Solenoidal Part Research Articles

Least Squares for the Perturbed Stokes Equations and the Reissner--Mindlin Plate

In this paper, we develop two least-squares approaches for the solution of the Stokes equations perturbed by a Laplacian term. (Such perturbed Stokes equations arise from finite element approximations of the Reissner--Mindlin plate.) Both are two-stage algorithms that solve first for the curls of the rotation of the fibers and the solenoidal part of the shear strain, then for the rotation itself (if desired). One approach uses L2 norms and the other approach uses H-1 norms to define the least-squares functionals. It is shown that the H-1 norm approach, under general assumptions, and the L2 norm approach, under certain H2 regularity assumptions, admit optimal performance for standard finite element discretization and either standard multigrid solution methods or preconditioners. These methods do not degrade when the perturbed parameter (the plate thickness) approaches zero. We also develop a three-stage least-squares method for the Reissner--Mindlin plate, which first solves for the curls of the rotation and the shear strain, next for the rotation itself, and then for the transverse displacement.

Exponential Decay Rates for a Full von Karman System of Dynamic Thermoelasticity

The full von Karman system accounting for in plane acceleration and thermal effects is considered. The main results of the paper are: (i) the wellposedness of regular and weak (finite energy) solutions, (ii) the uniform decay rates obtained for the energy function in the presence of mechanical damping affecting only the solenoidal part of the velocity field representing the horizontal displacements of the plate. The obtained decay rates are uniform with respect to the parameter γ which represents the momenta of inertia and whose presence distinguishes the “parabolic-like” from the hyperbolic character of the dynamics.

Interaction of an atom with superstrong laser fields

A theory of atomic interaction with a superstrong laser field has been developed. The specific feature of the suggested theory is that its small parameter is the interaction between the atom and the solenoidal part of the external field, whereas its interaction with the potential part is accurately taken into account. It follows from the reported investigation that, in calculating the interaction of atoms with superstrong fields, one must abandon calculations of multipole moments of transitions between unperturbed atomic levels, and calculate instead the atomic response, which comprises multipole moments of all orders and depends on the instantaneous field magnitude. The results are compared with calculations based on the perturbation theory in terms of the interaction Hamiltonian.

Similarity scaling of acceleration and pressure statistics in numerical simulations of isotropic turbulence

The scaling properties of one- and two-point statistics of the acceleration, pressure, and pressure gradient are studied in incompressible isotropic turbulence by direct numerical simulation. Ensemble-averaged Taylor-scale Reynolds numbers (Rλ) are up to about 230 on grids from 323 to 5123. From about Rλ 40 onwards the acceleration variance normalized by Kolmogorov variables is found to increase as Rλ1/2. This nonuniversal behavior is traced to the dominant irrotational pressure gradient contributions to the acceleration (whereas the much weaker solenoidal viscous part is universal). Longitudinal and transverse two-point correlations of the pressure gradient differ according to kinematic constraints, but both (especially the latter) extend over distances of intermediate scale size large compared to the Kolmogorov scale. These extended-range properties essentially provide the Eulerian mechanism whereby (as found in recent work) the accelerations of a pair of fluid particles can remain significantly correlated for relatively long periods of time even as they move apart from each other. Although a limited inertial range is attained in the energy spectrum, little evidence for classical inertial scaling is found in acceleration correlations and pressure structure functions. The probability density function (PDF) of pressure fluctuations has negatively skewed tails that exhibit a stretched-exponential form. Pressure gradient statistics show a rapid increase in intermittency with Reynolds number, characterized by widening tails in the PDF and large flatness factors. The practicality of computing acceleration correlations from velocity structure functions is also assessed using direct numerical simulations (DNS); within some resolution limitations good agreement is obtained with experimental data in grid turbulence at comparable Reynolds number.

Nature of the energy transfer process in compressible turbulence

Using a two-point closure theory, the eddy-damped-quasinormal-Markovian approximation, we have investigated the energy transfer process and triadic interactions of compressible turbulence. In order to analyze the compressible mode directly, the Helmholtz decomposition is used. The following issues were addressed: (1) What is the mechanism of energy exchange between the solenoidal and compressible modes, and (2) is there an energy cascade in the compressible energy transfer process? It is concluded that the compressible energy is transferred locally from the solenoidal part to the compressible part. It is also found that there is an energy cascade of the compressible mode for high turbulent Mach number. Since we assume that the compressibility is weak, the magnitude of the compressible (radiative or cascade) transfer is much smaller than that of the solenoidal cascade. These results are further confirmed by studying the triadic energy transfer function, the most fundamental building block of the energy transfer.

