Showing papers on "Velocity gradient published in 1971"

Particle motions in non-Newtonian media

Abstract: An experimental study of the behaviour of rigid and deformable particles suspended in pseudoplastic and elasticoviscous liquids undergoing slowCouette flow was undertaken. The velocity profiles deviated slightly from those obtained forNewtonian fluids, but the measured angular velocities of rigid spheres showed that the rotation of the field was equal to half the velocity gradient. While the measured angular velocities of rods and discs were in accord with theory applicable toNewtonian liquids, in both non-Newtonian media there was a steady drift in the orbit towards an asymptotic value corresponding to minimum energy dissipation in the flow. Furthermore, discs in elasticoviscous solutions of polyacrylamide at higher shear stresses aligned themselves in the direction of the flow and ceased to rotate. Migration of rigid particles across the planes of shear in the annul us of theCouette was also observed. In pseudoplastic liquids, the migration was towards the region of higher shear, whereas the opposite was true in elasticoviscous liquids. The deformation, orientation and burst of pseudoplastic drops inNewtonian liquids and that ofNewtonian drops in pseudoplastic fluids were similar to those previously in completelyNewtonian systems. With elasticoviscous drops, however, the deformation was smaller than given by theory. As in elasticoviscous fluids, two-body collisions of rigid uniform spheres in the pseudoplastic liquids were unsymmetrical and irreversible, thus differing from collisions inNewtonian systems where complete reversibility is observed. While some of the observed phenomena in elasticoviscous suspensions could be qualitatively interpreted, particle behaviour in the pseudoplastic liquids could not be explained in terms of the known rheological properties of the fluids.

Velocity-gradient paths in coagulation

Abstract: COAGULATION recognized as an important has long procbeen recognized as a important process for the removal of turbidity in surface-water treatment. Early designs of mechanically stirred coagulation basins commonly were based on detention time and peripheral tip velocities of the agitation paddles. However, the inconsistent performance of early units prompted a study by Camp and Stein.1 Their work established a design criterion based upon the concept of defining and specifying the velocity-gradient input during coagulation. According to Eq 1, the effective velocity gradient G can be calculated from the measured paddle speed N in rpm, the torque input T in dyne-cm, the volume V in cu cm, and the viscosity p in g/cm-sec :

A Turbulent Boundary Layer with Mass Addition, Combustion, and Pressure Gradients

Abstract: : A subsonic turbulent boundary layer with mass addition and combustion is studied to investigate the effects of combustion on the velocity field in constant pressure and accelerating flows. Particular attention is given to determining (1) the extent to which combustion alters the flow and (2) the mechanism whereby combustion interacts with the flow field. The experimental results demonstrate that combustion alters the velocity profiles in both constant pressure and accelerating flows. The velocity gradients at the surface in combusting flows differ markedly from those of corresponding isothermal flows and the velocity in the flame regions of accelerating flows actually exceed the freestream value. The results of analysis indicate that in a subsonic turbulent boundary layer with combustion the Reynolds stress is essentially kinematic and does not explicitly involve density fluctuations. This in turn indicates that the experimentally observed changes in the velocity profiles are attributable to the temperature dependence of the local mean density and molecular viscosity. Analytical results also indicate that the combustion-induced changes in velocity profile are strongly dependent on the axial pressure gradient. The consequence of combustion with regard to skin friction is also examined and it is found that effects on the skin friction coefficient are similar to those on the wall velocity gradient. A method of determining the velocity profiles in a combusting turbulent boundary layer is also presented. (Author)

On the computation of seismic ray travel times and amplitudes

Abstract: The usual methods of interpolating velocity-depth profiles cause discontinuities in the velocity gradient and anomalous geometrical ray amplitudes. Numerical methods are presented for interpolating a velocity profile and evaluating the ray integrals which avoid these difficulties. They allow smooth amplitude curves to be calculated directly. An example is given for the Herrin P velocity profile.

Measurement of Fluid Velocity Gradients Using Laser‐Doppler Techniques

Abstract: Two methods are developed for the measurement of fluid velocity gradients using a laser‐Doppler anemometer. In the first method the velocity gradient is found by a single measurement of the bandwidth of the laser‐Doppler signal spectrum. In the second, velocity gradients very close to a wall are calculated from the amount the signal spectrum shifts in frequency for a small change in the position at which the spectrum is measured. Both methods are examined theoretically and experimentally.

