Periodic Velocity Field Research Articles

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.

Velocity field of a cylinder in the wake of a rotor in forward flight

The rotor and the airframe of a rotorcraft can have severe effects on each other because of flowfield interactions. Data on flowfield characteristics have been obtained using laser velocimetry in an attempt to identify and interpret the features of such interactions. This paper summarizes previous results on the flowfield of the rotor and then describes the effects of the rotor wake on the flowfield around an airframe model. A circular cylinder with a hemispherical nose is used as the airframe model and is instrumented with static pressure taps and flush-mounted microphones. A two-bladed teetering rotor is suspended above it in a wind tunnel. The periodic and time-averaged velocity fields are measured along lines parallel to the airframe axis at a rotor advance ratio of 0.1 and a rotor tip Mach number of 0.29. The dominant features in the velocity distribution over the airframe are seen to be linked to vortex interactions with the airframe surface. The tip vortex and the trailing vortex sheet are seen to be displaced in opposite directions by the airframe. Secondary vortex generation phenomena are observed near the airframe surface during vortex interaction.

Anisotropic diffusion in fluids with steady periodic velocity fields

The authors study the effective diffusivity tensor D for a contaminant carried by a steady periodic two-dimensional velocity field, in the limit of small molecular diffusivity chi . They discuss a generic model where the transport process is strongly anisotropic for the presence of channels (where, if one neglects Brownian motion, the motion is ballistic) among the convection cells. It is shown that for the longitudinal (along the channels direction) diffusivity one has: D/sub /// varies as 1/ chi , while for the transverse diffusivity: Dperpendicular to varies as chi . The behaviour D/sub /// approximately Dperpendicular to varies as chi 12/, which is typical of the Rayleigh-Benard system, is found to hold at intermediate values of chi . The scaling arguments are supported by extended numerical simulations.

Eulerian kinematics of flow through spatially periodic models of porous media

For spatially periodic models of porous media, and incompressible fluids, we consider here in detail thephysical interpretation of averaged velocity fields in terms of a purely continuum-level Eulerian seepage flux relation involving the mass flow across an arbitrarily oriented ‘macroscale’ surface element. We prove that the appropriately averaged microscale velocities do indeed satisfy a macroscale Eulerian flux relation (albeit to some approximation), provided that the surface ‘element’ is large compared with the pore lengthscale (microscale). A quantitative discussion of an ‘intermediate scale’ (lying between the pore scale and a characteristic macroscale) is achieved through the use of locally periodic velocity fields. Within the context of seepage velocity kinematics, our developments provide a conceptual foundation that renders continuum theories of flow of incompressible fluids through (periodic) porous media entirely selfcontained on the macroscale—free from any details of the discrete microscale structure (inaccessible to macroscopic observers) from which this continuum theory sprang. The error bounds and scaling laws derived are of special interest when the macroscale/microscale disparity is not very ‘wide’. With reference to current rational mechanical theories of ‘immiscible mixtures,’ our analysis provides aconstructive existence proof, at least for periodic models of porous media, for phase-specific velocities; heretofore, such velocities were usually accepted, in a Lagrangian sense, on an axiomatic basis. Thus, clarification and quantification of thekinematical aspects of the continuum hypothesis for certain classes of multiphase systems can be effecteda priori by our arguments, independently of any subsequent macroscaledynamical modelling. Our analysis, whereby we systematically proceed from a discrete picture of the underlying geometry to a purely continuum picture, may be likened to that achieved in classical statistical mechanics. As in that case, we hope that the general results obtained transcend the underlying discrete (i.e. spatially periodic) model of the subcontinuum structure.

Velocity measurements of airframe effects on a rotor in a low-speed forward flight

Aerodynamic interactions between the rotor and the airframe of a rotorcraft can have severe effects and are difficult to predict analytically. To attack this problem, the velocity field of a two-bladed rotor has been measured in a wind tunnel with and without an airframe model in proximity. The periodic and time-averged velocity fields were measured using a laser velocimeter in planes both parallel to and above and below the rotor tip path plane at an advance ratio of 0.1 and a rotor tip Mach number of 0.29. The data were shown to be free of tunnel wall effects. The effect of including the cylindrical airframe model were measured. For the geometry studied, airframe influence on the rotor flowfield was mostly confined to the front half of the rotor disc. Hub effects were noticeable, even with the minimal-sized hub used. Strong vortex interaction effects were observed using strobed laser sheet flow visualization and measured using laser velocimetry.

Sedimentation and dispersion of non-neutrally buoyant Brownian particles in cellular circulatory flows simulating local fluid agitation.