Loop-star decomposition of basis functions in the discretization of the EFIE

A general and readily applicable scheme is presented for the determination of the basis functions that allow the decomposition of the surface current into a solenoidal part and a nonsolenoidal remainder. The proposed approach brings into correspondence these two parts with two scalar functions and generates the known loop and star basis functions. The completeness of the loop-star basis is discussed, employing the presented scheme; the issue of the irrotational property of the nonsolenoidal functions is addressed.

Three-dimensional thermal convection caused by spacecraft rotation in a rectangular enclosure with rigid walls

The influence of the spacecraft’s rotation on the convection in a nonuniformly heated fluid under microgravity conditions has been studied. This influence is determined by action of the buoyancy force generated by the resultant microacceleration vector and the inertia force related to variations in the angular rotational velocity of the spacecraft. It is shown that the solenoidal part of the resulting rotational force acting upon the nonuniformly heated fluid strongly depends on the angle between the temperature gradient at some point in the fluid and the radius vector of this point relative to the spacecraft’s centroid. To minimize the influence of the spacecraft’s rotation, it is necessary to make this angle close to zero. The thermal convection which could have arisen in a rectangular box with rigid walls under rotation of the Space Shuttle orbiter has been simulated numerically. It is shown that the intensity of this kind of convection may be rather appreciable.

Stochastic mechanics of nonequilibrium systems

A nonequilibrium system of classical particles interacting through nonconservative forces is studied by a stochastic Markovian process defined over the space of the particle positions. The velocities of the particles are treated as independent stochocastic variables with a given probability distribution. We deduce expressions for the rate in which energy is exchanged with the surrounding as well as the rate in which energy is dissipated. These results are applied to the special case in which the irrotational and solenoidal parts of the forces acting on the particles are orthogonal. We also solve exactly a model in which these two forces are linear functions of positions.

Elastic Herglotz Functions

Solutions of the spectral Navier equation in the linearized theory of elasticity that satisfy the Herglotz boundness condition at infinity are introduced. The leading asymptotic terms in a neighborhood of infinity provide the far-field patterns, and the Herglotz norm is expressed as the sum of the $L^2 $-norms of these patterns over the unit sphere. Basic integral representations that connect the solid spherical Navier eigenvectors to the vector spherical harmonics over the unit sphere are utilized to prove the fundamental representation theorem for the elastic Herglotz solutions in full space. It is shown that the longitudinal and the transverse Herglotz kernels are exactly the corresponding far-field patterns of the irrotational and the solenoidal parts of the displacement field. Particular methods to obtain the displacement field from the far-field patterns, and vice versa, are also described.

Rotational Representation of a Plate Made of Grioli-Toupin Material

A solenoidal part of displacement field appearing in a plate made of material with constraints has been determined. The semi-inverse method has been applied to description. The results being obtained together with the already known biharmonic representation (Jemielita, 1992) might be useful in a new micropolar plate theory formulation.

Compressibility effects on the passive scalar flux within homogeneous turbulence

Compressibility effects on turbulent transport of a passive scalar are studied within homogeneous turbulence using a kinematic decomposition of the velocity field into solenoidal and dilatational parts. It is found that the dilatational velocity does not produce a passive scalar flux, and that all of the passive scalar flux is due to the solenoidal velocity.

Kolmogorov-like spectra in decaying three-dimensional supersonic flows

A numerical simulation of decaying supersonic turbulence using the piecewise parabolic method (PPM) algorithm on a computational mesh of 5123 zones indicates that, once the solenoidal part of the velocity field, representing vortical motions, is fully developed and has reached a self-similar regime, a velocity spectrum compatible with that predicted by the classical theory of Kolmogorov develops. It is followed by a domain with a shallower spectrum. A convergence study is presented to support these assertions. The formation, structure, and evolution of slip surfaces and vortex tubes are presented in terms of perspective volume renderings of fields in physical space.

An analogue of Weyl's decomposition for first-order operators

One proves an analogue of Weyl's decomposition of an arbitrary function f into potential and solenoidal parts with respect to a first-order operator L that generalizes the divergence. The only constraint on the operator L, besides the natural smoothness of the coefficients, is the requirement of the ellipticity of the operator LL *.

Unique tomographic reconstruction of vector fields using boundary data

The problem of reconstructing a vector field v(r) from its line integrals (through some domain D) is generally undetermined since v(r) is defined by two component functions. When v(r) is decomposed into its irrotational and solenoidal components, it is shown that the solenoidal part is uniquely determined by the line integrals of v(r). This is demonstrated in a particularly simple manner in the Fourier domain using a vector analog of the well-known projection slice theorem. In addition, under the constraint that v (r) is divergenceless in D, a formula for the scalar potential phi(r) is given in terms of the normal component of v(r) on the boundary D. An important application of vector tomography, i.e., a fluid velocity field from reciprocal acoustic travel time measurements or Doppler backscattering measurements, is considered.

Improved scalar-vector potential formulations of 3-D compressible rotational flow

It is shown, that the vector potential Ψ of the solenoidal part of a 3-D vector field can be simplified by a term-condensing method.

