Second-order Correlation Tensors Research Articles

Theoretical Model Development for Energy Motion of Dusty Turbulent Flow of Fibre Suspensions in a Rotational Frame

Fibre suspension has garnered considerable attention in turbulent flows that are used in many industries. Solid particles, such as dust particles, notably affect the turbulent flow field in a rotational frame. In assessing their impacts, the dusty turbulent flow for fibre suspensions needs to be studied in a frame of rotation that can be substantially applied in many industries. This study, therefore, aims to build a theoretical model for the energy motion of dusty turbulent flow of fibre suspensions in a rotational frame. The turbulence momentum equation was considered to formulate the model in presence of dusty fluid rotating flow of fibre suspensions. The newly derived equation was derived in second-order correlation tensors Fi,j, Wi,j, Gi,j, Si,j, Xi,j, Yi,j, Qi,j, and Ri,j at any two points in the flow domain, where the tensors were expressed as space, time, and distance functions. The developed model is a considerable improvement because it takes into account all of the potential influential parameters that could affect the motion of turbulent energy, such as dust particles, suspending particles (fibres), and rotating frame. However, the impact of these parameters on turbulence energy motion must be evaluated in order to assess the performance of turbulence systems utilized in a variety of industries, such as paper manufacturing. The present theoretical development will contribute to open up experimental and numerical research opportunities for the advancement of the industry, science, and technology.

Turbulent energy motion of fiber suspensions in a rotating frame

Turbulent flows play a major role in many fields of science and industry. Noticeable attention is seen on turbulent flows of suspending fibers because of the sensitivity of the electrical, thermal, and mechanical properties of the connecting fiber composites to the spatial configuration and orientation of fibers. The involvement of fibers in the turbulent flow greatly affects the turbulent energy. It is more influenced when the turbulent flow occurs in a rotating system. The effect of fibers on the turbulent energy in the rotating frame must therefore be investigated. For turbulent energy with fiber suspension, a mathematical model can be built in a rotating system that is very important to enhance the quality of industrial goods. This paper, therefore, develops a mathematical model for turbulent energy motion in a rotating frame with a fiber suspension. The model was formulated using the averaging procedure. The momentum equation for incompressible and viscous fluid turbulent flow was considered to develop the model. The turbulent energy motion of the fiber suspensions was presented in the rotating frame in second-order correlation tensors,Wi,j,Si,j,Li,j,Fi,j,Gi,j,Di,j,Qi,j, andHi,j, where all the tensors are the function of time, distance, and space coordinates.

A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically averaged descriptors

A nonlinear spectral model in terms of spherically averaged descriptors is derived for the prediction of homogeneous turbulence dynamics in the presence of arbitrary mean-velocity gradients. The governing equations for the tensor $\hat{\unicode[STIX]{x1D619}}_{ij}(\boldsymbol{k},t)$, the Fourier transform of the two-point second-order correlation tensor, are first closed by an anisotropic eddy-damped quasinormal Markovian procedure. This closure is restricted to turbulent flows where linear effects induced by mean-flow gradients have no essential qualitative effects on the dynamics of triple correlations compared with the induced production effects in the equations for second-order correlations. Truncation at the first relevant order of spectral angular dependence allows us to derive from these equations in vector $\boldsymbol{k}$ our final model equations in terms of the wavenumber modulus $k$ only. Analytical spherical integration results in a significant decrease in computational cost. Besides, the model remains consistent with the decomposition in terms of directional anisotropy and polarization anisotropy, with a spherically averaged anisotropic spectral tensor for each contribution. Restriction of anisotropy to spherically averaged descriptors, however, entails a loss of information, and realizability conditions are considered to quantify the upper boundary of anisotropy that can be investigated with the proposed model. Several flow configurations are considered to assess the validity of the present model. Satisfactory agreement with experiments on grid-generated turbulence subjected to successive plane strains is observed, which confirms the capability of the model to account for production of anisotropy by mean-flow gradients. The nonlinear transfer terms of the model are further tested by considering the return to isotropy (RTI) of different turbulent flows. Different RTI rates for directional anisotropy and polarization anisotropy allow us to correctly predict the apparent delayed RTI shown after axisymmetric expansion. The last test case deals with homogeneous turbulence subjected to a constant pure plane shear. The interplay between linear and nonlinear effects is reproduced, yielding the eventual exponential growth of the turbulent kinetic energy.

Linear and angular coherence momenta in the classical second-order coherence theory of vector electromagnetic fields

A new concept of vector and tensor densities is introduced into the general coherence theory of vector electromagnetic fields that is based on energy and energy-flow coherence tensors. Related coherence conservation laws are presented in the form of continuity equations that provide new insights into the propagation of second-order correlation tensors associated with stationary random classical electromagnetic fields.

Second-order correlation tensor of rayleigh light scattered by statistically independent anisotropic microsystems

Dealing with the positions and orientations of the molecules as independent stochastic processes, the authors investigate the influence of translational and rotational fluctuations of anisotropically polarizable microsystems on the evolution in time of the second-order intensity correlation tensor of Rayleigh scattered light as well as on its spectral density. The notion of second-order depolarization is introduced and is shown to be a useful quantity for obtaining information on the translational and rotational motions of microsystems and their optical anisotropy.