High frequency wave propagation in the earth: theory and observation

Abstract: The wave-theoretical analysis of acoustic and elastic waves refracted by a spherical boundary across which both velocity and density increase abruptly and thence either increase or decrease continuously with depth is formulated in terms of the general problem of waves generated at a steady point source and scattered by a radially heterogeneous spherical body. A displacement potential representation is used for the elastic problem that results in high frequency decoupling of P-SV motion in a spherically symmetric, radially heterogeneous medium. Through the application of an earth-flattening transformation on the radial solution and the Watson transform on the sum over eigenfunctions, the solution to the spherical problem for high frequencies is expressed as a Weyl integral for the corresponding half-space problem in which the effect of boundary curvature maps into an effective positive velocity gradient. The results of both analytical and numerical evaluation of this integral can be summarized as follows for body waves in the crust and upper mantle: 1) In the special case of a critical velocity gradient (a gradient equal and opposite to the effective curvature gradient), the critically refracted wave reduces to the classical head wave for flat, homogeneous layers. 2) For gradients more negative than critical, the amplitude of the critically refracted wave decays more rapidly with distance than the classical head wave. 3) For positive, null, and gradients less negative than critical, the amplitude of the critically refracted wave decays less rapidly with distance than the classical head wave, and at sufficiently large distances, the refracted wave can be adequately described in terms of ray-theoretical diving waves. At intermediate distances from the critical point, the spectral amplitude of the refracted wave is scalloped due to multiple diving wave interference. These theoretical results applied to published amplitude data for P-waves refracted by the major crustal and upper mantle horizons (the Pg, P*, and Pn travel-time branches) suggest that the 'granitic' upper crust, the 'basaltic' lower crust, and the mantle lid all have negative or near-critical velocity gradients in the tectonically active western United States. On the other hand, the corresponding horizons in the stable eastern United States appear to have null or slightly positive velocity gradients. The distribution of negative and positive velocity gradients correlates closely with high heat flow in tectonic regions and normal heat flow in stable regions. The velocity gradients inferred from the amplitude data are generally consistent with those inferred from ultrasonic measurements of the effects of temperature and pressure on crustal and mantle rocks and probable geothermal gradients. A notable exception is the strong positive velocity gradient in the mantle lid beneath the eastern United States (2 x 10-3 sec-1), which appears to require a compositional gradient to counter the effect of even a small geothermal gradient. New seismic-refraction data were recorded along a 800 km profile extending due south from the Canadian border across the Columbia Plateau into eastern Oregon. The source for the seismic waves was a series of 20 high-energy chemical explosions detonated by the Canadian government in Greenbush Lake, British Columbia. The first arrivals recorded along this profile are on the Pn travel-time branch. In northern Washington and central Oregon their travel time is described by T = Δ/8.0 + 7.7 sec, but in the Columbia Plateau the Pn arrivals are as much as 0.9 sec early with respect to this line. An interpretation of these Pn arrivals together with later crustal arrivals suggest that the crust under the Columbia Plateau is thinner by about 10 km and has a higher average P-wave velocity than the 35-km-thick, 62-km/sec crust under the granitic-metamorphic terrain of northern Washington. A tentative interpretation of later arrivals recorded beyond 500 km from the shots suggests that a thin 8.4-km/sec horizon may be present in the upper mantle beneath the Columbia Plateau and that this horizon may form the lid to a pronounced low-velocity zone extending to a depth of about 140 km.

An Analysis of Blast Wave Laminar Boundary Layers and Particle Entrainment

Abstract: : In the first part of this report, the laminar boundary layer flowfields generated by plane, cylindrical, and spherical strong blast waves are studied. Solutions are obtained using two different methods. In the first method, the governing equations are reduced to ordinary differential equations by using a similarity transformation and a parametric integral technique. In the second method, the assumptions of quasi-steady state and local similarity are invoked. Results obtained using each method are presented. A method of solving the turbulent boundary layer equations is also outlined in detail. The second part of the report is on particle entrainment. Using a lift force generated by the boundary layer velocity gradient, the particle velocity is estimated. Sample particle trajectories in strong blast wave flowfields are illustrated, and the amount of soil erosion due to aerodynamic entrainment is assessed. (Author)

Finite-Difference Solution for Three-Dimensional Boundary Layers with Large Positive and Negative Crossflows

Abstract: This paper shows how finite-difference methods can he used to analyze three-dimensional boundary layers on axisymmetric bodies with large positive and negative crossflows. The particular problem considered is the flow around a sphere in a single layer of spheres. The external flow is given by the potential solution and is obtained by doing experiments in an electrolytic tank. The computational scheme was found to be stable provided that the components of the wall velocity gradient along the meridians were positive. The solution covered regions of both positive and negative crossflows so that the angle between the limiting steamlines and the meridians varied between -4~33° and —90°. These computational results are consistent with an analysis of the stability of the numerical scheme. INITE-DIFFERENCE methods are now widely used to solve the equations for two-dimensional boundary layers. This paper shows how such techniques can be applied to an axisymmetric body having a three-dimensio nal boundary layer with large positive and negative crossflows. We consider the laminar boundary layer on a sphere in a single layer of spheres for which the external flow is given by potential theory. Because of the very large crossflows, particular attention had to be paid to the stability of the calculations. The computational scheme developed is also of interest in that the external flow was determined from experiments in an electrolytic tank and therefore is given in tabular form. Boundary-layer separation can cause the flow external to the boundary layer to be different from that predicted by potential theory. Therefore the calculated results can be expected to disagree with measurements, particularly near separation. The two methods that have been most widely used for three-dimensional boundary layers involve series expansions or integral forms of the momentum equations. (Series solutions have been reviewed by Crab tree, Kuchemann, and Sowerby.1) They are limited in that they are conveniently applied only in a region close to the birthplace of the boundary layer. Momentum integral methods have usually been formulated in an "intrinsic" system of coordinates constructed from the projections of the external streamlines and their orthogonals. They have been most useful when a asmall crossflow approximation" can be made whereby the velocity components in the boundary layer perpendicular to the direction of the external streamlines can be assumed to be small. A review of this category of solutions is given by Cooke and Hall.2 Integral methods are not conveniently applied to the problem considered in this paper because equations for an intrinsic set of coordinates are not easily derived and because a small crossflow approximation would not be applicable. Very recently Der and Raetz,3 Hall,4 and Dwyer5 applied finite-difference methods to three-dimensional boundary layers. The difference approximations used are implicit in the direction normal to the solid surface and explicit in the tangential directions. The results of Hall and Dwyer are

Determination of the position of maximum velocity in turbulent shear flow

Abstract: In strongly asymmetric turbulent flows, shear stress is finite where the mean velocity gradient is zero, and vice versa. Since this is not describable in terms of an effective viscosity hypothesis, a more complete description of turbulent motion is required. The note describes the design and application of double pitot tube and an inductance type pressure transducer for determining the position of maximum velocity for turbulent flow.