Dispersion Tensor Research Articles

Particle Dispersion in Shear Turbulent Flows

Applying the time series analysis idea to the spatial and temporal fluid-velocity correlation functions, a Lagrangian model for particle motion in anisotropic turbulent flows has been established. As applications, particle motion is numerically investigated in two shear turbulent flows: one flow is homogeneous and the other is not. In the present study, the Saffman lift force due to flow gradient is accounted for. Quantities such as the variations of particle dispersion tensor, the cross-correlation coefficient, and the principal angle of the particle dispersion tensor with time are calculated for 60 and 140-μm particles. As a reference, the corresponding quantities are also calculated for fluid dispersion. Comparisons for the cross-correlation coefficient and the principal angle of the fluid dispersion tensor are made with the available direct numerical simulation results. A good agreement is observed. For particle motion in the homogeneous flow, where the stochastic process of particle motion can be als...

Method of homogenization applied to dispersion in porous media

The theory of homogenization which is a rigorous method of averaging by multiple scale expansions, is applied here to the transport of a solute in a porous medium. The main assumption is that the matrix has a periodic pore structure on the local scale. Starting from the pores with the Navier-Stokes equations for the fluid motion and the usual convective-diffusion equation for the solute, we give an alternative derivation of the three-dimensional macroscale dispersion tensor for solute concentration. The original result was first found by Brenner by extending Brownian motion theory. The method of homogenization is an expedient approach based on conventional continuum equations and the technique of multiple-scale expansions, and can be extended to more complex media involving three or more contrasting scales with periodicity in every but the largest scale.

Macrodispersion of sorbing solutes in heterogeneous porous formations with spatially periodic retardation factor and velocity field

Expressions for the macroscopic velocity vector and dispersion tensor for sorbing solute transport in heterogeneous porous formations whose hydrogeologic properties are repeated at intervals were derived via Taylor‐Aris‐Brenner moment analysis. An idealized three‐dimensional porous formation of infinite domain with spatially periodic retardation factor, velocity field, and microdispersion coefficients in all three directions was considered. Sorption was assumed to be governed by a linear equilibrium isotherm under local chemical equilibrium conditions. The analytical expressions presented are based on a perturbation method where all of the spatially periodic parameters employed were assumed to have “small” fluctuations. It was shown that the effective velocity vector is given by the volume‐averaged interstitial velocity vector divided by the volume‐averaged retardation factor, and the effective dispersion dyadic (second‐order tensor) is given by the volume‐averaged microdispersion dyadic divided by the volume‐averaged dimensionless retardation factor plus a dyadic expressing the increase in solute spreading caused by the spatial variability of the parameters.

On the continuum equations for dispersed particles in nonuniform flows

The continuum equations of a dispersed phase of solid, noncolliding particles in a nonuniform turbulent gas flow are derived from a kinetic equation for the transport of the average phase space density 〈W(v,x,t)〉 for particles with velocity v and position x at time t. The crucial feature of this equation is the form given for the phase space diffusion current j representing the net acceleration of a particle from interactions with turbulent eddies. This is based on Kraichnan’s Lagrangian history direct interaction approximation which gives j=−[(∂/∂v)⋅μ+(∂/∂x)⋅λ+γ]〈W(v,x,t)〉, where μ, λ, and γ are dispersion tensors dependent upon displacements in the velocity and position of a particle about v,x in times of order of the time scale of the fluctuating aerodynamic driving force. Most important these tensors are affected by spatial variations in the mean carrier flow velocity and external force as well as inhomogeneities in the carrier phase turbulence; γ is zero for homogeneous turbulence. This form for j is invariant under a random Galilean transformation. The dispersed phase momentum and energy equations deduced from the kinetic equation contain local gradient forms for the net fluctuating interphase force (per unit volume) and its rate of working. The former contains in general an asymmetric stress component (which adds to the particle Reynolds stress) as well as a body force dependent upon inhomogeneities in the turbulence. Conditions are examined under which the mass and momentum equations reduce to a convection–diffusion equation. The convection velocity in this case is the local carrier flow velocity plus a drift velocity proportional to local gradients of the turbulence (zero if the particles follow the flow). The diffusion coefficient is linearly related to the pressure/mean density of the dispersed phase via the particle response time. It is shown to be influenced by the local mean shearing of the carrier flow. Conditions are derived when this contribution can be ignored, the diffusion coefficient reducing to the form for homogeneous turbulence.

