for vector fields and applied to thisproblem, leading to an efficient solver of filteredbackprojection type. We prove convergence of the reconstructedsolution, if the number of data tends to infinity, which meansthe method is exact. Finally, numerical results are presentedfor a straight flow through a cylinder.

Random Taylor hypothesis and the behavior of local and convective accelerations in isotropic turbulence

The properties of acceleration fluctuations in isotropic turbulence are studied in direct numerical simulations (DNS) by decomposing the acceleration as the sum of local and convective contributions (aL=∂u/∂t and aC=u⋅∇u), or alternatively as the sum of irrotational and solenoidal contributions [aI=−∇(p/ρ) and aS=ν∇2u]. The main emphasis is on the nature of the mutual cancellation between aL and aC which must occur in order for the acceleration (a) to be small as predicted by the “random Taylor hypothesis” [Tennekes, J. Fluid Mech. 67, 561 (1975)] of small eddies in turbulent flow being passively “swept” past a stationary Eulerian observer. Results at Taylor-scale Reynolds number up to 240 show that the random-Taylor scenario 〈a2〉≪〈aC2〉≈〈aL2〉, accompanied by strong antialignment between the vectors aL and aC, is indeed increasingly valid at higher Reynolds number. Mutual cancellation between aL and aC also leads to the solenoidal part of a being small compared to its irrotational part. Results for spectra in wave number space indicate that, at a given Reynolds number, the random Taylor hypothesis has greater validity at decreasing scale sizes. Finally, comparisons with DNS data in Gaussian random fields show that the mutual cancellation between aL and aC is essentially a kinematic effect, although the Reynolds number trends are made stronger by the dynamics implied in the Navier–Stokes equations.

On nonlinear effects in inductively coupled plasmas

Nonlinear current and potential oscillations in low pressure inductively coupled plasmas are analyzed within the framework of electron magnetohydrodynamics. It is shown that both current and potential oscillations can be attributed, respectively, to the solenoidal and potential components of nonlinear Lorentz and inertial forces. Scaling of the nonlinear force with the phase shift between the electric current and the electric field is analyzed. It is demonstrated that the solenoidal part of the force that provides a source of nonlinear current vanishes in neglect of collisions and collisionless absorption, while oscillations of the electrostatic potential remain finite. It is shown that these oscillations are the result of plasma polarization due to Hall drifts.

Generalized Helmholtz Decomposition and Static Electromagnetics

The well-known decomposition of vector fields to solenoidal and irrotational parts, known as the Helmholtz decomposition, is generalized in terms of more general linear operators involving two arbitrary symmetric, positive-definite and complete (non-singular) dyadics. It is seen that, in terms of the generalized decomposition, potential expansions for the static electric and magnetic fields in anisotropic media can be formed in a straightforward manner. The decomposition theorem is further generalized in a form applicable to the static electromagnetic fields in the most general linear medium (bi-anisotropic medium) characterized by four medium dyadics. In this case, the decomposition theorem and potential expressions are presented in a compact form in terms of six-vectors.

Residual based a posteriori error estimators for eddy current computation

We consider H (curl ;Ω)-elliptic problems that have been discretized by means of Nedelec's edge elements on tetrahedral meshes. Such problems occur in the numerical computation of eddy currents. From the defect equation we derive localized expressions that can be used as a posteriori error estimators to control adaptive refinement. Under certain assumptions on material parameters and computational domains, we derive local lower bounds and a global upper bound for the total error measured in the energy norm. The fundamental tool in the numerical analysis is a Helmholtz-type decomposition of the error into an irrotational part and a weakly solenoidal part.