On the Build-Up of Oscillations in a Cylindrical Magnetron

A fully nonlinear theory of the magnetron is presented starting from the equations of classical electrodynamics. A crucial step in the development is an orthogonal-series expansion of the space charge and current densities using appropriate sets of cavity modes, the expansion coefficients being nonlinear functionals of the electron trajectories. Splitting the electric field vector into its solenoidal and irrotational (gradient of a scalar potential) parts, expanding the magnetic field and the solenoidal part of the electric field in a complete set of orthonormal solenoidal cavity modes, and substituting these expansions into Maxwell's equations, differential equations governing the time evolution of the mode amplitudes are derived. These differential equations for the mode amplitudes and the Poisson's equation for the scalar potential are solved in terms of the charge and current densities in the interaction region. Substitution of the explicit expression for the field components in terms of the charge an...

On the Build-Up of Oscillations in a Cylindrical Magnetron

A fully nonlinear theory of the magnetron is presented starting from the equations of classical electrodynamics. A crucial step in the development is an orthogonal-series expansion of the space charge and current densities using appropriate sets of cavity modes, the expansion coefficients being nonlinear functionals of the electron trajectories. Splitting the electric field vector into its solenoidal and irrotational (gradient of a scalar potential) parts, expanding the magnetic field and the solenoidal part of the electric field in a complete set of orthonormal solenoidal cavity modes, and substituting these expansions into Maxwell's equations, differential equations governing the time evolution of the mode amplitudes are derived. These differential equations for the mode amplitudes and the Poisson's equation for the scalar potential are solved in terms of the charge and current densities in the interaction region. Substitution of the explicit expression for the field components in terms of the charge and current densities into the equations of electron motion reduces the latter to a fixed-point problem for a vector nonlinear operator in an appropriate function space. The fixed point, and hence the solution for the oscillatory fields inside the magnetron, may be found by standard successive approximation techniques. For the special case of a smooth-anode magnetron (magnetron diode) the mode-amplitude equations are completely solved to yield the initial growth of the mode amplitudes. It is seen from this solution that as soon as the electrons are made to follow a curved path leading to a nonzero azimuthal current in the magnetron cavity, oscillations start to build up from a zero initial value. Thus it is seen, contrary to the conclusions of the current theories of the magnetron, that it is totally unnecessary to assume, a priori, the presence of any kind (random or deterministic) of high-frequency disturbances initially in the cavity to explain the build-up phenomenon. The detailed study of oscillation build-up based on the successive approximation solution for the electron trajectories will be deferred to the second part of this contribution.

On one property of Lamb vector in isotropic turbulent flow

It is shown that the potential part of Lamb vector λ=ω×u, ω=∇×u, where u is an isotropic Gaussian solenoidal random field, is much larger than its solenoidal part. This result shows that purely kinematic properties of turbulent flow can lead to essential reduction of its nonlinearity and thereby to impose considerable constraints on flow dynamics and topology.

Instantaneous and time-averaged energy transfer in acoustic fields

The fundamentals of energy transfer in an acoustic field are addressed and it is shown that describing the flux of energy in an acoustic field with the active intensity alone is inaccurate. A single active intensity vector describes only the time-average energy flux at a point in space, but not where the energy came from nor where it is going. Consequently, the instantaneous intensity must be used to properly describe energy flux as a time-dependent process. The phenomenon of the acoustic vortex is examined and, from the perspective of active intensity, it is seen to represent a resultant wave rotating around a zero pressure line or point at which the pressure phase is discontinuous. It is shown that this resultant wave travels with a phase speed cp, which is generally different than the plane-wave phase speed c. The instantaneous intensity, however, shows that energy is flowing through the vortex and not with the resultant waves. Although the complex intensity vector is normally separated into the active and reactive components (i.e., rotational and solenoidal parts), the active intensity can be further represented by two parts, one with zero and another with nonzero curl, which permits the representation of the coupling between sources or parts of a source. In general, this article emphasizes the physical meaning and interpretation of energy related quantities.

Technical memorandum. Rapidly converging expressions of dyadic green's functions for parallel-plate transmission line

The four dyadic Green's functions of electric and magnetic type for the parallel-plate transmission line are considered. From the expression of the dyadic Green's impedance in the form of a full modal expansion, two terms, diverging at the source point, are extracted and expressed in closed form. The first term represents the quasi-static approximation of the dyadic impedance and coincides with its irrotational part which contains the dominant singularity; the second term represents the lowfrequency approximation of the solenoidal part of the Green's impedance and is singular too. The remaining modal series, which represents a dyadic finite at the source point takes into account wave propagation and converges rapidly everywhere, even when the observation and the source points are very close to each other. Similar expressions are found for the other dyadic Green's functions.