An investigation of the validity of first‐order stochastic dispersion theories in isotropie porous media

Monte Carlo simulations are used to (1) investigate the accuracy of approximations that are implicit in first‐order stochastic dispersion theories and (2) identify the accuracy limits of first‐order dispersion theories in isotropic porous media. The Fickian theory of Gelhar and Axness (1983), as well as the Fickian and non‐Fickian theories of Dagan (1984) and Neuman and Zhang (1990) are investigated. All Monte Carlo simulations are in three dimensions. Confidence limits of ensemble‐averaged Monte Carlo results in isotropic porous media are established for 0.1 ≤ σY ≤ 1.5. These results showed that first‐order theoretical estimates of the Eulerian velocity covariance function are quite accurate for σY < 1; theoretical estimates of the non‐Fickian longitudinal dispersivity do not deviate significantly from theory for at least σY ≤ 1.5; theoretical estimation of the transverse dispersivity is limited to σY < 1; and, the Fickian longitudinal dispersivity is overestimated by the theory of Gelhar and Axness (1983). Of all first‐order dispersion theories, the theory of Dagan (1984) is most robust in estimating the dispersivity tensor.

Self-consistent two-component models of elliptical galaxies

Spherical self-consistent elliptical galaxy models, embedded in a dark matter halo, are constructed. Two different stellar distributions are considered : the King law, and another one reproducing the R 1/4 luminosity law; the stellar mass-to-light ratio is constant. The concentration and the amount of the stellar and dark matter distributions are determined by five parameters; two characteristic radii determine the radial trend of the anisotropy for the stellar and dark halo velocity dispersion tensors. The multicomponent decomposition of dynamical models of galaxies and a theorem concerning their consistency are introduced; then, the region of the parameter space where the models are self-consistently generated are determined

Mathematical theory of stationary miscible filtration

We consider a coupled system of two elliptic partial differential equations that models stationary incompressible two-component miscible displacement in porous media. We suppose that the dispersion tensor depends on the velocity (the so-called Peaceman model, see, e.g., [D. W. Peaceman, Soc. Pet. Eng. J. 6 (1966), 213–216]. Assuming that the mobility ratio is not too large, we prove the existence of at least one solution in W r 2(Ω) × W r 2(Ω) . Furthermore, supposing that the data are small, we also establish well-posedness in W r 2(Ω) × W l 1(Ω) .

On solute transport in a heterogeneous porous formation under saturated and unsaturated water flows

The results of a recent study of solute transport in a hypothetical scale‐heterogeneous soil under transient unsaturated flow conditions (Russo, this issue) suggested that the time‐dependent components of the spatial covariance tensor, based on a single realization of the solute concentration in unsaturated flow, are in relatively good agreement with the components of the spatial covariance tensor, based on the ensemble‐average solute concentration derived by Dagan (1984), for solute transport under steady saturated flow. Furthermore, the asymptotic value of the longitudinal component of the dispersivity tensor λzz, estimated from the horizontally averaged solute concentration in the unsaturated flow domain, was found to be in good agreement with the asymptotic λzz derived by Dagan (1982, 1984) and Gelhar and Axness (1983) for saturated flow. These findings are explored in a general Lagrangian formulation, using small‐perturbation, first‐order approximations. The results of the analysis of the velocity field in the unsaturated flow domain of the hypothetical scale‐heterogeneous soil of Russo (this issue) suggest that there was good agreement between the result of the Lagrangian kinematic analysis and the result of the analysis of the spatial moments of the concentration distribution, in agreement with the stochastic theory of transport in saturated flow. This is explained by the finding that under unsaturated flow conditions there is an increase in the variability of the hydraulic conductivity K (and concurrently an increase in the variability of the velocity Vz), accompanied by a decrease in the correlation scale of K (and concurrently a decrease in the correlation scale of Vz) relative to the saturated conditions. The increase in the variability of the unsaturated conductivity stemmed from the variability in the water content θ and the strong nonlinear dependence of K on θ, while the accompanying decrease in its correlation scale stemmed from the negative correlation between F = log Ks and θ, which persisted for a relatively large separation distance in the longitudinal vertical direction. Consequently, the product between the conductivity variance and its correlation scale is of similar magnitude for both saturated and unsaturated flow regimes. The results suggest that in spite of the complexity of unsaturated flow, as compared with saturated flow, it might be amenable to a similar analysis.