Spatially periodic fluid ``vortical'' velocity fields are utilized to model the role of microscale convection upon the sedimentation and dispersion of isolated spherical particles. In particular, generalized Taylor dispersion theory is employed to calculate the mean Stokes sedimentation velocity vector and dispersivity dyadic for a finite-size spherical Brownian particle settling in a laminar, spatially periodic fluid velocity field with zero mean flow (but nonzero ``circulation''). Surprisingly, such local cellular vortexlike motions are shown to have no effect whatsoever upon the long-time mean sedimentation velocity of the sphere, which is identical to its steady-state Stokes-law value occurring in a quiescent fluid. This conclusion is, inter alia, independent of the sphere size a relative to the spatial period l of the cellular flow (provided that a/l\ensuremath{\ll}1), as well as of the ratio of the particle settling velocity to the magnitude of the cellular fluid velocity or circulation. On the other hand, local convection does affect particle dispersion relative to the mean particle velocity in these systems, consistent with recent calculations for a particular cellular velocity field (Mofatt [Rep. Prog. Phys. 46, 621 (1983)], Sagues and Horsthemke [Phys. Rev. A 34, 4136 (1986)], Rosenbluth et al. [Phys. Fluids 30, 2636 (1987)]). We obtain a general expression for the dispersion coefficient for sedimenting, finite-size spheres in the limit of low Pecl\'et number; this general result is consistent with the specific formula derived by the latter authors for the particular cellular ``Taylor'' flow analyzed by them, applicable to the case of a nonsedimenting, point-size tracer particle.

Anomalous diffusion and anomalous stretching in vortical flows**Participation in this Conference has been possible thanks to the financial support of the Dutch foundation of Fundamental Research of Matter and the IUTAM Local Organizing Committee.

The trajectories of particles advected by non-random spatially periodic velocity fields may be chaotic, i.e., extremely sensitive to their initial positions. In the most general case, the area of a cloud of advected particles grows like (time)α where the exponent α ranges between 0 and 2 (shear stretching). These observations could explain many known facts about dispersion processes in shallow tidal seas.

Numerical simulation of the horizontal Bridgman growth. Part I: Two‐dimensional flow

AbstractWe study the generation of periodic velocity and temperature fields in a plane horizontal crucible of molten metal under the action of a horizontal temperature gradient. The geometry and the boundary conditions are similar to those encountered in the Bridgman growth process of semiconductor crystals, although the present paper is limited to two‐dimensional flows. We use transient finite difference and finite element algorithms which lead to identical results. We demonstrate the oscillatory mechanism in two different geometries.

Technique for Vorticity Measurement in Unsteady Flow

A technique is described for determining spatial vorticity distributions in nonstationary fluid flows exhibiting a periodic, time-varying, mean velocity field. The method requires the integration of velocity field data about a spatial array of closed contours to infer the distribution of vorticity using the generalized definition of circulation. A digital data acquisition scheme is suggested for the handling and processing of large quantities of data encountered in typical applications. The method was used to determine the flow characteristics in portions of an unsteady separated region generated by an oscillating spoiler on an airfoil surface. Several data display alternatives are discussed. ORTICITY is a significant physical variable in many real fluid flows, but it is a parameter that is often difficult to determine experimentally. The method to be described provides a means of measuring time-varying mean vorticity components and their distributions in nonstationary flows exhibiting a periodic mean velocity field. Through the years, a number of vorticity measurement techniques have evolved which have been successful to a greater or lesser degree. No single method, however, has proved satisfactory over the entire range of possible flow situations. Attempts to infer the streamwise vorticity component from velocity field measurements date back to the early part of the century.l Limitations on velocity measurements techniques at that time did not permit its application to time-varying fluid motions. A successful method for detecting the freestream-oriented vorticity component was discussed by Kovasznay2 and Kistler.3 A special purpose probe was constructed using an array of four hot-wire sensors of equal length and oriented at the same angle with respect to the freestream direction. The four sensors were operated as elements of a single constant current bridge circuit. In addition to the limitation on its directional capability, the wire array size must be of the order of the flow of Kolmogorov microscale for accurate spectral response.4 Vane-type mechanical vorticity meters have recently become popular, but these devices are also limited to measurement of the mean vorticity component in the freestream direction. The first use of these probes was discussed by May.5 The rotation frequency of the spinner and shaft is converted to an electrical signal using light from a bulb on one side of the shaft. A photocell on the opposite side converts this rotation to a train of pulses whose frequency is a function of the streamwise vorticity component. The resulting spatial resolution is limited by the vane array geometry, while accuracy and linearity are limited by friction in the shaft bearings. Effects of inertia and friction in unsteady flow applications have not yet been documented.

Sufficient Conditions for Stability of Fluid Motions Constitutively Described by the Infinitesimal Theory of Viscoelasticity

Energy method determining sufficient conditions for fluid motion stability described by infinitesimal theory of viscoelasticity, restricting analysis to confined fluids or periodic velocity field