Second-order correlation tensor of n-harmonically scattered light intensity

General formulas are derived for the time dependence of the second-order correlation tensor of the intensity of n-harmonically elastically scattered light in the model of free translational and rotational diffusion for motions of statistically independent particles.

Autocorrelation of the bounded thermal radiation field

The temporal and spatial coherence of the thermal radiation field in a finite domain bounded by perfectly conducting walls is studied. The second-order correlation tensors are expanded in powers of T -1 V -1 3 . Four terms of the complete autocorrelation function are given for the cube-shaped domain.

Stationarity, homogeneity and isotropy of the electromagnetic field in the presence of random sources

Some of the consequences of stationarity, homogeneity and isotropy and some of the simplifications on the forms and relations among the second-order tensor potentials and correlation tensors in the presence of random sources are studied. General forms of the second-order correlation tensors are derived for a fluctuating electromagnetic field which is stationary, homogeneous and isotropic. For such a field it is shown that the seventy-six components of the correlation tensors of the fields, charges and currents depend only on two linearly independent scalar functions. If the field is not isotropic then the correlation tensors depend on ten linearly independent functions.

New formulation of the quantummechanical analogue of the wiener-khintchine theorem for electromagnetic fields

A new formulation of the quantum-mechanical analogue of the Wiener-Khintchine theorem is presented. It is shown that for the radiation fields with null unpaired second-order correlation tensors and with the modes arbitrarily correlated, the trace of the paired secondorder correlation tensor between two arbitrary timest1 andt2 at the space pointr and the spectral energy density at the space-time point [r, (t1 + t2)/2] form a Fourier-transform pair. The condition of crossspectral purity is briefly discussed for fields obeying the new formulation of the Wiener-Khintchine theorem.

Some aspects of the relativisitc coherence theory of black-body radiation within a transparent medium

Coherence properties of black-body radiation within a transparent medium are examined in an arbitrary inertial frame. Expressions for some second-order correlation tensors are derived by using phenomenological quantum electrodynamics. The dependence of the temporal and spatial coherence on the refractive index of the medium is illustrated by some diagrams. The measurement of some correlation quantities in the medium is briefly discussed, and finally Bloch’s theorem concerning harmonic oscillators in thermal equilibrium is found to be valid also for electromagnetic-field oscillators in a dispersive medium.

Relativistic Correlation Theory of the Electromagnetic Field in the Presence of Random Sources*

Relativistic formulation is presented of the second-order classical coherence theory of the electromagnetic field in the presence of random sources. It is shown that the dynamics of the second-order correlation tensors may be reduced to a simpler form in terms of certain tensor potentials. The gauge transformations of these tensor potentials are considered. In particular, two gauges of special interest are found, analogous to the Lorentz gauge and the radiation gauge in Maxwell’s electromagnetic theory. Some interesting relations and conservation laws are presented.

Stationarity, homogeneity and isotropy in the coherence theory of the electromagnetic fields

The concepts of stationarity, homogeneity and isotropy are introduced in terms of the invariance of certain expectation values under time translations, under space translations and under space rotations respectively. Some consequences of homogeneity and isotropy on the forms and relations among second-order correlation tensors of the electromagnetic field are derived. It is shown that, if the second-order correlation tensors are invariant under space translations, they are also invariant under time translations. Further it is also shown that in such cases the trace of the electric correlation tensor is identical to the trace of the magnetic correlation tensor. General forms of the second-order electromagnetic correlation tensors are derived for fields which are both isotropic and homogeneous. Both quantum and classical descriptions of coherence are considered and the restrictions which stationarity, homogeneity or isotropy impose on the density operator in quantum description and on the probability distributions in classical description are deduced.

An Application of Green's Function in the Theory of Optical Coherence

The relation between the retarded Green's function and second-order correlation tensor of electromagnetic potential for a pure electromagnetic radiation in thermal equilibrium is derived. This relation holds also for a long wavelength electromagnetic radiation in media, so long as the radiation is in thermal equilibrium. The electromagnetic second-order correlation tensors for blackbody radiation is obtained, using the relation mentioned above. Upon following Abrikosov, Gorkov and Dzyaloshinski the retarded Green's function for a electromagnetic radiation interacting with matter is expressed in terms of the complex dielectric constant which is necessarily introduced for carrying out a macroscopic formulation. With the assumption of spatial homogeneity of media the electromagnetic second-order correlation tensors for an optical field are obtained. These correlation tensors are also expressed in tems of the complex dielectric constant. This retarded Green's function is simply related to the temperature Green's function, so that the diagram analysis will possibly be applied in the theory of optical coherence for an interacting radiation with matter.

Relativistic formulation of correlation theory of electromagnetic fields

The second-order correlation tensors for electromagnetic fields, described relativistically, are introduced. These correlation tensors characterize the correlations which exist between the electromagnetic field vectors at any two world points. With the help of these tensors, generalized energy-momentum tensors are introduced. It is shown that these correlation tensors satisfy certain differential equations and the generalized energy-momentum tensors satisfy certain conservation laws. For stationary and ergodic electromagnetic fields, the results obtained represent the relativistically covariant form of the known correlation theory. Some new invariants for stochastic electromagnetic field are also introduced.