Electron cyclotron damping in thermal and nonthermal plasma

The relativistic dispersion relation including temperature anisotropy and a parallel drift has been derived for electron cyclotron waves of frequency up to third harmonics, and solved for a perpendicularly stratified plasma around the fundamental and second harmonic resonances. For quasiperpendicular propagation, damping of fundamental modes is much more sensitively affected by T∥ than by T⊥. This is also true for the ordinary mode of oblique propagation. The theory has been extended to a bulk thermal distribution with a two-temperature or three-temperature tail distribution by a simple summation of each contribution to the dispersion tensor. The effect of a tail distribution is relatively small if the tail is localized in the resonant layer for the bulk electrons. For a nonlocal tail distribution, the existence of a tail becomes important.

Lagrangian and Eulerian statistics obtained from direct numerical simulations of homogeneous turbulence

Lagrangian statistics have been obtained from direct numerical simulations of isotropic turbulence and homogeneous shear flow. Quantities presented include properties of the dispersion tensor 〈Xi(t)Xj(t)〉, isoprobability contours of particle displacement, Lagrangian and Eulerian velocity autocorrelations and time scale ratios, and the eddy diffusivity tensor. The dispersion measurements from the simulations of isotropic turbulence are in good agreement with those of Warhaft [J. Fluid Mech. 144, 363 (1984)] and Stapountzis et al. [J. Fluid Mech. 165, 401 (1986)]. ‘‘Integral’’ time scales were defined as the time required for the temporal correlations to decrease to 1/e of their initial value. The ratio of TeL/TeE from the simulations of isotropic turbulence was approximately 0.8, in good agreement with the data of Sato and Yamamoto [J. Fluid Mech. 175, 183 (1987)]. The principal angle of 〈Xi(t)Xj(t)〉 from the shear flow simulations shows reasonable agreement with a similar study done by Riley (Ph.D. dissertation, The Johns Hopkins University, Baltimore, 1971). The Lagrangian time microscale was found to be consistently larger than the Eulerian microscale, presumably due to the advection of the small scales by the large scales in the Eulerian reference frame. A comparison made between the measured diffusivity tensor and measurements of Tavoularis and Corrsin [Int. J. Heat Mass Transfer 28, 265 (1985)] show reasonable agreement.

Deplacement miscible avec contrastes de densite et de viscosite en milieu poreux. Identification des parametres de deplacement et stabilite en regime de dispersion mecanique

The behaviour and estimation of pollutant transport with water in an aquifer are based on the determination of displacement parameters (velocity field, dispersion tensor, kinematic porosity, etc.). We consider the one-dimensional and isothermal displacement of two miscible fluids through an isotropic and homogeneous porous medium in mechanical dispersion. An in situ measurement technique coupled with an impulse response method is used to verify the theoretical solution in the error function (trace case). With viscosity and density gradients, experimental and numerical (application of instability criterion) results are in good agreement. For a quasi-stable displacement, we have shown an asymptotic dispersion regime. The localization of a pollutant front in the studied configuration, is discussed.

Models of elliptical galaxies: NGC 3379, 4261, 4278 and 4472

Summary We investigate whether the observed kinematics and surface photometry for the elliptical E1/E2 galaxies NGC 3379, 4261, 4278 and 4472 can be modelled under the assumptions of (i) axisyrnmetry, (ii) a distribution function of the form f = f (E, Lz,) and (iii) constant mass-to-light ratio. The methods used are an extension of the work of Binney, Davies & Illingworth (Paper I). Models satisfying the above assumptions fit the observations of all four galaxies remarkably well. For all galaxies, the rotation curves rule out isotropic velocity dispersion tensors. However, an excellent fit to the data can usually be obtained by including a measure velocity of anisotropy. Only in the case of NGC 4472 do the velocity dispersion profiles suggest that the radial and vertical dispersions are decoupled and thus the distribution function depends upon a third integral. In NGC 4278 the rotation velocity changes sign at R ⋍ 30 arcsec. Such behaviour is allowed by our models, but since it is associated with a large isophote twist, it is likely that it indicates a triaxial figure. We have developed a prolate model for NGC 4261 which accurately predicts the kinematics, although the comparison is limited to the rms line-of-sight velocities. We investigate how the predicted kinematics vary as a galaxy’s isophotes are varied from ‘boxy’ to ‘discy’ in shape merely by deforming the bulge rather than introducing a thin disc. A discy model rotates faster on the major axis than the equivalent boxy system, but the effect is too small to account for Bender’s suggested correlation of (v/σ)* with disciness (which exhibits a large scatter). In boxy galaxies the rotation velocity falls more slowly away from the equatorial plane than in the equivalent discy galaxies.

Thermal dispersion in a porous medium

The thermal dispersion conductivity tensor for convection in a porous medium is derived based on the method of volume averaging of the velocity and temperature deviations in the pores. The velocity and temperature deviations are obtained based on flow over a dilute array of spheres, incorporated with a scale analysis. A multiplying constant (i.e. the thermal dispersivity tensor) is introduced to account for the interaction of spheres. Separate considerations are given to the creeping flow at low Reynolds numbers, as well as boundary layer flow and wakes at high Reynolds numbers. It is found that the velocity and porosity dependencies in the thermal dispersion conductivity tensor are different for high Reynolds number and low Reynolds number porous media flows. The value of the transverse thermal dispersivity for a nearly parallel flow at high Reynolds numbers is determined by comparing the predicted heat transfer characteristics with existing experimental results for forced convection of water and air through heated packed channels and cylindrical packed tubes.

Diffusion, sedimentation, and Taylor dispersion of a Brownian cluster subjected to a time-periodic external force: A micromodel of ac electrophoretic phenomena

A generalized formulation of the Fokker–Planck equation is utilized to calculate the mean velocity and dispersivity of a flexible Brownian cluster of rigid particles which is acted upon by a time-periodic external force. It is shown that if the force consists of a nonzero mean part and a ‘‘fluctuating’’ (i.e., zero mean) part, their effects are decoupled. Similarly, if a Fourier expansion of the force is carried out, the effect of each term of the expansion can be treated independently of the others. A representative force term of the form Fn exp(iωnt) +F̂n exp(−iωnt) was selected to act upon a flexible dumbbell composed of two identical tethered spheres of radii a, with the inextensible tether acting as an ‘‘attractive’’ internal potential. The dispersion tensor is found to consist of a ‘‘parallel’’ contribution (directed along FnF̂n+F̂nFn) and a ‘‘hydrostatic’’ contribution. This dispersion tensor depends linearly upon the scalar (Fn⋅F̂n)a/24πμkT (μ=viscosity), approaches a constant asymptotic value for small nondimensional frequencies Ωn=12πμa3ωn/kT, and decreases asymptotically to zero for very large frequencies.

Two-dimensional shear dispersion for skewed flows in narrow gaps between moving surfaces

Flows transporting material between nearby moving surfaces are ubiquitous in machinery of all scales and with a variety of geometries. Here a general derivation is given of the effective two-dimensional mixing process in a narrow gap for a solute or miscible fluids. Explicit formulae are given for the shear dispersion tensor in laminar and turbulent (logarithmic velocity profile) flows. It is shown that if improved mixing is required, then the optimum direction for additional boundary motion or stress is at right angles to the primary flow direction.

Effective permeability and dispersion in locally heterogeneous aquifers

The global effective permeability and hydrodynamic dispersion tensors due to steady flow in locally heterogeneous and statistically anisitropic aquifers are analysed using the “self‐consistent approximation.” For small perturbations the results for the statistically anisotropic case are shown to be consistent with those derived using stochastic continuum theory. However, for larger perturbations the stochastic continuum theory has not yet produced rigorous results that can be compared with those in this paper.

Coupled groundwater flow and solute transport with fluid density strongly dependent upon concentration

The governing equations are discussed for the flow of salt solution with significant density variations through a porous medium. Equations for mass conservation, salt conservation, and momentum conservation are formulated in terms of the mass fraction weighted average velocity and the mass fraction of concentrated salt solution, and the underlying approximations are considered. These equations are incorporated in the finite element groundwater code NAMMU and applied to test case 5 of level 1 of the international HYDROCOIN project. This test case proposed to model the groundwater flow over a hypothetical salt dome. The nonlinearities arising from the density variation and from a velocity‐dependent dispersion tensor made the problem very hard: we had to use special techniques including mixed interpolation finite elements and parameter stepping. Starting from a related problem involving a large diffusivity we obtained a series of solutions at successively smaller values of dispersion. Our final solution is to a physically realistic problem very similar to that specified. We draw general conclusions relevant to modeling these flows and have built confidence in our numerical code.

Use of variable-scale pressure test data to estimate the log hydraulic conductivity covariance and dispersivity of fractured granites near Oracle, Arizona

Application of modern stochastic theories to nonuniform geologic media requires, among other things, knowing the principal directions and correlation lengths of the log hydraulic conductivity covariance. To infer this information by standard geostatistical methods directly from point measurements of hydraulic conductivity in three-dimensional space is seldom possible due to a ubiquitous scarcity of data. To show how this difficulty can be overcome we use an example of fractured granitic rocks near Oracle, Arizona. At this site, hydraulic conductivities were measured on a small scale by packer tests and on a larger scale by cross-hole tests. The test results suggest that the measured hydraulic conductivities can be viewed as quantities defined over a continuum which are scalar on a local scale but exhibit large scale anisotropy due to spatial nonuniformity and the effect of fracture orientations. Based on this observation, the two sets of hydraulic conductivity data are related theoretically and the resulting expression is used to estimate the anisotropic covariance function of log hydraulic conductivities at the Oracle site. We then employ these parameters to calculate a Fickian dispersivity tensor for the Oracle granite by relying on the recent theory of Neuman et al. (1987a). Our results highlight the need to extend available stochastic theories to the domain of strongly nonuniform media the properties of which fluctuate with large amplitudes.

Dispersion in heterogeneous porous media: 2. Predictions for stratified and two‐dimensional spatially periodic systems

The closure scheme which is presented in part 1 (Plumb and Whitaker, this issue) is solved for a stratified system and for a two‐dimensional spatially periodic system. By this we mean that the ω and η regions, which make up the heterogeneous media, are distributed in a spatially periodic manner. The solution to the closure problem yields the information necessary to determine the large‐scale dispersion tensor and the other coefficients that appear in the large‐scale dispersion equation. The theoretical results for the stratified system exhibit an increase in the longitudinal dispersion coefficient of several orders of magnitude over that which would be predicted using established correlations based on laboratory experiments with seemingly homogeneous porous media. For the cases studied, the coefficient of the mixed space‐time derivative appears to be significant and this coefficient increases for those conditions which yield increasing longitudinal dispersion coefficients. The results for the two‐dimensional spatially periodic system are in qualitative agreement with experimental observations, and we found that both the size of the heterogeneity and the difference in the hydraulic conductivity between the two regions had an important influence on the large‐scale dispersion tensor. As in the case of a stratified medium, the predicted values of the coefficient of the mixed space‐time derivative (the skewness vector) indicate that this term could be a significant factor in the prediction of mass transport in heterogeneous porous media.

Dispersion in heterogeneous porous media: 1. Local volume averaging and large‐scale averaging

The problem of diffusion and dispersion in porous media is considered in the absence of either adsorption or chemical reaction. For this situation, local volume averaging has been used to obtain both a convection‐dispersion equation and a method of closure that provides for a direct theoretical prediction of the dispersion tensor. This approach is associated with three length scales, lβ, the pore diameter, ro, the radius of the averaging volume, and L the macroscopic length scale, and makes use of a representative unit cell to determine the dispersion tensor. Previous work suggests that extremely complex unit cells must be used to capture the observed dispersion phenomena, and complex unit cells naturally introduce another length scale, i.e., the length scale associated with the local heterogeneities. To deal with this problem, we propose the use of large‐scale averaging to account for the influence of the local heterogeneities. This leads to a large‐scale convection‐dispersion equation that contains additional terms involving time derivatives which arise as a result of the presence of heterogeneities and which do not appear in the conventional local volume average convection‐dispersion equation. The first new term contains the second derivative with respect to time and is likely to be unimportant for time scales appropriate for groundwater transport processes; however, it may be important for small‐scale laboratory experiments. The second new term contains a mixed space‐time derivative which may be important for many cases of practical interest. A closure scheme is presented that allows for the theoretical determination of all the coefficients that appear in the large‐scale convective‐convection‐dispersion equation.